Tuesday, March 5, 2013

Anchor Ratios for the bored

Warning, This post contains maths, and other boring concepts, it some of the information has been superseded by better formulas and more information. another post on this will soon follow.
I have been playing around with Excel, you can do interesting things on it if you are bored enough (I was on a plane, planes are boring). Anyway I had been pondering a thread on Cruisers Forum regarding is a big anchor better than a smaller one?

I was keen to see how the law of Mechanical Similitude  applies to anchors and boat sizes.

I found it very interesting, much more interesting than the plane, although Karen, sitting next to me did not share my excitement...

Anyway, for me the discovery's were interesting, if you take the view that holding power is proportional to the anchors blade area (edit: I have since found that it needs to include depth of set as well, so it is normally proportional to the weight, or area times depth of set) then the benefits of going larger are not as great as you might at first think. In my case going from a 20 kg anchor to a 30 kg anchor then the weight increases by 1.5 but the anchors area only increases by 1.3. Useful, but not as big a benefit as at first you might think.

Now to get a bit controversial and way out of my depth, but thinking about what actually holds anchors into the seabed in some bottoms it occurred to me that the shear strength of the seabed might well play a part. and if that's the case then the circumference of the anchor would play a part in it's holding by defining the shearing boundary. Now the little bit I read on soil dynamics on Wikipedia confused the crap out of me so don't go thinking I am any sort of expert...  But the bad news for big anchors is that up-sizing the weight to 1.5 only gives you 1.09 more shear strength. Interesting, not sure how relevant it is, in soft mud area is critical, in clay or firm mud maybe shear strength comes into it? Any soil technicians out there? (edit:still havent found out anything about this, but probably not a big factor)

Thinking more about it the shear strength would also include the thickness of the substrate it needs to pull through, so it may increase as an area rather than a length, the same as the surface area of the anchor...(edit: now I am getting closer... see the next anchor post, comming soon)

I was also curious to see how the colossal anchors carried by some bigger cruising boats like Morgans Cloud and the Dashews stack up against the boat weights and windage. Now boats don't scale properly, no smart designer is going to directly scale up directly. What normally happens is boats get relatively narrower, lower and lighter as they get bigger, so my scaling is rather crude, but it is interesting to see what happens if I scale Sunburst up to 60 foot (18m) by a factor of 1.50. the displacement goes up to a whooping 33 tonnes, 3.37 times more to about 34 tonnes. The windage only goes up by a factor of 2.25. This would require a 70kg anchor to keep the ratios about right. (edit: I think I have this sorted out now, anchors scale in proportion to their weight. so not quite such a big anchor would be needed)

The biggest winner with a big anchor is the tip loadings which scale with the surface area, double the weight gives you 1.59 times the tip load, provided your dip is as sharp as the smaller anchor. This will help your setting in some bottom types since the extra weight of the anchor helps drive this sharp tip into seabed.

Now it also occurred to me in a light bulb moment during a restless nights sleep pondering exciting things like anchors that we compare yachts ratios with numbers such as the Sail Area/displacement ratio, and Displacement/Length ratio. So why not compare anchors to boat size the same way, this would stop half the internet arguments about if your anchor is to big or to small, we could exchange our numbers and we would know exactly where we stand on the continuum of permanent mooring to toy anchor.

So here are a few ratios to play with, they might make sense number wise or just be weird, maths isn't my strong point, I would be interested to hear your Ratios.(edit: I now have a better ratio Length squared/Anchor Wt, should be about 35 or less, 7 or less for metric)

Displacement to Anchor weight ratio = Disp (kg)/Anchor weight (kg)

Displacement to Anchor Area = Displacement (tonnes)/(Anchor fluke area (cm^2)/1000)^(3/2)

These are the easy ones, windage is another critical factor, probably much more important than vessel displacement, depending on the situation. I got a crude idea of my windage in m^2, took side windage and head on windage and averaged them to reflect the normal anchor ripping out angle of 45 degrees. made a bunch of crude assumptions about rigging drag and then even I got bored... So I have a crude guess really. keen for any input on better ways. I need to refer to a few yacht design books to see if I am ballpark but hey a guess done with maths is much better than a guess done by.. well just guessing, isn't it?

So the ratios for windage are

Windage to anchor area ratio = windage (m^2) / (Anchor fluke area (cm^2)/1000)

Windage to anchor circumference = windage (m^2)/(circumference (m))^2

Anyway, even I am getting bored now, My guess is the first few people with a real maths and engineering background will die laughing and probably make spot a heap of mistakes in my logic or my math, but if by some chance the laughter isn't fatal, please make comment so it can be improved or deleted.

Here are my results after some tweaking of the original formula to get the brackets in the right place.. DUH



PS can't work out how to add the spreadsheet to this blog. Any ideas?