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A cylindrical or a spherical pressure vessel, having inner diameter $d$ & thickness of wall $t$, is called thin if $\large \frac{t}{d}<\frac{1}{20}$.

$\bullet$ Thin cylindrical pressure vessel

hoop or circumferential stress $ \sigma_h=\large\frac{pd}{2t}$ & longitudinal stress $ \sigma_l=\large\frac{pd}{4t}$

$\bullet$ Thin spherical pressure vessel

hoop or circumferential stress $ \sigma_h=\large\frac{pd}{4t}$ & longitudinal stress $ \sigma_l=\large\frac{pd}{4t}$

In both the cases we consider only two stresses i.e. the system of plane stresses neglecting the radial stress $(\sigma_r)$.

The question is why do we neglect the radial stress for thin pressure vessels? Can it be explained by Lame's equation used for thick pressure vessels:

$$\sigma_h, \sigma_r=\frac{B}{x^2}\pm A\ \text{?}$$

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I've been doing large pressure vessel design for a while, and ultimately it comes down to what is negligible v. what is not. Especially in the realm of atmospheric storage vessels, scrubbers, or vessels within 10 atmospheres of pressure, there isn't a lot of radial stress compared to the other conservative factors.

In the realistic engineering world, by the time you deal with the following uncertainties (all within tolerance):

  1. Vessel is never round (typical tolerance is 0.5%)
  2. Vessel is never the correct diameter (typical tolerance is 1%)
  3. Environmental loading is only defined by a once in 50 year hazard (while vessel life is typically 20 years)
  4. Environmental loads are defined by 3-7 empirical factors, with errors in the 1%-15%, typically conservative.
  5. Buckling loads typically dominate on any vessel undergoing vacuum or handling atmospheric pressure - this is due to environmental axial buckling (i.e. the vessel is bending under the environmental loading, and may buckle at the inside throat).
  6. Factors of safety greater than necessary to ensure conservative design.

By the time the radial stress is added in and evaluated, the change in Von Mises stress is significantly lower in magnitude than these additional factors. In addition, by eliminating the radial stress, a linear relationship can be utilized between pressure, diameter, and thickness - allowing for simple, easy estimating solutions for use in deriving prices of vessels.

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There are several theories of failure in mechanics of materials. Among them we use shear theory and Distortion energy theory which arises from plasticity.

What causes failure is the difference between maximum and minimum stresses. If all the three are equal, materials will never fail. If we take the minimum radial stress as zero, it is conservative.Taking it into account the design is not much influenced.

EDIT1

Circumferential stress is like $ pr/t,$ radial stress is like $p,$ When stress ratio is $r/t\approx 15 $ ..negligible.

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