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

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

\sigma_1=\frac{p r}{t_1} \quad \text { and } \quad \sigma_2=\frac{p r}{2 t_1}

Circumferential strain considering biaxial stress situation is

\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}

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

For hemispherical ends, circumferential strain is

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

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

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

or        \frac{2-ν}{2 t_1}=\frac{1}{t_2}(1-ν)

or

\begin{aligned} \frac{t_2}{t_1} & =\frac{(1-ν) \times 2}{2-ν} \\ & =\frac{1-ν}{1-(ν / 2)} \end{aligned}

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

Note: The expression (1-ν)<1-(ν/2),t_2< t_1 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 t_2 / t_1 .