复杂条件下Winkler地基梁的解析解

Analytic solution of beams on Winkler foundation under complex conditions

基于梁的弯曲微分方程、有限元理论和最小势能原理(虚功原理),利用梁单元间的位移、转角、弯矩和剪力协调条件,地基梁的整体平衡条件,给出了复杂条件下Winkler地基梁的一种计算方法。此方法能对Winkler地基梁在受集中力、集中力偶和任意形式的分布荷载共同作用时进行解析解求解,而且地基梁的截面和弹性模量、地基的基床系数可以分段不同。此方法能得到各梁单元和整个地基梁的沉降位移表达式,且这一表达式与简单条件下的Winkler地基梁经典解有相同的结构形式;根据与沉降位移表达式之间的导数关系,可以进一步得到转角、弯矩和剪力表达式。算例表明,计算结果与一般有限元方法的计算结果完全吻合。另外,此方法虽然在概念上属于有限元方法的一种,但单元划分方法与一般有限元方法存在本质差别,单元个数也远远少于一般有限元方法的单元个数。

Based on the differential equation for curvature of beams, the theory of finite element and the principle of minimum potential energy, a method for beams on Winkler foundation under complex conditions was deduced with compatibility conditions of displacement, angular rotation, moment and shear of adjacent beam elements. The method could be used to compute beams on Winkler foundation under complex conditions, such as variable Young＇s modulus and section, complex foundation conditions, namely variable bedding modulus, and complex load consisting of concentrated force, moment and randomly distributed load. The displacement equations both for beam elements and for the whole beam, which had the same structure with the classic solution by Winkler, could be derived from the proposed method. Therefore, the equations for angular rotation, moment and shear of the beams could also be obtained from their differential relationship with displacement equation. It was shown by the computation of an example that the results obtaind by the proposed method were consistent with those by the classic finite element method when the elements were small enough. Furthermore, the element partition in the proposed method was thoroughly different from that in the classic finite element method.