Extreme value distribution and dynamic reliability estimation of high-pier bridges subjected to near-fault impulsive ground motions

Abstract

The seismic behaviour of high-pier and long-span bridges in fault zones is significantly influenced by the near-fault impulsive effect. In view of this, this study evaluates the extreme value distribution and dynamic reliability of high-pier bridges built in fault zones. First, based on the improved maximum entropy method constrained with fractional moments, a new method is proposed to evaluate the extreme value distribution of nonlinear structure seismic response. In the proposed method, the fractional moments are evaluated by the Latin hypercube sampling, and a linear system of equations that determine the initial value of fractional exponents and Lagrange multipliers is illustrated to improve the stability of the simplex algorithm. Second, based on the obtained extreme value distribution, the seismic reliability of structure is estimated by the simple numerical integration. Two numerical examples, including a nonlinear single-degree-of-freedom system and a three-storey nonlinear shear frame, are adopted to validate the proposed method. Finally, the seismic reliability of a typical high-pier and long-span continuous rigid frame bridge located in southwest of China is evaluated by the proposed method and some critical conclusions are drawn. The results obtained from this study not only indicate the accuracy and efficiency of the proposed method but also can provide some direct guidelines for the seismic design of the high-pier and long-span bridges in fault zones.

Keywords

dynamic reliability extreme value distribution fraction moments high-pier bridges near-fault impulsive ground motions

Introduction

Because of the special site topographies of southwestern region of China, numerous high-pier bridges have been built or are being constructed to cross deep valleys or rivers (Han et al., 2017; Jia et al., 2013). According to statistics, the bridges, which contain pier higher than 40 m, make a proportion more than 40% of all the bridges (Chen et al., 2018). As is known to all, the southwestern region of China is an active seismicity zone. Many strong earthquakes (e.g. the Wenchuan earthquake with magnitude 7.9, Lushan earthquake with magnitude 7.0, and Jiuzhaigou earthquake with magnitude 7.0) have occurred in recent 10 years. Earthquake has become one of the major factors that threaten the safety of bridges in this region.

Near-fault impulsive ground motions caused by the forward directivity as well as flipping step effect are commonly observed at the vicinity site of an earthquake rupture-fault (Bray and Rodriguez-Marek, 2004; Yang and Zhou, 2015). The most distinct characteristic of such ground motions is that the velocity time history contains a high-amplitude and long-period pulse. The pulse makes it pose significantly unfavourable structural seismic demands, especially for long-period structures. Many studies have been conducted on the effect of near-fault ground motions on the seismic response of structures, such as large-span cable-stayed bridges (Li et al., 2016), tall buildings (Salimbahrami and Gholhaki, 2018; Sehhati et al., 2011), dams (Zhang and Wang, 2013) and base-isolated structures (Rawat et al., 2018), but few on the seismic response of high-pier bridges. As for these high-pier bridges, the high and flexible piers lead to the longer fundamental period. Under the near-fault impulsive ground motion excitation, the fundamental period of bridge may be very close to the pulse period. This will lead to significantly resonate on the structure and force it to dissipate a large amount of energy during a limit displacement. Higher seismic demand of high-pier bridges is generated due to the impulsive effect. Studying the effect of near-fault impulsive ground motions on the seismic performance of high-pier bridges will help improve the aseismic design of them.

Extreme value distribution (EVD) of nonlinear seismic response of structures subjected to near-fault stochastic ground motions can be very helpful to the dynamic reliability and risk analysis of structures in fault zones (Chen and Li, 2007; Xu et al., 2018). According to the first-passage criterion, the fail probability of structure can be equivalent to the probability when the response exceeds a specific threshold value (Grigoriu and Samorodnitsky, 2014). In other words, once the EVD of structure response is determined, the dynamic reliability can be obtained. To evaluate the EVD of structure response, numerous methods have been developed during the past few decades, which can be roughly summarized as the following three categories: (1) the analytical methods, (2) the approximate methods and (3) the simulated methods. Among them, the analytical methods can obtain the exact solution of EVD, but it is only applicable for some very simple case. In this regard, they cannot be used in practical engineering. The approximate methods are, for example, level-crossing process-based method (Powell, 1958), equivalent linearization method (Tognarelli et al., 1997), moment closure method (Lutes and Sarkani, 2004) and so on. These methods may be effective when the nonlinearity is not strong, but they are usually developed based on some special assumptions, like the level-crossing process-based method, which is usually implemented based on the Rayleigh distribution of EVD (Powell, 1958). Different assumptions of level-crossing events may lead to a great difference in the dynamic reliability, especially for small failing probability case. The simulated method such as Monte Carlo simulation (MCS; Goller et al., 2013; Grigoriu and Samorodnitsky, 2014), subset sample method (Au and Beck, 2001) and important sample method (Liu and Yao, 2009) can obtain the accurate result of EVD, but the extensive amount of computation is usually unbearable, especially for small failure probability. Recently, Xu et al. (2018) proposed an efficient two-step method to evaluate the EVD of structure nonlinear dynamics response. The limitation of this approach is the selection of bandwidth of kernel density estimation (KDE), which should be specially considered in this approach. Hence, developing an efficient method with desirable accuracy to simulate the EVD of structure nonlinear dynamic response, especially at the low probability level, is still an extensive challenge.

To address this challenge, this study proposed a novel method by combining Latin hypercube sampling (LHS) and the improved maximum entropy method constrained with fractional moments (MEM-FM) to evaluate the EVD and seismic reliability of high-pier and long-span bridges in fault zones. The outline of this study is as follows: section ‘Problem formulation’ briefly elucidates the problem formulation. Then, the stochastic synthesis of near-fault impulsive ground motions on orientation of the strongest pulse is introduced in detail in section ‘Stochastic synthesis of near-fault pulse ground motion considering pulse directivity’. In section ‘The EVD estimation of structures’, a new method by combining LHS and MEM-FM is proposed to evaluate the EVD of structures subjected to near-fault impulsive ground motions, and the efficiency and accuracy of the proposed method is validated by a single-degree-of-freedom (SDOF) system and a three-storey nonlinear shear frame. Based on the proposed method, a typical high-pier continuous rigid frame bridge (CRFB) is taken as an example, and the EVD and dynamic reliability of the bridge subjected to near-fault impulsive ground motions are estimated in section ‘Practical application: reliability assessment of a high-pier bridge’. Finally, some conclusions are summarized in section ‘Conclusion’.

Problem formulation

Consider an n-DOF nonlinear system, the equations of motion under near-fault impulsive ground motions are governed by (Xu et al., 2018)

M (ζ) \overset{\cdot\cdot}{X} (t) + C (ζ) \overset{\cdot}{X} (t) + G [X (t) \overset{\cdot}{X} (t), ζ] = - M (ζ) I {\overset{\cdot\cdot}{x}}_{g} (ζ, t)

(1)

where M and C are mass and damping matrices of order n × n, respectively; $\overset{\cdot\cdot}{X} (t)$ , $\overset{\cdot}{X} (t)$ and $X (t)$ are the acceleration, velocity and displacement vector of order n × 1, respectively; G is the n × 1 nonlinear restore force vector; I is the unit vector; $ζ = {[ζ_{1}, ζ_{2}, \dots, ζ_{m}]}^{T}$ is the random vector that includes the random variables involved in both structural parameters and near-fault seismic records; ${\overset{\cdot\cdot}{x}}_{g} (ζ, t)$ is the external excitation, that is, the near-fault ground motion. Generally, ${\overset{\cdot\cdot}{x}}_{g} (ζ, t)$ can be generated by the superposition of high-frequency component and long-period pulse. It can be mathematically expressed as (Li et al., 2016; Yang and Zhou, 2015)

{\overset{\cdot\cdot}{x}}_{g} (ζ, t) = {\overset{\cdot\cdot}{x}}_{s} (ζ, t) + {\overset{\cdot\cdot}{x}}_{p} (ζ, t)

(2)

where ${\overset{\cdot\cdot}{x}}_{s} (ζ, t)$ and ${\overset{\cdot\cdot}{x}}_{p} (ζ, t)$ are acceleration history of high-frequency and long-period pulse.

Solving equation (1) by the numerical method, the nonlinear dynamic response of structure, which uniquely relies on the random vector $ζ$ , is determined, and it can be expressed as

X (t) = H (ζ, t), \overset{\cdot}{X} (t) = h (ζ, t)

(3)

where H(·) and h(·) are deterministic operators. As for general structures, H(·) and h(·) are usually implicit, and their mathematic expression cannot be written directly. In such conditions, the concerned physical quantity for reliability analysis, for example, displacement of piers and bearing, curvature and stress of piers and so on, can be written in the form

z (t) = φ [X (t), \overset{\cdot}{X} (t)] = R (ζ, t)

(4)

Similarly, φ(·) and R(·) are also deterministic operators. As for the seismic reliability analysis of structures, the extreme value of z(t) is usually concerned. When all the random variables are positive, it can be expressed as

z_{ext} (t) = max_{t \in [0, T]} | z (t) |

(5)

The seismic reliability R of bridge can be defined as the probability that seismic demand of bridge does not exceed its seismic capacity. If the extreme value of bridge nonlinear seismic response is adopted to represent the seismic demand, the reliability R can be expressed as

R = \Pr [z_{ext} (t) \leq z_{T}]

(6)

where Pr represents probability; Z_T is the seismic capacity of bridge, it can be determined by Pushover analysis. When the EVD of response is known, the reliability R can be obtained by numerical integration

R = \int_{0}^{z_{T}} p_{z_{ext} (t)} (z) dz

(7)

where $p_{z_{ext} (t)} (z)$ is the EVD of response. Thus, the failure probability P_f is accordingly expressed as

p_{f} = 1 - R

(8)

It can be seen from the earlier discussion that the most important step for the dynamic reliability evaluation of bridge subjected to near-fault impulsive ground motions is the estimation of EVD. In this regard, it is great desirable to develop an efficient method to evaluate the EVD of bridges subjected to stochastic ground motions.

Stochastic synthesis of near-fault pulse ground motion considering pulse directivity

The orientations of near-fault impulsive ground motions are of paramount importance for the seismic performance assessment of structures in fault zones. The peak parameters and pulse characteristic of ground motions in different directions may display significant difference, and the seismic response of structures under the excitation of ground motions in the strongest pulses direction may be significantly larger. When impulsive motions are taken as the critical excitations, the ground motion in the maximum principal orientation should be considered. More recently, based on the seismic record of Chi-Chi earthquake 1999 in Taiwan, several studies have been conducted to the simulation of stochastic ground motions (Huang and Wang, 2014, 2015; Liu and Hong, 2013; Yang and Zhou, 2015). Among them, the method proposed by Yang and Zhou, (2015) considers the effect of orientations of ground motions. In this study, this method will be adopted to generate near-fault ground motions, which will provide base of dynamic reliability of high-pier bridges.

Simulation of high-frequency component

Based on the statistic power spectral density (PSD) model of residual ground motions for near-fault impulsive seismic record, the widely used spectrum representation method (SRM) is adopted to model the high-frequency component. The time domain representation of the high-frequency component by the SRM can be expressed as

a_{u} (t) = e (t) \sum_{k = 0}^{N / 2} \sqrt{4 S (ω) Δ ω} \cdot \cos (ω_{k} t + φ_{k})

(9)

where $S (ω)$ is the PSD of high-frequency component; ω_k is the kth order discrete frequency; Δω is the frequency step, Δω =ω_k − ω_k−₁; φ_k is the random phase angles, which are random variables uniformly distributed over [0∼2π]; e(t) is the modulating function which is introduced to simulate the nonstationary characteristic of high-frequency components. It is mathematically expressed as

e (t) = {\begin{matrix} 0 & t \leq t_{0} \\ {(\frac{t - t_{0}}{τ})}^{α} & t_{0} \leq t \leq t_{0} + τ \\ e^{- β (t - t_{0} - τ)} & t \geq t_{0} + τ \end{matrix}

(10)

where t₀ is the initial period of no ground motion, it has no effect on the seismic response of structures. According to the suggestion of Yang and Zhou (2015), it can be ignored and the value of 0 can be taken; α, β and τ are shape control parameters of envelop function, the value of them are taken as 1.88, 0.31 and 12 s, respectively. It should be noted that the modulating function adopted in SRM only guarantees the nonstationarity in the time domain. If both the nonstationarity in time and frequency domain need to be considered, the energy-compatible and spectrum-compatible (ECSC) proposed by Huang and Wang (2017) can be employed.

Simulation of long-period pulse

For long-period impulse, the stochastic model is developed based on the Gabor wavelet function. Based on the seismic record dataset of Chi-Chi, Taiwan earthquake, the proposed stochastic pulse model Yang and Zhou (2015) can be mathematically expressed as

\begin{matrix} v_{p} (t) = PG V_{Rup} \cdot \exp \\ [σ_{\ln PGV} - \frac{π^{2}}{4} {(\frac{t - T_{pk}}{N_{c} T_{p}})}^{2}] \times \cos (2 π \frac{t - T_{pk}}{T_{p}} - φ) \end{matrix}

(11)

where v_p(t) is the time history of velocity pulse of long-period components; t is the time series; T_p denotes the pulse period; T_pk represents the location of the pulse in time axis; N_c is the number of circles in the pulse; φ represents the phase of the pulse; $σ_{\ln PGV}$ is the regression residuals of the peak ground velocity (PGV) attenuation model; PGV_rup is the amplitude of the predominant pulse, which will be determined by the PGV attenuation model as follows

PG V_{Rup} = \exp (4.47 - 0.17 \ln R_{rup})

(12)

where R_rup is the rupture distance. According to the statistical analysis of ground motions in the orientation of the strongest pulses conducted by Yang and Zhou (2015), the mean, standard deviation and type of probability distribution are summarized in Table 1.

Table 1.

Parameters of stochastic pulse model (Yang and Zhou, 2015).

Parameters	$σ_{\ln PGV}$	$T_{p}$ (s)	$φ$ (rad)	$T_{pk}$ (s)	$N_{c}$
Distribution	Normal	Normal	Normal	Normal	Lognormal
Mean	0	6.76	3.04	22.04	0.02
Standard deviation	0.257	1.63	1.70	6.52	0.296

Synthetic of nonstationary near-fault impulsive ground motions

As both the high-frequency component and long-period pulse have remarkable influence on the seismic performance of high-pier bridges, the near-fault impulsive ground motions are generated by combining the high-frequency components and the long-period pulses generated from equations (9) and (11), respectively. The main procedures of this algorithm are summarized as follows:

Determine the amplitude of the predominant pulse PGV_rup with specified rupture distance R_rup by equation (12).

Sample parameters $σ_{\ln PGV}$ , $T_{p}$ , $N_{c}$ , $T_{pk}$ and $φ$ , then combine them with PGV_rup determined in Step 1 to obtain the long-period random pulse v_p(t) by equation (11).

Sample the random phase angles φ_k by repeating N/2 + 1 times to obtain the random vector of random phase and generate acceleration time history of high-frequency component by equation (9).

Obtain the velocity time history $v_{u} (t)$ of high-frequency component by integrating $a_{u} (t)$ obtained in Step 3 in frequency domain, and adjust baseline by a low-pass filter with a cut-off frequency of 0.02 Hz to achieve zero initial and zero residual value.

Construct velocity time history v(t) of near-fault impulse ground motions by combining the long-period pulse v_p(t) and high-frequency component $v_{u} (t)$ .

Obtain the acceleration and displacement time history of near-fault impulse ground motions by integrating and differentiating of v(t) in frequent domain, respectively.

Furthermore, it should be noted that the spatial varying effect is important for the seismic performance assessment of long-span bridges. However, because of the particularity of near-fault impulse ground motions, both the high-frequency content and long-period velocity pulses, which play an important role on the seismic response of long-span bridges, are contained. There is still lack of simulation method of near-fault impulsive ground motions that consider the spatial varying effect. In this study, the spatial varying effect of the input ground motion is not considered, and the uniform excitation is adopted.

The EVD estimation of structures

As discussed in section ‘Problem formulation’, high-dimensional random variables are involved in the EVD evaluation of bridges subjected to near-fault impulsive ground motions, and the dynamic reliability estimation should deal with the high-dimensional numerical integration as follows

R = \int_{0}^{z_{T}} p_{z_{ext} (t)} (z) dz = \int_{Ω s} p_{z_{ext} (t)} (ζ) d ζ

(13)

where $Ω_{s} = {ζ | z_{ext} (t, ζ) < Z_{T}}$ denotes the domain in which the seismic demand of bridges less than its capacity; $p_{z_{ext} (t)} (ζ)$ is the EVD of nonlinear dynamic response of bridges subjected to near-fault impulsive ground motions, which is an implicit function of random vector ζ . As is known to all, it is impractical to obtain the analytical solution for practical bridge engineering. The numerical methods, for example, sparse grid integration method (Xiong et al., 2010), dimension reduction method (Xu and Rahman, 2004; Zhang and Pandey, 2013) and high-order unsent transform method (Xu and Kong, 2018) are also unrealistic because of ‘the curse of dimensionality’. The MCS (Goller et al., 2013) and its improved methods seem to be the only choice. It is well known that the extensive computation amount of MCS is time-consuming, which also makes it incapable in practical engineering. In order to reduce the computation burden, in this section, a new method combining the LHS (Stein, 1987) and improved MEM-FM is proposed to evaluate the EVD of bridges subjected to near-fault impulsive excitations.

Determination of nonlinear seismic response samples

As discussed earlier, the presented samples will be generated by the LHS strategy first, and the MEM-FM will be introduced to obtain the probability distribution of samples. Compared with the direct MCS method, this method requires fewer numbers of samples. The reason of adopting LHS instead of crude MCS is that the LHS always has superior performance than MCS and can result in faster convergence. The main steps to obtain the EVD of bridges are summarized as follows:

Sample the random phase angles $φ_{k, j} (k = 0, 1, \dots, N / 2)$ by the LHS strategy to obtain m phase angle random vectors

φ_{j} = {(φ_{0, j}, φ_{1, j}, φ_{2, j}, \dots, φ_{N / 2, j})}^{T}, j = 1, 2, \dots, m

(14)

where m is the required number of stochastic seismic record; N is the number of time steps, N = T/dt;

For each specified $φ_{j}$ , generate high-frequency component $a_{u, j} (φ_{j}, t), j = 1, 2, \dots, m$ of near-fault impulsive ground motion by equation (9).

Calculate the mean square deviation ${\tilde{σ}}_{a_{u}} (t)$ of m high-frequency components of near-fault impulsive ground motions and compare it with the exact one $σ_{a_{u}} (t)$ , which can be computed from the PSD of residual seismic record

σ_{a_{u}} (t) = \sqrt{2 \int_{0}^{\infty} S (ω) d ω}

(15)

where S(ω) is the PSD of high-frequency components. The relative error ε between simulated ${\tilde{σ}}_{a_{u}} (t)$ and the exact $σ_{a_{u}} (t)$ can be defined as

ε = \frac{‖ {\tilde{σ}}_{a_{u}} (t) - σ_{a_{u}} (t) ‖}{‖ σ_{a_{u}} (t) ‖}

(16)

where ‖·‖ is the deterministic operators of 2-norm. If the relative error ε between ${\tilde{σ}}_{a_{u}} (t)$ and $σ_{a_{u}} (t)$ is less than the tolerance e, go to Step 4, else increase m and repeat Steps 1 to 3 until ε is less than e;

Sample parameters $σ_{\ln PGV}$ , $T_{p}$ , $N_{c}$ , $T_{pk}$ and $φ$ , and repeat m times, then generate the long-period random pulse $v_{p, j} (t), j = 1, 2, \dots, m$ , by equation (11).

Obtain the velocity time history $v_{u, j} (φ_{j}, t)$ of high-frequency component by integrating $a_{u, j} (φ_{j}, t)$ in frequency domain and adjust the baseline by a low-pass filter with a cut-off frequency of 0.02 Hz to remove its nonzero residual velocity. Combine the velocity time history $v_{u, j} (φ_{j}, t)$ of high-frequency components and the long-period random pulses $v_{p, j} (t)$ , which are obtained in Step 4, to construct the velocity $v_{j} (t)$ of impulsive ground motions. Finally, determine the accelerations and displacements of random impulsive ground motions by differentiating and integrating $v_{j} (t)$ , respectively.

Sample the bridge parameters by the LHS strategy to obtain m nonlinear finite element model (FEM) of high-pier bridges. Combine the near-fault ground motions with the nonlinear FEM of bridges randomly and carry out the nonlinear time history analysis to obtain the concerned extreme value $z_{ext} (t) = max_{t \in [0, T]} | z (t) |$ of seismic response.

Determine the distribution of nonlinear seismic response of bridges by the improved MEM-FM method and compute the dynamic reliability R and failure probability P_f by equations (7) and (8), respectively.

EVD estimation of nonlinear seismic response of bridges by MEM-FM

For simplicity, let X represent the extreme value $z_{ext} (t)$ of nonlinear seismic response of bridge subjected to near-fault impulsive ground motions. Under the effect of model parameter uncertainty and the ground motion uncertainty, X is a positive random variable. The αth order moment (both integer and fractional moment are included) of X can be defined as

E [X^{a}] = \int_{X} x^{α} f_{X} (x) dx

(17)

where $f_{X} (x)$ is the probability density function (PDF) of X; α is a real number. Generally, $f_{X} (x)$ is unknown. The only known quantity are the m samples of X, {x₁, x₂,…, x_m}. Thus, the αth order moment can be computed by

E [X^{a}] = \frac{1}{m} \sum_{i = 1}^{m} x_{i}^{α}

(18)

According to the definition by Shannon, the entropy of random variable X can be expressed as

H (X) = - \sum_{i = 1}^{k} p_{i} \log p_{i}

(19)

where p_i is the probability of the state of random variable X. When X is a continuous random variable, $H (X)$ is defined as

H [X] = - \int_{X} f_{X} (x) \log [f_{X} (x)] dx

(20)

According to the principle of maximum entropy, the unbiased estimation of $f_{X} (x)$ can be obtained by maximizing the entropy H[X] with fractional moments of random variable as constraints. It can be mathematically expressed as (Zhang and Pandey, 2013)

f_{X} (x) \in \arg max H (X) = - \int_{X} f_{X} (x) \log [f_{X} (x)] dx

s . t . {\begin{matrix} \int_{X} f_{X} (x) dx = 1 \\ \int_{X} x^{α_{r}} f_{X} (x) dx = M^{α_{r}} (r = 1, 2, \dots, k) \end{matrix}

(21)

To solve this nonlinear optimization problem, the Lagrange function can be adopted, and the derived general form of $f_{X} (x)$ is given as

{\hat{f}}_{X} (x) = \exp (- \sum_{r = 0}^{k} λ_{r} x^{α_{r}})

(22)

where $λ = [λ_{0}, \dots, λ_{k}]^{T}$ is the k + 1 dimension vector of Lagrange multipliers, and $α = [α_{0}, \dots, α_{k}]^{T}$ is the vector of fractional exponents, respectively. According to the normalization condition that the PDF must integrate to one, it can be derive that α = 0 and

λ_{0} = \ln [\int_{X} \exp (- \sum_{r = 1}^{k} λ_{r} x^{α_{r}}) dx]

(23)

To implement the moment-constrained optimization problem, an alternative method is minimizing the Kullback–Leibler (K-L) divergence (Kullback, 1959) between ${\hat{f}}_{X} (x)$ and $f_{X} (x)$ as follows

\begin{matrix} K [f, \hat{f}] = \int_{X} f_{X} (x) \ln [f_{X} (x) / {\hat{f}}_{X} (x)] dx \\ = \int_{Y} f_{X} (x) \ln [f_{X} (x)] dx - \int_{Y} f_{X} (x) \ln [{\hat{f}}_{X} (x)] dx \\ = - H (f) - \int_{Y} f_{X} (x) \ln [{\hat{f}}_{X} (x)] dx \end{matrix}

(24)

Substituting the expression of ${\hat{f}}_{X} (x)$ into equation (24), the $K [f, \hat{f}]$ can be written as

K [f, \hat{f}] = - H (f) + λ_{0} + \sum_{r = 1}^{k} λ_{r} M_{X}^{α_{r}}

(25)

where $H (f)$ is the entropy of $f_{X} (x)$ which is a function of λ and α . The minimization of $K [f, \hat{f}]$ is equivalent to the following function (Zhang and Pandey, 2013)

I (λ, α) = K [f, \hat{f}] + H (f) = λ_{0} + \sum_{r = 1}^{k} λ_{r} M_{X}^{α_{r}}

(26)

Therefore, f_X(x) can be obtained when

f_{X} (x) \in \arg min I (λ, α) = λ_{0} + \sum_{r = 1}^{k} λ_{r} M_{X}^{α_{r}}

(27)

This un-constrained optimization can be carried out directly with the simplex search method in MATLAB, but a troublesome problem is the initial choice of λ and α . In the previous studies, λ and α are arbitrarily specified, which will lead to the instability of the solve progress. To improve the efficiency and accuracy of the simplex method, the initial values of λ and α will be determined by an estimator-update method in this study (Xu, 2016).

Initial value determination of MaxEnt distribution parameter

The main ideal of the estimator-update method is obtaining the initial value of λ and α by solving a linear system of equations instead of arbitrary specified. Based on this initial value, a more accurate result of equation (27) can be obtained by the simplex iteration progress. The details to determine λ and α can be illustrated as follows:

Give the dimension value of λ and α (according to the study by Zhang and Pandey (2013), three-order of fractional moments can accurately simulate the tail distributions, therefore, k equal to 3 is adopted in this study), starting the estimator-update progress.

Initialize the fractional exponent vector such that $α = [α_{1}, \dots, α_{k}]^{T}$ .

Determine the initial value of Lagrange multipliers $λ = [λ_{1}, \dots, λ_{k}]^{T}$ by solving the linear system equation as follows (Chen et al., 2010; Xu, 2016)

\frac{\partial I (λ, α)}{\partial λ_{i}} = M_{X}^{α_{i}} - M_{X}^{g_{i}} (λ)

(28)

where $(\partial I (λ, α)) / \partial λ_{i}$ is the gradient vector of objective function; $M_{X}^{g_{i}} (λ)$ takes the form of

M_{X}^{g_{i}} (λ) = \frac{\int_{X} g_{i} (x) \exp (- \sum_{r = 1}^{k} λ_{r} x^{α_{r}}) dx}{\int_{X} \exp (- \sum_{r = 1}^{k} λ_{r} x^{α_{r}}) dx}, (i = 1, \dots, k)

(29)

where $g_{i} (x) = x^{α_{i}}$ . Specifically, let $(\partial I (λ, α)) / \partial λ_{i} = 0$ , and integration by parts method is adopted, one can obtain

M_{X}^{α_{i}} = \sum_{r = 1}^{k} λ_{r} E [G_{i} (x) g'_{i} (x)] = \sum_{r = 1}^{k} λ_{r} ω_{ij}

(30)

where $E [\cdot]$ represents the expectation operator, $G_{i} (x) = \int_{X} g_{i} (x) dx$ . And $ω_{ij}$ can be represented as

ω_{ij} = E [\frac{α_{j}}{α_{i} + 1} x^{α_{i} + α_{j}}]

(31)

According to equations (30) and (31), Lagrange multipliers $\tilde{λ}$ can be easily obtained by solving the linear system equation as follows (Xu, 2016)

\tilde{λ} = ω^{- 1} M_{X}

(32)

where $ω = [ω_{ij}]$ , and $M_{X} = [M_{X}^{α_{1}}, M_{X}^{α_{1}}, \dots, M_{X}^{α_{k}}]$ .

According to equation (32), a solution with less accuracy, but easily calculated, can be obtained. It will provide an initial solution of estimated PDF ${\hat{f}}_{X} (x)$ of random variable X. However, it should be noted that in order to guarantee the uniqueness of solution of equation (32), the matrix $ω$ should be positive definite, otherwise go back to Step 2, and update fractional exponent α ;

4. Take the solution obtained by Step 3 as an initial value of un-constrain problem in equation (27) and obtain the final solution of λ and α by the simplex iteration method, and the final solution of EVD of bridge nonlinear dynamic response can be determined by equation (22).

Implementation and verification of the proposed method

According to the description from section ‘Determination of nonlinear seismic response samples’ to section ‘Initial value determination of MaxEnt distribution parameter’, the procedures to estimate the EVD of bridges subjected to near-fault impulsive ground motions are shown in Figure 1. To verify the proposed method, an SDOF system and a three-storey nonlinear shear frame structure will be used for EVD evaluation and compared with the crude MCS result. The efficiency and accuracy of the proposed method will be demonstrated.

Figure 1.

Flowchart for EVD estimation of bridge subjected to near-fault ground motions.

Test problem: nonlinear SDOF system

Consider a nonlinear SDOF dynamic system with the mass m = 1 kg, damping ratio $ζ_{0} = 0.05$ and initial period T₀ = 1.2 s. To estimate the maximum value of inelastic displacement demand, the approximate method proposed by Miranda et al. (2002) is adopted. The equivalent period and equivalent damping ratio are determined by the empirical equations derived by Iwan (1980) such that

{\begin{matrix} \frac{T_{eq}}{T} = 1 + 0.121 {(μ - 1)}^{0.939} \\ ζ_{eq} = ζ_{0} + 0.0587 {(μ - 1)}^{0.371} \end{matrix}

(33)

where the displacement ductility ratio μ is assumed to be equal to 4, in this study. The stochastic seismic excitations are generated by the method described in section ‘Stochastic synthesis of near-fault pulse ground motion considering pulse directivity’. The duration of ground motions is taken as 40.96 s, and time step dt is adopted as 0.02 s. According to the SRM in equation (9), the number of 1025 harmonic components of high-frequency should be taken into account, which means 1025 random variables will be involved. On the contrary, five stochastic parameters of the stochastic pulse model are also included in simulation of long-period component. As a result, a total number of 1030 random variables are involved in the dynamic reliability analysis.

To determine how many times nonlinear dynamic response analysis is sufficient in the proposed method, first, 800, 1000 and 1500 basic points of random variable are generated by the LHS strategy, respectively. Correspondingly, 800, 1000 and 1500 high-frequency components and long-period pulses of impulsive seismic records are generated by equations (9) and (11). The PSD and mean square deviation relative error between the simulated value and the exact value can be computed. When m = 800, 1000 and 1500, the relative errors of the mean square deviation ${\tilde{σ}}_{a_{u}} (t)$ to the exact one are 0.0242, 0.0214 and 0.0187, respectively. In this study, the tolerance value of e is adopted as 2%, the sample size m = 1500 is required in the analysis.

Figure 2 shows the comparison of the simulated PSD, mean and mean square deviation ${\tilde{σ}}_{a_{u}} (t)$ of high-frequency components with the exact value. It is shown in Figure 2 that the simulated nonstationary acceleration of high-frequency component with respect to PSD, mean, and mean square deviation accord quite well with the exact one. Figure 3 pictures the acceleration, velocity time history and acceleration response spectrum of a typical ground motion. It can be seen that the simulated ground motion contains an obvious long-period component in the velocity time history, indicating that the simulated ground motions can consider the impulse effect of the near-fault ground motions. It is shown in Figure 3(c) that the long-period pulse has a significant amplification effect on the seismic dynamic response of long-period structure. When the period is greater than 1 s, the response spectrum is significantly enlarged by the velocity pulse.

Figure 2.

Comparison of PSD, mean and mean square deviation between simulated and target: (a) PSD, (b) mean and (c) σ_{au (t)}.

Figure 3.

Typical sample of the generated near-fault seismic record: (a) acceleration, (b) velocity and (c) response spectrum.

Based on the proposed method, the EVD of the SDOF system displacement response can be obtained, which is shown in Figure 4(a), where the histogram of the results by MCS (10⁵ times run) is also pictured for comparison. It can be seen that the EVD of SDOF systems obtained by the proposed method is very close to the result of MCS, which demonstrates the efficiency and accuracy of the proposed method. Figure 4(b) gives the comparison of cumulative probability density functions (CDF) in the logarithmic scale. Likewise, the CDF of the SDOF system obtained by the method proposed in this article has very good agreement with the MCS result, even when the cumulative probability distribution is at the probability level of 10⁻⁵. The results show that the method proposed in this article can effectively simulate the EVD of the nonlinear seismic response of the structures under random earthquake excitations, and based on this method, the seismic reliability of the bridge can be effectively evaluated.

Figure 4.

EVD and CDF of SDOF system seismic response: (a) PDF and (b) CDF.

Test problem: three-storey shear frame structure

To exemplify the applications of the proposed method to reliability estimation of multi-DOF structures subjected to stochastic near-fault impulsive ground motions, consider a three-DOF two span of nonlinear planar share frame with uncertain parameters, which is shown in Figure 5. The lumped mass and lateral inter-story stiffness of the frame are assumed as independent uncertain variables that satisfy normal distribution. The mean value of mass is 2.71 × 10⁵ kg and coefficient of variation (COV) is equal to 10%. The mean value of lateral inter-storey stiffness is 2.42 × 10⁷ kN/m and the COV is assumed to be 15%. The Rayleigh damping C = α M + β K is considered, where α = 0.2191 and β = 0.0088.

Figure 5.

Three-storey shear frame structure.

The Bouc–Wen model (Wen, 1976; Xu and Kong, 2018) is adopted to simulate the nonlinear hysteretic behaviour, which is given as

f (z, x) = α_{B} kx + (1 - α_{B}) kz

(34)

where $f (z, x)$ is the restore force; x is the displacement; k is the initial stiffness; $α_{B}$ is the post- to pre-yield ratio; and z is the hysteretic displacement, which is given as

\overset{\cdot}{z} = (1 / η) [A_{B} \overset{\cdot}{x} - γ (β | \overset{\cdot}{x} | {| z |}^{n - 1} z + ν \overset{\cdot}{x} {| z |}^{n})]

(35)

where A_B describes the initial stiffness of the hysteretic component; γ and η are degradation parameters, which depend on the deformation history, and are defined as

γ = 1 + δ_{γ} ζ, η = 1 + δ_{η} ζ

(36)

where $δ_{γ}$ and $δ_{η}$ degradation rate; $ζ$ is the dissipated energy, which can be written as

ζ (t) = \int_{0}^{t} z \overset{\cdot}{x} dt

(37)

A total of seven parameters involved in this hysteretic model are adopted as: A_B = 2, α_B = 0.04, n = 1, $δ_{γ}$ = 0.02, $δ_{η}$ = 0.01 and $β = ν = 20$ . Under the excitation of the near-fault impulsive ground motions generated in the last section, a typical sample of the nonlinear seismic response of the frame is shown in Figure 6. It can be seen that strong nonlinearity has occurred during the excitation.

Figure 6.

Typical sample response of the second inter-storey drift and hysteretic curve.

According to the nonlinear seismic response, the EVD of the frame structure can be obtained by the proposed method, which is shown in Figure 7. For the frame structure, the 0–1 inter-storey is the most vulnerable location, therefore, the 0–1 inter-storey drift is taken as an example to validate. Furthermore, the result calculated by the KDE method is also pictured in Figure 7, and both the results are compared with the result obtained from MCS (4 × 10⁴ times run). It can be seen that the result obtained by the proposed method matches very well with the MCS result. However, significant errors are observed in both the main body and the tails obtained by the KDE method. Especially for the tail distribution, which is shown in Figure 7(b), about 10 times errors of the tails have been observed. This indicates that the KDE method is not accurate for the evaluation of small failure probability. However, the result calculated from the proposed method still accords very well with the result by MCS, which further demonstrates the accuracy of the proposed method for evaluating the PDF of the structure nonlinear dynamic response even when the uncertainty of the seismic ground motion, nonlinearity and uncertainty of the structures are all considered.

Figure 7.

EVD evaluation of inter-storey drift of shear frame structure: (a) PDF and (b) CDF.

Practical application: reliability assessment of a high-pier bridge

Description of the bridge and its FEM

This study adopts a typical high-pier CRFB as a practical engineering example. The EVD and dynamic reliability of the bridge under near-fault impulsive ground motions excitations are evaluated with the proposed method. As shown in Figure 8, the bridge has a main span and side span length of 170 and 90 m, and a total span of 350 m. The main girder of this bridge is a prestress concrete beam with variable single box and single chamber hollow cross-sections. These two bridge piers are numbered as Pier 1 and Pier 2, respectively, and with approximately height of 110 and 126 m. In each pier, the main girder is connected rigidly to restrict deformation in all directions, while a basin rubber bearing is set at the abutment to constrain vertical and transverse deformations. At the bottom of piers, Pier 1 and Pier 2 are both sited on a pile cap, which are supported by nine cast-in-place piles. The abutments are supported by rigid foundations, which are assumed to rest on stiff soil. The detailed configuration of this bridge is shown in Figure 8.

Figure 8.

Schematic view of the high-pier rigid frame bridge (m).

Three-dimensional nonlinear FEM of this bridge is developed in the OpenSees platform (McKenna et al., 2010) to simulate the nonlinear seismic response. The main girder is expected to remain linear elastic during earthquake. In this regard, a Displacement-Based Beam-Column Element connected with elastic section is adopted to simulate. As for the bridge piers, a significant nonlinearity will experience at the top and bottom during disastrous earthquake, a Force-Based Beam-Column Element with fibre section is used to simulate the nonlinearity and stochastic variation of axial load. The partition of fibre section of piers is schematically shown in Figure 9. Both the confined and unconfined concrete of piers are defined by the concrete02 material. The reinforcement bar in the piers is defined as Steel02 material. The basin rubber bearing, which will also suffer a serious nonlinearity during strong earthquake, is simulated by the zero-length element connected with the hardening uniaxial material. The expansion joints between the main girder and abutment are potential locations for pounding. To consider the energy dissipation and change in collision stiffness during the progress of pounding, a contact element of the hertz-damp model with a nonlinear damper (Muthukumar, 2003) is adopted to simulate the pounding behaviour. In addition, the soil–structure interaction is not included in this study. Both the foundations of pier and abutment are modelled as rigid. According to the study of Xu and Feng (2018), the effect of the randomness of the seismic ground motion is usually much more significant than the uncertainty of structure parameters. In this section, only the uncertainty of ground motion is taken into account.

Figure 9.

FEM of the high-pier CRF bridge.

Limit states definition of the bridge

In the performance-based earthquake engineering, four limit states (LS), namely slight (SL), moderate (MO), extensive (EX), and complete (CO) damage of structures are commonly defined (Wu et al., 2016). This study mainly considers the damage of bridge piers. Usually, the ductility factors, that is, displacement ductility factor and curvature ductility factor, are commonly defined as the damage index (DI) of concrete pier. For high-pier bridges, second and higher order mode have remarkable influence on seismic response of bridge structure, which will lead to the weak correlation between the curvature of the piers bottom and displacement of the piers peak. In view of this, in this study, both the ductility factor of displacement and curvature are adopted as the DI of piers. The failure probability of piers determined by displacement ductility factor and curvature ductility factor is compared with each other. Because the high-pier bridges are usually the key point of the transportation, extensive and complete damage should be prevented. The probability of the extensive and complete damage is very small. In this study, only the failure probability of SL and MO damage is considered. The quantified LS are summarized in Table 2.

Table 2.

Quantified limit states of bridge pier.

Pier	Axial force (kN)	$ϕ_{cy}$ (1/m)	$δ_{cy}$ (cm)	Slight (SL) damage		Moderate (MO) damage
Pier	Axial force (kN)	$ϕ_{cy}$ (1/m)	$δ_{cy}$ (cm)	$μ_{d}$	$μ_{ϕ}$	$μ_{d}$	$μ_{ϕ}$
Pier 1	110,985	1.60 × 10⁻⁴	32.64	1.0	1.0	1.28	1.28
Pier 2	175,385	1.52 × 10⁻⁴	40.26	1.0	1.0	1.24	1.24

EVD and dynamic reliability estimation

Figure 10 shows the comparison of PDF of curvature ductility factor between Pier 1 and Pier 2. From Figure 10, it can be observed that the difference in the PDF of curvature ductility factor between Pier 1 and Pier 2 is remarkable. The maximum value of PDF of Pier 1 is 4.07, the corresponding value of $μ_{ϕ}$ is 0.310, which is also the maximum probably response of Pier 1. However, as for Pier 2, the maximum value of PDF is 5.53, the corresponding value of $μ_{ϕ}$ is 0.274. The mean value and variability of curvature ductility factor $μ_{ϕ}$ of Pier 1 are obviously larger than those of Pier 2.

Figure 10.

PDF of the curvature ductility factor of Pier 1 and Pier 2.

Figure 11 shows the probability of exceedance (POE) curve of the curvature ductility factor of Pier 1 and Pier 2, which is the complementary part of the CDF. For comparison, the LHS-simulated result and the result obtained by KDE are also pictured in Figure 11. From these Figures, it can be observed that the result obtained by the proposed MEM-EM method matches very well with the LHS result, while a sudden drop of result obtained by KDE appeared, which further demonstrates the accuracy of the proposed method in large-scale practical engineering structure. Based on the POE curve and the LS presented in Table 2, the failure probability of piers, which is calculated from equation (8), can be obtained. Under the limited state of SL damage, the failure probability of Pier 1 and Pier 2 is 1.34 × 10⁻² and 0.32 × 10⁻², respectively. Under the LS of MO damage, the failure probability of Pier 1 and Pier 2 is 0.53 × 10⁻² and 0.16 × 10⁻², respectively. Obviously, it can be concluded that, under both the condition of SL and MO damage, the failure probabilities of Pier 1 are larger than those of Pier 2. This can be explained from the construction of bridge structure shown in Figure 8. As for the bridge presented in this study, the constraint of the pier and girder are rigid connection. Under the seismic excitation, the deformation of the Pier 1 and Pier 2 must be in conjunction with each other. Due to the difference in the height, the pier with the relative lower height (i.e. Pier 1) will suffer greater curvature on the bottom section of the pier. In this regard, it can be concluded that in the design of high-pier CRFB, more attention should be paid to the seismic performance improvement of piers with relative lower height.

Figure 11.

POE of the curvature ductility factor of Pier 1 and Pier 2: (a) POE of rotation ductile of Pier 1 and (b) POE of rotation ductile of Pier 2.

Moreover, to study the effect of DI on the failure probabilities of bridge piers, the failure probabilities of Pier 1 and Pier 2 at the LS of SL and MO damage are summarized in Table 3. It can be seen from Table 3 that the failure probabilities of piers, at both the damage state of SL and MO damage, defined by the curvature ductility factor $μ_{ϕ}$ are much greater than those defined by the displacement ductility factor μ_d. Take the failure probabilities of Pier 1 as an example, at the LS of SL damage, the failure probability defined by the DI of $μ_{ϕ}$ is 1.34 × 10⁻², whereas the failure probability defined by the DI of μ_d is only 4.22 × 10⁻⁵. Likewise, at the LS of MO damage, the failure probability defined by $μ_{ϕ}$ is 0.53 × 10⁻², whereas the failure probability defined by the DI of μ_d is only 6.21 × 10⁻⁷. The similar results of Pier 2 are also shown in Table 3. These results indicate that the displacement ductility factor μ_d is inapplicable to estimate the seismic reliability of high-pier bridges. If μ_d is taken as the DI, the estimated probability of bridge pier is still very small even when a serious damage has occurred at the bottom of bridge pier. It can be concluded from these results that compared with the displacement ductility factor μ_d, the curvature ductility factor $μ_{ϕ}$ is more reasonable to describe the damage of piers of high-pier bridges.

Table 3.

Comparison of failure probability of bridge pier with different damage indices.

Damage index	Pier 1		Pier 2
Damage index	Slight damage	Moderate damage	Slight damage	Moderate damage
$μ_{d}$	4.22 × 10⁻⁵	6.21 × 10⁻⁷	1.17 × 10⁻⁶	3.03 × 10⁻⁸
$μ_{ϕ}$	1.34 × 10⁻²	0.53 × 10⁻²	0.32 × 10⁻²	0.16 × 10⁻²

Conclusion

Based on the equivalent EVD and MEM-FM, this study developed an efficient and accurate method to evaluate the EVD and dynamic reliability of nonlinear structures, using which the seismic reliability of a typical high-pier and long-span CRFB subjected to near-fault impulsive ground motions is evaluated. The failure probabilities of piers with respect to different damage indices and different heights are investigated. The drawn conclusions can be summarized as follows:

The proposed method, which combines the LHS and improved MEM-FM, is an effective and accurate approach to evaluate the EVD and dynamic reliability of structure subjected to stochastic seismic excitations. Based on the method, both the main body and tails of EVD of the structure nonlinear dynamic response can be accurately evaluated within a small amount of computation. It can provide an effective approach to estimate the failure probability of structure at small probability level.

To CRFB with irregular high-piers, the failure probability of each pier with different heights varies greatly. Under the LSs of SL and MO damage, the failure probability of piers with the relative lower height is about four times larger than the failure probability of piers with higher height. In the design of high-pier CRFB, more attention should be paid to the seismic performance improvement of piers with relative lower height.

As for the high-pier and long-span CRFB, the difference in failure probability defined by curvature ductility coefficient and displacement ductility coefficient is quite large. The failure probability of bridge piers defined by curvature ductility factor is obviously higher than the probability defined by displacement ductility factor. In other words, to high-pier bridges, curvature ductility factor can more reasonably reflect the damage state of piers.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research presented in this study was supported by the National Natural Science Foundation of China (grant numbers: U1434205 and 51508473).

ORCID iD

Zhi-Qiang Chen

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