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对于平面单元同样也有分布力的处理
也是我们要把它等效成节点载荷
等效的原理还是由能量原理
能量原理里面
最小势能原理里面关于载荷的等效实际上是基于外力功
我们看看外力功的计算
外力功是等于体积力乘上相应的位移
还有面积力乘上相应的位移，进行积分
我们把它写成矩阵形式
体积力我们写成一个向量，当然要转置一下
位移我们也写成一个向量
我们把位移这个向量由我们基于节点描述的方式
也就是说位移场等于一个形状函数矩阵乘上节点位移
我们把这个关系代进去，我们就得到了这么一个表达
这个表达也就是说
外力功要表达成一个矩阵乘上一个节点位移
这个矩阵就是我们的等效节点力P
我们可以看看它的具体表达
就是一个N矩阵转置乘上一个体积力
对我们这个单元进行积分
同样对于分布的面力
我们也是要前乘一个形状函数矩阵N的转置
这样来进行表达
对于平面问题
我们把这个体积分变成面积分乘上一个厚度
这个面积分就变成一个线积分乘上一个厚度
我们把常用的几种分布力的情况列成一个表
第一种情况就是考虑单元的自重
我们平面单元的内部有一个自重
自重我们用ρ0表示密度
Ae是单元的面积，t为厚度
这样我们可以把这个单元的自重
分别等效到3个节点上
3个节点的节点力就是
节点1、节点2、节点3
分别在x方向的节点力和y方向的节点力
它一共有6个分量
对于侧边受均布压力的情况
同样我们也可以得到等效的节点外载的列阵
对于如图所示的情况
1号节点和2号节点之间的这条边
受到一个均布载荷
均布载荷的力是p0
我们通过前面的公式计算
也就是基于虚功原理或者是
最小势能原理里面的外力功的计算
我们可以得到等效的节点力
这样计算下来得到的也是6项
对于如图所示的情况
由于它的外力作用是在1、2连线的这条边
所以它等效出来是在1节点和2节点的
x方向和y方向的分量上
对于3号节点没有影响
我们可以看看3号节点上的力的分量为0
另一种情况是1号节点和2号节点这条边上
受x方向的均布侧压
这个时候我们同样可以按照
最小势能原理里面的外力功的计算
可以等效成等效节点外载
同样也有一个6X1的各个分量
这就是我们得到的位移分量的情况
我们可以看看
1号节点的x方向有，y方向没有
2号节点x方向有一个分量，y方向没有
3号节点两个方向都没有
同样对如图所示的1号节点、2号节点之间
受一个x方向三角形分布的侧压
我们也是按照外力功的计算等效原则
我们得到的等效力
我们看看1号节点x方向的等效力是2/3
y方向是0
2号节点x方向是1/3，y方向是0
3号节点没有
实际上我们可以看看
由于它是一个三角形分布，沿x方向
所以它分别对1号节点和2号节点
按照一个2/3和1/3的这么一个分布的
-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
