﻿422 Prof. E. L. Hancock on the Effect of Combined 



It is interesting at this time to see the results obtained by 

 substituting the results obtained from these tests in the various 

 formulae used for determining the unit stresses for the case 

 of combined stresses. The formulae generally used in designing 

 parts subjected to combined flexure and torsion, tension and 

 torsion, and compression and torsion, are as follows: — 



9 i = (l/2)[p + vV + W)]. • • • • (1) 



fc=(l/2)v/y + 4A , > ( 2 ) 



where q 1 represents the greatest unit tension on any internal 

 plane, q s the greatest unit shear on any internal plane, p the 

 unit simple tension, and p s the unit simple shear. The 

 formulae give the apparent unit tension and shear acting on 

 any particle of the body. To obtain the true maximum unit 

 stresses on any internal plane it is necessary to take account 

 of the lateral contraction as well as the elongation, L e. it is 

 necessary to use Poisson's ratio. Taking this ratio as 1/4 

 (Johnson, ' Materials of Construction '), the formulae giving 

 the maximum unit tension and the maximum unit shear on 

 any internal plane are 



T 3 = (3/8> + (hj$) vV + 4p s 2 ), .. . . (3) 



T.=(5/8XvV + 4ft s ) (4) 



Or using Poisson's ratio as 1/3 (Merriman, f Mechanics of 

 Materials/ 1905), the maximum unit stresses are 



T/=(l/3) P+ (2/3)v^M^?, ... (5) 



T s '=(2/3)vy + 4rf. ...... (6) 



Formulae (1), (2), (3), (4), (5), and (6) have been used in 

 constructing Table II. For this purpose p was taken as the 

 unit stress at the elastic limit in tension and p s the unit 

 torsional stress on the specimen. The unit stresses thus ob- 

 tained, whether apparent or real, which are greater than the 

 simple unit stress, are underlined. "For example, low carbon- 

 steel shows a unit stress in simple tension at tlie elastic limit 

 of 34,000 and a unit shear at the elastic limit in torsion of 

 30,000, when held in torsion at one.- third, the elastic limit 

 (corresponding to a fibre stress of 12-, 540), and tested in tension 



