Anchoring with dynamic loads a la Alain Fraysse

[MathiasW]

So, this took me a little while to compose…

I have read the very nice and seminal work of Alain again, who has calculated the anchor load - also in the presence of dynamic loads - in the force-time domain:

Forces

He models a gust as a stronger wind force that gets switched on at a certain point in time and then stays on “forever”.

He starts by discussing the linear case (like, approximately, a rope), where the gust leads to an overshooting over the position the boat would take, had the gust been a static wind. Because he discusses the linear case, the point of maximal overshooting (red) is exactly a mirror point of the initial static point (green), where the position where gust and rope tension balance out (yellow) is exactly in the middle.

The first diagram shows what happens. It is a plot of the displacement of the boat (away from the anchor) as a function of load - it is linear function, because we assumed a linear spring. When I integrate underneath the area (force times distance), I get directly the energy stored in the elastic rope. It is simply the area when looking at the graph sideways.

The areas in light red are energies that are not (yet) loaded into the rope (as opposed to the dark red area), but rather go into the kinetic energy of the boat (first), or are drawn from it (later). At the yellow point the pull back force of the rope equals the gust load. But as the boat has some velocity, it overshoots until it has reached the red point, the most extreme tension in the rope. The two areas labelled E must be the same, if no energy is lost somewhere.

Please note the loads at the green, yellow and red points. The difference between green and yellow is the same as the difference between yellow and red. This is the best possible scenario and leads to the smallest possible maximal overshooting tension.

Nonlinearity makes the result worse. The next diagram is the same as before, but now not with an ideal spring, but rather a chain acting as a spring. Again, the gust keeps blowing even when the boat has reached the extreme position with the highest displacement away from anchor. But now the distance between the yellow and the red point is much larger than the distance between the yellow and the green point. In fact, the most extreme point, the red point, now has about 4 times the load of the static wind. That is a lot, given that the gust load is only twice the static load!

There are two graphs in this diagram acually. One for deep water and one for shallow water. There is no big difference between the two. And in this scenario (in view of the discussion further below) the vessel’s mass does not matter at all.

Next I show the same graphs, but now assuming that the gust only blows for a few seconds, resulting in this “energy spike” to the right. So, in this case, the vessel is still moving backwards when the gust is stopping allready.

Overall, the red points are much more to the left than in the previous graphs, which is obvious, as the total energy transferred to the boat is less. But one can also see now that in shallow water the red point is much further to the right than in deep water.

Conclusion: Shallow water can be dangerous in the presence of gusts. But a very elastic rope can help to address this shortcoming of the chain, as its characteristics is much more linear, and its ability to absorb energy temporarily does not depend on anchor depth.

But, as we see, it depends on the nature of the gust. When it is simply a switching on of a higher wind force, then it does not matter whether it is shallow or not, the point of maximal overshooting will always be much higher than the point of the gust load. But when it is a sudden and short burst of wind, then it does matter.

The reality is - as so often - somewhere inbetween…

BTW - This last graph also explains why large commercial ships do not see this effect at all. Because of their large mass m, according to F = m * a, for the same load F, their acceleration a in a gust will be much smaller, and hence the displacement backwards away from the anchor in a given time window is also so much less. This results in a very small energy transfer E, and subsequently a very tiny overshooting.

In the very last graph I have shown as an example the case for a vessel twice as heavy, but with exactly the same windage area and exposed to the same gusty burst, hence same load F. The energy E the vessel has to absorb is only about half now, leading to less violent maximal loads.

Conclusion: Older pleasure crafts that are rather heavy for their windage area are less affected by these dynamic loads. They hardly move at all in a gust. But light-weight multihulls and modern "plastic" yachts with high windage area are strongly affected.

Cheers, Mathias

Anchor Chain Calculator

[jo_sail]

That is a lot of detailed stuff an food for some thought. Can you put units on you axes? I guess the displacement is in dimension of length (meters), horizontal, right? I like to add some thoughts for further discussion of your interesting results.

I much agree to most of your points and conclusions:
1. focus on dynamic load as the decisive/crucial load case
2. elastic line as key element for absorbing peak loads, (especially in low depth situations, when a chain has little chance to absorb energy by vertical movement)

I like to add

to 1.: Water has 1000 times more density compared to air. Thus a 50 cm or even higher wave hitting the boat in a stretched line/chain situation will put significant additional energy/load, which still must be absorbed safely.

to 2: The benefit of a well selected shallow water anchoring spot is a) (often) less chance for high waves and b) a small angle between rope / anchor shaft an seafloor even in the stretched rope or full load situation. The small angle will allow the maximum grip of the anchor in the ground. Every degree of angle uplift looses some of the anchors grip and its capacity to accept load without breaking. Of cause this will depend much on the quality of the ground and the characteristics of the anchor itself.

In deep water, anchor chains can absorb quite some energy by stretching and vertical movement. This is why less scope is often suggested for the deep water case and large ships.

So I think deciding for shallow or deep will always require some compromise. Elastic energy absorption in a rope can help to accept shallower water for anchoring. I try to use a not to thick long Nylon line as snubber going from the bow to the chain whenever possible.

In all that, protection from waves is in my opinion still the most important factor to minimize dynamic loads on the system (due to water density). After that, gusts will be quite important. And putting down the jib will be one of my first measures to reduce windage at the front where it matters most for the wind induced load.

[MathiasW]

Originally Posted by jo_sail

That is a lot of detailed stuff an food for some thought. Can you put units on you axes? I guess the displacement is in dimension of length (meters), horizontal, right? I like to add some thoughts for further discussion of your interesting results.

I much agree to most of your points and conclusions:
1. focus on dynamic load as the decisive/crucial load case
2. elastic line as key element for absorbing peak loads, (especially in low depth situations, when a chain has little chance to absorb energy by vertical movement)

I like to add

to 1.: Water has 1000 times more density compared to air. Thus a 50 cm or even higher wave hitting the boat in a stretched line/chain situation will put significant additional energy/load, which still must be absorbed safely.

to 2: The benefit of a well selected shallow water anchoring spot is a) (often) less chance for high waves and b) a small angle between rope / anchor shaft an seafloor even in the stretched rope or full load situation. The small angle will allow the maximum grip of the anchor in the ground. Every degree of angle uplift looses some of the anchors grip and its capacity to accept load without breaking. Of cause this will depend much on the quality of the ground and the characteristics of the anchor itself.

In deep water, anchor chains can absorb quite some energy by stretching and vertical movement. This is why less scope is often suggested for the deep water case and large ships.

So I think deciding for shallow or deep will always require some compromise. Elastic energy absorption in a rope can help to accept shallower water for anchoring. I try to use a not to thick long Nylon line as snubber going from the bow to the chain whenever possible.

In all that, protection from waves is in my opinion still the most important factor to minimize dynamic loads on the system (due to water density). After that, gusts will be quite important. And putting down the jib will be one of my first measures to reduce windage at the front where it matters most for the wind induced load.

Thanks for this detailed response!

I think we are in synch regarding our observations. You are right that in shallow water the pulling angle at the anchor shank can be kept low for longer than in deeper water. But, on the other hand, in bursty gusts (and in swell, as you rightly point out), the load on the anchor is much higher in shallow water compared to deeper water, so this advantage of better pulling angle is quickly eaten up.

As you say, it is always a compromise, and my thoughts here are meant as a supplement to help making an informed decision when anchoring.

Regarding units - it took me a while actually, to get rid of them in Igor Pro, which I had used to generate the graphs...

The horizontal unit is 'a', where 'a' is the catenary parameter as in cosh(x/a) for the catenary curve. 'a' is the current wind load divided by the weight (force) of one metre of chain.

The displacement away from anchor is given by a*asinh(x/a) - L, where L is the chain length with a horizontal pull at the anchor shank. So L = sqrt(Y*(Y+2*a)), where Y is the anchor depth (measured from the bow roller).

When using a*asinh(x/a) - L I make implicitly the assumption that there is always enough chain on the seabed to be able to pull horizontally at the anchor shank.

The two curves shown as 'shallow' and 'deeper' have a ratio of anchor depths as 1: 2.5, but both being smaller than 'a'.

Cheers, Mathias

PS: When anchoring close to mountains one may suffer from katabatic winds falling down the mountain. If the static wind was low before it hits, there will be quite some fetch in the chain or rope before the boat comes to rest again. Enormous energies can be picked up that way. This can be much worse than swell.

[skipperpete]

Hi Mathias, your PS is very true, during a cyclone in the Whitsunday islands I went for shelter in Gulnare inlet but arrived very late and all of the sensible space was taken by a large number of cruising and charter vessels and the only spot left was at the very end of the inlet in shallow water at the entrance of a creek. I payed out 100 metres of ½” chain and came to rest in 5’ depth right at the mouth of the creek in about 40 knots of wind and thought I was in a good place ( 19 tonne steel ketch, 4’6” draft)...... I was wrong! The creek was in flood and each time the wind dropped off a bit, the floodwater pushed the boat to windward until the next powerful gust pushed us back into the creek entrance with such speed that the entire length of chain became bar tight and visible for maybe 40 metres almost ripping the 4” steel mooring bits off the foredeck. The only way we had to stabilize the situation was to lay our second anchor attached to a very large plaited nylon anchor warp. This solved the shock load problem but not the mad surging and the anchor chain spent the night resting quietly on the bottom beneath a big timber charter boat that dragged until grounding out in the shallows. By morning all was calm and we waited for the “Plover” to get back into deep water so my main anchor and chain could be recovered. The heavy surging had deformed the chain to the point where it would no longer fit the gypsy.

[MathiasW]

Originally Posted by skipperpete

Hi Mathias, your PS is very true, during a cyclone in the Whitsunday islands I went for shelter in Gulnare inlet but arrived very late and all of the sensible space was taken by a large number of cruising and charter vessels and the only spot left was at the very end of the inlet in shallow water at the entrance of a creek. I payed out 100 metres of ½” chain and came to rest in 5’ depth right at the mouth of the creek in about 40 knots of wind and thought I was in a good place ( 19 tonne steel ketch, 4’6” draft)...... I was wrong! The creek was in flood and each time the wind dropped off a bit, the floodwater pushed the boat to windward until the next powerful gust pushed us back into the creek entrance with such speed that the entire length of chain became bar tight and visible for maybe 40 metres almost ripping the 4” steel mooring bits off the foredeck. The only way we had to stabilize the situation was to lay our second anchor attached to a very large plaited nylon anchor warp. This solved the shock load problem but not the mad surging and the anchor chain spent the night resting quietly on the bottom beneath a big timber charter boat that dragged until grounding out in the shallows. By morning all was calm and we waited for the “Plover” to get back into deep water so my main anchor and chain could be recovered. The heavy surging had deformed the chain to the point where it would no longer fit the gypsy.

What a nightmare!!!! Good that is worked out ok in the end.

Cheers, Mathias