Extreme Value Distribution Estimation Method for Nonlinear Seismic Response of Bridge Structures
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摘要: 为了研究近断层脉冲地震作用下桥梁非线性地震响应极值分布，进行小失效概率下的桥梁动力可靠度精确计算，提出了一种有效的近断层脉冲地震作用下桥梁结构非线性地震响应极值分布分析方法. 首先考虑桥梁结构的非线性和地震动的不确定性，采用拉丁超立方抽样对近断层脉冲地震动随机参数和结构随机参数进行随机抽样，通过模拟的高频地震动均方值和与精确值的相对误差确定出所需要的样本数量；其次以合成的近断层脉冲地震动作为地震激励，通过时程分析对结构非线性动力方程进行求解，从而得到结构非线性地震响应极值样本，再采用改进的分数阶矩最大熵原理获得结构非线性地震响应的极值分布；最后通过非线性单自由度系统和三层非线性剪切框架验证了该方法的有效性. 研究结果表明：该方法不仅能够有效的模拟近断层脉冲地震作用时，桥梁结构与地震动双重不确定性影响下的动力响应极值分布，更能在兼顾效率和计算精度时，精确估计桥梁结构非线性地震响应极值的尾部分布，能够为桥梁结构非线性动力可靠度评估提供一种有效的途径.Abstract: To study the extreme value distribution (EVD) of nonlinear seismic response of bridges subjected to near-fault impulse ground motion and accurately estimates the dynamic reliability of bridges at small failure probability level, an effective method is proposed and validated. Considering the nonlinearity of bridge structures and the uncertainty of ground motions, the random parameters of near-fault impulse ground motion and structure are sampled by Latin hypercube sampling (LHS). The required number of seismic samples is determined by the relative error between the mean square value of simulated high-frequency ground motion and the exact value. Using synthetic near-fault impulse ground motions as seismic excitation, the structural nonlinear dynamic equation is solved by time history analysis, and the structural nonlinear seismic response extreme value samples are obtained. Then, the improved maximum entropy principle with fraction moment is adopted to obtain the EVD of bridges. The accuracy and efficiency of the proposed method are verified by a single degree freedom system and a three-story shear frame structure in comparison with the Monte Carlo simulation. Results indicate that the proposed method can not only effectively simulate the EVD of dynamic response of bridge under the double uncertainties influence of structure and ground motion, but also accurately estimate the tail distribution of the extreme value of nonlinear seismic response of bridge structure, with a balanced between efficiency and calculation accuracy. This method provides an effective way for evaluating the non-linear dynamic reliability of bridge structures.
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图 2 高频分量均值和均方值与目标值的比较
Figure 2. Comparisons of simulated mean and standard deviation of high-frequency component with the exact one
图 3 SDOF系统地震响应极值分布和累积概率分布
Figure 3. PDF and CDF of the SDOF system subjected to near-fault impulse ground motions
表 1 单自由度系统地震响应估计值与MCS的比较
Table 1. Comparison of estimated SDOF system seismic response with MCS
方法 μ $\sigma ' $ E(X (0.25)) E(X (1.20)) LHS 12.17 3.454 1.854 3 20.250 0 MCS（105 次） 12.05 3.424 1.849 7 20.000 0 相对误差/% 1.04 0.88 0.25 1.26 
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表 2 单自由度系统地震响应极值最大熵分布参数
Table 2. MaxEnt distribution parameters of SDOF system
分布 r λr αr $M_X^{ {(\alpha _r)} } $ I（λ，α） 1 0 14.537 0 2 1 1.672 1 0.863 9 8.622 1 2.603 4 3 2 −13.279 3 0.303 9 2.119 4 4 3 86.647 8 −1.617 5 0.020 7
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表 3 框架结构的集中质量和层间刚度
Table 3. Lumped mass and lateral inter-story stiffness of the frame
变量 均值 变异系数 分布类型 M1 2.71 × 105 kg 正态 M2 2.71 × 105 kg 0.10 M3 2.71 × 105 kg k1 2.42 × 107 N/m k2 2.42 × 107 N/m 0.15 k3 2.42 × 107 N/m
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