层状自由场地固有频率的求解方法、特性及应用研究
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摘要
场地土层的固有频率是场地土的重要动力特性之一，大量震害观测资料表明，许多震害是由结构物与场地地基的共振或类共振效应引起的，因此，为了有效的估计和防止这类震害的发生，工程设施的固有频率应尽量避开场地地基的固有频率。本文视工程场地为半空间上的水平层状土层，对其固有频率的求解方法、特性及工程应用进行了深入系统的研究，在此基础上，对半空间上水平层状场地固有频率特性及影响因素进行了分析和总结。
对传统的基于集中质量矩阵的剪切质点系法(集中质量法)的质量矩阵、刚度矩阵，阻尼矩阵及振动方程的建立以及固有频率和振型的求解进行了说明，在此基础上，本文发展和建立了基于一致质量矩阵的剪切质点系法(一致质量法)，同时，引入半无限单元模拟半无限空间，建立了可考虑半空间弹性影响的改进一致质量法，并用其分析了半空间刚性假设对场地自振频率的影响。其次，对剪切质点系法求解精度的影响因素及其影响规律以及分层策略对一致质量法求解“收敛”速度的影响进行了分析，并比较了集中质量法和一致质量法的不同，发现了集中质量法的“振荡”缺陷以及一致质量法的良好求解稳定性。最后，应用剪切质点系法，分析了半空间上水平层状场地的固有频率影响因素及特性，通过实例分析，提出了场地固有频率计算深度的选取问题并简单进行了讨论。
基于匀直剪切梁横向自由振动微分方程及其振型函数解答，对刚性半空间上的直立悬臂剪切梁模型的自由振动微分方程进行了求解，获得了半空间上单层、双层以及三层覆盖情况下的超越方程隐式解析解答，并对适用于任意多层覆盖情况的递推解法进行了分析和推导。在此基础上，考虑半空间弹性，既考虑半空间与上覆土层之间的相互作用，建立和发展了改进弹簧一剪切梁模型，同样获得了单层、双层、三层以及任意多层覆盖情况的解析解答公式。其次，通过引入复材料常数，获得了考虑阻尼情况下的解析公式。最后，通过不同算例，用本章提出的弹簧-剪切梁模型分析了半空间弹性的对场地固有频率的影响，并用剪切梁法解析公式分析了“波速法”以及一致质量法在单层和多层情况下的求解精度。
基于平面体波波动方程，分平面内(P-SV波)和平面外(SH波)问题，分别获得了土层顶底面之间状态向量的转换矩阵，在此基础上得到了土层和半空间的动力刚度矩阵，通过组集所有土层及半空间动力刚度矩阵获得以整体刚度矩阵表示的体系振动方程，从而求解整体系的固有频率和振型。其次，考虑土层水平向纯剪切振动，推导了波垂直入射情况下的解答，并与前面剪切梁法解答进行了比较，分析了剪切梁法的求解精度以及利用改进弹簧-剪切梁模型来考虑半空间弹性的合理性。最后，通过引入复材料参数，获得了阻尼情况下的波动法解答，并通过算例，分析了上覆土层阻尼和半空间阻尼对场地固有频率的影响。
环境振动引起的地脉动能量以瑞利波为主，对脉动卓越频率所反映的影响深度和场地固有频率计算深度的关系进行了分析，获得了不同类型场地条件下瑞利波波长与计算深度之间的关系，同时对其特性和影响因素进行了总结。
The natural frequency of soil layers is one of the most important dynamic characteristics of site soils. It's indicated by large numbers of observation data that much seismic disasters is caused by the resonance vibration between structures and ground, so, in order to effectively estimate and prevent this kind of seismic disasters, the natural frequencies of engineering structures should keep away from that of the ground. In this dissertation, engineering sites are regarded as horizontally layered soils on half space, and the solving methods, characteristics and engineering applications are studied deeply and systematically. On the basis of these researches, the natural frequency and its influencing factors of horizontal soil layers on half space are analyzed.
The mass matrix, stiffness matrix, damping matrix, establishment of vibration equations and the solving of natural frequency and mode shapes of the shear mass point system method based on clump mass matrix(clump mass method) are explained firstly, based on it, another kind of shear mass point system method based on consistent mass matrix(consistent mass method) is developed. Through introducing semi-infinite element to simulate the half apace, a kind of improved consistent mass method is developed and the influence of the half space elasticity is analyzed by it. Then, the solving precision and its influencing factors of shear mass point system methods and the influence of delaminating strategy on the convergence speed of consistent mass method are analyzed. At the same time, the differences of two kinds of shear mass point system methods are compared. Finally, the influencing factors and their characteristics of the natural frequency of horizontally layered sites on half space are analyzed with shear mass point system methods and a problem about the computing depth selecting of sites' natural frequencies is put forward and discussed briefly.
Based on the horizontal vibration differential equations of straight beam and its mode shape function solution, the natural vibration differential equation of vertical shear cantilever model on rigid half space is solved and the analytic solutions with transcendental equation form of single layer and double, three layers on half space are obtained. At the same time, the recurrence solution which applicable for random multi-layer is deduced. Then, considering the elasticity of half pace, that is, take the reaction between half space and the soil layers on it considered, the spring-shear beam model is developed and the analytic solutions of single layer and double, three layers and random multi-layer on half space are obtained. Through introducing complex material parameters, the analytic solutions with damping are gained. Finally, the influence of the elasticity of half space on sites' natural frequencies and the solving precision of wave velocity method and consistent mass method are analyzed by spring-shear beam model solutions.
On the basis of plane body wave motion equations, dividing to in-plane(P-SV) and anti-plane(SH) problems, the conversion matrixes of state vectors between soil layer's top and under sides are deduced, and based on these, dynamic stiffness matrixes of soil layers and half space can also be gained separately. By assembling dynamic stiffness matrixes of all soil layers and half space, system vibration equation can be get, which is expressed with integral dynamic stiffness matrix, and hence natural frequency and vibration mode for the
whole system can be solved. Then, taking pure shear vibration in horizontal direction into considerations, solutions are deduced under the condition of wave injecting vertically, and taking contrast with solutions get by shear beam methods, accuracy of shear beam methods and rationality of using spring - shear beam model to consider half space elasticity are analyzed. Lastly, according to introducing complex material parameters, solutions with damping are obtained. Then through calculation cases, damping influences of soil layers and half space on natural frequency are studied.
On the assumption that Rayleigh wave is the main energy in microtremors caused by environmental vibrations, the relation between the reflecting depth of microtremors and the computing depth of site's natural frequency is analyzed and relations between Rayleigh wavelength and computing depth for different models with different site's conditions are obtained. On the basis of these, characteristics and influent factors of them are summarized.
对传统的基于集中质量矩阵的剪切质点系法(集中质量法)的质量矩阵、刚度矩阵，阻尼矩阵及振动方程的建立以及固有频率和振型的求解进行了说明，在此基础上，本文发展和建立了基于一致质量矩阵的剪切质点系法(一致质量法)，同时，引入半无限单元模拟半无限空间，建立了可考虑半空间弹性影响的改进一致质量法，并用其分析了半空间刚性假设对场地自振频率的影响。其次，对剪切质点系法求解精度的影响因素及其影响规律以及分层策略对一致质量法求解“收敛”速度的影响进行了分析，并比较了集中质量法和一致质量法的不同，发现了集中质量法的“振荡”缺陷以及一致质量法的良好求解稳定性。最后，应用剪切质点系法，分析了半空间上水平层状场地的固有频率影响因素及特性，通过实例分析，提出了场地固有频率计算深度的选取问题并简单进行了讨论。
基于匀直剪切梁横向自由振动微分方程及其振型函数解答，对刚性半空间上的直立悬臂剪切梁模型的自由振动微分方程进行了求解，获得了半空间上单层、双层以及三层覆盖情况下的超越方程隐式解析解答，并对适用于任意多层覆盖情况的递推解法进行了分析和推导。在此基础上，考虑半空间弹性，既考虑半空间与上覆土层之间的相互作用，建立和发展了改进弹簧一剪切梁模型，同样获得了单层、双层、三层以及任意多层覆盖情况的解析解答公式。其次，通过引入复材料常数，获得了考虑阻尼情况下的解析公式。最后，通过不同算例，用本章提出的弹簧-剪切梁模型分析了半空间弹性的对场地固有频率的影响，并用剪切梁法解析公式分析了“波速法”以及一致质量法在单层和多层情况下的求解精度。
基于平面体波波动方程，分平面内(P-SV波)和平面外(SH波)问题，分别获得了土层顶底面之间状态向量的转换矩阵，在此基础上得到了土层和半空间的动力刚度矩阵，通过组集所有土层及半空间动力刚度矩阵获得以整体刚度矩阵表示的体系振动方程，从而求解整体系的固有频率和振型。其次，考虑土层水平向纯剪切振动，推导了波垂直入射情况下的解答，并与前面剪切梁法解答进行了比较，分析了剪切梁法的求解精度以及利用改进弹簧-剪切梁模型来考虑半空间弹性的合理性。最后，通过引入复材料参数，获得了阻尼情况下的波动法解答，并通过算例，分析了上覆土层阻尼和半空间阻尼对场地固有频率的影响。
环境振动引起的地脉动能量以瑞利波为主，对脉动卓越频率所反映的影响深度和场地固有频率计算深度的关系进行了分析，获得了不同类型场地条件下瑞利波波长与计算深度之间的关系，同时对其特性和影响因素进行了总结。
The natural frequency of soil layers is one of the most important dynamic characteristics of site soils. It's indicated by large numbers of observation data that much seismic disasters is caused by the resonance vibration between structures and ground, so, in order to effectively estimate and prevent this kind of seismic disasters, the natural frequencies of engineering structures should keep away from that of the ground. In this dissertation, engineering sites are regarded as horizontally layered soils on half space, and the solving methods, characteristics and engineering applications are studied deeply and systematically. On the basis of these researches, the natural frequency and its influencing factors of horizontal soil layers on half space are analyzed.
The mass matrix, stiffness matrix, damping matrix, establishment of vibration equations and the solving of natural frequency and mode shapes of the shear mass point system method based on clump mass matrix(clump mass method) are explained firstly, based on it, another kind of shear mass point system method based on consistent mass matrix(consistent mass method) is developed. Through introducing semi-infinite element to simulate the half apace, a kind of improved consistent mass method is developed and the influence of the half space elasticity is analyzed by it. Then, the solving precision and its influencing factors of shear mass point system methods and the influence of delaminating strategy on the convergence speed of consistent mass method are analyzed. At the same time, the differences of two kinds of shear mass point system methods are compared. Finally, the influencing factors and their characteristics of the natural frequency of horizontally layered sites on half space are analyzed with shear mass point system methods and a problem about the computing depth selecting of sites' natural frequencies is put forward and discussed briefly.
Based on the horizontal vibration differential equations of straight beam and its mode shape function solution, the natural vibration differential equation of vertical shear cantilever model on rigid half space is solved and the analytic solutions with transcendental equation form of single layer and double, three layers on half space are obtained. At the same time, the recurrence solution which applicable for random multi-layer is deduced. Then, considering the elasticity of half pace, that is, take the reaction between half space and the soil layers on it considered, the spring-shear beam model is developed and the analytic solutions of single layer and double, three layers and random multi-layer on half space are obtained. Through introducing complex material parameters, the analytic solutions with damping are gained. Finally, the influence of the elasticity of half space on sites' natural frequencies and the solving precision of wave velocity method and consistent mass method are analyzed by spring-shear beam model solutions.
On the basis of plane body wave motion equations, dividing to in-plane(P-SV) and anti-plane(SH) problems, the conversion matrixes of state vectors between soil layer's top and under sides are deduced, and based on these, dynamic stiffness matrixes of soil layers and half space can also be gained separately. By assembling dynamic stiffness matrixes of all soil layers and half space, system vibration equation can be get, which is expressed with integral dynamic stiffness matrix, and hence natural frequency and vibration mode for the
whole system can be solved. Then, taking pure shear vibration in horizontal direction into considerations, solutions are deduced under the condition of wave injecting vertically, and taking contrast with solutions get by shear beam methods, accuracy of shear beam methods and rationality of using spring - shear beam model to consider half space elasticity are analyzed. Lastly, according to introducing complex material parameters, solutions with damping are obtained. Then through calculation cases, damping influences of soil layers and half space on natural frequency are studied.
On the assumption that Rayleigh wave is the main energy in microtremors caused by environmental vibrations, the relation between the reflecting depth of microtremors and the computing depth of site's natural frequency is analyzed and relations between Rayleigh wavelength and computing depth for different models with different site's conditions are obtained. On the basis of these, characteristics and influent factors of them are summarized.
引文
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