LAWS OF BONE ARCHITECTURE 209 



at the given point. These values of Q hold for any circular 

 cross section. As the cross section is symmetrical about A-A 

 (fig. 11) the factors Q are computed for only one-half of the 

 cross section. 



A similar application of the theory of horizontal shear will be 

 made for various cross sections of the femur to be taken up in 

 a subsequent section. 



62. Vertical shearing stress. The true vertical shearing stress 

 at any point in a beam acts at right angles to the direction of the 

 horizontal shearing stress, and has the same numerical value. 



53. Lines of stress in beams. In the preceding section it was 

 shown that at any point in a beam there is a horizontal and a 

 vertical shearing stress, Sh, the formula for which was also 

 given {50, p. 206). 



At the same point there is also a longitudinal compressive or 

 tensile unit stress, S, which can be computed from the beam 

 formula, S = Mc/I, and the principle that these longitudinal 

 stresses vary in proportion to their distances from the neutral 

 axis. 



It can be shown that these unit-stresses combine to produce 

 maximum and minimum normal stresses on planes at right 

 angles to each other, and maximum shearing stresses on the 

 planes that bisect these planes. 



From mechanics it is found that the angle 9, which the direc- 

 tion of a maximum or minimum normal tensile stress makes 

 with the neutral surface is given by the following formula. 



cotangent 29 = -^ 

 and the magnitude of this tensile stress is given by the formula. 



^n = 2^ + yj Si + i^S)- 



If the minus sign be placed before the radical, this formula 

 will give the amount of the compressive stress at right angles 

 to the maximmn tensile stress at the same point. 



THE AMEKICAN JOURNAL OF AXATOMV, VOL. '21, XO. 2 



