边坡滑动面构造的数学函数

在边坡稳定分析中，滑动面的自由度越小，搜索效率越高，但解的空间也变小了，可能难以覆盖真正的临界滑动面；增加顶点、增大自由度会扩大解的空间，但它会降低搜索效率，而且也不利于找到真实解。因此，如何构造滑动面是边坡稳定分析的关键问题之一。该文对滑动面的常用构造方法进行综述，在此基础上提出了构造滑动面的新的方法。包括基于极坐标的方法和基于logistic函数的方法，并且引入了参数方程。在此基础上，给出了滑动面的综合方程。从而把滑动面构造的几何问题转换为已知函数的参数选择问题，有望减少滑动面的自由度以及缩小相关参数的选择区间。

Mathematical functions for generating slip surfaces of slopes

In slope stability analysis, the efficiency of searching a critical slip surface increases as the degree of freedom of the solution space decreases, but the narrowed solution space may not be able to cover all the potentially true slip surfaces. In contrast, an increase in the degree of freedom by adding new vertexes of the surface curve can expand the solution space but lower searching efficiency, also undesirable for finding the true solution. Therefore, how to generate slip surfaces is a key question in slope stability analysis. Based on reviewing the traditional methods, this paper presents a new measure for generation of slip surfaces using polar coordinates, a logistic function, and parametric equations. And a comprehensive function for slip surfaces is constructed to simplify slip surface fitting. This measure converts the geometric problem of slip surface generation to a problem of determining the parameters of a known function. Thus it would help reduce the degrees of freedom of the slip surface spaces and narrow the searching ranges of the parameters.