一种有效的分析任意空间形状曲杆单元的位移函数

A SET OF EFFICIENT DISPLACEMENT FUNCTIONS FOR ARBTRARILY SPATIAL CURVED ROD ELEMENTS

利用经典弹性理论和微分几何、矩阵方法等数学理论,基于空间自然坐标系和随体坐标系,通过求解应变与位移之间关系的微分方程,得到了一种能完全反映任意空间形状圆截面曲杆单元刚体位移和常应变等模式的位移函数。给出了两个算例,通过将采用导出的位移函数建立的有限元解与解析解进行了比较,验证了它的正确性,同时通过将基于位移函数导出的有限元解与应用软件得到的解进行了比较,其计算精度,特别是应力计算精度大为改善,验证了它的有效性。由于计算效率高,提出的曲杆单元可望在三维大曲率井的钻柱非线性分析等工程实际中发挥作用。

For arbitrarily spatial elastic curved rod elements with circular cross-section, a set of displacement functions fully reflecting the rigid body modes is derived using the classical elasticity theory and mathematic theories of the differential geometry and matrix methods. Both natural (curvilinear) and intrinsic (Lagrangian) coordinate systems are used in the derivation. The displacement functions involve all rigid body and constant strain modes of the arbitrarily spatial elastic curved rods. To verify the formulation, two examples are analyzed using the curved rod element based on the displacement functions derived herein for a static situation. Numerical results are well compared with theoretical solutions. The convergence rate of the element based on the proposed displacement functions is better than that of the element in commercial codes. Due to its higher computational efficiency, the proposed curved element may find its practical use in the non-linear analysis of drill-strings confined in various three-dimensional curved wells.