基于多边形最佳平方逼近形式的边坡稳定上限有限元法

Upper Bound Finite Element of Slope Analysis Based on Optimal Square Polygon Approximation

边坡的极限分析上限法具有较严谨的理论基础和物理意义,且可以在得到安全系数的同时找到最危险的临界失稳速度场,因此具有较广阔的应用前景。针对目前上限有限元法中被广泛采用的外切多边形逼近摩尔库伦屈服圆所带来的收敛速度慢的缺点,放松边坡内任一点需严格满足上限性质的要求,将多边形采用最佳平方逼近的形式逼近屈服圆,并系统地推导出了最佳平方逼近形式的上限有限元法的整个计算模型。算例表明:该方法不仅继承了外接多边形从上方逼近解析解的优点,而且采用较少的多边形边数即可得到较为精确的结果,收敛速度大大提高。

The upper bound limit analysis has broad application prospect as it has rigorous theoretical basis and clear physical meaning,and could give the safety factor as well as the critical velocity field at the same time. But it has slow convergence speed because of circumscribed polygon approximation（ which is widely used） to Mohr-Coulomb yield circle. In view of this,we alleviated the requirement that all the points must strictly meet the limit properties,adopted the optimal square polygon approximation to the yield circle,and finally obtained the systematic calculation model of upper bound finite element based on optimal square polygon approximation. Numerical example shows that the method not only inherits the advantage circumscribed polygon have which approximates analytic solution from the upper,but also gets a precise result by less number of polygon edges,and also greatly improves the convergence speed.