24 



STEENGTH OF MATERIALS 



where \ is zero or an arbitrary integer, either positive or negative 

 Equation (6) gives the angles which the planes containing tin- maxi- 

 mum and minimum normal stresses make with the horizontal 

 From equation (5), 



sin 2 a = 



cos 2 a = 



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

 maximum and minimum values of the normal stress are found to be 



(?) 



ruin 



27. Principal stresses. Since X in equation (6) is an integer. 

 two values of a given by this equation differ by 90, and, conse- 

 quently, the planes containing the maximum and minimum normal 

 stresses are at right angles. The maximum and minimum nnual 

 stresses are called principal stresses, and the directions in which they 

 act, principal directions. 



From equation (3), the right member of equation (4) is equal t< 

 2 g 1 . But since equation (4) is the condition for a maximum <r min- 

 imum value of the normal stress, it is evident that tin* normal stress 

 is greatest or least when the shear is zero. 



The results of this article can therefore be summed up in 

 following theorem. 



Through each point of a body subjected to piano, strain there are 

 two principal directions at right angles, in each of which the shear is 



zero. 



Problem 24. Find the principal stresses and the principal directions at a ] 

 in a vertical cross section of a beam at which the unit normal i'o Ih./in.* 



and the unit shear is 250 lb./in. 2 . 



Solution. In this problem p x = 400 lb./in. 2 , p v = 0, and q = 250 lb./in. . 



Therefore, from equation (6), 



- 1 ~ + ^ = - 25 40.2', or + 04 19.8'; 



and from equation (7), 



= 520 lb./in., 



= - 120 Ib./inA 



\ 



