This shows how the strength of the tube is increased by the initial stress. When the initial stress is produced by tightly wound wire, a similar gain of strength accrues.
80. In the problem of determining the distribution of stress and strain in a circular cylinder, rotating about its axis, simple solutions have been obtained which are sufficiently exact for the two special cases of a thin disk and a long shaft.
Suppose that a circular disk of radius a and thickness 2l, and of density ρ, rotates about its axis with angular velocity ω, and consider the following systems of superposed stresses at any point distant r from the axis and z from the middle plane:
(1) uniform tension in all directions at right angles to the axis of amount 1⁄8 ω²ρa² (3 + σ),
(2) radial pressure of amount 1⁄8 ω²ρr² (3 + σ),
(3) pressure along the circular filaments of amount 1⁄8 ω²ρr² (1 + 3σ),
(4) uniform tension in all directions at right angles to the axis of amount 1⁄6 ω²ρ (l² − 3z²) σ (1 + σ) / (1 − σ).
The corresponding strains may be expressed as
(1) uniform extension of all filaments at right angles to the axis of amount
| 1 − σ | 1⁄8 ω²ρa² (3 + σ), |
| E |