RELATIONS r>ET\YEEX STRESS AND DEFORMATION 25 



28. Maximum shear. The condition that q r shall be a maximum or 

 a minimum is that - = 0. Applying this condition to equation (3), 



2 cos 2a - 2q sin 2a ; 

 whence 

 (8) 



By comparing equations (5) and (8) it is evident that tan 2 a, 



from (8), equals cot 2 a, from (5). Therefore the values of 2 a 



obtained from these equations differ by 90, and hence the values 



of a differ by 45. Therefore the planes of maximum and mini- 



shear are inclined at 4$ to the planes of maximum and 



<1 stress. 

 'in equation (8), 



d = 



Substituting these values of sin 2 a and cos 2 a in equation (3), 

 the maximum and minimum values of the shear are found to be 



(9) l * = 



mln 



It is to be noticed that the maximum and minimum values of the 

 shear given by equation (9) are equal in absolute amount and differ 

 only in sign, which agrees with the theorem stated in Article 23. 



Problem 25. Kind the maximum and minimum values of the shear in Prob- 

 lem 24, and their directions. 



29. Linear strain. If a body is strained in only one direction, the 



i is said to be linear. For instance, a vertical post supporting a 



weight, or a rod under tension, is subjected to linear strain. The 



normal stress and unit shear acting on any inclined section of 



a body strained in this way can be obtained by supposing the axes 



of coordinates drawn hi the principal directions and putting q = 



an<l p v in equations (2) and (3). These values can also be 



derived independently, as follows. 



