**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 Ïƒ_{t} is the tangential stress.

Note that Ïƒ_{t} has the same units of measurement as pressure.

F = P*D*L

T = Ïƒ_{t}*2*t*L

Equating the two, we come up with the formula for Ïƒ_{t}.

PDL = Ïƒ_{t}2tL

PD = Ïƒ_{t}2t

Ïƒ_{t} = PD/2t

Note that this formula can also be used for pipes.

**Longitudinal Stress**

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 Ïƒ_{l}Â .

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

*Photo credit to Supakitmod3 of freedigitalphotos.net*

## 2 comments

Join the conversation## aldwin - August 22, 2015

hello

## admin - August 22, 2015

Magandang araw sa iyo Kabayan!