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从 Levinson 高阶梁理论的一致变分到高次翘曲梁理论
引用本文:徐翔,郝际平. 从 Levinson 高阶梁理论的一致变分到高次翘曲梁理论[J]. 工程力学, 2008, 25(2): 56-61
作者姓名:徐翔  郝际平
作者单位:西安建筑科技大学土木工程学院,西安,710055;西安建筑科技大学土木工程学院,西安,710055
基金项目:国家自然科学基金项目(50378078)
摘    要:
将矩形截面梁的截面翘曲位移设定为3次Legendre多项式的形式,利用弹性力学平面应力问题分项的不完全的广义变分原理,导出高次翘曲梁理论,得到形式简单易求解的方程。由于引入轴向拉伸的情况,使梁的平面内变形问题得以统一;计及了梁表面剪切荷载的作用,并严格满足表面剪应力边界条件;通过引入轴向位移约束参考点间距离的概念对梁端翘曲约束作更精致地描述,且使得该理论包含了变分一致或者不一致的高阶剪切梁理论。该理论的推导还表明,Levinson梁理论的变分不一致仅仅局限于有转角约束的梁端。通过算例,将高次翘曲梁理论与弹性力学平面应力问题以及Timoshenko梁理论、Levinson梁理论进行比较,初步显示出该理论的优越性。

关 键 词:梁理论  高阶理论  广义H-R变分原理  高次翘曲  Legendre 多项式
文章编号:1000-4750(2008)02-0056-06
收稿时间:2006-06-17
修稿时间:2006-10-22

FROM CONSISTENT VARIATION OF LEVINSoN'S HIGHER ORDER BEAM THEORY TO HIGHER ORDER WARPING BEAM THEORY
XU Xiang,HAO Ji-ping. FROM CONSISTENT VARIATION OF LEVINSoN'S HIGHER ORDER BEAM THEORY TO HIGHER ORDER WARPING BEAM THEORY[J]. Engineering Mechanics, 2008, 25(2): 56-61
Authors:XU Xiang  HAO Ji-ping
Abstract:
A new beam theory which includes shear deformation is presented. By assuming the warping displacement of the rectangular cross-section in the form of 3 order Legendre polynomial function, the governing differential equations, which are simple and can be solved easily, was derived using incomplete generalized Hellinger-Reissner variational principle of sub-item in the scope of elasticity theory for plane stress problems. In the new theory, the nonzero shear stress boundary conditions on the lateral surfaces of the beam are strictly satisfied, and it enables analysis of the beams in a unified manner that incorporates tensile force. With definition of a distance between reference points of warping constraint, the warping boundary conditions at the ends of the beam is well-defined, and the theory covering high order shearing beam theory being or not being consistent with variational sense is available. It is also shown that the variational inconsistence of Levinsion’s beam theory is only restricted at the ends of the beam with rotation constraint. Examples compare the accuracy of the new beam theory with that of elasticity theory, Timoshenko beam theory and Levinson’s theory. The comparison shows the merit of the new theory.
Keywords:beam theory  high order beam theory  generalized HR variational principle  higher order warping  Legendre polynomial
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