FLEXURE OF BEAMS 97 



Problem 100. Three Carnegie I-beams, No. B 80, are placed 4 ft. apart across 

 an opening 25 ft. wide. Across their centers is placed another I-beam of the same 

 dimensions as the first, and upon the center of this cross beam there rests a load of 

 10 tons. Find the greatest stress which occurs in any member of the construction. 



Solution. Let the amount of the load at FT, which is carried by GK, be denoted 



by P (1 ii:. 77). Then the loads on AB and EF at G and K respectively are each 



p 

 equal to , and the load on CD is 20,000 Ib. - P. 



Now the work of deformation for a simple beam of length I bearing a single con- 

 centrated load P" at its center is, from Problem 91, 



W 



96 El 



p 



Therefore, since the load on A B or EF is , the work of deformation for either 



of these beams is 



(20,000 - 

 96 El 



Similarly the work of deformation for CD is 



and for GK is 



Hence the total work of deformation for the entire construction is 



P*l\ (20,000 -P?l\ 

 192^7 WEI 



~ trr 



By the principle of least work, -- = ; consequently 



dP 



dW = Pl\ (20,000 -P)ll PI* 

 dP WEI 4SEI 4SEI 



From the Carnegie handbook, I = 795.0 in. 4 , and from the figure, li = 300 in., 

 1% = 96 in. Inserting these numerical values in the above expression, and solving 



f rP ' P = 13,048 lb. 



Having determined P, the stress in the various members can easily be calcu- 

 lated. Thus it is found that the greatest stress occurs in CD, its amount being 

 p = 23,593 lb./in.-. 



