There are several ways of doing FEA matrix algebra by hand:

1. Design structures with three or less elements and use a slide rule for Gaussian Elimination, etc. calcs. :)

2. Seriously... My good old reliable HP-28S has excellent array capabilities, and can work with 100 x 100 matricies (although they'd better be sparse!). It's best, I've found, for checking; anything past and 8 x 8 matrix just takes too long to input. You can also program it to solve for {a} in[K]{a} = {f}, as {a} = [K]^-1{f}.

3. Even better, most of the commercial spreadsheets, except the bare-bones "works" packagages (you can still write a good macro, though) come with matrix algebra capabilities built-in. Back in 1990, Wingz and Quattro Pro and the one made by M$ had transpose options as a selection on the "paste special" menu item. I'm pretty sure that one made by M$ (...I'm not a fan, but I use their software at work...) has an "inverse" function built-in as well.

4. There are a lot of shareware programs out there for matrix algebra, though frankly the ones I've checked out (I looked at a couple for the Mac in 1995 and decided not to bother) are "clunky" in that the input is labor-intensive and the programs are surprisingly non-intuitive; it's as though the programming emphasis was put on *teaching* matrix manipulation rather than *using* it if you know what I mean.

Conclusion: I would first examine the spreadsheets available to me before looking to stand-alone programs (if you have an old VAX you might find MATLAB lying around, but...)

A very nice open source program that has many Matlab features in Scilab, http://www-rocq.inria.fr/scilab/.

They have an executable for Windows (and other OSs).

I would like to thank you (Dave Viking and rlnorton).
I will try your suggestions.

Another matlab-like program is Octave
(www.octave.org)

Years past, I used to surprise friends by doing a curve fit on a Hewlett-Packard calculator, a Vax VMS library function, and a Lotus 123 spreadsheet and get three obnoxiously different answers. ( The Hewlett-Packard was the most correct). The Mathworks Matlab folks, the Numerical Recipies in C folks, as well  Hewlett-Packard are still the top runners in my book. I've gotten good and bad results from Microsoft Excell depending on the problem being worked. A fundamental rule when using Commertial of the shelf( COTS) software : work a reasonable sized problem you know the correct answer to. The vendor will not always tell you which numerical routine they have included in their software. Numerical routines are designed to fit specific problem types and rarely do you find a perfect method that fits all problem types. No matter how smart computers get, the final responsibility for the problem results rests on the Engineer.

If you are programming your FEA from scratch, one way to assemble the global stiffness matrix is by using Locator Matrices.  The locator matix, L, is a non-square identity matrix formulated such that K = sum of (L'*k*L), where each local stiffness matrix k has its own locator matrix L.  For details on applying this procedure see Theory of Vibrations with Applications by Thomson.