基于剪切变形的矩形梁剪力滞求解方法

Timoshenko梁通过假设截面的剪切刚度和附加平均剪切转角变形的方式来近似修正初等梁中未考虑剪切变形能的问题,这与梁剪应力沿梁高变化的实际不符。本文基于材料力学剪应力计算式和相应的剪切变形理论,从剪切变形与梁的位移关系入手,导出矩形梁考虑剪切变形时的纵向位移沿梁高方向的函数关系式,证明该位移可分解为纯弯曲引起的位移和剪力引起的剪力滞翘曲位移之和。应用剪力滞广义坐标与广义力的概念,基于能量变分原理得到等截面梁剪力滞控制微分方程组及其通解形式。对均布荷载作用下矩形简支梁的算例分析表明,本文算法与弹性力学精确解对比,两者的应力和挠度剪力滞系数求解结果非常接近,本文算法有足够的精度,且比弹性力学简单。

Calculation method for shear lag of rectangular section beam based on the shear deformation

By assuming shear stiffness and additional average deformation of shear angle of cross-section in order to revise approximatively the problem that ignoring the shear deformation energy in elementary beam method,which is the main method in Timoshenko beam theory.But in fact,the shear stress along the beam height is variable.This article is based on material mechanics calculation formula of the shear stress and the corresponding shear deformation theory .Starting from the relationship between the shear deformation and displacement to deduce the function formula of longitudinal displacement along the height variation of rectangular beam when considering shear deformation,which has proved that the displacement can be decomposed into pure bending displacement and shear-induced displacement caused by shear lag warping.Based on energy variational principle,we can get shear lag differential equations and its general solutions of uniform section beam by applying concepts of shear lag generalized coordinates and generalized forces.The analysis of simply supported rectangular beam under uniform loads shows that stress and deflection of the shear lag coefficient are very close to exact solution of Elasticity.This algorithm has sufficient accuracy and is simpler than the Elasticity.