DELATIONS BETWKKX STKESS AND DEFORMATION 27 



components parallel to the axes instead of into normal and shearing 

 stresses as heretofore. Then, from Fig. 12, if d F denotes the area of 

 the inclined face, the conditions of equilibrium are 



p' x dF = p x dF sin a, 

 p' v dF = p y dF cos a ; 



whence 



^ 



Px 



P* 



cos a. 



Sjuarini: and adding, 



pi pl 



which is the equation of an ellipse with semi-axes p x and p , the 

 coordinates of any point on the ellipse being p' x and p' y . Conse- 

 quently, if the stress acting on the inclined face of the prism is 

 calculated for all values of a, and 



stresses are represented in 

 magnitude and direction by lines 

 radiating from a common center, 

 the locus of the ends of these 



- will be an ellipse called the 

 stress ellipse (Fig. 



31. Simple shear. If a body is 



pressed in one direction and 

 equally elongated in a direction ;t 

 ri^lit angles to the first, the strain is planar. In this case, if the axes 

 are drawn in the principal directions, q = 0, p x = p v , and the stress 

 ellipse becomes the circle p'* + p'f = p^ 



Moreover, the normal stress in the planes of maximum or mini- 

 mum shear is zero; for by substituting in equation (2) the values 

 of sin2 and cos 2 a obtained from equation (8), the normal stress 



/*J I /VJ 



in the planes of maximum or minimum shear is found to be v , 



and this is zero since p x = p v . 



Substituting q = and p x = p v in equation (9), Article 28, the 

 maximum or minimum value of the shear in the present case is 



Fi.;. 



