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Trusses and Transformations

In the previous sections all the forces and displacements were along the same line. However, real structures even those made up of rods are not usually 1 dimensional. We will now extend the approach to two-dimensional structures made up of rods (not Beams!). While the deformation of the rod is still along its axis, the rod itself may be in some general orientation. We can decompose the displacement into components along the global x and y axes. We can then write separate equations for tex2html_wrap_inline921 and tex2html_wrap_inline923 .

We begin by looking at a single rod element lying along the axis as in Fig. 1.8. We add two additional degrees of freedom vi,vj for components of displacement along the y-axis.

   figure250
Figure 1.8: General Rod/Truss element oriented along the global x axis. Note additional displacement variables vi,vj

The element level equations can be written as:

  eqnarray255

Now, let us orient the element at an angle tex2html_wrap_inline755 to the x axis. Fig. 1.9 shows the rotated rod with a local coordinate system x'-y' at an angle tex2html_wrap_inline755 to the global coordinate system x-y. In this local coordinate system x'-y' equation 1.13 holds.

  eqnarray269

or compactly,

displaymath879

To obtain the displacements and forces in coordinate system x-y we will have to resolve the dispalcement and force in terms of components along the x-y axis.

   figure281
Figure 1.9: General Rod/Truss element oriented at an angle tex2html_wrap_inline755 to the global x axis.

From basic trigonometric relations

eqnarray286

In matrix-vector notation

  eqnarray291

or compactly,

displaymath880

where [T] is called the transformation matrix. Applying this to equation 1.14 we get

displaymath881

Premultiplying both sides of the matrix with the transpose of [T] we get

  equation304

The matrix tex2html_wrap_inline951

eqnarray307

Applying in equation 1.17 we get

displaymath882

displaymath883

where tex2html_wrap_inline953 and tex2html_wrap_inline955 are the displacements and forces in global coordinate sytems. Now if we revisit our 5 step FEM process, we need to incorporate this process of transforming the stiffness matrix into the local approximation step.

Example

Let us now solve for the displacements in a small truss using the above approach (Fig. 1.10).

   figure321
Figure: 3 rod truss problem, discretized into a 3 element mesh

Step 1 FE Discretization

Fig. 1.10 shows the discretization into 3 truss elements with 3 nodes and 6 degrees of freedom.

Step 2 Local Approximation

Element 1

First we identify the element and it's orientation.

displaymath884

Then we obtain the transformed stiffness matrix in the global coordinate system.

eqnarray330

For this case tex2html_wrap_inline957 .

displaymath885

eqnarray339

eqnarray350

Now let us identify where the element matrix needs to assemble:

displaymath886

Element 2

First we identify the element and it's orientation.

displaymath887

Then we obtain the transformed stiffness matrix in the global coordinate system.

eqnarray356

displaymath888

displaymath889

eqnarray365

eqnarray376

Now let us identify where the element matrix needs to assemble:

displaymath890

Element 3

displaymath891

Then we obtain the transformed stiffness matrix in the global coordinate system.

eqnarray356

displaymath892

displaymath889

eqnarray391

eqnarray402

Now let us identify where the element matrix needs to assemble:

displaymath894

Step 3 Assembly

Now assemble the element matrices. First putting in element 1 contribution according to

displaymath886

eqnarray408

Now for the second element's contribution assembled accordin to

displaymath890

eqnarray416

Finally the third element contribution assembled according to

displaymath894

eqnarray424

Step 4 Loads and Constraints

Loads

The only load is the force P=-10 acting at node 1. Summing the terms in the above matrix and applying the load we get:

eqnarray434

Constraints

There are 3 constraints. U2=U3=V3=0 Applying these constraints yields:

eqnarray446

Solving U1= -.0150145, V1= -.0710172 and V2 = -.005.


next up previous
Next: The Energy Approach - Up: Basic Ideas - ``123 Previous: Application of Loads and

Abani Patra
Mon Mar 15 10:37:42 EST 1999