Solving Indeterminate beams with multiple supports

This is a math model that I developed to solve an optimization problem involving indeterminate beams of second and higher order.

Applications like distribution of supports in the design of water gates, bridges, and structures can be benefited by using this math model.

I already programed it into an excel file, to make easier its use in any of those design application.

THE PROBLEM:

Fig.1 - Indeterminate beam

During the time I worked on water gates design, I constantly had to dealt with the problem of optimizing the distribution of the total load over the gate to the roller supports. But, at that time I couldn't find any tools that help me to optimize the distribution of the load on supports. You may think I didn't try using parametric models in a Finite element software, but I can ensure you that I did, but they were so much time consuming. Therefore I come to the idea to

develop my own math model that help me achieve the result I wanted.

One of the biggest obstacles that I had to face was generating a model that helped me to study the appropriate position of supports by restricting the stress on them or by making them all have the same amount of stress. But, when dealing with indeterminate beams of second order and over the math becomes sometimes complex to solve this problem.

In my experience is not an easy task what I was looking. Commonly there are two paths:

1. The Integral equation Method. - Which is a maybe the most apropiate way to solve the problem, but It has to be said that as more unknown variables more and more complex became the equations that you need to solve.

2. Or modeling the beams as a structure by using the structure solution matrix method. - Which basically consist on cutting in small pieces the large beam, then each peace become a beam by itself, and you can write a stiffness matrix for each bea, integrating all those small matrixes in a supper matrix a solution can be obtained.

Well both paths are proven to give good results for finding the reactions, however you must re-write a new math problem “n” times as you evaluate the same model with more supports. And if you have try to solve a problem like this before you may already know that this is not easy work.

For me both methods were not enough for the simple task that I was asking, so I needed something new. Thefore, I decided to star studing the elastic equations to see if there was another way to get a solution to this case. Figure 1. shows a general case of what I'm trying to solve.

Fig. 2 - Modeling a gate as a indeterminate loaded beam

In my case, I know how the is going to be always the distributions of the load over my beam (in this case the gate), so I started to analyze a model like shown in figure 2. At the end I was able to find an extraordinary pattern on the equations that led me to develop of the following matrix:

Eq. 1 - Generalized Matrix for solving the gate model.

This matrix, wich I call the general matrix, is any capable of solving a indeterminate beam like shown in figure No.2 with "n" number of supports. The matrix is as big as the numbers of supports you have. And, because is based on the elastics equations the results of reactions are always exact.

I guess the next questions would be:

• How do I use that?

You can evaluate the model that I program in a excel file, it that file can be downloaded here.

• How do I know that the results of this models are reliable?

Lets put this equation on test! See the next post!

• How do you develop that matrix?

You can read a short description by downloading here this pdf file.