The determination of the wheel positions is given in Table XIV.
While many wheels on point 1 satisfy the condition, the greatest moment will occur when one of the large drivers is at the point, and it is therefore unnecessary to examine the point for other wheels. The same is true at the center point, 5, the maximum occurring under one of the heavy driver wheels. ' The reactions and the computations for the same are given in Table XV.
Table XVI gives the computations of the live-load moments at the tenth-points, the final results being in pound-feet.
Whenever any loads were off the left end of the bridge, the lines 7 to 16 of the engine diagram were used (Fig. 85). For example, with wheel 10 at 5, wheel 1 would be off the left end. By looking in the second space of line 8, there is found the quantity 13 904, which is the moment of wheels 2 to 18 inclusive about a point directly under wheel 18. Just to the right of the vertical line through wheel 18, is the value 284, which is the weight of wheels 1 to 18 in clusive; but this must be decreased by 10, the weight of wheel 1, as that wheel is off the span. As wheel 18 is 2 feet from the right end of the girder, the moment about the point is 13 904 + 274 X 2. By looking in the second space of line 16, the value 4 072 is found. This is the value of the moment of loads 2 to 9 inclusive about a point directly under wheel 10, and must be subtracted from the moment of the reaction in order to get the moment at 5 for this loading. See Articles 21 and 47 for further information regarding the use of the values in lines 7 to 16 of the engine diagram.
By the help of differential calculus it can be proved that the greatest possible moment does not occur at the middle of a beam loaded either with concentrated loads or with concentrated loads followed by a uniform load, but it occurs under the load nearest the middle of the beam when the loads are so placed that the middle of the beam is half way between the center of gravity of all the loads and the nearest load.
The wheel which produces this greatest moment is the same one which produces the maximum moment at the middle of the beam.
The exact solution of this problem involves the use of quadratic equations, but for all practical purposes the following rule will suffice : Place the loading so that the wheel which produces the maximum moment at the middle of the beam is at that point. Find the distance of the center of gravity of all the loads from the right end. Move the loads so that the middle of the beam is half way between the center of gravity as found above and the load which produced the maximum moment at the middle of the beam. Find the moment under that load, with the loads in the position just mentioned.
For the case in hand, wheel 12 at 5 gives the maximum moment. The moment at the right end of the span, wheel 12 being at 5, is: 9 514 + 10 X 214 + 2 X - 2 = 11 754 000 pound-feet.
The center o: gravity is 11 — 50.2 feet from the right sup port, or 0.2 foot to the left of the center of the girder. Now place wheel 12 one-tenth of a foot to the right of the center, and mint, the moment under it. The reaction will be: R = (9 514 + 9.9 X 214 + 2 X 9.9' - 2) - 100 = 117.306; and, M — 117.306 X 50.1 — 2 658 = 3 219 306 pound-feet.
In this particular case the difference between the greatest moment possible and the greatest moment at the middle is not sufficient to warrant the extra labor involved in computing it. In general it may be said that if the greatest moment possible occurs within six inches of the middle of the beam, it is not necessary to compute it, the moment at point 5 being taken.
5S. Shears in Plate.-Girders. in the case of through plate girders, the maximum live-load shears are determined by placing the wheels in such a position that Q = 11 — G is +, and q = — m In these equations, m is the number of panels into which the span is divided; and the other quantities are the same as given in Article 45, which should now be reviewed.
For example, let it be required to determine the dead and live load shears in the through plate-girder of Article 57, p. 111.