**Evolution Mechanism of Wind Vibration Coe**ffi**cient and Stability Performance during the Whole Construction Process for Super Large Cooling Towers**

### **Shitang Ke 1,2,\*, Peng Zhu 3, Lu Xu <sup>4</sup> and Yaojun Ge <sup>2</sup>**


Received: 28 August 2019; Accepted: 29 September 2019; Published: 9 October 2019

**Abstract:** Wind-induced damage during the construction process and the evolution of damage over time are important reasons for the wind-induced destruction of large cooling towers. In fact, wind vibration coefficient and stability performance will evolve with the construction height and material properties over time. However, the existing studies generally ignore the impact of wind load and structural performance during the construction period. In this study, we built the 3D physical model separately for all eight construction stages a super large cooling tower which is being currently constructed and stands 210 m. The dynamic characteristics of the cooling tower were analyzed in each stage. First, the flow field information and 3D time history of aerodynamic forces were obtained for the whole construction process using large eddy simulation (LES). Full transient dynamic finite element analysis was used to calculate the dynamic responses of the tower under the real-time changes of wind loads during the whole construction process. Five calculation methods were used to trace the evolution of wind vibration coefficient during the whole construction process of the super large cooling tower. Then the formula for wind vibration coefficient changing with the construction height was fitted. The differential values of wind vibration coefficient during the whole construction process of the cooling tower were discussed by taking the meridional axial force as the objective function. On this basis, the influence and working mechanism of wind vibration coefficient, concrete age, construction load, geometric nonlinearity, internal suction force on buckling stability, and ultimate bearing capacity of the cooling towers were investigated. This research provides an enhanced understanding on the evolution of wind-induced stability performance in super large cooling towers and a methodology to prevent wind-induced damage during the construction process.

**Keywords:** super large cooling tower; whole construction process; wind vibration coefficient; buckling stability; ultimate bearing capacity

#### **1. Introduction**

After Ferrybridge Cooling Tower failures in the UK in 1965 [1], the international wind engineering circle began to conduct studies in the following topics: influence of tower group and surrounding structures on wind pressure distribution on the surface of the tower body [2,3], buckling stability, and ultimate bearing capacity of the tower body under wind load [4,5], finite element analysis of responses of large cooling towers considering the tower defect and soil–structure interaction [6,7], and random dynamic responses of tower body induced by pulsating wind pressure [8]. Wind-induced damage and subsequent evolution of the damage during the construction process are considered [9] responsible for the collapse of three cooling towers (at Ardeer Power Station in Scotland in 1973, power plant in Bouchain, Franch in 1979, and Fiddlers Ferry Power Station in 1984). This is closely related to the wind loads, concrete performance, and crack evolution during the construction process of the cooling tower. We have also found through the overall and local stability performance of China's tallest exhaust cooling tower during the construction process that the wind vibration coefficient changes with the construction height and evolution of material properties. Moreover, dynamic wind pressure inside the cooling tower also has a non-negligible impact on the wind-induced stability performance during the construction process.

In China, the height of newly built thermal and nuclear power plants has far exceeded the upper limit of standard or broken the world's record. This directly leads to substantial 3D dynamic wind load effect [10,11]. The construction period of the main structure and the construction difficulty also increase [12]. For the template of the cooling tower, the concrete strength may be insufficient before the concrete pouring of the cooling tower is complete. The concrete, though having a relatively low strength during the construction process, is subjected to dead load, wind load, and construction load. The strength and modulus of elasticity of concrete will increase with the construction height, which results in the constant evolution of stiffness and stress performance of the overall tower. Changes in the morphology and mechanical performance of the cooling tower during this process will further lead to alterations of static and dynamic wind pressure distribution on tower surface, wind-induced response, and wind vibration coefficient. As a result, the calculation of internal force of the structure and analysis of stability performance and ultimate bearing capacity will be also affected. In light of this, it is of high importance to discuss the evolution and non-linear influence of wind-induced stability of super large cooling towers during the construction process.

Few studies have been devoted to the wind-induced stability performance of large cooling towers during the construction process so far. In one literature report [13], buckling failure and ultimate bearing capacity of cooling tower during the whole construction process were analyzed based on secondary development of the ANSYS and wind tunnel test. Ke employed self-written preprocessing and post-processing programs for checking computation of the local and overall ultimate bearing capacity of the exhaust cooling tower. The variation of critical wind speed with construction height was also discussed [14].

In this study, we focused on a 210 m super large cooling tower under construction, the tallest tower ever built in the world. We built the 3D physical model separately for all eight construction stages. The dynamic characteristics of the cooling tower were analyzed in each stage. First, the flow field information and 3D time history of aerodynamic force were obtained for the whole construction process using large eddy simulation. The wind pressure distributions of the constructed tower were compared against the standard and measured curves to validate the numerical simulation. Full transient dynamic finite element analysis was used to calculate the dynamic responses of the tower under the real-time changes of wind load during the whole construction process. Five calculation methods were used to trace the evolution of wind vibration coefficient during the whole construction process of the super large cooling tower. Then the formula for wind vibration coefficient changing with the construction height was fitted. The differential values of wind vibration coefficient during the whole construction process of the cooling tower were discussed by taking the meridional axial force as the objective function. On this basis, the influence and working mechanism of wind vibration coefficient, concrete age, construction load, geometric nonlinearity, and internal suction force on buckling stability and ultimate bearing capacity of the cooling tower were investigated.

#### **2. An Illustrative Example**

#### *2.1. An Overview of the Project*

This super large cooling tower stood 210.0 m, with a throat height of 157.5 m and air inlet height of 32.5 m. The top section had a diameter of 115.8 m, and the throat section in the middle portion had a diameter of 110 m. The zero-meter diameter was 180. The tower body was connected to the annular plate resting on the foundation with 52 pairs of X-shaped pillars. The X-shaped pillars had a rectangular cross section, which measured 1.2 m × 1.8 m. The foundation was a cast-in-situ reinforced concrete structure with a width of 12.0 m and a height of 2.5 m. The terrain category was B, with a basic wind velocity of 23.7 m/s. Table 1 shows the main parameters of the super large cooling tower.


**Table 1.** Main structural parameters of super large cooling tower.

#### *2.2. Modeling of the Whole Construction Process*

The construction process was divided into eight stages based on the progress of construction and calculation precision. Evolution of wind-induced stability performance was analyzed for each stage. Table 2 shows the parameters of each working condition.


**Table 2.** Parameters of super large cooling tower under typical working conditions.

**Table 2.** *Cont*.


#### **3. Numerical Simulation**

#### *3.1. Methodology*

Fluid is considered to be incompressible viscous flow in anti-wind design of structures. Spatial averaging of transient N–S equation can yield the governing equation in large eddy simulation (LES):

$$\frac{\partial \overline{\mu\_i}}{\partial \mathbf{x}\_i} = 0 \tag{1}$$

$$\frac{\partial \overline{\mu\_i}}{\partial t} + \frac{\partial (\overline{\mu\_i \mu\_j})}{\partial \mathbf{x}\_j} = -\frac{1}{\rho} \frac{\partial \overline{p}}{\partial \mathbf{x}\_i} + \upsilon \frac{\partial^2 \overline{\mu\_i}}{\partial \mathbf{x}\_j \partial \mathbf{x}\_j} - \frac{\partial \tau\_{ij}}{\partial \mathbf{x}\_j} \tag{2}$$

where ρ is air density, *t* is time, *v* is kinematic viscosity coefficient of air, μ*<sup>i</sup>* and μ*<sup>j</sup>* are velocities in three directions after filtering, and τ*ij* is non-closed term in the N–S equation after spatial averaging, i.e., subgrid scale stress.

$$
\pi\_{i\bar{j}} = \overline{\mu\_i \mu\_{\bar{i}}} - \overline{\mu\_i \mu\_{\bar{j}}} \tag{3}
$$

Boussinesq approximation was introduced according to Smagorinsky subgrid–scale model based on eddy viscosity assumption. Thus, the subgrid–scale stress is written as

$$
\pi\_{i\bar{j}} - \frac{1}{3}\pi\_{i\bar{j}}\delta\_{i\bar{j}} = -2\mu\_l \overline{\mathcal{S}\_{i\bar{j}}} = -\mu\_l(\frac{\partial \overline{\mu\_i}}{\partial \mathbf{x}\_{\bar{j}}} + \frac{\partial \overline{\mu\_{\bar{j}}}}{\partial \mathbf{x}\_{\bar{i}}}) \tag{4}
$$

*e* tensor of solvable scale, τ*kk* is the isotropic component of subgrid–scale stress, which is contained in the pressure item after filtering, δ*ij* is Kronecker delta function, μ*<sup>t</sup>* is subgrid–scale turbulence eddy viscosity coefficient, generally using the Smagorinsky assumption:

$$
\mu\_t = (\mathbb{C}\_s \Lambda)^2 \|\overline{\mathbb{S}}\|\tag{5}
$$

where *Cs* is Smagorinsky constant, generally taken as 0.1–0.23 and being 0.1 in this study. Strain rate tensor - - -*S* - - - = , 2*SijSij*. Δ is grid scale, Δ = (Δ*x*Δ*y*Δ*z*) 1/3, where Δ*x*, Δ*y*, and Δ*<sup>z</sup>* are the grid size in *x*, *y*, and *z* directions, respectively. This is the standard Smagorinsky subgrid–scale model. Some researchers propose dynamic determination of *Cs* value to better characterize collision, separation, free shear layer, and vortex shedding of the flow field around the blunt body. It is known as the dynamic Smagorinsky model.

However, both the Smagorinsky model and dynamic Smagorinsky model are algebraic models, which assume a local equilibrium between the generation and dissipation of subgrid–scale turbulent kinetic energy. These models are not fit for simulating structures with a high Reynolds number, such as the cooling towers. To solve this problem, we proposed a new form of subgrid–scale model based on literature [15]:

$$\frac{\partial \mathbf{k}\_{\rm g\rm gs}}{\partial t} + \frac{\partial \mu\_{\rm j} \mathbf{k}\_{\rm g\rm gs}}{\partial \mathbf{x}\_{\rm j}} = -\boldsymbol{\pi}\_{\rm ij} \mathbf{S}\_{\rm ij} - \mathbf{C}\_{\varepsilon} \frac{\mathbf{k}\_{\rm g\rm gs}^{3/2}}{\Delta} + \frac{\partial}{\partial \mathbf{x}\_{\rm j}} \left[ (\mathbf{C}\_{d} \Delta \boldsymbol{\nu} \sqrt{\mathbf{k}\_{\rm g\rm gs}} + \boldsymbol{\nu}) \frac{\partial \mathbf{k}\_{\rm g\rm gs}}{\partial \mathbf{x}\_{\rm j}} \right] - \boldsymbol{\varepsilon}\_{\rm w} \tag{6}$$

where, *Ksgs* is the kinetic energy of transportation equation SGS, *Cs* is Smagorinsky constant, and *v* is positive. Here, *Cs* will reduce the amplitude of *Ksgs*, and a Gaussian filter is needed for finite difference.

This model is better applied to engineering applications. There is no need for experimental filtering, and the computational load is small. So, it can be used to simulate structures with a high Reynolds number, such as cooling towers [10].

#### *3.2. Parameter Configuration and Grid Generation*

A physical model of the super large cooling tower was built according to original size, so that the Reynolds number used in numerical simulation would be comparable to that in the actual project. The size of the computational domain was X × Y × Z = 6000 m × 4000 m × 1000 m (X is across-wind direction, Y is along-wind direction, and Z is height direction). The blocking rate of the model was below 1%. The computational domain was divided into dense region and peripheral region so as to ensure both computational efficiency and precision. Non-structured grids with a high adaptability were used for the region near the cooling tower; for the peripheral space further away from the cooling tower, structured grids having a regular topology were used for the discretization. Therefore, the total number of grids was reduced, which improved the computational efficiency. The minimum grid size was 0.2 m in the core region. The model of the built-up tower had approximately 12.8 million grids.

Boundary conditions were defined using UDF (User define function) file. Inlet boundary condition was velocity inlet, and the outlet boundary condition was pressure outlet. The top and sides of the computational domain were equivalent to free slip walls, the symmetry boundary conditions (Symmetry). The floor and structure surface were equivalent to no-slip wall boundary condition. The wind field was considered as incompressible flow field. Discrete equations were solved using SIMPLEC. This calculation method has good convergence performance and applies to LES (Large eddy simulation) with small time step. The time step of LES was set to 0.05 s. Here, only the schematic for the computational domain and grid generation of the model of the built-up tower is provided, as shown in Figure 1.

(**c**) y-z plane

**Figure 1.** *Cont*.

**Figure 1.** Computational domain and grid generation for the super large cooling tower.

#### *3.3. Validation*

The design code for cooling towers [16–19] only provides the average and pulsating wind pressure distribution curves of the built-up towers. Therefore, we only validated the numerical simulation for the built-up tower.

Figure 2 shows the comparison of simulated average and pulsating wind pressure distributions of a cross section of the built-up tower against the measured and standard curves. It can be seen that the simulated average wind pressure distribution curve agreed well with the standard curve. The pressure coefficients in the upwind side, region of extreme negative pressure, and at the separation point on the leeward side were consistent with the standard curve. This validated the average wind pressures obtained by LES.

Moreover, the simulated pulsating wind pressure distribution curve basically coincided with the measured curve and the curve of wind tunnel test. The values lay between the results measured at home and abroad. Five curves are presented respectively in Figure 2b, among which three are the measured pulsating wind pressure curves in domestic and foreign regulations (VGB–R 610Ue 2005, Blanchette et al. 2013, DL/T 5339–2006, GB/T 50102–2014), and two are the measuring points of the third and ninth floors of the full-size cooling tower in this paper. Pulsating wind pressure distribution is closely associated with the terrain, incoming turbulent flow, and surrounding interference. The trend and values of pulsating wind pressure distribution estimated by LES were close or fell into the range of the measured values. Therefore, the pulsating wind pressures simulated by LES were reliable and suitable for subsequent analysis of wind-induced vibration and stability performance.

**Figure 2.** Comparison of numerical calculation, field measurement, and wind tunnel test.

#### *3.4. Simulation Results*

#### 3.4.1. Pulsating Wind Pressure

The same parameter configuration as the numerical simulation for the built-up tower was used for LES under the other seven working conditions. Thus, the flow field on the surface of the tower and the time history of pulsating wind pressure were obtained for the whole construction process. Due to limited space, Figure 3 only provides the time history of pressure coefficient at the upwind side, crosswind side, separation point and leeward side for the built-up model.

**Figure 3.** Time history of pressure coefficient at representative measuring points of the built-up tower.

Figure 4 provides the power spectral density curves of pulsating wind pressure at the representative measuring points. Comparison showed that for different positions, the peaks of the power spectral density function all occurred in the low frequency range. The energy of the pulsating wind pressure was mainly concentrated in low frequencies. The values of the power spectral density function were slightly higher on the upward side than at the separation point and leeward side.

#### 3.4.2. Pressure Coefficients on the Tower Surface

Figure 5 shows the nephograms of pressure coefficients on tower surface under eight typical working conditions. The distributions of pressure coefficient were basically consistent under different working conditions. The increase of construction height did not change the wind field characteristics of cylindrical structure. The flow separated in front of the tower body due to collision, which resulted in the separation bubble, and the bubble shedded off from the crosswind side. Consequently, there were positive and negative pressure distributions on the upward and crosswind sides of the tower body. The increase in the construction height greatly decreased the negative pressure on the crosswind and leeward sides, especially for the built-up tower.

**Figure 4.** Power spectral density curves of pulsating wind pressure at the representative measuring points.

**Figure 5.** *Cont*.

**Figure 5.** Nephograms of pressure coefficients on tower surface under eight typical working conditions.

#### 3.4.3. Turbulent Kinetic Energy

Figure 6 is the schematic for the turbulent kinetic energy distribution under the eight working conditions. There were significant differences between the eight working conditions. As the construction height increased, the scale component of maximum turbulent kinetic energy deviated from the upper end of the air inlet to the wake stream. In addition, as the construction height increased, the turbulent kinetic energy of fluid inside the tower decreased. This resulted in uniform distribution of internal pressure of the built-up tower along the circumferential and meridional directions.

**Figure 6.** *Cont*.

**Figure 6.** Schematic for turbulent kinetic energy distribution under working condition 8.

#### **4. Analysis of Dynamic Characteristics**

The integrated model of the tower body–pillar–circular foundation was built using ANSYS software. The tower body, pillar, circular foundation, and elastic foundation were simulated with shell element, beam element, and spring element, respectively. The connections of circular foundation to the tower body and pillars were simulated using multi-point constraint and rigid domain, respectively. Block Lanczos method was employed to analyze the dynamic characteristics of finite element (FE) model of the cooling tower through the whole construction process. Table 3 shows the fundamental frequencies and vibration mode distribution under each working condition. Figure 7 is the distribution curve of natural frequencies of the first fifty modes under different working conditions. It can be seen that construction height had a significant impact on lower-order frequencies, but a lesser impact on higher-order frequencies. As the construction height increased, the fundamental frequencies of the cooling tower decreased. The fundamental frequency was the largest under working condition 1, which was 0.963 Hz.


**Table 3.** Vibration modes at fundamental frequency under each working condition.

**Figure 7.** Frequency distribution of the first fifty modes under each working condition.

Figure 8 shows the changes of natural frequencies of the first ten modes with the number of circumferential harmonics under each working condition. Comparison shows that the minimum natural frequencies occurred when the number of circumferential harmonics was 4 under different working conditions. As the natural frequencies increased, the number of circumferential harmonics increased as well.

**Figure 8.** Changes of frequencies with the number of circumferential harmonics under each working condition.

Table 4 shows the capsize modes of the cooling tower for the whole construction process. As the construction height increased, the order of capsize mode also increased, while the frequency corresponding to the excitation mode decreased.

**Table 4.** Capsize modes of the cooling tower for the whole construction process.

#### **5. Analysis of Di**ff**erential Wind Vibration Coe**ffi**cient for the Whole Construction Process**

#### *5.1. Methods and Parameter Explanations*

Full transient dynamic analysis was performed to solve the dynamic equilibrium equations. The core principle was to use implicit methods, such as the Newmark method and HHT (Hilber–Hughes–Taylor) to directly solve the transient problems. The Newmark method uses the finite difference method, ad within one time interval, there are

$$[M]\{\ddot{\boldsymbol{\mu}}\} + [\mathbb{C}]\{\dot{\boldsymbol{\mu}}\} + [\mathbb{K}]\{\boldsymbol{\mu}\} = \{\boldsymbol{F}^{\boldsymbol{a}}\} \tag{7}$$

$$\left\{\dot{\mu}\_{n+1}\right\} = \left\{\dot{\mu}\_{n}\right\} + \left[(1-\delta)\left\{\dot{\mu}\_{n}\right\} + \delta\left\{\ddot{\mu}\_{n+1}\right\}\right] \Delta t \tag{8}$$

$$<\langle \mu\_{\boldsymbol{n}+1} \rangle = \langle \mu\_{\boldsymbol{n}} \rangle + \left\langle \dot{\boldsymbol{u}}\_{\boldsymbol{n}} \right\rangle \Delta t + \left[ (\frac{1}{2} - \alpha) \middle| \bar{\boldsymbol{u}}\_{\boldsymbol{n}} \right\rangle + \alpha \middle| \ddot{\boldsymbol{u}}\_{\boldsymbol{n}+1} \right] |\Delta t^2\tag{9}$$

where α and δ are integral parameters. However, the use of the Newmark method for the calculation of discrete spatial domain in FE model cannot satisfy the requirement (that is, numerical damping at high frequencies should not be introduced at the expense of precision, and not too many values of numerical damping should be generated at low frequencies). This defect can be compensated by combining with HHT method.

The basic HHT has the following expression:

$$\mathbb{E}\left[M\middle|\left\{\ddot{u}\_{n+1-\alpha\_{m}}\right\}+\left[\mathbb{C}\right]\middle|\dot{u}\_{n+1-\alpha\_{f}}\right\}+\left[\mathbb{K}\right]\middle|u\_{n+1-\alpha\_{f}}\right\}=\left\{F\_{n+1-\alpha\_{f}}^{\mathbf{u}}\right\}\tag{10}$$

where -.. *un*+1−α*<sup>m</sup>* . = (1 − α*m*) -.. *un*+<sup>1</sup> . + α*<sup>m</sup>* -.. *un* . ; - . *un*+1−α*<sup>f</sup>* . = (1 − α*f*) - . *un*+<sup>1</sup> . + α*<sup>f</sup>* - . *un* . ; *un*+1−α*<sup>f</sup>* . = (1 − α*f*) / *un*+<sup>1</sup> 0 + α*f*{*un*}; 1 *Fa n*+1−α*<sup>f</sup>* 2 = (1 − α*f*) - *Fa n*+1 . + α*<sup>f</sup>* / *Fa n* 0 .

In order to ensure unconditional stability of the second-order system without reducing the accuracy of time integral, four parameters α, δ, α*f*, and α*<sup>m</sup>* should satisfy the following relationships:

$$
\delta \ge \frac{1}{2}; \ a = \frac{1}{2}
\delta; \ \delta = \frac{1}{2} - a\_m - a\_f; \ a\_m \le a\_f \le \frac{1}{2} \tag{11}
$$

Combining Formulae (2), (4), and (6),

$$\begin{aligned} \left(a\_0[M] + a\_1[\mathbb{C}] + (1 - a\_f)[\mathbb{K}]\right)[u\_{n+1}] &= (1 - a\_f)\left\{\mathbf{F}\_{n+1}^a\right\} + a\_f[\mathbf{F}\_n^a] - a\_f\left\{\mathbf{F}\_n^{\text{int}}\right\} \\ + [M](a\_0[u\_n] + a\_2[\dot{u}\_n] + a\_3[\ddot{u}\_n]) &+ [\mathbb{C}](a\_1[u\_n] + a\_4[\dot{u}\_n] + a\_5[\ddot{u}\_n]) \end{aligned} \tag{12}$$

where *a*<sup>0</sup> = <sup>1</sup>−α*<sup>m</sup>* <sup>α</sup>Δ*t*<sup>2</sup> , *<sup>a</sup>*<sup>1</sup> <sup>=</sup> (1−α*f*)<sup>δ</sup> <sup>α</sup>Δ*<sup>t</sup>* , *<sup>a</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup>−α*<sup>m</sup>* <sup>α</sup>Δ*<sup>t</sup>* , *<sup>a</sup>*<sup>3</sup> <sup>=</sup> <sup>1</sup>−α*<sup>m</sup>* <sup>2</sup><sup>α</sup> <sup>−</sup> 1, *<sup>a</sup>*<sup>4</sup> <sup>=</sup> (1−α*f*)<sup>δ</sup> <sup>α</sup> <sup>−</sup> 1, *<sup>a</sup>*<sup>5</sup> = (<sup>1</sup> <sup>−</sup> <sup>α</sup>*f*)( <sup>δ</sup> <sup>2</sup><sup>α</sup> − 1)Δ*t*.

For terrain category B, the basic wind speed was 23.7 m/s and the damping ratio of the structure was 5%. Then the wind vibration coefficient was calculated as follows:

$$\beta\_{Ri} = \frac{R\_i}{\overline{R}\_i} = 1 + \frac{\mathcal{G}^{\sigma\_t}}{\overline{R}\_i} \tag{13}$$

where β*Ri* is the wind vibration coefficient of node *i*; *Ri*, *RI*, and σ*<sup>t</sup>* are the overall response, average response, and pulsating response of node *I*, respectively; *g* is peak factor of node *I*, taken as 3.0 (Ke et al. 2012).

#### *5.2. Distribution of Wind Vibration Coe*ffi*cient*

Based on the time history under eight working conditions, the wind vibration coefficient was calculated dynamically. The distributions of wind vibration coefficient for the whole construction process were discussed under five equivalent targets, as shown in Table 5.

**Table 5.** Five equivalent targets for value determination of wind vibration coefficient.


Note: Maximum pressure coefficient \* refers to wind pressure at the measuring point multiplied by the corresponding wind vibration coefficient.

The equivalent target 3–1 is the wind vibration coefficient and its mean value of the meridional axial force, the equivalent target 3–2 is the wind vibration coefficient and its mean value of the toroidal bending moment, and the equivalent target 3–3 is the wind vibration coefficient and its mean value of the radial displacement.

Equivalent goals 4–1, 4–2, 4–3 and equivalent goals 5–1, 5–2, 5–3 have the same representative meaning as above, which is not repeated here.

Figure 9 shows the distributions of wind vibration coefficient with construction height under five equivalent targets. It can be found that the wind vibration coefficient decreased with height under the eight working conditions. For the same construction model, the wind vibration coefficient was the largest under equivalent target 5, and that under equivalent target 1 was the smallest. Figure 10 shows the recommended values of wind fluttering coefficient under the eight working conditions for the five equivalent targets. Equivalent target 1 was the value of wind vibration coefficient, while all other equivalent targets were increments relative to target 1.

**Figure 9.** Wind vibration coefficient with meridional heights under five equivalent targets for each working condition.

**Figure 10.** Comparison of wind vibration coefficient under the five equivalent targets between eight working conditions.

#### *5.3. Fitting Formula of Wind Vibration Coe*ffi*cient*

Wind vibration coefficient of super large cooling towers is greatly affected by structural performance and wind pressure distribution during the whole construction process. However, the wind vibration coefficient does not linearly increase with the height significantly, and there is considerable discreteness of wind vibration coefficient for different equivalent targets. Therefore, the control of internal force, safety, and economic performance of the cooling towers were the major considerations for the determination of wind vibration coefficient in this study. We proposed the following formula of wind vibration coefficient by taking the meridional axial force as the target (equivalent target 1, GB/T 50102–2014, 2014):

$$y = \frac{m - \beta\_0}{1 + \left(\frac{y}{n}\right)^k} + \beta\_0 \tag{14}$$

where β<sup>0</sup> is the wind vibration coefficient of the built-up tower, β<sup>0</sup> = 1.74, *m*, *n,* and *k* are calculation parameters, *x* is the template number, and *y* is the wind vibration coefficient for the corresponding template number. After several iterations, the calculation parameters in the fitting formula were as follows: *m* = 2.526, *n* = 116.511, *k* = 1.320.

Figure 11 shows the fitted curve and the comparison of wind vibration coefficients under the five equivalent targets. The fitted curve could well reflect the differential values of wind vibration coefficient during the whole construction process when the meridional axial force was taken as the target. Table 6 shows the recommended values of wind vibration coefficient under each working condition.

**Table 6.** Recommended values of wind vibration coefficient under each working condition.


**Figure 11.** Fitted curve of wind vibration coefficients under the five equivalent targets for eight working conditions.

#### **6. Stability Performance of the Cooling Tower during the Whole Construction Process**

We then analyzed the evolution of wind-induced stability performance during the whole construction process of the super large cooling tower. The effects of wind vibration coefficient, concrete age, construction load, geometric non-linearity, and internal suction force on the buckling stability and ultimate bearing capacity of the cooling tower were discussed.

#### *6.1. Influence of Concrete Age and Construction Load*

The elastic modulus of concrete with different age was calculated under each working condition:

$$E\_{\mathbb{C}}(t) = E\_{\mathbb{C}} \sqrt{\beta\_t} \tag{15}$$

where *Ec*(*t*) is the elastic modulus (kPa) of C40 concrete with an age of *t* days, *Ec* is the elastic modulus of concrete shell with an age of 28 days, β*<sup>t</sup>* is a coefficient, β*t*= *e*s(1<sup>−</sup> √ 28/t), S depends on the type of concrete, the value is 0.25 for ordinary cement and rapid-hardening cement, and *t* is the age of concrete (day). The Poisson's ratio and linear expansion coefficient of concrete with an age of *t* days were the same as those of concrete with an age of 28 days. Shear modulus was 0.4 times that of the elastic modulus.

Figure 12 shows the distribution of elastic modulus of concrete for different template number under working condition 2 (30 templates).

The construction loads were determined based on the following criteria: (1) the uniformly distributed load imposed by templates, slidewalk, scaffold, hanging basket, railings, a-frame, and supporting system to the shell below along the circumferential direction was about 3.6 kN/m, (2) the newly cast concrete exerted a uniformly distributed load along the circumferential was calculated as 25 × template height (1.277 m) × average thickness of the plate (m) kN/m, (3) the uniformly distributed load exerted by the construction workers turning over the template to the shell below along the circumferential direction was about 0.75 kN/m, and (4) concentrated load would be generated by the reinforcing bars stacked on the slidewalk. The maximum concentrated load produced this way was 18 kN; (5) Concentrated load would be generated by the weight of electric welder and switchboard and acted on the slidewalk, reaching a level of about 3.6 kN.

**Figure 12.** Distribution of elastic modulus of concrete under working condition 2.

Figure 13 shows the typical construction conditions of cooling tower under two kinds of wind loads during the whole construction process (wind pressure of standard wind vibration coefficient and wind pressure of actual wind vibration coefficient).The buckling coefficients and displacements of cooling towers under typical construction conditions during the whole construction process are analyzed under the two conditions of concrete age change and concrete age change. Comparison reveals that:

(1) As the construction height increased, the buckling coefficient decreased, and the rate of decrease became smaller over time. Buckling displacement showed a discrete distribution, and no consistent variation trend was observed. Thus, wind vibration coefficient and whether the concrete age and construction load were considered had little impact on buckling mode and buckling displacement under the eight working conditions.

(2) Buckling coefficient decreased if the concrete age and construction load were considered. The influence of wind vibration coefficient referred from the specification and the actual wind vibration coefficient on the buckling stability was much smaller than the influence of whether the concrete age and construction load were considered or not.

**Figure 13.** Changes of buckling coefficient and buckling displacement under eight working conditions.

#### *6.2. Analysis of Geometric Non-Linearity*

Figures 14 and 15 respectively show the typical construction conditions of cooling tower under two kinds of wind loads (actual wind-induced coefficient wind load, buckling wind speed wind load).Considering the concrete age change and not considering the concrete age change, the linear and non-linear calculation of the typical construction conditions during the whole construction process of the cooling tower is carried out, and the variation law of the maximum displacement of the structure is analyzed.

**Figure 14.** Comparison of maximum displacement under linearity and non-linearity of the structure for basic wind speed during the whole construction process.

**Figure 15.** Comparison of maximum displacement under linearity and non-linearity of the structure for critical wind speed of buckling during the whole construction process.

It can be seen from the Figure 14, as the construction height increased, the maximum displacement of the tower increased constantly, but the amplitude decreased. The maximum displacement under working condition was smaller than that under working condition 7 due to the rigid ring constraint. The maximum displacement response was consistent under linearity and non-linearity of the cooling tower under different working conditions; the values differed very slightly.

It can be seen from the Figure 15 that the distribution pattern of maximum displacement changed under the critical wind speed of buckling when considering geometric non-linearity. Below is the linear analysis of wind-induced maximum displacement under some working conditions.

#### *6.3. Influence of Internal Suction Force*

The influence of internal suction force on stability performance of the cooling tower during the whole construction process was further analyzed. Figure 16 is the comparison of buckling coefficient and buckling displacement with or without internal suction force under different working conditions. There was an increment without internal suction force as compared with the condition with internal suction force. An absence of internal suction force caused a significant increment in the buckling coefficient during the whole construction process, but it had a lesser impact on buckling displacement. The maximum increment of buckling coefficient was 16.2%, and the maximum difference in buckling displacement was 1.2% in an absence of internal suction force without considering concrete age and construction load. The maximum increment of buckling coefficient was 16.6%, and the maximum difference in buckling displacement was −0.8% in an absence of internal suction force considering concrete age and construction load.

(**b**) Actual wind vibration coefficient

**Figure 16.** Comparison of buckling coefficient and displacement under standard and actual wind vibration coefficients.

Figures 17 and 18 are the comparisons of maximum displacement in linear and non-linear analyses under basic wind speed and critical wind speed of buckling for each working condition. It is easy to see that the internal suction force under the basic wind speed had less impact on the maximum displacement increment in the presence of internal suction force. In contrast, under the critical wind speed of buckling, the internal suction force caused a significant increment in maximum displacement under each working condition.

The comparison shows that when the wind load is the basic design wind speed, considering the influence of internal suction, the maximum displacement increment caused by wind has little influence on each working condition, and the influence is positive or negative.

However, when the wind load is buckling wind speed, considering the influence of internal suction, the maximum displacement of wind-induced displacement increases significantly under different working conditions.

(**a**) Linear analysis (**b**) Nonlinear analysis

**Figure 17.** Comparison of maximum displacement in linear and non-linear analyses under basic wind speed.

**Figure 18.** Comparison of maximum displacement in linear and non-linear analyses under critical buckling wind speed.

#### *6.4. Ultimate Bearing Capacity*

Figure 19 provides the curve of maximum displacement with wind speed during the whole construction period for each working condition. The histogram indicates the changes of maximum displacement with wind speed under the standard wind vibration coefficient. The displacement under actual wind vibration coefficient considering concrete age, construction load, geometric non-linearity, and internal suction force was the increment relative to the condition of standard wind vibration coefficient. Stepwise loading was performed using the initial wind speed of 23.7 m/s at the height of 10 m as the baseline. The step length was 1–20 m/s. Comparison indicated that the increase of construction height greatly reduced the ultimate bearing capacity of the cooling tower. The critical wind speed of buckling decreased from 350 (±20) m/s to 100 (±20) m/s, and the decrease rate became smaller over time. No consistent variation trend was observed for the maximum displacement upon buckling under each working condition.

Ultimate bearing capacity of the cooling tower increased when considering the geometric non-linearity, and decreased when considering the concrete age and construction load. The ultimate bearing capacity during the whole construction process was sensitive to wind vibration coefficient. The ultimate bearing capacity was much lower under actual wind vibration coefficient at a low construction height. As the construction height increased, the ultimate bearing capacity of the structure calculated with the actual wind vibration coefficient increased gradually.

**Figure 19.** Changes of maximum displacement and increment with wind speed under eight working conditions.

#### **7. Conclusions**

We discussed the evolution of wind-induced stability performance and performed parameter analysis for the whole construction process of super large cooling towers. The contents of research included dynamic characteristics, wind vibration coefficient, wind-induced response, buckling instability, ultimate bearing capacity, and geometric non-linearity of the tower. The following conclusions were reached:

(1) The fundamental frequency of the built-up tower was 0.57 Hz. As the construction height increased, the fundamental frequency decreased. Construction height had a significant impact on the lower-order frequencies, but a lesser impact on the higher-order frequencies. The order of capsize mode increased with the increase of construction height, while the frequency of the excitation mode decreased gradually.

(2) The wind vibration coefficient decreased with construction height during the whole construction process. For the same construction model, the wind vibration coefficient was the maximum under equivalent target 5, and it was the smallest under equivalent target 1. Based on the calculation results, we proposed the formula of wind vibration coefficient by taking the meridional axial force as the target for the tower, as shown below. In the formula, *x* is the template number, and *y* is the wind vibration coefficient for the corresponding template number.

$$y = \frac{0.786}{1 + \left(\frac{x}{116.511}\right)^{1.32}} + 1.74\tag{16}$$

(3) The buckling coefficient of the cooling tower decreased, and the maximum displacement increased gradually as the construction height increased. Buckling displacement showed a discrete distribution, without a consistent variation trend. In addition, the ultimate bearing capacity of the tower decreased with the construction height. The critical wind speed of buckling decreased from 350 (±20) m/s to 100 (±20) m/s, and the decrease trend slowed down over time.

(4) The buckling coefficient of the tower during the whole construction period decreased when considering the concrete age and construction load. Geometric non-linearity had mild impact on the maximum displacement under the basic wind speed, but the impact was higher under the critical wind speed of buckling. The presence of internal suction force caused a reduction in the buckling coefficient of the cooling tower. The buckling coefficients calculated from standard or actual wind vibration coefficient differed little. The influence factors of bucking stability of the cooling tower can be ranked in a decreasing order as follows: internal suction force > geometric non-linearity > concrete age and construction load > wind vibration coefficient. The degree of influence of these factors did not show a consistent variation trend over the construction height.

(5) The ultimate bearing capacity of the cooling tower during the construction period increased when considering geometric non-linearity, and decreased when considering concrete age and construction load. The ultimate bearing capacity during the whole construction process was sensitive to the wind vibration coefficient. As the construction height increased, the ultimate bearing capacity calculated with the actual wind vibration coefficient increased gradually.

To conclude, the checking computation of stability performance of the super large cooling tower during the whole construction period should consider the effect of differential values of wind fluttering coefficient, as well as the influence of concrete age, construction load, and internal suction force. The effect of geometric non-linearity is negligible.

**Author Contributions:** Conceptualization, S.K.; Data curation, S.K.; Formal analysis, P.Z.; Funding acquisition, S.K.; Investigation, P.Z.; Methodology, Y.G.; Software, L.X.; Writing–original draft, S.K.; Writing-Review & Editing, S.K.

**Funding:** This project is jointly supported by National Natural Science Foundation (51878351; 51208254; U1733129, 51878351, 51761165022), Jiangsu Province Natural Science Foundation (BK2012390), and Postdoctoral Science Foundation (2013M530255; 1202006B), which are gratefully acknowledged.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Research on Snow Load Characteristics on a Complex Long-Span Roof Based on Snow–Wind Tunnel Tests**

**Guolong Zhang 1,2, Qingwen Zhang 1,2,\*, Feng Fan 1,2 and Shizhao Shen 1,2**


Received: 2 September 2019; Accepted: 14 October 2019; Published: 16 October 2019

### **Featured Application: This study provides a simplified method that can be applied to help preliminarily estimate the snow load on a complex roof by combining the snow loads on several simple roofs, which have been provided in the load code or obtained by numerical studies.**

**Abstract:** A considerable number of studies have been carried out for predicting snowdrifts on roofs over the years. However, few studies have focused on snowdrifts on complex long-span roofs, as the complex shape and fine structure pose significant challenges. In this study, to simplify the calculation requirements of snow load on such roofs, work was conducted to decompose the snowdrift on a complex roof into snowdrifts on several simple roofs. First, the snow–wind tunnel test similarity criteria were investigated based on a combined air–snow–wind experimental system. Thereafter, with reference to the validated experimental similarity criteria, a series of snow–wind tunnel tests were performed for snowdrifts on a complex long-span structure under the conditions of different inflow directions. Finally, based on empirical orthogonal function (EOF) analysis, the snowdrifts on the complex roof were decomposed into basic characteristic distribution modes, including snowdrifts caused by the local and overall roof forms. The snow distribution under a specific inflow direction could be derived from the weighted combination of the basic characteristic modes, based on the wind direction coefficients. Therefore, it is possible for the snow load on a complex roof to be estimated preliminarily based on the snow distributions on several simple roofs.

**Keywords:** snow load; complex roof; snowdrift; EOF analysis; characteristics decomposition

#### **1. Introduction**

Snow-induced building collapse occurs frequently in long-span structures; for example, the collapse of the Katowice Trade Hall (Poland, 2006), the collapse of an ice rink in Bad Reichenhall (Germany, 2006), and the collapse of the Minnesota Vikings membrane stadium (USA, 2010). The sudden increase in the snowfall and unbalanced snow distribution, which leads to an excessive snow load in a local area, are the main reasons for such collapses [1]. Unfortunately, the snow load requirements are only provided for the design of buildings with certain simple roofs (e.g., flat roof, pitch roof, and gable roof, among others) in different national load codes [2,3], and few requirements are available for long-span structures. It is usually suggested that the snow load distribution coefficients for large or special-shaped buildings be defined following specific research or experiments in certain national codes [4,5].

In terms of the related special research methods, the computational fluid dynamics (CFD) method and wind tunnel experiment are usually adopted for the practical design of snowdrifts on or around buildings [6,7]. In the field of CFD research, Tominaga et al. [6] analyzed the snowdrift around an actual apartment building based on the revised *k-*ε model. In this study, the transport equation of snow concentration was solved only for snow particles suspended in the air. Thiis et al. [8,9] predicted the snow distribution on the curved roof of a sports hall under specific weather conditions. Generally, the overall snowdrift pattern fitted the measured result well and the location of end effects was reproduced close to the side edge of the roof. Beyers et al. [10] simulated the snowdrift development around a group of surface-mounted buildings and elevated structures, respectively. The analysis assisted the conceptual building design process to manage potential snowdrifts on and around these structures. The application of the CFD technique to snowdrift problems can provide detailed information on the relevant flow and phase variables in the entire calculation domain under well-controlled conditions and at a low cost. Unfortunately, the numerical technique is mainly applied to the snowdrift on small-scale roofs with usual shapes or simplified long-span roofs, as the large scale and fine structural details of complex roofs require higher resolution and higher quality grids. Cases with finer grids would be more time consuming, and the accuracy will be reduced if the grid quality is poor. Therefore, rare work was carried out on the snowdrift on complex long-span roofs by using the CFD method.

In the field of experimental research, Isyumov et al. [11] examined snowdrift formation on the lower level of a large-area two-level roof in different-surface shear stress and terrain roughness conditions. Delpech et al. [12] explored the hazard and alleviating measures of the snowdrift around the Concordia Antarctic research station by using real snow particles in the Jules Verne climatic wind tunnel. Flaga et al. [13] performed a series of snow load tests for three different complex roofs of sports facilities. Snow precipitation and wind-induced redistribution were simulated by using powdered polystyrene foam as artificial snow. The results of snow load distributions were presented for the practical structural design. Compared with the CFD method, this experiment can reproduce the snowdrift mechanism to the greatest extent and restore every detail of the building. Unfortunately, the required equipment is not always available and is usually expensive for experimental preparation and model creation. Therefore, it greatly limits the large-scale application of the experimental method in the study of snowdrift problems.

In order to analyze and summarize the snow load information on complex long-span roofs more efficiently and systematically, assisting the design process of the structure of a building, it is necessary to combine the advantages of CFD and experiment methods and avoid their limitations. Specifically, the distribution of snow load on a complex long-span roof should be analyzed in detail by using experiments. Based on the results, the distribution characteristics could be summarized systematically, before helping reduce the computing requirements for the snowdrifts on complex roofs and the large-scale analytical studies by using the CFD method. As a preliminary step, it is important to make clear the snow distribution characteristics on a complex long-span roof.

In order to express the distribution characteristics in detail, this study firstly presents the validation of the similarity criterion to figure out the experimental theory, based on an air–snow–wind combined experimental system. Thereafter, based on the similarity criterion, a description of the wind-induced snow drifting on a long-span structure with a membrane roof under different wind direction conditions is provided. Finally, according to the experimental results, the snowdrifts' characteristics on the complex membrane roof are analyzed. Three basic characteristic distribution modes and wind direction series are decomposed and derived from the results. The distribution pattern and wind direction coefficients of each mode are investigated with reference to the roof form.

#### **2. Validation of Experimental Approach**

#### *2.1. Experimental Facility*

Experimental works were carried out based on an air–snow–wind combined experimental system, which allows for the appropriate creation of natural wind velocity and turbulence profiles, as well as the precipitation environment. The snow–wind tunnel facility used belongs to the Key Laboratory of Structures Dynamic Behavior and Control of China Ministry of Education, Harbin Institute of Technology, Harbin, China, as shown in Figure 1. The facility was improved on the basis of previous research [14]. The snow–wind tunnel test chamber was closed, and the dimensions were 10 × 4.5 × 3 m. The temperature inside the chamber was the same as the outdoor air temperature, which ranged from −30 ◦C to −10 ◦C during winter. The wind velocity was measured with a hot-wire anemometer.

**Figure 1.** Schematic view of the experimental facility.

Artificial snow particles were used to simulate the snowdrift environment to deal with the combined wind–snow engineering problems (Figure 2a). The artificial snow particles were generated by spraying water droplets into freezing air with a snowmaker. A particle feeder was fixed at the top of the chamber to simulate precipitation. The snow particle feeder (Figure 2b) consisted of a stable steel frame and vibrating sieve with a perforated bottom (bottom size: 4.5 m long, 0.4 m wide; hole diameter: 5 mm). The sieve was moved by a motor with continuous speed regulation. The snowdrift flux profile was measured with a snow particle counter [15].

**Figure 2.** Snowfall simulator: (**a**) artificial snow particles; (**b**) snow particle feeder.

#### *2.2. Similarity Criterion*

The snowdrift phenomenon can be explained as solid particle transport. Depending on whether the friction velocity reaches a threshold, three particle transport mechanisms can be observed: creep, saltation, and suspension. Creep is a phenomenon in which snow particles move by rolling, sliding, or creeping at the surface; saltation is a process in which snow particles move with repeated leaping up or jumping and colliding with a snow surface; and at a higher wind velocity, particles are transported upwards by turbulent eddies and transported far downwind. Among the mechanisms, saltation has been identified as the major particle transport process, which causes approximately 67% of the total drifting mass [12]. For reliable modeling of these transport mechanisms, a reasonable similarity criterion is necessary for reduced-scale experiments. As the similarity criterions are incompatible with one another, compromises have to be made according to their relevance [12,16]. Even so, none of the models cited in the literature are able to provide a satisfying interpretation of experimental results.

In this study, considering that the experimental modeling was focused on the reliable preproduction of snowdrift on a long-span roof, the similarity number based on the similarity of the drifting snow flux in the saltation layer was selected. These similarity criteria were first introduced by Iversen [17] and also adopted by Delpech [12]. The similitude criterion based on the drifting flux is shown in Equation (1).

$$(\rho/\rho\_{\mathbb{P}})(\mathcal{U}^2/2\mathbb{g})(1-\mathcal{U}\_0/\mathcal{U})\tag{1}$$

where ρ and ρ<sup>p</sup> are the air and particle densities, respectively; *U* and *U*<sup>0</sup> are the reference wind velocity and threshold reference velocity, respectively; and *g* is the gravitational acceleration. Furthermore, similarity criteria related to geometric, dynamic, and kinetic criteria were also considered.

To validate the similarity criterion, snowdrift around a surface-mounted cubic model, employed in detailed field measurements carried out by Oikawa et al. [18] in Sapporo, Hokkaido, Japan, was adopted as the analyzed prototype. The snowdrift was observed for only one day by cleaning up the snow around the model following each drifting snow event. The model was 1.0 m on each edge. The averaged wind velocity was approximately 1.7 m/s at a 1.0 m height, and the maximum wind velocity was close to 4.3 m/s (Figure 3). The snow depth at a reference point was 20 cm. The other measurement parameters are summarized in Table 1. The smallest undulation of snowdrift due to the weak wind velocity and the large snowfall make it the most suitable measurement data for validation of the similarity criteria.

**Figure 3.** Wind speed *U*<sup>h</sup> and direction θ<sup>h</sup> at 1.0 m height for the entire measuring period on SN09 [18].

**Table 1.** Prototype drifting parameters.


The artificial snow particles were selected for the validation experiment. The air/water ratio in the snowmaker was set to make sure the artificial snow was dry enough to prevent the particles from sticking together. The accumulated snow density was measured with a scale and cylinder; whereas, the particle diameter was measured with a microscope. The snowfall velocity was obtained by measuring the artificial particle falling times from 1.0 m height [8]. The threshold friction velocity was estimated according to an empirical formula [1]. The parameters of the artificial snow particles are summarized in Table 2.


**Table 2.** Parameters relating to the physical properties of artificial snow.

The selected model scale of 1/2 satisfied the blockage effects. The vertical profiles of the flow field identified over the snow mantle in the test section were similar to these over the measured field in Sapporo. In order to assess the modeling reliability, the results of the typical similarity numbers for both the prototype and model are summarized in Table 3. The experimental wind velocity (2.3 m/s at the height of the model top) used for calculating the scaled model values was determined according to Equation (1). Requirement 2 in Table 3 represents the geometric similarity of both model and prototype. The correct modeling of the ejection process of particles is realized by satisfying requirements 3, 4, and 5. Requirement 5 is a roughness–height Reynold number, where ν is the fluid's kinematic viscosity. The basic requirements of dynamic similarity could be measured with densimetrical Froude numbers 6 and 7 (ratio between the inertia force and gravity force), and similarity number 8.


In order to model the fully rough saltation flow, it is desirable to guarantee the lower limit of the Reynolds number *U*<sup>∗</sup>3/2*g*ν > 30. If the saltation mechanism occurs, the fully rough flows will be satisfied if *U*∗<sup>3</sup> <sup>t</sup> /2*g*ν > 30 [16]. This lower limit was satisfied in this experimental case as shown in Table 3. Except for this, the noticeable mismatches are found in the particle ejection process scaling and dynamic similarity. For the particle ejection scaling, the gravitational force is overestimated in requirement 4 with a greater particle density, which is also observed in requirement 8 with a higher snowfall velocity. This indicates that the trajectory of the artificial snow particle is smaller than the natural snow particle one [19]. However, a higher particle/air density ratio is usually required [19] and the saltation trajectory for the scale model is small in comparison with the actual snowdrift observed [20]. For the dynamic similarity, the particulate Froude number weighted by the density ratio (requirement 7) is the parameter that allows for assessing the similarity of the transport mechanism of suspended particles [12]. The evaluation of the particulate Froude number based on the threshold friction velocity (requirement 6) is linked to surface transport. According to comparisons of requirement 6 and requirement 7, the saltation mechanism, which has been identified as the major particle transport process, is better reproduced than other transport processes [12].

The prototype snowstorm duration was assumed to be 24 h according to the observation duration [18]; whereas, the experimental snowstorm duration was set to approximately 8 h according to the definition for the dimensionless time used by Delpech [12]. As the experiment would have been excessively lengthy, the test was divided into four stages to allow the fan to rest and refill the snow particles into the feeder.

#### *2.3. Results and Discussion*

Figure 4 compares the dimensionless snow depth distributions obtained from the field measurement and experiment. The snow depths were normalized by the reference snow depth far from the model, which was not affected by the flow around the cube. The deep-colored part in the figure indicates greater snow coverage. The deposition areas in the upwind region ahead of the cube and erosion areas near the upwind corners of the model in the experiment correspond strongly to those of the prototype. However, as the snow particle feeder was set in the upwind region, the air–snow flow originated from the upwind direction. The incoming particles bypassed the model and moved downstream following the separating airflow. Few particles could enter the wake region behind the model with the aid of the vortex. Therefore, the deposition region behind the model was not reproduced in the experiment. Therefore, the building size should be limited to ensure that the entire building was covered by the stable air–snow flow field, on the premise of satisfying the drifting similitude.

**Figure 4.** Comparison of horizontal distributions of normalized snow depths: (**a**) prototype result [18]; (**b**) experimental result.

#### **3. Experimental Research on Snowdrifts on Complex Long-Span Roof**

#### *3.1. Experimental Setup*

#### 3.1.1. Test Model

For buildings with a special shape, complex separation and reattachment occur when airflow passes over the roof. The snow load on the roof will be redistributed easily, which may change the stress inside the roof structures, thereby leading to building collapse, especially for long-span lightweight roofs. Therefore, significant attention should be paid to the snow distribution on complex-shaped buildings. Based on the advantages of the experimental method and the validated similarity criterion, a series of snow–wind tunnel tests were conducted to investigate the snowdrift on such complex long-span roofs. In this study, Tongren Olympic Sports Center Stadium was selected as the target building. As the stadium is located on the Yunnan-Guizhou Plateau, China, the frequent snowfall requires additional consideration. The geometric shape of the roof is illustrated in Figure 5a. The cable membrane structure was adopted for the prototype roof. In order to avoid the excessive deformation of the roof, which may affect the safety of the structure and comfortableness of the space, the roof deformation should be strictly controlled at the structural design stage. Therefore, the roof deformation under the action of snow load was assumed to be negligible. Furthermore, considering that the unique roof shape substantially affects the shape of the snow cover, acrylonitrile butadiene styrene (ABS) plastic was used for creating the model roof to reproduce the complex shape as far as possible. Similar

hard material was also used by Flaga et al. [13] to make the membrane structure models. Perforated plates were adopted for the sidewall, with an air permeability of approximately 40% to realize the simulation of the wall air permeability (Figure 5b). Based on the above conclusion, the model scale of 1/150 was selected with model size limited to 1.81 (x) × 1.69 (y)0.33 × (z) m.

**Figure 5.** Model of membrane structure gymnasium: (**a**) Tongren Olympic Sports Center Stadium model (made of acrylonitrile butadiene styrene (ABS) plastic); (**b**) wall of membrane structure (40% air permeability).

The measuring points were designed according to the special shape of the roof (Figure 6a), combined with the distribution characteristics whereby snow would be deposited in the concave region more easily. Two types of concave regions were formed on the roof: along the radial cable and near the center of each membrane piece. For the concaves along the radial cables, 36 lines were set along the cables, with 16 measuring points set along each line (Figure 6b). For the concaves formed on the membrane, 36 lines were set along the circumferential direction, with an additional 9 measuring points set at the center of each membrane piece (Figure 6b). In total, 900 measuring points were set on the model roof. The snow depth at the measuring point was measured with a snow stick. Furthermore, a snowdrift experiment with an empty field was conducted to simulate the snow distribution on the ground. The shape coefficient μ of the snow distribution on the roof was calculated as the ratio of the snow depth on the model roof to the snow height on the ground.

**Figure 6.** Arrangement of measuring points for snow depth: (**a**) roof form; (**b**) arrangement of measuring points.

#### 3.1.2. Setup Parameters

The prototype stadium was constructed in a suburban area of Tongren City, Guizhou province, China. The reference wind pressure is 0.35 kN/m<sup>2</sup> (100 year return period); whereas, the reference snow pressure is 0.35 kN/m2 (100 year return period) [2]. The prototype wind velocity at a standard height (10 m) during a snowstorm is assumed as 0.45 of the reference wind velocity, calculated based on the reference wind pressure [21]. The air density there is about 1.22 kg/m3, and hence the reference wind velocity is close to 24 m/s. The snow density is about 150 kg/m3, and the maximum snow depth during a snowstorm could reach 0.23 m [2]. The threshold friction velocity and snowfall velocity are set to 0.15 m/s and 0.2 m/s, respectively. Regarding the snowfall duration, as no outdoor measuring record was available; the experimental duration was provided based on an eight-year-long outdoor measurement in China by the authors. This work was carried out from 2010 to 2017 and is ongoing. The measurements included the snowfall duration, interval time between two snowfalls, accumulated snow density, and snow depth. Based on the measurement data, a single snowfall usually lasted for nearly 6 to 7 h. The probability of snowfall lasting for less than 12 h was approximately 72% (Figure 7). Therefore, the prototype duration was set as 12 h.

**Figure 7.** Probability density curve of single snowfall duration, derived from an eight-year field measurement result.

The artificial snow particles were selected as the experimental particles. Prior to the experiment, the snow was sieved into uniform sphere particles. Based on the similarity criterion introduced in Section 2.2, the experiment reference velocity was 1.2 m/s at 0.06 m, and the experimental snowstorm duration was set to 0.6 h. A total of 700 L of artificial snow was poured into the sieve-like feeder during the experimental duration, and the snowfall flux was approximately 0.007 kg/m2s. As this stadium exhibited biaxial symmetry and the wind direction frequency was provided with a 22.5◦ interval in the annual wind rose diagram, five cases were designed and conducted for different inflow directions at 22.5◦ intervals, namely 0◦, 22.5◦, 45◦, 67.5◦, and 90◦, from the x-axis to y-axis (Figure 8).

**Figure 8.** Study cases conducted for different inflow directions at 22.5◦ intervals (0◦, 22.5◦, 45◦, 67.5◦, and 90◦).

#### *3.2. Results and Discussion*

The snowdrift shape coefficients μ on the membrane roof in different inflow directions are illustrated in Figure 9. The snowdrifts are extremely complex. For different inflow conditions, depositions occur on the windward and leeward surfaces along the air stream. The overall packing forms are approximately fan-shaped. When the inflow direction moved from 0◦ to 90◦, the angle between the windward and leeward depositions varies from 90◦ to 140◦. The results are similar to the unbalanced snow distribution on the umbrella-shaped roof, as indicated in the technical specification for cable structures (Figure 10a) [22]; whereas, in the experimental case, the angle between the depositions changed under asymmetrical conditions because of the non-central symmetrical roof form. Owing to the existence of the concave regions along the radial cables and at the center of each membrane piece, a large amount of snow is accumulated locally and a peak value of approximately 1.8 is generated. This is substantially larger than the peak value of 1.0 for the umbrella-shaped roof, but much closer to the maximum coefficient of the unbalanced distribution on the multi-span gable roof (Figure 10b) [2]. Therefore, the local accumulations that formed on the roof are likely caused by the local form of the roof itself. Evidently, the snow distribution pattern is closely related to the overall and local roof shapes.

(**a**) (**b**)

**Figure 9.** *Cont.*

(**e**)

**Figure 9.** Snow distribution shape coefficient μ: (**a**) 0◦; (**b**) 22.5◦; (**c**) 45◦; (**d**) 67.5◦; and (**e**) 90◦.

**Figure 10.** Unbalanced distributed shape coefficient on building roofs: (**a**) shape coefficient for umbrella-shaped roof indicated in the technical specification for cable structures; (**b**) shape coefficient for multi-span gable roof indicated in load code for design of building structures.

#### **4. Analysis of Snow Distribution Characteristics**

#### *4.1. Analytical Method*

To deeply analyze the relationship between the snow distribution pattern and the shape characteristic of the roof, the empirical orthogonal function (EOF) analysis was adopted. This approach is generally used to analyze the structural features in data and extract major data feature quantities. Specifically, EOF analysis can decompose a variable field matrix **X** that changes with time into a time-independent spatial matrix **EOF** and a time matrix **PC**, as shown in Equation (9). In this study, the time-dependent field matrix was replaced by a field matrix that changed with the wind direction, and a wind direction related matrix could be obtained instead of the time matrix. The spatial matrix **EOF** generalizes the geographical distribution characteristics of the field; whereas, the wind direction matrix **PC** is composed of the linear combination of coefficients of the spatial points of the field. As the main information of the original field **X** is concentrated in the first several components of the spatial matrix **EOF**, the study of the original field **X** changing with wind direction can be converted into the investigation of the first components of the spatial matrix **EOF** and their wind direction matrix **PC**, respectively.

$$\mathbf{X} = \mathbf{EOF} \times \mathbf{PC} \tag{9}$$

The specific method is illustrated in Equation (10)–(14). Firstly, a field matrix **X***m*×*<sup>n</sup>* is obtained by collating the measured data, where the subscript "*m*" represents the number of spatial points under a certain inflow direction condition, namely 900 and the subscript "*n*" represents the number of inflow cases, namely five. Based on the orthogonal decomposition theory, the spatial matrix **EOF** and its eigenvalue matrix **Λ** can be derived, as shown in Equation (11)–(13), where **C** is the cross-product of the field matrix **X** and its transpose matrix **X***T*. The *k*th column of the spatial matrix **EOF** represents the *k*th spatial mode **EOF***k*. Finally, the wind direction matrix **PC** can be derived from the spatial matrix **EOF** and original field matrix **X**, as indicated in Equation (14).

$$\mathbf{X}\_{m\times m} = \begin{bmatrix} \mathbf{x}\_{11} & \mathbf{x}\_{12} & \dots & \mathbf{x}\_{13} \\ \mathbf{x}\_{21} & \mathbf{x}\_{22} & \dots & \mathbf{x}\_{23} \\ \dots & \dots & \dots & \dots & \dots \\ \mathbf{x}\_{m1} & \mathbf{x}\_{m2} & \dots & \mathbf{x}\_{mm} \end{bmatrix} \tag{10}$$

$$\mathbf{C}\_{m \times n} = \frac{1}{n} \mathbf{X} \times \mathbf{X}^T \tag{11}$$

$$\mathbf{C}\_{m \times n} \times \mathbf{EOF}\_{m \times m} = \mathbf{EOF}\_{m \times m} \times \Lambda\_{m \times m} \tag{12}$$

$$\mathbf{A}\_{m \times m} = \begin{bmatrix} \lambda\_1 & 0 & \dots & 0 \\ 0 & \lambda\_2 & \dots & 0 \\ \dots & \dots & \dots & \dots \\ 0 & 0 & \dots & \lambda\_m \end{bmatrix} \tag{13}$$

$$\mathbf{PC}\_{\mathbf{m}\times\mathbf{n}} = \mathbf{EOF}\_{\mathbf{m}\times\mathbf{m}}^{\mathrm{T}} \times \mathbf{X}\_{\mathbf{m}\times\mathbf{n}} \tag{14}$$

#### *4.2. Spatial Matrix Empirical Orthogonal Function (EOF)*

According to the EOF theory, each spatial mode *EOFk* corresponds to an eigenvalue λ*k*. The contribution of *EOFk* to the total variance is determined by its eigenvalue λ*k*; therefore, the significance of *EOFk* can be defined by its eigenvalue. The larger the eigenvalue is, the more significant its influence is. Furthermore, if the eigenvalue error ranges (λ*<sup>k</sup>* − Δλ, λ*<sup>k</sup>* + Δλ) between two modes overlap, the mode characteristics will be similar. Here, Δλ*<sup>k</sup>* is determined according to Equation (15), where *N*<sup>∗</sup> represents the effective degrees of freedom of data.

$$
\Delta\lambda\_k = \lambda\_k \sqrt{\frac{2}{N^\*}}\tag{15}
$$

Figure 11 illustrates the eigenvalues of the first few spatial modes. The first three spatial modes (*EOF*1, *EOF*2, and *EOF*3) contribute the most to the total variance; therefore, only the characteristics of these three spatial modes are analyzed in the following. Compared with λ<sup>2</sup> and λ3, the λ<sup>1</sup> value for *EOF*<sup>1</sup> leaps significantly. Evidently, *EOF*<sup>1</sup> has an overwhelming influence on the snow distribution on the stadium roof. The eigenvalues for *EOF*<sup>2</sup> and *EOF*<sup>3</sup> are similar, but the significance of *EOF*<sup>2</sup> is slightly more pronounced. Furthermore, a relatively large overlap exists in the error ranges between *EOF*<sup>2</sup> and *EOF*3, indicating that their distribution patterns are similar.

The distribution characteristic of *EOF*<sup>1</sup> is illustrated in Figure 12. The values of *EOF*<sup>1</sup> are all positive. The distribution pattern is indicated by the strip of local accumulations formed in the concave regions along the radial cables, similar to the unbalanced snowpack formed in the eave region between multi-span gable roofs (Figure 10b). In general, *EOF*<sup>1</sup> further reflects the influence of the local roof shape on the snow load distribution on the building roof. The distribution characteristics of *EOF*<sup>2</sup> and *EOF*<sup>3</sup> are illustrated in Figure 13. The spatial characteristics of *EOF*<sup>2</sup> and *EOF*<sup>3</sup> are similar, as discussed previously; that is, the overall surface accumulations along the symmetry axes (0◦ or 90◦ inflow direction) and skew surface accumulations between the axes (45◦ inflow direction). These patterns are similar to the fan-shaped depositions on the umbrella-shaped roof (Figure 10a), reflecting the influence of the overall roof shape on the snow load distribution. Moreover, the magnitude of the shape coefficients for *EOF*<sup>2</sup> and *EOF*<sup>3</sup> are positive or negative in two perpendicular regions, taking on an opposite distribution pattern with a different inflow direction (the black or gray arrow in Figure 13). Therefore, *EOF*<sup>2</sup> and *EOF*<sup>3</sup> comprehensively reflect the influences of the inflow direction and overall roof shape on the snow load distribution.

**Figure 11.** Eigenvalues of each empirical orthogonal function (*EOF*) mode at 95% confidence level.

**Figure 12.** Distribution characteristics of the first spatial mode *EOF*1.

**Figure 13.** Distribution characteristics of the second and third spatial modes: (**a**) *EOF*2; (**b**) *EOF*3.

#### *4.3. Wind Direction Matrix PC*

As a group of the wind direction-dependent weighted coefficients for the spatial mode, the **PC** coefficient reflects the influence degree and the combination way of the spatial modes. The specific **PC** coefficients for *EOF*1, *EOF*2, and *EOF*<sup>3</sup> are illustrated in Figure 14. Under the five inflow direction conditions, which range from 0◦ to 90◦, the proportion of the **PC** coefficients for *EOF*<sup>1</sup> is the largest, indicating an overwhelming influence, as described earlier. The value fluctuates slightly depending on the wind direction and basically remains at 22.5. For *EOF*<sup>2</sup> and *EOF*3, the **PC** coefficients vary sharply with the wind direction, that is, the **PC** coefficients are sensitive to the wind direction. Although the coefficients fluctuate dramatically, the curves of **PC** coefficients are symmetrical along the *PC* = 0 axis with varying inflow directions. Specifically, when the inflow originates from the 0◦ direction, the inflow direction is consistent with that of the negative state of *EOF*<sup>2</sup> (gray inflow direction in Figure 13a). The overall snow distribution reflects the characteristics of *EOF*2, while the **PC** coefficient for *EOF*<sup>3</sup> is close to zero. When the inflow originates from the 45◦ direction, the inflow direction is consistent with that of the negative state of *EOF*<sup>3</sup> (gray inflow direction in Figure 13b). The overall snow distribution reflects the characteristics of *EOF*3, while the **PC** coefficient for *EOF*<sup>2</sup> is close to zero. Finally, when the inflow moves to 90◦, the overall distribution characteristics can be explained only by *EOF*2; whereas, the influence of *EOF*<sup>3</sup> could be neglected. Overall, the local roof shape has the greatest influence on the snowdrift on the complex roof; whereas, the contribution of the whole roof shape to the snowdrift depends on the dominant wind direction. The closer the wind direction represented by the spatial mode is to the dominant wind direction, the greater its contribution to the overall result.

**Figure 14.** Wind direction coefficients for *EOF*1, *EOF*2, and *EOF*3.

#### *4.4. Combination of EOF and PC*

Since the original field matrix **X** can be decomposed into a spatial matrix **EOF** and a wind direction matrix **PC**, conversely it is possible to generate the snow distribution on this complex membrane roof through the combination of the spatial matrix **EOF** and wind direction matrix **PC**. As the first three spatial modes made the greatest contribution to the snow distribution on this structure and reflected the influences of the local and overall roof shapes, respectively, the three modes were selected for combination. The combination method is shown in Equation (16). Each column of the generated matrix **X** represents the snow distribution under a specific wind direction condition.

$$\mathbf{X}'\_{m \times n} = \mathbf{EOF}\_{m \times 3} \times \mathbf{PC}\_{3 \times n} \tag{16}$$

The generated snowdrifts in different inflow directions are shown in Figure 15. Through comparison with the experimental results as shown in Figure 9, the snowdrift patterns, i.e., the overall fan-shaped deposition occurring on the windward and leeward surfaces along the air stream and the local packing formed in the concave regions along the radial cables and at the center of each membrane piece, show good correspondence with the experimental results. However, it should be noted that a significant underestimation of the shape coefficient values, especially the peak values, is observed. This underestimation was mainly caused by the fact that only the first three modes participated in the combination, and the contribution of the latter modes to the snow distribution was not considered

yet. In general, it is preliminarily verified that the snowdrift on a complex roof can be restored by the combination of snow distributions on several corresponding simple roofs.

**Figure 15.** Generated snow distribution shape coefficient μ: (**a**) 0◦; (**b**) 22.5◦; (**c**) 45◦; (**d**) 67.5◦; and (**e**) 90◦.

In comparison with the conventional methods, the existing national load codes only provide the snow load requirements for simple roofs, as mentioned in Section 1. The applications of the CFD and experimental methods in the prediction of snow loads on complex roofs are also limited, due to the characteristics of time and money consumption. However, this study provides a simplified idea of estimating the snow load on a complex roof. Specifically, in the structural design stage, it is possible for the snow load on a complex roof to be estimated preliminarily by the combination of the snow distributions on several simple roofs, which have been provided in the load code (e.g., dome, curved roof, pitch roof, and obstacle, among others) or can be obtained by numerical studies. This will save a lot of time and money. Unfortunately, as an initial exploration, there are still many problems to be solved. Firstly, the correct selection of simple roofs plays a decisive role in determining the accuracy of combinational results, therefore, the selection method of the corresponding simple roofs should be studied emphatically; secondly, the **PC** coefficients are closely related to roof forms, and a large number of in-depth and large-scale studies should be carried out to clarify the **PC** coefficient; finally, reliable measurement data of snowdrifts on actual complex roofs is indispensable for examining prediction results. However, there is no measurement data available now, due to a large number of works and high risks in the process of measurement. These problems should be considered in future investigations.

#### **5. Conclusions**

This study has demonstrated the feasibility of snowdrift reproduction based on the air–snow–wind combined experimental system and the experimental approach. A series of scaled tests were carried out to predict the wind–snow flow behavior on a complex membrane roof. The snow distribution characteristics were decomposed and analyzed, and the following results were obtained. (**i**) The similarity criterion based on the drifting snow flux in the saltation layer was proved to simulate the overall snow distribution effectively. Based on the validated similarity criterion, the snowdrift distributions on a complex long-span membrane roof under different inflow directions were reproduced by using artificial snow particles. The results preliminarily explained the influence of the roof shape and wind direction on the snowdrift, and illustrated the complexity of the snow distribution on the complex roof, compared with that on a simple roof. (**ii**) Based on the EOF analysis, the snow distribution on the complex roof under different inflow direction conditions was broken down into several spatial distribution modes and wind direction-dependent weighted coefficients. Through the analysis of spatial modes, it was proven that the snow load distribution on a complex roof can be broken down into an integral surface load and local concentrated load. Combined with the roof pattern, it was demonstrated that the snow distribution on a complex roof can be decomposed into the snow distributions on several simple roofs according to the specific roof form. (**iii**) Through the significance analysis of the main spatial modes, it was demonstrated that the local roof shape has the greatest influence on the snowdrift on the complex roof; whereas, the contribution of the whole roof shape to the snowdrift depends on the dominant wind direction. The closer the wind direction represented by the spatial mode is to the dominant wind direction, the greater its contribution to the overall result. In practice, when no reference is available in the load code, the actual roof snow load can be estimated preliminarily by reasonable combinations of the existing simple roof snow loads in the codes, referring to the specific roof form and the dominant wind direction.

**Author Contributions:** Conceptualization, F.F. and S.S.; methodology, G.Z.; software, G.Z.; validation, G.Z., Q.Z., and F.F.; formal analysis, G.Z.; investigation, G.Z.; resources, F.F.; data curation, Q.Z.; writing—original draft preparation, G.Z.; writing—review and editing, G.Z.; visualization, Q.Z.; supervision, F.F.; project administration, S.S.; funding acquisition, F.F.

**Funding:** This was funded by the Chinese National Natural Science Foundation project (51927813, 51978207) and the National Science Fund for Distinguished Young Scholars (51525802).

**Acknowledgments:** The authors are grateful to the members of the Space Structures Research Center at the Harbin Institute of Technology, for providing invaluable information and advice in this study.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Temperature Distribution Characteristics of Concrete during Fire Occurrence in a Tunnel**

#### **Seungwon Kim 1,\*, Jaewon Shim 2, Ji Young Rhee 2, Daegyun Jung <sup>1</sup> and Cheolwoo Park 1,\***


Received: 9 October 2019; Accepted: 3 November 2019; Published: 6 November 2019

**Abstract:** Fire in a tunnel or an underground structure is characterized by a rise in temperature above 1000 ◦C in 5–10 min, which is due to the characteristics of the closed space. The Permanent International Association of Road Congresses has reported that serious damage can occur in an underground structure as a consequence of high temperatures of up to 1400 ◦C when a fire accident involving a tank lorry occurs in an underground space. In these circumstances, it is difficult to approach the scene and extinguish the fire, and the result is often casualties and damage to facilities. When a concrete structure is exposed to a high temperature, spalling or dehydration occurs. As a result, the cross section of the structure is lost, and the structural stability declines to a great extent. Furthermore, the mechanical and thermal properties of concrete are degraded by the temperature hysteresis that occurs at high temperatures. Consequently, interest in the fire safety of underground structures, including tunnels, has steadily increased. This study conducted a fire simulation to analyze the effects of a fire caused by dangerous-goods vehicles on the tunnel structure. In addition, a fire exposure test of reinforced-concrete members was conducted using the Richtlinien für die Ausstattung und den Betrieb von Straßentunneln (RABT) fire curve, which is used to simulate a tunnel fire.

**Keywords:** RABT fire curve; fire simulation; tunnel fire; high temperature; fire safety; fire accident

#### **1. Introduction**

With the recent sharp increase in accidents involving transport vehicles carrying hazardous materials (e.g., explosive flammables), damage to highway infrastructure facilities, such as tunnels, has increased substantially [1,2]. Detailed summaries of road and rail tunnel fire events clearly show the importance of considering fire risk in the design of tunnels [3,4]. In particular, due to the extensions and elongations of tunnel structures and increasing passage access to popular town areas, it is urgently necessary to ensure the safety of tunnel structures against unexpected extreme disasters, such as fires [1,2]. For this reason, many developed countries are currently enhancing fire intensity standards and reviewing explosion resistance standards. In most countries, however, the assessment of, and response to, fire risks is still limited, and the maturity of the design goal is relatively low [1,2].

In 2001, two trucks collided in the Gotthard tunnel in Switzerland, resulting in a fire, as well as 11 deaths and many injuries [5]. The scale of the fire was approximately 120–200 MW, and the flame temperature was estimated at over 1000 ◦C [5]. The fire brigade experienced difficulty in accessing the fire scene for 48 h, causing damage over a length of 700 m inside the tunnel and spalling of up to 350 mm in depth [5]. The damage caused by this accident amounted to approximately \$31 billion, and restoration work lasted two months [5]. Even though the authorities were equipped with the latest

disaster prevention facilities at the time of the accident, the damage was considerable, and this clearly demonstrates the importance of the response as well as of the preparation of operational facilities [2].

This study analyzed fire intensity and the effects of fire on tunnel structures in terms of depth by simulating fire occurrences resulting from accidents of tank lorries loaded with inflammables in a tunnel environment. Furthermore, in order to verify the validity of the numerical analysis model, the effects of geometrical elements on fire were examined by a fire exposure test on reinforced-concrete members using the Richtlinien für die Ausstattung und den Betrieb von Straßentunneln (RABT) fire curve

#### **2. Characteristics of Fire in a Tunnel**

#### *2.1. Material Characteristics of Concrete Exposed to High Temperature*

Exposure to high temperatures results in spalling or destruction of the coating on members due to the water vapor pressure created inside the concrete [2,6–8]. At 100–400 ◦C, Al2O3-, Fe2O3-, and tobermorite-based hydrates are dehydrated, resulting in the collapse of gel and cement hydrates. In moderate-strength concrete that has been exposed to high temperatures, voids are generated as the vapor inside the concrete evaporates at approximately 200 ◦C [2,6–8]. The deformation recovery ability is drastically lowered in this temperature range, and the elastic modulus decreases significantly at temperatures of over 600 ◦C [2,6–8].

Concrete exhibits a tendency toward decreasing compressive strength and elastic modulus when exposed to high temperatures. Concrete exposed to high temperatures causes cross-sectional defects due to surface peeling or scattering [2,6–8]. This phenomenon is called spalling. The main cause of this phenomenon is the water vapor pressure that is generated when the water inside the concrete expands in response to high heat [2,6–8].

It is known that when a concrete structure is exposed to temperatures of approximately 650 ◦C or higher, it loses 50% of its original strength. When it is exposed to temperatures of approximately 850 ◦C or higher, it loses its structural performance [2,6–8].

#### *2.2. Temperature Distribution Characteristics Due to Vehicle Fire in a Tunnel*

According to a report published by the Permanent International Association of Road Congresses on fire and smoke control in road tunnels [9], in the event of a fire in a tunnel, when air flows in through the tunnel entrance at 6 m/s, the temperature of the ceiling reaches approximately 400 ◦C from the spot of the fire to a point approximately 100 m from the tunnel exit [9]. According to Dutch regulations, when a fire occurs in a large tank lorry with a loading capacity of 50 m3 or higher, the temperature rises to approximately 1400 ◦C.

#### **3. Tunnel Fire Simulation**

It is almost impossible to consider all possible fire situations involving dangerous-goods vehicles in experimental assessments and verifications for simulating fires that occur in a road-network system. Various costs and time-consuming limits exist in reality. Therefore, simulation analysis is typically used in such studies, and a real fire experiment is only conducted when necessary to complement the results of the simulation. Fire simulation is generally conducted through computational fluid dynamics (CFD) analysis. CFD analysis is actively applied to fire-modelling research at domestic and international facilities and to establish firefighting design and evacuation parameters [1]. It is the most important tool in fire engineering. Building upon the Field Model developed in the U.K., the National Institute of Standard and Technology (NIST) and the Building and Fire Research Laboratory in the United States have achieved continuous developments in this area since 2000 [1]. CFD analysis, in particular, can be used to examine the thermal-fluid flow phenomenon on a large scale, for which life-sized model experiments are impossible to conduct. It can also quantify the degree of damage, such as the size of the fire, smoke generation, toxic gas generation, and amount of radiation heat [1]. Furthermore, simulation is possible above the normal test-performance limits and can produce results according to specific scenarios, which enables the quantitative assessment of fire risks.

#### *3.1. Modelling*

The Fire Dynamic Simulator (FDS, version 6.5.3) [10], which has been developed by the NIST in the U.S., is the numerical analysis model used for the fire analysis in this study. The target space of the analysis was 45 m × 8 m × 5.5 m (length × width × height), and the length of the tunnel was assumed to be 45 m. It was believed that the length of the tunnel would not be affected by the temperature of the fire. As shown in Figure 1, a box-type cross section was applied to mimic the cross-sectional shape of the tunnel. The fire source was located between the center of the tunnel and its side. Tank lorries of 27 m3 (27,000 L) volume were used with diesel as the fuel. The open-boundary condition was applied at the tunnel entrance and exit. In addition, as in a real situation, the combustion rate method was used to simulate the fire in tank lorries.

**Figure 1.** Dimension of tunnel and position of fire in tunnel modelling.

#### *3.2. Analysis of Fire by Combustion Material Type*

Comparative analyses were carried out on the changes in the fire characteristics according to the duration of ignition sources that were used to simulate fires for different materials in the FDS. The simulations were conducted by modelling for materials with different properties (e.g., combustion heat, density, and ignition point). Based on the results, the appropriate duration of the ignition source was determined and applied to the fire simulation.

Diesel and octane (C8H18), which are the commonly used gasoline types, as well as heptane (C7H16) and ethanol (C2H6O), were the hazardous materials applied to the fire simulation analysis. The values of the heat of combustion for these hazardous materials are 44.80 MJ/kg, 47.89 MJ/kg, 48.07 MJ/kg, and 29.65 MJ/kg, respectively. To analyze the changes in fire intensity with respect to the types of hazardous materials, specific analysis conditions were set to verify the fire intensity inside the tunnel when the fire occurred due to a hazardous material with a similar or lower heat of combustion. Table 1 outlines the model and conditions used for the analysis. Table 2 lists the material properties of the combustion materials.




**Table 2.** Properties of combustion materials considered in this study.

Figure 2 shows the analysis results for different combustion materials. Except for ethanol, whose heat of combustion is 29.65 MJ/kg, diesel, octane, and heptane have similar values for heat of combustion, i.e., 44.80 MJ/kg, 48.07 MJ/kg, and 47.89 MJ/kg, respectively. These three fuels had a similar fire intensity level of approximately 160 MW. However, the results for diesel showed a longer fire duration by approximately 500 s. In comparison to these three fuels, the heat of combustion of ethanol is approximately 60%, and this fuel also resulted in a lower fire intensity. However, the duration of its fire was longer than 1 h, much longer than the fire durations of the other combustion materials.

**Figure 2.** Heat release rates of the tested combustion materials.

The above analysis revealed that different types of hazardous materials on a highway can cause fires with different characteristics (i.e., the fire intensity and duration are likely to differ depending on the material type). The analysis of the simulation results showed that a fire caused by dangerous-goods vehicles carrying diesel results in the most serious conditions of fire intensity and fire duration.

#### *3.3. Analysis of Vehicle Fire in a Tunnel*

A simulation was conducted for a fire caused by dangerous-goods vehicles carrying diesel in a tunnel. The tunnel model used for this simulation was the same as that in Figure 2. Table 3 lists the analysis conditions used for this simulation.


**Table 3.** Vehicle fire analysis conditions.

#### **4. Fire Exposure Experiment on Reinforced-Concrete Member**

In order to verify the analysis of the fire and heat-transfer characteristics of concrete in the event of a fire in a tunnel structure using the CFD analysis program FDS [10], the heat-transfer characteristics of a concrete member were examined through a fire experiment. The results were then compared with those of the CFD analysis. The fire exposure experiment was conducted using the RABT fire curve, which can simulate a tunnel fire for reinforced-concrete members.

#### *4.1. RABT Fire Curve*

The RABT fire curve was developed by the Road Construction Department of the German Ministry of Transportation under the Eureka Project [11]. In a simulated scenario, the temperature sharply rises to 1200 ◦C within 5 min after the beginning of the fire. The durations of fires involving trains and cars at a temperature of 1200 ◦C are 55 min and 25 min, respectively. The fire then cools down for 110 min. The RABT fire curve is known to have a shape similar to that of a real tunnel fire [11]. Figure 3 shows the RABT fire curves for railways and highways.

**Figure 3.** Richtlinien für die Ausstattung und den Betrieb von Straßentunneln (RABT) fire curves of highways and railways [11].

#### *4.2. Dimensions of the Specimen and Experimental Method*

The arrangement of reinforcement bars used in the RABT fire curve fire exposure experiment is the same as the arrangement in real road tunnels. Furthermore, Figure 4 shows the horizontal heating furnace for the high-temperature test used in the fire exposure experiment. Only the bottom surface of the specimen was exposed in this experiment. Ceramic fibers that can endure temperatures of up to approximately 1400 ◦C were installed to provide insulation between the top of the furnace and the specimen. The horizontal heating furnace was designed to allow installation of a rectangular specimen with size of 1400 mm (length) × 1000 mm (width). The actual heating area was 1100 mm (length) × 700 mm (width). The design standard compressive strength of the concrete specimen that was used in this experiment was 27 MPa. Figure 5 shows the experimental setup for the fire exposure test using the RABT fire curve.

**Figure 4.** Horizontal heating furnace for the high-temperature test.

**Figure 5.** Experimental setup for the fire exposure test using the RABT fire curve.

To examine the heat-transfer characteristics using the RABT fire exposure experiment, thermocouples were installed at the center at 0, 20, 40, 60, 80, and 100 mm from the heating surface. Table 4 lists the positions of the thermocouples.



#### **5. Experimental Results and Analysis**

#### *5.1. Analysis of Vehicle Fire in the Tunnel*

When a fire source corresponding to 27,000 L of diesel was located at the center of the tunnel, the fire intensity was approximately 150 MW, as shown in Figure 6. Figure 7 shows the results of the temperature measurements on the concrete surface and at depths of 20, 40, 60, and 100 mm. The analysis results for the concrete hydrothermal temperatures at each depth showed that for a fire of such a scale (27,000 L), the surface temperature rose to approximately 1000 ◦C at approximately 300 s after the occurrence of the fire. This high temperature was maintained for approximately 1800 s.

**Figure 7.** Temperature distribution by depth based on vehicle fire analysis.

The peak temperature of the concrete surface was approximately 1380 ◦C. The corresponding temperatures at depths of 20, 40, 60, and 100 mm were 1005 ◦C, 750 ◦C, 405 ◦C, and 110 ◦C, respectively. The International Tunneling Association (ITA) [12] specifies that the maximum critical temperatures for concrete and reinforcement bars should be 380 ◦C and 250 ◦C, respectively. On the basis of these results, the maximum critical temperature standard could not be met at concrete depths higher than 60 mm.

Figure 8 shows the temperature distribution inside the tunnel. Furthermore, the fire exposure experiment was conducted for real concrete members by applying the RABT fire curve where the temperature rose sharply to 1200 ◦C within 5 min. The result was similar to the that of the empirical verification of the tunnel structures' characteristics after exposure to high temperatures [12].

**Figure 8.** Temperature distribution inside the tunnel.

#### *5.2. Fire Exposure Experiment of Reinforced-Concrete Member*

Figures 9 and 10 compare the specimen surface before and after the fire test based on RABT-ZTV (highways) 30 min fire curves and RABT-ZTV (railways) 60 min fire curves. Spalling is a complex process, which occurred in the concrete specimen due to the rapid temperature increase in the furnace. On the basis of the temperature changes in Figures 11 and 12, the result for the left thermocouple shows an abnormal pattern of temperature change with depth. On the basis of the right thermocouple, section loss appeared from the heating surface to 60 mm, and no section loss occurred above 80 mm. The temperature inside the concrete increased sharply depending on the generation of spalling. The temperature measured at approximately 60 mm, where the spalling occurred, was similar to that inside the furnace. At a depth of 80 mm, the temperature was approximately 420 ◦C during the rising period and approximately 540 ◦C during the descending period. However, some differences were evident because the spalling by heat varied depending on the condition of the concrete heating surface. The range of section loss can also be seen in Figures 9 and 10, which show the heating surface after the fire test was completed. Considering the arrangement depth of the reinforcement bars inside the specimen, the section loss was found to be 60–80 mm.

**Figure 9.** Specimen surface before and after the RABT-ZTV (highways) fire experiment.

**Figure 10.** Specimen surface before and after the RABT-ZTV (railways) fire experiment.

**Figure 11.** Temperature measurement results for the RABT-ZTV (highways) fire experiment.

**Figure 12.** Temperature measurement results for the RABT-ZTV (railways) fire experiment.

On the basis of the results of the fire experiments, the appropriate coating thickness of the tunnel structure should be at least 80 mm in order to protect the concrete and internal reinforcement bars from fire. This contradicts the the simulation result that assumed that there had been no spalling in real concrete.

#### **6. Conclusions**

This study analyzed the effects of fire in a tunnel due to an accident involving dangerous-goods vehicles using fire simulation. Furthermore, a fire exposure test of a reinforced-concrete member was conducted using the RABT fire curve, which is employed to simulate a tunnel fire. The conclusions of this study are as follows:


**Author Contributions:** Conceptualization, S.K., J.S., J.Y.R., C.P.; methodology, S.K., D.J.; writing and draft preparation, S.K., C.P.

**Funding:** This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) and funded by the Ministry of Education (grant no. 2017R1A2B4012678) and the Korea Expressway Corporation Research Institute as part of the research project "Simulation and Experimental Simulation for verification of fire and explosion safety measures for vehicle fires".

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Experimental Assessment of the E**ff**ect of Vertical Earthquake Motion on Underground Metro Station**

#### **Zhiming Zhang 1,2, Emilio Bilotta 2,\*, Yong Yuan 3,\*, Haitao Yu <sup>4</sup> and Huiling Zhao <sup>5</sup>**


Received: 27 September 2019; Accepted: 26 November 2019; Published: 29 November 2019

**Featured Application: This research has proven that the ratio of vertical to horizontal peak ground acceleration (RVH) has a significant influence on dynamic soil-structure interaction. It is believed that under extreme earthquake loading, such as near fault zones, RVH is a parameter of paramount importance and should be accounted for in the seismic analyses and seismic performance assessment of underground structures, especially for those with zero or near-zero buried depth, such as an atrium-style metro station. The conclusions of this research are expected to contribute to the revision of codes for seismic design of underground structures.**

**Abstract:** This paper presents experimental assessment of the effect of the ratio of vertical to horizontal peak ground acceleration (RVH) on underground metro station. An atrium-style metro station embedded in artificial soil subjected to earthquake loading is examined through shaking table tests. The experimental results for three different RVH, including soil acceleration, soil-structure acceleration difference, dynamic soil normal stress (DSNS), and structural dynamic strain, are presented and the results are compared with the case of horizontal-only excitation. It is found that for an atrium-style metro station, the differences in horizontal acceleration amplitude between the structure and the adjacent soil rise with increasing RVH, which are different at different depths. The most significant differences occur at the depth of the ceiling slab. It is also observed that both the amplitude and distribution of peak DSNS have obvious differences between the left and right side walls at all levels. It is therefore concluded that the RVH has a significant influence on dynamic soil-structure interaction. It is believed that under extreme earthquake loading, such as near fault zones, RVH is a parameter of paramount importance and should be accounted for in the seismic analyses and seismic performance assessments of underground structures, especially for those with zero or near-zero buried depth, such as atrium-style metro stations.

**Keywords:** vertical earthquake motion; seismic response; atrium-style metro station; shaking table test

#### **1. Introduction**

The three-dimensional behavior of underground structures during earthquakes is a matter of interest for improving the seismic analysis and design in civil engineering under extreme loads. Although extensive studies were carried out on the effects of the horizontal component of the ground shaking, only limited work has been done in the analysis of the effect of the vertical component. This topic deserves attention since some field evidence and investigations on the near-field ground motions have demonstrated that vertical components caused destructive consequences, even possible collapse, for some existing structures [1–4]. The latest developments in the underground vibration or deformation monitoring techniques such as the fiber-optic sensors [5,6] are expected to facilitate both the acquisition of the field evidence and the research work concerning the vertical earthquake motion.

In view of the fact that the ratio of vertical to horizontal peak ground acceleration (RVH) is extremely high in the vicinity of the active faults, it seems essential to study the influence of the RVH on the seismic responses of the underground structures, such as metro stations in the seismic active regions. Through numerical analyses based on the finite element-finite difference method, Uenishi and Sakurai [7] discussed the effect of the vertical oscillations and overburden on the damage-concentration mechanism of the columns at several specific underground station sections in the 1995 Hyogo-ken Nanbu (Kobe) earthquake. On the other hand, An et al. [8] discussed the effect of the vertical motion on the failure mode and dynamic response of the metro station, and concluded that in this case the vertical ground motion was not the primary cause of the collapse.

Research had also been carried out to investigate the influencing factor of RVH. Moore and Guan [9] investigated numerically the dynamic interaction of the twin tunnels subjected to the incident seismic waves. They have shown how the dynamic response of the twin tunnels depended on the incident angle of the earthquake wave, together with the spacing between two adjacent tunnels and the modulus ratio of the tunnel to the surrounding soil. Stamos and Beskos [10,11] provided insights into the dynamic responses of the underground infinitely long lined tunnels buried into an elastic or viscoelastic half-space subjected to the obliquely incident waves using the boundary element method (BEM), and proved the influence of the incident directions of the seismic waves on the dynamic responses of the tunnels. Lee and Trifunac [12] proved that the amplitudes of the stresses and deformations near the tunnel were dependent upon the angle of incidence, which determined the overall trends of the amplitudes. Lee and Manoogian [13,14] adopted the weighted residual method to study the scattering and diffraction of the plane SH-waves by an underground cavity of the arbitrary shape in a two-dimensional elastic half-space, and demonstrated same observations as for circular cavities that ground amplifications depended on the orientation of the incident waves. Wong et al. [15] studied the dynamic response of a cylindrical pipe embedded in an elastic semi-infinite medium using hybrid finite element and eigenfunction expansion techniques. They found that dynamic amplification was significantly dependent on the angle of incidence and the depth of embedment. Huang et al. [16] investigated the impact of the incident angles of earthquake shear waves (SV and SH-waves) on the seismic responses of a long lined tunnel, and numerical results indicated that non-linear seismic responses of the long lined tunnels were highly affected by the incident angles of the S-waves, which should be taken into consideration in mapping the seismic risk of the tunnels. Sun and Wang [17] investigated numerically the seismic responses of the underground rectangular tunnel under the vertical seismic excitation. Numerical results show that under vertical seismic excitation the tunnel experienced greater vertical stress from time to time and vertical compressive deformation, and sometimes it completely separated from the above soil layer.

The effect of the vertical shaking during an earthquake has been experimentally investigated previously through just a few shaking table tests. In terms of the experimental study on the tunnels, Xu et al. [18] investigated the mechanism and effect of the seismic measures of the mountain tunnel through the three-dimensional shaking table tests, where the peak ground acceleration (PGA) of the vertical input motion was taken to be 2/3 of the value corresponding to the horizontal input motion. Experimental results show that with 63% probability of exceedance in 50 years, including vertical components of the earthquakes increased the peak dynamic soil pressures on the tunnel sidewall, while the opposite was true for 1% probability of exceedance in 50 years. In both cases there was no significant change in the peak dynamic strain of the inverted arch on two sides of the flexible joints. Wang et al. [19] also adopted the above-mentioned definition of the horizontal and vertical strategies and investigated experimentally two shallow-bias tunnels with a small clear distance

subjected to the horizontal–vertical earthquake excitations with different levels. Results show that the acceleration amplification factors of the vertical-direction were generally larger (1.02–3.94 times) that that of the horizontal-direction and the earthquake intensity presented different influence on the acceleration responses in the horizontal and vertical directions. In addition, Zhao et al. [20] investigated experimentally the dynamic responses of a tunnel subjected to near-fault pulse-like earthquake motions, where both vertical excitations and transverse-longitudinal-vertical excitations were included.

With respect to the underground metro stations, for example, Zhao et al. [21] studied the vibration behavior of a framed metro station under horizontal and horizontal–vertical seismic excitation using a shaking table. Experimental results show that for the lower shaking intensity the structural relative displacements under the horizontal seismic excitation were lower than those under the horizontal–vertical seismic excitation, while the opposite was true for the greater shaking intensity. It is believed that in the latter case, the vertical seismic excitation increased soil densification and ground subsidence, which in turn imposed greater constraints on the station. Chen et al. [22] investigated the effect of the vertical ground motion on the dynamic response characteristics of the central columns in a six-story metro station and the test results revealed that with the increasing ratio of the vertical/horizontal acceleration amplitude, the central columns would undertake much more vertical dynamic axial forces compared to side walls. Che et al. [23] experimentally investigated the dynamic behaviors of a single-story subway station embedded in the dry sand excited by the vertical sinusoidal and vertical random waves, respectively. They found that in both cases there was difference in the acceleration amplitude of the ground between the free-field model and soil-station model. Some attempts were also made to explore the relationship between seismic soil pressures and shear strains of the surrounding ground based on the regression analysis of the experimental data.

Numerical and analytical studies of the effect of RVH on underground tunnels dominate most of the previous work. Extremely limited studies have been dedicated to study the influence of the RVH or incident angle of earthquake waves on the seismic responses of the underground metro station, especially for the experimental studies. In terms of the underground metro station, the existing several experimental studies only focus on the effect of the RVH on certain kinds of structural responses, such as structural relative displacement and dynamic internal force. Experimental studies of the effect of RVH on more comprehensive seismic responses also including the acceleration response and dynamic interaction of soil-station system, dynamic soil normal stress (DSNS) along the side wall of station, and structural dynamic tensile strain (DTS), are not available in the literature. Due to the limited number of the existing experiments, further experimental studies of the seismic behavior of the soil and metro station under different vertical earthquake actions are needed to draw more general conclusion. Therefore, in this study the effect of RVH on the seismic responses of an atrium-style metro station including the acceleration, dynamic soil normal stress DSNS, and DTS are the primary foci. Through series of 1 g shaking table tests, this paper presents the key experimental findings, and an attempt is made to evaluate the effect of the RVH both qualitatively and quantitatively. In addition, this paper also provides insights into the seismic behavior of the atrium-style underground structures, which will help researchers and practitioners to better understand the seismic aspects of these kind of structures to improve their seismic design and construction.

#### **2. Shaking Table Tests**

#### *2.1. Experimental Facility*

The series of shaking table tests were conducted using the MTS Company shaking table facility at the State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University. The shaking table (Figure 1) has six degrees of freedom: two horizontal (X and Y), one vertical (Z), and three rotational, and the main technical parameters are listed in Table 1. The operating frequency range, maximum input accelerations, and maximum loading capacity all satisfy the experimental demands.

**Figure 1.** Photos of the shaking table and flexible-wall container.


**Table 1.** Main technical parameters of the shaking table system.

A cylindrical flexible-wall container (Figure 1) with diameter 3.0 m and height 1.8 m was adopted. It was made of 5 mm thick rubber membrane. Reinforced bars with a diameter of 4 mm and a spacing of 60 mm were arrayed circumferentially around the exterior of the rubber membrane. The top-ring, base-ring, and four columns were all manufactured from H-shaped steel members. The top-ring was supported by four columns with universal joints, providing the container with full translational and rotational freedom. Furthermore, the rubber membrane was designed to have similar shear stiffness with model soil, in order to minimize any soil-container interactions. The container had been proven to present good lateral shear-type deformations and to be reliable in terms of the negligible influence of the vertical boundary on the specified experiment [24,25].

#### *2.2. Model*

In view that the similitude ratio placed a severe limitation on the choice of suitable model materials, the similitude ratio design would be conducted before choosing the model materials. Based on the capability of the existing facility and equipment, the similitude ratio design of the experiment was accomplished on the basis of the dimensional analysis, which was adopted extensively in the design of the shaking table tests by other researchers. Meymand [26] designed and performed a series of scale model shaking table testing of the piles in clay in a 1 g scale model environment [27], in which he summarized that the method of the governing equations involved the differential equation describing the process and the formation of the similarity variables that related the model to the prototype. The similitude ratio design for this research herein closely follows the aforementioned method.

With respect to dynamics problems, the differential equation of motion expressed through displacement can be stated as (take the equation of x-direction as an example)

$$(\lambda + G)\frac{\partial \varepsilon}{\partial \mathbf{x}} + G\nabla^2 u + X = \rho \frac{\partial^2 u}{\partial t^2} \tag{1}$$

where *u*, *t*, ρ, and ε are displacement, time, density, and strain, respectively. *x* and *X* refer to the coordinate position and the force per unit volume in the x direction, respectively. λ and *G* are the Lamé constant and shear modulus, respectively. Hence, Equation (1) is also called the Lamé equation. The problem-solving process of general dynamics problems is reduced to solve the Lamé equation under the given boundary conditions. Then, Equation (1) can be expressed as

$$G\_{\parallel} = \frac{\rho \frac{\partial^2 u}{\partial t^2} - \lambda \frac{\partial \epsilon}{\partial x} - X}{\nabla^2 u + \frac{\partial \epsilon}{\partial x}} \tag{2}$$

The most common sets of basic quantities are those of mass M, length L, and time T [28]. In Equation (2), ε is a dimensionless quantity and thus the dimension of ∂ε <sup>∂</sup>*<sup>x</sup>* is L−1. The dimension of ρ is ML−3. The dimension of <sup>∂</sup>2*<sup>u</sup>* <sup>∂</sup>*t*<sup>2</sup> , which represents acceleration, a, is LT−2. The three items in the numerator have the same dimensions. The two items in the denominator also have the same dimensions. Since ε is a dimensionless quantity for both prototype and model soil, the similitude ratio for <sup>ε</sup> must equal unity (1). The similitude ratios for <sup>∇</sup>2*<sup>u</sup>* + ∂ε <sup>∂</sup>*<sup>x</sup>* and <sup>ρ</sup>∂2*<sup>u</sup>* <sup>∂</sup>*t*<sup>2</sup> <sup>−</sup> <sup>λ</sup>∂ε <sup>∂</sup>*<sup>x</sup>* <sup>−</sup> *<sup>X</sup>* are *<sup>S</sup>*−<sup>1</sup> *<sup>l</sup>* and *S*ρ*Sa*, respectively. The similitude ratio for *G* is *SGd* , where the subscript *Gd* represents dynamic shear modulus. Then, the similitude relations between the model soil and the prototype soil can be deduced from Equation (2) as:

$$\frac{G\_d^m}{G\_d^p} = S\_{G\_d} = S\_\beta S\_d S\_l \tag{3}$$

where *G<sup>m</sup> <sup>d</sup>* and *<sup>G</sup><sup>p</sup> <sup>d</sup>* are dynamic shear modulus of model and prototype, respectively. *S* with the subscripts refers to the similitude ratio of the quantities corresponding to those subscripts, i.e., *SGd* , *S*ρ, *Sa*, and *Sl* represent the similitude ratios for soil dynamic shear modulus, soil density, soil acceleration, and soil geometry size, respectively.

The similitude ratios corresponding to other quantities can also be obtained in a similar way. The similitude ratio of the length is set to 1/30 for both the model soil and model station. Table 2 shows the main parameters adopted in defining the similitude relations in this study.


**Table 2.** Similitude relations.

To satisfy the aforementioned similitude relations, an artificial model soil made of sand mixed with sawdust was selected, which had been studied and used by other researchers [29]. By targeting the curves of both the shear modulus and damping ratio decaying with the shear strain for the artificial model soil and prototype soil [30], the optimum mass ratio of sawdust to sand was determined to be 1:2.5 through a series of cyclic triaxial tests performed in the laboratory. The comparison of the curves between artificial model soil and prototype soil can be found in [24].

The prototype of the atrium-style metro station had the dimension of 21.3 × 17.7 m (width × height) and had two stories, including the station hall floor and platform floor. No columns on the station hall floor and thin-walled columns with spacing 7.6 m on the platform floor were adopted. Both the ceiling and middle slabs were mostly replaced with the flat-beams, directly resulting in about 50% opening area of the floorage for both slabs. The prototype station was a cast-in-place reinforced concrete structure and the strength grades of concrete and steel rebar were C35 and HRB400, respectively. Galvanized steel wire and micro-concrete were adopted in the model station to represent two practical materials mentioned above. To satisfy the similitude relations, an optimum mass ratio of micro-concrete was determined to be as cement:sand:lime powder:water = 1:5:0.64:1.18 through a series of material tests, and the compressive yield strength and elastic modulus were measured to be 10.68 MPa and 1.32 <sup>×</sup> 104 MPa, respectively. Figure <sup>2</sup> shows the details of the model station, including three observation planes. The observation plane-1 was located within the central symmetrical plane of the whole container-soil-structure system, while observation plane-2 and plane-3 were both 50 mm away from it.

**Figure 2.** Dimensions of the model station: (**a**) perspective view; (**b**) cross-section; (**c**) locations of three observation planes (unit: mm).

For the whole soil-structure system, various real-time data were collected, including the acceleration and displacement in soil and on structure, the dynamic soil normal stress on both the left and right side walls, and the strain produced on the surface of the structural members.

#### *2.3. Instrumentation*

The locations of these instruments were all based on numerous prior numerical analyses using finite element method [31,32], which are shown in Figure 3.

**Figure 3.** Layouts of sensors for soil-station model: (**a**) accelerometers of side wall and soil across the whole depth in observation plane-1; (**b**) accelerometers of soil adjacent to station in observation plane-2; (**c**) soil pressure cells in observation plane-3; (**d**) strain gauges; (**e**) photos of strain gauge, accelerometers, and soil pressure cells (all dimensions in mm).

Five accelerometers (AM-1, AM-2, AM-3, AM-4, and AM-5), as shown in Figure 3a, were placed in the middle between the center of the model soil and the wall of the flexible-wall container to capture the acceleration responses with soil depth. In considering the symmetry, Figure 3a only shows one-half of the whole container-soil-structure system while the accelerometers are completely displayed.

As displayed in Figure 3a,b three accelerometers (AN-1, AN-2, and AN-3) were arrayed in the model soil adjacent to the side wall, which are set to investigate the effects of the underground metro station on the soil acceleration.

Three more accelerometers (AW-1, AW-2, and AW-3) were installed near the previous ones on the side wall to record its acceleration. The depths of the accelerometers AW-1, AW-2, and AW-3 correspond to the depths of the base slab, middle slab, and ceiling slab, respectively. The comparison of the acceleration responses between the two groups of accelerometers is expected to investigate the effect of RVH on the dynamic interaction, which is a most interesting topic for the soil-structure system.

Soil pressure cells PL1–PL11 on the left side wall and PR1–PR11 on the right side wall were arrayed to investigate the amplitude and distribution of the dynamic soil normal stress on the two walls, Figure 3c. The interval between any two adjacent cells was 48 mm, which allowed to capture the essential data and to represent the complex stress distribution along the side wall as stated by the early finite element analyses.

Strain gauges, plotted in Figure 3d, were arrayed to investigate the values and spatial distribution of the structural strain, especially focusing near the joints between different structural components as the ends of the beam, column, side wall, and slab. In Figure 3d the strain gauges S23 and S24 were located on the top of the column (Z1 in Figure 2b), which was just below the longitudinal beam (ZL2 in Figure 2b).

Figure 3e shows the selected photos of various sensors. The experimental procedure is shown in Figure 4.

**Figure 4.** Photos of the experimental procedure.

#### *2.4. RHV Inputs*

The shaking table was accelerated incrementally and the earthquake motion in the horizontal and vertical directions were applied synchronously. The acceleration amplitude of all the horizontal input motions was adjusted to 0.1 g and corresponding vertical acceleration amplitudes were adjusted to 0 g, 0.05 g, 0.1 g, and 0.15 g, respectively. Hence, different RVH that equals to 0, 0.5, 1, and 1.5 are obtained to investigate their influence on the seismic behavior of the soil and buried metro station. For convenience, a strong-motion record of the Loma Prieta, California, earthquake of 18 October 1989 from the PEER

Strong Motion Database [33] was selected as the input motion for both types of the loading mentioned above, whose acceleration time history and corresponding Fourier amplitude spectra, together with target design spectrum of prototype ground concerning the experiment, are shown in Figure 5. It is seen that the acceleration response spectra of the selected 1989 Loma Prieta earthquake matched the target design spectrum well. Table 3 shows the test cases in this study.

**Figure 5.** The details of Loma Prieta ground motion: (**a**) accelerogram; (**b**) spectral acceleration compared with target design spectrum [34].


**Table 3.** Test cases for the shaking table tests.

The targeted base motions were converted from prototype to model scale units. The base accelerations recorded on the shaking table were refered to as achieved motions. Figure 6 shows the comparison between targeted base motions and achieved base motions both in horizontal and vertical components, including all test cases in Table 3. For any test case, the whole acceleration time history was consistent and minor differences in amplitude were observed between achieved and targeted base motions. These minor differences were acceptable, in view of shaking table performance and such large model specimens.

**Figure 6.** *Cont.*

**Figure 6.** Comparison between targeted base motions and achieved base motions both in the horizontal and vertical components for all test cases.

It is worth mentioning that some details of the experimental program, verification of the free-field test, and interpretation of some preliminary results can be found in prior papers [24,25].

#### **3. Test Results and Discussion**

#### *3.1. Influence of RVH on Soil Acceleration*

To investigate the influence of the RVH on the vertical propagation of the waves in the ground, as an example, Figure 7 compares the horizontal acceleration time histories of the soil at different measuring points (AM-1, AM-2, AM-3, AM-4, and AM-5) between RVH = 0 and RVH = 1. It is seen that at any measuring point their horizontal acceleration always behaves consistently in the trend, and the differences mainly lie in the amplitude. For both RVH = 0 and RVH = 1, the horizontal acceleration of the soil increases with decreasing depth.

**Figure 7.** Comparison of horizontal acceleration time histories of soil between RVH = 0 and RVH = 1. RVH: ratio of vertical to horizontal peak ground acceleration.

To better evaluate the trend of the horizontal acceleration amplitude, the acceleration amplification factor (AAF) is obtained by dividing the peak horizontal acceleration at each measuring point by the peak horizontal acceleration of input motion. Therefore, the AAF at the base of the model (at soil depth of 1.6 m in Figure 3a) is equal to unity (1). This definition also applies to the following parts. Figure 8 shows the AAF of the ground along the depth under different RVH. The horizontal acceleration amplitude of the soil tends to increase from the bottom to the ground surface. Under the four conditions the AAFs on the ground surface are about 2.3–3.1 and the soil presents significant amplification. This is reasonable since the amplitude of all the input motions are relatively low.

**Figure 8.** Acceleration amplification factor of ground under different RVH.

In terms of the acceleration levels in this study, with increasing RVH, soil horizontal acceleration increases and the increase in the amplitude grows significantly with increasing RVH on the whole. It demonstrates that soil horizontal acceleration is more sensitive to higher RVH or higher vertical acceleration.

#### *3.2. Acceleration Di*ff*erence between the Side Wall and Adjacent Soil*

To investigate the influence of the RVH on the acceleration response difference between the side wall and adjacent soil, Figure 9 illustrates the comparison of the acceleration time histories between side wall and adjacent soil at three different depths, which are ceiling slab (Figure 9a), middle slab (Figure 9b), and base slab (Figure 9c), respectively.

**Figure 9.** *Cont.*

**Figure 9.** *Cont.*

**Figure 9.** Comparison of the acceleration time histories between side wall and adjacent soil. at three different depths: (**a**) comparison at depth of ceiling slab; (**b**) comparison at depth of middle slab; (**c**) comparison at depth of base slab.

As can be seen from Figure 9, the phases of the side wall and adjacent soil are always consistent at three depths. The acceleration difference between the side wall and the adjacent soil mainly lies on the amplitude. For the side wall and the adjacent soil, the amplitude of them is almost the same at the depth of the middle slab, where the amplitude differences are about 0–0.025 g at the peak-strain moment. The amplitude of them has minor differences at depth of base slab, where the amplitude differences are about 0.023–0.035 g at the peak-strain moment. It is worth mentioning that some larger differences between the side wall and adjacent soil can be observed at the depth of the ceiling slab, where the amplitude differences reach about 0.045–0.084 g at the peak-strain moment. Under RVH = 1.5 and horizontal input acceleration of 0.1 g, the aforementioned amplitude difference even reaches 0.084 g. It reveals larger differences in the horizontal acceleration between the side wall and the adjacent soil at the depth of the ceiling slab. The differences can be attributed to that the ceiling slab lies too close to the ground surface and there is a significant amplification of the acceleration responses since fewer soil constraints exist there. It reveals that for the underground structures with zero or near-zero buried depth, such as atrium-style metro stations, special attention should be paid to the acceleration response difference between the structure and adjacent soil.

In order to investigate the differences of the horizontal acceleration amplitude between the side wall and the adjacent soil with increasing RVH, the results at three different depths are depicted in Figure 10. It is found that at any depth, the acceleration differences between side wall and adjacent soil appear a linear increase in the amplitude as RVH increases from 0 to 1.5. With regard to the amplitude, the acceleration differences at the depth of the ceiling slab are larger than those at the depth of the base slab. The acceleration differences at the depth of the middle slab are the smallest ones among the three depths. With regard to the increasing rates, the increasing rate at the depth of the ceiling slab is about 2.56%, which is larger than that of 1.65% at the depth of the middle slab. The smallest one is 0.82% at the depth of the base slab.

In conclusion, for an atrium-style metro station, the differences in the horizontal acceleration amplitude between the structure and the adjacent soil rise with increasing RVH, which are different at different depths. The most significant differences occur at the depth of the ceiling slab. The RVH has a significant influence on the dynamic soil-structure interaction, especially for higher RVH.

**Figure 10.** The difference of horizontal acceleration amplitude with increasing RVH between side wall and adjacent soil at three different depths.

#### *3.3. Influence of RVH on Dynamic Soil Normal Stress*

Dynamic soil normal stress (DSNS) is defined as the difference between the total stress and initial static stress along the side wall of the model station. As an example, Figure 11 compares the two time histories of the DSNS between RVH = 0 and RVH = 1. It is found that the time histories of the DSNS resemble that of the input motion. The essence of this phenomenon is also revealed by Kramer, who points out that the input seismic excitation in the form of an acceleration can be converted into a stress wave [35]. With different RVH their phases are basically consistent at all measuring points, though the amplitudes of the DSNS behave differently at different depths of the side wall. For example, the amplitudes are almost the same near the bottom of the side wall (e.g., PR1 and PR2) while the amplitudes under RVH = 1 are always larger than those under RVH = 0 at other depths of the side wall (e.g., PR3~PR11).

**Figure 11.** *Cont.*

**Figure 11.** *Cont.*

**Figure 11.** Comparison of time histories of dynamic soil normal stress (DSNS) along right side wall between RVH = 0 and RVH = 1.

To figure out the influence of the RVH on the DSNS along the side wall, Figure 12 displays the peak DSNS along the left and right side walls under different RVH.

**Figure 12.** Distribution of peak DSNS under different RVH along (**a**) left side wall, and (**b**) right side wall.

As can be seen from Figure 12a, under the input motion of Loma Prieta, peak DSNS along the left side wall follows an approximate L–shaped distribution. With increasing RVH, on the whole, there is no significant change in the amplitude of the peak DSNS. Compared to other RVH, a relatively large increase at the depth of the platform floor can be observed when RVH equals to 1.5.

With respect to the peak DSNS along the right wall, as shown in Figure 12b, at the depth between the top of the right side wall and the middle of the station hall floor, they all follow an approximately linear distribution. At the depth between the middle of the station hall floor and the bottom of the right side wall, they all follow an approximately –shaped distribution. It is worth mentioning that the maximum stress occurs at the bottom of the side wall basically. With increasing RVH, peak DSNS along the right wall increase on the whole, except for the bottom of the side wall (base slab level). Compared to the results under only horizontal input motion (RVH = 0), the maximum increase in peak DSNS under RVH = 0.5, 1, and 1.5 reach about 148%, 159%, and 199%, respectively. For the bottom of the side wall (base slab level), with increasing RVH, the peak DSNS has no significant change, which is different

from the situation at other position of the side wall and has also been concluded above from Figure 11. It might be attributed to the sharp transition in the corner and a possible stress concentration there.

When comparing the distribution of the peak DSNS between the left and right side walls, some obvious differences between them can be found. A similar finding was also drawn from a numerical study of a shallow buried rectangular underground structure under earthquake loading with both the horizontal and vertical components [36]. The differences in stress distribution of the peak DSNS between the left and right side walls can be explained by the shadow effect, which is caused by the entrapment of the waves between the ground and the underground structure [37]. With increasing RVH, the above-mentioned differences between the left and right side walls also increase. From the perspective of dynamic soil normal stress, the experimental results have proven that RVH has significant influence on the dynamic soil-structure interaction.

#### *3.4. Influence of RVH on Structural Dynamic Strain*

The dynamic strain is defined as the difference between the total strain during shaking and initial static strain before shaking. As an example, Figure 13 compares the time histories of the dynamic strain between RVH = 0 and RVH = 1 from one side of every cross section (Figure 3d). Just as the characteristics of the DSNS mentioned earlier, the time histories of the dynamic strain also resemble that of the input motion. In terms of the two different RVH, their phases are extremely consistent for all the measuring points and their amplitudes have minor differences for most of the measuring points.

Figure 14 shows the variation of the peak dynamic tensile strain (DTS) with different RVH for different measuring points of the model station. To easily observe the rate of change in the dynamic strain over different RVH, the value of the y–coordinate in Figure 14 represents the ratio of the peak DTS under RVH = i (i = 0, 0.5, 1, and 1.5) to that under RVH = 0. Since two strain gauges are arranged for every cross section (Figure 3d), the strain results on one side are displayed in Figure 14a and the opposite ones are shown in Figure 14b, where the layouts of corresponding strain gauges are plotted again. As can be seen from Figure 14a, with increasing RVH, the DTS will change in an undulating fashion (decrease, increase, and decrease again), which is different from the response characteristics of the soil, in that the soil acceleration at any depth increases rapidly with increasing RVH. It means the influence of the RVH on the structural DTS is rather complex.

**Figure 13.** *Cont.*

**Figure 13.** Comparison of time histories of dynamic strain on one side of structural components between RVH = 0 and RVH = 1.

**Figure 14.** Peak dynamic tensile strain (DTS) of model station under different RVH: (**a**) strains on one side; and (**b**) strains on another side.

The soil and structure behave differently in the seismic responses with increasing RVH. It can also be seen from Figure 14a that the DTS with bidirectional input motion may be less than those with only horizontal input motion (the value of the y–coordinate is less than 1). This conclusion is verified by the numerical simulation conducted on a prototype atrium-style metro station [31], and can also be found in [38]. This phenomenon can be explained as follows: the presence of a cavity can cause, under certain conditions, intense and selective de-amplification of the free-filed motion, which is referred to as "shadow zone" [37]. When comparing the results of the DTS in Figure 14a,b, for every cross section, the DTS on one side is not equal to the opposite one. It reveals that the stresses in the cross section of every structural element are uneven under bidirectional input motion.

#### **4. Conclusions**

An atrium-style metro station is a favorite choice owing to its excellence in providing a larger and clearer space for the passengers and commercial clients at the station hall floor. However, concern for the seismic safety of such a station is growing since its hall slabs are replaced with the flat-beams. Tests are needed for the evaluation of its seismic safety. A series of 1 g shaking table tests were conducted to investigate the seismic responses of an atrium-style metro station.

In this study, extensive experimental results were presented to investigate the influence of the RVH (ratio of vertical to horizontal peak ground acceleration) on the seismic responses of both the soil and the underground metro station. Results show that the acceleration amplification factors (AAFs) on the ground surface are about 2.3–3.1, and that the soil presents significant amplification under several considered conditions. Soil horizontal acceleration is more sensitive to the higher RVH or higher vertical acceleration. Significant differences in horizontal acceleration amplitude between the side wall and the adjacent soil at the depth of the ceiling slab are found, and attention should be paid to these kind of differences for the underground structures with zero or near-zero buried depth, such as atrium-style metro stations. With increasing RVH, the differences in the horizontal acceleration amplitude between the side wall and the adjacent soil also rise. The RVH has a significant influence on the dynamic interaction, especially for higher RVH. Under the input motion of Loma Prieta, the distribution of the peak DSNS (dynamic soil normal stress) along the left and right side walls represents significant difference. The peak DSNS along the left side wall follows an approximate L-shaped distribution, while it is different for the right side wall. With increasing RVH, there is no significant change in the amplitude of the peak DSNS along the left side wall on the whole, while it increases a lot (the maximum increase reach about 148%, 159%, and 199% under RVH = 0.5, 1, and 1.5 when comparing with that under RVH = 0) for the right side wall. The soil and structure behave differently in the seismic responses with increasing RVH. The stresses in the cross-section of every structural element are uneven under horizontal–vertical earthquake excitation.

In conclusion, from the perspective of the soil acceleration, structural acceleration, and dynamic soil normal stresses along the side wall of station, experimental results have proven that ratio of vertical to horizontal peak ground acceleration (RVH) has significant influence on the dynamic soil-structure interaction. It is believed that under extreme earthquake loading, such as near fault zones, RVH is a parameter of paramount importance and should be accounted for in the seismic analyses and seismic performance assessments of the underground structures, especially for those with zero or near-zero buried depth, such as atrium-style metro stations.

It is worth mentioning that further investigation of the effect of the vertical earthquake motion is essential through a series of numerical analyses using a minimum number of the earthquake records specified by a certain code. In this way, a cross-check of the findings from the multiple records could be expected to accomplished and this is a work in progress.

**Author Contributions:** Test scheme, Z.Z., Y.Y., H.Z., and H.Y.; experimental works, Z.Z.; discussion and analysis, E.B. and Y.Y.; writing—original draft preparation, Z.Z.; writing—review and editing, E.B. and Y.Y.; funding acquisition, Y.Y. and H.Z.

**Funding:** This research was funded by the National Key Research and Development Plan of China (Grants Nos. 2018YFC1504305, 2018YFC0809600, 2018YFC0809602, and 2017YFC1500703), the State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University (Grant No. SLDRCE15-02), and the China Scholarship Council (Grant No. 201706260143).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

#### *Article*

## **An Analytical Framework for the Investigation of Tropical Cyclone Wind Characteristics over Di**ff**erent Measurement Conditions**

### **Lixiao Li 1,2, Yizhuo Zhou 1, Haifeng Wang 3, Haijun Zhou 1,2, Xuhui He <sup>4</sup> and Teng Wu 3,\***


Received: 1 November 2019; Accepted: 6 December 2019; Published: 9 December 2019

**Abstract:** Wind characteristics (e.g., mean wind speed, gust factor, turbulence intensity and integral scale, etc.) are quite scattered in different measurement conditions, especially during typhoon and/or hurricane processes, which results in the structural engineer ambiguously determining the wind parameters in wind-resistant design of buildings and structures in cyclone-prone regions. In tropical cyclones (including typhoons and hurricanes), the inconsistent wind characteristics may be in part ascribed to the complex flow structure with the coexistence of both mechanical and convective turbulence in the boundary layer of tropical cyclones. Another significant contribution to the scattered wind characteristics is due to various measurement conditions (e.g., terrain exposure and height) and data processing schemes (e.g., averaging time). The removal of the inconsistency in the field-measurement system may offer a more rational comparison of measured wind data from various observation platforms, and hence facilitates a better identification scheme of the wind characteristics to guide the urban planning design and wind-resistant design of buildings and structures. In this study, an analytical framework was firstly proposed to eliminate the potential observation-related effects in wind characteristics and then the wind characteristics of seven field measured tropical cyclones (four typhoons and three hurricanes) were comparatively investigated. Specifically, field measurements of wind characteristics were converted to a standard reference station with a roughness length of 0.03 m, observation duration of 10 min for mean wind and averaging time of 3 s for gusty wind at a 10 m height. The differences of the measured wind characteristics between the typhoons and hurricanes were highlighted. The standardized turbulent wind characteristics under the analytical framework for typhoons and hurricanes were compared with the corresponding recommendations in standard of American Society of Civil Engineers (ASCE 7-10) and Architectural Institute of Japan Recommendations for Loads on Buildings (AIJ-RLB-2004).

**Keywords:** wind characteristics; boundary layer; typhoon; hurricane; field measurement

#### **1. Introduction**

Wind characteristics (e.g., mean wind speed, gust factor, turbulence intensity and integral scale, etc.) are the critical factors for wind-resistant design of the wind-sensitive infrastructures and urban planning. Resisting wind effects and reducing wind-induced damage in tropical cyclones is the challenge for the wind sensitive buildings and structures in cyclone-prone regions, as these regions are normally in the economically developed areas with crowded populations and large-scale landmark buildings and structures. Therefore, a rational analytical framework for investigating wind characteristics in tropical cyclones is essential to understand the nature of winds, calibrate codes of practice for wind-resistant design of the large-scale structures and enhance wind tunnel simulations and numerical modeling [1]. Oncoming winds of buildings and bridges are usually simplified as the steady flow part featured by mean wind speed and corresponding vertical profile, and the fluctuating flow part characterized by turbulence intensity, integral scale, gust factor, peak factor, probability distribution, and power spectrum.

Tropical cyclones are characterized by the asymmetric helical flow structure and complex turbulence driven mechanism (both convective and mechanical turbulence). The spatial distribution of the flow structure also varies significantly in the footprint of tropical cyclones. Due to the limited measurements in the lower boundary layer of tropical cyclones, a basic premise of the existing codes and standards is that the turbulent wind characteristics in tropical cyclones are similar to those observed in the boundary layer winds of extratropical storms. However, it is well known that the downward transport of convective cells generated at higher levels together with the boundary layer rolls could modulate the wind structure and turbulence in the lower tropical cyclone boundary layer. These thermodynamics-related activities may lead to the turbulent wind characteristics of the hurricanes/typhoons different from those of the extratropical winds [2–4].

A direct and reliable approach to examine the turbulent wind characteristics is based on the field observations in the paths of landfalling tropical cyclones. Thus, a number of field measurement programs were initiated in the tropical cyclone-prone regions to monitor the hurricane/typhoon winds [5–14]. The field-measured wind characteristics from different observation stations for various tropical cyclones are quite scattered and hard-to-reach unified conclusions to guide the wind-resistance design of buildings and structures in the cyclone-prone regions. The inconsistent wind characteristics of tropical cyclones may be attributed in part to the complexity of turbulence driven mechanisms, e.g., shear (namely roll and streak structures near the surface), convection, rotation, blocking and sheltering effects at the boundary layer, and also the interactive motions of multi-scale eddies in the flow fields of tropical cyclones [15,16]. On the other hand, the underlying surface and the employed schemes to obtain the turbulence parameters may also significantly influence on the variability of wind characteristics. Since the tracks of tropical cyclones are random, most of the field observations were conducted by installing anemometers and accelerometers on structures or observation towers, which were built in the regions frequently attacked by tropical cyclones. The measured turbulent wind characteristics from these observation stations are quite different from one another because of the underlying surrounding terrain conditions and the lack of well-established guidelines for an appropriate documentation of the near surface wind filed in tropical cyclones. In the China wind codes, wind characteristics are specified over standard terrain with roughness length of 0.03 m, averaging time of 10 min for mean wind and duration time of 3 s for gusty wind at 10 m height. Accordingly, it is essential to convert the turbulence characteristics obtained from various stations to the standard terrain and investigate the wind nature in a unified analytical framework. The "standardized" wind characteristics due to their universality could be useful in instructing the structural design in cyclone-prone regions.

This study first presented an analytical framework in which the mean wind speed, turbulence intensity, integral scale, gust factor, and peak factor measured at various terrains, heights and averaging times were properly standardized. Then, the typhoon and hurricane wind data analyzed here were briefly described. Finally, field-observed turbulent wind characteristics of four typhoons and three hurricanes were converted to the standard condition and comparatively investigated. The standardized results were also compared with the corresponding recommendations in ASCE7-10 [17] and AIJ-RLB-2004 [18]. The difference of the wind characteristics in hurricanes and typhoons were also highlighted.

#### **2. Analytical Framework**

In this section, an analytical framework will be proposed in which the turbulent characteristics measured in various terrain conditions, heights and averaging times could be converted to a standard station. The standardization of the wind characteristics is based on the assumption of the equilibrium boundary-layer theory [19]. The atmosphere stratification in the boundary layer of tropical cyclones is assumed to be neutrally stable, which implies that the turbulent structure within this region is driven mainly by the local surface roughness effects [20]. In the non-equilibrium boundary-layer, this analytical framework may need further investigations subjecting to specific terrain conditions.

#### *2.1. Mean Wind Speed*

To analyze the wind characteristics, an essential step is to convert the wind speeds measured at different station conditions (i.e., various exposures, heights, and averaging times) to the standard condition. The standardization of the mean wind speed in this study follows the three steps: (1) determine the exposure type of the observation station; (2) calculate the gradient wind speeds over the observation exposure; and (3) calculate the mean wind speed at the reference height (10 m high) over standard exposure (open flat terrain) by assuming the gradient wind speeds are equal at the gradient height over different exposures.

#### 2.1.1. Logarithmic Law Wind Profile

Normally, the observations by Global Position System (GPS) dropsonde and Doppler radar show that the variation of mean wind speed with height follows the logarithmic law in the lower part of tropical cyclone boundary layer [21–25]. Thus, the logarithmic law can be used to describe lower boundary layer and the outer-vortex regions of a tropical cyclone:

$$\mathcal{U}L\_{\mathfrak{s}}(z\_{\mathfrak{s}}) = \frac{u\_{\mathfrak{s}\_{\mathfrak{s}}}}{k} \ln \left(\frac{z\_{\mathfrak{s}}}{z\_{0\mathfrak{s}}}\right) \tag{1}$$

where *Us*(*zs*) represents the mean wind speed at height *zs* over the standard exposure. Specifically, the standard exposure in this study corresponds to the roughness length *z*0*<sup>s</sup>* = 0.03 m, the reference height is 10 m, and the time scale for the average value is 10 min. *u*∗*<sup>s</sup>* denotes the friction velocity over the standard exposure, and *k* ≈ 0.40 is the von Kármán constant.

According to Equation (1), the key procedure to standardize the mean wind speed is to determine the relationship of the friction velocities in various terrains. As all anemometers used in this study are set between 10–60 m height, it is reasonable to assume that the friction velocity in the lower tropical cyclone boundary layer is a height-independent constant [25–27]. Based on the assumption of local equilibrium conditions, the transition model in Engineering Sciences Data Unit (ESDU) [28], which has been applied to convert the 3 s peak speed over open-terrain and the 1-min mean wind speed above open water in hurricane by Simiu et al. [29], is employed:

$$\frac{\mu\_{\ast\_{S}}}{\mu\_{\ast\_{M}}} = \frac{\ln\left(\frac{10^5}{z\_{0\_s}}\right)}{\ln\left(\frac{10^5}{z\_{0\_{0}}}\right)},\tag{2}$$

where *u*∗*<sup>m</sup>* is the friction velocity over the field measured exposure with roughness length of *z*0*m*. Then the relation of mean wind speeds with different terrains can be accordingly expressed as:

$$\frac{dL\_s(z\_s)}{dL\_m(z\_m)} = \frac{\ln\left(\frac{10^5}{z\_{0s}}\right)}{\ln\left(\frac{10^5}{z\_{0m}}\right)\ln\left(\frac{z\_m}{z\_{0m}}\right)}\tag{3}$$

where *Um*(*zm*) is the mean wind speed measured at experiment station with height *zm* and roughness *z*0*m*. In this model, the gradient balance assumption, which has been demonstrated to be valid at a sufficiently high altitude [30], is adopted. The super-gradient flows were observed in the boundary layer of some tropical cyclones, however, it is not systematic in tropical cyclones, especially in overland conditions [31,32]. The case of super-gradient flows in tropical cyclones will discussed in next section.

The relation among mean wind speeds of various averaging times can be expressed as [33]:

$$\mathcal{U}\_{\pi}(z) = \mathcal{U}\_{3600}(z) \left[ 1 + \frac{\beta^{0.5} c(\pi)}{2.5 \ln\left(\frac{z}{z\_0}\right)} \right] \tag{4}$$

where τ denotes the averaging time; *U*τ(*z*) and *U*3600(*z*) are respectively τ-s mean and 1-h mean wind speeds; β represents the ratio of the fluctuating wind speed variance to the square of friction velocity; *c*(τ) is an averaging time-related parameter that determined by statistical characteristics of wind speed measurements.

#### 2.1.2. Super-Gradient Wind Profile

The field measurements show the existence of super-gradient wind over ocean surface and the sea land transition regions in tropical cyclones, and the variation of mean wind with height following a logarithmic-quadratic profile [34]. Based on the field measurements in hurricanes over land and ocean surface, Snaiki and Wu [35] proposed a semi-empirical model to depict the mean wind profile. As the empirical model is convenient and accurate, it is adopted here to convert the mean wind speed in landfalling typhoons. The power law-based wind profile is used as follows:

$$\mathcal{U}I\_s(z\_s) = \mathcal{U}I\_{10\_s} \left[ \left(\frac{z\_s}{10}\right)^{\alpha\_s} + \eta\_1 \sin\left(\frac{z\_s}{\delta\_s}\right) \exp\left(-\frac{z\_s}{\delta\_s}\right) \right] \tag{5}$$

where *Us*(*zs*) and *U*10*<sup>s</sup>* are the mean wind speed at height *zs* and 10 m over the standard exposure; α*<sup>s</sup>* is the power law exponent over the standard exposure; δ*<sup>s</sup>* is the height of the wind maximum over the standard exposure; η1*<sup>s</sup>* is derived to be:

$$\eta\_{1s} = \frac{\left(\frac{\delta\_s}{10}\right)^{\alpha\_s} \alpha\_s \varepsilon}{\sin 1 - \cos 1'} $$

Analogously, the field measured mean wind over the experiment exposure is:

$$\mathcal{U}l\_{m}(z\_{m}) = \mathcal{U}l\_{10\_{m}} \left[ \left(\frac{z\_{m}}{10}\right)^{a\_{m}} + \eta\_{1m} \sin\left(\frac{z\_{m}}{\delta\_{m}}\right) \exp\left(-\frac{z\_{m}}{\delta\_{m}}\right) \right] \tag{6}$$

where *Um*(*zm*) and *U*10*<sup>m</sup>* are the mean wind speed at height *zm* and 10 m over the measured exposure; α*<sup>m</sup>* is the power law exponent over the measured exposure; δ*<sup>m</sup>* is the height of the wind maximum over the measured exposure; η1*<sup>s</sup>* is:

$$
\eta\_{1m} = \frac{\left(\frac{\delta\_m}{10}\right)^{\alpha\_m} \alpha\_m e}{\sin 1 - \cos 1}
$$

By adopting the assumption that the wind speeds at the wind maximum height ( δ*<sup>s</sup>* and δ*m*) are equal, the following expression can be deduced:

$$\frac{dL\_s(z)}{dL\_m(z)} = \frac{\left(\frac{\delta\_m}{10}\right)^{a\_m} \left[ \left(\frac{z\_s}{10}\right)^{a\_s} + \eta\_{1s} \sin\left(\frac{z\_s}{\delta\_s}\right) \exp\left(-\frac{z\_s}{\delta\_s}\right) \right]}{\left(\frac{\delta\_s}{10}\right)^{a\_s} \left[ \left(\frac{z\_m}{10}\right)^{a\_m} + \eta\_{1m} \sin\left(\frac{z\_m}{\delta\_m}\right) \exp\left(-\frac{z\_m}{\delta\_m}\right) \right]},\tag{7}$$

Equation (7) could be used to convert the field wind speeds to the standard exposures in the tropical cyclones with super-gradient flow. Actually when the wind speed measured in the lower regions following the logarithmic law, the Equation (7) will merge into the logarithmic law or power law.

#### *2.2. Turbulence Intensity*

It is conventional to treat the turbulence ratio (the ratio of the standard deviation of longitudinal wind velocity component σ*<sup>u</sup>* to the friction velocity *u*∗) as terrain-independent in the equilibrium boundary layer [36,37]. On the other hand, Harris and Deaves [38] proposed an empirical model to consider the variation of turbulence ratio with height as:

$$\frac{\sigma\_{\rm II}}{\mu\_{\star}} = 2.63\eta \left[ 0.538 + 0.090 \ln \left( \frac{z}{z\_0} \right) \right]^{\eta^{16}} \text{ .} \tag{8}$$

where <sup>η</sup> = 1 <sup>−</sup> *<sup>z</sup>*/*h*; *<sup>h</sup>* = *<sup>u</sup>*∗/(<sup>6</sup> *<sup>f</sup>*); *<sup>f</sup>* = 1.458 <sup>×</sup> 10−<sup>4</sup> sin <sup>φ</sup> is the Coriolis parameter; and <sup>φ</sup> denotes the latitude of the observation site. Due to the fact that the derivation of Equation (8) was partly based on non-equilibrium-condition data, the estimation of maximum turbulence ratio, [σ*u*/*u*∗]*max* obtained from Equation (8) is dependent on the terrain roughness length, which is in contradiction to the equilibrium assumption. To correct this issue, the empirical variation of σ*u*/*u*<sup>∗</sup> with respect to terrain roughness is introduced in ESDU [39] to obtain approximately a constant [σ*u*/*u*∗]*max* for various terrain roughness lengths:

$$
\left[\frac{\sigma\_u}{\mu\_\*}\right](z\_0) = 1 + 0.156 \ln\left(\frac{\mu\_\*}{fz\_0}\right) \tag{9}
$$

Since the field measurements give [σ*u*/*u*∗]*max* = 2.85 for the terrain with a roughness length of 0.03 m, Equation (8) can be corrected by factoring 2.85/ / <sup>1</sup> + 0.156 ln[*u*∗/(*f z*0)]<sup>0</sup> , resulting in an improved model to calculate turbulence ratio as in ESDU [39]:

$$\frac{\sigma\_{\rm II}}{\mu\_{\rm \ast}} = \frac{7.496 \eta \left[0.538 + 0.090 \ln\left(\frac{z}{z\_0}\right)\right]^{\eta^{16}}}{1 + 0.156 \ln\left(\frac{\mu\_{\rm \ast}}{f^z z\_0}\right)}. \tag{10}$$

As expected, Equation (9) gives a maximum turbulence ratio [σ*u*/*u*∗]*max* of approximately 2.85 for various roughness lengths. However, field measurements show that turbulence ratios in tropical cyclones are usually greater than the values in extratropical storms [12,24,40], making the selection of [σ*u*/*u*∗]*max* = 2.85 inapplicable to tropical cyclones. On the other hand, a height-independent relation between turbulence ratio σ*u*/*u*<sup>∗</sup> and underlying surface roughness length *z*<sup>0</sup> was proposed by Li et al. [13] based on the analysis of field measurements in typhoons. In this study, this height-independent relation proposed by Li et al. [13] is adopted as:

$$
\left[\frac{\sigma\_u}{\mu\_\*}\right]\_{\text{max}} = 2.72 - 0.25 \log z\_0. \tag{11}
$$

As a result, the turbulence ratio in tropical cyclone will be corrected by multiplying Equation (8) by the following factor:

$$\frac{2.72 - 0.25 \log z\_0}{1 + 0.156 \ln[\mu\_\*/(f z\_0)]}. \tag{12}$$

Then the turbulence ratio could be expressed as:

$$\frac{\sigma\_{\rm II}}{\mu\_{\star}} = \frac{2.63\eta \left[0.538 + 0.009\ln\left(\frac{z}{z\_0}\right)\right]^{\eta^{16}} \left[2.72 - 0.25\log(z\_0)\right]}{1 + 0.156\ln\left(\frac{\mu\_{\rm I}}{f\bar{z}\_0}\right)}.\tag{13}$$

For a standard terrain condition (*z* = 10 m; *z*<sup>0</sup> = 0.03 m; assuming *u*<sup>∗</sup> = 1 m/s and φ = 25◦), the turbulence ratios estimated by Equations (9) and (11) are 2.55 and 2.78, respectively. In this

study, Equation (13) is utilized to convert the measured turbulence ratio to the standard condition. Accordingly, the longitudinal turbulence intensity in a standard exposure can be calculated as:

$$[I\_u]\_s = \frac{2.63\eta\_s [0.538 + 0.090 \ln\left(\frac{z}{z\_0}\right)]^{\eta\_s^{1.6}} [2.72 - 0.25 \ln(z\_{0t})] [u\_\*]\_s}{\left[1 + 0.156 \ln\left(\frac{u\_\*}{f^z z\_0}\right)\right] [L]\_s}. \tag{14}$$

#### *2.3. Integral Scale*

The approach of integrating correlation function by invoking Taylor's hypothesis is frequently used to estimate the integral scale as it has a clear physical meaning [24,33]:

$$L\_u^x = \frac{\mathcal{U}}{\sigma\_u^2} \int\_0^{\mathcal{R}\_{\text{int}} = 0.05r\_u} R\_{uu}(\tau) \,\text{d}\tau,\tag{15}$$

where *Ruu* is the autocorrelation function of the longitudinal fluctuating component.

The integration of autocorrelation function, however, usually overestimates the value of integral scale and will result in a deviation of inertial sub-range in the estimated von Kármán-type spectrum [41] compared to that in the field-measured spectra. To improve the accuracy, Harris and Deaves [38] suggested the following model to estimate the longitudinal integral scale:

$$L\_u^x = \frac{A^{\frac{3}{2}} \left(\frac{\sigma\_u}{u\_\*}\right)^3 z}{2.5 K\_z^{\frac{3}{2}} \left(1 - \frac{z}{h}\right)^2 \left(1 + 5.75 \frac{z}{h}\right)},\tag{16}$$

where *z* is the height from the ground.

$$A = 0.115 \left( 1 + 0.315 \eta^6 \right)^{\frac{2}{3}} \tag{17}$$

and

$$K\_z = 0.19 - (0.19 - K\_0) \exp\left[ -B \left( \frac{z}{h} \right)^N \right] \tag{18}$$

in which

$$K\_0 = \frac{0.39}{R\_\phi^{0.11'}}\tag{19}$$

$$B = 24R\_o^{0.155} \text{\AA} \tag{20}$$

$$N = 1.24 R\_{\vartheta} \stackrel{0.008}{\text{ \\$}},\tag{21}$$

$$R\_{\theta} = \frac{u\_{\ast}}{f z\_{0}}.\tag{22}$$

The longitudinal integral scale over standard exposure [*Lx <sup>u</sup>*]*<sup>s</sup>* can be estimated according to Equation (16) by introducing the corresponding values of [*u*∗]*<sup>s</sup>* and [*z*0]*s*.

#### *2.4. Peak Factor*

Peak factor *gu* is defined as the ratio of maximum wind speed fluctuation in a duration τ to the standard deviation of the fluctuating wind speed within an observation period of *T:*

$$g\_{\boldsymbol{u}}(\boldsymbol{\tau},T) = \frac{\max[\boldsymbol{u}(\boldsymbol{\tau},T)]}{\sigma\_{\boldsymbol{u}}(\boldsymbol{\tau},T)} \frac{\sigma\_{\boldsymbol{u}}(\boldsymbol{\tau},T)}{\sigma\_{\boldsymbol{u}}}.\tag{23}$$

For a stationary stochastic process following Gaussian distribution, the peak factor with τ → 0 and *T* ≥ 3600 *s* could be calculated as [42]:

$$g\_{\mu}(\tau, T) = \sqrt{2\ln[\nu(\tau, T)T]} + \frac{0.5772}{\sqrt{2\ln[\nu(\tau, T)T]}},\tag{24}$$

where υ(τ, *T*) is the zero up-crossing rate. It can be calculated by [42,43]:

$$\nu^2(\tau, T) = \frac{\int\_0^\infty n^2 S\_\mathrm{u}(n) \chi^2(n, \tau, T) \mathrm{d}n}{\int\_0^\infty S\_\mathrm{u}(n) \chi^2(n, \tau, T) \mathrm{d}n},\tag{25}$$

in which *Su*(*n*) represents the wind velocity spectrum; *n* denotes the frequency in Hertz, and χ2(*n*, τ, *T*) is a filter function used to consider the influence of sampling frequency, averaging time and response characteristics of the anemometer. In this study, the von Kármán-type spectrum is employed as:

$$\frac{nS\_{\rm II}(n)}{\sigma\_{\rm u}^{2}} = \frac{4\left(\frac{nL\_{\rm u}^{x}}{\ell I}\right)}{\left[1 + 70.8\left(\frac{nL\_{\rm u}^{x}}{\ell I}\right)^{2}\right]^{\frac{5}{6}}},\tag{26}$$

The filter function is chosen as following for sonic anemometers [44]:

$$\chi^2(n,\tau,T) = \left[\frac{\sin(\pi n \tau)}{\pi n \tau}\right]^2 - \left[\frac{\sin(\pi n T)}{\pi n T}\right]^2,\tag{27}$$

For propeller anemometers, the following filter function, which takes the mechanical features of propeller anemometers into consideration, is adopted [43]:

$$\chi^2(n,\tau,T) = \left\{ \left[\frac{\sin(\pi n \tau)}{\pi n \tau}\right]^2 - \left[\frac{\sin(\pi n T)}{\pi n \tau}\right]^2 \right\} \frac{1}{1 + \left(\frac{2\pi n \lambda}{lI}\right)^2},\tag{28}$$

where λ is the distance constant of the propeller anemometer.

Equation (24) is valid for calculating the average of instantaneous peak factor ( τ → 0) from a long enough wind speed record (e.g., *T* ≥ 1 h). With a finite averaging time, τ, and a finite observation period, *T*, the estimation of standard deviation in Equation (26) could be biased since the measured spectrum is truncated in both high-frequency and low-frequency regions, and might eventually lead to an inaccurate estimation of the peak factor. In the case that these conditions are not satisfied, the following relation is necessary to be introduced to consider the effects of the variance reduction due to the truncation of the velocity spectrum:

$$\frac{\sigma\_{\rm u}(\tau, T)}{\sigma\_{\rm u}} = \frac{\int\_{0}^{\infty} S\_{\rm u}(n) \chi^{2}(n, \tau, T) \mathrm{d}n}{\int\_{0}^{\infty} S\_{\rm u}(n) \mathrm{d}n},\tag{29}$$

The 3-s peak factor, [*gu*] *<sup>s</sup>*, in time scale [*T*]*<sup>s</sup>* = 600 s in the standard terrain can be estimated by introducing *Us* and *Lx us*, which were respectively calculated through Equations (3) or (7) and (16).

#### *2.5. Gust Factor*

Gust factor, *Gu*(τ, *T*), herein is defined as the ratio of gust speed with gust duration τ to the mean wind speed *U*(*T*) with an observation period of *T*:

$$G\_{\rm u}(\tau, T) = 1 + \frac{\max[\boldsymbol{u}(\tau, T)]}{\sigma\_{\rm u}(\tau, T)} \frac{\sigma\_{\rm u}(\tau, T)}{\sigma\_{\rm u}(\Delta t, T)} \frac{\sigma\_{\rm u}(\Delta t, T)}{\mathcal{U}(T)},\tag{30}$$

where Δ*t* is the sampling interval.

Substituting the peak factor and turbulence intensity into the corresponding terms of Equation (30), the gust factor can be re-expressed as:

$$G\_{\rm u}(\tau, T) = 1 + g\_{\rm u}(\tau, T) I\_{\rm u} \frac{\sigma\_{\rm u}(\tau, T)}{\sigma\_{\rm u}(\Delta t, T)},\tag{31}$$

where σ*u*(τ, *T*)/σ*u*(Δ*t*, *T*) can be calculated by:

$$\frac{\sigma\_{\rm u}(\tau, T)}{\sigma\_{\rm u}(\Delta t, T)} = \frac{\int\_0^\infty S\_{\rm u}(n) \chi^2(n, \tau, T) \mathrm{d}n}{\int\_0^\infty S\_{\rm u}(n) \chi^2(n, \Delta t, T) \mathrm{d}n},\tag{32}$$

As a result, the gust factor in the standard exposure, [*Gu*(τ, *T*)]*s*, can be estimated by:

$$[G\_{\rm u}(\tau, T)]\_{\rm s} = 1 + [g\_{\rm u}(\tau, T)]\_{\rm s} [I\_{\rm u}]\_{\rm s} \left[\frac{\sigma\_{\rm u}(\tau, T)}{\sigma\_{\rm u}(\Delta t, T)}\right]\_{\rm s} \tag{33}$$

#### **3. Data Sources**

#### *3.1. Tropical Cyclones and Instruments*

In this study, the data of four typhoons (0601 typhoon Chanchu, 0606 typhoon Prapiroon, 0812 typhoon Nuri, and 0814 typhoon Hagupit) and three hurricanes (0504 hurricane Katrina, 0510 hurricane Rita, and 0512 hurricane Wilma) were comparatively analyzed. The detailed descriptions of the observation site exposures and the observation tower configurations for the four typhoons and three hurricanes were presented in Li et al. [16] and Masters et al. [12], respectively. The GPS coordinates of the observation stations were listed in Table 1. As the latitudes of all observation stations are around 25◦, the latitude of the standard condition is set to be 25◦ for the convenient of calculation.

**Table 1.** The GPS coordinates of observation towers. Reproduced with permission from [16], Elsevier, 2019.


It is noted that the distance constant λ of the propeller anemometer is an important factor to calculate the peak factor and gust factor as this type of anemometer mechanically filters the amplitudes of gusts with wavelengths less than 2πλ due to the mechanical limitations [10]. In this study, the propeller anemometers of models R.M. Young 05103L and R.M. Young 27106R were respectively used to measure typhoons and hurricanes. The specifications of these two propeller anemometers are listed in Table 2. Based on the parameters in Table 2, the data measured by propeller anemometers were corrected according to Equation (24). In addition to propeller anemometers, the sonic anemometers were also utilized in the field measurement of typhoons. Specifically, two 3-D ultrasonic anemometers (WindMaster™ Pro., Gill Instruments Ltd., Lymington, UK) were installed on tower RBT and one 3-D ultrasonic anemometer (HD2003, Delta Ohm Srl, Selvazzano Dentro, Italy) were setup on tower OT for the measurements of Typhoon Chanchu; one HD2003 anemometer were installed on tower BT to acquire data from Typhoon Prapiroon, and towers DIT and ST were equipped with Gill WindMasterTM Pro. anemometers. The specifications of the utilized sonic anemometers are also listed in Table 2. For the data obtained from sonic anemometers, the filter function presented by Equation (23) were used for the correction.


**Table 2.** Specifications of anemometers.

#### *3.2. Data Quality Control and Data Source*

Tropical cyclones are characterized by strong winds accompanied by torrential rain, ocean waves, and storm surge. The representative wind records are usually located in the eyewall regions of tropical cyclones. In the field measurements of tropical cyclones, however, the anemometers in heavy rain bands usually present some spikes and errors. Therefore, the data quality-control procedure is a necessary step before the analysis of the wind characteristics. In this study, the data quality-control schemes and the criteria for the selection of samples were referred to Li et al. [13]. Specifically, the spikes and errors in the data were first identified and replaced by the five-point weighted averages. Then, the reverse arrangement test [45] and run test [46] with a 95% significance level were employed to test the stationarity of the recorded winds. The datasets that failed to pass both two types of stationary tests were removed from the analysis. The stationarity test ensured that the analyzed data could satisfy the local equilibrium boundary-layer assumption, where the friction velocity is independent of the location in the along-wind direction and the Reynolds number [37].

Table 3 briefly summarized the datasets utilized in this study, together with the observation heights, types of the anemometers used, number of the runs in each group (sample size) and average of the 10 min mean wind speeds. The detailed analysis of the turbulent wind characteristics of the original datasets both in typhoons and hurricanes were presented in Li et al. [16].



#### **4. Results and Discussions**

The selected datasets from the four typhoons and the three hurricanes summarized in the preceding section were investigated in the analytical framework presented in Section 2. Specifically, both the datasets in typhoons and hurricanes were converted to the standard exposure with a roughness length of 0.03 m at 10 m height and an observation time scale of 10 min. The latitude of the standard terrain is assumed to be 25◦.

#### *4.1. Turbulence Intensity*

The turbulence intensities were extracted based on the analytical framework and shown in Figures 1a and 2a respectively for investigated typhoons and hurricanes. The corresponding probability density functions (PDFs) are shown in Figures 1b and 2b. For typhoons, the average value of longitudinal turbulence intensities is 0.1952 and the standard deviation is 0.0032. For hurricanes, the average value of longitudinal turbulence intensities is 0.1906 and the standard deviation is 0.0022. The longitudinal turbulence intensity of these four typhoons presents slightly higher values in terms of both mean and standard deviation compared to those three hurricanes. This observation can be further demonstrated by comparing Figures 1b and 2b, where the probability distribution of the longitudinal turbulence intensities in both typhoons and hurricanes follows the normal distribution quite well. One can easily conclude that in these four typhoons the turbulence intensity has a larger value than that in those three hurricanes under the same probability of exceedance.

In ASCE7-10, the longitudinal turbulence intensity is given by:

$$I\_{\rm II} = c \left(\frac{10}{z}\right)^{\frac{1}{b}},\tag{34}$$

where *c* equals to 0.30, 0.20, and 0.15 for category B, C (corresponding to the standard terrain in this study), and D exposures, respectively. In AIJ-RLB-2004 code, the longitudinal turbulence intensity over flat terrain categories is given by:

$$I\_{\rm li} = 0.1 \left(\frac{z}{z\_G}\right)^{-\alpha - 0.05},\tag{35}$$

in which α and *zG* are parameters reflecting the category of exposures. In category II exposure, which is the closest to the standard terrain in this study, α and *zG* are respectively 0.15 and 350. The longitudinal turbulence intensities obtained from ASCE7-10 and AIJ-RLB-2004 are 0.2000 and 0.2036, respectively, for the standard exposure. As depicted in Figures 1a and 2a, both ASCE7-10 and AIJ-RLB-2004 present a slightly higher estimation of longitudinal turbulence intensity for hurricanes and typhoons. Generally, the estimation of ASCE7-10 is relatively better compared to that of AIJ-RLB-2004.

**Figure 1.** Turbulence intensities and their probability density functions (PDF) for typhoons.

**Figure 2.** Turbulence intensities and their PDF for hurricanes.

#### *4.2. Integral Scale*

The integral scales and the corresponding probability distributions obtained are presented in Figures 3 and 4 for typhoons and hurricanes, respectively. The longitudinal integral scale in typhoons has an average value of 146.4482 and a standard deviation of 9.2143. The length scales extracted from hurricane measurements have a slightly higher average value of 157.5796 and a significantly lower standard deviation of 3.8106 compared to those of typhoons. As shown in Figures 3a and 4a, the range of integral scales extracted from typhoon measurements is significantly larger than that of hurricanes, which can be better illustrated by the probability distributions of longitudinal integral scales in typhoons and hurricanes. The probability distribution of typhoons follows the Weibull distribution with scale parameter of 150.368 and shape parameter of 20.0454, while the probability distribution of longitudinal integral scales for hurricanes follows the generalized extreme value distribution with scale parameter of −0.3259, shape parameter of 3.8960 and location factor of 156.311. Compared with Figure 4b, where the observed hurricane length scale shows a narrow distribution, the probability distribution of the observed typhoon integral scale of Figure 3b has a significantly wider range. This phenomenon may be in part attributed to the exposures of the observation station for the original datasets. The observation station in those three hurricanes are located in homogeneous open flat terrain as stated in Masters et al. [12]. However, in the observation of those four typhoons, the exposures of the measured stations are a little bit inhomogeneous. Another influence could be ascribed to the differences of turbulent structures of those typhoons and hurricanes. As noted in Li et al. [16], at the same roughness regime, the field measured integral scales in these four typhoons were greater than that in those three hurricanes. The different distributions of the observed hurricane and typhoon length scales might indicate that energy-containing eddies in the observed typhoons have various representative length-scales while those of the observed hurricanes are concentrated around the mean value. The multiple-scale eddy interactions in typhoons and hurricanes need further investigations before fully understanding the observed difference.

In ASCE7-10, the longitudinal integral scale is computed by:

$$L\_u^{\mathbf{x}} = l \left(\frac{z}{10}\right)^{\overline{7}},\tag{36}$$

where *l* and ε are respectively 152.4 m and 0.2 for category C exposure (standard exposure in this study). In AIJ-RLB-2004 code, the turbulence integral scale is defined independently of the terrain categories and is given by:

$$L\_u^x = \begin{cases} 100 \left(\frac{z}{30}\right)^{0.5} & 30 \text{ m} < z < z\_G\\ 100 & z \le 30 \text{ m} \end{cases} \tag{37}$$

where *zG* equals to 350 m for category II exposure, corresponding to the standard exposure in this study. Accordingly, the longitudinal integral scales obtained from ASCE7-10 and AIJ-RLB-2004 are respectively, 152.4 m and 100 m, for the standard exposure. It is noted that ASCE7-10 presents a reasonable estimation of the longitudinal integral scales for both typhoons and hurricanes. However, AIJ-RLB-2004 underestimates the longitudinal integral scales for both typhoons and hurricanes, suggesting that the usage of AIJ-RLB-2004 may lead to an inaccurate estimation of the power spectrum.

**Figure 3.** Longitudinal integral scales and their PDF for typhoons.

**Figure 4.** Longitudinal integral scales and their PDF for hurricanes.

#### *4.3. Peak Factor*

The peak factor is usually utilized for the estimation of gust factor, which plays an important role in determining the wind load on structures [43]. The estimated peak factors of typhoons and hurricanes are respectively presented in Figures 5 and 6, together with the corresponding probability distributions. For typhoons, the average value of the peak factor is 2.5211 and the standard deviation is 0.0198. The fitted PDF is shown in Figure 5b. For hurricanes, the average value of peak factor is 2.5123, slightly smaller than that of typhoons, and the standard deviation of peak factor is 0.0102, significantly smaller than that of typhoons. The probability distribution of peak factors in hurricanes follows t location-scale distribution with location parameter of 2.5174, scale parameter of 0.0022 and shape parameter of 0.9845.

Neither the expression nor the value of peak factor is explicitly prescribed in ASCE7-10. By matching the gust factor over open terrain ( *Gu* = 1.53) and the turbulence intensity *Iu* of

0.2 in Equation (34), a peak factor of 2.65 could be obtained. It should be noted that this calculation is based on an averaging time of 1 h. For a duration of *tg* , the gust wind speed can be expressed as:

$$
\mathcal{U}\{t\_{\mathcal{G}'}T\} = \overline{\mathcal{U}}(T) + \mathcal{g}\_{\mathfrak{u}}(t\_{\mathcal{G}'}T)\sigma\_{\mathfrak{u}}.\tag{38}
$$

Suppose the turbulent wind fluctuations follow the Gaussian distribution, the peak factor will be associated with the exceedance probability of the standard normal distribution. The probability of exceedance of wind gust with a duration of *tg* within an observation period of *T* could be calculated as [47,48]:

$$P\left[\mathcal{U} > \mathcal{U}\left(t\_{\mathcal{S}'}\right)\right] = \frac{t\_{\mathcal{S}}}{T}.\tag{39}$$

Thus, the peak factor should satisfy:

$$\log \left( t\_{\mathcal{S}'} \, T \right) = \Phi^{-1} \left( 1 - \frac{t\_{\mathcal{S}}}{T} \right) . \tag{40}$$

With the gust duration of 3 s and the averaging time of 10 min, the peak factor is around 2.575, which is slightly higher than the measured values in typhoons and hurricanes. AIJ-RLB-2004 carries out a performance-based wind resistant design procedure. Accordingly, the peak factor is included in the required performance of wind load level and the return period of wind speed. Hence, the comparison of the measurement results with AIJ-RLB-2004 is not discussed here.

**Figure 5.** Peak factors and their PDF for typhoons.

**Figure 6.** Peak factors and their PDF for hurricanes.

#### *4.4. Gust Factor*

The gust factor in steady wind conditions depends on several wind characteristics, such as the intensity and integral scale, hence, it is a basic representation of the dynamic properties of wind loads [49]. Figure 7 depicts the gust factors and the corresponding probability distribution of typhoon winds obtained based on the unified analysis framework. For typhoons, the average value of gust factors is 1.4919 and the standard deviation is 0.0069. The gust factors of hurricanes are presented in Figure 8a, where the gust factors have a mean of 1.4787 and a standard deviation of 0.0071. The mean of gust factors for typhoons are slightly higher than that for hurricanes, while the standard deviations of gust factors for typhoons and hurricanes are almost identical with similar probability distribution shapes. The probability distribution of gust factors for typhoons follows the extreme value distribution with a location parameter of 1.4948 and a scale parameter of 0.0051, while the probability distribution of gust factors for hurricanes follows the generalized extreme value distribution with shape parameter of −0.5535, location parameter of 1.4771, and scale parameter of 0.0076.

In ASCE7-10, the calculation of gust factor is referenced to the gust factor curve proposed by Durst [50]. The averaging time of the mean wind speed is 1 h in the Durst gust factor curve, while the gust factor is calculated based on an averaging time of 10 min in this study. Therefore, the conversion scheme for the gust factors with different averaging times presented in Vickery and Skerlj [20] was utilized here. Gust factor with a duration of 3 s and an averaging time of 10 min can be expressed as:

$$G\_{\mu} = 1 + (SLI)[SD(600, 3)],\tag{41}$$

where *SU* is the value of the standard normal deviation associated with the exceedance probability of 0.5% and equals to 2.575. The *SD*(600, 3) could be estimated by the following formula:

$$SD(600, \,\,3) = \left[SD^2(3600, \,\,3) - SD^2(3600, \,\,600)\right]^{1/2},\tag{42}$$

where *SD*(3600, 3) and *SD*(3600, 600) can be interpolated as indicated in Vickery and Skerlj [20] and equal to 0.1617 and 0.0650, respectively. The gust factor with duration of 3 s and averaging time of 10 min based on Equations (37) and (38) is around 1.3814, which indicates that the gust factors for both typhoons and hurricanes are greater than those for extratropical storms.

**Figure 7.** Gust factors for typhoons.

**Figure 8.** Gust factors for hurricanes.

#### **5. Concluding Remarks**

An analytical framework was introduced in this study to standardize the turbulent wind characteristics, namely turbulence intensity, integral scale, peak factor and gust factor for various terrain conditions, heights, and averaging times in tropical cyclones. This analytical framework is based on the equilibrium boundary-layer theory and the assumption that the lower tropical cyclone boundary layer is neutrally stable. Field-measured data of the four typhoons and three hurricanes were standardized to the reference exposure with roughness length of 0.03 m, height of 10 m, and averaging time of 10 min, and then utilized for the extraction of wind characteristics under standard exposure. The differences of obtained wind characteristics between typhoons and hurricanes were highlighted, which may be attributed to the basins and latitudes of the genesis of hurricanes and typhoons, the influence of local topography and sea-land transition zone and the differences in turbulent flow structures of typhoons and hurricanes that need further investigations. More specifically, the wind characteristics of these observed typhoons typically present larger values compared to those of observed hurricanes, except for the turbulence integral scale. The turbulence integral lengths of typhoons have a wider distribution compared with those of hurricanes. The obtained turbulent wind characteristics based on the unified analysis framework were comparatively investigated together with the recommendations in ASCE7-10 and AIJ-RLB-2004. The difference between the standardized turbulent characteristics and the corresponding suggested values in the standards (ASCE7-10 and AIJ-RLB-2004) indicates that the tropical cyclone-induced wind loads need be taken into consideration in standards for tropical cyclone-prone regions. It is noted that the ASCE7-10 presents good estimations of the longitudinal turbulence intensity and integral scale for both typhoons and hurricanes, while the peak factor was slightly overestimated and the gust factor was underestimated. The AIJ-RLB-2004 makes a slightly higher estimation of the longitudinal turbulence intensity and a lower estimation of the longitudinal integral scale for both typhoons and hurricanes. The potential reason may be ascribed to the limitation of datasets which used to specify the wind characteristics, although it includes both tropical and extratropical winds. As noted in the AIJ-RLB-2004, the integral scale was treated to be terrain independent. However, the scales of wind eddies are strongly affected by the local roughness.

**Author Contributions:** Conceptualization, L.L. and T.W.; methodology, L.L. and H.Z.; software, Y.Z. and H.W.; validation, T.W. and H.W.; formal analysis, Y.Z.; investigation, Y.Z. and H.W.; resources, X.H.; data curation, Y.Z.; writing—original draft preparation, L.L. and T.W.; writing—review and editing, L.L., X.H. and T.W.; visualization, Y.Z.; supervision, L.L., H.Z., X.H. and T.W.; project administration, L.L.; funding acquisition, L.L. and T.W.

**Funding:** This research was funded by National Natural Science Foundation of China (Grant No. 51778373), the Knowledge Innovation Project of Shenzhen (Grant No. JCYJ20170302143625006), Natural Science Foundation of SZU (Grant no. 082017), Natural Science Foundation Grant # CMMI 15-37431 and National Key Research and Development Program of China (2017YFB1201204).

**Acknowledgments:** The authors gratefully acknowledge K. Gurley of the University of Florida for providing the hurricane data and Lili Song of China Meteorological Administration for providing the typhoon data for this study.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Full-Scale Train Derailment Testing and Analysis of Post-Derailment Behavior of Casting Bogie**

#### **Hyun-Ung Bae 1, Jiho Moon 2, Seung-Jae Lim 3, Jong-Chan Park <sup>4</sup> and Nam-Hyoung Lim 4,\***


Received: 10 November 2019; Accepted: 18 December 2019; Published: 19 December 2019

**Abstract:** In this study, a full-scale train bogie derailment test was conducted. For this, test methodologies to describe the wheel-climbing derailment of the train bogie and to obtain accurate test data were proposed. The derailment test was performed with the casting bogie for a freight train and a Rheda 2000 concrete track. Two different derailment velocities (28.08 km/h and 55.05 km/h) were considered. From the test, it was found that humps in the concrete track affected the post-derailment behavior of the bogie when the derailment velocity was 28.08 km/h. For a higher derailment velocity (55.05 km/h), significant lateral movement of the derailed bogie was observed. This lateral movement was first controlled by wheel–rail contact, followed by contact with the containment wall. Finally, the train was returned to the track center.

**Keywords:** train derailment; derailment containment provisions; collision testing; post-derailment behavior

#### **1. Introduction**

In Korea, there were 33 train accident cases in total during the five-year period 2012–2016. Among these, derailment accidents accounted for 78.8% (26 cases) [1], and derailment occurred more often than other types of train accidents. Derailment accidents can cause catastrophic damage to a community. It is hard to prevent 100% of derailment accidents since there are always unexpected factors that can cause derailment, such as human error and natural disasters. Thus, it is necessary to develop technology to reduce damage due to derailment. This technology can be categorized as derailment protection [2–4].

To reduce the damage from derailment accidents, protection facilities can be installed in the railway track. In Korea, guard rails to prevent derailment are used at sharp curves, bridges, and switches, as shown in Figure 1a. Containment walls are also installed on bridges for high-speed railways (where the minimum speed of the line is 200 km/h), as shown in Figure 1b [5–7].

In European countries, three different types of derailment containment provisions (DCPs) are used (DCP types I, II, and III), as shown in Figure 2 [8]. The guard rail is one example of a DCP type I facility, where the DCP is installed inside the track gauge. The wheel of the derailed train comes into direct contact with the facility. DCP type II is similar to DCP type I, but is installed outside of the track gauge. DCP type III facilities are installed outside of the track, similar to DCP type II. However, they are different from DCP type II since the axis of the wheel or bogie of the derailed train impacts this type of DCP.

**Figure 1.** Examples of derailment protection facilities: (**a**) guard rail and (**b**) containment wall.

**Figure 2.** Concept of derailment containment provisions (DCPs).

To verify the performance of such derailment protection facilities, tests and simulations must be conducted, including investigation of the post-derailment behavior of the train. Some researchers have conducted derailment and post-derailment simulations by using 3D finite element analysis. Researchers in Sweden [9,10] analyzed the post-derailment behavior of the wheel of a derailed train colliding with a concrete railroad sleeper. Some researchers in China [11–13] investigated the post-derailment dynamic behavior of a railway vehicle under earthquake excitations. Also, there have been studies simulating collision with a protective facility after the derailment of a high-speed train [2–4].

Full-scale derailment testing is the most reliable method to evaluate the post-derailment behavior of a train and the performance of derailment protection facilities. However, it is very difficult, and only a few such studies can be found in the literature. Wu et al. [14] performed derailment testing, but it was lab-scale testing and the speed was limited to 16 km/h. The test methods relating to train derailment, such as the derailing method and data filtering, are not well established and must be investigated.

In this study, full-scale train derailment testing was conducted. For realistic simulation of the derailment situation and reliable data acquisition from the test, a derailment device and data acquisition system were proposed. Then, from the test, the post-derailment behavior of the train was analyzed.

#### **2. Experimental Methodologies**

#### *2.1. Test Site and Track*

Full-scale train derailment testing requires a large test site. The test site consisted of acceleration, test, and braking regions, as shown in Figure 3. To increase the speed of the train at the point of derailment, a sufficient acceleration region is needed. Also, adequate test and braking regions should be provided to investigate the post-derailment behavior and to ensure safety, respectively. The lengths of the acceleration, test, and braking regions were 1200 m, 400 m, and 400 m, respectively. In the 400 m test region, a region of 100 m was used to construct a concrete track. The total length of the test line was about 2000 m. Once the target speed of the bogie is reached, a test bogie is released at the end of the acceleration region. The test bogie is derailed at the start of the test region and the post-derailment behavior is observed in the test region. A braking region is provided at the end of the test region. This is a margin region to provide safety after the unexpected behavior of derailed bogie or train. The derailment test site was constructed using a closed railway line to save on costs and replicate actual railway operating conditions.

**Figure 3.** Overview of the test site.

In this study, the focus was on the post-derailment behavior on a concrete track. A Rheda 2000 concrete track, shown in Figure 4, was constructed in the test region where the post-derailment behavior was observed. The Rheda 2000 concrete (ballastless) track was used for the first time in Germany in 2000 as a pilot project on the new rail line between Erfurt and Halle-Leipzig. It was also installed on the high-speed railway in Korea in 2004. Apart for this 100-m concrete track, ballast tracks were used. It should be noted that the derailment containment wall was installed on the left side of the track, as shown in Figure 4. The distance between the wall and the center of the track was 2700 m, considering the geometric condition of the axle of the test vehicle after the derailment.

**Figure 4.** The concrete track used in the test region.

#### *2.2. Test Bogie and Acceleration Method*

In this study, a bogie-level test was conducted. The bogie used in this study was a casting bogie for a freight train, as shown in Figure 5. The total weight of the test bogie was 40.91 kN and the specifications of the test bogie are listed in Table 1. The frame structure was made of three pieces of cast steel. The fixed wheel base and wheel diameter were 1676 mm and 860 mm, respectively. A suspension system with coil springs was used between the bolster and side frame.

**Figure 5.** Test bogie (casting bogie for a freight train).


**Table 1.** Specifications of the test bogie.

In order to accelerate the test bogie, several methods can be used, such as a reverse towing system, push system, or remote-controlled system. In this study, the push system shown in Figure 6 was used. The power car was linked to the test bogie with a connector. The test car was accelerated by the power car. Then, the test car was released after reaching the target speed. For this, the releasing system of the connector and braking system of the power car were designed to be controlled by air pressure and a remote controller, as shown in Figure 7. The power car used in this study is shown in Figure 8. The power car had a traction power of 147 kN.

**Figure 6.** Concept of the push system to accelerate the test bogie.

**Figure 7.** Release and braking system for the test.

**Figure 8.** Power car to accelerate the test bogie.

#### *2.3. Derailment Device*

A device to induce derailment was needed for the test. Wheel-climbing derailment is one of the most frequent types of derailment. In this study, a derailment device that can induce wheel-climbing derailment was developed and installed at the derailment point, as shown in Figure 9 [15]. The device consisted of a wheel entrance part and a derailment part. In the wheel entrance part, there is an upward slope, and the entered wheel flange reaches the same height as the surface of a rail head. Then, the wheel climbs the surface of a rail head in the derailment part, shown in Figure 9c. Thus, the train or bogie is derailed due to wheel-climbing derailment.

**Figure 9.** Derailment device: (**a**) overview, (**b**) wheel entrance part, and (**c**) derailment part.

#### *2.4. Data Acquisition System*

The position of the derailed train bogie and related data have to be accurately measured during the test. The acceleration and angular velocity of the bogie were measured. For this, an accelerometer and an angular velocity sensor were installed on the geometrical bogie center as shown in Figure 10b, where the capacities of the accelerometer and angular velocity sensor were 2000 g and ±1500 deg/s full-scale range, respectively. Large impact forces are expected after derailment. Thus, the sensors must be securely fixed to the test bogie. To measure the impact force, special accelerometers such as MEMS accelerometers [16] can be used. However, it is hard to determine the exact impact point after derailment. Thus, the impact force to the rail was estimated from the acceleration data of the test bogie. A data logger was installed on the test bogie, as shown in Figure 10c. The data logger used was a shock-resistance data logger (high shock rating of 500 g) for collision testing and it has high sampling capabilities (max. sampling rate of 100 k samples/s/channel). The sampling rate of the data was 1/10,000 s (10,000 Hz) in this test.

**Figure 10.** Data acquisition system for the bogie: (**a**) layout, (**b**) accelerometer and angular velocity sensor, and (**c**) a shock-resistance data logger for collision testing.

The velocity of the test bogie at the derailment point is important information for the analysis of post-derailment behavior. Since the test bogie was released before the derailment point, it was necessary to set up an additional velocity measurement system at the derailment point, as shown in Figure 11. In this study, a photoelectric tube speedometer was used. Two lights and two receiver sensors were installed at a specific distance. As the test bogie passes, the passing time between the two lights can be obtained. Then, the initial derailment velocity of the test bogie can be calculated.

**Figure 11.** Speed measuring system: (**a**) system conceptual diagram; (**b**) photoelectric tube speedometer.

In this study, a high-speed camera was also used to investigate the post-derailment behavior of the test bogie in depth. Image data are very useful to evaluate the trace of the bogie after derailment. The high-speed camera used in this study is shown in Figure 12. According to its specifications, it can store 3.6 s at 1000 fps (frame/s) when the image format is 1024 × 1024. In total, three high-speed cameras were used. One camera was used to take the top view with a crane, as shown in Figure 12. The others were used to take the side views. For each second, 500 frames were obtained (500 fps). Also, some ordinary digital cameras were installed to obtain image data from various viewpoints.

**Figure 12.** High-speed cameras: (**a**) side view; (**b**) top view.

#### *2.5. Simultaneous Trigger System*

Trigger systems are used to synchronize several different pieces of measurement equipment. By synchronizing the equipment, data analysis is convenient since the initial time of data recording is the same for all equipment. In this study, the three high-speed cameras and data logger were synchronized by a trigger system with a tape switch, as shown in

The tape switch method is simple and free from spurious operation. The trigger for the high-speed cameras was set up on the test track at the derailment point, as shown in Figure 13a. The trigger for the data logger, shown in Figure 13b, was installed on the test bogie. When the test bogie passes the derailment point, the trigger on the test bogie contacts the trigger for the high-speed camera. Then, data recoding is started. It should be noted that the trigger on the bogie was made of flexible materials since the post-derailment behavior could be affected by high stiffness and contact force between the triggers. Figure 13.

**Figure 13.** Simultaneous trigger system by a tape switch: (**a**) trigger for the high-speed cameras; (**b**) trigger for the data logger.

#### *2.6. Running Diagram of the Power Car and Test Bogie*

It was necessary to determine the running diagram of the power car and test bogie to achieve the target speed of the test bogie and to ensure the safety of the power car after releasing the test bogie. Figure 14 shows a schematic view of the running diagram of the power car and test bogie. The blue solid line and red dashed line represent the running diagrams of the power car and the test bogie, respectively. Before releasing the test bogie, the power car and the test bogie move together. The velocity at the release time is larger than the target speed. After releasing the test bogie, the power car reduces its speed and stops before the derailment point. The test bogie reaches the target velocity at the derailment point and is derailed.

**Figure 14.** A schematic view of the running diagram of the power car and test bogie.

The distance to the release point depends on the target speed and acceleration capacity of the power car. The running diagram was constructed from the results of several preliminary tests. In this study, the target speeds of the test bogie were 30 km/h and 55 km/h. Figure 15 shows a representative running diagram of the power car through the preliminary tests. An acceleration distance of about 480 m was required for the stationary power car to accelerate to about 56 km/h, and a braking distance of about 150 m was required for subsequent stops. The deceleration to the derailment point after the release of the test bogie was approximately 1–2 km/h. Therefore, the running distance of the power car to ensure safety during the experiment with a target speed of 50 to 55 km/h was determined to be 650 m. In addition, an acceleration distance of about 140 m was required for the power car to accelerate to about 32 km/h, and a braking distance of about 90 m was required for subsequent stops. The deceleration to the derailment point after the release of the test bogie was approximately 2–3 km/h. Thus, the running distance of the power car to ensure safety with a target speed of 25 to 30 km/h was determined to be 250 m.

**Figure 15.** Examples of running diagrams of the power car.

#### *2.7. Post-Processing of Data*

It is necessary to conduct a filtering process on raw data since raw data may contain unexpected noise, especially in the case of impact or collision testing. In the case of car collision testing, the data are analyzed after filtering the raw data obtained at a 1000 Hz or 10,000 Hz frequency. Usually, the moving average method is used for the filtering [17,18]. The moving average method is used not only for the smoothing of the data, but also for the evaluation of equivalent static design force. A previous study reported that the moving average method gave a reasonable estimation of the equivalent static design load [19]. Thus, the 50 ms moving average method was applied with 10,000 Hz data sampling in this study.

#### **3. Experimental Results**

#### *3.1. Test Cases*

After derailment, wheel–rail interaction disappears, and the train runs along the top surface of the track. In this case, various contact and impact conditions, such as gear box–rail, wheel–rail fastener, and wheel–sleeper, arise. These various conditions affect the post-derailment behavior. The post-derailment behavior is an important consideration for the design of derailment protection facilities. In this study, full-scale train bogie derailment tests were conducted. The test bogie and test methodologies are detailed in Section 2. The main test parameter of this study was the derailment velocity. Two different derailment velocities were considered. For Cases #1 and #2, the derailment speeds were 28.08 km/h and 55.05 km/h, respectively.

#### *3.2. Case #1 (Derailment Velocity of 28.08 km*/*h)*

For Case #1, the target speed range was 25–30 km/h. The measured derailment velocity was 28.08 km/h. Figure 16 shows a front view of the post-derailment behavior in Case #1. It can be seen that the bogie ran in the right lateral direction after derailment (0.410–1.810 s), and the left wheel continuously contacted the hump of the concrete track (1.810–3.570 s). Then, the bogie moved to the left direction without contact between the right rail and left wheel (3.570–5.630 s). The direction was changed due to the effect of the hump in this case. In this study, the Rheda 2000 type concrete track was used, and the hump in this track affected the post-derailment behavior.

The acceleration data were analyzed to calculate the velocity and traveling distance of the test bogie. The analysis results were compared with the results obtained in the image data from the high-speed cameras for cross-validation of the data. It is known that the image analysis offers a certain potential for the dynamic investigation [20]. The velocity and traveling distance of the test bogie were also calculated by target mark tracking from the image data, as shown in Figure 17. The results are shown in Figure 18 as a solid line. It can be seen that the velocity obtained from the high-speed camera showed initial fluctuation. However, the overall trend of the velocity was similar to that from the accelerometer. The velocity decreased almost linearly, as shown in Figure 18a. The slope of Figure 18a represents the deceleration, and it was approximately <sup>−</sup>0.97 m/s2. By integrating the acceleration data twice, the traveling distance could be calculated, as shown in Figure 18b. The traveling distances obtained from the accelerometer and high-speed cameras were similar to each other. The discrepancy may come from accumulated error during the integration or distortion of the image by the use of a wide-angle lens.

**Figure 16.** Post-derailment behavior of the bogie for Case #1, front view.

**Figure 17.** Example of image analysis by target mark tracking.

**Figure 18.** Data analysis results, Case #1: (**a**) velocity, (**b**) traveling distance.

The longitudinal, lateral, and vertical acceleration data of the test bogie were analyzed with the video images together. The first impact was observed at 0.410 s after derailment, as shown in Figure 19. It can be seen that there were sudden changes in acceleration at the time of the first impact, as shown in Figure 19a. The maximum acceleration was found in the vertical direction, and it was 5.07 g (49.78 m/s2). If the total mass of the bogie is included in the calculation of the impact force, it is equivalent to approximately 207 kN of impact force. The first impact occurred between the left wheel and hump, as shown in Figure 19b. Figure 19c shows the damage to the hump and rail fastener due to the first impact.

**Figure 19.** First impact, Case #1: (**a**) acceleration vs. time (0.3–0.6 s); (**b**) test bogie at first impact; (**c**) damage to the hump and rail fastener due to the first impact.

Similar to Figure 19, information about the second and third impacts is presented in Figure 20. The second and third impacts were observed at 0.750 s and 0.810 s, respectively. In the second impact, the right and left rear wheels impacted the fourth hump, as shown in Figure 20b,c. The maximum vertical acceleration was approximately 5.57 g—larger than that from the first impact. This may be attributed to the pitching moment of the bogie after the first impact. In the case of the third impact, the front right wheel contacted the sixth and seventh humps in succession. At the third impact, the peak of the vertical acceleration was markedly decreased.

**Figure 20.** Second and third impacts, Case #1: (**a**) acceleration vs. time (0.65–0.95 s); (**b**) test bogie at the second and third impacts; (**c**) damage to the hump and rail fastener due to the second and third impacts.

After the third impact, the test bogie ran the track without any further significant impact with track components. However, considerable lateral displacement was observed. For example, at 1.170 s, the front left wheel was located at almost the center of the track, as shown in Figure 21a. The lateral displacement continuously increased at 1.170–1.810 s, and the front left wheels contacted the humps in

the left rail at 1.810 s. Then, the test bogie returned to the center of the track at 3.750–5.630 s, as shown in Figure 21b.

(**b**)

**Figure 21.** Position of the test bogie: (**a**) 1.170–1.810 s, (**b**) 2.150–5.630 s.

In summary, for Case #1, where the derailment velocity was 28.08 km/h, the derailed test bogie ran the track with considerable lateral movement. The lateral movement was restrained by contact with the humps in the concrete track. Then, the test bogie returned to the center of the track. The whole trace of the test bogie is shown in Figure 22. The maximum vertical acceleration was approximately 5.57 g at the second impact with the humps. The maximum lateral acceleration was approximately 2.62 g at the first impact with the humps. From the results, it can be seen that the effect of the humps on the post-derailment behavior was significant when the Rheda 2000 track system was used. Thus, the type of track must be considered when evaluating the post-derailment behavior of a derailed train.

**Figure 22.** Trace of the left and right wheels for Case #1.

#### *3.3. Case #2 (Derailment Velocity of 55.05 km*/*h)*

In Case #1, the derailed wheel was not guided by the rail or containment wall. However, it was guided by the humps on the track. To identify the effects of the rail and containment wall, the derailment velocity was increased for Case #2. The target derailment velocity range was 50–55 km/h. The measured derailment speed was 55.05 km/h. Figure 23 presents the variation in the velocity and longitudinal traveling distance after derailment for Case #2. The data were obtained by the accelerometer as well as the high-speed cameras. However, the data from the high-speed cameras cannot be interpreted well after 2 s since the test bogie left the camera range due to the increased speed of the test bogie. The deceleration of the test bogie was approximately <sup>−</sup>1.48 m/s2. The traveling length was approximately 38 m at 3 s.

**Figure 23.** Data analysis results, Case #2: (**a**) velocity, (**b**) traveling distance.

Figure 24 shows the test bogie at derailment. It was found that the test bogie seemed to jump, bypassing the derailment device as shown in Figure 24a. Then, the front left wheel contacted with the third hump and rail fastener at 0.240 s due to the roll of the test bogie, as shown in Figure 24b. When the test bogie passed through the derailment device, the vertical acceleration in Case #2 was much greater than that in Case #1 before the first wheel contact with the track (0.242 s), as shown in Figure 24c. If higher-speed experiments are carried out, the derailment device should be improved because excessive jumping is expected.

**Figure 24.** Test bogie at derailment for Case #2: (**a**) 0.124 s; (**b**) 0.242 s (first impact); (**c**) vertical acceleration when derailed (0–0.20 s).

The main impact to the track was observed around from 0.4 s to 0.7 s, as shown in Figure 25. At 0.460 s, the front right wheel came into contact with the eighth rail fastener. Then, the front right wheel impacted the ninth hump and the rear right wheel immediately came into contact with the seventh rail fastener. The maximum vertical acceleration was found at this time, and it was approximately 3.53 g. The maximum acceleration in Case #2 was smaller than that in Case #1. This is because increased velocity is related with the longitudinal direction and vertical impact force is mainly a function of the velocity in the vertical direction.

**Figure 25.** Impact of the test bogie after derailment for Case #2: (**a**) acceleration vs. time (0.4–0.7 s), (**b**) 0.460 s; (**c**) 0.534 s.

After 0.7 s, the test bogie ran the track with considerable lateral movement, similar to Case #1. However, for Case #2, the wheel flange was in contact with the rail as shown in Figure 26b. Then, the test bogie impacted the containment wall at 1.910 s, as shown in Figure 26c. The lateral acceleration was increased due to the impact with the rail and containment wall, as shown in Figure 26a. The lateral acceleration values were approximately 1.17 g (11.51 m/s2) and 0.75 g (7.36 m/s2) for the rail and containment wall impact, respectively. Thus, it was expected that the impact force on the containment wall would be reduced approximately 36% by the first guide by the rail for Case #2. Finally, the test bogie returned to the track center direction after containment wall impact, as shown in Figure 27.

**Figure 26.** Impact of the test bogie after derailment for Case #2: (**a**) acceleration vs. time (1.7–2 s), (**b**) 1.780 s; (**c**) 1.910 s.

**Figure 27.** Trace of the left and right wheels for Case #2.

As mentioned in the introduction, the basic concept of DCP type III is that the derailed train is first guided by the rail and the containment wall, preventing excessive lateral movement [8,21,22]. The containment wall installed in this test is DCP type III, and it was seen that the derailed train was effectively guided by DCP type III in test Case #2.

#### **4. Conclusions**

This study presented full-scale train bogie derailment test methodologies and post-derailment behavior based on the test results. The derailment tests were conducted at the bogie level, and the test speeds were 28.08 km/h and 55.05 km/h. The major findings of this study are as follows:


**Author Contributions:** Investigation, Conceptualization, Methodology, Data Curation, Project Administration, Supervision, Writing (original draft, review & editing) from H.-U.B., J.M. and N.-H.L.; Experiment, Data processing from S.-J.L. and J.-C.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by a grant (19RTRP-B122273-04) from the Railway Technology Research Program, funded by the Ministry of Land, Infrastructure, and Transport of the Korean government.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Prediction of Friction Resistance for Slurry Pipe Jacking**

#### **Yichao Ye, Limin Peng, Yang Zhou, Weichao Yang \*, Chenghua Shi and Yuexiang Lin**

School of Civil Engineering, Central South University, Changsha 410075, China; tunnelye@163.com (Y.Y.); lmpeng@csu.edu.cn (L.P.); csu.zy@csu.edu.cn (Y.Z.); csusch@163.com (C.S.); csulyx2010@foxmail.com (Y.L.)

**\*** Correspondence: weic\_yang@163.com; Tel.: +86-137-8723-2438

Received: 18 November 2019; Accepted: 23 December 2019; Published: 26 December 2019

### **Featured Application: The new approach established in this paper can provide accuracy prediction of friction resistance for slurry pipe jacking with various soil conditions, which lays a good foundation for better future design and less construction costs.**

**Abstract:** Friction resistance usually constitutes one of the two main components for the calculation of required jacking force. This paper provides a new approach to predict the friction resistance of slurry pipe jacking. First, the existing prediction equations and their establishment methods and essential hypotheses used were carefully summarized and compared, providing good foundations for the establishment of the new model. It was found that the friction resistance can be uniformly calculated by multiplying an effective friction coefficient and the normal force acting on the external surface of the pipe. This effective friction coefficient is introduced to reflect the effect of contact state of pipe-soil-slurry, highly affected by the effect of lubrication and the interaction of pipe-soil-slurry. The critical quantity of pipe-soil contact angle (or width) involved may be calculated by Persson's contact model. Then, the equation of normal force was rederived and determined, in which the vertical soil stress should be calculated by Terzaghi's silo model with parameters proposed by the UK Pipe Jacking Association. Different from the existing prediction models, this new approach has taken into full consideration the effect of lubrication, soil properties (such as internal friction angle, cohesion, and void ratio), and design parameters (such as buried depth, overcut, and pipe diameter). In addition, four field cases and a numerical simulation case with various soils and design parameters were carefully selected to check out the capability of the new model. There was greater satisfaction with the measured data as compared to the existing models and the numerical simulation approach, indicating that the new approach not only has higher accuracy but is also more flexible and has a wider applicability. Finally, the influence of buried depth, overcut, and pipe diameter on the friction resistance and lubrication efficiency were analyzed, and the results can be helpful for the future design.

**Keywords:** slurry pipe jacking; friction resistance; effective friction coefficient; pipe-soil-slurry interaction; lubrication efficiency

#### **1. Introduction**

In many parts of the world, the numerous constructions of municipal tunnels are creating unforeseen problems, such as blocking of roads, existing pipelines failure, and buildings subsidence. This has motivated attempts at the development of trenchless construction technology, such as pipe jacking, especially in metropolitan cities [1,2]. Pipe jacking is defined as a trenchless excavation technique, which employs hydraulic jacks to thrust specially made pipes through the ground behind a jacking machine, from a drive shaft to a reception shaft, as illustrated in Figure 1. It has many technical merits, such as a short time limit, high security, low environmental effect, and little traffic

disturbance [3–6]. Because of that, pipe jacking has been widely used in the construction of infrastructure for traffic and transportation systems in cities [7,8].

In pipe jacking, the jacking force is a critical factor that determines the thickness of pipe and reaction wall, selection or design of jacking machine and lubricant requirements [9]. The accuracy of prediction of jacking force is directly related to the structural safety and construction cost.

The main component of the jacking force is due to frictional resistance. Application of a lubricant such as bentonite slurry in pipe jacking (so-called 'slurry pipe jacking') is essential to reduce the friction resistance and, therefore, the jacking force [3,10–13]. However, the use of slurry makes it more complex to calculate or predict the friction resistance because of the change in contact conditions between the pipe and soil and lubricant slurry. The new contact state, which is due to the pipe-soil-slurry interaction, is affected by factors such as pipe diameter, soil properties, overcut [3,9], lubrication efficiency [9], pipeline misalignments [10,14–16], and stoppages [9,14,17–19]. The existing prediction models have not fully taken these factors into consideration, leading to an overestimation or underestimation of the friction resistance [9,20–23]. It is therefore obvious that a new prediction approach or model is imperatively needed to be established to solve the problem in slurry pipe jacking [24].

**Figure 1.** Schematic of slurry pipe jacking.

#### **2. Overview of the Existing Prediction Models of Friction Resistance**

Numerous models that calculate the friction resistance of pipe jacking have been proposed by authors from all over the world. The superposition principle is usually used, which holds that the comprehensive outcome of two or more linear factors of a system is equal to the accumulation of the effect of each factor. In pipe jacking, the linear factors to generate friction resistance are due to the weight of pipe (*fW*), soil pressure (*fs*), slurry pressure (*fm*), pipe-soil cohesion (*fsc*), and pipe-slurry cohesion (*fmc*). Their equations can be expressed as [5,6,9,10,14,19,21,22,25,26].

$$f\_W = \mu\_s \mathbf{W} \tag{1}$$

$$f\_s = \mu\_s N\tag{2}$$

$$f\_m = \mu\_m \text{N} \tag{3}$$

$$f\_{\text{sc}} = c\_s B\_s \tag{4}$$

$$f\_{mc} = c\_m B\_m \tag{5}$$

where μ*<sup>s</sup>* and μ*<sup>m</sup>* are the kinematic friction coefficient of pipe-soil and pipe-slurry, respectively; *W* is the weight of pipe per unit length, kN/m; *N* is the total normal force acting on the pipe, kN/m; *cs* and *cm* are the pipe-soil cohesion resistance and pipe-slurry cohesion resistance, respectively, kPa; *Bs* and *Bm* are the pipe-soil contact width and pipe-slurry contact width, respectively, m.

Some hypotheses have been made to establish the prediction models, by which one or some of the items listed above should be considered. Typical hypotheses are:

**Hypothesis 1.** *The excavated tunnel is self-stable, the pipeline simply slides along the bottom of the tunnel due to its own weight (see Figure 2a) [9,10,14].*

**Hypothesis 2.** *The angular space due to overcut is completely filled with lubricant slurry, and the excavated tunnel is stable under the slurry pressure (see Figure 2b) [22,25].*

**Hypothesis 3.** *The excavated tunnel is unstable, the surrounding soil collapses and is in full contact with the whole area of the jacking pipes (see Figure 2c) [5,6,9,21,26].*

**Hypothesis 4.** *The excavated tunnel is stable under the pressure of slurry, and part of the pipe comes in contact with the surrounding soil (see in Figure 2d) [3].*

**Figure 2.** The models to calculate friction resistance according to different hypotheses: (**a**) Hypothesis 1; (**b**) Hypothesis 2; (**c**) Hypothesis 3; (**d**) Hypothesis 4.

According to Hypothesis 1, there are two kinds of prediction models, in which the item of *fw* has to be taken into consideration. The first one assumes that the friction resistance is only due to the weight of pipe [9,21]. It is given as

$$f = \mu\_s \mathcal{W} \tag{6}$$

$$
\mu\_s = \tan(q/2) \tag{7}
$$

The second one also takes the pipe-soil cohesion resistance (item of *fsc*) into account [10,14,21], which is given by

$$f = \mu\_s \mathcal{W} + \mathfrak{c}\_s B\_s \tag{8}$$

where the pipe-soil contact width *Bs* is calculated by Hertzian contact model, as [10,14,21]

$$B\_s = 1.6(\mathrm{Pk}\_d \mathbb{C}\_\varepsilon)^{1/2} \tag{9}$$

$$k\_d = \frac{D\_c D\_p}{D\_c - D\_p} \quad , \quad \mathbb{C}\_c = \frac{1 - v\_p^2}{E\_p} + \frac{1 - v\_s^2}{E\_s} \tag{10}$$

where *Dc* and *Dp* are the internal diameter of cavity and external diameter of pipe, respectively, m; ν*<sup>p</sup>* and ν*<sup>s</sup>* are the Poisson's ratio for pipe and soil material, respectively; *Ep* and *Es* are the elasticity modulus of the pipe and soil, respectively, kPa; and *P* is the effective external force acting on the center of the pipe, kN/m, usually it is considered as equal to the weight of pipe per unit length *W*.

According to Hypothesis 2, the friction resistance is related only to the properties of lubricant slurry, and the only model found was one that takes both the items of *fm* and *fmc* into account [22,25].

$$f = \mu\_m \mathcal{N} + \mathfrak{c}\_m B\_m \tag{11}$$

$$N = \pi D\_p P\_m \tag{12}$$

$$B\_m = \pi D\_p \tag{13}$$

where *Pm* is the mud slurry pressure.

It is noted that most of the studies completed to date have focused exclusively on the prediction models established by Hypothesis 3. This may be attributed to the assumption of full contact of the pipe and soil that leads to a large prediction value of friction resistance, or in other words, Hypothesis 3 is conservative. Because of that, this kind of model is widely accepted by authorities and standards from all over the world, such as Japan Sewage Association (JSA) [22], UK Pipe Jacking Association (PJA) [27], Chinese Trenchless Technology Association (CTTA) [28], and Germany Standard (AVT A-161) [29]. This kind of model can be summarized and divided into the following four categories.

$$f = \mu\_{\text{s}} \mathcal{N} \tag{14}$$

$$f = \mu\_s \mathcal{N} + \mu\_s \mathcal{W} \tag{15}$$

$$f = \mu\_s \mathcal{N} + \mu\_s \mathcal{W} + \mathcal{c}\_s B\_s \tag{16}$$

$$f = \beta \mu\_s \mathcal{N} + \mu\_s \mathcal{W} + c\_s B\_s \tag{17}$$

The fourth kind of model (Equation (17)) introduces an empirical constant β (smaller than 1) on the basis of the third model (Equation (16)) to reflect the effect of lubrication.

For the models summarized above, the item of *fs* (=μ*sN*) has to be taken into consideration. Thus, the key problem for this kind of model is exactly focused on the calculation of soil pressure *N.* CTTA suggests *N* to be calculated by Rankine's formula, which gives that [28]

$$N = 2(1+K)D\_p \sigma\_v \tag{18}$$

$$
\sigma\_{\mathcal{V}} = \gamma h \tag{19}
$$

$$K = \tan^2(\pi/4 - \phi/2)\tag{20}$$

where σ*<sup>v</sup>* is the vertical soil stress; *K* is the coefficient of soil pressure above the pipe; γ is the unit weight of soil; *h* is the overburden depth of the pipeline; and ϕ is the internal friction angle of soil.

However, JSA suggests using Terzaghi's silo model, the expression of *N* is then expressed as [22]

$$N = \pi D\_p \sigma\_v \tag{21}$$

$$\sigma\_{\upsilon} = \frac{b\gamma - 2c\_s}{2K \tan(\delta)} \left( 1 - e^{-2K \tan(\delta) \hbar/b} \right) \tag{22}$$

where *cs* is the cohesion of soil; δ is the friction angle between the pipe and soil; *b* is the influencing silo width of soil above the pipe, and the other symbols have the same meanings as before.

Although a lot of prediction equations of friction resistance have been proposed and even some of them have been applied to engineering practice, it is obvious that their hypotheses are quite different, and even for the same hypothesis, models and parameters can be different too. Thus, it is bound to make the prediction friction resistances vary greatly. Furthermore, apart from Equations (11) and (17), the other models completely ignore the effect of lubrication, which is very important to slurry pipe jacking.

In the design philosophy of slurry pipe jacking, the angular space due to overcut is expected to be completely filled with lubricant slurry, to reduce the friction resistance with maximum efficiency, creating a 'filter cake' layer around the cavity and is then pressurized to the support pressure required for the soil (see Figure 2b) [3,10]. In this case, the friction resistance should be only related to the slurry pressure and the friction coefficient between slurry and the pipe. From this point of view, the expression Equation (11) seems a convincing explanation here.

However, the more general case is that the excavated tunnel is stable under pressure of slurry and part of the pipe inevitably comes in contact with the soil (see Figure 2d) [3]. The reasons for the occurrence of pipe-soil contact can be complex, such as insufficient design and control of grouting amount of slurry, the pipeline deviates from the intended line and level, irregular deformation of the surrounding soil, and interpenetration between the soil and slurry. Thereby, the state of contact can change from 'pipe-slurry' into 'pipe-soil-slurry' (Figure 2b,d). In such a case, a simple form of Equation (17) seems more suitable to reflect the effect of lubricant slurry. However, it is logical that the greater the contact width between the pipe and the soil, the smaller the effect of lubrication will be, and, therefore, the greater the friction resistance will be. Thus, the value of β should be highly affected by the pipe-soil contact width. For different soils, grouting amount of slurry and design parameters (such as buried depth and overcut) is bound to lead to completely different contact widths of pipe-soil. Thus, β should be in a large range, and it would be rather difficult to pick out a value to use in application.

In fact, a successful prediction model of friction resistance should not only consider the effect of lubrication but also needs to be able to reflect the effect of pipe-soil-slurry interaction. It is based on this understanding that the following model comes into being.

#### **3. Calculation of Friction Resistance for Slurry Pipe Jacking**

The general contact state of pipe-soil-slurry due to the interaction of the pipe, surrounding soil, and lubricant slurry is shown in Figure 3. In the picture, the position of pipe-soil contact is arbitrary, with a contact width of *Bs* and the corresponding contact angle of 2ε.

**Figure 3.** The general contact state of pipe-soil-slurry.

From Figure 3, it is obvious that to calculate friction resistance *f*, both of the items of *fs* and *fm* have to be taken into account, which can be expressed as

$$f = \mu \mathbf{N} = f\_{\mathbf{s}} + f\_{m} = \mu\_{\mathbf{s}} \mathbf{N}\_{\mathbf{s}} + \mu\_{m} \mathbf{N}\_{m} \tag{23}$$

where μ is an effective friction coefficient introduced to reflect the effect of lubrication and the influence of pipe-soil contact width. It is generally accepted that μ*<sup>s</sup>* = tan(ϕ/2) for the coefficient of kinematic friction between soil and the pipe [21,22]; μ*<sup>m</sup>* for the coefficient of kinematic friction between mud slurry and the pipe can be taken as 0.01, according to the test result reported by Guo [30]. *Ns* and *Nm* are the total normal force of pipe-soil and pipe-slurry in contact, respectively.

To calculate *Ns* and *Nm* precisely, the location of pipe-soil contact and the magnitude of contact angle (or contact width) and the contact force have to be determined. For various reasons leading to the occurrence of pipe-soil contact, it seems impossible to calculate these quantities in a target section of the pipeline. However, if taking the whole pipeline into consideration, and assuming that the pipe-soil contact can occur at any position of a section of the pipeline with a same probability and the contact force is approximately equal to the soil pressure in the contact area, this problem can be greatly simplified. In this case, we have the following equations:

$$P = N\_s = \frac{B\_s}{\mathbb{C}} N = \frac{\varepsilon}{\pi} N \tag{24}$$

$$N\_m = \frac{B\_m}{C} N\_c \tag{25}$$

where *C* (=π*Dp*) is the external circumference of pipe and ε is the semi-angle of contact (Figure 3). By substituting Equations (24) and (25) in Equation (23), after some algebra, giving that

$$\begin{cases} \mu = \mu\_{\text{s}} \lambda\_{\text{s}} + \mu\_{\text{m}} \lambda\_{\text{m}}\\ \lambda\_{\text{s}} = \frac{B\_{\text{s}}}{C} = \frac{\varepsilon}{\pi} \end{cases}, \quad \lambda\_{\text{m}} = \frac{B\_{\text{m}}}{C} \tag{26}$$

$$B\_m = C - \frac{B\_s}{1+c} \tag{27}$$

where *e* is the void ratio of soil.

By substituting Equation (27) in Equation (26), the expression of μ can be further rewritten as

$$
\mu = \mu\_s \frac{\varepsilon}{\pi} + \mu\_m \left( 1 - \frac{\varepsilon}{\pi(1+\varepsilon)} \right) \tag{28}
$$

According to Equation (28), the calculation of ε is essential to calculate μ*.* Hertzian model provides a simple way for the calculation of the width of contact (or contact angle) as we have mentioned before (see Equations (8) and (10)); however, the Hertzian contact problem is approached only when the applied force is small, or the large radial clearance is large, and the limited angle of contact is smaller than about 30◦ [31]. Due to the technical limitations, most of the pipe jacking projects encounter clay or sandy soils and with a small overcut, it is therefore important that the applicability of the Hertzian contact model should be extremely limited here. Actually, the Hertzian contact model is just a special case of the Persson's contact model with a small contact width (or angle) [31]. If a large contact angle (larger than 30◦) occurs, the more general contact model proposed by Persson should be taken as the first choice. The following singular integro-differential governing equation of contact angle is derived by Persson, as [31,32]

$$B = 4(1 - \beta) - 2(1 - \alpha) \int\_{-\xi}^{+\xi} q(t) \frac{dt}{1 + t^2} - \frac{\pi}{2}(1 + \alpha) \frac{E\_p \Delta R}{(1 - v\_p^2)P} \tag{29}$$

or

$$\frac{\pi(1+a)E\_p\Delta R}{(1-v\_p^2)P} = 4(1-\beta) - 2(1-a)\int\_{-\xi}^{+\xi} q(t)\frac{dt}{1+t^2} - B \tag{30}$$

The involved auxiliary variables are defined as [31]

$$\begin{array}{ll} \Delta \mathsf{R} = \frac{D\_{\mathsf{r}} - D\_{\mathsf{p}}}{2} \\ \Delta \mathsf{R} = \frac{1 - \eta}{1 + \eta} \quad , \quad \mathsf{\beta} = \frac{\lambda}{2(1 + \eta)} \\ \eta = \frac{E\_{\mathsf{p}}}{E\_{\mathsf{s}}} \frac{1 - v\_{\mathsf{s}}^{2}}{1 - v\_{\mathsf{p}}^{2}} \quad , \quad \lambda = \frac{1 - 2v\_{\mathsf{p}}}{1 - v\_{\mathsf{p}}} - \eta \frac{1 - 2v\_{\mathsf{s}}}{1 - v\_{\mathsf{s}}} \end{array} \tag{31}$$

After some approximate treatments, the key terms of Equation (30) have been solved by Michele [32], as follows:

$$\int\_{-\xi}^{+\xi} q(t) \frac{dt}{1+t^2} = \frac{1}{2\pi} \frac{I\_b}{\xi^2(\xi^2+1)} + \frac{\xi^2}{\xi^2+1} \tag{32}$$

$$I\_b = \pi \log(\xi^2 + 1)\tag{33}$$

$$B = \frac{2\xi^4 + 2\xi^2 - 1}{\xi^2(\xi^2 + 1)}\tag{34}$$

By substituting these equations into Equation (29), Michele obtained an approximate form of general contact angle relation.

$$\frac{\pi(a+1)E\_p\Delta\mathcal{R}}{(1-v\_p^2)P} = \frac{(a-1)[\ln(\xi^2+1)+2\xi^4]+2}{(\xi^2+1)\xi^2} - 4\beta\tag{35}$$

As compared with Equation (9), Equation (35) is a far more complex nonlinear equation. It can be further simplified with respect to that the elastic modulus of soil *Es* is generally much smaller than that of pipe *Ep* (the difference between the two can be three orders of magnitude). Thus, from Equation (30), the magnitude of auxiliary variable η should be very large, and, therefore, the following approximate relations can be obtained: <sup>π</sup>(α+1)*Ep*

$$\begin{cases} \frac{\pi (a+1)E\_p}{(1-v\_p^2)} \approx \frac{2\pi E\_s}{(1-v\_s^2)'}\\ \alpha \approx -1\_\prime\\ \beta \approx \frac{1-2v\_s}{2(1-v\_b)} \end{cases} \tag{36}$$

Using Equation (36), Equation (35) can be then simplified as

$$\frac{\pi E\_{\text{s}} \Delta R}{(1 - \upsilon\_{\text{s}}^2)P} + \frac{1 - 2\upsilon\_{\text{s}}}{1 - \upsilon\_{\text{s}}} = \frac{1 - [\ln(\xi^2 + 1) + 2\xi^4]}{(\xi^2 + 1)\xi^2} \tag{37}$$

From Equation (37), it is essential to calculate *P*, which requires one to calculate the total normal force *N*. It can be gained by integrating the normal stress σ*<sup>n</sup>* on an element of the pipe surface and is determined on the basis of vertical and horizontal soil stresses.

$$
\sigma\_n = \sigma\_\upsilon \sin \theta + \sigma\_d \cos \theta \tag{38}
$$

$$N = 4\int\_0^{\pi/2} \sigma\_n \frac{D\_p}{2} d\theta \tag{39}$$

where θ is defined as the angle between the corresponding radius line and the horizontal line at each point of the pipe (Figure 4).

By substituting Equation (38) in Equation (39), it is easy to obtain the equation of *N*, which has the same form as Equation (18).

To calculate Equation (18), the vertical soil stress σ*<sup>v</sup>* has to be first determined. It is noted that at the present time, by far the most commonly used model for soil pressure calculation is Terzaghi's silo model (Equation (22)) [5,6,9,19,21,26]. According to Equation (22), the calculation of the vertical soil stress requires some physical parameters that may be determined with some accuracy, such as the height of cover *h*, the cohesion *cs*, and the unit weight of soil γ, but also some empirical parameters, such as *b*, δ, and *K*. The definition of these empirical parameters varies from one author to another. Here, typical approaches of Terzaghi, Germany Standard ATV-A 161 E-90 [29], Chinese Standard GB 50332-2002 [33], UK Standard BS EN 1594-09 [34], US Standard ASTM F 1962-11 [35], UK PJA [27], Japan JMTA [36], and Japan JSA [22] would be discussed and compared.

For the calculation of silo width *b*, three kinds of boundary planes of wedge failure assumed by different authors and the corresponding equations are clearly illustrated in Figure 5. The width of the boundary plane is related to the 'vault' effect of soil. Generally, a smaller *b* means a lower 'vault' effect of soil, leading to a larger vertical soil stress.

**Figure 4.** The earth pressure and the normal stress acting on the pipe.

**Figure 5.** Boundary planes of wedge failure assumed by different authors. JSA: Japan Sewage Association; PJA: UK Pipe Jacking Association.

For the determination of δ, most of the guidelines, such as PJA, JSA, JMTA, BS EN 1594-90, and GB 50332-2002 assume shear planes as perfectly rough and take an angle of friction in the shear planes δ equal to the soil internal friction angle ϕ*.* However, ATV-A 161 E-90 and ASTM F 1962-11 make a more cautious assumption and only takes into account half the internal friction angle ϕ/2.

For the lateral pressure coefficient *K* above the tunnel, Terzaghi assumes *K* coefficient is equal to 1, which corresponds to the range of values encountered in clayey soils. PJA, ASTM F 1962-11, and GB 50332-2002 suggest *K* = *Ka* (calculated by Rankine's formula of active soil pressure coefficient), while BS EN 1594 and ATV-A 161 assume *K* = *K*<sup>0</sup> (calculated by Rankine's formula of soil pressure coefficient at rest). Moreover, according to ATV-A 161, this *K* coefficient is equal to 0.5, which corresponds to an internal soil friction angle of 30◦, a typical value for sandy soils.

Parameters of *b*, δ, and *K* chosen by the different authors have been summarized in Table 1.


**Table 1.** Definition of the empirical parameters in Terzaghi's silo model by different authors [37].

Note: α = π/4 − ϕ/2; β = π/4 + α/2.

It is noted that none of the approaches use the same parameters. Consequently, the vertical soil stress calculated by these approaches would be quite different. Thus, it is not convincing to pick out an approach to use without checking the field data. This work will be carried out in the next section.

Thus far, all the equations needed to calculate friction resistance have been determined. If the parameters needed for the prediction equations are quantified, by using Equations (18) and (22) the total normal force *N* can be determined, then together with Equations (23), (24), (27), (31), and (37), the contact angle 2ε, the effective friction coefficient μ, and the friction force *f* now can be uniquely identified. The flow chart is shown in Figure 6.

**Figure 6.** Flow chart of friction resistance prediction.

Apparently, the effective friction coefficient here is not just related to the interfriction angle of soil ϕ but also the state of pipe-soil contact and the effect of lubrication.

#### **4. The Verification of the E**ff**ectiveness of the Proposed New Approach**

*4.1. Comparison between the Predicted Friction Resistances and the Field Data*

Before comparison analysis, the existing prediction equations were numbered from M 1 to M 13 (Table 2). For M 7~M 13, the Equations (7), (14), (18), and (22) should be used and the parameters (*b*, δ, and *K*) of Terzaghi's silo model (Equation (22)) in the calculation of earth pressure should be determined by their approaches that are shown in Table 1.


**Table 2.** The numbering of the existing friction resistance prediction equations.

As has been mentioned before, the new approach of this paper introduces an effective friction coefficient μ (Equation (28)) to replace the original pipe-soil friction coefficient μ*<sup>s</sup>* (Equation (7)). The calculation of μ (Equation (28)) should follow the flow chart shown in Figure 2. Equations (18) and (22) are also used to calculate the normal force *N* and vertical earth pressure σ*v*, respectively. The new prediction equations were numbered from M 7 \* to M 13 \* (Table 3).

**Table 3.** The numbering of the new friction resistance prediction equations.


The superscript "\*" is used to distinguish from the number of models in Table 2.

Some parameters required by the predicted equations (such as μ*s*, *Pm*, and *W*) might not be given in the literature. In principle, during the calculation, the parameters given in the field case should be used, and the missing parameters can be evaluated by the following rules:


6. The average values of soil parameters (such as γ*s*, *cs*, ϕ, and *e*) can be obtained from the Geological Engineering Handbook.

Parameters in each of the four cases were finally determined, which are given in Table 4 [2,9,22]. These cases encountered some representative soils, such as silt, clay, sand, and gravels. Furthermore, they have different overburden depths of 2.72~8.5 m, overcut of 0~20 mm, and pipe diameters of 0.66~4.06 m. All of these characteristics provide good foundations for identifying the capability of the prediction models.


**Table 4.** Parameters that are needed to calculate the prediction equations in each case.

In Table 5, for each of the drives, measured frictional force values are presented and compared to values calculated by the approaches of the existing models. One can see that, for most cases, the prediction results of the models (M 1 and M 2) established based on Hypothesis 1 are generally too small. This is because it is correct only when the overcut is stable and the pipeline slides on its base inside the annular gap remaining open. The same problem is encountered in M3 (established based on Hypothesis 2), which is probably more due to ignoring the occurrence of pipe-soil contact decreasing the magnitude of the effective friction coefficient. Obviously, for slurry pipe jacking, the prediction friction based on these two assumptions may be insufficient and unsafe.

**Table 5.** Comparison of friction resistances calculated by the existing models and the measured data.


Note: *f* mea. is the measured friction resistance; *f* cal. is the calculated friction resistance.

The calculated results of M4 are much larger than the measured data, which presumably result from ignoring the 'vault effect' of soil, so that the soil pressure has been overestimated. Apart from M4, the other models established based on Hypothesis 3 (M5~13) have shown some applicability in case 4, in which the overcut of C4 is equal to zero, which is exactly in line with the assumption of the full contact of pipe-soil of Hypothesis 3. Except for C4, the prediction results of other cases are much larger, and the amplitude may be up to 30 times the measured data. Therefore, for slurry pipe jacking, Hypothesis 3 is generally over-conservative. Although it can ensure the structural safety of the design, it may also cause a much higher construction cost.

From what we have analyzed above, the existing models have good prediction results only when the field case is consistent with the basic hypothesis of each model. However, it is also these hypotheses that determine their limited applicability.

Table 6 presents the friction resistances calculated by the approach of this paper, by using the parameters of Terzaghi silo model chosen from different authors (Table 1). Better agreements with the field data in each of these cases were obtained, which indicate that compared with the existing prediction models, the approach of this paper that considers the effect of lubrication and interaction of pipe-soil-slurry not only has higher accuracy but is also more flexible and has wider applicability.


**Table 6.** Comparison of the friction resistances calculated by the new approach and the measured data.

The superscript "\*" is used to distinguish from the number of models in Table 2.

Moreover, the approaches of PJA (M9), ATV A 161 E-90 (M11), ASTM F 1962-11 (M12), and GB 50332-2002 (M13) provide higher prediction accuracy than the other ones (Table 6). However, it is noted that ATV A 161 E-90, ASTM F 1962-11, and GB 50332-2002 suggest the effect of soil cohesion *c* should be neglected (Table 1), but it is because the cohesion of soil has been considered in the calculation that the good prediction results can be obtained, as shown in Table 6. From this point of view, UK PJA (consider the cohesion of soil) provides the best choice of parameters of Terzaghi' silo model in the calculation of earth pressure. Therefore, the UK PJA approach is supposed to be used in the next analyses.

#### *4.2. Comparison between the Predicted Friction Resistances and the Numerical Simulation Result*

Some excellent work of numerical simulation has been conducted for the estimation of the jacking force (or friction resistance) of pipe jacking. For example, Ji et al. [5,40] and Barla and Camusso [41] presented a novel discrete 2D numerical model to evaluate the normal force acting on the pipe. Then, the friction resistance is determined by multiplying the interface friction coefficient by the normal force. Three different drives in a pipe jacking projects were analyzed, where the calculated pattern of jacking force was compared with the measured data, which demonstrated the effectiveness of the proposed approach. Yen and Shou [25] used a model coupling finite element method and a displacement control method to estimate the required jacking force in pipe jacking. The displacement control option in the numerical analysis software ABAQUS (Abaqus Inc., (Palo Alto, CA, USA) 2012) was used to designate the displacement at the end cross section of the pipe in the launch shaft. Accounting for the contact property and the contact range between the pipes and the soil during the jacking process, the stresses exerted on the pipes were used to back-calculate the jacking forces [25].

The numerical simulation strategy of Yen and Shou [25] is quite consistent with the analytical approach of this paper. Therefore, the numerical simulation approach of Yen and Shou will be described in the next section and its estimated result of friction resistance will be discussed and compared with that calculated by the analytical approach of this paper.

One of the analyses focused on a case of slurry pipe jacking in the Taichung Science Park, the basic parameters of this case were summarized in Table 7. In the numerical simulation, the lateral boundaries were fixed by roller, and hinges were used to constrain the bottom boundary. The three dimensional hexahedron element (C3D8R) with eight nodes was used in the simulation. To verify the jacking force obtained from the displacement control simulation, overcut and lubrication were included in the model by setting the contact range (1/3, 1/2, and 1 pipe-soil contact) and the frictional coefficient (Figure 7a–c). The results suggested the 1/3 contact case can estimate the jacking force with a better accuracy towards the middle and the final stage of the pipe jacking process (Figure 8).

**Table 7.** The details of the slurry pipe jacking in Taichung Science Park.

**Figure 7.** The setting of pipe-soil contact range and frictional coefficient in the numerical simulation. (**a**) 1/3 of the pipe-soil contact; (**b**) 1/2 of the pipe-soil contact; (**c**) 1 of the pipe-soil contact.

In Figure 8, the slope of the linear regression for the scattered points represents frictional resistance, and the intercept represents face resistance. For the calculated result of this paper, the measured face resistance is used, and the frictional resistance (*f* cal. = 57.7 kN/m) is calculated by the new approach (Section 3) using parameters of UK PJA in Terzaghi's silo model (Table 1). For the result of numerical simulation, the decrease of jacking force in the initial stage of drives (before 62.5 m) is explained by authors that the weight of the pipe jacking machine (the material of which is steel, featuring a larger unit weight) directly pressing on the soil, causing a larger face resistance. After the drives of 62.5 m, the influence exerted by the machine weight decreased gradually and the increase of jacking force is caused by the accumulation of friction resistance [25]. In other words, the slope of regression from drives of 62.5 to 100 m represents the friction resistance estimated by numerical simulation (*f* num. = 64.4 kN/m).

It is obvious that both the predicted friction resistances of the analytical equations (*f* cal. = 57.7 kN/m) and the numerical simulation (*f* num. = 64.4 kN/m) are acceptable as compared to the measured data (*f* mea. = 41.5 kN/m). However, better accuracy (especially towards the first 80 m of drives) is obtained by the approach of this paper.

**Figure 8.** The comparison of jacking forces from monitoring, the numerical simulation, and the predicted equations of this paper.

The predicted accuracy of friction resistance by the numerical simulation approach is highly affected by the setting of pipe-soil contact angle. The pipe-soil contact angle of 71 degrees is that calculated by the analytical equations for this field case. Thus, it seems that the pipe-soil contact angle of 120 degrees (1/3 contact) in the simulation is set too large, resulting in the calculated result slightly larger than the measured data. From this point of view, the numerical simulation approach can accurately predict the friction resistance, but the contact angle of pipe-soil in the simulation needs to be set reasonably to obtain good prediction results. Conversely, in the approach of this paper, the pipe-soil contact angle is theoretically calculated with respect to soil property, overcut, pipe diameter, etc., while human factors or empirical factors can be eliminated as far as possible.

#### **5. Influence of Design Factors on Lubrication E**ffi**ciency and Friction Resistance**

For a better design of slurry pipe jacking in the future, it is meaningful to study the influence of design parameters (such as buried depth *h*, pipe diameter *Dp*, and the overcut Δ*R*) on lubrication efficiency and friction resistance. To achieve this objective, a set of reference parameters is used, and then, by changing a target parameter according to the design rules, the effect of that parameter on friction resistance and lubrication efficiency can be obtained. The designed cases were shown in Table 8.



#### *5.1. Influence of Design Factors on Friction Resistance*

The influences of design factors (*h*, *Dp*, and Δ*R*) on the critical quantities of effective friction coefficient μ, normal force acting on the pipe *N*, and friction resistance *f* are respectively shown in Figure 9a–c, Figure 9d–f, Figure 9g–i.

**Figure 9.** The influence of factors of *h*, *Dp*, and Δ*R* on the quantities of μ (**a**–**c**), *N* (**d**–**f**)*,* and *f* (**g**–**i**).

It is evident that increasing buried depth and pipe diameter led to a double action for the increasing of friction resistance. Firstly, this increase then increased the possibility of contact between the pipe and soil, and, therefore, increase the effective friction coefficient μ on the interface. Secondly, the increasing of buried depth increases the vertical soil stress, while the increasing of pipe diameter increases the contact area, both effects of them increase normal force *N* acting on the pipes. The main difference between the two is that buried depth causes both of the effective friction coefficient and normal force to slightly increase and then gradually stabilize, while they approximately linearly increase with pipe diameter. Especially for the normal force induced by pipe diameter, which is strongly increased from 29.83 to 991.34 kN, this leads to a notable increase of friction resistance from 0.34 to 48.16 kN/m. Thus, the additional friction is strongly affected by the pipe diameter but appears not to be greatly affected by the buried depth.

Different from the buried depth and pipe diameter, the overcut with small values has no effect on the normal force (Figure 9h). However, it has a strongly negative effect on the effective friction coefficient on the interface (Figure 9g). In fact, it does determine the volume of injected lubricant slurry, which has a significant influence on the occurrence of the pipe-soil contact, and, therefore, determine the magnitude of the effective friction coefficient. Thus, it has a strongly negative effect on the friction resistance.

#### *5.2. Influence of Design Factors on Lubrication E*ffi*ciency*

Except for friction resistance, engineers are also concerned about the lubrication efficiency [9,42,43]. According to Equation (27), the magnitude of μ is between the pipe-slurry friction coefficient μ*<sup>m</sup>* and the pipe-soil friction coefficient μ*s*. If there is no contact between the pipe and the soil, the angular space due to overcut is completely filled with lubricant slurry, the effective friction coefficient is equal to μ*m*; and if the soil is in full contact with the external surface of the pipe, the effective friction coefficient mainly depends on the pipe-soil nature and approximately equals to μ*s*. Thus, the lubrication efficiency can be defined as

$$
\chi = \left(1 - \frac{\mu - \mu\_m}{\mu\_s}\right) \times 100\% \tag{40}
$$

By substituting Equation (28) in Equation (31), and considering that μ*<sup>m</sup>* is far smaller than μ*s*, the χ can approximately be expressed by Equation (41).

$$\chi = \left(1 - \frac{\varepsilon}{\pi}\right) \times 100\% \tag{41}$$

If pipe-soil contact angle ε = 0, χ = 100% for maximum lubrication efficiency, and if ε = π, χ = 0% for minimum lubrication efficiency. It is noted that χ = 0% is not going to happen. According to the Passon model (Equation (37)), the result calculated by the left terms of Equation (37) should be not less than zero, while the right term of Equation (37) is a monotonically decreasing function of pipe-soil contact angle. In other words, when the right term of Equation (37) is equal to zero, the pipe-soil contact angle reaches its maximum value, which is solved by ε = ε*max* = 72◦ = 0.4π, corresponding to the minimum lubrication efficiency of 60% (i.e., χ is theoretically between 60% and 100% on the basis of Passon contact model). Although it is not theoretically correct as compared to that counted by Pellet and Kastner as between 45% and 90% [9] and tested by Zhou as between 47.8% and 78.6% [5], it seems practical to estimate the efficiency of lubrication by the approach of this paper.

The pipe-soil contact angle and the corresponding lubrication efficiency calculated by Equation (41) has been shown in Figure 10a–c, Figure 10d–f, respectively. It is found that, as compared to the low effect of overburden depth, special attention should be paid to the effect of pipe diameter and the overcut.

Figure 10f shows that the lubrication efficiency strongly increases from 64% to 91%, while the overcut increases only from 0 to 15 mm, and after that, the effect of overcut is significantly reduced. This observation confirms the importance of overcut, which has to be sufficiently wide so that the decrease of tunnel diameter induced by the elastic ground unloading does not lead to the closure of the annular space. Moreover, Figure 10e shows that the increasing of pipe diameter from 1 to 4 m causes obvious efficiency losses of lubrication from 99% to 82%. Thus, one can conclude that the larger the pipe diameter, the lager the overcut is needed.

The buried depth *h* is determined by the intended line and level of the tunnel, which is often limited by geological conditions and distributions of the existing buildings and structures, while the pipe diameter *Dp* highly depends on the practical use or traffic requirements. It seems that there are not many options for the buried depth *h* and pipe diameter *Dp* in the design. From this point of view, both for lubrication efficiency and friction reaction, more attention should be paid to the design of overcut.

Parts of the conclusions analytically discussed above have been verified by authors from field observations [9,20], which in turn again confirm the feasibility of the approach of this paper.

**Figure 10.** The influences of factors of *h*, *Dp*, and Δ*R* on the pipe-soil contact angle 2ε (**a**–**c**) and lubrication efficiency χ (**d**–**f**).

#### **6. Conclusions**

Some typical prediction models of friction resistance have been presented and detailed comparisons and analyses have also been made. Then, a new approach considering both the effect of lubrication and the interaction of pipe-soil-slurry, by introducing an effective friction coefficient, has been established. Values of friction resistance calculated using them have been compared with values measured in four field cases and a numerical simulation case with various soils and design parameters. Better agreements are obtained, which indicate a more flexible and wider applicability of the approach in this paper as compared to the existing prediction models and numerical simulation approach. Explanations have also been sought for limited use of the existing models that may be attributed to their hypotheses not that suitable for slurry pipe jacking. The numerical simulation approach can accurately predict the friction resistance, but it is hard to determine the contact angle of pipe-soil reasonably with respect to various soils, overcut, and other conditions to obtain good prediction results.

Using the approach of this paper, for higher prediction accuracy, the cohesion of soil has to be taken into account in the calculation for drives in clayey soils. The Terzaghi's silo model together with parameters determined by the approach of UK PJA is verified as the most well-considered to calculate earth pressure.

For better design in the future, the influences of design factors (buried depth, pipe diameter, and overcut) on friction resistance and lubrication efficiency have been analyzed too. The increase of pipe diameter has a strong influence on the increase of friction resistance; however, the friction amplitude appears not to be greatly affected by the buried depth. As the selection of pipe diameter and buried depth are limited by various objective conditions, special attention should be paid on the design of overcut. The overcut has to be sufficiently wide. When the overcut is small (for example smaller than 15 mm), the decrease of overcut strongly affects the decrease of lubrication efficiency, and, therefore, leads to a notable increase in friction resistance. Moreover, pipe diameter has an obvious influence on the effect of overcut on lubrication efficiency, the larger the pipe diameter the larger the overcut needed.

**Author Contributions:** L.P. and C.S. put forward the methodology; Y.Z. and Y.L. collected cases' data; Y.Y. derived the formulas and wrote the paper; W.Y. reviewed and edited the paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors acknowledge the financial support of the National Natural Science Foundation of China (No. 51878670).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**


#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Fragility Curves for RC Structure under Blast Load Considering the Influence of Seismic Demand**

#### **Flavio Stochino \*, Alessandro Attoli and Giovanna Concu**

Department of Civil Environmental Engineering and Architecture, University of Cagliari, 09123 Cagliari, Italy; ale-attoli@tiscali.it (A.A.); gconcu@unica.it (G.C.)

**\*** Correspondence: fstochino@unica.it; Tel.: +39-070-675-5115

Received: 10 December 2019; Accepted: 4 January 2020; Published: 8 January 2020

#### **Featured Application: The fragility curves can be useful for the early design of strategic RC buildings under blast load.**

**Abstract:** The complex characteristics of explosion load as well as its increasingly high frequency in the civil environment highlight the need to develop models representing the behavior of structures under blast load. This work presents a probabilistic study of the performance of framed reinforced concrete buildings designed according to the current Italian NTC18 and European EC8 technical standards. First, a simplified single degree of freedom model representing the structural system under blast load has been developed. Then, a probabilistic approach based on Monte Carlo simulation analysis highlighted the influence of seismic demand on the behavior of Reinforced Concrete RC buildings subjected to blast load.

**Keywords:** concrete; blast load; Monte Carlo analysis; seismic demand; pushover

#### **1. Introduction**

In the last years, structural safety under blast load has become a dramatic problem. Extreme events, such as impacts, explosions, etc., can occur in everyday life with unexpectedly high frequency [1,2]. In fact, the problem of terrorist attacks, important for strategic and military building design [2,3], can be put side by side with civil building explosion accidents [4,5].

Recently, many studies were aimed at assessing the performance of new and advanced materials under blast load: glass [6], fiber reinforced polymer [7,8], layered composite materials [9], and foam [10]. On the other hand, the structural design itself is evolving to a more general framework in which structural elements are designed and assembled to obtain general properties like robustness. The latter is the ability of a structure to withstand extreme loads without being damaged to an extent disproportionate to the cause. When an extreme load is concerned, structural damages are common and robustness is of paramount relevance. See [11] for a current state of the art review and [12] for a detailed analysis of the problem and of the available quantitative indexes.

RC structures designed and built in seismic zones should be robust in order to withstand the extreme earthquake load and many studies on this topic have been developed in the last years: [13–17]. Fewer studies deal with the interaction between earthquake and blast load. Abdollahzadeh and Faghihmaleki [18] evaluated the robustness of a seismic designed RC structure under blast load with deterministic, probabilistic and risk-based methods but did not investigated the influence of seismic demand. The latter risk-based approach has been developed in [19] considering a multi-hazard analysis for seismic and blast critical events.

The uncertainties due to blast load imply the need of a probabilistic approach in order to have an accurate estimation of the structural behavior and integrity [20,21]. Performance based fragility estimates have been adopted to assess the reliability of structures under impact in [22] showing how it is possible to formulate a Bayesian physical model for these kinds of problems.

This paper reports on a probabilistic analysis of the effect of a seismic demand on the structural fragility in case of blast load. Starting from the capacity curves of framed structures designed for different seismic loads in the Italian territory, an equivalent single degree of freedom model is developed in order to perform a probabilistic analysis based on Monte Carlo approach. Fragility curves and performance analysis are obtained with a general methodology that can be extended to many other structures. After this introduction, in Section 2 the blast load model is presented, while Section 3 describes the selected structure. Section 4 depicts the structural model while Section 5 presents the probabilistic framework. Results are in Section 6, while some concluding remarks and prospective developments are stated in Section 7.

#### **2. Load Model**

In this work, the case of hemispheric explosion load was considered. The stand-off pressure *Pso* in MPa was estimated using the Mills' approach [23]:

$$P\_{sO} = 1.772 \left(\frac{1}{Z^3}\right) - 0.114 \left(\frac{1}{Z^2}\right) + 0.108 \left(\frac{1}{Z}\right) \tag{1}$$

where *z* is the scaled distance representing the ratio between the distance from the explosive charge to the building and the cubic root of the explosive charge, it is expressed by:

$$z = \frac{R}{\mathcal{W}^{\frac{1}{2}}} \tag{2}$$

where *R* is the stand-off distance and *W* is the mass of explosive in kg of equivalent TNT [24,25]. The incident impulse is represented by Held's [24] equation:

$$I\_{sO} = B \frac{W^{2/3}}{R} \tag{3}$$

where *<sup>B</sup>* is a numerical coefficient that has been considered equal to 4.5 <sup>×</sup> <sup>10</sup><sup>5</sup> for *<sup>R</sup>* <sup>&</sup>gt; 10 m and 3.5 <sup>×</sup> 105 for *R* ≤ 10 m following the indications reported in [24,25]. Instead, the reflected pressure peak can be expressed as [21]:

$$Pr = 2 \cdot P\_{sO} \left( \frac{7P\_{atm} + 4P\_{sO}}{7P\_{atm} + P\_{sO}} \right) \tag{4}$$

where Patm = 0.1 MPa. The positive phase duration *td* can be expressed assuming a triangular impulse:

$$t\_d = \frac{2\ I\_{sO}}{P\_{sO}}\tag{5}$$

The blast load time history is usually expressed with an exponential function of time *t* as proposed by Friedlander [26] considering β = 1.8:

$$P\_r(t) = P\_r \left(1 - \frac{t}{t\_d}\right)^{\frac{-\beta t}{t\_d}} \tag{6}$$

In this work, in order to reduce the computational cost, the nonlinear Equation (6) can be simplified with an equivalent triangular time-history, as shown in Figure 1:

$$P\_r(t) = P\_r \left(1 - \frac{t}{t\_d}\right) \tag{7}$$

WVHF 7LPH

**Figure 1.** Blast load time-histories: exponential and triangular.

In this linear case, the positive phase duration is obtained by equating the area underneath the two curves in order to have an equivalent impulse for the two models.

#### **3. Case Study**

A framed RC structure with squared cross section has been considered as a case study, see Figure 2a for the geometrical sizes. Beam and column characteristics are detailed in Figure 2b. This kind of structure can serve as watchtower in a military environment.

**Figure 2.** Structure lateral view and cross section (**a**), beam and column cross section (**b**).

In order to study the influence of the seismic demand in the structural design, the same structure has been designed considering four different locations in Italy: L'Aquila, Catania, Bari and Cagliari characterized by different seismic load, from the highest to the lower, see Table 1 and Figure 3. In this way, the reinforcements distributions will be different for each construction site. Clearly, the city with the highest seismic demand is L'Aquila and the highest reinforcement ratio is obtained when the structure is located in this city. At the same time, the lowest reinforcement ratio is obtained for Cagliari, that is the location with the lowest seismic demand.


**Figure 3.** Italian map of seismic PGA (peak ground acceleration), taken from [27] and geographical locations of the four design sites.

For the sake of synthesis, the same structure was designed in order to fulfill four different seismic demands corresponding to different Italian locations, see Tables 1 and 2. Consequently, four different structural models are considered in the next sections in order to investigate how the seismic demand can influence also the blast resistance.

**Table 2.** Materials characteristics.


#### **4. Structural Model**

In case of blast load the structural behaviour of a mechanical system can be represented by a single degree of freedom (SDOF) model characterized by a spring denoting the stiffness and a mass expressing the inertia, see Figure 4. Indeed, in this kind of problem damping can be disregarded because the maximum displacement is obtained in the first cycle of loading, see [28]. In fact, the aim of this structural model is to evaluate the maximum displacement of the structure at collapse.

**Figure 4.** Hemispherical aboveground blast (**left**) and structural model (**right**).

If the mechanical non-linearities are taken into account the SDOF constitutive law can be simplified with load-displacement bilinear diagram, as shown in Figure 5.

**Figure 5.** SDOF constitutive law, *Py* and *uEy* respectively are the yielding load and displacement, while *Pu* and *uEu* are the corresponding ultimate ones. *KE,el* represents the elastic stiffness while *KE,pl* the plastic one.

For simple structures like beams or columns, it is possible to obtain the bilinear force-displacement diagram quite easily, just by identifying the collapse mechanism and, consequently, yielding and ultimate displacement values. In the case of a complex structure, this process becomes difficult, and in general cases it is not always possible to represent the structural behavior with an equivalent SDOF system. However, in the present case, the framed structure is quite slender and a simple modal analysis (the modal analysis was performed with the numerical model presented in Section 4.2) showed that the first eigenmode is characterized by the 85% of participant mass. For this reason, it is possible to assume that the dynamic behavior of the structure under a uniform blast load pressure can be represented by an equivalent SDOF system. Push-over analysis [29–32] can produce the force-displacement diagram known as a capacity curve. From this capacity curve it is possible to obtain an equivalent bilinear

force–displacement diagram that represents the SDOF constitutive behavior [33–35] as shown in Section 4.2.

#### *4.1. Materials and Strain Rate E*ff*ects*

The materials constitutive laws and characteristics are shown in Figure 6 and Table 2.

**Figure 6.** Materials constitutive law: (**a**) concrete, (**b**) steel.

 The time dependency of the mechanical characteristics of concrete and steel on strain rate is already known. Indeed, in case of blast or impulsive load the characteristics of materials can be strongly influenced by strain rate, see [36–39]. In the literature, it is possible to find quite advanced analytical models for the strain rate effects [40], but obviously they would increase the computational cost and the complexity of the model. In order to simplify the problem and reduce the computational cost in this paper the approach proposed in [25] has been applied. A set of Dynamic Increase Factors (DIF) equal to the ratio between a dynamic mechanical characteristic *fd* and the equivalent static one *f* has been defined as reported in Table 3.


**Table 3.** Dynamic increase factor (DIF) for RC elements, extracted from [25].

Thus, given the critical internal force for each structural component, the appropriate DIF has been chosen from Table 3 in order to modify the mechanical characteristics of the structural model.

#### *4.2. Capacity Curves*

The structure presented in Section 3 has been designed for permanent, service and earthquake load following the Italian [41] and European Standard [42] in each location, see Table 4. Then a finite element (FE) model of the framed structure has been developed using the commercial software JASP 6.5 [43]. This FE model was used to perform a static non-linear analysis with a uniform horizontal load. The lumped plastic hinge behavior has been modelled by standard approach [42–44], as illustrated in Figure 7.

**Table 4.** Reinforcements details for columns in each structure, *Ac* represents the concrete area, *As* the tensile reinforcement, *A's* the compressive reinforcement, *r* is the reinforcement ratio (*r* = *As*/*Ac*) while the letter *x* or *y* denotes the reinforcements for the bending moment around x axis or y axis respectively, see Figure 2.


**Figure 7.** Moment-rotation constitutive behavior for a frame member-end considered in JASP 6.5, where *Mu* is the ultimate bending moment, *Mcr* is the first cracking moment, θ*y'* is the yielding rotation and θ*u'* is the ultimate rotation.

The results in terms of horizontal force and top horizontal displacement (capacity curves) are reported in Figure 8 for the structures designed in the selected locations. The equivalent bilinear SDOF constitutive curves have been calculated equating the area underneath bilinear and capacity curves assuming that the ending point and the first elastic slope should be the same for the two curves.

**Figure 8.** Capacity curves and SDOF constitutive law for each structure location. The total horizontal load Vb is plotted versus the horizontal displacement of the top floor.

#### *4.3. Analytical Model*

The equations of motion of the equivalent SDOF oscillator in the elastic and plastic regimes have the following forms:

$$\mathbf{M}\_{E,el}\frac{d^2\upsilon\_E(t)}{dt^2} + \mathbf{K}\_{E,el}u\_E(t) = P\_E(t) \quad \text{for } 0 \le u\_E \le u\_{Ey} \tag{8a}$$

$$\mathbf{M}\_{\mathrm{E},\mathrm{pl}}\frac{d^2\mathbf{v}\_{\mathrm{E}}(t)}{dt^2} + \mathbf{K}\_{\mathrm{E},\mathrm{pl}}\boldsymbol{u}\_{\mathrm{E}}(t) + \left[\mathbf{K}\_{\mathrm{E},\mathrm{el}} - \mathbf{K}\_{\mathrm{E},\mathrm{pl}}\right]\boldsymbol{u}\_{\mathrm{E},\mathrm{y}} = \boldsymbol{P}\_{\mathrm{E}}(t) \quad \text{for } \boldsymbol{u}\_{\mathrm{E},\mathrm{y}} < \boldsymbol{u}\_{\mathrm{E}} \le \boldsymbol{u}\_{\mathrm{E},\mathrm{el}} \tag{8b}$$

where *uE*(*t*) is the model displacement, *PE*(*t*) is the total load on the structure. M*E*,*el* and M*E*,*pl* denote the equivalent mass of the oscillator respectively for the elastic and plastic field. In general: M*<sup>E</sup>* = *KLMM* where *M* is the structural mass and *KLM* is a coefficient which accounts for the boundary conditions of the structural element, the type of load and the regime considered (elastic or plastic). The structure presented in Section 3 can be considered similar to a vertical cantilever fixed in the bottom part. Thus, in case of a uniformly distributed load for a cantilever structure M*E,el* = 0.65·*M* and M*E,pl* = 0.66·*M*, see [45]. *vEy* and *vEu* are the equivalent yieding and ultimate model displacement, while *KE*,*el* and *KE*,*pl* respectively are the equivalent stiffnesses in the elastic and plastic range.

Given the analytical expression of the load time history it is possible to find the close form solution of Equation (8), see [46]. In this work, *PE*(*t*) is approximated by the linear expression presented in Equation (7).

#### *4.4. Load Scenario*

In this paper, only external explosion produced by a terrorist attack has been taken into account. Stewart et al. [47] described some of the possible scenarios that can generate an external hemispheric explosion. It is interesting to distinguish theme by the ways in which a mass of explosives could be transported near the object of the attack: 5 kg body explosive; 25 kg suitcase explosive; 200 kg car explosive.

In this work the load scenario obtained with 200, 300, 400, and 500 kg of TNT has been considered. These situations can be easily obtained considering a car or a truck containing the explosives. Various stand-off distances have been investigated studying the effects of the explosives for the structure described in Section 3.

#### *4.5. Damage Thresolds*

In order to measure the structural performance under blast load the drift values proposed by [48] have been adopted, see Table 5. It is important to point out that this approach considers the whole structural response given that the stand-off distance is sufficiently large to obtain a planar blast wave acting on the building, see Figure 4. Thus, for the sake of simplicity, localized column or beam collapse has not been considered, in addition, also the harmful damages on secondary elements which can lead to a loss of life are neglected. Instead, the top floor maximum displacement *uMAX* related to the building height *h*=12 m has been considered to define the relative drift:

$$X = \frac{u\_{MAX}}{h} \tag{9}$$



This simplified approach is clearly limited to its assumptions but can be useful in case of preliminary or early design because it can easily provide a synthetic parameter describing the damage condition of a building after the blast load.

#### **5. Probabilistic Analysis**

Fragility curves describe the conditional probability of exceedance (P(X > x0|Z)) of the response parameter X (drift in this case) given a demand intensity measure (scaled distance Z in this case). Thus, the structural fragility can be expressed as the cumulative distribution of the probability that a damage threshold *x*<sup>0</sup> is exceeded [21,49]:

$$P(X > \mathbf{x}\_0) = \int\_{-\infty}^{+\infty} P(X > \mathbf{x}\_0 | Z) \, p(Z) dz \triangleq \sum\_{i=0}^{\infty} P(X > \mathbf{x}\_0 | Z)\_i \, p(Z)\_i \Delta Z\_i \tag{10}$$

where the discretization of the integral calculation is represented by a discrete sum of conditions in which the scaled distance is varied with a given step Δ*Z*.

In this paper, the structural characteristics have been considered deterministic while the uncertainties of the load have been modelled considering the explosive mass and stand-off distance as stochastic variables characterized by lognormal distributions whose characteristics are shown in Table 6.



A Monte Carlo analysis has been developed with the above described SDOF model in order to obtain fragility curves presented in Section 6.2 and the probability of thresholds exceedance shown in Section 6.3. The coefficient of variation (COV) of the maximum drift has been checked in order to define the convergence condition of the analysis.

#### **6. Results and Discussion**

#### *6.1. Maximum Drift*

In the first load scenario, a 500 kg TNT bomb was blown up at various distance from the structure. The maximum drift values have been plotted as a function of the scaled distance in Figure 9. In the same picture the above-mentioned damage thresholds have been plotted in order to easily find the safety scaled distance for each structure. As expected, the structure designed in L'Aquila with the highest seismic load produced the best performance reaching the damage thresholds with smallest scaled distance in comparison with the other structures. This can be explained given the higher reinforcement ratio (see Table 4) and robustness of this structure in comparison with the others. At the same time, it is clear that 500 kg of TNT represents a huge amount and it is necessary to reach very high scaled distance (higher than 10 /kg1/3) to avoid any damage.

**Figure 9.** Maximum drift for the structure under blast load for each location, 500 kg TNT.

#### *6.2. Fragility Curves*

The fragility curves for the structures designed in the four locations are presented in Figures 10–12. In this case the TNT mass has been varied considering a Lognormal distribution with different mean values: 200, 300, 400, and 500 kg. The three damage thresholds have been considered: slight (Figure 10), moderate (Figure 11) and severe (Figure 12). Instead, the stand-off distance has been varied in a deterministic way in order to analyze all the scaled distance values.

**Figure 10.** Fragility curves for slight damage condition for each structure location.

**Figure 11.** Fragility curves for moderate damage condition for each structure location.

**Figure 12.** Fragility curves for severe damage condition for each structure location.

As a representative case, the COV of the maximum displacement for the severe damage threshold in case of 500 kg bomb for each structure location has been shown in Figure 13. Also, the other cases present similar trends.

**Figure 13.** Maximum drift COV variation for severe damage condition in case of 500 kg bomb for each structure location.

In Figures 10–13, the curves corresponding to 500 kg are the ones placed on the far right of each figure highlighting that this is the most severe condition in each case. Instead, if the structure designed in the site with less seismic demand (Cagliari) is compared to the one with the highest (L'Aquila) it is possible to note how the curves are shifted to the left in the latter case and to the right in the former one. This is another proof of the better structural performance of the structure designed in L'Aquila.

#### *6.3. Probability of Thresholds Exceedance*

The Monte Carlo analysis has been developed considering both stand-off distance *R* and mass weight *W* as stochastic variables following a Lognormal probability distribution as described in Section 5. Tables 7–10 present the probability of exceedance for each different threshold and for each structure designed in the different location.

Also in this case, it is possible to highlight the better structural performance of the structure designed in L'Aquila in comparison with the others. In addition, it is possible to note how, as the stand-off distance increases, the probability of exceedance for each threshold is reduced. In most cases the safe stand-off distance for a 500 kg TNT bomb is about 40–45 m. Instead, in the case of 200 kg, it is 25 m for L'Aquila, 30 m for Catania and Bari, and 35 m for Cagliari.


**Table 7.** Percentage probability of thresholds exceedance for L'Aquila location.

**Table 8.** Percentage probability of thresholds exceedance for Catania location.



**Table 9.** Percentage probability of thresholds exceedance for Bari location.

**Table 10.** Percentage probability of thresholds exceedance for Cagliari location.


#### **7. Conclusions**

This paper presented a simplified procedure to evaluate the safety of a framed RC structure under blast load highlighting the seismic design influence. The considered framed building has been designed considering four different seismic demand corresponding to specific locations in Italy characterized by different PGA. Then, a simplified SDOF system has been obtained from a pushover analysis in order to perform several Monte Carlo analyses aimed at highlighting what is the performance of the considered building under blast load in both a deterministic and probabilistic framework.

The structure designed to withstand the highest seismic load (L'Aquila) has been proven to have better performance also in the case of blast load. Thus, the influence of the seismic demand in the building design is evident. Indeed, the structure designed with the lowest seismic demand (Cagliari) presents the worst structural behavior under the blast load. This can be clearly seen from the fragility curves Figures 10–12, probability Tables 7–10, and also from the deterministic maximum drift presented in Figure 9.

The obtained results can be useful for safety evaluation in the case of a terrorist attack. Indeed, the fragility curves and the probability of threshold exceedance can help the designer in evaluating what should be a "safe" distance for the given structure that can be obtained with fence system or bollards.

Further development of this work is expected, applying this method to existing structures, like those described in [50,51]. It is also interesting to merge the proposed method with other assessment approaches and retrofitting techniques considering other kind of structures [52–54].

**Author Contributions:** F.S. conceived the illustrated strategy and the theoretical formulation contributing also to the numerical analysis; A.A. developed the numerical and analytical analysis; G.C. analyzed the numerical results and contributed to the theoretical formulation. All the authors wrote the paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** The financial support of the Autonomous Region of Sardinia under grant PO-FSE 2014–2020, CCI: 2014-IT05SFOP021, through the project "Retrofitting, rehabilitation and requalification of the historical cultural architectural heritage (R3-PAS)" is acknowledged by Flavio Stochino.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **The Seepage and Soil Plug Formation in Suction Caissons in Sand Using Visual Tests**

#### **Liquan Xie, Shili Ma \* and Tiantian Lin**

College of Civil Engineering, Tongji University, Shanghai 200092, China; xie\_liquan@tongji.edu.cn (L.X.); gotobest8@163.com (T.L.)

**\*** Correspondence: ma\_shili@163.com; Tel.: +86-191-2176-3067

Received: 4 December 2019; Accepted: 8 January 2020; Published: 13 January 2020

**Abstract:** The rapid development of offshore wind energy in China is becoming increasingly relevant for movement toward green development. This paper presents the results of visual tests of a suction caisson used as foundation for offshore wind turbines. The distribution of hydraulic gradients of sand at the mudline in the caisson was obtained to find out the relationship with the heights of soil plugs. The relationship equation was proposed and obtained by using quadratic regression, guiding project designs, and construction. It was found that there was no soil plug in the caisson when small suction was applied during the suction penetration. The relationship between the heights of the soil plugs and the hydraulic gradient of the soil was proposed and obtained by using quadratic regression to predict (roughly) the height of soil plugs in suction caissons in sand during suction penetration. The influence of settlement outside caissons on the soil plug was found to decrease as the buried depth rose.

**Keywords:** suction caisson; suction penetration; soil plug; hydraulic gradient; visual tests

#### **1. Introduction**

In the past few decades, it has become increasingly important to rapidly develop the offshore wind industry, which provides practical sources of energy with a low carbon footprint [1,2]. Foundations play an important role in guaranteeing the safe operation of offshore wind turbines [3]. The suction caisson, being installed economically and efficiently into soil deposits, has been increasingly used as a competitive foundation for offshore wind turbines in deep water [4,5]. A suction caisson is a large cylindrical structure that is typically made of steel, open at the bottom and closed at the top [6]. To set up a perfect offshore wind turbine, two aspects need to be considered for the engineering design of this foundation: suction installation and in-service performance [7]. The capacity of suction caissons as the foundation for offshore wind turbines is enhanced by means of peripheral embedded thin walls, which confine the internal soil [8]. A caisson is installed by penetrating the seabed under its own weight, and then by pumping water out of the caisson to create suction that forces the foundation into the seabed [9].

During suction penetration, the induced seepage flow through highly permeable sand into the caisson interior can create some negative effects. To investigate these effects, a number of studies have been completed, and they show that the seepage facilitates the installation process at the caisson tip and along the inner wall [10–13]. Erbrich and Tjelta [14] presented a series of finite element analyses and found that the suction forced water to migrate through the soil from outside to the inner caisson. Previous research [15,16] still has certain issues, in that the seepage around the suction was assumed to follow Darcy's law, which requires that we carry on further discussion. Experimental investigations in sand have revealed that soil plugs are likely to occur during suction-assisted installation [15,17,18]. A series of centrifuge tests on the installation of suction caissons were carried out by Tran et al. [19,20],

who found that the soil plugs heave up to 20% of the caisson penetration as excessive suction, and they developed a void ratio–permeability relationship to check the sand heave against the plugs. There is no previous study on the relationship between soil plugs and the seepage velocity of sand around suction caissons.

This paper presents the results of visual tests of a suction caisson used as a foundation for offshore wind turbines. The process of suction installation of the caisson foundations with axisymmetric geometry is simplified to a plane problem. In order to study their composition, the heights of the final soil plugs in the caisson were measured. The distribution of the hydraulic gradient of sand at the mudline in the caisson was obtained to find out its relationship with the heights of the soil plugs. The relationship between the heights of soil plugs and the hydraulic gradient of soil was proposed and obtained by using quadratic regression, guiding project designs, and construction.

#### **2. Experimental Program**

The process of suction installation of the caisson foundations (shown in Figure 1a) with axisymmetric geometry is simplified to a plane problem in this paper. Due to the fact that the soil was replaced by the walls of the soil tank in tests, it is assumed that the forces of water adhesion to the walls are equal to that of the sand around the suction caissons. A soil tank (shown in Figure 1), as used in tests, has dimensions of 20 × 0.5 cm in its plan view, and a depth of 28 cm. Transparent fiberglass was used to make the soil tank, with a thickness of 1.0 cm. The two pumping outlets in the soil tank were connected to the suction loading system by a thin pipe and a drainpipe, respectively. Figure 1b shows the diagram of the test device, with a constant suction *S* in the caisson caused by the suction loading system creating the different head *H*. The value of the applied constant suction can be expressed as *S* = γw*H*, where γ<sup>w</sup> is the unit weight of water (10 kN/m3).

(**a**) Suction caisson. (**b**) Diagram of test devices.

The suction caisson with an external diameter *D* of 12 cm had a height of 17.5 cm *L* and a thickness *T* of 0.5 cm. In order to create an enclosed compartment between the soil tank, the suction caisson, and the soil, UV glue was used to stick the soil tank and the suction caisson together. The suction caisson model was located at a distance 7.5 cm from the bottom of the soil tank.

The siliceous sand shown in Figure 2 was used in tests because it is commercially available and shows a deep contrast with the carmine stain. The void ratio *e* of the soil was determined according to the standard soil testing methods, and can be written as *e* = ρw*G*s/ρd-1, where ρ<sup>d</sup> is the dry density of the sand and *G*s is the specific gravity of the sand. Table 1 shows the properties of the siliceous sand in tests. The siliceous sand particles had an average radius of 0.748 mm, and they obeyed uniform distribution,

with a ratio of maximum to minimum radii of 2.0. The pluviation method and the compaction method were used to prepare uniform sand specimens in layers in the soil tank. The permeability coefficient of the siliceous sand was obtained by using an empirical equation, *k* = 2*d*102*e*<sup>2</sup> [21], where *d*<sup>10</sup> is the effective size of the sand. Due to the uniform particle size of the sand, it is suggested that *d*<sup>10</sup> is equal to the average particle size (0.718 mm). Before carrying out every test, the sand specimen was left to stand for 24 h.

**Figure 2.** The siliceous sand in tests.

**Table 1.** Properties of siliceous sand.


The seepage field around the suction caisson model was visualized by the tracer titration system, which consisted of an infusion system and carmine stain. The infusion system was made of a medical syringe with a thin pipe (radius of 2 mm), shown in Figure 1b. Three holes were drilled in the soil tank at points A, B, and C, connecting to the infusion system with the thin pipes. At points A, B, and C, the carmine stain was released with pinpoint accuracy by the infusion system. To reduce the effect of turbulence, the carmine stain was injected into the soil used in tests as slowly as possible.

The suction caisson model had a buried depth *h* (5, 10, and 15 cm) before each test, to simulate the process of suction installation. The influence of the temperatures of the environment and the water were not considered in the tests. A video camera was used to collect images of the experimental phenomena of seepage and the soil plug in the suction caisson. At the end of the trial, the height of the soil plug did not seem to change. The testing programs are listed in Table 2 in tests. The pre-test results show that seepage failure took place in the sand when the suction in the caisson with buried depths of 5, 10, and 15 cm were greater than 2.0, 3.5, and 4.0 kPa, respectively.

**Table 2.** List of visual tests on suction caissons.


#### **3. Test Results and Discussion**

#### *3.1. Visual Seepage Paths*

The seepage field can be visualized where the carmine stain flows along streamlines in the soil around the suction caisson. Figure 3 shows the visualization of the seepage flow at point B as *h* = 15 cm and *S* = 0.5 kPa. It was observed that the visual tests achieved good results to study the seepage of sand around a suction caisson during suction penetration. It took 500 s for the carmine stain to move from point B to the mudline, and the length of the seepage path was 3.21 cm. Tests revealed that the seepage paths of soil with different *S* in suction caissons had the same motion path. The results indicated that the seepage path of soil is unrelated to the *S* applied in the caisson and is affected by the penetration depth during the installation of the foundation. The seepage paths of the carmine stain in sand are plotted with a plane coordinate system, shown in Figure 4. It can be seen that there is an obvious trend: the streamline dyed by carmine stain moves toward the wall of the caisson model. Additionally, the larger the penetration depth, the more obvious was the streamline–adherent trend. It was observed that the seepage path lengths were 1.95 and 1.34 *h* when the penetration depths were equal to 5 and 15 cm, respectively. The distance between the carmine stain in the mudline and the inner wall of the caisson model was 0.375 times the diameter of the suction caisson, for *h* = 0.42, 0.83, and 1.25 *L*, respectively.

**Figure 3.** The visualization of seepage flow at point B with *h* = 15 cm and *S* = 0.5 kPa.

**Figure 4.** The seepage path of the carmine stain in sand.

#### *3.2. Hydraulic Gradient Analysis*

In this paper, the test results prove that Darcy's law is not applicable, and this phenomenon was also observed from the large test pressure difference found in the literature [22]. For the seepage velocity *v*, it was assumed that the streamlines around the caissons were stable and not affected by each other. The path lengths of the carmine stain were measured from test images. All results are presented in terms of dimensionless forms. The seepage velocity *v* of the sand was assumed to be affected by *x*/*D* and *S*/γ*'h*. The hydraulic gradient *v*/*k* of the sand in tests at the mudline was proposed by using the regression function, and can be expressed as follows:

$$\frac{v}{k} = 0.163 \left[ \frac{0.94DS}{(0.5D - x)\gamma\nu h} \right]^{0.76} \tag{1}$$

where *x* is the abscissa value in Figure 1 and γ*'* is the effective unit weight of soil.

The comparison of actual values and calculated values is shown in Figure 5. It can be seen that the fitted values agree well with the actual values. Compared with *v*/*k* obtained from tests, the fitted values have a residual sum of squares of about 1.467.

**Figure 5.** The comparison of actual values and fitted values for *v*/*k.*

Figure 6 shows the hydraulic gradient of sand around a suction caisson with different *x*/*D* when the suction in the caisson is equal to 1.5 kPa. It can be seen that the hydraulic gradient *v*/*k* increases with the increase of *x*/*D*. The results indicated that the seepage velocity is larger as the streamline moves toward the wall of the suction caissons. The hydraulic gradient *v*/*k*, with *x*/*D* = 0.2675 and *h*/*D* = 1.25, is shown in Figure 7. It can be observed that the *v*/*k* increases with the increase of *S*/γ*'h* and the fitted values of the *v*/*k* agree well with the test results. The *v*/*k* when *S*/γ*'h* = 2.451 is 5.85 times greater than when *S*/γ*'h* = 0.351 in tests. The minimum relative error is 5.05% for Equation (1) when *S*/γ*'h* = 5.478. When *h*/*D* = 0.42, the streamline around the suction becomes disordered as a result of the larger applied suction (*S* = 1.5 kPa). Due to the assumption that the streamlines are stable and not affected by each other, the seepage velocity obtained is smaller than the actual situation, leading to large deviations as compared with Equation (1).

**Figure 6.** The hydraulic gradient with *S* = 1.5 kPa.

**Figure 7.** The hydraulic gradient with *x*/*D* = 0.2675 and *h*/*D* = 1.25.

#### *3.3. Soil Plug and Settlement Formation*

During suction penetration, a part of the soil in the caissons gets into the open-ended hollow caisson cavity, forming a soil plug [23]. The final soil plugs in the caisson at the buried depth of 5 cm (caused by the caisson) are shown in Figure 8. The test results show that there is no soil plug in the caissons with small suction applied during suction penetration. The maximum heights of soil plugs in the suction caisson models with buried depths of 5, 10, and 15 cm are 0.169, 0.085, and 0.087 times the buried depths, respectively. It was observed that the suction caused settlement of the surrounding soils outside the caisson. The maximum soil settlement appeared a certain distance from the outer wall of a suction caisson due to caisson–soil friction.

**Figure 8.** The soil plug in the caisson.

Figure 9 shows the curves of the heights of soil plug *h*sp and of soil settlement *h*settlement outside the caisson. Other test conditions not drawn in Figure 9 did not have the obvious soil plug and soil settlement around the suction caisson model. There is no obvious soil settlement outside the suction caisson with a buried depth of 15 cm. The height of soil plug *h*sp increases with the increase of *x* as a result of the distribution of seepage velocity, which is greater as it moves toward the wall of suction caissons.

**Figure 9.** The height of soil plug and soil settlement outside the caisson.

The volume of soil plug *V*sp and that of soil settlement *V*settlement were measured by using the volume formula after obtaining the size of the soil plug and settlement from the images. The comparison between the volume of soil plug *V*sp and that of soil settlement *V*settlement in tests is shown in Table 3. The soil plug consists of the soil heaved by seepage force into the caisson, and the soil from outside into the caisson, leading to soil settlement around the foundation. It can be seen that the influence of settlement outside the caisson on the soil plug decreases as the buried depth rises. The soil plug is mainly influenced by the soil from outside into the caisson, and *V*settlement/*V*sp = 1.581 due to the decrease in the void ratio of soil around the caisson caused by the seepage force during suction penetration when *h* = 5 cm and *S* = 1.0 kPa. The influence of the soil heaved by the soil plug increases with the increase of *h*. The volume of soil heaved begins to rise as the velocity of seepage reaches a certain value.


**Table 3.** The volume of soil plug and soil settlement in tests.

#### *3.4. Prediction of Soil Plug Height*

The relationship between the height of the soil plug *h*sp and the hydraulic gradient *v*/*k* was proposed by using quadratic regression, and can be expressed as follows:

$$\frac{h\_{sp}}{h} = A + B\frac{S}{\gamma\nu h} + C\left(\frac{S}{\gamma\nu h}\right)^2 + D\frac{\upsilon}{k} + E\left(\frac{\upsilon}{k}\right)^2 + F\frac{S\upsilon}{\gamma\nu hk'}\tag{2}$$

where *A*–*F* are constant coefficients (given in Table 4). Figure 10 shows the comparison of actual values and fitted values for the dimensionless soil plug *h*sp/*h*. Fitted values are evenly distributed around the actual values. Compared with the actual values, the fitted values have an average relative error of about 28.74%.

**Table 4.** Optimal value of constant coefficients for *hsp*/*h.*


**Figure 10.** The comparison of actual values and fitted values for the dimensionless soil plug *h*sp/*h.*

Figure 11 shows the relationship between *h*sp/*h* and *v*/*k* when *h* = 15 cm. It is shown that the dimensionless soil plug *h*sp/*h* first increases and then decreases with the increase of *v*/*k*. The reason behind the decreased trend is that the soil plug is subjected to downward friction applied by the inner wall of the suction caisson. The combination of Equations (1) and (2) is used to predict roughly the height of soil plugs in suction caissons in sand during suction penetration, guiding project designs, and construction.

**Figure 11.** The relationship between *h*sp/*h* and *v*/*k* when *h* = 15 cm.

#### **4. Conclusions**

A series of model tests were conducted in this study to investigate the visualization of suction caisson penetration in sand. The following conclusions can be drawn:

(1) The seepage field can be visualized while the carmine stain flows along a streamline in the soil around the suction caisson. The seepage velocity is larger as the streamline moves toward the wall of the suction caissons. The pre-test results show that the seepage failure took place in the sand when the suction in the caisson with buried depths of 5, 10, and 15 cm was greater than 2.0, 3.5, and 4.0 kPa, respectively. The results indicated that the seepage path of the soil is unrelated

to *S* applied in the caisson and is affected by the penetration depth during the installation of the foundation.


**Author Contributions:** Conceptualization, S.M. and L.X.; methodology, L.X. and S.M.; validation, T.L.; formal analysis, T.L. and S.M.; investigation, T.L. and S.M.; resources, data curation, T.L. and S.M.; writing original draft preparation, T.L. and S.M.; visualization, T.L. and S.M; supervision L.X.; project administration, L.X.; funding acquisition, L.X. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was financially supported by the Chinese National Natural Science Foundation (No. 51479137).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Notation**


#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

*Article*

## **Experimental and Numerical Investigation of Wind Characteristics over Mountainous Valley Bridge Site Considering Improved Boundary Transition Sections**

### **Xiangyan Chen 1, Zhiwen Liu 1,2,\*, Xinguo Wang 3, Zhengqing Chen 1,2, Han Xiao <sup>1</sup> and Ji Zhou <sup>3</sup>**


Received: 17 December 2019; Accepted: 16 January 2020; Published: 21 January 2020

**Featured Application: The results of research in this paper can provide a certain design basis for the correction of an inlet boundary for both wind tunnel test and CFD simulation over complex terrain. By adopting the improved transition sections recommended in this work, engineers will have access to more precise assessment of wind characteristics over mountainous terrain, which contributes to wind-related issues such as wind actions of long-span bridges, transmission lines, wind farm site selection, and the prediction of pollutant dispersion.**

**Abstract:** To study wind characteristics over mountainous terrain, the Xiangjiang Bridge site was employed in this paper. The improved boundary transition sections (BTS) were adopted to reduce the influence of "artificial cliffs" of the terrain model on the wind characteristics at the bridge site over the mountainous terrain. Numerical simulation and experimental investigations on wind characteristics over mountainous terrain with/without BTS were conducted for different cases, respectively. The research results show that the cross-bridge wind speed ratios and wind attack angles at the main deck level vary greatly along the bridge axis, which can be roughly divided into three parts, namely the mountain (I, III) and central canyon areas (II). The cross-bridge wind speed ratios at the main deck level with BTS is generally larger than that without BTS in the central canyon area (II) for most cases, while the opposite trend can be found in wind attack angles. The longitudinal wind speed ratios of the terrain model with BTS at L/4, L/2, and 3L/4 of the bridge length are larger than that of the terrain model without BTS for most cases. In general, the maximum relative error between numerical results and experimental results is about 30% for most cases.

**Keywords:** mountainous valley; bridge site; boundary transition section (BTS); wind characteristics; numerical simulation; wind tunnel test

#### **1. Introduction**

Wind characteristics over mountainous valleys are critical to many wind-related issues, such as wind actions of long-span bridges and transmission lines, wind farm site selection, prediction of pollutant dispersion, and so on. In particular, wind characteristics at mountainous valley bridge sites such as design wind speed, wind yaw angles, wind attack angles, and turbulence spectra play a critical role in wind-resistant design of long-span bridges. The wind characteristics over complex mountainous terrain are significantly different from those of open areas. Therefore, reasonable determination of wind parameters at the bridge sites over mountainous terrain is important to the balance between wind-resistance safety and the economy of long-span bridges. However, the current wind-resistant design codes for bridges generally make simple corrections for mountain wind fields based on wind characteristics of flat terrain (Chock et al., [1]). Therefore, it is especially necessary to investigate wind characteristics over complex mountainous terrain precisely.

The main methods for studying the wind characteristics over complex mountainous terrain bridge sites include theoretical study, field measurement, experimental study, and numerical study. Theoretical studies were used to predict the wind speed-up effects over simple topographic features with slopes low enough to avoid flow separation (Jackson and Hunt [2]; Hunt et al. [3]), which cannot be adopted to study the wind characteristics over complex mountainous terrains. Field measurement is regarded as the most reliable approach to investigate the wind characteristics in the ABL (Li, et al. [4]), which can accurately measure the wind speed and direction data at different measurement points. During the past two decades, numerous field measurements of wind characteristics over mountainous valley were conducted (Harstveit [5]; Hannesen et al. [6]; Lubitz et al. [7]; Sharples et al. [8]; Abiven et al. [9]; Risan et al. [10]; Lystad et al. [11]; Peng et al. [12]; Zhang et al. [13]; Yu et al. [14]; Jing et al. [15]). However, field measurements of wind characteristics over mountainous bridge site are usually expensive, time-consuming, and can only provide wind data at a limited number of measurement points, which are easily affected by the terrain.

Experimental studies of wind characteristics over complex mountainous terrain were conducted by many researchers over the last three decades (Cermak [16], Chock et al. [1], and Xu et al. [17]; Kozmar et al. [18]; Yan et al. [19]; Mattuella et al. [20]; Muhammad et al. [21]; Kozmar et al. [22]; Chen et al. [23]; Flay et al. [24]). Bowen [25] pointed out that the geometric scales of complex terrain model should not be less than 1:2500~5000, and the Reynolds number should be larger than Reh = 105 (h is the model hill height) for wind tunnel tests. However, for engineering structures located at complex mountainous terrain, such as long-span bridges, power transmission towers, and wind turbines, it is especially important, and an urgent matter, to carry out numerical simulation of wind characteristics since complex terrain makes it difficult for scholars to propose a common wind characteristic model. When the wind tunnel test model or numerical simulation model are adopted to study wind characteristics of mountainous terrain, it is inevitable to take a certain range of areas from mountainous terrain, causing "artificial cliffs" (shown in Figure 1) on the edge of terrain models. These artificial cliffs can deviate practical wind characteristics at the inlet boundary from these of theoretical ones, such as wind speed, wind attack angles, and turbulence characteristics. To cope with the negative influence of an artificial cliff on wind tunnel tests, some scholars have put forward with different solutions. Maurizi et al. [26] and Pang et al. [27] used an inclined boundary transition section (BTS) to connect the ground and the top of the terrain model. Hu et al. [28] proposed a curved transition section for complex terrain and concluded that this kind of transition section has a better flow transition performance compared with the traditional inclined one. Li et al. [29] combined a terrain model with three-dimensional transition sections in the wind tunnel. Huang et al. [30] evaluated speed-up ratios, mean vertical attack angles, and mean exceeding turbulence intensities of different transition curves to determine the optimal transition section for terrain models. Hu et al. [31] established two different BTS in the computational domain for comparison purposes and the results show that the updated curved BTS has a better flow transition efficiency than those reported previously. Liu et al. [32] proposed an improved BTS to modify the inlet boundary by combining the Witozinsky curve and straight line.

With the development of computational fluid dynamics (CFD), many scholars (Uchida and Ohya [33]; Tong et al. [34]; Deleon et al. [35]; Risan et al. [10]; Tamura et al. [36] and Cassiani et al. [37]) investigated wind characteristics over complex terrain with CFD method. Maurizi et al. [26], Kim et al. [38] and Castellani et al. [39] confirmed the availability of the *k* − ε model and RNG *k* − ε model. Although there is some research focusing on wind characteristics over complex terrain by CFD simulation, few studies on transition sections on the edge of terrain models have been found recently. Due to diversity and complexity of complex terrain, it is necessary to conduct further research on the BTS of mountainous terrain models.

**Figure 1.** Schematic diagram of Artificial Cliffs.

The Xiangjiang Bridge in Guizhou, China, was employed as an engineering example to study the wind characteristics at bridge site over complex terrain and effects of the improved BTS. The mid-span of the cable-stayed bridge was regarded as the origin. The terrain model with a geometric scale of 1:1500 (as suggested by Bowen [25]) was fabricated to simulate a region with a diameter of 8 km. Wind tunnel tests were conducted with and without the proposed BTS in order to investigate the BTS effects on wind characteristics (Liu et al. [32]). To verify the feasibility of the improved BTS numerically, wind characteristics over the scaled mountainous terrain model with and without the proposed BTS were predicted by CFD simulation. The wind characteristics, such as mean cross-bridge wind speed and wind attack angles at the main deck level, longitudinal wind speed profiles at L/4, L/2 and 3L/4 of the bridge length were studied in detail to verify the accuracy of CFD simulation and effects of improved BTS.

This paper is organized as follows: Section 2 introduces the numerical method, including governing equations of fluids, turbulence model, optimization of terrain model with BTS, the and numerical model. Section 3 describes the details of wind tunnel test setup. Section 4 investigates the cross-bridge wind speed ratios and wind attack angles at main deck level, longitudinal wind speed ratios profiles at L/4, L/2 and 3L/4 of bridge length of the terrain model with/without BTS under different cases. Section 5 presents conclusions of the study.

#### **2. Numerical Method**

#### *2.1. Governing Equations of Fluids*

Air flow passing through mountainous terrain can be approximately considered to be incompressible viscous fluid, which can be expressed based on the Reynolds-averaged Navier-Stokes (RANS) equations as follows:

$$\frac{\partial u\_i}{\partial \mathbf{x}\_i} = 0 \tag{1}$$

$$\frac{\partial u\_i}{\partial t} + u\_j \frac{\partial u\_i}{\partial \mathbf{x}\_j} = -\frac{1}{\rho} \frac{\partial p}{\partial \mathbf{x}\_i} + \frac{\partial}{\partial \mathbf{x}\_j} \left[ \nu \left( \frac{\partial u\_i}{\partial \mathbf{x}\_j} + \frac{\partial u\_j}{\partial \mathbf{x}\_i} \right) - \overline{u\_i' u\_j'} \right] \tag{2}$$

where *t* and *xi* are time and Cartesian coordinates, respectively; *ui* and *ui* are the time-averaged and fluctuating flow velocity components, respectively; *p* is the time-averaged pressure; ρ and *v* are the fluid density and kinematic viscosity; *ui uj* is the Reynolds stress tensor.

#### *2.2. Turbulence Model*

The Large eddy simulation (LES) has inherent advantages over the steady-state and unsteady RANS turbulence models in physical simulation, and is well suited for the simulation of turbulence and nonlinear features of complex terrain wind. The use of LES for complex mountainous terrain requires large computing resources. However, for wind field modeling over complex terrain, RANS remains the main turbulence model adopted in CFD simulation up to present, where it is often being applied with a satisfactory degree of success. Many researchers (Bitsuamlak et al. [40]; Kim et al. [38]; Mohamed et al. [21]; Yassin, et al. [41] and Yan et al. [42]) found that results obtained by RNG *k* − ε agree well with these of experiments. Therefore, the RNG *k* − ε model is adopted to numerical simulation in this paper. The transport equations for RNG *k* − ε model are given as follows,

$$\frac{\partial}{\partial t}(\rho k) + \frac{\partial}{\partial \mathbf{x}\_i}(\rho k u\_i) = \frac{\partial}{\partial \mathbf{x}\_j}(a\_k \mu\_{eff} \frac{\partial \mathbf{k}}{\partial \mathbf{x}\_j}) + \mathbf{G}\_k - \rho \varepsilon + \mathbf{S}\_k \tag{3}$$

$$\frac{\partial}{\partial t}(\rho \boldsymbol{\varepsilon}) + \frac{\partial}{\partial \mathbf{x}\_i}(\rho \boldsymbol{\varepsilon} \boldsymbol{u}\_i) = \frac{\partial}{\partial \mathbf{x}\_j}(\alpha\_\varepsilon \mu\_{eff} \frac{\partial \boldsymbol{\varepsilon}}{\partial \mathbf{x}\_j}) + \mathbf{C}\_{1\varepsilon} \frac{\boldsymbol{\varepsilon}}{k} \mathbf{G}\_k - \mathbf{C}\_{2\varepsilon} \rho \frac{\boldsymbol{\varepsilon}^2}{k} - \mathbf{R}\_\varepsilon + \mathbf{S}\_\varepsilon \tag{4}$$

where *Gk* is the turbulent kinetic energy caused by mean speed gradient whose definition is *Gk* = −ρ*u i u j* ∂*uj*/∂*xi* and the equation is *Gk* = μ*iS*<sup>2</sup> using Boussinesq hypothesis where *S* is the strain rate tensor of mean speed, i.e., *S* = , 2*SijSij*. α*<sup>k</sup>* and αε are reciprocals of turbulent kinetic energy, *k* and dissipation rate, ε respectively. *Sk* and *S*<sup>ε</sup> are source terms defined by users. *R*<sup>ε</sup> is the addition item of RNG *k* − ε model compared with *k* − ε model and its expression is

$$R\_{\varepsilon} = \frac{\mathbb{C}\_{\mu} \rho \eta^{3} (1 - \eta / \eta\_{0})}{1 + \beta \eta^{3}} \frac{\varepsilon^{2}}{k} \tag{5}$$

where η = *Sk*/ε is the ratio of turbulence and mean flow time scale. η<sup>0</sup> is the typical value of η in uniform shear flow. *C*<sup>μ</sup> and β are constants of the turbulence model. ρ is the air density.

#### *2.3. Optimization of Terrain Model BTS*

#### 2.3.1. Form of Boundary Transition Section Curve

The Witozinsky curve was obtained in the case of an ideal axis of uncompressed axisymmetric flow. Wind tunnel operation results show that this kind of contraction curve can achieve good flow quality in test sections. Many low-speed wind tunnels constructed in 1960s by Chinese scholars were designed based on the Witozinsky formula (Wang et al. [43]). Evaluating the results of different contraction curves in terms of average wind speed reduction factors, vertical wind attack angles, and turbulence intensity, Huang et al. [44] drew a conclusion that the performance of the Witozinsky curve is the best and proposed an optimal curve expression of the Witozinsky curve at the same time. In his research, the terrain model boundary is directly connected to the BTS as shown in Figure 2a, which may lead to insufficient development of the inflow. In order to make the inflow develop fully, BTS in the present paper is improved by combining the Witozinsky curve with a horizontal line as shown in Figure 2b. The formula of the Witozinsky curve is shown as follows:

$$y = H \left\{ 1 - \frac{\left[1 - \left(\mathbf{x} / L\right)^2\right]^2}{\left[1 + A\left(\mathbf{x} / L\right)^2\right]^3} \right\} \tag{6}$$

where *A* = 1/3, *H* is the vertical distance from the highest point of the curve to the ground, *L* is the horizontal projection length of the whole curve and the corresponding coordinates of any point on the curve is (*x*, *y*).

**Figure 2.** Terrain model BTS with/without horizontal straight line (Liu, et al., 2019). (**a**) BTS without horizontal straight line; (**b**) BTS with horizontal straight line.

#### 2.3.2. Weight Allocation of Evaluation Indexes

Because the evaluation of BTS at the entrance of the terrain model mainly focuses on speed-up effect, the equivalent wind attack angle and the increase ratio of turbulence intensity, the specific evaluation indexes of effects of the terrain model BTS are defined as follows:

$$V\_{\chi} = \sum\_{k=1}^{N} \frac{v\_k - v\_{0k}}{v\_{0k}} \cdot \frac{\Delta h\_k}{H\_w} \tag{7}$$

$$A\_x = \sum\_{k=1}^{N} \frac{a\_k \cdot \Delta h\_k}{H\_w} \tag{8}$$

$$I\_x = \sum\_{k=1}^{N} \frac{I\_k - I\_{\text{Ox}}}{I\_{\text{Ox}}} \cdot \frac{\Delta h\_k}{H\_{\text{w}}} \tag{9}$$

where *N* is the total amount of monitoring points. *vk* is the downward wind speed of the *kth* monitoring point from the ground and *v*0*<sup>k</sup>* is the wind speed at the *kth* monitoring point of the wind speed entrance. Δ*hk* is the vertical distance between the *kth* monitoring point and the (*k* − 1) *th* monitoring point. *Hw* is the total height of the monitoring points. *ak* is the wind attack angle of the *kth* monitoring point from the ground. *Ik* is the turbulence intensity at the *kth* monitoring point and *I*0*<sup>k</sup>* is the turbulent intensity at the *kth* monitoring point from the ground at the wind speed entrance.

The two-dimensional CFD simulation was used to optimize the parameters of the Witozinsky curve and horizontal line. For the indexes such as the speed-up ratio of wind speed, equivalent wind attack angle, and turbulence intensity increase ratio at the boundary of the terrain model, the weight allocation based on the correlation degree proposed by Tang (Tang et al. [45]) was adopted to determine the BTS curve parameters for a given total length of the transition section (Liu, et al. [32]).

There are *m* = 13 schemes and *n* = 4 indexes that have an influence on the comprehensive evaluation of each scheme. They can compose the matrix *X* = (*xij*) *<sup>m</sup>*×*n*. In this paper, the evaluation indexes are all cost indexes, which means that the smaller the index is, the better the result is. Standardize values in each column of *X* = (*xij*) *<sup>m</sup>*×*<sup>n</sup>* using min-max standardization (shown in formula (10) to map each index of the matrix into [0,1]. After conversion, the dimensionless matrix is *X* = (*xij*) *<sup>m</sup>*×*n*. Then, select the maximum in each column of *X* = (*xij*) <sup>8</sup>×<sup>6</sup> to obtain the optimal objects *<sup>S</sup>*<sup>+</sup> <sup>0</sup> and select the minimum in each column to get the worst object *S*− <sup>0</sup> . The correlation coefficient of each scheme with the optimal object and the worst object in each index can be calculated by Equations (11) and (12),

$$\alpha' = \frac{\text{x} - \text{min}}{\text{max} - \text{min}} \tag{10}$$

$$
\zeta\_{0^+i}(j) = \frac{1}{2\Delta\_{0^+i\bar{j}} + 1} \tag{11}
$$

$$
\zeta\_{0^{-j}}(j) = \frac{1}{2\Delta\_{0^{-j}j} + 1} \tag{12}
$$

where <sup>Δ</sup>0+*ij* <sup>=</sup> - - *x*0<sup>+</sup> *<sup>j</sup>* − *xij* - - and <sup>Δ</sup>0−*ij* <sup>=</sup> - - *x*0<sup>−</sup> *<sup>j</sup>* − *xij* - - -. The larger ζ0+*i*(*j*) is, the larger the correlation coefficient between scheme to be evaluated and the optimal scheme is in terms of index *j*. On the other hand, the performance of ζ0−*i*(*j*) is absolutely opposite.

Assume the weight vectors of *n* indexes are ω = (ω1, ω2, ...... , ω*n*−1, ω*n*) respectively and *<sup>f</sup>*(ω) = *<sup>n</sup> j*=1 ω2 *j* [<sup>1</sup> <sup>−</sup> <sup>ζ</sup>0+*i*(*j*)]<sup>2</sup> <sup>+</sup> *<sup>n</sup> j*=1 ω2 *j* ζ0−*i*(*j*) <sup>2</sup> represents the square sum of the weighted distance between the optimal solution and the worst one. Therefore, the smaller *f*(ω) is, the better scheme *i* is. Establish the multi-objective programming model as follows:

$$\min f(\boldsymbol{\omega}) = \left( f\_1(\boldsymbol{\omega}), f\_2(\boldsymbol{\omega}), \dots, f\_{m-1}(\boldsymbol{\omega}), f\_m(\boldsymbol{\omega}) \right)^T \\ \text{s.t.} \begin{cases} \min & \min\_{\boldsymbol{\beta}} \\ & \text{s.t.} \begin{cases} \min \sum\_{j=1}^m f\_j(\boldsymbol{\omega}) \\ \boldsymbol{\omega} = 1 \\ \boldsymbol{\omega} = 1, 2, \dots, n-1, n \\ \end{cases} & \text{s.t.} \begin{cases} \min \sum\_{j=1}^m f\_j(\boldsymbol{\omega}) \\ \sum\_{j=1}^n \boldsymbol{\omega}\_j = 1 \\ \boldsymbol{\omega}\_j = 1 \\ \boldsymbol{\omega}\_j \ge 0, j = 1, 2, \dots, n-1, n \\ \end{cases} \end{cases} \tag{13}$$

Since *fi*(ω) ≥ 0(*i* = 1, 2, ... , *m* − 1, *m*), the multi-objective programming model is transformed into single-objective programming model. Establish Lagrange Function,

$$F(\omega, \lambda) = \sum\_{i=1}^{m} \sum\_{j=1}^{n} \omega\_j^2 \left\{ \left[ 1 - \zeta\_{0^+i}(j) \right]^2 + \zeta\_{0^-i}(j)^2 \right\} - \lambda \left( \sum\_{j=1}^{n} \omega\_j - 1 \right) \tag{14}$$

$$\begin{cases} \frac{\partial F}{\partial \omega\_j} = 2\omega\_j \sum\_{i=1}^m \left\{ [1 - \zeta\_{0^+i}(j)]^2 + \zeta\_{0^-i}(j)^2 \right\} - \lambda = 0\\ \frac{\partial F}{\partial \lambda} = \sum\_{j=1}^n \omega\_j - 1 = 0 \end{cases} \tag{15}$$

Solve Equation (15), the weight of *n* indexes can be calculated by

$$w\_{\bar{j}} = \frac{1}{u\_{\bar{j}} \sum\_{i=1}^{n} \frac{1}{u\_{\bar{j}}}} \tag{16}$$

where *uj* is an intermediate variable and can be calculated by the following equation,

$$\mu\_j = \sum\_{i=1}^{m} \left\{ \left[ 1 - \zeta\_{0^+i}(j) \right]^2 + \zeta\_{0^-i}^2(j) \right\} \tag{17}$$

The final calculation results of all the wind profile positions are shown in Table 1. The virtual angle is defined as the angle between the horizontal line and the line connecting the ground and the end of the Witozinsky curve. The horizontal length refers to the horizontal straight line length after the Witozinsky curve. Since the improved BTS proposed in this paper aims to provide an effective corrections for terrain models of wind tunnel test and CFD simulation, the total length of the improved BTS need to be sufficiently short to reduce blockage ratio while ensuring the accuracy of the wind characteristics over complex terrain; therefore, the total length of the improved BTS is regarded as the major consideration. If the total length of the horizontal projection is smaller than 1.0 m, the minimal final result is obtained at the combination of a virtual angle of 57◦ and a horizontal length of 0.8 m. Different optimal solutions can be obtained by changing the total horizontal projection length.

**Table 1.** Final results concerning weights at different wind profiles.


#### *2.4. Numerical Terrain Model*

#### 2.4.1. Survey of the Xiangjiang Bridge

The Xiangjiang Bridge is a four-span three-pylon cable-stayed bridge with span of 120 m + 235 m + 235 m + 120 m = 710 m. The main deck is 252.2 m above the bottom of the valley, as shown in Figure 3a. The balanced cantilever construction method are adopted to build the bridge, and the wind loads of the bridge with maximum double cantilever is the focus of wind-resistant design. Note that the bridge site is located in the mountainous terrain, as shown in Figure 3b. It can be seen from Figure 3b that the river approximately runs southwest to northeast, and the bridge is located in the center of the terrain. In order to determine the wind characteristics parameters for wind-resistant design of the bridge, numerical simulations and wind tunnel tests on wind characteristics over the mountainous terrain with/without BTS were carried out, respectively.

**Figure 3.** Elevation of the Xiangjiang Bridge and the mountainous terrain (Unit: cm). (**a**) Elevation of the Xiangjiang Bridge; (**b**) Complex terrain at the bridge site.

#### 2.4.2. Terrain Model Construction

With reference to the existing literatures on mountainous terrain wind characteristics, as shown in Table 2, the bridge site is taken as the center, and an area with a diameter of 8.0 km was selected to establish the mountainous terrain model. The computational domain with 32.34 m long, 8.0 m wide, 2.0 m high, and the geometry scale of 1:1500 were adopted according to the value of blockage ratio of the terrain model. The lowest elevation of the terrain model was taken as the bottom of the computational domain. The left side of the computational domain is defined as velocity-inlet boundary condition. The right side of the computational domain is defined as pressure-outlet boundary condition. The front, back, and top sides of the computational domain are defined as symmetry boundary conditions. The bottom of the computational domain is defined as non-slip wall boundary condition. The boundary conditions and computational domain of the terrain model without BTS are given in Figure 4. As can be seen in Figure 4, the distance between the center of the terrain model and the front and back walls of the computational domain is 4.0 m, respectively. The center of the terrain model is 9.67 m and 22.67 m away from the left and right sides of the computational domain.


**Table 2.** Summary of main parameters of mountainous terrain models.

**Figure 4.** Computational domain, boundary conditions, and grid of terrain model without BTS.

The Power Law Profile Model is currently the only model of boundary layer wind profile [46]. The wind speed at the velocity inlet boundary after scaling of the computational domain are defined as follows:

$$\begin{array}{l} V(z) = V\_{0.007} \Big( \frac{z}{0.007} \Big)^{a}, 0 \le z \le 0.233 \text{m} \\ V(z) = V\_{ref}, z > 0.233 \text{m} \end{array} \tag{18}$$

where *z* is the distance between the monitoring point and the ground; *a* is the wind profile index and *a* = 0.16 when the ground surface belongs to class-B wind field; *V*(*z*) is the wind velocity of the monitoring point which is *z* away from the ground; *V*0.007 is the basic wind speed at the level of *z*<sup>0</sup> = 0.007m in the power law profile. *Vre f* = *V*0.233 is the wind velocity when *z* = 0.233m.

In order to investigate the effects of the improved BTS on wind characteristics over mountainous terrain model, numerical simulation of the terrain model with/without BTS were conducted, respectively. With reference to the results of two-dimensional numerical simulation of the terrain model BTS, it is determined that the virtual inclination of the transition section is θ = 57◦ and the length of the horizontal line is *L* = 0.80m (Liu et al. [32]).

#### 2.4.3. Grids of the Terrain Model

To ensure the numerical simulation accuracy of the terrain model grid, the grid independence checking was performed firstly. The mesh is divided into different parts. Unstructured prism was used to find grids near the ground, and the tetrahedral grid was adopted to reflect the undulating changes of the terrain. The near-wall grids of the terrain model are given in Figure 4. The detailed parameters of the three grids for independence checking are shown in Table 3. The computational domain and grid arrangement of the terrain model with BTS are shown in Figure 5.


**Table 3.** Detailed setups of three grids of terrain model without BTS.

**Figure 5.** Computational domain, boundary conditions and grid of terrain model with BTS.

The flow solver used in this paper to simulate the wind characteristics over complex terrain is based on the finite-volume discretization in the space and the SIMPLEC (Semi implicit method for pressure linked equation consistent) pressure-correlation algorithm on a non-staggered grid arrangement (Kim, et al. [38]). The second-order cell-centered is adopted to deal with spatial discrete. The second-order upwind is used for time discrete. The time step is 0.015 s and the total amount of time steps is 10,000. According to the main parameters of wind-resistant design of the bridge, the mean cross-bridge wind speed and wind attack angles at the main deck level, longitudinal wind speed profiles at L/4, L/2 and 3L/4 of the bridge length were monitored to evaluate the calculation accuracy of different grids. The wind speed monitoring points are given in Figure 6.

**Figure 6.** Schematic diagram of monitoring points. (**a**) Monitoring points at the main deck level; (**b**) Monitoring points profiles at L/4, L/2 and 3L/4 of the bridge length.

The grid sensibility was conducted by taking case 1 without BTS as an example. Numerical results of mean cross-bridge wind speed and wind attack angles of grid G1, G2, and G3 at the main deck level over the terrain model without BTS are given in Figure 7a,b, respectively. The longitudinal wind speed profiles at L/4, L/2 and 3L/4 of the bridge length of grid G1, G2, and G3 over the terrain model without BTS are shown in Figure 7c. It can be found in Figure 7a that the mean cross-bridge wind speed of G2 agrees well with that of G3 with a relative error of 23% while the result of G1 departs from the results of the other two grid generations to a great extent with relative error 116%. Note that the same tendency can be found in Figure 7b (relative error 41% for terrain model with BTS while 160% without BTS) and (c), which means that relative sparse grid generation in the terrain model and computational domain will cause significant discrepancies in terms of average wind characteristics at bridge site. Therefore, the grid generation, G2, is considered to be the optimal grid since its amount of grids is less compared to G3 and the calculation accuracy can be guaranteed.

**Figure 7.** Comparisons for average wind characteristics with different grid generations. (**a**) Mean cross-bridge wind speed at main deck level; (**b**) Wind attack angle at main deck level; (**c**) Longitudinal wind speed profiles at L/4, L/2 and 3L/4 of the bridge length.

#### **3. Wind Tunnel Test Set-Up**

The wind tunnel tests were conducted in the third test section of HD-2 wind tunnel of Hunan University, Changsha, Hunan, China. HD-2 wind tunnel of Hunan University is 53 m long and 18 m wide, including three test sections. The first test section is 17 m long, 3 m wide, and 2.5 m high, and the wind speed of the test section was 0~58 m/s, which can be continuously adjusted. The second test section is 15 m long, 5.5 m wide and 4.4 m high, and the maximum wind speed was 18 m/s. The third test section was 15 m long, 8.5 m wide and 2 m high, and the maximum wind speed was 15 m/s. Considering the size of the third section of HD-2 wind tunnel, the terrain model centered on the mid-span of the bridge with a geometric scale of 1:1500 was designed and fabricated to simulate a region with a diameter of 8 km. The average height of the terrain model was roughly 0.2 m. The blockage ratio is defined as the ratio of cross-sectional area of the terrain model and the wind tunnel test section. If the blockage ratio of the terrain model is too large, the inflow cannot develop fully, leading to inaccurate prediction of the wind field over complex terrain. The blockage ratio of the terrain model is about 6.27%, which is within the range of 5.0% and 10% required in Design Rules for Aerodynamic Effects on Bridges published by British.

To investigate the BTS effects on wind characteristics over mountainous terrain model, the wind tunnel tests of terrain model with/without BTS were conducted, respectively. The geometric parameters of the BTS are consistent with these of the CFD simulation model. Figure 8 shows the terrain model with/without BTS.

**Figure 8.** Mountainous terrain model with/without BTS in wind tunnel. (**a**) Terrain model without BTS; (**b**) Terrain model with BTS.

Before the wind tunnel test of terrain model was carried out, the inflow wind speed and turbulence intensity profiles in the empty wind tunnel section was conducted first. The total height of the monitoring points was 0.6 m, and the interval between two adjacent monitoring points was 0.03 m. The first monitoring point was 0.03 m away from the bottom of the wind tunnel section. The wind speed was measured by Cobra probes from TFI (Turbulent Flow Instrumentation Pty Ltd., Victoria, Australia). Figure 9 shows the comparison of the wind profile of the incoming wind speed in the empty wind tunnel section and the wind profile of terrain category B suggested in wind-resistant design specification for highway bridges (JTG/T 3360-01-2018). As shown in Figure 9a, the measured inflow wind speed profile in the empty wind tunnel section was closed to the corresponding wind speed profile of category B. The main reason is that the geometric scale of the terrain model is 1:1500, the height of the boundary layer of the terrain category B was 0.233 m, which was approximately close to the boundary layer height in the wind tunnel test section under a uniform inflow condition. It can be seen from Figure 9b that the longitudinal turbulence intensity was about 3.0% when the height of the monitoring points is larger than 0.2 m.

**Figure 9.** Wind speed and turbulence intensity profiles at different height. (**a**) Wind speed profile; (**b**) Longitudinal turbulence intensity profile.

To find out the influence of wind yaw angles on wind characteristics over mountainous terrain, four cases of wind from the west, south, east, and north were investigated by numerical simulations and wind tunnel tests, respectively. The wind yaw angle was 0◦ when the inflow enters the terrain model along the west. Counterclockwise rotation was positive for the wind yaw angle and the detailed test cases are shown in Figure 3b.

#### **4. Verification of Improved BTS**

Due to space limitations, only wind speed, turbulence kinetic energy, and wind attack angle of case 1 as well as 2, without and with improved BTS are compared to verify its effect.

It can be seen from Figure 10 that when there is improved BTS, wind speed distribution is more uniform and closer to that of the inflow wind speed profile. Besides, there is no flow separation near the boundary of the terrain model.

**Figure 10.** Wind speed distribution along the inflow direction near the inlet of the terrain model without or with improved BTS from CFD simulation (unit: m/s). (**a**) Case 1 without BTS; (**b**) Case 2 without BTS; (**c**) Case 1 with BTS; (**d**) Case 2 with BTS.

Turbulence kinetic energy (TKE) is a measure of the intensity of turbulence and determines the ability of the flow to maintain turbulence or become turbulence, thus indicating flow stability. From Figure 11, it can be concluded that at the inlet of the terrain model without improved BTS, the TKE with BTS is smaller than that of the terrain model without BTS, which indicates that the improved BTS is able to reduce turbulence intensity induced by artificial cliffs and uplift the inflow gradually.

**Figure 11.** Turbulence kinetic energy distribution along the inflow direction near the inlet of the terrain model without or with improved BTS from CFD simulation (unit: m2s−2). (**a**) Case 1 without BTS; (**b**) Case 2 without BTS; (**c**) Case 1 with BTS; (**d**) Case 2 with BTS.

Wind attack angles induced by artificial cliffs will have a significant influence on the wind attack angle at the bridge site, thus reducing wind attack angles at the inlet boundary hold a decisive place in wind characteristic prediction over complex terrain. As shown in Figure 12, the wind attack angle at the inlet boundary with the improved BTS is much smaller than that without BTS and is closer to zero, indicating that the improved BTS can reduce the wind attack angle caused by artificial cliffs to a great extent.

**Figure 12.** Wind attack angle distribution along the inflow direction near the inlet of the terrain model without or with improved BTS from CFD simulation (unit: ◦). (**a**) Case 1 without BTS; (**b**) Case 2 without BTS; (**c**) Case 1 with BTS; (**d**) Case 2 with BTS.

In conclusion, by comparing wind speed, turbulence kinetic energy, and wind attack angle near the inlet of the terrain model without and with improved BTS, the BTS can effectively reduce the impact of artificial cliffs on wind characteristics after the inlet boundary.

#### **5. Numerical and Experimental Results**

According to the wind-resistance design requirements of the bridge, the cross-bridge wind speed, wind attack angles at main deck level, and longitudinal wind speed profiles at L/4, L/2, and 3L/4 of the bridge length were analyzed for numerical and experimental cases. In this paper, the positive direction of cross-bridge wind is from southwest to northeast, and the positive of attack angle is from underside to upside of the main deck. Here, cross-bridge wind speeds are all normalized by the horizontal component of the gradient wind speed of inflow wind, namely

$$\overline{V}\_{H} = \frac{V\_{H}}{V\_{ref} \cdot \cos \beta} \tag{19}$$

where *VH* is the non-dimensional cross-bridge wind speed, *VH* is the cross-bridge wind speed, *Vre f* is the gradient wind speed of inflow above which the wind speed remains unchanged with the increase of altitude, *Vre f* = 3.70m/s, and β is the angle between inflow wind direction and perpendicular of the bridge axis.

Wind attack angle at the main deck level is defined as follows:

$$\alpha = \arctan\left(\frac{V\_V}{V\_H}\right) \tag{20}$$

where α is the wind attack angle at main deck level, *VV* is the vertical wind speed at main deck level, and *VH* is the cross-bridge wind speed at main deck level.

#### *5.1. Cross-Bridge Wind Speed at Main Deck Level*

The numerical and experimental results of the non-dimensional cross-bridge wind speed at main deck level under different cases are given in Figure 13. Considering the span layout characteristics large-span bridges, it can be roughly divided into three parts along bridge axis; namely, mountain areas (I, III) and the central canyon area (II).

**Figure 13.** Non-dimensional cross-bridge wind speed at main deck level under different cases. (**a**) Case 1; (**b**) Case 2; (**c**) Case 3; (**d**) Case 4.

From Figure 13a, it can be seen that for case 1 (wind blows from the west), the non-dimensional cross-bridge wind speed at the main deck level varies greatly along the bridge axis. The non-dimensional cross-bridge wind speed in part III was relatively large, while that in part I was relative small. For experimental results, the cross-bridge wind speed of the terrain model with BTS was generally larger than that of the terrain model without BTS. For case 1, due to the topographic effects, there was a certain deviation between the numerical simulation results of the cross-bridge wind speed at the main deck level and the experimental results.

From Figure 13b, it can be seen that for case 2 (wind blows from the south), the non-dimensional cross-bridge wind speed at the main deck level varies along the bridge axis. The non-dimensional cross-bridge wind speed in part III is relatively large, while the cross-bridge wind speed in part I was relative small. For experimental results, the cross-bridge wind speed of the terrain model with BTS was generally larger than that of the terrain model without BTS. The numerical simulation results also show the same trend.

From Figure 13c, it can be seen that for case 3 (wind blows from the east), the non-dimensional cross-bridge wind speed at the main deck level varies little along the bridge axis. The non-dimensional cross-bridge wind speed in part III was relatively large, while the non-dimensional cross-bridge wind speed in part I is relatively small. For both the experimental results and numerical results, the cross-bridge wind speed of the terrain model with BTS was generally larger than that of the terrain model without BTS. Besides, it can be seen from Figure 13c that little difference of the wind characteristics with and without improved BTS was found, which indicated that the effect of improved BTS depends on the inflow direction. The altitude in the east over the complex terrain was high. When

the height difference between the inlet boundary and the bridge site is large, the flow separation is severe and the effect of the improved BTS on flow separation is limited. Thus, the relation between the height difference and the improved BTS can be evaluated in future research.

From Figure 13d, it can be seen that for case 4 (wind blows from the north), the non-dimensional cross-bridge wind speed at the main deck level varies little along the bridge axis. For experimental results, the cross-bridge wind speed of the terrain model with BTS is generally smaller than that of the terrain model without BTS in part II. For numerical simulation results, the cross-bridge wind speed of the terrain model with BTS was generally larger than that of the terrain model without BTS in part II.

In conclusion, in cases 1 to 3, the wind tunnel test results of the terrain model show that the cross-bridge wind speed ratio at the main deck level of the terrain model with the improved BTS was relatively larger than that of the terrain model without BTS. A similar tendency can be found from numerical results of the terrain model with/without BTS. Thus, if the there is no improved BTS in front of the terrain model, the cross-bridge design wind speed of the bridge deck may be underestimated. It is recommended to use the improved BTS for terrain model boundary correction.

The statistical values of numerical results and experimental results of non-dimensional cross-bridge wind speed at main deck level under different cases are shown in Figure 14. It can be seen from Figure 14 that though there are some discrepancies between the wind tunnel tests results and numerical simulation results, the majority of the data is located within relative errors of 30%. This kind of error is indeed unavoidable because the scale of terrain model was small which made it a challenge to place the model in the exact center of the test section and find the accurate monitoring point positions as CFD simulation did. Thus, the wind tunnel test results can be regarded as accurate ones, and the results from CFD simulations can be used for qualitative analysis.

**Figure 14.** Numerical results vs. experimental results of non-dimensional cross-bridge wind speed at main deck level under different cases. (**a**) Case 1 and 2; (**b**) Case 3 and 4.

#### *5.2. Wind Attack Angles at Main Deck Level*

The numerical and experimental results of the wind attack angle at main deck level of the bridge for different cases are given in Figure 15. From Figure 15a, it can be seen that for case 1 (wind blows from the west), the wind attack angle at the main deck level of the bridge varied greatly along the bridge axis, which can be roughly divided into three parts; namely, mountain areas (I, III) and the central canyon area (II). For experimental results, the wind attack angles at the main deck level of the terrain model with BTS is about −15◦~−10◦ in part II, while the wind attack angles at the main deck level of the terrain model without BTS is −20◦~−5◦ in part II. For numerical results, the wind attack angles at the main deck level of the terrain model with BTS was about −10◦~0◦, while the wind attack angles at the main deck level of the terrain model without BTS was −7.5◦~5◦.

**Figure 15.** Wind attack angle at main deck level of wind tunnel tests and CFD simulation. (**a**) Case 1; (**b**) Case 2; (**c**) Case 3; (**d**) Case 4.

From Figure 15b, it can be seen that for case 2 (wind blows from the south), the wind attack angle at the main deck level of the bridge changes along the bridge axis. For experimental results, the wind attack angles at the main deck level of the terrain model with BTS was about −5◦~0◦ in part II, while the wind attack angles at the main deck level of the terrain model without BTS was −5◦~10◦ in part II. For numerical results, the wind attack angles at the main deck level of the terrain model with BTS was about −12◦~2◦, while the wind attack angles at the main deck level of the terrain model without BTS was about 0◦~10◦.

From Figure 15c, it can be seen that for case 3 (wind blows from the east), for experimental results, the wind attack angles at the main deck level of the terrain model with BTS were about 5◦~12◦in part II, while the wind attack angles at the main deck level of the terrain model without BTS were 1◦~12◦ in part II. For numerical results, the wind attack angles at the main deck level of the terrain model with BTS are about 0◦~3◦, while the wind attack angles at the main deck level of the terrain model without BTS were about −7◦~−2◦.

From Figure 15d, it can be seen that for case 4 (wind blows from the north), the experimental results of the wind attack angles at the main deck level of the terrain model with BTS were about −10◦~7◦, while the wind attack angles at the main deck level of the terrain model without BTS were about 3◦~ 5◦. For numerical simulation results, the wind attack angles at the main deck level of the terrain model with BTS were about −0.5◦~7.5◦, while the wind attack angles at the main deck level of the terrain model without BTS were about −0.5◦~10◦.

In general, both numerical simulation and experimental results under different cases show that the range of the wind attack angles at the main deck level in part II with BTS is smaller than that of the terrain model without BTS. The reason is that the existence of the BTS effectively reduces the influence of artificial cliffs. Thus, the turbulence development of wind field with BTS is more sufficient than that without BTS; the wind attack angles with BTS are smaller than that without BTS.

The statistical values of numerical and experimental results of the wind attack angles at main deck level under different cases are shown in Figure 16. It can be seen from the Figure 16 that though there are some discrepancies between the wind tunnel tests and numerical simulation, the majority of the deviation is located within relative errors of 30%.

**Figure 16.** Numerical results vs. experimental results of wind attack angles at main deck level under different cases. (**a**) Case 1 and 2; (**b**) Case 3 and 4.

#### *5.3. Wind Profiles at L*/*4, L*/*2 and 3L*/*4 of Bridge Length*

The longitudinal wind speed ratio is defined as the measured longitudinal wind speed above the terrain model divided by longitudinal wind speed of the corresponding inflow boundary layer. The numerical and experimental results of longitudinal wind speed ratio profiles at L/4, L/2 and 3L/4 of the bridge length are given in Figures 17–19. As shown in Figures 17–19, the longitudinal wind speed ratio at different positions, namely L/4, L/2 and 3L/4 of the bridge length, generally increase with height.

As shown in Figure 17, the experimental results of longitudinal wind speed ratio were larger than the numerical results at L/4 of bridge length for case 1 (wind blows from the west) and case 4 (wind blows from the north), respectively. The experimental results of longitudinal wind speed ratio were slightly larger than the numerical results at L/4 of bridge length for case 2 (wind blows from the south) and case 3 (wind blows from the east), respectively. The experimental results of the longitudinal wind speed ratio at L/4 of the bridge length shows that the longitudinal wind speed ratio of terrain model with BTS were larger than that of the terrain model without BTS for different cases. The numerical results at L/4 of the bridge length shows that the longitudinal wind speed ratio of terrain model with BTS were slightly larger than that of terrain model without BTS under cases 2 to 4, except for case 1.

As shown in Figure 15, the experimental results of longitudinal wind speed ratio at L/2 of bridge length were larger than the numerical results for case 1 (wind blows from the west) and case 2 (wind blows from the north), respectively. The experimental results of longitudinal wind speed ratio were close to the numerical results for case 3 (wind blows from the east) and case 4 (wind blows from the north), respectively. The experimental results of the longitudinal wind speed ratio at L/2 of the bridge length shows that the longitudinal wind speed ratio of terrain model with BTS were larger than that of the terrain model without BTS for cases 1 to 3, except for case 4. The numerical results of the longitudinal wind speed ratio at L/2 of the bridge length shows that the longitudinal wind speed ratio of terrain model with BTS were slightly larger than that of terrain model without BTS under cases 1, 2, and 4, except for case 3.

**Figure 17.** Numerical and experimental results of longitudinal wind speed ratio profiles at L/4.

**Figure 18.** Numerical and experimental results of longitudinal wind speed ratio profiles at L/2.

**Figure 19.** Numerical and experimental results of longitudinal wind speed ratio profiles at 3L/4.

As shown in Figure 16, the experimental results of longitudinal wind speed ratio at 3L/4 of bridge length are close to the numerical results for cases 2 to 4. The experimental results of longitudinal wind speed ratio at 3L/4 of bridge length were smaller than the numerical results for case 1. The experimental results shows that the longitudinal wind speed ratio of terrain model with BTS were slightly smaller than that of the terrain model without BTS for case 1 while the longitudinal wind speed ratios of the terrain model with BTS were slightly larger than that of the terrain model without BTS for cases 2 to 4.

The statistical values of the numerical and experimental results of longitudinal wind speed ratio at L/4, L/2 and 3L/4 under different cases are given in Figures 20–22. From Figure 20a, it can be seen that the relative error between numerical and experimental results of longitudinal wind speed ratios at L/4 of the bridge length under case 1 with/without BTS were about 50%, while the relative error between numerical and experimental results of longitudinal wind speed ratios at L/4 of the bridge length under case 2 with/without BTS were about 30%. From Figure 20b, it can be seen that the relative error between numerical and experimental results of longitudinal wind speed ratio at L/4 of the bridge length under case 3 and 4 with/without BTS were about 30%. As shown in Figure 18, the relative error between numerical and experimental results of longitudinal wind speed ratios at L/2 of the bridge length under different cases with/without BTS were less about 30% except for case 3 with BTS. From Figure 22, the relative error between numerical and experimental results of longitudinal wind speed ratios at 3L/4 of the bridge length under different cases with/without BTS were less than 30%. In general, it can be seen from the Figures 20–22 that though there were some discrepancies between the wind tunnel tests and numerical simulation, the majority of the data are located within relative errors of 30%.

**Figure 20.** Numerical results vs. experimental results of longitudinal wind speed ratio at L/4 under different cases. (**a**) Case 1 and 2; (**b**) Case 3 and 4.

**Figure 21.** Numerical results vs. experimental results of longitudinal wind speed ratio at L/2 under different cases. (**a**) Case 1 and 2; (**b**) Case 3 and 4.

**Figure 22.** Numerical results vs. experimental results of longitudinal wind speed ratio at 3L/4 under different cases. (**a**) Case 1 and 2; (**b**) Case 3 and 4.

#### **6. Conclusions**

To study the wind characteristics over mountainous terrain, the Xiangjiang Bridge site was employed in this paper. The improved boundary transition sections were adopted to reduce the influence of the artificial cliffs on the edge of the terrain model on the wind characteristics at the bridge site over the mountainous terrain. The mean cross-bridge wind speed and wind attack angles at the main deck level, longitudinal wind speed profiles at L/4, L/2, and 3L/4 of the bridge length were investigated in detail with the assistance of numerical simulations and wind tunnel tests. The main conclusions are summarized as follows:


**Author Contributions:** Conceptualization, Z.C., Z.L. and X.C.; methodology, Z.L. and X.C.; software, X.C.; validation, Z.L., X.W. and J.Z.; formal analysis, H.X.; investigation, X.C.; resources, Z.L., X.W. and J.Z.; data curation, H.X.; writing—original draft preparation, X.C.; writing—review and editing, Z.L.; visualization, H.X.; supervision, Z.C.; project administration, X.W. and J.Z.; funding acquisition, Z.L. All authors have read and agree to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China, grant number 51478180 and 51778225 and China Railway Siyuan Survey and Design Group Co., Ltd.

**Acknowledgments:** We deeply appreciate the assistance provided by Shuqiong Li, Ruilin Zhang, Yuefei Chen, Beisong Sun, Yafeng Li, Jianzhong Wang and Jingsi Shen in the process of wind tunnel test and the administrative and technical support by the Hunan Provincial Key Lab of Wind Engineering and Bridge Engineering.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders of China Railway Siyuan Survey and Design Group Co., Ltd. provided authors with the access to the engineering example, Xiangjiang Bridge site and collection, analyses or interpretation of data.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Study on the Horizontal Axis Deviation of a Small Radius TBM Tunnel Based on Winkler Foundation Model**

#### **Shifan Qiao 1, Ping Xu 1, Ritong Liu <sup>2</sup> and Gang Wang 3,\***


Received: 9 December 2019; Accepted: 19 January 2020; Published: 22 January 2020

**Abstract:** During the construction stage of the small radius TBM (tunnel boring machine) interval, the improper control of the boring parameters and the boring posture can cause the horizontal axis deviation of the shield tunnel. In order to address this issue, the TBM segments lining structure of the small radius interval is simplified as the continuous circular curved beam based on the longitudinal equivalent continuous model and Winkler elastic foundation beam theory. The theoretical model is solved through the transfer matrix method, and its applicability is verified by comparing it with the field monitoring data. It is found that the horizontal axis deviation of the completed tunnel increases with the total jack thrust, and the lateral displacement tends to be stable when the distance between the ring and the tail is far. The horizontal axis deviation has a negative relationship with the thrust difference or path difference when the jack thrust in the outside of the shield curve is larger than that of inside the shield curve. The horizontal axis deviation has a positive relationship with the thrust difference or path difference when the jack thrust in the outside of the shield curve is smaller than that of inside the shield curve.

**Keywords:** small radius TBM interval; equivalent continuous model; Winkler elastic foundation beam theory; transfer matrix method; horizontal axis deviation

#### **1. Introduction**

Tunnel boring machine (TBM) technology has been widely used for the construction of the urban subway tunnel. The longitudinal performance research of the TBM tunnel based on the equivalent continuous model and elastic foundation theory is a hotspot. The longitudinal affects the safe operation of the subway, which has also been widely studied through field monitoring, numerical analysis and analytical solution [1–5]. The research on the tunnel longitudinal performance focuses on the longitudinal settlement and horizontal axis deviation, which are caused by the coupling effects of multi-load [6–9].

Koizumi et al. [10] and Shiba et al. [11] were the pioneers developing the analytical solution for calculating the longitudinal settlement based on longitudinal bending stiffness. Elastic foundation models such as the Winkler model [12], the Pasternak model [13], and the Kerr model [14] are the most widely used theoretical calculation models in the longitudinal deformation research of tunnels. After that, the generalized longitudinal equivalent continuous model was proposed by Zhang et al. [15]. Li et al. [16] proposed the model considering the mechanical behavior of blots. Shiba et al. [17] and Talmon and Bezuijen [18] used the continuous elastic beam to simulate the longitudinal structure model

of the shield tunnel, which can analyze the longitudinal property of the shield tunnel. Yu et al. [19] derived an analytical solution for the longitudinal bending stiffness of a segmental liner, which was mainly utilized in the shield tunnels. This method is verified by the simulation results from the finite element program.

The longitudinal settlement mechanism of the tunnel can also be analyzed by the two-dimensional and three-dimensional numerical simulation methods [20–22]. Huang et al. [23] developed a finite element method for analyzing the longitudinal performance of shield tunnels considering the longitudinal variation of geotechnical parameters. Yuan et al. [9] analyzed the deformation of the shield tunnel and the surrounding soil by building a three-dimensional model of shield tunnel. A numerical model was established to simulate the earth pressure balance (EPB) in excavation processes—the simulation results were compared with those obtained by the field measurement [24,25].

Also, field monitoring is an important method to analyze the longitudinal settlement of the shield tunnel. Ocak [26] studied the relationship between shield parameters and the shield surface settlement through field surface settlement measurements. Fargnoli et al. [27] collected the measurements of tunneling-induced settlements during the construction of the new Milan underground line 5, which were back analyzed with the method of the classical Gaussian empirical expression. Then, the detailed description of the EPB tunneling performance in the transverse and longitudinal directions could be achieved.

The research on the longitudinal performance of shield tunnels is mainly focused on the longitudinal settlement of tunnels. Limited research has been conducted on the problem of horizontal axis deviation of the small radius TBM interval construction.

In the design of urban subway lines, the sharp curve of the shield tunnel with a small curvature radius is usually adopted to avoid the adverse geological conditions and surrounding existing buildings [28,29]. When the curvature radius is less than 40 times of the TBM's diameter, the construction difficulty and quality risk will be increased significantly. The damage caused by the horizontal axis deviation of the constructed tunnel is particularly prominent, which can easily affect the assembly quality of the segments. The horizontal axis deviation can result in tunnel mud leakage and local structural damage, which pose a severe threat to the safety of metro operation. Therefore, it is of great significance to study the effect of the construction load on the horizontal axis deviation of a small radius TBM interval tunnel.

The objective of this paper is to propose a solution of the tunnel horizontal axis deviation of small radius TBM interval with the transfer-matrix method, which is based on generalized longitudinal equivalent continuum model and Winkler elastic foundation beam theory. The applicability of the calculation model is verified through the monitored data in the field. The factors such as the jack thrust and shield posture affecting the tunnel horizontal axis deviation are analyzed.

#### **2. Generalized Longitudinal Equivalent Continuous Model**

#### *2.1. Model Description*

The prefabricated segment lining structure used in the TBM tunnel has a large number of longitudinal joints, which makes the deformation mechanism of the lining structure complicated. Various theoretical models for longitudinal deformation calculation have simplified the joint structures. Among these models, the longitudinal equivalent continuous model proposed by Shiba et al. [17] has the most extensive application. The tunnel cross-section is assumed as a homogeneous ring, and the tunnel longitudinal integral rigidity is also reduced when considering the joints. The tunnel is simplified as a uniform continuous beam with the equivalent stiffness on the elastic foundation.

Traditional models ignore the impacts of deformation of the tunnel cross-section and the limitation of the influential region of the longitudinal joints, which leads to the lower longitudinal effective rigidity ratio compared with the measured value. The generalized longitudinal equivalent continuous model [15] is adopted to calculate the longitudinal equivalent bending rigidity of the TBM tunnel segment lining structure in this paper. This model takes account of the influential region of the joints and lateral stiffness.

#### *2.2. Basic Assumptions*

Some assumptions are proposed in the generalized longitudinal equivalent continuous model:


$$k\_{\dot{j}} = \frac{nk\_{\dot{j}\dot{i}}}{2\pi Rt} = \frac{nE\_{\dot{j}}A\_{\dot{j}}}{2\pi Rtl} \,\tag{1}$$

where *kji* is the elastic stiffness coefficient of a single bolt, *n* is the number of bolts inside the ring, *R* is the average of the internal and external diameter of the segment, *t* is the thickness of the segment, *Ej* is the elastic modulus of the bolt, *Aj* is the cross-section area of the bolt, and *l* is the interaction length of the bolt.


#### *2.3. Longitudinal Equivalent Bending Sti*ff*ness*

Under the longitudinal bending, when the tensions of the bolts are all smaller than the elastic limit *Py*, the whole segment is in the elastic state. Figures 1 and 2 show the longitudinal deformation and stress distribution of the segment inside and outside the influence scope of annular gaps. *a* and *b* are the long half axis and short half axis of the ellipse respectively, the ellipse is the cross-section of the segment under surrounding rock pressure, φ shows the position of neutral axis, *c* is the distance between neutral axis and y-axis, *ds* is the length of the micro-segmentation, *d*β is the corresponding angle and *x* is the distance between neutral axis and the micro-segmentation, ε*<sup>t</sup>* is the tension strain of the segment and ε*<sup>c</sup>* is the compressive strain of the segment, *M* is the bending moment, *Ec* is the elastic modulus of tunnel reinforced concrete segment and δ is the displacement; *ls* is the segment length between the centerline of the two-segment rings and θ is the rotation angle.

**Figure 1.** Diagram of stress and deformation of the lining within the influence range of annular gaps [16].

**Figure 2.** Diagram of stress and deformation of the lining outside the influence range of annular gaps [16].

Based on the basic assumptions, the equilibrium equation, the deformation compatibility equation, and the tunnel equivalent bending stiffness (*EI*)*eq* can be obtained [31].

$$(EI)\_{eq} = \frac{Ml\_s}{\theta} = \frac{Ml\_s}{\frac{M}{\frac{E\_4 Z\_1}{\frac{E\_4 Z\_1}{\lambda I} + \frac{E\_4 k\_f Z\_2}{\lambda E\_e + \lambda k\_f^3}}} + \frac{Ml\_s - Ml}{\frac{E\_4 I\_s}{\frac{E\_4 Z\_1}{\lambda I} + \frac{E\_4 k\_f Z\_2}{\lambda I}}} + (l\_s - \lambda I) \tag{2}$$

The longitudinal stiffness effective rate η is [32],

$$
\eta = \frac{(EI)\_{cq}}{E\_c I\_c} \tag{3}
$$

where [16],

$$\begin{array}{l} Z\_{1} = \frac{(a+b)b^{2}t\left(\frac{\pi}{2}-\wp\right)}{2} + \frac{(a+b)^{3}t\sin^{2}\wp\left(\frac{\pi}{2}-\wp\right)}{4} + \frac{(a+b)b^{2}t\sin2\wp}{4} - \frac{(a+b)^{2}bt\sin2\wp}{2} \\ Z\_{2} = \frac{(a+b)b^{2}t\left(\frac{\pi}{2}+\wp\right)}{2} + \frac{(a+b)^{3}t\sin^{2}\wp\left(\frac{\pi}{2}+\wp\right)}{4} - \frac{(a+b)b^{2}t\sin2\wp}{4} + \frac{(a+b)^{2}bt\sin2\wp}{2} \end{array}$$

*Ic* is the inertia moment of tunnel segment cross-section; *t* is the thickness of lining ring.

#### **3. Elastic Foundation Beam Model Used in the Small Radius TBM Interval**

#### *3.1. Model Description*

The elastic foundation model is the most widely used theoretical model in the study of tunnel longitudinal deformation, mainly it has three categories: (a) Winkler foundation model [12], the foundation is regarded as a series of independent springs. The soil properties are manifested by the stiffness of the spring. (b) Pasternak model [13], an incompressible shear layer, only having horizontal shear deformation, is added in the Winkler foundation model. The shearing between springs is considered in the Pasternak model. (c) Kerr model [14], the spring layer, is added based on the Pasternak model. In contrast, the parameters in the Winkler foundation model are straightforward. When the compressible strata are thin and a hard layer exists, the calculation results based on the Winkler foundation model agree well with the real present situation. Therefore, the Winkler foundation model is selected to calculate the horizontal axis deviation of a small radius TBM tunnel subjected to construction loading. The lining structure of TBM segments in the small radius interval is simplified as the continuous circular curved beam on the Winkler foundation based on the equivalent elastic foundation beam theory (Figure 3). *qr* is the radial linear load on the curved beam; *qt* is the tangential linear load on the curved beam. Φ is the arbitrary angle on the circular curved beam; *p* is the concentrated force.

**Figure 3.** Schematic diagram of the circular curved beam on the Winkler elastic foundation [33].

#### *3.2. Basic Assumptions*

Some basic assumptions are:


#### *3.3. Model Solution*

The uniform cross-section circular curved beam on the Winkler elastic foundation is depicted in Figure 4. The microelement body of the circular curved beam on the Winkler elastic foundation is shown in Figure 5. The equivalent equation of the microelement body, ignoring the high order, is shown below [32].

$$\frac{dQ}{dx} = p - \frac{N}{R} - q\_{r\prime} \tag{4}$$

$$\frac{dN}{d\mathbf{x}} = \frac{\mathbf{Q}}{R} - q\_{t\nu} \tag{5}$$

$$\frac{dM}{dx} = \mathbb{Q}\_{\prime} \tag{6}$$

where *Q* is the shear force; *N* is the axial force.

For simplification, the effect of axial force on the deformation is neglected. Then, the deflection differential equation of the circular curved beam is obtained [34],

$$
\left(\frac{d^2y}{dx^2} + \frac{y}{R^2}\right) + \frac{M}{EI} = 0,\tag{7}
$$

Combining Equations from (4) to (7), we can get deflection differential equation of the circular curved beam, which ignores the tangential deformation [34],

$$2\frac{d^5y}{d\Phi^5} + 2\frac{d^3y}{d\Phi^3} + \mu^2 \frac{dy}{d\Phi} = \frac{R^4}{EI} \left(\frac{dq\_r}{d\Phi} + q\_t\right) \tag{8}$$

where *EI* is the bending stiffness of the tunnel, μ<sup>2</sup> is the coefficient, μ<sup>2</sup> = *1* + *R*4*DK*/*EI*, *K* is the coefficient of foundation bedding, *D* is the width of the curved beam cross-section, *R* is the curvature radius of the curved beam.

When there is no load on the beam span, the Equation (8) can be changed into,

$$\frac{d^5y}{d\Phi^5} + 2\frac{d^3y}{d\Phi^3} + \mu^2 \frac{dy}{d\Phi} = 0.\tag{9}$$

This is the homogenous fifth-order differential equation with constant coefficients, whose general solution is as below [34],

$$y(\Phi) = \mathbb{C}\_1 + (\mathbb{C}\_2 \text{char}\Phi + \mathbb{C}\_3 \text{char}\Phi) \cos \beta \Phi + (\mathbb{C}\_4 \text{char}\Phi + \mathbb{C}\_5 \text{char}\Phi) \sin \beta \Phi,\tag{10}$$

where,

 $a = \sqrt{\frac{(\mu - 1)}{2}}$ , $\beta = \sqrt{\frac{(\mu + 1)}{2}}$ 

*C*1–*C*<sup>5</sup> are undetermined coefficients, obtained by loading and boundary conditions.

With the successive derivation of Equation (10), we can obtain analytical expressions of the rotation angle θ, the bending moment *M*, the shear force *Q*, the axial force *N*. With the initial condition of the beam, we can get the solutions of undetermined coefficients *C*1–*C*5, the general solution of the defection differential equation can be converted into the below equation [34],

$$y(\Phi) = y\_0 F\_{y1}(\Phi) + \theta\_0 F\_{y2}(\Phi) + M\_0 F\_{y3}(\Phi) + Q\_0 F\_{y4}(\Phi) + N\_0 F\_{y5}(\Phi),\tag{11}$$

where

$$\begin{cases} \begin{aligned} &F\_{y1}(\Phi) = \frac{2a\beta}{1+2a\beta} + \frac{1}{1+2a\beta}W\_{1}(\Phi) + \frac{R^{4}kD}{EI(1+2a\beta)}W\_{2}(\Phi) \\ &F\_{y2}(\Phi) = \frac{a}{\alpha^{2}+\beta^{2}}W\_{2}(\Phi) + \frac{\beta}{\alpha^{2}+\beta^{2}}W\_{3}(\Phi) \\ &F\_{y3}(\Phi) = \frac{R^{2}}{(1+2a\beta)EI}W\_{1}(\Phi) - \frac{R^{6}kD}{2a\beta(1+2a\beta)(EI)^{2}}W\_{4}(\Phi) \\ &F\_{y4}(\Phi) = \frac{R^{3}}{2EIa(\alpha^{2}+\beta^{2})}W\_{2}(\Phi) - \frac{R^{3}}{2EI\beta(\alpha^{2}+\beta^{2})}W\_{3}(\Phi) \\ &F\_{y5}(\Phi) = -\frac{R^{3}}{2a\beta EI}W\_{4}(\Phi) \end{aligned} \\ &\left\{ \begin{aligned} &W\_{1}(\Phi) = ch(\alpha\Phi)\cos(\beta\Phi) \\ &W\_{2}(\Phi) = sh(\alpha\Phi)\sin(\beta\Phi) \\ &W\_{3}(\Phi) = ch(\alpha\Phi)\sin(\beta\Phi) \\ &W\_{4}(\Phi) = sh(\alpha\Phi)\sin(\beta\Phi) \end{aligned} \right. \end{cases}$$

When the arbitrary point on the curved span has the concentrated moment *Mi*, the concentrated force *Pi*, the radial distribution load *qri*, the tangential distribution load *qti*, the external load on the beam span can be regarded as the partial initial parameters. Therefore, additional deflection caused by the concentrated moment *Mi* is as below [34],

$$(y(\Phi))\_{M\_i} = M\_i F\_{y\mathcal{G}}(\Phi - \Phi\_{M\_i})(\Phi \ge \Phi\_{M\_i})\_{\prime} \tag{12}$$

The additional defection caused by the concentrated force *Pi* when Φ ≥ Φ*Pi* is shown below [34],

$$g(\Phi)\_{P\_i} = -P\_i F\_{y4} (\Phi - \Phi\_{P\_i}) (\Phi \ge \Phi\_{P\_i})\_\prime \tag{13}$$

The additional defection caused by the radial distribution load *qri* when Φ ≥ Φ*ai* can be obtained [34],

$$\log(\Phi)\_{q\_{\rm ii}} = -D \int\_{\Phi\_{\rm ci}}^{\Phi} q\_t(\delta) F\_{y\mathbb{S}}(\Phi - \delta) d\delta(\Phi \ge \Phi\_{\rm ci}),\tag{14}$$

When Φ ≥ Φ*bi*, the integral upper limit Φ = Φ*bi* is selected.

The additional defection caused by the tangential distribution load *qti* when Φ ≥ Φ*ci* can also be obtained [34],

$$\left(y(\Phi)\right)\_{\mathfrak{q}\_{\ell}} = -D \int\_{\Phi\_{\mathfrak{c}}}^{\Phi} q\_{\ell}(\delta) F\_{\mathfrak{Y}}(\Phi - \delta) d\delta(\Phi \ge \Phi\_{\mathfrak{c}})\_{\mathfrak{r}} \tag{15}$$

When Φ > Φ*di*, the integral upper limit Φ = Φ*di* is selected.

With the initial conditions while Φ = 0 and the above external load, the deflection of the curved beam at any points can be achieved [34],

$$\begin{split} y(\Phi) &= \quad y\_0 F\_{y1}(\Phi) + \partial\_0 F\_{y2}(\Phi) + M\_0 F\_{y3}(\Phi) + Q\_0 F\_{y4}(\Phi) + N\_0 F\_{y5}(\Phi) + \sum\_{i=1}^{n\_{\text{M}}} M\_i F\_{y5}(\Phi - \Phi\_{\text{M}\_i}) \\ &- \sum\_{i=1}^{n\_{\text{P}}} P\_i F\_{y4}(\Phi - \Phi\_{P\_i}) - \sum\_{i=1}^{n\_{\text{S}}} D \int\_{\Phi\_{\text{si}}}^{\Phi} q\_{\text{ri}}(\delta) F\_{y4}(\Phi - \delta) d\delta - \sum\_{i=1}^{n\_{\text{G}}} D \int\_{\Phi\_{\text{si}}}^{\Phi} q\_{\text{ti}}(\delta) F\_{y5}(\Phi - \delta) d\delta \end{split} \tag{16}$$

With the same procedure, we can obtain the analytical expressions of rotation angle θ, bending moment *M*, shear force *Q*, and axial force *N* when some loads are acting on the beam span.

The TBM tunnel segment experiences leaving the shield tail, pea gravel backfilling, and grouting in turn. There are differences in the stiffness of the medium near the segment and the deformation of segment cross-section with loading. Therefore, the bending of the limited length variable cross-section circular curved beam on the inhomogeneous Winkler foundation needs further discussion.

**Figure 4.** Uniform cross-section circular curved beam on the Winkler elastic foundation [35].

**Figure 5.** Diagram of force analysis on the microelement body [35].

Figure 6 shows the equal section circular curved beam subjected to the distributed load on the homogeneous Winkler foundation. The analytical expressions of the displacement and the internal force can be expressed in the following matrix form [34].

$$A(\Phi)\_{5\times 1} = B(\Phi)\_{5\times 5} A(0)\_{5\times 1} + C(\Phi)\_{5\times 1} + D(\Phi)\_{5\times 1} \tag{17}$$

where,

$$B(\Phi)\_{5\times5} = \begin{bmatrix} y(\Phi)\_r \theta(\Phi), M(\Phi), Q(\Phi), N(\Phi) \end{bmatrix}^T;$$

$$B(\Phi)\_{5\times5} = \begin{bmatrix} F\_{y1}(\Phi) & F\_{y2}(\Phi) & F\_{y3}(\Phi) & F\_{y4}(\Phi) & F\_{y5}(\Phi) \\ F\_{\theta1}(\Phi) & F\_{\theta2}(\Phi) & F\_{\theta3}(\Phi) & F\_{\theta4}(\Phi) & F\_{\theta5}(\Phi) \\ F\_{M1}(\Phi) & F\_{M2}(\Phi) & F\_{M3}(\Phi) & F\_{M4}(\Phi) & F\_{M5}(\Phi) \\ F\_{Q1}(\Phi) & F\_{Q2}(\Phi) & F\_{Q3}(\Phi) & F\_{Q4}(\Phi) & F\_{Q5}(\Phi) \\ F\_{N1}(\Phi) & F\_{N2}(\Phi) & F\_{N3}(\Phi) & F\_{N4}(\Phi) & F\_{N5}(\Phi) \end{bmatrix};$$

$$A(0)\_{5\times1} = \begin{bmatrix} y\_0, \theta\_0, M\_0, Q\_0, N\_0 \end{bmatrix}^T;$$

$$D(\Phi)\_{5\times1} = \begin{bmatrix} C\_{\mathcal{Y}}(\Phi), C\_{\mathcal{O}}(\Phi), C\_{\mathcal{M}}(\Phi), C\_{\mathcal{O}}(\Phi), C\_{\mathcal{N}}(\Phi) \end{bmatrix}^T;$$

$$D(\Phi)\_{5\times1} = \begin{bmatrix} D\_{\mathcal{Y}}(\Phi), D\_{\mathcal{O}}(\Phi), D\_{\mathcal{M}}(\Phi), D\_{\mathcal{O}}(\Phi), D\_{\mathcal{N}}(\Phi) \end{bmatrix}^T;$$

The elements in the matrix *C(*Φ*)*5×<sup>1</sup> are as follows [34],

$$\begin{cases} \begin{aligned} \mathcal{C}\_{\mathcal{Y}}(\Phi) &= -D \int\_{\Phi\_{\mathcal{Y}}}^{\Phi} q\_{\mathcal{r}}(\delta) F\_{y4}(\Phi-\delta) d\delta \\ \mathcal{C}\_{\mathcal{O}}(\Phi) &= -D \int\_{\Phi\_{\mathcal{Y}}}^{\Phi} q\_{\mathcal{r}}(\delta) F\_{04}(\Phi-\delta) d\delta \\ \mathcal{C}\_{M}(\Phi) &= -D \int\_{\Phi\_{\mathcal{Y}}}^{\Phi} q\_{\mathcal{r}}(\delta) F\_{M4}(\Phi-\delta) d\delta \\ \mathcal{C}\_{\mathcal{O}}(\Phi) &= -D \int\_{\Phi\_{\mathcal{Y}}}^{\Phi} q\_{\mathcal{r}}(\delta) F\_{Q4}(\Phi-\delta) d\delta \\ \mathcal{C}\_{N}(\Phi) &= -D \int\_{\Phi\_{\mathcal{Y}}}^{\Phi} q\_{\mathcal{r}}(\delta) F\_{N4}(\Phi-\delta) d\delta \end{aligned} \end{cases}$$

The elements in the matrix *D(*Φ*)*5×<sup>1</sup> are as follows [34],

$$\begin{cases} \begin{aligned} D\_{\mathcal{Y}}(\Phi) &= -D \int\_{\Phi\_{\mathcal{S}}}^{\Phi} q\_{t}(\delta) F\_{\mathfrak{Y}\mathfrak{S}}(\Phi-\delta) d\delta \\\ D\_{\mathcal{O}}(\Phi) &= -D \int\_{\Phi\_{\mathcal{S}}}^{\Phi} q\_{t}(\delta) F\_{\mathfrak{O}\mathfrak{S}}(\Phi-\delta) d\delta \\\ D\_{M}(\Phi) &= -D \int\_{\Phi\_{\mathcal{S}}}^{\Phi} q\_{t}(\delta) F\_{M\mathfrak{S}}(\Phi-\delta) d\delta \\\ D\_{\mathcal{Q}}(\Phi) &= -D \int\_{\Phi\_{\mathcal{S}}}^{\Phi} q\_{t}(\delta) F\_{\mathcal{Q}\mathfrak{A}}(\Phi-\delta) d\delta \\\ D\_{N}(\Phi) &= -D \int\_{\Phi\_{\mathcal{S}}}^{\Phi} q\_{t}(\delta) F\_{N\mathfrak{A}}(\Phi-\delta) d\delta \end{aligned} \end{cases}$$

The curved beam is divided into n sections into the abruptly changed sections of bending stiffness and foundation bedding coefficient, in the action points of the concentrated force and moment, and the starting action points of the distributed loading. There are concentrated moments *Mi* and concentrated forces *Pi* at the junctures Φ = Φ*<sup>i</sup>* (i = 0, 1, 2, ... , *n*) of the adjacent curved beams. Each curved beam section has radial distributed load *qri*(Φ) and tangential distributed load *qti*(Φ).

For curved beam *i*, in the interval Φ*i*−<sup>1</sup> <sup>+</sup> <sup>≤</sup> <sup>Φ</sup> <sup>&</sup>lt; <sup>Φ</sup>*<sup>i</sup>* −,

$$\begin{array}{rcl} A\_i(\Phi^-)\_{5\times 1} &= B\_i(\Phi - \Phi\_{i-1})\_{5\times 5} A\_i(\Phi^+\_{i-1})\_{5\times 1} + \mathbb{C}\_i(\Phi - \Phi\_{i-1})\_{5\times 1} \\ &+ D\_i(\Phi - \Phi\_{i-1})\_{5\times 1} (i = 1, 2, \dots, n-1, n) \end{array} \tag{18}$$

where the α, β, *K*, *EI* in the factor expression should be corresponding to the α*i*, β*i*, *Ki,* and (*EIeq*)*<sup>i</sup>* in section *i*.

The interface between section *i*−1 and section *i*, the cross-section Φ*i*−1, the Equation based on the deformation consistency and force equilibrium is as follows,

$$A\_i \mathbf{(}\Phi\_{i-1}^+\big|\_{5\times 1} = A\_{i-1} \big(\Phi\_{i-1}^-\big)\_{5\times 1} + E\_{i-1}(i = 2, 3, \dots, n - 1, n),\tag{19}$$

where,

$$E\_{i-1} = [0, 0, M\_{i-1}, P\_{i-1}, 0]^T (i = 2, 3, \dots, n - 1, n);$$

The calculation results at the end of curved beam *i*−1 can be regarded as the initial condition of the curved beam *i* (2 ≤ *i* ≤ *n*), then we can obtain,

$$A\_i(\Phi^-)\_{5\times 1} = \widetilde{B}(\Phi^-)\_{5\times 5} A\_1(\Phi^+\_0)\_{5\times 1} + \widetilde{C}(\Phi^-) + \widetilde{D}(\Phi^-)(i = 2, 3, \dots, n - 1, n),\tag{20}$$

where,

$$\begin{split} B(\Phi) &= B\_i(\Phi - \Phi\_{i-1})B\_{i-1}(\Phi\_{i-1} - \Phi\_{i-2})\dots B\_1(\Phi\_1 - \Phi\_0); \\ \widetilde{C}(\Phi) &= \begin{array}{c} \mathbb{C}\_i(\Phi - \Phi\_{i-1}) + B\_i(\Phi - \Phi\_{i-1})[\mathbb{C}\_{i-1}(\Phi\_{i-1} - \Phi\_{i-2}) + E\_{i-1}] \\ + B\_i(\Phi - \Phi\_{i-1})B\_{i-1}(\Phi\_{i-1} - \Phi\_{i-2})[\mathbb{C}\_{i-2}(\Phi\_{i-2} - \Phi\_{i-3}) + E\_{i-2}] \\ + \dots \\ + B\_i(\Phi - \Phi\_{i-1})B\_{i-1}(\Phi\_{i-1} - \Phi\_{i-2})\dots B\_3(\Phi\_3 - \Phi\_2) \times [\mathbb{C}\_2(\Phi\_2 - \Phi\_1) + E\_2] \\ + B\_i(\Phi - \Phi\_{i-1})B\_{i-1}(\Phi\_{i-1} - \Phi\_{i-2})\dots B\_3(\Phi\_3 - \Phi\_2)B\_2(\Phi\_2 - \Phi\_1) \end{array}$$

*Appl. Sci.* **2020**, *10*, 784

$$\begin{array}{ll} \overline{D}(\Phi) = & D\_{\bar{i}}(\Phi - \Phi\_{\bar{i}-1}) \\ & + B\_{\bar{i}}(\Phi - \Phi\_{\bar{i}-1}) [D\_{\bar{i}-1}(\Phi\_{\bar{i}-1} - \Phi\_{\bar{i}-2}) + E\_{\bar{i}-1}] \\ & + B\_{\bar{i}}(\Phi - \Phi\_{\bar{i}-1})B\_{\bar{i}-1}(\Phi\_{\bar{i}-1} - \Phi\_{\bar{i}-2}) [C\_{\bar{i}-2}(\Phi\_{\bar{i}-2} - \Phi\_{\bar{i}-3}) + E\_{\bar{i}-2}] \\ & + \dots \\ & + B\_{\bar{i}}(\Phi - \Phi\_{\bar{i}-1})B\_{\bar{i}-1}(\Phi\_{\bar{i}-1} - \Phi\_{\bar{i}-2})\dots B\_{3}(\Phi\_{3} - \Phi\_{2}) \times [D\_{2}(\Phi\_{2} - \Phi\_{1}) + E\_{2}] \\ & + B\_{\bar{i}}(\Phi - \Phi\_{\bar{i}-1})B\_{\bar{i}-1}(\Phi\_{\bar{i}-1} - \Phi\_{\bar{i}-2})\dots B\_{3}(\Phi\_{3} - \Phi\_{2})B\_{2}(\Phi\_{2} - \Phi\_{1}) \end{array}$$

When *i* = *n* and Φ<sup>0</sup> <sup>+</sup> = 0, Φ*n*<sup>−</sup> = Φ*max*, the following equation can be achieved based on Equation (20),

$$A(\Phi\_{\text{max}})\_{5\times 1} = \overline{B}(\Phi\_{\text{max}})\_{5\times 5} A(0)\_{5\times 1} + \overline{C}(\Phi\_{\text{max}}) + \overline{D}(\Phi\_{\text{max}}).\tag{21}$$

The unknown initial parameters in matrix *A*(*0*)5×<sup>1</sup> can be solved with Equation (21) based on the boundary conditions of the two ends of the curved beam. Putting all the initial parameters at the initial section into the Equation (20), we can obtain the deflection, rotation angle, moment, shear force and axial force at the arbitrary angle Φ of the circular curved beam when the foundation bedding coefficient and bending stiffness stepped change.

**Figure 6.** Sectional schematic diagram of the circular curved beam.

#### **4. Analysis of Engineering Example**

#### *4.1. Engineering Description*

The entrance/exit tunnel MRDK0+457.8~688.2 of Min-Le parking lot belongs to the Shenzhen rail transit line 6 phase II, which is a small radius TBM internal with the curvature radius R = 300 m (as illustrated in Figure 7).

**Figure 7.** TBM construction interval with the small radius tunnel.

The reinforced concrete (C50) universal wedge-shaped segments with a ring width of 1.2 m are utilized for the tunnel lining structure. The outer and inner diameters are 6200 mm and 5400 mm, respectively, and the thickness is 400 mm. The rings are connected by ten longitudinal bending bolts (M24). There are a total of 192 rings in the interval. The No. 1 and 2 segments are wrapped by the shield tail brush and sealing materials, the elastic resistance coefficient of foundation at the shield tail *k*<sup>1</sup> = 100 kPa/m; The No. 3 to 5 segments have finished the backfilling of the bottom part by pea gravel, then the elastic resistance coefficient of foundation *k*<sup>2</sup> = 292 MPa/m; The No. 6 to 10 segments have already finished the backfilling of the sidewall by pea gravel, then the elastic resistance coefficient of foundation *k*<sup>3</sup> = 1701 MPa/m; The No. 11 to 96 segments are in the slightly weathered cataclasite formation, then the elastic resistance coefficient of foundation *k*<sup>4</sup> = 1993 MPa/m; The No. 97 to 120 segments are in the slighted weathered granite formation, the elastic resistance coefficient of foundation *k*<sup>5</sup> = 4770 MPa/m; The No. 121 to 192 segments are in the section where the grout is not solidified, the elastic resistance coefficient of foundation *k6* is linearly changed, and finally reaching to 10,309 MPa/m. The tangential resistance in the model is provided by the friction between the segments and the surrounding medium. The frictional coefficient between the shield tail brush and the segment is 0.3, and the frictional coefficient between the pea gravel and the segment is 0.5. The action sphere of the tangential resistance can be solved through condition *N(*Φ*qt)* = 0.

The influence range coefficient of the annular gap λ is set as 0.2 in this paper, and the generalized longitudinal equivalent continuous model is utilized to calculate the longitudinal equivalent bending stiffness *(EI)eq* in the tunnel. The grouting pea gravels haven't formed the ring in the wall back of No. 1 to 10 segments, and the segments haven't undertaken the surrounding rock pressure, then the bending stiffness *(EI)eq*<sup>1</sup> = 162.7 GPa·m4; The vertical uniform pressure of No. 11 to 96 segments is 107.7 kPa, the base uniform reaction is 110.8 kPa, the horizontal uniform pressure is 32.3 kPa, and the bending stiffness *(EI)eq*<sup>2</sup> = 163.1 GPa·m4; The vertical uniform pressure of No. 97 to 192 segments is 54.9 kPa, the base uniform reaction is 58.0 kPa, the horizontal uniform pressure is 8.2 kPa, and the bending stiffness *(EI)eq*<sup>3</sup> <sup>=</sup> 162.8 GPa·m4.

The signs of the force and moment are stipulated for the convenience of further study. When the horizontal jack thrust on the outside of the shield curve is larger than that on the inside of the shield curve, the horizontal force couple is positive; when the path of the shield on the outside of the shield curve is larger than that on the inside of the shield curve, the jack thrust deviation angle is positive When the segment's lateral displacement points to the circle center of the circular curve, this direction is positive, and vice versa. Through the monitoring data, the average of the total auxiliary cylinder jack thrust is 5931.33 kN, the average of the jack thrust on the outside of the shield curve is larger than that on the inside of the shield curve. The average of the force couple caused by the difference of the jack thrust in the horizontal direction is 1708.20 kN·m. The average of the lateral component caused by the path difference of the auxiliary cylinder jack is 40.36 kN.

#### *4.2. The Accumulative Value of the Segments' Lateral Displacements*

The calculated lateral displacements of the ring segments and the monitored lateral displacements of the ring segments from the field are shown in Figure 8, the parameters used in this calculation are shown in Table 1. From Figure 8, we understand the following.


The first ring leaving the tail is most influenced by this effect. From the field data, we can know that the lateral displacement increases rapidly when the ring leaves the tail.


**Figure 8.** Comparison of horizontal accumulated displacements of the segments.



Finally, the simulated displacements agree well with the field data; the simulated results are more conservative. Therefore, this model is applicable in the prediction of the horizontal accumulated displacement of the shield segments in the small radius interval.

#### **5. Single-Factor Influence Analysis of the Segments' Horizontal Displacements Based on the Model**

#### *5.1. The E*ff*ect of the Jack Total Thrust*

The total jack thrust is set as N = 6000 kN, 7000 kN, 8000 kN, 9000 kN, and 10,000 kN, respectively, during the calculation in the model while other parameters are kept the same. The parameters in the theoretical model are summarized in Table 2. Then, the evolution law of the total thrust on the lateral accumulated displacements of a single ring is analyzed.


**Table 2.** The initial parameters under different total jack thrust.

Figure 9 shows that the lateral accumulated displacement increases with the increasing distance between the ring and the shield tail. With the increase of the leaving distance (completed rings), the lateral accumulated displacements tend to be stable. The segment results in the deformation along the axis under the jack thrust. When the outside of the segment contacts with the surrounding medium, the friction is produced to resist the jack thrust. When the frictional resistance of the first ring inside the shield tail cannot resist the jack thrust, the remaining thrust can be delivered to the next rings. The transverse component of the remaining thrust can cause a horizontal axis deviation. Therefore, with the increase of the total jack thrust, the remaining thrust acting on the segments increase, and the horizontal axis deviation also increases.

**Figure 9.** Lateral accumulated displacement curves of the first 20 assembled rings under different total thrusts.

Based on the stable criteria that the displacement difference of the adjacent ring is no more than 0.1 mm, the stable ring number that was leaving the tail increase from the 12th to the 17th when the total thrust increases from 6000 kN to 10,000 kN. In the stable state, the minimum lateral accumulated displacement is −15.347 mm, while the maximum lateral accumulated displacement is −28.321 mm, increasing by 84.54%.

#### *5.2. The E*ff*ect of the Jack Thrust Misalignment*

The lateral force couple *M* (2000 kN·m, 1000 kN·m, 0 kN·m, −1000 kN·m, and −2000 kN·m) caused by the thrust difference of the right and left jacks is selected as the variable parameters. The parameters for calculated are summarized in Table 3. The lateral accumulated displacement of a single ring is simulated based on the model.


**Table 3.** The initial parameters under different jack thrust misalignments.

Figure 10 illustrates that the lateral accumulated displacement increases with the distance of the shield tail, and tends to be stable when leaving the rail around 12 rings. The effect of the jack thrust on the shield ring will decay with the increasing distance, causing the decreasing lateral accumulated displacement. The direction of displacements is leaving the circular center; the displacement when the inside force > the outside force is larger than that when the outside force > the inside force. When the force couple is positive, the increasing force couple will prevent the lateral deviation of the shield efficiently.

**Figure 10.** Lateral accumulated displacement curves of the first 20 assembled rings under different thrust misalignments.

#### *5.3. The E*ff*ect of the Jack Thrust Deviation Angle*

The deviation angle α caused by the path difference of lateral jacks is set as control variables. The lateral accumulated displacements of a single ring with different deviation angles (3◦, 2◦, 1◦, 0◦, −1◦, −2◦, and −3◦) are simulated with the model, and the parameters used in the calculation are summarized in Table 4.

**Table 4.** The initial parameters under different jack thrust deviation angles.


As depicted in Figure 11, the lateral accumulated displacement increases with the distance between the ring and the tail and tends to be stable when the tail leaves the 12th ring. In the construction of the small radius TBM tunnel, the path of the external jack is always larger than that of the internal jack, which causes the deviation angle is negative. Therefore, the axial of the ring will have an external deviation unavoidably. The situation that the external path is larger than the internal path should be avoided in construction, especially in the process of a regripping cycle.

**Figure 11.** Lateral accumulated displacement curves of the first 20 assembled rings under different thrust deviation angles.

#### **6. Conclusions**


**Author Contributions:** Conceptualization, S.Q.; methodology, S.Q. and R.L.; data curation, R.L. and G.W.; formal analysis, R.L. and G.W.; validation, R.L.; investigation, P.X. and G.W.; writing—original draft preparation, G.W. and P.X.; resources, S.Q. and P.X.; writing—review and editing, G.W. and P.X.; supervision, S.Q. and G.W.; funding acquisition, S.Q. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by China Railway No.5 Engineering Croup Co. Ltd.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Review* **Seismic Design of Timber Buildings: Highlighted Challenges and Future Trends**

#### **Mislav Stepinac 1,\*, Iztok Šušteršiˇc 2, Igor Gavri´c <sup>2</sup> and Vlatka Rajˇci´c <sup>1</sup>**


Received: 15 December 2019; Accepted: 14 February 2020; Published: 19 February 2020

**Abstract:** Use of timber as a construction material has entered a period of renaissance since the development of high-performance engineered wood products, enabling larger and taller buildings to be built. In addition, due to substantial contribution of the building sector to global energy use, greenhouse gas emissions and waste production, sustainable solutions are needed, for which timber has shown a great potential as a sustainable, resilient and renewable building alternative, not only for single family homes but also for mid-rise and high-rise buildings. Both recent technological developments in timber engineering and exponentially increased use of engineered wood products and wood composites reflect in deficiency of current timber codes and standards. This paper presents an overview of some of the current challenges and emerging trends in the field of seismic design of timber buildings. Currently existing building codes and the development of new generation of European building codes are presented. Ongoing studies on a variety topics within seismic timber engineering are presented, including tall timber and hybrid buildings, composites with timber and seismic retrofitting with timber. Crucial challenges, key research needs and opportunities are addressed and critically discussed.

**Keywords:** seismic design; tall timber buildings; timber composites; seismic retrofitting; Eurocode 8

#### **1. Introduction**

In the past century, extensive demand for steel, concrete and masonry as construction materials pushed the development and significant advancement of building codes, standards and guidelines for structural systems based on these materials [1–3]. In seismically-prone areas around the globe special attention had to be paid to ensuring seismic resistance of structures as well. Seismic design of structures differs from "regular" structural design in several aspects; structural response to strong earthquakes is dynamic, nonlinear and random, whilst almost all the rest actions and responses are static, linear and deterministic. Due to globalization, seismic design of structures has recently become part of the regular structural engineering curriculum and practice, even in the areas where earthquakes are not so relevant. Past, present and future trends in analyses in seismic provisions for buildings are very well explained by Fajfar [4].

On the other hand, although serious studies on earthquakes and seismic activity began about a century ago, intense research in the field of seismic design of timber structures started only a couple of decades ago, with the advancement of engineered wood products (EWP), which enabled more complex and ambitious timber construction. Global tendency towards more sustainable, energy efficient and environmentally-friendly building solutions has further popularized timber as principle structural material.

Wood in its nature differs significantly from concrete, masonry and steel, as it is considerably lighter compared to them and it is an anisotropic natural material, while the other ones are isotropic man-made materials. These material characteristics influence significantly the overall structural and seismic performance of timber buildings. Recent technological developments and exponentially increased use of engineered wood products and wood composites reflected in deficiency of current timber norms and standards. This paper focuses on some open questions and recent developments in timber engineering regarding the use of timber in seismically active regions, on seismic design of timber structures and normative acts in Europe, and especially on the lack of information in the Eurocode 8.

Recently completed COST Action FP1402 has contributed to a better understanding and overview of broadly available scientific results and the specific information needed by the code-writers, authorities, designers and end-users in the safe, durable and efficient use of timber in structures and, consequently, increase its acceptance and use in the design of buildings. Significant progress has been made with respect to the cross-laminated timber (CLT) structures [5–9], timber–concrete composites [10–12] and understanding of the connections in timber structures [13–16]. As a result, input data for the improvement and future development of EN 1995 are given [17]. However, several topics on seismic design of timber structures still need further investigation [18].

Future trends in timber construction will require major development and research on topics of: Tall timber and hybrid buildings, new engineered wood products and connection systems related to the new technologies, modular construction with timber, composites with wood, assessment of existing timber buildings and retrofitting of historical buildings with timber (Figure 1). Due to rapid development of new timber technologies, and especially due to taller and taller timber buildings, precisely defined impact of earthquakes on these structures needs to be investigated.

**Figure 1.** Future need for a research in timber engineering.

#### **2. European Seismic Design Norms, Standards and Guidelines**

Recent developments in timber engineering and exponential intense of timber in construction required the evolution of currently existing standards. Materials like cross-laminated timber (CLT) proved that timber can be an excellent material for mid- and high-rise buildings even in seismic areas [19,20]. Cross-laminated timber (CLT) is used for floors and walls and can be considered as floor diaphragms and shear walls in seismic design [21–23]. Significant development has also been achieved at the connection level. Self-tapping screws [24,25], glued-in rods [26–28] and a lot of different innovative systems [29–32] show improved behavior in seismic applications than traditional

dowel-type connections. At the system level, hybrid systems with concrete [33], EWPs [34,35], steel [36,37], polymers [38] and glass [39–41] are in development and are intensely researched.

Current seismic design approaches in building codes around the world (e.g., Eurocode (EC8) [42], NBCC [43], ASCE-7 [44]) follow force-based design methods. At the moment there are only a few norms on seismic assessment of existing structures [45–48] and future development in research shall be focused in this direction as well. Nevertheless, timber as a structural material is poorly represented in the current norms (i.e., EC8 has only four pages related to seismic design of timber structures). In the current Eurocode 8—Section 8: "Specific rules for timber buildings", no information is provided for seismic design of widely used structural systems such as cross-laminated timber structures. In addition, no provisions are given regarding capacity design methods for different types of timber structural systems, which proved to be crucial in seismic design of timber buildings [49–51], as well as provisions and rules for transfer zones for continuity of shear walls along the building's height in multi-story timber buildings. The revision process of the Eurocodes began in 2015 and the final updated version is expected to be released sometime after 2020. The new proposal of timber part, prepared by Work Group 3 of CEN Technical Committee 250 (CEN/TC 250/SC 8/WG 3), is explained by Follesa et al. [18] and is based on following modifications and recommendations:


Above-mentioned new provisions and concepts of timber structures seismic design will demand additional information on mechanical properties of timber connection systems such as connection ductility under cyclic loading, overstrength factors, elastic and plastic stiffness, strength degradation properties under cyclic loading, energy dissipation properties, etc. Therefore, in the near future European technical assessment documents (ETA) for timber connections shall include more information on mechanical properties of connections under cycling loading, defined in EN 12512 standard [52].

Current trend of exponential growth of new timber buildings, larger, taller and more complex projects not only requires higher volume of engineered wood products production, but also higher demand for skilled carpenters and tradespeople with proper education and training on timber construction. Thus, in addition to the updates of the current building codes, also regulation and guidelines in the area of execution and construction supervision of timber buildings shall be improved, where contracting companies shall obtain certifications as a proof of being competent to execute such buildings. Further, regulation on periodic monitoring of structural health of timber buildings, especially tall timber buildings, shall also be addressed.

#### **3. Timber Buildings—Future Trends and Challenges in Seismic Design**

In this section, current and emerging challenges in seismic-related topics in the field of timber engineering are presented, compared and critically discussed. Due to the rapid advancement in the development of engineered wood products (EWP) and structural connections, presented earlier in this paper, more and more new applications of EWPs in timber engineering and other engineering fields have emerged. These applications extend from possibilities of building taller timber buildings, to combining timber structural systems with structural systems based on other materials such as

concrete and steel, forming so called hybrid timber buildings, exhibiting even higher potential for high-rise construction but also additional challenges to overcome. Further, a combination of advanced EWPs and structural connections can also serve for new composite load-bearing assemblies, such as timber–glass composite wall systems. Finally, advanced EWPs have also been proposed for seismic retrofitting of existing buildings, not only for existing buildings with timber structure, but also existing buildings with stone, masonry and concrete frame structure. Based on the current state-of-the art, key future research and development needs and trends in these selected topics are identified and presented.

#### *3.1. Tall Timber Buildings*

European strategy for a sustainable growth and sustainable society acknowledges the importance of EU research framework programs for increasing the offer of new high-quality products and services [53,54]. Sustainable construction embraces a number of aspects, such as design and management of buildings and constructed assets, choice of materials, building performance as well as interaction with urban and economic development and management. Timber as a material and as a structural system could be an ideal material for a new era of construction. Without compromising architectural requirements, the transfer of part of the on-site construction activity to off-site production, independent from weather conditions, will ensure a more continuous activity, a better quality of the finished products and an improved control of their environmental characteristics, as discussed, for example, by [55–58].

Building with timber, in recent years, has become a huge trend among the construction sector around the world. With products like CLT, the possibilities are enormous. It is obvious that the market is changing and wood as a structural material will become more relevant and more in use in the following years. It is now well understood that the concrete and steel industries are highly energy intensive and contribute to a significant portion of global carbon emissions, and wood is becoming an obvious solution to reduce it. If sustainably sourced, timber is undoubtedly one of the most environmentally-friendly materials currently available, being a natural carbon sink and truly renewable [59].

Tall timber buildings are built almost on a monthly rate and more and more are planned in the near future. Short overview of the existing tall timber building and buildings under construction are given in the Figure 2 and one erected building is shown in Figure 3.

**Figure 2.** Tall timber buildings around the world—combined data [60–62].

**Figure 3.** Tall hybrid timber building Brock Commons, Vancouver, Canada.

Although, timber has a good reputation in earthquake-prone regions, in the new era of timber skyscrapers, seismic actions and detailing require additional attention in design and construction of such structures. Tendency to build taller multi-residential and non-residential buildings with more occupants will require additional precautions. In the previous chapter it was pointed out what should change and how the design of such buildings should be done [18]. Holistic approach must be appointed to a design of such structures. Namely, in addition to conventional actions such as permanent loads, imposed loads, snow and wind, there are several additional actions which can significantly affect structural behavior of timber buildings, such as thermal actions, moisture, mold, fire, etc. Therefore, special attention has to be paid to detailing in tall timber buildings (architectural, structural and seismic detailing, special acoustics details, etc.). To prevent any potential structural damage in tall timber buildings due to aforementioned actions, continuous monitoring of buildings during their lifetime could be beneficial (installation of moisture sensors in critical places, vibration monitoring, etc.). Additional potential problems which should be addressed regarding the seismic provisions of tall timber structures are:


#### *3.2. Hybrid Systems with Timber*

Combining timber with other materials, to achieve composite action, enables the strength of timber to be enhanced and its shortcomings to be strengthened or eliminated. Timber is most commonly combined with concrete and steel, but recently also with glass and polymers. Pre-stressed timber buildings [82], timber–steel hybrid systems [83,84] and timber–concrete hybrid systems [33] are already in use and are proven as earthquake-resistant structural systems. Nevertheless, due to tendency towards taller and larger timber construction, and consequently higher structural performance demand, these topics continue to be furtherly explored in terms of experimental, numerical and analytical studies [36,85,86]. In this paper, hybrid systems with structural glass will be briefly addressed. While timber–concrete and timber–steel hybrid structural systems are already widely applied to mid-rise construction in practice, the timber–glass hybrid structural system is still emerging. In this section, current state-of-the-art of this hybrid structural system is presented and key open questions and challenges for its future possible launch to the construction market are discussed.

Timber and glass composite systems are lately intensely investigated due to extremely high-aesthetic and -ecological value in addition to their cost-effectiveness and the possibility of significant load transfer [40]. Nevertheless, design models and current European standards include the usage of glass panels as the secondary elements [42], which means the positive impact of these elements when transferring transverse loads caused by the earthquake have to be ignored [39]. According to the EC8, it is necessary to calculate the primary structural elements within the allowed displacements regarding the protection of the secondary elements. If the problem is approached as defined in Eurocode 5 (EC5) [87], timber wall diaphragms shall be designed to resist both horizontal and vertical actions imposed upon them, and shall be adequately restrained to avoid overturning and sliding. Racking resistance is provided by in-plane plane stiffness of board materials, diagonal bracing or moment connections. Method A from EC5 defines that shear diaphragms with windows and doors do not contribute to stability of the structure. Method B is less restrictive and it proposes to regard panel parts from each side of the opening as separate panels. Since the openings decrease the racking resistance and significantly reduce the horizontal stiffness of precast elements, glass helps to enhance these properties. In recent years several articles on quasi-static and dynamic tests of timber–structural glass composite systems were published [39,40,88–94] (Figure 4). The researchers have concluded that timber–structural glass load-bearing systems can be used in various construction applications, depending on the required bearing or ductility levels. Žarni´c et al. [95] investigated deformation capacity, lateral strength, stiffness and strength deterioration and energy dissipation capacity in order to provide data for the future development of computational models and design guidance for the new codes. Nevertheless, it is possible to devise a combined system with timber and structural glass in which each material could transfer load, and in mutual interaction of constitutive elements could be resistant to earthquake. Several extensive tests of composite systems timber–structural glass have been conducted with various types of bonding timber and glass. Bonding glass on timber proved as a good example for accomplishing high-load-bearing of composites, but deficiencies are noticed in the level of ductility along with possible problems with the durability of the structure. Further research on timber–glass composites is needed to get the structural response in seismic actions. In addition, factor q (force reduction factor) of structures constructed with hybrid laminated glass–CLT structural panels need be derived. These results may contribute to future upgrading of EC 8 where glass-based structures are not yet addressed [95]. Besides the load-bearing characteristics and development of structural design tools the special attention need to be paid to energy efficiency of developed building components [96].

**Figure 4.** An example of a timber–glass composite system [39,40,88–94].

#### *3.3. Seismic Retrofitting with Timber*

Seismic retrofitting can be defined as the modification of existing structures to make them more resistant to seismic activity, ground motion or soil failure due to earthquakes. Seismic retrofit strategies have been developed in the past few decades following the introduction of new seismic provisions and the availability of advanced materials. Different techniques and methods are applied for a seismic retrofitting of different structures (e.g., concrete structures [97–99], timber structures [35,100,101], historical and heritage structures [102–104] and especially masonry structures [3,105–107]).

In this chapter, retrofitting of existing structures with CLT is presented. Several researchers around the world are investigating the possibility to seismically strengthen existing structures with engineered wood products (EWPs), mostly CLT. Japanese researchers [108] proposed RC frames strengthening with several narrow CLT elements bonded onto the RC frame with epoxy resin. In their strengthening method, CLT panels are infilled in the RC frame and are acting as shear walls. An Italian group of researchers [109] is developing a novel integrated retrofit solution based on the use of CLT shear walls encased as infill in existing RC framed structures (Figure 5). The idea was to increase the overall lateral stiffness of the concrete structure and to reduce the lateral drift values. The main conclusions of preliminary experimental and numerical work are that the CLT infill allowed the RC frame to reach a lower drift value and a higher peak load with respect to common masonry infills. Numerical modeling and optimal seismic retrofit design with CLT was proposed by [110]. Bahmani et al. [111] were retrofitting four-story-soft-story timber buildings with CLT. They concluded that a retrofit, in accordance with the FEMA P-807 guidelines, using CLT panels is suitable for achieving life safety performance levels during 50% MCE level earthquakes, when retrofit of all story levels is not possible due to one or more constraints.

**Figure 5.** Retrofitting with CLT proposed by Sustersic [95].

Extensive research on seismic strengthening of different types of existing buildings with CLT panels was performed by Sustersic and Dujic [112]. The experimental work and numerical study were focused on typical non-earthquake-resistant older masonry and concrete structures from Southern European countries. A new outer shell is added onto existing buildings (Figure 5). It is made from CLT plates that serve as the load-bearing layer and are attached to the existing buildings with special connections. The basic idea of the proposed CLT outer envelope is that it is light (it does not contribute much to seismic forces and is easier to install), strong (timber has one of the best strength:weight ratios) and can be produced as a prefabricated system so little on-site work is necessary. Depending on the building type (concrete or masonry), the elements are anchored into buildings on floor levels, either into concrete slabs or masonry walls. The CLT is connected to the existing building with special steel brackets that offer high strength and stiffness and have a "control fuse" that enables a brittle yet predictable failure of the connection. For the unreinforced masonry walls, the most efficient mechanically-attached strengthening panel resulted in a 33% increase of idealized strength and a 166% higher idealized ultimate displacement. A substitute frame model, with concentrated plasticity for modeling URM and linear elastic shell elements combined with nonlinear springs for modeling the CLT strengthening system, could satisfactorily describe the behavior of the tested wall specimen. Finite element models of the tested RC frame, with or without masonry infill, showed that the damage in both RC frame and masonry infill is lower at equal ground acceleration when the CLT strengthening plates are installed on the structure.

The state-of-the-art research have; therefore, shown that a CLT seismic strengthening system has the potential to be used in design practice, but currently there is still a lot of ''unknowns" and further research is needed. A holistic approach should be applied—seismic and energy retrofitting—as both are very important. In addition to the currently existing regulation on energy certificates of buildings, which are informing the general public, owners and potential buyers as to which condition a building is regarding the energy consumption, a similar concept shall be established for seismic evaluation of buildings—"seismic certificates". This will, on one hand raise awareness of current owners about the seismic health of the buildings where they live, and on the other hand additionally define and position a real estate worth on the market, and; therefore, influence the demand on the seismic retrofit of existing buildings in order to be competitive on the market.

#### **4. Conclusions**

Due to rapid development of high-performance engineered wood products and new timber technologies, resulting in taller and larger timber buildings with applications spanning through all building types, current building codes and standards reflect a deficiency of provisions for contemporary seismic design. State-of-the-art research in various fields of timber seismic design are presented in this paper and crucial challenges, research needs and opportunities are discussed.

The new generation of Eurocode 8—timber part—will address many topics which are not present in the current version. An updated list of timber-based structural systems with definitions of dissipative and non-dissipative zones in structures, which are needed for newly-introduced capacity design rules and overstrength factors for each type of structural system, will be included. Further, adapted q-behavior factors values for different ductility classes will be defined and a new procedure for application of non-linear static (pushover) analysis will be included.

Tall timber buildings with more than ten stories are already present in moderate- and high-seismic zones around the world. Further, due to climate and economic reasons, more and more conceptual architectural designs for taller timber buildings, including timber skyscrapers, are being proposed, which poses several additional engineering challenges to overcome. A holistic design approach including architectural, structural, durability, fire and acoustic designs as an integrated process is crucial, as all these topics are interrelated. Challenges in terms of numerical modeling of timber and hybrid structural systems, ensuring lateral stability due to wind and seismic actions, high-performing

energy dissipating connections, acoustic insulation vs. seismic design philosophy, execution and building monitoring need to be addressed more in depth.

In addition to traditional timber composites with steel and concrete, recent research and developments have shown potential for timber–glass and timber–polymers composites as well. A timber–glass seismic-resistant structural system consists of a timber frame and a structural glass infill bonded together with adhesive, or based on friction contact forming a lateral-resisting wall system. The main advantage of this structural system is increasing lateral stability of buildings with high proportion of facades with glass surfaces by avoiding diagonal bracings or moment connections. Experimental and numerical studies have shown encouraging results in terms of load-bearing and stiffness, whereas the durability aspect needs further examination.

Cross-laminated timber (CLT) has proved to be a great solution for new mid- and high-rise timber buildings. Recently, CLT has also been studied for seismic and energy retrofitting of existing older masonry and concrete buildings, which do not meet current seismic design and energy efficiency criteria. In terms of seismic performance, increased strength and stiffness were observed, yet the research is still ongoing and needs additional investigation of connections between the CLT strengthening panels and the existing structure, with its application to a wider range of existing buildings.

New technologies and knowledge in timber engineering opened many new possibilities in timber application, not only for new timber buildings, but also in combination with other conventional building materials forming hybrid and composite assemblies and structural systems, and also for retrofitting of existing buildings. In this paper, in addition to an overview of some of the current challenges and emerging trends of seismic behavior of timber structures, the focus was set on three topics of advanced engineered wood products applications (tall timber buildings, composites with timber and seismic retrofitting with timber), which are representing new trends of timber engineering and push the boundaries of timber for the use in sustainable construction. All three discussed topics have shown lots of potential for their application in seismic areas, yet there are still several research challenges which need to be addressed in terms of seismic performance and seismic design.

**Author Contributions:** Conceptualization and development of the main idea of the paper, M.S., I.G. and I.Š.; methodology, M.S., I.G. and I.Š.; validation, all authors; formal analysis, M.S., I.G. and I.Š.; investigation, M.S., I.G. and I.Š.; resources, M.S., I.G. and I.Š.; photo credit, M.S.; literature review and manuscript writing related to Section 3.1., M.S. and I.G.; literature review and manuscript writing related to Section 3.2, M.S and V.R.; literature review and manuscript writing related to Section 3.3, I.G. and I.Š.; writing—original draft preparation, M.S. and I.G.; writing—review and editing, M.S., I.G., I.Š. and V.R.; visualization, M.S., I.G., I.Š.; supervision, I.Š. and V.R.; project administration, M.S.; funding acquisition, M.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Unity through Knowledge Fund (UKF): Seismic behavior of multi-storey buildings, UKF Grant Agreement No. 18/19. Authors Iztok Sustersic and Igor Gavric gratefully acknowledge receiving funding from programme Horizon 2020 Framework Programme of the European Union; H2020 WIDESPREAD-2-Teaming: (#739574) and the Republic of Slovenia. Iztok Sustersic would also like to thank the Slovenian Research Agency ARRS for funding the bilateral project BI-US/19-21-014.

**Conflicts of Interest:** The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## **Methods for the Assessment of Critical Properties in Existing Masonry Structures under Seismic Loads—The ARES Project**

#### **Mislav Stepinac 1,\*, Tomislav Kisicek 1, Tvrtko Reni´c 1, Ivan Hafner <sup>1</sup> and Chiara Bedon <sup>2</sup>**


Received: 12 December 2019; Accepted: 19 February 2020; Published: 25 February 2020

**Abstract:** Masonry structures are notoriously vulnerable to horizontal actions caused by earthquakes. Given the high seismicity of the European region, and that the European building stock comprises a lot of masonry buildings, knowledge about their structural response to seismic excitation is particularly important, but at the same time difficult to determine, due to the heterogenous nature of materials and/or constructional techniques in use. An additional issue is represented by the current methods for mechanical properties assessment, that do not provide a reliable framework for accurate structural estimations of existing buildings characterized by different typological properties. Every structure, in other words, should be separately inspected in regard to its mechanical behaviour, based on dedicated approaches able to capture potential critical issues. In this review paper, an insight on the Croatian ARES project is presented (Assessment and Rehabilitation of Existing Structures), including a state-of-the-art of the actual building stock and giving evidence of major difficulties concerning the assessment of existing structures. The most commonly used techniques and tools are compared, with a focus on their basic features and field of application. A brief overview of prevailing structural behaviours and Finite Element numerical modelling issues are also mentioned. As shown, the general tendency is to ensure "sustainable" and energy-efficient building systems. The latter, however, seem in disagreement with basic principles of structural maintenance and renovation. The aim of the ongoing ARES project, in this context, is to improve the current knowledge regarding the assessment and strengthening of structures, with a focus on a more reliable design and maintenance process for existing masonry buildings.

**Keywords:** structural assessment; masonry buildings; earthquakes; seismic loads; existing structures; reliability; rehabilitation; risk

#### **1. Introduction**

According to the literature and recent events, it is well established that masonry is one of the most commonly used materials across the world, due to its simplicity and high quality characteristics.

Even though the use of masonry for construction in earthquake-prone regions gave evidence of its intrinsic limitations (due especially to its limited tensile resistance, relevant mass and stiffness), extensive research has been carried out in the last few decades, with a focus on the material characteristics and structural behaviour, even under extreme loading conditions such as earthquake events. These efforts enabled engineers to design masonry structures on sound and safety principles, with progressively greater exactitude, economy and confidence. Accordingly, a huge number of existing buildings in the European region are composed of masonry. Given that most of the so-called "strategic" buildings of

cultural significance and high historical importance are built using masonry, such a condition is the first motivation, suggesting that the assessment and rehabilitation of existing masonry structures must be conducted on a very high level.

As is known, the main goal of seismic design is to protect property—and thus life in buildings and infrastructure—in the case of earthquake events. However, an appropriate seismic design approach must necessarily develop on knowledge and feedback from existing structures. Compared to other constructional typologies and materials, past events showed that seismic loads usually cause significant damage, especially in masonry buildings, due to their large mass and stiffness. This represents an intrinsic risk for personnel, given that most people in several European countries (especially in the urban areas) work and live in masonry buildings. In addition, a huge number of masonry structures were built—over decades—before any seismic codes were developed, thus no confining elements or reinforcement members exist.

Based on all the above motivations, it is thus clear that a concise strategy for masonry buildings must be activated.

The seismic behaviour of buildings generally depends on several important factors, such as material properties, the geometry of the structure, additional non-linear effects, conceptual design and stiffness properties. The issue of seismic vulnerability assessment and rehabilitation of under-performing existing buildings is hence a complex problem [1]. "Seismic vulnerability" can be conventionally defined as a measure of the inadequacy of a given structure to resist to seismic actions [2]. In modern assessment methods, the seismic vulnerability is represented by design curves which express the physical vulnerability as a function of the intensity of the process and the degree of loss [3]. For individual cases only, some structural characteristics of the affected buildings are considered [3]. Throughout the decades, various methods have been developed to evaluate the vulnerability of buildings, and they can be divided into *empirical* and *analytical* (and thus *hybrid*) methods, and an approach based on engineering judgment by experts. Methods for vulnerability assessments mainly model damage to a discrete scale, where damage itself is commonly grouped using three to six categories [4]. However, no unified approaches on a European level are available.

The seismic vulnerability of masonry buildings is particularly difficult to assess, and notoriously requires a multitude of specialized technical skills [5]. But actually, how accurate are the methods in use for the assessment of the seismic vulnerability of existing buildings?

Unfortunately, this question generally remains unanswered, given that more extended research needs to be carried out on the topics of seismic risk and seismic vulnerability assessment. Focusing only on "visible" structural/material parameters cannot solve such an open issue. Traditional assessment methods, in most of the cases, are in fact well-known to allow the assessment of only the actual condition of a given existing structure, once its stability has already been compromised. In this paper, selected traditional assessment methods are thus discussed, pointing out some possibilities related to the use of newer technologies.

The surge of buildings of higher consequence class (and the global goals towards sustainable development) typically demands higher levels of reliability, and a more sustainable use of raw materials. That is why it the aim is the modification or extension of existing buildings rather than the demolition and substitution. There are several important aspects that have a fundamental role in the assessment of existing masonry structures, namely assessment, deterioration and damage, inspection and investigation, updating, verification, repair, rehabilitation and reinforcement and maintenance. In this paper, assessment methods of critical properties (structural and material parameters) are presented for masonry structures. A focus is set on the available methods able to provide crucial data and feedback for preventing failure mechanisms and collapses under extreme design loads.

#### **2. The Croatian Scenario**

According to the results of systematic research of Statistical Yearbooks, Croatia's national building stock consists of approx. 800,000 residential and 125,000 non-residential buildings [6]. More than 75% of the building stock is older than 30 years, thus corresponding to a life-time requiring at least some renovation or modification of primary structural components. More than 40% of the building stock is then older than 50 years, meaning that the service life of a given structure is fully expired. In the Croatian building sector, finally, it is recognized that up to 40% of the expenses are dedicated to the rehabilitation, modification and demolition of existing structures (Figure 1a).

**Figure 1.** (**a**) Investment (values in HRK) for the renovation of Croatian national building stock (data derived from [6]); (**b**) Typical masonry building in Zagreb, with a graffiti meaning "a mess" (photo by M. Stepinac).

The main characteristic of Croatian buildings constructed in the period before the 1970s is represented by the use of traditional constructional techniques and materials, such as masonry and timber (Figure 1b). Buildings were built as full-brick masonry structures, with mostly wooden ceilings, and 30–60 cm thick walls, thus resulting in statically satisfactory structural assemblies. In the 1960s, reinforced concrete (in combination with timber and steel) started to progressively replace traditional constructional materials. In fact, most of the residential structures are still built as a combination of concrete, masonry and timber load-bearing components.

One of the most important HORIZON 2020 programme objectives is the acquisition of necessary knowledge and skills by all stakeholders in the process of energy-efficient building renovation. European Union directives clearly suggest the energy renovation of existing structures, but structural aspects are somewhat ignored and/or disregarded. At the moment, a number of existing structures are in fact under energy renovation and/or energy upgrading processes (Figure 2a). Besides such a consideration for energy performances, however, structural assessment and/or structural upgrading is mostly disregarded. According to an estimate of total investments in Europe, for the period between 2014 and 2049 (including initial investment expenditures, maintenance and replacement of worn-out equipment), around 3 <sup>×</sup> <sup>10</sup><sup>9</sup> Euros will be dedicated to energy processes. In such an expected scenario, it is thus clear that structural updating and retrofitting can (and must) represent an additional value for the energy renovation of buildings. This is especially the case in existing masonry structures, which, in most cases, need robust seismic strengthening interventions.

In the Croatian framework, it is in fact recognized that the majority of residential buildings older than 50 years consists of masonry structures (i.e., Figure 2b) without appropriate bonding elements to connect floors and walls [7].

**Figure 2.** (**a**) Renovation of facades and energy upgrading of existing structures (**b**) Typical building blocks in Zagreb (photos by M. Stepinac).

#### **3. Existing Masonry Structures and Maintenance Issues**

The need for maintaining the built environment is supported by the global policy of the Kyoto protocol 1997 and all further World Climate Summits on existing buildings and engineering works. Sustainable development is a long-term goal of the global policy which results in modifications, substitutions or extensions of existing buildings.

In simple words, existing structures can be distinguished with regard to their value between economical (*monetary*) values and cultural (*non-material*) value. Modern existing structures commonly have a higher economical value, compared to heritage structures (with a dominating cultural value).

Accordingly, the assessment of an existing structure can be performed through different steps, with increased precision. The degree of precision thereby depends on the amount and on the quality of available information, as well as on the importance of the building being assessed. This can be reached by breaking down the assessment into different phases. The number of required phases is dependent on the remaining level of doubt, and the feasibility and simplicity of repair/strengthening, always in combination with economic considerations [8].

Advances in technology and sustainability requirements, and requirements for preservation of existing structures provoked an increased interest in scientific and professional community for assessment methods. Regarding the masonry structures, a wide variety of methods exist'; however, their frequency and scope, the decision-making approach concerning safety and the necessary interventions are far from being agreed upon. The need for an assessment of an existing structure can be based upon a multitude of reasons. The most typical are briefly explained in [8]. In situations where doubts may be raised in regard to the design assumptions, a re-assessment of the structure may be also necessary, such as [9,10]:


With regard to economy and sustainability, finally, it is certainly of high interest for the building owners (as well as society) to maintain existing structures, rather than demolish and rebuild them.

#### **4. European Standards, Norms and Guidelines**

Most of the current design standards and guidelines are based on reliability-based design, as specified in ISO 2394 [11] or JCSS [12] documents. The majority of them regulate the design of new structures, such as the Eurocodes. Nevertheless, the intention is to make the Eurocodes applicable also for existing structures. Work on new technical rules for the assessment and retrofitting of existing structures is currently very intense, with special attention being given to heritage structures, aiming at elaborating the new Eurocode part for existing structures.

One of the few international guidelines for the assessment of existing structures is the ISO 13822 document [13]. Although different guidelines for the maintenance of existing structures exist in other European countries, only a few of them have issued standards for the assessment of existing structures (e.g., Switzerland [14] and the Netherlands [15]). More in detail, SIA 462 [14] standard, in combination with the SIA 469 [16] document, specify the general fundamentals for the assessment of the load-carrying capacity of existing buildings, and regulate the professional and economical maintenance of buildings according to their cultural value, respectively.

Nonetheless, the daily basis for the assessment and rehabilitation of existing structures is still based on a rather rudimentary scheme, mainly replacing and reinforcing defective members. More sophisticated procedures are needed, in addition to those available in seismic codes (such as Eurocodes). Eurocode EN 1998-3 [17] provides some suggestions for the assessment of masonry structures, but it is only informative and lacks of detailed practical suggestions. ISO 13822:2010 [13] gives general recommendations for a rough assessment of existing structures, but does not take materials into account. Similar to EN 1998-3 [17], it is mainly informative and lacks depth. Neither of these standards are very practical. The American documents ASCE 41-13 [18,19] and ASTM [20–25], in this regard, constitute a more comprehensive guide, including suggested equipment, procedures of assessment, number of tests, as well as analysis and strength calculation methods. From the perspective of an engineer, this standard is hence more practical and easily applicable.

#### **5. Selected Assessment Methods**

Numerous technical documents have been published by national and international authorities and focus on systematic and scientific methods that can be used to accurately assess the residual strength, durability and reliability of structural materials, assemblies and systems in existing buildings. Nevertheless, especially in the scope of masonry structures, all these documents need to be continuously revised, expanded and enhanced. Major issues are related to the increasing knowledge in the field of material sciences, as well as to the technological advancement in the field of the structural assessment and monitoring of structural systems, or to the practical experience and feedback derived from professional engineering activities on existing structures. The key role and relevance of research, in this context, lays in the preservation of existing building stock, restoration of objects, towards the support and development of reliable and consistent guidelines/norms.

Masonry structures are composite systems, whose main components are masonry units and mortar layers. Both can be made of various materials, with different mechanical characteristics. The geometry of masonry units (length, width, height, amount and direction of holes), the thickness of mortar and the area it covers can also strongly affect the overall mechanical behaviour. Recent research studies have shown that the inclusion of masonry infills leads to a significant increase in structural stiffness, thus affecting the overall probability of structural collapse under dynamic loads (such as those induced during an earthquake). Specifications of compressive strength, a function of parameters of brick and mortar, are often required. Assessing the variability of these properties and the uncertainty in the modelling of the masonry compressive strength is hence a topic of great importance [26].

For the assessment of existing structures, the strength classes of used materials are often unknown. Nevertheless, major advantage and benefit can derive from the fact that the reliability of structures directly depends on the actual properties of the elements in use. Assessment methods should therefore aim at identifying these properties to the highest degree of achievable certainty, in order to reduce the

uncertainty with regard to the resistance of the structure as a whole. In contrast, for the design of new structures, possible uncertainties on material properties are taken into account in the partial safety factors for strength properties.

Parameters which are always measured when assessing masonry structures are thus represented by (i) the compressive strength of masonry units, (ii) the compressive strength of masonry mortar, (iii) the compressive strength of concrete infill (if any), (iv) the strength of reinforcing steel bars (if any), (v) the compressive, shear and flexural strength of masonry, (vi) the Modulus of Elasticity (MoE) and the geometry of the masonry structure (size and location of bearing walls, location and size of openings).

The most important Non Destructive Testing (NDT) methods for existing masonry structures are summarized in Table 1 and Figures 3 and 4.


**Table 1.** Available Non Destructive Testing (NDT) assessment methods for existing masonry structures.


**Table 1.** *Cont*.

Although each method mentioned in Table 1 has both advantages and disadvantages, not all of them are needed for adequate assessment. For an adequate assessment, further methods may be applied (although other combinations are possible).

Visual inspection presents a basic tool for the assessment of existing buildings. It is a significant method which should always be implemented to determine further actions. The type of masonry, building location specifics and overall state of the structure can be determined and rough estimations may be made. One of the negatives is that it requires an experienced person to determine the important parameters.

A rebound hammer is a tool that may be used to indirectly determine the compressive strength of masonry. Since this is one of the most important parameters and considering both its low cost and ease of applicability, this method seems to be a valuable tool. Although some methods may be better at determining compressive strength, they are either more expensive and complex, or not non-destructive.

Ground penetrating radar can detect the location of steel, thickness of the wall and possible voids in it. It is a versatile tool with which the geometry of a wall can be determined precisely.

Flat-jack systems can be used to assess the stresses in existing walls, strength and deformability parameters, as well as shear strength of a wall. These are all important values to obtain for a more precise assessment. One of the downsides is the fact that slits need to be cut in a wall. Either rigid or flexible flat-jacks can be used, depending on the type of wall that is being tested.

In addition to assessment, continuous Structural Health Monitoring (SHM) may assist engineers for a better understanding of the actual structural behaviour of a given assembly. Together with structural risk and reliability research and development, SHM specialists figure a comprehensive research community. Accordingly, it is generally recognized that SHM represents an important field of today's infrastructure engineering. It is a goal to enhance the benefit of SHM by the novel utilization of applied decision analysis on how to assess the value of SHM—even before it is implemented.

**Figure 3.** Example of ultrasonic velocities distribution (values in m/s) for a portion of the "P1MF" case-study building (box detail). Reproduced from [25] with permission from Elsevier® (Copyright© Agreement license n. 4726430416591, December 2019).

**Figure 4.** Example of NDT assessment of existing masonry: (**a**) GPR testing instruments and results (reproduced from [27] with permission from Elsevier® (Copyright© Agreement license n. 4726440628105, December 2019)); (**b**) pendulum rebound hammer, to evaluate masonry hardness; with (**c**) borescope investigation (figures (**b**,**c**) are reproduced from [31] with permission from John Wiley & Sons® (Copyright© Agreement license n. 4726520295075, December 2019)).

Knowing the intrinsic value of SHM techniques, the decision basis for the design, operation and life-cycle integrity management of structures can thus be improved and can finally facilitate more cost-efficient, reliable and safe strategies for maintaining and developing the built environment to the benefit of society [43]. All of the above-mentioned methods (see also Figure 5) are in fact eligible to offer support for better understanding the structural behaviour, risks and costs for the preservation of existing masonry structures.

**Figure 5.** Graphical illustration of the typical assessment procedures for masonry structures.

#### *Seismic Actions and Masonry Structures*

In order to strengthen existing weak structures, assessment must be necessarily designed and carried out. In the past, most of the initial work on seismic assessment was based only on the visual inspection of buildings [27], NDT methods (see, for example, [32,41,44–46]), as well as the experience of engineers. The last decade, however, has seen a growth in the technological development of various tools which can greatly support the prediction of structural safety and seismic behaviour of existing structures, i.e., thermography [36], photogrammetry [47], unmanned aerial vehicles [48], etc.

The first key step is visual inspection. A structure's location, both in respect to seismic hazards [49] and with respect to micro-location (e.g., hill, valley, etc.) matters. Neighbouring structures can also influence structural behaviour. For example, different storey heights may lead to the collision of a slab with a neighbouring structure, as can be seen in Figure 6a.

While newer buildings tend to have lower stories than old buildings, this is not uncommon when a new structure is built inside a block of existing structures. Even if their storey height is the same, neighbouring structures, according to Figure 6b, may also collide.

The layout of a building's structural element defines its dynamic response, so it should be adequately measured or characterized [50]. It is not uncommon that access to the inside of a building is restricted. The time of construction can also represent another valuable piece of information, since it gives insight into the most probable material characteristics (both masonry units and mortar), used building technology and design codes. In terms of floor systems, the most commonly used solutions are timber in older constructions and reinforced concrete in newer constructions. Timber floors, as is known, are more flexible than stiff concrete slabs. This not only leads to a different distribution of seismic actions, but also means that walls are more susceptible to potential out-of-plane failure mechanisms. Concrete slabs are, in fact, recognized to ensure a box-type behaviour of walls. In structures with timber floors, horizontal steel rods are usually added to connect the walls (when they do not already exist) and overcome such an issue.

Besides the need of such a box-type behaviour and interlocking effect, the mortar and masonry quality of load-bearing elements typically degrades with time. Since different materials were used in

history, it is advantageous to know the time period of construction. In addition, visual inspection of material degradation should be necessarily performed. From visual inspection, the position, direction and width of cracks can be measured and can indicate weak spots in a given structure, as well as poor soil conditions, or poor craftsmanship. In some cases, cracks may also be structurally negligible. The geometry of a wall, as well as shape, size and position of openings, can significantly influence crack formation and propagation. Cracks cause a reduction in stiffness, which in turn changes the dynamic response of a structure. To adequately model stiffness, the MoE needs to be known, which can be done with a flat-jack system or rebound hammer. Since stiffness distribution is also relevant, measurements should be done on all walls. When stiffness is correctly assessed, seismic forces can be determined. In order to check if a failure will occur, the strength of a material is needed. Compressive strength can be appraised (with careful laboratory calibration) by using the rebound hammer, which measures wall hardness. If reinforcement exists, its distribution and amount influence the crack width. Usually, reinforcement cannot be detected by visual inspection, so additional tools like Ground Penetrating Radar (GPR) are needed. Some of the reinforcement might be corroded or poorly anchored, and such an issue should be properly checked. Other important parameters of a wall can be assessed with GPR, such as the thickness of each wythe and a complete wall, or void location and size. Poor connection between different wythes may cause the falling out of a part of wall, which can potentially be dangerous for personnel. The dynamic response of a structure should be then properly checked using a Finite Element (FE) model. In the latter, however, the previously measured material parameters (or appraised on-site) should be carefully calibrated.

In addition to previously mentioned data, an important role is also that of soil, whose stiffness and strength should be carefully taken into account. The variation of soil conditions under the building should also be considered. Based on a structural model, stresses and strains may be calculated in each point of a wall. The vulnerability of a structure under different earthquake records may also be calculated and, in combination with hazard risk, can be defined. Based on risk, expected cost can be estimated. When the expected risk is unacceptable, the structure should be properly strengthened. This can be done by adding steel reinforcement or some sort of non-metallic material with high tensile resistance, such as Fiber-Reinforced Polymers (FRP) or Textile Reinforced Mortar (TRM) solutions. Based on the expected critical failure mechanisms, different configurations of reinforcement may be adopted. Strengthening can also be effective if maximum deformations either cause the failure of non-load-bearing elements or stability problems. Since stiffness changes with adding reinforcement, however, additional changes in the reference models should be necessarily implemented.

**Figure 6.** (**a**) Reconstruction works in urban areas—building of new facilities which directly influence the seismic safety of existing structures. (**b**) Typical street in the old city of Zagreb, where the majority of buildings are made of masonry (photos by M. Stepinac).

#### **6. Open Challenges in the Framework of the ARES Project**

The preservation of the building stock tackles societal challenges, environmental issues, and resource efficiency and represents the complex process of engineering and technical work.

The broad range of available methods shows the extensive experience in the measurement techniques gained in recent years. However, this experience, so far, has not led to a considerable advance in the quality of assessment with regard to the evaluation of the reliability of structures. Most methods only allow for the determination of the level of degradation and the localization of damages or the determination of localized material properties that do not represent the overall properties and load-carrying capacity of the entire structure. So far, it is often not possible to relate and interpret these measurements with regard to strength and stiffness properties required for the design and evaluation of reliability of the structure with sufficient certainty. Advanced methods for the assessment and probabilistic evaluation of existing structures make use of updated information and are already successfully applied in different other fields [51] but are still scarce in masonry engineering [29,52–56].

At the moment, there is a huge knowledge gap in the assessment methods and design checks of existing structures. Without appropriate guidelines, it is difficult to approach the problem, and this often leads to the wrong interpretation of collected data, and, thus, wrong decision scenarios for the reinforcement and rehabilitation of existing structures. The development of investigation techniques for the updating of material properties will help in reducing the uncertainty associated with the prediction of the structural behaviour of existing structures.

The ARES project, in this context, currently tries to fill the mentioned gaps. The project is financially supported by the Croatian Science Foundation and aims at delivering the basis for advanced assessment and design of existing structures, allowing a more economic design and a more accurate analysis of the consequences of failure. The ARES project aims to improve the way assessment is actually carried out. Standard methods will be compared, and necessary procedures will be determined to simplify assessment for practical use. General guidelines will also be developed, and safety factors will be reviewed.

The research project will provide knowledge about the building stock in Croatia, assessment of existing masonry structures and prediction of material properties from NDT. In addition, one of the current project goals is to evaluate updating methods of properties for the application of the assessment and verification methods for structures, including consideration of time-dependent behaviour and the influence of environmental conditions.

Furthermore, it is expected that in the current revision of the Eurocodes the consideration of existing structures will be more prominent, but, so far, no adequate international rules for the assessment, reuse and rehabilitation of existing structures exist.

In the most developed societies, as they progress, the feeling grows that it is necessary to maintain the existing building stock to preserve cultural identities. Preserving old buildings also benefits businesses and the local economy of societies. Preservation reduces waste, demolition energy use and new construction. Adaptive re-use concepts, renovations for less energy use, maintenance and type of use also affect building sustainability.

The following main challenges for future research and development in the field of assessment of existing structures can be identified:


• enhancement of the communication with decision-makers.

Solving the above-mentioned issues can be of valuable use and precious help for consultants and designers, as well as for researchers and scholars dealing with the assessment of masonry structures.

#### **7. Conclusions**

European building stock comprises a large number of masonry structures which are vulnerable to seismic excitation. To reduce vulnerability, they usually need to be strengthened with either steel or non-metallic reinforcement. Configuration and the amount or reinforcement need to be determined after the assessment is done. Since sustainability is becoming a prevailing issue, the general aim seems to be shifting from building new structures to the maintenance of existing ones. Thus, the evaluation of structural behaviour and strengthening techniques is becoming more important than it was before. Many questions regarding seismic vulnerability assessment of masonry structures remain unanswered, although many tools and techniques for estimation of material properties exist. Most of the historic structures and a lot of residential buildings are built with masonry, so both economic and cultural aspects are at risk. In many cases, access to the interior of a structure is limited. At the same time, energy efficiency is currently being improved in a lot of structures without any seismic or even structural considerations. Some simple solutions may lead to a large improvement in seismic behaviour, such as adding horizontal steel rods to ensure a box-type behaviour of a structure.

The majority of existing design standards regulate the design of new structures but provide poor information for their maintenance and repair. Although some guidelines and technical documents exist, they need to be improved and adapted to different solutions. Further research on material properties (and also its variability), risk assessment and modelling is, hence, necessary. Although there is a lot of experience regarding existing structures, no significant advancement in the quality of assessment can actually be perceived. The interpretation of measurements and their relation to calculation parameters still represents a weak aspect of the overall process. Currently, there is a huge knowledge gap, especially in the assessment methods and design checks for existing structures, in every aspect of the assessment process.

Firstly, it is important to clearly define which parameters must be tested, and which may be calculated (i.e., is it necessary to measure mortar characteristics or are they similar for the Croatian building stock? Is it necessary to measure shear strength or will the well-known expressions approximate the values appropriately?). In this regard, the plan is to make case studies on existing structures to develop the exact procedure.

Secondly, the focus should be spent on how to properly model such a series of structures, considering economy. This is in contrast with the current practice for existing structures, which is usually either overly complex or overly simplified. The plan is thus to model structures with different input parameters, and afterwards try to assess them before testing. The values will be compared to the ones provided by tests and models will be calibrated. The procedure will be repeated until a stable system (i.e., where no further calibration is needed) will be obtained.

Thirdly, gaps exist in the way strengthening is ensured (which safety factors can be expected for a specific strengthening technique, masonry and standard practice characteristic of Croatia? How does strengthening influence the parameters for modelling? etc.).

The ARES project, in this context, tries to fill these gaps, allowing for a more economic design and a more accurate analysis of the consequences of failure. In order to achieve sustainability, SHM should be implemented to ensure continuous tracking of structural behaviour and provide owners with information important for maintenance.

**Author Contributions:** Conceptualization, M.S. and C.B.; methodology, M.S., C.B. and T.K.; validation, M.S. and T.K.; formal analysis, M.S., T.R. and I.H.; investigation, M.S., T.R. and I.H.; resources, M.S., C.B., T.R. and I.H.; photo credit, M.S.; writing—original draft preparation, M.S. and T.R.; writing—review and editing, M.S., C.B.; visualization, M.S. and I.H.; supervision, M.S. and T.K.; project administration, M.S.; funding acquisition, M.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Croatian Science Foundation, grant number UIP-2019-04-3749.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

#### *Article*

## **Numerical Evaluation of Dynamic Responses of Steel Frame Structures with Di**ff**erent Types of Haunch Connection Under Blast Load**


Received: 16 December 2019; Accepted: 12 February 2020; Published: 6 March 2020

**Abstract:** This research is aimed at investigating the dynamic behaviour of, and to analyse the dynamic response and dynamic performance of steel frames strengthened with welded haunches subjected to a typical hydrocarbon blast loading. The structural dynamic analysis was carried out incorporating the selected blast load, the validated 3D model of the structures with different welded haunch configurations, steel dynamic material properties, and non-linear dynamic analysis of multiple degree of freedom (MDOF) structural systems. The dynamic responses and effectiveness of the reinforced connections were examined using ABAQUS finite element software. Results showed that the presence of the welded haunch reinforcement decreased the maximum frame ductility ratio. Based on the evaluation of the results, the haunch reinforcements strengthened the selected steel frame and improved the dynamic performance compared to the frame with unreinforced connections under blast loading, and the biggest haunch configuration is the "best" type.

**Keywords:** blast loading; welded haunch connection; steel frame structures; non-linear dynamic analysis; ABAQUS; multiple degree of freedom (MDOF); frame ductility ratio

#### **1. Introduction**

The record of oil and gas industry accidental events shows that the historical offshore disasters have generally caused very significant losses in term of human lives, economy, and environmental pollution. Steel frames, as shown in Figure 1, are typical structures used in the offshore industry. The main functions of these frames is to support mechanical equipment, electrical and instrument cables, and hydrocarbon pipelines. Its function as a pipe rack is one of the reasons for selecting the frame because a ruptured pipeline which contains hydrocarbon liquid or gases may escalate the damage in a blast accident by intensifying the fire from the explosion. Therefore, this frame type was considered in our research.

**Figure 1.** Typical steel frames used in offshore structures. (**a**) Pipe rack; (**b**) Mechanical Equipment Support.

During the period from 1993–2013, the worldwide petroleum and chemical insurance market reported that about 1100 insurance claims on major incidents were made [1]. Among these accidents, hydrocarbon explosions are the most hazardous events. Although a hydrocarbon explosion is a very rare accidental event, totally eliminating such incidents is difficult. Therefore, efforts should be made to keep the risk as low as possible and reduce the amount of structural damage if an explosion incident occurs. One of the approaches to minimize the risk to people and the facility is to utilize a blast resistant design or structural strengthening. The frames must have adequate stiffness, strength, and ductility capacity to resist blast loading. The connections of the structural component need to provide a significant contribution to maintain the integrity of the structural system. In many cases, connection strengthening can be implemented to enhance the structural dynamic performance.

In recent years, a few studies [2–7] were conducted on the investigation of the steel connection performance subjected to blast loading. Several steel joint reinforcement methods which strengthen and provide a better performance were proposed. An additional plate attached to beam flanges can improve the connection performance under blast loading [3]. Blast events are dynamic phenomena which are similar to earthquakes. Both blast resistant structure design and seismic design involve the ductile behaviour of the structures. Under seismic loading, the performance of the welded haunch connection reinforcements was studied by many researchers [8–11]. In seismic design, the welded triangular haunch, as shown in Figure 2, is one of the steel connection reinforcement methods that has been effectively implemented for enhancing steel moment connection performance as recommended by several seismic design guidelines [12–14]. Since seismic and blast loading characteristics are different, design for seismic load does not guarantee adequate performance for blast loading. Therefore, seismic design recommendations on connection reinforcement, and using welded haunches, cannot be directly implemented in blast design.

**Figure 2.** Typical sketch of a haunch connection configuration.

This study investigates the dynamic behaviour of a simple single-bay two-storey steel frame structure subjected to blast loading caused by hydrocarbon explosions. To achieve these objectives, a typical hydrocarbon explosion dynamic pressure was considered and the selected structural configuration including the connection details were modelled by numerical simulation using the finite element analysis (FEA) computer program ABAQUS/Explicit. To facilitate the model validation, a single-bay two-storey steel frame configuration from Chan and Chui [15] was adopted and developed as a 3D model. Before modifying the model for further analysis, a validation exercise was performed by comparing the numerical simulation responses with the responses as reported by Chan and Chui [15]. The model linear response was also compared with the theoretical approach/method computed using MATLAB Toolbox.

#### **2. Non-Linear Finite Element Analysis**

An explosion event typically happens in a very short duration of less than 1 s which indicates that the loading is time-dependent and requires a dynamic approach in the structural analysis. To solve the dynamic problem, a few techniques for structural dynamic analysis for blast resistant design have been developed; namely, the conventional equivalent static load or an elastic approach, a single degree of freedom (SDOF), and non-linear FEA. Offshore steel structures or petrochemical facility structures are generally constructed as a structural module with complex configurations and highly occupied by equipment and pipelines. This complexity leads to the classification of the structure as a multiple degree of freedom (MDOF) system and requires the non-linear MDOF analysis to provide the most comprehensive approach in the structural response computation.

The basic concepts and problem-solving techniques for MDOF systems can be found in many references [16–19]. A simple structure consisting of a MDOF system with the general dynamic equation of motion given in Equation (1).

$$\mathcal{M}\ddot{u}(t) + \mathcal{C}\dot{u}(t) + F\_{int}(t) = F\_{ext}(t) \tag{1}$$

*M* is the mass matrix, *C*is the damping matrix, *u*, *ú, ü* are the displacement, velocity, and acceleration vectors, respectively, *Fint* is the internal forces vector, and *Fext* is the external forces vector. For dynamic events like the explosion case with a very high loading rate within short duration, the calculation requires small time increments to obtain a high-resolution for accurate solutions. Additionally, this technique is stable for small-time increment steps and requires relatively small computational cost per increment [20]. Therefore, the explicit method is efficient and suitable to be used in transient dynamic cases such as blasts, explosions, and impacts [21].

#### *2.1. Finite Element Modeling*

#### 2.1.1. Geometry

Pipe rack framings are usually symmetric with uniformly distributed loadings. The braces support any lateral loadings in the longitudinal direction and restrain lateral movements whereby relatively small deflections will be experienced in this direction. This is a reasonable assumption to simplify the model by considering only a single-bay two-storey moment frame in the analysis.

To provide sufficient space for an access walkway and escape route, a minimum clear space of 1.0m width and 2.1 m height must be included within the structure [22]. Therefore, a column distance of 4 m that represents a typical offshore steel pipe racks framing configuration is considered sufficient. Regarding the requirement for vertical space between beams, the first author found no restrictions as long as the space for pipelines is sufficient in accordance with the piping engineers design, and the structural integrity can fulfil the design criteria. Therefore, a single-bay two-storey frame [15] was adopted and modified. The selected frame configuration is shown in Figure 3. This selected configuration facilitates the final model development after conducting the validation.

**Figure 3.** The steel frame model description.

The haunch length, a, and angle, θ, were dimensioned in accordance with the provision given in Gross et al. [13]. The haunch is usually fabricated by cutting a structural beam sections or plates with standard thickness. In this research, plates with a standard thickness of 10 mm were considered for the haunch and stiffener plates. The thickness was selected because it is the maximum standard plate thickness that is less than the beam and column web/flange thicknesses. The haunch flanges are attached to the beam and column flanges by groove welding, and the webs are then fillet welded to the beam and column flanges as described in Figure 4. The haunches are placed only on the lower side of the beams to avoid obstruction for the pipelines on the top of the beams.

**Figure 4.** The welded triangular haunch configuration.

Varying details of beam-column connections were considered in this study. In addition to the joint type based on the configuration shown in Figure 4, a connection without reinforcement and a connection with a haunch slope angle of 45◦ were also considered in the analyses, as presented in Figure 5. The latter is a typical connection that is usually used in many offshore frame structures. The FE model of the steel frame was developed using ABAQUS/Explicit. The model geometry was

modelled in accordance with the single-bay two-storey frame configuration, as depicted in Figure 3. The cross section of W8 × 48 was modelled and assigned to both the beams and columns.

**Figure 5.** Different details of joint configurations with reinforcement. (**a**) Case 1—Beam-column connection without haunch; (**b**) Case 2—Haunch\_01 Beam-column connection with haunch by Chopra [16] but no web stiffener plate on the beam and no continuity plates for haunch flange; (**c**) Case 3—Haunch\_02 Beam-column connection with haunch by Chopra [16]. Web stiffener plates and haunch continuity plates were provided on the beam and column, respectively; and (**d**) Case 4—Haunch\_03 Beam-column connection with haunch but the haunch configuration is modified by providing the slope angle of 45◦.

#### 2.1.2. Constraint and Boundary Conditions

The beams-columns, plates, and haunches were connected using the tie constraint that connects two surfaces regardless of the mesh size of each surface. The tie constraint is equivalent to a welded joint that prevents penetration, separating or sliding in the interaction between the modelled surface relative to another surface [20]. The application of this constraint in the model is illustrated in Figure 6. The column bases were considered as fixed supports which were created by restraining all the degrees of freedom of the nodal points concerned in the model boundary conditions.

**Figure 6.** Typical tie constraint at beam-column connection.

#### 2.1.3. Material Properties

For large deformation cases, the cross-sectional area undergoes a significant reduction and therefore true stress-strain indicating true material deformation must be considered. The following Equations (2) and (3) are the nominal stress-strain and true stress-strain relationships [23].

$$A = \pi r^2 \sigma\_{true} = \sigma\_{\text{eng}} \left( 1 + \varepsilon\_{\text{eng}} \right) \tag{2}$$

$$
\varepsilon\_{\text{true}} = \ln(1 + \varepsilon\_{\text{eng}}) \tag{3}
$$

where σ*true* is the true stress, σ*eng* is the engineering stress, ε*true* is the true strain and ε*eng* is the engineering strain. It is recommended that the material properties for nonlinear analysis is based on the actual test results. However, in many cases this data is unavailable. In the absence of actual tensile test results, Det Norske Veritas (DNV) [23] established idealized material curves according to the European Standards. The elasto-plastic material properties for steel grade S235 with isotropic hardening was selected for the non-linear properties of steel [23]. ABAQUS defines the rate-dependent behaviour in term of plastic strain rate . ε *pl* expressed as:

$$
\overline{\dot{\varepsilon}}^{pl} = D(R-1)^q \tag{4}
$$

where *R* is the ratio of the dynamic yield stress to the static yield stress, *D* and *q* are material constants. To match the Cowper-Symonds constitutive equation, the material constants for structural steel in Equation (4) were specified as *D* = 40.4 s−<sup>1</sup> and *q* = 5 [24] which were included in the power law rate dependence definition input.

Typically, Gurson's porous model is considered to model material with relative density greater than 0.9. Amadio et al. [25] considered Gurson's porous model in the material attributes to overcome the limitation of the Von Mises constitutive law. However, Gurson's damage model on material plasticity is not considered in this study. The study is more concentrated on the structural performance based on global response without investigation of the material constitutional law, sensitivity of the local effect in the beam-column connections, and detail failure mechanism.

#### 2.1.4. Element Type and Mesh

The required numerical results obtain from the FE analysis are depending on the selection of element type for the FE model. The accuracy of FE results is relatively influenced by the type of elements defined onto the FE model [26]. Solid element is significantly useful to obtain numerical stress components of the FE model. However, in this study, only the displacement component of the FE model is required to estimate the dynamic behaviour of the structure considered under blast load. Both solid and shell elements can significantly produce the same results of the displacement component of the element under static and even extreme load such as blast load [26,27]. All structural components were modelled using deformable four-node doubly curved with reduced integration S4R shell element. The S4R element is suitable for large-rotation problems because it includes finite membrane strains and arbitrarily large rotations [20]. A fine mesh size of 25 mm was used in the connection region and a coarse mesh size of 100 mm for the regions other than the connection area were selected as shown in Figure 7.

**Figure 7.** Finer mesh size at used in the connection region.

#### 2.1.5. Loadings

All loadings associated with in-place actions that contribute to the dynamic masses must be included in the dynamic analysis [28]. The dead loads not contributing to the overall structural stiffness were incorporated as inertia masses input because these masses, including the weight of all pipes, contribute to the inertia mass in the dynamic behaviour of the overall structure. The inertia mass of pipes was determined by assuming all pipe size as 400 mm diameter × 12.7 mm thickness with a span length of 5 m. The weight of each pipe was 1.2 tonnes. At the same time, the miscellaneous dead loads such as smaller pipes, cable tray, electrical lines, grating, and handrails were included in the model. A total inertia mass of 10.2 tonne (100 kN) at each pipe location were assumed and included in the model. The total mass based on the assumed inertia masses gives a conservative scenario, besides maintaining the same total mass as in the original steel frame presented by Chan and Chui [15]. The weight of the pipes on the rack was modelled by applying mass inertia at four prescribed locations as shown in Figure 8.

**Figure 8.** Pipe inertia masses and locations.

During the gas explosion process, the atmospheric pressure increases dramatically to a maximum pressure with the propagation of the blast wave, and then it slowly decreases to a negative value with respect to the standard atmospheric pressure. The negative phase pressure occurs because the shockwave forces the air to move as it spreads outward from the explosion centre and creates a lack of air behind, causing a partial vacuum or negative pressure phase. This negative phase can be ignored in structural design [29]. However, the negative phase of the blast pressure is important in order to accurately predict the responses of blast loading on structures [30], and it is unconservative if the negative phase is ignored [31]. The original pressure-time history output of blast simulation is not practicable to be used in structural dynamic analysis and needs to be idealized [32]. Mohamed et al. reference [30] summarised typical nominal blast overpressures for offshore structure according to industrial standard guidelines. The durations of these loads were reported to be between 50 and 200 msec and considered to be close to typical offshore structure natural periods (between 300 to 1100 msec). Yasseri et al. [33] also proposed an overpressure load of 0.9 bar and 2.25 bar as the lower and upper level events for hydrocarbon explosions, respectively.

During an explosion event, the following three fundamental consequences can occur; namely, the blast overpressures, dynamic pressures (drag loads) and projectiles, missiles, and shrapnel. Among these consequences, the explosion overpressures are generally considered to be the most critical measurement. In the absence of project-specific data, it has been suggested in a Chevron Engineering Standard that a load equal to one-third of the positive phase load can be considered for the negative phase pressure. The negative pressures are usually within the range 10–30% of the maximum pressure [34].

The explosion loads on open frame structures, structural components, equipment items and pipework are usually caused by dynamic pressure loads [35] containing drag loads, inertia loads, and a pressure difference load. The first two loadings are similar to the fluid force terms in the Morrison equations (refer Equation (5)) [36].

$$F\_D = \frac{\pi}{4} \mathcal{C}\_m \rho D^2 \frac{\partial \mathcal{U}}{\partial t} + \frac{1}{2} \rho v\_{\text{gas}}^2 \mathcal{C}\_D A \tag{5}$$

*FD* is the drag force, *Cm* and *CD* are the inertial and drag coefficients. For small objects and typical gas velocities, the contribution of the inertia load is less than 1% of the force hence it may be neglected [37]. Therefore, the Morrison's equation can be reduced and the magnitude of the drag forces on the steelwork elements can be calculated using Equation (6).

$$F\_D = \frac{1}{2} \rho v\_{\text{gas}}^2 \mathcal{C}\_D A \tag{6}$$

where *CD* is the drag coefficient of the object which is dependent on the shape of the structure (projected area), ρ is density of gas (kg/m3), *vgas* is velocity of the unburned gas mixture (m/s) and *A* is structures projected area (m2). Since the prediction of gas density and velocity in an explosion event are very difficult, Mohamed et al. Reference [30] reported that the dynamic pressure (drag load) calculation in Equation (6) can be simplified to the following empirical expression given by Equation (6).

$$q\_D(t) = \mathbb{C}\_D \times p(t) \times OD \tag{7}$$

where *qD*(*t*) is the line load on a pipe function with respect to time, *p*(*t*) is maximum overpressure time-history and *OD* is outer pipe diameter. According to Equation (7), the dynamic pressure can be calculated by using the maximum overpressure values. The maximum dynamic pressure *Pmax* of 2.5 bar was selected based on Mohamed et al. [30] whereas the peak negative pressure *Pmin* of −0.83 bar was taken as one-third of the maximum dynamic pressure as suggested by in the Chevron Engineering Standard and Hansen et al. [34]. The pressure duration of 0.136 s for pressures *Pmax* of 2.5 bar and the duration of 0.24 s for *Pmin* of −0.83 bar were determined using the curve in Figure 9.

**Figure 9.** Pressure durations relationship based on API RP 2FB [24].

The dynamic pressure wave profile can be developed using the pressures and durations information. The idealized triangular waveform shown in Figure 10 was considered to describe the dynamic pressure history that was applied in the FE models.

**Figure 10.** Dynamic pressure history.

In the analyses, a uniform distributed pressure of 2.5 bar was applied on the column flanges. To simplify the problem, the drag loads on the unmodeled pipes and non-structural items on the frame were ignored. Since the blast direction and the beam longitudinal direction are parallel, the beams are to be minimally affected by the lateral pressures, and thus no blast loads were applied to the beams.

In a blast event, the structures might buckle and undergo large geometry changes, hence the necessity to also consider geometric non-linearity [20]. In the initial step, initial conditions, boundary conditions, and predefined fields that are applicable with the analysis can be specified. In addition to the initial step, three steps were specified to simulate the blast analysis, namely; self-weight (quasi-static analysis with structural weight and dead load), blast (dynamic analysis using dynamic pressure history with self-weight propagation) and post-blast (only dead load is set active, no propagation of blast load from previous analysis step). In each analysis step, the incrementation settings were set to automatic incrementation so that ABAQUS automatically adjusts the time increment size depending on the numerical stability limit [20].

Before applying dynamic pressure, the initial condition of the structure was in a static condition under self-weight consisting of structural dead load and piping masses. Unfortunately, ABAQUS cannot directly combine static analysis with dynamic explicit analysis in the same model. When dynamic explicit analysis is being used, all the following analysis in the next steps must also be in the dynamic explicit mode. Therefore, the self-weight member forces also need to be analysed using dynamic explicit analysis. When the self-weight is applied instantaneously in the dynamic analysis, oscillatory vertical reaction force occur as shown in Figure 11. This oscillation does not represent the correct static condition in which the structure should be before the blast is initiated. This oscillation will create incorrect stress variations during and after the actual blast dynamic analysis. To overcome this problem, the ramp loading was implemented in the self-weight dynamic analysis whereby the load was applied slowly to avoid the dynamic effect. The basis of this method is related to the concept of the dynamic case of "step force with finite rise time" [16–18] in which if the rising time of the self-weight dynamic loading is relatively long and greater than three times the structural natural period, the dynamic response is quasi-static or like a static force. Therefore, a rise time of 2.15 s (=3 × 0.717 sec) was used in the self-weight dynamic step as shown in Figure 12. The result of ramp loading implementation in the self-weight step is presented in Figure 13 whereby no oscillation has occurred in the self-weight dynamic step.

**Figure 11.** Vertical reaction force for instantaneously applied self-weight.

**Figure 13.** Vertical reaction force after ramp loading implementation in the self-weight step.

In the "blast" analysis step, the dynamic pressure was introduced. The duration of the blast analysis step was specified in line with the total duration of the blast amplitude definition as 0.376 sec. During this analysis step, the self-weight propagating from the previous self-weight analysis step was included in the analysis. After the blast analysis step, the last analysis step, called the post blast, was created with the purpose of studying the post-blast response of the structure. No load was applied in this step except the self-weight propagation from the previous step. Since the transient dynamic response is the primary interest of the study, a step duration of 2 s was considered in the post-blast analysis. In the explosion dynamic analysis, the maximum dynamic response usually occurred within

the transient response duration. Therefore, the duration of 2 s is considered adequate to describe the steady state response of the structure. Moreover, a longer duration needs to be avoided to minimize the total computation duration.

#### *2.2. Structural Assessment*

#### 2.2.1. Ductility Ratio of the Steel Frame

The level of structural performance can be measured by quantifying its stresses, strains, displacements, ductility, functionality, and similar parameters. Like seismic design, blast resistant structural design also considers inelastic response under dynamic load. In terms of the structural response, the impulsive blast load yields higher response frequency than the response due to seismic load [38] and the inter-story drifts due to the blast loading on a building are generally higher than the drifts due to earthquakes [39]. In the case of blast loading, the design is based on the member's plastic behaviour and the structural component is governed by its ductility and rotation [40]. The ductility is defined as the structural capability of plastic deformation without fracturing. The maximum plastic displacement of a structure can be measured by the ductility ratio (μ) relative to first yield [41] as given in Equation (8).

$$
\mu = \frac{\delta\_u}{\delta\_y} \tag{8}
$$

where δ*<sup>u</sup>* is the maximum plastic displacement, and δ*<sup>y</sup>* is the displacement at first yield. The displacement at first yield is not obvious for most material. It can be determined by using graphical methods with several alternative definitions [41]. The method based on initial yielding, also called the Tangent Method, is usually implemented to approximate the yield point which is defined as the intersection between the tangential lines of elastic and plastic behaviour [42].

In design practice, ductility ratio values and rotation criteria are used for designing structure particularly topside offshore structures. The ductility ratio and the plastic strain are related, and the failure of the structure is assumed to occur when the plastic strain approaches the material failure strain [43]. The allowable range of the structural member's plastic deformations is measured based on the calculated ductility ratio, whereas the limit on the support rotation makes sure that tension membrane action, that may develop in a member, will be in a safe range where no connection failures occur.

#### 2.2.2. Evaluation Criteria

In industrial practice, various design codes and guidelines [24,31,40,44–46] are used for blast design evaluation criteria. Specifically, for a petroleum facility, API RP 2FB [24] defines two important levels of assessment namely ductility level blast (DLB) and the strength level blast (SLB). These levels are associated with the explosion loading risks by analogy with earthquake assessment. A low-probability high-consequence event representing the extreme design event may be assessed using DLB. While a higher-probability lower-consequence event, such an explosion case with low overpressure, can be assessed using SLB.

Louca et al. [6] summarised the ductility values that were used in the offshore structure design practice. By correlating the ductility ratio and damage levels, Yasseri [47] proposed the ductility ratio criteria. Several design guidelines in the industry present the response criteria [40,46].

#### **3. Results**

#### *3.1. Model Validations Results*

A single-bay two-storey steel framing configuration as presented in Figure 14 was chosen to develop the model. The selected steel frame was considered appropriate for the object in the study because a simple and small structure does not require excessive computation resources. Chan and

Chui [15] investigated the transient dynamic response of the inelastic steel frames with nonlinear connections. Geometric imperfections and residual stress with a maximum magnitude of 50% of yield stress were included in their study.

**Figure 14.** A two-storey steel frame under dynamic loading [15].

The model of the selected steel frame was developed in ABAQUS. Before starting the simulations for the research, two different analyses namely elastic and elastic-perfectly-plastic analyses were used for validating the model by comparing the dynamic responses from numerical analyses against the dynamic responses presented by Bathe [19]. To differentiate the analysis, two different material properties were created representing the elastic and elastic-perfectly plastic with a yield stress of 235 MPa. The elastic and elastoplastic analyses were carried out. To verify the dynamic characteristic of the frame using theoretical calculations, elastic dynamic analysis was also carried out using CALFEM which is a MATLAB computational toolbox for teaching FEM developed at Lund University [48]. The structural lateral displacement responses are presented in Figures 15 and 16.

**Figure 15.** Elastic analysis dynamic response ABAQUS, Chan and Chui [15] and CALFEM/MATLAB.

**Figure 16.** Elastoplastic analysis dynamic response ABAQUS and Chan and Chui [15].

Since the response based on CALFEM calculation matches the elastic response presented by Chan and Chui [15], their dynamic characteristics are considerably identical. This implies that the dynamic characteristic based on CALFEM calculations represents the similar dynamic characteristic of the frame presented by Chan and Chui [15]. According to the dynamic response in Figures 15 and 16, the maximum response from the ABAQUS analysis is 42% higher than the maximum response given by Chan and Chui [15]. In addition to this, the displacement response wave period of the ABAQUS results are shorter than the results from Chan and Chui [15].

To investigate these discrepancies, the natural frequencies that represent the dynamic characteristic of the frames were calculated using ABAQUS and CALFEM. As shown in Figure 17, the structure natural frequency of 1.4978 Hz extracted from ABAQUS is 5.4% higher than the CALFEM result (1.4214 Hz). Tedesco et al. [18] showed that the effect of mass and load distribution on the dynamic response of a structural system in which a structural system with distributed mass is stiffer than a system where the entire mass is concentrated at midspan points. Unlike beam finite element formulation, the structural masses in the ABAQUS model is distributed throughout the structure and not concentrated at points. This implies that the ABAQUS model should be stiffer than the CALFEM calculation, hence it will have a higher natural frequency.

**Figure 17.** Structural natural frequency from (**a**) ABAQUS and (**b**) CALFEM/MATLAB.

The effect of the nonlinearity of the system was also investigated. As shown in Figure 18, the Von Mises stress reaches the yield stress value. This situation is considered overstress, in which

the validated model undergoes yielding and buckling. In this case, analysis using ABAQUS does have capability to consider inelastic moment redistribution [49] and P-delta effects by activating the geometric non-linearity option in the dynamic analysis. These features allow the analysis using ABAQUS to capture the more realistic dynamic behaviour of the structure.

**Figure 18.** Yielding and buckling in the validated model.

The preceding section explained possible factors affecting the higher dynamic responses obtained from ABAQUS compared to the dynamic response presented by Chan and Chui [15].

#### *3.2. Eigenmodes and Eigenvalues*

The natural period of the structure was obtained by eigenvalue analysis. To perform this, the analysis model was copied and modified by removing the blast loading and maintaining the structural and piping masses. The dynamic analysis step was replaced by linear perturbation frequency analysis type. Ten values of the eigenvalues were requested for the analysis output. The mode shape which is parallel with the direction of blast loading was selected as the representation of the dynamic characteristics of the structure. Therefore, the structural natural period associated with this mode shape was used in the step duration of self-weight dynamic analysis. By relating the structural natural period to the dynamic loading duration, the structural dynamic response was classified into impulsive td/Tn < 0.3, dynamic 0.3 < td/Tn < 3, or quasi-static 3 < td/Tn categories [50]. Where td is the dynamic load duration and Tn is the structural natural period. The structural dynamic characteristic was identified by undertaking the eigenvalue analysis. Four mode shapes considered to to be the most important shapes are presented in Figure 19. The first mode as the highest eigenvalue is usually considered as the critical case. However, mode shapes 1 and 2 in this study were considered not to be the critical cases because the columns, bracings, and lateral beams provide lateral restrains in the longitudinal direction of the actual structure. The mode shape-3 that deflects in the same direction as the blast excitation loading was selected to represent the dynamic characteristic for the structure. Subsequently, the eigenvalue based on mode shape-3 was used in all calculations that are relevant with the structural dynamic characteristics.

By considering the dynamic load duration of td = 0.376 s and the structural natural period of 0.717 s, the ratio of the dynamic loading duration to the natural period is determined as td/Tn = 0.52. The ratio is greater than 0.3 and less than 3, hence falls into the dynamic category. Therefore, dynamic analysis was solved using numerical integration of the dynamic equations of equilibrium that is already implemented in the ABAQUS numerical computational technique.

**Figure 19.** Structural eigen modes and eigen periods. (**a**) Mode shape 1; (**b**) Mode shape 2; (**c**) Mode shape 3; and (**d**) Mode shape 4.

#### *3.3. Lateral Displacement Response and Ductility Ratio*

Structural ductility was calculated using Equation 8 and the value of structural displacement at first yield was determined using initial yield or the tangent method [41]. A new model was developed by modifying the analysis model. The static pushover analysis was performed with only lateral loading applied at the top corner of the frame considered.

The force-displacement resulting from the analysis is presented in Figure 20. By using the curve, the structural displacement at first yield was determined by estimating the yield point location which is defined as the intersection between the tangential lines of elastic and plastic behaviour, where the displacement at first yield is 71 mm. Subsequently, the structural ductility ratio was calculated as the ratio of peak displacement under blast load to the displacement at first yield.

**Figure 20.** Structural displacement at first yield approximation.

Figure 21 describes the general displacement direction of the frame swaying horizontally in the same direction as the blast excitation load. Two different points depicted in the figure that represents structural height and beam elevations were selected as the node locations to extract the global lateral displacement responses. The lateral displacement at the top right corner (Point A) was considered as a good measurement point for the frame system deformation, whereby the maximum lateral displacements at Point B were used in structural ductility ratio calculation.

**Figure 21.** Selected points for structural displacement location.

The sway histories for these two observation points are presented in Figure 22a,b for all frame analysis with different beam-column connection. The maximum sway responses at lower and upper beam elevations, points A and B, were observed during the blast duration. The maximum responses of the frame without reinforcement are higher than the frames that are reinforced with haunches.

**Figure 22.** (**a**) Lateral displacement response history at top right corner of the frame; (**b**) Lateral displacement response history at right end of the lower beam.

To measure the frames dynamic performance based on the sway responses, the extracted maximum sway displacements at the top elevation for each frame during blast and post blast are presented in Table 1 and Figure 23. Subsequently these maximum sway responses were used to calculate the ductility ratios as tabulated in Table 1. The maximum lateral displacement of 94.6 mm occurred in the frame without haunch reinforcement. Since the ductility ratio is proportional with the maximum displacement, the maximum ductility ratio of 1.33 also occurred in the same frame without haunch. The calculated results as presented in Table 1 show that all maximum sway responses and maximum ductility ratios are less than the maximum criteria recommended [40,46]. The allowable deflection and ductility ratio of frame structures are 1.5 and 240 mm (Height/25), respectively [40,46].


<sup>1</sup> From Equation (8).

**Table 1.** Summary of maximum displacement at all beam elevations for connection types of all cases considered in this study.

**Figure 23.** Maximum lateral displacement response.

To evaluate the relative differences between frames performance using ductility criteria, the ductility ratios are presented in Figure 24. The trends show that the frames with haunches performed better compared to the frame without haunches. Qualitatively, the Haunch\_03 impact on the frame sway response is the highest among all the haunch types. The Haunch\_03 connection reinforcement reduced the ductility ratio by 12% against the frame without haunches, while in the case of the Haunch\_02 the reduction is 5% and for the Haunch\_01 is 4%. The results demonstrated that the Haunch\_03 provides a enhanced dynamic performance. It was also observed that the plate stiffeners on beam-column joints (Haunch\_02) provided slightly better ductility ratio than the case of the haunch without the plate stiffeners (Haunch\_01). In general, the biggest haunch configuration provided the best frame dynamic performance. This finding is relevant to the beam-to-column connections flexibility whereby connection strengthening increases the stiffness of the frame.

**Figure 24.** Frames ductility.

#### **4. Discussion**

The results of this research are limited and cannot be extrapolated to cover the structural dynamic performance of steel frames with connection haunch reinforcement because this research considered only four cases of frames with different joint haunch configurations. To further understand the dynamic performance aspects of typical steel frame subjected to blast loading, the following points are recommended to be undertaken:


#### **5. Conclusions**

The structural eigenvalue was extracted and compared to the dynamic loading duration in order to characterize the dynamic response. The obtained ratio of 0.52 that is greater than 0.3 and less than 3 has classified the dynamic response under the influence of dynamic category. Therefore, the dynamic analysis was solved using numerical integration of dynamic equations of equilibrium that is already implemented in ABAQUS. The structural maximum ductility ratio achieved using haunches was 1.33 and less than the allowable criteria of 1.5. Generally, the structural ductility ratios decreased due to the presence of the haunch reinforcement. The ductility ratio of the frame with Haunch\_03 reinforcement was reduced by 12% compared with the frame without haunches, while in the case of the Haunch\_02 the reduction was 5% and for Haunch\_01 was 4%. According to the evaluation results, the haunch reinforcements have strengthened the selected steel frame and improved dynamic performance compared to the frame with unreinforced connections under blast loading. The Haunch\_03 as the biggest reinforcement configuration performed better compared to other connection configurations.

**Author Contributions:** Conceptualization, supervision, project administration, visualization, M.M.Y; methodology, software, validation, resources, data curation, writing—original draft preparation, M.Y. and J.S; formal analysis, investigation, J.H.S.; supervision, project administration M.M.Y.; writing—review and editing, M.M.Y., M.K.K., P.-S.C. and G.A.R.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the Universiti Sains Malaysia bridging grant (304/PAWAM/6316571).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

1. Nolan, D.P. *Handbook of Fire and Explosion Protection Engineering Principles: In For Oil, Gas, Chemical and Related Facilities*, 3rd ed.; Elsevier: Amsterdam, The Netherlands, 2014.


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