External Force Analysis - How External Forces are Transmitted through a Bridge

Both the external and internal force analysis was done by following the principles on this site.  We start by drawing a free body diagram of the structure showing the external load and support forces. The Bridge is drawn to scale, and accurately shows the 39.8 degree angle at the corner.

Next we break all the external forces down into their x and y components. We'll take advantage of Katherine's observation about how Beam Bridges sag when load is transmitted through a pressure plate, and assume that the pressure is put on only two points "B" and "C". In addition, we assume the bridge is supported in just 2 places: "A" and "D". Finally we assume the bridge is supporting 100 pounds of weight distributed evenly: 50 pounds on point "B" and 50 pounds on point "C".  Fortunately, the equations scale linearly, so once we find the forces for 100 pounds of load, we can multiply all results by 2 to find the forces for 200 pounds and so on.

Since the structure doesn't move, we can balance the external forces on the structure using equations for torque, force in the X direction and force in the Y direction.

1. ∑ τ_p = 0; The sum of the torques about any point must be zero. τ_{A +} τ_{E +} τ_{F +} τ_D = 0.  or A_y x 0 – 9 x 50 – 27 x 50 + 36 x D_y = 0. So D_y = 50.
2. ∑ F_x = 0; The sum of the forces in the X direction must be zero. Here, there are no forces in the X direction.
3. ∑ F_y = 0; The sum of the forces in the Y direction must be zero.  A_y + D_y – B_y – C_y = 0.  or A_y + D_y – 50 – 50 = 0.  Since D_y = 50, then A_y = 50.

Clearly, these results make sense because the structure is symmetrical.  That is,  if we have 50 pounds of force coming down on point "B" and 50 pounds of force coming down on point "C", this should be transmitted into 50 pounds of reaction force coming up corner "D" and 50 pounds of reaction force coming up corner "A".

Now that we know the external forces, we can calculate the internal forces.  We will do that next.