考虑注浆加固作用的非圆形隧道应力和变形解析解

为了得到考虑注浆加固作用的非圆形隧道应力和变形解析解，基于复变函数理论引入一种新的考虑注浆加固作用的非圆形隧道应力和变形的求解方法。首先，为克服非圆形隧道断面、注浆圈几何形状和考虑衬砌支护造成的计算困难问题，引入了保角变换及复变函数幂级数解法。通过采用最优化解法确定保角变换中各项系数，得到计算模型映射函数。其次，通过幂级数复变函数法和弹性力学连续性条件克服隧道衬砌以及围岩注浆圈带来的多连通域问题，确定应力函数各项系数。随后，将得到的应力函数代入应力、位移方程求解考虑注浆加固作用的非圆形隧道应力和变形值。最后，将新方法所得结果分别与未考虑注浆作用的非圆形隧道应力及位移解和数值模拟计算结果进行对比分析。研究结果表明：注浆后注浆圈环刚度增大，整体性提升；衬砌变形减小，衬砌受到的围岩压力减小；围岩注浆有效改善了衬砌受力状态，使衬砌拱顶下沉减小约21.8%，拱底隆起减小约18.1%，拱脚附近法向应力减小约19.9%，环向应力减小约8.9%；围岩注浆可以有效加固岩体，封闭隧道周边岩体裂隙，改善衬砌受力状态，提高隧道抵抗变形和破坏的能力；解析解与数值解吻合较好，所得规律符合工程实际规律。研究结果可为考虑注浆加固作用下的非圆形隧道开挖问题提供一种新的快速、准确的计算方法，并为考虑注浆加固作用的非圆形隧道数值计算和安全运行提供参考依据。

Analytical Solution of Stress and Deformation of Noncircular Tunnel Considering Grouting Circle

A new complex variable method is presented for solving stress and displacement problems of noncircular deep tunnels with certain boundary conditions at infinity, and obtaining analytical solutions of stress and deformation of noncircular tunnel under grouting reinforcement. The new method uses complex variables with power series and conformal transformations, and thus, is able to overcome several difficulties arising from the noncircular geometric configurations, lining supports, and grouting circle. Furthermore, multiconnected domain problems caused by the tunnel lining and surrounding rock grouting ring were solved using the power series complex function method and continuity condition of elastic mechanics. The ultimate stress function was incorporated into the stress and displacement equations to obtain the stress and deformation values of the noncircular tunnel under grouting reinforcement. The results obtained using the new method were compared with numerical results as well as results obtained without taking the grouting circle into consideration. The conclusions are as follows:The grouting circle stiffness increases after grouting, thereby improving the stress state of the tunnel. The vault subsidence decreases by approximately 21.8% due to the integral stress of the grouting circle of the lining. The arch bottom uplift and normal stress in the lining decrease by approximately 18.1% and 19.9%, respectively. The peak value of the circumferential stress in the lining decreases by 8.9% after the grouting. The surrounding rock grouting can effectively reinforce the rock mass, seal the cracks, improve the stress state of the lining, and improve the ability of the tunnel to resist deformation and damage. The analytical solutions are in good agreement with the corresponding numerical solutions, and the obtained law conforms to engineering practices. The new method provides an alternative route for obtaining rapid and accurate solutions of noncircular tunnel excavation problems taking grouting circle into account. The results are expected to provide a reference for numerical computation of noncircular tunnels taking grouting circle into account.