l'i|;s|o\ OF CKKTAIN FORMS OF SHAITIXG. 885 



7. IiHin-tnn<;- .i/' //,< .M'lriiu'iin .sv/v.w. Application to 



1 now pass cm tn the numerical determination of the torsion inoinent and stresses, 

 and in particular \' i lie maximum Mi> 



SAINT-YI.N \vr has shown, in his memoir on torsion,* that if we assume an 

 ellipsoidal distribution of limiting stretch (i.e., stretch such that, when it is exceeded, 

 tin- elasticity of the material is impaired), then, in the case of a shaft under torsion, 

 \\r have at any point 



./"= vK + 



where o- K , <r t . are the shearing strains in the planes yz, xz respectively, a- is the value 



of the limiting shearing .strain of the material, and s/s is the maximum value of the 

 ratio of the stretch in any direction to the limiting stretch in that direction. 



The condition that there should he no failure of elasticity is therefore that .</* < 1. 

 Therefore 



a*,. + o*, < oi 



and, since <r, t = i/z/p, <r._, xz/p, 



xz- + yz- < /roi 



The points where this condition will first be broken are called by SAINT-VENAXT 

 the fail-points ("points dangvreux"). They are clearly the points where xz- + yi l is 

 a maximum, i.e., where the resultant stress across an element of the section is a 

 maximum. Hence the importance of determining the points of maximum stress. 

 Strictly speaking, the latter give us no certain information as to where, or how, 

 rupture will actually take place : all that they tell us is where linear elasticity begins 

 to fail. But they will, in general, give us a useful clue to the regions where breaking 

 may be expected to occur, and, in the absence of any definite theory of plastic 

 deformation and rupture, we must l>e content to be guided by the results of elastic 

 theory. 



I have worked out numerically the values of the stresses at the points = , 

 r) = atid = 0, 77 = ft. These give the four points in which the axes meet the 

 boundary of the cross-section. I have denoted them by A and B resjxjctively. The 

 lioundary is convex at A and concave at B From considerations of symmetry it 

 follows that A and B must l>e point* of maximum or minimum stress, and the stress 

 being zero both at the corners and at the centre, it will often happen that they are 

 points of maximum stress. When this is the case those of the points A and B, 

 where the stress is numerically greater, will give us the fail-jtoint*. But there is an 

 obvious exception, when there are two points of maximum stress on either side of the 

 mid-|oiiit, and this is a case which, we shall see, does occur in these sections. 



* ' M^moires ties Savante Strangers, 1 1855, vol. riv., pp. 278-288. See also ToDHUNTER and PEARSON, 



1 Hist. Ela>t.,' vol. ii., part i., pp. 7-10. 



