FLEXURE OF BEAMS 93 



78. Castigliano's theorem. Consider a beam bearing any number 

 of concentrated loads P lf P 2 , ,P 1I , acting either vertically upward 



or downward, and let W 



\f 



denote the work of defor- 

 mation due to these loads 

 74). Then if one of the 

 loads, say P., is increased 



R, 



R, 



by a small amount rfP,., the 



deflection of P t is increased 



by the amount </, t dP,., that FlG ' 74 



of P 2 by the amount J 3i dP., etc., where e7 lt , 7 2 ,., etc., are influence 



numbers. Therefore the work of deformation is increased by the 



amount 



Ml' = P^dP. + P^dP, + + P n J nl dP, ; 



whence ,,,- 



- PA + PA +-+ PA 



In funning tliis expression the work done by dP i itself has been 

 neglected, since it is infinitesimal in comparison with that done by 

 / P,,etc. 



Now, from Maxwell's theorem, J ik = J ti . Therefore the above expres- 

 sion becomes ,,,- 



The right member of this equality, however, is the total deflection 

 D t at the point /, due to all the loads. Consequently the above expres- 

 sion may be written ,, 



dF t = /) - 



Suice the work of deformation W is a function of all the loads and 

 not of P,. only, this latter expression should be written as a partial 

 derivative; thus 



and hi this form it is the algebraic statement of Castigliano's theorem. 

 Expressed in words, the theorem is : The deflection of the point of 

 application of an external force acting on a beam is equal to the par- 

 tial derivative of the work of deformation with respect to this force. 



