28 



STRENGTH OF MATERIALS 



that is to say, the intensity of the shear in the planes of zero normal 

 stress is equal to the maximum value of the normal 



To give a geometrical represen- 

 tation of the conditions of tin- 

 problem, suppose a small cube cut 

 out of the body with its i 

 inclined at 45 to the principal 

 directions. Then tin- uiily stresses 

 acting on the inclined faces of 

 this cube are shears equal in 

 amount to the principal stresses. 

 The strain in this case is called 

 simple shear. 



Conversely, if a small cube is 

 subjected to simple shear, as indi- 

 cated in Fig. 15, tensile stresses equal in amount to this shear < 

 in the diagonal plane AC of the cube, and compr 

 like amount in the diagonal plane BD. 



FIG. 14 







D 



Problem 27. The steel propeller shaft of a 

 steamship is subjected to a shearing stress of 

 10,000 lb./in. 2 . Find the maximum tensile stress 

 in the shaft. 



32. Coefficient of expansion. Consider 

 an infinitesimal prism of dimensions dx, 

 dy, dz, and suppose that under strain 

 these dimensions become dx + s x dx, 

 dy + s v dy, dz + s z dz, where s x , s y , s t are ' " i& 



the unit deformations in the directions of the edges of the prism, 

 Then the volume of the prism becomes 



V+dV=(dx + s x dx) (dy + s,dy) (dz + / 

 or, neglecting infinitesimals of an order higher than the : 



Let 7f = s x + s y + Sg . Then the change in the volume of the prism 

 due to the strain is 



dV=Kdxdydz. 



