Dynamical Principle of Least Action. 301 



equivalent to the one referred to. Next, let a beam of uniform 

 section be under the action of a transverse load of uniform in- 

 tensity w per unit of length, and also of bending moments 

 Mj, M 2 acting on its extremities against the load ; then, if the 

 bending moment at a point distant x from one end be ex- 

 pressed in terms of M„ M 2 , iv, x, and c s the half span of the 

 beam, the result on being squared and integrated between and 

 2c gives for U, the work done, 



P-gSjW-t M x M 2 + M 2 2 -^ (M, + MJ + ?w 2 c 4 }. 



By aid of this formula the work done in a uniform beam, 

 acted on by any transverse forces in one plane and a uniformly 

 distributed transverse load, can be estimated by a process of 

 addition in terms of the bending moments at the points of ap- 

 plication of the forces. If the forces are not transverse, then 

 the following theorem is necessary : — The work done in a beam 

 by a thrust and a bending moment is the sum of the works 

 done by each of them acting separately. 



For the bending moment does not at all affect the length 

 of the line drawn through the centres of gravity of its several 

 transverse sections, since that line lies in the neutral surface ; and, 

 on the other hand, the work done by the thrust, considered as 

 uniformly distributed over the sectional area, is equal to the 

 work done by its resultant which passes through the centres of 

 gravity, and consequently that work is unaffected by the bend- 

 ing moment. It follows, therefore, that if H be the thrust and 



Cf M 2 H 2 1 



A the sectional area, U = l 4 nW + 9 ^ A r dx. 



The hypotheses employed in these formulae are those which 

 are usually employed in treating of the strength of beams, 

 namely, that the strain is within the limits of perfect elasticity, 

 and that the effect of the tangential stress is inconsiderable. 

 1 shall now give some examples of the method of applying the 

 principle to the questions I referred to ; but as it is very uni- 

 form, two will suffice. 



1. A beam is fixed horizontally at both ends, and loaded with 

 a vertical load distributed uniformly ; find the bending moments 

 at the fixed ends. Here, if w be the intensity of the load, 

 M, M 2 the required bending moments, and c the half span, the 

 work done in the beam is given above, and we have simply to 

 make it a minimum by variation of the unknown quantities 

 Mj, M 2 , whence 



2M 1 + M 2 -m;c 2 = 0, 2M 2 + M 1 -wc 2 = 0, 



M 1 = M 2 =i«;c 2 . 



This result agrees exactly with that given by the ordinary 



