深梁弯曲解的一种经典化表示

Classical Expressions for Bending Solutions of Thick Beams

经典Euler-Bernoulli (E-B)梁理论简单好用,对于细长梁的变形预测具有良好的精度.但是,由于其中忽略了横向剪切变形影响,在预测深梁的变形时会有明显误差.基于精细化的梁理论,引入精确满足自由表面切应力为零的切应力形函数,将挠度分解为弯曲变形部分和剪切变形部分.采用Levinson高阶剪切变形梁理论,建立了矩形截面弹性梁静力弯曲位移形式的控制微分方程.利用载荷互等关系,推导出了用相同端部约束和载荷工况下E-B梁的挠度表示的Levinson梁的弯曲通解.从而实现了高阶剪切变形梁理论下弯曲解的经典化表示,即将弯-剪耦合问题的求解简化为两个无量纲系数的计算以及在具体边界条件下积分常数的确定,因为不同载荷工况和端部约束下E-B梁的解答在材料力学教科书中已经给出.通过两个算例,定量分析了梁的细长比和不同切应力形函数对挠度的影响程度.结果表明,不同的切应力形函数对应的弯曲挠度值差别很小.本文给出的解答可以方便地应用于深梁的静力弯曲分析.

The classical Euler-Bernoulli(E-B) beam theory is simple and easy to use, and has a good accuracy in predicting the deformation of thin beams. However, it will have a significant error in estimating the deformation of thick beams because it neglects the effects of transverse shear deformations on the bending solutions. Based on the refined beam theory, a general shear stress shape function is introduced, and the deflection of the beam is divided into the bending and shearing parts. Using the Levinson beam theory, the governing equations in terms of displacements for static bending of thick beams with rectangular cross sections are established. Then, by using the reciprocal relation of loads, the general solutions for the Levinson beams expressed by the deflections of the related E-B beams with the same loadings and end constraints are derived. As a result, the classical expression of solutions based on the higher-order shear deformation beam theories is realized, which simplifies the solution of complex bending-shearing coupling problem into the calculations of two dimensionless coefficients together with the determination of the integrating constants under the specified boundary conditions, since the solutions of E-B beam for different loadings and end constraints can be easily found in the text books of Strength of Materials. Two examples are given to quantitatively analyze the influence of the slenderness ratio and the shear stress shape function on the deflection. The results show that the deflection values corresponding to different shear stress shape functions have little difference from each other. Solutions presented in this paper can be conveniently applied to the static bending analysis of thick beams.