﻿ 含圆形衬砌的直角平面区域受出平面动力时地表位移幅值

# 含圆形衬砌的直角平面区域受出平面动力时地表位移幅值The Surface Displacement Amplitudes of Rectangular Planar Region Including Circular Lining When Enduring Out-Plane Dynamic

Abstract: The dynamic response of rectangular planar infinite region including circular lining while bearing out-of-plane dynamic incidenting in an arbitrary angle is investigated. The out-plane wave field acting on the rectangular planar infinite region is expanded into the out-plane wave field acting on semi-infinite space region, and the field of stress and displacement is deduced to solve the problems of the model, the solution of which is a set of infinite algebraic equations. The results of numerical example show that, the surface displacement amplitude of the model is affected by the incident wave frequency, Angle of the incident wave and parameters such as the location of the lining and the thickness of the lining.

1. 引言

2. 力学模型

$\frac{{\partial }^{2}W}{\partial z\partial \stackrel{¯}{z}}+\frac{1}{4}{k}^{\text{2}}W=0$ (1)

Figure 1. The model of rectangular planar region including circular lining

$\begin{array}{l}{\tau }_{xz}=\mu \left(\frac{\partial W}{\partial z}+\frac{\partial W}{\partial \stackrel{¯}{z}}\right)\\ {\tau }_{yz}=\mu \left(\frac{\partial W}{\partial z}-\frac{\partial W}{\partial \stackrel{¯}{z}}\right)\end{array}$ (2)

$\begin{array}{l}{\tau }_{rz}=\mu \left(\frac{\partial W}{\partial z}{\text{e}}^{i\theta }+\frac{\partial W}{\partial \stackrel{¯}{z}}{\text{e}}^{-i\theta }\right)\\ {\tau }_{\theta z}=i\mu \left(\frac{\partial W}{\partial z}{\text{e}}^{i\theta }-\frac{\partial W}{\partial \stackrel{¯}{z}}{\text{e}}^{-i\theta }\right)\end{array}$ (3)

3. 振动引起的位移函数

3.1. 直角域内的入射波位移函数和反射波位移函数

${W}^{\left(i\right)}={W}_{0}\mathrm{exp}\left\{\frac{ik}{2}\left[{z}_{h}{\text{e}}^{-i\left(\text{π}-{\alpha }_{0}\right)}+{\stackrel{¯}{z}}_{h}{\text{e}}^{i\left(\text{π}-{\alpha }_{0}\right)}\right]\right\}$ (4)

${W}^{\left(i,e\right)}={W}_{0}\left\{\mathrm{exp}\left[\frac{ik}{2}\left({z}_{h}{\text{e}}^{-i{\beta }_{0}}+{\stackrel{¯}{z}}_{h}{\text{e}}^{i{\beta }_{0}}\right)\right]+\mathrm{exp}\left[\frac{ik}{2}\left({{z}^{\prime }}_{h}{\text{e}}^{-i{\alpha }_{0}}+{\stackrel{¯}{{z}^{\prime }}}_{h}{\text{e}}^{i{\alpha }_{0}}\right)\right]\right\}$ (5)

${W}^{\left(r,e\right)}={W}_{0}\left\{\mathrm{exp}\left\{\frac{ik}{2}\left[{z}_{h}{\text{e}}^{i{\beta }_{0}}+{\stackrel{¯}{z}}_{h}{\text{e}}^{-i{\beta }_{0}}\right]\right\}+\mathrm{exp}\left\{\frac{ik}{2}\left[{{z}^{\prime }}_{h}{\text{e}}^{i{\alpha }_{0}}+{\stackrel{¯}{{z}^{\prime }}}_{h}{\text{e}}^{-i{\alpha }_{0}}\right]\right\}\right\}$ (6)

Figure 2. The mirror model

3.2. 直角域内的散射波位移函数

${W}^{\left(s\right)}\left(z,\stackrel{¯}{z}\right)=\underset{n=-\infty }{\overset{\infty }{\sum }}{A}_{n}{H}_{n}^{\left(1\right)}\left(k|z|\right)\cdot {\left[z/|z|\right]}^{n}$ (7)

${W}^{\left(s,e\right)}\left(z,\stackrel{¯}{z}\right)=\underset{n=-\infty }{\overset{\infty }{\sum }}{A}_{n}\underset{j=1}{\overset{4}{\sum }}{S}_{n}^{\left(j\right)}$ (8)

$\left\{\begin{array}{l}{S}_{n}^{\left(1\right)}={H}_{n}^{\left(1\right)}\left(k|z|\right){\left[z/|z|\right]}^{n}\\ {S}_{n}^{\left(2\right)}={H}_{n}^{\left(1\right)}\left(k|z-2ih|\right){\left[\left(z-2ih\right)/|z-2ih|\right]}^{-n}\\ {S}_{n}^{\left(3\right)}={\left(-1\right)}^{n}{H}_{n}^{\left(1\right)}\left(k|{z}^{\prime }|\right){\left[\left({z}^{\prime }\right)/|{z}^{\prime }|\right]}^{-n}\\ {S}_{n}^{\left(4\right)}={\left(-1\right)}^{n}{H}_{n}^{\left(1\right)}\left(k|{z}^{\prime }-2ih|\right){\left[\left({z}^{\prime }-2ih\right)/|{z}^{\prime }-2ih|\right]}^{n}\end{array}$

3.3. 衬砌内的散射波位移函数

${{W}^{\left(s\right)}|}_{r=b}=\underset{n=-\infty }{\overset{\infty }{\sum }}{B}_{n}{H}_{n}^{\left(2\right)}\left({k}_{\text{2}}|z|\right)\cdot {\left[z/|z|\right]}^{n}$ (9)

${{W}^{\left(s\right)}|}_{r=a}=\underset{n=-\infty }{\overset{\infty }{\sum }}{C}_{n}{H}_{n}^{\left(1\right)}\left({k}_{\text{2}}|z|\right)\cdot {\left[z/|z|\right]}^{n}$ (10)

3.4. 内含的圆形衬砌平面直角域模型的边界条件

$\left\{\begin{array}{l}{\Gamma }_{3}:{\tau }_{rz}^{\left(i,e\right)}+{\tau }_{rz}^{\left(r,e\right)}+{\tau }_{rz}^{\left(s\right)}={{\tau }_{rz}^{\left(s\right)}|}_{r=b}+{{\tau }_{rz}^{\left(s\right)}|}_{r=a}\\ {W}^{\left(i,e\right)}+{W}^{\left(r,e\right)}+{W}^{\left(s\right)}={W}^{\left(s\right)}{}_{r=b}+{{W}^{\left(s\right)}|}_{r=a}{W}_{rz}^{Ι}\\ {\Gamma }_{4}:{{\tau }_{rz}^{\left(s\right)}|}_{r=b}+{{\tau }_{rz}^{\left(s\right)}|}_{r=a}=0\end{array}$ (11)

$\left\{\begin{array}{l}\underset{n=-\infty }{\overset{\infty }{\sum }}{A}_{n}{\xi }_{1n}^{\left(1\right)}+{B}_{n}{\xi }_{2n}^{\left(1\right)}+{C}_{n}{\xi }_{3n}^{\left(1\right)}={\eta }_{1}\\ \underset{n=-\infty }{\overset{\infty }{\sum }}{A}_{n}{\xi }_{1n}^{\left(2\right)}+{B}_{n}{\xi }_{2n}^{\left(2\right)}+{C}_{n}{\xi }_{3n}^{\left(2\right)}={\eta }_{2}\\ \underset{n=-\infty }{\overset{\infty }{\sum }}{B}_{n}{\xi }_{2n}^{\left(3\right)}+{C}_{n}{\xi }_{3n}^{\left(3\right)}=0\end{array}$ $\left(n=0,±1,\cdots \right)$ (12)

$\left\{\begin{array}{l}{\xi }_{1n}^{\left(2\right)}=\underset{j=1}{\overset{4}{\sum }}{S}_{n}^{\left(j\right)}\\ {\xi }_{2n}^{\left(2\right)}={H}_{n}^{\left(2\right)}\left({k}_{1}|z|\right)\cdot {\left[z/|z|\right]}^{n}\\ {\xi }_{3n}^{\left(2\right)}={H}_{n}^{\left(1\right)}\left({k}_{1}|z|\right)\cdot {\left[z/|z|\right]}^{n}\end{array}$

$\left\{\begin{array}{l}{\eta }_{1}=-i{\tau }_{0}\mathrm{cos}\left(\theta -{\beta }_{0}\right)\mathrm{exp}\left\{\frac{ik}{2}\left({z}_{h}{\text{e}}^{-i{\beta }_{0}}+{z}_{h}{\text{e}}^{i{\beta }_{0}}\right)\right\}-i{\tau }_{0}\mathrm{cos}\left(\theta -{\alpha }_{0}\right)\mathrm{exp}\left\{\frac{ik}{2}\left({{z}^{\prime }}_{h}{\text{e}}^{-i{\alpha }_{0}}+{\stackrel{¯}{{z}^{\prime }}}_{h}{\text{e}}^{i{\alpha }_{0}}\right)\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-i{\tau }_{0}\mathrm{cos}\left(\theta +{\beta }_{0}\right)\mathrm{exp}\left\{\frac{ik}{2}\left({z}_{h}{\text{e}}^{i{\beta }_{0}}+{z}_{h}{\text{e}}^{-i{\beta }_{0}}\right)\right\}-i{\tau }_{0}\mathrm{cos}\left(\theta +{\alpha }_{0}\right)\mathrm{exp}\left\{\frac{ik}{2}\left({{z}^{\prime }}_{h}{\text{e}}^{i{\alpha }_{0}}+{\stackrel{¯}{{z}^{\prime }}}_{h}{\text{e}}^{-i{\alpha }_{0}}\right)\right\}\\ {\eta }_{2}={w}_{0}\left\{\mathrm{exp}\left[\frac{ik}{2}\left({z}_{h}{\text{e}}^{-i{\beta }_{0}}+{\stackrel{¯}{z}}_{h}{\text{e}}^{i{\beta }_{0}}\right)\right]+\mathrm{exp}\left[\frac{ik}{2}\left({{z}^{\prime }}_{h}{\text{e}}^{-i{\alpha }_{0}}+{\stackrel{¯}{{z}^{\prime }}}_{h}{\text{e}}^{i{\alpha }_{0}}\right)\right]\right\}\\ \text{\hspace{0.17em}}+{w}_{0}\left\{\mathrm{exp}\left\{\frac{ik}{2}\left[{z}_{h}{\text{e}}^{i{\beta }_{0}}+{\stackrel{¯}{z}}_{h}{\text{e}}^{-i{\beta }_{0}}\right]\right\}+\mathrm{exp}\left\{\frac{ik}{2}\left[{{z}^{\prime }}_{h}{\text{e}}^{i{\alpha }_{0}}+{\stackrel{¯}{{z}^{\prime }}}_{h}{\text{e}}^{-i{\alpha }_{0}}\right]\right\}\right\}\end{array}$

4. 地表位移幅值

${W}^{\left(t\right)}\left(z,\stackrel{¯}{z}\right)={W}^{\left(i,e\right)}\left(z,\stackrel{¯}{z}\right)+{W}^{\left(r,e\right)}\left(z,\stackrel{¯}{z}\right)+{W}^{\left(s\right)}\left(z,\stackrel{¯}{z}\right)$ (16)

${W}_{d}=|{W}^{\left(t\right)}/{W}_{0}|$ (17)

${d}^{\ast }\left({x}_{h}\right)=|{W}^{\left(t\right)}\left({x}_{h}\right)/\left({W}^{\left(i,e\right)}+{W}^{\left(r,e\right)}\right)|-1$ (18)

${d}^{\ast }\left({x}_{h}\right)>0$ 说明衬砌对地表位移有增强作用； ${d}^{\ast }\left({x}_{h}\right)<0$ 说明衬砌的对地表位移有减弱作用， ${d}^{\ast }\left({x}_{h}\right)=0$ 说明衬砌对地表位移没有影响。

5. 算例与结果分析

(a) (b) (c) (d)

Figure 3. The change distribution of surface displacement amplitudes “Wd

(a)(b)

Figure 4. The influence coefficient of surface displacement

6. 结论

[1] Liu D.K. and Han, F. (1990) Scattering of Plane SH-Waves by Cylindrical Canyon of Arbitrary Shape in Anisotropic Media. Acta Mechanica Sinica, 6, 256-260.
https://doi.org/10.1007/BF02487648

[2] Liu, D.K. and Han, F. (1991) Scattering of Plane SH-Waves by a Cylindrical Canyon of Arbitrary Shape. International Journal of Soil Dynamics and Earthquake Engineering, 10, 249-255.
https://doi.org/10.1016/0267-7261(91)90018-U

[3] 刘殿魁, 许贻燕. 各向异性介质中SH波与多个半圆形凹陷地形的相互作用[J]. 力学学报, 1993, 25(1): 93-102.

[4] 崔志刚, 邹永超, 刘殿魁. SH波对圆弧形凸起地形的散射[J]. 地震工程与工程振动, 1998, 18(1): 140-146.

[5] 刘殿魁, 曹新荣, 崔志刚. 多个半圆形凸起地形对平面SH波散射[J]. 固体力学学报(特刊), 1998: 178-185.

[6] 袁晓铭, 廖振鹏. 任意圆弧形凸起地形对平面SH波的散射[J]. 地震与工程振动, 1996, 16(2): 1-13.

[7] 袁晓铭, 廖振鹏. 圆弧形凹陷地形对平面SH波散射问题的级数解答[J]. 地震与工程振动, 1993, 13(2): 1-11.

[8] 袁晓铭, 廖振鹏. 圆弧形沉积盆地对平面SH波的散射[J]. 华南地震, 1995, 15(2): 1-8.

[9] 袁晓铭, 廖振鹏. 自由表面圆弧形不规则边界对SH波的散射[J]. 科学通报, 1996, 42(3): 262-264.

[10] Yuan, X.M. and Men, F.L. (1992) Scattering of Plane SH-Waves by a Semi-Cylindrical Hill. Earthquake Engineering & Structural Dynamics, 21, 1091-1098.
https://doi.org/10.1002/eqe.4290211205

[11] 梁建文, 张郁山, 顾晓鲁, 等. 圆弧形层状凹陷地形对平面SH波的散射[J]. 振动工程学报, 2003, 16(2): 158-165.

[12] 梁建文, 罗昊, Lee, V.W. 任意圆弧形凸起地形中隧洞对入射平面SH波的影响[J]. 地震学报, 2004, 26(5): 495-508.

[13] 杨彩红, 梁建文, 张郁山. 多层沉积凹陷地形对平面SH波散射问题的解析解[J]. 岩土力学, 2006, 27(12): 2191-2196.

[14] 梁建文, 巴振宁. 弹性层状半空间中凸起地形对入射平面SH波的放大作用[J]. 地震工程与工程振动, 2008, 28(1): 1-10.

[15] 梁建文, 巴振宁. 弹性层状半空间中沉积谷地对入射平面SH波的放大作用[J]. 地震工程与工程振动, 2007, 27(3): 1-9.

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