Hoop stress
Encyclopedia
Circumferential stress is a type of mechanical stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

of a cylindrically shaped part as a result of internal or external pressure.

The classic example of circumferential stress is the tension
Tension (mechanics)
In physics, tension is the magnitude of the pulling force exerted by a string, cable, chain, or similar object on another object. It is the opposite of compression. As tension is the magnitude of a force, it is measured in newtons and is always measured parallel to the string on which it applies...

applied to the iron bands, or hoops, of a wooden barrel. In a straight, closed pipe
Pipe (material)
A pipe is a tubular section or hollow cylinder, usually but not necessarily of circular cross-section, used mainly to convey substances which can flow — liquids and gases , slurries, powders, masses of small solids...

, any force applied to the cylindrical pipe wall by a pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

differential will ultimately give rise to hoop stresses. Similarly, if this pipe has flat end caps, any force applied to them by static pressure will induce a perpendicular axial stress on the same pipe wall. Thin sections often have negligibly small radial stress, but accurate models of thicker-walled cylindrical shells require such stresses to be taken into account.

## Definitions

One such type of stress is the hoop stress, which is defined for rotationally-symmetric objects as the average force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

exerted circumferentially (perpendicular both to the axis and to the radius of the object) in both directions on every particle in the cylinder wall. It can be described as:
where:
• F is the force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

exerted circumferentially on an area of the cylinder wall that has the following two lengths as sides:
• t is the radial thickness of the cylinder
• l is the axial length of the cylinder

An alternative to hoop stress in describing circumferential stress is wall stress or wall tension (T), which usually is defined as the total circumferential force exerted along the entire radial thickness:

Along with axial stress and radial stress, circumferential stress is a component of the stress tensor
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

in cylindrical coordinates.

It is usually useful to decompose
Vector decomposition
Vector decomposition refers to decomposing a vector of Rn into several vectors, each linearly independent .-Vector decomposition in two dimensions:...

any force applied to an object with rotational symmetry
Rotational symmetry
Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has...

into components parallel to the cylindrical coordinates r, z, and θ. These components of force induce corresponding stresses: radial stress, axial stress and hoop stress, respectively.

### Thin-walled assumption

For the thin-walled assumption to be valid the vessel must have a wall thickness of no more than about one-tenth (often cited as one twentieth) of its radius. This allows for treating the wall as a surface, and subsequently using the Young–Laplace equation
Young–Laplace equation
In physics, the Young–Laplace equation is a nonlinear partial differential equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or wall tension, although usage on the...

for estimating the hoop stress created by an internal pressure on a thin wall cylindrical pressure vessel:
(for a cylinder)
(for a sphere)

where
• P is the internal pressure
• t is the wall thickness
• r is the inside radius of the cylinder.
• is the hoop stress.

The hoop stress equation for thin shells is also approximately valid for spherical vessels, including plant cells and bacteria in which the internal turgor pressure may reach several atmospheres.

Inch-pound-second system (IPS) units for P are pounds-force per square inch (psi). Units for t, and d are inches (in).
SI units for P are pascals
Pascal (unit)
The pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...

(Pa), while t and d=2r are in meters (m).

When the vessel has closed ends the internal pressure acts on them to develop a force along the axis of the cylinder. This is known as the axial stress and is usually less than the hoop stress.

Though this may be approximated to

Also in this situation a radial stress is developed and may be estimated in thin walled cylinders as:

### Thick-walled vessels

When the cylinder to be studied has a r/t ratio of less than 10 (often cited as 20) the thin-walled cylinder equations no longer hold since stresses vary significantly between inside and outside surfaces and shear stress through the cross section can no longer be neglected.

In order to calculate the stresses and strains here a set of equations known as the Lamé
Lamé
In fencing, a lamé is an electrically conductive jacket worn by foil and sabre fencers in order to define the scoring area . Foil lamés, although traditionally a metallic grey, are becoming more and more popular in an array of colors. In foil, the lamé extends on the torso from the shoulders to...

equations must be used.

where
• A and B are constants of integration, which may be discovered from the boundary conditions
• r is the radius at the point of interest (e.g. at the inside or outside walls)

A and B may be found by inspection of the boundary conditions. For example, the simplest case is a solid cylinder:

if then and a solid cylinder cannot have an internal pressure so

### Engineering

Fracture is governed by the hoop stress in the absence of other external loads since it is the largest principal stress. Note that since the hoop stress is largest when r is smallest, cracks in pipes should theoretically start from inside the pipe. This is why pipe inspections after earthquakes usually involve sending a camera inside a pipe to inspect for cracks.
Yielding is governed by an equivalent stress that includes hoop stress and the longitudinal or radial stress when present.

### Medicine

In the pathology
Pathology
Pathology is the precise study and diagnosis of disease. The word pathology is from Ancient Greek , pathos, "feeling, suffering"; and , -logia, "the study of". Pathologization, to pathologize, refers to the process of defining a condition or behavior as pathological, e.g. pathological gambling....

of vascular
Blood vessel
The blood vessels are the part of the circulatory system that transports blood throughout the body. There are three major types of blood vessels: the arteries, which carry the blood away from the heart; the capillaries, which enable the actual exchange of water and chemicals between the blood and...

or gastrointestinal walls
Gastrointestinal tract
The human gastrointestinal tract refers to the stomach and intestine, and sometimes to all the structures from the mouth to the anus. ....

, the wall tension represents the muscular tension
Muscle contraction
Muscle fiber generates tension through the action of actin and myosin cross-bridge cycling. While under tension, the muscle may lengthen, shorten, or remain the same...

on the wall of the vessel. As a result of the Law of Laplace, if an aneurysm
Aneurysm
An aneurysm or aneurism is a localized, blood-filled balloon-like bulge in the wall of a blood vessel. Aneurysms can commonly occur in arteries at the base of the brain and an aortic aneurysm occurs in the main artery carrying blood from the left ventricle of the heart...

forms in a blood vessel wall, the radius of the vessel has increased. This means that the inward force on the vessel decreases, and therefore the aneurysm will continue to expand until it ruptures. A similar logic applies to the formation of diverticuli in the gut
Gut (zoology)
In zoology, the gut, also known as the alimentary canal or alimentary tract, is a tube by which bilaterian animals transfer food to the digestion organs. In large bilaterians the gut generally also has an exit, the anus, by which the animal disposes of solid wastes...

.

## Historical development of the theory

The first theoretical analysis of the stress in cylinders was developed by the mid-19th century engineer William Fairbairn
William Fairbairn
Sir William Fairbairn, 1st Baronet was a Scottish civil engineer, structural engineer and shipbuilder.-Early career:...

, assisted by his mathematical analyst Eaton Hodgkinson
Eaton Hodgkinson
Eaton A. Hodgkinson was an English engineer, a pioneer of the application of mathematics to problems of structural design.-Early life:...

. Their first interest was in studying the design and failure
Boiler explosion
A boiler explosion is a catastrophic failure of a boiler. As seen today, boiler explosions are of two kinds. One kind is over-pressure in the pressure parts of the steam and water sides. The second kind is explosion in the furnace. Boiler explosions of pressure parts are particularly associated...

s of steam boilers. Early on Fairbairn realised that the hoop stress was twice the longitudinal stress, an important factor in the assembly of boiler shells from rolled sheets joined by riveting. Later work was applied to bridge building, and the invention of the box girder
Box girder
A box or tubular girder is a girder that forms an enclosed tube with multiple walls, rather than an or H-beam. Originally constructed of riveted wrought iron, they are now found in rolled or welded steel, aluminium extrusions or pre-stressed concrete....

. In the Chepstow Railway Bridge, the cast iron
Cast iron
Cast iron is derived from pig iron, and while it usually refers to gray iron, it also identifies a large group of ferrous alloys which solidify with a eutectic. The color of a fractured surface can be used to identify an alloy. White cast iron is named after its white surface when fractured, due...

pillars are strengthened by obvious bands of wrought iron
Wrought iron
thumb|The [[Eiffel tower]] is constructed from [[puddle iron]], a form of wrought ironWrought iron is an iron alloy with a very low carbon...

. The vertical, longitudinal force is a compressive force, which cast iron is well able to resist. The hoop stress though is tensile, and so wrought iron, a material with better tensile strength is added.