Mathematic approach to anchoring scope

[CarinaPDX]

This is good work for understanding the catenary effect. That is just one aspect of anchoring with all-chain, but a very important one. I like to consider the extreme case where the rode (of whatever material) is straight, or nearly so, and thus is not pulling horizontally - more likely for rope than chain but useful to think about. As long as the shank is not lifted too high the palms or scoop will cause the anchor to be pulled deeper; but once these surfaces are horizontal or higher then the anchor will pull out. The angle between shank and holding surface is similar on all anchors, and IIUC relates to about a 6:1 scope, which is what I think is a recommended minimum for rope plus short chain. The point of the catenary is to keep the pull angle below this limit while also absorbing shock loads.

I read Alain's article a long time ago, and found its insights useful. His calculation for an optimal chain/rope rode had/has me considering using longer chain on my rope rodes. The reality of cruising is that most cruisers use all chain and boats with rope/chain rodes don't play well with the neighbors as they swing differently with wind/current, and need more room. When I started out I had a manual windlass so was often tempted to use the rope/chain rode, and found it to be very reliable and a whole lot easier to recover. But in many places the all-chain was necessary in order to fit in; even when there was room to swing separately the occasional "chain snob" would feel the need to lecture me about the need to use all chain. With the addition of an electric windlass and chain counter/controls in the cockpit I now rarely use rope except as a second anchor, or a stern anchor for bow-to. But I always remember that the "best" rode, given the swing room, is chain and rope.

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.

Greg

[MathiasW]

Thank you Greg!

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.

Yes, excellent point, this is an additional complication, and in fact partially the reason why I had opted for a horizontal pulling at the anchor shank in the first place. If I do not know the seabed well enough, I am easily wrong by a few degrees right at the anchor position. So, with that accuracy in mind, I might as well go for a horizontal pull.

But when applying these results to any real-world scenario, one does need to factor in critical slopes that might exist at the seabed. In the longer German essay there is a small section on how to deal with that, but I did not take this over to the digests for now.

As you say, these are all different models, and it is up to us to interpolate between them as we see fit in a given situation.

[MathiasW]

Hi again,

Thanks for all the feedback provided so far! Really much appreciated!

In an effort to return to this planet , I have slightly updated the web page with an additional graph that hopefully explains the model around the potential energy as being the absorbing mechanism for incoming swell.

I have also tried to make it clearer that snubber or bridle are needed in the anchor gear and I am only working out the worst case here. It should be easy to calculate the effect of a snubber / bridle, if I know its physical characteristics such as the equivalent spring constant. With the formula for the force at the bow before and after the swell impact, I can work out the additional energy stored in the snubber / bridle. This is then on top of my Delta E used for all the graphs. Example: If the snubber can absorb 1000 J, then my graph for 500 J actually corresponds to 1500 J when using a snubber. So, the graphs will remain the same, they only correspond to different amounts of kinetic energy that can be absorbed.

[Montanan]

From numerous personal experiences when attempting to anchor in deep water I have found that when my rode is say 100 feet or meters short of the anchor touching the ground and the ratio of scope being below one, [and / or if the ratio by defacto approaches towards dividing by zero], things remains problematic. It would not matter if I used heavy chain, light chain, rope or a snubber, the damn thing will not set properly regardless of wind or current induced loading.

But then I have never actually dragged an anchor under those no ground contact, low scope ratio, deep water conditions, so maybe the author is onto something with all that math.

[barnakiel]

Originally Posted by carstenb

(...)



Amongst other conclusions is that in deeper water, you need less rode than in shallow water and vice versa of course.


(...)

Absolute? Relative? ;-)



ABSOLUTE "less":

Water 2 meters, rode 20 meters.
Water 20 meters, rode 10 meters (sic) ?????


RELATIVE "less":
Water 2 meters, rode 5x, rode 100 meters.
Water 20 meters, rode 4x, rode 80 meters, OK


I bet this meaning of less rode is more rode.


And only now .. I will go and see what you say ... ;-)))


I have this nice french xls sheet that has all the calculations. I bet you have it too.



;-)
b.

[barnakiel]

BTW


Following Alain's xls sheet I modified two things in our kit:


- replaced our anchor size for 1.5 x size - to get extra SURFACE (not weight),
- used 8mm chain in place of 10mm, went for 1.5x the length too.



We never noticed any differences in holding but we did notice now the boat sails about less. I lay this on the chain creating plenty of side drag when the boat tries to sail about. I think soft line creates x less side drag then.



From my experience, more SURFACE and LONGER chain both count.


b.

[MathiasW]

HI Barnakiel,

Yes, the very first post had indeed been a bit confusing...

Originally Posted by barnakiel

Absolute? Relative? ;-)

I bet this meaning of less rode is more rode.

;-)
b.

Normally, the deeper the water is, the more chain / rope you need. That is pretty normal and expected.

However, when swell is really strong, it can happen in shallow water that an inversion situation occurs. With that I mean that in more shallow water you need actually more chain than in deeper water. The reason for this is that in a chain-only situation - so no snubber - the chain finds it hard to absorb huge amounts of energy when it is already hanging almost horizontally. The two end points of the chain are fixed, it is tightly pulled already, and so it cannot really get raised much more.

This is a situation to avoid. At this point the anchor load is much higher than the mere wind load.

If I now relocate to deeper water, then the chain is not as horizontal anymore and it is easier for the chain to absorb energy. As a result, the required minimal chain length may even be a little less than in more shallow water.

But this is not a regime you want to be in to begin with. With a good snubber / bridle this situation can be diffused somewhat.

So, I guess my point here is that fleeing to the most shallow location when a severe storm is approaching, may not be the best possible tactics. If the swell is the same at two different locations, the deeper-water one may be preferable to give the chain a better chance to absorb energy. It can be shown that for a given fixed length of chain, the scope at which the chain can absorb energy best is close to 2.5:1.

As to the French Excel sheets - very nice, but they do not include the effect of swell as far as I know.

[MathiasW]

Originally Posted by barnakiel

BTW
- replaced our anchor size for 1.5 x size - to get extra SURFACE (not weight),
- used 8mm chain in place of 10mm, went for 1.5x the length too.

Yes, I fully support this conclusion!

Originally Posted by barnakiel

From my experience, more SURFACE and LONGER chain both count.

Exactly our design choice for our SAN trimaran.

[RichMac]

Originally Posted by carstenb

Amongst other conclusions is that in deeper water, you need less rode than in shallow water and vice versa of course.

That interpretation is not correct, of course you need more rode (chain length) i deeper water. His point is that the scope multiplier (rode/depth) should be different. You need a bigger multiplier in shallow water, because the chain is very flat, maybe 8:1. In deep water, the chain is more vertical, and the scope multiplier can be 3:1 or less.

I don't have a German mathematician or supercomputer on board, so I use the really simple formula:- 15m (50ft) plus double the depth. This is reasonably close to the ideal length, then adjust the 15m up or down to suit the conditions. Always let out an extra 10m (30ft) in strong winds.

[rondelais]

Interestingly, given a bunch of down time recently (guess why) I did some calculations like these. (Credentials - MIT engineering degree). I looked at chain (with snubber) and a chain/rope mix. He is right that in shallow water “scope” in its usual sense gives too short an answer, and in deeper water it gives too long. In fact, just multiplying the depth by some number is a pretty poor way to decide on anchoring strategy. First, you need to determine the force on the boat due to wind and perhaps current, which obviously depends on the geometry of the boat (hull, rig, and tendency to yaw at anchor) and the wind and current velocities. The catenary of the chain can be easily calculated both for the case of keeping the pull on the anchor horizontal and at some angle. Most anchors permit some angle; I have seen data that indicates that a modern anchor like the Rocna maintains 85% of its holding power at 3 degrees from horizontal (I have no data on the composition of the bottom for this) and pulls out at about 8 degrees. Given a straight line from boat to anchor, a “scope” of 10 gives an angle of 5.7 degrees. For 3 degrees, the straight line scope required is 19. Thus, in extreme conditions, where the rode is pulled taut, nothing but a long rode works. However, in more reasonable conditions (say under 30 kt wind), chain works well to reduce the angle. One interesting result I found was that a pretty reasonable rode for my boat (36 foot ketch, bow height 6 feet) is 100 feet of 5/16 chain plus 250 feet of nylon. I need 200 feet out to hold in 25 feet at 40 kt; all chain would require 180 feet, only 20 feet less. A kellett also works quite well to reduce the angle, and is more effective the closer it is to the anchor. This also means that for a given total ground tackle weight you are better off to use a shorter, heavier chain and more line since that concentrates the weight toward the anchor. Anyway, the bottom line is that for my boat and chain/rope rode I am OK anchoring in 15 to 35 feet using a “scope” of 4 up to 25kt, 6 up to 35 kt, and 10 up to 60 kt. Not sure the anchor holds at that point, though...I would be glad to share the math if anyone cares.

[MathiasW]

Originally Posted by rondelais

I would be glad to share the math if anyone cares.

Oh yes, most definitely! Perhaps as a PM?

And yes, the C down time is why I had (and still have) so much time at my hand. Have been at anchor here for almost 10 weeks now, and only once off the boat...

Needed something to stay sane... Can be argued, of course, whether I managed...

[MathiasW]

Hi RichMac

Originally Posted by RichMac

I don't have a German mathematician or supercomputer on board, so I use the really simple formula:- 15m (50ft) plus double the depth. This is reasonably close to the ideal length, then adjust the 15m up or down to suit the conditions. Always let out an extra 10m (30ft) in strong winds.

As far as further approximations are concerned, I do like your approach of using an offset. For example, when you take the catenary without swell, S = square_root(Y(Y+2a)), and then approximate this for 2a << Y, you get L =~ Y + a. So, here we have an offset...

This is the formula the folks from blauwasser.de are promoting. Problem is that this only holds for 2a << Y and hence not in a severe storm.

Now, your formula is L = 2Y + b.

I only did a brief comparison with the catenary results, but it would appear to me that I preferred the value of 'b' to be somewhat larger than you choose, even beyond adding another 10 metres. But if it has always worked for you, then ok.

[mayberrybfd]

Achems Razor !!! The simplest way is the best way !!!!

[waterman46]

Thank goodness nobody hijacked the thread by bringing up the subject of which type of anchor is best. OOPS, sorry!

[StuM]

Originally Posted by mayberrybfd

Achems Razor !!! The simplest way is the best way !!!!

Occam's ?


(Guess you're american and hear it pronounced it ark-em, not ock-am )