当前课程知识点：Finite Element Method (FEM) Analysis and Applications > 9、Finite element analysis of continuum structure (2) > 9.5 MATLAB programming for typical 2D problems > Video 9.5
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下面我们给出一个典型空间问题的MATLAB编程
我们看一下这是一个空间的块体
实际上看起来就是一个悬臂梁
左端是固定，右端是作用集中力F
我们用MATLAB来进行节点位移、支座反力
以及单元应力的计算
那它的几何形状是这样
长0.8m，宽0.2m，高0.6m
所施加的力F为100000N
弹性模量是1e10Pa
泊松比是0.25，宽度是0.2m
我们对这个结构首先划分节点
这个节点是1,2,3,4,5,6,7,8
当然这个问题比较简单
也可以用一个8节点正六面体单元来进行分析
那么我们下面用MATLAB程序
但是由5个4节点的四面体来进行分析
那么这5个四面体的单元
单元的编号为1,2,3,4,5
比如1号单元对应的节点号是1,4,2,6
5号单元是1,4,6,7
那么总的节点位移
由于我们有8个节点，每个节点有3个自由度
所以有24个自由度
节点外载荷，我们看看
节点外载荷在1号节点、2号节点
外力没有施加在上面
实际上它是一个支座
3号节点和4号节点是我们施加的外力
当然我们这里的情况呢
3号节点和4号节点给的外力是0
5号节点、6号节点是支座
7号节点、8号节点是所施加的外力
我们把每一个节点的3个分力分别写出来
比如3、4号节点3个分量
我们这里是自由的，没有施加外力
所以3个分量分别是0
7号节点、8号节点的3个分量也一样
x方向为0，y方向为0
z方向是-100000N
我们看看支座反力
恰好刚才1、2、5、6号
恰好就是我们的支座反力
我们用R1,R2,R5,R6来表达
其中R1有3个分力
R2也有3个分力
R5,R6分别都有3个分力
我们把施加的节点外载和支反力合到一块
就得到总的节点力
也就是P=F+R，我们把它合到一块来写
首先在MATLAB环境下
我们要输入弹性模量、泊松比
然后针对单元1到5分别调用5次Stiffness
这样我们就可以分别得到k1,k2,k3,k4,k5
当然每一个k都是6X6的矩阵
由于该结构有8个节点
所以总的自由度是24
那么我们总的刚度阵就是KK（24X24）
首先对KK清零，然后5次调用Assembly
也就是刚度矩阵的组装
这个就是MATLAB窗口的输入和输出情况
KK清零
然后5次调用
把它组装到KK里面去
然后我们要进行边界条件的处理和方程的求解
那么1、2、5、6节点的三个方向的位移为0
所以相应位移我们就把它划掉
所以只针对节点3、4、7、8的位移进行求解
我们作一个MATLAB的编程
就直接从总的刚度阵KK里面
分别把对应着1、2、5、6的位移划掉
然后把对应着3、4、7、8的位移提出来
赋给k矩阵
然后再生成对应的载荷列阵p
采用高斯消元法进行计算
这样我们就可以得到所计算的节点位移
这就是MATLAB的输入、输出情况
这就是我们所得到的节点位移
我们看看，3、4、7、8节点的位移情况
我们得到了所求的节点位移以后
我们再把整体的节点列阵进行一个组合
也就是说把刚才有位移约束的地方
和我们已经求过的地方进行一个组合
组合以后就得到了整个的位移列阵
它是24X1的
然后再代回原来的方程进行计算
这样求出所有的节点力
这个节点力也包含我们所施加的外力
也包含我们的支反力
那么我们要找支反力
就可以找对应的位置就可以找到
我们得到的支反力的结果
也就是1、2、5、6节点分别的3个方向的支反力
我们把它列在这里
同样，我们要用计算单元应力的函数Stress
可以分别计算各个单元的应力分量
这是我们所得到的各个单元的应力分量的情况
1-5号单元，每个单元分别有6个分量
3个主方向的，还有3个剪应力
那么后面我们要基于ANSYS平台
对同样的这个例题进行分析
-Finite element, infinite capabilities
--Video
-1.1 Classification of mechanics：particle、rigid body、deformed body mechanics
--1.1 Test
-1.2 Main points for deformed body mechanics
--1.2 Test
-1.3 Methods to solve differential equation solving method
--1.3 Test
-1.4 Function approximation
--1.4 Test
-1.5 Function approximation defined on complex domains
--1.5 Test
-1.6 The core of finite element: subdomain function approximation for complex domains
--1.6 Test
-1.7 History and software of FEM development
--1.7 Test
-Discussion
-Homework
-2.1 Principles of mechanic analysis of springs
--2.1 Test
-2.2 Comparison between spring element and bar element
--2.2 Test
-2.3 Coordinate transformation of bar element
--2.3 Test
-2.4 An example of a four-bar structure
--2.4 Test
-2.5 ANSYS case analysis of four-bar structure
--ANSYS
-Discussion
-3.1 Mechanical description and basic assumptions for deformed body
--3.1 Test
-3.2 Index notation
--3.2 Test
-3.3 Thoughts on three major variables and three major equations
--3.3 Test
-3.4 Test
-3.4 Construction of equilibrium Equation of Plane Problem
-3.5 Test
-3.5 Construction of strain-displacement relations for plane problems
-3.6 Test
-3.6 Construction of constitutive relations for plane problems
-3.7 Test
-3.7 Two kinds of boundary conditions
- Discussion
-- Discussion
-4.1 Test
-4.1 Discussion of several special cases
-4.2 Test
-4.2 A complete solution of a simple bar under uniaxial tension based on elastic mechanics
-4.3 Test
-4.3 The description and solution of plane beam under pure bending
-4.4 Test
-4.4 Complete description of 3D elastic problem
-4.5 Test
-4.5 Description and understanding of tensor
-Discussion
-5.1 Test
-5.1Main method classification and trial function method for solving deformed body mechanics equation
-5.2 Test
-5.2 Trial function method for solving pure bending beam: residual value method
-5.3 Test
-5.3How to reduce the order of the derivative of trial function
-5.4 Test
-5.4 The principle of virtual work for solving plane bending beam
-5.5 Test
-5.5 The variational basis of the principle of minimum potential energy for solving the plane bending
-5.6 Test
-5.6 The general energy principle of elastic problem
-Discussion
-6.1Test
-6.1 Classic method and finite element method based on trial function
-6.2 Test
-6.2 Natural discretization and approximated discretization in finite element method
-6.3 Test
-6.3 Basic steps in the finite element method
-6.4 Test
-6.4 Comparison of classic method and finite element method
-Discussion
-7.1 Test
-7.1 Construction and MATLAB programming of bar element in local coordinate system
-7.2 Test
-7.2 Construction and MATLAB programming of plane pure bending beam element in local coordinate syste
-7.3 Construction of three-dimensional beam element in local coordinate system
-7.4 Test
-7.4 Beam element coordinate transformation
-7.5 Test
-7.5 Treatment of distributed force
-7.6 Case Analysis and MATLAB programming of portal frame structure
-7.7 ANSYS case analysis of portal frame structure
-8.1 Test
-8.1 Two-dimensional 3-node triangular element and MATLAB programming
-8.2 Test
-8.2 Two-dimensional 4-node rectangular element and MATLAB programming
-8.3 Test
-8.3 Axisymmetric element
-8.4 Test
-8.4 Treatment of distributed force
-8.5 MATLAB programming of 2D plane rectangular thin plate
-8.6 Finite element GUI operation and command flow of a plane rectangular thin plate on ANSYS softwar
-Discussion
-9.1 Three-dimensional 4-node tetrahedral element and MATLAB programming
-9.2 Three-dimensional 8-node hexahedral element and MATLAB programming
-9.3 Principle of the isoparametric element
-9.4Test
-9.4Numerical integration
-9.5 MATLAB programming for typical 2D problems
-9.6 ANSYS analysis case of typical 3Dl problem
-Discussion
-10.1Test
-10.1Node number and storage bandwidth
-10.2Test
-10.2 Properties of shape function matrix and stiffness matrix
-10.3Test
-10.3 Treatment of boundary conditions and calculation of reaction forces
-10.4Test
-10.4 Requirements for construction and convergence of displacement function
-10.5Test
-10.5C0 element and C1 element
-10.6 Test
-10.6 Patch test of element
-10.7 Test
-10.7 Accuracy and property of numerical solutions of finite element analysis
-10.8Test
-10.8 Error and average processing of element stress calculation result
-10.9 Test
-10.9 Error control and the accuracy improving method of h method and p method
-Discussion
-11.1 Test
-11.1 1D high-order element
-11.2 Test
-11.2 2D high-order element
-11.3 Test
-11.3 3D high-order element
-11.4 Test
-11.4 Bending plate element based on thin plate theory
-11.5 Test
-11.5 Sub-structure and super-element
-12.1Test
-12.1 Finite element analysis for structural vibration: basic principle
-12.2 Test
-12.2 Case of finite element analysis for structural vibration
-12.3 Test
-12.3 Finite element analysis for elastic-plastic problems: basic principle
-12.4 Test
-12.4 Finite element analysis for elastic-plastic problems: solving non-linear equations
-Discussion
-13.1 Test
-13.1 Finite element analysis for heat transfer: basic principle
-13.2 Test
-13.2 Case of finite element analysis for heat transfer
-13.3 Test
-13.3 Finite element analysis for thermal stress problems: basic principle
-13.4 Test
-13.4 Finite element analysis for thermal stress problems: solving non-linear equation
-Discussion
-2D problem: finite element analysis of a 2D perforated plate
-3D problem: meshing control of a flower-shaped chuck
-Modal analysis of vibration: Modal analysis of a cable-stayed bridge
-Elastic-plastic analysis: elastic-plastic analysis of a thick-walled cylinder under internal pressur
-Heat transfer analysis: transient problem of temperature field during steel cylinder cooling process
-Thermal stress analysis: temperature and assembly stress analysis of truss structure
-Probability of structure: Probabilistic design analysis of large hydraulic press frame
-Modeling and application of methods: Modeling and analysis of p-type elements for plane problem
