Mathematic approach to anchoring scope

[Dockhead]

Originally Posted by CarinaPDX

One thing I find missing in these discussions is the angle of the sea floor - it is always assumed to be level. On a steeply angled bottom the geometry aspect (as opposed to the shock loading) is favorable with the wind blowing towards the shallows, and often massively unfavorable in the opposite direction. My first (but sadly not even close to last) experience with a boat dragging down on me happened in Friday Harbor; the Hans 38 was anchored on a steep slope in 30' with 3:1 chain (the harbor is a flat 50' except for a steep rise to the shore). When the thunderstorm hit the wind came off the land and he had no chance of holding. He did a lot of damage and left without dealing with it. Anyway, a complete analyses would have to consider how level the bottom is. Among other things. Much of this just comes with experience but I do find the various models to be useful in understanding what is happening.

Slope of the bottom is one of the most crucial factors in anchoring. I suppose most people understand that pretty well.



I am pretty passionate about finding a flat area of seabed to put my anchor into. I have learned with experience that sloping seabed is more than one kind of trouble. Not only do you have the risk of getting angulation on the anchor all out of proportion to your scope, but I find the seabed itself is rarely very good, on a slope -- silt runs downhill, leaving different kinds of crap on the slope.


My father was a scope fetishist, and was always looking for shallow water so that with the limited amount of chain on board our old boat, he could get what he thought was better scope. He refused to anchor with less than 6:1 and was only really happy with 10:1. It meant we were often anchored on a sloping bottom and we often dragged (well, we were using a CQR too, so doubly cursed). It took me years, nay decades, to figure out that this is all wrong -- a fundamental error. Now I typically anchor in the deepest part of a cove, even if it means short scope, and results are much better. The deepest part will be flat, and besides that, that's where all the silt has rolled down to, so typically holding is best too. With a big enough anchor 3:1 scope is plenty, at least it is in deep water, and all the more in a good quality and flat bottom.


Even with all the catenary pulled out, which gets harder and harder to do as the water gets deeper (which our German math wizz has just proven ), 3:1 should still give about 50% of the maximum possible holding power of the anchor, according to Alain's table.



According to another one of Alain's tables, a typical variation in holding power due to bottom quality is 7x. So you will lose a lot more holding power by compromising on the bottom quality, than you will by making a small compromise with scope. And if you throw a sloping bottom into the equation, it's even much, much worse.



My biggest challenge with this was in Greenland, in Scorseby Sund, where the shore is mountainous and the bottom slopes very steeply. Behind Milne Land the narrow sound, just a few cables across, is supposed to be 1000 meters deep. It was really hard to find any place to anchor, and never in less than 30 meters of water. The logical thing to do is shore ties, but I don't like shore ties.

[valhalla360]

After reading thru this thread, I keep coming back to the old computer saying...garbage in = garbage out.

Not suggesting the math is wrong but when you don't account for all the variables, you often come to incorrect conclusions.
- In shallow water, you often are relying on friction between the chain and seabed as much as cantenary to absorb loads.
- Snubbers are commonly used on cruising boats (the math scales up differently on large ships, so it's not directly comparable).
- In storm force conditions, there really isn't an option to assume the pull is parallel to the sea floor (or that the sea floor is perfectly level).
- Etc...

Empirical evidence is often used over mathematical calculations because if it passes the empirical tests over a long period of time, it works.

Not suggesting an understanding of the math is a bad idea just that once you set aside a lot of variables to simplify the calculations, you can't make statements about "the best".

[Lodesman]

Regarding the British Admiralty formula - as was noted in the paper, it gives a very conservative amount of scope; what is not noted is the BA formula is designed to hold a ship in 60 kt winds. And it is based on having all-chain in a size and weight that is commensurate to the size of ship. It is worth noting that there is a different formula for aluminum-bronze chain which is heavier, and gives a shorter scope.
It was noted by the author that this is not suitable in tight anchorages; this is true, so I understand his thought process in trying to determine the minimum scope required for specific conditions. I think this is generally a flawed approach to anchoring. You shouldn't have the absolute minimum amount of chain out, as that won't hold if conditions suddenly worsen without warning (not that this ever happens). It is generally preferable to have enough chain out to handle the foreseeable conditions, plus a reasonable "safety factor." I actually use a modified version of the BA formula. Sq rt(depth in feet) x 30 = feet of chain. I tend to anchor away from the crowds.

On a separate note, I cringe at peoples' notions of catenary disappearing completely. At the point that the last link comes off the seabed it is still pulling horizontally, and there is still a catenary in the chain, even though it will appear bar-taught and straight from your vantage on the bow. I think Alain's site has a formula to calculate the force required to get to that point. To pull the chain to a straight line requires considerably more force. This is of course dependent on the length and weight of the chain, but it is certainly possible for many cruising boats to have sufficient chain, such that catenary would never disappear. IIRC, this is around 350 feet of chain - the force required to pull this into a straight line would part the chain - assuming the anchor held fast and the chain held fast to the bow.

[rslifkin]

Of interesting note, I found with Alain's calculators, that even with a mixed rode in deep water, where adding scope doesn't change the amount of chain, adding more scope increases the force required to lift the last link of chain. Seems to be a function of angle of pull against the chain, where you have less lifting leverage with longer scope.

[Dockhead]

Originally Posted by Lodesman

. . . On a separate note, I cringe at peoples' notions of catenary disappearing completely. At the point that the last link comes off the seabed it is still pulling horizontally, and there is still a catenary in the chain, even though it will appear bar-taught and straight from your vantage on the bow. I think Alain's site has a formula to calculate the force required to get to that point. To pull the chain to a straight line requires considerably more force. This is of course dependent on the length and weight of the chain, but it is certainly possible for many cruising boats to have sufficient chain, such that catenary would never disappear. IIRC, this is around 350 feet of chain - the force required to pull this into a straight line would part the chain - assuming the anchor held fast and the chain held fast to the bow.

I used to think this, too, until others proved otherwise beyond a doubt, in this thread: https://www.cruisersforum.com/forums...in-215250.html, "Myth of the Bar-Tight Chain".

Catenary DOES go away at a certain force. We found the limits of this -- 200m of 1/2" chain will still have meaningful catenary at its breaking strength; but 100m will not.

Alain's calculators (have you played with them?) will give you the exact force required to lift the last link of chain at various combinations of depth and scope. In many cases it corresponds to not all that horrendous conditions. Once the last link is up, you are not far from the point where the pull is practically straight.

It is true that you will never get an absolutely straight line before the chain breaks, but this is not relevant. "Absolute" is not needed. The ability of chain to absorb energy decreases, the more you pull the catenary out, to the point that the chain behaves very much like a bar, and in most cases, these are not survival conditions. If this were not so, then we wouldn't need snubbers at all.

For that reason, we need to consider whether scope is enough to limit angulation to an acceptable level even with no catenary. In my own case, I feel fine in a good bottom on 3:1 in deepish water. and have been through a bad blow on 2.5:1. Obviously prefer 5:1 or 6:1 if I have room. But more scope also dramatically increases the risk of fouling something on the bottom (swept area of your swinging circle is pi r^2, obviously). So as I anchor a lot in rocky bottoms, I often shorten up scope when I'm afraid of this. It's really good to know the limits of this.

[conachair]

Originally Posted by Dockhead

The ability of chain to absorb energy decreases, the more you pull the catenary out, to the point that the chain behaves very much like a bar,

Which is actually just plain wrong, though very often stated..

So with a perfect anchor perfectly set which doesn't budge a mm under load, after the boat shoots off in big gust then is brought up quickly by the chain where does the energy go?
There's nowhere to go but into the chain (less a bit of warming to water and noise).
What you should be interested in is how quickly the energy transfers , which in the case of a chain which already pretty tight is very quickly, just a few feet for the force to get huge!

So the force is high, put a snubber in the energy transfer is slower, less force. F = MA. Energy still has to go somewhere.

[Dockhead]

Originally Posted by conachair

Which is actually just plain wrong, though very often stated..

So with a perfect anchor perfectly set which doesn't budge a mm under load, after the boat shoots off in big gust then is brought up quickly by the chain where does the energy go?
There's nowhere to go but into the chain (less a bit of warming to water and noise).
What you should be interested in is how quickly the energy transfers , which in the case of a chain which already pretty tight is very quickly, just a few feet for the force to get huge!

So the force is high, put a snubber in the energy transfer is slower, less force. F = MA. Energy still has to go somewhere.

You are just quibbling about terminology, and I know you know what I'm talking about. This is just obfuscation.

The way I use "absorb" is not even wrong -- energy IS asborbed -- temporarily.



"Absorbing energy", the way I'm using the phrase, to be more precise (but not actually more correct), is reduction of the peak forces by spreading out the transfer of energy between boat and anchor in time. Of course the energy doesn't disappear, but it is absorbed temporarily and transferred over a longer period of time. OK, are you happy now? The point is that as the chain tightens, the useful reduction of peak forces by the action of catenary fades and soon disappears. It does start to behave like a bar. THAT is why we need snubbers.

[MathiasW]

Originally Posted by valhalla360

- In shallow water, you are often relying on friction between the chain and seabed as much as cantenary to absorb loads

Now, that is a myth! If you simply work out the force a chain lying on the seabed can withstand, you will be disappointed. It is friction coefficient times weight in water. The friction coefficient is less than one. So, the force required to pull this chain on the seabed is less than the weight of the chain.

Assuming a 10 mm chain, which is 2 kp/m weight in water, and 30 m of chain on the seabed, this is just 60 kp, which is a small fraction of the holding power a, say, 30 kg anchor could develop. And it is only twice the weight of the anchor. So, you seem to feel safe if three anchors are just lying on the ground, not dug in at all???

So, this chain lying on the seabed really does not improve holding power by a lot.

What it does, though, is to add room for the chain to gain more potential energy when needed and thereby reduce shock loads. But this is an effect due to potential energy, and not due to friction.

And this is precisely, what I had calculated with my formulas...

Originally Posted by valhalla360

- Snubbers are commonly used on cruising boats (the math scales up differently on large ships, so it's not directly comparable).

Why should the maths not scale? The physics does. Some variables will scale differently than others, but all scale. If that were not the case, why would small models in a water tank be of any use to study? But professional vessel designers do this all over the world! Some folks claim catenaries only work for small vessels, but not for large vessels, others claim the opposite. I happen to have a bulletin of the Japanese Loss Prevention and Ship Inspection Department, which works with the catenary formula. So, at least they seem to believe in it. And for sure there is a lot of money at stake there, so they had better get it right...

And yes, on our vessels we can and must use snubbers / bridles. They help a great deal. But they will not absorb 100%, it is always a sharing of the load between the snubber and the chain, as otherwise the snubber would not engage. Think of it like a voltage divider in an electrical circuit created by two resistors in series. Consequently, it is good to know how well the chain can cope with its part in the deal...

Originally Posted by valhalla360

- In storm force conditions, there really isn't an option to assume the pull is parallel to the sea floor (or that the sea floor is perfectly level).

Again, the catenary formula will ALWAYS yield a chain length where a horizontal pull at the anchor is achieved, no matter how hard the wind blows. This is not an assumption, but simply physics. If you pay out enough chain, the pull will be parallel. Period. Now, of course, it can happen that you do not have enough chain for doing that, but it is still interesting to know how much chain would have been needed. And, funnily enough, it turns out it is less chain than one might have thought, at least when anchoring in deep water. There, it is often less than a scope 9:1 would suggest.

Some say they have SEEN the catenary to be gone. I doubt that. In a serious storm, you may have 100 m of chain out. With the anchor perhaps at 10 m. So, this is already a scope that is hard to tell with the naked eye. A difference of as little as one degree angle at the bow, with which the chain is pointing into the water, would make about 15-20 of meters difference were the chain hits the seabed. So, it may still be lying on the seabed and you just don't know it. And how can one tell as little as one degree angle difference on a rocking boat?

Originally Posted by valhalla360

- Etc...

No arguing with that... I rest my case!

I have said it before: I am not trying to imply my work has solved all problems and gives answers to everything. But it does show what chain length is minimally needed when using certain models for the real world. It helps at least me to make an informed decision when anchoring. I will adapt my formulas to account for factors at my anchorage that are not included in the model as best as I can, but at least I have a reference point to start from. And if I have more than one model, I can see how compatible their results are. And yes, I will pay out a little more chain than what the formula suggests.

Anybody is free to use whatever works for them, as long as they do not bump into me at an anchorage...

BTW - The hands the old Tasmanian Crayfisherman from St Helens known as "Big Bob" has - they are tiny compared to the hands of a German mathematician!

[conachair]

Originally Posted by Dockhead

You are just quibbling about terminology, and I know you know what I'm talking about. This is just obfuscation.

Completely disagree, like 11/10 - it's a very fundamental mistake in some fairly basic physics and doesn't help anyone getting a rough handle in what's going on. Nothing to do with terminology at all.

You should be focusing on how quickly the energy of a moving boat transfers , that's why a snubber helps.
Saying no more energy can be absorbed by the chain is just plain wrong and will confuse anyone trying to get and handle of the physics of what's happening, it has to go somewhere. Though to be fair this basic fundamental mistake even makes it into published articles. But if you're going to talk physics you really should get it right, it makes so much more sense that way and you look better than getting it wrong... .....

[MathiasW]

Originally Posted by Lodesman

It is worth noting that there is a different formula for aluminum-bronze chain which is heavier, and gives a shorter scope.

Thanks, I was not aware of this. Can you share this, please?

Originally Posted by Lodesman

It was noted by the author that this is not suitable in tight anchorages; this is true, so I understand his thought process in trying to determine the minimum scope required for specific conditions. I think this is generally a flawed approach to anchoring. You shouldn't have the absolute minimum amount of chain out, as that won't hold if conditions suddenly worsen without warning (not that this ever happens).

Absolutely, could not agree more! My point was to understand what the minimally required length of chain is at the worst anticipated condition, and then you pay out a bit more...

The nice thing about the catenary formula is that it allows you to plug in the value of the peak wind, and so, when you do that, you have already gone some way towards accounting for the worst, and you are, in fact, making a conservative judgement. Perhaps it should be called the minimally required amount of chain in the worst gust conditions that you anticipate?

What is harder to judge, I agree, is how much energy a swell will have. In this there is a larger uncertainty, and consequently the safety margin to apply will be bigger.

[MathiasW]

Originally Posted by conachair

So with a perfect anchor perfectly set which doesn't budge a mm under load, after the boat shoots off in big gust then is brought up quickly by the chain where does the energy go?
There's nowhere to go but into the chain (less a bit of warming to water and noise).

Well, actually there is: In a gust or swell, the anchor gear will pull the bow deeper into the water, meaning that more water will have to be displaced by the vessel. This is a temporary energy transfer, as - hopefully - the vessel will bounce back, eventually.

And, obviously, this will be tough on the cleats at the bow and so one would want to minimise this. Hence the snubber.

[conachair]

Originally Posted by MathiasW

Well, actually there is: In a gust or swell, the anchor gear will pull the bow deeper into the water, meaning that more water will have to be displaced by the vessel. This is a temporary energy transfer, as - hopefully - the vessel will bounce back, eventually.

Less a little warming of the water & noise Plus at high loads the chain stretch gets to be a more significant factor. All energy transfer is temporary.... eventually

[MathiasW]

Originally Posted by conachair

Less a little warming of the water & noise Plus at high loads the chain stretch gets to be a more significant factor. All energy transfer is temporary.... eventually

Correct!

[Lodesman]

Originally Posted by MathiasW

Thanks, I was not aware of this. Can you share this, please?

An approximate formula for

forged steel cable

is:

Amount of cable to veer in shackles is one-and-a half times the square root of

the depth of water in metres

, and for copper based cable (Aluminum bronze),

which is heavier and larger than forged steel, the formula is:

Amount of cable to veer

in shackles is equal to the square root of the depth of water in metres.

That's clipped from the 1995 edition. I don't think earlier editions mentioned it, but OTOH they stated something to the effect that the resultant scope would hold to 60 kts (iirc).

[Lodesman]

Originally Posted by conachair

All energy transfer is temporary.... eventually

Well first there's gravity, then added to that is elastic stretch - both temporary, but pull a bit more and you add deformity, which would come with some heating, that would dissipate to the surrounding water - lost energy, that you're not getting back