A MATHEMATICAL MODEL FOR THE GRIDS BRACING PROBLEM
Suppose that you are constructing a rectangular grid out of rigid beams.
For example the framework for a wall of a multistory structure.
Such a simple grid can be easily deformed:
Diagonal braces are often used to make such a grid rigid:
Are all of these braces necessary? Clearly not, one can easily see that, if the
brace in one of the corner cells is removed, the grid remains rigid. In fact all four
corner braces could be removed. How many others can be removed?
The fundamental questions that we wish to explore and ultimatly answer are:
- How many braces are actually necessary to make the grid rigid?
- How should they be placed?
To aid in this exploration we have built a computer simulation for this problem. To use this simulation you should:
- read this list then click on
computer simulation;
- select the dimensions of the grid (using the pull down menu)
[by the dimensions of a grid we mean the number of rows and columns of cells
in the grid - the above grid is a 3 by 4 grid.];
- click on "new";
- click on the cells you wish to brace (braced cells will be "bricked in");
- click on the "check" button to see if the bracing is rigid.
To investigate this problem you might wish to experiment with some small grids. Then formulate some conjectures and try these out on slightly larger grids. You might try to solve the problem completely for all 1 by n grids (n=1, 2, 3, ....); then for all 2 by n grids; and so on.
The grid bracing problem is a very special case of the general rigidity problem in the plane: given the design of any planar structure consisting of rods bolted together at their ends, can you predict, before you actually construct it, whether or not it will be rigid? If the "framework" is made up of triangles with common edges - like the left hand framework below, we would all agree that the answer is yes. But what about the right hand framework?
There is a general theory that will help us decide. It is based on the concept of "degrees of freedom." We say that a point in the plane has 2 degrees of freedom. If you like, it takes two numbers (coordinates) to identify its position. A larger rigid structure, like a segment, has three degrees of freedom, it takes three numbers to identify its exact location: two positional numbers - say the coordinate of one of its end points and a number to indicate its orientation - say the angle it makes with the positive x-axis.
We analyse a framework with j joints and r rods by noting that the joints alone
have 2j degrees of freedom and noting that each edge may reduce the total degrees of freedom of the framework by one. Thus 2j-r is a lower bound on the degrees of freedom of the framework. If 2j-r>3, the framework cannot be rigid; if 2j-r=3 and each edge has actually reduced the degrees of freedom, the framework is rigid. For example a simple quadrilateral has 4 joints and 4 rods giving 2x4-4=4 degrees of freedom. This simple framework is not rigid. But, if we add a diagonal rod, the count, 2x4-5=3, indicates that the quadrilateral with diagonal is rigid. Both of the above frameworks have 6 joints and 9 rods. Since 2x6-9=3, we expect that they are both rigid. A NOTE OF CAUTION: this degree of freedom approach can be in error if
the joints are in very special positions. Nevertheless, it is usually a very good predictor of planar rigidity. In the bracing of grids problem, the degrees of freedom analysis will accurately predict just how many braces will be needed but it will be of little help in deciding how to place them.
We have also included a mathematical model for the grids bracing problem. To explain this model, we must introduce some concepts from a part of mathematic called graph theory. A graph is an abstract mathematical object consisting of vertices and edges. Any pair of vertices may or may not be joined by an edge. To each braced grid, we associate a graph as follows: the vertices of the graph are associated with the rows and columns of the grid and each brace is represented an edge between the vertices representing the row and column containing it. For example:
To use this mathematical model click on "computer simulation" above and set up a braced grid. Then to see the graph of that bracing, just click on the "graph" button.
Experimenting with this mathematical model you should soon see that the rigidity of the braced grid seems to correspond with a simple property of the graph. This leads to another set of tasks:
- Find the property of the graph which corresponds to the rigidity of the bracing.
- Reformulate our original questions in terms of the graph.
- Using the graph answer these questions.
The Applet was written by Mehmet Sen
Comments and suggestions can be emailed to: Jack Graver
Web Address: http://lsb.syr.edu/projects/grids/
Date Created: June 5, 1997