THE EQUILIBRIUM OF ELASTIC SOLIDS. 65 



CASE XII. 



When a long beam is bent into the form of a closed circular ring (as in 

 Case V.), all the pressures act either parallel or perpendicular to the direction 

 of the length of the beam, so that if the beam were divided into planks, there 

 would be no tendency of the planks to slide on one another. 



But when the beam does not form a closed circle, the planks into which it 

 may be supposed to be divided will have a tendency to slide on one another, 

 and the amount of sliding is determined by the linear elasticity of the sub- 

 stance. The deflection of the beam thus arises partly from the bending of the 

 whole beam, and partly from the sliding of the planks ; and since each of these 

 deflections is small compared with the length of the beam/ the total deflection 

 will be the sum of the deflections due to bending and sliding. 



Let A = Mc = E\xy*dy (65). 



A is the strfihess of the beam as found in Case V., the equation of the 

 transverse section being expressed in terms of x and y, y being measured from 

 the neutral surface. 



Let a horizontal beam, whose length is 2l, and whose weight is 2w, be 

 supported at the extremities and loaded at the middle with a weight W. 



Let the deflection at any point be expressed by ty, and let this quantity 

 be small compared with the length of the beam. 



At the middle of the beam, S,?/ is found by the usual methods to be 



8$ = -. (-faVw + %l 3 W) (66). 



A 



Let B= \xdy = - f r (sectional area) (67). 



2 J 2i 



B is the resistance of the beam to the sliding of the planks. The de- 

 flection of the beam arising from this cause is 



cs /i T/T7A / fi Q ^ 



VOL. I. 9 



