变分极限平衡法对土质边坡稳定性的精确分析

Exact analysis of soil slope stability by using variational limit equilibrium method

基于变分法和极限平衡法的基本原理,对土质边坡的稳定性分析进行了理论研究。首先,通过极限平衡法建立了问题的基本平衡方程,通过引入拉格朗日乘子构造出约束条件下等周问题的辅助泛函;其次,联合欧拉方程得到了以潜在滑动面、滑面正应力、滑动体内力、安全系数以及拉格朗日乘子为基本未知量的一阶常微分方程组,并通过引入辅助变量将边坡稳定性分析转化为固定边界条件下的两点边值问题;最后,利用打靶法对算例进行了数值求解,得到了滑动面位置、滑动面上正应力分布以及边坡稳定性系数的精确解答,并与条分法和有限元强度折减法得到的结果进行了比较,验证了所提方法的有效性。分析结果表明,提出的土质边坡稳定性分析方法可有效避免传统计算中对滑面位置的搜索优化过程,能精确得到边坡滑面形状、滑面上的法向应力以及条间力分布,可为边坡的稳定性分析提供科学合理的计算方法。

Based on the basic principles of the variational method and the limit equilibrium method,the stability analysis of soil slopes is conducted theoretically.Firstly,the basic equilibrium equation for the problem is established by using the limit equilibrium method,and the auxiliary functional for the isoperimetric problem under constraints is constructed by introducing the Lagrange multiplier.In addition,the first-order ordinary differential equations with potential sliding surface,normal stress of sliding surface,internal force of sliding body,safety factor and Lagrange multiplier as the basic unknowns are obtained by combining the Euler equation,and the slope stability analysis is transformed into a two-point boundary value problem under fixed boundary conditions by introducing the auxiliary variables.Finally,the numerical solution of the example is carried out by using the shooting method,and the accurate solutions of the position of the sliding surface,the normal stress distribution on the sliding surface and the slope stability coefficient are obtained.The results are compared with those obtained by the slice method and the finite element strength reduction method to verify the effectiveness of this method.The analysis results show that the proposed analysis method for soil slope stability can effectively avoid the search and optimization process of the sliding surface position in the traditional calculation,and can accurately obtain the shape of the slope sliding surface,the normal stress on the sliding surface and the inter-strip force distribution,which may provide a scientific and reasonable calculation method for the stability analysis of slopes.