26 



STRENGTH OF MATERIALS 



Consider an elementary triangular prism, and let the axis of X be 

 drawn in the direction of the linear strain. The stresses acting on 



the prism will then be as shown in 

 Fig. 11. Let dF denote the are 

 the inclined face. Then the area of 

 the vertical face" is dF sin a. Resolv- 

 ing p x into components parallel to p r 

 and q f respectively, the conditions of 

 equilibrium are 



p x sin a (dF sin a) = j> 

 p x cosa(dFsina) = 7 



FIG. 11 



or, dividing by 

 (10) 



/ = 



dq' 



From the condition = 0, it is found that the maximum shear 

 da 



occurs when a = 45, and its value is 



For a = or 90, q' = 0. Consequently, there is no shear in 

 planes parallel or perpendicular to the direction of the linear strain. 



Problem 26. A wrought-iron bar 4 in. wide and fin. thick is subjected to a 

 pull of 10 tons. What is the unit shear and unit normal stress on a plane inclined 

 at 30 to the axis of the strain ? Also 

 what is the maximum unit shear in 

 the bar ? 



30. Stress ellipse. Suppose 



that an elementary triangular * 



prism is cut out of a body sub- 

 jected to planar strain, so that 

 two sides of the prism coincide 

 with the principal directions. 

 Then, by Article 27, the shears 

 in these two sides are zero. Now 



let the axes of coordinates be drawn in the principal directions, and 

 resolve the stress acting on the inclined face of the prism into 



