So, this took me a little while to compose…
I have read the very nice and seminal work
of Alain again, who has calculated the anchor
load - also in the presence of dynamic loads - in the force-time domain:
He models a gust as a stronger wind
force that gets switched on at a certain point in time and then stays on “forever”.
He starts by discussing the linear case (like, approximately, a rope), where the gust leads to an overshooting over the position the boat
would take, had the gust been a static wind
. Because he discusses the linear case, the point of maximal overshooting (red) is exactly a mirror point of the initial static point (green), where the position where gust and rope
tension balance out (yellow) is exactly in the middle.
The first diagram shows what happens. It is a plot of the displacement
of the boat
(away from the anchor) as a function of load - it is linear function, because we assumed a linear spring. When I integrate underneath the area (force times distance), I get directly the energy stored in the elastic rope
. It is simply the area when looking at the graph sideways.
The areas in light red are energies that are not (yet) loaded into the rope (as opposed to the dark red area), but rather go into the kinetic energy of the boat (first), or are drawn from it (later). At the yellow point the pull back force of the rope equals the gust load. But as the boat has some velocity, it overshoots until it has reached the red point, the most extreme tension in the rope. The two areas labelled E
must be the same, if no energy is lost
Please note the loads at the green, yellow and red points. The difference between green and yellow is the same as the difference between yellow and red. This is the best possible scenario and leads to the smallest possible maximal overshooting tension.
Nonlinearity makes the result worse. The next diagram is the same as before, but now not with an ideal spring, but rather a chain acting as a spring. Again, the gust keeps blowing even when the boat has reached the extreme position with the highest displacement
away from anchor
. But now the distance between the yellow and the red point is much larger than the distance between the yellow and the green point. In fact, the most extreme point, the red point, now has about 4 times the load of the static wind. That is a lot, given that the gust load is only twice the static load!
There are two graphs in this diagram acually. One for deep water
and one for shallow water
. There is no big difference between the two. And in this scenario (in view of the discussion further below) the vessel’s mass does not matter at all.
Next I show the same graphs, but now assuming that the gust only blows for a few seconds, resulting in this “energy spike” to the right. So, in this case, the vessel is still moving backwards when the gust is stopping allready.
Overall, the red points are much more to the left than in the previous graphs, which is obvious, as the total energy transferred to the boat is less. But one can also see now that in shallow water the red point is much further to the right than in deep water.
Conclusion: Shallow water can be dangerous in the presence of gusts. But a very elastic rope can help to address this shortcoming of the chain, as its characteristics is much more linear, and its ability to absorb energy temporarily does not depend on anchor depth
But, as we see, it depends on the nature of the gust. When it is simply a switching on of a higher wind force, then it does not matter whether it is shallow or not, the point of maximal overshooting will always be much higher than the point of the gust load. But when it is a sudden and short burst of wind, then it does matter.
The reality is - as so often - somewhere inbetween…
BTW - This last graph also explains why large commercial
ships do not see this effect at all. Because of their large mass m
, according to F
, for the same load F
, their acceleration a
in a gust will be much smaller, and hence the displacement backwards away from the anchor in a given time window is also so much less. This results in a very small energy transfer E
, and subsequently a very tiny overshooting.
In the very last graph I have shown as an example the case for a vessel twice as heavy, but with exactly the same windage area and exposed to the same gusty burst, hence same load F
. The energy E
the vessel has to absorb is only about half now, leading to less violent maximal loads.
Conclusion: Older pleasure crafts that are rather heavy for their windage area are less affected by these dynamic loads. They hardly move at all in a gust. But light-weight multihulls and modern "plastic" yachts with high windage area are strongly affected.
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