考虑横向剪切变形矩形中厚板的后屈曲研究

Study on Post-Buckling of Moderately Thick Rectangular Plates Considering Transverse Shearing Deformation

基于考虑横向剪切变形中厚板的非线性几何方程、本构关系及非线性平衡方程,建立关于一个中面位移和两个中面转角为独立变量的中厚板后屈曲的非线性基本方程,从而获得中厚板后屈曲的非线性控制微分方程。中厚板后屈曲的非线性控制微分方程退化为薄板后屈曲的控制微分方程的正确性说明推导过程的正确性及一般性。中厚板后屈曲的非线性控制微分方程是一个六阶耦合微分方程,对其简支边条使用双重三角级数并作为广义坐标,将两个中面转角解耦为中面挠度的表达式,然后运用伽辽金方法进一步建立中厚板后屈曲的特征方程,从而得到简支中厚板后屈曲的平均应力表达式,最后应用MATLAB工具通过平均应力表达式获得矩形中厚板的后屈曲平衡路径曲线,同时给出了中厚板小挠度屈曲的临界荷载表达式及临界荷载数值表。整个求解过程简便,并且其曲线退化后符合经典的薄板后屈曲平衡路径曲线。

The nonlinear fundamental equations of moderately thick plates large deflection buckling concern three independent variables,i.e.one middle surface displacement and two middle surface intersection angles were established based on the nonlinear geometric equations,constitutive relations and nonlinear equilibrium equations of moderately thick plates considering transverse shearing deformation,and the nonlinear governing differential equations of moderately thick plates large deflection buckling can be obtained further.The equations degenerate to the governing differential equations of the thin plates large deflection buckling,demonstrate the validity and generality of the solving process.The nonlinear governing differential equations of moderately thick plates post-buckling are sixth-order coupled differential equations decoupled by using double trigonometric series as generalized coordinates,two middle surface intersection angles were decoupled into the functions of middle surface displacement,then the characteristic equation of moderately thick plates post-buckling was obtained by Galerkin method to get the average stress expression of moderately thick plates large deflection buckling.Finally,the curves of buckling paths were obtained by MATLAB through the average stress expression.The whole solving process is simplified and convenient,and the degenerative curves are in line with the classical curves of the thin plates large deflection buckling paths.