水平随机地震激励下弹性圆拱的半解析法
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摘要
结合虚拟激励法与Galerkin法,研究弹性圆拱在水平随机地震作用下随机响应的半解析解.在建立圆拱平面内动力平衡微分方程的基础上,通过选取适当的试函数,应用Galerkin法将动力平衡微分方程转化为线性常微分方程组.通过设定虚拟荷载,采用确定性方法求解响应量的功率谱密度函数的近似解,得到圆拱随机振动问题的闭合解.该方法无须计算拱的振型,对非正交阻尼同样适用.通过算例分析和与有限元计算结果的比较,验证了该方法的计算精度.当采用的试函数与圆拱振型接近时,采用较少的试函数就能获得较高的精度,该方法是一种简便、高效的方法.
Combining pseudo-excitation method and Galerkin method,the semi-analytical solution of elastic circular arch's random responses under horizontal random seismic excitation was studied.The dynamical equilibrium differential equations of circular arch were established and then transformed into a system of linear ordinary differential equations by choosing appropriate trial functions and using Galerkin method.The closed-form solution of random vibration and the approximate solution of power spectrum density of response were solved by assuming pseudo-excitation load and using the deterministic method.The method is also applicable to the non-orthogonal damping due to unnecessarily calculating the mode shape of arch.A numerical example was given to demonstrate the accuracy of the proposed method by comparing the results with those of finite element method.The method is highly efficient and convenient by using a few trial functions when the trial functions are very close to the mode shape of the circular arch.
Combining pseudo-excitation method and Galerkin method,the semi-analytical solution of elastic circular arch's random responses under horizontal random seismic excitation was studied.The dynamical equilibrium differential equations of circular arch were established and then transformed into a system of linear ordinary differential equations by choosing appropriate trial functions and using Galerkin method.The closed-form solution of random vibration and the approximate solution of power spectrum density of response were solved by assuming pseudo-excitation load and using the deterministic method.The method is also applicable to the non-orthogonal damping due to unnecessarily calculating the mode shape of arch.A numerical example was given to demonstrate the accuracy of the proposed method by comparing the results with those of finite element method.The method is highly efficient and convenient by using a few trial functions when the trial functions are very close to the mode shape of the circular arch.
引文
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[3]赵灿晖,周志祥.大跨度钢管混凝土拱桥在多维多点地震激励作用下的平稳随机响应[J].世界地震工程,2007,23(4):66-71.ZHAO Can-hui,ZHOUZhi-xiang.Stationary stochastic response of long-span CSFTarch bridge under the multi-component and multi-support seismic excitation[J].World Earthquake Engineering,2007,23(4):66-71.
[4]LIN J H,WILLIAMS F W,BENNETT P N.Direct Ritz method for random seismic responses for non-uni-form beams[J].Structure Engineering and Mechanics,1994,2(3):285-294.
[5]林家浩,张亚辉.随机振动的虚拟激励法[M].北京:科学出版社,2004:42-49.
[6]项海帆,刘光栋.拱结构的稳定与振动[M].北京:人民交通出版社,1991:5-12.
[7]林家浩.随机地震响应的确定性算法[J].地震工程与工程振动,1985,5(1):89-94.LINJia-hao.Adeterministic methodfor the computation of stochastic earthquake response[J].Earthquake Engineering and Engineering Vibration,1985,5(1):89-94.
[8]LIN J H.A fast CQC algorithm of PSD matrices for randomseismic responses[J].Computers&Structure,1992,44(3):683-687.
[9]DAVENPORT A G.Note on the distribution of the lar-gest value of a randomfunction with application to gust loading[J].Proceeding of the Institution of Civil Engi-neers,1964,28:187-196.
[10]李国豪.工程结构抗震动力学[M].上海:上海科学技术出版社,1980:284-297.
[2]向天宇,赵人达.大跨度拱在空间变化的非平稳随机地震动作用下的随机响应[J].中国公路学报,2001,14(1):39-41.XIANG Tian-yu,ZHAO Ren-da.Random response of long span arch under spatially variable seismic excitation[J].China Journal of Highway and Transport,2001,14(1):39-41.
[3]赵灿晖,周志祥.大跨度钢管混凝土拱桥在多维多点地震激励作用下的平稳随机响应[J].世界地震工程,2007,23(4):66-71.ZHAO Can-hui,ZHOUZhi-xiang.Stationary stochastic response of long-span CSFTarch bridge under the multi-component and multi-support seismic excitation[J].World Earthquake Engineering,2007,23(4):66-71.
[4]LIN J H,WILLIAMS F W,BENNETT P N.Direct Ritz method for random seismic responses for non-uni-form beams[J].Structure Engineering and Mechanics,1994,2(3):285-294.
[5]林家浩,张亚辉.随机振动的虚拟激励法[M].北京:科学出版社,2004:42-49.
[6]项海帆,刘光栋.拱结构的稳定与振动[M].北京:人民交通出版社,1991:5-12.
[7]林家浩.随机地震响应的确定性算法[J].地震工程与工程振动,1985,5(1):89-94.LINJia-hao.Adeterministic methodfor the computation of stochastic earthquake response[J].Earthquake Engineering and Engineering Vibration,1985,5(1):89-94.
[8]LIN J H.A fast CQC algorithm of PSD matrices for randomseismic responses[J].Computers&Structure,1992,44(3):683-687.
[9]DAVENPORT A G.Note on the distribution of the lar-gest value of a randomfunction with application to gust loading[J].Proceeding of the Institution of Civil Engi-neers,1964,28:187-196.
[10]李国豪.工程结构抗震动力学[M].上海:上海科学技术出版社,1980:284-297.
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