ENGN1300: Mechanics of Structures
 FEM Truss Analysis Program.

 

Truss.mws is a two-dimensional finite element truss analysis code, written in MAPLE. It is quite general, in that it can handle an arbitrary number of joints, members, loads, and boundary conditions. The program illustrates how simple the matrix truss analysis is to implement. The worst parts of the program are those that involve the graphics!

The code reads an input file which specifies the coordinates of all the joints, the connectivity of the members and  the relevant  material and geometrical data for each and the boundary conditions and load conditions. The input file must have the following format:

Run Title

Number of joints (nodes)  N,
coordinate of node 1, coordinate of node 1,
coordinate of node 2, coordinate of node 2,
…
coordinate of node N, coordinate of node N,
Number of members (elements) M,
For element 1: 1st node number, 2nd node number , value of EA

For element 2: 1st node number, 2nd node number, value of EA

…

For element M: 1st node number, 2nd node number , value of EA

Number of prescribed displacements L,
Node number of 1st prescribed displacement, component (1 or 2), value of the displacement,
Node number of 2nd prescribed displacement, component (1 or 2), value of the displacement,
…
Node number of Lth prescribed displacement, component (1 or 2), value of the displacement,
Number of loaded nodes K,
Number of first loaded node, force component , force component
Number of second loaded node, force component , force component
…

Number of Kth loaded node, force component , force component
 

On the screen, you are asked the pathname of the input file. A picture of the truss with node and element numbers shown, is displayed, as is the truss with vectorial representations of load and boundary conditions. If these look correct, hit return and the computer will display a deformed truss and plot of the truss showing member forces. If you then provide a pathname for an output file, the numerical values for nodal displacement components, member strains, and member forces are printed to it.


In the ABAQUS truss tutorial, you analyzed the following simply supported statically determinate truss:


An input file for analyzing this problem with the Maple program is shown below. 

Tutorial
3
0.0 0.0
1.0 1.0
2.0 0.0
3
1  2  1.0
2  3  1.0
3  1  1.0
3
1  1  0.0
1  2  0.0
3  2  0.0
1
2  0.0  1.0


Here are the graphics Maple shows on the screen once it has read the input file:
 


After the computations are made, these graphic results are displayed:

The output file produced after running the program looks like this:

Tutorial
Number of Nodes and Elements:   3   3
Node#    x1     x2
    1    0.0000     0.0000
    2     .7070      .7070
    3    1.4100     0.0000
El#   Node 1  Node 2     EA
   1          1           2    1.0000
   2          2           3    1.0000
   3          3           1    1.0000
Fixed Node#  DOF  Value
        1            1     0.0000
        1            2     0.0000
        3            2     0.0000
Loaded Node#    P1       P2
             2        0.0000    1.0000

*********************************

Nodal Displacements:
 Node    u1       u2
  1   0.0000   0.0000
  2   -.3490   1.3461
  3   -.7029   0.0000
Strains and Forces:
Element   Strain     Force
  1     .7051     .7051
  2     .7071     .7071
  3    -.4985    -.4985

The code itself is divided into 10 main sections:

1)     Read the input file and set up data structures to store nodal coordinates, element connectivity and boundary conditions

2)     Graphically display the problem as understood by Maple.

3)     Define the element stiffness matrix

4)     Define the element residual vector

5)     Assemble the global stiffness matrix

6)     Assemble the residual force vector

7)     Modify the stiffness to account for constrained degrees of freedom

8)     Solve the FEM equations

9)     Display the results graphically on the screen.

10)Write results to a file.

To run the program: Place your curser in the top comment area, and hit return. Before hitting return, however, be sure that you have given give the full pathname of the input file in the line indicated by the comments. Backslashes must be used in place of front slashes in the pathname. The file name and path should be something like /u1/blumeja/tutorial.inp.txt. Maple will then print the output to a file called /u1/blumeja/tutorial.out.txt. This output file will be created if it isn't already present. If it is, the output will overwrite the existing file.

You can download a copy of this Maple file by right clicking here. You should use "save target as..." to save the file as Truss.mws.  The input file for the tutorial truss is available by right clicking here.

You can try a number of tests to understand how the program works. You can add print() commands to print out any intermediate variables you like – for example, you could examine each element stiffness matrix as it is added to the global stiffness, or print the final global stiffness matrix. You can also do tests like computing the determinant of the global stiffness (The determinant should be zero before the boundary constraints are added. Can you see why?)

Here is a full listing of the program. Those sections devoted to graphics are marked as such. It is a good idea to look at the listing for the non-graphics version of the program. This will enable you to concentrate on the nuts and bolts of coding finite element calculations. To download the actual non-graphics maple worksheet, right-click here.