Mechanics Question: Simply Supported Beam with Load

  1. 1. The problem statement, all variables and given/known data

    A bridge is idealized as a simply supported beam AB of length L, as shown in Figure 2. A
    four-wheeled vehicle of weightW now drives across the bridge. Sketch the shear force diagram
    for the case illustrated, where the rear wheel is a distance a from the left support. Notice that
    the two wheels are a distance d apart and the center of gravity of the vehicle is a distance d/3
    behind the front wheel. Use your results to determine the value of a at which the maximum
    bending moment reaches its greatest value, and find this value.

    http://i43.tinypic.com/11ry5us.png


    2. Relevant equations

    Equilibrium equations, M(x)= ∫V(x), V(x) being shear force

    3. The attempt at a solution

    Drew a free body diagram of the car and bridge to find the reaction forces, one at each end and at the contact points of the car and the wheel. R1=W(1-(a+2d/3)/L), R2=W/3, R3=2W/3,R4= W(a+2d/3)/L from left to right. I then sketched a shear force diagram starting at
    V(0) = -R1
    V(a) = -R1 + R2
    V(a+d) = V(a)+R3
    V(L) = 0

    I tried to use the areas under this curve (3 rectangles) and have them equal 0 and try to solve for a, but I couldn't get anywhere with that. any help/direction would be awesome.
     
  2. jcsd
  3. Never mind, got it. Same approach but more perseverance.
     
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