FEA code in Matlab for a Truss Structure

By: Shiwei Zhou

Outline:

MATLAB programs for 2D trussVerification of the MATLAB code by

Strand72D Truss transmitter towerStrand7 example for 2D truss

transmitter tower

Input Geometrical Model

1. Node coordinate2. Element connection3. Force vector4. Displacement vector5. Boundary conditoins

Element Stiffness Matrix

Element Stiffness Matrix

Assemble Element Stiffness Matrix to Globe Matrix

Boundary Conditions

Calculate Ax=b

Calculate Node Force

Internal force

Internal Force

Exercise

Node x y1 4 02 4 33 0 04 0 3

Elem P1 P21 1 22 2 33 4 2

Boundary Conditions:Node 1, 3 ,4 are fixed:D=[0

-0.025D2xD2y0000];

F=[F1xF1y00F3xF3yF4xF4y]

2 kN

D=[00 D2xD2y000

0]

Q=[Q1xQ1y 20Q3xQ3y

Q4xQ4y]

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) = -0.025; U(2) = -0.025;syms U3 U4U =[0 -0.025 U3 U4 0 0 0 0].';%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 matrixKe = 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

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 forcefor 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*NNfprintf(['U(%i) = %5.3f\n'],i,D(i));

end

for i=1:2*NNfprintf(['F(%i) = %5.3f\n'],i,F(i));

end

for i=1:NEfprintf(['q(%i) = %5.3f\n'],i,q(i));

end

Flowchart for Exercise

Go to Computer Lab

Develop a MATLAB code for the Exercise!

Verify Homework in Strand7

Create Node Create Beam Set BCs Apply load

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

Linear Static Analysis Log File

Model

Results

/ ______________________________________________________________________________/ Strand7 MODEL EXCHANGE FILE/ TIMESTAMP: 10:50:08 am, 18 August 2014

/ ______________________________________________________________________________/ MODEL INFORMATION

FileFormat Strand7.2.4.4ModelName "HOMEWORK"Title ""Project ""Author ""Reference ""Comments ""

/ ______________________________________________________________________________/ UNITS

LengthUnit mMassUnit kgEnergyUnit JPressureUnit PaForceUnit NTemperatureUnit K

/ ______________________________________________________________________________/ GROUP DEFINITIONS

Group 1 16711680 "\\Model"

/ ______________________________________________________________________________/ FREEDOM CASE DEFINITIONS

FreedomCase 1 0 1 "Freedom Case 1"

Output Strand7 Model in Txt Format

/ ______________________________________________________________________________/ LOAD CASE DEFINITIONS

LoadCase 1 0 "Load Case 1"LCInclude 3

/ ______________________________________________________________________________/ COORDINATE SYSTEM DEFINITIONS

CoordSys 1 "Global XYZ" GlobalXYZ

/ ______________________________________________________________________________/ NODE COORDINATES

Node 1 0 4.00000000000000E+0 0.00000000000000E+0 0.00000000000000E+0 Node 2 0 4.00000000000000E+0 3.00000000000000E+0 0.00000000000000E+0 Node 3 0 0.00000000000000E+0 0.00000000000000E+0 0.00000000000000E+0 Node 4 0 0.00000000000000E+0 3.00000000000000E+0 0.00000000000000E+0

/ ______________________________________________________________________________/ BEAM ELEMENTS

Beam 1 0 1 1 1 2 Beam 2 0 1 1 2 3 Beam 3 0 1 1 4 2

/ ______________________________________________________________________________/ NODE RESTRAINTS (ROTATION AS RADIAN)/ Freedom Case 1

NdFreedom 1 1 1 DX NdFreedom 1 3 1 DX DY NdFreedom 1 4 1 DX DY

/ ______________________________________________________________________________/ NODE FORCES/ Load Case 1

NdForce 1 2 -1.92000000000000E+4 -1.44000000000000E+4 0.00000000000000E+0

/ BEAM PROPERTIESTrussProp 1 16737843 "Beam Property 1"MaterialName "Unknown Material - Modified"Modulus 8.00000000000000E+10 UsePoisson TRUEInstantAlpha FALSEArea 1.00000000000000E-4 MomentJ 1.40700000000000E-9 SectionType SolidRectB 1.00000000000000E-2 D 1.00000000000000E-2 CT FALSE

IncludeTorsion FALSENonLinType ElasticplasticHardening Isotropic

________________/ LINEAR STATIC SOLVER DATA

LoadFreedomSetLSA 1 ON1

/ LINEAR BUCKLING SOLVER DATABuckNumModes 4 BuckShift 0.00000000000000E+0

/ LOAD INFLUENCE SOLVER DATA

LoadFreedomSetLIA 1 ON1

/ GENERAL SOLVER DATA

SolverTempDependence None

SolverLoadCaseTempDependence 0

SolverActiveStage 0

SturmCheck FALSE

SolverFreedomCase 1

A Matlab Code for 2D Transmitter tower

Flow Chart clc;clear;close all;

% read node and element information

N = textread('node.txt'); E = textread('elem.txt');

FN = [11 12 17 18 23 24]; 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 = [1 2 3 4]; % 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;

Displacement

Strain Distribution

Result Comparison Between Matlab and Strand 7

1.000000000000000 0 02.000000000000000 0 03.000000000000000 -0.056705187259539 -0.0576968260678554.000000000000000 -0.020358528001983 0.0125597822448235.000000000000000 -0.045999647171697 -0.1727331924041316.000000000000000 -0.053109054594353 0.0302022273735127.000000000000000 -0.263157875158521 -0.2377748339796678.000000000000000 -0.251268324026876 0.0588360635482129.000000000000000 -0.521945256702415 -0.290906572903267

10.000000000000000 -0.530747504449517 0.07596093293729111.000000000000000 -0.693619429698391 -1.00875692279681712.000000000000000 -0.681789204532035 0.52192282179440313.000000000000000 -0.848748038328659 -0.33209434637371014.000000000000000 -0.825251678313140 0.08427213350532615.000000000000000 -1.165271291297121 -0.36005349681014516.000000000000000 -1.183802187560931 0.08753814231897217.000000000000000 -1.358028554905666 -1.54354170320105018.000000000000000 -1.352392698081489 0.67629163903515919.000000000000000 -1.540139025658684 -0.37879029354935520.000000000000000 -1.514214220162403 0.08688591697876921.000000000000000 -1.877909261927200 -0.38969398160077122.000000000000000 -1.894365603525534 0.08378851873987423.000000000000000 -2.069597330694618 -1.18092861598961124.000000000000000 -2.067850950761376 0.61204378049250725.000000000000000 -2.254750932371863 -0.39417439839425926.000000000000000 -2.236519903204394 0.081969337109746

Strand7 Matlab

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)