高阶剪切理论下梯度直梁在热环境中的静动态响应分析

The Static and Dynamic Response Analysis for Functionally Graded Materials Based on High Order Shear Under Heated

基于高阶剪切变形梁理论研究了两端不可移简支功能梯度梁在横向非均匀升温下的热屈曲和自由振动问题。首先依据高阶剪切变形梁理论和Hamilton原理建立了功能梯度梁受热-机载荷共同作用下的几何非线性动力学控制方程;在研究静态热屈曲问题时,把方程退化成强非线性边值问题,采用打靶法数值求解该边值问题,获得了横向非均匀升温下梁的屈曲构型,绘出了梁的变形随温度载荷及材料梯度参数变化的特征关系曲线;研究动态响应时,采用Navier方法数值求解所建立的动力学控制方程,获得了横向非均匀升温下梁的自由振动响应,数值比较了不同剪切理论下梁的前3解固有频率随跨高比、材料梯度参数变化的规律。结果表明,剪切变形、梁的跨高比、材料的非均匀性、温度变化对于高阶剪切功能梯度材料梁的变形及固有频率有很显著的影响。

On the basis of the high order shear deformation beam theory,thermal buckling and free vibration of functionally graded beams with two-point boundary conditions were studied. Firstly,based on the high order shear deformation beam theory and the Hamilton principle,the geometric nonlinear dynamic control equations of the functionally graded beam were established. For the static thermal buckling problem,the equation was reduced into a nonlinear boundary value problem;numerical solution of the boundary value problem were obtained by using the shooting method; transversely non-uniform temperature rise buckling beam characteristic curves were plotted; and the law for deformation of the beam changing with the parameters of temperature gradient loading and material were obtained. Studying on dynamic response,dynamic control equations were solved by using Navier numerical method,and the free vibration response of the beam subjected to transverse nonuniform heating is obtained. Numerical comparison of the natural frequencies of the top 3 solutions with the span height ratio and the variation of the gradient parameters of the beam under different shear theory were made. The results showed that the shear deformation of beam,non-homogeneous materials,temperature change have a significant influence on the shear deformation and natural frequency of the beam.