# Thin and Thick cylinders

### Contents

**Define Pressure Vessel**1**List various types of Pressure Vessels**2**Classify Stresses in Pressure Vessels**3**Describe the Circumferential and Longitudinal Stresses for Thin Cylindrical Shell subjected to an Internal Pressure**4**Explain the Maximum Shear Stress in Thin Cylindrical Shell**5**Describe Circumferential and Longitudinal Strain for Thin Cylindrical Shell subjected to an Internal Pressure**6**Explain Volumetric Strain of Thin Cylindrical Shell**7**Explain the phenomena of Thin Spherical Shell**8**Explain The Volumetric Strain in a Thin Spherical Shell**9**Explain the Stresses in Hemispherical Portion of the Shell**11**Describe Lame’s Equation to find out Stresses in Thick Cylinder and assumptions made in Lame’s Theory**12**Describe Lame’s Equation to find out the Stresses in Thick Spherical Shell**13

Define Pressure Vessel

A pressure vessel is a container or vessel designed to hold fluids or gases at a pressure significantly different from the ambient pressure. These vessels are typically used in industrial processes where it is necessary to store or transport materials under high pressure, such as compressed air or gases, steam, and various chemicals.

Pressure vessels are commonly found in a wide range of applications, including power generation, chemical and petrochemical processing, oil and gas production, and pharmaceutical manufacturing. Examples of pressure vessels include storage tanks, boilers, and heat exchangers.

The design of pressure vessels is subject to a variety of national and international standards and codes, which are intended to ensure their safety and reliability. These standards address factors such as material selection, structural integrity, and pressure testing procedures.

Due to the high pressures involved, pressure vessels must be designed and manufactured with great care and attention to detail. Any defects or weaknesses in the vessel can result in catastrophic failures, which can lead to injuries, deaths, and extensive damage to property and the environment. As such, pressure vessels are subject to rigorous inspection and maintenance procedures throughout their service life.

Overall, pressure vessels play a critical role in a wide range of industrial applications, and their safe and reliable operation is essential for the continued growth and development of many industries.

List various types of Pressure Vessels

There are several different types of pressure vessels, each designed to meet specific industrial needs and operating conditions. Some of the most common types of pressure vessels include:

- Storage tanks: These are the simplest and most common type of pressure vessel. They are designed to store gases or liquids at a fixed pressure and temperature. Storage tanks can be vertical or horizontal, and are commonly made of steel or other metals.
- Boilers: These pressure vessels are designed to heat water or other fluids to produce steam. Boilers are commonly used in power generation and industrial processes, such as in the production of chemicals, paper, and textiles.
- Heat exchangers: These pressure vessels are used to transfer heat from one fluid to another. They are commonly found in industrial processes that involve heating or cooling, such as in refrigeration systems or chemical processing plants.
- Reactors: These pressure vessels are used in chemical and petrochemical processes to contain and control chemical reactions. They are typically made of stainless steel or other high-performance alloys, and are designed to withstand high temperatures and pressures.
- Separators: These pressure vessels are used to separate fluids based on their physical properties, such as density or viscosity. They are commonly used in the oil and gas industry to separate oil, gas, and water during production.
- Distillation columns: These pressure vessels are used to separate and purify mixtures of liquids. They are commonly used in the chemical and petrochemical industries to produce high-purity chemicals and fuels.
- Autoclaves: These pressure vessels are used in the medical and pharmaceutical industries to sterilize equipment and materials. They are typically made of stainless steel, and are designed to withstand high temperatures and pressures.

Overall, the specific type of pressure vessel used will depend on the application, and will be designed to meet the specific requirements of the process in which it is used.

Classify Stresses in Pressure Vessels

Pressure vessels are subjected to a range of different stresses that can affect their structural integrity and performance. These stresses can be classified into three main categories:

- Primary stresses: Primary stresses are the result of the internal pressure that the vessel is designed to contain. These stresses can be further divided into circumferential stresses, which act perpendicular to the longitudinal axis of the vessel, and longitudinal stresses, which act parallel to the longitudinal axis. The magnitude of primary stresses is determined by the operating pressure of the vessel, as well as its dimensions, shape, and material properties.
- Secondary stresses: Secondary stresses are caused by external loads or constraints that are applied to the vessel, such as weight, thermal expansion or contraction, or pressure fluctuations. These stresses are typically much lower than primary stresses, but they can still be significant and must be taken into account during the design process.
- Residual stresses: Residual stresses are the result of the manufacturing process and are typically present in all pressure vessels. These stresses can be caused by welding, rolling, or forming the material, and can be influenced by factors such as temperature, material properties, and manufacturing techniques. Residual stresses can affect the performance of the vessel over time, particularly if they are not properly accounted for during the design and manufacturing process.

In addition to these categories, stresses in pressure vessels can also be classified by their distribution, such as uniform or non-uniform stresses. Non-uniform stresses can occur at points of stress concentration, such as at sharp corners, changes in cross-section, or where components are attached to the vessel. These points of stress concentration can be particularly vulnerable to cracking or other types of damage over time.

Overall, the classification of stresses in pressure vessels is important for ensuring their safe and reliable operation over their service life. Proper consideration and analysis of the various stress types and distributions is critical to the design and manufacturing process, as well as to the ongoing maintenance and inspection of the vessel.

Describe the Circumferential and Longitudinal Stresses for Thin Cylindrical Shell subjected to an Internal Pressure

When a thin cylindrical shell is subjected to an internal pressure, it experiences both circumferential and longitudinal stresses. These stresses can be calculated using the following equations:

Circumferential Stress:

σc = pd/2t

where σc is the circumferential stress, p is the internal pressure, d is the diameter of the shell, and t is the thickness of the shell.

Longitudinal Stress:

σl = pd/4t

where σl is the longitudinal stress.

The circumferential stress is the stress that acts perpendicular to the longitudinal axis of the cylinder and is highest at the inner surface of the cylinder. This is because the circumference of the inner surface is smaller than that of the outer surface, resulting in a greater stress. The longitudinal stress is the stress that acts parallel to the longitudinal axis of the cylinder and is highest at the ends of the cylinder. This is because the ends of the cylinder are free to expand, resulting in a greater stress.

It is important to note that these equations are only valid for thin-walled cylinders, where the thickness of the wall is small compared to the diameter of the cylinder. If the wall thickness is not negligible, more complex stress calculations are required.

The maximum stress occurs at the inner surface of the cylinder, and it is important to ensure that this stress does not exceed the yield strength of the material. If it does, the cylinder may undergo plastic deformation or even rupture, which can result in serious safety issues.

In addition to these equations, other factors must be considered when designing a cylindrical pressure vessel, such as the effects of external forces, the properties of the material, and the type of end closures used. Proper consideration of all of these factors is critical to the safe and reliable operation of the vessel.

Explain the Maximum Shear Stress in Thin Cylindrical Shell

When a thin cylindrical shell is subjected to an internal pressure, it experiences both circumferential and longitudinal stresses. In addition to these stresses, there is also a maximum shear stress that occurs in the shell.

The maximum shear stress occurs at a 45-degree angle to the longitudinal axis of the cylinder and can be calculated using the following equation:

τmax = pd/4t

where τmax is the maximum shear stress, p is the internal pressure, d is the diameter of the shell, and t is the thickness of the shell.

This equation shows that the maximum shear stress is proportional to the internal pressure and the diameter of the shell, and inversely proportional to the thickness of the shell. The maximum shear stress is typically lower than the circumferential and longitudinal stresses, but it is still an important factor to consider in the design and analysis of pressure vessels.

If the maximum shear stress exceeds the shear yield strength of the material, the shell may experience permanent deformation or failure. Therefore, it is important to ensure that the maximum shear stress is kept below the allowable stress limit for the material being used.

Overall, the maximum shear stress is an important consideration when designing and analyzing thin cylindrical shells that are subjected to internal pressure. Proper calculation and analysis of this stress, in addition to the circumferential and longitudinal stresses, can help ensure the safe and reliable operation of pressure vessels over their service life.

Describe Circumferential and Longitudinal Strain for Thin Cylindrical Shell subjected to an Internal Pressure

When a thin cylindrical shell is subjected to an internal pressure, it not only experiences circumferential and longitudinal stresses, but also circumferential and longitudinal strains. Strain is the deformation of a material caused by stress, and it is expressed as a change in length or volume relative to the original length or volume.

The circumferential strain, also known as hoop strain, is the strain that acts perpendicular to the longitudinal axis of the cylinder. It can be calculated using the following equation:

ε_{c} = pd/2tE

where εc is the circumferential strain, p is the internal pressure, d is the diameter of the shell, t is the thickness of the shell, and E is the modulus of elasticity of the material.

The longitudinal strain is the strain that acts parallel to the longitudinal axis of the cylinder. It can be calculated using the following equation:

εl = νpd/4tE

where εl is the longitudinal strain, ν is the Poisson’s ratio of the material.

The Poisson’s ratio is a material property that relates the strain in one direction to the strain in another direction perpendicular to it.

It is important to note that these equations are only valid for thin-walled cylinders, where the thickness of the wall is small compared to the diameter of the cylinder. If the wall thickness is not negligible, more complex strain calculations are required.

The circumferential and longitudinal strains are important to consider when designing cylindrical pressure vessels, as they can affect the overall deformation and stability of the vessel. Proper consideration of these strains.

Overall, the circumferential and longitudinal strains are key factors to consider in the design and analysis of thin cylindrical shells subjected to internal pressure, and proper calculation and analysis of these strains can help ensure the safe and reliable operation of pressure vessels.

Explain Volumetric Strain of Thin Cylindrical Shell

When a thin cylindrical shell is subjected to an internal pressure, it not only experiences circumferential and longitudinal strains, but also a volumetric strain. Volumetric strain is the change in volume of a material per unit volume and is expressed as a change in volume relative to the original volume.

The volumetric strain in a thin cylindrical shell can be calculated using the following equation:

ε_{v} = 3pd/4tE

where εv is the volumetric strain, p is the internal pressure, d is the diameter of the shell, t is the thickness of the shell, and E is the modulus of elasticity of the material.

This equation shows that the volumetric strain is proportional to the internal pressure, the diameter of the shell, and the Poisson’s ratio of the material. The Poisson’s ratio is a material property that relates the strain in one direction to the strain in another direction perpendicular to it. The volumetric strain is also inversely proportional to the thickness of the shell.

The volumetric strain is an important consideration when designing cylindrical pressure vessels, as it can affect the overall deformation and stability of the vessel. Excessive volumetric strain can cause the vessel to deform or even rupture under pressure, leading to catastrophic failure. Therefore, it is important to ensure that the volumetric strain is kept within safe limits for the material being used.

Overall, the volumetric strain is a critical factor to consider in the design and analysis of thin cylindrical shells subjected to internal pressure, and proper calculation and analysis of this strain can help ensure the safe and reliable operation of pressure vessels.

Explain the phenomena of Thin Spherical Shell

A thin spherical shell is a hollow, spherical object with a relatively small wall thickness compared to its diameter. When a thin spherical shell is subjected to internal or external pressure, it experiences stresses and strains that can affect its mechanical behaviour.

When a thin spherical shell is subjected to internal pressure, it experiences circumferential or hoop stresses that act tangential to the surface of the shell. The hoop stress is given by the following equation:

σ_{h} = pd/4t

where σh is the hoop stress, p is the internal pressure, d is the diameter of the shell, and t is the thickness of the shell. The hoop stress is highest at the equator of the sphere and decreases towards the poles.

In addition to hoop stress, the thin spherical shell experiences radial stress that acts normal to the surface of the shell. The radial stress is given by the following equation:

σ_{r} = pd/2t

where σr is the radial stress. The radial stress is highest at the poles of the sphere and decreases towards the equator.

The circumferential and radial stresses can cause the spherical shell to deform, leading to circumferential and radial strains. The circumferential or hoop strain is given by the following equation:

ε_{h} = σh/E

where εh is the circumferential strain, and E is the modulus of elasticity of the material. The radial strain is given by the following equation:

ε_{r} = σr/E

where εr is the radial strain.

In addition to hoop and radial stresses and strains, thin spherical shells are also subject to volumetric strain, which is the change in volume of a material per unit volume. The volumetric strain is given by the following equation:

ε_{v} = 3p/2E

where εv is the volumetric strain.

The mechanical behaviour of a thin spherical shell under pressure is critical in the design and analysis of pressure vessels and other spherical objects. Proper consideration of the stresses and strains can help ensure the safe and reliable operation of the object over its service life..

Explain The Volumetric Strain in a Thin Spherical Shell

The volumetric strain is a measure of the change in volume of a material due to deformation. In a thin spherical shell, the internal or external pressure causes the shell to deform, resulting in a change in volume. The volumetric strain in a thin spherical shell is given by the following equation:

ε_{v }= 3ΔV / (4πr³)

where εv is the volumetric strain, ΔV is the change in volume of the shell, and r is the radius of the shell.

The change in volume of the shell is given by:

ΔV = V₂ – V₁

where V₂ is the final volume of the shell under pressure and V₁ is the initial volume of the shell before the pressure is applied.

For a thin spherical shell, the initial and final volumes are given by:

V₁ = 4/3πr³₀t₀ and V₂ = 4/3πr³t

where r₀ is the radius of the shell before deformation, t₀ is the thickness of the shell before deformation, and t is the thickness of the shell after deformation.

By substituting these equations into the expression for ΔV and simplifying, we obtain:

ΔV = 4/3πr³t – 4/3πr³₀t₀

Substituting this expression into the equation for εv, we get:

ε_{v} = 3(4/3πr³t – 4/3πr³₀t₀) / (4πr³)

Simplifying, we obtain:

ε_{v} = (4t – 4t₀) / (3r)

This equation shows that the volumetric strain in a thin spherical shell is directly proportional to the change in thickness and inversely proportional to the radius of the shell. This means that the volumetric strain increases as the thickness of the shell decreases, and as the radius of the shell increases.

Proper consideration of the volumetric strain is important in the design and analysis of thin spherical shells, as it can affect the mechanical behaviour and reliability of the shell under pressure.

Explain the Stresses in Hemispherical Portion of the Shell

When a thin hemispherical shell is subjected to internal or external pressure, it experiences stresses that are a function of the radius of the shell, the thickness of the shell, and the applied pressure.

The stresses in the hemispherical portion of the shell can be analyzed using the theory of thin shells, which assumes that the thickness of the shell is small compared to the radius of the shell. The theory also assumes that the deformation of the shell is small, and that the material of the shell is homogeneous and isotropic.

Under these assumptions, the circumferential and longitudinal stresses in the hemispherical portion of the shell can be calculated using the following equations:

σ_{θ} = Pr / 2t

σ_{z} = P(1 – r² / 4t²) / 2t

where σθ is the circumferential stress, σz is the longitudinal stress, P is the applied pressure, r is the radius of the shell, and t is the thickness of the shell.

The circumferential stress is maximum at the equator of the hemispherical shell and is equal to Pr/2t. This stress is tensile on the outer surface and compressive on the inner surface of the shell.

The longitudinal stress is maximum at the crown of the hemispherical shell and is equal to P(1-r²/4t²)/2t. This stress is tensile on the outer surface and compressive on the inner surface of the shell.

In addition to these stresses, there is also a radial stress that acts perpendicular to the surface of the shell. This stress is negligible in comparison to the circumferential and longitudinal stresses, as it is proportional to the square of the thickness of the shell.

The stresses in the hemispherical portion of the shell are important to consider in the design and analysis of pressure vessels and other structures that incorporate hemispherical components. Proper selection of the thickness and material of the shell can help to ensure that the stresses remain within safe limits and that the shell performs as intended under pressure.

Describe Lame’s Equation to find out Stresses in Thick Cylinder and assumptions made in Lame’s Theory

Lame’s equation is used to determine the radial and tangential stresses in a thick-walled cylinder subjected to an internal pressure. The equation is derived from the theory of elasticity, and it assumes that the cylinder is homogeneous, isotropic, and linearly elastic.

Lame’s equation is given by:

σ_{r} = [(A-B)/2 + (A+B)/2(R_{2}/R_{1})^{2}]P

σ_{θ} = [(A-B)/2 – (A+B)/2(R_{2}/R_{1})^{2}]P

where σ_{r} is the radial stress, σ_{θ} is the tangential stress, P is the applied internal pressure, R_{1} is the inner radius of the cylinder, R_{2} is the outer radius of the cylinder, and A and B are constants that depend on the material properties of the cylinder.

The assumptions made in Lame’s theory are:

- The cylinder is homogeneous: The material properties of the cylinder are assumed to be the same throughout the cylinder.
- The cylinder is isotropic: The mechanical properties of the cylinder are assumed to be the same in all directions.
- The cylinder is linearly elastic: The deformation of the cylinder is assumed to be linearly proportional to the applied stress.
- The cylinder is subjected to internal pressure only: No external loads or forces are applied to the cylinder.
- The cylinder is thick-walled: The thickness of the cylinder is assumed to be greater than the radius.

Lame’s equation is widely used in the design and analysis of pressure vessels and other structures that incorporate thick-walled cylinders, such as pipes and boilers. The equation provides a means to calculate the radial and tangential stresses in the cylinder, which are important considerations in ensuring that the cylinder can withstand the applied pressure without failing.

Describe Lame’s Equation to find out the Stresses in Thick Spherical Shell

Lame’s equation can also be used to determine the stresses in a thick-walled spherical shell subjected to an internal pressure. The equation is derived from the theory of elasticity and is similar to the equation for a thick-walled cylinder, with some modifications to account for the spherical shape of the shell.

Lame’s equation for a thick spherical shell is given by:

σ_{r} = [(A-B)/2 + (A+B)/2(R_{2}^{3}/R_{1}^{3})]P

σ_{θ} = [(A-B)/2 – (A+B)/2(R_{2}^{3}/R_{1}^{3})]P

where σ_{r} is the radial stress, σ_{θ} is the tangential stress, P is the applied internal pressure, R_{1} is the inner radius of the shell, R_{2} is the outer radius of the shell, and A and B are constants that depend on the material properties of the shell.

Similar to the case of a thick-walled cylinder, Lame’s equation for a thick spherical shell assumes that the material of the shell is homogeneous, isotropic, and linearly elastic. The equation also assumes that the shell is subjected to internal pressure only and that the thickness of the shell is greater than the radius.

The radial stress in a thick spherical shell is tensile in nature, meaning it tends to pull the shell apart, while the tangential stress is compressive, meaning it tends to crush the shell. The magnitude of the stresses increases with increasing internal pressure and decreasing thickness of the shell.

Lame’s equation is useful in the design and analysis of pressure vessels and other structures that incorporate thick-walled spherical shells, such as tanks and containers. By calculating the stresses in the shell, engineers can ensure that the structure is strong enough to withstand the applied pressure without failing.