686 ON FORM AND MECHANICAL EFFICIENCY [ch. 



Fig. 338. 



Therefore ^a/cos = pa sin 6, therefore q = p sin 6 cos 6, = ^p sin 26. 



Therefore when sin 2^ = 1, that is, when d = 45°, g- is a maximum, and 

 =p/2; and when sin 2^ = 0, that is when ^ = 0° 

 or 90°, then q vanishes altogether. 



This is as much as to say, that a 

 shearing stress vanishes altogether along 

 the hnes of maximum compression or 

 tension ; it has a definite value in all 

 other positions, and a maximum value 

 when it is incUned at 45° to either, or 

 half-way between the two. 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 as a rule it disrupts by shearing, and along 

 some plane approximately at 45° to the axis of compression. 

 Again, in the beam which we have already considered 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 line 

 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 completely got rid of when we arrange the 

 materials of our construction wholly 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-Hnes, for instance, it will be in 

 a position of comparative equihbrium, or minimal disturbance; 

 but if it be incHned obUquely to the pressure-lines, the shearing 

 force will at once tend to act upon it and move it away. This 

 is neither more nor less than what happens when we comb our 



