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 and .

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.

**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:

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

or compactly,

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.

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

From basic trigonometric relations

In matrix-vector notation

or compactly,

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

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

The matrix

Applying in equation 1.17 we get

where and 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).

**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.

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

For this case .

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

**Element 2**

First we identify the element and it's orientation.

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

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

**Element 3**

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

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

*Step 3 Assembly*

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

Now for the second element's contribution assembled accordin to

Finally the third element contribution assembled according to

*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:

**Constraints**

There are 3 constraints. *U*2=*U*3=*V*3=0
Applying these constraints yields:

Solving *U*1= -.0150145, *V*1= -.0710172 and *V*2 = -.005.

Mon Mar 15 10:37:42 EST 1999