远场剪切波作用下深埋矩形隧洞地震响应解析解

Analytical solution of deep buried rectangular tunnel under the action of far field shear wave

既有理论针对土体与结构相互作用的地震响应研究一直集中在圆形隧洞截面,而就矩形隧洞的分析往往会增加动力计算的复杂性。基于拟静力法,采用复变函数理论和保角变换方法,推导获得远场剪切波作用下深埋矩形隧洞的地震响应解析解。该复变函数解答考虑了土体与隧洞间相互作用的剪应力和法向正应力,并针对保角转化函数引入修正系数,提高了在复平面转化时矩形隧洞截面表示函数的计算精度。结合数值模型验证解析方法的正确性,并对矩形隧洞归一化位移比和土体应力场变化进行参数分析。研究表明,通过引入修正系数,使得解析解精度得到一定提高,与数值模拟解答更为接近。矩形隧洞归一化位移比值随着隧洞衬砌与土体的相对刚度比值增大而减小,而随着截面尺寸比率的增大而增大。在矩形隧洞截面尺寸的比率和相对刚度比值一定的情况下,矩形隧洞的归一化位移比随着土体泊松比的减小而增大。当截面尺寸比率为1时(正方形隧洞),土体泊松比的变化对于隧洞归一化位移比值影响较小。矩形隧洞周围土体应力随着土体泊松比的增大而增大,随着土体弹性模量的增大而减小,而随着衬砌剪切模量的增大而减小。

The existed solutions on the interaction between the seismic responses of soil and structure have focused on the tunnels of circular cross section. The dynamic analysis for the rectangular tunnel tends to increase in complexity of calculation. An analytical solution of the seismic response of deep buried rectangular tunnel under the action of far field shear waves is derived based on the pseudo static method by using the complex analysis and conformal transformation. The shear and normal stresses in the interaction between soil and tunnel are considered in the solution of the complex function. The accuracy of calculating the conformal transformation function for rectangular tunnel is improved by introducing a correction factor. The accuracy of the analytical method is verified with the numerical model. The parameters influencing the normalized displacement ratio and the variation of the soil stress field for the rectangular tunnel are analyzed as well. The study shows that the accuracy of the analytical solution is improved by introducing the correcting coefficient, which is more close to the numerical solution. The normalized displacement ratio of the rectangular tunnel decreases with the increase of the relative stiffness ratio of tunnel to soil. The normalized displacement ratio increases as the increasing of the ratio of the cross section size. When the cross section size ratio of the rectangular tunnel and the relative stiffness ratio are unchanged, the normalized displacement ratio of the rectangular tunnel increases with the decrease of the Poisson＇s ratio of the soil. When the cross section size ratio is 1 （Square tunnel）, the change of Poisson＇s ratio of soil has little influence on the normalized displacement ratio of the tunnel. The soil stress field around the rectangular tunnel increases with the increase of the Poisson＇s ratio of soil. The soil stress field decreases with the increase of the elastic modulus of soil. The soil stress field decreases with the increase of the shear modulus of the liner.