**Buildings and Structures under Extreme Loads**

Editors

**Chiara Bedon Flavio Stochino Daniel Honfi**

MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin

*Editors* Chiara Bedon University of Trieste Italy

Flavio Stochino University of Cagliari Italy

Daniel Honfi RISE Research Institutes of Sweden Sweden

*Editorial Office* MDPI St. Alban-Anlage 66 4052 Basel, Switzerland

This is a reprint of articles from the Special Issue published online in the open access journal *Applied Sciences* (ISSN 2076-3417) (available at: https://www.mdpi.com/journal/applsci/special issues/Buildings Structures Extreme Loads).

For citation purposes, cite each article independently as indicated on the article page online and as indicated below:

LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. *Journal Name* **Year**, *Article Number*, Page Range.

**ISBN 978-3-03943-569-2 (Hbk) ISBN 978-3-03943-570-8 (PDF)**

Cover image courtesy of Joey Banks.

c 2020 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications.

The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND.

### **Contents**


### **About the Editors**

**Chiara Bedon** (1983), Assistant Professor. M.Sc. in Civil Engineering and Ph.D. in Structural Engineering (University of Trieste, Italy). Her research activity includes the analysis of structural materials and systems under extreme design loads, with a focus on buckling-related phenomena, blast, fire, earthquakes. She has been involved in numerous European projects and networks (JRC-ERNCIP, NATO-SPS, COST, etc.) since 2009.

**Flavio Stochino** (1985), Assistant Professor. M.Sc. in Civil Engineering and Ph.D. in Structural Engineering (University of Cagliari, Italy). His scientific research deals with extreme loads on RC structures with a special focus on blast/impulsive loading and fire design of structures. He has extensive experience regarding reliability and seismic analysis of existing structures, isogeometrical analysis, hybrid/mixed finite elements, laminate mechanics, structural optimization, and structure sustainability.

**Daniel Honfi** (1977), Senior Researcher. M.Sc. in Civil Engineering (Budapest University of Technology and Economics), Ph.D. in Structural Engineering (Lund University). His research topics include structural serviceability and robustness, resilience of infrastructure systems, engineering decision making, and assessment of existing structures.

### *Editorial* **Special Issue on "Buildings and Structures under Extreme Loads"**

#### **Chiara Bedon 1,\*, Flavio Stochino <sup>2</sup> and Daniel Honfi <sup>3</sup>**


Received: 17 July 2020; Accepted: 5 August 2020; Published: 15 August 2020

#### **1. Introduction**

Exceptional loads on buildings and structures may have different causes, including high-strain dynamic effects due to natural hazards, man-made attacks, and accidents, as well as extreme operational conditions (severe temperature variations, humidity, etc.). All these aspects can be critical for specific structural typologies and/or materials that are particularly sensitive to unfavorable external conditions. In this regard, dedicated and refined methods are required for their design, analysis, and maintenance under the expected lifetime. However, major challenges are usually related to the structural typology and materials object of study, with respect to the key features of the imposed design loads. Further issues can be derived from the need for the mitigation of adverse effects or retrofit of existing structures, as well as from the optimal and safe design of innovative materials/systems. Finally, in some cases, no appropriate design recommendations are currently available in support of practitioners, and thus experimental investigations (both on-site or on laboratory prototypes) can have a key role within the overall structural design and assessment process. This Special Issue presents 19 original research studies and two review papers dealing with the structural performance of buildings and structures under exceptional loads, and can represent a useful answer to the above-mentioned problems.

#### **2. Contents**

A first set of papers reports on earthquake structural design of structures and buildings [1–5]. Various kinds of structures have been considered under the effects of seismic loads, including steel frames [1], liquid storage tanks [2], and an experimental prototype of atrium-style underground metro station [3], but also existing masonry structures [4] or new timber buildings [5], presenting a perspective review on their seismic design. Among others, a extreme natural event is certainly represented by windstorms. In this Special Issue, wind load modelling and design is mainly addressed by [6–8], while [9] describes the results of a visual test carried out on a suction caisson that support offshore wind turbines. Finally, the last natural hazard analyzed in the Special Issue is snowdrift. Actually, the effects of snowdrift and snow loads in cold regions have been investigated by [10] and [11], with the proposal of a novel calculation approach and a case-study application, respectively.

The knowledge of material properties and characteristics, as known, represents the first influencing parameter for the load-bearing performance assessment of a given structure. In this regard, the knowledge on the topic has been improved by two interesting research contributions focused on composite concrete-steel shear walls [12] and structural glass members [13], respectively, with the support of laboratory/on-site experiments and numerical analyses.

Another interesting group of papers dealing with soil properties and structures–soil interaction phenomena further extends the research fields covered in this Special Issue. In particular, [14] deals with the determination of Young modulus in bored piles, while [15] presents an investigation on the horizontal axis deviation of a small radius Tunnel Boring Machine (TBM). In this context, it is important to also mention the study in [16], and reporting on the friction resistance for slurry pipe jacking. Finally, an interesting analysis on the effects of derailment and post-derailment of trains is presented in [17], with the support of full-scale testing.

In conclusion, it is known that both man-made attacks and accidents can yield to explosions and fire loads that could push the constructional materials, and thus the structures, to their capacity limits. Blast loads analyses, in this regard, are reported in [18–20], while fire effects on a tunnel structure are analyzed in [21].

**Acknowledgments:** This Special Issue would not be possible without the contributions of various talented authors, hardworking and professional reviewers, and dedicated editorial team members of the *Applied Sciences* journal. We would like to take this opportunity to record our sincere gratefulness to all the involved scientists, both authors and reviewers, for their valuable contribution to this collection. Finally, we place on record our gratitude to the editorial team of *Applied Sciences*, and special thanks to Felicia Zhang, Assistant Managing Editor for *Applied Sciences*.

**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* **Seismic Design of Steel Moment-Resisting Frames with Damping Systems in Accordance with KBC 2016**

#### **Seong-Ha JEON 1, Ji-Hun PARK 2,\* and Tae-Woong HA <sup>1</sup>**


Received: 26 April 2019; Accepted: 31 May 2019; Published: 5 June 2019

**Abstract:** An efficient design procedure for building structures with damping systems is proposed using nonlinear response history analysis permitted in the revised Korean building code, KBC 2016. The goal of the proposed procedure is to design structures with damping systems complying with design requirements of KBC 2016 that do not specify a detailed design method. The proposed design procedure utilizes response reduction factor obtained by a limited number of nonlinear response history analyses of the seismic-force-resisting system with incremental damping ratio substituting damping devices. Design parameters of damping device are determined taking into account structural period change due to stiffness added by damping devices. Two design examples for three-story and six-story steel moment frames with metallic yielding dampers and viscoelastic dampers, respectively, shows that the proposed design procedure can produce design results complying with KBC 2016 without time-consuming iterative computation, predict seismic response accurately, and save structural material effectively.

**Keywords:** damping device; seismic design; design base shear; nonlinear response history analysis

#### **1. Introduction**

There were no seismic design provisions for the application of damping systems in the Korean Building Code (KBC) 2009, therefore, Korean engineers encountered many difficulties in the practical application of damping devices [1]. The KBC was revised in 2016 with the addition of design criteria for structures with damping systems [2]. The design provisions for structures with damping systems in KBC 2016 adopted only nonlinear response history procedure. In the case of ASCE 7, both equivalent lateral force procedure and response spectrum procedure are allowed. However, the use of those two procedures is restricted for strict conditions and nonlinear response history procedure is adopted major design procedure in ASCE 7-16 [3].

In spite of being adopted as a major design procedure, nonlinear response history analysis procedure requires much more computational efforts compared to linear analysis. It is difficult to design damping devices by trial and error using nonlinear response history analysis. Therefore, many design procedures adopt equivalent linearization technique to take into account the nonlinear characteristics of either damping device or building structure [4–10]. Some design procedures adopt nonlinear static analysis to take into account the inelastic behavior of the structure more directly [11,12], which utilizes the equivalent linearization technique to determine the performance point. Many equivalent linearization techniques for different nonlinear damping devices have been proposed [13–15]. Most of them utilize damping correction factors, which represent response reduction for a specified amount of damping ratio developed for linear or nonlinear systems [13,14,16,17]. However, those design procedures based on equivalent linearization technique have limitations in that they assume deformed shape based on elastic analysis and are difficult to identify localized

nonlinear behavior of a structure, such as weak story mechanism and force or deformation demands on structural components and damping devices with sufficient accuracy for detailed design. Besides, proposed optimization procedures have been developed. Some of those procedures adopt stochastic analysis [18,19] or linear response history analysis [20]. The others make use of nonlinear response history for evaluation of objective function or boundary conditions [21–23]. However, the latter methodologies repeat nonlinear response history analysis for every iterative step during optimization and, as a result, are computationally demanding in the case of actual building structures with many degrees of freedom

This study is to propose and validate an efficient and systematic design procedure for inelastic multi-degree-of-freedom (MDOF) structures with damping systems complying with KBC 2016, which requires nonlinear response history analysis but does not provide methodology for detailed design. Two design examples for steel moment frames using nonlinear response history analysis are presented. The proposed design procedure makes use of nonlinear response history analysis of an MDOF structure directly to determine design parameters of damping devices and captures complex inelastic behavior of building structures more realistically. Differently from the existing optimization procedure, the design procedure performs only a small number of nonlinear response history analyses, which is usually three to five times. As a result, the proposed design procedure requires only a small amount of computational efforts compared to the existing optimization-based design procedures that perform nonlinear response history analysis repeatedly until convergence. The proposed design procedure can be implemented using commercial structural analysis software and can be applied to design code except for KBC 2016.

#### **2. KBC2016 Seismic Design Provisions for Structures with Damping Systems [2]**

#### *2.1. Damping Systems*

The damping system is intended to reduce the seismic demand to a structure and refers to a subsystem that includes both damping devices and structural elements that transmit forces from the damping devices to seismic-force-resisting systems or foundations of the structure. The damping device is a structural element that dissipates energy by relative motion between two ends of the device and includes all the elements such as pins, bolts, gusset plates, braces, etc. necessary to install the damping device. The damping device may be installed in a separate structure out of the seismic-force-resisting system or in the seismic-force-resisting system. Figure 1 shows examples of configurations of damping devices and damping systems connected to a seismic-force-resisting system.

**Figure 1.** Damping system (DS) and seismic-force-resisting system (SFRS) configurations.

A damping device is classified into a velocity-dependent damping device whose force response depends on the relative velocity between the two ends of the device and a displacement-dependent damping device of which force response is determined by the relative displacement between the two ends of the devices. A mathematical model of the velocity-dependent damping system shall include the velocity coefficient corresponding to the test data. Displacement-response characteristics of the displacement-dependent damping device shall be modeled considering the dependence of seismic force response on the frequency, amplitude, and duration of ground motion clearly.

The components constituting the damping system shall be designed so that the damping device works normally without interruption. Thus, structural elements in the damping system are designed to remain elastic when subjected to design earthquake including forces transmitted from the damping device. The forces from the damping devices shall not be calibrated by the intensity reduction factor or the response correction factor. Moreover, the damping device shall be designed so as not to break when subjected to the maximum considered earthquake.

#### *2.2. Seismic-Force-Resisting System*

A building structure to which damping systems are applied shall have a seismic-force-resisting system defined in the KBC in each direction. Table 1 shows design factors of steel moment-resisting frame systems, which are appropriate to install damping devices due to relatively low stiffness and used in design examples of this study. At the initial stage, the seismic-force-resisting system of a building structure with damping systems are designed in order to resist the minimum base shear *Vmin* independently. *Vmin* is calculated by Equations (1) and (2).

$$V\_{\min} = \eta V \tag{1}$$

$$
\eta \ge 0.75\tag{2}
$$

where *V* is the design base shear calculated by equivalent lateral force procedure and η is the expected damping correction factor representing the degree of seismic force reduction acting on the structure obtained by damping systems. The expected damping correction factor η shall be validated through nonlinear response history analysis of the seismic-force-resisting systems combined with damping systems as described in the next section.


**Table 1.** Design factors for steel moment-resisting frame systems.

#### *2.3. Damping Performance*

The expected damping correction factor η applied to the minimum base shear *Vmin* of the seismic-force-resisting system shall be higher than or equal to the actual damping correction factor η*<sup>h</sup>* shown in Equation (3).

$$
\eta\_h = \frac{V\_h}{V\_{hc}}\tag{3}
$$

where *Vh* and *Vhe* are the base shear calculated from the analysis of the structure with damping systems and from the analysis of the structure in which velocity-dependent components of damping devices are removed and displacement-dependent components of the damping devices are substituted into the effective stiffness, respectively.

The effective stiffness for the displacement-dependent component of the damping device is calculated based on the peak displacement and corresponding force of the damping device obtained from the analysis to calculate *Vh* as follows.

$$k\_{eff} = \frac{\left|F^+\right| + \left|F^-\right|}{|\Delta^+| + |\Delta^-|}\tag{4}$$

where Δ<sup>+</sup> and Δ<sup>−</sup> is the peak displacement of each damping device in positive and negative directions, respectively, and *F*<sup>+</sup> and *F*<sup>−</sup> are corresponding forces, respectively.

#### **3. Damping System Design Procedure**

An efficient and systematic seismic design procedure for damping systems using nonlinear response history analysis is proposed in this section. The goal of the proposed procedure is to design structures with damping systems complying with the design requirements of KBC 2016 that does not specify a detailed design method.

#### *3.1. Elastic Design of Seismic-Force-Resisting Systems*

Seismic-force-resisting systems are designed to meet strength requirement for the minimum base shear with a target value of η. Then, the story drifts are checked using allowable story drifts corresponding to the seismic risk category. If some story drifts exceed the allowable story drifts, it is necessary to use damping systems in order to reduce the demand for those story drift. The allowable story drifts are 1.0%, 1.5%, and 2.0% for seismic risk category *S*, I, and II, respectively.

#### *3.2. Target Damping Ratio Calculation*

In order to estimate the target damping ratio, nonlinear response history analysis of the seismic-force-resisting system designed in Section 3.1 is performed repeatedly with a damping ratio raised at a constant increment. The peak story drift and peak base shear of the seismic-force-resisting systems are computed for each analysis until those peak responses reach their target values, respectively. Both peak story drift and peak base shear are determined as average responses obtained from nonlinear response history analysis using seven or more ground motion records. The target base shear *V*\* is determined as the minimum base shear corrected by the demand-capacity ratio (DCR), which is determined in the elastic design of the seismic-force-resisting system and represent redundancy in the design.

$$V^\* = \frac{V\_{\text{min}}}{DCR} \tag{5}$$

The damping ratio of the seismic-force-resisting system is composed of inherent damping and damping added by damping systems. In this study, the damping ratio of the structure is assumed to be 5%. The target damping ratio β*<sup>t</sup>* is calculated by Equation (6).

$$
\beta\_t = \beta\_{\rm IH} + \beta\_v \tag{6}
$$

where, β*IH* is the inherent damping ratio and β*<sup>v</sup>* is the damping ratio added by damping system. In the nonlinear response history analysis, β*<sup>v</sup>* is increased at a constant increment, and the increment need not be too small because the response can be interpolated numerically. Thus, three to five times of nonlinear response analysis for a given ground motion set is adequate.

Response reduction ratio with respect to effective damping ratio can be plotted in order to determine the target damping ratio and design of damping devices. An illustrative example of such plots is represented in Figure 2, where response reduction ratios for both base shear and maximum story drift are plotted in broken lines.

**Figure 2.** Design process of damping device.

#### *3.3. Damping Device Design*

Damping device design is to determine design parameters that can provide damping parameters. Both displacement-dependent damping device and velocity-dependent damping device are addressed for illustrative design examples in this study. Metallic yielding dampers, friction dampers and many other types of dampers belong to displacement-dependent damping devices. The TADAS (triangular-plate added damping and added stiffness) system is adopted as an example of displacement-dependent damping devices in this study. Velocity-dependent damping devices include viscous dampers and viscoelastic dampers of which the latter provides a displacement-dependent force component. The viscoelastic damper is adopted as an example of velocity-dependent damping devices in this study.

#### 3.3.1. Effective Damping of Displacement-Dependent Damping Devices

The force-displacement relationship of the TADAS system is defined as a bi-linear model with a post-yield stiffness ratio of 0.02. The strength and stiffness characteristics of the TADAS system including deformation capacity are calculated by Equations (7) to (9) using metal plate dimensions and material strengths of which details can be found in Ramirez et al. [24].

$$
\delta\_y = \frac{3}{2} \left( \frac{\varepsilon\_y h^2}{t} \right) \tag{7}
$$

$$V\_y = \frac{F\_y b t^2}{4h} \tag{8}$$

$$
\delta\_{\text{max}} = \frac{\varepsilon\_{\text{max}} h^2}{t} \tag{9}
$$

where δ*y*, *Vy* and δ*max* are yield deformation, yield strength and deformation capacity, respectively, and *b*, *h* and t are width, height and thickness of the triangular metal plate, ε*<sup>y</sup>* is the yield strain of the material, ε*max* is the strain limit of the damping device and can be calculated by the following equation [25–28].

$$
\varepsilon\_{\max} = A N\_f^{-B} \tag{10}
$$

where *Nf* is the number of deformation cycles and assumed to be 100, *A* and *B* are constant and assumed to be 0.08 and 0.3, respectively [24]. The peak deformation demand on the damping device is calculated as follows.

$$
\delta\_u = h\_i \theta\_i \tag{11}
$$

where θ*<sup>i</sup>* and *hi* are the peak story drift angle (rad) and the story height of the i-th story.

For given peak deformation demands, the effective damping ratio can be calculated by Equations (12) in accordance with ASCE 7-10 based on the energy dissipation due to cyclic deformations illustrated in Figure 3 [29].

$$
\beta\_v = \frac{\sum E\_{dj}}{4\pi E\_s} \tag{12}
$$

where *Es* is the strain energy stored in the structure *Edj* is the energy dissipated by the j-th damping device. *Es* and *Edj* are calculated by the following equations.

$$E\_s = \frac{1}{2} \sum F\_i \delta\_i \tag{13}$$

$$E\_{dj} = 4(F\_{yj}\delta\_{uj} - F\_{uj}\delta\_{yj})\tag{14}$$

where *Fi* and δ*<sup>i</sup>* are the peak lateral force and peak lateral displacement at the i-th story, respectively, δ*yj*, δ*uj*, *Fyj* and *Fuj* are the yield displacement, peak displacement, yield strength and peak force of the i-th damping device.

**Figure 3.** Energy dissipation of metallic damping device.

#### 3.3.2. Effective Damping of Velocity-Dependent Damping Devices

The characteristics of the viscoelastic damping device adopted in this study are based on the experimental results of Soong and Dargush (1997) represented in Table 2 [30]. Kelvin model illustrated in Figure 4 is adopted for numerical modeling of the viscoelastic damping device. Effective stiffness and damping coefficients of the model are given by the following equations

$$K\_d = \frac{AG'}{t\_d} \tag{15}$$

$$\mathbf{C}\_d = \frac{\eta\_d}{\alpha \nu} \mathbf{K}\_d \tag{16}$$

where *G* and η*<sup>d</sup>* are storage modulus and loss factor of the viscoelastic material, respectively, A and *td* are the shear area and thickness of the viscoelastic damper, respectively, and ω is the excitation frequency, which is taken as the fundamental frequency of the structure with effective stiffness of damping devices. Added damping ratio β*<sup>v</sup>* can be calculated by the following equation on the basis of the modal strain energy method [31].

$$\beta\_v = \frac{\eta\_d}{2} \left( 1 - \frac{\omega\_n^2}{\overline{\omega}\_n \overline{\omega}\_n^2} \right) \tag{17}$$

where ω*<sup>n</sup>* and ω*<sup>n</sup>* are the natural frequency before and after installation of damping systems, respectively. Although the modal strain energy method is applicable to linear elastic structures, it is assumed that nonlinear damping devices is linearized using effective stiffness. Thus, ω*<sup>n</sup>* is calculated by eigenvalue analysis of the structure with the effective stiffness of damping systems added. The thickness of the viscoelastic damper *tVED* can be calculated as follows.

$$t\_{VED} = \frac{\Delta\_{\rm u}}{\gamma\_{\rm max}}\tag{18}$$

where Δ*<sup>u</sup>* is the maximum shear deformation of the damping device and γ*max* is the maximum strain capacity


**Table 2.** Properties of the viscoelastic damping device [20].

**Figure 4.** Kelvin model of viscoelastic damping devices.

#### 3.3.3. Design of Damping Devices Considering Change of the Natural Frequency

Once damping device properties are determined, it is necessary to update the effective damping ratio β*<sup>v</sup>* because the effective damping ratio β*<sup>v</sup>* given by Equations (12) or (17) is dependent on the stiffness and/or strength of the damping device. A rational method to update the effective damping ratio considering response reduction and stiffening effects of damping devices is described in this section and Figure 2 illustrates the procedure for updating the effective damping ratio conceptually.

(Step 1) Base shear and maximum story drift reduction factors are plotted with respect to β*t*, broken lines in Figure 2.

(Step 2) The target damping ratio β*t*<sup>1</sup> corresponding to the initial base shear reduction factor (*V*/*Vo*)1 is interpolated from the plot 'Initial *V*/*Vo*' (thick broken line) in Figure 2.

(Step 3) The maximum story drift reduction factor (*D*/*Do*)1 is interpolated from the plot 'Initial *D*/*Do*' (thin broken line) in Figure 2.

(Step 4) Damping devices are designed to achieve the target damping ratio Equation (12) or (17) subjected to deformations corresponding to (*D*/*Do*)1. Those deformations can be approximated by multiplying (*D*/*Do*)1 to the initial response for only β*IH*.

(Step 5) Effective stiffness of damping devices is calculated and added to the structure without damping devices. Then, the fundamental frequencies of the structure are updated.

(Step 6) Response reduction factors are updated corresponding to change of the fundamental frequencies. The peak story drift reduction factor is reduced on the basis of displacement design spectrum, as shown in Figure 5a due to fundamental period shortening. The modified peak story drift reduction factor is plotted as 'Updated *D*/*Do*' (thin solid line) in Figure 2.

**Figure 5.** Modification of response due to change of the fundamental period: (**a**) Displacement spectrum and (**b**) pseudo-acceleration spectrum.

(Step 7) The effective damping ratio is updated to β*t*<sup>2</sup> considering the decrease of deformations in (Step 6), and corresponding maximum story drift reduction factor is interpolated from the plot 'Updated *D*/*Do*' in Figure 2.

(Step 8) The base shear reduction factor is determined on the basis of pseudo-acceleration design spectrum and increased due to shortening of the fundamental period, as shown in Figure 5b. The modified base shear reduction factor is represented as 'Updated *V*/*Vo*' in Figure 2 (thick solid line).

(Step 9) Damping correction factor η for β*t*<sup>2</sup> is calculated as a ratio between *V*/*Vo*'s at β*t*<sup>2</sup> and β*IH* on 'Updated *V*/*Vo*' in Figure 2.

(Step 10) Adjust (*V*/*Vo*)1 and repeat (Step 2) to (Step 9) until η becomes sufficiently close to the target.

(Step 11) If η converges to the target, nonlinear response history analysis is performed in order to confirm whether the actual response reduction factor satisfies design requirements or not.

The design procedure proposed above utilizes nonlinear response history analysis only at Step 1. Additional response prediction is performed using the elastic design spectrum. Thus, the proposed design procedure is computationally efficient compared to the trial-and-error method based on fully nonlinear response history analysis.

#### **4. Design Example**

Two design examples based on the proposed damping system design procedure are presented. The first example is a three-story steel moment-resisting frames with metallic yielding dampers and the second example is a six-story steel moment-resisting frames with viscoelastic dampers.

#### *4.1. Nonlinear Modeling of Structural Elements*

Nonlinear modeling of beams and columns of the steel moment frames is performed in accordance with ASCE 41-13 [32]. Common load-deformation relationship for beams and columns subjected to flexure is represented in Figure 6, where the yield rotation angle of beams and columns is calculated by Equations (19) and (20), respectively.

$$\text{Beams}: \ \theta\_y = \frac{ZF\_y l\_b}{6EI\_b} \tag{19}$$

$$\text{Columns}: \ \theta\_y = \frac{ZF\_y l\_c}{6EI\_c} \left( 1 - \frac{P}{P\_{yc}} \right) \tag{20}$$

where *lb*, *lc*, *Z* and l are the moment of inertia for beams and columns, the plastic section modulus and member length, respectively. In addition, *E*, *Fy*, *P* and *Pye* are the modulus of elasticity, yield strength of steel, the axial force acting on the member and the axial strength of member, respectively.

**Figure 6.** Load-deformation relationship of beams and columns [12].

The panel zone of the steel moment frame was explicitly modeled with Krawinkler's model [33] of which configuration is represented in Figure 7, and the load-deformation relationship is shown in Figure 8. The characteristics of the panel zone model were calculated by the following Equations.

$$K\_c = 0.95d\_bd\_ct\_pG\tag{21}$$

$$K\_p = 1.04 b\_{fc} t\_{fc}^2 G \tag{22}$$

$$M\_{\!\!\!F} = 0.55 F\_{\!\!\!F} t\_{\!\!\!P} 0.95 d\_{\!\!\!u} d\_{\!\!\!u} \tag{23}$$

$$
\theta\_p = 4\theta\_y \tag{24}
$$

where *db*, *dc*, *bf c*, *tp*, *tf c*, *G* and α are beam depth, column depth, column flange width, panel zone thickness, column flange thickness, shear modulus and strain-hardening ratio (0.02), respectively. Perform-3D software was used for modeling and nonlinear response history analysis.

**Figure 7.** Krawinkler's model for panel zone.

**Figure 8.** Load-deformation relationship for panel zone [22].

*4.2. Three-Story Steel Moment-Resisting Frames with Metallic Yielding Dampers (3F-OMRF-MD)*

4.2.1. Initial Design of Seismic-Force-Resisting System

An example building with displacement-dependent damping devices is designed based on the KBC 2016, which is composed of three stories, five spans in X direction, three spans in Y direction. Figure 9a,b shows a three-dimensional view and plan view of the building. All the X-directional internal frames are identical and only one frame is used in this design example and represented in Figure 9c. The dead and live loads of the structure applied to building floors are 5.0 kN/m2 and 3.5 kN/m2, respectively, and identical for all stories. The building is assumed to be located in seismic zones I and belong to seismic risk category 'Special'. Site class *SD* is assumed for the building. The seismic-force-resisting system of the building is designed as an ordinary moment-resisting frame of which design factors are listed in Table 1. Allowable story drift of 1.0 % for seismic risk category 'Special' is adopted.

**Figure 9.** Three-story steel moment resisting frame with metallic yielding dampers: (**a**) Isometric view, (**b**) plan, (**c**) elevation of internal frame.

The initial design of the moment-resisting frame was performed for the minimum base shear with η = 0.75. The properties of columns and beams of the designed frame are listed in Table 3. The same section is used for each member in all stories. SM490 material was applied to all the members. The DCR of the initial design result is 0.95. Thus, the target base shear reduction factor required for damping systems is moderated to be 0.79, considering the DCR.


**Table 3.** Properties of moment-resisting frames.

From linear dynamic analysis using the response spectrum method, the maximum story drift without damping devices was 1.32%, which occurs at the first story and is higher than the allowable story drift of 1.0%. Therefore, it is necessary to reduce the story drift as well as base shear using damping devices. TADAS damping devices are installed at the center span using brace members listed in Table 3. Thus, the center frame and the damping devices shown in Figure 9c comprises a damping system.

#### 4.2.2. Design of Displacement-Dependent Damping Devices

In order to achieve target reduction factors for base shear and maximum story drift, those two response values are recorded from nonlinear response history analyses repeated with incremental damping of 0.05. Thus, the nonlinear response history analysis was performed only five times. Response reduction factors obtained from the nonlinear response history analysis are plotted in Figure 10.

**Figure 10.** Base shear and peak story drift reduction factors.

Target damping ratio corresponding to the base shear reduction factor 0.79 is interpolated to be 19.1%. The damping devices were designed to achieve an added damping of 0.141 except 0.05 inherent damping of the moment-resisting frame. To design each damping device, the total dissipated energy *Edj* was calculated from Equation (12) in combination with Equation (13). Then *Edj* was distributed to each story in proportion to the story shear force. It is taken into account that yield strength or friction force of displacement-dependent damping devices are distributed based on the distribution of story shear force to maximize energy dissipation [34,35]. First estimation of target β*<sup>v</sup>* and corresponding β*<sup>t</sup>* were 0.141 and 0.191. Characteristics of damping devices determined to achieve the target β*<sup>v</sup>* on the basis of Equation (14) are given in Table 4. The post-yield stiffness ratio of the damping device was assumed to be 0.02 in the calculation of dissipated energy.


**Table 4.** Characteristics of TADAS damping devices to achieve target damping ratio.

The fundamental period of the structure without damping devices was 1.04 sec. The fundamental period of the moment-resisting frame with damping devices substituted by secant stiffness thereof at respective peak deformations of them was reduced to 0.92 sec. Considering the change of the fundamental period, the story drift reduction factor was decreased, as shown in Figure 10. Then, β*<sup>v</sup>* and corresponding β*<sup>t</sup>* were modified into 0.12 and 0.17, respectively, using updated damping device deformations corresponding to the adjusted story drift reduction factor. As a result, the corresponding maximum story drift response is reduced to 64% compared to the structure with 0.05 damping ratio. Using those updated damping device deformations, the fundamental period based on the secant stiffness of damping devices was calculated to be 0.91 second. Then, the base shear reduction factor was elevated corresponding to 0.91 second period as shown in Figure 10. Finally, damping device yield strengths are modified in order to compensate reduced deformations due to period change and to achieve an added damping ratio β*<sup>v</sup>* of 0.12. The final damping device properties are listed in Table 4 for each story. Based on the modified base shear reduction factor represented by the thicker solid line in Figure 10, the expected damping correction factor is 0.93/1.14 = 0.82, which is slightly higher than the target η = 0.79. However, the damping performance obtained from the results of the nonlinear response history analysis is 0.79, which mean that the expected damping performance goal was achieved with a sufficiently accurate prediction of performance.

The average maximum story drift ratios for seven ground motion records representing design earthquake were 0.66%, 0.65% and 0.48% for the first, second, and third story, respectively, as summarized in Table 5 and all of those values are much smaller than the allowable story drift ratio 1.0%. This is because the base shear reduction factor 0.79 for seismic-force-resisting system governs the design rather than story drift reduction in this design example. The average maximum TADAS damping device deformation for seven ground motion records representing the maximum considered earthquake was maximum at the first story and calculated to be 41.2 mm. The deformation capacity of the example TADAS damping device is 60.3 mm. Therefore, damping devices can maintain the damping performance even under the maximum considered earthquake.


**Table 5.** Comparison of drift ratio and structural weight.

KBC 2016 requires structural elements comprising a damping system to remain elastic subjected to both seismic loads and forces induced by damping devices for design earthquake. DCRs for the frame members and panel zones were computed in terms of rotation angle ductility from nonlinear response history analysis and represented in Figure 11. In the case of columns and braces, higher DCR among bending moment DCR and axial force DCR in a member is given in Figure 11a. All the members that belong to the damping system at the central bay remain elastic since DCRs are lower than 1.0. Therefore, the design result obtained by the proposed procedure satisfies all the requirements of KBC 2016.

**Figure 11.** Demand–capacity ratio (DCR) of three-story steel moment-resisting frames with metallic yielding dampers (3F-OMF-MD): (**a**) Frame members, (**b**) panel zones.

To examine the effect of structural steel material reduction by damping devices, a bare ordinary moment resisting frame is designed to achieve story drifts similar to the frame with damping devices. Steel sections and story drifts of two models with and without damping devices are summarized with the respective total weights in Table 5, where 3F-OMRF-SD represents the seismically designed bare frame. The total weight of steel sections for the frame with damping devices is 149 kN, which is 67% of 221 kN for the frame without damping devices. Thus, the proposed procedure can yield efficient structural material-saving design. Ramirez et al. [5] provide similar design example, in which three-story and three-bay frames are designed without and with metallic yielding damping devices using equivalent lateral force procedure although the seismic-force-resisting system is a special moment-resisting frame and target story drift ratio is set to 2% differently from this study. In the comparative design example, the frame with damping devices has 76% of the weight for frames without damping devices. In spite of several different conditions, this comparison supports the ability of the proposed design procedure to reduce seismic demand on the seismic-force-resisting system with supplementary energy dissipation relying on more accurate response prediction by nonlinear response history analysis.

#### *4.3. Six-Story Steel Moment Frames with Viscoelastic Dampers (6F-SMRF-VED)*

#### 4.3.1. Initial Design of Seismic-Force-Resisting System

A six-story steel moment-resisting frame is designed in this example. Velocity-dependent damping devices are added for seismic response reduction. The steel moment frame has five spans in the longitudinal direction, and three spans in the transverse direction. The building is assumed to be located in Seismic zone I of KBC 2016 and belong to seismic risk category 'Special' of which importance factor is 1.5. Site class was assumed to be *SD*. The overall design was performed under conditions similar to the example building with displacement-dependent damping systems. However, a special moment-resisting frame was adopted for the seismic-force-resisting system. The numerical model of the example building is represented in Figure 12.

**Figure 12.** Six-story steel moment-resisting frames with viscoelastic damping devices: (**a**) Isometric view, (**b**) plan, (**c**) elevation of internal frame for design example.

In the transverse direction, only two special moment-resisting frames placed at the outermost part of the building plan play a role of seismic-force-resisting system. Considering geometrical symmetry, only one moment-resisting frame is modeled in the example for simplicity. In addition, the *P-*Δ effect due to gravity loads at the center of the plan was taken into account using the leaning column as shown in Figure 12c.

The initial design of the special moment-resisting frame was performed for the minimum base shear with η = 1.0. The DCR of the initial design result is 0.88. Table 3 summarizes sections of members used in the designed frame. Since η assumed in the design equals 1.0, it is unnecessary to confirm whether a target base shear reduction factor is achieved. The damping devices are installed at the center span of the planar frame with braces and illustrated in Figure 12c. Thus, the frame in the central bay and the damping devices shown in Figure 12c comprises a damping system.

From linear dynamic analysis based on response spectrum method with response modification factor and deflection amplification factor defined in KBC 2016, it was observed that the highest peak story drift was 1.75% and observed in the third and fourth stories. It is necessary to reduce the story drift by the damping device because it does not satisfy the allowable story drift of 1.0%.

#### 4.3.2. Design of Velocity-Dependent Damping Devices

The natural frequency of the first mode was 0.5 Hz from the eigenvalue analysis of the structure with only the stiffness component of the viscoelastic damping devices. Viscoelastic damper characteristics corresponding to an excitation frequency of 1.0 Hz, which is the closest one to 0.5 Hz, was adopted among those dependent on excitation frequencies. The stiffness and damping coefficients of each damping device were calculated using Equation (15) and (16).

The average maximum story drift ratios from nonlinear response history analysis for seven ground motion records representing design earthquake are listed in Table 6. The maximum story drift ratio is found to be 1.48% for the third story. Target story drift reduction factor is 1.0%/1.48% = 0.68. The maximum story drift reduction factor was obtained from the nonlinear response history analysis of the moment-resisting frame with an incremental damping ratio of 0.05 that substitute damping devices. Thus, the nonlinear response history analysis was performed only five times and the maximum story drift reduction factor was plotted in Figure 13 and the target damping ratio β*<sup>t</sup>* interpolated from the plot is 0.20. To achieve the target damping ratio, β*<sup>v</sup>* = 0.15 excluding β*IH* = 0.05 is necessary to be added by damping devices.


**Table 6.** Peak story drifts of six-story steel moment frames with viscoelastic dampers (6F-SMRF-VED).

**Figure 13.** Normalized response vs. incremental damping ratios.

The damping ratio added by viscoelastic damping devices is calculated by Equation (17), in which the added damping ratio is dependent on the fundamental frequency of the structure with effective stiffness of damping devices. In order to design damping devices, the stiffness *Kd* of damping devices represented by Kelvin model are increased until the fundamental frequency becomes the target value corresponding to the target damping ratio. This work is conducted by eigenvalue analysis of the linear elastic model and does not require additional nonlinear response history analysis. The same stiffness was applied to all the damping devices in this design example, but more efficient distribution may be investigated [18]. When *Kd* is determined, a corresponding *Cd* can be calculated using Equation (16).

However, change of the fundamental frequency due to damping devices affects the maximum story drift reduction factor. As a result, the maximum story drift reduction factor in Figure 13 is updated repeatedly. For each update of the maximum story drift reduction factor, the target damping ratio changes correspondingly. Four times of update were performed and updated parameters including target damping ratios and target frequencies are summarized in Table 7. The final fundamental frequency converged to 1.89 sec and *Ka* and *Cd* reached 6800 kN/m and 2718 kN·sec/m, respectively.



Nonlinear response history analysis was performed using the final stiffness and damping coefficient of the damping devices. It is unnecessary to examine design base shear because the damping correction factor was set to 1.0. The base shear reduction factor was 0.92 which is smaller than 1.0. The peak story drifts of the final design are summarized in Table 8. Compared to Table 6, the maximum peak story drift was reduced to 0.90%, which is slightly lower than the allowable story drift ratio of 1.0%. As a result, the proposed design methodology can design damping systems with a sufficiently accurate prediction of performance.

**Table 8.** Comparison of peak drift ratio and structural weight.


As with the displacement-dependent damping system design, the structural elements comprising the damping system must be both elastic against the loads including seismic loads and forces induced by damping devices for design earthquake. Braces to install damping devices transmitting damping device force to the seismic-force-resisting force and the column at the right-hand side of the first story damping device transmitting vertical component of damping device force to the foundation comprises the damping system of the structure. DCRs for the frame members and panel zones were computed in terms of rotation angle ductility from nonlinear response history analysis for design earthquake and represented in Figure 14. In the case of columns and braces, higher DCR between bending moment DCR and axial force DCR is given in Figure 14a. All the members that belong to the damping system in the central bay remain elastic with DCRs lower than 1.0.

**Figure 14.** DCR of 6F-SMRF-VED. (**a**) Frame members, (**b**) panel zones.

Finally, damping device safety subjected to the maximum considered earthquake was checked. The maximum shear strain of damping devices was 0.455 from the response analysis for the maximum considered earthquake. The experimental data of Soong and Dargush used in the design of the damping device does not provide the deformation capacity of the damping device [20]. Therefore, it is necessary to ensure whether or not the damping device is broken for the strain demand subjected to the maximum considered earthquake. Therefore, the design result satisfies all the requirements of KBC 2016 under the premise that the deformation capacity requirement for the damping device can be met.

Like the three-story frame example, a bare special moment-resisting frame is designed to achieve story drifts similar to the frame with damping devices. Steel sections and story drifts of two models with and without damping devices are summarized with the respective total weights in Table 8 where 6F-SMRF-SD represents the seismically designed bare frame. The total weight of steel sections for the frame with damping devices is 297 kN, which is about 50% of 590 kN for the frame without damping devices. Thus, the proposed procedure can yield efficient structural material-saving design in case of viscoelastic damping devices. Ramirez et al. [5] provide similar design example, in which six-story and three-bay frames are designed without and with viscous damping devices using equivalent lateral force procedure although the damping device does not have stiffness component and target story drift ratio is set to 2% differently from this study. In the comparative design example, the frame with damping devices has 60% of the weight for frames without damping devices. Similar to the preceding design example, the proposed design procedure can design damping devices effectively to reduce seismic demand on the seismic-force-resisting system with better efficiency, which is owing to more accurate response prediction by nonlinear response history analysis.

#### **5. Conclusions**

This study proposed an efficient seismic design procedure for building structures with damping systems subjected to requirements of the revised Korean building code, KBC 2016, using nonlinear response history analysis. The proposed design procedure was validated by two design examples of steel moment-resisting frame with metallic yielding dampers and viscoelastic dampers, respectively. The conclusions from this study are summarized as follows.


**Author Contributions:** Formal analysis, S.-H.J.; writing—original draft, J.-H.P.; writing—review & editing, T.-W.H.

**Acknowledgments:** This work was supported by Incheon National University Research Grant in 2016.

**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* **Dynamic Responses of Liquid Storage Tanks Caused by Wind and Earthquake in Special Environment**

#### **Wei Jing 1,\*, Huan Feng <sup>2</sup> and Xuansheng Cheng <sup>1</sup>**


Received: 24 April 2019; Accepted: 3 June 2019; Published: 11 June 2019

**Abstract:** Based on potential flow theory and arbitrary Lagrangian–Eulerian method, shell–liquid and shell–wind interactions are solved respectively. Considering the nonlinearity of tank material and liquid sloshing, a refined 3-D wind–shell–liquid interaction calculation model for liquid storage tanks is established. A comparative study of dynamic responses of liquid storage tanks under wind, earthquake, and wind and earthquake is carried out, and the influences of wind speed and wind interference effect on dynamic responses of liquid storage tank are discussed. The results show that when the wind is strong, the dynamic responses of the liquid storage tank under wind load alone are likely to be larger than that under earthquake, and the dynamic responses under wind–earthquake interaction are obviously larger than that under wind and earthquake alone. The maximum responses of the tank wall under wind and earthquake are located in the unfilled area at the upper part of the tank and the filled area at the lower part of the tank respectively, while the location of maximum responses of the tank wall under wind–earthquake interaction is related to the relative magnitude of the wind and earthquake. Wind speed has a great influence on the responses of liquid storage tanks, when the wind speed increases to a certain extent, the storage tank is prone to damage. Wind interference effect has a significant effect on liquid storage tanks and wind fields. For liquid storage tanks in special environments, wind and earthquake effects should be considered reasonably, and wind interference effects cannot be ignored.

**Keywords:** liquid storage tank; earthquake; wind; dynamic response; fluid–solid interaction

#### **1. Introduction**

With the development of economy and society, more and more liquid storage tanks are built in seismically active areas, in extreme cases, these areas may also belong to strong wind areas, which leads to the threat of wind and earthquake to large-scale liquid storage tanks in the whole life cycle. Moreover, earthquake and wind-induced damage cases of liquid storage tanks are very common [1–3], two cases corresponding to earthquake and wind are shown in Figure 1. The destruction of the liquid storage tank not only involves the structure itself, but it will also cause huge economic losses, environmental pollution, fire, and so on, and even threaten people's safety.

**Figure 1.** Failure cases of liquid storage tank. (**a**) Earthquake. (**b**) Wind [1].

Dynamic responses of liquid storage tanks during earthquakes involves shell–liquid interaction, Rawat et al. [3] used a coupled acoustic–structural (CAS) approach in the FEM for the analysis of the tanks with rigid and flexible walls with varying parameters. Kotrasov et al. [4] simulated the interaction between structure and liquid on the contact surface based on the bidirectional fluid–solid coupling technique and studied the dynamic responses of liquid storage tanks by finite element method. Gilmanov et al. [5] proposed a numerical method to simulate the shell–liquid interaction of elastic thin plate with arbitrary deformation in incompressible fluid. In addition, a large number of studies and post-earthquake investigations show that the failure modes of liquid storage tanks under earthquake basically include liquid overflow, bottom lifting, circumferential tension, and instability. Ishikawa et al. [6] proposed a practical analytical model for shallow excited tank, which exhibited complex behavior because of nonlinearity and dispersion of the liquid. Moslemi et al. [7] conducted nonlinear sloshing analysis of liquid storage tanks and found that the sloshing nonlinearity had a significant effect on the seismic performance of liquid containing structures. Miladi and Razzaghi [8] performed numerical analysis of oil tank by using ABAQUS software, and carried out parametric study to evaluate the effect of amount of stored liquid on seismic behavior and performance of the studied tank. Ormeño et al. [9] performed shake table experiments to investigate the effects of a flexible base on the seismic response of a liquid storage tank, results showed that the axial compressive stresses decreased after a flexible base was considered. Sanapala et al. [10] performed shake table experiments to study the fluid structure interaction effects between the sloshing liquid and the internal structure, and found that when the partially filled storage tank was subjected to seismic excitation, spiky jet-like features were observed over the free surface. Rawat et al. [11] investigated three-dimensional (3-D) ground-supported liquid storage tanks subjected to seismic base excitation by using finite element method based on coupled acoustic–structural and coupled Eulerian–Lagrangian approaches. Generally speaking, dynamic responses of liquid storage tank involves complex fluid–structure interaction, and numerical simulation is an effective means to solve this problem.

Researchers have made certain explorations on the behavior of liquid storage tanks under wind load. Flores and Godoy [12] used numerical methods to study the buckling problem of liquid storage tanks under typhoon, and obtained that bifurcation buckling analysis could better evaluate the critical state of liquid storage tanks. Portela and Godoy [13] used computational model to evaluate the buckling behavior of steel tanks under wind loads. Zhang et al. [14] studied the dynamic responses of flexible liquid storage structure under wind load by multi-material ALE finite element method. Yasunaga et al. [15] used wind tunnel testing and finite element method to study the buckling behavior of thin-walled circular liquid storage tanks, and discussed the effect of wind load distribution on the buckling of liquid storage tanks by comparing it with a static wind load. Chen and Rotter [16] used finite element method to study the buckling of anchored liquid storage tanks with equal wall thickness under wind load. Zhao et al. [17] and Lin et al. [18] used wind tunnel tests to study the distribution of wind pressure and the stability of liquid storage tanks under wind loads.

In view of the structural dynamic response under the combined action of wind and earthquake, Hong and Gu [19] found that for high-flexible structures whose horizontal loads are controlled by wind load, the combined total loads after considering wind and earthquake loads may be more

disadvantageous than those when considering wind loads in seismic design. Ke et al. [20] obtained that the structure responses of super-large cooling tower varied significantly along with height under wind load, earthquake, and wind–earthquake. Peng et al. [21] used the method of combining theoretical analysis with numerical simulation to get the position of maximum stress under wind load and earthquake action is different. Sapountzakis et al. [22] studied the nonlinear responses of wind turbine under wind load and earthquake. Mazza [23] synthesized velocity time history of wind based on equivalent spectrum technology, and studied the dynamic responses of steel frame structures under wind load and earthquake action.

To sum up, the dynamic responses of structures under earthquake and wind are obviously different, and the combined action of wind and earthquake will have more adverse effects on the structures, but the research on dynamic responses of liquid storage tanks under wind and earthquake is rare. In this paper, the shell–liquid and the shell–wind interactions are considered, and a refined calculation model of the liquid storage tank is established. The dynamic responses of the liquid storage tank under wind, earthquake, and wind and earthquake are studied in many aspects, which is of great significance to the rationality of the design and the reliability of the operation of the liquid storage tank.

#### **2. Wind Field Control Equations**

Large eddy simulation (LES) is used to calculate the wind field, and its control equation is

$$\frac{\partial \overline{\dot{u}\_i}}{\partial t} + \frac{\partial \overline{\dot{u}\_i} \overline{\dot{u}\_j}}{\partial \mathbf{x}\_j} = -\frac{1}{\rho} \frac{\overline{p}}{\partial \mathbf{x}\_i} - \nu \frac{\partial^2 \overline{\dot{u}\_i}}{\partial \mathbf{x}\_j \partial \mathbf{x}\_j} + \frac{\partial \overline{\pi}\_{ij}}{\partial \mathbf{x}\_j} \tag{1}$$

$$\frac{\partial \overline{\dot{u}\_i}}{\partial \mathbf{x}\_i} = 0 \tag{2}$$

where <sup>τ</sup>*ij* <sup>=</sup> . *ui* . *uj* <sup>−</sup> . *ui* . *uj*, τ*ij* is subgrid-scale stress, namely, SGS stress, which reflects the influence of the motion of small-scale vortices on the motion equation.

If the equations consisting of Equations (1) and (2) are closed, then according to Smagorinsky's basic SGS model, it is assumed that the SGS stress satisfies the following requirements

$$
\overline{\pi}\_{ij} - \frac{1}{3} \overline{\pi}\_{kk} \delta\_{ij} = -2\mu\_t \overline{S}\_{ij} \tag{3}
$$

$$
\mu\_t = (\mathbb{C}\_s \Lambda)^2 |\overline{\mathbb{S}}| \tag{4}
$$

where *Sij* <sup>=</sup> <sup>1</sup> 2 ∂*ui* ∂*xj* + ∂*uj* ∂*xi* , - - -*S* - - - = <sup>2</sup>*SijSij*, <sup>Δ</sup> <sup>=</sup> Δ*x*Δ*y*Δ*<sup>z</sup>* 1/3, <sup>μ</sup>*<sup>t</sup>* is turbulent viscosity at sublattice scale, Δ is filtration scale of large eddy model, Δ*<sup>i</sup>* represents the grid size along the *i*-axis, *Cs*Δ is equivalent to

#### **3. Structure Control Equations**

mixing length, *Cs* is SGS constant.

The structure equation of motion is

$$\mathbf{M}\_{\rm ss}\ddot{\mathbf{u}}\_{\rm s} + \mathbf{C}\_{\rm ss}\dot{\mathbf{u}}\_{\rm s} + \mathbf{K}\_{\rm ss}\mathbf{u}\_{\rm s} = \mathbf{F}\_{\rm ss} \tag{5}$$

where **M***ss*, **C***ss*, and **K***ss* are mass, damping and stiffness matrices of structures, respectively; **F***ss* is load vector acting on structure, which includes liquid pressure; .. **<sup>u</sup>***s*, . **u***s*, and **u***<sup>s</sup>* are vectors of acceleration, velocity, and displacement of structure, respectively.

Newmark method is used to solve the dynamic Equation (5), and the first assumption is

$$
\dot{\mathbf{u}}\_{\mathbf{s}(i+1)} = \dot{\mathbf{u}}\_{\mathbf{s}(i)} + \left[ (1 - \beta) \ddot{\mathbf{u}}\_{\mathbf{s}(i)} + \beta \ddot{\mathbf{u}}\_{\mathbf{s}(i+1)} \right] \Delta t \tag{6}
$$

*Appl. Sci.* **2019**, *9*, 2376

$$\mathbf{u}\_{s(i+1)} = \mathbf{u}\_{s(i)} + \dot{\mathbf{u}}\_i \Delta t + \left[ \left( \frac{1}{2} - \gamma \right) \ddot{\mathbf{u}}\_i + \gamma \ddot{\mathbf{u}}\_{i+1} \right] \Delta t^2 \tag{7}$$

where β and γ are adjustment coefficients for accuracy and stability.

The incremental forms <sup>Δ</sup> . **<sup>u</sup>***<sup>s</sup>* and <sup>Δ</sup>**u***<sup>s</sup>* of velocity . **u***<sup>s</sup>* and displacement **u***<sup>s</sup>* can be obtained from Equations (6) and (7), respectively

$$
\Delta \dot{\mathbf{u}}\_{\mathbf{s}(i)} = \dot{\mathbf{u}}\_{\mathbf{s}(i+1)} - \dot{\mathbf{u}}\_{\mathbf{s}(i)} = \left( \ddot{\mathbf{u}}\_{\mathbf{s}(i)} + \beta \Delta \ddot{\mathbf{u}}\_{\mathbf{s}(i)} \right) \Delta t \tag{8}
$$

$$
\Delta \mathbf{u}\_{\mathbf{s}(i)} = \mathbf{u}\_{\mathbf{s}(i+1)} - \mathbf{u}\_{\mathbf{s}(i)} = \dot{\mathbf{u}}\_{\mathbf{s}(i)} \Delta t + \frac{1}{2} \ddot{\mathbf{u}}\_{\mathbf{s}(i)} \Delta t^2 + \gamma \Delta \ddot{\mathbf{u}}\_{\mathbf{s}(i)} \tag{9}
$$

Acceleration increment <sup>Δ</sup>.. **<sup>u</sup>***<sup>i</sup>* can be obtained by transforming Equation (9), then taking <sup>Δ</sup>.. **u***<sup>i</sup>* into Equation (8)

$$
\Delta \ddot{\mathbf{u}}\_i = \frac{1}{\gamma \Delta t^2} \Delta \mathbf{u}\_i - \frac{1}{\gamma \Delta t} \dot{\mathbf{u}}\_i - \left(\frac{1}{2\gamma} - 1\right) \ddot{\mathbf{u}}\_i \tag{10}
$$

$$
\Delta \dot{\mathbf{u}}\_{i} = \frac{\beta}{\gamma' \Delta t} \Delta \mathbf{u}\_{i} + \left(1 - \frac{\beta}{\gamma'}\right) \dot{\mathbf{u}}\_{i} + \left(1 - \frac{\beta}{2\gamma'}\right) \Delta t \ddot{\mathbf{u}}\_{i} \tag{11}
$$

The incremental form corresponding to Equation (5) is

$$\mathbf{M}\_{\rm ss} \Delta \ddot{\mathbf{u}}\_{\rm s(i)} + \mathbf{C}\_{\rm ss} \Delta \dot{\mathbf{u}}\_{\rm s(i)} + \mathbf{K}\_{\rm ss} \Delta \mathbf{u}\_{\rm s(i)} = \Delta \mathbf{F}\_{\rm ss(i)} \tag{12}$$

Taking Equations (9)–(11) into Equation (12)

$$
\overline{\mathbf{K}} \Delta \mathbf{u}\_{\mathbf{s}(i)} = \overline{\mathbf{F}} \tag{13}
$$

where **K** = **K** + 1 <sup>γ</sup>Δ*t*<sup>2</sup> **<sup>M</sup>** <sup>+</sup> <sup>β</sup> γΔ*t* **C**; **F** = Δ**F***ss*(*i*) + **M** 1 γΔ*t* . **u***<sup>i</sup>* + 1 <sup>2</sup><sup>γ</sup> <sup>−</sup> <sup>1</sup> .. **u***i* + **C** β <sup>γ</sup> <sup>−</sup> <sup>1</sup> . **u***<sup>i</sup>* + β <sup>2</sup><sup>γ</sup> <sup>−</sup> <sup>1</sup> Δ*t* .. **u***i* .

The displacement increment <sup>Δ</sup>**u***s*(*i*) can be obtained by Equation (13), velocity increment <sup>Δ</sup> . **u***s*(*i*) can be obtained by substituting displacement increment Δ**u***s*(*i*) into Equation (11). As a result, the displacement **<sup>u</sup>***s*(*i*+1) and velocity . **u***s*(*i*+1) of *i* + 1 time step can be obtained

$$\mathbf{u}\_{\mathbf{s}(i+1)} = \mathbf{u}\_{\mathbf{s}(i)} + \Delta \mathbf{u}\_{\mathbf{s}(i)} \tag{14}$$

$$
\dot{\mathbf{u}}\_{s(i+1)} = \dot{\mathbf{u}}\_{s(i)} + \Delta \dot{\mathbf{u}}\_{s(i)} \tag{15}
$$

The acceleration .. **u***s*(*i*+1) of time step *i* + 1 can be obtained by substituting Equations (14) and (15) into Equation (5)

$$\ddot{\mathbf{u}}\_{\mathbf{s}(i+1)} = \mathbf{M}\_{\mathbf{s}\mathbf{s}}{}^{-1} \cdot \left[ \mathbf{F}\_{\mathbf{s}\mathbf{s}} - \mathbf{C}\_{\mathbf{s}\mathbf{s}} \cdot \dot{\mathbf{u}}\_{\mathbf{s}(i+1)} - \mathbf{K}\_{\mathbf{s}\mathbf{s}} \cdot \mathbf{u}\_{\mathbf{s}(i+1)} \right] \tag{16}$$

#### **4. Fluid–Solid Interaction**

In order to overcome the defects of large calculation amount and low calculation efficiency, the potential flow theory is used to solve the shell–liquid interaction, and the arbitrary Lagrangian–Eulerian method is used to solve the shell–wind interaction.

#### *4.1. Shell–Liquid Interaction*

Because the calculation process involves a large number of nonlinearities, the exact solution of each response can be obtained through multiple equilibrium iterations. Δφ is used to express the increment of the unknown velocity potential φ, and Δ**u** is used to express the increment of the unknown displacement **u**. The shell–liquid interaction dynamic equation based on potential fluid theory is [24]

$$
\begin{bmatrix}
\mathbf{M}\_{\otimes\varsigma} & \mathbf{0} \\
\mathbf{0} & \mathbf{M}\_{\mathcal{I}l}
\end{bmatrix}
\begin{bmatrix}
\Delta\ddot{\mathbf{u}} \\
\Delta\ddot{\boldsymbol{\phi}}
\end{bmatrix} +
\begin{bmatrix}
\mathbf{C}\_{\mathbf{u}\mathbf{u}} + \mathbf{C}\_{\varsigma s} & \mathbf{C}\_{\mathbf{u}l} \\
\mathbf{C}\_{\mathbf{l}\mathbf{u}} & -(\mathbf{C}\_{\mathcal{I}l} + (\mathbf{C}\_{\mathcal{I}l})\_{\mathcal{S}}) \\
\mathbf{K}\_{\mathbf{l}\mathbf{u}} & -(\mathbf{K}\_{\mathcal{I}l} + (\mathbf{K}\_{\mathcal{I}l})\_{\mathcal{S}})
\end{bmatrix}
\begin{bmatrix}
\Delta\dot{\mathbf{u}} \\
\Delta\dot{\boldsymbol{\phi}}
\end{bmatrix} + \\
\begin{bmatrix}
\mathbf{K}\_{\mathbf{u}\mathbf{u}} \\
\end{bmatrix}
\begin{bmatrix}
\Delta\mathbf{u} \\
\Delta\boldsymbol{\Phi}
\end{bmatrix} = 
\begin{bmatrix}
\mathbf{F}\_{\mathbf{s}s} \\
\mathbf{0}
\end{bmatrix} - 
\begin{bmatrix}
\mathbf{F}\_{p} \\
\mathbf{F}\_{l} + (\mathbf{F}\_{l})\_{\mathcal{S}}
\end{bmatrix}
\end{bmatrix}
\tag{17}$$

where **M***ll* is the liquid mass matrix; **C***uu*, **C***lu*, **C***ul*, and **C***ll* are the damping matrices of the structure itself, the liquid contributed by the structure, the structure contributed by the liquid and the liquid itself, respectively; and **K***uu*, **K***lu*, **K***ul*, and **K***ll* are the stiffness matrices of the structure itself, the liquid contributed by the structure, the structure contributed by the liquid and the liquid itself, respectively; **F***p*, **F***l*, and (**F***l*)*<sup>S</sup>* are the forces acting on the structural boundary caused by the liquid pressure, volume force, and area force, respectively; **F***<sup>l</sup>* is obtained by the volume integral of Equation (18), and (**F***l*)*<sup>S</sup>* is obtained by surface integral of Equation (19) [24]

$$\mathbf{F}\_{l} = \int\_{V} \left( \frac{\partial \rho\_{l}}{\partial l} \dot{h} \delta \phi - \rho\_{l} \nabla \phi \right) dV \tag{18}$$

$$(\mathbf{F}\_l)\_S = \int\_S -\rho\_l \mathbf{u} \cdot \mathbf{n} \delta \phi dS \tag{19}$$

where ρ*<sup>l</sup>* is the liquid density; *V* is the liquid domain; *S* is the liquid domain boundary; **n** is the internal normal direction vector of *<sup>S</sup>*; and . **u** is the moving speed of the boundary surface *S*.

The boundary surface adjacent to the structure is represented as *S*1, and the force acting on structure boundary **F***p* caused by the liquid pressure can be expressed as Equation (20)

$$-\delta \mathbf{F}\_p = -\int\_{S\_1} p n \cdot \delta \mathbf{u} dS\_1 \tag{20}$$

where δ**F***<sup>U</sup>* is differentiation of additional forces caused by liquid; **n** is normal vector of adjacent interface. Liquid pressure *p* is calculated by Equation (21)

$$p = p(h) = p\left[\Omega(\mathbf{x} + \mathbf{u}) - \dot{\phi} - \frac{1}{2}\mathbf{v}\_n \cdot \mathbf{v}\_n - \frac{1}{2}\mathbf{v}\_\mathbf{\tau} \cdot \mathbf{v}\_\mathbf{\tau}\right] \tag{21}$$

where Ω is volume acceleration potential energy; **v***<sup>n</sup>* and **v**<sup>τ</sup> are liquid normal and tangential velocities on the interaction boundary.

#### *4.2. Shell–Wind Interaction*

The wind field equation and the structure equation are expressed by *Gw <sup>w</sup>*, . *w* = 0 and *Gs* **u**, . **u**, .. **u** = 0, respectively, subscript *w* denotes wind field variables, and subscript *s* denotes structure variables.

Firstly, the velocity and acceleration of wind field are expressed as [25]

$$\begin{aligned} v^{t+a\Delta t}v &= \frac{t + a\Delta t\_{\mathcal{U}} - t\_{\mathcal{U}}}{t} = {}^{t+\Delta t}v\alpha + {}^{t}v(1-\alpha) \\ v^{t+a\Delta t}a &= \frac{t + a\Delta t\_{\mathcal{U}} - t\_{\mathcal{U}}}{t} = {}^{t+\Delta t}a\alpha + {}^{t}a(1-\alpha) \end{aligned} \tag{22}$$

where α is stability conditions of compatible time integral.

*Appl. Sci.* **2019**, *9*, 2376

Velocity and acceleration of Equation (22) at *t* + Δ*t* can be expressed as functions of unknown displacement

$$\begin{aligned} \upsilon^{t+\Delta t} \upsilon &= \frac{1}{\alpha \Delta t} \Big( ^{t+\Delta t}u - ^t u \Big) - ^t \upsilon \Big( \frac{1}{\alpha} - 1 \Big) = ^{t+\Delta t} dm + ^t \zeta \\ \upsilon^{t+\Delta t} a &= \frac{1}{\alpha^2 \Delta t^2} \Big( ^{t+\Delta t}u - ^t u \Big) - ^t \upsilon \frac{1}{\alpha^2 \Delta t} - ^t a \Big( \frac{1}{\alpha} - 1 \Big) = ^{t+\Delta t} dn + ^t \eta \end{aligned} \tag{23}$$

Taking Equations (22) and (23) into wind field equation *Gw <sup>w</sup>*, . *w* = 0 and structure equation *Gs u*, . *u*, .. *u* = 0

$$\begin{aligned} t^{t+a\Delta t} \mathbf{G}\_{\mathbf{w}} &\approx \mathbf{G}\_{\mathbf{w}} \Big[ {}^{t+a\Delta t} \boldsymbol{w}, \Big( {}^{t+a\Delta t} \boldsymbol{w} - {}^{t} \boldsymbol{w} \Big) / \alpha \Delta t \Big] = 0 \\\ t^{t+\Delta t} \mathbf{G}\_{\mathbf{s}} &\approx \mathbf{G}\_{\mathbf{s}} \Big[ {}^{t+\Delta t} \boldsymbol{u}, {}^{t+\Delta t} d m + {}^{t} \boldsymbol{\xi}, {}^{t+\Delta t} d n + {}^{t} \boldsymbol{\eta} \Big] = 0 \end{aligned} \tag{24}$$

In order to solve the coupled system, Equation (24) is discretized. Assuming that the solution vector of the coupled system is **X** = **X**(X*w*, X*s*), **X***w*, and **X***s* represents solution vectors of wind field and structure nodes. Therefore, *us* = *us*(X*s*) and τ*<sup>w</sup>* = τ*w*(X*w*), and the shell–wind coupling equation can be expressed as [25]

$$\begin{aligned} \, \, \, \, \, \, \, G\_f \left[ \mathbf{X}\_{w\prime}^k \, \lambda\_d \boldsymbol{u}\_s^k + (1 - \lambda\_d) \boldsymbol{u}\_s^{k-1} \right] &= 0 \\ \, \, \, \, \, G\_s \left[ \mathbf{X}\_{s\prime}^k \, \lambda\_\tau \tau\_w^k + (1 - \lambda\_\tau) \tau\_w^{k-1} \right] &= 0 \end{aligned} \tag{25}$$

where λ*<sup>d</sup>* and λτ are displacement and stress relaxation factors.

The above solving process can be illustrated by Figure 2.

**Figure 2.** Shell–wind interaction solution.

#### **5. Boundary Conditions**

#### *5.1. Wind Field Boundary Conditions*

For high Reynolds number incompressible steady flow, velocity-inlet is chosen as the boundary condition at the entrance; pressure-outlet without backflow is chosen as the boundary condition at the outlet, that is, at the exit boundary of the flow field, the diffusion flux of the physical quantity of the flow field along the normal direction of the exit is 0; the non-slip wall boundary is used as boundary condition on the structure surface and ground. Symmetry is chosen as the boundary on both sides and on the top. The boundary conditions for wind field simulation are shown in Figure 3.

**Figure 3.** Boundary conditions for wind field simulation.

#### *5.2. Shell–Liquid Interaction Boundary Conditions*

The conditions of displacement continuity and force balance need to be satisfied at the shell–liquid interaction interface, namely

$$\mathbf{u}\_{\sf s} = \mathbf{u}\_{l\sf l} \mathbf{F}\_{\sf s} = \mathbf{F}\_{l} \tag{26}$$

where **u***<sup>s</sup>* and **u***<sup>l</sup>* are structure and liquid displacement vectors; **F***<sup>s</sup>* and **F***<sup>l</sup>* are structure and liquid dragging forces.

$$\mathbf{F}\_{\\$} = \mathbf{o}\_{\\$} \cdot \mathbf{n}\_{\\$} \tag{27}$$

where **n***<sup>s</sup>* and **n***<sup>l</sup>* are interface normal vector; σ*<sup>s</sup>* and σ*<sup>l</sup>* are structure and liquid stress vectors.

#### **6. Numerical Example**

#### *6.1. Calculation Model*

The diameter and height of the tank are 21 m and 16 m, liquid storage height is 8 m. The wall thickness from the bottom to the top is as follows: 0–2 m is 14 mm; 2–4 m is 12 mm; 4–6 m is 10 mm; 6–10 m is 8 mm; and 10–16 m is 6 mm. Bilinear elastic-plastic material and shell elements are used to simulate a liquid storage tank, potential fluid material model and 3D solid element are used to simulated liquid, and liquid free surface is defined to reflect liquid sloshing behavior. El-Centro wave is selected as the ground motion input for time-history analysis.

Since there are a large number of liquid storage tanks in actual oil depots, it is necessary to study the influence of wind interference effect. By comparing the dynamic responses of single tanks and double tanks under wind load, the influence of wind interference effect on liquid storage tanks can be preliminarily discussed. Wind field is simulated by using 8-node 6-hedral FCBI-C element and large-eddy-simulation material. The calculation model material parameters are shown in Table 1, and the calculation model are shown in Figures 4 and 5.


**Table 1.** Material parameters.

**Figure 4.** Calculation model of shell–liquid interaction. (**a**) Single tank. (**b**) Double tank.

**Figure 5.** *Cont.*

**Figure 5.** Wind field calculation model. (**a**) Single tank. (**b**) Double tank.

#### *6.2. Comparison of Dynamic Responses under Di*ff*erent Actions*

In view of the possibility that the liquid storage tank may be damaged under the action of wind and earthquake, and the combined action of wind and earthquake will have more adverse effects on the structure, a comparative study on the dynamic response of the liquid storage tank under the action of wind, earthquake, and wind and earthquake is carried out, and the specific results are shown in Figures 6–8 and Table 2.

**Figure 6.** Comparison of tank effective stress (unit: Pa). (**a**) Wind. (**b**) Earthquake. (**c**) Wind and earthquake.

**Figure 7.** Comparison of tank displacement (unit: m). (**a**) Wind. (**b**) Earthquake. (**c**) Wind and earthquake.

*Appl. Sci.* **2019**, *9*, 2376

**Figure 8.** Comparison of tank base shear force (unit: N). (**a**)Wind. (**b**) Earthquake. (**c**) Wind and earthquake.


**Table 2.** Absolute maximum dynamic response under different actions.

As shown in Figures 6–8, the maximum effective stress and displacement of tank under wind load is located in the unfilled area of the upper part of the liquid storage tank, while the maximum effective stress and displacement of the structure under earthquake is located in the filled area of the lower part of the liquid storage tank, and the maximum of base shear force appears near the contact position between the tank wall and foundation.

As shown in Table 2, it can be seen that when the wind speed is larger, the effective stress and displacement under wind load is greater than that under earthquake. However, a large number of researches on liquid storage tanks have been carried out on the basis of considering only the earthquake action, so there are some defects. Besides, the dynamic responses of tank under the combined action of wind and earthquake are obviously greater than those under the separate action of wind and earthquake. The effective stress, displacement, and base shear force obtained by SRSS are 130.04 Mpa, 7.65 mm and 257.73, respectively, which are very different from considering the interaction of wind and earthquake at the same time.

Therefore, when the wind speed is large, the influence of wind load on the liquid storage tank cannot be ignored. The location of maximum dynamic responses of liquid storage tank under wind and earthquake is different, and the combined effect of wind and earthquake will have a more adverse impact on the liquid storage tank. Therefore, for the liquid storage tank in special areas (such as coastal areas), the combined effect of wind and earthquake should be reasonably considered in its design.

#### *6.3. Influences of Wind Speed on Dynamic Responses*

Through the above analysis, it has been found that the influence of wind load on the liquid storage tank cannot be ignored. In order to further discuss the responses of the liquid storage tank under wind load, a comparative study is carried out under the wind speed of 10 m/s and 20 m/s. The nephograms of the effective stress, displacement, and base shear force are shown in Figures 9–11 and Table 3.

**Figure 9.** Effect of wind speed on tank effective stress (unit: Pa).

**Figure 11.** Effect of wind speed on tank base shear force (unit: N).

**Table 3.** Effect of wind speed on absolute maximum dynamic responses.


As shown in Figures 9–11, under the combined action of wind and earthquake, when the wind speed is 10 m/s, the location of maximum effective stress and displacement of the tank is located in the liquid filled area at the bottom of the liquid storage tank, but when the wind speed increases to 20 m/s, the location of maximum effective stress and displacement of the structure shifts to the unfilled area at

the upper part of the liquid storage tank. That is to say, when the wind speed is lower, the responses of liquid storage tanks are dominated by earthquake, on the contrary, when the wind speed is higher, the responses of liquid storage tanks will be dominated by wind.

As shown in Table 3, when the wind speed is increased by 2 times, the effective stress, displacement, and base shear force are significantly increased, especially the effective stress and displacement are approximately increased by 2 times.

Therefore, when the wind speed is high, the probability of damage to the liquid storage tank is relatively high. For liquid storage tanks built in special areas, sufficient attention should be paid to the adverse effects of wind load.

#### *6.4. Wind Interference E*ff*ect*

Significant wind disturbance effect exists in group structures, Zhao et al. [26] obtained that amplification effect caused by wind disturbance reaches to 20–40% through wind tunnel test. Zhang et al. [27] obtained that the unfavorable influence of double-row arrangement of towers is obviously larger than that of single-row arrangement.

Through the research on the influence of wind speed on the dynamic response of the structure, it is found that the larger the wind speed, the more unfavorable it is to the tank. Therefore, taking the wind speed of 20 m/s as an example, single tanks and double tanks are selected as research objects to study the influence law of wind interference effect on the dynamic responses of liquid storage tanks. The comparisons of effective stress, displacement, base shear force, and velocity field are shown in Figures 12–15 and Table 4.

#### **Figure 12.** Effect of wind interference on effective stress (unit: Pa).

**Figure 13.** Effect of wind interference on displacement (unit: m).

(**b**)

**Figure 15.** Effect of wind interference on wind velocity field (unit: m/s). (**a**) Single tank. (**b**) Double tank.


**Table 4.** Effect of wind interference on absolute maximum responses.

As shown in Figures 12–15 and Table 4, without considering and considering the wind interference effect, the maximum absolute values of effective stress of the tank are 218.96 MPa and 233.81 MPa, respectively; the maximum absolute values of the displacement of the tank are 13.13 mm and 18.02 mm, respectively; the maximum absolute values of the base shear force of the tank are 951.11 kN and 1263.77 kN, respectively; and the maximum absolute values of the wind field speed are 35.59 m/s and 52.13 m/s, respectively. The difference ratios corresponding to effective stress, displacement, base shear force and wind velocity are 9.07%, 37.24%, 32.85%, and 46.47%.

For a single tank, the maximum dynamic response is located on the tank axis and the maximum wind speed is located near the tank walls on both sides; while for a double tank, the maximum dynamic response shifts to the side between the two tanks, and the maximum wind speed is located in the area between the two tanks.

It can be seen that the wind interference effect has a great influence on the dynamic response and wind field of liquid storage tanks. Liquid storage tanks in actual oil storage facilities are basically arranged side by side. In order to ensure their safety, it is necessary to consider the wind interference effect.

#### **7. Conclusions**

Considering shell–liquid interaction and shell–wind interaction, calculation model of liquid storage tanks is established. The dynamic responses of liquid storage tanks under wind, earthquake, and wind and earthquake are studied comparatively. Besides, the influences of wind speed and wind interference effect on the dynamic responses of the liquid storage tank are discussed. The main conclusions are as follows:


(5) In order to consider combination of wind and earthquake, the effective stress, displacement, and base shear force obtained by conducting SRSS for wind and earthquake alone conditions are very different from considering wind and earthquake at the same time.

**Author Contributions:** Methodology, W.J.; software, W.J. and H.F.; formal analysis, W.J. and H.F.; writing—original draft preparation, W.J.; writing—review and editing, X.C.

**Funding:** This paper is a part of the China Postdoctoral Science Foundation (grant no. 2018M633652XB), a part of the National Natural Science Foundation of China (grant no. 51368039), a part of the Hongliu Outstanding Young Talents Support Program of Lanzhou University of Technology (grant no. 04-061807) and a part of the Open Foundation of International Research Base on Seismic Mitigation and Isolation of Gansu Province (grant no. TM-QK-1904).

**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* **Experimental Study of High-Strength Concrete-Steel Plate Composite Shear Walls**

### **Dongqi Jiang 1,\*, Congzhen Xiao 2, Tao Chen <sup>2</sup> and Yuye Zhang <sup>1</sup>**


Received: 17 June 2019; Accepted: 8 July 2019; Published: 15 July 2019

**Abstract:** Shear walls are effective lateral load resisting elements in high-rise buildings. This paper presents an experimental study of the seismic performance of a composite shear wall system that consists of high-strength concrete walls with the embedded steel plate. Two sets of wall specimens with different aspect ratios (height/width, 1.5 and 2.7) were constructed and tested under quasi-static reversed cyclic loading, including five reinforced concrete shear walls (RCSW) and six reinforced concrete-steel plate shear walls (RCSPSW). The progression of damage, failure modes, and load-displacement responses of test specimens were studied and compared based on experimental observations. The test results indicated that high-strength (HS) RCSPSW system showed superior lateral load strength and acceptable deformation capability. The axial compressive load was found to have an indispensable effect on the ductility of both RCSW and RCSPSW, and an upper limit of axial compression ratio (0.5) is recommended for the application of HS RCSPSW in engineering practices. In addition, the design strength models were suggested for predicting the shear and flexure peak strength values of RCSPSW systems, and their applicability and reliability were verified by comparing with test results.

**Keywords:** composite shear wall; seismic behavior; quasi-static test; design strength model

#### **1. Introduction**

Shear walls serve as an effective structural element for resisting lateral load in tall buildings during seismic events. Conventional reinforced concrete shear walls (RCSW) are considered to be a cost-effective way for preventing structure collapse and ensure life-safety, which has been widely used in the design of low- and medium-rise building structures in seismic regions [1]. With the increase of building height, vertical axial load demands at lower floors exponentially grow, resulting in much thicker RCSW with the consideration of code-specified axial compression ratio limits [2–4]. Under such circumstance, excessively thick walls reduce the usable floor areas and increase the gravity load intensity, causing more seismic forces and severer structural damage. A review of past research work indicated that RCSW were mainly tested under low axial compressive loadings [1,5–7]. The first explanation is that reinforced concrete buildings are usually designed with a low axial compression ratio in high seismicity zones due to code limits. The second possibility is that conventional RCSW undergo significant structural damage in the form of undesirable failure patterns and deteriorated strength and ductility performances when subjected to high axial loadings [8]. Su [9] observed an abrupt out-of-plane compressive failure mode on experimentally tested RCSW under high axial compression ratios, and concluded that significant strength degradation and ductility deterioration would occur with the increase of the axial compression ratio. Moradi [10] conducted a comprehensive research study to set up a library of critical parameters that affect the behavior of shear walls based on

a large amount of existing experimental tests, from which a predictive meta-model was developed to forecast the responses of desired shear walls.

Researchers explored the composite steel-concrete wall system by adding structural steel in the wall section, which potentially takes advantages of two materials' mechanical properties in order to improve the seismic performance of concrete shear walls under high axial loading and decrease the wall thickness. Dan [11] tested composite shear walls with I-shape or box-shape steel profiles that were placed at the extremities, and found that the lateral resistance and displacement ductility can be effectively improved by encased steel profiles. Tong and Hajjar [12,13] studied RCSW partially restrained by the steel frame at boundaries. It is concluded from the experimental results that this composite structural system provided adequate strength and stiffness to resist lateral forces and it would be applicable for low- and medium-rise buildings. Some researchers investigated the mixed use of RCSW and circular or rectangular concrete-filled steel tubular (CFST) columns [14–16]. The existing experimental results indicated that the strength and ductility performances were effectively improved by using CFST columns as the boundary elements. Previous research studies show that the structural steel is usually arranged at the extremities of reinforced concrete wall sections to form composite boundary elements; Cho [17], Esaki [18], and Zhou [19] conducted similar research work. For such a structural system, RCSW mainly serve as a shear-resisting component, while the function of composite boundary elements is to resist the overturning moment.

Apart from the use of composite boundary elements, researchers began to make use of the structural steel plate in the wall panel to further improve the lateral load performance in recent years. The concrete-steel plate composite shear wall system can be classified into two types: concrete-filled steel plate (CFSP) shear wall and steel-plate-embedded concrete shear wall. Hu and Nie [20,21] experimentally studied the CFSP composite shear walls and concluded that this wall system has larger lateral strength and deformation capacity under high axial compressive loadings. Hossain [22] and Rafie [23] reported similar findings on the shear wall system that consists of two skins of profiled steel sheeting with an in-fill of concrete. However, corrosion-protection and fire-protection layers are necessary for CFSP walls, because the steel plate is directly exposed to the environment, which increases the difficulty and complexity of construction. Therefore, the steel-plate-embedded concrete shear wall has advantages in engineering practice and has attracted the attention of researchers and practitioners.

Zhao [24] proposed a composite shear wall system comprising an infill steel plate with concrete panels that are attached on one side or both sides, and it is included in AISC Seismic Provisions and denoted as concrete stiffened steel plate shear wall (CSPSW) [25]. Note that a gap exists between concrete panel and the steel frame, thus the concrete panel merely works as a stiffer to prevent the bulking of the infill steel plate. The China Academy of Building Research (CABR) systematically studied the monolithic cast-in-place (CIP) RCSW with embedded steel plate, named as reinforced concrete-steel plate shear walls (RCSPSW). Figure 1 presents configurations of conventional RCSW and RCSPSW. The latter wall system consists of three major components, as depicted in Figure 1b. Concrete and steel reinforcement are identical to RCSW, while the I-shaped steel profiles are encased in boundary elements and the steel plate is embedded in the wall web. Sun [26] and Chen [27] investigated the shear and flexure behavior of normal-strength concrete-SP shear walls and demonstrated their good performances in both strength and deformation capabilities. Xiao [28] studied the effects of aspect ratios on lateral load performances of RCSPSW based on the previous research outcome. Subsequently, RCSPSW was successfully applied in the construction of high-rise buildings in China, such as: Shanghai Tower (632 m in height), Guangzhou East Tower (530 m in height), etc. [29].

**Figure 1.** The Concept of Reinforced Concrete–Steel Plate Composite Shear Wall System.

Not content with the benefits that RCSPSW brought, practitioners claimed that there is still a need to further reduce the wall thickness for super-tall buildings. In response to their expectation, the use of high-strength (HS) concrete in RCSPSW system becomes a potential option, because the high compressive strength of HS concrete can be an advantage when the walls are subjected to high axial compressive loadings. Meanwhile, whether the brittleness of HS concrete affects the ductility of RCSPSW remains in question. Jiang [30] and Xiao [31] investigated the compression-bending behavior of HS concrete shear walls and explore viable structural steel arrangements in the wall section to improve the deformability. The research outcome demonstrated that the flexure strength and ductility were effectively improved by the steel profiles encased in boundary elements. However, the overall seismic performances of HS RCSPSW system remain unknown to researchers. Additionally, the progressions of damage and failure modes of HS RCSPSW with different aspect ratios are not clear. The authors have conducted a comprehensive experimental investigation to study both shear and flexure behavior of HS RCSPSW in order to verify the reliability of HS RCSPSW system in engineering practices. Eleven high-strength concrete shear walls with two different aspect ratios (1.5 and 2.7) were constructed and tested under quasi-static reversed cyclic loading, including five conventional RCSW

and six RCSPSW. Lateral load performance and failure modes are compared and discussed based on experimental observations. This paper emphasizes the experimental investigation and the design strength models of HS RCSPSW. Analytical and numerical models, as well as ways to incorporate axial–shear-flexure interaction, will be presented in a follow-up paper.

#### **2. Experimental Investigation**

#### *2.1. Specimen Properties*

Two batches of specimens were designed, constructed, and tested under quasi-static cyclic loadings, including RCSW and RCSPSW. Table 1 lists key information for test specimen configurations. Specifically, the specimens in Batch No.1 (RCSW 1-X and RCSPSW 1-X) with the aspect ratio (height/width, *h*/*b*) of 1.5 were designed to study the shear behavior, while the specimens in Batch No.2 (RCSW 2-X and RCSPSW 2-X) with a higher aspect ratio (2.7) were mainly designed for the flexure behavior investigation. These two aspect ratios were selected based on available references [1,8–10,14,19,32–35] and past engineering experience in high-rise building design to purposely achieve shear failure mode and flexure failure mode, respectively. Apart from the aspect ratio, the vertical axial load is another critical parameter that affects the seismic performance of shear wall specimens. As listed in Table 1, a wide range of vertical load values were applied on shear specimens (900–1500 kN) and flexure specimens (2180–3050 kN) in the quasi-static test to simulate different levels of axial loads that would be expected in the design of high-rise buildings. An axial compression ratio, *n*, is defined as the ratio of the applied vertical load to the compression capacity of the wall section and it is given by

$$m = \frac{N}{f\_{\text{C}}A\_{\text{C}} + f\_{\text{A}\text{Y}}A\_{\text{A}} + f\_{\text{P}\text{Y}}A\_{\text{P}}} \tag{1}$$

where *N* represents the vertical load that is applied by the hydraulic jack, *f* c is the compressive strength of concrete, *f* ay and *f* py are yielding strength values of I-shape steel profiles and steel plate, and *A*c, *A*a, and *A*<sup>p</sup> indicate the area of concrete section, boundary I-shaped steels, and embedded steel plate. For given vertical loads, axial compression ratio ranges from 0.33 to 0.50 for shear specimens and 0.42 to 0.70 for flexure specimens. Note that these values are relatively high for shear walls in building structures, and very few past research work has been performed while using such high axial compression ratios.

Figure 2 shows the cross-sectional dimensions and reinforcing detailing for RCSW and RCSPSW. All of the specimens were designed with the target concrete cubic compressive strength of *f* c,cube = 80 MPa. The use of high-strength concrete not only effectively increases the lateral capacity, but it also significantly reduces the required wall thickness due to code-specified axial compression ratio limits in the design of tall buildings [3,4]. The geometrical sizes and steel reinforcement were carefully designed, so that desirable failure mechanisms would take place in the test for different batches of specimens. For instance, longitudinal boundary reinforcements with relatively smaller diameters (ϕ8 and ϕ10) were purposely implemented in specimens with the aspect ratio of 2.7, so as to observe a flexure mode of failure prior to achieving a shear mode of failure. The geometric dimensions and reinforcement arrangement were identical for RCSW and RCSPSW specimens with specified aspect ratios. The shear specimens consist of an 800 mm wide by 80 mm thick wall web and two 100 mm wide by 120 mm thick boundary elements. As for flexure specimens, the thickness values for the web and boundary elements are 150 mm and 190 mm, respectively. The distances that were measured from the top surface of the footing to the lateral loading point are 1.2 m and 2.16 m for shear and flexure specimens, as shown in Figure 2. For the RCSPSW specimens, I-shaped steels in boundary elements were welded to the 5 mm thick steel plate that was embedded in the wall web, and steel studs or tie bars were used to ensure the bonding strength between the concrete and steel plate. Longitudinal reinforcement was placed in

boundary elements, and steel ratios of 16.9% and 1.35% were used for shear and flexure specimens, respectively. Table 1 lists the steel reinforcement ratios in wall web.


**Table 1.** Test Specimen Configurations.

RCSW #-# represents conventional reinforced concrete shear walls and RCSPSW #-# represents new reinforced concrete–steel plate composite shear walls.

Table 2 lists key material properties of test specimens. The 28-day cubic compressive strength, *f* ck,cube, for shear specimens and flexure specimens are 72.3 MPa and 84.1 MPa, respectively. Two types of steel reinforcements (HPB235 and HRB 335) were used in the wall specimens. Table 2 summarizes the measured yield strength *f* yk and ultimate strength *f* tk of steel reinforcements and steel plates.


**Table 2.** Material Properties.

**Figure 2.** Cross-sectional Dimensions and Reinforcement Details of RCSW and RCSPSW.

#### *2.2. Test Setup, Instrumentation and Load Protocol*

The quasi-static load test was conducted on shear and flexure specimens to evaluate the seismic performance of the HS RCSPSW system. Figure 3 presents the experimental setup details. Wall specimen was anchored onto the laboratory rigid floor with fasteners through the footing. One 1500 kN servo-controlled hydraulic actuator along the east-west direction was installed on the reaction wall to provide racking loads on the specimen during the reversed cyclic load test. The vertical load was applied through a 4000 kN capacity hydraulic jack that was installed beneath the rigid steel frame, and the vertical load is uniformly spread along the top surface of the wall specimen by means of two steel distribution beams. Out-of-plan steel bracings were used to keep the specimen movement in-plane and avoid twisting during testing, as can be seen in the setup photograph (Figure 3d).

**Figure 3.** Details of Experimental Setup (**a**) Plan View; (**b**) Front View (Section B-B); (**c**) Side View (Section A-A); and, (**d**) Photograph of Experimental Setup.

Figure 4 depicts the instrumentation layout of test specimens. The load cells were installed inside the hydraulic jack and hydraulic actuator to measure the vertical and lateral forces during the test. A Linear Variable Differential Transformer (LVDT) was attached to the surface of the loading beam to measure the displacement under racking loads. As shown in Figure 4c, the strain rosettes with 60-mm gauge length were attached in diagonals on the concrete surface of shear specimens to capture

the shear cracks. In terms of flexure specimens, individual concrete strain gauges with 60-mm gauge length were installed on the bottom part of wall panel to capture the flexure-tension cracks. Similar strain rosette and strain gauge arrangements were adopted for the embedded steel plate to measure the strain values, but the gauge length is much smaller (5 mm). Additionally, individual strain gauges with 5-mm gauge length were used to monitor the longitudinal strains in the vertical and horizontal reinforcements, as well as the I-shape steel profiles.

**Figure 4.** Instrumentation Layout of Test Specimens (**a**) Strain Gauges on Steel Reinforcements; (**b**) Strain Gauges on I-shaped Steel Profiles and Steel Plate; and. (**c**) Strain Gauges, Linear Variable Differential Transformer (LVDT) & Load Cells on Concrete.

Figure 5 shows the cyclic lateral displacement history that was used in the quasi-static test. The specified axial compressive load was constantly applied on the top surface of wall specimen throughout the test. The two phase mixed lateral force/displacement control cyclic loading scheme was used herein, as specified in Chinese Specification for Seismic Test of Buildings [36] and suggested

by Zhang [8], Zhou [19], and Nie [21]. Force control cycles were adopted before yielding occurs in shear wall specimens, and only one cycle is applied in each load step. It is possible to determine the accurate lateral forces corresponding to the first visible crack initiation and crack pattern propagation with dividing the first phase into different levels of load cycles. Once the yielding has commenced, displacement-control tests were performed by gradually increasing reversed cyclic displacements in the form of triangular waves and two repeated cycles of the same displacement amplitude were applied for each displacement increment. The magnitude of displacement in each level equals multiple times of the displacement at yield, *y* = *n*Δy, which also indicates the increasing ductility level at each cycle. The reversed cyclic load test ends until the lateral force drops below 85% of the maximum lateral load capacity. The yield dispalcemnt value, Δy, is taken as 2 mm and 6 mm for shear and flexure specimens, respectively, in this series of experimental tests based on a trial test recording.

**Figure 5.** Cyclic Displacement History for Quasi-static Test of Shear Wall Specimens. Δ*y* indicates the lateral displacement when the wall specimen shows yielding behavior.

#### *2.3. Analysis Methods for Experimental Observations*

Key information was inferred from the instrumentation recordings in the reversed cyclic load test to evaluate the seismic performance of shear walls, including hysteresis curve, skeleton curve, ductility, energy dissipation ability, etc. Figure 6a presents a typical hysteresis curve that is directly obtained by the lateral force and displacement readings from the load cell and LVDT, and the skeleton curve is determined from the hysteresis curve by joining the peak-load tips of each primary loop.

Ductile structures are preferable in the earthquake-resistant design. Structural members are required to undergo large amplitude lateral deformations without the substantial loss of strength and also dissipate significant amounts of energy in those cyclic deformations. A displacement ductility factor, μ, is defined as the ultimate displacement, Δu, to the displacement at yield, Δy, and it is given by

$$
\mu = \frac{\Delta\_{\rm u}}{\Delta\_{\rm y}} \tag{2}
$$

where the ultimate displacement (Δu) is defined as displacement value that corresponds to 85% of peak load on descending branch of the skeleton curve as specified in Chinese Specification for Seismic Test of Buildings [36] and suggested by Dan [11], Tong [12], and Liao [37]. The yield displacement (Δy) is determined with the method that Priestley recommended [38], which will be described in detail later.

The definition of the yield point in the skeleton curve often causes difficulty in the calculation of ductility factors. Figure 6b illustrates three alternative definitions of the yield displacement and the corresponding yield strength. Priestley [38] suggested that the yield displacement be determined by the equivalent elasto-plastic system with the secant stiffness at 75% of the peak lateral load, *Pmax* (Figure 6(b1). Mahin [39] proposed the use of the equivalent elasto-plastic system with the same energy absorption from the origin to the peak lateral load level, as shown in Figure 6(b2)). In ASTM E2126 [40], an equivalent energy elastic-plastic (EEEP) is defined as an elastic-plastic curve that circumscribes an area that is equal to the area enclosed by the skeleton curve between the origin, the displacement axis, and the ultimate displacement (Figure 6(b3)). Park [41] claimed that the first definition is the most realistic option to determine the yield displacement for reinforced concrete structures, which will be used herein to calculate the displacement ductility factors of test shear wall specimens.

(**c**) Definition for Equivalent Viscous Damping Coefficient

**Figure 6.** Analysis Methods for Experimental Observations (**a**) Hysteresis Curve and Skeleton Curve; (**b**) Yield Strength and Yield Displacement; and, (**c**) Equivalent Viscous Damping Coefficient.

As shown in Figure 6a, structural members have some deformation capacity beyond the peak lateral load, *Pmax*, with undergoing a small reduction of strength. Thus, it is reasonable to recognize part of the post-peak deformation capacity, and the ultimate displacement is defined herein as the point when the load carrying capacity drops below 85% of the peak strength on descending branch of the skeleton curve, or when severe collapse failure takes place in the specimen, whichever occurs first.

The ductility factor is defined in terms of deformation, which provides no indication regarding the energy dissipation capacity of the structural members. The equivalent viscous damping (EVD) coefficient is considered to be a critical parameter in evaluating the energy dissipation ability of given structural members. For shear wall specimens subjected to reversed cyclic loadings, the EVD coefficient can be obtained by a function of the dissipated energy of the structural member, *E*Di, and the elastic energy stored in an equivalent viscous system, *E*Si, in cycle *i* [42]. The energy that is dissipated by the structural member can be expressed as the area enclosed by the hysteresis loop (*S*ABCD) and the stored energy is measured by areas within two triangles of the first and third quadrants (*S*AOF and *S*COF), as depicted in Figure 6c. The formal equation is given by

$$\zeta\_{\rm ei} = \frac{1}{2\pi} \frac{E\_{\rm Di}}{E\_{\rm Si}} \tag{3}$$

where ζei is the EVD coefficient for the cycle *i*.

#### **3. Experimental Results and Discussions**

A total of 11 shear wall specimens were tested under reversed cyclic loadings. The progressions of damage and failure modes for all specimens were observed throughout the test procedure. Performance parameters, including lateral load capacity, ultimate displacement, ductility factor, and EVD coefficient, were determined.

#### *3.1. Progression of Damage and Failure Modes*

Specimen RCSW 1-1 was conceived to represent a conventional RCSW with low aspect ratio (1.5). The first inclined crack was observed on the web surface during the 200 kN load step, and then horizontal tension cracks formed at the lower part of boundary elements during the 300 kN load step. Afterwards, transverse steel bars in the web and vertical steel bars at the extreme end of the boundary toe yielded at the load step of 400 kN, and the corresponding lateral displacement value is 2 mm. In the following steps, displacement-controlled cycles were adopted and the inclined cracks gradually extended. The quantity of boundary horizontal cracks and the web inclined cracks significantly increased and the crack width reached up to 0.35 mm during the 4 mm load step (0.33% drift ratio). The specimen achieved the maximum lateral load capacity at the 8 mm load step (0.67% drift ratio) and the concrete cover started spalling during that load step. Significant concrete spalling was observed at both wall web and boundary toes, and the lateral load strength substantially reduced at the 16 mm load step (1.33% drift ratio), when the test ended.

In general, specimen RCSW 1-1 showed a typical shear damage pattern as the representative of shear specimens. On the contrary, the flexural damage pattern is primarily observed in specimens with the aspect ratio of 2.7 and the damage progression of RCSW 2-1 is described in detail as an example. The horizontal flexure-tension crack is initiated at the 350 kN load step in the bottom of the boundary elements, and it was then extended to the web surface with the crack width expanding up to 0.35 mm at the 400 kN load step. Longitudinal steel bars in the boundary elements yielded at the 450 kN load step and the lateral displacement reaches 11 mm (0.51% drift ratio). Vertical cracks in boundary toes were observed at the 15 mm load step (0.69% drift ratio) with little concrete cover spalling, and the existing cracks were further diagonally extended to the web surface with the crack widths increasing up to 1.4 mm. Subsequently, the specimen reached the maximum positive and negative lateral loading capacities with more severe concrete spalling at +21 mm (0.97% drift ratio) and −27 mm (1.25% drift

ratio) load steps. The test of specimen RCSW 2-1 ended after the cycle of 39 mm (1.81% drift ratio) when the boundary steel rebar fractured and concrete at wall toes also crushed.

Figures 7 and 8 presents crack distributions at different inter-story drifts (ISD) and photographs of failure patterns for all of the specimens. Three ISD values (0.1%, 0.33%, and 1.0%) were selected to evaluate in-service performances of shear walls when subjected to minor, moderate, and major earthquakes. It can be seen from Figures 7a and 8a that no cracks or very few cracks were observed at the ISD of 0.1% and all of the specimens remain in the elastic range at that stage. As the level of ISD increased, more web inclined cracks and boundary horizontal cracks occurred in the shear specimens with the crack width expanding. In terms of flexure specimens, the quantity of boundary horizontal cracks increased, some of which diagonally extended to the web, forming flexure-shear cracks. When the ISD value reached 1%, very dense cracks were distributed on the wall surface. In contrast to crack-riddled shear specimens (Figure 7a), the cracks were mainly distributed in the lower half of flexure specimens, and the upper half almost kept undamaged (Figure 8a). As compared to the RCSW specimens, more densely-distributed cracks were shown in the RCSPSW specimens, but the crack widths are much smaller based on observations. In addition, with the increase of the axial compressive load, the quantity of distributed cracks becomes less and the crack width is smaller due to the compaction of compression forces.

Figure 7b shows the failure modes for shear specimens. Significant spalling of concrete cover along the shear cracks in wall web and concrete crushing in boundary toes were observed in the RCSW shear specimens. As for RCSPSW shear specimens, the embedded steel plate tended to buckle at the failure point, accompanied by severe concrete spalling in the lower part of the wall web. When it comes to flexure specimens, concrete crushing at the toe and longitudinal steel rebar facture usually occurred in the boundary elements, as shown in Figure 8b. Severe spalling of concrete cover in the bottom of wall web is also observed in specimen RCSPSW 2-1. In general, the RCSPSW specimens show a denser crack distribution but the crack width is smaller. Moreover, the damage observed in RCSPSW specimens is not as severe as RCSW.

The failure modes of shear wall specimens were controlled by axial-flexure-shear interactions that were affected by a variety of parameters, including aspect ratio, axial compression ratio, structural steel arrangement, etc. With the variation of key parameters, aspect ratio, in particular, the failure characteristics transit from 'shear failure' to 'flexural failure'.

Figure 9a shows a photograph of shear failure patterns referred to past research studies [1,19,43]. Shear critical walls mainly exhibit a diagonal cracking pattern on the wall surface. The first inclined crack at an angle of approximately 45◦ is initiated by the shear stress at the corner of the wall and then extends to the mid portion. Upon further loading cycles, new inclined cracks occur and existing cracks extend; the strains of reinforcing bars and the widths of intercrossing main diagonal cracks are developed until the peak lateral load is attained. Afterwards, the spalling and crushing of concrete take place at corners and the diagonal cracking becomes severer until the wall fails. In general, shear-critical walls usually fail by diagonal tension or diagonal compression. Typical shear failure characteristics include the formation of diagonal cracks, yielding of reinforcing bars, and spalling of the concrete cover at the intercrossing diagonal cracks region or wall toes, as illustrated in Figure 9b.

**Figure 7.** Crack Distributions and Failure Patterns of Specimens with Aspect Ratio of 1.5 (**a**) Crack Distributions at Different Drifts and (**b**) Photograph of Failure Patterns.

**Figure 8.** *Cont.*

**Figure 8.** Crack Distributions and Failure Patterns of Specimens with Aspect Ratio of 2.7 (**a**) Crack Distributions at Different Drifts and (**b**) Photograph of Failure Patterns.

**Figure 9.** Failure Modes of Shear Wall Specimens.

Figure 9c presents the flexure failure patterns and associated crack distribution excerpt from references [8,9,32]. Flexure-dominant failure starts with horizontal cracks that occur at the wall base on the tensile zone. With the load increasing, new horizontal cracks appear along the wall height and existing cracks gradually expand and propagate inwards to the core of the section. These cracks eventually form an inclined cracking pattern in the web. Upon further loading cycles, vertical reinforcing bars at the wall toe yield and significant inclined flexure-shear cracks form on the lower portion of the wall. Vertical cracks appear at the bottom edge of the compression zone with continuing loading. After the lateral load decreases, the concrete cover at the toe in compression spalls off, finally the failure occurs with the crushing of concrete and buckling of reinforcing bars and steel profiles. The typical flexure failure characteristics include horizontal flexure-tension cracks initiated at wall boundaries, inclined flexure-shear cracks on the wall web, longitudinal reinforcement yielding, vertical cracks and concrete cover spalling at the wall toe, concrete crushing in the compression zone, and buckling of reinforcing bars, as shown in Figure 9d.

In the experimental study that is presented herein, specimens with the aspect ratio of 1.5 exhibited a shear failure mode with significant diagonal cracks appearing in the wall web. As for specimens with the aspect ratio of 2.7, the flexure-dominant failure mode is identified with horizontal cracks initiating at the ends, concrete spalling, and tensile fracturing of steel bars. A special characteristic that is observed in RCSPSW specimens is that the embedded steel plate tended to buckle when failure occurred, which was accompanied by severe concrete spalling in the lower portion of the wall. In general, more characteristics of shear failure were observed on the specimens with the decrease of the aspect ratio. A higher axial compression ratio restrained the development of inclined cracks in the web for the reason that the principle tensile stress would be reduced with the increase of the axial load based on experimental observations. Additionally, the "distance" between concrete spalling and wall failure drastically reduces as the axial compression ratio increasing, which results in a more brittle failure mode.

The wall panel in RCSPSW is divided into two halves by the embedded steel plate when compared with conventional RCSW, which results in relatively weaker mechanical collaboration between concrete and steel. It is very important to ensure the effective bonding between two parts to support the engineering application of RCSPSW, although no debonding failure was observed in the quasi-static test. Potential approaches include: (1) using ribbed steel bars with rough surface to increase the mechanical adhesion and friction; (2) specifying sufficient concrete cover to effectively confine steel reinforcement; (3) connecting reinforcement mesh to the embedded steel plate with steel ties, so that the RCSPSW works as a monolithic system; and, (4) adding steel fibers in the concrete to increase the bond strength, as suggested by Dancygier [44] and Harajli [45]. Besides, the use of fiber composite elements is helpful in strengthening the bonding between concrete and steel, and the techniques that were developed by Gattesco [46] using glass fiber reinforced polymer (GFRP) can be deployed.

#### *3.2. Force-Displacement Responses*

Figure 10 presents the hysteresis curves (blue line) and skeleton curves (red line) for the shear specimens. Significant "pinching" effects were observed in the hysteresis loops of RCSW shear specimens. With the embedment of the steel plate, the hysteresis curves appear in a plumper shape with higher peak lateral load capacities, which indicated that RCSPSW specimens have better seismic performances. Figure 11 shows comparisons among the skeleton curves of all shear specimens. The blue, red, and green lines in Figure 11a,b represent skeleton curves of specimens under the axial compression ratio of 0.33, 0.45, and 0.50, respectively. The dash and solid lines in Figure 11c show the curves of the RCSW and RCSPSW specimens. It is evident that the use of embedded steel plate significantly increased the lateral load capacity, but a severer and quicker post-peak strength and stiffness degradation was observed in the RCSPSW specimens as compared to RCSW specimens under same axial compression ratios. In terms of the RCSPSW 1-1 specimen, it shows better deformation capability than other RCSPSW specimens, which is attributed to a lower axial compression ratio. Additionally, the positive lateral load re-increased after the strength decay in the previous step, which may result from the hardening of the steel plate.

Figure 12 shows the hysteresis and skeleton curves for flexure specimens. Overall, the RCSW specimens show a S-shaped hysteretic behavior with a "pinching" phenomenon. The hysteresis curves of RCSPSW specimens appear in a full bow shape, which indicates that mixed flexure-shear failure mechanisms exist in the damage progression of flexure specimens. From the results that are presented in Figure 12, it is evident that "fatter" hysteresis loops were obtained for specimens when they were subjected to lower axial compressive loads. Figure 13 presents the comparative results of skeleton curves for different flexure specimens. As compared to the RCSW specimens (dash lines), the RCSPSW specimens (solid lines) show higher lateral load capacities. The skeleton curves with the same color represent specimens subjected to the same axial compressive load. With the increase of the axial compressive load, the peak lateral load increased, but the ultimate lateral displacement decreased. A severer and quicker strength and stiffness degradation was observed for RCSPSW specimens with an axial compression ratio higher than 0.50.

**Figure 10.** Hysteresis and Skeleton Curves for Specimens with Aspect Ratio of 1.5 (**a**) RCSW Specimens and (**b**) RCSPSW Specimens.

**Figure 11.** Comparison among the Skeleton Curves for Specimens with Aspect Ratio of 1.5 (**a**) RCSW Specimens; (**b**) RCSPSW Specimens; and, (**c**) All Shear Specimens.

The key characteristics were summarized for all shear wall specimens with considerations of the damage progression and force-displacement responses, although differences do exist among their skeleton curves. Figure 14a shows the skeleton curve of RCSW 1-1 specimen as a representative example, where five critical points are defined, including crack point, yield pint, peak point, failure point, and collapse point. Each critical point represents an event when the specimen's behavior is significantly altered. The crack point corresponds to the load step when the first web inclined shear crack or boundary horizontal flexure-tension crack is observed. The determination of yield point, peak point, and failure point has been described in the previous section. The collapse point indicates the tip of last load cycle if the post-ultimate load capacity (0.85*Pmax*) exists. It is worth mentioning that the yield point and failure point in the load-displacement skeleton curve (Figure 14b) can be determined while using alternative approaches. Smarzewski and Pan [47,48] suggested the equivalent elastoplastic line determine the yield displacement with the secant stiffness at two-thirds of the peak lateral load. As for the failure load, Smarzewski and Lim [47,49] recommended the use of 80% of the peak lateral load, resulting in a relatively larger ultimate displacement as compared to the 0.85*Pmax* defined herein. Based on trial calculations, the yield displacement determined with the approach that was suggested by Smarzewski is slightly lower than the value that was calculated using the Priestley's method, which is adopted in current research work. Therefore, the ductility factors of shear wall specimens determined with Smarzewski's approach would be larger than the values that are presented in this paper.

59

**Figure 12.** Hysteresis and Skeleton Curves for Specimens with Aspect Ratio of 2.7 (**a**) RCSW Specimens and (**b**) RCSPSW Specimens.

**Figure 13.** Comparison among the Skeleton Curves for Specimens with Aspect Ratio of 2.7 (**a**) RCSW Specimens; (**b**) RCSPSW Specimens; and, (**c**) All Flexure Specimens.

**Figure 14.** Definition of Critical Points and Working Stages of Shear Wall Specimens.

Based on critical load points, the entire loading procedure was divided into five stages: elastic stage, cracking stage, yielding stage, failure stage, and collapse stage, as shown in Figure 14b. A summary of structural behavior of shear wall specimens in each stage is briefly described, as follows.

(a) Elastic stage (origin to crack point): the specimen keeps intact or minor damage is observed. The structural behavior remains in the elastic range and the load-displacement curve keeps linear.

(b) Cracking stage (crack point to yield point): as the lateral load increasing, more web inclined cracks and boundary horizontal cracks occur in the shear specimens. As for flexure specimens, the quantity of boundary horizontal cracks increases and some of them diagonally extended to the wall web, forming flexure-shear cracks. Crack width expands, but the damage is repairable. The nonlinearity develops in the load-displacement curve.

(c) Yielding stage (yield point to peak point): web transverse steel bars and boundary longitudinal steel bars are gradually yielding in shear and flexure specimens. The crack distribution becomes denser and the crack width further increases. Vertical cracks and concrete cover spalling occasionally take place in the boundary toe. The damage becomes irreparable in this stage. The lateral load continues increasing, but the stiffness value decreases until the peak load is observed.

(d) Failure stage (peak point to failure point): the lateral load capacity starts to reduce. Significant concrete spalling occurs in boundary toes or web panels. Embedded steel plate tends to buckle out-of-plane. The damage is severe and the cycle load test may end in this stage.

(e) Collapse stage (failure point to collapse point): the damage is extremely severe in this stage. The later load capacity continues reducing until the collapse point occurred.

#### *3.3. Lateral Load and Dispalceent Capacity*

Table 3 lists lateral load, lateral displacement, and corresponding drift values for the shear specimens at yield point, peak point, and failure point. It is seen that the positive and negative values that were obtained from the cyclic test are asymmetrical, which is also observed in the hysteresis and skeleton curves (Figures 10 and 12). The explanation is when the specimen undergoes damage in one direction; its lateral load capacity is slightly weakened, as it is racked in the opposite direction. The average values are used herein for comparative studies. As compared to RCSW specimens, RCSPSW specimens own approximately 100% higher yield loads, peak loads, and ultimate loads, proving the efficiency of the embedded steel plate in the improvement of shear load capacities. As for the ultimate displacement, the RCSPSW specimens do not show obvious superiority over RCSW specimens, except for RCSPSW 1-1 under lower axial compression ratio (0.33). In general, RCSPSW shear specimens are capable of withstanding an ultimate drift value of 1.0% approximately, showing acceptable deformability for design purpose. The last column lists ductility factors for all shear specimens that were determined by Equation (2). The RCSW specimens own relatively higher ductility factors than RCSPSW specimens under the same axial compression ratio.

Table 4 summarizes the lateral load and displacement capacities of flexure specimens. In general, shear wall specimens with a higher aspect ratio show better ductility performances, because the flexural failure mode governs. Similar with shear specimens, the embedment of steel plate is able to increase the lateral load capacities of the flexure specimens. In particular, the peak load capacities of RCSPSW are 20–30% higher than RCSW under the same axial compressive loads. The RCSW flexure specimens show good deformability with the ultimate drift value of around 1.5% and the ductility factor higher than 4. As for RCSPSW specimens, the deformability is satisfactory for design purposes when the axial compression ratio is lower than 0.50 (ultimate drift is larger than 1.0% and the ductility factor is around 4). As the axial compression ratio increases to 0.58, the ductility factor substantially decreases to 2.61 and the ultimate drift is lower than 1.0%.

Figure 15 presents the effects of the axial compression ratio on peak lateral load capacities, ductility factors, and ultimate drift values of all specimens. Blue lines and red lines represent values for shear specimens and flexure specimens; dash and solid lines are results for RCSW specimens and RCSPSW specimens. It is seen from Figure 15a that all the specimens' lateral load capacities increase with the axial compression ratio, except for conventional RCSW shear specimens (RCSW1 series). For the RCSPSW specimens, the ultimate drift decreases with the axial compression ratio. In particular, the value drops below 1% when it is subjected to the highest axial compressive loads (1500 kN for shear specimens and 3050 for flexure specimens). The ductility factor generally decreases with the axial compression ratio for RCSPSW specimens, but the trend is not as straightforward as the ultimate drift. The relationship between the deformability and axial compression ratio for RCSW specimens (blue lines) is not clear, as seen in Figure 15b,c, but their ultimate drift and ductility factor values are larger than those of the RCSPSW specimens (red lines) when subjected to the axial compression ratio higher than 0.5.

**Figure 15.** Effects of Axial Compression Ratio on (**a**) Peak Lateral Load; (**b**) Ultimate Drift; and, (**c**) Ductility Factor.

It is observed from the experimental test that the RCSPSW specimens show a relatively lower deformability under high axial compression ratios, which could be attributed to the brittleness of high strength concrete. Another explanation is the weak concrete confinement effect on the embedded steel plate due to the small specimen size. The thickness values of the concrete web on both sides are only 40 mm and 75 mm for shear and flexure specimens. The concrete cover was easily spalled under higher axial compressive loads, and the embedded steel plate tended to buckle when severe concrete spalling took place. Under such circumstance, the strength and stiffness decay quickly and substantially decay. In addition, shear studs were only used in the footing and the connection between the steel plate and concrete wall web is weak, which exacerbate the spalling and strength reduction. Therefore, it is suggested that the shear wall structural component should be designed with caution when subjected to high axial compressive loads and an upper limit of axial compression ratio (0.5) should be set for RCSPSW. Besides, shear studs or steel ties should be used to strengthen the connection between the steel plate and concrete on both sides. Additionally, a higher transverse reinforcement ratio is recommended for further improving the concrete confinement effect. It is worth mentioning that the wall is much thicker in the practical design of high-rise buildings. Thus, the confinement effect and deformability of RCSPSW in future applications could be better than the experimental observations.



#### *Appl. Sci.* **2019** , *9*, 2820


**Table 4.** Lateral Load and

Displacement

 Capacity of Flexure Specimens.

*Appl. Sci.* **2019**

, *9*, 2820

#### *3.4. Energy Dissipation Ability*

The EVD coefficients for all specimens were calculated with Equation (3) to compare their energy dissipation abilities. Table 5 lists hysteresis loop areas and EVD coefficients of all the test specimens for test cycles at the peak lateral load (peak point).


**Table 5.** Equivalent Viscous Damping (EVD) Coefficients of All Specimens.

In general, the energy dissipation capacities that were found in shear mode of failure are weaker than those in flexure-controlled failure, and the embedment of steel plate effectively improves the energy dissipation ability of RCSW. For the RCSPSW specimens, the hysteresis loop area decreases with the increase of axial compressive load. Specifically, for the RCSPSW 2-3 specimen that was subjected to the highest axial compressive load (3050 kN), its EVD coefficient is significantly lower than other RCSPSW flexure specimens.

#### **4. Design Models for RCSPSW**

#### *4.1. Shear Strength Model*

The form of the shear strength equation refers to the design model that was suggested by Chinese Code for Design of Composite Structures [50], given as:

$$V = V\_{\odot} + V\_{\sf s} + V\_{\sf a} + V\_{\sf p} \tag{4}$$

where the shear strength, *V*, is contributed by four components: concrete shear-resisting component, *V*c, horizontal reinforcement shear-resisting component, *V*s*,* boundary steel profiles shear-resisting component, *V*a*,* and embedded steel plate shear-resisting component, *V*p. Equation (5) provides the expressions for different shear strength components.

$$\begin{cases} V\_{\rm c} = 0.67 f\_l b\_{\rm w} h\_0 + 0.2 N \frac{A\_{\rm w}}{A} \\\ V\_{\rm s} = f\_{\rm yh} \frac{A\_{\rm uh}}{s} h \\\ V\_{\rm a} = \frac{0.3}{\lambda} f\_{\rm A} A\_{\rm a} \\\ V\_{\rm P} = \frac{0.6}{\lambda - 0.5} f\_{\rm P} A\_{\rm P} \end{cases} \tag{5}$$

where *f* <sup>t</sup> is concrete tensile strength; *b*<sup>w</sup> is the web width; *h*<sup>w</sup> is the web depth; *h* and *h*<sup>0</sup> are the depth and effective depth of the shear wall section; *A*<sup>w</sup> and *A* represent the area of concrete web and entire section, respectively; *N* indicates the axial compressive load, *N* ≤ 0.2 *f*c*b*w*h*w; *f* yh, *fa,* and *f* <sup>p</sup> mean the yield strength of transverse web reinforcement, boundary I-shape steel profiles, and embedded steel plate; *A*yh, *A*a*,* and *A*<sup>p</sup> denote the area of those three components; *s* is the spacing of transverse reinforcement; and, λ is the aspect ratio.

Table 6 summarizes shear strength capacities that were measured from the quasi-static cyclic load test and calculated by shear strength models. Note that the measured characteristic concrete strength values were used in the calculation. It is seen from Table 6 that design models provide 10–15 percent conservative shear strength capacity values for RCSW specimens as compared to the test results. In terms of RCSPSW specimens, shear strength capacities that are estimated by design models are very close to the average experimental data, but 5–10 percent lower shear strength values were observed in the positive direction. A correction factor, *k*s, for design models is taken as 0.9 to determine the shear strength capacity in order to achieve a reasonable degree of conservatism. Figure 16a presents the comparison of design strength capacities and experimental results for the shear specimens. Dash red line shows the design values determined by the modified design model with the correction factor. It is evident that the modified design shear capacities are generally larger than the test values in both positive and negative directions, and a reasonable degree of conservatism (approximate 10–20 percent) is obtained by using the shear correction factor.

**Table 6.** Comparisons of Experimental and Design Strength Capacities for Shear Specimens.


(**a**) Shear Specimens (**b**) Flexure Specimens

**Figure 16.** Comparisons of Design Strength Capacities and Experimental Results for Shear Specimens and Flexure Specimens.

According to Equation (5), the shear strength capacities contributed by reinforced concrete (*V*<sup>c</sup> + *V*s), I-shape steel profiles (*V*a), and the steel plate (*V*p) were quantified and are summarized in Table 7. It is seen that the embedded steel plate takes up approximately 50 percent of the design shear strength capacity. For comparisons, shear strength provided values by each component in the cyclic test were inferred from strain gauge readings. Figure 17a shows the strain gauge arrangement in boundary I-shape steel profiles and the embedded steel plate. Shear strength that was provided by boundary I-shape steel profiles was calculated based on strain values of SG16–SG19 by following Equation (6).

$$V\_{\mathfrak{a}} = \frac{W\_{\mathfrak{a}} E\_{\mathfrak{a}} \left| \frac{\varepsilon\_{\mathrm{SCG1}} + \varepsilon\_{\mathrm{SCG1}} \gamma}{2} - \frac{\varepsilon\_{\mathrm{SCG18}} + \varepsilon\_{\mathrm{SCG19}}}{2} \right| / 2}{H} \tag{6}$$

where *E*<sup>a</sup> is measured Young's Modulus of I-shape steels; *W*<sup>a</sup> is the section modulus; and, *H* is the height of shear specimens.


**Table 7.** Design Strength Capacities Contributed by Different Components in RCSPSW.

**Figure 17.** Shear Strength of I-shape Steels and Steel Plate Inferred from Strain Gauge Readings (**a**) Strain Gauge Arrangement in the I-shape Steels and Steel Plate; (**b**) Strain Rosette; (**c**) Principle Stress State; and, (**d**) Required Stress State.

Strain rosettes were attached on the embedded steel plate. Strain values of *SG*7, *SG*8, and *SG*9 are recorded and used to infer principle stress state and horizontal shear stress state and further determine the shear strength by using Equation (7).

$$\left\{ \begin{array}{c} \sigma\_{1} \\ \sigma\_{2} \end{array} \right\} = \frac{E(\varepsilon\_{0^{0}} + \varepsilon\_{90^{0}})}{2(1 - \nu)} \pm \frac{\sqrt{2}\varepsilon\_{\text{P}}}{2(1 + \nu)} \sqrt{(\varepsilon\_{0^{0}} - \varepsilon\_{45^{0}})^{2} + \left(\varepsilon\_{45^{0}} - \varepsilon\_{90^{0}}\right)^{2}} \tag{7}$$
 
$$V\_{\text{P}} = A\_{5}\tau\_{\text{Y}\text{x}}$$

where ε0<sup>0</sup> , ε45<sup>0</sup> , and ε900 represent strain rosette readings; *E*<sup>p</sup> is the Young's Modulus of steel plate; ν is the Poisson's ratio; *As* is the area of steel plate; and, τyx is the horizontal stress. Note that the shear strength of steel plate is considered to be unchanged after yielding, and shear-resisting component of reinforced concrete is taken as the lateral load strength of RCSW specimens under the same axial compression ratios.

Figure 18 shows the shear-resisting strength that was contributed by different components in the cyclic load test. The blue line, red line, and green line show the capacity trend of reinforced concrete, I-shape steel profiles, and steel plate. The magenta line presents the experimental skeleton curve for a comparative study. It is observed that the shear strength of all the components increases in the elastic stage. When the specimen goes to the crack stage, the strength increase of reinforced concrete component becomes slower. Moreover, the shear strength of the reinforced concrete component decreases before the specimen reaches the peak lateral load, while the steel plate's strength continues increasing. At the failure point, the summation of shear-resisting strength that is provided by different components is close to the value that is presented in the skeleton curve, showing that the calculation of different components' shear strength is reliable.

**Figure 18.** Shear Strength Capacities of RCSPSW Specimens Provided by Different Components.

#### *4.2. Flexural Strength Model*

The strain gauges were attached on the bottom part of the embedded steel plate in flexure specimens, as shown in Figure 4b. Figure 19a presents the measured strain distribution along the wall depth at the peak point. It is observed that the strain distribution keeps close to a linear fashion when the specimen reaches the peak lateral load capacity. Under such circumstance, the design flexure strength model is established based on the plain section assumption and the form refers to the bearing capacity equations of eccentrically-compressed members in Chinese Code for Design of Composite Structures [50], as represented in Figure 19b and given in Equation (8):

$$\begin{cases} \begin{aligned} N \leq N\_{\text{c}} + f'\_{\text{a}}A'\_{\text{a}} + f'\_{\text{y}}A'\_{\text{s}} - \sigma\_{\text{a}}A\_{\text{a}} - \sigma\_{\text{s}}A\_{\text{s}} + N\_{\text{sw}} + N\_{\text{pw}} \\ N\big{(}e\_{0} + \frac{h}{2} - a\big{)} \leq M\_{\text{c}} + f'\_{\text{a}}A'\_{\text{a}}\big{(}h\_{0} - a'\_{\text{a}}\big{)} + f'\_{\text{y}}A'\_{\text{s}}\big{(}h\_{0} - a'\_{\text{s}}\big{)} + M\_{\text{sw}} + M\_{\text{pw}} \\ M = N\mathfrak{e}\_{0} \\ V\_{f} = \frac{M}{H} \end{aligned} \tag{8}$$

where *M* and *N* are the design moment and axial force values; *Vf* is the design shear strength; *e*<sup>0</sup> is the eccentricity of the axial compression force; *H* is the shear wall height; *h* is the depth of wall section; *a* is the distance from the extreme tension fiber to centroid of resultant tensile force in tensile I-shape steels and steel reinforcement; *h*<sup>0</sup> is the effective depth of wall section; *f* <sup>a</sup> and *f* <sup>y</sup> are the yield strength of compressive I-shape steels and steel reinforcement; *A* <sup>a</sup> and *A* <sup>y</sup> are the areas of compressive I-shape steels and steel reinforcement; σ<sup>a</sup> and σ<sup>s</sup> are stresses of tensile I-shape steels and steel reinforcement; *A*<sup>a</sup> and *A*<sup>s</sup> are areas of tensile I-shape steels and steel reinforcement; and, *a* <sup>a</sup> and *a* <sup>s</sup> are distances from the extreme compression fiber to centroid of resultant compressive force in tensile I-shape steels and steel reinforcement. *M*c, *M*sw, and *M*pw are the design moment values provided by concrete section, longitudinal steel reinforcement distributed in the wall web, and embedded steel plate, and *Nc*, *Nsw*, and *Npw* are the design axial force values that are provided by those three components concrete section, which can be determined by Equations (9)–(12):

⎪⎪⎨ ⎪⎪⎩

when *x* ≥ *h* <sup>f</sup> <sup>⎧</sup>

$$\begin{aligned} N\_{\rm c} &= \alpha\_1 f\_{\rm c} \Big[ \xi b\_{\rm w} h\_0 + (b\_{\rm f}' - b\_{\rm w}) h\_{\rm f}' \Big] \\\ M\_{\rm c} &\leq \alpha\_1 f\_{\rm c} \Big[ \xi (1 - 0.5 \xi) b\_{\rm w} h\_0^2 + (b\_{\rm f}' - b\_{\rm w}) h\_{\rm f}' (h\_0 - 0.5 h\_{\rm f}') \Big] \end{aligned} \tag{9}$$

when *x* < *h*

$$\begin{cases} \begin{aligned} N\_{\mathbb{C}} &= a\_1 f\_{\mathbb{C}} \xi b\_{\mathbb{f}}' h\_0 \\ M\_{\mathbb{C}} &\le a\_1 f\_{\mathbb{C}} \xi (1 - 0.5 \xi) b\_{\mathbb{f}}' h\_0^2 \end{aligned} \tag{10}$$

when *<sup>x</sup>* <sup>≤</sup> <sup>β</sup>1*h*<sup>0</sup> <sup>⎧</sup>

$$\begin{cases} N\_{\rm sw} = (1 + \frac{x - \beta\_1 h\_0}{0.5 \beta\_1 h\_{\rm pw}}) f\_{\rm yw} A\_{\rm sw} \\ N\_{\rm pw} = (1 + \frac{x - \beta\_1 h\_0}{0.5 \beta\_1 h\_{\rm pw}}) f\_{\rm PW} A\_{\rm PW} \\ M\_{\rm sw} = [0.5 - \left(\frac{x - \beta\_1 h\_0}{\beta\_1 h\_{\rm sw}}\right)^2] f\_{\rm yw} A\_{\rm sw} h\_{\rm sw} \\ M\_{\rm pw} = [0.5 - \left(\frac{x - \beta\_1 h\_0}{\beta\_1 h\_{\rm pw}}\right)^2] f\_{\rm PW} A\_{\rm PW} h\_{\rm PW} \end{cases} \tag{11}$$

when *x* > β1*h*<sup>0</sup>

$$\begin{cases} N\_{\rm sw} = f\_{\rm yw} A\_{\rm sw} \\ N\_{\rm pw} = f\_{\rm P} A\_{\rm P} \\ M\_{\rm sw} = 0.5 f\_{\rm yw} A\_{\rm sw} h\_{\rm sw} \\ M\_{\rm pw} = 0.5 f\_{\rm P} \ast A\_{\rm Pw} h\_{\rm Pw} \end{cases} \tag{12}$$

 

where *x* is the depth of the compression zone, and ξ = *x*/*h*0; α<sup>1</sup> is the concrete stress block factor that is related to equivalent rectangular concrete compressive stress block intensity; *f*c is the concrete compressive strength; *b* <sup>f</sup> and *b*<sup>w</sup> are the width of wall flange and web; *h* <sup>f</sup> is the depth of wall flange; β<sup>1</sup> is the stress block factor that is related to concrete strength; *h*sw and *h*pw are the depth of distributed longitudinal reinforcement and embedded steel plate in concrete web; *f*yw and *f*pw are the yield strength of distributed longitudinal reinforcement and embedded steel plate; and, *A*sw and *A*pw are the areas of distributed longitudinal reinforcement and embedded steel plate.

(**a**) Strain Distribution along Wall Depth Distance along Wall Depth (mm)

**Figure 19.** *Cont.*

(**b**) Flexure Design Models and Notations

**Figure 19.** Flexure Design Models Developed based on Plane Section Assumption.

Table 8 summarizes the flexure strength capacities measured from the quasi-static cyclic load test and calculated by flexure strength models. It is seen that the flexure strength capacities predicted by the design models are approximately 20–35 percent lower than the measured values in both positive and negative directions. Similar with the shear model, a correction factor, *kf*, of 1.1 is added to achieve a reasonable degree of conservatism. Figure 16b compares the design strength capacities with experimental values for flexure specimens, in which the line and symbol types are the same with Figure 16a. It is observed that the modified design flexure capacities are generally 10–20 percent as compared to the test values, verifying the reliability of the suggested design models.


**Table 8.** Comparisons of Experimental and Design Strength Capacities for Flexure Specimens.

#### **5. Summary and Conclusions**

This paper introduces a high-strength reinforced concrete-steel plate composite shear wall system as lateral load resisting elements the design of high-rise building structures. A total of 11 RCSW and RCSPSW specimens were tested under quasi-static cyclic lateral loading to investigate the effects of critical factors on their seismic performances, including embedment of steel plate, aspect ratio, axial compression ratio, etc. Throughout the test procedure, the progression of damage and failure modes were observed and the key parameters were determined, such as lateral load capacity, ultimate displacement, ductility factor, and EVD coefficient. In addition, the design models were suggested to determine the shear and flexure strength of RCSPSW. By analyzing the data collected from experimental tests, the following conclusions are drawn.

1. Shear and flexure modes of failure dominate in specimens with the aspect ratio of 1.5 and 2.7, respectively. Inclined cracks were observed on the shear specimens, whilst flexure-tension cracks and flexure-shear cracks were mainly distributed in the lower half of boundary elements and wall web in flexure specimens. As compared to the RCSW specimens, more densely-distributed, but finer, cracks took place in the RCSPSW specimens. With the increase of axial compressive loads, the crack quantity becomes less and the crack width is smaller due to the compaction of compression forces.


**Author Contributions:** Conceptualization, D.J. and C.X.; methodology, D.J., C.X. and T.C.; validation, Y.Z.; formal analysis, D.J. and T.C.; investigation, D.J.; C.X. and T.C.; writing—original draft preparation, D.J.; writing-review and editing, C.X. and T.C.; supervision, C.X.; project administration, D.J.; funding acquisition, D.J.

**Funding:** This project was supported by Natural Science Foundation of Jiangsu Province under Grant No. BK20180487, National Natural Science Foundation of China (NSFC) under Grant Nos. 51808292 and 51508276, and Nanjing University of Science and Technology Start-up Grant AE89958.

**Acknowledgments:** The authors are grateful to collaborators from China Academy of Building Research (CABR) who provided invaluable assistance in the experimental test of high-strength concrete-steel plate composite shear walls.

**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**


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#### *Article*
