Seismic Fragility Analysis of Aqueduct Structural Systems Based on G-PCM Method

Abstract

In order to accurately predict the seismic fragility of an aqueduct system, the General Product of Conditional Marginal (G-PCM) method was applied to the seismic fragility analysis of the aqueduct structural system, consisting of interrelated components such as the aqueduct body, pier, and support. First, a finite element dynamic analysis model of a three-span aqueduct with an equidistant simply-supported beam was established, based on the OpenSees platform. The uncertainties of structure, ground motion, and structural capacity were considered, and then the incremental dynamic analysis (IDA) method was used to calculate the seismic fragility of the three individual components, such as the aqueduct pier, the plate rubber bearing at the cap beam, and the PTFE sliding plate bearing at the aqueduct platform. Subsequently, seismic fragility curves of the aqueduct system were established using the G-PCM method and were compared with the traditional second-order bound method. The results showed that the two bearings of the aqueduct are more likely to be damaged than the pier; the failure probability of the aqueduct system is higher than that of any single component; and the seismic fragility curves of the aqueduct system acquired via the G-PCM method were all within the range of the failure probability obtained by the second-order bound method and had a better accuracy, which is suitable for the seismic fragility analysis of multi-failure mode aqueduct systems.

1. Introduction

The aqueduct, as an overhead water-conveying structure across rivers, roads, valleys, etc., has played a vital role in solving the problems of the uneven spatial and temporal distribution of water resources. Under the action of strong earthquakes, the damage or failure of an aqueduct may result in the interruption of the water-conveying process, which will cause economic losses and inconvenience to the cities and residents along the water line. It is of great significance to research the seismic performance of aqueduct structures in earthquake-prone areas, for their seismic design and post-disaster recovery and reconstruction [1]. Seismic fragility analysis can reflect the probability that a structure will reach or exceed a certain damage state under a given ground motion intensity. It quantitatively describes the seismic performance of engineering structures, in the sense of probability and has been widely used in the field of the seismic performance analysis of hydraulic structures [2,3,4].

An aqueduct structure is a structural system consisting of interrelated components, such as the aqueduct body, pier, and support; and using the seismic fragility of individual components to represent that of the aqueduct system would overestimate its seismic capacity [5]. Some researchers have studied methods relating the structural system fragility to component fragility. The core process of the system fragility analysis methods based on the traditional reliability theory [6] lies in the solution of the multidimensional standard normal distribution function. Although the exact solution of the distribution function can be obtained by the direct integration method, the computational efficiency is too low for a complex structural system with many components. The first-order bound estimation method proposed by Cornell [7] can estimate the results, by calculating the upper and lower bounds of the structural failure probability, but this method cannot consider the interconnection between different components, resulting in a wide range of bounds and large errors. The second-order bounds method proposed by Hunter [8] considers the correlation between different components, which improves the shortcomings of the first-order bounds method and can obtain a relatively narrow estimation interval. Nielson et al. [9] proposed a fragility analysis method for systems based on Monte Carlo simulation, but this method requires a large number of samples for sampling, and the calculation efficiency is low, which is not suitable for practical engineering with various failure modes. As an approximate solution method, to study the fragility of complex structural systems, the product of conditional marginal (PCM) [10] is widely used in mechanical and civil engineering fields, owing to its simplicity and efficiency. Yuan and Pandey [11] analyzed the deficiencies of the PCM method in calculating the series systems and made improvements, proposing an improved product of conditional marginal (I-PCM). The G-PCM method proposed by Tu [12] further improved the I-PCM method in the calculation of the conditional normal quantile and the correlation coefficient, which effectively improved the accuracy of the original method in the reliability calculation of the series system. In addition, some researchers have studied the fragility of structural systems combining different failure modes [13,14,15].

At present, the boundary estimation method and Copula function are mostly employed for seismic fragility analysis of aqueduct systems, and some scholars only evaluate the seismic fragility from the perspective of the components. Zhang [16] performed a seismic fragility analysis for single-column reinforced concrete piers, and on this basis, carried out time-varying reliability analysis and seismic risk analysis of an aqueduct. Zhang et al. [17,18] introduced the Copula function to describe the component correlation of the aqueduct structure, established the system fragility curves under series and parallel systems based on the fragility curves of frame column and rubber bearing, and performed a validated analysis through the bounds estimation method.

In this paper, a three-span aqueduct with an equidistant simply-supported beam was taken as the research object. Based on the finite element dynamic analysis model established by the OpenSees platform, the fragility curves of three individual components, such as the pier, the plate rubber bearing at the cap beam, and the PTFE sliding plate bearing at the aqueduct platform, were established using the IDA method and the demand-capacity ratio curve fitting method. Then, the seismic fragility curves of the aqueduct system were established using the G-PCM method and compared with the second-order bound method.

2. Seismic Fragility Analysis Method

2.1. Component Fragility

The seismic fragility of an aqueduct structure refers to the probability that the seismic demand reaches or exceeds the structural capacity under a given ground motion intensity. For a single component, the seismic fragility under different damage states can be expressed as

$$P_f=P\left. [S_d/S_c\ge 1\right]$$

(1)

where P_f is the failure probability of the structure or component, i.e., the probability that the seismic demand reaches or exceeds the structure or component capacity; P represents probability density function; S_d represents seismic demand of the structure or component; and S_c denotes seismic capacity of structure or component.

The procedure of IDA method-based seismic fragility analysis of aqueduct structure is as follows [19]: first, the finite element model of the aqueduct is built, and several aqueduct samples are obtained through Latin hypercube sampling (LHS). Then, the appropriate ground motion records are selected. The ground motion intensity index is determined, and amplitude modulation is performed. After that, a dynamic time history analysis is employed, by randomly combining the amplitude modulated ground motion samples with the aqueduct samples. Finally, according to the relationship between the seismic response results and damage index, the seismic fragility curves of different components are established based on the selected fragility function.

The demand–capacity ratio curve fitting method assumes that the seismic fragility curve conforms to a log-normal distribution, and its fragility function is expressed as [20]

$$P_f=1-\mathsf{\Phi }\left(\frac{\ln (1)-\lambda }{\sigma }\right)=\mathsf{\Phi }\left(\frac{\lambda }{\sigma }\right)$$

(2)

where $\mathsf{\Phi }\left(\cdot \right)$ is the standard normal distribution function; and λ and σ represent the mean and standard deviation of $\ln \left(S_d/S_c\right)$ in the logarithmic regression analysis of the capacity demand ratio, respectively, with the following expressions

$$\lambda =a{\left(\ln \left(IM\right)\right)}^2+b\ln \left(IM\right)+c$$

(3)

$$\sigma =\sqrt{S_r/\left(N-2\right)}$$

(4)

where IM is the ground motion intensity index; a, b, and c are the parameters of regression analysis; S_r is the residual sum of squares of discrete points; and N is the number of discrete points.

2.2. System Fragility

As a complex structural system, an aqueduct consists of many components, such as the body, pier, support, etc. Under the action of strong earthquakes, any component may be damaged to different degrees. The seismic fragility of the aqueduct as a whole cannot be reasonably assessed without considering the interconnection between the components. Therefore, it is necessary to adopt the system fragility to analyze the seismic performance of the aqueduct structure under different ground motion intensities. Based on the reliability theory, the aqueduct system is assumed to conform to the series model [17,21,22].

2.2.1. PCM Method and I-PCM Method

The product of conditional marginal (PCM) is taken as an approximate solution to the multi-dimensional standard normal cumulative distribution function ${\mathsf{\Phi }}_n(\beta ,\rho )$. The basic idea is to convert ${\mathsf{\Phi }}_n(\beta ,\rho )$ into the form of a product of a series of one-dimensional standard normal distribution functions, according to conditional probability theory, to obtain the approximate solution, and its expression is as follows [10]:

$$\begin{array}{l}{\mathsf{\Phi }}_n\left(\beta ,\rho \right)=P\left. [X_n\le {\beta }_n\left|{\displaystyle \underset{k=1}{\overset{n-1}{\cap }}\left(X_k\le {\beta }_k\right)}\right.\right]\times P\left. [X_{n-1}\le {\beta }_{n-1}\left|{\displaystyle \underset{k=1}{\overset{n-2}{\cap }}\left(X_k\le {\beta }_k\right)}\right.\right]\times \cdots \times P\left(X_1\le {\beta }_1\right)\\ \approx \mathsf{\Phi }\left({\beta }_{n\left|\left(n-1\right)\right.}\right)\times \mathsf{\Phi }\left({\beta }_{\left(n-1\right)\left|\left(n-2\right)\right.}\right)\times \cdots \times \mathsf{\Phi }({\beta }_1)\\ ={\displaystyle \prod _{k=1}^n\mathsf{\Phi }\left({\beta }_{k\left|(k-1)\right.}\right)}\end{array}$$

(5)

where n is the number of components that may fail; X represents a random variable corresponding to the failure mode of the structure or component; $\beta =\left. [{\beta }_1,{\beta }_2,{\beta }_3,\dots ,{\beta }_n\right]$ is the reliability index vector of n components; ρ is the correlation coefficient matrix; and ${\beta }_{k\left|(k-1)\right.}$ is the conditional normal quantile. The conditional normal quantile and the correlation coefficient matrix can be calculated using the following equations:

$${\beta }_{i\left|k\right.}=\frac{{\beta }_{i\left|\left(k-1\right)\right.}+{\rho }_{ik\left|\left(k-1\right)\right.}{A}_{k\left|\left(k-1\right)\right.}}{\sqrt{1-{\rho }_{ik\left|\left(k-1\right)\right.}^2{B}_{k\left|\left(k-1\right)\right.}}}$$

(6)

$${\rho }_{ij\left|k\right.}=\frac{{\rho }_{ij\left|\left(k-1\right)\right.}-{\rho }_{ik\left|\left(k-1\right)\right.}{\rho }_{jk\left|\left(k-1\right)\right.}{B}_{k\left|\left(k-1\right)\right.}}{\sqrt{\left(1-{\rho }_{ik\left|\left(k-1\right)\right.}^2{B}_{k\left|\left(k-1\right)\right.}\right)\left(1-{\rho }_{jk\left|\left(k-1\right)\right.}^2{B}_{k\left|\left(k-1\right)\right.}\right)}}$$

(7)

where the condition parameters $A_{k\left|(k-1)\right.}$ and $B_{k\left|(k-1)\right.}$ are computed by the following formula:

$$A_{k\left|(k-1)\right.}=\varphi \left({\beta }_{k\left|(k-1)\right.}\right)/\mathsf{\Phi }\left({\beta }_{k\left|(k-1)\right.}\right)$$

(8)

$$B_{k\left|(k-1)\right.}=A_{k\left|(k-1)\right.}\left({\beta }_{k\left|(k-1)\right.}+A_{k\left|(k-1)\right.}\right)$$

(9)

where $k=1,\dots ,n-1$; $i=k+1,\dots ,n$; $j=k+1,\dots ,n$.

Yuan et al. [11] further studied the PCM based on the direct numerical integration method. The research showed that the PCM has a high calculation accuracy for parallel systems, with calculation error for the series system. This can attribute to the fact that the values of $A_{k\left|\left(k-1\right)\right.}$ and $B_{k\left|\left(k-1\right)\right.}$ tend to 0 when the component reliability β is large, which results in the calculation of the conditional normal quantile in Equation (6) ignoring the influence of component correlation, thus overestimating the failure probability of the series system. To this end, Yuan et al. proposed an improved product of conditional marginal, with the following computational procedure:

$${\beta }_{i\left|k\right.}={\mathsf{\Phi }}^{-1}\cdot \left. [1-\frac{\mathsf{\Phi }\left(-{\beta }_{i\left|\left(k-1\right)\right.}\right)-\mathsf{\Phi }\left(-{\beta }_{i\left|\left(k-1\right)\right.},-{\beta }_{k\left|\left(k-1\right)\right.},{\rho }_{ik\left|\left(k-1\right)\right.}\right)}{\mathsf{\Phi }\left({\beta }_{k\left|\left(k-1\right)\right.}\right)}\right]$$

(10)

where the two-dimensional normal distribution function can be calculated according to Equations (11)–(14).

$$\mathsf{\Phi }\left(-{\beta }_{i\left|\left(k-1\right)\right.},-{\beta }_{k\left|\left(k-1\right)\right.},{\rho }_{ik\left|\left(k-1\right)\right.}\right)\approx \mathsf{\Phi }\left(c_{i\left|k\right.}\right)\mathsf{\Phi }\left(-{\beta }_{k\left|\left(k-1\right)\right.}\right)$$

(11)

$$c_{i\left|k\right.}=\frac{\left(-{\beta }_{i\left|\left(k-1\right)\right.}+{\rho }_{ik\left|\left(k-1\right)\right.}{D}_{k\left|\left(k-1\right)\right.}\right)}{\sqrt{1-{\rho }_{ik\left|\left(k-1\right)\right.}^2{E}_{ik\left|\left(k-1\right)\right.}}}$$

(12)

$$D_{k\left|\left(k-1\right)\right.}=\phi \left(-{\beta }_{k\left|\left(k-1\right)\right.}\right)/\mathsf{\Phi }\left(-{\beta }_{k\left|\left(k-1\right)\right.}\right)$$

(13)

$$E_{ik\left|\left(k-1\right)\right.}=D_{k\left|\left(k-1\right)\right.}\left(-{\beta }_{i\left|\left(k-1\right)\right.}+D_{k\left|\left(k-1\right)\right.}\right)$$

(14)

2.2.2. G-PCM Method

According to the calculation process of the I-PCM method, it can be seen that the calculation of $\mathsf{\Phi }\left(-{\beta }_{i\left|\left(k-1\right)\right.},-{\beta }_{k\left|\left(k-1\right)\right.},{\rho }_{ik\left|\left(k-1\right)\right.}\right)$ in the conditional normal quantile adopts an approximate solution formula, and there are still methods for improvement. If a more accurate calculation method is instead employed to solve the two-dimensional normal distribution function, the calculation accuracy of the I-PCM method can be further improved, while ensuring efficiency. Related research [23] showed that the two-dimensional normal distribution function can be transformed into the following one-dimensional integral expression.

$$\mathsf{\Phi }\left(-{\beta }_{i\left|\left(k-1\right)\right.},-{\beta }_{k\left|\left(k-1\right)\right.},{\rho }_{ik\left|\left(k-1\right)\right.}\right)=\mathsf{\Phi }\left(-{\beta }_{i\left|\left(k-1\right)\right.}\right)\mathsf{\Phi }\left(-{\beta }_{k\left|\left(k-1\right)\right.}\right)+{\displaystyle {\int }_0^{{\rho }_{ik\left|\left(k-1\right)\right.}}\varphi \left(-{\beta }_{i\left|\left(k-1\right)\right.},-{\beta }_{k\left|\left(k-1\right)\right.},t\right)}dt$$

(15)

Based on the above equation, Tu [12] further improved the formula for the conditional normal quantile.

$$\begin{array}{l}{\beta }_{i\left|k\right.}={\mathsf{\Phi }}^{-1}\left. [1-\frac{\mathsf{\Phi }\left(-{\beta }_{i\left|\left(k-1\right)\right.}\right)-\mathsf{\Phi }\left(-{\beta }_{i\left|\left(k-1\right)\right.},-{\beta }_{k\left|\left(k-1\right)\right.},{\rho }_{ik\left|\left(k-1\right)\right.}\right)}{\mathsf{\Phi }\left({\beta }_{k\left|\left(k-1\right)\right.}\right)}\right]\\ ={\mathsf{\Phi }}^{-1}\left. [\begin{array}{l}\frac{\mathsf{\Phi }\left({\beta }_{i\left|\left(k-1\right)\right.}\right)+\mathsf{\Phi }\left({\beta }_{k\left|\left(k-1\right)\right.}\right)+\mathsf{\Phi }\left(-{\beta }_{i\left|\left(k-1\right)\right.}\right)\mathsf{\Phi }\left(-{\beta }_{k\left|\left(k-1\right)\right.}\right)}{\mathsf{\Phi }\left({\beta }_{k\left|\left(k-1\right)\right.}\right)}\\ +\frac{{\displaystyle {\int }_0^{{\rho }_{ik\left|\left(k-1\right)\right.}}\varphi \left(-{\beta }_{i\left|\left(k-1\right)\right.},-{\beta }_{k\left|\left(k-1\right)\right.},t\right)}dt-1}{\mathsf{\Phi }\left({\beta }_{k\left|\left(k-1\right)\right.}\right)}\end{array}\right]\end{array}$$

(16)

where ${\displaystyle {\int }_0^{{\rho }_{ik\left|\left(k-1\right)\right.}}\varphi \left(-{\beta }_{i\left|\left(k-1\right)\right.},-{\beta }_{k\left|\left(k-1\right)\right.},t\right)}dt$ can be calculated using Simpson’s numerical integration method. The modified $B_{k|(k-1)}$ can be obtained by substituting the $B_{i\left|k\right.}$ of Equation (16).

$$B_{k|(k-1)}=\frac{1}{{\rho }_{ik\left|(k-1)\right.}^2}-\frac{{\left({\beta }_{i\left|(k-1)\right.}+{\rho }_{ik\left|(k-1)\right.}{A}_{k\left|(k-1)\right.}\right)}^2}{{\rho }_{ik\left|(k-1)\right.}^2{\beta }_{i\left|k\right.}^2}$$

(17)

The corrected correlation coefficient matrix of Equation (7) can be obtained by substituting the corrected $B_{k|(k-1)}$ of Equation (17). In contrast to the I-PCM method, the general product of conditional marginal (G-PCM) has improvements in terms of both the conditional normal quantile and the calculation of correlation coefficients.

2.2.3. Error Analysis of Approximate Solution Methods

To further verify the applicability and superiority of the G-PCM method in the approximate solution of system fragility analysis, the failure probability of the aqueduct structural system under different ground motion intensities is calculated using the G-PCM method and the other PCM and I-PCM methods, respectively. Comparing the calculated results with those obtained by the direct integration method, which is closer to being true, the accuracy of the three approximate solution methods is reflected by the error value.

$$Error=\frac{P_f-P_f^{\ast }}{P_f}\times 100\%$$

(18)

where, $P_f^{*}$, $P_f$ are the failure probability of the aqueduct structural system obtained by the approximate solution and the direct integration methods, respectively.

Figure 1 shows the relative error curves of the PCM, I-PCM, and G-PCM methods under the four damage states and different ground motion intensities. It can be seen from the figure that the three approximate solution methods have a high accuracy, and the error values are all less than 2%. There is no significant difference in the relative error of the same method under different damage states. The relative error of the PCM method is most distinct when the ground motion intensity is low. With the increase of ground motion intensity, the relative error of the three methods first increases and then decreases. The order of relative error is the PCM method > I-PCM method > G-PCM method.

3. Example Calculation of Aqueduct

3.1. Structural Finite Element Model

This paper takes a three-span aqueduct with an equidistant simple-supported beam, in the middle route of the South-to-North Water Diversion Project in China, as an example. The superstructure of the aqueduct is a rectangular aqueduct body composed of the bottom plate, web, and tension rod, with 28 m per span. The designed water level in the aqueduct is 2.21 m. The substructure adopts an H-shaped frame structure, composed of a double-column pier, cap beam, and cross-girder. The cross-section of the pier is a circular section 0.8 m in diameter, and the bottom of the pier is consolidated to the foundation. The height and width of the rectangular section of the cross-girder are both 0.6 m. The cap beam is equipped with a plate rubber bearing, and the support at the aqueduct platform is a PTFE slide bearing. The numerical calculation model of the aqueduct is shown in Figure 2a.

The aqueduct body is made of concrete, with a designed compressive strength of 50 MPa and an elastic modulus of 3.45 × 10⁴ MPa; the concrete compressive strength of the H-shaped frame structure is 30 MPa, and its elastic modulus is 3.00 × 10⁴ MPa; the reinforcements are smooth steel bars with a yield strength of 300 MPa and elastic modulus of 2.1 × 10⁵ MPa. The rubber layer thickness of the plate rubber bearing on the cap beam type GJZ 250 × 500 × 63 and the PTFE slide bearing at the aqueduct platform type GJZF 4 250 × 500 × 65 are both 45 mm.

The finite element dynamic analysis model was established based on the OpenSees platform. For an aqueduct structure with a simply-supported beam, the seismic capacity of the lower support structure affects the overall collapse performance of the aqueduct. As an important substructure of the aqueduct, the double-column pier has a high seismic fragility under strong earthquakes [24,25], so nonlinear beam-column elements were adopted to simulate the aqueduct pier to consider its elastic-plastic failure. The plate rubber bearing was simulated using a zero-length element based on a linear model, and the PTFE sliding plate bearing was simulated using an ideal elastic-plastic bearing element, based on a bilinear model. The cross-girder and cap beam of the frame, as well the aqueduct body, were simulated by elastic beam-column elements. The designed water depth of the aqueduct is 2.21 m, and the water in the aqueduct was simulated using the additional mass model [26]. The constitutive model of each component of the aqueduct is shown in Figure 2b. The concrete of the aqueduct pier is made of Concret02 material, based on the modified Kent-Park model, and the steel bar of the aqueduct pier is made of Steel02 material based on the Menegotto-Pinto model, which considers the strain hardening and the Bauschinger effect. For simplification, the did not consider the pinching effect of smooth steel. The fiber section of the aqueduct pier was divided into the two parts of the concrete cover and the core area, for simulating the effect of stirrups and improving the compressive strength of the core concrete.

3.2. Uncertainty of the Structure

The uncertainty of the aqueduct structure was mainly considered from the two perspectives of the material parameters and the structural dimensions. The structural uncertainty parameters considered in this study mainly included the unit weight of the concrete, the diameter of longitudinal reinforcement, the diameter of the pier column, the thickness of the concrete cover, and the damping ratio of the aqueduct structure. The uncertainties of these five parameters cannot represent all uncertainties. This method can consider various uncertainties of the structure according to the actual situation. It was assumed that these five parameters conform to a normal distribution [5], as shown in

Table 1. Model uncertainty parameters and their distribution.

Parameters	Unit	Distribution Form	Average Value	COV
Unit weight of concrete	kN/m3	Normal distribution	25	0.1
Diameter of longitudinal reinforcement	mm	Normal distribution	16	0.035
Diameter of pier column	m	Normal distribution	0.8	0.05
Concrete cover	m	Normal distribution	0.03	0.05
Damping ratio		Normal distribution	0.05	0.28

. Ten samples of the aqueduct were obtained using Latin hypercube sampling, to account for the uncertainty of the structural parameters.

3.3. Selection of Ground Motion

Seismic fragility analysis needs to select a sufficient number of seismic waves to account for the uncertainty of ground motion [27,28]. Based on the seismic fragility analysis of structures with the IDA method, selecting 10 to 20 ground motion records can achieve a certain accuracy [29]. The basic principle of the IDA method is to modulate the ground motion parameters to a specified intensity, so as to obtain the structural demands under different ground motion intensities [30,31].

The aqueduct project is located at a Class II site, and the seismic fortification intensity is VII degrees. The design response spectrum was obtained according to the “Seismic Design Standards for Hydraulic Buildings” (GB 51247-2018), and this was used as the target spectrum. Twenty far-field ground motion records were selected from the PEER Ground Motion Database considering an earthquake magnitude of 5.5~8.0 M and shear wave velocity 260~550 m/s [32,33]. The response spectrum and mean spectrum of the selected 20 seismic waves, as well the design response spectrum, are shown in Figure 3. It can be seen that the selected seismic waves are conformed well with the designed response spectrum. Using the IDA method to generate enough ground motion samples can make up for the problem of a large discreteness of peak ground acceleration (PGA), to a certain extent. In addition, PGA is easier to obtain, and the corresponding research method is more mature. Thus, peak ground acceleration (PGA) was selected as the ground motion intensity index, and the 20 ground motion records were uniformly modulated to 0.1~1 g, with a step size of 0.1 g, and a total of 200 ground motion samples were established. At present, research on the seismic performance of aqueduct structure normally includes the transverse and longitudinal directions [34]. In this paper, the horizontal ground motion was employed to study the seismic fragility of the aqueduct structure in the transverse direction.

3.4. Definition of Damage Index

Under earthquake action, the degree of damage of an aqueduct can be divided into four grades: slight damage, moderate damage, extensive damage, and complete damage [35]. Defining an appropriate damage index for different components is essential, to reasonably evaluate the seismic fragility of each component under different damage states.

In previous studies, various damage indexes were used to describe the seismic damage of reinforced concrete piers, such as drift ratio at the pier top, displacement ductility, and curvature ductility. It has been shown that the displacement ductility is greatly affected by the shear–span ratio of the pier, and both the curvature ductility and drift ratio have a good consistency for piers with different shear–span ratios [36,37,38,39]. To eliminate the influence of pier height, the dimensionless parameter drift ratio at the pier top was selected as the damage index in this paper, which is defined as follows:

$$\delta =\frac{\Delta }{L}$$

(19)

where δ is the drift ratio of the pier top; Δ is the displacement of the pier top; L is the height of the pier.

Referring to a related study [40], the double-column pier was converted into a single-column pier, and the drift ratio δ at the pier top under different damage states were calculated according to the deformation capacity model of the pier column [41]. The drift ratio of the pier top, as an index of pier damage, varies with the pier parameters, rather than being a constant (e.g., 5%). Thus, the pier capacity was accurately evaluated, and the corresponding uncertainty was considered. The results are shown in

Table 2. Damage index under different damage states.

Components	Damage Index	Slight Damage	Moderate Damage	Extensive Damage	Complete Damage
Double-column Pier	δ	0.0089	0.0142	0.0252	0.0454
Plate rubber bearing	λ	100%	150%	200%	250%
PTFE sliding plate bearing	u/m	0.09	0.15	0.20	0.30

.

The bearing of the aqueduct structure plays the role of supporting the superstructure. For different types of bearings, different scholars have adopted different damage indexes. The shear strain γ [42] and sliding displacement u [43] were selected as the damage indexes of the plate rubber bearing and the PTFE slide plate bearing, respectively. The specific parameters are shown in Table 2.

4. Fragility Analysis of the Aqueduct Structural System

4.1. Component Fragility Curve

The 200 seismic wave samples after amplitude modulation and 10 aqueduct samples were randomly matched, to carry out a nonlinear time history analysis, and the seismic demands of each component of the aqueduct were obtained. A logarithmic regression analysis was carried out on the ratio of seismic demand to structural capacity (S_d/S_c) and the ground motion parameters (PGA), according to Equation (3), and the logarithmic IDA curve of the demand capacity ratio of each component was obtained, taking the complete damage state as an example, as shown in Figure 4.

Table 3. Logarithmic regression analysis of the demand–capacity ratio.

Components	Damage State	Fitting Function	Standard Deviation
Double-column pier	Slight damage	$\ln \left(S_d/S_c\right)=0.0109\ln {\left(\mathrm{PGA}\right)}^2+1.7319\ln \left(\mathrm{PGA}\right)+1.7743$	0.7033
	Moderate damage	$\ln \left(S_d/S_c\right)=0.0109\ln {\left(\mathrm{PGA}\right)}^2+1.7319\ln \left(\mathrm{PGA}\right)+1.3071$
	Extensive damage	$\ln \left(S_d/S_c\right)=0.0109\ln {\left(\mathrm{PGA}\right)}^2+1.7319\ln \left(\mathrm{PGA}\right)+0.7335$
	Complete damage	$\ln \left(S_d/S_c\right)=0.0109\ln {\left(\mathrm{PGA}\right)}^2+1.7319\ln \left(\mathrm{PGA}\right)+0.1449$
Plate rubber bearing	Slight damage	$\ln \left(S_d/S_c\right)=-0.1586\ln {\left(\mathrm{PGA}\right)}^2+0.2539\ln \left(\mathrm{PGA}\right)+1.064$	0.2559
	Moderate damage	$\ln \left(S_d/S_c\right)=-0.1586\ln {\left(\mathrm{PGA}\right)}^2+0.2539\ln \left(\mathrm{PGA}\right)+0.6586$
	Extensive damage	$\ln \left(S_d/S_c\right)=-0.1586\ln {\left(\mathrm{PGA}\right)}^2+0.2539\ln \left(\mathrm{PGA}\right)+0.3709$
	Complete damage	$\ln \left(S_d/S_c\right)=-0.1586\ln {\left(\mathrm{PGA}\right)}^2+0.2539\ln \left(\mathrm{PGA}\right)+0.1478$
PTFE sliding plate bearing	Slight damage	$\ln \left(S_d/S_c\right)=-0.0025\ln {\left(\mathrm{PGA}\right)}^2+1.1519\ln \left(\mathrm{PGA}\right)+1.976$	0.4963
	Moderate damage	$\ln \left(S_d/S_c\right)=-0.0025\ln {\left(\mathrm{PGA}\right)}^2+1.1519\ln \left(\mathrm{PGA}\right)+1.465$
	Extensive damage	$\ln \left(S_d/S_c\right)=-0.0025\ln {\left(\mathrm{PGA}\right)}^2+1.1519\ln \left(\mathrm{PGA}\right)+1.117$
	Complete damage	$\ln \left(S_d/S_c\right)=-0.0025\ln {\left(\mathrm{PGA}\right)}^2+1.1519\ln \left(\mathrm{PGA}\right)+0.7720$

enlists the seismic demand fitting functions and standard deviations of three components under four damage states. The calculated results were substituted into Equation (2), to obtain the failure probability of each component under different damage states; and the seismic fragility curves of three aqueduct components were established as shown in Figure 5.

It can be seen from Figure 5 that with the increase of PGA, the failure probability of different components gradually increases. When the PGA is smaller than 0.1 g, the aqueduct pier is basically not damaged, and the plate rubber bearing and the PTFE sliding plate bearing are most likely to be in an intact or slightly damaged state, which basically meets the seismic design requirements. When the PGA is between 0.1 g and 0.2 g, the probability of slight damage to the two supports increases rapidly, but almost no complete damage occurs. When the PGA is between 0.2 g and 0.4 g, the probability of slight damage to the aqueduct pier gradually increases, and the other two supports are basically slightly damaged. When the PGA is greater than 0.4 g, all three components may be damaged to different degrees. Under the same ground motion intensity and damage state, the two types of bearings of the aqueduct are more prone to damage than the aqueduct pier. Under the two states of slight and moderate damage, the fragility of the plate rubber bearing is greater than that of the PTFE sliding plate bearing, and under the two states of extensive and complete damage, the PTFE sliding plate bearing is more likely to incur damage than the plate rubber bearing.

4.2. System Fragility Curve

To analyze the effect of the correlation between components on the system failure probability described in Section 2.2, taking the extensive damage state as an example, the failure probability of the aqueduct structural system in an extensive damage state for a given ground motion intensity was computed, as shown in Figure 6.

It can be seen that the different levels of correlation between components had little impact on the failure probability of the aqueduct structural system. Therefore, this paper assumes that the correlation coefficients between components are ρ = 0.5 (mean between 0.1 and 1.0).

In this paper, the aqueduct system is assumed to conform to the series model. According to the failure probability of three aqueduct components, namely, the double-column pier, the plate rubber bearing, and the PTFE sliding plate bearing, a fragility analysis of the aqueduct structural system was carried out using the G-PCM method and compared with the seismic fragility curves of components, as shown in Figure 7.

As can be seen from Figure 7, the failure probabilities of the aqueduct components under four damage states were all less than that of the structural system based on G-PCM method. Generally, the seismic fragility of an aqueduct pier is commonly used to represent that of a structural system. Considering the fragility of pier is the smallest among these three components, the fragilities of the aqueduct system compared with that of piers under four damage states were analyzed. For slight damage, the difference between the failure probability of the aqueduct system (87.9%) and pier (8.1%) was the largest when PGA = 0.2 g; for moderate damage, the difference between the failure probability of aqueduct system (79.4%) and pier (13.9%) was the largest when PGA = 0.3 g; for extensive damage, the difference between the failure probability of the aqueduct system (81.2%) and pier (18.1%) was the largest when PGA = 0.45 g; for complete damage, the difference between the failure probability of aqueduct system (79.5%) and pier (19.7%) was the largest when PGA = 0.65 g. It follows that the seismic capacity of the aqueduct, which is a complex structural system containing multiple components, will be overestimated if the failure probability of a single component is used to represent that of the aqueduct structure. The aqueduct system is more likely to be damaged than a single component.

4.3. Evaluation of G-PCM Method

Ma Ying et al. [22] established seismic fragility curves of an aqueduct structural system through the first-order and second-order bound methods, respectively, and made a comparative analysis. The research showed that it is more reasonable to use the system fragility curve established by the second-order bound method to evaluate the seismic performance of the aqueduct structure than the first-order bound method. To verify the accuracy of the system fragility curve established using the G-PCM method, it was compared with the calculation results of the second-order bound method.

The second-order bound method can estimate the maximum and minimum failure probability of the structural system, to determine the maximum bound interval of the failure probability, which is as follows [8]:

$$P_{f1}+{\displaystyle \sum _{i=2}^n\max \left(P_{fi}-{\displaystyle \sum _{j=1}^{i-1}{P}_{fij},0}\right)\le P_{sys}}\le {\displaystyle \sum _{i=1}^n{P}_{fi}-}{\displaystyle \sum _{i=2}^n\underset{j<i}{\max }\left(P_{fij}\right)}$$

(20)

where n is the number of components that may be damaged; P_sys represents the damage probability of the structural system; P_f1 denotes the probability of failure of a single component; P_fi stands for the probability of failure of the i-th component (except P_f1); and P_fij denotes the probability of simultaneous failure of the i-th and j-th components, $i,j=2,3,\dots ,n$, which can be expressed as

$$P_{fij}=P\left(F_i\cap F_j\right)$$

(21)

where F_i and F_j are the i-th and j-th failure events. Assuming that all variables conform to a normal distribution, P_fij can be expressed as $\mathsf{\Phi }\left(-{\beta }_i,-{\beta }_j,\rho \right)$. The two-dimensional normal distribution function can be solved directly using numerical integration, or it can be calculated according to the approximate formula of the relevant literature. Studies [44] show that the order of component failure mode arrangement has a great influence on the system failure probability calculated by the second-order bound method, and a relatively ideal narrow interval can be obtained when the component failure probability P_fi is arranged in the order of probability magnitude.

For the convenience of comparison, the fragility curve of the aqueduct system established via the G-PCM method, and the upper and lower bounds of system failure probability obtained by the second-order bound method are compared in Figure 8. The corresponding data are displayed in

Table 4. Failure probability of the aqueduct system for a given ground motion intensity.

System Fragility Analysis	Damage State	0 g	0.2 g	0.4 g	0.6 g	0.8 g	1.0 g
G-PCM	Slight damage	0%	87.91%	99.89%	100.00%	100.00%	100.00%
	Moderate damage	0%	36.87%	93.97%	99.41%	99.92%	99.99%
	Extensive damage	0%	11.01%	72.44%	94.16%	98.74%	99.70%
	Complete damage	0%	1.77%	37.26%	73.90%	90.31%	96.46%
Second-order upper bound	Slight damage	0%	88.00%	99.91%	100.00%	100.00%	100.00%
	Moderate damage	0%	37.15%	94.81%	100.00%	100.00%	100.00%
	Extensive damage	0%	11.04%	73.30%	95.23%	99.28%	99.92%
	Complete damage	0%	1.77%	37.51%	74.73%	91.19%	97.05%
Second-order lower bound	Slight damage	0%	87.86%	99.89%	100.00%	100.00%	100.00%
	Moderate damage	0%	36.66%	93.70%	99.32%	99.89%	99.98%
	Extensive damage	0%	10.96%	72.02%	93.82%	98.52%	99.60%
	Complete damage	0%	1.76%	37.01%	73.28%	89.62%	95.93%

.

It can be seen from Figure 8 that the maximum differences of the upper and lower bounds obtained via the second-order bound method were 0.3%, 1.4%, 1.8%, and 2.0%, respectively, which indicates that the calculation results of the second-order bound method were ideal and a narrow failure probability interval can be obtained. Meanwhile, the aqueduct system fragility curves based on G-PCM method were all located between the upper and lower bounds obtained by the second-order bound method. This illustrated that the G-PCM method can obtain accurate values of the failure probability of the aqueduct system under different ground motion intensities.

The results of the above comparative analysis show, that the G-PCM method can calculate the seismic fragility of the aqueduct system quickly and accurately compared with the second-order bound method, and has a higher applicability to actual aqueduct project with complex components and more failure modes.

5. Conclusions

In this study, a three-span aqueduct with an equidistant simply-supported beam was taken as an example, and a finite element analysis model of the aqueduct structure was established based on the OpenSees platform. The LHS was adopted to consider the uncertainty of the aqueduct structure, the PGA was selected as the ground motion intensity index, and the drift ratio of the pier top, calculated using a probabilistic model of deformation capacity, was used to consider the uncertainty of the structural capacity. Based on the IDA and demand capacity ratio curve fitting method, the lateral seismic fragility curves of the three individual components; namely, the aqueduct pier, the plate rubber bearing at the cap beam, and the PTFE slide bearing at the aqueduct platform were established. After that, the aqueduct system was simplified to a series model, based on reliability theory. On this basis, the seismic fragility curves of the aqueduct system were established via the G-PCM method and were compared with the second-order bound method, and the following conclusions could be drawn:

(1) The two bearings of the aqueduct are more likely to be damaged than the aqueduct pier under the same ground motion intensity and damage state; the fragility of the plate rubber bearing is greater than that of the PTFE sliding plate bearing under the two states of slight and moderate damage; the PTFE sliding plate bearing is more likely to incur damage than the plate rubber bearing under the two states of extensive and complete damage.

(2) The failure probability of the aqueduct system calculated using the G-PCM method is higher than that of any single member, and the seismic fragility of the aqueduct system represented by any single component will overestimate the seismic capacity during earthquakes.

(3) Although the second-order bound method can obtain an ideal and narrow failure probability interval, it cannot obtain the exact failure probability of the aqueduct system. In comparison, the G-PCM method can quickly and accurately establish the fragility curves of the aqueduct system and is more suitable for the fragility analysis of a complex structural system that needs to consider multiple failure modes.