挡土墙上被动土压力的变分求解方法

基于滑楔体整体极限平衡方程,根据变分法原理推导了被动土压力泛函极值的变分模型,并引入拉格朗日乘子,将等周变分模型转化为含有两个函数自变量的泛函极值模型。依据欧拉方程、边界条件和横截条件,得到了滑裂面函数和滑裂面上的应力函数,函数泛函极值模型转化为两个未知量的函数优化模型。算例表明,对于一般土体,在作用点位置系数下界限处,滑裂面呈现对数螺旋曲面,此时被动土压力最小;当作用点位置上移时,被动土压力呈非线性增长,在作用点位置系数上界限处,滑裂面为平面,被动土压力达到最大,与库仑土压力理论解完全一致,但作用点在墙体的相对位置并非在墙高的1/3处。结果表明,被动土压力大小和作用点位置受坡面的起伏和坡面超载的不均匀性影响比较明显。

Variational method for computation of passive earth on retaining 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 passive earth pressure is deduced based on the force equilibrium equations of the sliding mass. Then, the model can be transferred into a functional extreme-value problem with two undetermined functions by introducing Lagrange undetermined multiplier. According to Euler equations, Logarithmic spiral slip surface and normal stress distribution along the slip surface are obtained. Combined with the boundary conditions and transversality conditions, the conditional functional extremum problem of passive earth pressure involves searching the minimum of unconstrained optimizations of function with two unknown Lagrange multiplier. Results show that the passive earth pressure resultant force is minimal when the point of resultant force is on the lower bound and it increase nonlinearly as the point of resultant force moved up to the upper bound for general soil. Accordingly, the slip face evolves from logarithmic spiral face to plan. Although the magnitude of passive earth pressure reaches maximal that is the same with result calculated from Coulomb''s theory, the application point of earth pressure is not at 1/3 height of the retaining wall. In addition, curvilinear fill and uneven surface load have significant effect on both the magnitude and location of application point of passive earth pressure.