Stresses and Strains in tivo Dimensions. 



457 



On OX take OM = q in magnitude and direction. From 

 M draw MU=fi in magnitude and direction (i.e. upwards if 

 tensile, downwards if compressive), and on UM take MV = £> 2 



Fig. 4. 



+ x 



in such way that UY=^ 1 +^» 2 - [In the figure p x and p 2 are 

 both tensile, and g is positive.] 



Bisect UV in T, and with centre T and radius TO describe 

 a circle cutting UV in A, B. 



Then OA, OB are the directions of the principal planes, 

 and UA, UB the corresponding principal stresses (p,^)- 



For, if AOK=0, tan 2(9=-^-; 

 and Pi-JPa 



UA, UB = Ki?i+i> 2 )±i Vlih-^ + V, 

 which are the equations for the directions and magnitudes of 

 the principal stresses. 



The principal strains i, i' can also be shown by an extension 

 of the same figure ; for if E is Young's modulus for the ma- 

 terial, and 7] the ratio of lateral compression to longitudinal 

 strain for a single stress, the following relations hold : — 



E(t+O=(l-^)'0P+y)5 



Eow)=(i+^)(i>-y). 



