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前面讲解梁单元的时候
关于力都是集中力
也就是说我们都是作用在节点上的
要么是集中力，要么是弯矩
很多工程上的情况，在单元上还有很多分布力
那么对于梁单元来说
我们怎么来进行分布力的计算
这就要用到等效的节点载荷
等效的节点载荷也是基于能量原理
那么我们用最小势能原理来进行推导
最小势能原理里面的外力功
它是分布力乘上相应的挠度
在整个梁单元上进行积分
那么我们把这个挠度用我们前面
基于节点的形状函数的描述，把它代入
那么我们知道v(x)它是等于N乘上个qe
那么qe是我们的节点位移
这个N就是形状函数矩阵
我们代入以后就可以分别得到四项
我们把四项分别作积分就可以得到
对应于平面梁单元的四个节点的等效载荷
也就是说它是1号节点y方向的载荷
1号节点的等效弯矩
2号节点的y方向载荷以及2号节点的弯矩
我们把常用的平面梁单元的等效节点载荷
列到一个表里
我们看看
这是作用在梁单元中点的这么一个情况
那么它的等效载荷我们看看
在1号节点，也就是A节点
它的力是-P/2
那在1号节点，A节点上它的等效弯矩是-PL/8
在2号节点，B节点处
它的等效节点力是-P/2
在2号节点，它的等效弯矩是PL/8
那么我们可以看一下
我们很多同学有一种直觉
就是把这个力好像用静力等效
直接等效到节点上就可以了
比如对于刚才这个中点集中力为P的情况
我们一般凭一种直觉就是把这个P
有一半作用在左边的这个梁上
另一半作用在右边的这个梁上
我们以左边的为例
那么下面我们把这个梁取一半L/2
那么这个P也是P/2
对于左边A点的在y方向的等效力为-P/2
这是对的
那对于它的弯矩是P/2再乘上它的力臂L/2
应该是-PL/4
但是我们这里列出来的是-PL/8
显然是不一样的
所以直接用静力平衡等效的方式是不正确的
一定要用外力功等效的方式所得到的等效节点力
这才是正确的
那么下一种情况就是
不是作用在中点，分别作用在a,b位置的集中力
这样我们也可以得到A节点y方向的等效力
以及A节点的弯矩
同样B节点的等效节点力和等效弯矩
对于受均布载荷p0的情况
我们通过外力功计算等效的原理
得到的等效节点载荷
那么在A节点得到的y方向的等效力是
在A节点得到的力矩是
那么在B节点情况也一样
那么这个例子大家也可以验证一下
如果是按照静力平衡等效
那就是p0有一半作用在L/2的梁上
左边一半，右边一半
那么我们可以看见y方向的等效力
由静力平衡，它是对的
但是弯矩就不对了
大家可以验证一下
对于三角形分布的等效力的情况
我们得到的A节点它的等效力是这样的
同样这是它的弯矩
在B节点得到的等效力和等效弯矩
对于一个梁单元只在一个局部承受有分布载荷的情况
同样也得到A节点的y方向的等效力和等效弯矩
B节点等效的力和等效弯矩
对于房顶形的这么一个三角形分布
同样我们也把它列出来
A节点的等效力还有等效弯矩
B节点的等效力和等效弯矩
另外还有，我们在梁单元中间
任意位置承受有一个力矩的情况M0
如果按照我们一种直觉
在A节点和B节点按照杠杆的等效原理
那是否也是a和b这么一个直接的分配呢
这样似乎A节点和B节点就没有y方向的等效力了
只有A节点和B节点的等效弯矩
但是我们按照外力功得到的等效载荷
我们y方向的等效力以及等效的弯矩都有
-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
