| 2σ | p1r1² − p0r0² | . | |
| E | r0² − r1² |
For a shell subject only to internal pressure p the greatest extension is the circumferential extension at the inner surface, and its amount is
| p | ( | r0² + r1² | + σ ); |
| E | r0² − r1² |
the greatest tension is the hoop tension at the inner surface, and its amount is p (r0² + r1²) / (r0² − r1²).
78. When the ends of the tube, instead of being free, are closed by disks, so that the tube becomes a closed cylindrical vessel, the longitudinal extension is determined by the condition that the resultant longitudinal tension in the walls balances the resultant normal pressure on either end. This condition gives the value of the extension of the longitudinal filaments as
(p1r1² − p0r0²) / 3k (r0² − r1²),
where k is the modulus of compression of the material. The result may be applied to the experimental determination of k, by measuring the increase of length of a tube subjected to internal pressure (A. Mallock, Proc. R. Soc. London, lxxiv., 1904, and C. Chree, ibid.).
79. The results obtained in § 77 have been applied to gun construction; we may consider that one cylinder is heated so as to slip over another upon which it shrinks by cooling, so that the two form a single body in a condition of initial stress.
We take P as the measure of the pressure between the two, and p for the pressure within the inner cylinder by which the system is afterwards strained, and denote by r′ the radius of the common surface. To obtain the stress at any point we superpose the system consisting of radial pressure p (r1²/r²) · (r0² − r²) / (r0² − r1²) and hoop tension p (r1²/r²) · (r0² + r²) / (r0² − r1²) upon a system which, for the outer cylinder, consists of radial pressure P (r′²/r²) · (r0² − r²) / (r0² − r′²) and hoop tension P (r′²/r²) · (r0² + r²) / (r0² − r′²), and for the inner cylinder consists of radial pressure P (r′²/r²) · (r² − r1²) / (r′² − r1²) and hoop tension P (r′²/r²) · (r² + r1²) / (r′² − r1²). The hoop tension at the inner surface is less than it would be for a tube of equal thickness without initial stress in the ratio
| 1 − | P | 2r′² | r0² + r1² | : 1. | ||
| p | r0² + r1² | r′² − r1² |