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从滑动土体整体静力平衡方程出发,推导了坡面起伏,且有不均匀超载,墙背倾斜,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。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。通过算例表明,对于一般土体,在作用点位置系数下界限处,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,在作用点位置系数上界限处,主动土压力达到最大。在作用点位置系数上下界限处所对应的主动土压力构成的数值范围,包含了各种挡墙变位模式下的主动土压力。通过算例对比分析,采用库仑土压力理论进行的抗倾覆设计存在安全系数偏低的缺点,应该根据主动土压力和作用点位置的区间估计来指导设计。
Based on the overall static equilibrium equation of sliding soil, an isochron model of the functional extremes of active earth pressure under general conditions such as non-uniform overload, wall back inclination, and cohesive backfill is deduced. Based on this, the Lagrange multiplier is introduced to transform the active earth pressure problem into a functional extreme problem with two independent variables. According to the Euler equation which must be satisfied when the extremes of the functionals are taken, the function of the sliding surface of the logarithmic spiral and the normal stress distribution along the sliding surface are obtained. Combining boundary conditions and transversal conditions, the functional extreme value of active earth pressure is further transformed into a function optimization problem with two Lagrangian constants as unknowns. The example shows that for the general soil, the active earth pressure is the smallest and the slip plane is flat at the boundary of the position coefficient of the point of action. As the position of the action point moves up, the active earth pressure increases nonlinearly and the corresponding slip occurs. The surface is a logarithmic spiral and the active earth pressure reaches its maximum at the boundary of the position coefficient of the action point. The numerical range of the active earth pressure corresponding to the upper and lower limits of the position coefficient of the action point includes the active earth pressures in the various retaining modes of the retaining wall. According to the comparative analysis of examples, the design of the anti-overturn design based on the Coulomb earth pressure theory has the disadvantage of low safety factor. The design should be guided by the interval estimation of the active earth pressure and the point of action.