45. Position of Wheel Loads for Maximum Shear. By methods of differential calculus, it can be proved that, for any system, either of wheel loads or wheel loads followed by a uniform load (see Fig. 91), the correct wheel that should be at the panel point b in order to If a load is directly over the panel point a, it is not to be included in the weight G; neither is P included in the weight G. If a wheel load should come directly over the right end of the truss, it should not be considered in the quantity W.
The only way to determine which wheel is the correct one, is to try wheel 1, then wheel 2, and so on, until the wheel or wheels are reached that will give the Q and q signs of an opposite character.
The process should not be stopped there, but the next succeeding wheels should be tried until Q and q again have the same sign.
As an example, let it be required to determine the position of the wheel loads to produce the maximum positive shears in a 6-panel 120-foot Pratt truss. This work should be arranged in tabular form, and Table V is found to be convenient.
A study of Table V shows the fact that wheel 1 can never produce a maxi mum. It also shows that there are in some cases two positions which will give large values of the moment. In these cases the shears for each position of the engines must be determined in order to tell which wheel at the panel point in reality gives the greatest. In practical work it is customary to use the first position found, as the difference in the shears resulting from the use of the two positions is not large enough to affect the final design.
Fig. 92 shows the engine diagram on the truss in the correct position to give the maximum shear in the second panel. The weight of wheel 16 is not included in the weight IV, as it is directly over the right support.
46. Position of Wheel Loads for Maximum Moments. In this case the methods of differential calculus are em ployed to determine which wheels will, if placed at a point, give a maximum mo ment at that point. For any system,
either of wheel loads or of wheel loads followed by a uniform load, that wheel which will cause K = n — L to be positive, and k = Wn m m(L P) to be negative, is the wheel. Here n is the number of the panel under consideration, and is to be reckoned from the left end; L = the load to the left of the point under consideration; and the remainder of the letters signify the same as they do in Article 45. In some cases there will be more than one position of the loads which will satisfy the above condition. It is then necessary to work out the actual moments created by the loads in each position, in order to find out which is the largest. The position of the loads for the greatest mo ments should be de termined for all panel points except the one on the extreme right, as the greatest moment possible maybe caused by wheels of the rear engine being on the point on the right hand side of the truss, instead of the wheels of the front engine be ing at the correspond ing point on the left hand side.
In general, it may be said that there will be a number of wheels which, if placed at the panel point in the cen ter of the span, will satisfy the given con ditions. In this par ticular case, it is not necessary to determine all of the moments. The greatest moment possible will occur when that one of the heaviest wheels of the second locomotive which gives the heav iest load upon the truss is at the point. In case several of the heavy wheels give the same maximum load W, use the first wheel which gives this max imum W.
Let it be required to determine the position of the wheel loads for maximum moments at the lower panel points of the 0-panel 120 foot Pratt truss of Article 45. The necessary work can be con veniently arranged in the form of a table, as is done in Table VI.