Calculating Sag

This thread will be a discussion of how to calculate sag in a stayed sail. Not just to determine the maximum sag, but to determine the magnitude and vector of that load on the stay, mast and chainplate.

If anyone out there knows how to do this, Please, Please, post the method. I don’t know how. But I will step forward, put my ignorance on the line, and let you watch me fumble around, trying to figure it out. I can see how understanding sag could be a trade secret and such a competitive advantage that one who knows may be reluctant to post it.

I’m pretty sure that more than one person has experimental data about sag, or who know how to deal with sag through experience and empirical trials. I hope you will share your knowledge to help me stay on the path so the mathematical models and formulas do not get too far from the real world.

With each posting, I hope to advance a bit more toward the answer. I will explain my methods and conclusions in each step, and am willing to discuss any part that’s unclear or questionable or wrong. I will try not to lecture, which my wife says I tend to and you have already witnessed here.

I am an engineer who works for a company that designs electrical substations and transmission lines. We deal with sag every day, but ours is a 2-dimensional black-and-white cartoon sag when compared with the 3-D color action-movie sag of sailing.

 

This depends on the flexibility of the mast, the elasticity of the stays and the rigidity if the hull girder. You need to specify these if you want to accurately predict the sag in the stay. Alternatively if there is a tensioner you work with a given stay tension.
In that case do you just want to know the load distribution on the luff or do you want the full monty ?

 

Getting Started

Sag is the result of a complex interaction of aerodynamic forces applied to solid, stranded and woven materials of varying strength and composition. So I’ll start with some SIMPLIFYING ASSUMPTIONS to get the situation into my cartoon world, where wind blows on a sail.

1. Air is at standard conditions at sea level, has a velocity of 21 knots, and is incompressible. The aerostatic and dynamic works of Newton, Bernoulli and Marchaj are applicable. The works of NACA above this velocity are not applicable.
21 knots is the top of Beaufort 5 and the recommended first reef, so it is a reasonable maximum. When designing your rig from the loads eventually determined here, you would need to apply the appropriate safety factor.
2. The sail is made of a perfect material, without bias or stretch, without seams or cuts, without thickness or permeability. Reality will be a refinement introduced by section, row and seat once we get in the right mathematical ballpark.
3. The luff is a straight line with no sag. Remember? Reality later.
4. The sail, when laid flat, has a perfect triangular shape with 3 straight sides and 3 angles that total 180 degrees.
5. The sail has a cylindrical curvature which has its axis parallel to the luff.

 

MikeJohns
I agree that all those things affect sag. I hope this thread determines a method to caclulate the sag with rigid mast, etc., and the forces in the stay/halyard.

Once determined, then these forces can be applied in calculations to determine the deflections in the rest of the rig. This can be used to calcualate additional sag.

Larry

 

Larry

In the following thread Tony Pearce has posted Gutelles musings on sail and rig interaction. Reading them may be a good start for you.

http://www.boatdesign.net/forums/sailboats/sail-loading-rig-rig-loading-vessel-2293-3.html#post35427

look at all of Tony's posts for all the chapters.

Then you might want to sketch some diagrams and specifically show what you want to determine .

cheers

 

Mike
Thank you for referring me to those posts. They clearly show the need to accurately determine the stay loads in designing a rig. They show how much guesstimating there is in the current design methods used to determine the load and deflection on the various parts of the rigging.

They also show the least understood design factor is how the force of the wind apecifically acts on the rigging. And example one of the posts gave was that the maximum shroud loads are determined by the righting moment of the hull. This is an adequate method that uses reactions to the wind loads rather than the applied wind loads. But it does not help design the halyard bearing, or determine the safety factor in the forestay, or the torque on the mast.

My limited and perhaps unattainable goal is to determine the magnitude and vectored direction of the loads in the forestay at a first reef wind speed. Designers may appropriate safety factors to this maximal loading when designing the rest of the rig. Designers may calculate deflections of the rigging knowing that in the more saggy deflected contition the loads will be less and the safety factor more.

In accomplishing this goal, the shape of the sag curve ma have to be determined. The shape of that curve (whether, parabolic, catenary, or something else) is dependent on the loads applied along its length. It is the end angle of the curve that determines the relationship between sag and tension.

In getting to that goal, initiallt the loads will be determined assuming a straight luff. This happens to be the conditions ossuring at a mast, so the initial loads may be helpful to designers and riggers in determining the bending and twisting forces in masts.

These loads then are used to determine the shape of the stay sag and the loads adjusted to the new curve until some reasonable balance is achieved. Then the vector and magnitude of the force applied to the mast and hull determined.

Or so I hope.
Larry

 

Sail curvature

CALCULATING THE SAIL SHAPE

It is wind pressure acting on the sail that generates the forces that cause sag. At 21 knots, according to the dionysis-sail-aerodynamics(1).xls, this pressure, q=1.50 lb/ft2. This spreadsheet is available in this forum and of which a tiny part is inserted here,

KNOTS M/SEC FT/SEC Beaufort dyn press
21 10.811 35.469 5 1st reef 1.50
From: dionysis-sail-aerodynamics(1).xls

Marchaj’s Table 2.3, at V=35 ft/sec, lists the pressure as q= 1.4565 psf. And the formula listed with the table at V=35.469 gives a pressure of q=1.497 psf. That’s such an astonishing agreement, I’ll use it eagerly.


A uniform pressure means a circular curvature to the sail, and both NACA and Marchaj used uniformly curved plates in their work. The tension in such a curved plate can be easily calculated using the hoop tension formula, which is used to calculate the tension in pipes, tanks, balloons, soap bubbles, and such.

According to the hoop tension equation, T=Rp. The tension in a curved surface is equal to the radius of the curve times the pressure. The tension is the same all around a circular hoop and in each part of it, like in a balloon. Taking a half circle, along any diameter, removes the trig, and shows the formula is valid for every point in the whole circle. And why small soap bubbles are spherical. Large soap bubbles can be non-spherical, if they are large enough to have air currents within them

View attachment Hoop tension.xls

The same is true in a triangular sail. The tension is equal from luff to clew along any line parallel to the luff. But, unlike a bubble or cylindrical tank, those lines get smaller as they approach the clew. So, there is less area for the pressure to act on and therefore a larger radius.


Next, will be an analysis of that varying radius and the curvature of the sail.

 

Sail Curvature

I’ll divide a sail into triangles by dividing the luff into equal segments as in the next illustration. The tension at the luff or base of each triangle is t, and at the clew these t’s add up to T which is also the force in the sheet, or at least the component perpendicular to the luff.

[Insert Subdivide Sail worksheet from Diagrams for Sag Calcs.xls]

At the midpoint of the sail, the area and force from the air pressure is half, therefore to keep T the same in T=Rp, the radius of the sail at that point is double what the radius is at the luff. At the ¾ point the area is 1/4 of that at the luff and the radius is 4x.

The radius at any point n depends on its distance from the luff is:

Rn=R/(1-cn),

where R is the radius at the luff and cn is the location along the chord with cn=0 at the clew and cn=1 at the luff.

[Insert Radius Graph worksheet from Diagrams for Sag Calcs.xls]

Unlike a sheet of metal, a sheet of sail cloth does not have a circular curvature, but a spiral. The use of circular curvature may be acceptable for aircraft wings, even delta wings on aircraft, but it is poor modeling for triangular sails.

 

Ignoring Sail Shape

IGNORING SAIL SHAPE

Until I had consulted with a few cohorts, I had considered the shape of the sail and therefore to angle between the luff and clew as the key factor in calculating the tension in the sail (T) for a given wind pressure (p). This is illustrated in the Vector Diagram (1).


VECTOR DIAGRAM 1

It shows the force parallelogram (a rhumbus, since it is equal sided) formed by the two vectors for the sail tension (T). one is tangent to the luff and the other to the clew, with the vector for the wind force (P) as a diagonal. In practice the tangent to the clew may vary 30 degrees or so from pointing to running. whereas the luff may vary by three times that angle, from well forward the beam on a dead run, to well aft the beam while pinching. And the sum of those angles never exceeds 180 degrees, unless the sail is backwinded, and as it approaches 180 degrees, the sail loses camber and becomes so flat it begins to wave like a flag. It is this angle (B) between the sail tension vectors that determines relationship between P and T:

P=2Tcos(B/2) T=P/(2cos(B/2))

The wind force (P) vector is the wind pressure times the sail area (pSA).
-------------------------------
At this point my analysis turns to sand running through my fingers because both B and p vary with heading, and SA may also.

How the wind pressure (p) varies with heading is shown in the dionysis-sail-aerodynamics(1).xls spreadsheet available in this Forum. So how it varies has been determined, and could be mathematically modeled. But it is not uniform and varies across the surface of the sail.

How B, the angle between tangents, depends on what shape the sail takes, which in turn depends on the pressure distrubution across its surface. With the assumption of uniform pressure distribution, the shape is not a parabola, but a spiral of a sort not determine here. Yet, we know that the pressure is not uniform. Marchaj has measured the pressure and found at the luff it is 10 times that near the clew (approximating from his graph). My mathematical model of my measurements of his graph gave wierd results: the leading third was parabolic which transitioned in less than 5% of the chord to a curcular arc in the middle third, and then transitioned in the trailing third to a curve with a radius so large that the relatively short segment was equally mismatched by all linear, exponential and trignometric approximations tried. All I can say is that it is not a straight line.

In addition, the sail area (SA) varies with heading. Well not the sail area as such, but the projected sail area. As in the hoop-tension illustration, the sail area used in class rules or handicapping is the circumference curve, whereas the projected sail area which the pressure is acting on is the diameter. If you put the clew and luff together, the projected sail area would be zero, even though there is an unchanged number of square yards or meters of cloth.

And if that wasn't enough, the angle of attack varies with height.

If you know of a rational method or have experimental data that can pin these variations down, PLEASE SHARE it with us.
-----------------------------------------
For now, I set B = 120 degrees. This means that T=P/1

 

This is not meant to be tongue in cheek at all, but

If it were me, I'd take the 100% practical approach.
Hire a good NA.
He will in all likelihood choose an existing boat with a great reputation that is very similar to yours and copy that rig, then based more on experience and intuition than math, adjust the rig specs accordingly. And you will never have to worry about "in theory, there is no difference between theory and practice....."

Some things are not best solved with pure math. Some things require art and experience.

 

PortTacker
I agree that a professional knows how to calculate sag.

So, again I ask if one will please show us amatuers how to do it.

Lacking that, I will continue to try, hoping for knowledgable inout, your input.
Larry

 

Is sag really this simple?

The answer to the title question seems to be a little yes and a lot no.

The conversion from wind speed to sail forces started to get very complex, but suddenly fell out into an astonishingly simple approximation:

T = P = pSA

The max load on the sheet equals the max wind pressure times the sail area which also equals the uniform load along the stay.

This max load is when pointing in a 25k wind, and where T is the component of the sheet load perpendicular to the stay.

There have been dialogs of whether the shape of the sagging stay is catenary or parabolic. My view is that the sag/span ratio of a stay (say 3" in 30' or 8" in 60') puts us very close to the nose of the two curves and the mathematical difference between the two curves in this region is very small. Even up to and beyond 6' sag in a 60' stay the difference negligible, and the numbers of either approach gives valid results.

My position in the debate is: 1) that most foresails are not attached to the stay in a continuous manner, so the shape of the curve in really is that of a segmented arch which is more accurately solved with vector analysis, and 2) both the catenary and parabolic formulas are approximations of this real-world condition

For calculating sag in a stay, I don't see the catenary equations, with hypebolic trigonometry and transendental mathematics, giving results that are significantly superior to the algebraic equations for a parabola.

I'm attempting to derive a formula that will calculate sag and stay tension, given the stay length and sail area. I've found that another factor separation, is needed. This is the straight line distance between the ends of the sagging stay.

PortTacker will be happy to know that this 'separation' is where a NA earns
his keep.

The separation is close enough to stay length when at rest, to say they are the same. But, 1) the stay stretches underload, and 2) the separation gets shorter under load. It gets shorter because the steel backstay stretches allowing the mast head to pivot forward. It gets shorter because the aluminum mast gets shorter under the compressive loads. It gets shorter because the fibreglass (or wood or metal) hull distorts under the loads from the rigging and wave action. It gets shorter, or longer, because of the geometry of the rigging.

MikeJohns, you are so right.

I am hoping to derive a reasonably workable formula for stays up to 200' or more with sag up to 1/50 the stay length. With this formula one should be able to determine stay size and a minimum sag. One could improve the accuracy of the sag value by determining the stretch in the stay, plus by applying the forces to the rest of the rigging and hull determining the change in separation, and then use these values in the formula. This can be repeated until the changes are small enough for the degree of accuracy you want.

Anybody have any advice on this?
Larry Modes

 

I do not think that is right as it takes additional tension in the sheet to maintain the camber from increasing.

So basically that particular simplification will vastly underestimate the actual sheet tension.