Non-Linear Analysis of Three-Segment Stiffness Compressive Bar Based on Direct Stiffness Method
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摘要: 为分析考虑二阶效应的分段刚度压杆内力及位移，根据位移控制方程，建立了变刚度压杆位移和转角方程；根据杆端位移边界条件和变刚度截面处连续条件，得到了位移系数；根据压杆内力方程，建立了以矩阵形式表达的刚度平衡方程，变换得到了变刚度压杆刚度矩阵模型. 将本模型用于分段刚度压杆分岔失稳临界荷载计算，并与解析解、插值形函数单元模型结果进行对比与分析，验证了模型的精度和效率. 结果表明：采用插值形函数法计算压杆临界荷载时，若只划分一个单元，其计算结果与理论解的相对误差最高可达43.24%，随着划分单元数量增加，相对误差降为0.023%；采用基于直接刚度法得到的变刚度压杆单元刚度矩阵计算压杆临界荷载时，只需划分一个单元，即可保证计算结果与理论解一致，该矩阵可用于压杆的非线性分析中，得到压杆内力及位移的精确解.Abstract: To study the internal forces and displacement of a piecewise stiffness compressive bar with a consideration of the second-order effect, displacement and rotation equations were established based on the displacement governing equation. The displacement coefficients were obtained according to the displacement boundary condition at the ends of the bar and the continuity condition at the variable-stiffness section. The stiffness equilibrium equation in the matrix was established in line with the equations of internal forces. After matrix transformation, a stiffness matrix model for a piecewise stiffness compressive bar was developed. To validate the precision and efficiency of the developed model, the critical load for piecewise stiffness compression was calculated by using the proposed model and the interpolation shape function method. The maximum relative error between the results calculated from the interpolation shape function method and the theoretical solution is 43.24% for one element; as the number of elements increased, the relative error decrease to 0.023%. The results calculated by the stiffness matrix model are consistent with the analytical solutions for a single element. The proposed stiffness matrix model can be applied in non-linear analysis to obtain analytical solutions of internal forces and displacement.
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图 1 分段刚度压杆局部坐标系及参数正方向
Figure 1. Local coordinate system and parameter positive direction of segmented stiffness compressive bar
表 1 压杆临界荷载计算结果对比
Table 1. Comparison of calculated buckling loads of compressive bar
构件类型 理论解/MN 直接刚度法 整体插值形函数法 临界荷载/MN 误差/% 临界荷载/MN 误差/% 简支压杆 5.522 6 5.522 6 0 7.692 7 39.29 悬臂压杆 0.858 3 0.858 3 0 1.229 5 43.24 一端固支一端铰支杆 11.990 6 11.990 6 0 17.148 8 43.02 一端固支一端滑动杆 3.850 3 3.850 3 0 4.168 6 8.27 
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表 2 不同单元数量下压杆临界荷载对比
Table 2. Buckling load of bar with different numbers of element
约束情况 理论解 /MN 直接刚度法 经典插值形函数法 单元数/个 临界荷载/MN 误差/% 单元数/个 临界荷载/MN 误差/% 简支压杆 5.522 6 1 5.522 6 0 6 5.522 9 0.005 悬臂压杆 0.858 3 1 0.858 3 0 9 0.858 1 0.023 一端固支一端铰支杆 11.990 0 1 11.990 0 0 9 11.993 0 0.020 一端固支一端滑动杆 3.850 3 1 3.850 3 0 9 3.850 7 0.010
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