90 STRENGTH OF MATERIALS 



Maxwell's theorem, when modified so as to apply to beamsj si 

 that if unit loads rest on a beam at two points I and K, the deflection 

 at I due to the unit load at K is equal to the deflection at K due to 

 the unit load at I. The following simple proof of the theorem is due 

 to FoppL* 



Consider a simple beam bearing unit loads at two points / and K 

 (Fig. 71). Let the deflection at K due to a unit load at / le denoted 

 by J ki) the deflection at / due to a unit load at / by J iit etc., the 

 second subscript in each case denoting the point at which the unit 

 load is applied, and the first subscript the point for which the number 

 gives the deflection. Thus J ik denotes the influence of a unit load 



at K on the deflection at /. 



For this reason the quantity 



J ik is called an influence num- 



ber. 



If the load at / is of 



amount P,, the deflection at 

 FIG. 71 / is J u P it that at A' i- -/ /' . 



etc. 



Now suppose that a load P t . is brought on the beam gradual 1\ 

 the point I. Then its average value is J-P,.,the deflection undrr tin- 

 load is J it P it and consequently the work of deformation is i /',(./ / . 

 After the load P i attains its full value suppose that a load P k is 

 brought on gradually at K. Then the average value of this load is 

 \P k , but since P. keeps its full value during this second deflection, 

 the work of deformation in this movement is P.^P^) -f J- P k (J kk P k ). 

 Therefore the total work of deformation from both deflections is 



Evidently the same amount of work would have been done if the 

 load P k had first been applied, and then P,.. The expression for the 

 total work obtained by applying the loads in this order is 



Therefore, equating the two expressions for the work of deformation, 

 which proves the theorem. 





* Festigkeitslehre, p. 197. 



