## Introduction

Towlines connect a tug to the vessel being towed and are defined by multiple characteristics like Weight, Diameter, and Stiffness. The tension in the towline during the towing operation is not static but keeps varying with the distance between the tug and the vessel being towed. It is quite obvious that when the distance between the vessels is high, the tension shall be higher, and vice versa.

When the tug and the vessel are closer, the towline becomes slack and a sag in the towline is expected. The sag is also caused due to the self-weight of the towline, thus leading to the formation of a ‘catenary with a sag’ in the towline, instead of a straight line.

Towlines have a significant weight of their own, and due to their self-weight, the towline may not be in the shape of a straight line when in use. Despite the tension in the towline that makes the towline straight, the self-weight of the towline makes it sag. This sagging is not a desirable feature if the water depth available is low, as the towline may touch the seabed which will lead to its wear.

In this article we’ll explore a theoretical method to calculate the catenary and sag of a towline. It is to be noted that this method is highly simplified in approach, and assumes the towline as a single rope with uniform weight and cross section. However, in reality, the end of the towline which connects with the towed vessel carries the towing bridle which makes that end heavier and makes the catenary asymmetric. The complication arising out of the towing bridle is not covered in the current article, and will be a subject of future articles.

## The sag of a Towline

Since the tension in the towline is a dynamically changing parameter, so the sag of the towline also keeps changing during the course of towing. A sudden or temporary increase in the sag of the towline may lead to sudden sag in the towline which may lead to fouling of the towline or dragging on the seabed. This is an undesirable situation, and so it is important to be able to calculate the sag in the towline by knowing the parameter of the tow.

Basically, the sag will depend upon the following parameters:

- The tension in the towline
- The length of the towline
- The self-weight of the towline

The figure above can be used to represent the catenary of a towline. We can see the following important parameters form the above figure:

- The tension in the towline is T
_{0}. This is a variable parameter and is dynamically changing throughout the towing operation. A higher tension in the towline will lower the sag - L – is the length of the towline. The towline length should be sufficient to maintain a safe distance between the tug and the vessel towed. As the distance between the two vessels changes, so does the sag. A smaller distance between the two vessels will lead to greater sag.
- Z – this is the sag in the towline at any point along its length. Z
_{m}is the maximum sag which is experienced at the mid-towline-length - s – this is the coordinate measured along the length of the towline (and not along the X-axis). It is to be noted that s will vary from -L/2 to +L/2 along the length of the towline

## Defining a Catenary

As we can see in the figure above, the catenary is characterized by two coordinates:

- X-coordinate – runs along the horizontal axis, and measures the distance between the tug and the towed vessel. X is measured not along the catenary’s length, but along the horizontal
- Z-coordinate – runs along the vertical axis, and measures the depth of the catenary at any location X

While the value of ‘s’, the coordinate along the towline, varies from -L/2 to +L/2, the range of the values of ‘x’ will depend upon the tension in the towline, T_{0}. At higher tensions, the catenary is flatter and the range of ‘x’ will be higher. At lower tensions, the catenary is deeper and the range of ‘x’ is lower. In the ideal case, when T0 is infinite, the range of ‘x’ becomes equal to L (length of the towline).

The catenary will be plotted with a range of x(s) and z(s) values, with each coordinate being a function of the catenary length ‘s’. DNV-RP-H-103 provides the formulations for each coordinate in Sec 7.2.3. These are presented below. These are applicable when the tension of the cable is much larger than the weight of the cable (and gives good estimates even when they are comparable).

From the above formula, we can see that it depends on the following additional parameters:

- w –
**submerged**weight per unit length of the towline. Here, we emphasize the word ‘submerged’. This is the weight in water while factoring for the buoyancy which the towline will experience inside water. To calculate ‘w’, we use a simple formulation:

w = weight of towline in air – buoyancy of towline inside water

The weight of the towline in air is obtained from the towline’s specifications. If the mass per unit length is ‘m’, then the weight per unit length is (m x g), where ‘g’ is the acceleration due to gravity.

To get the buoyancy of the towline inside water, we follow the fundamental principle of calculating buoyancy: Buoyancy = volume of submerged body x density of water

How do we calculate the submerged volume of the towline? Since it is cylindrical in shape, the formula for volume is

**Volume = Section area x Length**

The volume per unit length will equal to the section area of the towline.

Let’s take a look at the typical section of a towline. It is shown below. We can see that the section area is not fully circular, as it is made of individual strands of ropes put together. Thus, no the entire area of the circumscribing circle is filled up with steel ropes. The percentage of the area of the circumscribed circle which is actually filled with steel ropes is called the ‘Fill Factor’. The fill factor C_{F} can be obtained from the towline manufacturer, or by looking into the towing rope specifications

The net area of the steel rope can be found out as

**A = C _{F} x S**

Where C_{F} is the fill factor, and S is the total circumscribed area. ‘A’ is also the submerged volume per unit length of the towline.

Thus, the buoyancy of the submerged towline is given by

**Buoyancy = ρ _{w} x A x g = ρ_{w} x C_{F} x S x g**

The submerged weight per unit length is given by

w = (m – ρ_{w} x C_{F} x S) x g

- EA – The term ‘EA’ is the Modulus of Elasticity of the towline multiplied by the nominal sectional area of towline (A).

Modulus of elasticity can be measured in units Pa, MPa, or GPa. The modulus of elasticity of a steel wire towline is typically 40% of that of solid steel. The Modulus of Elasticity of solid steel is 206000 MPa, and the value for steel towline generally comes to 80000 MPa.

## Plotting the Catenary Curve

Once we have all the inputs with us, we can use the catenary equations provided above to plot the catenary of the towline at any given tension T_{0}.

This requires calculating x(s) and z(s) at different values of s. The value of ‘s’ ranges from 0 to L (L = length of towline), and we can tabulate the values of s, x(s), and z(s) at fixed intervals of s. It may look like below:

We can see from above that the catenary properties are listed in the table on the left, while catenary coordinates are listed in the table on right.

The catenary can also be plotted in Excel using the coordinates above, and may look like below:

## Maximum Sag

The most interesting data from the catenary curve above is the maximum sag in the catenary, which is at the mid-catenary. The maximum sag is the most critical parameter to know for safe towing operation. Situations of towline dragging on the sea bed will lead to fouling of the towline and can be avoided by knowing the maximum sag.

It is quite obvious that the sag in the towline will vary depending on the towline tension. The higher the tension, the lower is the sag, and vice versa. The formula for maximum sag in the towline is given by below:

Since all parameters except T_{0} are fixed, we can vary T_{0} to see how it affects the maximum sag. The minimum tension is for the case when the towline is just hanging under its own weight, and the maximum tension shall be when the towline tension is equal to the maximum bollard pull of the tug. By varying T_{0} between these two values, we can calculate z_{m} using the formula above, and put it in a tabular form (see below). A graph can also be plotted to demonstrate how the maximum sag varies with the towline tension

That brings us to the end of this article. We have seen how the catenary of a towline can be computed, and also how the maximum sag can be computed. These calculations are useful in knowing the towline clearance from the seabed, and also to ensure a safe tow.

## References

- US Navy Towing Manual
- DNV RH-H-103, 2011

https://www.thenavalarch.com/towline-catenary-calculator/

*Disclaimer: This post is not meant to be authoritative writing on the topic presented. thenavalarch bears no responsibility for the accuracy of this article, or for any incidents/losses arising due to the use of the information in this article in any operation. It is recommended to seek professional advice before executing any activity which draws on information mentioned in this post. All the figures, drawings, and pictures are property of thenavalarch except where indicated, and may not be copied or distributed without permission.*

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