# Part1: Result and summary

## Length of anchor rode as a function of depth and wind force

The purpose of the rode is to connect the anchor to the stem-head in such a way that the anchor can give of its best and not be plucked off the bottom by the forces needed to hold the boat stationary. The standard 'rules of thumb' are obviously wrong: a moment's thought shows that 5 x depth for instance can't possibly be correct, and besides the MCA recommendation for commercial vessels prescribes letting out chain proportional to the square-root of depth. Whence this divergence between recreational and commercial practice?

A correct relationship must have at least the following characteristics:
• as depth tends to infinity the first derivative of length with respect to depth must tend to 1
• it must be an explicit function of wind force
• it must vary with weight of chain, and proportion of chain and rope

This document derives the formulae and plots graphs of the minimum amount of chain or rope which is sufficient given the depth of water, taking into account the weight of the chain, type of anchor and the wind strength. The results are far more rigorous than the traditional 'rules of thumb' and can be applied over a wider range of vessel types, depths and conditions.

The default parameters used are those for the author's boat, a 12.8m (42 feet) heavy monohull, but the parameters can be changed and the graphs replotted for any other boat using the 'Update Graph' button below.

The horizontal forces as a function of wind force are derived from the ABYC standard formulae for mono-hulled sailing yachts, with corrections made for the typical windage of catamarans and power boats. See the web site of Alain Fraysse for a rationale. The forces given by this formula have also been verified by measurement as correct for the author's boat at least.

The author would be grateful if anyone who spots an error in his maths or JavaScript would let him know, and hopes the document is useful, informative or interesting, but makes no warranty that the results are correct: the mariner must at all times rely on his or her own judgement. As an aside, it should be noted that the mathematics contained herein is not new, it's just hoped that the presentation is in a format convenient for practical use.

Figure 1: Rode length (in m) on y axis, Water depth (in m) on x axis, Wind in Beaufort force

Chain and anchor parameters:

 Size of chain Weight Length of chain Allowable angle of chain at anchor 9.5 mm 6.5 mm 8.0 mm 10 mm 11 mm 12.5 mm 13 mm 14 mm 16 mm 19 mm kg/m metres °

Boat parameters: forces in daN, wind in Beaufort scale

 Type of boat LOA in meters Force 4 Force 6 Force 8 Force 10 Cruising monohull Low windage monohull Catamaran Motor Cruiser

# Part 2: Mathematics

## The catenary

Consider the forces acting on a small element of chain, length δs. The horizontal force is the same along the whole chain since the only horizontal forces acting are equal and opposite ones at the anchor and at the stem-head. The vertical force is entirely due to the weight of the chain: thus it will be zero at the anchor (provided the chain is horizontal there) and at any point will be equal to the weight of the chain between this point and the anchor. In the equations below Fx, is the (constant) horizontal component of force, Fy the vertical component of force and m the weight (in water) of chain per metre. This is the familar catenary equation and is plotted in

Figure 2: Shape taken by chain in a variety of wind strengths The four coloured lines represent the curve of the chain for a variety of forces and weights of chain. Note that even in F10 the chain is not completely straight and in principle if one had enough it could be made horizontal at the anchor

## Length as a function of depth

The catenary equation above relates x and y, ie vertical and horizontal distances, but we want to relate y and s, ie depth and length. We can rearrange the above equations to suit: This can immediately be seen to have the right characteristics: it has the correct dimensions (of length), the derivative with respect to depth approaches 1 for very deep water and it calls for proportionally greater scope in shallow waters and / or high winds. Using this equation we could plot immediately the relationship between depth and length of rode as a function of wind force, under the assumption of all-chain rode. To see what this looks like, make length of chain very large, eg 100m, angle at anchor zero and click Update Graph

## Mixed chain and rope

However most people have only a limited amount of chain, 60m in my case, so if I need to let out more scope it has to be rope. As an aside let me point out here that having some rope rather than an all-chain rode is beneficial anyway in very strong winds because it provides some stretch which can damp out snubbing forces. The angle at the top of the chain is a very simple formula which can be calculated from the previous equation for p: In the graphs to follow I have assumed 60m of chain and the rest rope. The curves can be recalculated for any mix of chain and rope, and for any weight of chain. If you look at them carefully you can see that they are curved for length below 60m and straight lines for greater lengths.

## Effect of anchor weight and type

The foregoing has assumed that the chain at the anchor has to be horizontal. But actually this isn't quite true: in reality anchors can tolerate some upwards pull, and a heavy anchor obviously more than a lighter one (an extreme case of a heavy anchor is a mooring, for which clearly one uses a very short scope). In addition, modern anchors have been conceived to be particularly tolerant of some lifting of the rode at the shank of the anchor - much more so than a previous generation. Nonetheless the upwards pull on the anchor has to be limited or it will gradually work out of the bottom. The allowed departure from horizontal can be entered as an angle. It's usually only a few degrees, upto about 5°. If set to 0 then the pull has to be horizontal, and increasingly larger values imply that one can tolerate more upwards pull: Suggested values are 5° for a modern style such as a Rocna, Spade or Manson, 2 or 3° for slightly older anchors such as a Delta and 0° for the CQR.

The graphs are actually quite affected by the value of this angle; changing it between 0 and 5° can be seen to make significant difference. However this is borne out be a wealth of user's experience. Note that this is not to state that there is necessarily a difference to holding power, but rather that there are differences in the amount of scope required for a particular anchor type to achieve its best performance, whatever that may be.

## Practical Length / Depth curves

Putting all the above factors together gives the curves here:

Figure 3: Practical length - depth curves This is the graph the author has stuck to the inside of his chart table.

## Comparison with rules of thumb

These curves can be compared with L = 10m + 3 x depth and L = 5 x depth formulae which are overlaid, in purple and yellow respectively, on the calculated curves. Neither of these are very good approximations even if one takes maximum wind strength to be between F4 and F6 as typical. They suggest one lets out too little chain in shallow water and too much in deep. They also show that anchor trials are usually dubious because inappropriate scope is used for the forces being applied, which is one factor in their rather random and possibly misleading results.

Figure 4: Comparison with rules of thumb ## Critical Angle

A new result - at least the author has never seen this before - is that one can easily tell if one has let out enough scope just by measuring the angle of the rode at the stem-head. The result (ie the answer to the question "have I let out enough for the actual conditions?") is independant of the boat and of the weight of chain, and so is universal. This slightly unexpected result comes solely from the geometry of the catenary, which is fully determined if one knows the scope that is already out and the water depth over the anchor.

Returning to the formula for angle we can re-arrange it to eliminate λ by substitution. This formula is plotted below, where the critical angle θ' is the angle from the vertical.

Figure 5: Critical angle from vertical How to use this graph

Consider what happens to the rode as wind increases from zero: At first the chain hangs vertically from the stem-head and the rest of the chain lies on the bottom. As the wind increases the force on the boat pulls the boat back and the rode will no longer be vertical, more chain being off the bottom, but much of it nonetheless still lying there. As the wind becomes stronger the angle of the rode at the stem head will become less and less vertical, and at some point the force will be just enough to lift all the chain. The angle of the chain at the stem head at this point is the Critical Angle. For forces higher than this the force on the anchor will no longer be horizontal and the angle at the stem-head gets less and less vertical until in the limit the chain becomes bar taut (or, more likely, the anchor drags).

Method 1. Measure the actual angle and, knowing the depth, use the graph to look up on the y-axis the length of rode. This figure is the minimum length for the actual conditions such that the force on the anchor will be horizontal.

Method 2. Knowing the depth and the amount of rode you have let out look up the critical angle. Now compare that to the actual angle; if the rode is hanging more vertically than this critical angle there is enough scope let out. Conversely if the actual angle is greater (ie more horizontal) than the critical angle you should let out more scope.

Since angle is easily measured this provides a trivial way to determine if one has let out enough scope given the actual conditions and actual depth. Note that this depends on the geometry alone and so always works: it is the same for all boats and weights of chain and depends solely on the mix of chain and rope.

## All chain versus mixed rode

The question "is it better to have an all-chain or a mixed rode?" can be answered using the interactive graph plotted above by varying the weight of chain and the length of chain while keeping the total weight constant: on the author's boat that is 120kg made up of 60m of chain weighing 2kg/m. Trying this, varying the weight between 1 and 4 kg/m and seeing the scope required for 10m water depth gives:

Chain weight    Chain length    Rode length for F6    Rode length for F8
1kg/m 120m 62m 97m
2kg/m 60m 46m 71m
4kg/m 30m 34m 57m

One can see that it is actually better to have a shorter length of heavier chain, even 'tho that means that you have a mixed rode more often. Instead of changing the conditions manually and updating the interactive graph one can plot the total rode length versus chain weight for a variety of wind forces, as below.

Also plotted, in purple, is the total length of chain given the constraint of constant total weight, so any situation above and to the right of the purple line is a mixed chain-rope rode.

Figure 6: Effect of varying chain size while keeping total weight consstant Despite heavier chain being always better from the perspective of minimising total scope, the effect is perhaps marginal. In any event, the size, and thus weight per metre, of chain is selected for holding power, never for strength. What is perhaps a useful conclusion is that one is better with a relatively short but heavy length of chain on the kedge.

# Part 3: Force as a function of wind speed

## Experimental measurement of force on boat

It has proved nigh impossible to find measured, empirical, data on the static forces on anchor cables, and actually impossible to find any data on the dynamic forces. While each boat is likely to behave differently, on the author's boat at least, the static forces are lower than expected and dynamic forces not that much bigger than static ones; nowhere near the ‘up to three times’ often stated. It would be extremely useful to gather vastly more data and correlate this with cable type and hull form; the author feels sure that many yachtsmen would volunteer to collect this data over a season’s sailing were a respected person or organisation to undertake to collate and analyse it.

## Horizontal forces as a function of wind speed

Comperhensive measurements to back up theoretical predictions are yet to be made. The author has only a few data points taken in winds of F4 to F8 (for instance 64kg horizontal force in F4, which gives λ = 32 at 13kts). For winds above F8 the forces are extrapolations, assuming force is proportional to the square of wind velocity. Using this we get the table:

Beaufort Force    Mean velocity in kts    Horizontal force in kg
4 13 64
6 24 218
8 37 518
10 52 1024

## Means of measuring force

One could, and probably should, use a load cell, and by measuring the tension and the angle of the rode, calculate the horizontal component of force. However there is an easy and a zero cost method! Using the graph below, which is for chain weighing 2.0kg/m, simply by knowing depth and angle of rode seen at the stem-head one can derive the force.

The steps are:

Step 1 Make sure enough scope is out, by using the critical angle method described above. Err on the generous side

Step 2 Using the graph below and the measured angle from vertical read off the tension in the rode 'T'

Step 3 Again using the measured angle, calculate the horizontal component of the force Fx = T . sin(θ')

Step 4 Scale the result for the weight of chain you have; the graph below is for 10mm chain, ie 2.0kg/m in water

Figure 7: Force on rode as a function of angle and depth 