4 BOVEY ON SHEAE PEODUCED 



The reaction R., at A with the new distribution of weights is given by the equation, 



B2.l = w,t^\.{ai,+i+x)+Wj,+2-(.ap+2+x')+ • • • +«'«■(«„+-'■) + w«+! •««+!+ • • • »«,.+»■««+« 

 = (R, — R,^.l+x.{W„— W^.)+BJ 



. •. (R,— B,yi =z (E^, — Br)-l — x-OV,.—W,.). 

 The shear S., at the same point P as before, is given by the equation, 



S, = R, — (Wj,+ l+U'j,+2-h .... W,+ lV,.+l+Wr+2+ .... +Wr+,j) 



= R,-iW^-W,+ T). 



Hence the shear at P with the first distribution of load is greater or less than the shear at 

 the same point with the second distribution, according as Si > < 8-2 



or R^ — W, >< i?^ — ( Wr — W^ + T) 



or R, — R,><Wp—T 



oc 



or i?, — i?, ( TF„ — W„-) ><W^—T 



I 



or R,, -R,- W^,+ T>< - ( TF„ - W,). 



b 



Corollary. — If no weights advance upon or leave the girder, R^„ i?,, and W,, become 

 severally nil, and the last relation reduces to the simple form, 



T W„ 

 — X — 

 X I 



A. — In words, the shear at P with the Jlrst distribution will be greater or less than 

 the shear at the same point with the second distribution, according as the weight trans- 

 ferred, divided by the distance of transfer, is greater or less than the total weight divided 

 by the span. 



Again, the bending moment iJf, at P with the first distribution is given by the 

 equation, 



M^= R^{1 — z) — u\(_a^ — z) — w.,{a., — z) — .... — «',.(»,. — ~) 



= R,(l — z) — R,.l+z.W, 

 z being the distance of P from B. 



