That’s how I arrived at the 20 shots in my post above.
I got confused by “l” being denominated as “minimum required length of anchor cable.” On reading a bit more closely, it appears that “l” is indeed the portion of the chain on the bottom. The “chain holding factor” also fits neatly with published figures of the coefficient of friction of chain on sand or mud, which sits in the 0.7 - 1.0 range.
So, in a steadily increasing wind, the force applied to the ground tackle increases exponentially, and the holding power decreases linearly.
In a big swell and wind (variable force - sort of like the difference between pushing on a nail or driving it with a hammer), the holding power of more chain also increases exponentially.
Simply put, as we all know, more chain is better.
In modest conditions the chain provides enough holding power (the anchor just starts in running out).
Similarly, when a tug and barge “lays on the wire,” the wire (and chain bridles and surge gear on the barge) provide all the holding power (there is no anchor).
The relationship between wind speed and force is exponential. Not sure exactly about adding more chain. Doubling the amount of chain does not double the holding power because only some of the extra chain lies on the bottom. The rest is added to catenary.
The extra catenary does absorb more energy for gust, surges etc but the calculation doesn’t take that into account.
At any constant force and depth, the catenary stays the same. Any extra chain that is paid out just lays on bottom increasing holding power.
If there are gusts and/or swells causing surging on the chain, as is typically the case, the catenary increases temporarily and then decreases. The increased amount of chain off the bottom in the catenary is proportional to the shock force being absorbed, up to the point where the anchor starts to drag.
The points on the diagram where the exponential force curve intersects the straight lines showing the linearly decreasing holding power of different set quantities of chain suggests to me that putting out extra chain also increases holding power exponentially to match the force, up to the point where holding power is exceeded (dragging).
I might be reading it wrong, but the paper seems to be dealing with increasing wind force as additional cable is put out: “…by the increased critical wind speed.”
It’s worded somewhat confusingly. “Critical wind speed” is the result of a calculation, not actual wind speed. Laying out extra chain increases calculated “critical wind speed” because more chain is laid on the bottom, just not the full amount paid out.
As a practical matter in the example not much is in catenary. One extra shackle (27.5 meters) laid out, 23.6 meters on the bottom and 3.9 meters in catenary.
If there is no chain laying on the bottom, then there is an upward vector to the force applied to the anchor, which would reduce its holding power.
Yeah, when the shank lifts, in a way it could be thought of as acting more like part of the chain.
There’s some diagrams somewhere but IIRC rule of thumb lifting 15 degrees cuts holding power of the anchor roughly by half.
The OP was about the accuracy of the video. I agree with @Klaveness in that it’s a bit confusing because it’s very uneven, good in places, not so much in others.
It starts out with a depiction of the scene in the movie “Battleship” with an anchor stopping the ship, he does say the movie take artistic license which is true but it does hurt the credibility of the video. Not least because in the movie the anchor was used to turn the ship, not stop it, which is better but still not credible. Pay out all the chain at full speed something’s likely going to let go, not sure what exactly.
Perhaps the greatest movie scene ever where the anchor is used as a ship handling aid, with all the usual caveats of course. 
The second issue is saying that the anchor doesn’t hold the ship in place but only holds the end of the chain. Again this is confusing but it’s just playing with words.
If the anchor was replaced with the required 20 shots of chain saying that the last 20 shots only hold the first 12 shots but not the ship doesn’t make sense.
If the narrator had instead said something along the lines of “it’s helpful to think of the chain and anchor as having different functions”, in that the catenary of the chain mostly absorbs energy and while the anchor provides holding power that would have been more clear.
Top speed of most battleships is over 3o knots so the anchor is going to skip along the water and bang into the side of the hull. If it was a commercial ship it would likely pierce the hull.
I’m not sure anything would let go because they were really far from shore in the Pacific Ocean and I doubt they had enough chain to even come close to the bottom.
Should have said something is likely to go wrong. Either way, wrt the scene in the movie, “artistic license” covers a lot of ground, especially in that movie!
Gents. Are we really using the movie Battleship as an even remotely realistic depiction of this subject? Artistic license or not, it is like using the movie Contraband as an example of life at sea. It’s preposterous.
Hence my original post about the YouTube video using it to teach anchor handling and coming to the wrong conclusions. We know it’s all wrong but the next generation is using YouTube to teach themselves and this video has millions of views.
My post wasn’t clear but when I mentioned letting go at full speed I was referring to actual practice, not the movie.
Is this wrong?
I do have actual experience with problems from too much strain on ground tackle, take to long to explain the context here to avoid the “coulda, shoulda, woulda” comments.
Don’t have any good references with actual numbers etc. that I can think of off-hand.
Just this from “Shiphandling for the Mariner” .
As a young 2nd mate on a VLCC anchoring off Bahrain we let go the anchor as ordered but our speed over the ground was more than the bridge realised. Playing a fire hose on the brake did control the flames but did nothing to slow the cable. At 8 shots I think it was we beat the feet and the bitter end toured around part of the forecastle before making its way down the hawse pipe.
The other was as a lowly sprog on the bridge of a frigate entering the harbour at Portland, UK., at night.
This was before walkie-talkies and at night lights were used, green for stbd anchor red for port, during the day semaphore flags were used.
The ship was doing about 14 knots and this red light appeared on the forecastle. The CO was angrily demanding what was going on and a signalman was making inquiries over a sound powered telephone to his opposite number on the forecastle. The OOW turned to the CO and said “ the lights gone out”, and there was a fully audible roar of anchor chain from forward as the port anchor was let go.
Where I was standing I could see the navigating officer had been consulting his notebook with a red torch.
Full astern was ordered and there was no damage.
I followed Sargent Shultz advice. “I saw. nothing!”
Old greek ferries steam into harbours full speed, drop the anchor, slow, spin the boat on the pick, restart the direct reversing engines to the dock.
When they leave, full speed out and get the anchor on the way past.
Well rehearsed maneuver
Imagine if you were to try to get a heavy boat moving ahead just by pushing with your foot or whatever. It would take more force get the boat to slide ahead (or astern) than it would if you were to push the bow or stern to get the boat to pivot.
It takes more energy to get the entire boat moving than just to get the boat to pivot.
It takes less energy to turn a ship at speed than to stop one.
The very first line in the video in the OP is “Do you remember the movie Battleship?”. The narrator does say that in the movie that the Missouri is turning but in his demo later he talks about using the anchor to slow down without turning.
It’s an important point that the video misses. If the anchor of a large, heavy ship is dropped at high speed there are very large forces at play and thus little margin of error.
On the other hand if the anchor is used instead to assist in a turn the ship, forces are lower and there is a greater margin for error.
The energy of longitudinal vs yaw motion is also the principle behind anchoring with the wind on the beam.
You mean like this?:
