Art. 86. 



DEFLECTION OF A BEAM OR GIRDER. 



135 





If -prf is taken as the unit load (w) at any point of a beam, 



then y will be the moment (M') at that point. Graphically, y 

 will be the ordinate in the string polygon, for this imaginary 

 load, if the pole distance is unity ; if the pole distance is made 

 equal to El, then the ordinate of the string polygon will corres- 

 pond to a unit load equal to M. It follows that */ a string poly- 

 gon be drawn for a distributed load represented by the moment 

 diagram, this polygon will represent the elastic line, if the pole 

 distance is equal to El. 



In a girder with flange plates there will be several values of 

 El, and hence the string polygon must be constructed with 

 different pole distances 1 . 



87. Deflection of a Beam Due to Shear. In the preced- 



ing discussion of the deflection of beams, only the bending 

 stresses were taken into account; it will now be shown that the 

 influence of the shear is negligible except in very short beams, 

 and in plate girders, in certain cases. 



Fig. 92. 



Fig. 92 shows the cross section of a girder of 24 ft. span, 

 carrying a load of 40000 Ibs. at its middle. Each reaction is 

 20000 Ibs. The shear in the girder, between the support and th? 

 load, is constant, and is equal to 20000 Ibs. Upon the usual 

 assumption, this shear is resisted by a shearing stress in the web 

 only, which is uniformly distributed over its cross section. The 



unit stress is therefore s s =^ 



13.5 



= 1480 Ibs. per sq. in. By 



equation (3), the deflect ionjper mcjL_of_Jength of girder is 

 inches; the deflection in 12 ft. (144 in.) 



1480 



1480 



12000000 



is ?/ 8 = 



1480 



12000000 



144=0.0178 inches. 



Mueller-Breslau's Graphteche &q,tik Vol. JI, Part 2. 



