![Circumferential stress](https://upload.wikimedia.org/wikipedia/commons/1/1f/Circumferential_stress.svg)
In
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
, a cylinder stress is a
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
distribution with rotational symmetry; that is, which remains unchanged if the stressed object is rotated about some fixed axis.
Cylinder stress patterns include:
* circumferential stress, or hoop stress, a normal stress in the tangential (
azimuth
An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north.
Mathematicall ...
) direction.
* axial stress, a normal stress parallel to the axis of cylindrical symmetry.
*
radial stress
Radial stress is stress towards or away from the central axis of a component.
Pressure vessels
The walls of pressure vessels generally undergo triaxial loading. For cylindrical pressure vessels, the normal loads on a wall element are:
* the lon ...
, 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
Tension may refer to:
Science
* Psychological stress
* Tension (physics), a force related to the stretching of an object (the opposite of compression)
* Tension (geology), a stress which stretches rocks in two opposite directions
* Voltage or el ...
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 staves and bound by wooden or metal hoops. The word vat is often used for large containers for liquids, ...
. In a straight, closed
pipe
Pipe(s), PIPE(S) or piping may refer to:
Objects
* Pipe (fluid conveyance), a hollow cylinder following certain dimension rules
** Piping, the use of pipes in industry
* Smoking pipe
** Tobacco pipe
* Half-pipe and quarter pipe, semi-circula ...
, 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 e ...
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 change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
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 the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
.
It is usually useful to
decompose any force applied to an object with
rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
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 algebraic 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 ...
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
The pound per square inch or, more accurately, pound-force per square inch (symbol: lbf/in2; abbreviation: psi) is a unit of pressure or of stress based on avoirdupois units. It is the pressure resulting from a force of one pound-force applied t ...
(psi). Units for ''t'', and ''d'' are inches (in).
SI units for ''P'' are
pascals
The pascal (symbol: Pa) is the unit of pressure in the International System of Units (SI), and is also used to quantify internal pressure, stress, Young's modulus, and ultimate tensile strength. The unit, named after Blaise Pascal, is defined as ...
(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:
:
However, 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: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...
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.
:
:
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 hoop stress is obtained:
:
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 the causes and effects of disease or injury. 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 ...
of
vascular
The blood vessels are the components of the circulatory system that transport blood throughout the human body. These vessels transport blood cells, nutrients, and oxygen to the tissues of the body. They also take waste and carbon dioxide away f ...
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 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 be a nidus (s ...
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 medicine or biology, a diverticulum is an outpouching of a hollow (or a fluid-filled) structure in the body. Depending upon which layers of the structure are involved, diverticula are described as being either true or false.
In medicine, t ...
in the
gut.
[E. Goljan, ''Pathology, 2nd ed.'' Mosby Elsevier, Rapid Review Series.]
Historical development of the theory
![Wrought iron straps, Chepstow Railway Bridge](https://upload.wikimedia.org/wikipedia/commons/d/d5/Wrought_iron_straps%2C_Chepstow_Railway_Bridge.jpg)
The first theoretical analysis of the stress in cylinders was developed by the mid-19th century engineer
William Fairbairn
Sir William Fairbairn, 1st Baronet of Ardwick (19 February 1789 – 18 August 1874) was a Scottish civil engineer, structural engineer
Structural engineers analyze, design, plan, and research structural components and structural systems ...
, assisted by his mathematical analyst
Eaton Hodgkinson
Eaton A. Hodgkinson FRS (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, Ch ...
. Their first interest was in studying the design and
failure
Failure is the state or condition of not meeting a desirable or intended objective (goal), objective, and may be viewed as the opposite of Success (concept), success. The criteria for failure depends on context, and may be relative to a parti ...
s of
steam boiler
Steam is a substance containing water in the gas phase, and sometimes also an aerosol of liquid water droplets, or air. This may occur due to evaporation or due to boiling, where heat is applied until water reaches the enthalpy of vaporization. ...
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 to the head is called the ''tail''. On installation, the rivet is placed in a punched o ...
. Later work was applied to bridge-building and the invention of the
box girder. In the
Chepstow Railway Bridge
Chepstow Railway Bridge was built to the instructions of Isambard Kingdom Brunel in 1852. The "Great Tubular Bridge" over the River Wye at Chepstow, which at that point forms the boundary between Wales and England, is considered one of Brunel' ...
, the
cast iron
Cast iron is a class of iron–carbon alloys with a carbon content more than 2%. Its usefulness derives from its relatively low melting temperature. The alloy constituents affect its color when fractured: white cast iron has carbide impuriti ...
pillars are strengthened by external bands of
wrought iron
Wrought iron is an iron alloy with a very low carbon content (less than 0.08%) in contrast to that of cast iron (2.1% to 4%). It is a semi-fused mass of iron with fibrous slag Inclusion (mineral), inclusions (up to 2% by weight), which give it a ...
. 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
The Great Molasses Flood, also known as the Boston Molasses Disaster, was a disaster that occurred on January 15, 1919, in the North End neighborhood of Boston, Massachusetts.
A large storage tank filled with of molasses, weighing approximat ...
**
Boiler explosion
**
Boiling liquid expanding vapor explosion
A boiling liquid expanding vapor explosion (BLEVE, ) is an explosion caused by the rupture of a vessel containing a pressurized liquid that has reached temperature above its boiling point. Because the boiling point of a liquid rises with p ...
* Related engineering topics:
**
Stress concentration
In solid mechanics, a stress concentration (also called a stress raiser or a stress riser) is a location in an object where the stress is significantly greater than the surrounding region. Stress concentrations occur when there are irregularitie ...
**
Hydrostatic test
A hydrostatic test is a way in which pressure vessels such as 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, usually water, wh ...
**
Buckling
**
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 uses stored rocket propellants as the reaction mass for forming a high-speed propulsive jet of fluid, usually high-temperature gas. Rocket engines are reaction engines, producing thrust by ejecting mass rearward, in accordanc ...
**
Flywheel
A flywheel is a mechanical device which 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, ass ...
** The dome of
Florence Cathedral
Florence Cathedral, formally the (; in English Cathedral of Saint Mary of the Flower), is the cathedral of Florence, Italy ( it, Duomo di Firenze). It was begun in 1296 in the Gothic style to a design of Arnolfo di Cambio and was structurally c ...
References
*
{{Refimprove, date=March 2012
Mechanics