当前课程知识点：Finite Element Method (FEM) Analysis and Applications > 7、Finite element analysis of bar and beam structures > 7.3 Construction of three-dimensional beam element in local coordinate system > Video 7.3
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对于一般情况下的平面梁
它同样也是两个节点
1号节点和2号节点
那么除了刚才平面的纯弯梁
也就是说它的挠度和转角以外
还有x方向，也就是沿轴向的位移
那么我们把这种问题分解成两个
一个是平面纯弯梁，刚才我们已经分析了
另外再加上一个杆单元，沿着轴向的u1和u2
我们分别把这两个单元集成以后
就可以得到一般平面梁的梁单元
那么集成后的情况是这样的
单元的节点位移是u1,v1,θ1
其中这个u1就是我们1号节点的杆单元的节点位移
v1,θ1是我们平面纯弯梁1号节点的挠度和转角
对于2号节点也是一样，它是u2,v2,θ2
对应着节点位移有相应的节点力
那么同样我们进行组装以后也可以得到
一般平面梁的刚度方程
它是一个6X6的刚度矩阵乘上一个6X1的节点位移列阵
等于6X1的节点力
那么我们看看这个6X6的总的刚度阵
也就是说平面问题一般梁单元的刚度阵
它怎么组装
平面梁我们对应着v1,θ1,v2,θ2
我们有一个4X4的单元刚度阵的描述
对于杆单元u1,u2这两个节点位移
我们也有相应的2X2的刚度阵的描述
我们分别把这两个刚度阵按照
一般梁单元的u1,v1,θ1,u2,v2,θ2
分别把它装配到相应的位置
就可以得到平面一般梁单元的刚度矩阵
我们看看，比如这个u1,u2这四个
就是我们杆单元的四个元素
剩下的这些就是我们平面纯弯梁的
4X4的那些元素
对于一般空间梁单元
我们空间梁单元同样也是有两个节点
节点1、节点2
那么空间梁单元每个节点上有三个平移的位移
我们分别叫u1,v1,w1
同样，绕三个轴还有三个转动的转角
我们分别叫θx1,θy1,θz1
那对于2号节点也是同样
这样每一个节点就有六个自由度
那么我们整个单元有两个节点
所以总共有12个自由度
那么我们节点位移就是一个12X1的节点位移qe
对应着节点位移我们有节点力的表达
那同样，也有对应着三个集中力
还有三个力矩
那对于空间的梁单元
我们怎么来建立它的总体刚度矩阵呢
我们把它作一个分解
我们把它分解成四个部分
第一个部分我们沿着x轴的转角分别是θx1,θx2
我们把它分出来
另外沿着x轴的移动的自由度，也就是u1和u2
另外在xoy平面，我们有v1,θz1和v2,θz2
我们把这个叫作xoy平面上的纯弯梁的自由度
另外在xoz平面我们同样也有w1,θy1,w2,θy2
在这个平面内，同样也是一个平面纯弯梁的情况
那么我们对四种情况分别建立它的单元刚度阵
然后把它按照整体位移的位置进行组装
就可以得到一般梁单元也就是说空间梁单元的刚度矩阵
那么我们看看
第一种情况，梁的扭转情况
(θx1,θx2)这一对转角
扭转我们没有专门来讲它的有限元构造
它实际上和拉压的情况一样
因为我们把扭矩用右手法则可以把它变成一个向量
也就是说它相当于是变成与u1,u2对应的
沿着轴向的这么一个位移的向量
那么唯一不同的是
由于它是扭转，所以我们拉压杆的弹性模量
要把它改成剪切模量
我们拉压杆里面的横截面面积要变成极惯性矩
也就是说它是x平方加y平方括起来
对横截面dA进行积分
这叫极惯性矩，它也是一个2X2的这么一个刚度阵
沿着轴向的拉伸，也就是u1,u2这一对自由度
它是一个2X2的矩阵，这个以前我们已经讨论过了
对于xOy平面内的弯曲
它的自由度是v1,θz1,v2,θz2
那么这个平面纯弯梁的情况呢
我们4X4的这个刚度阵我们已经构建了
值得注意的是，这个里面的截面的惯性矩
它是绕着z轴旋转的，所以它是Iz
对于第四个就是xOz平面内的弯矩
它对应的自由度是w1,θy1,w2,θy2
同样，它也是一个纯弯梁的4X4的刚度矩阵
值得注意的是它的惯性矩是Iy
我们把刚才的四种情况按照空间梁单元的自由度
也就是说u1,v1,w1,θx1,θy1,θz1
还有2号节点的6个自由度
把它列成一个12X12的大矩阵
把前面的四种情况下的每个元素
按照12X12这个矩阵里面对应的自由度进行装配
这样就得到一个12X12的空间梁单元的刚度矩阵
-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
