随机激励下自复位结构首次穿越失效研究

梁霄，陈林聪，韩琳

引 en

随机激励下自复位结构首次穿越失效研究^*

梁霄¹
机构：
1. 中船第九设计研究院工程有限公司,上海 200090
×
，陈林聪²
机构：
2. 华侨大学土木工程学院,厦门 361021
×
，韩琳³
机构：
3. 山西省古建筑与彩塑壁画保护研究院,太原 030012
×

1. 中船第九设计研究院工程有限公司,上海 200090；
2. 华侨大学土木工程学院,厦门 361021；
3. 山西省古建筑与彩塑壁画保护研究院,太原 030012；

First-Passage of Self-Centering System Under Random Excitation^*

Liang Xiao¹
Affiliation：
1. China Shipbuilding NDRI Engineering Co.,Ltd,Shanghai 200090 ,China
×
，Chen Lincong²
Affiliation：
2. College of Civil Engineering,Huaqiao University, Xiamen 361021 ,China
×
，Han Lin³
Affiliation：
3. Shanxi Academy of Ancient Building and Painted Sculpture & Fresco Preservation, Taiyuan 030012 ,China
×

1. China Shipbuilding NDRI Engineering Co.,Ltd,Shanghai 200090 ,China；
2. College of Civil Engineering,Huaqiao University, Xiamen 361021 ,China；
3. Shanxi Academy of Ancient Building and Painted Sculpture & Fresco Preservation, Taiyuan 030012 ,China；

通讯作者:

陈林聪,E-mail:lincongchen@hqu.edu.cn

中图分类号:O324

文献标识码:A

文章编号:1672-6553-2023-21(5)-093-008

DOI:10.6052/1672-6553-2023-055

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目录contents

摘要 Abstract
关键词 Keywords
1 理论分析
2 首次穿越失效
3 算例分析
4 结论
参考文献

摘要

自复位结构遭受地震或强风荷载时,会发生显著的非线性随机振动,这可能导致结构的运行性能大大降低,甚至完全失效.本文旨在研究随机激励下双自由度自复位结构的首次穿越失效问题.应用广义谐波平衡技术,将自复位恢复力进行分解,获得等效随机系统;利用随机平均原理推导出关于幅值的平均Itô随机微分方程;求解后向Kolmogorov(BK)方程得到首次穿越时间的条件可靠性函数和条件概率密度函数通过.作为算例,选用金井清滤波白噪声模型,分析了激励强度D及土层阻尼比ξ_g取值变化时对条件可靠性函数和条件概率密度函数的影响.通过与Monte Carlo模拟得到的结果进行对比,验证了解析解的有效性.

Abstract

When the self-centering structure is subjected to earthquake or strong wind load, significant nonlinear random vibration will occur, which may lead to the structural performance greatly reduced or even complete failure.In this paper, the first-passage failure problem of a self-centering structure with two-degrees-of-freedom under random excitation is studied.The generalized harmonic balance technique is used to decompose the self-centering restoring force and obtain the equivalent random system.The average Itô stochastic differential equation with respect to amplitude is derived by means of stochastic average method.The backward Kolmogorov (BK) equation is solved to obtain the conditional reliability function (CRF) and conditional probability density function of the first pass time (PDF).As an example, Kanai-Tajimi filtered white noise model is used to analyze the influence of the change of excitation strength D and soil damping ratio ξ_g on conditional reliability function and conditional probability density function.The validity of the analytical solution is verified by comparing with Monte Carlo simulation results.

关键词

自复位结构；双自由度；随机平均法；首次穿越失效；金井清过滤白噪声激励

Keywords

self-centering structure ； two degrees of freedom ； stochastic average method ； first-passage failure ； Kanai-Tajimi filtered white noise excitation

引言
地震作为一种常见的自然灾害，给人类社会带来的损害难以想象.传统的抗震设计方法确保结构具有足够的延性和消能能力，但对于控制地震后的残余变形及其带来的损坏还未能达到预期的目标^[1，2].因此，在实际工程中迫切需要提高工程结构的恢复能力和抗震性能.自复位结构作为一种新型抗震结构，在过去十几年中得到了广泛研究^[3-10].研究结果表明，自复位结构在地震作用下可以提高结构的复位能力，减小结构的残余变形，降低结构的修复成本.
首次穿越失效是非线性随机振动中最主要同时也是最困难的问题之一，它是研究结构可靠度以及预测结构寿命的基础.到目前为止，已经提出了多种方法来研究首次穿越失效问题.例如，Zeng等^[11]采用蒙特卡洛模拟法研究了泊松白噪声和随机激励下滞回系统的平稳响应和首次穿越失效; Chen等^[12]利用随机扩散原理，得到了多自由度非线性结构的随机动力响应概率密度函数，进而得到单一失效模式的结构动力可靠度; Engelund等^[13]运用级数展开方法，得到了结构首次穿越失效概率的上下界.以上这些方法虽能有效研究首次穿越失效问题，但仍存在计算过程复杂、计算量大、效率低等不足.因此，发展一种可行且高效的方法来研究结构的首次穿越失效问题成为目前的关注重点.随机平均法作为研究随机强非线性系统的有效方法之一，目前得到了广泛的关注和应用^[14-19].一方面，经过随机平均处理后，随机微分方程的维数大大降低，便于求解后向Kolmogorov方程; 另一方面，经过平均处理后，随机系统对非高斯白噪声激励的响应在一定条件下可近似为扩散的Markov过程，扩大了这一理论分析方法的适用范围.近年来，张雷等^[20]运用随机平均法，研究了高斯白噪声激励下多自由度强非线性随机振动系统的首次穿越; Zhu等^[21，22]应用随机平均法，研究了随机地震激励下多自由度拟可积/拟不可积哈密顿系统的首次穿越.研究表明，随机平均法的应用大大减小了计算量，使得效率得到了很大的提升.因此，这一方法已成为近几十年来研究首次穿越失效问题的主要方法.例如，曾岩等^[23]、Chen等^[24]、Wu等^[25]运用随机平均法分别研究了非高斯激励下双线性滞迟系统和Bouc-Wen滞迟系统这两类经典滞迟系统、具有分数阶导数阻尼的单自由度非线性振子、谐和与真实噪声激励下强非线振子的首次穿越失效问题，并均与Monte Carlo模拟结果进行对比，验证了所用理论分析方法的可行性.
本文研究了随机激励下双自由度自复位结构的首次穿越失效问题.在理论分析中，首先利用广义谐波平衡技术将旗帜形恢复力解耦为等效阻尼和等效刚度，得到与原系统等效的非线性随机系统，再经过随机平均进一步简化为扩散过程.随后，求解由平均伊藤方程建立的后向Kolmogorov（BK）方程，得到条件可靠性函数和条件概率密度函数.最后，研究了激励强度D及土层阻尼比ξ_g对系统可靠性的影响.通过与Monte Carlo模拟数据的对比，验证了所提方法的有效性和实用性.
1 理论分析
考虑随机激励下双自由度自复位系统，其运动微分方程可表示为:

\begin{matrix} m_{1} {\ddot{Y}}_{1} + (c_{1} + c_{2}) {\dot{Y}}_{1} + k_{2} Y_{1} + F (Y_{1}, {\dot{Y}}_{1}) - \\ c_{2} {\dot{Y}}_{2} - k_{2} Y_{2} = - m_{1} ξ (t) \\ m_{2} {\ddot{Y}}_{2} + c_{2} {\dot{Y}}_{2} + k_{2} Y_{2} - c_{2} {\dot{Y}}_{1} - k_{2} {\dot{Y}}_{1} = - m_{2} ξ (t) \end{matrix}

(1)

其中，k_i为质量m_i的刚度系数; c_i为质量m_i的阻尼系数; ξ（t）为随机激励; F（Y₁， ${\dot{Y}}_{1}$ ）为自复位恢复力; Y₁， ${\dot{Y}}_{1}$ ， ${\ddot{Y}}_{2}$ 分别为位移、速度、加速度.
图1 旗帜形恢复力分解
Fig.1 The decomposition of flag-shaped restoring force
大量试验证明地震作用下自复位结构的力-位移关系曲线呈旗帜形^[26-28]，如图1所示.其中Y_y为系统的屈服位移，F_y为系统的屈服力; α表示消能单元的屈服后刚度系数.
旗帜形恢复力可分解为弹性力（图1（b））和滞回力（图1（c））:

F (Y, \dot{Y}) = α Y + (1 - α) Z

(2)

其中，Z为旗帜形模型的滞变位移（图2），α为能量耗散系数.
滞变位移Z可表示为以下的分段函数形式:
图2 自复位体系滞变位移
Fig.2 The hysteretic displacement of the self-centering system

(3)

其中，A、Y_y、β分别表示系统位移幅值、屈服位移和能量耗散系数.
将式（3）代入式（2）中，可得旗帜形恢复力的表达式为:

(4)

由图1可知，恢复力不仅对阻尼有贡献，对系统刚度也有贡献.引入广义谐波平衡技术，即:

\begin{matrix} Y (t) = A (t) c o s Φ^{'} (t) \\ \dot{Y} (t) = - A (t) ω (A) s i n Φ^{'} (t) \end{matrix}

(5)

其中，Φ′（t）≈ωt+Φ.旗帜形恢复力可解耦为幅值依赖的等效刚度和等效阻尼.结果为:

F (Y, \dot{Y}) \approx K_{1} (A) Y + C (A) \dot{Y}

(6)

式中，K₁（A）表示等效刚度系数，C（A）表示等效阻尼系数，表达式分别为:
$K_{1} (A) = \frac{1}{π A} \int_{0}^{2 π} (A c o s Φ^{'}, - A ω s i n Φ^{'}) c o s Φ^{'} d Φ^{'} =$

(7)

式中， $ω = \sqrt{K_{1} （ A ）}$ 为系统平均频率，Ω₁-Ω₄及φ₁-φ₃的具体表达式如下:

\begin{matrix} Ω_{1} = π - φ_{2} - φ_{1} + \frac{1}{2} s i n 2 (π - φ_{2}) - \frac{1}{2} s i n (2 φ_{1}) \\ Ω_{2} = \frac{1}{2} α [s i n (2 φ_{2}) + 2 φ_{2}] - \frac{2}{A} (1 - α) Y_{y} s i n φ_{2} \\ Ω_{3} = \frac{1}{2} α [s i n (2 φ_{1}) - s i n (2 φ_{3})] + \frac{2}{A} (1 - α) \cdot \\ (1 - β) (s i n φ_{1} - s i n φ_{3}) Y_{y} + α (φ_{1} - φ_{3}) \\ Ω_{4} = \frac{1}{2} s i n (2 φ_{3}) + φ_{3} + \frac{2}{A} (1 - α) (Y_{y} - A) s i n φ_{3} \\ φ_{1} = a r c c o s [\frac{(1 - β) Y_{y}}{A}] \\ φ_{2} = a r c c o s (\frac{Y_{y}}{A}) \\ φ_{3} = a r c c o s (\frac{A - β Y_{y}}{A}) \end{matrix}

(8)

考虑等效刚度和等效阻尼，系统（1）可重新写为:

\begin{matrix} m_{1} {\ddot{Y}}_{1} + [c_{1} + c_{2} + C (A_{1})] {\dot{Y}}_{1} + [k_{2} + \\ K_{1} (A_{1})] Y_{1} - c_{2} {\dot{Y}}_{2} - k_{2} Y_{2} = - m_{1} ξ (t) \\ m_{2} {\ddot{Y}}_{2} + c_{2} {\dot{Y}}_{2} + k_{2} Y_{2} - c_{2} {\dot{Y}}_{1} - k_{2} Y_{1} = - m_{2} ξ (t) \end{matrix}

(9)

设系统（9）有如下形式的解:

\begin{matrix} Y_{i} (t) = A_{i} (t) c o s Φ_{i} (t) \\ {\dot{Y}}_{i} (t) = - A_{i} (t) ω_{i} (A_{i}, Θ_{i}) s i n Φ_{i} (t) \end{matrix}

(10)

式中，Φ_i（t）≈ω_i（A）t+Θ_i（t）; A_i（t）、Φ（t）、Θ_i（t）均为随机过程.
式（10）可看作 ${[Y_{i} ， {\dot{Y}}_{i}]}^{T}$ 到[A_i，Θ_i]^T的广义范德波变换，将式（10）代入式（9），得到关于[A_i，Θ_i]^T的随机微分方程组，

\begin{matrix} \frac{d A_{1}}{d t} = F_{11} (A_{1}, Φ_{1}) + G_{11} (A_{1} + Φ_{1}) ξ (t) \\ \frac{d Φ_{1}}{d t} = F_{21} (A_{1}, Φ_{1}) + G_{21} (A_{1} + Φ_{1}) ξ (t) \\ \frac{d A_{2}}{d t} = F_{12} (A_{1}, Φ_{1}) + G_{12} (A_{1} + Φ_{1}) ξ (t) \\ \frac{d Φ_{2}}{d t} = F_{22} (A_{1}, Φ_{1}) + G_{22} (A_{1} + Φ_{1}) ξ (t) \end{matrix}

(11)

式中
$\begin{matrix} G_{11} = \frac{\sin Φ_{1}}{ω_{1}}, G_{21} = \frac{\cos Φ_{1}}{A_{1} ω_{1}} \\ G_{12} = \frac{\sin Φ_{2}}{ω_{2}}, G_{22} = \frac{\cos Φ_{2}}{A_{1} ω_{2}} \\ F_{11} = - A_{1} ω_{1} s i n Φ_{1} c o s Φ_{1} - \end{matrix}$
$\begin{matrix} \frac{[c_{1} + c_{2} + C (A_{1})] A_{1} {s i n}^{2} Φ_{1}}{m_{1}} + \\ \frac{c_{2} A_{2} ω_{2}}{m_{1} ω_{1}} s i n Φ_{1} s i n Φ_{2} + \\ \frac{[k_{2} + K_{1} (A_{1})] A_{1}}{m_{1} ω_{1}} s i n Φ_{1} c o s Φ_{1} - \\ \frac{k_{2} A_{2}}{m_{1} ω_{1}} s i n Φ_{1} c o s Φ_{2} \end{matrix}$
$\begin{matrix} F_{21} = - ω_{1} {c o s}^{2} Φ_{1} - \\ \frac{[c_{1} + c_{2} + C (A_{1})] s i n Φ_{1} c o s Φ_{1}}{m_{1}} + \\ \frac{c_{2} A_{2} ω_{2}}{m_{1} A_{1} ω_{1}} s i n Φ_{2} c o s Φ_{1} + \frac{[k_{2} + K_{1} (A_{1})]}{m_{1} ω_{1}} {c o s}^{2} Φ_{1} - \\ \frac{A_{2} k_{2}}{m_{1} A_{1} ω_{1}} c o s Φ_{1} c o s Φ_{2} \end{matrix}$
$\begin{matrix} F_{12} = - A_{2} ω_{2} s i n Φ_{2} c o s Φ_{2} + \frac{c_{2} A_{1} ω_{1}}{m_{2} ω_{2}} s i n Φ_{1} s i n Φ_{2} - \\ \frac{c_{2} A_{2} {s i n}^{2} Φ_{2}}{m_{2}} - \frac{A_{1} k_{2}}{m_{2} ω_{2}} c o s Φ_{1} s i n Φ_{2} + \\ \frac{k_{2} A_{2}}{m_{2} ω_{2}} s i n Φ_{2} c o s Φ_{2} \end{matrix}$

\begin{matrix} F_{22} = - ω_{2} {c o s}^{2} Φ_{2} + \frac{c_{2} A_{1} ω_{1}}{m_{2} A_{2} ω_{2}} s i n Φ_{1} c o s Φ_{2} - \\ \frac{c_{2} s i n Φ_{2} c o s Φ_{2}}{m_{2}} - \frac{k_{2} A_{1}}{m_{2} A_{2} ω_{2}} c o s Φ_{1} c o s Φ_{2} + \\ \frac{k_{2}}{m_{2} ω_{2}} {c o s}^{2} Φ_{2} \end{matrix}

(12)

根据Stratonovich-Khasminskii极限定理^[29]，[A_i，Θ_i]^T可以近似为四维Markov扩散过程.但需注意，经过随机平均和确定平均后，A_i（t）的平均微分方程独立于Θ_i（t）.因此，支配极限过程A_i（t）的平均伊藤随机微分方程如下，

d A_{i} = {\bar{m}}_{i} (A) d t + {\bar{σ}}_{i} (A) d B (t), i = 1,2

(13)

其中，B（t）为单位维纳过程，

\begin{matrix} {\bar{m}}_{i} = ⟨F_{i i} + \int_{- \infty}^{0} [{{\frac{\partial G_{1 i}}{\partial A_{i}}|}_{t} G_{1 i}|}_{t + τ} R (τ) + \\ {{{\frac{\partial G_{1 i}}{\partial Φ_{i}}|}_{t} G_{2 i}|}_{t + τ} R (τ)] d τ⟩}_{Φ} \\ {\bar{σ}}_{i}^{2} (A) = {⟨{{\int_{- \infty}^{\infty} G_{1 i}|}_{t} G_{1 i}|}_{t + τ} R (τ) d τ⟩}_{Φ} \end{matrix}

(14)

式中， $⟨ \cdot ⟩_{Φ} = \frac{1}{（ 2 π ）^{2}} \int_{0}^{2 π} \int_{0}^{2 π} ⟨ \cdot ⟩ d Φ_{1} d Φ_{2}; Φ = [Φ_{1} ， {Φ_{2}]}^{T}$ ; R（τ）表示系统激励的自相关函数，相应的功率谱函数为 $S （ ω ） = \frac{1}{π} \int_{- \infty}^{0} R （ τ ） c o s ω τ d τ$ .
为进一步得到漂移系统与扩散系数的显式表达式，将式（14）中的G_ik展开成Fourier级数，

\begin{matrix} {\bar{m}}_{1} = - \frac{[c_{1} + c_{2} + C (A_{1})] A_{1}}{2 m_{1}} - \\ \frac{π S_{2} (ω)}{4 {[k_{2} + K_{1} (A_{1})]}^{2}} \frac{d K_{1} (A_{1})}{d A_{1}} + \\ \frac{π S_{2} (ω)}{2 A_{1} [k_{2} + K_{1} (A_{1})]} \\ {\bar{m}}_{2} = \frac{- c_{2} A_{2}}{2 m_{2}} + \frac{π S_{2} (ω)}{2 A_{2} k_{2}} \\ {\bar{σ}}_{1}^{2} = \frac{π S_{2} (ω)}{k_{2} + K_{1} (A_{1})} \\ {\bar{σ}}_{1} {\bar{σ}}_{2} = 0, {\bar{σ}}_{2}^{2} = \frac{π S_{2} (ω)}{k_{2}} \end{matrix}

(15)

2 首次穿越失效
对建筑结构系统而言，相空间内系统的状态可用振幅A来表示.假设存在某个开区间Ω_s:（0，A_c）作为式（9）的安全域，系统（9）的位移振幅A（t）=[A₁，A₂]^T可在[0，∞）内变化，一旦其达到或超过临界值A_c，系统就发生首次穿越而破坏.因此，扩散过程（15）的条件可靠性函数可定义为:

R (t ∣ A_{0}) = P \{A (τ) \in Ω_{s}, τ \in (0, t] ∣ A_{0} \in Ω_{s}\}

(16)

其中A₀=[A₁₀，A₂₀]^T表示系统初始幅值.
根据扩散的Markov过程理论，控制条件可靠性函数的后向Kolmogorov方程可表示为:

\begin{matrix} \frac{\partial R (t ∣ A_{0})}{\partial t} = {\bar{m}}_{1} (A_{0}) \frac{\partial R}{\partial A_{10}} + {\bar{m}}_{2} (A_{0}) \frac{\partial R}{\partial A_{20}} + \\ \frac{1}{2} {\bar{σ}}_{1}^{2} (A_{0}) \frac{\partial^{2} R}{\partial A_{10}^{2}} + \frac{1}{2} {\bar{σ}}_{2}^{2} (A_{0}) \frac{\partial^{2} R}{\partial A_{20}^{2}} \end{matrix}

(17)

其中，用A₀代替A得到可得到漂移系数与扩散系数.后向Kolmogorov方程还满足如下的初始条件与边界条件，

R (t ∣ A_{0}) = \{\begin{matrix} 0, A_{10} = A_{1 c} 或 A_{20} = A_{2 c} \\ f i n i t e A_{0} = 0 \\ 1, A_{0} \in Ω_{s} \end{matrix}

(18)

在Crank-Nicolson隐式格式的有限差分法的帮助下，通过初始条件与边界条件求解后向Kolmogorov方程，得到系统的条件可靠性函数.首次穿越时间的条件概率密度函数可对式（16）进行一阶求导获得:

p (τ ∣ A_{0}) = - {\frac{\partial R (t ∣ A_{0})}{\partial t}|}_{t = τ}

(19)

3 算例分析
高斯白噪声激励是特殊的宽带噪声激励，且较为理想化，易于进行数学处理.但在实际工程中，较为常见的是随机激励的情况，需将其化为滤波白噪声激励才更为合理.与高斯白噪声激励有差别的地方是，此时的功率谱密度S（ω）不再看作常数.对于滤波白噪声激励情形，式（9）中随机激励采用金井清滤波白噪声模型，其功率谱密度为:

S_{2} (ω) = \frac{ω_{g}^{4} + 4 ξ_{g}^{2} ω_{g}^{2} ω^{2}}{(ω_{g}^{2} - ω^{2}) + 4 ξ_{g}^{2} ω_{g}^{2} ω^{2}} S_{0}

(20)

其中，ω_g与ξ_g分别为地震波传播过程中所经历土层的特征频率和阻尼比.
图3 不同激励强度D下的计算结果
Fig.3 Calculation results under different excitation intensity D
通过无量纲化处理，选取其他系统参数m₁=1，m₂=1.16，β₁=0.005，β=0.5，α=0.1，D=0.01，Y_y=0.5，c₁=0.000763s-1，c₂=0.0763 s-1，k₂=12.57，ω_g=10，ξ_g=0.5.图3~图4给出了通过激励强度D、土层阻尼比ξ_g取值的变化对条件可靠性函数和条件概率密度函数的影响，通过近似解析结果（实线）与数值模拟结果（符号（○，◇，△））的对比，可以看出两者较为吻合，证明本节求解方法有效.此外，从图中可看出，条件可靠性函数均随时间的增加而减小.
图4 不同土层阻尼比ξg下的计算结果
Fig.4 Calculation results of damping ratio of different soil layers
图3（a）与（b）中，考察激励强度D对系统可靠性的影响，其取值为:D=0.01，D=0.02，D=0.04.从图3（a）中可看出，随着激励强度D的减小，系统可靠性函数R（t）=R（t|A₀=0）增大，意味着系统可靠性的提高; 图3（b）中条件概率密度函数p（t）=p（τ|A₀=0）的峰值随激励强度D的减小而减小，且变化明显，同样说明系统可靠性随激励强度D的减小而提高.图4（a）与（b）中土层阻尼比ξ_g取值为:ξ_g=0.3，ξ_g=0.7，ξ_g=1.0.从图中可看出，与激励强度的影响效果不同，随着土层阻尼比ξ_g取值的减小，系统可靠性逐渐降低.
图5 时间t=0.01s时条件可靠性函数随初始幅值A₁₀和A₂₀的变化
Fig.5 Change of conditional reliability function with initial amplitudes A₁₀ and A₂₀ at time t=0.01s
图5是系统的条件可靠性函数随初始幅值A₁₀和A₂₀变化的三维图像，可见随着A₁₀和A₂₀靠近阈值边界，系统的可靠性是降低的，且变化较平缓; 此外，A₁₀和A₂₀对条件可靠性函数的影响效果较为一致.
4 结论
本文对随机激励下双自由度自复位结构的首次穿越失效进行研究.首先，应用广义谐波平衡技术，将旗帜形恢复力解耦为等效刚度系数和等效阻尼系数，得到原系统的等效随机系统; 然后，应用随机平均法，得到关于幅值的平均伊藤微分方程，建立并求解相应的后向Kolmogorov（BK）方程，得到系统的条件可靠性函数和条件概率密度函数; 最后，讨论了激励强度D与土层阻尼比ξ_g发生变化时对系统可靠性的影响; 利用Monte Carlo数值模拟验证理论解析解的正确性.研究表明:
（1）在滤波白噪声激励下，当选用金井清滤波白噪声模型时，系统的可靠性随着激励强度的减小以及土层阻尼比的增加而增大;
（2）随着初始幅值A₁₀和A₂₀靠近安全域，系统的可靠性降低，变化平缓;
（3）本文所用的理论分析方法得到的结果与Monte Carlo数值模拟结果吻合得很好.
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基本信息

中图分类号: O324
文献标识码: A
DOI: 10.6052/1672-6553-2023-055
文章编号: 1672-6553-2023-21(5)-093-008

基金信息

国家自然科学基金(12072118)；
福建省杰出青年科学基金(2021J06024)；
厦门青年创新基金(3502Z20206005)；
National Natural Science Foundation of China (12072118)；
the Natural Science Funds for Distinguished Young Scholar of the Fujian Province of China (2021J06024)；
the Project for Youth Innovation Fund of Xiamen, China (3502Z20206005)；

引用信息

稿件历史

收稿日期: 2022-11-12

参考文献

[1] RAMIREZ C M,MIRANDA E.Significance of residual drifts in building earthquake loss estimation [J].Earthquake Engineering and Structural Dynamics,2012,41(11):1477-1493.
[2] ROSHANFEKR R Z,GHOBADI M S,YAKHCHALIAN M.Probabilistic seismic collapse and residual drift assessment of smart buildings equipped with shape memory alloy connections [J].Engineering Structures,2019,197:109375.
[3] 张国伟,赵紫薇,孙祚帅.自复位结构抗震性能研究综述 [J].建筑结构,2018,48(S2):463-470.ZHANG G W,ZHAO Z W,SUN Z S.Review on the seismic behavior of self-centering structure [J].Building Structure,2018,48(S2):463-470.(in Chinese)
[4] GU A Q,ZHOU Y,XIAO L,et al.Experimental study and parameter analysis on the seismic performance of self-centering hybrid reinforced concrete shear walls [J].Soil Dynamics and Earthquake Engineering,2019,116:409-420.
[5] XU L H,XIAO S J,LI Z X.Behaviors and modeling of new self-centering RC wall with improved disc spring devices [J].Journal of Engineering Mechanics,2020,146(9):04020102.
[6] XU G,GUO T,LI A Q.Equivalent linearization method for seismic analysis and design of self-centering structures [J].Engineering Structures,2022,271:114900.
[7] YOU T,WANG W,TESFAMARIAM S.Effects of self-centering structural systems on regional seismic resilience [J].Engineering Structures,2023,274:115125.
[8] 吕西林,陈云,毛苑君.结构抗震设计的新概念——可恢复功能结构 [J].同济大学学报(自然科学版),2011,39(7):941-948.LU X L,CHEN Y,MAO Y J.New concept of structural seismic design:earthquake resilient structures [J].Journal of Tongji University(Natural Science),2011,39(7):941-948.(in Chinese)
[9] 吕西林,武大洋,周颖.可恢复功能防震结构研究 [J].建筑结构学报,2019,40(2):1-15.LU X L,WU D Y,ZHOU Y.State-of-the-art of earthquake resilient structures [J].Journal of Building Structures,2019,42(2):1-15.(in Chinese)
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随机激励下自复位结构首次穿越失效研究^*

First-Passage of Self-Centering System Under Random Excitation^*

摘要

Abstract

关键词

Keywords

1 理论分析

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

2 首次穿越失效

(16)

(17)

(18)

(19)

3 算例分析

(20)

4 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

WeChat

Mobile website

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随机激励下自复位结构首次穿越失效研究*

First-Passage of Self-Centering System Under Random Excitation*

摘要

Abstract

关键词

Keywords

1 理论分析

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

2 首次穿越失效

(16)

(17)

(18)

(19)

3 算例分析

(20)

4 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

WeChat

Mobile website

随机激励下自复位结构首次穿越失效研究^*

First-Passage of Self-Centering System Under Random Excitation^*