How to Solve a Truss Problem

In the field of mechanical and civil engineering, trusses are a major subject due to the inherent stability of triangles. Trusses are used in bridges, roofs, and even bicycles and aircraft, which shows how useful they are as stable structures.

This Instructable will help one solve for the forces in a truss problem so that the internal forces can be seen. It should take about fifteen to twenty minutes to complete. It is helpful to be knowledgeable about both forces and moments and how they work and how to use the functions sine and cosine before reading this Instructable.

First, the reaction forces at points A and D must be found. These reaction forces are the forces that the two supports at A and D exert on the truss in order to keep it stationary. These forces must be found first before the internal forces can be found.

• The left support is a roller support, which can only have a vertical y-direction force applied to it because it moves freely horizontally. This force should have the variable name R_A_y (A and y are subscripts to R).
• The right support is a pin support, which can have both a vertical y-direction force and a horizontal x-direction force applied to it because it is firmly planted to the ground. These forces should have the variable names R_D_y and R_D_x.

In order to get the reaction forces at the supports, a moment must be taken about A or D and the forces in the horizontal and vertical directions must add up to zero.

• Summing moments about a point involves looking at forces that don't go through the point in question and multiplying their distance away from the point by the amount of force perpendicular to a line from the point to the force.
• Since the upper 450N force is going through point D, it is easier to take the moment at Point D compared to Point A because that force does not apply in the moment equation.
• R_A_y is obtained from this moment equation, which can be used in the sum of forces in the y-direction equation to obtain R_D_y.
• R_D_x is obtained from the sum of forces in the x-direction equation. Since there is only one other force with an x-component, R_D_x is equal to that x-component in the opposite direction, so the force at the pin support will be to the left.

Now that the external forces on the truss are known, one can solve for the internal forces within the truss. When solving for internal forces, it is easiest to start at one of the supports with the least amount of beams connected to it. Point A or D works for this, but for this example, Point A will be looked at first.

• Since there are only two forces in the y-direction, and one is known, the sum of forces in the y-direction equation should be used first.
• The y-direction equation yields the value of F_AB.
• F_AB is pointing away from the rest of the AB beam. This means it is in compression (C) and the force is compressing the beam inward.
• With the value of F_AB known, the x-component of that force is equal and opposite to the F_AC force.
• Solving this equation yields the value of F_AC.
• F_AC is pointing towards the rest of the AC beam. This means it is in tension (T) and the force is stretching the beam outward.

After finding the forces of Beam AB and Beam AC, it is easiest to move on to Point C and obtain the forces in the beams that meet there.

• Draw the forces that are applied at Point C.
• The external force at C, 450 N, applies here like the force of the support did in Step 3.
• Setting the sum of the forces in the x-direction equal to 0, F_CD must be equal and opposite to F_AC. Since the force is going toward the rest of Beam AC, the beam is in tension.
• Setting the sum of the forces in the y-direction equal to 0, F_BC must be equal and opposite to the downward external force of 450 N. Since the force is going toward the rest of Beam BC, the beam is in tension.

Using this method of using sum of forces in the x-direction and y-direction being equal to 0, the rest of the internal forces can be found. The easiest way to do this after starting at Point A is to move from left to right across the truss. The final answers are shown below:

• F_AB = 450 N in compression
• F_AC = 389.71 N in tension
• F_BC = 450 N in tension
• F_BD = 900 N in compression
• F_CD = 389.71 N in tension

The method used to solve truss problems is to:

• Find the forces at the supports by using force and moment equations with given external forces.
• Calculate the internal forces of beams connected to a support, keeping in mind which are in compression and which are in tension.
• Find the other internal forces by moving across the beam using the forces of the beams obtained in other steps and the relevant external forces.

Some links that are relevant to those who have an interest in trusses are:

• http://www.bridgeweb.com - A website about bridge design, most of which use trusses in their construction.

• http://www.littfintruss.com - A website that manufactures and sells trusses.