478 X. STATICS OF DEFORMABLE BODIES. 



183. Problem of de Saint -Veiiant. We shall now treat a 

 problem dealt with by the distinguished elastician Barre de Saint- 

 Venant, in two celebrated memoirs on the torsion and flexion of 

 prims, published in 1855 and 1856. 1 ) For the full treatment of this 

 subject the reader is referred to Clebsch, Theorie der Elasticitdt, or 

 especially to the French translation of the same work, edited with 

 notes by de Saint -Venant himself. The problem is thus defined by 

 Clebsch, whose analysis is here followed. 



What are the circumstances of equilibrium of a cylindrical body, 

 on whose cylindrical surface no forces act, and whose interior is not 

 subjected to external forces, under which the longitudinal fibres 

 composing the body experience no sidewise pressure? 2 ) What forces 

 must act on the free end surfaces, in order to bring about such 

 circumstances? 



We have already treated a special case of this problem in 3, 

 176. If the ^-axis be taken parallel to the generators of the 

 cylindrical surface of the body, the conditions that adjacent fibres 

 exercise no stress on each other perpendicular to their length are 



67) 

 The conditions at the cylindrical surface are 



X n = X x cos (nx) H- X y cos (ny) + X z cos (nz) = 0, 



68) Y n = T x cos (nx) + Y y cos (ny) + Y z cos (ne) = 0, 

 Z n = Z x cos (nx) + Z y cos (ny) + Z z cos (nz) = 0, 



of which the first two are satisfied identically, since cos nz = 0, and 

 the last is simply, 



69) . Z x cos (nx) + Z y cos (ny) = 0. 



In order to remove the possibility of a displacement of the 

 cylinder like a rigid body involving six freedoms we will suppose 

 fixed: a point, a line, and an infinitesimal element of surface of one 

 orthogonal end or cross -section. Take the fixed point as origin, so 

 that U Q '= v = W Q = 0. Take the fixed element of surface for the 

 XT- plane, and the fixed line for the X-axis. For a point near the 

 origin, the shifts are 



1) Memoire sur la torsion des prismes, avec des considerations sur leur 

 flexion. Memoires des Savants etr -angers, 1855. 



Memoire sur la flexion des prismes. Journ. de Math, de Liouville, 2 me Serie, 

 T. I, 1856. 



2) That is, the stress on a plane parallel to a generator is only tangentij 

 and in the direction of the generator. 



