The pressure vessels, according to their dimensions, may be classified as thin shell or thick shell. When the wall thickness of the shell is greater than 1/10 of the diameter of the shell, then it is said to be a thick shell. Another criterion to classify the pressure vessels as thin shell or thick shell is the internal fluid pressure (p) and the allowable stress (). If the internal fluid pressure is greater than 1/6 of the allowable stress, then it is said to be a thick shell.

When the wall is thick, the tangential stress at the inside surface is much higher than that the outside surface. A thickness too small is too dangerous; too large uneconomical dimensions. A more accurate expression for the tangential stress is obtained by assuming that the thick shell is composed of series of thin shell is differential thickness for each of which the stress is uniform, and the plane transverse sections remain plane.

Fig. 4.1 Thick-walled cylinder

Consider a thick walled cylinder with open ends as shown above. It is loaded by internal pressure Pi and external pressure Po as seen below. It has inner radius ri and outer radius ro.

Fig. 4.2 Thick-walled cylinder

When a cylindrical shell of a pressure vessel, hydraulic cylinder, gun barrel and a pipe is subjected to a very high internal fluid pressure, then the walls of the cylinder must be made extremely heavy or thick. In thin cylindrical shells, we have assumed that the tensile stresses are uniformly distributed over the section of the walls. But in the case of thick wall cylinders as shown in Fig. 4.3 (a), the stress over the section of the walls cannot be assumed to be uniformly distributed. They develop both tangential and radial stresses with values which are dependent upon the radius of the element under consideration. The distribution of stress in a thick cylindrical shell is shown in Fig. 4.3 (b) and (c). We see that the tangential stress is maximum at the inner surface and minimum at the outer surface of the shell. The radial stress is at its maximum at the inner surface and zero at the outer surface of the shell.

In the design of thick cylindrical shells, the following equations are mostly used:1. Lames equation;2. Clavarinos equation;3. Birnies equation; and 4. Barlows equation. The use of these equations depends upon the type of material used and the end construction.

Fig. 4.3 Stress distribution in thick cylindrical shells subjected to internal pressure.

Now consider and element at radius r and defined by an angle increment and a radial increment dr. By circular symmetry, the stresses and are functions of r only, not and the shear stress on the element must be zero. For an element of unit thickness, radial force equilibrium gives:

Ignoring second order terms gives:

. (1)Assuming that there are no body forces. Now consider strains in the element. By symmetry there is no displacement v. there is only a radial displacement u given by line aa. Point c is displaced radially by (u + du) given by line cc. As the original radial length of the element is dr (line ac), the radial strain is:

Line ab has length rd and line ab has a length (r + u)d. The tangential strain is thus:

As the ends are open, and we thus have planes stress conditions. From Hookes law we get:

Solving for the stresses gives:

and

Substituting into equation above yields:

This has solution:Giving the stresses as:

. (2) . (3)

The boundary conditions are: and This yields the integration constants:

And

Giving the stresses as a function of radius:

These are known as Lames equations.ro =outer radius of cylinder, in.ri = inner radius of cylinder, in.Pi, Po =internal and external pressure, respectively, psit =wall thickness, ro ri = Poissons ratio = tangential stress, psi = radial stress, psi = axial stress, psi = radial strain = tangential strain = longitudinal or axial strain = yield point stress = ultimate stressFrom equation 2 and 3 above we can see that the sum of the radial and tangential stresses is constant, regardless of radius:

Hence the longitudinal strain is also constant since: Constant. Hence we get constant = c

If the ends of the cylinder and open and free we have FZ = 0, hence Or c = as we assumedIf the cylinder has closed ends, the axial stress can be found separately using only force equilibrium considerations as was done for the thin walled cylinder. The result is then simply superimposed on the above equations.The pressure Pi acts on are given by The pressure Po acts on are given byThe axial stress acts on an are given byForce equilibrium then gives:The following is a summary of the equations used to determine the stresses found in thick walled cylindrical pressure vessels. In the most general case the vessel is subject to both internal and external pressures. Most vessels also have closed ends -this results in an axial stress component.Principal stresses at radius r :

:

And, if the ends are closed,

Where: :

(a) Internal Pressure only (Po= 0): The most common case dealt with in machine design is a cylinder subjected to internal pressure only. In this case Po is zero.

At inside surface, r = ri: ::

At outside surface, r = ro:: :

(b) External Pressure only (Pi= 0):: :

At inside surface, r = ri:: :

At outside surface, r = ro: ::

Fig. 4.4 Example of cylinder with Pi = 1000 psi, ri= 2 and ro=4

Note that in all cases the greatest magnitude of direct stress is the tangential stress at the in-side surface. The maximum magnitude of shear stress also occurs at the inside surface.(c) Press and shrink fitsWhen a press or shrink fit is used between 2 cylinders of the same material, an interface pressure pi is developed at the junction of the cylinders. If this pressure is calculated, the stresses in the cylinders can be found using the above equations. The pressure is:

Nomenclatures:E = Youngs Modulus= radial interference between the two cylindersa = inner radius of the inner cylinderb = outer radius of inner cylinder and inner radius of outer cylinderc = outer radius of outer cylinderIt is assumed that is very small compared to the radius b and that there are no axial stresses. Thus we have = binner bouter. Note that this small difference in the radii is ignored in the above equation.

This equation is also based on the maximum-strain theory of failure, but it is applied to closed-end cylinders (or cylinders fitted with heads) made of ductile material. According to this equation, the thickness of a cylinder,

In this case also, the value of may be taken as 0.8 times the yield point stress (y).

If the cylinder is considered to be open at the end so that no direct axial stress is possible, an analysis similar to that for Clavarinos equations leads to the following equations for equivalent stress:

and

The equation for the wall thickness is:

In case of open-end cylinders (such as pump cylinders, rams, gun barrels etc.) made of ductile material (i.e. low carbon steel, brass, bronze, and aluminum alloys), the allowable stresses cannot be determined by means of maximum-stress theory of failure. In such cases, the maximum-strain theory is used. The value of may be taken as 0.8 times the yield point stress (y).

This equation is generally used for high pressure oil and gas pipes. According to this equation, the thickness of a cylinder,

For ductile materials, = 0.8 y and for brittle materials = 0.125 u, where u is the ultimate stress.

1. A hydraulic press has a maximum capacity of 1000 kN. The piston diameter is 250 mm. Calculate the wall thickness if the cylinder is made of material for which the permissible strength may be taken as 80 MPa. This material may be assumed as a brittle material.

Solution:

Given: W = 1000 kN = 1000 103 N; d = 250 mm; t = 80 MPa = 80 N/mm2

First of all, let us find the pressure inside the cylinder (p). We know that load on the hydraulic press (W),

Let ri = Inside radius of the cylinder = d / 2 = 125 mmWe know that wall thickness of the cylinder,

Answer: 37 mm

2. The cylinder of a portable hydraulic riveter is 220 mm in diameter. The pressure of the fluid is 14 N/mm2 by gauge. Determine suitable thickness of the cylinder wall assuming that the maximum permissible tensile stress is not to exceed 105 MPa.

Solution: Given : di = 220 mm or ri = 110 mm ; p = 14 N/mm2 ; t = 105 MPa = 105 N/mm2

Since the pressure of the fluid is high, therefore thick cylinder equation is used.Assuming the material of the cylinder as steel, the thickness of the cylinder wall (t) may be obtained by using Birnies equation. We know that,...(Taking Poissons ratio for steel, = 0.3)

Answer: 16.5 mm

3. The hydraulic cylinder 400 mm bore operates at a maximum pressure of 5 N/mm2. The piston rod is connected to the load and the cylinder to the frame through hinged joints.Design: cylinder.The allowable tensile stress for cast steel cylinder and end cover is 80 MPaSolution: Given : di = 400 mm or ri = 200 mm ; p = 5 N/mm2 ; t = 80 MPa = 80 N/mm2 Design of cylinderLet do = Outer diameter of the cylinder.We know that thickness of cylinder,

Answer: 12 mm

Outer diameter of the cylinder,do = di + 2t = 400 + 2 12 = 424 mmAnswer: 424 mm

4. A cylinder having an internal diameter of 20 in. and an external diameter of 36 in. is subjected to an internal pressure of 10000 psi and an external pressure of 2000 psi. Determine the hoop stress at the inner and outer surface of the cylinder.Solution:

Where: ri = 10Pi = 10000 psiro = 18Po = 2000 psi

Answer: Sti = 13143 psi

Answer: 5143 psi

5. The work cylinder of a hydraulic system is acted by a hydraulic pressure of 750 psi while the maximum load of the piston is 5500 lbs. If the allowable tensile stress is 2000 psi, what is the required wall thickness of the cylinder?

Solution:

Force = Pressure x Area5500 = 370 x (/4)D = 4.35 in

Assume thick-wall cylinder ,

t/D = 0.448/4.35 = 0.103Answer: t = 0.448 in

Republic of the PhilippinesBATAAN PENINSULA STATE UNIVERSITYMain Campus, Province of Bataan, City of Balanga C-2100

COLLEGE OF ENGINEERING AND ARCHITECTURE DEPARTMENT OF MECHANICAL ENGINEEERING

Chapter 4: Thick-Walled Pressure Vessels

IAAngle back-pressure valve. An angle back-pressure valve is a valve with its outlet opening at right angles to its inlet opening.

DAutoclave. A process vessel used any time a vacuum needs to be pulled on a product.Design pressure. The pressure used in calculating the minimum thickness or design characteristics of a boiler or pressure vessel.Design stress. A permissible maximum stress to which a machine part or structural member may be subjected, which is large enough to prevent failure in case the load exceeds the expected value.Digester. A pressure vessel that acts much like a big stomach to break a product down.

PHHeat Exchanger. A process vessel used to either add or remove heat from a product. It works much like the radiator on a car, trying to remove heat from the engine.Hyperbaric chamber. A specially equipped pressure vessel used in medicine and physiological research to administer oxygen at elevated pressure.Integral-type flange. A flange which is forge or cast with, or butt-welded to a nozzle neck, pressure vessel or piping wall.Ion Exchanger. A process vessel used in separating a product at the molecular level, such as ethanol or bio diesel.

LJJacketed. Processed vessel where material is added to the outside of a pressure vessel to maintain an acceptable temperature.Lined. A process vessel where material is added to the inside of a pressure vessel to prevent rust and erosion or to reduce cost

NNormal operation. Normal operation is the operation of boiler or pressure vessel at or below the conditions of coincident pressure and temperature for which the vessel has been designed.Paint Pots. A specialized storage pressure vessel normally used to store and transfer paint to a spray gun. For instance, Thermatech provides paint pots used in the transportation industry for striping roads and highways.Pressure. Pressure is a type of stress which is exerted uniformly in all directions.Pressure dye test. Leak detection method in which a pressure vessel is filled with liquid dye and is pressurized under water to make possible leakage path visible.Pressure gage. An instrument used to measure pressure.Pressure Measurement. Measurement of the internal force of a process vessel, tank, or piping caused by pressurized gas or liquid.Pressure-relief valve. Pressure relief valve is a valve which relieves pressure beyond a specified limit and re-closes upon returning to normal operating condition.

UPressure-retaining member. A part of pressure-relieving device loaded by the restrained pressurized fluid.Pressure storage. The storage of a volatile liquid or liquefied gas under pressure to prevent evaporation.Pressure vessel. A pressure vessel is a metal container, generally cylindrical or spherical in shape which is capable to withstand bursting pressure.Pressurize. To maintain normal atmospheric pressure in a chamber subjected to high or low external pressure.

RRadiography (RT). Is used to obtain an image (on film) of the weld metal through a seam. A lighted view is required to see the film and interpret its results. It is used to detect cracks, slag lines, porosity, lack of fusion, and suck back, among other things.Required thickness. The thickness calculated by recognized formulas for boiler or pressure vessel construction before corrosion allowance is added.Resealing pressure. The inlet pressure at which leakage stops after a pressure relief valve is closed.

SSaturation Diving System. A type of pressure vessel (PVHO) used for deep sea dives. A series of living chambers are fitted together for a multi-day project. For example, workers can compress just once on the way down and once on the way up for a 30-day deep sea construction project.Spring-loaded regulator. A pressure-regulator valve for pressure vessels or flow system.Stub tube. A short tube welded to a boiler or pressure vessel to provide for the attachment of additional parts.Super Duplex Stainless Steel. A specialized material often called an exotic metal. Super duplex stainless steel is only manufactured properly in two places in the world. Thermatech is one of only a handful of U.S. manufacturers with the expertise to fabricate pressure vessels using this unique material.

TTemperature. A property of an object which determines the direction of heat flow when the object is placed in thermal contact with another object.Ultrasonic Testing (Angle UT). Is a process by which ultrasonic waves are introduced by a transducer into the weld (typically from the side of the weld). A machine will convert the waves that bounce back into a visible pattern that a trained person can use to detect discontinuities.

VUnfired pressure vessel. A pressure vessel that is not in direct contact with a heating flame.

YVessel. A container or structural envelope in which materials are processed, treated, or stored.Yield temperature. The temperature at which a fusible plug device melts and is dislodge by its holder and thus relieves pressure in a pressure vessel.