Question 7.4: Design of a Cylindrical Pressure Vessel for Fluctuating Load...
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 p_{\min} to p_{\max}. Determine the thickness t for an ultimate strength S_{u}, modified endurance limit S_{e}, and a safety factor of n.
Given: D = 1.5 m, p_{\min} = 0.8 MPa, p_{\max} =4 MPa
S_{y} = 300 MPa, S_{u} = 400 MPa, S_{e} = 150 MPa.
Design Decision: The Goodman theories, based on maximum normal stress and a safety factor of n = 2, are used.
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
\sigma_m=\frac{p_m r}{t}, \quad \sigma_a=\frac{p_a r}{t} (a)
where
\begin{array}{l} p_m=\frac{1}{2}\left(p_{\max }+p_{\min }\right)=\frac{1}{2}(4+0.8)=2.4 MPa \\ p_a=\frac{1}{2}\left(p_{\max }-p_{\min }\right)=\frac{1}{2}(4-0.8)=1.6 MPa \end{array}
Since stresses are proportional to pressures, we have
\frac{\sigma_a}{\sigma_m}=\frac{p_a}{p_m}=\frac{1.6}{2.4}=\frac{2}{3}
Substitution of the given data into Equation 7.20 gives 400/2
\sigma_m=\frac{S_u / n}{\frac{\sigma_a}{\sigma_m} \frac{S_u}{S_e}+1} (7.20)
\sigma_m=\frac{400 / 2}{\frac{2}{3} \frac{400}{150}+1}=72 MPa
Using Equation (a), we have
t=\frac{p_m r}{\sigma_m}=\frac{2.4(750)}{72}=25 mm
This is the minimum safe thickness for the pressure vessel.
Alternatively, a graphical solution of \sigma _{m} 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 \sigma _{m} and \sigma _{a} 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 \sigma _{m} \sigma _{a} for the minimum safe value of the thickness t. The corresponding value of the mean stresses is \sigma _{m} = 72 MPa.
