Fellenius法的解析解法

Fellenius 法是一种经典的边坡稳定性分析方法。由于此法更加安全，故而在工程施工中被广泛应用。现行方法多为数值计算方法，通过大量分条和滑弧的遍历来确定最小安全系数 Ks 和最危险滑弧的位置（圆心横坐标 x0，圆心纵坐标 y0，半径 R）。但数值计算方法计算量浩大且精度不及解析计算方法。通过对 Fellenius 法的连加形式的数学模型进行分析，得到了 Fellenius 法积分形式的数学模型，进而得到了安全系数的表达式 K（x0，y0，R）。然后，将求解最小安全系数 Ks 的问题转化为 K（x0，y0，R）求极值问题，并导出了 K（x0，y0，R）取最小值 Ks 的时候须满足的方程 gradK ＝（0，0，0）。最后，基于 gradK ＝（0，0，0）无根式解这一基本事实，利用麦克劳林展开将 K（x0，y0，R）简化，并利用费拉里法对 gradK ＝（0，0，0）进行解答，得到了最危险滑弧的位置（xs ，ys ，Rs ）的表达式。将（xs ，ys ，Rs ）代入 K（x0，y0，R），得到了最小安全系数 Ks ＝ K（xs ，ys ，Rs ）。继而，只需要得知土坡的相关参数，便可得到 Ks ，无需试算与解方程且具有较高的效率。

The Analytical Solution of Fellenius Method

Fellenius method is a classical solution which hasbeen widely used in engineering construction .Numerical methods are more and more popular at the moment ,which divide a lot of stripes and traverse the sliding surface to calcu-late the minimum safety factor and the location of the most dangerous sliding surface (center coordinates and radius) . However ,the numerical methods need a large amount of computation time and the accuracy of the numerical methods is less than the analytical calculation methods .This work try to get the analytical formula of safety factor Ks which is the function of the center coordinates (x0 ,y0 ) and the radius R .By soloving the equation gradKs = (0 ,0 ,0) ,we can get the center coordinates (x0 ,y0 ) and the radius R and we can also get the minimum safety factor Ksmin .A program “LC -SLOPE” was developed to calculate gradKs = (0 ,0 ,0) analytic solution .This program “LC - SLOPE” does not distin-guish between homogeneous slope and stratified slope and has no theoretical error .In order to get a result which is easy for the engineering staff to use and accurate ,for homogeneous slope ,it can be simplified as grad Ks = (0 ,0 ,0) by taylor formula and get the Ksmin ,(x0 ,y0 ) and R .