牛顿法计算Sarma法边坡安全系数

The Computation of Safety Factor of Sarma Method by Newton Method

根据Sarma法的基本假定即条块侧面与底面抗剪强度按同一比例调用，将斜条块侧面的推力分解为分别与摩擦力和凝聚力有关的2个分量，由条块力的平衡条件，推导出更为简洁的隐含安全系数的条块推力递推方程。为了加速收敛，采用牛顿法迭代法计算安全系数，并推导出计算中所需有关导数的解析表达式。同时，利用所得推力递推方程重新推导出了临界地震加速度系数Ke的显式表达式，该式与原始的Sarma法等效，但形式上更为简明且便于应用。算例表明，本文的改进的Sarma法算法收敛迅速，迭代3～5步即可达到工程所需精度，计算结果与经典算例Sarma法解答及塑性力学理论解均非常接近。

By employing the same assumption used in the Sarma method, that is,the shear strengths along the interface between slices are mobilized to the same degree as that along the slip surface, the lateral thrusts of oblique slices are decomposed into two components related to friction and cohesion respectively. According to the force equilibrium of a slice a more concise recursive equation is derived in terms of the safety factor. The Newton method is employed for computing the safety factor in order for fast convergence. Relevant derivatives needed in the computation are derived that are in recursive form and of analytical nature. Meanwhile, by using the present recursive equation an explicit expression of critical seismic coefficient is obtained which is equivalent to the original equation of the Sarma method, but is in simpler form and easy to apply. Example studies demonstrate that the modified algorithm of the Sarma method converges rapidly and only 3-5 iterations are needed for achieving the precision required by practical engineering. The results of computations are in close agreement with both the classic solution of the Sarma method and the theoretical solution based on the mechanics of plasticity.