Design of a Cylindrical Pressure Vessel for Fluctuating Loading A thin-walled cylindrical pressure vessel of diameter d is subjected to an internal pressure p varying continuously from pmin to pmax. Determine the thickness t for an ultimate strength Su, modified endurance limit Se, and a safety

Question

Design of a Cylindrical Pressure Vessel for Fluctuating Loading

A thin-walled cylindrical pressure vessel of diameter [latex]d[/latex] is subjected to an internal pressure p varying continuously from [latex]p_{\min}[/latex] to [latex]p_{\max}.[/latex] Determine the thickness [latex]t[/latex] for an ultimate strength [latex]S_{u},[/latex] modified endurance limit [latex]S_{e},[/latex] and a safety factor of [latex]n[/latex].

Given: [latex]D[/latex] = 1.5 m, [latex]p_{\min}[/latex] = 0.8 MPa, [latex]p_{\max}[/latex] =4 MPa

[latex]S_{y}[/latex] = 300 MPa, [latex]S_{u}[/latex] = 400 MPa, [latex]S_{e}[/latex] = 150 MPa.

Design Decision: The Goodman theories, based on maximum normal stress and a safety factor of [latex]n[/latex] = 2, are used.

Accepted Answer

The state of stress on the cylinder wall is considered to be biaxial (see Section 3.4). Maximum principal stress, that is, tangential stress, in the cylinder has the mean and range values

[latex] \sigma_m=\frac{p_m r}{t}, \quad \sigma_a=\frac{p_a r}{t} [/latex] (a)

where

[latex] \begin{array}{l} p_m=\frac{1}{2}\left(p_{\max }+p_{\min } ight)=\frac{1}{2}(4+0.8)=2.4 MPa \ p_a=\frac{1}{2}\left(p_{\max }-p_{\min } ight)=\frac{1}{2}(4-0.8)=1.6 MPa \end{array} [/latex]

Since stresses are proportional to pressures, we have

[latex] \frac{\sigma_a}{\sigma_m}=\frac{p_a}{p_m}=\frac{1.6}{2.4}=\frac{2}{3} [/latex]

Substitution of the given data into Equation 7.20 gives 400/2

[latex] \sigma_m=\frac{S_u / n}{\frac{\sigma_a}{\sigma_m} \frac{S_u}{S_e}+1} [/latex] (7.20)

[latex] \sigma_m=\frac{400 / 2}{\frac{2}{3} \frac{400}{150}+1}=72 MPa [/latex]

Using Equation (a), we have

[latex] t=\frac{p_m r}{\sigma_m}=\frac{2.4(750)}{72}=25 mm [/latex]

This is the minimum safe thickness for the pressure vessel.

Alternatively, a graphical solution of [latex]\sigma _{m}[/latex] by the modified Goodman criterion is obtained by plotting the given data to scale, as shown in Figure 7.14. We observe from the figure that the locus of points representing [latex]\sigma _{m}[/latex] and [latex]\sigma _{a}[/latex] for any thickness is a line through the origin with a slope of 2/3. Its intersection with the safe stress line gives the state of stress [latex]\sigma _{m}[/latex] [latex]\sigma _{a}[/latex] for the minimum safe value of the thickness t. The corresponding value of the mean stresses is [latex]\sigma _{m}[/latex] = 72 MPa.

Question 7.4: Design of a Cylindrical Pressure Vessel for Fluctuating Load...

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