杆系结构位移及应力灵敏度分析的解析法研究
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摘要
结构优化设计中的灵敏度分析是指约束函数对优化设计变量的导数,它是最优准则法等许多结构优化算法能够高效迭代收敛计算的基础。本文首先阐述了有限差分法及基于虚功原理的简化法在约束函数灵敏度分析上的不足,然后从有限单元法求解的逆过程出发,根据线弹性分析中的空间梁单元刚度矩阵元素的组成特点,提出了基于通用有限元软件计算结果的空间梁单元杆端结点内力、结点位移及杆端应力等灵敏度分析的解析表达式,最后通过两个算例对本文提出的算法进行了验证。结果表明:利用本文所提出的相关灵敏度计算的解析表达,只需进行一次线弹性结构分析,并将有限元分析结果当作已知量,即可方便且准确地得到线性结构系统的位移及应力灵敏度。另外,本文利用大型通用有限元软件SAP2000基于MATLAB系统的应用程序接口(API)开发环境,实现了两款软件的数据交互及无缝链接进行灵敏度分析的高效平台,本文提出的方法适用于大型复杂结构的灵敏度分析并可为其他工程结构优化提供参考。
Sensitivity analysis in structural optimum design refers to the derivation calculation of design constraint function with the optimum design variables, it is the basis for efficient iterative convergence in Optimal Criterion(OC) and many other structural optimization methods. The deficiencies in efficiency and accuracy for Finite Difference(FD) method and simplified method based on virtual work principle are firstly described in this paper. The design sensitivity analysis of structural responses with respective to the design variables are investigated by adopting the finite element analysis results and considering the unique feature of the element stiffness matrix for the three dimensional beam in linear structural system. The analytical solution for design sensitivity of element nodal internal force, nodal displacement and stress for linear structural system are proposed in this paper by utilizing the finite element analysis results from SAP2000 software. Finally the accuracy and effectiveness of the proposed formula are verified by two examples. The comparing results show that the proposed analytical formulas for design sensitivity are both accurate and efficient since it could reduce the numerical analysis burden by adopting only one round of finite element analysis results for design displacement and stress sensitivity analysis. Furthermore an integrated design sensitivity analysis system is proposed by combining the SAP2000 Application Program Interface(API) for MATLAB environment. It could be used as a good reference for structural design sensitivity analysis for other large complex linear structural structures.
Sensitivity analysis in structural optimum design refers to the derivation calculation of design constraint function with the optimum design variables, it is the basis for efficient iterative convergence in Optimal Criterion(OC) and many other structural optimization methods. The deficiencies in efficiency and accuracy for Finite Difference(FD) method and simplified method based on virtual work principle are firstly described in this paper. The design sensitivity analysis of structural responses with respective to the design variables are investigated by adopting the finite element analysis results and considering the unique feature of the element stiffness matrix for the three dimensional beam in linear structural system. The analytical solution for design sensitivity of element nodal internal force, nodal displacement and stress for linear structural system are proposed in this paper by utilizing the finite element analysis results from SAP2000 software. Finally the accuracy and effectiveness of the proposed formula are verified by two examples. The comparing results show that the proposed analytical formulas for design sensitivity are both accurate and efficient since it could reduce the numerical analysis burden by adopting only one round of finite element analysis results for design displacement and stress sensitivity analysis. Furthermore an integrated design sensitivity analysis system is proposed by combining the SAP2000 Application Program Interface(API) for MATLAB environment. It could be used as a good reference for structural design sensitivity analysis for other large complex linear structural structures.
引文
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[6] Chan C M. Optimal lateral stiffness design of tall buildings of mixed steel and concrete construction[J]. Structural Design of Tall Buildings, 2010, 10(3):155-177
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[8] 赵若红, 傅继阳, 吴玖荣,等. 一种改进的最优准则法及其在结构优化设计中的应用[J]. 建筑科学, 2012, 28(11):66-71 Zhao Ruohong, Fu Jiyang, Wu Jiurong, et al. The modified optimality criteria and application in structural design[J]. Building Science, 2012, 28(11):66-71(in Chinese)
[9] 张旭明, 王德信. 结构灵敏度分析的解析方法[J]. 河海大学学报(自然科学版), 1998(5):44-49Zhang Xuming, Wang Dexin. An analytical method for structural sensitivity analysis[J]. Journal of Hohai University, 1998(5):44-49(in Chinese)
[10] Wu C C. Design optimization of geometric and material nonlinear problems[D]. University of Iowa, 1986
[11] SAP2000 Application Programming Interface Documentation[M]. Copyright Computers and Structures, Inc.
[2] 唐文艳, 顾元宪. 遗传算法在结构优化中的研究进展[J]. 力学进展, 2002, 32(1):26-40Tang Wenyan, Gu Yuanxian. Advances of genetic algorithms in structural optimization[J]. Advances in Mechanics, 2002, 32(1):26-40(in Chinese)
[3] 傅继阳, 钟亮, 黄友钦,等. 基于量子粒子群算法的门式刚架结构抗风优化[J]. 西南交通大学学报, 2013, 48(5):845-850Fu Jiyang, Zhong Liang, Huang Youqin et al. Wind-resistant optimization of portal frames based on quantum-behaved particle swarm algorithm[J]. Journal of Southwest Jiaotong University, 2013, 48(5):845-850(in Chinese)
[4] Zou X K, Chan C M. An optimal resizing technique for seismic drift design of concrete buildings subjected to response spectrum and time history loadings[J]. Computers & Structures, 2005, 83:1689-1704
[5] 吴玖荣, 董超超, 徐安,等. 大跨度变截面门式刚架抗风优化设计研究[J]. 武汉理工大学学报, 2012, 34(2):69-74Wu Jiurong, Dong Chaochao, Xu An, et al. Wind-resistant optimized design of long span portal-rigid frame with web-tapered section[J]. Journal of Wuhan University of Technology, 2012, 34(2):69-74(in Chinese)
[6] Chan C M. Optimal lateral stiffness design of tall buildings of mixed steel and concrete construction[J]. Structural Design of Tall Buildings, 2010, 10(3):155-177
[7] Chan C M, Chui J K L. Wind-induced response and serviceability design optimization of tall steel buildings[J]. Engineering Structures, 2006, 28(4):503-513
[8] 赵若红, 傅继阳, 吴玖荣,等. 一种改进的最优准则法及其在结构优化设计中的应用[J]. 建筑科学, 2012, 28(11):66-71 Zhao Ruohong, Fu Jiyang, Wu Jiurong, et al. The modified optimality criteria and application in structural design[J]. Building Science, 2012, 28(11):66-71(in Chinese)
[9] 张旭明, 王德信. 结构灵敏度分析的解析方法[J]. 河海大学学报(自然科学版), 1998(5):44-49Zhang Xuming, Wang Dexin. An analytical method for structural sensitivity analysis[J]. Journal of Hohai University, 1998(5):44-49(in Chinese)
[10] Wu C C. Design optimization of geometric and material nonlinear problems[D]. University of Iowa, 1986
[11] SAP2000 Application Programming Interface Documentation[M]. Copyright Computers and Structures, Inc.