Doug - it bugs me because lots of small things bug me
and they really aren't the same. Yes FEA does use Matrix Methods to solve for its answers, but the fundamental set ups are different.
For structural members, the stiffness matrix is comprised of the terms that relate the internal forces in the member with joint displacements - such as PA/L, 12EI/L^3, 4EI/L, etc. Then the structure is solved like a complex spring where [P] = [K][d]. Joint displacements are solved for first and then member internal forces are found using the assembled stiffness matrix and the applied loads.
For finite element analysis my interpretation is; the relative displacements of the nodes on a particular element are defined with respect to one another (i.e. it's internal element stiffness); an overall stiffness matrix is developed, the displacements of nodes are solved for; and then the internal stresses of the elements are solved for using the internal stiffness fomulations that were used to define the element.
Here is a much better answer from thread earlier this year (thread507-323635)
PowersPE80 (Civil/Environmental) 10 Jun 12 9:59
Hello, Can someone please explain to me, without getting overly technical, what exactly a finite element analysis is?
Thanks.
. . .
JAE (Structural)
10 Jun 12 12:33
Powers - very tough to do that well without knowing what you do know.
Are you are familiar with frame analysis programs where you have beams and columns represented by "stick" members - a member that has a joint at either end? You connect all these into a frame or other representation of a structure and by knowing the deflection-force relationship between the two joints you can analyze the structure.
These two-joint "sticks" are two-joint finite elements. They are essentially a member/entity that creates a relationship between two joints. If you move one joint, the other responds with a defined force.
Each joint, typically - in 3D models - represents six degrees of freedom - X, Y, and Z direction translation and X, Y and Z direction rotations.
Now if you expand that two-joint member into a three or more joint member, the member now becomes either a planar flat element (triangle or rectangle) or perhaps even a solid element with multiple joints that all are related to each other in terms of displacement and force.
The analysis/solution is similar to the two-joint member except there is a problem. The two-joint beam type members can be analyzed such that both equilibrium and deformation compatibility are satisfied (Sum of forces/moments = 0 and the deflection of element A at a common joint with element B is the same). However, with more than two joints, you can't satisfy both of these - only one. Most all finite element solutions choose to satisfy the deformation compatibility and then by keeping the size of the elements small, minimize the sum of forces error in the model.
And then the mesh can be refined to determine if you are honing in on the approximate answer.
gjc