Alexander — Maximum Bendimj-moments on Short Girders. 25 



FiEST Graphical Method with one Parabola only. 



For the 42-ton locomotive on fig. 5, after finding the centre of gravity G, 

 choose the smallest span which will accommodate the loco, standing with G 

 and the ruling-wheel Wi equidistant from the two abutments. That span is 

 40 feet ; but we have taken 42 feet, which is only slightly greater, and makes 

 the steps of the construction more evident, and simplifies any arithmetical 

 checks. 



Construction, fig. 5. — Lay down the span AB = 42 feet on a suitable scale 

 of feet, and divide it into five fields proportional to the weights on the wheels 

 of the locomotive. On the left side draw the load-line equal to 42 tons, the 

 weight of the locomotive to a scale of tons judiciously chosen, so that with a 

 polar distance of 10 feet a well-conditioned isosceles triangle is formed. Join 

 the pole to the ends of the load-line. Construct the apex of an isosceles 

 triangle standing on the base AB, and having its sides parallel to the extreme 

 vectors from the pole. The height of this apex above the middle of the base 

 should measure 441 foot-tons, being a fourth of the product of 42 tons and 

 42 feet on a scale ten times finer than the ton scale, the polar distance being 

 ten. Construct the scale of foot-tons by renumbering the ton scale accord- 

 ingly. Taking the apex of the isosceles triangle as a vertex, construct the 

 parabolic right segment standing on the span AB. Construct the points close 

 together near the vertex, but more sparsely well out from it. A construction 

 is shown in the appendix. It will be seen at this stage that the shape of this 

 parabolic segment depends on the choice of scales. 



To construct the polygon ACDEFB, draw vertically upwards the lines of 

 action of four fictitious forces through the junctions of the five fields. On the 

 right side draw upwards their load-line 7, 10, 8, and 5 tons, being numericallj' 

 the distances between the wheels multiplied by unity, the ratio of the weight 

 of the locomotive to the length of the span. To a trial pole draw the dotted 

 link-polygon among the four upward forces and the two holding-down forces 

 at A and B. A vector from the trial pole parallel to the closii;ig side gives 

 the junction between the magnitudes of the two holding- down forces. From 

 this junction lay off horizontally 10 feet for the true pole, and construct the 

 link-polygon ^CZ)^^-?!^. Otherwise the height of i^'may belaid up directly 

 to scale as 9 tons x 13 feet = 117 foot-tons, being the product of the 9-ton 

 wheel-load and its distance from O. Then the height of E is 189 foot-tons, 

 being 12 x 6 greater than the last, while the height of D is to be 145 or 

 11 X 4 less than the last, also the height of C is 85, or 5 x 12 lesser again. 

 In this way by taking the moments of the weights of the wheels about G, their 



