解析型Timoshenko梁有限单元

为提高深梁结构内力及变形的计算精度和效率，以Timoshenko梁理论为基础，建立了深梁位移控制方程，进而构造了深梁挠度、截面弯曲转角和剪切角的解析位移形函数. 采用势能原理建立了深梁的势能泛函，利用势能变分原理得到了解析型单元列式，进而给出了解析型单元总刚度矩阵，将其与理论解、插值多项式深梁单元进行对比分析. 结果表明:构造的解析型单元只需划分为一个单元即可保证计算的深梁挠度和转角与理论解一致，采用插值多项式单元确定的挠度和转角与理论解的相对误差最大可达到19.785%. 同时，为验证剪切变形对深梁位移影响，将构造的单元与Euler梁单元的计算结果进行对比. 对比表明:对于承受均布荷载作用的悬臂梁，基于Euler梁计算的位移与基于Timoshenko梁理论构造的解析型单元计算的位移偏差可达到50%；对于承受端部集中弯矩作用的简支梁，基于Euler梁计算的位移与基于Timoshenko梁理论构造的解析型单元计算的位移偏差可达到10.769%. 本文构造的单元满足了高精度、高效率的要求；该解析型梁单元可适用于浅梁分析，且不存在剪切闭锁的问题. 

Analytical Finite Element for Timoshenko Beams

To improve the calculation accuracy and efficiency of structural force and deformation of deep beams, the deflection control equation of a deep beam was built by Timoshenko beam theory, and analytical displacement shape functions for deflection, section flexural angle and shear angle of deep beam were constructed. Then, potential energy functions for the beam model were established using the potential energy principle; analytical element formulations for beams and the total element stiffness matrix were obtained via the variational principle of potential energy stationary value. Finally, the proposed analytical finite element method was applied to calculate the end deflections of a cantilever deep beam and a simply supported deep beam; and the calculation results were compared with those by theoretical solution and interpolation polynomial method. The results show that the solutions of end deflection and rotation obtained from the proposed analytical element by one element number is in accordance with the theoretical solutions; the maximum relative error between the results calculated from interpolation shape function method and the theoretical solution is 19.785%. To verify the influence of shear deformation on the deflection, the proposed element was also compared with the Euler beam element. The comparison results show that, for cantilever beams subjected to distributed load, the relative error between the results calculated from the Euler beam theory and the proposed element derived by the Timoshenko beam theory is 50%. For simply supported beams subjected to a concentrated bending moment at the end, the relative error is 10.769%. It is proved that the proposed analytical beam element can satisfy the high accuracy and efficiency requirement and avoid shear locking problems.