拉格朗日函数法在静不定结构内力计算中的应用

Application of Lagrange functions for internal force calculation of statically indeterminate structure

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.