FEA code in Matlab for a Truss Structure. By: Shiwei Zhou

Transcription

1 FEA code in Matlab for a Truss Structure By: Shiwei Zhou

2 Outline: MATLAB programs for 2D truss Verification of the MATLAB code by Strand7 2D Truss transmitter tower Strand7 example for 2D truss transmitter tower

3 Input Geometrical Model 1. Node coordinate 2. Element connection 3. Force vector 4. Displacement vector 5. Boundary conditoins

4 Element Stiffness Matrix

5 Element Stiffness Matrix

6 Assemble Element Stiffness Matrix to Globe Matrix

7 Boundary Conditions

8 Calculate Ax=b

9 Calculate Node Force

10 Internal force

11 Internal Force

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13 Exercise

14 Node x y Elem P1 P

15 D=[0 0 D2x D2y ] Boundary Conditions: Node 1, 3,4 are fixed: D=[ D2x D2y ]; F=[F1x F1y 0 0 F3x F3y F4x F4y] 2 kn Q=[Q1x Q1y 2 0 Q3x Q3y Q4x Q4y]

16 Flowchart for Exercise clc;clear; close all; AE=8e6; % Determine the node coordinates % Determine the truss element N=[]; E=[]; %The definitation of loads F=zeros(2*NN,1); D=zeros(2*NN,1); D(2) = ; U(2) = ; syms U3 U4 U =[ U3 U ].'; %Assembling the global matrix K=zeros(2*NN,2*NN); for n=1:ne L = sqrt((n(e(n,1),1)-n(e(n,2),1))^2 + (N(E(n,1),2)-N(E(n,2),2))^2); lx = (N(E(n,2),1) - N(E(n,1),1))/L; ly = (N(E(n,2),2) - N(E(n,1),2))/L; %element matrix Ke = AE*[ lx*lx lx*ly -lx*lx -lx*ly lx*ly ly*ly -lx*ly -ly*ly -lx*lx -lx*ly lx*lx lx*ly -lx*ly -ly*ly lx*ly ly*ly]/l; i = E(n,1); j = E(n,2); K([2*i-1 2*i 2*j-1 2*j],[2*i-1 2*i 2*j-1 2*j]) = K([2*i-1 2*i 2*j-1 2*j],[2*i-1 2*i 2*j-1 2*j]) + Ke; end

17 Flowchart for Exercise T=K*U; Q=solve(T(3),T(4)); D(3) = subs(q.u3,'x',1); D(4) = subs(q.u4,'x',1); F=K*D; %Calculate internal force for n=1:ne L = sqrt((n(e(n,1),1)-n(e(n,2),1))^2 + (N(E(n,1),2)- N(E(n,2),2))^2); lx = (N(E(n,2),1) - N(E(n,1),1))/L; ly = (N(E(n,2),2) - N(E(n,1),2))/L; P1 = E(n,1); P2 = E(n,2); q(n) = AE*[-lx -ly lx ly]*d([2*p1-1 2*P1 2*P2-1 2*P2])/L; end for i=1:2*nn fprintf(['u(%i) = %5.3f\n'],i,D(i)); end for i=1:2*nn fprintf(['f(%i) = %5.3f\n'],i,F(i)); end for i=1:ne fprintf(['q(%i) = %5.3f\n'],i,q(i)); end

18 Go to Computer Lab Develop a MATLAB code for the Exercise!

19 Verify Homework in Strand7 Create Node Create Beam Set BCs Apply load

20 Set up Young s Modulus Set up area of cross section

21 Linear Static Analysis Log File

22 Model

23 Results

24 Output Strand7 Model in Txt Format / / Strand7 MODEL EXCHANGE FILE / TIMESTAMP: 10:50:08 am, 18 August 2014 / / MODEL INFORMATION FileFormat Strand ModelName "HOMEWORK" Title "" Project "" Author "" Reference "" Comments "" / / UNITS LengthUnit m MassUnit kg EnergyUnit J PressureUnit Pa ForceUnit N TemperatureUnit K / / GROUP DEFINITIONS Group "\\Model" / / FREEDOM CASE DEFINITIONS FreedomCase "Freedom Case 1"

25 / / LOAD CASE DEFINITIONS LoadCase 1 0 "Load Case 1" LCInclude 3 / / COORDINATE SYSTEM DEFINITIONS CoordSys 1 "Global XYZ" GlobalXYZ / / NODE COORDINATES Node E E E+0 Node E E E+0 Node E E E+0 Node E E E+0 / / BEAM ELEMENTS Beam Beam Beam / / NODE RESTRAINTS (ROTATION AS RADIAN) / Freedom Case 1 NdFreedom DX NdFreedom DX DY NdFreedom DX DY / / NODE FORCES / Load Case 1 NdForce E E E+0

26 / BEAM PROPERTIES TrussProp "Beam Property 1" MaterialName "Unknown Material - Modified" Modulus E+10 UsePoisson TRUE InstantAlpha FALSE Area E-4 MomentJ E-9 SectionType SolidRect B E-2 D E-2 CT FALSE IncludeTorsion FALSE NonLinType Elasticplastic Hardening Isotropic / LINEAR STATIC SOLVER DATA LoadFreedomSetLSA 1 ON 1 / LINEAR BUCKLING SOLVER DATA BuckNumModes 4 BuckShift E+0 / LOAD INFLUENCE SOLVER DATA LoadFreedomSetLIA 1 ON 1 / GENERAL SOLVER DATA SolverTempDependence None SolverLoadCaseTempDependence 0 SolverActiveStage 0 SturmCheck FALSE SolverFreedomCase 1

27 A Matlab Code for 2D Transmitter tower

28 Flow Chart clc;clear;close all; % read node and element information N = textread('node.txt'); E = textread('elem.txt'); FN = [ ]; F(2*FN-1) = -920*sin(20/180*pi); F(2*FN) = -920*cos(20/180*pi); %Assembling the global matrix K=zeros(2*NN,2*NN); for n=1:ne L = sqrt((n(e(n,1),1)-n(e(n,2),1))^2 + (N(E(n,1),2)-N(E(n,2),2))^2); i = E(n,1); j = E(n,2); K([2*i-1 2*i 2*j-1 2*j],[2*i-1 2*i 2*j-1 2*j]) = K([2*i-1 2*i 2*j-1 2*j],[2*i-1 2*i 2*j-1 2*j]) + Ke; end fixeddofs = [ ]; % Fix end supports with zero deformation alldofs = [1:2*NN]; freedofs = setdiff(alldofs,fixeddofs); U(freedofs,:) = K(freedofs,freedofs)\F(freedofs,:); U(fixeddofs,:)= 0; for n=1:ne P1 = E(n,1); P2 = E(n,2); P(n) = A0*E0*[-lx -ly lx ly]*u([2*p1-1 2*P1 2*P2-1 2*P2])/L; end %Calculate the stress and strain stress = P/A0; strain = stress/e0;

29 Displacement

30 Strain Distribution

31 Result Comparison Between Matlab and Strand 7 Strand7 Matlab

32 Debug and Helps for MATLA Code clc;clear;close all; % read node and element information N = textread('node.txt'); E = textread('elem.txt'); %the numbers of nodes and elements NN = length(n); NE = length(e); %Young's Modulus and cross section area E0 = 2.1e6; A0 = 6.91; %plot the element hold on; for i=1:ne plot([n(e(i,1),1),n(e(i,2),1)],[n(e(i,1),2),n(e(i,2),2)],'k','linewidth',2); text((n(e(i,1),1)+n(e(i,2),1))/2,(n(e(i,1),2)+n(e(i,2),2))/2,num2str(i),'color','b','fontsize',12); end %plot the nodes for i=1:nn if i<3 plot(n(i,1),n(i,2),'r^','markersize',12,'markerfacecolor','r'); %plot fixed supports else plot(n(i,1),n(i,2),'ko','markersize',8,'markerfacecolor','k'); end t t(n(i 1) N(i 2) 2 t (i) ' l ' ' ' 'f t i ' 12)

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