Dear All,
Thanks so much for this wealth of replies that the initial post generated. Really much appreciated and I think it helps me to make my
pitch clearer and easier to understand. As a result, I have updated the web page already and more updates will come.
If I may, here are my main take-aways from the above discussion:
- I am still on a different planet, but making an effort to return to Earth!
- The catenary form of the chain is not disputed as such, but it may only be partially developed, meaning that the anchor 'hangs' somewhere in the slope of the catenary. The discussion of a perfectly straight line is not leading anywhere.
- It is easier to develop a full catenary in deeper water and with thicker chain. As the chain can absorb a lot of potential energy in such a case, this scenario is also less stressful to the anchor
gear and the vessel.
- In shallow water, the catenary may well be there, but it has little room to get further tightened up in a gust or swell, possibly resulting in only a partial catenary when it most matters. This has to do with the inability of the chain to absorb further potential energy when it is hanging almost horizontally, no matter how long the chain is.
- Any energy that needs to be transferred in such a situation in shallow water can only go into
- elasticity of
steel of the chain, which is minimal (think of chain load x stretch of bare steel), and you really do not want to rely on this.
- the bow of the vessel being pulled deeper into the water and thereby temporarily displacing more water. Very stressful for the
cleats and anchor
gear.
- good long snubbers or bridles. They cannot do anything if the chain is permanently overloaded, but they do help a great deal to take most of the blow in a gust or swell. But they need to be of good length. Short snubbers of perhaps only 2 metres length will not help a lot. In shallow water they are most needed, but they should be always used, no matter what the water depth is.
- For deep water, the catenary equation will eventually increase less per every additional meter water depth than the scope approach does (in the limit, only one metre chain per one metre additional water depth). Hence, eventually, the scope approach will be more conservative than the catenary approach for very deep water.
- Going for a partial catenary to reduce the load at the anchor is the opposite of what one should be doing. All other things being equal, the catenary always has a smaller load than any partial catenary has.
- Chain working loads permitting, and also swinging circles permitting, it is better to use longer thinner chains than shorter thicker chains (both having the same total weight in the locker). You get more accessible water depth out of them.
- There is still some confusion as to what correct windage area to choose in all these calculations. Whilst some, like myself, work out the windage area from the geometry of the vessel or better, make a measurement with a gauge when connected to a pole, others choose the windage based on ABYC rules - a pointer to which would be useful to have for this community. At the moment, the difference between these two approaches may be as large as a factor of 4 as far as windage area is concerned. This difference may well be the reason why some believe catenary is difficult to establish for small recreational vessels, whilst others maintain it is almost always possible.
- Trying to model the effect of swell and other impacts on the vessel other than wind and gusts by using a larger effective windage area are likely to over estimate the effect in deep water. Such an approach will predict the anchor load to be the same regardless of water depth. In contrast, if the swell is accounted for as kinetic energy that needs to be absorbed as potential energy of the chain, the anchor load will decrease as the water depth increases. In an extreme case it can thus be that a massive swell head-on in shallow water creates a load that the anchor cannot bear any longer, and the only resolution is to relocate to deeper water, provided the swell is not even worse there.
- If a scope approach is chosen, then it needs to be adjusted to the water depth. This is of course known in the
trade, but it is difficult to know how to adjust when you are a beginner, or your vessel is vastly different to your previous one.
- Some argue that such an approach is flawed, because one should not cut corners and try to get away with the minimal chain length. This is a misunderstanding. The approach is to
plug in the values for the worst case that is assumed to hit me, like the forecasted gusts, add swell as it is forecasted, add to this a
safety margin, and only THEN
plug it into the formulas. Then the
safety margin is already built in. I rather know what the minimal chain length is, rather than guessing it. I can always pay out more chain. Well, not always, but then I know I need extra anchor watch.
- Some argue that all this is nonsense, anyway, as it does not cover all the variables needed to make a full assessment. Well, the latter is true, but that was never the intention. It is about model building that reflects reality as best as possible, and about knowing the limitations of these models. Then, with the full knowledge of those limitations, and accounting for particularities of the anchorage site at hand, one can make an informed decision.
- It has been criticised multiple times that snubbers / bridles are not part of my model. I do agree that they are a vital part of any good anchor gear, no question about that. But their effect can be modelled as a dampening of the effect that swell has on the chain. So, for instance, a swell of 1000 J energy may get absorbed to 90% by the snubber, and only 10% by the chain. In this case, all I have to do is to look up the curves for 100 J for the chain, and I have accounted for the snubber perfectly. How this split is done, 9:1 or whatever, is determined by the matching of the chain force at the bow with the force applied to the snubber. Others have done this, like Bjarne, but I have not.
- Seabed not being flat. Yes, this needs accounting for. If one does want to see the maths for that, then at least in the absence of swell it is straight forward. A graph has been added to my web page explaining this. In essence, one needs to subtract two catenaries from each other, to have the anchor 'hang' in the slope of one of the two catenaries. The same approach can also be used to deal with an imperfectly developed catenary.
- It has been argued that there is a lot of dynamics at the anchorage, the vessel is possibly tossing around quite violently, but the model is only about the static case with steady wind and everything else being at rest. Well, that is not quite true. My approach had been to include gusts in the standard catenary formula already, so that accounts for the worst case wind-wise. And, secondly, I include swell via the kinetic energy the vessel is picking up whilst at anchor. By requiring this kinetic energy to be absorbed as an increase in the potential energy of the chain, I do not need to follow all the details of the vessel's motion. It is an energy-balance approach. As long as the chain is essentially a catenary also when it moves back and forth, this approach seems reasonable.
- Dissipative energy transfers, like the drag of the anchor chain when it is moving through the water, or the vessel itself dragging through the water, have not been accounted for. Nor the anchor dragging...
Except for the latter - all these help towards reducing the required amount of chain.
- What has NOT really been commented upon in this thread is my basic approach to account for swell: I model it as kinetic energy of the vessel, which then subsequently needs to be transferred somewhere else. This could be a couple of places, but I picked the chain as
storage medium of energy. So, its potential energy. The gain in potential energy is simply calculated by taking the difference of two chains of same length: One, before the swell, which has some slack lying on the seabed, and the other after having absorbed all the kinetic energy, and being a perfect catenary with just the last chain link about to get raised from the seabed. As I had been able to derive explicit formulas for the potential energy of a general catenary, this analysis could be carried out with little numerical effort. In some sense one can argue that this analysis proves why snubbers are so extremely useful. Bjarne's calculator includes this effect, but Alain's spread
sheets to my knowledge not. HOWEVER, the fact that this part of the model had not been discussed in this thread only shows that I had done a rather poor job at explaining it. Apologies for that. I hope the web page is improved in this regard.
Hmm, this post got ever so slightly longer than I had thought...
Once again, thanks to all for their contributions so far!
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