解析型Winkler弹性地基梁单元构造

ELEMENT FOR BEAM ON WINKLER ELASTIC FOUNDATION BASED ON ANALYTICAL TRIAL FUNCTIONS

该文采用Winkler弹性地基梁理论确定了弹性地基梁的挠度方程解析通解;根据最小势能原理建立了解析型Winkler弹性地基欧拉梁及铁摩辛柯梁的单元刚度及等效节点荷载;得到了解析型弹性地基欧拉梁单元AWFB-E及铁摩辛柯梁单元AWFB-T。同时,论文还采用传统里兹法求得了相应的Winkler弹性地基欧拉梁及铁摩辛柯梁单元刚度矩阵,得到了里兹法弹性地基欧拉梁单元RWFB-E及铁摩辛柯梁单元RWFB-T。对该文构建的两类单元与一般梁-基体系有限元分析结果及理论解析解进行了对比。对比结果表明,传统里兹法由于其多项式形函数无法精确模拟弹性地基梁变形,因此其结果与理论解析解有误差,但随着单元数量增多其误差减小;采用解析型单元进行计算时,无论单元数量多少,得到的均为"真实"解,说明解析试函数法求得的位移形函数比一般的多项式形函数精确,得到的弹性地基梁单元具备解析型、精确性的特点,可应用于解决实际工程问题。

Based on Winkler elastic foundation beam theory, the general analytical solution of a deflection equation for an elastic foundation beam is deduced; the element matrixes and equivalent load arrays of Winkler elastic foundation Euler and Timoshenko beams based on analytical trial functions are established by applying the principle of minimum potential energy. The element stiffness matrixes of Winkler elastic foundation Euler and Timoshenko beams via Ritz method are also created. Through calculating, the results are obtained by two types of elements, and then are compared with results via general beam-foundation finite element model and analytical solutions. The comparisons show that there is a noticeable difference between the results by Ritz method and analytical solution due to that the polynomial shape function is unable to accurately simulate the actual deformation of an elastic foundation beam, but the difference becomes smaller with the increasing number of elements in the analysis of an elastic foundation beam; the results are identical with analytical solutions when applying analytical trial functions regardless of the number of elements, which means the displacement shape function based on an analytical trial function is much more accurate than a general polynomial shape function. It is proved that the Winkler elastic foundation beam element based on an analytical trial function can satisfy the accuracy and efficiency requirement. It would be applied in practice engineering.