COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 299 
REFERENCES. 
_ (1) ‘Stress Distributions in Engineering Materials.” B.A. Report, 1914. 
(2) ‘ Photo-Elastic Measurements of the Stress Distribution in Tension Members 
used in the Testing of Materials.’ By Prof. E. G. Coker, F.R.S. Min. 
Proc. Inst.C.E. Vol. ceviii. (1918-19). Part ii. 
(3) ‘Contact Pressures and Stresses.’ Prof. E. G. Coker, K. C. Chakko, and 
M. S. Ahmed, Proc. Inst. Mech. Engrs. March 1921. 
(4) ‘ Photo-Elastic and Strain Measurements of the Effects of Circular Holes on 
the Distribution of Stress in Tension Members.’ By Prof. E. G. Coker, 
K. C. Chakko, and Y. Satake. Zrans. The Inst. of Engrs. and Ship- 
builders in Scotland. Vol. 1xiii. Part i. 
II. 
The Distribution of Stress in a Flanged Pipe. 
By Gisert Coor, D.Sc. 
The problem of the elastic deformation of a thin cylindrical shell subjected 
to internal pressure when certain boundary conditions restricting the displace- 
ment at the ends are to be satisfied has been investigated theoretically by Love,’ 
and as applied to the case of a short boiler, in more detail by Nicolson.? 
The particular types of end constraint considered were those in which the ends 
of the tube remain circular and suffer no radial displacement. In a discussion 
of certain experiments on short boilers? it has been shown that the stress 
A 8 
L-—.— _§ -—_-4 
| 
mes 
FIG. 7. 
distribution in the vicinity of the end constraints is of a1 somewhat peculiar 
nature, inasmuch as over a certain region the radial deformation and the 
corresponding circumferential stress are greater than that which would obtain 
in an infinitely long tube, but little experimental evidence of this fact appears 
to have been forthcoming. A third type of constraint which possesses consider- 
able practical interest is that produced by a flange or collar at any point in 
the length of the tube. At this point the inclination of the generators of the 
cylinder to the axis is, by symmetry, zero. The radial displacement is not zero, 
but is determinate, and the constraint is of a type which may be more accurately 
and conveniently reproduced experimentally than those referred to above. 
The geveral equation of equilibrium for any type of end constraint may be 
very simply obtained as follows : 
Let ¢ be the thickness of the tube (fig. 7), assumed small in comparison with 
R, the mean radius. Let z be the radial displacement, q the circumferential 
* Mathematical Theory of Elasticity. Third Ed. Art. 340. 
2 Nicolson, ‘The Strength of Short Flat-ended Cylindrical Boilers.’  7'rans. 
N.E. Coast Inst. Engineers and Shipbuilders. Vol. vii., p. 205. 
% Engineering. Vol. 51 (1891), p. 468. 
x 2 
