Catenary

Kette Kettenkurve Catenary 2008 PD

PylonsSunset-5982

SpiderCatenary

GaudiCatenaryModel

Analogy between an arch and a hanging chain and comparison to the dome of St Peter's Cathedral in Rome

Catenary is the term used to describe the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends. The word "catenary" is derived from the Latin word catena, which means "chain". This curve has a distinct U-shaped profile, distinct from the parabolic shape that is often assumed for similar problems. The catenary curve is significant in the fields of engineering, architecture, and physics, due to its properties and the efficiency it provides in various structures.

Definition

The mathematical description of a catenary curve is given by the hyperbolic cosine function: \(y = a \cosh\left(\frac{x}{a}\right)\), where \(a\) is a constant that determines the curve's steepness and \(\cosh\) is the hyperbolic cosine function. The parameter \(a\) is related to the physical properties of the chain, including its weight per unit length and the horizontal tension at the lowest point of the curve.

History

The study of the catenary curve dates back to the 17th century, with significant contributions from mathematicians such as Galileo Galilei, Johann Bernoulli, and Christiaan Huygens. Initially, Galileo mistakenly believed that the curve of a hanging chain was a parabola. However, it was later proven by Huygens and Bernoulli that the actual shape is a hyperbolic cosine function, marking an important development in the understanding of this phenomenon.

Applications

The catenary curve finds applications in various engineering and architectural projects. One of the most notable examples is in the design of arches and bridges, where the catenary shape can be inverted to provide an optimal load-bearing structure. This principle is evident in the design of the famous Gateway Arch in St. Louis, Missouri, which is a monumental example of a catenary arch.

In the field of electrical engineering, overhead power lines are often suspended in a catenary curve, which allows for efficient transmission of electricity over long distances with minimal sagging. Similarly, the design of suspension bridges, such as the Golden Gate Bridge, incorporates catenary principles to ensure stability and structural integrity.

Mathematical Properties

The catenary curve possesses several unique mathematical properties. It is the graph of the hyperbolic cosine function, which makes it symmetric about the vertical axis. The curve also represents the solution to the problem of the shape that minimizes potential energy for a flexible chain of uniform density and cross-section, hanging under gravity.

See Also

• Hyperbolic function
• Suspension bridge
• Arch (architecture)
• Structural engineering

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