Catenary calculations

A catenary curve describes the shape the displacement cable takes when subjected to a uniform force such as gravity. This curve is the shape of a perfectly flexible chain suspended by its ends and acted on by gravity. The equation was obtained by Leibniz and Bernoulli in in response to a challenge by Bernoulli and Jacob. The equation of a catenary curve can be derived by examining a very small part of a cable and all forces acting on it see Figure 2. Here h is the sag the cable gets under the action of gravitational force.

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To simplify, we will examine two points on the cable: points 1 and 2. Let the distance between point 1 and 2 be so small, that cable segment is linear. Let dx and dy be projections of section length to X and Y axes respectively. A tightening force is acting at every point of cable. It is directed at a tangent to cable curve and depends only on the coordinates of cable point.

Let P be the weight of cable section Weight is directed downwards, parallel to Y axis. For cable section to be at rest and equilibrium with the rest of cable, forces acting on this section need to balance each other.

The sum of these forces need to equal to zero. Projections of sum of all forces acting at section to X and Y axes should look like formula 1.

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These equations give us the value for cable weight P formula 2. We see from Figure 2 that the ratio of tighting force projections N is found to be a slope ratio of the force N see formula 3. At the same time, cable weight P is cable weight per unit length q mutliplied by differential of arc dS formula 5.

Using formula 2, we can see that first derivative of projecting of tightening force to Y axis can be showed by the differential of arc formula 6. Finally we get formula 10where C1 and C2 are coefficients that are defined by point of origin in concerned system. Hence the equation of cable form looks like formula Please refer to CalQlata's Catenary technical help page for a detailed description of the properties of a catenary.

A single catenary always occupies a single vertical plane it hangs vertically down as its shape is created by the force of gravity Fig 1. The fact that these catenaries are both independent means that the distribution of 'F' between them is not self-evident it is not shared equally in any direction. Should you apply extreme conditions to your catenary before applying 'F': e.

How to Calculate Catenary

As such, applying a force of e. This catenary calculator has been designed to apply a point-load to a predefined unloaded catenary. Owing to the complex nature of this calculation, it is always wise to carry out independent verification of a program's calculations in order to generate confidence in its output data.

Step 1. Check the resolution of 'F' into its three dimensional axis forces by hand as follows:. Step 2. Depending upon whether you prefer to use 'Catenary x,y ' or 'Catenary Fx,Fy ' calculation options in Catenary. Step 3.

Catenary - The Hanging Cable Problem -- Mathematics All Around Us

Depending upon whether you prefer to use Catenary x,y or Catenary Fx,Fy calculation options in Catenary. Step 4. For example:. We have used a hybrid sign convention in order to maximise compatibility with our Catenary calculator. In general:. Catenary Calculator Point Load Fig 1. Catenary Before Applying 'F'. Fig 2. Catenary After Applying 'F'.The catenary curve has a U-like shape, superficially similar in appearance to a parabolic archbut it is not a parabola.

The curve appears in the design of certain types of arches and as a cross section of the catenoid —the shape assumed by a soap film bounded by two parallel circular rings. The catenary is also called the alysoidchainette[1] or, particularly in the materials sciences, funicular.

catenary calculations

Mathematically, the catenary curve is the graph of the hyperbolic cosine function. The surface of revolution of the catenary curve, the catenoidis a minimal surfacespecifically a minimal surface of revolution. A hanging chain will assume a shape of least potential energy which is a catenary. Catenaries and related curves are used in architecture and engineering e. In the offshore oil and gas industry, "catenary" refers to a steel catenary risera pipeline suspended between a production platform and the seabed that adopts an approximate catenary shape.

In the rail industry it refers to the overhead wiring that transfers power to trains. This often supports a lighter contact wire, in which case it does not follow a true catenary curve. In optics and electromagnetics, the hyperbolic cosine and sine functions are basic solutions to Maxwell's equations. The English word "catenary" is usually attributed to Thomas Jefferson[9] [10] who wrote in a letter to Thomas Paine on the construction of an arch for a bridge:.

Differential Equations

It appears to be a very scientifical work. I have not yet had time to engage in it; but I find that the conclusions of his demonstrations are, that every part of the catenary is in perfect equilibrium.

It is often said [12] that Galileo thought the curve of a hanging chain was parabolic. The application of the catenary to the construction of arches is attributed to Robert Hookewhose "true mathematical and mechanical form" in the context of the rebuilding of St Paul's Cathedral alluded to a catenary.

InHooke announced to the Royal Society that he had solved the problem of the optimal shape of an arch, and in published an encrypted solution as a Latin anagram [16] in an appendix to his Description of Helioscopes, [17] where he wrote that he had found "a true mathematical and mechanical form of all manner of Arches for Building.

InGottfried LeibnizChristiaan Huygensand Johann Bernoulli derived the equation in response to a challenge by Jakob Bernoulli ; [12] their solutions were published in the Acta Eruditorum for June Euler proved in that the catenary is the curve which, when rotated about the x -axis, gives the surface of minimum surface area the catenoid for the given bounding circles. Catenary arches are often used in the construction of kilns. To create the desired curve, the shape of a hanging chain of the desired dimensions is transferred to a form which is then used as a guide for the placement of bricks or other building material.

The Gateway Arch in St.The catenary is a plane curve, whose shape corresponds to a hanging homogeneous flexible chain supported at its ends and sagging under the force of gravity. So it was believed for a long time.

However, a rigorous proof was obtained only half a century later after Isaak Newton and Gottfried Leibniz developed a framework of differential and integral calculus. As a result we obtain the differential equation of the catenary :. The order of this equation can be reduced. Integrating once more gives the final nice expression for the shape of the catenary:.

Thus, the catenary is described by the hyperbolic cosine function. Catenaries are often found in nature and technology. For example, the square sail under the pressure of the wind takes the form of a catenary this problem has been considered by Jacob Bernoulli. They have a high stability because the internal compression forces are ideally compensated and do not cause sagging. The catenary has another interesting feature. When revolved about the x -axis, the catenary gives the surface called catenoid.

Catenary Curve (Arch) Graphing Calculator

This surface has minimum surface areai. In particular, the soap film between two circles trying to minimize the free energy takes the form of a catenoid.

catenary calculations

Differential Equations. Figure 1. Figure 4. Example 1 Determine the shape of the cable supporting a suspension bridge. Example 2 Determine the shape of a nonuniform catenary of equal strength. Page 1. Page 2. This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept Reject Read More.

catenary calculations

Necessary Necessary.A Catenary is the natural curve generated by a cable or chain with zero bending stiffness and infinite axial stiffness when strung between two fixed points. Whilst the theory used for the analysis of a Catenary is strictly valid for cables with zero bending stiffness and infinite axial stiffness, you can also use it to obtain approximate configurations in cables that exhibit a resistance to bending, if sufficiently long.

As with many of our calculators you could use the catenary calculator for pre-analysis and optimisation work instead of expensive and time-consuming analytical software. For instance, you could use Catenary to configure closely, say, a small diameter pipeline relative to water depth hung off a vessel or platform in fairly still water limiting the expensive and time-consuming analytical software to final analysis.

Local Properties can be found for any point on the catenary with respect to its anchored end using either calculation option. For help using this calculator, including its internal self-checking facility, see Technical Help.

One end of the cable is anchored and the other end is free to move. Either you apply a force Fx,Fy to the free end or you move it to a specified position relative to the anchor x,y. Catenary will provide the properties of the resulting catenary along with an image of its configuration. Check minimum system requirements. Catenary Calculator v1 Back to product. Calculator Description.Catenaries with different scaling factors Wikimedia Commons A catenary is the shape that a cable assumes when it's supported at its ends and only acted on by its own weight.

It is used extensively in construction, especially for suspension bridges, and an upside-down catenary has been used since antiquity to build arches. The curve of the catenary is the hyperbolic cosine function which has a U shape similar to that of a parabola. The specific shape of a catenary may be determined by its scaling factor.

The scaling factor may be though of as the ratio between the horizontal tension on the cable and the weight of the cable per unit length. A low scaling factor will therefore result in a deeper curve. This equation describes a bouncing spring instead of a hanging cable.

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Louis Arch where the measurements are in units of feet. Eagle Line Toolswww. Delivered On Time. Wind Load Calculationswww. The silk on a spider's web forming multiple elastic catenaries. In physics and geometry, the catenary is the curve that an idealised hanging chain or cable assumes when supported at its ends and acted on only by its own weight.

The curve is the graph of the hyperbolic cosine function, and has a U-like shape, superficially similar in appearance to a parabola though mathematically quite different. Its surface of revolution, the catenoid, is a minimal surface and is the shape assumed by a soap film bounded by two parallel circular rings. Huygens first used the term catenaria in a letter to Leibniz in However, Thomas Jefferson is usually credited with the English word catenary.

A careful reading of his book Two new sciences[5] shows this to be an oversimplification. Galileo discusses the catenary in two places; in the dialog of the Second Day he states that a hanging chain resembles a parabola. But later, in the dialog of the Fourth Day, he gives more details, and states that a hanging cord is approximated by a parabola, correctly observing that this approximation improves as the curvature gets smaller and is almost exact when the elevation is less than 45o.

That the curve followed by a chain is not a parabola was proven by Joachim Jungius — and published posthumously in Some much older arches are also approximate catenaries. InHooke announced to the Royal Society that he had solved the problem of the optimal shape of an arch, and in published an encrypted solution as a Latin anagram[9] in an appendix to his Description of Helioscopes,[10] where he wrote that he had found "a true mathematical and mechanical form of all manner of Arches for Building.

David Gregory wrote a treatise on the catenary in Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. November Hooke discovered that the catenary is the ideal curve for an arch of uniform density and thickness which supports only its own weight. When the centerline of an arch is made to follow the curve of an up-side-down i. Catenary arches are often used in the construction of kilns. In this construction technique, the shape of a hanging chain of the desired dimensions is transferred to a form which is then used as a guide for the placement of bricks or other building material.

The Gateway Arch in St. Louis, Missouri, United States is sometimes said to be an inverted catenary, but this is incorrect. While a catenary is the ideal shape for a freestanding arch of constant thickness, the Gateway Arch is narrower near the top.

According to the U. National Historic Landmark nomination for the arch, it is a "weighted catenary" instead. Its shape corresponds to the shape that a weighted chain, having lighter links in the middle, would form.

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Free-hanging chains follow the catenary curve, but suspension bridge chains or cables do not hang freely since they support the weight of the bridge. In most cases the weight of the cable is negligible compared with the weight being supported.You can reset your password hereAfter submitting this form you'll receive an email with the reset password link. If you still can't access your account please contact our customer service. Play Play Play NowPlay a game against a human or computer opponent LobbyFind other players, chat, and watch games in progress TournamentsCompete for trophies in the ultimate multiplayer challenge LeaderboardSee who's on top and how you compare.

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Catenary curve

After submitting this form you'll receive an email with the reset password link. Sports betting is a very important part of the gambling industry. Oddset is sports betting with fixed odds where you compete against the betting operator. If the predictions are correct the odds are first multiplied with each other and then with the amount of the stake.

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catenary calculations

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In Europe, select betting is not as popular as 1-X-2- betting. The same principle is applied in most Lotto games.


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