Consider a thin-walled cylindrical pressure vessel with hemispherical ends as shown in Figure 3.12. Internal diameter of the cylinder is d and thicknesses t1 and t2 as shown. Find out the requisite relationship between t1 and t2 to avoid any distortion of the vessel.

Question

Consider a thin-walled cylindrical pressure vessel with hemispherical ends as shown in Figure 3.12. Internal diameter of the cylinder is d and thicknesses [latex] t_1 ext { and } t_2 [/latex] as shown. Find out the requisite relationship between [latex] t_1 ext { and } t_2 [/latex] to avoid any distortion of the vessel.

Accepted Answer

Circumferential strain develops at the cylindrical portion as well as hemispherical ends. To avoid distortion, we note that these two strains must be equal. For cylindrical portion, circumferential or hoop stress and longitudinal stress, respectively are

[latex] \sigma_1=\frac{p r}{t_1} \quad ext { and } \quad \sigma_2=\frac{p r}{2 t_1} [/latex]

Circumferential strain considering biaxial stress situation is

[latex] \begin{aligned} ∈_{ C } & =\frac{\sigma_1}{E}-\frac{\sigma_2}{E} ν \ & =\frac{p r}{t_1 E}-\frac{p r}{2 t_1 E} ν \end{aligned} [/latex]

or [latex] ∈_C=\frac{p r}{2 t_1 E}(2-ν) [/latex] (1)

For hemispherical ends, circumferential strain is

[latex] ∈_C=\frac{p r}{t_2 E}(1-ν) [/latex] (2)

Equating Eqs. (1) and (2), we get

[latex] \frac{p r}{2 t_1 E}(2-ν)=\frac{p r}{t_2 E}(1-ν) [/latex]

or [latex] \frac{2-ν}{2 t_1}=\frac{1}{t_2}(1-ν) [/latex]

or

[latex] \begin{aligned} \frac{t_2}{t_1} & =\frac{(1-ν) imes 2}{2-ν} \ & =\frac{1-ν}{1-(ν / 2)} \end{aligned} [/latex]

Therefore, [latex] t_2 / t_1=(1-ν) /(1-ν / 2) [/latex] which is the desired relationship.

Note: The expression [latex] (1-ν)<1-(ν/2),t_2< t_1[/latex] means that to avoid distortion, thickness of hemispherical ends is always lower than thickness of cylindrical part. Suppose in one situation with a similar pressure vessel, maximum stress at cylindrical portion and hemispherical portion is to be same. Find out the requisite ratio [latex] t_2 / t_1 [/latex].

Question 3.6: Consider a thin-walled cylindrical pressure vessel with hemi...

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