雙向地震下非對稱房屋結構的黏彈塑性分析

Abstract

本研究旨在研究房屋結構在雙向地震作用下的受震反應，提出了一個非對稱房屋結構的黏彈塑性模式以分析其受震反應，此模式考慮兩向層間剪力（位移）與扭矩（扭角）的關係，包含：各向勁度不對稱、走動硬軟化、非等向降伏面與阻尼現象。為了要能夠精確計算出房屋結構的地震反應歷時，本研究使用狀態空間表示法，根據精確的判斷準則，將複雜的黏彈塑性動態系統細分為一個彈性項與多個塑性相動態系統，接著再針對每一相求解對應的房屋結構動態反應。本研究發現，在有樓層進入塑性態時，可以再進一步將計算塑性相動態系統反應分解成：先計算樓層塑性內變數，再計算整棟房屋的動態反應；並且發現模式具有李群（Lie group）和李代數(Lie algebra)的內在對稱性，因此可推得塑性內變數單步解析解，進而獲得整棟房屋的動態反應單步解析解。此計算方法也經由誤差分析，驗證其精確性。接下來，本研究根據提出之非對稱房屋結構黏彈塑性模式，考慮921集集地震事件全台灣所有地震測站歷時，分析房屋結構其受震後的反應。先以原始地震歷時分析比較單向地震分析及雙向地震分析之反應；接著經由計算出全台灣地震測站歷時所得之房屋塑性當量圖，以分析房屋結構的塑性損傷。此外，也進行地震力入射角對於房屋結構反應的影響；並且進行最大地表加速度分析探討房屋結構隨著譜加速度增加，其速度脈衝型地震與非速度脈衝型地震作用下的反應。研究結果顯示，房屋結構若以單向地震分析，確實相較雙向地震力分析有錯估的情勢；並且使用單向地震力分析建築結構時其塑性損傷會嚴重的低估，其總層間位移反應及總層間剪力反應平均來看也是相對低估，由此可知以雙向水平地震分析更能貼近房屋結構的受震情況。再者，經由地震力入射角分析也可以看出非對稱房屋對於地震力入射角之考量有一定的影響。此外，非對稱房屋結構經由最大地表加速度分析，顯示某些速度脈衝地震確實造成較大的房屋結構反應。

This study aims to investigate the seismic response of asymmetric building structures under bidirectional earthquake excitations. A viscoelastoplastic model of the building structures is proposed taking into account an anisotropic stiffness between the interstory two-directional shears, torque and the interstory two-directional displacement and rotation; a anisotropic yield surface; kinematic hardening and softening; and the damping effects. To accurately calculate the time history of seismic responses for asymmetric building structures, the conventional formulation of the model is rearranged; the state-space representation approach is employed; and the internal symmetry of the model is explored in this study. According to careful derivation about the switching of the plastic mechanism, the complex viscoelastoplastic dynamical system is decomposed into an viscoelastic phase and multiple viscoelastoplastic phases, and then the dynamical responses of the building structure for each part are solved separately. In each viscoelastoplastic phase, the response of the building is decomposed into two steps including the calculation of the plastic internal variables based on the internal symmetry, and then determinate of the total responses via the state-space representation. Since the internal symmetry of the Lie group SO(3,1) and the Lie algebra so(3,1), provides the closed-form solution for the plastic internal variables in each single step, and consequently the closed-form response of the building structure in each single step is obtained. The accuracy of this calculation method is also verified through error analysis. Next, based on the proposed viscoelastoplastic model for asymmetric building structures and considering the ground motion time histories recorded at all seismic stations in Taiwan during the 1999 Chi-Chi earthquake event, the seismic responses of building structures were analyzed. First, the original ground motion time histories were used to compare the responses from unidirectional and bidirectional seismic analyses. Then, the plastic ductility demand contours for building structures in Taiwan were computed from the recorded ground motions to assess the plastic damage. Furthermore, the influence of the incidence angle of seismic excitations on the structural responses was investigated. A peak ground acceleration analysis was also performed to study the responses of building structures under velocity pulse-like and non-velocity pulse-like earthquakes as the peak ground acceleration increases. The results show that if building structures are analyzed using unidirectional seismic excitations, their responses would indeed be underestimated compared to those from bidirectional seismic analyses. The plastic damage would be severely underestimated when using unidirectional seismic analysis, and the total interstory drift and shear responses would also be underestimated on average. This indicates that bidirectional horizontal seismic analysis can better capture the actual seismic responses of building structures. Additionally, the analysis of the incidence angle of seismic excitations also reveals that the consideration of incidence angles has a certain influence on the responses of asymmetric building structures. Moreover, the peak ground acceleration analysis for asymmetric building structures shows that some velocity pulse-like earthquakes indeed cause larger structural responses.