面内变形曲梁的显式单元刚度矩阵

Explicit Element Stiffness Matrix for Curved Beam with In-Plane Deformation

目的针对平面内变形的微圆段进行研究,给出等曲率梁的显式单元刚度矩阵,以便对含等曲率梁的结构进行分析.方法求解等曲率梁的微圆段平衡微分方程,获得等曲率梁的杆端力计算表达式.利用截面内力与应力关系、本构方程以及应变与位移关系,推导等曲率梁的任意截面内力与位移的关系,进而推导任意截面的位移与杆端位移之间的关系.根据等曲率梁的杆端力表达式及杆端位移表达式得到等曲率梁的杆端力与杆端位移的关系式.结果通过对平面内变形等曲率梁研究,给出了等曲率梁的任意截面处内力的计算公式以及杆端力表达式;得到了等曲率梁的任意截面处的内力与位移之间的关系式;给出了等曲率梁的任意截面处位移的计算表达式和杆端位移表达式;得到了等曲率梁的杆端力与杆端位移关系式.结论针对平面内变形的等曲率梁,笔者给出了一种解析的显式等曲率梁的单元刚度矩阵,该单元刚度矩阵可用于含等曲率梁的杆系结构的有限元分析.

The explicit element stiffness matrix of curved beam with constant curvature is derived by studying an infinitesimal circle element. This kind of element stiffness matrix can be easily used in the analysis of structure with curved beams in it. The equilibrium differential equations of the internal forces have been es- tablished and then the computational formulations of the forces of end points are given. Based on the rela- tionship between the internal forces and stresses, the constitutive equations and the strain-displacement rela- tionships, the internal force-displacement expressions for any cross-section are formulated and then the dis- placement relationships between any cross-section and the end points as well as the computational formula- tions of the displacements of end points are provided. The relationships between the end-point forces and the end-point displacements are given by using the computational formulations for the end-point forces and the end-point displacements mentioned above. Finally the analytic explicit expression of the element stiffness matrix is presented for constant curvature beam.