基于辛体系的正交各向异性地基梁解析解
Symplectic system for analytical solutions of orthotropic foundation beam
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摘要: 基于二维弹性理论, 利用Hellinger-Reissner变分原理, 通过引入对偶变量, 推导了双参数地基上正交各向异性梁平面应力问题的辛对偶方程组; 采用分离变量法和本征展开方法, 将原问题归结为求解零本征值本征解和非零本征值本征解, 得到了适用于任意横纵比的梁的解析解. 由于在求解过程中不需要事先人为地选取试函数, 而是从梁的基本方程出发, 直接利用数学方法求出问题的解, 使得问题的求解更加合理化. 其中, 地基对梁的力学行为的影响看作是侧边边界条件, 类似于外载, 可通过零本征解的线性展开来评价, 非零本征值本征解对应圣维南原理覆盖的部分. 还利用哈密顿变分原理, 给出了两端固支梁的一种新的改进边界条件. 编程计算了细梁和深梁等算例, 研究了地基上梁的变形沿着厚度方向的变化特性, 验证了辛方法的有效性.Abstract: Based on the two-dimensional theory of elasticity (2DET),Hamiltonian system is introduced to solve the bending of orthotropic BeamsResting on Pasternak Elastic Foundations and the original problems come downto solve the eigensolutions of zero eigenvalue and non-zeroes eigenvalue.Elastic foundation is treated as the side boundary conditions similar to theapplied load and their contributions to the solutions of beams areapproximated by linear expansion of all eigensolutions of zero. Thesymplectic concept makes no hypothesis of deformation along the thicknessdirection and shows a rational derivation. Thus, the current method canprecisely analyze foundation beams with arbitrary depth-to-length ratio, andcan deal with arbitrary end conditions. In additional, a new improvedboundary condition for fixed ends beam is presented. Numerical examples incomparison with other methods are given to illustrate the accuracy of thepresent symplectic approach.