Fall 2012: Project 3: Bridge

Objective: Construct a bridge that has the largest load to bridge weight ratio.

Materials:

Corrugated Cardboard 

Glue

Rules:

1.   The bridge must span a 4 foot gap.

2.   The bridge can be no more than 6 inches wide and 3 inches below the height of the testing surface.

3.   There is no limit to it being longer than 4 feet and there is no limit to its height above the testing surface.

4.   The bridge will be loaded with a rope attached to a metal plate that must sit on the bed of the bridge. The bridge must have a hole 1 inch in diameter and a flat surface of 3 x 3 inches to support the plate. The plate will be placed on the bridge at the center of the 4 foot gap and at the height of the test surface

5.      Calculation of load to bridge weight ratio:

R = (weight supported by bridge before collapsing)

(weight of finished bridge)

Side-view of Testing Rules

3rd Place Winner

2nd Place Winner

1st Place Winner

Truss Analysis

Building bridges can be very complicated, but one of the simplest designs to build and analyze is the truss bridge. The truss bridge consists of straight members (ie., steel plates) and joints (ie., rivets). Usually, like the bridge above, the members are jointed in a vertical plane to form a repeating pattern. The members and joints can be jointed into very complex patterns, but regardless of the pattern, truss bridges can be analyzed using the same method. The analysis takes advantage of the fact that each member is in tension or compression, and this force acts along the length of the member. This means that each member either pulls or pushes on the joints it is attached to and this force lies on a line going through the member. Also, the entire bridge is in static equilibrium; the net forces in all directions and the net torques about all points in space are zero. The bridge is not rotating or moving. 

Steps:

1. Create a drawing of the members and joints. Label the angles, lengths of the members, and external loads. Here is an example of one side of a truss bridge made from three isosceles triangles, with a load applied in the center and supports on either end.  

2. Determine the external forces. We already know the external load. There is a support on either end with unknown forces. Since the bridge is in static equilibrium, the net torque about any point must equal zero and the net forces in any direction must also equal zero. Assume the members have no mass, and solve for F₁ and F₂ by first solving for the forces in the vertical direction and then taking the torques about point A.

3. Determine the internal forces. In order to solve for the internal forces, we will use the cutting method. Image cutting the bridge in half along the joints and saving only one of the halves. The members we cut through had internal forces in them, but we did not draw them because the force of the member on the joint was balanced by an equal and opposite force of the joint on the member. Although the member has been imaginatively removed, the force of the member on the joint is still there and must be accounted for, so we will draw this force in our diagram. Here we removed the left half of the bridge, and placed unknown forces at points A and B in the direction of the members that would have connected there. The members are arbitrarily assumed to be in tension, so F₁ and F₂ point away from the joints. If they are really in compression, we will get a negative value for F₁ or F₂. Again, since the bridge is in static equilibrium, the net torque about any point must equal zero and the net forces in any direction must also equal zero. Assume the members have no mass, and solve for F₁ and F₂ by first solving for the torques about point A and then about point B.

F₁ represents the force of the member on this joint. Since F₁ is negative, it really points the opposite direction to the one drawn. The force of the joint on the member will be equal and opposite to F₁. So, F₁ pushes on the joint while joint A pushes on the member, meaning the member is in compression.

4. Repeat Step 3 as many times as are needed. Cut the bridge at different locations, so the forces in different members may be “exposed” and solved for. When completed, it should look like this:

From this analysis, the members on the outside are in compression while the members on the inside are in tension. Since we are working with cardboard, there is a greater chance that the cardboard will buckle under compression than rip under tension. Members 1, 2, and 5 should be made thicker and with “fins” to prevent buckling.

Calculating the forces can be complicated, so here is a website by Johns Hopkins University which allows you to draw and automatically analyze the truss bridge: Virtual Laboratory Bridge Designer.