Coordinates Transformation in FEM

In this article, we will be discussing how the coordinate transformation is carried out to transform the element information i.e. element stiffness, element nodal displacements and element nodal forces, in the local/element coordinate system oriented at an angle to the global coordinate system.

This is a necessary step to be perform before starting the element assembly process. We need to check the orientation of local/element coordinate system xy-CS, used for building the element model with reference to the global coordinate system XY-CS. If the xy-CS is aligned with the XY-CS, then, we can begin the assembly of elements by constructing the element connectivity table as discussed in the previous article. Otherwise, it is essential to do coordinate transformation to align the element local information to the global coordinate system before the element assembly if the xy-CS is oriented at a certain angle  to the XY-CS.

We need to transform all information of an element model i.e. nodal displacements, nodal forces and element stiffness matrix. In the below text, we discuss each of them one by one.

Transformation of Element Nodal Displacements

Let consider a linear spring element model with 2 nodes, as shown in Figure 1. Each node has 02 displacement degrees of freedom, which are represented with blue colored vectors along the element/local coordinate system xy-CS.

Please pay attention to the fact that the xy-CS is NOT aligned to the global coordinate system XY-CS, and is oriented at a certain angle. Therefore, we have to transform the blue colored nodal displacements to red colored nodal displacements which are aligned to XY-CS.

Figure 1. Transformation of element nodal displacement

Now, we discuss how to perform this transformation of the element nodal displacements. Here is the geometrical representation of the transformation between the components of nodal displacements in xy-CS and XY-CS in Figure 2.

Figure 2. Geometrical representation of the nodal displacement transformation

In terms of mathematical equations, we can write the same geometrical representation of nodal displacement transformation as following in Figure 3.

Figure 3. Mathematical representation of the nodal displacement transformation

We prefer writing all nodal displacements in the following matrix form as a matter of convenience.

Where [T] is the transformation matrix, {u'} is the nodal displacement vector in the xy-CS oriented an angle, and  {u} is the nodal displacement vector in aligned with the XY-CS.

Transformation of Element Nodal Forces

In a similar fashion for the nodal displacements, we can develop transformation relation for the nodal forces as shown below in Figure 4.

Figure 4. Transformation of element nodal forces

The above information can be also present in the matrix form as below:

Where [T] is the same transformation matrix, {f'} is the nodal displacement vector in the xy-CS oriented an angle, and  {f} is the nodal displacement vector in aligned with the XY-CS.

Transformation Matrix [T]

For conversion of nodal displacements and forces, we developed a transformation matrix [T] with entries of trigonometric sine and cosine ratios, shortly denoted by symbols l and m. This transformation matrix is a very special matrix because if we take its transpose or try to invert it, we get the same result. The matrix demonstrating such special property is called orthogonal matrix. Hence, the transformation matrix in our case is an orthogonal matrix as its transpose is equal to its inverse, as shown below.

The orthogonality of transformation matrix is very useful in numerical computation involving the matrix inversion. Whenever we need to inverse an orthogonal matrix, then, rather than taking its inverse we just take its transpose as inverting a matrix is a computationally expensive task as compare to taking its transpose which is merely the switching of its rows with columns. Please do remember that if a matrix is not an orthogonal matrix then we need takes its inverse as usual.

Transformation of Element Stiffness Matrix

So for we are able to develop the transformation relationships for the nodal displacement and load vectors using the transformation matrix [T], as shown below.

The remaining is the element stiffness matrix that also needs to be transformed from the xy-CS to the XY-CS. However, in this case use the previous transformation equations for the nodal displacements and forces, and manipulating them to derive the transformation relation for the element stiffness matrix as following.

Summary

Here is the summary of complete transformation relations for the element information in xy-CS oriented at an angle to XY-CS. 

Once all element information is fully transformed from -CS to -CS, then, we proceed to the assembly process by constructing the element connectivity table. For further details on the element assembly process, please visit following article.

About the Author: Dr. Khazar Hayat is a professional engineer with almost 15+ year of experience in research, design, analysis and development of products made of fiber reinforced plastics composites (FRPCs). Currently, he is working as an Associate Professor at Mechanical Engineering Department, The University of Lahore, Pakistan, can be reaching by emailing at khazarhayat@gmail.com.