机械和热载荷共同作用下梁的非线性弯曲和稳定性

基于Euler-Bernoulli梁的精确的几何非线性理论,建立弹性直梁在横向分布机械载荷和均匀升温共同作用下的几何非线性静平衡控制方程.应用打靶法分别数值求解两端不可移简支和两端固定支承梁在上述复合载荷作用下的非线性弯曲和弹性稳定性问题,给出梁的变形与机械载荷和热载荷之间的特性关系曲线,讨论载荷参数对变形和内力的影响.数值结果表明:当无机械载荷作用时,所得的曲线为热过屈曲平衡路径;当同时作用机械载荷和热载荷时,所得的曲线为非线性弯曲曲线,且随着温度载荷的增大,热膨胀所引起的变形逐渐占据主要地位,而机械载荷对变形的影响逐渐减弱.在热过屈曲前,轴力大小单调线性增加,而在热过屈曲后,轴力的大小随升温有微小减小.

Nonlinear bending and stability of beams subjected to mechanical and thermal loads

Based on an exact geometrically nonlinear theory of Euler-Bernoulli beams,the governing equation of geometrically nonlinear static equilibrium of an elastic straight beam under combined action of both transversely distributed mechanical load and uniform thermal rising load was set up. By using shooting method,the problems of nonlinear bending and elastic stability of the beam with both pined-pined and fixed-fixed ends were numerically solved,respectively.The characteristic curves of the beam deformation versus the mechanical loads and the thermal loads were presented and the effects of load parameters on the deformation and inner forces were also discussed.The numerical result indicated that the variation curve obtained in the case of absence of mechanical load was a thermal post-buckled equilibrium path,and it was a non-linear bending curve when there were both the mechanical load and the thermal load.Meantime,as the thermal rising load increasing,the deformation caused by thermal expansion took gradually a major position while the influence of mechanical load on the deformation decreased gradually.Before thermal buckling,the axial force linearly and monotonically increased.But after that,its magnitude decreased a little with temperature rising.