弹性力学基础及有限元法.ppt
内蒙古科技大学 机械工程学院 刘学杰 2015-03,弹性力学方程组的求解The solution of elasticity equationsFEM&ANSYS-3,有限元法及实践-3Finite Element Method(FEM)&the Practice,Outline 概要,1.Review 回顾2.Solution of the set of equations 求解2.1.Methods 方法2.2.Problem 问题3.From IFEM to FEM 从无限元到有限元吗？变分虚位移最小能量原理,Review,The set of equations1.The equilibrium equations（平衡方程）2.The strain-displacement relationship（几何方程）3.Stress-strain relationship（物理方程）The solution of the set of equations,The equilibrium equations:Stress&loads,The equilibrium equations,Strain-Displacement Relation(几何方程),Stress-strain relationship物理方程（广义胡克定律）,The set of equations,Question 1:How to make the loading&constrains into the equations？,Elemental volume,位移/displacements u,v,w,应变/strains,应力/stress,Question 2:what are the exact unknown values?What is the form of solution?,Elemental volume,位移/displacements u,v,w u(x,y,z),v(x,y,z),w(x,y,z),应变/strains,应力/stress,The solution of the equations：How to get the solution?,1.The required known values(what must to be known?)1)the geometrical values:shape and size（几何模型）2)the materials properties:E,G（材料性质）3)The loads（载荷）4)The constrains（约束）2.The solution plan(how to resolve the equations?)1)the transition of the set of equations（微分方程化简）the displacement method the stress method2)the integration field（积分域）3)the loads matrix（载荷矩阵）4)the constrains the displacement matrix（位移矩阵）,The solution of elasticity equations:a great challenge,The solution methodology1.The analytical method（解析法）2.The calculus of variations（变分法，能量法）3.The difference method（差分法）微分方程 代数方程（等距差分 偏微分与函数值联结）4.The finite element method（有限元法）连续体 离散化结构,The analytical method（解析法）:Elimination method（消元法）,The displacement method（位移法）The elimination of strains from the strain-displacement equation The elimination of stress from the strain-stress equation The equilibrium equation to obtain the displacement u,v,wTo get the strainsTo get the stressesThe stress method（应力法）反逆解法、半逆解法Problems,The set of equations,The problem of equation solutions,1.求解的问题1）高阶微分方程2）载荷处理3）边界处理4）解的性质：坐标的函数2.解决方法1）探索-深度认识-精确解2）简化处理 数值方法3.应该怎么样取向？学会欣赏、学会尊重、学会坚持。,The calculus of variations（变分法，能量法）,变分的概念(Concepts)函数与泛函(Function&Functional)y=f(x)=y(x)微分与变分(Differential&Variation)极值(Extreme value),The calculus of variations（变分法，能量法）,函数(Function)：f(x)是变量x的实函数，即在其定义域内，任一x值都有一个实数f(x)与之对应。泛函(Functional)：(y)是函数y(x)的泛函，即在其定义域内，任一函数y(x)都有一个实数(y)与之对应。变分命题：寻找y(x)使得泛函(y)取极值。变分方法：设使泛函取得极值的函数y(x)存在，通过变分法求得这个极值函数y(x)所需满足的微分方程。,The calculus of variations（变分法，能量法）,虚位移原理(The principle of virtual displacement)外力F方向，虚位移，虚应变外力F所做虚功W：整个结构虚应变能U：虚位移原理：最小势能原理结构势能：=U-W最小势能原理：什么意思？,=U W=0,The model of elasticity problem弹性力学问题的模型,Elemental volume,位移/displacements u,v,w u(x,y,z),v(x,y,z),w(x,y,z),应变/strains,应力/stress,Two-dimensional problems两类平面问题,平面应力问题：,物理方程：应变矩阵D：弹性矩阵,Two-dimensional problems两类平面问题,平面应变问题：,Let:,Two-dimensional problems两类平面问题,从无限元法走向有限元法From InFEM to FEM,为什么从无限元法走向有限元法？采用有限元法就能够解决问题吗？为什么？有限元法解决问题的基本思路是什么？,The concept of FEM有限元概念,Homework,1、简述弹性力学方程组的解析法求解方法并讨论其中存在的问题。2、简述泛函、变分、虚位移与最小能量原理。3、为什么要采用有限元法？有限元法如何解决问题？4、分组？,谢谢！Thanks for your attention!问题和讨论（The questions and discussion),内蒙古科技大学 机械工程学院 刘学杰 2015-03,
