## Stress Distribution In Thick Cylinders

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For compound tubes the resultant hoop stress is the algebraic sum of the hoop stresses resulting from shrinkage and the hoop stresses resulting from internal and external pressures.

For force and shrink fits of cylinders made of different materials, the total interference or shrinkage allowance (on radius) is l£H0-£H^r where eH and eH are the hoop strains existing in the outer and inner cylinders respectively at the common radius r. For cylinders of the same material this equation reduces to

For a hub or sleeve shrunk on a solid shaft the shaft is subjected to constant hoop and radial stresses, each equal to the pressure set up at the junction. The hub or sleeve is then treated as a thick cylinder subjected to this internal pressure.

### Wire-wound thick cylinders

If the internal and external radii of the cylinder are Rl and R2 respectively and it is wound with wire until its external radius becomes R3, the radial and hoop stresses in the wire at any radius r between the radii R2 and R3 are found from:

radial stress = (jJ^T^j hoop stress - r{l - (^)log. (^f )}

where T is the constant tension stress in the wire.

The hoop and radial stresses in the cylinder can then be determined by considering the cylinder to be subjected to an external pressure equal to the value of the radial stress above when r = R2.

When an additional internal pressure is applied the final stresses will be the algebraic sum of those resulting from the internal pressure and those resulting from the wire winding.

Plastic yielding of thick cylinders

For initial yield, the internal pressure P, is given by:

For yielding to a radius Rp, and for complete collapse,

10.1. Difference in treatment between thin and thick cylinders-basic assumptions

The theoretical treatment of thin cylinders assumes that the hoop stress is constant across the thickness of the cylinder wall (Fig. 10.1), and also that there is no pressure gradient across the wall. Neither of these assumptions can be used for thick cylinders for which the variation of hoop and radial stresses is shown in Fig. 10.2, their values being given by the Lamé equations:

Development of the theory for thick cylinders is concerned with sections remote from the  Fig. 10.2. Thick cylinder subjected to internal pressure.

ends since distribution of the stresses around the joints makes analysis at the ends particularly complex. For central sections the applied pressure system which is normally applied to thick cylinders is symmetrical, and all points on an annular element of the cylinder wall will be displaced by the same amount, this amount depending on the radius of the element. Consequently there can be no shearing stress set up on transverse planes and stresses on such planes are therefore principal stresses (see page 331). Similarly, since the radial shape of the cylinder is maintained there are no shears on radial or tangential planes, and again stresses on such planes are principal stresses. Thus, consideration of any element in the wall of a thick cylinder involves, in general, consideration of a mutually prependicular, tri-axial, principal stress system, the three stresses being termed radial, hoop (tangential or circumferential) and longitudinal (axial) stresses.

10.2. Development of the Lamé theory

Consider the thick cylinder shown in Fig. 10.3. The stresses acting on an element of unit length at radius r are as shown in Fig. 10.4, the radial stress increasing from ar to ar + dar over the element thickness dr (all stresses are assumed tensile), For radial equilibrium of the element:

(or + dor)(r + dr)d6 x 1 — ar x rdO x 1 = 2<X// x dr x 1 x sin — Unit length w/r

Unit length

For small angles:

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