Matrix Analysis of Variable Cross-Section Curved Beam Elements Based on Energy Method
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摘要:
为得到2节点6自由度显式变截面曲梁单元刚度矩阵的解析解公式，将变截面曲梁单元纳入常用杆系结构有限元系统. 基于卡氏第二定理推导变截面空间曲梁单元刚度矩阵，利用弹性体的势能驻值原理推导变截面曲梁单元柔度矩阵；通过逆运算得到变截面曲梁单元悬臂端刚度矩阵；根据静力平衡条件与虚功原理获得整体变截面曲梁单元刚度矩阵，且本文变截面曲梁单元有限元列式可退化为等截面等曲率梁单元列式及等截面直线梁单元标准列式；利用MATLAB编制变截面曲线梁桥的静力计算程序，并和ANSYS实体有限元模型对比，验证变截面曲梁静力分析理论. 研究结果表明：本文理论基于有限元理论分析了变截面曲梁的弯扭耦合，变截面曲梁理论与ANSYS实体有限元模型挠度最大误差为3.72%，与ANSYS梁单元模型最大误差不足0.5%；变截面曲梁理论退化后的变截面直梁与ANSYS实体有限元梁模型挠度最大误差为1.72%，与ANSYS梁单元模型最大误差不足0.1%.
Abstract:To obtain the analytical solution formula of the stiffness matrix of a 2-node 6-degree-of-freedom explicit variable cross-section curved beam element, the variable cross-section curved beam element was incorporated into the widely used finite element system of rod structures. Based on Castigliano’s second theorem, the stiffness matrix of the spatial variable cross-section curved beam element was derived, and the flexibility matrix of the variable cross-section curved beam element was deduced by using the principle of stationary potential energy of elastic bodies; then, the cantilever end stiffness matrix of the variable cross-section curved beam element was obtained through inverse calculation. According to the static equilibrium condition and the principle of virtual work, the element stiffness matrix of the overall variable cross-section curved beam was obtained. In addition, the finite element formulation of the variable cross-section curved beam element could be degenerated into the formulation of uniform cross-section and uniform curvature beam elements and the standard formulation of uniform cross-section straight beam elements. A static calculation program for variable cross-section curved beam bridges was developed using MATLAB, and it was compared with the ANSYS solid finite element model to verify the static analysis theory of variable cross-section curved beams. The results indicate that the developed theory analyzes the bending-torsion coupling of the variable cross-section curved beam based on the finite element theory. The maximum deflection error between the variable cross-section curved beam theory and the ANSYS solid finite element model is 3.72%, and the maximum error compared with the ANSYS beam element model is less than 0.5%; furthermore, the maximum deflection error between the variable cross-section straight beam degenerated from the developed theory and the ANSYS solid finite element beam model is 1.72%, and the maximum error compared with the ANSYS beam element model is less than 0.1%.
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图 1 变截面曲梁单元结构坐标系
Figure 1. Structural coordinate system of variable cross-section curved beam element
图 3 变截面直线连续梁三维实体模型
Figure 3. Three-dimensional solid model of variable cross-section straight continuous beam
图 8 变截面曲线连续梁三维实体模型
Figure 8. Three-dimensional solid model of variable cross-section curved continuous beam
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