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1、2022/11/23,1,第三章 MATLAB有限元分析与应用,3-1 弹簧元,结构分析编程及软件应用,3-2 线性杆元,3-3 二次杆元,3-4 平面桁架元,3-5 空间桁架元,3-6 梁元,2022/11/23,2,3-1 弹簧元,结构分析编程及软件应用,1、有限元方法的步骤：,离散化域,形成单刚矩阵,集成整体刚度矩阵,引入边界条件,求解方程,后处理,2022/11/23,3,结构分析编程及软件应用,2、基本方程,3-1 弹簧元,弹簧元是总体和局部坐标一致的一维有限单元,每个弹簧元有两个节点（node）,单刚矩阵为：,总刚矩阵：,结构方程：,单元节点力：,2022/11/23,4,结构分析
2、编程及软件应用,3、MATLAB函数编写,3-1 弹簧元,%SpringElementStiffness This function returns the element stiffness %matrix for a spring with stiffness k. %The size of the element stiffness matrix is 2 x 2.,3.1 单元刚度矩阵的形成,y = k -k ; -k k;,function y = SpringElementStiffness(k),2022/11/23,5,结构分析编程及软件应用,3、MATLAB函数编写,3-1
3、弹簧元,%SpringAssemble This function assembles the element stiffness% matrix k of the spring with nodes i and j into the% global stiffness matrix K.% This function returns the global stiffness matrix K % after the element stiffness matrix k is assembled.,3.2 整体刚度矩阵的形成,K(i,i) = K(i,i) + k(1,1);K(i,j) =
4、K(i,j) + k(1,2);K(j,i) = K(j,i) + k(2,1);K(j,j) = K(j,j) + k(2,2);y = K;,function y = SpringAssemble(K,k,i,j),2022/11/23,6,结构分析编程及软件应用,3、MATLAB函数编写,3-1 弹簧元,%SpringElementForces This function returns the element nodal force% vector given the element stiffness matrix k % and the element nodal displace
5、ment vector u.,3.3 节点载荷计算,y = k * u;,function y = SpringElementForces(k,u),2022/11/23,7,结构分析编程及软件应用,4、实例计算分析应用,3-1 弹簧元,如图所示二弹簧元结构，假定k1=100kN/m，k2=200kN/m，P=15kN。求：系统的整体刚度矩阵； 节点2、3的位移； 节点1的支反力； 每个弹簧的内力,解：,步骤1：离散化域,2022/11/23,8,结构分析编程及软件应用,4、实例计算分析应用,3-1 弹簧元,步骤2：形成单元刚度矩阵,k1=SpringElementStiffness(100)
6、;,k1 = 100 -100 -100 100,k2=SpringElementStiffness(200)；,k2 = 200 -200 -200 200,调用 function y = SpringElementStiffness(k)函数,2022/11/23,9,结构分析编程及软件应用,4、实例计算分析应用,3-1 弹簧元,步骤3：集成整体刚度矩阵,调用 function y = SpringAssemble(K,k,i,j)函数,n=3; K = zeros(n,n);,K = SpringAssemble(K,k1,1,2),K = 0 0 0 0 0 0 0 0 0,K = S
7、pringAssemble(K,k2,2,3),K = 100 -100 0 -100 100 0 0 0 0,K = 100 -100 0 -100 300 -200 0 -200 200,2022/11/23,10,结构分析编程及软件应用,4、实例计算分析应用,3-1 弹簧元,步骤4：引入边界条件,已知边界条件：,2022/11/23,11,结构分析编程及软件应用,5、实例计算分析应用,3-1 弹簧元,步骤5：解方程,U=zeros(2,1);F=0;15;K = K(2:3,2:3);U=KF,U=inv(K)*F,K(1,:)=;K(:,1)=;,U = 0.1500 0.2250,2
8、022/11/23,12,结构分析编程及软件应用,5、实例计算分析应用,2-1 弹簧元,步骤6：后处理,U=0;U,U = 0 0.1500 0.2250,F=K*U,F = -15.0000 0.0000 15.0000,u1=U(1:2);f1=SpringElementForces(k1,u1);,f1 = -15.0000 15.0000,u2=U(2:3);f2=SpringElementForces(k2,u2);,f2 = -15.0000 15.0000,2022/11/23,13,结构分析编程及软件应用,5、实例计算分析应用,3-1 弹簧元,k1=SpringElementS
9、tiffness(100);k2=SpringElementStiffness(200);n=3;K=zeros(n,n);K=SpringAssemble(K,k1,1,2);K=SpringAssemble(K,k2,2,3);U=zeros(2,1);F=0;15;K = K(2:3,2:3);KK=K;U=KFU=0;U;F=K*U;u1=U(1:2);f1=SpringElementForces(k1,u1)u2=U(2:3);f2=SpringElementForces(k2,u2),2022/11/23,14,结构分析编程及软件应用,1、基本方程,3-2 线性杆元,线性杆元也是总
10、体和局部坐标一致的一维有限单元，用线性函数描述,每个线性杆元有两个节点（node）,单刚矩阵为：,总刚矩阵：,结构方程：,单元节点力：,2022/11/23,15,结构分析编程及软件应用,2、MATLAB函数编写,%LinearBarElementStiffness This function returns the element % stiffness matrix for a linear bar with % modulus of elasticity E, cross-sectional % area A, and length L. The size of the % elemen
11、t stiffness matrix is 2 x 2.,2.1 单元刚度矩阵的形成,y = E*A/L -E*A/L ; -E*A/L E*A/L;,function y = LinearBarElementStiffness(E,A,L),3-2 线性杆元,2022/11/23,16,结构分析编程及软件应用,2、MATLAB函数编写,%LinearBarAssemble This function assembles the element stiffness% matrix k of the linear bar with nodes i and j % into the global
12、stiffness matrix K.% This function returns the global stiffness % matrix K after the element stiffness matrix % k is assembled.,2.2 整体刚度矩阵的形成,K(i,i) = K(i,i) + k(1,1);K(i,j) = K(i,j) + k(1,2);K(j,i) = K(j,i) + k(2,1);K(j,j) = K(j,j) + k(2,2);y = K;,function y =LinearBarAssemble(K,k,i,j),3-2 线性杆元,202
13、2/11/23,17,结构分析编程及软件应用,2、MATLAB函数编写,%LinearBarElementForces This function returns the element nodal % force vector given the element stiffness % matrix k and the element nodal% displacement vector u.,2.3 节点载荷计算,y = k * u;,function y = LinearBarElementForces(k,u),3-2 线性杆元,2022/11/23,18,结构分析编程及软件应用,2、
14、MATLAB函数编写,%LinearBarElementStresses This function returns the element nodal % stress vector given the element stiffness % matrix k, the element nodal displacement % vector u, and the cross-sectional area A.,2.4 节点应力计算,y = k * u/A;,function y = LinearBarElementStresses(k, u, A),3-2 线性杆元,2022/11/23,1
15、9,结构分析编程及软件应用,3、实例计算分析应用,如图所示二线性杆元结构，假定E=210MPa，A=0.003m2，P=10kN， 节点3的右位移为0.002m。求：系统的整体刚度矩阵； 节点2的位移； 节点1、3的支反力； 每个杆件的应力,解：,步骤1：离散化域,3-2 线性杆元,2022/11/23,20,结构分析编程及软件应用,3、实例计算分析应用,步骤2：形成单元刚度矩阵,k1=LinearBarElementStiffness(E,A,L1),k2=LinearBarElementStiffness(E,A,L2),调用 function y = LinearBarElementSt
16、iffness(E,A,L)函数,3-2 线性杆元,2022/11/23,21,结构分析编程及软件应用,3、实例计算分析应用,步骤3：集成整体刚度矩阵,调用 function y = LinearBarAssemble(K,k,i,j)函数,n=3; K = zeros(n,n),K = LinearBarAssemble (K,k1,1,2),K = 0 0 0 0 0 0 0 0 0,K = LinearBarAssemble (K,k2,2,3),3-2 线性杆元,2022/11/23,22,结构分析编程及软件应用,3、实例计算分析应用,步骤4：引入边界条件,已知边界条件：,3-2 线性
17、杆元,2022/11/23,23,结构分析编程及软件应用,3、实例计算分析应用,步骤5：解方程,U=zeros(1,1);U3=0.002F=-10;K = K(2,2) 105000K0 = K(2,3); -630000U=K(F-K0*U3),U =0.0012,3-2 线性杆元,2022/11/23,24,结构分析编程及软件应用,3、实例计算分析应用,步骤6：后处理,U=0;U;0.002,U = 0 0.0012 0.0002,F=K*U,F = -500.0000 -10.0000 510.0000,u1=U(1:2);f1= LinearBarElementForces(k1,u
18、1)sigma1=LinearBarElementStresses(k1, u1, A),u2=U(2:3);f2= LinearBarElementForces(k2,u2)sigma2=LinearBarElementStresses(k2, u2, A),3-2 线性杆元,2022/11/23,25,结构分析编程及软件应用,3、实例计算分析应用,E=210E6;A=0.003;L1=1.5;L2=1;k1= LinearBarElementStiffness(E,A,L1);k2= LinearBarElementStiffness(E,A,L2);n=3; K = zeros(n,n)
19、;K = LinearBarAssemble (K,k1,1,2);K = LinearBarAssemble (K,k2,2,3);U=zeros(1,1);U3=0.002;F=-10;,3-2 线性杆元,KK=K;K=K(2,2);K0=K(2,3);U=K(F-K0*U3);U=0;U;U3;F=KK*Uu1=U(1:2);f1= LinearBarElementForces(k1,u1)sigma1=LinearBarElementStresses(k1, u1, A)u2=U(2:3);f2= LinearBarElementForces(k2,u2)sigma2=LinearBa
20、rElementStresses(k2, u2, A),2022/11/23,26,结构分析编程及软件应用,1、基本方程,3-3 二次杆元,二次杆元也是总体和局部坐标一致的一维有限单元，用二次方程描述,每个线性杆元有三个节点（node）,单刚矩阵为：,总刚矩阵：,结构方程：,单元节点力：,2022/11/23,27,结构分析编程及软件应用,2、MATLAB函数编写,%QuadraticBarElementStiffness This function returns the element % stiffness matrix for a quadratic bar % with modulu
21、s of elasticity E, % cross-sectional area A, and length L. % The size of the element stiffness % matrix is 3 x 3.,2.1 单元刚度矩阵的形成,y = E*A/(3*L)*7 1 -8 ; 1 7 -8 ; -8 -8 16;,function y = QuadraticBarElementStiffness(E,A,L),3-3 二次杆元,2022/11/23,28,结构分析编程及软件应用,2、MATLAB函数编写,%QuadraticBarAssemble This functi
22、on assembles the element stiffness% matrix k of the quadratic bar with nodes i, j % and m into the global stiffness matrix K.% This function returns the global stiffness % matrix K after the element stiffness matrix % k is assembled.,2.2 整体刚度矩阵的形成,K(i,i) = K(i,i) + k(1,1);K(i,j) = K(i,j) + k(1,2);K(
23、i,m) = K(i,m) + k(1,3);K(j,i) = K(j,i) + k(2,1);K(j,j) = K(j,j) + k(2,2);,function y =QuadraticBarAssemble(K,k,i,j,m),3-3 二次杆元,K(j,m) = K(j,m) + k(2,3);K(m,i) = K(m,i) + k(3,1);K(m,j) = K(m,j) + k(3,2);K(m,m) = K(m,m) + k(3,3);y = K;,2022/11/23,29,结构分析编程及软件应用,2、MATLAB函数编写,%QuadraticBarElementForces
24、This function returns the element nodal % force vector given the element stiffness % matrix k and the element nodal % displacement vector u.,2.3 节点载荷计算,y = k * u;,function y = QuadraticBarElementForces(k,u),3-3 二次杆元,2022/11/23,30,结构分析编程及软件应用,2、MATLAB函数编写,%QuadraticBarElementStresses This function re
25、turns the element % nodal stress vector given the element % stiffness matrix k, the element nodal % displacement vector u, and the % cross-sectional area A.,2.4 节点应力计算,y = k * u/A;,function y = QuadraticBarElementStresses(k, u, A),3-3 二次杆元,2022/11/23,31,结构分析编程及软件应用,3、实例计算分析应用,如图所示双二次杆元结构，假定E=210MPa，
26、A=0.003m2求：系统的整体刚度矩阵； 节点2、3、4、5的位移； 节点1的支反力； 每个杆件的应力,解：,3-3 二次杆元,2022/11/23,32,结构分析编程及软件应用,3、实例计算分析应用,E=210E6;A=0.003;L=2;k1= QuadraticBarElementStiffness(E,A,L);k2= QuadraticBarElementStiffness(E,A,L);n=5; K = zeros(n,n);K =QuadraticBarAssemble(K,k1,1,3,2);K =QuadraticBarAssemble(K,k2,3,5,4);U=zero
27、s(4,1);F=5;-10;-7;10;,KK=K;K=K(2:n,2:n);U=KF;U=0;U;F=KK*U;u1=U(1);U(3);U(2);f1= QuadraticBarElementForces(k1,u1);sigma1=QuadraticBarElementStresses(k1, u1, A);u2=U(3);U(5);U(4);f2=QuadraticBarElementForces(k2,u2);sigma2=QuadraticBarElementStresses(k2, u2, A);,3-3 二次杆元,2022/11/23,33,结构分析编程及软件应用,1、基本方
28、程,3-4 平面桁架元,平面桁架元是既有局部坐标又有总体坐标二维有限元，用线性函数描述,每个平面桁架元有二个节点（node）,单刚矩阵为：,总刚矩阵：,结构方程：,单元节点力：,2022/11/23,34,结构分析编程及软件应用,2、MATLAB函数编写,%PlaneTrussElementLength This function returns the length of the% plane truss element whose first node has % coordinates (x1,y1) and second node has % coordinates (x2,y2).,
29、2.1 计算单元长度,y = sqrt(x2-x1)*(x2-x1) + (y2-y1)*(y2-y1);,function y = PlaneTrussElementLength(x1,y1,x2,y2),3-4 平面桁架元,2022/11/23,35,结构分析编程及软件应用,2、MATLAB函数编写,%PlaneTrussElementStiffness This function returns the element % stiffness matrix for a plane truss % element with modulus of elasticity E, % cross-
30、sectional area A, length L, and% angle theta (in degrees).% The size of the element stiffness % matrix is 4 x 4.,2.2 单元刚度矩阵的形成,x = theta*pi/180;C = cos(x);S = sin(x);y = E*A/L*C*C C*S -C*C -C*S ; C*S S*S -C*S -S*S ; -C*C -C*S C*C C*S ; -C*S -S*S C*S S*S;,function y = PlaneTrussElementStiffness(E,A,L
31、, theta),3-4 平面桁架元,2022/11/23,36,结构分析编程及软件应用,2、MATLAB函数编写,%PlaneTrussAssemble This function assembles the element stiffness% matrix k of the plane truss element with nodes% i and j into the global stiffness matrix K.% This function returns the global stiffness % matrix K after the element stiffness
32、matrix k is assembled.,2.3 整体刚度矩阵的形成,K(2*i-1,2*i-1) = K(2*i-1,2*i-1) + k(1,1);K(2*i-1,2*i) = K(2*i-1,2*i) + k(1,2);K(2*i-1,2*j-1) = K(2*i-1,2*j-1) + k(1,3);K(2*i-1,2*j) = K(2*i-1,2*j) + k(1,4);K(2*i,2*i-1) = K(2*i,2*i-1) + k(2,1);K(2*i,2*i) = K(2*i,2*i) + k(2,2);K(2*i,2*j-1) = K(2*i,2*j-1) + k(2,3);
33、K(2*i,2*j) = K(2*i,2*j) + k(2,4);,function y =PlaneTrussAssemble(K,k,i,j),K(2*j-1,2*i-1) = K(2*j-1,2*i-1) + k(3,1);K(2*j-1,2*i) = K(2*j-1,2*i) + k(3,2);K(2*j-1,2*j-1) = K(2*j-1,2*j-1) + k(3,3);K(2*j-1,2*j) = K(2*j-1,2*j) + k(3,4);K(2*j,2*i-1) = K(2*j,2*i-1) + k(4,1);K(2*j,2*i) = K(2*j,2*i) + k(4,2);
34、K(2*j,2*j-1) = K(2*j,2*j-1) + k(4,3);K(2*j,2*j) = K(2*j,2*j) + k(4,4);y = K;,3-4 平面桁架元,2022/11/23,37,结构分析编程及软件应用,2、MATLAB函数编写,%PlaneTrussElementForce This function returns the element force% given the modulus of elasticity E, the % cross-sectional area A, the length L, % the angle theta (in degrees)
35、, and the % element nodal displacement vector u.,2.4 节点载荷计算,x = theta * pi/180;C = cos(x);S = sin(x);y = E*A/L*-C -S C S* u;,function y = PlaneTrussElementForce(E,A,L,theta,u),3-4 平面桁架元,2022/11/23,38,结构分析编程及软件应用,2、MATLAB函数编写,%PlaneTrussElementStress This function returns the element stress% given th
36、e modulus of elasticity E, the % the length L, the angle theta (in % degrees), and the element nodal % displacement vector u.,2.5 节点应力计算,x = theta * pi/180;C = cos(x);S = sin(x);y = E/L*-C -S C S* u;,function y = PlaneTrussElementStress(E,L,theta,u),3-4 平面桁架元,2022/11/23,39,结构分析编程及软件应用,3、实例计算分析应用,如图所
37、示平面桁架结构，假定E=210MPa，A=0.0004m2求：系统的整体刚度矩阵； 节点2的水平位移； 节点3的水平竖向位移； 节点1、2的支反力； 每跟杆件的应力,3-4 平面桁架元,2022/11/23,40,结构分析编程及软件应用,1、基本方程,3-5 空间桁架元,空间桁架元是既有局部坐标又有总体坐标三维有限元，用线性函数描 述。各单元之间通过铰接系统连接，只能传递力，而不能传递弯矩,每个桁架元有二个节点（node）,2022/11/23,41,结构分析编程及软件应用,1、基本方程,3-5 空间桁架元,总刚矩阵：,结构方程：,单元节点力：,单刚矩阵为：,2022/11/23,42,结构分
38、析编程及软件应用,2、MATLAB函数编写,%SpaceTrussElementLength This function returns the length of the% space truss element whose first node has % coordinates (x1,y1,z1) and second node has % coordinates (x2,y2,z2).,2.1 计算单元长度,y = sqrt(x2-x1)*(x2-x1) + (y2-y1)*(y2-y1) + (z2-z1)*(z2-z1);,function y = SpaceTrussEleme
39、ntLength(x1,y1,z1,x2,y2,z2),3-5 空间桁架元,2022/11/23,43,结构分析编程及软件应用,2、MATLAB函数编写,%SpaceTrussElementStiffness This function returns the element % stiffness matrix for a space truss % element with modulus of elasticity E, % cross-sectional area A, length L, and% angles thetax, thetay, thetaz % (in degrees
40、). The size of the element % stiffness matrix is 6 x 6.,2.2 单元刚度矩阵的形成,x = thetax*pi/180;u = thetay*pi/180;v = thetaz*pi/180;Cx = cos(x);Cy = cos(u);Cz = cos(v);w = Cx*Cx Cx*Cy Cx*Cz ; Cy*Cx Cy*Cy Cy*Cz ; Cz*Cx Cz*Cy Cz*Cz;y = E*A/L*w -w ; -w w;,function y = SpaceTrussElementStiffness(E,A,L,thetax,th
41、etay,thetaz),3-5 空间桁架元,2022/11/23,44,结构分析编程及软件应用,2、MATLAB函数编写,%SpaceTrussAssemble This function assembles the element stiffness% matrix k of the space truss element with nodes% i and j into the global stiffness matrix K.% This function returns the global stiffness % matrix K after the element stiffn
42、ess matrix % k is assembled.,2.3 整体刚度矩阵的形成,K(3*i-2,3*i-2) = K(3*i-2,3*i-2) + k(1,1);K(3*i-2,3*i-1) = K(3*i-2,3*i-1) + k(1,2);K(3*i-2,3*i) = K(3*i-2,3*i) + k(1,3);K(3*i-2,3*j-2) = K(3*i-2,3*j-2) + k(1,4);K(3*i-2,3*j-1) = K(3*i-2,3*j-1) + k(1,5);K(3*i-2,3*j) = K(3*i-2,3*j) + k(1,6);K(3*i-1,3*i-2) = K(
43、3*i-1,3*i-2) + k(2,1);K(3*i-1,3*i-1) = K(3*i-1,3*i-1) + k(2,2);K(3*i-1,3*i) = K(3*i-1,3*i) + k(2,3);K(3*i-1,3*j-2) = K(3*i-1,3*j-2) + k(2,4);K(3*i-1,3*j-1) = K(3*i-1,3*j-1) + k(2,5);K(3*i-1,3*j) = K(3*i-1,3*j) + k(2,6);,function y =SpaceTrussAssemble(K,k,i,j),3-5 空间桁架元,2022/11/23,45,结构分析编程及软件应用,2、MA
44、TLAB函数编写,2.3 整体刚度矩阵的形成,3-5 空间桁架元,K(3*j-1,3*i-2) = K(3*j-1,3*i-2) + k(5,1);K(3*j-1,3*i-1) = K(3*j-1,3*i-1) + k(5,2);K(3*j-1,3*i) = K(3*j-1,3*i) + k(5,3);K(3*j-1,3*j-2) = K(3*j-1,3*j-2) + k(5,4);K(3*j-1,3*j-1) = K(3*j-1,3*j-1) + k(5,5);K(3*j-1,3*j) = K(3*j-1,3*j) + k(5,6);K(3*j,3*i-2) = K(3*j,3*i-2) +
45、 k(6,1);K(3*j,3*i-1) = K(3*j,3*i-1) + k(6,2);K(3*j,3*i) = K(3*j,3*i) + k(6,3);K(3*j,3*j-2) = K(3*j,3*j-2) + k(6,4);K(3*j,3*j-1) = K(3*j,3*j-1) + k(6,5);K(3*j,3*j) = K(3*j,3*j) + k(6,6);y = K;,K(3*i,3*i-2) = K(3*i,3*i-2) + k(3,1);K(3*i,3*i-1) = K(3*i,3*i-1) + k(3,2);K(3*i,3*i) = K(3*i,3*i) + k(3,3);K
46、(3*i,3*j-2) = K(3*i,3*j-2) + k(3,4);K(3*i,3*j-1) = K(3*i,3*j-1) + k(3,5);K(3*i,3*j) = K(3*i,3*j) + k(3,6);K(3*j-2,3*i-2) = K(3*j-2,3*i-2) + k(4,1);K(3*j-2,3*i-1) = K(3*j-2,3*i-1) + k(4,2);K(3*j-2,3*i) = K(3*j-2,3*i) + k(4,3);K(3*j-2,3*j-2) = K(3*j-2,3*j-2) + k(4,4);K(3*j-2,3*j-1) = K(3*j-2,3*j-1) +
47、k(4,5);K(3*j-2,3*j) = K(3*j-2,3*j) + k(4,6);,2022/11/23,46,结构分析编程及软件应用,2、MATLAB函数编写,%SpaceTrussElementForce This function returns the element force% given the modulus of elasticity E, the % cross-sectional area A, the length L, % the angles thetax, thetay, thetaz% (in degrees), and the element nodal
48、 % displacement vector u.,2.4 节点载荷计算,x = thetax * pi/180;w = thetay * pi/180;v = thetaz * pi/180;Cx = cos(x);Cy = cos(w);Cz = cos(v);y = E*A/L*-Cx -Cy -Cz Cx Cy Cz*u;,function y = SpaceTrussElementForce(E,A,L,thetax,thetay,thetaz,u),3-5 空间桁架元,2022/11/23,47,结构分析编程及软件应用,2、MATLAB函数编写,%SpaceTrussElement
49、Stress This function returns the element stress% given the modulus of elasticity E, the % length L, the angles thetax, thetay, % thetaz (in degrees), and the element % nodal displacement vector u.,2.5 节点应力计算,x = thetax * pi/180;w = thetay * pi/180;v = thetaz * pi/180;Cx = cos(x);Cy = cos(w);Cz = cos
50、(v);y = E/L*-Cx -Cy -Cz Cx Cy Cz*u;,function y = SpaceTrussElementStress(E,L,thetax,thetay,thetaz,u),3-5 空间桁架元,2022/11/23,48,结构分析编程及软件应用,3、实例计算分析应用,如图所示空间桁架结构，假定E=210MPa，A14=0.001m2 A24=0.002m2，A34=0.001m2，P=12kN求：系统的整体刚度矩阵； 节点4的水平位移； 节点3的水平竖向位移； 节点1、2、3的支反力； 每跟杆件的应力,3-5 空间桁架元,2022/11/23,49,结构分析编程及软
