University of Florida/Eml4500/f08.FEABBQ/HW1/Notes

Lecture Notes - Weeks 1,2
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The Following is a summary of the topics and information discussed during the first two weeks of EML4500 - Finite Element Analysis (FEA) at the University of Florida in the fall of 2008. A summary of all administrative information can be found  here .

Wikiversity
Wikiversity is just one of many projects formed under the parent organization of Wikimedia. Since Wikiversity is the platform in which the Professor, Dr. Loc Vu Quoc, plans on using for group collaboration and homework submission, most of the first week was dedicated to familiarizing the students with its structure.

Free Body Diagrams
Although a review from statics, the free body diagram (FBD) is an essential tool in FEA. In addition, because the systems for which FEA is used can be complex, it is necessary to form a rigorous convention for the labeling of all FBDs. Figures 1-3 below are the examples used in class to illustrate this convention.

Labeling Convention
In breaking a structure apart, it is important to differentiate between global FBDs (Fig. 2) and local FBDs (Fig. 3). For this purpose a circle is used when describing global nodes, while a square is used for local nodes. In addition, it is necessary to distinguish each piece, or element. A triangle is used for this purpose.

It is also important to assign a convention for the naming of forces and displacements. In global FBDs, as shown in Fig. 2, each force is titled R where the subscript denotes which node the force is being applied to, and the superscript shows the direction of the force. For local FBDs, as shown in Fig. 3, f denotes a force while d indicates a displacement. Here the subscript defines the particular local force or displacement, while the superscript indicates the element it corresponds to.

Force-Displacement Relation
The relationship between forces and displacements can be modeled by a combination of springs. The most simple example is a 1-D spring with one fixed end. In this case, the relationship between the force applied to the free end and the displacement of that end of the spring is given by:


 * $$f = k \cdot d$$

To illustrate how matrices can be used for systems with more degrees of freedom (dof), a simple example is a 1-D spring with both ends being free. Here the force-displacement relation is given by:


 * $$\left[\begin{array}{c} f_1\\ f_2 \end{array}\right] = \left[\begin{array}{cc} k & -k\\ -k & k \end{array}\right] \left[\begin{array}{c} d_1\\ d_2 \end{array}\right]$$

Recipe: Steps to Solving Simple Truss System Problems

 * 1) The first step of analyzing a truss system is to look at the entire system as a whole, also known as the global picture. The preliminary procedures in this analysis are first creating the global coordinate system, numbering the nodes in the system, as well as noting the global forces and displacement degrees of freedom (dofs) in the system.  As a convention, both the global forces and displacement dofs should be named starting with the first global node in the directions of the global coordinate system (x and y directions) and continued in this fashion for each successive node.  The aforementioned FBDs are good tools for the visualization of systems and the forces and dofs acting upon them.  For both the global dofs and the global forces, each contain some unknown and some known quantities.  The known values are usually given in the problem as contraints or parameters, such as a fixed node (zero displacement) or an applied force.  The unknown values are the values that will be solved for using finite element method (FEM).
 * 2) After the system is correctly identified, the next step is to analyze each element in the system separately. A FBD should be created for each element of the system, identifying the dofs and forces acting upon each particular element.  The nodes, dofs, and forces should be identified by the same convention mentioned in step 1, however either global or local element coordinates may be used.
 * 3) Once all the dofs and forces are identified for each element in the system, the next step is to create the element stiffness and force matrices. These matrices are assembled into the global force-displacement (FD) relation:
 * $$\underline{K} \cdot \underline{d} = \underline{F}$$
 * where K is a nxn stiffness matrix, d is a nx1 displacement dof matrix, F is a nx1 force matrix, and n is the number of both known and unknown dofs.


 * 1) Next, the known dofs are eliminated to reduce the global force displacement relations. The K,d, and F matrices above - that were originally nxn, nx1, and nx1 matrices, respectively - become mxm, mx1, and mx1, respectively, where m is the number of unknown only dofs and is less than n.  Assuming the new stiffness, K, matrix is an invertible matrix, the above equation can be rearranged to solve for the displacement matrix.
 * 2) Compute the element forces from the now known dofs and from this also calculate the element stress.
 * 3) Use these to compute the remaining unknown (reaction) forces.