Stresses in thin circular cylinder and spherical shell

Thin wall pressure vessels are in fairly common use. We would like to consider two specific types. Cylindrical pressure vessels, and spherical pressure vessels. By thin wall pressure vessel we will mean a container whose wall thickness is less than 1/10 of the radius of the container. Under this condition, the stress in the wall may be considered uniform.

We first look at a cylindrical pressure vessel shown in Diagram 1, where we have cut a cross section of the vessel, and have shown the forces due to the internal pressure, and the balancing forces due to the longitudinal stress which develops in the vessel walls. (There is also a transverse or circumferential stress which develops, and which we will consider next.)

The longitudinal stress may be found by equating the force due to internal gas/fluid pressure with the force due to the longitudinal stress as follows:**P(A) = ****(A’); or P(3.1416 * R ^{2}) = (2 * 3.1416 * R * t)**, then canceling terms and solving for the longitudinal stress, we have:

**= P R / 2 t ; where**

P = internal pressure in cylinder; R = radius of cylinder, t = wall thickness

To determine the relationship for the transverse stress, often called the hoop stress, we use the same approach, but first cut the cylinder lengthwise as shown in Diagram 2.

We once again equate the force on the cylinder section wall due to the internal pressure with the resistive force which develops in walls and may be expressed in terms of the hoop stress, . The effective area the internal pressure acts on may be consider to be the flat cross section given by (2*R*L). So we may write:**P(A) = ****(A’); or P(2*R*L) = (2*t*L)**, then canceling terms and solving for the hoop stress, we have:**= P R / t ; where**P = internal pressure in cylinder; R = radius of cylinder, t = wall thickness

We note that the hoop stress is twice the value of the longitudinal stress, and is normally the limiting factor. The vessel does not have to be a perfect cylinder. In any thin wall pressure vessel in which the pressure is uniform and which has a cylindrical section, the stress in the cylindrical section is given by the relationships above.

**Example A:** A thin wall pressure vessel is shown in Diagram 3. It’s cylindrical section has a radius of 2 feet, and a wall thickness of the 1″. The internal pressure is 500 lb/in^{2}. Determine the longitudinal and hoop stresses in the cylindrical region.

**Solution:**We apply the relationships developed for stress in cylindrical vessels.**= P R / 2 t = 500 lb/in ^{2} * 24″/2 * 1″ = 6000 lb/in^{2}.**

**= P R / t = 500 lb/in**

^{2}* 24″/ 1″ = 12, 000 lb/in^{2}.Next we consider the stress in thin wall spherical pressure vessels. Using the approach as in cylindrical vessels, in Diagram 4 we have shown a half section of a spherical vessel. If we once again equal the force due to the internal pressure with the resistive force expressed in term of the stress, we have:

**P(A) = ****(A’); or P(3.1416 * R ^{2}) = **

**(2 * 3.1416 * R * t)**, then canceling terms and solving for the stress, we have:

**= P R / 2 t ; where**

P = internal pressure in sphere; R = radius of sphere, t = wall thickness

Note that we have not called this a longitudinal or hoop stress. We do not do so since the symmetry of the sphere means that the stress in equal in what we could consider a longitudinal and/or transverse direction.

**Example B**. In Example A, above, if the radius of the spherical section of the container is also 2 feet, determine the stress in the spherical region.

**Solution: **We apply our spherical relationship:**= P R / 2 t = 500 lb/in ^{2} * 24″/2 * 1″ = 6000 lb/in^{2}.**Thus in the container in Example A, the limiting (maximum) stress occurs in the in the cylindrical region and has a value of 12,000 lb/in

^{2}.

**Pressure Vessels – Problem Assignment 1**

1. A welded water pipe has a diameter of 8 ft. and a wall of steel plate 3/4 in. thick. After fabrication, this pipe was tested under an internal pressure of 230 psi. Calculate the circumferential stress developed in the walls of the pipe. (14,700 psi)

2. Calculate the tensile stresses developed (circumferential and longitudinal) in the walls of a cylindrical boiler 5 ft. in diameter with a wall thickness of 1/2 in. The boiler is subjected to an internal gage pressure of 155 psi. (9300 psi, 4650 psi)

3. Calculate the internal water pressure that will burst a 15 in. diameter cast-iron water pipe if the wall thickness is 1/2 in. Use an ultimate tensile strength of 62,500 psi. for the pipe. (8333 psi)

4. Calculate the wall thickness required for a 5 ft. diameter cylindrical steel tank containing gas at an internal gage pressure of 600 psi. The allowable tensile stress for the steel is 17,500 psi. (1.03″)

5. A spherical gas container, 50 ft. in diameter, is to hold gas at a pressure of 40 psi. Calculate the thickness of the steel wall required. The allowable tensile stress is 20,000 psi. (.3″)

6. Calculate the tensile stresses (circumferential and longitudinal) developed in the walls of a cylindrical pressure vessel. The inside diameter is 15 in. The wall thickness is 1/4 in. The vessel is subjected to an internal gage pressure of 450 psi. and a simultaneous external axial tensile load of 45,000 lb. (13,500 psi, 10,570 psi)

7. A thin wall pressure vessel is composed of two spherical regions and a cylindrical region is shown in Diagram 5.

The larger spherical region has a radius of 3 ft and a wall thickness of 1.25″. The smaller spherical region has a radius of 2.5 ft and a wall thickness of 3/4″. The cylindrical region has a radius of 2 ft and a wall thickness of 1/2″.

Determine the axial and hoop stress in the cylindrical region, and the wall stresses in the two spherical regions. Which stress is the largest? (1: 5,760 psi, 2: 19,200 psi, 9600 psi, 3: 8,000 psi)

8. A thin wall pressure vessel is composed of two spherical regions and is shown in Diagram 4. The larger spherical region has a radius of 3 ft , and the smaller spherical region has a radius of 2.5 ft. The vessel is made of steel with an allowable tensile stress of 24,000 lb/in^{2}.

If we wish the maximum stress in both spherical sections to at the allowable stress, Determine wall thickness for each spherical region.(.45″, .375″)

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