quality:corruption, structural:scholar_edition
Model of chains and arches

In the mathematical model the chain (or cord, cable, rope, string, etc.) is idealized by assuming that it is so thin that it can be regarded as a curve and that it is so flexible any force of tension exerted by the chain is parallel to the chain. The analysis of the curve for an optimal arch is similar except that the forces of tension become forces of compression and everything is inverted. An underlying principle is that the chain may be considered a rigid body once it has attained equilibrium. Equations which define the shape of the curve and the tension of the chain at each point may be derived by a careful inspection of the various forces acting on a segment using the fact that these forces must be in balance if the chain is in static equilibrium.

Let the path followed by the chain be given parametrically by r = (x, y) = (x (s), y (s)) where s represents arc length and r is the position vector. This is the natural parameterization and has the property that

where u is a unit tangent vector.

A differential equation for the curve may be derived as follows. Let c be the lowest point on the chain, called the vertex of the catenary. The slope dy / dx of the curve is zero at c since it is a minimum point. Assume r is to the right of c since the other case is implied by symmetry. The forces acting on the section of the chain from c to r are the tension of the chain at c, the tension of the chain at r, and the weight of the chain. The tension at c is tangent to the curve at c and is therefore horizontal without any vertical component and it pulls the section to the left so it may be written (− T, 0) where T is the magnitude of the force. The tension at r is parallel to the curve at r and pulls the section to the right. The tension at r can be split into two components so it may be written T u = (T cos φ, T sin φ), where T is the magnitude of the force and φ is the angle between the curve at r and the x -axis (see tangential angle). Finally, the weight of the chain is represented by (0, − ws) where w is the weight per unit length and s is the length of the segment of chain between c and r.

The chain is in equilibrium so the sum of three forces is 0, therefore

and

and dividing these gives

It is convenient to write

which is the length of chain whose weight is equal in magnitude to the tension at c. Then

is an equation defining the curve.

The horizontal component of the tension, T cos φ = T is constant and the vertical component of the tension, T sin φ = ws is proportional to the length of chain between r and the vertex.