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Video 2.3课程教案、知识点、字幕

杆单元的描述我们已经有了

它是刚度阵乘上一个节点位移等于节点力

相应的节点位移的描述、节点力的描述

还有刚度阵的描述我们都有了

大家要注意一下

我们这个杆单元是基于局部坐标的描述

也就是说我们这个坐标系

是取的这个杆单元的轴线

这对于平面杆单元或者是对于

一个水平的复杂结构应该是没问题

那么还有大量的工程结构

比如像这个桁架结构里面

有一些斜着的杆

这个斜着的杆我们也可以看作局部坐标x

斜着的沿着杆的轴线的坐标

现在有一个问题

沿着斜着这个方向的位移

和其它杆件的位移怎么进行组合

这是一个问题

所以说我们要进行一个绝对坐标系

或者叫整体坐标系下的一个坐标的变换

也就需要把我们局部坐标系下的位移

分解成整体坐标系上

我们把所有的杆单元都按照整体坐标来进行分解

来进行描述

这样的话,我们才有一个组装的基础

我们现在假定局部坐标系是x

那么整体坐标系是x一横和y一横

局部坐标系和整体坐标系之间相差α角度

局部坐标的位移叫u1,u2

那整体坐标系下叫u1一横,v1一横

就分解成两个方向了

同样2号节点也是分解成两个方向

那么这两个坐标系

一个是局部坐标系ox

还有一个整体坐标系

这两个坐标系我们假定它的夹角是α

我们把局部坐标系的节点位移写成

把整体坐标系的节点位移写成

我们可以看出来局部坐标系是两个节点位移u1,u2

整体坐标系每一个点上有两个分量

那么我们把每一个节点上的整体坐标系上的位移

向局部坐标系ox进行投影

我们就可以分别写出来

局部坐标系的合成以后的位移的量

这样我们就得到两个坐标系之间的

节点位移的转换关系

我们把两个坐标系之间的节点位移的转换关系

再进一步写成矩阵的形式

并且把节点的位移用我们前面所定义的qe来表达

这边是整体坐标系下节点位移的分量

我们把前面这个矩阵叫作坐标变换矩阵

整个的坐标变换就用这么一个

矩阵变换的方程来描述

我们有了节点位移的坐标变换

我们把它代入原来局部坐标系的平衡方程

也叫刚度方程,代进去

再在前面乘上一个单元转换阵的转置

我们把前面这一项写成

这一项继续保留

把这两个合到一块写成

这样我们就得到整体坐标系下的

平面杆单元的刚度方程或者平衡关系

大家看一下,这上面都带一横

表明它是在整体坐标系下的描述

其中里面的K矩阵

它是原来局部坐标系下的

前乘一个转置的坐标变换阵

后乘一个坐标变换阵

同样,整体坐标系下的节点力也等于

原来局部坐标系下的节点力

前乘一个坐标变换阵的转置

我们把刚度阵的计算分别做出来

就得到这么一个基于α的整体刚度矩阵

它是一个4×4的矩阵

对于空间杆单元其实也是类似的

假定我们局部坐标系还是ox

三维的空间杆单元的整体坐标系为

同样局部坐标系的节点位移还是u1,u2

那么整体坐标系下每一个节点位移有三个分量

所以说它是有六个分量

假定我们两个坐标系之间的方向余弦

我们可以用两个节点的坐标来进行计算

也就是cos这三个方向余弦

分别把它写出来

整体坐标系的分量向

局部坐标系的节点位移进行投影

同样,跟刚才二维情况类似

可以写出这样一个转换关系

那么它的转换阵就是这样一个

基于方向余弦的表达

它是一个2×6的矩阵

同样,我们得到整体坐标系三维情况下的

刚度方程是这样的

它的规模是6×6的

这是6×1的

这也是6×1的

它的整体刚度阵的计算和刚才类似

也就是说在局部坐标系下的刚度阵上

前乘一个空间坐标系下的转换阵的转置

后乘一个转换阵

整体坐标系下的节点力也是一样

在局部坐标系下的节点力前面

前乘一个空间坐标系下的坐标转换阵的转置

Finite Element Method (FEM) Analysis and Applications课程列表:

0、Course summary

-Finite element, infinite capabilities

--Video

1、Introduction

-1.1 Classification of mechanics:particle、rigid body、deformed body mechanics

--Video 1.1

--1.1 Test

-1.2 Main points for deformed body mechanics

--Video 1.2

--1.2 Test

-1.3 Methods to solve differential equation solving method

--Video 1.3

--1.3 Test

-1.4 Function approximation

--Video 1.4

--1.4 Test

-1.5 Function approximation defined on complex domains

--Video 1.5

--1.5 Test

-1.6 The core of finite element: subdomain function approximation for complex domains

--Video 1.6

--1.6 Test

-1.7 History and software of FEM development

--Video 1.7

--1.7 Test

-Discussion

--Discussion

-Homework

2、Finite element method of bar system based on direct stiffness method

-2.1 Principles of mechanic analysis of springs

--Video 2.1

--2.1 Test

-2.2 Comparison between spring element and bar element

--Video 2.2

--2.2 Test

-2.3 Coordinate transformation of bar element

--Video 2.3

--2.3 Test

-2.4 An example of a four-bar structure

--Video 2.4

--2.4 Test

-2.5 ANSYS case analysis of four-bar structure

--Video 2.5

--ANSYS

-Discussion

--Discussion

3、Mechanical description of deformed bodies with complex geometry 1

-3.1 Mechanical description and basic assumptions for deformed body

--Video 3.1

--3.1 Test

-3.2 Index notation

--Video 3.2

--3.2 Test

-3.3 Thoughts on three major variables and three major equations

--Video 3.3

--3.3 Test

-3.4 Test

-3.4 Construction of equilibrium Equation of Plane Problem

--Video 3.4

-3.5 Test

-3.5 Construction of strain-displacement relations for plane problems

--Video 3.5

-3.6 Test

-3.6 Construction of constitutive relations for plane problems

--Video 3.6

-3.7 Test

-3.7 Two kinds of boundary conditions

--Video 3.7

- Discussion

-- Discussion

4、Mechanical description of deformed bodies with complex geometry 2

-4.1 Test

-4.1 Discussion of several special cases

--Video 4.1

-4.2 Test

-4.2 A complete solution of a simple bar under uniaxial tension based on elastic mechanics

--Video 4.2

-4.3 Test

-4.3 The description and solution of plane beam under pure bending

--Video 4.3

-4.4 Test

-4.4 Complete description of 3D elastic problem

--Video 4.4

-4.5 Test

-4.5 Description and understanding of tensor

--Video 4.5

-Discussion

--Discussion

5、Principle of trial function method for solving mechanical equations of deformed body

-5.1 Test

-5.1Main method classification and trial function method for solving deformed body mechanics equation

--Video 5.1

-5.2 Test

-5.2 Trial function method for solving pure bending beam: residual value method

--Video 5.2

-5.3 Test

-5.3How to reduce the order of the derivative of trial function

--Video 5.3

-5.4 Test

-5.4 The principle of virtual work for solving plane bending beam

--Video 5.4

-5.5 Test

-5.5 The variational basis of the principle of minimum potential energy for solving the plane bending

--Video 5.5

-5.6 Test

-5.6 The general energy principle of elastic problem

--Video 5.6

-Discussion

--Discussion

6、Classic implementation and finite element implementation based on trial function method

-6.1Test

-6.1 Classic method and finite element method based on trial function

--Video 6.1

-6.2 Test

-6.2 Natural discretization and approximated discretization in finite element method

--Video 6.2

-6.3 Test

-6.3 Basic steps in the finite element method

--Video 6.3

-6.4 Test

-6.4 Comparison of classic method and finite element method

--VIDEO 6.4

-Discussion

--Discussion

7、Finite element analysis of bar and beam structures

-7.1 Test

-7.1 Construction and MATLAB programming of bar element in local coordinate system

--Video 7.1

-7.2 Test

-7.2 Construction and MATLAB programming of plane pure bending beam element in local coordinate syste

--Video 7.2

-7.3 Construction of three-dimensional beam element in local coordinate system

--Video 7.3

-7.4 Test

-7.4 Beam element coordinate transformation

--Video 7.4

-7.5 Test

-7.5 Treatment of distributed force

--Video 7.5

-7.6 Case Analysis and MATLAB programming of portal frame structure

--Video 7.6

-7.7 ANSYS case analysis of portal frame structure

--Video 7.7

8、Finite element analysis of continuum structure (1)

-8.1 Test

-8.1 Two-dimensional 3-node triangular element and MATLAB programming

--Video 8.1

-8.2 Test

-8.2 Two-dimensional 4-node rectangular element and MATLAB programming

--Video 8.2

-8.3 Test

-8.3 Axisymmetric element

--Video 8.3

-8.4 Test

-8.4 Treatment of distributed force

--Video 8.4

-8.5 MATLAB programming of 2D plane rectangular thin plate

--Video 8.5

-8.6 Finite element GUI operation and command flow of a plane rectangular thin plate on ANSYS softwar

--Video 8.6

-Discussion

--Discussion

9、Finite element analysis of continuum structure (2)

-9.1 Three-dimensional 4-node tetrahedral element and MATLAB programming

--Video 9.1

-9.2 Three-dimensional 8-node hexahedral element and MATLAB programming

--Video 9.2

-9.3 Principle of the isoparametric element

--Video 9.3

-9.4Test

-9.4Numerical integration

--Video 9.4

-9.5 MATLAB programming for typical 2D problems

--Video 9.5

-9.6 ANSYS analysis case of typical 3Dl problem

--Video 9.6

-Discussion

--Discussion

10、Basic properties in finite element method

-10.1Test

-10.1Node number and storage bandwidth

--Video 10.1

-10.2Test

-10.2 Properties of shape function matrix and stiffness matrix

--Video 10.2

-10.3Test

-10.3 Treatment of boundary conditions and calculation of reaction forces

--Video 10.3

-10.4Test

-10.4 Requirements for construction and convergence of displacement function

--Video 10.4

-10.5Test

-10.5C0 element and C1 element

--Video 10.5

-10.6 Test

-10.6 Patch test of element

--Video 10.6

-10.7 Test

-10.7 Accuracy and property of numerical solutions of finite element analysis

--Video 10.7

-10.8Test

-10.8 Error and average processing of element stress calculation result

--Video 10.8

-10.9 Test

-10.9 Error control and the accuracy improving method of h method and p method

--Video 10.9

-Discussion

--Discussion

11、High-order and complex element

-11.1 Test

-11.1 1D high-order element

--Video 11.1

-11.2 Test

-11.2 2D high-order element

--Video 11.2

-11.3 Test

-11.3 3D high-order element

--Video 11.3

-11.4 Test

-11.4 Bending plate element based on thin plate theory

--Video 11.4

-11.5 Test

-11.5 Sub-structure and super-element

--Video 11.5

12、Introduction to the application of finite element analysis (1)

-12.1Test

-12.1 Finite element analysis for structural vibration: basic principle

--Video 12.1

-12.2 Test

-12.2 Case of finite element analysis for structural vibration

--Video 12.2

-12.3 Test

-12.3 Finite element analysis for elastic-plastic problems: basic principle

--Video 12.3

-12.4 Test

-12.4 Finite element analysis for elastic-plastic problems: solving non-linear equations

--Video 12.4

-Discussion

--Discussion

13、Introduction to the application field of finite element analysis (2)

-13.1 Test

-13.1 Finite element analysis for heat transfer: basic principle

--Video 13.1

-13.2 Test

-13.2 Case of finite element analysis for heat transfer

--Video 13.2

-13.3 Test

-13.3 Finite element analysis for thermal stress problems: basic principle

--Video 13.3

-13.4 Test

-13.4 Finite element analysis for thermal stress problems: solving non-linear equation

--Video 13.4

-Discussion

--Discussion

14、Project

-2D problem: finite element analysis of a 2D perforated plate

--Video I-1

-3D problem: meshing control of a flower-shaped chuck

--Video I-2

-Modal analysis of vibration: Modal analysis of a cable-stayed bridge

--Video I-3

-Elastic-plastic analysis: elastic-plastic analysis of a thick-walled cylinder under internal pressur

--Video I-4

-Heat transfer analysis: transient problem of temperature field during steel cylinder cooling process

--Video I-5

-Thermal stress analysis: temperature and assembly stress analysis of truss structure

--Video I-6

-Probability of structure: Probabilistic design analysis of large hydraulic press frame

--Video I-7

-Modeling and application of methods: Modeling and analysis of p-type elements for plane problem

--Video I-8

Video 2.3笔记与讨论

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