弹性地基上矩形加肋板自由振动分析的无网格法

Free vibration analysis of ribbed plates on elastic foundation with a meshfree method

应用移动最小二乘无网格法研究弹性地基上矩形加肋板的自由振动问题。假设弹性地基与加肋板紧密接触,以弹簧模拟弹性地基,将弹性地基上的加肋板视为板与肋条组合的结构。基于一阶剪切理论,用无网格伽辽金法推出了板和肋条各自的动能与势能;再通过位移协调条件将两者的能量叠加,得到了弹性地基上整个加肋板的动能与势能。由Hamilton原理导出了弹性地基上加肋板自由振动的控制方程。采用完全转换法引入边界条件,求解自由振动方程,并编制了计算程序,给出了算例。将算例与ABAQUS有限元解及已有文献结果进行了比较分析,其相对误差均在5%以内,验证了该方法计算弹性地基上矩形加肋板结构自振频率的有效性。

A moving least square meshfree method is introduced to study the free vibration of a rectangular ribbed plate which rests on elastic foundation. The Winkler foundation is assumed to closely contact with the ribbed plate, and the elastic foundation is simulated by springs. The ribbed plate on the elastic foundation is simulated as a composite structure of plate and ribs. Based on the first-order shear deformation theory, the moving least square approximation is used to derive the kinetic energy and potential energy of the plate and the ribs. By applying the displacement compatibility conditions, the superposition of the two kinds of energy is obtained, and the total kinetic energy and potential energy of the ribbed plate is derived. According to the Hamilton's principle, a governing equation that describes the free vibration of ribbed plates on Winkler foundation is deduced. The full transformation method is used to introduce the essential boundary conditions. And the corresponding computer program is compiled to solve the equation. The equation is solved, and corresponding computer program is compiled. Numerical examples are given to compare the solutions from the proposed method with those from ABAQUS and literatures, which proved the validity of the proposed method.