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MATLAB CODE FOR 3D TRUSS

A matlab code is developed to solve the 3D truss problem as given in D.A Hutton Fundamental of finite element analysis. The problem is analyzed in ansys to verify the results.

Problem

The three member truss is connected by a ball and socket joint and fixed at nodes 1,3 and 4 as shown in the figure below. A 5000-lb is force is applied in negative Y direction at node 4.The area, length and modulus of elasticity of each element is 0.5 in2,50 in and 30x106 psi. Compute the displacements at node 4.

Fig (a) shows the coordinates of nodes in global coordinate system and the load acting on the structure.

Fig(b) Shows the nodal and element numbering

Matlab code clear all; clc;

% NCA(Nodal collection array) is 2d array consisting of element number ,node

% numbers,area,Young's modulus, coordinates of nodes corresponding to the

% each element in row wise.

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NCA=[1 1 4 0.5 30*10^6 50 0 0 30 40 0 0;2 2 4 0.5 30*10^6 50 0 0 -30 40 0 0 ;3 3 4 0.5 30*10^6 50 0 -30 0 40 0 0]; for EN=1:3 A=NCA(EN,4); EM=NCA(EN,5); L=NCA(EN,6); K=A*EM/L; % LSTIFF is stiffness matrix(2x2) element LSTIFF=[K -K;-K K]; % TM is transformation matrix of elements TM(:,:,EN) =[(NCA(EN,10)-NCA(EN,7))/NCA(EN,6) (NCA(EN,11)- NCA(EN,8))/NCA(EN,6) (NCA(EN,12)-NCA(EN,9))/NCA(EN,6) 0 0 0;0 0 0 (NCA(EN,10)-NCA(EN,7))/NCA(EN,6) (NCA(EN,11)-NCA(EN,8))/NCA(EN,6) (NCA(EN,12)-NCA(EN,9))/NCA(EN,6)]; % ESTIFF is stiffness matrix stored as 3D matrix ESTIFF(:,:, EN)=(TM(:,:,EN))'*LSTIFF*(TM(:,:,EN)); % Row numbering and column numbering relating element stiffness matrix and global stiffness matrix through nodal numbering nodal numbering

for NN=1:2 RN(EN,3*NN-2)=3*NCA(EN,NN+1)-2; RN(EN,3*NN-1)=3*NCA(EN,NN+1)-1; RN(EN,3*NN)=3*NCA(EN,NN+1); end

for NN=1:2 CN(EN,3*NN-2)=3*NCA(EN,NN+1)-2; CN(EN,3*NN-1)=3*NCA(EN,NN+1)-1;

CN(EN,3*NN)=3*NCA(EN,NN+1); end

end % GSTIFF Global stiffness matrix % GSTIFF initialized to zero GSTIFF=zeros(12); for EN=1:3 for I=1:6 for J=1:6 GSTIFF(RN(EN,I),CN(EN,J))=GSTIFF(RN(EN,I),CN(EN,J))+ESTIFF(I,J,EN); end end end % boundary conditions % The constrained degree of freedom is assigned as 1 and free degree of freedom as 0

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stat=[1 1 1 1 1 1 1 1 1 0 0 0]; % KK – Stiffness matrix after imposing boundary conditions p=0; for i=1:12 if(~stat(i))

p=p+1; q=0;

for j=1:12 if(~stat(j)) q=q+1; KK(p,q)=GSTIFF(i,j); end end end end % Force acting on the truss corresponding to displacements to be determine FF=[0 -5000 0]'; % solving the equation for displacements disp('displacements'); qq=KK\FF; disp(qq); % stress and forces calculation E=30*10^6; for EN=1:3 % Stresses in elements stress(:,EN)=E*[-1/NCA(EN,6) 1/NCA(EN,6)]*TM(:,:,EN)*[0 0 0 qq(1) qq(2) qq(3)]'; % Forces at nodes Force(:,EN)=ESTIFF(:,:,EN)*[0 0 0 qq(1) qq(2) qq(3)]'; % Resultant force at nodes force(:,EN)= TM(:,:,EN)*ESTIFF(:,:,EN)*[0 0 0 qq(1) qq(2) qq(12)]'; end EN=1:3; disp('Stresses'); fprintf('%f\n\n',stress(:,EN)); disp('Resultant forces at nodes'); fprintf('%f\n\n',force(:,EN)); disp('Forces at nodes'); fprintf('%f\n\n',Force(:,EN));

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ANSYS procedure

The key steps involved in analysis the model using the ANSYS is detailed below

Preprocessing

1.Specify title

2.Specify preferences

3.Define elements

4.Define real contants

5.Define material properties

6.Define nodes

7. Define elements

Solution

8. Apply displacements

9. Apply loads

10.Solve

Post processing

11.Read results

12. Print deformed results

13. Exit ansys

Preprocessing

1.Specify title

Utility menu: File Change title

a. Enter the title ‘Static analysis of 3D truss’

b. Click on ok

2. Specify preferences

Main menu: Preferences

a. Turn on Structural

b. Click on ok

3. Define element type

Main menu: Preprocessor Element type Add/edit/delete

a. Click on Add

b. Select Structural LINK family elements

c. Select 3D spar 8.

d. Click on ok

e. Click on close

4. Define Real constants

Main menu: Preprocessor Real constants add/edit/delete

a. Click on add

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b. Click on OK

c. Enter 0.5 for area

d. Click on ok and close

5. Define material properties

Main menu: preprocessor Material Properties Material models Structural

Linear elastic isotropic

a. Click on ok for material type 1

b. Enter 30e6 and 0.25 for PRXY

c. Click on ok

6. Define nodes

Main menu: Preprocessor Modelling_create Nodes In active CS

a. Enter 1 for node number

b. Enter (0,0,30) for X,Y,Z locations

c. Enter 2 for node number

d. Enter (0,0,-30) for X,Y,Z locations

e. Enter 3 for node number

f. Enter (0,-30,0) for X,Y,Z locations

g. Enter 4 or node number

h. Enter (40,0,0) for X,Y,Z locations

7. Define elements

Main menu: Preprocessor modelling Create Elements Auto numbered

Through nodes

a. Pick the nodes 1 and 4.

b. Click on apply

c. Pick the nodes 2 and 4

d. Click on apply

e. Pick the nodes 3 and 4

f. Click on apply

Solution

8. Apply displacements

Main menu: Solution Define loads Apply Structural displacement On nodes

a. Pick the node 1,2 and 3.

b. Click on apply

c. Select UX,UY,UZ

d. Enter 0 against value

e. Click on ok

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9.Apply loads

Main menu: Solution Loads Apply Structural force/moment on nodes

a. Pick the node 4.

b. Click on apply.

c. Select FY .

d. Enter -5000 for value.

e. Click on ok.

10. Solve

Min menu: Solution Solve Current LS

a. click ok and close.

Post Processing

11. Read results

Main menu: General postproc Read results First set

12. Print deformed results

Main menu: General postproc List results Nodal solution

a. Select DOF solution.

b. Select displacement sum

c. Click on ok

13. Plot member forces

Main menu: General postproc Element table Define table

a. Add element table data

b. Enter Labs as MEMBER

c. Scroll data and choose sequence number

d. Choose SMISC

e. Enter 1 in the selection field after SMISC

f. Click on ok

g. Close

13. Print deformed shape

Main menu:General postproc Plot results

a. Click on deformed + undeformed

b. Click ok

14. Exit

Toolbar :quit

a. Click on quit

b. Click on ok.

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Results

The results of the matlab code agrees with the solution obtained by solving the problem in

ANSYS.

The nodal displacements as obtained from the ansys and matlab code are tabled as follows.

Node UX UY UZ 1 0 0 0 2 0 0 0 3 0 0 0 4 0.01736 -0.06944 0

The member forces as obtained from the ansys and matlab code are tabled as follows

Element no Forces(lb) Stresses(lb/in2) 1 4166.7 8333.33 2 4166.7 8333.33 3 -8333.33 -16666.66

Fig 1 The loading and boundary conditions in 3d truss

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Fig 2 The deformed geometry of the 3D truss

Fig 3 The maximum and the minimum forces acting on the 3d truss