悬臂梁固定端不同位移边界条件下解的对比

Comparison of solutions from different displacement boundary conditions at fixed end of cantilever beams

为了获得不同的悬臂梁固定端位移边界处理方式对结果的影响,针对悬臂梁承受3种载荷的情况:自由端受切向力,上表面受均布载荷和线性分布载荷,给出悬臂梁固定端利用传统边界条件和最小二乘法处理边界时,Timoshenko梁理论、Levinson梁理论和弹性力学理论的解析解,与有限元计算结果对比.结果表明,Timoshenko梁理论采用传统位移边界和最小二乘法处理边界的结果一致,采用最小二乘法处理边界获得的Levinson梁理论和弹性力学理论的解明显优于传统位移确定方法,且这种优势随着载荷阶次的增加而越加明显.

To obtain the influence of different displacement boundary conditions for the fixed end on analyt- ical solutions of a cantilever beam, three load cases for a cantilever beam were investigated, which were a transverse shear force at the free end, a uniformly distributed load at the top surface, and a linearly dis- tributed load at the top surface, respectively. Analytical solutions were given for Levinson theory, Timo- shenko theory, and the elastic theory by using the conventional displacement boundary condition and the boundary condition through least squares method at the fixed end of the beam, and were compared with the solutions by finite element method. It is shown that the solutions from Timoshenko theory by using both the conventional displacement boundary condition and the condition through least squares method are the same; Levinson theory and the elastic theory by using the boundary condition through least squares meth- od provide better results than those by using the conventional boundary condition. With an increase in the order of the load, the superiority becomes more and more obvious.