finite element method evaluates the stress/strain of a structural at defined points, called nodes. nodes are interconnected to make a structure by elements. Elements can represent linear elements (rods or beams), planar elements (plates or membranes), or solids (3D pieces). Loads are applied at nodes as required. Constraints are applied to nodes to "ground" the model (ie provide the reaction loads to keep the model in equilibrium).
Do you understand how a frame analysis works? Such as a beam element with various degrees of freedom at each end like x, y, z translation and x, y, z rotation?
Well, think of this as a 2 node finite element.
You can also develop stiffness/force relationships between 3 node, 4 node, etc. elements. Essentially the same thing except that you now have to make assumptions as to how the forces distribute between the "other" nodes. For a 2 node element - the force simply reacts to the other singular node.
Because of the multiple nodes and the assumption as to how the stiffness and forces relate, you find that you can only satisfy one of the two following aspects of the structure:
1. Force equilibrium
2. Boundary condition compatibility (i.e. - the nodes from one element deflect the same way as the adjacent element sharing the same node).
Since you can't satisfy both, and you really don't want a disjointed structure after you apply the forces (i.e. you want to satisfy no. 2 above). You allow for approximation in the force equilibrium.
So using many smaller elements reduces the error in force equilibrium and you get a pretty close answer.
A couple of examples that I often use it for in building design are:
Areaway wall analyzed as a plate with pinned bottom, fixed sides and a free top. I apply a triangular earth load to it (plus a uniform surcharge if adjacent to traffic or parking).
Concrete mat foundation with multiple column loads
Concrete balcony supported on 3 sides
I am sure there are many other examples. With today's software programs, most of these problems can be modeled and solved within 5 minutes.
The finite element method is an approximate method used to solve differential equations. In structural engineering, these equations are those governing stress and strain relationships. The solutions taken at node locations are considered "exact". The FEM uses these solutions at nodes and interpolates for the values between the nodes. As you can imagine, the type of interpolation will govern just how exact your solution at a given point may be. In order to reduce error introduced by this interpolation, you can use more nodes, or make smaller elements.
The interpolation functions are built in to the type of element you want. For the constant strain triangle, strain is assumed to be constant across the element from node to node. With the linear strain triangle, strain varies linearly across the triangle. Now we know that in many cases, the strain is not constant nor does it vary linearly through an element, but if you make each increment small, so that you have more nodes where the solution is better, then your results will also become better.
FEM is used for heat transfer analysis as well as fluid dynamics, among other things. What's important to remember is that it is a method to solve differential equations, and it is an approximate solution.
FEM is an approximate method used to analyze complex indeterminate structures or elements of a stucture with a computer.
A simple explanation is that each element is discretized (that is broken down into smaller elements that are determinate). The forces, strains, and stresses are then solved for each discretized element. The discretized elements are then reassembled to give results through compatibility. The more discretized elements that are made (the more pieces that the original element is broken up in) the more convergence on the the "real" solution, i.e. the more accurate the solution.
This method can only be accomplished by a computer because the computations are immense.
