Enter any 3 of the horizontal force, distance between supports, weight per unit length, and sag into the calculator to determine the missing catenary value (assumes supports are at the same height and the load is the cable/chain’s own uniform weight).

Catenary Sag Calculator (Exact)

Enter any 3 values to calculate the missing variable

Related Calculators

Catenary Equation Formula

The following formula is used to calculate the midspan sag of a true catenary for a cable/chain with uniform weight per unit length (along the cable) hung between supports at the same height.

\begin{aligned}
S &= \frac{H}{w}\left(\cosh\left(\frac{w\,d}{2H}\right)-1\right) \\
&\approx \frac{w\,d^2}{8H}\quad(\text{small-sag approximation})
\end{aligned}

Variables:

  • S is the sag (vertical drop) from the support level to the lowest point of the cable (unit of length)
  • w is the weight per unit length of the cable/chain (force per unit length, assumed uniform)
  • d is the horizontal distance between the supports (length)
  • H is the constant horizontal component of the cable tension (force)

For an exact catenary, first compute a = H / w, then compute the sag using S = a(\cosh(d/(2a)) – 1). For small sag (when the curve is shallow), a common approximation is S \approx (w d^2)/(8H).

What is a Catenary Curve?

A catenary curve describes the shape of a flexible chain or cable that hangs between two points under its own weight and is acted upon by gravity. The exact catenary shape is described using hyperbolic cosine functions (cosh), and it is relevant in various engineering and architectural applications, such as the design of suspension bridges and overhead power lines. Understanding sag is important for clearance, tension, and overall structural performance.

How to Calculate Sag in a Catenary Curve?

The following steps outline how to calculate the sag in a Catenary Curve.

  1. First, determine the horizontal force (H), i.e., the constant horizontal component of the cable tension.
  2. Next, determine the horizontal distance between the supports (d).
  3. Next, determine the weight per unit length (w) of the cable or chain (assumed uniform).
  4. Next, use the exact catenary relations: compute a = H / w, then compute S = a(\cosh(d/(2a)) – 1).
  5. Finally, calculate the sag (S) in the catenary curve.
  6. After inserting the variables and calculating the result, check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

horizontal force (H) = 500 N

distance between supports (d) = 100 m

weight per unit length (w) = 2 N/m (exact catenary sag S ≈ 5.0167 m)