FLEXUKE OF BEAMS 



91 



Problem 97. A beam bears a load of 15 tons at a certain point A, and its 

 deflections at three other points, B, C, D, are measured and found to be .30 in., 

 . 15 in., and .09 in. respectively. If loads of 5, 12, and 8 tons are brought on at #, C, 

 and D respectively, find the deflection at A. 



Solution. The deflections at 1?, C, and D due to a unit load (one ton) at A are 



= .02 in., = .01 in., and = .006 in. respectively. Therefore, by Maxwell's 

 !) 15 lo 



theorem, the deflection at A is 



D a = .02 x 5 + .01 x 12 + .006 x 8 = .268 in. 



77. Influence line for reactions. The most important application 

 of Maxwell's theorem is to the determination of the unknown reac- 

 tions for a continuous beam. 



Consider a beam continuous dVer three supports, as shown in 

 Fig. 72. Suppose the middle support removed and a unit load (say 1 

 ton) placed at this point. A I B c 



Then, if the elastic curve 

 is plotted, the ordinate to 

 this curve at any point / 

 is the deflection at / due 

 to the unit load at B, or, 

 in other \vnnls this ordi- 

 nate is the influence num- 

 ber J ifi . Similarly, the ordinate to the elastic curve at B is the influence 

 number -/,. 



N "W y/ 2 , the unknown reaction at B, must be of such amount as 

 to counteract the deflection at B due to a load P at any point /. 

 Therefore 



Fi... 



But, by Maxwell's theorem, J bi = J^ ; consequently 



The influence numbers J^ and J^ are known as soon as the elastic 

 curve for unit load at B is plotted. Therefore, in this case, the con- 

 struction of one elastic curve gives sufficient data for all further 

 calculations. 



