对Janbu普遍条分法计算方法的改进

Janbu普遍条分法(或称严格Janbu法)是第一个基于任意形状滑面且考虑滑体所有平衡条件的边坡稳定性计算方法。该方法存在的严重不收敛问题一直困扰着边坡理论和工程界近半个世纪。本文基于Janbu普遍条分法的基本原理和假设,重新推导出数学上更为简洁的安全系数积分表达式,指出传统计算格式不易收敛性是推力二阶导数的不合理数值误差所致,应用数值平滑技术可减少这个误差,使计算过程稳定收敛。研究表明,改进后的Janbu普遍条分法不失为理论相对严格、工程较为实用的边坡稳定性计算方法。

A Modified Algorithm for Calculating Slope Stability Using Janbu's Generalized Procedure of Slices

The Janbu's Generalised Procedure of Slice (GPS), i. e. , the rigorous Janbu method, is the first method which takes into account all the equilibrium conditions for calculating factors of safety of slopes with arbitrary-shaped failure surfaces. But the problem of frequently encountered non-convergence associated with this method has frustrated slope researchers and practitioners for nearly half a century. In this paper, on the basis of the fundamental principle and assumptions inherent in the Janbu's generalized procedure of slices, a more mathematically concise formula of factor of safety has been derived. By assuming the location of the line of thrust across the sliding body and considering the moment equilibrium condition for an infinitesimal slice, the shear thrust force can be expressed in terms of the normal thrust force and its first-order derivative. By considering the vertical force equilibrium condition as well as the above relation between the normal and shear thrust forces, the normal stress on the failure surface is expressed in terms of the normal thrust force and its first-and second-derivatives as well as other known parameters. By considering the horizontal force equilibrium condition, the factor of safety for the sliding mass is derived in terms of the normal stresses over the failure surface. A simple iterative procedure is developed for calculating the factor of safety: first, calculating the normal stresses on the failure surface (initially assuming zero thrust force); second, calculating the improved factor of safety; third, calculating the thrust forces which is then used for calculating the improved normal stresses on the failure surface. The process is repeated until the difference in value of factor of safety between two adjacent steps is within tolerance. Either present or conventional computation procedure associated the Janbu's generalized procedure involves the second-order derivative of thrust force, which cannot be reasonably determined by conventional finite difference method, particularly with large number of slices, say over 100 slices. It is the unreasonable errors in the second-order derivative of thrust forces that leads to the non-convergence associated with the conventional procedure. By employing numerical smoothing technique such errors can be eliminated, thus rendering the computation process being converged. It is demonstrated that the modified Janbu's Generalised Procedure of Slices is still a theoretically rigorous and practicable method for analysing slope stability.