58 Isotropic Elastic Solid Ellipsoids. 



A material line parallel to an axis 2b, perpendicular to the axis of 

 rotation, is exposed to two opposing actions. The components of 

 "centrifugal " force in its own direction tend to lengthen it, while those 

 in the perpendicular direction 2c tend to shorten it. On an average 

 the former action will prevail so long as the mean dimension parallel 

 to 2& bears to that parallel to 2c a ratio exceeding v/>; : 1. 



Approximate Methods. 



14. Suppose a body symmetrical with respect to the coordinate 

 planes, of great length, 2c, in the direction of z, to be acted on by the 

 bodily forces whose components are P#, Qy, Rz. The stresses over 

 the plane z = z must balance the force whose components are 



1 1|/> Px dx dy dz, &c. 



Since every section perpendicular to z has its e.g. on that axis, it is 

 clear the integrals vanish, which give the components parallel to x 

 and y. If now we assume that when there is no rapid variation in 



the cross section, <r, as z alters, zz is large compared to the other 

 stresses, and is approximately uniform over a, then we may take as a 

 first approximation 



^ Rr rr 



\zdzdy dx. 



For an elongated ellipsoid this gives, as in (51), 



2 2 2 ). 



Taking this as the sole stress, we obtain from the ordinary stress- 

 strain relations the same values of s x , s y , s z as in (ol), and thence by 

 integration the correct first approximation values of the displace- 

 ments. 



For a thin elliptic disc in which one of the axes, 2c, of the elliptic 

 section is very large compared to the other or to the thickness, we 

 find as first approximations 



= ply = |-R^(c 2 - z 2 )/E, > .......... (77). 



If, for instance, the disc be rotating about its thickness, i.e., the 

 axis of the cylinder, then R = u? and 



c 2 -z 2 E-| 



