# Dictionary Definition

catenary n : the curve theoretically assumed by a
perfectly flexible and inextensible cord of uniform density and
cross section hanging freely from two fixed points

# User Contributed Dictionary

## English

### Noun

- The curve described by a flexible chain or a rope if it is supported at each end and is acted upon only by no other forces than a uniform gravitational force due to its own weight.
- The curve of an anchor cable from the seabed to the vessel; it should be horizontal at the anchor so as to bury the flukes

#### Translations

curve

- Finnish: ketjukäyrä
- French: chaînette
- German: Kettenlinie
- Swedish: kedjekurva

curve of an anchor cable

# Extensive Definition

In physics, the catenary is the
shape of a hanging
flexible chain or cable when supported at its ends
and acted upon by a uniform gravitational force (its own
weight). The chain is steepest near the points of suspension
because this part of the chain has the most weight pulling down on
it. Toward the bottom, the slope of the chain decreases
because the chain is supporting less weight.

## History

The word catenary is derived from the Latin word catena, which means "chain". The curve is also called the "alysoid", "funicular", and "chainette". Galileo claimed that the curve of a chain hanging under gravity would be a parabola, but this was disproved by Jungius in a work published in 1669.In 1691, Leibniz, Christiaan
Huygens, and Johann
Bernoulli derived the equation in response to a
challenge by Jakob
Bernoulli. Huygens first used the term 'catenaria' in a letter
to Leibniz in 1690, and David
Gregory wrote a treatise on the catenary in 1690. However
Thomas
Jefferson is usually credited with the English word
'catenary'.

The application of the catenary to the
construction of arches is ancient, as
described below; the modern rediscovery and statement is due to
Robert
Hooke, who discovered it in the context of the rebuilding of
St
Paul's Cathedral, possibly having seen Huygen's work on the
catenary. In 1671, Hooke announced
to the Royal
Society that he had solved the problem of the optimal shape of
an arch, and in 1675 published an
encrypted solution as a Latin anagram in an appendix to his
Description of Helioscopes, where he wrote that he had found "a
true mathematical and mechanical form of all manner of Arches for
Building," He did not publish the solution of this anagram in his
lifetime, but in 1705 his executor
provided it as: meaning

## Mathematical Description

### Derivation

To derive the equation for the shape of a catenary in a uniform gravitational field \vec, we begin with the condition of static equilibrium for a link at position s along the catenary; the sum of all forces must be 0.- \vec=F_0+\int_0^s ds'\lambda(s')\hat+\vec(s),

where F_0 is the anchoring force holding up the
catenary at s=0, \lambda(s) is the mass per unit length at position
s along the catenary, and \vec(s) is the tension at s. Taking the
derivative with
respect to s and assuming a constant \lambda(s)=\lambda_0
yields

- \vec=\lambda_0\vec+\frac.

Breaking this into its constituent x and y
components and assuming \vec points in the -y direction
yields

- \frac=0,\qquad\quad\qquad(1)
- \frac=\lambda_0 g,\qquad\qquad(2)

where \theta is the angle from the horizontal x
axis.

From equation (1) we see that \tau\cos=c, where c
is a constant, implies that

- \tau=\frac.

Substituting into equation (2),

- \frac=\frac.\qquad\qquad(3)

Now substituting the arc
length

- ds=\sqrtdx

into equation (3) yields the differential
equation

- \frac=\frac\sqrt,\qquad\qquad(4)

where y'=dy/dx. The solution to equation (4)
is

- y=\frac\cosh+\beta,

where \alpha and \beta are constants to be
determined, along with c, by the boundary
conditions of the problem.

### General Equation

The intrinsic equation of the shape of the catenary with both ends anchored at equal height is given by the hyperbolic cosine function or its exponential equivalent- y = a \cdot \cosh \left ( \right ) = \cdot \left (e^ + e^ \right ),

in which

- a =\frac.

where T_o is the horizontal component of the
tension (a constant) and \lambda is the weight per length
unit.

If you roll a parabola along a straight line, its
focus
traces out a catenary (see roulette).
(The curve traced by one point of a wheel (circle) as it makes one
rotation rolling along a horizontal line is not an inverted
catenary but a cycloid.) Finally, as proved by Euler in 1744, the
catenary is also the curve which, when rotated about the x axis,
gives the surface of minimum surface area
(the catenoid) for the
given bounding circle.

Square
wheels can roll perfectly smoothly if the road has evenly
spaced bumps in the shape of a series of inverted catenary curves.
The wheels can be any regular polygon save for a triangle, but one
must use the correct catenary, corresponding correctly to the shape
and dimensions of the wheels .

A charge in a uniform electric
field moves along a catenary (which tends to a parabola if the charge velocity
is much less than the speed of
light c).

## Suspension bridges

Free-hanging chains follow the curve of the hyperbolic function above, but suspension bridge chains or cables, which are tied to the bridge deck at uniform intervals, follow a parabolic curve, much as Galileo originally claimed (derivation).When suspension bridges are constructed, the
suspension cables initially sag as the catenaric function, before
being tied to the deck below, and then gradually assume a parabolic
curve as additional connecting cables are tied to connect the main
suspension cables with the bridge deck below.

## The inverted catenary arch

The catenary is the ideal curve for an arch which supports only its own weight. When the centerline of an arch is made to follow the curve of an up-side-down (ie. inverted) catenary, the arch endures almost pure compression, in which no significant bending moment occurs inside the material. If the arch is made of individual elements (eg., stones) whose contacting surfaces are perpendicular to the curve of the arch, no significant shear forces are present at these contacting surfaces. (Shear stress is still present inside each stone, as it resists the compressive force along the shear sliding plane.) The thrust (including the weight) of the arch at its two ends is tangent to its centerline.In Antiquity, the
curvature of the inverted catenary was intuitively discovered and
found to lead to stable arches and vaults. A spectacular example
remains in the Taq-i Kisra
in Ctesiphon, which
was once a great city of Mesopotamia. In
ancient Greek and Roman cultures, the less efficient curvature of
the circle was more commonly used in arches and vaults. The
efficient curvature of inverted catenary was perhaps forgotten in
Europe from the fall of Rome to the Middle-Ages and the
Renaissance, where it was almost never used, although the pointed arch was
perhaps a fortuitous approximation of it.

The Catalan architect Antoni
Gaudí made extensive use of catenary shapes in most of his
work. In order to find the best curvature for the arches and ribs
that he desired to use in the crypt of the
Church of Colònia Güell, Gaudí constructed inverted scale
models made of numerous threads under tension to represent stones
under compression. This technique worked well to solve angled
columns, arches, and single-curvature vaults, but could not be used
to solve the more complex, double-curvature vaults that he intended
to use in the nave of the church of the Sagrada
Familia. The idea that Gaudi used thread models to solve the
nave of the Sagrada Familia is a common misconception, although it
could have been used in the solution of the bell towers.

The
Gateway Arch in Saint
Louis, Missouri, United
States follows the form of an inverted catenary. It is 630 feet
wide at the base and 630 feet tall. The exact formula

- y = -127.7 \; \textrm \cdot \cosh() + 757.7 \; \textrm

is displayed inside the arch.

In structural
engineering a catenary shell is a structural form, usually made
of concrete, that
follows a catenary curve. The profile for the shell is obtained by
using flexible material subjected to gravity, converting it into a
rigid formwork for
pouring the concrete and then using it as required, usually in an
inverted manner.

A kiln, a
kind of oven for firing pottery, may be made from
firebricks
with a body in the shape of a catenary arch, usually nearly as wide
as it is high, with the ends closed off with a permanent wall in
the back and a temporary wall in the front. The bricks (mortared
with fireclay) are
stacked upon a temporary form in the shape of an inverted catenary,
which is removed upon completion. The form is designed with a
simple length of light chain, whose shape is traced onto an end
panel of the form, which is inverted for assembly. A particular
advantage of this shape is that it does not tend to dismantle
itself over repeated heating and cooling cycles — most
other forms such as the vertical cylinder
must be held together with steel bands.

## Anchoring of marine vessels

The catenary form given by gravity is made advantage of in its presence in heavy anchor rodes which usually consist mostly of chain or cable as used by ships, oilrigs, docks, and other marine assets which must be anchored to the seabed.Particularly with larger vessels, the catenary
curve given by the weight of the rode presents a lower angle of
pull on the anchor or mooring device. This assists the performance
of the anchor and raises the level of force it will resist before
dragging. With smaller vessels it is less effective.

The catenary curve in this context is only fully
present in the anchoring system when the rode has been lifted clear
of the seabed by the vessel's pull, as the seabed obviously affects
its shape while it supports the chain or cable. There is also
typically a section of rode above the water and thus unaffected by
buoyancy, creating a slightly more complicated curve.

## Towed cables

When a cable is subject to wind or water flows, the drag forces lead to more general shapes, since the forces are not distributed in the same way as the weight. A cable having radius a and specific gravity \sigma , and towed at speed v in a medium (e.g., air or water) with density \rho _, will have an (x,y) position described by the following equations (Dowling 1988):\frac=-\rho _\left( \right) \pi a^g\sin \phi
-\rho _v^\pi aC_\cos \phi ;

- T\frac=-\rho _\left( \right) \pi a^g\cos

\frac=\cos \phi ;

- \frac=-\sin \phi .

Here T is the tension, \phi is the incident
angle, g=9.81 / ^, and s is the cable scope. There are three drag
coefficients: the normal drag coefficient C_ (\approx 1.5 for a
smooth cylindrical cable); the tangential drag coefficient C_
(\approx 0.0025), and C_ (=0.75C_).

The system of equations has four equations and
four unknowns: T, \phi , x and y, and is typically solved
numerically.

### Critical angle tow

Critical angle tow occurs when the incident angle does not change. In practice, critical angle tow is common, and occurs far from significant point forces.Setting \frac=0 leads to an equation for the
critical angle:

\rho _\left( \right) \pi a^g\cos \phi =\rho
_av^\left[ \right] \sin \phi .

If \pi C_, the formula for the critical angle
becomes

\rho _\left( \right) \pi a^g\cos \phi =\rho
_av^^

or

\left( \right) \pi ag\cos \phi =v^^v^ \left(
1-\cos ^\phi \right) ;

or

\cos ^\phi +\frac\cos \phi -1=0;

leading to the rule-of-thumb formula

\cos \phi =-\frac+\sqrt.

The drag coefficients of a faired cable are more
complicated, involving loading functions that account for drag
variation as a function of incidence angle.

## Other uses of the term

- In railway engineering, a catenary structure consists of overhead lines used to deliver electricity to a railway locomotive, multiple unit, railcar, tram or trolleybus through a pantograph or a trolleypole. These structures consist of an upper structural wire in the form of a shallow catenary, short suspender wires, which may or may not contain insulators, and a lower conductive contact wire. By adjusting the tension in various elements the conductive wire is kept parallel to the centerline of the track, reducing the tendency of the pantograph or trolley to bounce or sway, which could cause a disengagement at high speed.
- In semi-rigid airships, a catenary curtain is a fabric and cable internal structure used to distribute the weight of the gondola across a large area of the ship's envelope.
- In conveyor systems, the catenary is the portion of the chain or belt underneath the conveyor that is traveling back to the start. It is the weight of the catenary that keeps tension in the chain or belt.

## References

A.P. Dowling, The dynamics of towed flexible cylinders. Part 2. Negatively buoyant elements (1988). Journal of Fluid Mechanics, 187, 533-571.## External links

- Hanging With Galileo - mathematical derivation of formula for suspended and free-hanging chains; interactive graphical demo of parabolic vs. hyperbolic suspensions.
- Catenary Demonstration Experiment - An easy way to demonstrate the Mathematical properties of a cosh using the hanging cable effect. Devised by Jonathan Lansey
- Horizontal Conveyor Arrangement - Diagrams of different horizontal conveyor layouts showing options for the catenary section both supported and unsupported
- Catenary curve derived - The shape of a catenary is derived, plus examples of a chain hanging between 2 points of unequal height, including C program to calculate the curve.
- Cable Sag Error Calculator - Calculates the deviation from a straight line of a catenary curve and provides derivation of the calculator and references.

catenary in Afrikaans: Kettinglyn

catenary in Catalan: Catenària

catenary in Czech: Řetězovka

catenary in German: Katenoide

catenary in Spanish: Catenaria

catenary in French: Chaînette

catenary in Galician: Catenaria

catenary in Italian: Catenaria

catenary in Hebrew: קו השרשרת

catenary in Hungarian: Láncgörbe

catenary in Dutch: Kettinglijn

catenary in Japanese: カテナリー曲線

catenary in Polish: Krzywa łańcuchowa

catenary in Portuguese: Catenária

catenary in Russian: Цепная линия

catenary in Swedish: Kedjekurva

catenary in Chinese: 悬链线