拉格朗日函数法在静不定结构内力计算中的应用
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摘要
由于静不定结构在外荷载作用下的平衡状态是一个稳定的平衡状态,因此静不定结构在外荷载作用下的应变能存在极小值,是满足结构静力平衡方程或节点平衡方程的条件极值问题.引入拉格朗日乘数,结合结构静力平衡方程或节点平衡方程,构造了拉格朗日函数,对采用拉格朗日函数求解静不定结构内力的极值问题进行了数学证明.通过算例,阐述了拉格朗日函数在求解静不定结构内力中的应用.结果表明,采用拉格朗日函数求解静不定结构内力,不但克服了求解桁架内力需利用几何关系建立协调方程的缺陷,而且计算量小于力法、位移法,还具有力学概念直观、计算过程简便、求解模式固定等优点,便于工程技术人员在实际工程中掌握应用.
The equilibrium state of statically indeterminate structure is a stable equilibrium state on the condition of external load,so there is a minimum strain energy in statically indeterminate structure under the external load and this is the conditional extremum for the static equilibrium equations or node balance equations.In this paper,by introducing Lagrange's multiplier and combination of structure static equilibrium equations or node balance equations,the Lagrange function is constructed and mathematical justification is provided for the solution of extremum problem of the internal force of statically indeterminate structure.The numerical example provides the application of Lagrange functions for the internal force of statically indeterminate structure.Results show that the method which takes Lagrange functions to solve the internal force of statically indeterminate structure,it not only overcomes the shortcoming of the geometric relationship of coordination equation in the solution of truss internal force,but also decreases the calculation quantity which is less than the method of force and displacement method.This method features intuitive mechanical concept,simple calculation process and fixed solutions model and it is valuable for engineers to master.
The equilibrium state of statically indeterminate structure is a stable equilibrium state on the condition of external load,so there is a minimum strain energy in statically indeterminate structure under the external load and this is the conditional extremum for the static equilibrium equations or node balance equations.In this paper,by introducing Lagrange's multiplier and combination of structure static equilibrium equations or node balance equations,the Lagrange function is constructed and mathematical justification is provided for the solution of extremum problem of the internal force of statically indeterminate structure.The numerical example provides the application of Lagrange functions for the internal force of statically indeterminate structure.Results show that the method which takes Lagrange functions to solve the internal force of statically indeterminate structure,it not only overcomes the shortcoming of the geometric relationship of coordination equation in the solution of truss internal force,but also decreases the calculation quantity which is less than the method of force and displacement method.This method features intuitive mechanical concept,simple calculation process and fixed solutions model and it is valuable for engineers to master.
引文
[1]S铁摩辛柯,J盖尔.材料力学[M].北京:科学出版社,1990:237-242.
[2]刘鸿文.材料力学(第三版,下册)[M].北京:高等教育出版社,1999:57-60.
[3]张晓月.基于敏度分析的不同模量桁架正反问题求解[D].大连:大连理工大学,2008.ZHANG Xiao-yue.Sensitivity Analysis Based Numerical Solutions of Normal and Inverse Problems of Elastic Bimodular Truss Structure[D].Dalian:Dalian University of Technology,2008.
[4]杨海天,张晓月,何宜谦.基于敏度分析的拉压不同模量桁架问题的数值分析[J].计算力学学报,2011,28(2):237-242.YANG Hai-tian,ZHANG Xiao-yue,HE Yi-qian.Sensitivity analysis based numerical solution for truss structures with bi-modulus[J].Chinese Journal of Computational Mechanics,2011,28(2):237-242.
[5]姚顺忠.一般杆系结构节点位移计算方法的探讨[J].西南林学院学报,2002,22(1):61-63.YAO Shun-zhong.Analysis of methods calculating displacement of a truss joint[J].Journal of Southwest Forestry College,2002,22(1):61-63.
[6]陈平,陈国良,杨绪普.载荷作用下多杆汇交问题的通解[J].力学与实践,2014,36(3):348-350.CHEN Ping,CHEN Guo-liang,YANG Xu-pu.The general solution of the multi-rod junction problems[J].Mechanics and Engineering,2014,36(3):348-350.
[7]冯贤桂.结点位移计算的一种简单方法[J].力学与实践,2002,24(1):49-50.FENG Xian-gui.A simple method for calculating node displacement[J].Mechanics and Engineering,2002,24(1):49-50.
[8]朱伊德.静定和静不定杆系结构中节点位移的一种计算方法[J].上海应用技术学院学报,2007,7(1):33-35.ZHU Yi-de.A method for calculating node displasement of statically determinate and statically indeterminate structure of bar system[J].Journal of Shanghai Institute of Technology,2007,7(1):33-35.
[9]倪尔有.超静定桁架中建立变形几何方程的解析法[J].鞍山钢铁学院学报,1991,14(3):56-59.NI Er-you.Analytic method of setting up deformation geometry equations on statically indeterminate truss problems[J].Journal of Anshan Institute of Iron and Steel Technology,1991,14(3):56-59.
[10]边文凤,董正筑.超静定桁架变形协调方程的新方法[J].计算力学学报,2002,19(2):250-252.BIAN Wen-feng,DONG Zheng-zhu.A new method for problem of hyperstatic truss[J].Chinese Journal of Computational Mechanics,2002,19(2):250-252.
[11]高金华.利用微分方法推导静不定桁架变形协调方程[J].力学与实践,1998,20(5):67-68.GAO Jin-hua.Using differential method to derive deformation compatibility equation of statically indeterminate truss[J].Mechanics and Engineering,1998,20(5):67-68.
[2]刘鸿文.材料力学(第三版,下册)[M].北京:高等教育出版社,1999:57-60.
[3]张晓月.基于敏度分析的不同模量桁架正反问题求解[D].大连:大连理工大学,2008.ZHANG Xiao-yue.Sensitivity Analysis Based Numerical Solutions of Normal and Inverse Problems of Elastic Bimodular Truss Structure[D].Dalian:Dalian University of Technology,2008.
[4]杨海天,张晓月,何宜谦.基于敏度分析的拉压不同模量桁架问题的数值分析[J].计算力学学报,2011,28(2):237-242.YANG Hai-tian,ZHANG Xiao-yue,HE Yi-qian.Sensitivity analysis based numerical solution for truss structures with bi-modulus[J].Chinese Journal of Computational Mechanics,2011,28(2):237-242.
[5]姚顺忠.一般杆系结构节点位移计算方法的探讨[J].西南林学院学报,2002,22(1):61-63.YAO Shun-zhong.Analysis of methods calculating displacement of a truss joint[J].Journal of Southwest Forestry College,2002,22(1):61-63.
[6]陈平,陈国良,杨绪普.载荷作用下多杆汇交问题的通解[J].力学与实践,2014,36(3):348-350.CHEN Ping,CHEN Guo-liang,YANG Xu-pu.The general solution of the multi-rod junction problems[J].Mechanics and Engineering,2014,36(3):348-350.
[7]冯贤桂.结点位移计算的一种简单方法[J].力学与实践,2002,24(1):49-50.FENG Xian-gui.A simple method for calculating node displacement[J].Mechanics and Engineering,2002,24(1):49-50.
[8]朱伊德.静定和静不定杆系结构中节点位移的一种计算方法[J].上海应用技术学院学报,2007,7(1):33-35.ZHU Yi-de.A method for calculating node displasement of statically determinate and statically indeterminate structure of bar system[J].Journal of Shanghai Institute of Technology,2007,7(1):33-35.
[9]倪尔有.超静定桁架中建立变形几何方程的解析法[J].鞍山钢铁学院学报,1991,14(3):56-59.NI Er-you.Analytic method of setting up deformation geometry equations on statically indeterminate truss problems[J].Journal of Anshan Institute of Iron and Steel Technology,1991,14(3):56-59.
[10]边文凤,董正筑.超静定桁架变形协调方程的新方法[J].计算力学学报,2002,19(2):250-252.BIAN Wen-feng,DONG Zheng-zhu.A new method for problem of hyperstatic truss[J].Chinese Journal of Computational Mechanics,2002,19(2):250-252.
[11]高金华.利用微分方法推导静不定桁架变形协调方程[J].力学与实践,1998,20(5):67-68.GAO Jin-hua.Using differential method to derive deformation compatibility equation of statically indeterminate truss[J].Mechanics and Engineering,1998,20(5):67-68.