Testing the General Matrix for solving Indeterminate Beams

Ok, but the main question remains:

How reliable are the results of this General Matrix?

Becuase is a math model derived from the general integration method the results should be exact. To demonstrate this I decided run a model on AutoCAD Mechanical and compare it's results with the ones obtained from the General Matrix. I decided to use AutoCAD Mechanical because it's solution algorithm is based on a matrixial method for solving structures, which by the way is a highly trusted method employed extensively by civil engineers.

I desided to evaluate a short "I" beam of 1 meter of length, using 10 supports equally spaced, loaded with a equally distributed load of 75 N/mm [75 kN/m] all over its length as shown in the screen-cap below. (you can also download a copy of this CAD file here).

The results of maximum stress and deflections are shown in the big blue box in the left. But right now we are interested only in the reactions. Each one appear below its respective support, as you can see in the figure below.

Now, loading the same specifications of the beam modeled in the General Matrix (download file here), as you see in the next screen-cap:

The Answer would be obtained in the page n10, because we are modeling 10 supports equally spaced.

In the last picture you can notice two matrices. The upper matrix is the General Matrix formulation that represents the geometry, next to it there is a vector which are the elements that represents the load. The matrix below is the inverse of the general Matrix and next to it we have the results computed. They are amplified in the picture below:

By comparing this results with the ones obtained in AutoCAD it is possible to notice that there is no difference in the results withing the first 8 digits.

As a conclusion:

The General Matrix is a reliable method for solving Indeterminate beams reactions.

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.

Blog Foreword

Hi Everyone! and welcome to my personal blog web page.

I plan to use this blog site as my personal journal and a space for posting useful information of some of my design projects. Projects that I have worked on as a bachelor student and/or I'm currently working on.

There are six main reason's why I decided to do this:

1. Create an interactive journal, to stay in touch with my friends and family.
2. Find people related to the mechanical design field and share experiences with them.
3. Share personal project ideas with the intention of discuss about them.
4. Create a profile because I think that now a days is very important.
5. Improve my English and Chinese writing skills.
6. Lastly but not least, force me to study more!