弹性矩形悬臂中厚板弯曲问题积分变换解

Integral transform solution for bending problem of elastic rectangular cantilever moderately thick plates

利用二维有限域积分变换的方法推导出了矩形悬臂中厚板挠度的精确解.采用Mindlin三变量理论,直接对弹性矩形厚板控制方程进行二维有限域积分变换,将高阶偏微分方程组化为简单的线性方程组,从而在变换域内进行求解,然后进行相应的积分逆变换得到实际问题的精确解.其较叠加法、傅里叶级数法概念清晰,计算简便,而且在求解过程中不需要预先人为选取位移函数,仅用有限域积分变换的数学方法推导出了完全满足其边界条件的精确解,使得问题的求解更加合理,对于不同边界的矩形中厚板问题具有较好的通用性.最后通过计算实例验证了所采用方法及所推导公式的正确性.

The exact deflection solutions for rectangular cantilever moderately thick plates are derived by the double finite integral transform method.The governing equations are based on Mindlin theory including three variables.The double finite integral transform method is directly applied to the equations.The higher-order partial differential equations are reduced to linear simultaneous equations so that the problem is solved first in the transform domain.After that,the inverse transform is carried out to obtain the exact solutions for the problem.Compared with the superposition method and Fourier method,the current method is clear in concept and simple in calculation.Besides,there is no need to preselect the deformation function and only the mathematical method,the finite integral transform method,is used to derive the exact solutions satisfying all the boundary conditions.Therefore,the present solution is more reasonable and the method can be generally used for different boundary conditions.The results of the numerical examples validate the proposed method and the formulations.