Page:EB1911 - Volume 09.djvu/174

The corresponding strains consist of

(1) uniform extension of all lines of the material at right angles to the axis of amount

1 − σ		p1r 12 − p0r 02	,
E		r 02 − r 12

(2) radial contraction of amount

1 + σ		p1 − p0		r 02r 12	,
E		r 02 − r 12		r 2

(3) extension along the circular filaments numerically equal to this radial contraction,

(4) uniform contraction of the longitudinal filaments of amount

2σ		p1r 12 − p0r 02	.
E		r 02 − r 12

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	(	r 02 + r 12	+ σ );
E		r 02 − r 12

the greatest tension is the hoop tension at the inner surface, and its amount is p (r ₀² + r ₁²) / (r ₀² − r ₁²).

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

(p₁r ₁² − p₀r ₀²) / 3k (r ₀² − r ₁²),

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 (r ₁²/r ²) · (r ₀² − r ²) / (r ₀² − r ₁²) and hoop tension p (r ₁²/r ²) · (r ₀² + r ²) / (r ₀² − r ₁²) upon a system which, for the outer cylinder, consists of radial pressure P (r ′²/r ²) · (r ₀² − r ²) / (r ₀² − r ′²) and hoop tension P (r ′²/r ²) · (r ₀² + r ²) / (r ₀² − r ′²), and for the inner cylinder consists of radial pressure P (r ′²/r ²) · (r ² − r ₁²) / (r ′² − r ₁²) and hoop tension P (r ′²/r ²) · (r ² + r ₁²) / (r ′² − r ₁²). 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		r 02 + r 12	: 1.
	p		r 02 + r 12		r ′2 − r 12

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 ω²ρr ² (3 + σ),

(3) pressure along the circular filaments of amount 1/8 ω²ρr ² (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

(2) radial contraction of amount

1 − σ2	3/8 ω2ρr 2,
E

(3) contraction along the circular filaments of amount

1 − σ2	1/8 ω2ρr 2,
E

(4) extension of all filaments at right angles to the axis of amount

1	1/6 ω2ρ (l2 − 3z2) σ (1 + σ),
E

(5) contraction of the filaments normal to the plane of the disk of amount

2σ	1/8 ω2ρa2 (3 + σ) −	σ	1/2 ω2ρr 2 (1 + σ) +	2σ	1/6 ω2ρ (l2 − 3z2) σ	(1 + σ)	.
E		E		E		(1 − σ)

The greatest extension is the circumferential extension near the centre, and its amount is

(3 + σ) (1 − σ)	ω2ρa2 +	σ (1 + σ)	ω2ρl2.
8E		6E

Fig. 32.

The longitudinal contraction is required to make the plane faces of the disk free from pressure, and the terms in l and z enable us to avoid tangential traction on any cylindrical surface. The system of stresses and strains thus expressed satisfies all the conditions, except that there is a small radial tension on the bounding surface of amount per unit area 1/6 ω²ρ (l² − 3z²) σ (1 + σ) / (1 − σ). The resultant of these tensions on any part of the edge of the disk vanishes, and the stress in question is very small in comparison with the other stresses involved when the disk is thin; we may conclude that, for a thin disk, the expressions given represent the actual condition at all points which are not very close to the edge (cf. § 55). The effect to the longitudinal contraction is that the plane faces become slightly concave (fig. 32).

81. The corresponding solution for a disk with a circular axle-hole (radius b) will be obtained from that given in the last section by superposing the following system of additional stresses:

(1) radial tension of amount 1/8 ω²ρb² (1 − a²/r ²) (3 + σ),

(2) tension along the circular filaments of amount

1/8 ω²ρb² (1 + a²/r ²) (3 + σ).

The corresponding additional strains are

(1) radial contraction of amount

3 + σ	{ (1 + σ)	a2	− (1 − σ) } ω2ρb2,
8E		r 2

(2) extension along the circular filaments of amount

3 + σ	{ (1 + σ)	a2	+ (1 − σ) } ω2ρb2.
8E		r 2

(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 − r ²) (3 − 2σ) / (1 − σ),

(2) circumferential tension of amount 1/8 ω²ρ {a² (3 − 2σ) / (1 − σ) − r ² (1 + 2σ) / (1 − σ)},

(3) longitudinal tension of amount 1/4 ω²ρ (a² − 2r ²) 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 1/2 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).

The value of any theory of the strength of long rotating shafts founded on these formulae is diminished by the circumstance that at sufficiently high speeds the shaft may tend to take up a curved form, the straight form being unstable. The shaft is then said to whirl. This occurs when the period of rotation of the shaft is very nearly coincident with one of its periods of lateral vibration. The lowest speed at which whirling can take place in a shaft of length l, freely supported at its ends, is given by the formula

ω²ρ = 1/4 Ea² (π/l)⁴.

As in § 61, this formula should not be applied unless the length of the shaft is a considerable multiple of its diameter. It implies that whirling is to be expected whenever ω approaches this critical value.

83. When the forces acting upon a spherical or cylindrical body are not radial, the problem becomes more complicated. In the case of the sphere deformed by any forces it has been completely solved, and the solution has been applied by Lord Kelvin and