Mathematic approach to anchoring scope

[Lodesman]

Originally Posted by Dockhead

In my experience, or at least where I sail, really bad weather is usually accompanied by a significant wind shift, which can turn good shelter into a lee shore, which is really dangerous. And can do so in a way not predicted by the forecasts. That's why for really bad weather, we normally try to get into a cove with shelter from all or most sides. Swing room is often, maybe usually, limited in such places. Even if you have the swing room, you would be loath to be dragging the whole bottom of the whole cove with 130m of chain out, as the wind shifts. Even 100m is too much in many such places, which is why I'm often on short scope.

That is why the idea of just finding a wide open space and putting out a massive amount of chain, doesn't work for me. I guess maybe storms might work differently in different latitudes, without such wind shifts as we get here, so YMMV, perhaps.

Well I'm used to anchoring a 450' ship with 1000' of chain available, so finding room to swing 300' is relatively easy.
IMO, storms work the same in most places, although local effects (katabatic winds, etc) might add complexity to a given situation. Last place we "took cover" from a storm was a rather large bay behind a fairly low-lying cay. There was a 180º windshift, which was entirely predictable, but didn't stop the vast majority of the little-boat crowd from bunching up along the beach. We were one of a few boats, further out and spaced farther apart. We rode it out comfortably and enjoyed the entertainment watching a bunch of the sheep madly scramble to relocate. Luckily no-one ended up on the shore. IIRC I had 200' out; water depth was 10-11'.

In really high latitudes, shore ties are kind of SOP for bad weather. That is a technique I have used in the Aegean, but didn't use in the Arctic. It's very awkward to do, if you don't have drums on deck with very long lines. The problem with shore ties is they prevent you from swinging, so the anchor may be pulled out if you end up abeam to the wind. The answer is shore ties from more than one direction, and to do that takes a massive amount of rope, hence the drums you see on the decks of true high latitude boats.

The drums are fairly common in BC. I don't have much personal experience with shore ties - only done them a couple times. To deal with the beam wind, you could go to a side-bridle (swell/winch/Pardey-bridle) on the anchor and take the shore-tie to the opposite beam or even the bow. Not sure how well it would work, but should reduce the windage.

[MathiasW]

Originally Posted by Rothblum

"Have you ever done scale modeling? I was referring to your simplistic analogy that implies a 30ft cruising boat and a 1000ft oil tanker react in the same way."

I spent 31 years testing ship models at the US Navy's David Taylor Ship Model Basin, and the answer is yes, you can scale the reaction of a 30 ft model to that of a gigantic oil tanker. The trick is to express the significant variables as dimensionless numbers. Then, if the large scale version of the model has same dimensionless parameters as the model, then the behavior will be the same. The problem is that not all significant parameters can be duplicated simultaneously, or that some variables might be significant at model scale and not at full scale -- surface tension, for example.

And yes, this is exactly what the catenary equations are doing - they scale and work with dimensionless parameters! Wether it is the windage area, the chain length, the chain thickness, you will see that only certain ratios enter. For instance, it is the force (as applied to the windage area) divided by the chain weight per meter that matters - my parameter 'a'.

So, yes, catenary theory does apply also to the larger vessels. Certainly, some parameters might be less important than others, compared to our recreational vessels. For instance, I do not know whether huge vessels will ever move a lot whilst at anchorage. Under normal conditions they don't, but in a severe storm, I guess they will. And their speed does not need to be large to amount to a very substantial kinetic energy.

And btw - the British Admiralty guidance IS based on a catenary formula. They cut out the wind dependency and made it work for up to 60 kn of wind, resulting in a very conservative guidance for normal conditions.

I am not claiming to have hit the right values for all parameters of huge vessels, as also indicated by me. But if you know the correct values, you can apply this theory. And it is even simpler in some sense, as you do not have to work out what snubbers or bridles would do...

[MathiasW]

Originally Posted by MathiasW

Thank you Dockhead!

If one takes the formula for a chain's potential energy above and asks, what are the conditions where the capabilities of the chain are best to store additional potential energy for a given, FIXED length L of chain, then one needs to differentiate this formula with respect to the anchor depth Y and set this to zero.

A bit tedious, but the maximum of this capability is reached when - in addition to being a fully developed catenary - the chain also fulfils the relation L =~ 2.2 .. 2.5 Y, where the precise factor depends slightly on the anchor load. It is 2.5 in the limit of vanishing anchor load.

I need to correct myself here, I am afraid. Not sure what my brain was doing when I wrote this , but it is the differentiation with respect to anchor load and not anchor depth, that is needed here....

So, dE_pot/dF is what one is looking at. It is the elasticity of the chain, so how easy can the chain absorb more potential energy when being pulled a bit harder at the anchor load (or the vessel).

It turns out the result is rather universal, as the attached plot shows. Here I have kept the anchor depth Y constant and varied the strength of the anchor load, so the parameter a. To make the plot easy to read I have used as x axis the scope L:Y. But don't be fooled by me talking about scope here. At any point I do require the chain to be a perfect catenary, so its length needs to be adjusted as the scope increases. (And yes, not practical in reality, this is a Gedankenexperiment... )

What does this graph show?

Extremely small scopes are no good, so when the chain is almost vertically hanging down, it is hard to store additional energy in the chain. But extremely large scopes are also not nice. There is a sweat spot somewhere in-between where the chain works best.

For large scopes, as one might have them in shallow water in a storm, the elasticity of the chain is poor. The chain does not work well and good snubbers are vital. I can improve on that by relocating to deeper water, where I need to use more chain, yes, but it will reduce the effective scope. So, I get closer to the peak in this graph, coming from the right. Also, as in absolute terms the peak in this graph is proportional to the anchor depth Y, I benefit doubly from this move - the peak gets even larger.

So, this helps perhaps to explain why some folks like Dockhead prefer to anchor in deeper water at rather smaller scopes. Their chain simply works better there and the need for snubbers is less. (And btw - the peak is also proportional to the chain's weight per metre.)

I have also updated my web page correspondingly.

[Dockhead]

Originally Posted by MathiasW

I need to correct myself here, I am afraid. Not sure what my brain was doing when I wrote this , but it is the differentiation with respect to anchor load and not anchor depth, that is needed here....

So, dE_pot/dF is what one is looking at. It is the elasticity of the chain, so how easy can the chain absorb more potential energy when being pulled a bit harder at the anchor load (or the vessel).

It turns out the result is rather universal, as the attached plot shows. Here I have kept the anchor depth Y constant and varied the strength of the anchor load, so the parameter a. To make the plot easy to read I have used as x axis the scope L:Y. But don't be fooled by me talking about scope here. At any point I do require the chain to be a perfect catenary, so its length needs to be adjusted as the scope increases. (And yes, not practical in reality, this is a Gedankenexperiment... )

What does this graph show?

Extremely small scopes are no good, so when the chain is almost vertically hanging down, it is hard to store additional energy in the chain. But extremely large scopes are also not nice. There is a sweat spot somewhere in-between where the chain works best.

For large scopes, as one might have them in shallow water in a storm, the elasticity of the chain is poor. The chain does not work well and good snubbers are vital. I can improve on that by relocating to deeper water, where I need to use more chain, yes, but it will reduce the effective scope. So, I get closer to the peak in this graph, coming from the right. Also, as in absolute terms the peak in this graph is proportional to the anchor depth Y, I benefit doubly from this move - the peak gets even larger.

So, this helps perhaps to explain why some folks like Dockhead prefer to anchor in deeper water at rather smaller scopes. Their chain simply works better there and the need for snubbers is less. (And btw - the peak is also proportional to the chain's weight per metre.)

I have also updated my web page correspondingly.

That is REALLY interesting. That starts to explain something which many of us know exists from experience, but which I guess none of us ever understood before. Fascinating. So the same quantity of chain works THAT much better in deeper water. That explains a lot.

[MathiasW]

Originally Posted by Dockhead

That is REALLY interesting. That starts to explain something which many of us know exists from experience, but which I guess none of us ever understood before. Fascinating. So the same quantity of chain works THAT much better in deeper water. That explains a lot.

Thanks. I am most glad that I seem to be able to advance the community's understanding a little...

The graph above showed chain elasticity as a function of scope, L:Y, with anchor depth Y fixed, and thus the chain length L had to vary. The graph below is again chain elasticity as a function of scope, L:Y, but now with a fixed chain length L, and consequently the anchor depth Y has to vary instead. The sweat-spot peak is at a slightly different position and the slope to the right is somewhat steeper compared to the previous graph. This is due to the different boundary conditions. In one case I keep Y constant, whereas in the other I keep L constant. The arguments why seeking deeper water in a storm makes a lot of sense run in exactly the same manner, though, also for this graph.

In order to confuse all even more, I decided to plot this graph at double log scale...

Perhaps one should rephrase the saying of "when you need the catenary most - in a storm - it is gone" to "when you need the chain's elasticity most - in a storm - it is gone".

[MathiasW]

With reference to my graph shown in post #198, where I plotted the chain's elasticity as a function of scope L:Y, with anchor depth Y kept constant:

Here is a real situation where this graph applies:

Suppose you are anchoring in shallow water and there is only a light breeze. But forecast has it that a storm is expected for the night, and in anticipation of that you have already paid out a LOT of chain.

For now, in the light breeze, most of that chain lies unused on the seabed. The effective scope of your chain - so the ratio of the chain that is actually off the seabed to the anchor depth Y - is small. For the sake of the argument let's assume this effective scope to be 1.4. The chain thus operates in the peak of the elasticity curve, so all is fine and smooth.

Now the wind starts to pick up and more and more chain is lifted off the seabed. The anchor depth Y stays the same, but the effective scope becomes larger, as more and more chain is being used.

With this, we move to the right in the graph, so away from the peak. At an effective scope of 3.7 we have only 50% of the chains optimal elasticity left. And to be clear - this optimal elasticity is the elasticity of the effectively much shorter chain when there was only a light breeze! So, despite of the fact that more chain is now being used, it has less elasticity. It is much less efficient at absorbing energy than the much shorter chain at scope 1.4 was.

As the wind further develops, even more chain gets lifted off the seabed. At an effective scope of 20:1 only 10% of the chain's original elasticity has remained and the cleats at the bow start to moan... It is getting uncomfortable. Snubber to the rescue... Or - a relocation to deeper water well before the storm sets in.

[MathiasW]

If I repeat the above, but now at twice the anchor depth, 2Y, I will initially be slightly before the peak at a scope of 1.22 (one can work this out using the equation L/Y = sqrt((Y+2a)/Y) and the parameter a being the same as before). Eventually, when the storm has fully developed, we find us at a scope of only 14.2, or 13.5% of the peak elasticity. But since we doubled the anchor depth, in absolute terms, this peak is also twice of what it was before! So, we have 2 x 13.5% = 27% of the 'old' optimal elasticity at anchor depth Y. This is almost a factor 3 difference! And I needed only 41.6% more chain. Doing the same exercise with a relocation to three times the original anchor depth, 3Y, one ends up at a scope of 11.6 and as much as 49.4% of the 'old' peak elasticity at the original anchor depth Y - a gain by a factor 5! The chain will have to be increased by 73.6% only.

[MicHughV]

I was flipping thru' Youtube and came across this...

"How To Solve Amazon's Hanging Cable Interview Question"...if you just type this into the YT search bar you will be led to the video...not sure if this is helpful or not....but give it a look..

[MathiasW]

Originally Posted by MicHughV

I was flipping thru' Youtube and came across this...

"How To Solve Amazon's Hanging Cable Interview Question"...if you just type this into the YT search bar you will be led to the video...not sure if this is helpful or not....but give it a look..

Thanks. Exactly the formulas I have been using. He even calls the catenary parameter 'a', as I do... It holds a = f/(m g), where F ist the anchor load, m the weight of the chain per metre, and g = 9.81 m/s^2.

I am tempted to post a challenge to him, though...

[Dockhead]

Originally Posted by MathiasW

With reference to my graph shown in post #198, where I plotted the chain's elasticity as a function of scope L:Y, with anchor depth Y kept constant:

Here is a real situation where this graph applies:

Suppose you are anchoring in shallow water and there is only a light breeze. But forecast has it that a storm is expected for the night, and in anticipation of that you have already paid out a LOT of chain.

For now, in the light breeze, most of that chain lies unused on the seabed. The effective scope of your chain - so the ratio of the chain that is actually off the seabed to the anchor depth Y - is small. For the sake of the argument let's assume this effective scope to be 1.4. The chain thus operates in the peak of the elasticity curve, so all is fine and smooth.

Now the wind starts to pick up and more and more chain is lifted off the seabed. The anchor depth Y stays the same, but the effective scope becomes larger, as more and more chain is being used.

With this, we move to the right in the graph, so away from the peak. At an effective scope of 3.7 we have only 50% of the chains optimal elasticity left. And to be clear - this optimal elasticity is the elasticity of the effectively much shorter chain when there was only a light breeze! So, despite of the fact that more chain is now being used, it has less elasticity. It is much less efficient at absorbing energy than the much shorter chain at scope 1.4 was.

As the wind further develops, even more chain gets lifted off the seabed. At an effective scope of 20:1 only 10% of the chain's original elasticity has remained and the cleats at the bow start to moan... It is getting uncomfortable. Snubber to the rescue... Or - a relocation to deeper water well before the storm sets in.

So, summary -- you can't get much usable catenary effect in shallow water even with massive scope. Fair summary?


You have shown that catenary effect, that is energy absorption (or damping) capacity, is proportionate to gross quantity of chain out but reduced with increasing scope due to shallower water.



So if you want maximum catenary effect, by all means put out all your chain, but find water deep enough that the scope is not more than the minimum you are comfortable with.


Fair summary?


I'm very surprised at how catenary effect falls so much with increasing scope, but it really does account for subjective experience about deep water, experienced by me over decades and something a lot of people say.


So deep water, all your chain out, but deep enough to get maybe 4:1 scope would be the sweet spot, with 50% of the maximum possible catenary effect available but most of maximum possible holding power of the anchor available even if all the catenary is pulled out. Depending on how much and how heavy chain you have, it starts to become impossible to to pull all the catenary out, so maybe it becomes optimal to reduce scope further (by going into deeper water with all your chain). This explains why so many of us have such good luck on quite short-sounding scope (3:1 or even less) in very deep water.


As I wrote, I went through a storm in Greenland on about 2.5:1 in very deep water with all my chain out (40m of water, 100m of 12mm chain) and felt very secure.



We used to have a very focal "scope fetishist" on here who was adamant about how foolish it is to try to go through bad weather on scope of less than 10:1, and who said he'd rather anchor off a lee shore if necessary, to get into water shallow enough to get the scope up to 10:1. Now we see concretely why that is such a bad idea.



Well done, Mathias, you have really advanced our understanding

[MathiasW]

Originally Posted by Dockhead

So, summary -- you can't get much usable catenary effect in shallow water even with massive scope. Fair summary?


You have shown that catenary effect, that is energy absorption (or damping) capacity, is proportionate to gross quantity of chain out but reduced with increasing scope due to shallower water.



So if you want maximum catenary effect, by all means put out all your chain, but find water deep enough that the scope is not more than the minimum you are comfortable with.


Fair summary?


I'm very surprised at how catenary effect falls so much with increasing scope, but it really does account for subjective experience about deep water, experienced by me over decades and something a lot of people say.


So deep water, all your chain out, but deep enough to get maybe 4:1 scope would be the sweet spot, with 50% of the maximum possible catenary effect available but most of maximum possible holding power of the anchor available even if all the catenary is pulled out. Depending on how much and how heavy chain you have, it starts to become impossible to to pull all the catenary out, so maybe it becomes optimal to reduce scope further (by going into deeper water with all your chain). This explains why so many of us have such good luck on quite short-sounding scope (3:1 or even less) in very deep water.


As I wrote, I went through a storm in Greenland on about 2.5:1 in very deep water with all my chain out (40m of water, 100m of 12mm chain) and felt very secure.



We used to have a very focal "scope fetishist" on here who was adamant about how foolish it is to try to go through bad weather on scope of less than 10:1, and who said he'd rather anchor off a lee shore if necessary, to get into water shallow enough to get the scope up to 10:1. Now we see concretely why that is such a bad idea.



Well done, Mathias, you have really advanced our understanding

Hi Dockhead, yes, this is a perfect summary!

And thank you, and thanks to all in this forum asking questions, making suggestions and challenging the original pitch. It helped me to sharpen my analysis considerably!

[MathiasW]

And if circumstances have it that one is forced to stay in shallow water and accept a large scope, then one is better served with a long elastic rope and a short lead chain to avoid chafing on the seabed. The shape of this combination will be almost identical to that of a pure chain at large scope, and hence the pulling angle at the anchor virtually identical, but its performance in terms of elasticity is so much better.

I think I need to have a look how to make a nice splice between chain and rope...

[MicHughV]

the solution...in my opinion...is to go with the best of both worlds....

nylon rode and chain....I have already mentioned here, that my preferred anchoring method is 75' chain followed by nylon rode....length depending on water depth...

[Lodesman]

Originally Posted by MicHughV

I was flipping thru' Youtube and came across this...

"How To Solve Amazon's Hanging Cable Interview Question"...if you just type this into the YT search bar you will be led to the video...not sure if this is helpful or not....but give it a look..

I think the question isn't designed to test your knowledge of catenary equations, but to see if you can see the obvious answer - an 80-metre cable suspended at both ends at a height of 50 m must be folded in half and hanging straight down to end 10m (50-40m) above the ground.

[MathiasW]

Originally Posted by MicHughV

the solution...in my opinion...is to go with the best of both worlds....

nylon rode and chain....I have already mentioned here, that my preferred anchoring method is 75' chain followed by nylon rode....length depending on water depth...

Oh yes, absolutely!

And now that I understand better what works in which circumstances, I should sleep better at night whilst at anchor...