XVI 



OF SHEARING STRESS 



Q upon CD = q . area of CD, which is = g . a/cos 6. Therefore qa/ cos 6= pa sin By 

 therefore q=p sin 6 cos 6 = ^p sin 2^. Therefore when sin 20= 1, that is, when 

 6 = 45°, g is a maximum, and =p/2; and when sin 20 = 0, that is when = 0° or 

 90°, then q vanishes altogether. 



This is as much as to say, that under this form of loading there is 

 no shearing stress along or perpendicular to the lines of principal 

 stress, or along the Unes of maximum compression or tension; but 

 shear has a definite value on all other planes, and a maximum value 

 when it is inclined at 45° to the cross-section. This may be further 

 illustrated in various simple ways. When 

 we submit a cubical block of iron to 

 compression in the testing machine, it 

 does not tend to give way by crumbling 

 all to pieces, but always disrupts by 

 shearing, and along some plane approxi- 

 mately at 45° to the axis of compression ; 

 this is known as Coulomb's 'Theory of 

 Fracture, and, while subject to many 

 qualifications, it is still an important 

 first approximation to the truth. Again, 

 in the beam which we have already con- 

 sidered under a bending moment, we 

 know that if we substitute for it a pack of cards, they will be strongly 

 sheared on one another ; and the shearing stress is greatest in the 

 "neutral zone," where neither tension nor compression is manifested: 

 that is to say in the Hne which cuts at equal angles of 45° the 

 orthogonally intersecting lines of pressure and tension. 



In short we see that, while shearing stresses can by no means 

 be got rid of, the danger of rupture or breaking-down under shearing 

 stress is lessened the more we arrange the materials of our con- 

 struction along the pressure-lines and tension-lines of the system; 

 for along these lines there is no shear*. 



To apply these principles to the growth and development of 

 our bone, we have only to imagine a little trabecula (or group of 

 trabeculae) being secreted and laid down fortuitously in any direction 

 within the substance of the bone. If it lie in the direction of one of 

 the pressure-lines, for instance, it will be in a position of comparative 



Fig. 467. 



It is also obvious that a free surface is always a region of zero-shear. 



