| 3 + σ | { (1 + σ) | a² | + (1 − σ) } ω²ρb². |
| 8E | r² |
(3) contraction of the filaments parallel to the axis of amount
| σ (3 + σ) | ω²ρb². |
| 4E |
Again, the greatest extension is the circumferential extension at the inner surface, and, when the hole is very small, its amount is nearly double what it would be for a complete disk.
82. In the problem of the rotating shaft we have the following stress-system:
(1) radial tension of amount 1⁄8 ω²ρ (a² − r²) (3 − 2σ) / (1 − σ),
(2) circumferential tension of amount
1⁄8 ω²ρ {a² (3 − 2σ) / (1 − σ) − r² (1 + 2σ) / (1 − σ)},
(3) longitudinal tension of amount ¼ ω²ρ (a² − 2r²) σ / (1 − σ).
The resultant longitudinal tension at any normal section vanishes, and the radial tension vanishes at the bounding surface; and thus the expressions here given may be taken to represent the actual condition at all points which are not very close to the ends of the shaft. The contraction of the longitudinal filaments is uniform and equal to ½ ω²ρa²σ / E. The greatest extension in the rotating shaft is the circumferential extension close to the axis, and its amount is 1⁄8 ω²ρa² (3 − 5σ) / E (1 − σ).