一般条件下主动土压力的变分求解

基于库仑土压力理论的假设，主动土压力是由墙后填土在极限平衡状态下出现的滑动体产生，从墙后滑动体整体静力平衡方程出发，推导出坡面起伏且有不均匀超载、倾斜墙背、黏性填土等一般情况下的主动土压力泛函极值的等周模型。在该基础上，引入拉格朗日乘子，将主动土压力问题转化为确定含有两个函数自变量的泛函极值问题。依据泛函取极值时必须满足的欧拉方程,得到了线性的滑面函数和沿滑面线性分布的法向应力函数。结合边界条件和横截条件，主动土压力泛函极值问题进一步转化为单个未知量的一维方程问题。通过算例，土压力计算结果与库仑土压力理论解完全一致，但土压力作用点在墙体的相对位置却并非总是作用在墙高的1/3 处。通过算例进一步表明，坡面的起伏和坡面超载的不均匀性对主动土压力大小和作用点位置有显著的影响。

A variational method for computing of active earth pressure under general conditions

On the basis of Coulomb’s concept that the active earth pressure against the back of a retaining wall is due to the thrust force exerted by a sliding wedge of soil between the back of the wall. Considering an inclined rough retaining wall under the general conditions such as curvilinear fill, cohesive soil and uneven surface load, the functional extreme-value isoperimetric model about active earth pressure is deduced based on the force equilibrium equations of the sliding mass. Then the problem of active earth pressure is transcribed as the functional extreme-value problem of two undetermined functions by means of Lagrange undetermined multiplier. According to Euler equations, linear sliding surface function and linear normal stress function are obtained. Combined with the boundary conditions and transversality conditions, the problem of active earth pressure turned to be the solution of a one-dimensional equation. The magnitude of active earth pressure calculated by variational method is the same as that of Coulomb’s theory; and the location of application point of earth pressure is not always at 1/3 height of the retaining wall by case studies. In addition, curvilinear fill and uneven surface load also have significant effect on the magnitude and the location of application point of active earth pressure.