22 



STRENGTH OF MATERIALS 



24. Planar strain. If no stress occurs on one pair of opposite 

 faces of the cube, the stresses on the other faces all lie in one of the 

 diametral planes. This is called the planar condition of strain. 



Suppose the Z-axis is drawn in the direction in which no stress 

 occurs, as shown in Fig. 8. Then the stresses all lie in the plane 

 parallel to XOY, and the relation between them is as represented in 

 Fig. 7 of the preceding article. 



25. Stress in different directions. As an application of planar 

 stress, consider a triangular prism on which no stress occurs in 

 the direction of its length. Let the -axis be drawn in the direction 

 in which no stress occurs, and let a denote the angle which the 



FIG. 9 



inclined face of the prism makes with the horizontal, as shown in 

 Fig. 9. Then if dF denotes the area of the inclined face ABCD, 

 and /, q' denote the normal and shearing stresses on this face 

 respectively, / and cf can be expressed in terms of p^ p, an.l 

 means of the conditions of equilibrium. Thus, from 2 hor. . . .nips. = 6, 



p'dFsma + q'dFcosa -p x dFsma - qdFcosa = 0. 

 Similarly, from 2 vert, comps. = 0, 



p f dFcosa - q'dFsma-p x dFcosa - qdFsina = 0. 

 Dividing by dF, these equations become 



f 

 \ 



p f sina + q r cosa - p x sina - q cosa = 0, 

 p' cosa - q' since -p x cosa - q sma = 0. 



