考虑巷道周边有非均匀作用力的基尔希解修正

Corrected Solution of Kirsch under Heterogeneous Force around Roadway

地下工程中的巷道(隧道)围岩大部分情况下会受到两向不等压的地应力作用,通常都需要进行支护。已有研究结果表明,围岩受到两向不等压的地应力作用时会导致巷道周边不同点处的径向位移不再相等,导致支护结构对围岩的反作用力也是非均匀的。原基尔希解为仅仅考虑了围岩受到两向不等压地应力作用的解析结果,因此,有必要研究支护结构的反作用力对巷道围岩的应力场、位移场产生的影响,对原基尔希解进行修正与完善。采用了与基尔希解相同的求解思路,首先,对巷道周边非均匀作用力的分布形式进行了合理假设与分解,将其分解为对称条件和非对称条件的叠加。其中,对称条件下的应力解采用弹性力学中拉梅解的求解方法,非对称条件下的应力解采用弹性力学中的半逆解法求解。将二者进行叠加即求得非均匀力作用于巷道围岩周边的弹性应力解的解析式,再与原基尔希解进行叠加即可得到基尔希解的修正解。其次,利用巷道围岩的几何方程和本构方程推导出了巷道周边处的径向位移和环向位移。最后,通过工程实例的计算,对比分析了λ和q0的变化对σr、σθ和巷道周边径向位移ur的影响,为下一步塑性区的求解提供理论基础。

The roadway（tunnel） surrounding rock in underground engineering are subjected to unequal stress from two directions in most cases, so it is usually necessary to be supported. The results show that the radial displacement at different points around roadway will be heterogeneous when the surrounding rock is subjected to non-equal stress from two directions,which leads to the heterogeneously reacting force of the supporting structure on the surrounding rock. The original Kirsch Solution only took into consideration on the analytical results of the surrounding rock subjected to two directions of unequal geostress. Hence, it is essential to study the influence of the reacting force of the supporting structure on the stress field and displacement field of the surrounding rock, and to correct and improve the original Kirsch solution. With the same solution way as the Kirsch solution, the distribution of nonuniform forces around the tunnel is assumed and decomposed firstly, and then the forces are decomposed into the superposition of symmetric and asymmetric conditions. In the symmetric condition, the stress solution is gained by the way of the lame solution in elastic mechanics. In the asymmetric condition, the stress solution is solved by the semi inverse method in elastic mechanics. The analytic formula of the elastic stress solution around the surrounding rock of roadway under heterogeneous forces can be obtained by superposition of the two ones, and the corrected solution of the Kirsch solution can be obtained with the superposition of the original Kirsch solution. Secondly, the radial displacement and circumferential displacement around the roadway are derived from the geometric equation and constitutive equation of roadway surrounding rock. Finally, through the calculation in the engineering example, the influence of the variation of λ and q0 on σr,σθ and the radial displacement ur around the tunnel is analyzed, which provides the theoretical basis for the solution of the plastic zone next.