# 23 Facts on Hoop Stress: The Complete Beginner’s Guide

In this article, the topic, “hoop stress” with 23 Facts on Hoop Stress will be discussed in a brief portion. In the outer radius or inner radius portion of a tube hoop stress is remains maximum.

The calculation of the hoop stress is estimate the stress which is acted on a thin circumference pressure vessel. For estimate the hoop stress in a sphere body in some steps. The steps are listed below,

• The internal diameter of the yard and internal pressure should be multiply at the beginning of the process.
• In the next step the resultant should be divided four times with the thickness of the shell.
• In final stage divide the resultant with the joint efficiency.

## What is hoop stress?

The hoop stress can be explain as, the stress which is produce for the pressure gradient around the bounds of a tube. The maximum amount of hoop stress is appearing in the outer radius and inner radius of the tube. The hoop stress depends upon the way of the pressure gradient.

The reason behind the hoop stress is, when a cylinder is under the internal pressure is two times of the longitudinal stress. In a tube the joints of longitudinal produced stress is two times more than the circumferential joints. If pressure is applied in a tube uniformly then the hoop stress in the length of the pipe will be uniform. Image – Cast iron pillar of Chepstow Railway Bridge, 1852. Pin-jointed wrought iron hoops (stronger in tension than cast iron) resist the hoop stresses; Image Credit – Wikipedia

## What is hoop stress in pressure vessel?

The hoop stress in a pressure vessel is acted perpendicular to the direction to the axis. Hoop stresses are generally tensile. The hoop stress is appearing for resist the effect of the bursting from the application of pressure.

Mathematically can written for hoop stress in pressure vessel is,

$\sigma _\theta = \frac{P.D_m}{2t}$

Where,

$\sigma _\theta$ = Hoop stress

P = Internal pressure of the pressure vessel

$D_m$ = Mean diameter of the pressure vessel

t = Thickness of the wall of the pressure vessel

For thin walled pressure vessel the thickness will be assumed as one tenth of the radius of the vessel not more than of it.

In the system of the Inch – pound – second the unit for the internal pressure of the pressure vessel express as ponds – force per square inch, unit for Mean diameter of the pressure vessel is inches, unit for thickness of the wall of the pressure vessel inches and, In the system of the S.I. unit for the internal pressure of the pressure vessel express as Pascal, and unit for Mean diameter of the pressure vessel is meter, unit for thickness of the wall of the pressure vessel meter.

## What is hoop stress in pipelines?

Hoop stress in pipelines can be explain as, the stress in a wall of a pipe operable circumferentially in a profile perpendicular to the axis of the longitudinal of the tube and rose by the tension of the fluid substance in the pipe.

The hoop stress actually is a function which is go about to tension the pipe separately in a direction of the circumferential with the tension being created on the wall of the pipe by the internal pressure of the pipe by natural gas or other fluid.

The hoop stress increases the pipe’s diameter, whereas the longitudinal stress increases with the pipe’s length. The hoop stress generated when a cylinder is under internal pressure is twice that of the longitudinal stress.

## Hoop stress formula:

The formula of the Barlow’s is used for estimate the hoop stress for the wall section of the pipe.

The formula for the hoop stress can be written as,

$\sigma _\theta = \frac{P.D}{2t}$

Where,

$\sigma _\theta$ = Hoop stress

P = Internal pressure of the pipe

D = Diameter of the pipe

t = Thickness of the pipe

In S.I. unit, P (the internal pressure of pipe) expresses as Pascal, and unit for D (diameter of the pipe) is meter, unit for t (thickness of the wall of the pipe) is meter. In the system of the Inch – pound – second unit, P (the internal pressure of pipe) expresses as ponds – force per square inch, and unit for D (diameter of the pipe) is inches, unit for t (thickness of the wall of the pipe) is inches.

## Hoop stress formula for thick cylinder:

Tangential stress and radial stress in a cylinder with thick walled tubes or cylinder with internal pressure, external pressure with closed ends.

Hoop stress formula in the case of thick cylinder three sections. The three sections are listed below,

• Hoop stress in the direction of the axial
• Hoop stress in the direction of the circumferential
• Hoop stress in the direction of the radial

### Hoop stress in the direction of the axial:-

The hoop stress in the direction of the axial at a particular point in the wall of the cylinder or tube can be written as,

$\sigma _a = \frac{(p_ir_i^2 – p_or_o^2)}{r_o^2 – r_i^2}$

Where,

$\sigma _a$ = Hoop stress in the direction of the axial and unit is MPa, psi.

$p_i$ = Internal pressure for the cylinder or tube and unit is MPa, psi.

$r_i$ = Internal radius for the cylinder or tube and unit is mm, in.

$p_o$ = External pressure for the cylinder or tube and unit is MPa, psi.

$r_o$ = External radius for the cylinder or tube and unit is mm, in.

### Hoop stress in the direction of the circumferential:-

The hoop stress in the direction of the circumferential at a particular point in the wall of the cylinder or tube can be written as,

$\sigma _c = \frac{(p_ir_i^2 – p_or_o^2)}{r_o^2 – r_i^2} – \frac{r_i^2r_o^2(p_o – p_i)}{r^2(r_o^2 – r_i^2)}$

Where,

$\sigma _c$ = The hoop stress in the direction of the circumferential and unit is MPa, psi.

$p_i$ = Internal pressure for the cylinder or tube and unit is MPa, psi.

$r_i$ = Internal radius for the cylinder or tube and unit is mm, in.

$p_o$ = External pressure for the cylinder or tube and unit is MPa, psi.

$r_o$ = External radius for the cylinder or tube and unit is mm, in.

r = Radius for the cylinder or tube and unit is mm, in.  $(r_i < r < r_o)$

Maximum hoop stress for the cylinder or tube is, $r_i = r$

### Hoop stress in the direction of the radial:-

The hoop stress in the direction of the radial at a particular point in the wall of the cylinder or tube can be written as,

$\sigma _r = \frac{(p_ir_i^2 – p_or_o^2)}{r_o^2 – r_i^2} + \frac{r_i^2r_o^2(p_o – p_i)}{r^2(r_o^2 – r_i^2)}$

Where,

$\sigma _r$= The hoop stress in the direction of the radial circumferential and unit is MPa, psi.

$p_i$ = Internal pressure for the cylinder or tube and unit is MPa, psi.

$r_i$ = Internal radius for the cylinder or tube and unit is mm, in.

$p_o$ = External pressure for the cylinder or tube and unit is MPa, psi.

$r_o$ = External radius for the cylinder or tube and unit is mm, in.

## Hoop stress formula for pipe:

The formula of the Barlow’s is used for estimate the hoop stress for the wall section of the pipe.

The formula for the hoop stress can be written as,

$\sigma _\theta = \frac{P.D}{2t}$

Where,

$\sigma _\theta$ = Hoop stress

P = Internal pressure of the pipe

D = Diameter of the pipe

t = Thickness of the pipe

In S.I. unit, P (the internal pressure of pipe) expresses as Pascal, and unit for D (diameter of the pipe) is meter, unit for t (thickness of the wall of the pipe) is meter.

In the system of the Inch – pound – second unit, P (the internal pressure of pipe) expresses as ponds – force per square inch, and unit for D (diameter of the pipe) is inches, unit for t (thickness of the wall of the pipe) is inches.

## Hoop stress formula for sphere:

The hoop stress formula for the sphere is discussed in below section,

• Hoop stress formula for sphere in thin walled section
• Hoop stress formula for sphere in thick walled section
• Hoop stress formula for sphere in thick walled section (Only for internal pressure)
• Hoop stress formula for sphere in thick walled section (Only for external pressure)

### Hoop stress formula for sphere in thin walled section:-

Thin walled portions of a spherical tube or cylinder where both internal pressure and external pressure acted can be express as,

$\frac{Pr}{2t}$

### Hoop stress formula for sphere in thick walled section:-

Thick walled portions of a spherical tube and cylinder where both internal pressure and external pressure acted can be express as,

$\sigma _h = \frac {(P_ir_i^2 – P_or_o^2)}{r_o^2 – r_i^2} – \frac{r_i^2r_o^2(P_o – P_i)}{r^2(r_o^2 – r_i^2)}$

### Hoop stress formula for sphere in thick walled section (Only for internal pressure):-

Thick walled portions of a tube and cylinder where only internal pressure acted can be express as,

$\sigma _h = \frac {(P_ir_i^2 )}{r_o^2 – r_i^2}(1 – \frac{r_o^2}{r^2})$

### Hoop stress formula for sphere in thick walled section (Only for external pressure):-

Thick walled portions of a tube and cylinder where only external pressure acted can be express as,

$\sigma _h = \frac {(P_or_o^2 )}{r_o^2 – r_i^2}(1 + \frac{r_i^2}{r^2})$

Where,

$\sigma _h$ = The hoop stress and unit is MPa, psi.

P = Pressure under consideration and unit is MPa, psi.

$p_i$= Internal pressure for the cylinder or tube and unit is MPa, psi.

$r_i$ = Internal radius for the cylinder or tube and unit is mm, in.

$p_o$ = External pressure for the cylinder or tube and unit is MPa, psi.

$r_o$ = External radius for the cylinder or tube and unit is mm, in.

r = Radius for the cylinder or tube and unit is mm, in.

t = Wall thickness for the cylinder or tube and unit is mm, in.

## Hoop stress formula for conical cylinder:

Hoop stress formula for conical cylinder can be express for two conditions. The conditions are listed below,

• When the liquid substance is stays at the surface of below y (y < d)
• When the liquid substance is stays at the surface of above y or equal to y (y >d, y = d)

### Case: 1: When the liquid substance is stays at the surface of below y (y < d):-

Meridional stress:

$\sigma _1 = \frac{\delta ytan \alpha }{2t cos \alpha }(d – \frac{2y}{3})$

Hoop stress or circumferential stress:

$\sigma _2 = \frac{y(d – y)\delta tan\alpha }{t cos \alpha }$

$\Delta R = \frac{\delta y^2 tan^2 \alpha }{E tcos \alpha }[d(1 – \frac{v}{2}) – y (1 – \frac{v}{3})]$

Change in the height of dimension y:

$\Delta y = \frac{\delta y^2sin\alpha }{Etcos^4\alpha }{\frac{d}{4}(1 – 2y) – \frac{y}{9}(1 – 3v) – sin^2 \alpha [\frac{d}{2}(2 -y) – \frac{y}{3}(3 – v)]}$

Turning of a meridian out of its unloaded condition:

$\psi = \frac{\delta ysin^2\alpha }{6Et cos^3 \alpha } (9d – 16y)$

### Case: 2: When the liquid substance is stays at the surface of above y or equal to y (y >d, y = d):-

Meridional stress:

$\sigma _1 = \frac{\delta d^3tan\alpha }{6tycos\alpha }$

Hoop stress or circumferential stress:

$\sigma _2$ = 0

$\Delta R = \frac{-v \delta d^3tan^2\alpha }{6Etcos\alpha }$

Change in the height of dimension y:

$\Delta y = \frac{\delta d^3 sin \alpha}{6Et cos^4 \alpha }[\frac{5}{6} – v(1 – sin^2 \alpha ) + ln \frac{y}{d}]$

Turning of a meridian out of its unloaded condition:

$\psi = \frac{-\delta d^3 sin^2 \alpha }{6 Etcos^3 \alpha }\frac{1}{y}$

Where,

$\sigma _1$ = Hoop stress and unit is $lbs/in^2$.

$\sigma _2$ = Hoop stress and unit is $lbs/in^2$.

E = Modulus of Elasticity and unit is $lbs/in^2$.

$\psi$ = Turning of a meridian out of its unloaded condition.

v = Poisson’s ratio and it is unit less.

$\delta$ = Liquid density and unit is $lbs/in^3$.

d = Liquid fill level and unit is in.

t = Wall thickness unit is in.

$\alpha$ = Angle and unit degree.

y = Pointing a level of a cone and unit is in.

## Hoop stress formula derivation:

Another term for the cylindrical tube is pressure vessel. In various fields of engineering the pressure vessels are used such as, Boilers, LPG cylinders, Air recover tanks and many more.

### Derivation of the hoop stress formula:-

Cylindrical shell bursting will take place if force due to internal fluid pressure will be more than the resisting force due to circumferential stress or hoop stress developed in the wall of the cylindrical shell.

Let consider the terms which explaining the expression for hoop stress or circumferential stress which is produce in the cylindrical tube’s wall.

P = Internal fluid pressure of the cylindrical tube

t = Thickness for the cylindrical tube

L= Length for the cylindrical tube

d = Internal diameter for the thin cylindrical tube

$\sigma _H$ = Hoop stress or circumferential stress which is produce in the cylindrical tube’s wall

Force produce for the internal fluid pressure = Area where the fluid pressure is working * Internal fluid pressure of the cylindrical tube

Force produce for the internal fluid pressure = $(d \times L) \times P$

Force produce for the internal fluid pressure =$P \times d\times L$ …….eqn (1)

Resulting force for the reason of hoop stress or circumferential stress =$\sigma _H \times 2Lt$ …….eqn (2)

From the …….eqn (1) and eqn (2) we can write,

Force produce for the internal fluid pressure = Resulting force for the reason of hoop stress or circumferential stress

$P \times d\times L = \sigma _H \times 2Lt$

$\sigma _H = \frac{Pd}{2t}$

## How to calculate hoop stress?

For calculating the hoop stress for a sphere body the steps are listed below,

• The internal diameter of the yard and internal pressure should be multiply at the beginning of the process.
• In the next step the resultant should be divided four times with the thickness of the shell.
• In final stage divide the resultant with the joint efficiency.

## How to calculate hoop stress in pipe?

For calculating the hoop stress just need to multiply the internal diameter (mm) of the pipe with internal pressure (MPa) of the pipe and then the value need to divided with the thickness (mm) of the pipe with 2.

Formula for estimate the hoop stress in a pipe is,

Hoop stress = $\frac{Internal diameter \times Internal pressure}{2 \times Thickness}$

Mathematically hoop stress can be written as,

$\sigma _\theta = \frac{P.D}{2t}$

Where,

$\sigma _\theta$ = Hoop stress

P = Internal pressure

D = Diameter of the pipe

t = Thickness of the pipe

## How to calculate hoop stress of a cylinder?

Hoop stress can be explained as; the mean volume of force is employed in per unit place. The hoop stress is the capacity is applied circumferentially in both ways on every particle in the wall of the cylinder.

Formula for estimate the hoop stress of a cylinder is,

Hoop stress = $\frac{Internal diameter \times Internal pressure}{2 \times Thickness}$

Mathematically hoop stress can be written as,

$\sigma _\theta = \frac{P.D}{2t}$

Where,

$\sigma _\theta$ = Hoop stress in the direction of the both and unit is MPa, psi.

P = Internal pressure of the pipe and unit is MPa, psi.

D = Diameter of the pipe and unit is mm, in.

t = Thickness of the pipe and unit is mm, in.

## Hoop stress vs. radial stress:

The major difference between hoop stress and radial stress are describe in below section,

## Hoop stress vs. axial stress:

The major difference between hoop stress and axial stress are describe in below section,

## Hoop stress vs. tangential stress:

The major difference between hoop stress and tangential stress are describe in below section,

## Hoop stress vs. yield strength:

The major difference between hoop stress and yield strength are describe in below section,

## Is hoop stress shear stress?

No, hoop stress or circumference stress is not a shear stress. In the theory of pressure vessel, any given element of the wall is evaluated in a tri-axial stress system, with the three principal stresses being hoop, longitudinal, and radial. Therefore, by definition, there exist no shear stresses on the transverse, tangential, or radial planes.

## Is hoop stress tensile?

Hoop stress can be explained as; the stress is developed along the circumference of the tube when pressure is acted.

Yes, hoop stress is tensile and for this reason wrought iron is added to various materials and has better tensile strength compare to cast iron. Hoop stress is works perpendicularly to the direction of the axial. Hoop stresses are tensile, and developed to defend the effect of the bursting that appears from the movement of pressure.

## Is hoop stress a principal stress?

Yes, hoop stress is the principal stresses. To estimate the longitudinal stress need to create a cut across the cylinder similar to analyzing the spherical pressure vessel. The form of failure in tubes is ruled by the magnitude of stresses in the tube.

If there is a failure is done by the fracture, that means the hoop stress is the key of principle stress, and there are no other external load is present.

## Is hoop stress normal stress?

Yes, hoop stress or circumferential stress is a normal stress in the direction of the tangential. Stress is termed as Normal stress when the direction of the deforming force is perpendicular to the cross-sectional area of the body. The length of the wire or the volume of the body changes stress will be at normal.

## How to reduce hoop stress?

The method is to reducing the hoop stress is control a strong wire made with steel under tension through the walls of the cylinder to shrink one cylinder over another.

The most efficient method is to apply double cold expansion with high interference along with axial compression with strain equal to 0.5%. This technique helps to reduce absolute value of hoop residual stresses by 58%, and decrease radial stresses by 75%.

## Conclusion:

• A normal stress in tangential direction horizon of the cylinder surface.
• Hoop stress also called as, Circumferential stress.
• Hoop stress acts along $\phi$.

Indrani Banerjee

Hi..I am Indrani Banerjee. I completed my bachelor's degree in mechanical engineering. I am a enthusiastic person and I am a person who is positive about every aspect of life. I like to read Books and listening to music.