# Analysis of Thick Pressure Vessels

In this tutorial, you will learn about the intricacies involved in the design of thick pressure vessels and the modifications associated with them for them to be of functional use. In short, you will learn about the stresses developed in thick pressure vessels, their enclosures and openings, and the use of compound cylinders.

## What are Thick Pressure Vessels?

A pressure vessel usually has a cylindrical or spherical shape. A pressure vessel is said to be a thick pressure vessel when the ratio of the inner diameter to the wall thickness is less than 15. Some applications of Thick Pressure Vessels include Hydraulic Cylinders and Gun Barrels.

• Like a thin pressure vessel, the thick vessels also experience circumferential and longitudinal stresses in the wall.
• Also, due to the thickness of the wall, the vessel develops greater circumferential stress on the inside surface of the vessel and reduces towards the outside diameter.
• Thick-walled pressure vessels see an application in engineering, oil and gas, structural, petrochemical, nuclear, and pressure vessel industries.

## Stresses in a Thick Vessel

For high-pressure hydraulic lines and other thick walled pressure vessels, the stress variation along the thickness is significant. This radial stress is neglected in thin-walled vessels but plays an important role in pressure vessels with thick walls.
As such, thick-walled vessels undergoes Tangential Stress, which is tensile, Radial Stress, which is compressive, and Axial stress which is in the longitudinal direction and is tensile when experiencing internal pressure.
The diagram below shows the stress variation in the radial direction.

In the above diagram, assume unit thickness in the axial direction.
Analyzing forces in equilibrium,
tdr + 2(r+dr)(σr+dσr)=2rσr
Simplifying and neglecting very small terms
σtdr + rσr + rdσr + σrdr=rσr
Rearranging the terms
σt + σr + r$$\frac{dσ_r}{dr}$$=0
We use this equation to establish the relation between the three stresses.

## Brittle Material based Pressure Vessels

Experimental investigations suggest that the maximum principal stress theory gives a good prediction for brittle materials. For a pressure vessel shaped like a cylinder and made out of a brittle material such as cast iron or cast steel, we use Lame’s equation to determine the wall thickness.
Lame’s Equation
t= $$\frac{D_i}{2} \left[ \sqrt{\frac{\sigma_t + P_i}{\sigma_t – P_i}} – 1 \right]$$
where σt = $$\frac{S_{ut}}{fs}$$
Here
Sut = Ultimate tensile strength
Pi = Internal Pressure
Di = Internal Diameter
fs = Factor of Safety
This equation is based on the maximum principal stress theory. The theory takes into account only the maximum principal stress and disregards any other stresses in their entirety.

## Ductile Material based Pressure Vessels

When Pressure vessels are made of Ductile materials like steel, the maximum strain theory of failure is used to determine the failure criteria. The maximum strain theory, also called St. Venant’s Theory, suggests that material failure begins when the maximum strain at any point equals the yield point strain value for the material in a simple tension test.
The Principle Stresses on the Inner surface of the cylinder
σr = – Pi
σt = $$\frac{P_i (D_o^2 + D_i^2)}{D_o^2 – D_i^2}$$
σl = $$\frac{P_i D_i^2}{D_o^2 – D_i^2}$$

The maximum principal strain is given by
εt = $$\frac{1}{E}$$ [σt-μ(σrl)] = $$\frac{σ}{E}$$
Equating the Stresses
σ = [σt-μ(σrl)]

We use this equation to analyze the two scenarios.
For cylinders with Closed Ends, substitute the values of the three principal stresses and solve for t. The resulting equation is called Clavarino’s equation.
t = $$\frac{D_i}{2} \left[ \sqrt{\frac{\sigma + (1 – 2\mu) P_i}{\sigma – (1 + \mu) P_i}} – 1 \right]$$

For a cylinder with open ends, the axial stress is 0. Substituting the principal stresses now into the stress equation, we solve for t. The resulting equation is called Birnie’s equation.
t = $$\frac{D_i}{2} \left[ \sqrt{\frac{\sigma + (1 – \mu) P_i}{\sigma – (1 + \mu) P_i}} – 1 \right]$$

Using these two equations, we can calculate the thickness of thick-walled pressure vessels required.

## Cylinders under External Pressure

A cylinder under external pressure is frequently used in conditions where environment pressure exceeds the internal pressure of the cylinder, like a submarine or a deep dive tank. The cylinder is under similar stresses as a case of internal pressure, with one difference.
The figure below shows the external pressure acting on a cylinder.

Here, pressure is reversed, and the equations of the stress at the outer surface are as follows
σr = -Po
σt = $$-\frac{P_o (D_o^2 + D_i^2)}{D_o^2 – D_i^2}$$
σl = $$=\frac{P_o D_i^2}{D_o^2 – D_i^2}$$

## What are Compound Cylinders?

A compound cylinder is made from two concentric cylinders with one shrunk onto the other. This results in compressive stresses on the inner cylinder, resulting in prestressing. When the cylinder is loaded in service, the compressor stresses at the inner surface begin to decrease before they become zero and convert to tensile stresses, increasing the pressure capacity of the vessel.
The figure below shows the cross-section of a compound cylinder.

• The outer cylinder is called the jacket. The inner diameter of the jacket is slightly smaller than the outer diameter of the inner cylinder.
• When the jacket is heated it expands sufficiently to place forward the inner cylinder. On cooling jacket contracts and this results in residual compressive stresses.
• As a result, there exists a shrinkage pressure between the cylinder and a jacket which tends to contract the cylinder and expand the jacket as shown in the figure below.
• The interference (δ) for a compound cylinder is the sum of the increase in the inner diameter of the jacket and the decrease in the outer diameter of the cylinder.
• The shrinkage pressures can be calculated from the equation given below for a given amount of interference. The resultant stresses in the compound cylinder can be found by imposing the two types of stresses, stresses due to shrink fit and those due to internal pressure
δ = $$\frac{P D_2}{E} \left[ \frac{2D_2^2 (D_3^2 – D_1^2)}{(D_3^2 – D_2^2)(D_2^2 – D_1^2)} \right]$$

## End Enclosures in Pressure Vessels

A Pressure vessel is usually a closed container and hence needs enclosures designed to handle internal and external pressure at various operating temperatures. They are attached to the vessel by either welding or by bolts. These enclosures are of various shapes and are selected for the properties with which they handle loads and stresses.

• Flat Enclosures are used for small diameter vessels and are very simple in construction. They are often used are manholes and are found in low-pressure vessels frequently.
• Plain Formed Head is used for horizontal cylindrical storage vessels operating under atmospheric pressures. They are also used in vertical cylindrical vessels resting on the ground.
• Tori-spherical dished heads are used for vertical or horizontal pressure vessels in the range of 0.1 – 1.5 N/mm2. They are used in low-thick vessels.
• The Semi-elliptical dished heads are used for pressure vessels above 1.5 N/mm2. The ratio of the major to minor axis is taken as 2.
• A Hemispherical pressure vessel, as the name suggests, has a hemispherical enclosure. They are considered the strongest among all categories of enclosures, but cannot be used where there are space constraints.
• Conical heads are widely used as bottom heads and facilitate the removal and draining of materials. They work well under lower pressure.

## Openings in a Pressure Vessel

Often, Openings are provided in pressure vessels, such as for an inlet and outlet pipe connections, a manhole or a had hole cover, connection for pressure gauges, temperature gauges, and safety valves. The openings are circular, elliptical, or obround. Such areas are designed by the area compensation method.
The figure below shows the area compensation method for openings in pressure vessels.

• The basic principle of the area compensation method is illustrated in the figure above. When the opening is cut in the pressure vessel, the area is removed from the shell. It must be reinforced by an equal amount of area near the opening.
• The area is added by providing a reinforcing pad in the form of an annular circular plate around the opening. The diagram displays this in the form of a rectangular strip.
• It is not always a necessity to replace the removed area of the metal. It is a norm to make the plate of the shell and nozzle thicker than would be required to withstand the pressure.

## Key Points to Remember

Here is the list of key points we need to remember about “Analysis of Thick Pressure Vessels”.

• Thick Pressure Vessels have an inner diameter to wall thickness ratio of less than 15.
• In addition to circumferential and longitudinal stresses, thick-walled cylinders experience radial stresses with maximum values at the surface of greater pressure.
• The Lami’s equation is used to calculate the thickness of a thick-walled pressure vessel made of Brittle material.
• For pressure vessels made of ductile materials, the Clavarino’s equation is used to calculate thickness for vessels with closed ends, and Birnie’s equation is used for vessels with open ends.
• For vessels facing external pressure like submarines, the direction of the stress along the circumference changes direction compared to those in internal pressure.
• A compound cylinder system is frequently used to increase the pressure capacity of high-pressure systems.
• Pressure vessels have enclosures of various shapes depending on the application and pressure condition of the vessel when in use.
• Openings are provided in pressure vessels for accessibility and need to be suitably compensated lest they propagate failure during application.

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