Computation of the forces in the members of a truss

by Zach Dunbar
Statics 223 H (Fall 2000)

A truss is a complicated and potentially frustrating statics problem to solve. The more members a truss contains, the longer it takes to solve. It can be overwhelming to attempt to solve these trusses via the method of joints.  Using the method of sections, this process can be sped up a great deal, but it requires a good deal of thought to figure out where to begin, and how to proceed.
A way around this is to utilized the power of computers to assist in the computation of the method of joints.  Since this method is a fairly straight forward and mindless task, the computer can perform these masses of computations quicker and more accurately (and with less frustration) than a person.  Therefore, it would be useful if there where a computer program to perform these operations for you.

The method my program will use for determining the forces in a truss member is as follows:

I. The program will receive input from the user regarding the various components of the truss, such as

A. the coordinates of the vertices
B. the loads at those vertices
C. which vertices are connected to one another by various members.

II. The program will then use the input data to determine the loads in each of these members.

A simple example of a truss that this program will evaluate is the one shown at left.  It consists of several vertices, one of which is a hinge (vertex C, exerting forces only in the X and Y direction) and another which is set on rollers (vertex B, exerting forces only in the Y direction) as shown in the free body diagram below.

The Program

This program will utilize the method of joints to determine the forces present in its various members.  Actually writing the code to accomplish this is somewhat difficult. First, we find a vertex that is connected to only two other vertices. Next, we solve for the components of force in this vertex.  It is usually easiest to start with the vertex on the roller, since there is only one force component to worry about, besides the forces applied by the two other members connected to it.  We are specifically looking for a the one of the components (either the i or j) in the member to equal zero while the same component isn't zero in the other unknown member. Then, taking into account the force applied at the vertex, we can sum forces in either the x or y direction, and solve for the force in that member.  Depending on if these force is positive or negative, the member will be in tension or compression, respectively.

Once the forces in these first 2 members are solved, we move on to another vertex which has only two unknown member forces, and solve for these forces in a similar way. Then we simply repeat this process until all the members of the truss have been found.  Complications arise when a vertex has more than just 2 members attached to it, and if the force on the vertex is in any direction besides the vertical.  This is the none computer code explanation of what the program is doing to find this information.  For a full computer code explanation of how I programmed the program, please download the source code available at the bottom of this page.

Equations

The following are some key equations that the program uses to compute various aspects of the truss.  These are the same equations that a person would use, the key is laying them out in an organized and formatted way that the program can use.  There are no real complex equations that make the method of joints difficult, it is just the large volume of calculations, many of which relay on some of the following basic equations:

# of members =(2*#of vertices-3)
Force B=Force A/sin theta; if F is some vertical force and B is some angled force, and there are no other vertical forces at the joint.
Force C = Force B*cos phi; if FC is the purely horizontal member that is attached to the same point as B, and phi is the angle between them.
DX=X2 /X1;
DY=Y2 /Y1;
a2 + b2 = c2

All the force at the various vertices are required before the program can be successfully run.  However, these forces can be calculated manually, using various mathematical techniques.  This is not an overall limitation to computer aided truss calculation, but merely a limitation in my programing skill.

The limitations of the program are as follows:

 The forces at all the vertices must be known. Why? The user must have a good knowledge of the physical layout of the truss structure, i.e. the coordinates of each vertex. Why? If the force isn't given for a particular vertex, the user must have the necessary skills of how to calculate it. Why? The truss must be of a limited number of members, only a 3 member truss can be calculated using this program. Why? The forces applied at the vertices most be applied only in the vertical direction, not horizontal or at an angle. Why?

All of these program limitations aren't set in stone.  With sufficient time and programming skill, all of them could eventually be overcome, and a truss program could be written what could solve any type of mathematically solvable truss problem.