Influence of lining tunnel on sub-ground motion for incident plane SH wave excitation
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摘要: 以地下隧道对附近场地动力特性的影响为研究目标，基于弹性波动理论，利用波函数展开法和镜像法，分析了弹性半空间中圆形衬砌隧道对平面SH波入射产生的散射问题，得到了地下圆形衬砌隧道附近场地位移的级数解答。通过数值算例分析了地下圆形衬砌隧道对场地动力响应的影响，重点考察了SH波入射角度、入射频率和隧道埋深、衬砌刚度对隧道周围土体动力响应随深度变化的影响规律。结果表明，地下隧道对沿线场地的地下地震动影响显著。Abstract: Taking the influence of underground tunnel on the dynamic characteristics of the nearby site as the researched objective, this paper analyzed the scattering of incident plane SH wave resulted from circular lining tunnel in elastic half space by using the wave function expansion method and the image theory based on the elastic wave theory, and then obtained the series solution of the site displacement near the underground circular lining tunnel. Numerical examples are used to analyze the influence of the underground circular tunnel on the dynamic response of the site, i.e., the influence of SH wave incidence angle, incident frequency, tunnel depth and lining stiffness on the dynamic response of soil around the tunnel with depth. The results show that underground tunnel has a significant impact on sub-ground motion along the site.
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Keywords:
- plane SH wave /
- circular lining tunnel /
- image method /
- sub-ground motions
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引言
随着城镇化建设的发展，土地资源日渐紧缺，对地下空间的开发和利用不断深化。然而，地下结构的建设破坏了土体的局部整体性，改变了原场地的动力特性，因而对附近地上和地下工程结构的抗震安全造成了影响。因此，系统深入地分析由于地下结构的存在而引起的沿线地震动随深度的变化规律，对地下结构沿线的结构抗震设计具有十分重要的意义。
地下结构对附近场地地震动影响的实质是地下结构对地震波的散射。Mow和Pao （1973）最早运用波函数展开法研究了无限空间中的隧道在弹性波入射下的动应力集中问题；随之，Lee和Trifunac （1979）运用该方法分析了半空间中衬砌隧道对SH波的动力响应。Lee和Karl （1992，1993）通过理论分析给出了半空间中单个无衬砌隧道对P波和SV波散射问题的解析解。基于单个衬砌隧道的研究，Liang等（2003）得到了半空间中双衬砌隧道对P波和SV波散射问题的解析解。Liu等（2016）分析了弹性半空间中平面波作用下双垂直衬砌隧道的动力相互作用。Xu等（2011）采用傅里叶-贝塞尔级数展开方法计算了半空间中圆形衬砌隧道对P波入射的动力响应。考虑不同种类弹性波的入射情形，梁建文等（2005a，b）研究了地下圆形隧道对地表运动幅值的影响。Liu等（2013）考察了弹性半空间中隧道处于浅埋时平面 P-SV 波和瑞雷波的动力响应。Luco和de Barros （2010）以及de Barros和Luco （2010）计算了水平层状半空间中埋置隧道在入射体波下的三维动力响应。对于平面SV波和P波垂直入射的情形，Oliaei和Alitalesh （2015）分析了由于地下圆形和椭圆形隧道的存在而引起的地面位移被放大的现象。利用四阶有限差分方法，Narayan等（2015）探讨了瑞雷波入射下地下无衬砌隧道和有衬砌隧道对其周围应变和黏弹性地基地表位移的影响。Alielahi和Adampira （2016）应用边界元法给出了P波和SV波入射时双平行隧道对其周围垂直平面内地震动响应的影响。Liu和Liu （2015）利用间接边界元法讨论了弹性楔形空间中隧道在SH波入射下对附近表面地震动的影响。Parvanova等（2014）利用数值模拟方法探讨了局部地形对隧道动力响应的影响。
目前，大部分研究成果仅针对SH波入射情况下地下隧道对地表面地震动的影响（Liang et al，2012，2013；付佳等，2016），而且对考虑地下隧道周围土体在一定深度范围内的动力响应研究也较少。为此，本文拟以含有圆形衬砌隧道的弹性半空间为研究对象，分析地下圆形隧道对场地动力响应的影响，重点研究隧道埋深、衬砌刚度、入射角度以及入射频率对地下隧道周围土体位移振动幅值随深度的变化规律，以期为定量评估地下隧道对既有地下建筑物地震安全性提供理论依据。
1. 地下位移幅值的求解
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1.1 入射波场
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$ w_1^i({r_1}{\text{，}} {\theta _1}) {\text{＝}} \sum\limits_{m {\text{＝}} 0}^{ {\text{＋}} \infty } {{\varepsilon _m}} {({\text{－}}{\rm{i}})^m}{{\rm J}_m}(k{r_1})(\cos m\gamma \cos m{\theta _1} {\text{＋}} \sin m\gamma \sin m{\theta _1}){\text{，}} $
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$ w_2^i({r_2}{\text{，}} {\theta _2}) {\text{＝}} \sum\limits_{m {\text{＝}} 0}^{ {\text{＋}} \infty } {{\varepsilon _m}} {({\text{－}} {\rm{i}})^m}{{\rm J}_m}(k{r_2})(\cos m\gamma \cos m{\theta _2} {\text{＋}} \sin m\gamma \sin m{\theta _2}){\text{.}} $
(2) 1.2 散射波场
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$ w_1^r\left({{r_1}{\text{，}} {\theta _1}} \right) {\text{＝}} \sum\limits_{m {\text{＝}} 0}^{ {\text{＋}} \infty } {{\rm H}_m^{\left(2 \right)}} \left({k{r_1}} \right)\left({{A_m}\cos m{\theta _1} {\text{＋}} {B_m}\sin m{\theta _1}} \right){\text{，}} $
(3) $ w_2^r\left({{r_2}{\text{，}} {\theta _2}} \right) {\text{＝}} \sum\limits_{n {\text{＝}} 0}^{ {\text{＋}} \infty } {{\rm H}_n^{\left(2 \right)}} \left({k{r_2}} \right)\left({{A_n}\cos n{\theta _2} {\text{＋}}{B_n}\sin n{\theta _2}} \right){\text{，}} $
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$ w_1^f\left({{r_1}{\text{，}} {\theta _1}} \right) {\text{＝}} \sum\limits_{m {\text{＝}} 0}^{ {\text{＋}} \infty } {{\rm H}_m^{\left(2 \right)}} \left({{k_1}{r_1}} \right)\left({C_m^{(2)}\cos m{\theta _1} {\text{＋}} D_m^{(2)}\sin m{\theta _1}} \right){\text{，}} $
(5) $ w_2^f\left({{r_1} {\text{，}} {\theta _1}} \right) {\text{＝}}\sum\limits_{m {\text{＝}} 0}^{ {\text{＋}} \infty } {{\rm H}_m^{\left(1 \right)}} \left({{k_1}{r_1}} \right)\left({C_m^{(1)}\cos m{\theta _1} {\text{＋}} D_m^{(1)}\sin m{\theta _1}} \right) {\text{，}} $
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因此，在SH波入射的情形下，半空间中波的势函数为
$ {w^d} {\text{＝}} w_1^i {\text{＋}} w_2^i {\text{＋}} w_1^r {\text{＋}} w_2^r{\text{，}} $
(7) 衬砌介质中波的势函数为
$ {w^f} {\text{＝}} w_1^f {\text{＋}} w_2^f{\text{.}} $
(8) 1.3 引入边界条件求解问题
引入衬砌隧道的边界条件：
$ {\sigma _{rz}} {\text{＝}} {\mu _1}\frac{{\partial {w^f}}}{{\partial {r_1}}} {\text{＝}} 0 {\text{，}}{r_1} {\text{＝}} b {\text{，}} $
(9) $ {w^d}{\text{＝}} {w^f} {\text{，}} {r_1} {\text{＝}} a {\text{，}} $
(10) $ {\mu _0}\frac{{\partial {w^d}}}{{\partial {r_1}}} {\text{＝}} {\mu _1}\frac{{\partial {w^f}}}{{\partial {r_1}}}{\text{，}} {r_1}{\text{＝}} a{\text{，}} $
(11) 即可求得式（3）—（6）中所有待定系数，从而确定式（7）中半空间介质内波的势函数。在SH波作用下，沿深度方向的位移可通过求解以上边界条件得出，从而求得隧道周围沿深度方向的位移幅值，具体求解过程不再赘述。
1.4 解的验证
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2. 算例与分析
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表 1 距地表6a深度范围内隧道左右两侧最大地下位移幅值Table 1. The maximum amplitude of underground displacement on both sides of tunnel within a depth of 6a from surface入射角/° 地下位移幅值 x/a=−3.0 x/a=−1.5 x/a=1.5 x/a=3.0 0 2.66 2.79 2.79 2.66 30 2.90 2.83 2.31 2.45 60 3.30 2.37 2.64 2.61 90 3.31 3.94 3.37 3.01 ![]()
图 4 隧道两侧SH波从不同角度入射时的地下位移幅值变化曲线Figure 4.This page contains the following errors:
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图 5 不同隧道埋深时隧道两侧的SH波地下位移幅值变化Figure 5. Variation of underground displacement amplitude with D/a for SH waves on both sides of the tunnelThis page contains the following errors:
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图 6 不同入射频率时隧道两侧的SH波地下位移幅值变化曲线Figure 6. Variation of underground displacement amplitude with η for SH waves on both sides of the tunnelThis page contains the following errors:
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图 7 不同隧道衬砌刚度隧道两侧的SH波地下位移幅值变化Figure 7. Variations of underground displacement amplitude with lining stiffness for SH waves on both sides of the tunnel3. 讨论与结论
本文应用波函数展开法和镜像法，得到了平面SH波作用下含圆形衬砌隧道的弹性半空间中散射波场的级数解答。通过数值算例分析，研究了平面SH的入射角度、入射频率和隧道埋深、衬砌刚度对沿线地下地震动的影响。结果表明，地下隧道的存在对其周围地震动具有显著的影响，并具有如下规律：
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2） 随着入射频率的增加，隧道周围土体的位移幅值有逐渐增大的趋势，但是在隧道左侧近处存在异常区，该区随着频率的增加地下位移振幅逐渐减小。
3） 在隧道周围近距离处，衬砌刚度的变化对地下位移幅值的影响显著，而在隧道远距离处，衬砌刚度对地下位移幅值的影响较弱。
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图 4 隧道两侧SH波从不同角度入射时的地下位移幅值变化曲线
Figure 4.
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图 5 不同隧道埋深时隧道两侧的SH波地下位移幅值变化
Figure 5. Variation of underground displacement amplitude with D/a for SH waves on both sides of the tunnel
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图 6 不同入射频率时隧道两侧的SH波地下位移幅值变化曲线
Figure 6. Variation of underground displacement amplitude with η for SH waves on both sides of the tunnel
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图 7 不同隧道衬砌刚度隧道两侧的SH波地下位移幅值变化
Figure 7. Variations of underground displacement amplitude with lining stiffness for SH waves on both sides of the tunnel
表 1 距地表6a深度范围内隧道左右两侧最大地下位移幅值
Table 1 The maximum amplitude of underground displacement on both sides of tunnel within a depth of 6a from surface
入射角/° 地下位移幅值 x/a=−3.0 x/a=−1.5 x/a=1.5 x/a=3.0 0 2.66 2.79 2.79 2.66 30 2.90 2.83 2.31 2.45 60 3.30 2.37 2.64 2.61 90 3.31 3.94 3.37 3.01 
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