28 Proceedings of the Uoyal Irish Academy. 



To find the equivalent uniform rate of loading to give the maximum 

 bending-moment on each span at the same point of the span, we must double 

 the moment, and divide by the difference of the squares of the half -span and 

 the displacement from the middle of the span of the point at which the 

 maximum bending occurs. 



Span. 



Moment. 



Displacement. 



Rate of Load. 



Feet. 



Foot-tons. 



Feet. 



Tons per foot. 



42 



261 



3-00 



1-209 



37 



214-3 



1-85 



1-265 



32 



170 



•73 



1-331 



23 



83 



2-40 



1-312 



It will be seen that the rate of loading on the 23- foot span is less than 

 that on the 32-foot span. But the rate should constantly increase as the 

 span decreases. By inspection it will be found that the two driving-wheels, 

 11 and 12 tons, can be accommodated on a span of 15 feet, instead of 23 feet. 



To interpolate spans, slightly smaller or larger than those given hy dropping 

 off parts of the span proportional to the loads on the ivheels dropped off the 

 locomotive. 



Thus to determine the max. bending-moment for a span of 20 feet loaded 

 in the most trying way by the group of wheels 11, 12, 9 tons. Consider the 

 parabolic segment and polygon standing on the oblique base D'B, the 

 horizontal projection of which is 32 feet. By adopting a coarser scale for feet 

 we can make this horizontal projection measure 20 feet instead of 32. Now, 

 however, the height of the parabolic segment must also be measured on a 

 coarser vertical scale in the like ratio assumed to save the trouble of 

 re-drawing the parabola. But the polygon must be re-drawn. On fig. 5, then, 

 the ruling-side of that polygon is shown by a hatched line with its two ends 

 set up higher from the oblique base D'B' in the ratio of 32 to 20, and 

 produced each way to h and k ; then hk is bisected by a black spot the height 

 from which to the parabola measures 119 on the old scale. This is to be 

 decreased in the ratio 20 to 32, when we have 74-4 foot-tons, and the horizontal 

 distance of the black spot from the ring at the middle of the oblique base 

 B'B measures on the old scale 1-19, but when altered in the ratio of 20 to 32, 

 we get -734 feet. 



Again, if the original locomotive is to ride on a span of 50 feet instead of 

 42 feet, it is only necessary to lower the side of the polygon EF in the ratio 

 of 42 to 50, shown by a hatched line ; produce it each way to m and n ; then 

 bisecting mw in the black spot, and reading the height to the parabola, we have 

 288-6, which is to be increased in the ratio 50 to 42, giving 343-56 foot-tons. 



