弹性曲梁几何非线性精确模型及其数值解

EXACT GEOMETRICALLY NONLINEAR MATHEMATICAL FORMULATION AND NUMERICAL SIMULATION OF CURVED ELASTIC BEAMS

基于直法线假设,采用轴线可伸长梁的几何非线性理论,建立了弹性曲梁在任意荷载(保守和非保守)作用下的静态大变形数学模型。其中包含了轴线弧长、轴线位移、横截面转角、内力等七个独立未知函数。通过引进变形后的弧长为未知函数,使得问题的求解区间为未变形梁的轴线长度。该模型不仅考虑了轴线伸长,同时精确地考虑了梁的初始曲率对变形的影响以及轴向变形与弯曲变形之间的相互耦合效应。作为应用,采用打靶法计算了悬臂半圆形曲梁在沿轴线均布的切向随动载荷作用下的非线性平面弯曲问题,给出了随载荷参数大范围变化的平衡路径曲线及平衡构形。

Based on the assumption of straight normal line of beams and by employing geometrically nonlinear theory for axially extensible beams, governing equations of large static deformations of curved elastic beams subjected to arbitrarily distributed loads (conservative or non-conservative) are derived. The equations contain seven independent unknown functions such as the arc length, the displacements of the central line, the rotational angle and the resultant internal forces at a cross section. By introducing the deformed arc length as one of the unknown functions, it makes the range of the spatial variables of the problem still within the undeformed length of the beam. In the mathematical model, not only the effects of the axial elongation and the initial curvature of the curved beam on the deformation are accurately taken into account but also the coupling between elongation and bending is considered. As a numerical example, the nonlinear plane bending of a cantilever semicircle beam subjected to tangentially distributed follower force along the axial line is analyzed by a shooting method. The equilibrium paths and configurations of the deformed beams, varying with the load parameter in a large range, are presented.