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From: (no name) (no email)
Date: Thu Jun 28 2007 - 17:54:18 EDT
In Part 1 we looked at general considerations of anchopring. Now let us
calculate the strain on the rode and the forces on the anchor. We will start with
an all chain rode, then progress to elastic chain and nylon rodes. Don't be
put off by the equations that follow. The math in the first part of the essay
is used to justify the conclusions reached in the latter part. There is no
penalty for skipping the calculations. Of course if you like this sort of stuff,
you can work through the equations for your own boat. There will be no exam.
First, we must determine the horizontal force on the boat attempting to pull
the anchor loose.
WIND DRAG in lbs = .00339 * Cd * V * V * A
where: Cd = drag coefficient, a dimensionless number determined by wind
tunnel tests. For most pleasure boats this number is approximately .80. V = wind
speed in Knots. A = total frontal area in sq. feet. This includes hull, deck
house, mast and rigging.
Thus for an 80 sq. ft. area pleasure boat in a 60 kt. wind, the total wind
drag = 781 lbs. This frontal area approximates that of most pleasure boats in
the 30 to 40 foot range.
Current drag is equal to the thrust necessary to move the boat ahead in still
water at the same speed as the current. It can be found from propulsion data
by this equation:
CURRENT DRAG in lbs. = 325.5(P x RPM - 1216 x V) / (V x P x RPM)
where: P = propeller pitch in inches, RPM = propeller revolutions per minute,
V = velocity in knots, H = engine HP delivered at the desired speed. If you
know this, great. If not, it can be estimated by Keith's formula which comes
next.
Thus a 40 ft. yacht, estimated to require 20 HP to go 5 kt. with a prop pitch
of 8" at 1000 shaft RPM would have a drag of 312.5 lbs. in a 5 kt. current.
Keith's formula for (roughly) estimating required HP is:
HP = Displacement in lbs. x (V / 11.8 x LWL^.5)^3
Wave action loads are difficult to estimate since they depend both on the
length of the boat and the period of the waves. Basically wave action imposes
severe loads when the boat is pitching in resonance with the waves. Fortunately
in strong winds wave lengths quickly grow to the point where resonance is
unlikely for modestly sized pleasure boats. In most cases, the boat can be
reasonable well insulated from wave action if the weight of an anchor chain is
supported by a buoy and a pennant led to the boat. Nylon rodes are easier on the boat
because of their lightness and elasticity.
The horizontal anchoring load is determined by the sum of wind drag and
current drag, with occasional shock loads imposed by wave action. I usually
increase my estimate of the horizontal load by 20% to provide a safety margin for the
unpredictability of wave and other transient loads. We can calculate the
horizontal anchor rode requirements using the following equations:
RODE TENSION (in lbs.) FOR CHAIN
T = Th + (w * d)
where: T = maximum tension in line in lbs. Th = horizontal load in lbs.
(i.e. the sum of wind and current and wave drags) w = underwater weight of rode
in lbs/ft. d = depth of water in feet (including max. tides and wave heights)
Next we must determine the vertical load. This is a major consideration for
chain rodes where the weight of the chain impedes the ability of the bow to
rise in waves.
VERTICAL LOAD (in lbs.) FOR CHAIN
Tv = sq. rt. of ((T * T) - (Th * Th))
where: Tv = vertical load in lbs.
LENGTH OF RODE (in ft.) FOR CHAIN
Length = Tv/w
Let's see what it takes to anchor our hypothetical 80 sq. ft. area boat in a
60kt. wind.
The underwater weight of steel chain is .87 of its weight in air. If we
assume a horizontal load of 1000 lbs, 30 ft. of water and 5/16" chain (which weighs
1.0 lbs./ft. underwater) we find that: Horizontal load = T = 1000 + (30 *
1.0) = 1030 lbs. Vertical load = Tv = 247 lbs. Rode length = 247 ft. Scope =
Length to depth ratio = 8.23
Now if we increase the chain diameter to 1/2" (underwater weight is 2.57
lbs./ft.), we get: Horizontal load = T = 1000 + (30 * 2.57) = 1077 lbs. Vertical
load = Tv = 400 lbs. Rode length = 156 ft. Scope = Length/d = 5.2
Increasing the chain diameter lets us cut the length of the rode by 91 ft.,
about a third, however it almost doubles the vertical weight on the bow and
would materially decrease its ability to lift over waves. A effect similar to
using heavier chain can be achieved by fixing a weight roughly equivalent to the
difference in weights of the heavier over the lighter chain near the midpoint
of the rode. In both cases the chain is far stronger than necessary to handle
the load, the scope reduction attributed to the larger chain comes simply from
its increased weight. The anchor, of course, has to be capable of holding the
horizontal load. Anchor makers generally give very optimistic figures for the
holding power of their products. For example, if the Danforth catalog is to
be believed, a 5H anchor, just about key chain sized, holds 2700 lbs in hard
sand. A muddy bottom would require a 20H to 35H anchor. My own approach is to
use the biggest anchor my wife can lift and the longest nylon rode that I can
let out in a given anchorage.
There are several basic problems when using a nylon rode. The first is to
assure adequate elasticity. New, three strand standard laid nylon rope can
stretch up to 50% before breaking. The stretch at lower tensions is almost directly
proportional to the strain. An elastic nylon rode should be designed to
stretch about 15% to 20% between maximum and no load conditions. If the rope is too
thick, it will not stretch sufficiently and you have the equivalent of a chain
rode without the catenary effect of chain. The best way of determining the
optimum diameter for the nylon rope is to calculate the maximum expected tension
on the line, double it, then consult a table of nylon rope strengths to
determine the required diameter. This assumes, of course, that the rope is
comparatively new, has no knots or abrasions, and has a well formed and thimbled eye
splice at the anchor.
If you don't have a table of nylon ropes available, the following equation
works pretty well for determining rope diameter:
Diameter = sq. rt. of ((3.1416 * Tension)/18000)
The second main problem of nylon rodes is abrasion. Most nylon rodes fail
because they rub against rough surfaces and chafe through. One solution is to use
chafe protection at vulnerable points. The elasticity of nylon works against
it in extreme conditions. The stretching and relaxation of dry nylon can cause
enough internal friction to actually melt the fibers and weaken the rope.
This doesn't happen if the rope is kept wet. The chafe protection must be porous
enough to let water penetrate in storm conditions. An alternative plan is to
use a dacron rode. Dacron is highly resistant to chafe. But while an all dacron
rode will not wear through, it has very little elasticity. A composite rode
was proposed by MIT researchers. A nylon rode of the appropriate strength and
length was spliced to a dacron pennant which was attached to the cleats of the
boat. The splice, consisting of interlocking eye splices at the junction, was
under water. Tests to destruction showed that the rode never failed at the
splice.
Thus our hypothetical pleasure boat, anchored in a 60 kt. wind, with an
anchor rode tension of 1030 lbs. would require a nylon anchor rode .424 inches in
diameter. In this case I would use a 1/2" rope. In a 40 kt. wind, a 3/8" rope
would be more than sufficient.
Most modern lightweight anchors are designed to work with the pull on the
shank being no more than 8 degrees above the horizontal. This includes Danforths,
Fortresses, Ploughs, Deltas, and their variants. The sine of 8 degrees =
.139. To achieve the required angle with an unweighted line, the line must be
(DEPTH/.139) long or about 7.2 times the depth of the water. This is where the
famous 7:1 scope requirement comes from. Any weight placed near the shank
lowers this requirement. So a length of chain placed next to the shank of a
lightweight anchor satisfies two requirements at once. It lessens the slope of the
line and protects the vulnerable nylon from abrasion. Since the anchor rode is
only as strong as its weakest link, the chain should have at least the proof
strength of the breaking strength of the line. This condition is usually
satisfied by chain one size smaller than the line size. Thus a 1/2" line with a
breaking strength of 7100 lbs. should have no less than a 7/16 chain with a proof
strength of 7200 lbs. A 3/8" line can use a 5/16" chain, etc.
Calculations for the optimum length of chain on a combined nylon chain rode
are complex. However William van Dorn in "Oceanography and Seamanship"; Dodd,
Mead (1974), presents a graph based on calculations for anchoring
oceanographic vessels in storm conditions. It suggests that the optimum chain/nylon
combination for anchoring vessels < 50ft. in 30 ft. of water under storm conditions
is a 20% chain, 80% nylon rode with an overall scope of 6:1. Assuming that
the boat's bow chock is 6 feet above the water and that the waves are 4 feet (8
feet peak to trough) this works out to a 240 foot total rode comprised of 48
feet of chain and 192 feet of nylon. Clearly these are extreme conditions. In
shallower water the rode could be reduced proportionately. However, the length
of chain required approximates one boat length and a good working rule for a
combined rode is a boat length of chain plus whatever nylon is required to give
a 6:1 scope. In shallower water, the scope should be increased, within swing
limitations, to 7:1 to permit the bow to lift more easily to the choppy waves
near the shore.
Although a 60 kt. wind is well into gale force levels, hurricane winds are
much stronger. Even a minimum force hurricane will produce forces on the anchor
and rode 50% greater than given in the above calculations. If you want to cope
with 120 kt. winds, anchor and rode will have to be four times stronger and
it is unlikely that a single anchor of a size usually carried aboard a trawler
will be sufficient. Multiple anchors and rodes will be required in these
conditions. And for God's sake get off the boat and spend the next couple of days
ashore. Trust the Force and your insurance company.
In summary, at a fixed anchoring depth, the longer the rode, the less chain
required. The shortest rodes are achieved with all chain, the heavier the
better, but the penalty is increased weight and handling difficulty, and the
slightly increased possibility of catastrophic failure if the chain stretches taut.
Don't drag!
Larry Zeitlin
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