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Mathematics and Fly Fishing

The Garrison's Stress Curve:

A superfluous curve or Garrison's gag?

Everett Garrison was born in another time - just 27 years after Sitting Bull defeated General Custer. Wyatt Earp was still living, when Garrison built his first rods. Knowledge was not so easy to obtain at that time, which made it precious. When a man's reputation depended on his knowledge he was not going to give it away. We have the stories of how rod makers guarded their rod tapers, as were they the Holy Grail. Garrison continued that tradition in a very sophisticated way.
It is easy to get the impression, that Garrison told all he knew about rod design to Carmichael.
But a few things do not add up.
The explanation of how to draw the stress curve is unsatisfactory.

It is postulated in Garrison's book, that you start with a blank sheet of paper, using some approx. values at the tip and the rod bottom, then you drawn a curve of the shape similar to what is shown in the book, (if you don't have French curves, use the edge of your shoe). Now that is not the way a professional engineer works.
Given the fact that Garrison was a structural engineer and calculated the fly rods that some describe as second to none. It doesn't seem likely, that a curve drawn as described should be the secret of the design.
Calculating the stresses in the deflected rod is not nearly as difficult as mounting spokes on a bike wheel.
A calculation of the stresses in a deflected rod shows that Garrison used the same stress distribution for all his rods.
This is really amazing, when we know that Garrison did all his calculations with a slide rule and a logarithmic table. It means that Garrison had but one idea of how to design a rod. An idea which could be explained in a simple words:

A rod should be designed to give a constant stress when loaded.

That was Garrison's main idea - his secret.

It seems like Garrison used the stress curve as a red herring; he obviously did not want to share his knowledge of rod calculations with his fellow rod builders.
When Garrison used words as 'progressive action' and 'semi-parabolic' it was not to enlighten us but to lead us astray.
Withholding that information and presenting the stress curve as a starting point for rod design, Garrison put an effective brake on the study of the theory of rod design. Who could hope to come up with a better method than the Master? Admittedly, Garrison was the master of rod design and he shared his knowledge of how to build a rod with us, he gave us the tapers for many of his rods, but he kept the fundamental idea of how he had arrived at his tapers a secret.
That must have been in his mind when he repeatedly told Carmichael, that he was a rod designer first, and then a builder of rods.
I am not a rod builder myself - had I been - I might not have shared my findings.
The Garrison stress curve is by many rod builders considered the basic tool for changing the action of a rod or for designing a new rod.
Rod builders designing rods by experimenting with small changes in taper usually calculate the Garrison stress curve for the final design. They look for similarities to alphabetic letters as if the curve was a Rorschach test for evaluation of the rod's character.

We do not know why Garrison drew his stress curves; but he must have needed them for some purpose.
We will probably never know, how Garrison achieved his goal; but a good guess is that he - with his knowledge as a structural engineer - was capable of calculating the deflection and the corresponding stresses for his fly rods.

It is a lengthy and time-consuming process, if you work with logarithmic tables and slide rule. To reduce time Garrison might have reduced the number of points and calculated the deflection at 10" intervals.
See note #1 at end of The Curve Geometry
The Garrison stress curve drawn by plotting points at 10" intervals could then be used as a tool for interpolation to calculate the intermediary points. Ordinary interpolation procedures do not work in this case.

Calculating the tapers on 1" spacing does not increase the accuracy on the first four digits!
A more accurate reading of the stress curve would accomplish mush more.

What is a Garrison stress curve?

A Garrison stress curve is a diagram, that shows the stresses we would get in a rod, loaded as suggested by Garrison, and made from a material with a modulus of elasticity so large, that the deflection would be virtually zero.

It is a consequence of Hooke's Law, that if the deflection was exactly zero, we would have no stresses. Except to calculate the diameters for the rod used to originate the stress curve, there is not much information we can get from the stress curve.
It is not possible to explain the idea behind the design of a rod in simple terms by looking at a Garrison stress curve. It is implied by Carmichael that two rods having the same Garrison stress curve also have the same action. But.... there is no theory to support that view. The stresses on the curve are fictive, they do not exist.
Garrison never said they did.
Note: The Garrison stress curves are applicable only to rods with hexagonal sections.

So what we should do - and what Garrison probably did - is to calculate the stresses in the deflected rod. The mathematics involved is fairly simple -more about that later.

The stresses calculated for a deflected Garrison rod is thought provoking in several ways. First they show, that Garrison must have been a real wizard with a slide rule. All his rods have an unbelievable similarity in the stress curves for the deflected rods. Secondly, the stress curves reveals the very simple and logic idea behind his design, that a rod basically should be designed to have a uniform stress distribution.
It is interesting that Garrison never mentions the working stresses except in relation to allowable stresses on the Garrison stress curve.

To avoid any misunderstandings let it be understood that I regard Garrison as a very competent engineer; but in his article on the Theory of the Six Strip Rod, he is arguing like a man, who does not tolerate other view points. His technichal explanation is overdone. He is clearly counting on "the technical education of the old-time rodmakers" to let him get away with it.

Misinterpretations

It is common misunderstanding that it is possible to calculate a rod for a specific length of line. For instance if you want to change your favorite class 6 rod to cast a shorter line.
Assume the rod was calculated to cast 45 feet of line; but you want to make a new rod to cast 35 feet of line. The new rod you calculate using the Garrison stress curve is no longer a class 6, but a class 5 rod.
Nothing wrong with that, as long as you know what you are doing. And if most of your casts are short, it is not a bad idea to use a line one class higher than the rod is intended for. But don't kid yourself into believing that your new rod is a class 6 rod.

Garrison defines a class 6 rod as a rod that is working optimal with a 'line out' of 45´ and a calculated tip impact of 2.5. The same 'tip impact' could be calculated using 65´ of class 4 line, 55´ of class 5 line or 40´ of class 7 line. It is the value of the 'tip impact'; not the way you calculate it that determines the classification of the rod.
The tip impact may be taken as a measure for the weight applied to the rod. If we accept Garrison's numbers, it means that a rod designed for a tip impact of 2.5 is a class 6 rod and a tip impact of 2.0 is a class 5 rod. If we want to calculate rods for tip impact between the standards, how should we designate the rod? Class 5.85?

Most of Garrison's diagrams have a note of the size of the tip factor used. On the diagrams called tip impact. It indicates that the tip impact is an important information in relation to the curve. If the curve was valid for any size of the tip impact, it would not be necessary to include it in the diagram.
The unit for the tip impact is oz.; 1 oz. = 0.2780 Newton.

There is no way we can predict the action of the rod from the Garrison stress curve, without comparing it to a curve for another rod (same length and same line class).
Carmichael did not have the theoretical knowledge to ask the right questions and had to accept the master's explanations.
Carmichael writes, that the curves can be elongated or foreshortened to correspond to any rod length. This is evidently as far as some rod builders have read. Had they read a few pages further, they would have noticed the contradiction: As the rod length varies from 6'9" to 10'0" the stress in the butt should be decreased from approximately 150,000 to 130,000 oz. per square inch. Likewise, as the length increases from 7'0" to 10'0", the stresses in the tip, at the five-inch mark, decreases from 195,000 to 180,000 oz. per square inch for a regular tip.

Note: I started this project without having read Garrison, therefore in all my diagrams the bottom of the rod is located to the left and the tip to the right.

Some results using UniRod

Let us change a typical Garrison design:


 Old rod  			     New rod

 Rod length: 7'6"              Rod length: 7'6"
 Weight of ferr: 3.5 gram      Weight of ferrule: 7.65 gram
 AFTM #4                       AFTM #6
 Tip diameter 0.0625"          Tip diameter 0.0700"

We will use the Garrison stress curve to calculate the new rod.
Then we use Unirod to calculate the stress profiles and plot the result.

Diagram : Garrison stress curve and stress profile

The Garrison stress curve is common for the two rods, as assumed.

The red curve is the stress profile for the old rod and the blue curve for the new rod. The higher location of the new rod is not surprising – with a higher load (heavier line) we must expect a higher stress. The old rod had a uniform stress on the lower part of the rod, whereas the new rod is relative stiffer on the lower part.

The deflection for the old rod is 599 mm and for the new rod 660 mm.

Diagram : Deflection

We get a better understanding of the deflection by studying the change in radii of curvature.

Diagram : Radii

The old rod (red) has a smaller radius at all point – it does not bend as much as the new rod. The increase in the radii at the tip corresponds to the stresses at the tip going to zero.

Finally we may look at the energy distribution. The old rod has the energy evenly distributed except at the tip, which is modified for reasons explained in Rod Tip. The new rod has more energy stored due to the larger load. But the bottom does not carry its share being stiffer then the rest of the rod.

Diagram : Energy distribution

The potential energy per unit of length is ½EI/R².
The diagram shows that the constant stress distribution results in a uniform energy distribution.
Further investigation would show that the new rod would match the action of the original rod with a reasonable accuracy, when loaded with a class 7 line. Garrison 'forgot' to tell that the curve should be modified, when you increase the load (change to a heavier line). In the same way as when you increase the rod length.

Calculations by NewUni

Let us look at the alternative and calculate the new rod to match stress profile.

The red curves are for the original rod and the blue curves for the new rod.

We notice that the stress profiles are identical except for the level. Identical is not the same as they geometrical congruent, but in the meaning that the stress reduction is the same in per cent.

The Garrison stress curves are not identical. And the Garrison stresses for the new rod is higher.
It explains why we get a stiffer rod, when we use a Garrison stress curve to calculate a rod for a higher line class.

Accuracy of calculations

The equation we use to calculate the stress is:

The equation shows that a small change in the diameter of for instance 4% results in a change in the stress of 12%. The implication is that the Garrison stress curve we may calculate from the measurements of a rod may not be accurate enough to give a correct picture of the stress distribution in the rod.

Garrison’s stress curves are all smooth. To a structural engineer - as Garrison - an uneven transfer of loads would not have been acceptable.

Reading from an original Garrison stress curve, even if taken from an enlarged Xerox copy are ‘wavy’. Now I am being very meticulous, but we have to be in order to finding a consistent method for reproducing Garrison’s calculations.

The easiest way to solve that problem is to read the values from the curve as accurate as possible, plot the curve in Excel and make the program calculate a trendline for a polynomial of the order 3; mark display equation.

Then you calculate the new Garrison stress curve from the equation and run a ‘New Diameter’ using ‘UniRod’. The new diameter will be saved in the worksheet ‘New_data’.

It may not work if the stress curve is very irregular.

You can remove obvious irregularities in a Garrison stress curve chart in ‘UniRod’ simply by dragging the point with the mouse.

Important:

A misreading of 1000 lbs./sq. in. on the diagram results in an error of 0.00049" on a diameter of 0.200". The error is so small that the rod builders don’t have to consider it, but it is the key to find Garrison’s design conception.

The equation shows that an error in the stress of 4% gives an error in diameter of 1%. That explains the fact, that we can calculate the diameters from a Garrison stress curve, even if we cannot get accurate readings from the diagrams.

Now, let us look at the problems connected with calculation of the Deflection Curve for a rod.