Plot the circumferential residual stress σθ when the autofrettaging pressure in Example 15.1 is reduced to zero. Also plot σθ when a design pressure of 45,000 psi is applied.

Question

Plot the circumferential residual stress [latex]σ_θ[/latex] when the autofrettaging pressure in Example 15.1 is reduced to zero. Also plot [latex]σ_θ[/latex] when a design pressure of 45,000 psi is applied.

Accepted Answer

The maximum autofrettaging pressure of 60 ksi is less than twice the lower bound pressure P[latex]^{*}[/latex]. Accordingly, the stress distribution resulting from a pressure drop of 60 ksi is in the elastic range, as shown in Fig. 15.4. From Eq. 15.1,

[latex]\begin{aligned}&\sigma_{ heta}=P^{\prime}\left(1+\frac{r_{o}^{2}}{r^{2}} ight) \&\sigma_{r}=P^{\prime}\left(1-\frac{r_{o}^{2}}{r^{2}} ight) \&\sigma_{l}=P^{\prime}\end{aligned}[/latex] (15.1)

where

[latex]P^{\prime}=P\left(\frac{r_{i}^{2}}{r_{o}^{2}-r_{i}^{2}} ight)[/latex]

[latex]P^{\prime}=-60,000\left(\frac{8^{2}}{17.6^{2}-8^{2}} ight)[/latex]

[latex]P^{\prime}=-15,625 psi[/latex]

[latex]\sigma_{ heta}=-15,625\left(1+\frac{17.6^{2}}{r^{2}} ight)[/latex]

The stress distribution given by this equation is superimposed with that in Fig. 15.3 for [latex]\sigma_{ heta}[/latex] at P = 60 ksi. The resultant residual stress is shown in Fig. 15.5.

The internal pressure of 45,000 psi gives a stress of

[latex]\sigma_{ heta}=11,720\left(1+\frac{17.6^{2}}{r^{2}} ight)[/latex]

and the total stress distribution due to this and residual stress is given in Fig. 15.5.

Question 15.2: Plot the circumferential residual stress σθ when the autofre...

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