104 Mr. J. J. Guest on the Strength of 



where », = — — X fluid pressure ; and a radial stress 



1 area 



varying from the value of the fluid pressure at the interior 



surface to zero at the exterior surface. 



Since in a cylinder under internal pressure the circuni- 



/ r 2\ 



ferential stress is given by the equation p> — Q 1 1 H — g )> 



where r is the external radius and r the radius at which the 

 stress is p, the circumferential stress at the interior bears to 



(r 2 \ t 



1 + -. — 9-^s 1= 1 + - nearly. 

 {r -t) 2 J r J 



Hence the variation from the mean value is about 2 or 3 per 



cent., as in the case of the shearing. stress due to torque. 



In the calculations the mean values of the radius have been 

 used, and no notice taken of the variation of the stress. In 

 the hydraulic tests the actual amount of variation in pounds 

 per sq. inch is equal to the value of the fluid pressure, since 

 the sum of the radial and circumferential stresses is the same 

 throughout the material of the cylinder. 



The value of the axial stress due to the fluid pressure at 

 the yield-point in the test is tabulated in the column headed 

 pi. In the case of a simultaneous tension-load producing an 

 additional stress p , this is simply added on, so that the 

 principal stresses are then p +p x axially, 2pi circumferentially, 

 and jtp radially. 



To find the principal stresses when torsion is combined with 

 tension or internal pressure, notice first that the radial stress 

 is always perpendicular to the other components and to the 

 plane of the shear, and hence it is a principal stress, and the 

 case reduces to one of two dimensions and the stress ellipsoid 



to jha; 2 + 2qxy +p 2 y 2 = 1, 



where p u p 2 are the component axial and circumferential 

 stresses and q the shearing-stress. The principal stresses ta Xs 

 -BTg are then the roots of the discriminant 



|j>!— ta q I =0; 



q P2 — 1B- J 



and hence are the values of 



Pi+p 2 ± slpx-pi + ^q 1 



OT - 2 



From this equation are obtained the tabulated values of t^i 

 and ■BT 2 . The angle at which the principal stresses are inclined 

 to the generators may be found from its algebraic value 



tan-i-?^-. 

 P2-P1 



