Mathematic approach to anchoring scope

[Lodesman]

Originally Posted by MathiasW

Very true, but once it is all lifted off the seabed - and this will happen quickly in severe conditions - there is little additional energy that can be stored in the chain.

Where are you getting all this additional energy? If I assume a 5m depth (bow to seabed) with my 10mm chain. A 3:1 scope gives me 15m out, which will be fully lifted (just) with 40 daN of "pull." If I had 50m of chain out, that same 15m of chain would be suspended with 40 daN of pull, but there'd be 35m of chain sitting on the seabed. To get that last link of chain just off the seabed requires 505 daN of pull - that's nearly a 13-fold increase. If we're suspecting "severe conditions", I put out all 135m of chain. Again only 15m is suspended with 40 daN of pull, or 50m is suspended under 505 daN of pull, but to get all of that chain just off the seabed requires 3712 daN of pull - that is a lot of stored energy. I don't know what sort of sustained severe conditions you plan to be experiencing, that require you to ride out worse than that, but good luck.

[Dockhead]

Originally Posted by Lodesman

. . . but to get all of that chain just off the seabed requires 3712 daN of pull - that is a lot of stored energy. I don't know what sort of sustained severe conditions you plan to be experiencing, that require you to ride out worse than that, but good luck.

Don't assume that all that is usable for energy absorption. As the OP posted, we have to look into the time dimension. At a certain point the force required to straighten the chain out by another x cm spikes up so high that you might as well be dealing with a bar. Whether that is happening at the point where the last link is lifting, I don't know, but I wouldn't assume it without some analysis.


Force * distance = work. As the distance vanishes, the work vanishes.


If we think of catenary as a spring -- it's a spring which gets stiffer and stiffer, the more you pull on it, and the stiffness rises sharply. As it gets stiffer and stiffer, more and more energy is passed straight through to the anchor, and less and less is getting absorbed in the catenary. Eventually the catenary effect vanishes even if the chain is still slightly curved.

[rslifkin]

On top of that, if anyone thinks they can realistically put out 135m (442 feet) of chain in 5m (16.4 feet) of water, they're absolutely insane. In many places, you just won't find enough space to do that. And even if you do, you'll be sailing around so wildly if the wind gets a bit gusty and shifty that it's all going to be a big mess and you risk sailing off fast enough to put a heck of a lot of load on the anchor when you pull things tight.

[MathiasW]

Originally Posted by Lodesman

Where are you getting all this additional energy? If I assume a 5m depth (bow to seabed) with my 10mm chain. A 3:1 scope gives me 15m out, which will be fully lifted (just) with 40 daN of "pull." If I had 50m of chain out, that same 15m of chain would be suspended with 40 daN of pull, but there'd be 35m of chain sitting on the seabed. To get that last link of chain just off the seabed requires 505 daN of pull - that's nearly a 13-fold increase. If we're suspecting "severe conditions", I put out all 135m of chain. Again only 15m is suspended with 40 daN of pull, or 50m is suspended under 505 daN of pull, but to get all of that chain just off the seabed requires 3712 daN of pull - that is a lot of stored energy. I don't know what sort of sustained severe conditions you plan to be experiencing, that require you to ride out worse than that, but good luck.

Ah, ok, if you use THAT much chain, ok. Clearly, if you pay out enormous amounts of chain, you can always make sure some part of it will remain on the seabed, no matter how high the anchor load is. No arguing with that.

My point was more that a chain of given length can absorb less energy when it is spanned in a more horizontal way.

To illustrate this, here is the formula for the potential energy stored in a perfect catenary of length L where the last chain link is about to come off the seabed, and with the seabed as the reference point of zero energy:

E = m g /2 (L Y - a (L - X))

with m being the mass of the chain in water, g being 9.81 m/s^2, L the length of the chain, Y the water depth at anchor, and X = a asinh(L/a) the swinging radius (calculated to the bow roller point only). And we have a = F/(m g), with F being the load at the anchor.

You will notice that the first term in this equation is exactly the potential energy of this chain at a perfect scope of L:Y - its entire weight m g L at half the height, Y/2. The second term is clearly always negative, as compared to this perfect scope the chain will hang down a little - and it has to . Just double checking that it all makes sense.

In a severe storm we will have a >> Y, and if the chain is not too long, we can even assume a >> L. If you then make a Taylor expansion of the 2nd term, you get to lowest order

E =~ m g /2 (L Y - 1/6 L^3/a)

Now do the same Taylor expansion for the first term, using Y = sqrt(L^2 + a^2) - a, and you get

E =~ m g /2 (1/2 L^3/a - 1/6 L^3/a) = m g L^3 / (6 a)

Now the final approximation, again for Y << a. In this case we have

L = sqrt(Y(Y+2a)) =~ sqrt(2 a) sqrt(Y)

(which, by the way, is the basis of the British Admiralty guidance)

leading to

E =~ m g Y^(3/2) sqrt(2 a) / 3

Here you already see that as the anchor depth Y gets smaller and smaller, the total potential energy of the chain gets smaller and smaller - never mind differences of potential energies, they are even tinier.

But anyway, just to press on, assume you have a base anchor load F1, leading to a1 = F1/(m g). Then this load increases to F2, with corresponding a2.

Then the potential energy the chain gained is

E2 - E1 =~ m g Y^(3/2) sqrt(2) / 3 (sqrt(a2) - sqrt(a1))

Thus, the potential energy that can be gained when increasing the anchor load from F1 to F2 gets smaller when the anchor depth Y gets smaller.

So, that was my point. If the chain is not arbitrarily long but short on the scale of the parameter a = F/(m g), and if the water is shallow, the chain has more and more problems to store additional energy in the form of potential energy.

And at some point you will run out of chain...

[MathiasW]

And here an ever so slightly less involved argument...

Consider a perfect scope of a chain of length L and a water depth Y. Its potential energy is total mass times half the height,

E = m g L Y / 2

Now, clearly, the DIFFERENCE of potential energies between any two catenaries, which is the useful energy that can be stored by a chain by changing its shape from one catenary to another, has to be (much) smaller than that.

As the water depth Y decreases, so does the upper bound of the potential energy above, and with this the potential differences between any two catenaries hanging 'underneath' the perfect scope chain will get smaller.

This is the dilemma of a chain in shallow water, and this is why snubbers are needed.

[barnakiel]

Nice formulas!


I hope you have noticed, that it all boils down to two maybe 3 choices:


- how big is your anchor's surface (or be it a concrete block),
- how much chain can the boat and crew carry & manage.


And since the first thing is out of scope of this conversation, we are limited to only one decision:


Q: How much chain can / will you carry in your boat?


As long as you carry the longest chain you can, that's about that.


So the whole story confirms some lore where people say "there is no such a thing as too much chain and too big an anchor".



b.

[MathiasW]

Originally Posted by barnakiel

Nice formulas!


I hope you have noticed, that it all boils down to two maybe 3 choices:


- how big is your anchor's surface (or be it a concrete block),
- how much chain can the boat and crew carry & manage.


And since the first thing is out of scope of this conversation, we are limited to only one decision:


Q: How much chain can / will you carry in your boat?


As long as you carry the longest chain you can, that's about that.


So the whole story confirms some lore where people say "there is no such a thing as too much chain and too big an anchor".



b.

Thanks!

And, oh yes, couldn't agree more on your conclusions!!!

To me, before doing all this exercise, I had somewhat naively assumed it is always better to flee to as shallow water as possible to ride a storm. But now, I know this is not necessarily the case...

[MathiasW]

Originally Posted by MathiasW

To illustrate this, here is the formula for the potential energy stored in a perfect catenary of length L where the last chain link is about to come off the seabed, and with the seabed as the reference point of zero energy:

E = m g /2 (L Y - a (L - X))

with m being the mass of the chain in water, g being 9.81 m/s^2, L the length of the chain, Y the water depth at anchor, and X = a asinh(L/a) the swinging radius (calculated to the bow roller point only). And we have a = F/(m g), with F being the load at the anchor.

You will notice that the first term in this equation is exactly the potential energy of this chain at a perfect scope of L:Y - its entire weight m g L at half the height, Y/2. The second term is clearly always negative, as compared to this perfect scope the chain will hang down a little - and it has to . Just double checking that it all makes sense.

Ah, I only just noticed! With a = F / (m g), you can rewrite this formula for the potential energy as

E = m g L Y/2 - F (L - X)/2

This has a nice interpretation: As stated already, the first term is the potential energy of a perfect L:Y scope chain. From this one has to subtract a term that is given by the product of the anchor load F and half of the difference between chain length L and swinging radius X.

Rather nice, isn't it?

[MicHughV]

I was anchored in the Bahama's one time, when a chartered cat pulled up near me.

They lowered the anchor until it hit bottom....about 10' down...and then stopped. No formula required here.

They were " anchored" as far as they were concerned...

Off course, 30 minutes later....they were waaaayyyy over there.

We had to dinghy over to give them some impromptu anchoring advice..but were paid with a six pack of beer..

[MathiasW]

Originally Posted by MicHughV

I was anchored in the Bahama's one time, when a chartered cat pulled up near me.

They lowered the anchor until it hit bottom....about 10' down...and then stopped. No formula required here.

They were " anchored" as far as they were concerned...

Off course, 30 minutes later....they were waaaayyyy over there.

We had to dinghy over to give them some impromptu anchoring advice..but were paid with a six pack of beer..

That is a nice story - and it has an excellent ending!

And it is situations like this, why I am kind of on a crusade to make folks think more about anchoring. If they come up with different formulas than me, fine, as long as they work and as long as they do not bump into me at anchorage.

I am always worried when I see a chartered vessel trying to anchor close to me...

To be fair to these charter folks. If a big sailing association like the German Sailing Association DSV lets you sit exams where the correct answer is to use a scope of 3:1, regardless of the situation, then you know you should be particularly worried when it is a vessel chartered by Germans. it is not them to blame - they get taught nonsense!

Consequently, I have reached out to DSV as well and try to get them to change their answers to their anchoring questions...

[MicHughV]

I think part of the problem is that the nautical world is full of nautical terms generally unknown to the landlubber.

I have tried to explain this to a landlubber, by telling them there are no "ropes" on a boat...a "rope" is a "rope" as long as it is in the store...but once it comes aboard, it ceases to be a " rope" and adopts a name particular to it's use...halyard...downhaul...sheet...topping lift...etc..etc..but no " rope" anywhere to be found on a boat.

A landlubber will look at the myriad of " ropes" on a sailboat at a dock and wonder what the hell I'm talking about as all they can see is a craft festooned with " ropes"

At this point I get glazed eyes and folk refer to the above as " that rope thingy that holds that doodad"..

Trying to explain " scope" to a yacht charterer is perhaps beyond their ability to grasp.

Heck, trying to explain "scope" to a newbie yachtsman is equally frought with blank stares.

Even such basics as "port" and " starboard" are beyond the grasp of a landlubber...

Tell a landlubber to turn the helm....will get a response..."say what again"....

Anchor catenary topics is best left to the professional .....

[MathiasW]

Originally Posted by MicHughV

I think part of the problem is that the nautical world is full of nautical terms generally unknown to the landlubber.

I have tried to explain this to a landlubber, by telling them there are no "ropes" on a boat...a "rope" is a "rope" as long as it is in the store...but once it comes aboard, it ceases to be a " rope" and adopts a name particular to it's use...halyard...downhaul...sheet...topping lift...etc..etc..but no " rope" anywhere to be found on a boat.

A landlubber will look at the myriad of " ropes" on a sailboat at a dock and wonder what the hell I'm talking about as all they can see is a craft festooned with " ropes"

At this point I get glazed eyes and folk refer to the above as " that rope thingy that holds that doodad"..

Trying to explain " scope" to a yacht charterer is perhaps beyond their ability to grasp.

Heck, trying to explain "scope" to a newbie yachtsman is equally frought with blank stares.

Even such basics as "port" and " starboard" are beyond the grasp of a landlubber...

Tell a landlubber to turn the helm....will get a response..."say what again"....

Anchor catenary topics is best left to the professional .....

All very true, you made a huge grin to appear on my face...

Still, there is an amazing interest in anchoring correctly, as also this thread shows. Even though it is controversial at times. The paper I did some months back at blauwasser.de attracted many comments - many more than others on that platform. And we are now covering this topic in two back-to-back issues of Die Yacht, the largest German speaking sailing magazine. The magazine said it is attracting the largest number of feedbacks they have seen in some 16 years. So, it seems worth it to tell the story...

[MicHughV]

yes, I've been involved in large ship moorings, which is a science unto itself..

prior to the computer age, I had a mooring manual developed by the US Navy..
It was filled from front to back with formula's, diagrams, charts, graphs, etc, covering every possible eventuality...coming up with a specific mooring design required pages and pages of calculations...as tying together all the different " variables" for a particular application was mind numbing work. Besides, ocean moorings, it also delves into mooring a large ship alongside a dock. Try figuring out, simultaneously, the strain on the dozen or so lines attaching a ship to a dock. Often, each line of a different diameter, different length, different stretch, different tension, often tied to a mooring that also moves, etc....

somebody, somewhere, put this manual together...bless him (or her). I still have it somewhere, but I wonder if it is available online.

it is a treasure trove of useful data and would be worth your while to chase down to supplement your current mathematical approach to anchoring.

Anchoring can be made into quite the science. I don't think there is a fixed solution. There are too many variables, and the anchoring situation needs to be modified or adapted to its own particular situation.

[MathiasW]

Originally Posted by MicHughV

ysomebody, somewhere, put this manual together...bless him (or her). I still have it somewhere, but I wonder if it is available online.

it is a treasure trove of useful data and would be worth your while to chase down to supplement your current mathematical approach to anchoring.

Aha, most interesting! You would not happen to have the title of this manual, would you?

[Dockhead]

Originally Posted by MicHughV

. . . I have tried to explain this to a landlubber, by telling them there are no "ropes" on a boat...a "rope" is a "rope" as long as it is in the store...but once it comes aboard, it ceases to be a " rope" and adopts a name particular to it's use...halyard...downhaul...sheet...topping lift...etc..etc..but no " rope" anywhere to be found on a boat.. . ..

A myth common among landlubbers with a recently developed interest in boating and a desire to make those who are one-half step behind, feel ignorant. Sorry, it's a pet peeve. Nothing wrong with the word "rope". There are a few miles of rope on board my boat.