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从基坑柔性支护结构后的滑动土楔体的整体静力平衡方程出发,推导了考虑作用点位置的主动土压力泛函极值等周模型。在此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。对于一般黏性土,依据泛函取极值时必须满足的欧拉方程,得到了对数螺旋线的滑裂面函数和沿滑裂面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以两个拉格朗日常数为未知量的函数优化问题。同时,讨论了滑裂面为平面和圆弧面两种特殊情况。通过算例表明,对于一般土体,在作用点位置系数下限处,主动土压力最小,滑裂面为平面;随着作用点位置的上移,主动土压力呈非线性增长,相应滑裂面为对数螺旋面。对于砂性土,位置系数上限值随内摩擦角的增大而增大,其相应的土压力值也随之增加。对于软黏土,滑裂面为圆弧面,随着作用点位置的上移,主动土压力呈非线性下降,滑裂面背离基坑方向移动。
Based on the global static equilibrium equation of the sliding earth wedge after the flexible supporting structure of the foundation pit, the equivalent model of the active earth pressure extremum is deduced considering the position of the action point. On this basis, Lagrange multiplier is introduced to convert the active earth pressure problem into the functional extreme value problem with two function arguments. For general cohesive soils, the sliding surface function of logarithmic spiral and the normal stress function along the slip surface are obtained according to the Euler equation which must be satisfied when the extreme value of functional is satisfied. Combining the boundary conditions and the cross-section conditions, the active-earth pressure extremism problem is further transformed into a function optimization problem with two Lagrange constant as the unknown. In the meantime, two special cases of plane and arc are discussed. The results show that for the general soil, the active earth pressure is the smallest and the slip surface is plane at the lower limit of the position coefficient of the action point. The active earth pressure increases nonlinearly with the upward movement of the action point, and the corresponding slip surface Logarithmic spiral surface. For sandy soil, the upper limit value of position coefficient increases with the increase of internal friction angle, and the corresponding earth pressure value also increases. For soft clay, the slip surface is arc-shaped. With the upward movement of the point of action, the active earth pressure decreases nonlinearly and the slip surface moves away from the foundation pit.