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 同济大学学报(自然科学版)  2019, Vol. 47 Issue (1): 1-8.  DOI: 10.11908/j.issn.0253-374x.2019.01.001 0

### 引用本文

QU Yang, LUO Yongfeng, HUANG Qinglong, ZHU Zhaochen. An Improved Modal Pushover Analysis Procedure for Estimating Seismic Responses of Latticed Arch[J]. Journal of Tongji University (Natural Science), 2019, 47(1): 1-8. DOI: 10.11908/j.issn.0253-374x.2019.01.001

### 文章历史

An Improved Modal Pushover Analysis Procedure for Estimating Seismic Responses of Latticed Arch
QU Yang , LUO Yongfeng , HUANG Qinglong , ZHU Zhaochen
College of Civil Engineering, Tongji University, Shanghai 200092, China
Abstract: The static pushover analysis(SPA) procedure has been widely applied to the estimate of the nonlinear seismic performance of the multi-story structure subjected to horizontal earthquakes. However, SPA could hardly obtain a satisfying result in accuracy when applied to the evaluation of seismic responses of the large-span latticed arch under vertical earthquake ground motions, whose lateral and vertical displacements were coupled. Accordingly, an improved modal pushover analysis(IMPA) procedure was proposed, based on the eigen-stiffness, introduced from the static stability analysis of structures. Firstly, equivalent single degree of freedom(ESDOF) system was established based on eigen-stiffness by modal pushover analysis for each mode selected. Secondly, the seismic responses of the ESDOFs were solved respectively and combined. Finally, the seismic responses of the structure were obtained by pushover analysis once more according to the load formula derived. The seismic responses of a latticed arch under a series of vertical earthquake waves on firm and soft sites, were obtained by IMPA. Compared with the results given by response history analysis, the node vertical displacements and the most unfavorable stresses of the arch were consistent in distribution, with errors about 10.5% and 33.2% respectively, and 25% computing consumption. Along with the increase in the number of the vertical modes selected of the arch, the accuracy of IMPA could improve.
Key words: latticed arch    modal pushover analysis    vertical earthquake    eigen-stiffness

1 基于特征刚度的IMPA方法基本理论 1.1 第一阶段推覆分析与ESDOF体系

 $\mathit{\boldsymbol{M\ddot u}}\left( t \right) + \mathit{\boldsymbol{C\dot u}}\left( t \right) + {\mathit{\boldsymbol{F}}_{\rm{s}}}\left( t \right) = - \mathit{\boldsymbol{M}}{\mathit{\boldsymbol{l}}_{\rm{z}}}{{\ddot u}_{{\rm{gz}}}}\left( t \right)$ (1)

 $\mathit{\boldsymbol{u}}\left( t \right) = \sum\limits_{l = 1}^N {{\mathit{\boldsymbol{\phi}} _l}{q_l}\left( t \right)}$ (2)

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\phi}} _n^{\rm{T}}\mathit{\boldsymbol{M}}{\mathit{\boldsymbol{\phi}} _n}{{\ddot q}_n}\left( t \right) + \mathit{\boldsymbol{\phi}} _n^{\rm{T}}\mathit{\boldsymbol{C}}{\mathit{\boldsymbol{\phi}} _n}{{\dot q}_n}\left( t \right) + \mathit{\boldsymbol{\phi}} _n^{\rm{T}}{\mathit{\boldsymbol{F}}_{\rm{s}}}\left( t \right) = }\\ { - \mathit{\boldsymbol{\phi}} _n^{\rm{T}}\mathit{\boldsymbol{M}}{\mathit{\boldsymbol{l}}_{\rm{z}}}{{\ddot u}_{{\rm{gz}}}}\left( t \right)} \end{array}$ (3)

Mn=ϕnTMϕnCn=ϕnTCϕnLn=ϕnTMlz，并考虑到Cn=2Mnζnωn，其中ωn为第n阶竖向振型的圆频率，整理得

 ${M_n}{{\ddot u}_n}\left( t \right) + 2{M_n}{\zeta _n}{\omega _n}{{\dot q}_n}\left( t \right) + \mathit{\boldsymbol{\phi}} _n^{\rm{T}}{\mathit{\boldsymbol{F}}_{\rm{s}}}\left( t \right) = - {L_n}{{\ddot u}_{{\rm{gz}}}}\left( t \right)$ (4)

qn(t)=ΓnDn(t)，考虑到振型参与系数Γn=Ln/Mn，整理得

 ${{\ddot D}_n}\left( t \right) + 2{\zeta _n}{\omega _n}{{\dot D}_n}\left( t \right) + \frac{{\mathit{\boldsymbol{\phi}} _n^{\rm{T}}{\mathit{\boldsymbol{F}}_{\rm{s}}}\left( t \right)}}{{{L_n}}} = - {{\ddot u}_{{\rm{gz}}}}\left( t \right)$ (5)

 $k_j^ * = \frac{{\Delta \mathit{\boldsymbol{U}}_j^{\rm{T}}{\mathit{\boldsymbol{K}}_{{\rm{T}},j}}\Delta {\mathit{\boldsymbol{F}}_j}}}{{\Delta \mathit{\boldsymbol{U}}_j^{\rm{T}}\Delta {\mathit{\boldsymbol{U}}_j}}}$ (6)

 ${\mathit{\boldsymbol{F}}_{\rm{s}}}\left( t \right) = {\mathit{\boldsymbol{K}}_{\rm{T}}}{\mathit{\boldsymbol{\phi}} _n}{\mathit{\Gamma }_n}{D_n}\left( t \right) = \omega _n^2\mathit{\boldsymbol{M}}{\mathit{\boldsymbol{\phi}} _n}{\mathit{\Gamma }_n}{D_n}\left( t \right)$ (7)

 ${F_{n,j}} = \omega _n^2\mathit{\boldsymbol{M}}{\mathit{\boldsymbol{\phi}} _n}{\mathit{\Gamma }_n}{D_{n,j}} = {\chi _j}\mathit{\boldsymbol{M}}{\mathit{\boldsymbol{\phi}} _n}$ (8)

 $\Delta {A_{n,j}} = \frac{{\mathit{\boldsymbol{\phi}} _n^{\rm{T}}\left( {{\mathit{\boldsymbol{F}}_{n,j}} - {\mathit{\boldsymbol{F}}_{n,j - 1}}} \right)}}{{{L_n}}} = \frac{{{\chi _j} - {\chi _{j - 1}}}}{{{\mathit{\Gamma }_n}}}$ (9)
 $\Delta {D_{n,j}} = \frac{{\Delta {A_{n,j}}}}{{\omega _n^2}} = \frac{{{m_{n,eq}}\left( {{\chi _j} - {\chi _{j - 1}}} \right)}}{{{\mathit{\Gamma }_n}k_{n,1}^ * }}$ (10)

 $\Delta {F_{n,j}} = {m_{n,eq}}\left( {{A_{n,j}} - {A_{n,j - 1}}} \right)$ (11)
 $\Delta {d_{n,j}} = \frac{{\Delta {F_{n,j}}}}{{k_{n,j}^ * }} = \frac{{{m_{n,eq}}\left( {{\chi _j} - {\chi _{j - 1}}} \right)}}{{{\mathit{\Gamma }_n}k_{n,j}^ * }}$ (12)

1.2 第二阶段推覆分析与推覆荷载

 $\mathit{\boldsymbol{u}}\left( t \right) = {\mathit{\boldsymbol{\phi}} _1}{q_1}\left( t \right) + {\mathit{\boldsymbol{\phi}} _2}{q_2}\left( t \right)$ (13)

 ${\mathit{\boldsymbol{u}}^{\rm{t}}}\left( t \right) = {\mathit{\boldsymbol{\phi}} _1}{q_1}\left( t \right) + {\mathit{\boldsymbol{\phi}} _2}{q_2}\left( t \right) + {\mathit{\boldsymbol{u}}_g}$ (14)

 ${\mathit{\boldsymbol{u}}_{\max }} = {\mathit{\boldsymbol{\phi}} _1}{q_1}\left( {{t_{\max }}} \right) + {\mathit{\boldsymbol{\phi}} _2}{q_2}\left( {{t_{\max }}} \right)$ (15)

 ${\mathit{\boldsymbol{P}}_j} = \mathit{\boldsymbol{Ku}}_j^P$ (16)

 ${\left( {\mathit{\boldsymbol{u}}_j^P} \right)^{\rm{T}}}\mathit{\boldsymbol{Ku}}_j^P = \chi _j^2\left( {q_1^2\omega _1^2\mathit{\boldsymbol{\phi}} _1^{\rm{T}}\mathit{\boldsymbol{M}}{\mathit{\boldsymbol{\phi}} _1} + q_2^2\omega _2^2\mathit{\boldsymbol{\phi}} _2^{\rm{T}}\mathit{\boldsymbol{M}}{\mathit{\boldsymbol{\phi}} _2}} \right)$ (17)

Pj=χjM(q1ω12ϕ1+q2ω22ϕ2)，则整理式(16)左边得到

 ${\left( {\mathit{\boldsymbol{u}}_j^P} \right)^{\rm{T}}}{\mathit{\boldsymbol{P}}_j} = \chi _j^2\left( {q_1^2\omega _1^2\mathit{\boldsymbol{\phi}} _1^{\rm{T}}\mathit{\boldsymbol{M}}{\mathit{\boldsymbol{\phi}} _1} + q_2^2\omega _2^2\mathit{\boldsymbol{\phi}} _2^{\rm{T}}\mathit{\boldsymbol{M}}{\mathit{\boldsymbol{\phi}} _2}} \right)$ (18)

 ${\mathit{\boldsymbol{P}}_j} = {\chi _j}\mathit{\boldsymbol{M}}\left( {{q_1}\omega _1^2{\mathit{\boldsymbol{\phi}} _1} + {q_2}\omega _2^2{\mathit{\boldsymbol{\phi}} _2}} \right)$ (19)

 ${\mathit{\boldsymbol{P}}_j} = {\chi _j}\mathit{\boldsymbol{M}}\sum\limits_{i = 1}^n {{q_i}\omega _i^2{\mathit{\boldsymbol{\phi}} _i}}$ (20)
1.3 IMPA方法计算步骤

(1) 进行结构模态分析，选取主振型，确定各阶主振型的荷载空间分布模式；

(2) 根据各主振型的荷载空间分布模式，对结构分别进行第一阶段非线性模态推覆分析，得到各主振型荷载模式下的特征刚度变化；

(3) 基于各主振型的特征刚度，建立对应各主振型的等效单自由度(ESDOF)体系；

(4) 针对对应各主振型的ESDOF进行时程分析，得到各ESDOF的位移响应时程；

(5) 将各ESDOF位移响应时程按振型组合，得到结构整体位移响应时程u(t)；

(6) 取u(t)的最不利位移作为变形模式，以式(20)中的Pj为推覆荷载，对结构整体进行第二阶段推覆分析，得到结构在地震作用下的总响应.

2 数值算例

2.1 格构拱结构模型

 图 1 格构拱结构模型及杆件截面(单位：m) Fig.1 Structural layout and member sections of the arch(Unit: m)

 图 2 格构拱结构竖向主振型 Fig.2 Fundamental vertical vibration modes of the latticed arch
2.2 ESDOF体系

 图 3 前两阶竖向主振型A-D和k*-D关系曲线 Fig.3 A-D and k*-D curves for the modal ESDOFs
2.3 结构动力响应 2.3.1 地震动输入

 图 4 地震波伪加速度反应谱 Fig.4 Pseudo acceleration response spectra of the selected seismic waves
2.3.2 节点位移响应

 图 5 IMPA方法和RHA方法节点位移计算结果对比 Fig.5 Nodal displacement responses given by IMPA and RHA
2.3.3 单元最不利应力响应

 图 6 IMPA方法和RHA方法单元最不利应力计算结果对比 Fig.6 The most unfavorable stresses of elements given by IMPA and RHA
2.4 振型截取阶数的精度影响

2.5 计算耗时对比

3 结论