AN IMPROVED REDUCED-ORDER NUMERICAL SUBSTRUCTURE METHOD BASED ON NEWTON ITERATIVE ALGORITHM
-
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.
-
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 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 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 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 -
[1] Clough R W, Wilson E L. Dynamic analysis of large structural systems with local nonlinearities [J]. Computer Methods in Applied Mechanics and Engineering, 1979, 17(79): 107 − 129. [2] Wong K K, Yang R. Inelastic dynamic response of structures using force analogy method [J]. Journal of Engineering Mechanics, 1999, 125(10): 1190 − 1199. doi: 10.1061/(ASCE)0733-9399(1999)125:10(1190) [3] Li G, Yu D. Efficient inelasticity-separated finite-element method for material nonlinearity analysis [J]. Journal of Engineering Mechanics, 2018, 144(4): 4018008. doi: 10.1061/(ASCE)EM.1943-7889.0001426 [4] 李钢, 吕志超, 余丁浩. 隔离非线性分层壳有限单元法[J]. 工程力学, 2020, 37(3): 18 − 27. doi: 10.6052/j.issn.1000-4750.2019.01.0189Li Gang, Lü Zhichao, Yu Dinghao. The finite element model for inelasticity-separated multi-layer shell [J]. Engineering Mechanics, 2020, 37(3): 18 − 27. (in Chinese) doi: 10.6052/j.issn.1000-4750.2019.01.0189 [5] 苏璞, 李钢, 余丁浩. 基于子结构的Woodbury非线性分析方法[J]. 工程力学, 2020, 37(5): 26 − 35. doi: 10.6052/j.issn.1000-4750.2019.07.0419Su Pu, Li Gang, Yu Dinghao. A Woodbury nonlinear analysis approach based on the substructure method [J]. Engineering Mechanics, 2020, 37(5): 26 − 35. (in Chinese) doi: 10.6052/j.issn.1000-4750.2019.07.0419 [6] 陆新征, 林旭川, 叶列平. 多尺度有限元建模方法及其应用[J]. 华中科技大学学报(城市科学版), 2008, 25(4): 76 − 80.Lu Xinzheng, Lin Xuchuan, Ye Lieping. Multiscale finite element modeling and its application in structural analysis [J]. Journal of Huazhong University of Science and Technology (Urban Science Edition), 2008, 25(4): 76 − 80. (in Chinese) [7] Sun B, Li Z. Multi-scale modeling and trans-level simulation from material meso-damage to structural failure of reinforced concrete frame structures under seismic loading [J]. Journal of Computational Science, 2016, 12: 38 − 50. doi: 10.1016/j.jocs.2015.11.003 [8] Sun B, Huang X, Zheng Y, et al. Multi-scale lattice method for mesoscopic crack growth simulation of concrete structures [J]. Theoretical and Applied Fracture Mechanics, 2020, 106: 102475. doi: 10.1016/j.tafmec.2020.102475 [9] 陈宇, 李忠献, 李宁. 钢筋混凝土柱地震破坏分析的多尺度建模方法[J]. 工程力学, 2016, 33(6): 46 − 53. doi: 10.6052/j.issn.1000-4750.2014.09.0801Chen Yu, Li Zhongxian, Li Ning. Multi-scale modeling for seismic failure analysis of reinforced concrete columns [J]. Chinese Journal of Computational Mechanics, 2016, 33(6): 46 − 53. (in Chinese) doi: 10.6052/j.issn.1000-4750.2014.09.0801 [10] 孙宝印, 古泉, 张沛洲, 等. 钢筋混凝土框架结构弹塑性数值子结构分析方法[J]. 工程力学, 2016, 33(5): 44 − 49. doi: 10.6052/j.issn.1000-4750.2015.07.ST08Sun Baoyin, Gu Quan, Zhang Peizhou, et al. Elastoplastic numerical substructure method of reinforced concrete frame structures [J]. Engineering Mechanics, 2016, 33(5): 44 − 49. (in Chinese) doi: 10.6052/j.issn.1000-4750.2015.07.ST08 [11] Sun B, Gu Q, Zhang P, et al. A practical numerical substructure method for seismic nonlinear analysis of tall building structures [J]. The Structural Design of Tall and Special Buildings, 2017, 26(16): e1377. [12] 张沛洲, 孙宝印, 古泉, 等. 基于数值子结构方法的低延性RC框架结构抗震性能精细化分析[J]. 工程力学, 2017, 34(增刊): 38 − 48. doi: 10.6052/j.issn.1000-4750.2016.04.S018Zhang Peizhou, Sun Baoyin, Gu Quan, et al. Refined aseismic analysis of low-ductility RC frame building based on numerical substructure method [J]. Engineering Mechanics, 2017, 34(Suppl): 38 − 48. (in Chinese) doi: 10.6052/j.issn.1000-4750.2016.04.S018 [13] 孙宝印, 古泉, 张沛洲, 等. 考虑P-Δ效应的框架结构弹塑性数值子结构分析[J]. 工程力学, 2018, 35(2): 153 − 159. doi: 10.6052/j.issn.1000-4750.2016.10.0773Sun Baoyin, Gu Quan, Zhang Peizhou, et al. Elastoplastic numerical substructure method for frame structure considering P-Δ effect [J]. Engineering Mechanics, 2018, 35(2): 153 − 159. (in Chinese) doi: 10.6052/j.issn.1000-4750.2016.10.0773 [14] 孙爱伏, 欧进萍. 高层钢结构地震失效模式控制的失效路径修正法[J]. 地震工程与工程振动, 2010, 30(1): 22 − 28.Sun Aifu, Ou Jinping. Seismic failure mode control of tall steel buildings by changing the failure path [J]. Journal of Earthquake Engineering and Engineering Vibration, 2010, 30(1): 22 − 28. (in Chinese) [15] 孙宝印, 张沛洲, 古泉, 等. 基于数值子结构方法的结构弹塑性分析[J]. 计算力学学报, 2015, 32(4): 465 − 472. doi: 10.7511/jslx201504004Sun Baoyin, Zhang Peizhou, Gu Quan, et al. Numerical substructure method for nonlinear structural analysis [J]. Chinese Journal of Computational Mechanics, 2015, 32(4): 465 − 472. (in Chinese) doi: 10.7511/jslx201504004 -