变曲率曲梁的单元等效节点荷载分析

Equivalent Nodal Load Analysis of the Variable Curvature Curved Beam

研究目的:变曲率曲线梁在工程中得到广泛的应用,对其进行数值分析时,需建立变曲率曲线梁的单元刚度矩阵和节点荷载矩阵。因跨间集中荷载和均布荷载无法直接参与计算,需将其转换为等效节点荷载。本文主要研究目的是获得变曲率曲线梁跨中集中荷载的等效荷载矩阵。在极坐标系下,依据虚功原理,推导出一端固定的悬臂梁在跨间荷载作用下的悬臂端位移,再依据位移等效和内力等效得到曲梁单元一端的等效节点力,进而根据静力平衡条件得到单元另一端的等效节点力与等效节点荷载矩阵。研究结论:以两端固定的曲线梁在跨间集中荷载作用下为例,利用MATLAB编程与ANSYS有限元计算结果进行了对比,算例中竖向位移和扭转角误差较小,两者相差都在5%以内,得到如下结论:(1)等效节点荷载矩阵是正确的,验证了对变曲率曲线梁计算的有效性;(2)等效节点荷载矩阵的元素可用带参数的显函数表达,且所有参数都可直接引用;(3)推导的变曲率曲梁等效节点荷载矩阵可用于含等曲率梁的杆系结构的有限元分析。

Research purposes： Variable curvature curved beam is widely used in engineering practice,and it is necessary to investigate their element stiffness matrixes and nodal loads matrix in the numerical calculation. Because the concentrated loads and uniformly distributed loads cannot be directly involved in the calculation,it is necessary to obtain the equivalent nodal loads of concentrated loads and uniformly distributed loads which are between the beam units. In this paper,the main research purpose is obtained the equivalent nodal loads matrix. In polar coordinates,based on the principle of virtual work,it is achieved that the displacement of the cantilever end of the cantilever beam which is loaded between beam units. Based on displacement equivalent and internal force equivalent principle,the equivalent nodal force at the end of the curved beam element is obtained. Then according to conditions of static equilibrium,the equivalent nodal force of the other end of the unit and the equivalent nodal loads matrix are obtained.Research conclusions： Taking curved girder with two clamped ends which is loaded by concentrated load between beam and four-span continuous beam which is loaded by uniformly distributed load between beams for example,the calculation results by using MATLAB programme compare with ANSYS finite element calculation results. In the example,vertical displacement and torsion angle difference is within 5% and the error is smaller. The conclusion is as follows：（1） The correctness of the equivalent nodal loads matrix is verified and proves that it can be used in the calculation of variable curvature curved beam.（2） The elements in the matrix can be presented by explicit function withparameters and all the parameters can be directly referenced.（3） The equivalent nodal load matrix of variable curvature curved beam can be used for the finite element analysis of framed structure with equal curvature beam.