高效的三维曲梁单元

三维井眼中延伸数千米的三维细长圆截面钢钻柱应力分析问题是一个复杂的力学问题，通常使用有限元数值分析方法对其进行受力分析。而在进行有限元分析时，现有的圆弧曲粱单元和空间直粱单元在几何上都不能很好地模拟三维曲线形状的钻柱。为了确保计算精度．其单元划分势必不能过大，结果是计算时间长，收敛性差。为了解决这一问题，显然必须构建一种新的较有效的曲梁单元。基于自然坐标系，依据圆截面空间曲粱单元节点有6个自由度——3个线位移和3个角位移，利用包含全部刚体位移模式和常应变的形函数，忽略剪切变形，假设变形后的梁轴线的弯曲曲率改变为线性变化，建立起了保证收敛性的具有12个自由度的有初始曲率和挠率的圆截面空间曲梁的有限元模型。为了证明给出的有限元模型的高效性，分析了几个静态问题，并与现有文献中的解析解或数值结果进行了比较。基于所给出的结果，可望该有限元模型可以作为分析三维空间曲粱结构的有效工具。

An efficient finite element of spatial curved beams

Stress analysis of a drill-string, usually thousands meters long slender steel tube with circular cross-section, is a complex problem in a general three-dimensional well bore. Thus numerical method, such as commonly used finite element method, has to be resorted to. When carrying out finite element analysis, the element size of the existing beam elements should be small enough to capture the curve shape on geometry of drill string in three-dimensional well bore as well as to ensure the accuracy of the calculated results. This in turn results in considerably long computational time. A special three-dimensional curved and twisted beams with circular cross-section, which posses 12 degrees of freedom, is proposed herein. The key to success for the proposed element is that shape functions are obtained by ensuring all rigid body displacements and rotations as well as the constant strain state, thus the convergence is guaranteed. The natural (curvilinear) coordinate system is used in the derivations and detailed formulations are given. To verify the formulations, several examples are analyzed by using the proposed element for the static situation. Numerical results are compared well with existing theoretical and/or numerical data in the literature. Based on the results reported herein, one may conclude that the proposed curved element may be used for the analysis of three-dimensional curved beam structures.