Timoshenko-Euler楔形梁有限元

Tapered Timoshenko-Euler Beam Element

本文首先建立楔形梁包含轴力和剪切变形效应的平衡微分方程。由于该方程是二阶变系数微分方程,其解析解很难得到。本文通过将该方程中的变系数和方程的解用Chebyshev多项式逼近得到了Timoshenko-Euler楔形梁的单元刚度方程。最后通过算例检验了所得单元刚度方程的对称性,以及验证了计算悬臂梁挠度和悬臂柱弹性临界力的正确性及其收敛性。本文提出的方法可适用于任意变截面Timosnenko-Euler梁单元刚度方程的求解。运用此方法,除可以考虑轴力和剪切变形的影响外,还可以减少结构分析中的单元数和自由度,提高包含楔形构件的结构分析的精度和速度。

In this paper, the equilibrium differential equation including the effects of constant axial force and shear deformation was established for tapered beams. Since this equation is a second-order differential equation with variable coefficients, it is hard to find the closed soultion. To solve the problem, Chebyshev polynomial series was proposed for the solution to the equation. And the expression of the stiffness equation could be obtained for tapered Timoshenko-Euler beam elements. The symmetry of the stiffness matrix was checked, and the accuracy and convergence of the solution were verified by the numerical applications on predicting the deformations and elastic critical loads of tapered cantilever beams. The proposed strategy is suitable for any element with variable sections. The utility of this approach in structural analysis not only consider the effects of axial force and shear deformation, but also reduce the number of element and degree of freedom. In other words, it can improve the accuracy and speed of structural analysis comprising tapered members.