Several methods to derive accurate Timoshenko beam finite elements are presented and compared. Two application problems are examined: linear elastostatics and linearized prebuckling (LPB) stability analysis. Accurate elements can be derived for both problems using a well known technique that long preceeds the Finite Element Method: using homogeneous solutions of the governing equations as shape functions. An interesting question is: can accurate elements be derived with simpler assumptions? In particular, can linear-linear interpolation of displacements and rotations with one-point integration reproduce those elements? The answers are: no if standard variational tools based on classical functionals are used, but yes if modified functionals are introduced. The connection of modified functionals to newer methods, in particular templates, modified differential equations and Finite Increment Calculus (FIC) are examined. The results brings closure to a 50-year conumdrum centered on this particular finite element model. In addition, the discovery of modified functionals provides motivation for extending these methods to full geometrically nonlinear analysis while still using inexpensive numerical integration.

提出并比较了几种推导准确的Timoshenko束有限元的方法。研究了两个应用问题：线性弹性静力学和线性预屈曲（LPB）稳定性分析。使用早于有限元方法的众所周知的技术，可以得出两个问题的精确元素：使用控制方程的齐次解作为形状函数。一个有趣的问题是：可以用更简单的假设得出精确的元素吗？特别是，通过一点积分对位移和旋转进行线性-线性插值是否可以重现这些元素？答案是：如果使用基于经典功能的标准变体工具，则为否，但如果引入了修改的功能，则为是。修改后的功能与较新方法（尤其是模板）的连接，研究了改进的微分方程和有限增量演算（FIC）。结果使以该特定有限元模型为中心的50年难题得以终结。另外，发现修改后的功能为将这些方法扩展到完整的几何非线性分析提供了动力，同时仍使用廉价的数值积分。