I was interested in replacing the steel cables used in the Weekender standing rigging as specified in the Weekender Plans with a modern synthetic rope that has a traditional old school three stranded hemp rope look.
The question that immediately arises is "How strong a rope do we need for the shrouds in the standing rigging?"
The steel cable specified in the plans is ridiculously strong (many thousands of pounds of working strength), and a single cable is more than strong enough to suspend the entire 600 pound Weekender hull plus any reasonable amount of crew from a crane without risk of breaking. Does the cable need to be that strong, or is that simply the strength of the cable most commonly found to be in stock at the local hardware store or home center? Inquiring minds want to know.
I read an interesting book titled "The Rigger's Apprentice" in which the author Brion Toss reveals the following astonishing facts, which I have paraphrased almost beyond recognition ...
The breaking strength required of the lines used in the standing rigging have almost nothing to do with the strength of the wind encountered by the boat, or the size of the spars used to spread the canvas. Instead, the standing rigging needs to only be strong enough to stand up to the Righting Moment of the hull that it sits on.
Consider a sailboat with all sails set in the midst of a hurricane on a beam
reach with the wind perpendicular to the hull. Obviously, the wind is trying to
knock the boat down and the hull will heel right over. In lesser winds, the
boat will heel over until the heeling torque of the wind acting on the sails
equals the righting moment caused by the sideways displacement of the hull's
center of buoyancy with respect to the center of gravity. The red dot in the
animation
represents the center of gravity of the hull, while the blue area
shows the lateral displacement of buoyancy as the hull heels over. The
horizontal difference between the center of gravity and the center of buoyancy
creates a lever arm that tries to twist the boat upright. The entire weight of
the boat and crew rests on this lever arm, so considerable forces are involved.
The windward shrouds in the standing rigging must be able to transfer that
righting moment from the chain plate attachment point on the edge of the hull
back up to the top of the mast to resist the wind.
Using an Excel spreadsheet, I constructed a Reinman-sum approximation of the integral calculus needed to calculate the Righting Moment of the Weekender hull. The math is not very complicated, just extraordinarily tedious. First a cross sectional area model of the hull at each station line was constructed on a one inch by one inch grid with the stations all stacked on top of each other like a loaf a bread. The densities of the materials occupying each square at all the stations, multiplied by the one foot length between the stations, ( density x dimension = wieght ) were added together in each cell on a "stations" worksheet. Next the cells were "rotated" around the center of gravity on another worksheet using the standard ( x' = x cos a + y sin a, | y' = -x sin a + y cos a ) equations for coordinate transformations in a Cartesian plane. Finally, the weights of the materials in each of the rotated cells were summed vertically while subtracting the density of fresh water ( 62.5 lbs/ft^3 ) for any cells below the inclined waterline to account for buoyancy. Each vertical sum was multiplied by the horizontal distance from the center of gravity, and the products summed horizontally to determine the final righting moment at that particular angle of heel.
The graph on the right summarizes the righting moments of the Weekender hull at
various loads and angles of heel. The lowest green curve shows the righting
moments of the empty hull. The other two green curves are for 200 lbs ballast
and 400 lbs ballast. The blue curves are 84 lbs, 319 lbs, 567 lbs, and 827 lbs
ballast and represent the one inch, two inch, three inch, and four inch load
waterlines, which puts the pounds per inch immersion (ppi) of the hull at about
250 lbs per inch. The red curve shows the righting moments of 400 lbs of captain
and crew sitting on the windward seats with the maximum righting moment marked
with the red diamond at 1,400 ft lbs.
The advantages of moveable ballast (crew) in a small sailboat are brutally
visible in the graph on the right.
The red diamond illustrates two 200 lb sailors sitting on opposite sides of
the boat with the boat sailing at 18 degrees of heel and generating 800 ft
lbs of righting moment. Moving the sailor from the lee seat to the windward
seat increases the righting moment of the boat and allows the following
events to occur. We can bring the heel back to a more comfortable 4 degrees
of tilt (horizontal arrow) and still generate the same 800 ft lbs to
balance the wind. We can also maintain the same heel angle (vertical arrow)
and generate 1300 ft lbs of righting moment and sail in a much stronger
wind. Or we can do anything between these two extremes, ie. faster winds and
less heel. Clearly, 400 lbs of movable crew weight on the windward rail
(red curve) makes the boat stand up to the wind much more stiffly for
comfortable angles of heel than 800 lbs of evenly spread ballast
(the upper blue curve).
In way of explanation, the kinkiness or jaggedness of the righting moment data points in the graphs is caused by the one cubic inch resolution of the mathematical model of the hull. As the hull heels, the location of the waterline needs to move slightly up or down so that the total buoyancy of the immersed volume balances the weight of the boat. Unfortunately, the location of the waterline is constrained in the model to one inch increments, which allows an error term of nearly ½ of the amount of the ppi value. This error term maxed out around 118 lbs, so 6th order polynomial trend lines were added to smooth out the curves. It is also interesting to note that the hard-chined shape of the hull is visible in the curves. As the hull is progressively loaded down with ballast, the curves trend more and more like the ideal stability curves described in design texts. It looks like the classic method of determining maximum righting moment by inclining the hull by 10 degrees (500 ft lbs) and multiplying the result by four (equivalent to 40 degrees of heel) leads to a result of 2000 ft lbs, which I believe is overly optimistic considering the small size of the boat and the limited seating options for the crew.
According to the spreadsheet, the maximum Righting Moment of the empty weekender hull peaks out at about 700 foot pounds at 43 degrees of heel. Adding crew weight to the windward rail of the boat increases the righting moment by about 2 ft lbs per pound of crew and decreases the maximum heel to about 28 degrees as it's hard to get the crew much farther than two feet away from the center of gravity of the five foot wide hull. So adding 400 pounds of crew sitting on the windward seats at 28 degrees of heel brings the maximum Righting Moment up to about 1,400 ft pounds.
The shrouds are attached to the hull via chain plates located about two feet out from the centerline of the boat, so we take the 1,400 ft lbs and divide by 2 ft and we get 700 lbs of force in the shrouds. Two shrouds per side means 350 lbs are carried by each shroud.
A quote of the Cordage Institute on the New England Ropes web site says the safe working load of rope should be 1/5th to 1/10th of the tensile strength of the rope for non critical loads and 15th for life line applications. Since we don't want the mast to fall on our heads, we need a line with between 3,500 and 5,000 lbs of tensile strength. Half inch MultiLine II has a breaking strength of 5,800 lbs so that should be sufficient.
But all of this is pure conjecture and turns out to be a monumental waste of
time because steel wire has two other properties that should not be ignored.
First, it has a very thin diameter compared to rope rigging which cuts down
on widage and allows the boat to perform better. Second, it does not stretch
as much as other cordage so the rig will not be bent out of shape and come
toppling down on our heads.
These two factors alone were responsible for the rapidity and completeness
with which wire rigging replaced its predecessors.
Oh well. But these are the kinds of things we have to waste our time with
during the winter months when we live above the 42nd parallel and the
temperature in the garage is below freezing and the outside wind chills approach
-30°F and it's just too damn cold to work on the boat.
