Yielding of a thin-walled shell

A thin-walled spherical shell is under internal pressure, p. The shell is 0.5 m in diameter and 2.5 mm thick. It is made of a perfectly plastic material with a yield stress of 140 MPa. Calculate the pressure required to cause yielding of the shell according to both yield criteria.

Question

Yielding of a thin-walled shell

A thin-walled spherical shell is under internal pressure, p. The shell is 0.5 m in diameter and 2.5 mm thick. It is made of a perfectly plastic material with a yield stress of 140 MPa. Calculate the pressure required to cause yielding of the shell according to both yield criteria.

Accepted Answer

For this shell under internal pressure, the membrane stresses are given by

[latex]\sigma_{1}=\sigma_{2}=\frac{p r}{2 t}[/latex] (2.38)

where r = 254 mm and t= 2.54 mm The stress in the thickness direction, [latex]σ_3[/latex], is negligible because of the high r/t ratio of the shell. Thus, according to the maximum-shear-stress criterion,

[latex]\sigma_{\max }-\sigma_{\min }=Y[/latex]

or

[latex]\sigma_{1}-0=Y[/latex]

and

[latex]\sigma_{2}-0=Y[/latex]

Hence [latex]σ_1= σ_2= 140[/latex] MPa. The pressure required is then

[latex]p=\frac{2 t Y}{r}=\frac{(2)\left(2.54 imes 10^{-3} ight)(140)}{0.254}=2.8 \mathrm{MPa}[/latex]

According to the distortion-energy criterion,

[latex]\left(\sigma_{1}-\sigma_{2} ight)^{2}+\left(\sigma_{2}-\sigma_{3} ight)^{2}+\left(\sigma_{3}-\sigma_{1} ight)^{2}=2 Y^{2}[/latex]

or

[latex]0+\sigma_{2}^{2}+\sigma_{1}^{2}=2 Y^{2}[/latex]

Hence [latex]σ_1= σ_2= Y[/latex]. Therefore, the answer is the same, or p = 2.8 MPa.

Question 2.5: Yielding of a thin-walled shell A thin-walled spherical shel...

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