Problem

The closed cylinder in Example 11.6 is made of a Tresca material instead of a van Mises ma...

The closed cylinder in Example 11.6 is made of a Tresca material instead of a van Mises material. Obtain the solution for the Tresca material.

Example 11.6

A closed cylinder has an inner radius of 20 mm an d an outer radius o f .4 0 In m • It is made of steel that has a yield stress of Y = 450 MPa and obeys the von M1ses yield criterion.

(a) Determine the fully plastic internal pressure pP for the cylinder.
(b) Determine the maximum circumferential and axial residual stresses when the cylinder is uploaded from pp, assuming that the values based on linear elastic unloading are decreased by 50 % because of inelastic deformation during unloading.
(c) Assuming that the elastic range of the octahedral shear stress has not been altered by the inelastic deformation, determine the internal pressure p1 that can be applied to the cylinder based on a factor of safety SF = l.80. For SF = l.80, compare this result with the pressure p1 for a cylinder without residual stresses.

Solution:

(a) The shear yield stress τY for the von Mises steel is obtained using the octahedral shear-stress yield Criterion

The magnitude of pp is given by Eq. 11.29. Thus, we find

The circumferential and axial stresses at the inner radius for fully plastic conditions are given by Eqs. 11.30 and 11.31. They are

( b) Assuming linearly elastic unloading, we compute the circumferential and axial residual stress at r = a as

The actual residual stresses may be as much as 50% less than these computed values. Thus,

(c) Yielding is initiated in the cylinder at a pressure (SF)p1 = 1.80p1. If the residual stresses are neglected, the stresses at the inner radius caused by pressure (SF)p1are

The actual stresses at the inner radius are obtained by adding the residual stresses given by Eq (a) to those g1ven by Eqs. (b). Thus,

The octahedral shear-stress yield condition requires that

Substituting the values for the stress components given by Eqs. (c) into Eq. (d), we find that

p1 = 154.2 MPa

i. the working internal pressure for the cylinder that was preloaded to the fully plastic pressure. Substituting the values for the stress components given by Eqs. (b) into Eq. (d), we obtain the working internal pressure for the cylinder without residual stresses:

p1 = l08.3 MPa

Hence, the working pressure for the cylinder that is preloaded to the fully plastic pressure is 42.4% greater than the working pressure for the elastic cylinder without residual stresses.

(11.30)