# Beam problem followup and Radial Basis Functions

To follow up to my recent post on optimal beam supports, I am now working on a little Maxima script to approximate a solution to this problem. It's an interesting problem from a practical perspective. I actually want to know something about the solution because I plan to fix some poor construction practices at the entrance to my garage. My house was built in 1948 and the beam that spans the garage opening could be much stiffer, and tied together with the surrounding wall framing to prevent it from sagging and reduce the chance of earthquake damage...

But in general, the problem is a nice one because it poses a straightforward problem that can be solved by variational principles, and can be attacked by computer algebra pretty easily without requiring any fancy numerical FEM solutions or what have you.

To model the forces of support I would like to use little narrow gaussian curves, and then to calculate the deflection curve it makes sense to represent the solution in a way that can easily represent the effect of the narrow gaussian curves being moved about from place to place. It turns out that "Radial Basis Functions" seem to be an excellent general purpose interpolation technique for representing such things.

For a 1D situation, a radial basis function is some function which is determined by distance from some location. In other words, the basis is formed by translation and scaling of some "mother function". It sounds a bit like "wavelets" which were a big topic in the 90's but RBFs need not have compact support, growth at infinity can be controlled by destructive interference, since all of the functions have the same order of growth.

My plan is to solve the problem for the simply supported beam with constant loading in terms of a polynomial (the natural choice since it's exact for the simple case), and then add a perturbation represented by a set of radial basis functions which represents the change due to the new support force. I'll try to work it out and post the code, perhaps on the plane flight home.

Practically speaking, can you rip out the veneer of the garage, exposing the existing beam, cut it out, and replace it with two 2x12s bolted together with a 3/8" steel plate between them? (The concept is easy, but this will require some time and some help!) That will fix your sag and should be a lot stronger than your existing structure.

The beam would remain simply supported, but the construction should be more than adequate for the load bearing job of a garage header.

Not much theory or modeling there, but it works.

Also you will need a "stand" consisting of a temporary beam supported with some 2x4s to bear the load while you are cutting and installing the new beam. Using that temporary structure you could even jack up the existing construction to "de-sag" it.

For the practical aspect, it's actually easier than that. I want to jack up the beam using temporary stands to de-sag it, and then apply OSB sheathing which will tie the existing garage header to an upper beam about 2 feet above it. currently there is a little cripple wall of 2x4 studs in between the lower beam and upper beam. The OSB will turn the whole cripple wall assembly into a single much deeper beam that will be much stronger than the current cripple wall assembly, especially since the nailing is inadequate in the cripple wall. Sure there's a little torsional issue of applying sheathing to one side only, but i don't think it will be a major problem.

So the practical question is how to do the jacking to get a nice straight line, but the modeling is itself an interesting problem, and the garage is functional so I'm not in a hurry to get it done right away before doing a little analysis.