线性分布基床系数弹性地基梁有限单元法改进

弹性地基梁法常用于研究土和结构的相互作用，对于均布荷载和边界条件简单的弹地基梁，采用理论解即可方便地进行计算。侧向荷载作用下桩体、嵌入式挡墙一般根据弹性地基梁理论进行分析，并假定基床系数随深度增加。对于基床系数呈线性分布或呈均匀分布但边界条件复杂的弹性地基梁理论求解困难，通常采用有限差分法或有限单元法近似求解。采用有限单元法计算线性分布基床系数弹性地基梁时，若单元划分数量不够，就存在计算精度不足的问题。采用加权余量法推导了更为精确的2节点5次位移函数和相应的单刚矩阵，得出了线性分布荷载作用下挠度的5次多项式近似解，从而实现只需划分很少的单元数，节点位移及单元内位移的分布即可达到较高的计算精度，极大地提高了计算效率，单元内力的分布可直接由位移函数导出，简化了后处理计算程序。

Improvement of finite element method for beam on elastic foundation with linearly distributed coefficient of subgrade reaction

The elastic foundation beam method is usually used to analyze the soil-structure interaction. For elastic foundation beams under uniform loads and simple boundary conditions, their theoretical solutions can be obtained from the previous research. Piles under lateral loads and embedded retaining walls are generally analyzed according to the theory of beam on elastic foundation and their coefficients of subgrade reaction are assumed to increase with the increase of the depth. For elastic foundation beams with linearly distributed coefficient of subgrade reaction or with uniform coefficient of subgrade reaction but complex boundary conditions, it is difficult to obtain their theory solutions. In this case, finite difference method or finite element method is used to obtain their approximate solution. When the beam on elastic foundation with linearly distributed coefficient of subgrade reaction is analyzed by finite element method, the calculation accuracy is not ideal if the number of elements are not enough. A 5 times polynomial displacement function and corresponding stiffness matrix of beam element with 2 nodes are presented by adopting weighted residual method. An approximate solution of the deflection of beams under linear distribution of the force is expressed with quintic polynomial. And then more precise nodal displacements and the distribution of element displacements are achieved by the fewer elements. Therefore, the computational efficiency of the presented method is improved. Furthermore, the postprocessing is simplified because the internal force distributions of element can be derived from the displacement function.