In this article:
– we will talk about how to derive the formulas of stresses for thin-walled pressure vessels
– We will also look at the formulas for thick-walled pressure vessels
– Afterwards, we will compare with the formulas in ASME VIII Div 1.
Thin-Walled Pressure Vessels
In thermodynamics, we learned that pressure equals force per unit area.
P = F/A
We will use this concept in deriving the formulas of stresses.
Note that a pressure vessel is considered thin-walled when the radius to thickness ratio is greater than 10.
Thus, r/t > 10.
Tangential Stress (Circumferential Stress or Hoop Stress)
From the illustration, F and T are in equilibrium where F is the force exerted by pressure while T is the tangential force exerted by the vessel. Calculating the forces using F = P*A based on the illustration, where Ïƒt is the tangential stress.
Note that Ïƒt has the same units of measurement as pressure.
F = P*D*L
T = Ïƒt*2*t*L
Equating the two, we come up with the formula for Ïƒt.
PDL = Ïƒt2tL
PD = Ïƒt2t
Ïƒt = PD/2t
Note that this formula can also be used for pipes.
We calculate following the same concept as above, where L is the longitudinal force and ÏƒlÂ is the longitudinal stress.
The area of a circle is Ï€D2/4 while the circumference is Ï€D.
Equating the two, we come up with the formula for ÏƒlÂ .
Note that this formula is used for cylindrical and spherical vessels. From the two formulas, we can say that the tangential stress is twice as much as longitudinal stress.
Thick-Walled Pressure Vessels
There is a third type of stress which is the radial stress, sr. This stress is acting radially toward the axis. Note that we did not consider sr in thin-walled pressure vessels since this is negligible compared to tangential stress.
The formulas for stresses in thick-walled pressure vessels at any given radius are as follows:
The maximum shear stress is:
If the pressure vessel is closed, the longitudinal stress is:
ASME VIII Div 1
Thickness of Shells Under Internal Pressure
where: E = joint efficiency
S = maximum allowable stress
R = inside radius
P = internal design pressure
t = minimum required thickness of shell
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