In
mechanics
Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ...
, a cylinder stress is a
stress distribution with rotational symmetry; that is, which remains unchanged if the stressed object is rotated about some fixed axis.
Cylinder
A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infinite ...
stress patterns include:
* circumferential stress, or hoop stress, a normal stress in the tangential (
azimuth
An azimuth (; from ) is the horizontal angle from a cardinal direction, most commonly north, in a local or observer-centric spherical coordinate system.
Mathematically, the relative position vector from an observer ( origin) to a point ...
) direction.
* axial stress, a normal stress parallel to the axis of cylindrical symmetry.
*
radial stress, a normal stress in directions coplanar with but perpendicular to the symmetry axis.
These three principal stresses- hoop, longitudinal, and radial can be calculated analytically using a mutually perpendicular tri-axial stress system.
The classical example (and namesake) of hoop stress is the
tension applied to the iron bands, or hoops, of a wooden
barrel
A barrel or cask is a hollow cylindrical container with a bulging center, longer than it is wide. They are traditionally made of wooden stave (wood), staves and bound by wooden or metal hoops. The word vat is often used for large containers ...
. In a straight, closed
pipe, any force applied to the cylindrical pipe wall by a
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
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 considered.
In thick-walled pressure vessels, construction techniques allowing for favorable initial stress patterns can be utilized. These compressive stresses at the inner surface reduce the overall hoop stress in pressurized cylinders. Cylindrical vessels of this nature are generally constructed from concentric cylinders shrunk over (or expanded into) one another, i.e., built-up shrink-fit cylinders, but can also be performed to singular cylinders though autofrettage of thick cylinders.
Definitions
Hoop stress

The hoop stress is the force over area exerted circumferentially (perpendicular to the axis and 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
In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
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 in cylindrical
coordinates
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as ...
.
It is usually useful to
decompose
Decomposition is the process by which dead organic substances are broken down into simpler organic or inorganic matter such as carbon dioxide, water, simple sugars and mineral salts. The process is a part of the nutrient cycle and is essen ...
any force applied to an object with
rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape (geometry), shape has when it looks the same after some rotation (mathematics), rotation by a partial turn (angle), turn. An object's degree of rotational s ...
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.
Relation to internal pressure
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 Diameter / t > 20) of its radius. This allows for treating the wall as a surface, and subsequently using the
Young–Laplace equation
In physics, the Young–Laplace equation () is an 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 tensi ...
for estimating the hoop stress created by an internal pressure on a thin-walled cylindrical pressure vessel:
:
(for a cylinder)
:
(for a sphere)
where
*''P'' is the internal pressure
*''t'' is the wall thickness
*''r'' is the mean 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
Turgor pressure is the force within the cell that pushes the plasma membrane against the cell wall.
It is also called ''hydrostatic pressure'', and is defined as the pressure in a fluid measured at a certain point within itself when at equilibri ...
may reach several atmospheres. In practical engineering applications for cylinders (pipes and tubes), hoop stress is often re-arranged for pressure, and is called
Barlow's formula.
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 (Pa), while ''t'' and ''d''=2''r'' 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
:
There is also a radial stress
that is developed perpendicular to the surface and may be estimated in thin walled cylinders as:
:
In the thin-walled assumption the ratio
is large, so in most cases this component is considered negligible compared to the hoop and axial stresses.
Thick-walled vessels
When the cylinder to be studied has a
ratio of less than 10 (often cited as
) the thin-walled cylinder equations no longer hold since stresses vary significantly between inside and outside surfaces and
shear stress
Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...
through the cross section can no longer be neglected.
These stresses and strains can be calculated using the ''Lamé equations'', a set of equations developed by French mathematician
Gabriel Lamé
Gabriel Lamé (22 July 1795 – 1 May 1870) was a French mathematician who contributed to the theory of partial differential equations by the use of curvilinear coordinates, and the mathematical theory of elasticity (for which linear elasticity ...
.
:
:
where:
:
and
are constants of integration, which may be found from the boundary conditions,
:
is the radius at the point of interest (e.g., at the inside or outside walls).
For cylinder with boundary conditions:
:
(i.e. internal pressure
at inner surface),
:
(i.e. external pressure
at outer surface),
the following constants are obtained:
:
,
:
.
Using these constants, the following equation for radial stress and hoop stress are obtained, respectively:
:
,
:
.
Note that when the results of these stresses are positive, it indicates tension, and negative values, compression.
For a solid cylinder:
then
and a solid cylinder cannot have an internal pressure so
.
Being that for thick-walled cylinders, the ratio
is less than 10, the radial stress, in proportion to the other stresses, becomes non-negligible (i.e. P is no longer much, much less than Pr/t and Pr/2t), and so the thickness of the wall becomes a major consideration for design (Harvey, 1974, pp. 57).
In pressure vessel theory, 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.
In thick-walled cylinders, the maximum shear stress at any point is given by half of the algebraic difference between the maximum and minimum stresses, which is, therefore, equal to half the difference between the hoop and radial stresses. The shearing stress reaches a maximum at the inner surface, which is significant because it serves as a criterion for failure since it correlates well with actual rupture tests of thick cylinders (Harvey, 1974, p. 57).
Practical effects
Engineering
Fracture is governed by the hoop stress in the absence of other external loads since it is the largest principal stress. Note that a hoop experiences the greatest stress at its inside (the outside and inside experience the same total strain, which is distributed over different circumferences); hence 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 absent.
Medicine
In the
pathology
Pathology is the study of disease. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in the context of modern medical treatme ...
of
vascular Vascular can refer to:
* blood vessels, the vascular system in animals
* vascular tissue
Vascular tissue is a complex transporting tissue, formed of more than one cell type, found in vascular plants. The primary components of vascular tissue ...
or
gastrointestinal walls, the wall tension represents the
muscular tension on the wall of the vessel. As a result of the
Law of Laplace, if an
aneurysm
An aneurysm is an outward :wikt:bulge, bulging, likened to a bubble or balloon, caused by a localized, abnormal, weak spot on a blood vessel wall. Aneurysms may be a result of a hereditary condition or an acquired disease. Aneurysms can also b ...
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.
[E. Goljan, ''Pathology, 2nd ed.'' Mosby Elsevier, Rapid Review Series.]
Theory development
The first theoretical analysis of the stress in cylinders was developed by the mid-19th century engineer
William Fairbairn, assisted by his mathematical analyst
Eaton Hodgkinson
Eaton Hodgkinson (26 February 1789 – 18 June 1861) was an English engineer, a pioneer of the application of mathematics to problems of structural design.
Early life
Hodgkinson was born in the village of Anderton, near Northwich, Cheshire, ...
. Their first interest was in studying the design and
failure
Failure is the social concept of not meeting a desirable or intended objective, and is usually viewed as the opposite of success. The criteria for failure depends on context, and may be relative to a particular observer or belief system. On ...
s of
steam boiler
file:Dampfkessel für eine Stationärdampfmaschine im Textilmuseum Bocholt.jpg, An industrial boiler, originally used for supplying steam to a stationary steam engine
A boiler or steam generator is a device used to create steam by applying heat ...
s.
Fairbairn realized that the hoop stress was twice the longitudinal stress, an important factor in the assembly of boiler shells from rolled sheets joined by
riveting
A rivet is a permanent mechanical fastener. Before being installed, a rivet consists of a smooth cylindrical shaft with a head on one end. The end opposite the head is called the ''tail''. On installation, the deformed end is called the ''s ...
. Later work was applied to bridge-building and the invention of the
box girder
A box girder or tubular girder (or box beam) is a girder that forms an enclosed tube with multiple walls, as opposed to an i-beam, - or H-beam. Originally constructed of wrought iron joined by riveting, they are now made of rolled steel, rolled ...
. In the
Chepstow Railway Bridge, the
cast iron
Cast iron is a class of iron–carbon alloys with a carbon content of more than 2% and silicon content around 1–3%. Its usefulness derives from its relatively low melting temperature. The alloying elements determine the form in which its car ...
pillars are strengthened by external bands of
wrought iron
Wrought iron is an iron alloy with a very low carbon content (less than 0.05%) in contrast to that of cast iron (2.1% to 4.5%), or 0.25 for low carbon "mild" steel. Wrought iron is manufactured by heating and melting high carbon cast iron in an ...
. The vertical, longitudinal force is a compressive force, which cast iron is well able to resist. The hoop stress is tensile, and so wrought iron, a material with better tensile strength than cast iron, is added.
See also
* Can be caused by cylinder stress:
**
Boston Molasses Disaster
**
Boiler explosion
A boiler explosion is a catastrophic failure of a boiler.
There are two types of boiler explosions. One type is a failure of the pressure parts of the steam and water sides. There can be many different causes, such as failure of the safety val ...
**
Boiling liquid expanding vapor explosion
A boiling liquid expanding vapor explosion (BLEVE, ) is an explosion caused by the rupture of a Pressure vessel, vessel containing a Compressed fluid, pressurized liquid that has attained a temperature sufficiently higher than its boiling po ...
* Related engineering topics:
**
Stress concentration
In solid mechanics, a stress concentration (also called a stress raiser or a stress riser or notch sensitivity) is a location in an object where the stress (mechanics), stress is significantly greater than the surrounding region. Stress concentra ...
**
Hydrostatic test
A hydrostatic test is a way in which pressure vessels such as Pipeline transport, pipelines, plumbing, gas cylinders, boilers and fuel tanks can be tested for strength and leaks. The test involves filling the vessel or pipe system with a liquid ...
**
Buckling
In structural engineering, buckling is the sudden change in shape (Deformation (engineering), deformation) of a structural component under Structural load, load, such as the bowing of a column under Compression (physics), compression or the wrin ...
**
Blood pressure#Relation_to_wall_tension
**
Piping#Stress_analysis
* Designs very affected by this stress:
**
Pressure vessel
A pressure vessel is a container designed to hold gases or liquids at a pressure substantially different from the ambient pressure.
Construction methods and materials may be chosen to suit the pressure application, and will depend on the size o ...
***
Rocket engine
A rocket engine is a reaction engine, producing thrust in accordance with Newton's third law by ejecting reaction mass rearward, usually a high-speed Jet (fluid), jet of high-temperature gas produced by the combustion of rocket propellants stor ...
**
Flywheel
A flywheel is a mechanical device that uses the conservation of angular momentum to store rotational energy, a form of kinetic energy proportional to the product of its moment of inertia and the square of its rotational speed. In particular, a ...
** The dome of
Florence Cathedral
Florence Cathedral (), formally the Cathedral of Saint Mary of the Flower ( ), is the cathedral of the Catholic Archdiocese of Florence in Florence, Italy. Commenced in 1296 in the Gothic style to a design of Arnolfo di Cambio and completed b ...
References
*
{{Refimprove, date=March 2012
Mechanics