地球物理学报  2010, Vol. 53 Issue (9): 2129-2143 PDF

Analytical solution to dynamic response of circular-arc-shaped multi-layered valley due to incidence of Rayleigh wave
ZHANG Yu-Shan
China Earthquake Disaster Prevention Center, Beijing 100029, China
Abstract: By the method of Fourier-Bessel series expansion of wave functions, the paper presents an analytical solution to the two-dimension stationary dynamic response of alluvial valley containing arbitrary number of circular-arc-shaped layers, excited by incident Rayleigh waves. And not only the convergence of the proposed series-form analytical solution with the truncation number of series terms is analyzed, but also the influence of the number of the terms of the finite Fourier series which determine the free-field displacement and stress of the incident Rayleigh wave, as well as that of the radius of the big arc which simulates approximately the flat half-space surface, on the solution is discussed. The results show that the proposed analytical solution can converge in a very broad frequency band. At last, by the proposed solution, the layering effects of the deposit in the valley, such as the number of alluvial layers, the existence of soft interlayer and its thickness, etc., on the ground motion are investigated in a broad frequency band..
Key words: Alluvial valley      Layering      Scattering      Analytical solution      Rayleigh wave
1 引言

2 模型

 图 1 圆弧状多层沉积谷地的模型 Fig. 1 The model of alluvial valley

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3 波的势函数

 图 2 波的散射图 Fig. 2 The scattering of waves

3.1 半空间介质中波的势函数

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3.2 沉积介质中散射波的势函数

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l=0时，

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4 位移与应力的级数展开

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5 边界条件的引入与待定系数的求解 5.1 地表零应力边界条件的引入

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5.2 连续条件的引入

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5.3 待定系数的求解

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6 结果分析

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6.1 级数解收敛性分析

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 图 3 级数解随截断项数的收敛性 Fig. 3 The convergence of series solution with the truncation number

 图 4 谷地几何形状对级数解收敛性的影响 Fig. 4 The influence of valley geometry on the convergence of series solution

 图 5 自由场有限Fourier级数的项数N0对解的影响 Fig. 5 Influence of the number of terms of free-field finite Fourier series on the solution

 图 6 模拟水平地表的大圆弧的半径只对解的影响 Fig. 6 Influence of the radius of the big arc simulating the levfl surface on the solution
6.2 谷地沉积介质的成层性对地面运动的影响

 图 7 谷地沉积层数对地面运动的影响 Fig. 7 Influence of the number of valley layers on the ground motion

 图 8 谷地软弱夹层的存在及其刚度对地面运动的影响 Fig. 8 Influence of the existence of soft interlayer in the valley and its stiffness on the ground motion

 图 9 谷地软弱夹层的厚度对地面运动的影响 Fig. 9 Influence of the thickness of soft interlayer in the valley on the ground motion
7 结论

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