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梁单元的坐标变换包含两种情况
一个是平面坐标变换
还有一个是空间坐标变换
我们首先考察一下平面梁单元的坐标变换
我们看一下局部坐标系是ox
整体坐标系是xoy上面带一横的
那么相应的节点位移
局部坐标系ox这个轴上有
u1,v1,转角θ1,u2,v2,θ2
对于整体坐标系
我们是u1一横，v1一横，转角也是θ1
对于2号节点，u2一横，v2一横，转角是θ2
那么我们考察一下它们的各个分量之间的投影转换关系
对于局部坐标系下的u1
它由整体坐标系下的u1一横和v1一横进行投影
那么投影关系就是乘上相应的cosα和sinα
对于局部坐标系的v1
它也是由整体坐标系下的u1一横和v1一横进行投影
这个时候的投影是乘上相应的负sinα还有cosα
对于两个坐标系的θ角，θ1和θ2
它都是一样的，因为转角是一样的
那对于2号节点在局部坐标系下的u2
它也是由整体坐标系的u2一横和v2一横来进行投影
同样局部坐标系上的v2也是由
整体坐标系下的u2一横和v2一横进行投影得到的
我们把这个投影关系写成一个矩阵形式
就得到局部坐标系节点的位移，6X1的列阵
等于转换阵6X6的，乘上q一横
它也是一个6X1的
那q一横是整体坐标系下的
这个坐标转换阵6X6的矩阵分别是
注意θ1对应的位置是1，因为它是相等的
这样就得到6X6的坐标转换阵
那么和平面杆单元的坐标变换类似
同样我们得到整体坐标系的梁单元的刚度方程为
这个K一横就是整体坐标系的刚度阵
它等于局部坐标系的刚度阵6X6
前乘一个坐标转换阵6X6的转置
再后乘一个坐标转换阵T
对于整体坐标系的节点力的列阵
它等于局部坐标系的节点力的列阵
前乘一个坐标转换阵的转置
对于空间梁单元的坐标变换
同样我们首先给出局部坐标系中的节点位移
它是在xoy坐标系里面的节点位移的列阵
它是个12X1的
它分别是u1,v1,w1,θx1,θy1,θz1
那对于2号节点也是一样，也有6个自由度
对于整体坐标系中间的节点位移
它是在xoy一横的坐标系里面
它也是一个12X1的节点位移的列阵
它分别是u1,v1,w1上面带一横的
同样，转角是θx1,θy1,θz1，也是带一横的
对于2号节点，同样也是6个自由度
我们分别对1号节点的平移的自由度
也就是u1,v1,w1进行一个投影的转换
我们可以得到局部坐标系的u1
它分别由整体坐标系的u1,v1,w1一横
进行cos方向余弦的相乘的这么一个投影
对于v1和w1也是这样的
对于转角，这个转角不太好理解
如果我们把转角用右手法则变为一个矢量
那么这个矢量的投影同样也满足我们刚才提到的
节点位移的那个投影关系
所以说转角的投影关系也是一样的
最后都可以写成一个转换阵λ3X3的
分别乘上整体坐标的位移
不管是平移位移还是转角，都是一样的
那么这个节点坐标的转换阵λ，它是一个3X3的矩阵
它是关于局部坐标轴和整体坐标轴
方向余弦组成的这么一个3X3的转换阵
对于节点2来说，和节点1类似
它也是不管是平移的位移还是转角的位移
它都满足λ3X3的这么一个坐标转换
那么我们把刚才的1号节点、2号节点
平移自由度和转动自由度全部合成到一块
写成一个总的转换关系，它就是
这个坐标转换阵T它是12X12
分别由λ3X3在对角线的3X3的子矩阵块上
其它都是0
这样组成的一个空间梁单元的坐标转换阵
有了它以后
我们就可以得到梁单元在整体坐标系中的刚度方程
它分别是
它的自由度是12
同样，整体坐标系的刚度阵也是等于
局部坐标系的刚度阵前乘一个转置
后乘一个坐标变换阵
对于节点力的列阵
它也是等于局部坐标系的节点力
前乘一个坐标变换阵的转置
-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
