AN IMPROVED REDUCED-ORDER NUMERICAL SUBSTRUCTURE METHOD BASED ON NEWTON ITERATIVE ALGORITHM
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摘要: 充分利用结构在地震作用下的局部非线性特征，数值子结构方法将原本复杂的结构非线性分析转化为以初始弹性刚度迭代的主结构等效线弹性分析和屈服构件隔离子结构非线性分析。由于主结构采用常刚度迭代分析收敛速度较慢，尚有一定局限性，于是该文提出一种改进的降阶牛顿迭代数值子结构方法。在主结构系统中，将塑性自由度位移场作为基本未知量，设计牛顿算法进行非线性迭代分析，并由隔离子结构跨平台非线性分析计算得到屈服单元的内力和切线刚度。对一平面15层3跨钢结构进行地震弹塑性时程分析，模拟结果表明：该文提出的方法是准确、可靠的，接近传统牛顿算法的二次收敛，且对于局部非线性结构系统，需要集成和分解的矩阵规模远小于传统方法。Abstract: Making use of the local nonlinearities of the structures under earthquakes, the numerical substructure method (NSM) transforms the original complex structural nonlinear analysis into the equivalent linear elastic analysis of a master structure based on a fixed-point iterative algorithm and nonlinear analyses of isolated substructures for yield components. However, the NSM still has a limitation due to the low convergence speed for the master structure based on the initial elastic stiffness. In this study, an improved reduced-order NSM based on a Newton algorithm is presented. In the master structure, the displacements of nonlinear degrees of freedom are taken as unknown quantities, and the nonlinear analysis process using the Newton algorithm is conducted, in which resisting forces and tangent stiffnesses of nonlinear elements are obtained from isolated substructures. Seismic elastoplastic time-history analyses of a plane 15-storey three-bay steel structure are carried out. The numerical analysis results show that: the present method is accurate and efficient, closing to the second-order convergence of the traditional Newton algorithm, and only needs to form and decompose a much smaller matrix than that in the traditional algorithm for a local nonlinear system.
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图 2 不动点迭代数值子结构方法迭代次数时程曲线
Figure 2. Iteration number time-history curves using the numerical substructure method based on fixed-point iteration
图 3 关键构件隔离子结构精细化模型边界处理
Figure 3. Boundary treatment of isolated refined substructures for key components
图 6 8度罕遇地震钢结构顶层位移时程曲线
Figure 6. Time-history curves of top displacement for the steel structure under 8 degree rare earthquakes
图 7 9度罕遇地震钢结构顶层位移时程曲线
Figure 7. Time-history curves of top displacement for the steel structure under 9 degree rare earthquakes
图 8 8度罕遇地震钢结构基底剪力时程曲线
Figure 8. Time-history curves of base reaction for the steel structure under 8 degree rare earthquakes
图 9 9度罕遇地震钢结构基底剪力时程曲线
Figure 9. Time-history curves of base reaction for the steel structure under 9 degree rare earthquakes
图 10 8度罕遇地震迭代步数时程曲线
Figure 10. Iteration number time-history curves under an 8-degree rare earthquake
图 11 9度罕遇地震迭代步数时程曲线
Figure 11. Iteration number time-history curves under a 9-degree rare earthquake
表 1 不动点迭代数值子结构方法迭代信息统计表
Table 1. Table of iteration information using the numerical substructure method based on fixed-point iteration
工况 收敛误差/mm 最大收敛次数 总迭代次数 平均迭代次数 8度罕遇 1.0×10-5 24 12257 4.6 1.0×10-8 41 18882 7.1 9度罕遇 1.0×10-5 26 12828 6.9 1.0×10-8 44 18735 11.1 
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表 2 传统方法与本文方法的误差
Table 2. Errors using the traditional and proposed methods
工况 最大响应 传统方法 本文方法 相对误差/(%) 8度罕遇 顶层位移/mm 292.7 293.2 0.2 基底剪力/kN 2424.9 2429.1 0.2 9度罕遇 顶层位移/mm 490.2 490.4 0.1 基底剪力/kN 2970.2 2971.9 0.1
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表 3 不同方法迭代信息统计表
Table 3. Table of iteration information using various methods
工况 收敛误差/
mm算法 最大收敛
次数总迭代
次数平均迭代
次数8度罕遇 1.0×10−5 传统方法 4 7282 2.7 原方法[10-13] 24 12 257 4.6 本文方法 6 8409 3.2 1.0×10−8 传统方法 5 8610 3.2 原方法[10-13] 41 18 882 7.1 本文方法 6 10 157 3.9 9度罕遇 1.0×10−5 传统方法 5 8267 3.1 原方法[10-13] 26 12 828 6.9 本文方法 7 9743 3.7 1.0×10−8 传统方法 6 9437 3.5 原方法[10-13] 44 18 735 11.1 本文方法 7 11 340 4.4
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表 4 传统方法与本文方法的计算信息统计表
Table 4. Computation information using the traditional and proposed methods
工况 算法 K集成及
分解次数Hs集成及
分解次数总耗时/min 8度罕遇 传统方法 7282 − 313 本文方法 1 8409 157 9度罕遇 传统方法 8267 − 398 本文方法 1 9743 332
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