The location of the truck (or point loads) that will maximize the moment for a particular load are found first finding the resultant force and its location with respect to that load. In the diagram on the left, load P_{2} is used to illustrate the calculation process. Assuming R and d are known, the reaction at the left end is
ΣM_{B} = 0 = -LR_{A} + R(L - d - x)
R_{A} = R (L - d - x)/L
Next, summing moment at the cut at P_{2} (remember, the maximum is under the load), gives
M_{2} = x R_{A} - a P_{1}
= x R (L - d - x)/L - a P_{1}
The maximum location will occur when
dM_{2}/dx = 0 = R (L - d - 2x)/L
Therefore, the maximum is when
This is true for any particular load and for any number of loads. Even though three loads were used in this example, the number of loads does not effect the maximum location. |