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Chapter 4 of the textbook Finite Element Analysis With Mathematica and Matlab Computations visits the Finite Element Method approach to understanding trusses, beams, and frame systems. Often, engineers and other problem solvers are faced with statically indeterminate systems. In other words, using standard knowledge of statics and mechanics of materials will not be able to understand these systems. The Finite Element Method is a different way of understanding these systems with the use of matrix algebra. Basically, the method rather than analyzing the system as a whole, parts of the system are analyzed separately. This chapter first treats plane truss systems. Unlike beams and frames, trusses are only subjected to axial loads. It is assumed that they are elastic bars that are deformable.

When faced with a problem dealing with truss systems, it is important to clearly understand the background of the problem and what is being asked. Diagrams are especially useful in visualizing the problem when they are properly labeled. Starting with the entire system itself, a clearly and properly marked global free body diagram (FBD) is necessary. Included in this diagram would be the global coordinate system, loads, reactions, degrees of freedom, and labels for each node and member of the system. Examples of degrees of freedom are the known, fixed degrees of freedom and constraints, or the unknown, often displacements. It is standard to label the global nodes using numbers with circles encircling the numbers. The members of the system should be labeled with numbers encircled by triangles. From this, one can determine whether or not the system is statically indeterminate.

Once the system has been determined statically indeterminate, one may proceed with Finite Element Analysis. Since the members of the system will be analyzed separately, element free body diagrams should be created for each of the members of the system. This process also has its own standard labeling method. Although the nodes should already have been labeled encircled by circles, labeling the local nodes encircled with a square helps clarify the diagram. The reactions, loads, and degrees of freedom should be written the same except for the global nodes that experience the loads, once should choose one member of the system that undergoes the load while the other member doesn't. The local coordinate system may be the same as the global coordinate system, or one may be chosen to further simplify the problem.

Recalling back to fundamental physics, the equation that describes the force (F) of a spring with a spring constant (k) where one end is fixed and the other goes through a displacement (x) is F=kx. However, this single equation can not describe the motion of a spring where both ends are not fixed. This is a statically indeterminate system, therefore the Finite Element Method must be used. The motion of this scenario can be described

$$\begin{Bmatrix} F_1 \\ F_2 \end{Bmatrix} = \begin{bmatrix} k & -k \\ -k & k \end{bmatrix}\begin{Bmatrix} d_1 \\ d_2 \end{Bmatrix}$$

where Fi is the force of node i=1,2, k is the stiffness matrix, and di is the displacement of node i=1,2. This can also be applied to truss systems in a similar manner. However many degrees of freedom determines the size of the matrix. In the spring scenario, there were two degrees of freedom creating a 2X1 force matrix, a 2X1 displacement matrix and a 2X2 stiffness matrix. In truss systems that are unconstrained the stiffness matrix will be of dimensions mXn, where m is the number of unknown displacement degrees of freedom and n is the number of both known and unknown degrees of freedom. Once the known degrees of freedom are eliminated the force and displacement matrix will be mX1, and the stiffness matrix will be mXm. For consistency purposes, the element stiffness and force matrices will be situated in the global coordinate system. Once the stiffness and displacement matrices are established, one may calculate the reactions (the unknown forces).