MYOG Technical Note – Catenary Curves

Roger,

As an advocate of catenary curves and using spreadsheet calculations for laying them out, I appreciate your article.

I've tried the 'hang and trace' method and found it far more trouble. Finding a wall long enough and free of windows, doors and furniture; taping pattern paper or fabric to the wall straight, level and without stretching; finding, hanging and adjusting a cord heavy enough to hang smoothly and light enough to tape up; tracing the line; and returning everything to my work area was much more time consuming than just transferring measurements directly from a spreadsheet.

When using CAD software, I simply plot a circular arc. Small arcs of the conical shapes(circle, elipse, parabola, hyperbola) and their derivatives (catenary curve) are indistinguishable from each other. Test arcs I've plotted were less than 1/64" different from the spreadsheet catenary curve.

However, I am a bit confused by your spreadsheet. Shouldn't the calculated maximum deflection at mid span (cell D37) be the same as the desired maximum deflection (variable 'b',cell B13)?

Thanks

I am pretty sure that is a GoLite Shangri 2 not 4.

See http://www.golite.com/Product/ProdDetail.aspx?p=SH6122&mc=154&t=&lat=

Hi Lance

You are right that variable b is NOT the same as the maximum deflection. It CONTROLS the deflection. To be honest, I forget the exact explanation, but I *think* b may be the extra length of the curve over a taut straight line. But I could easily be wrong here.

Hum – I could redo the spreadsheet to make b the actual deflection. That might be more useful. Thanks for the idea.

Cheers

Hi All

The comment about the 'b' parameter made me consider more carefully how to present the data. The V1 spreadsheet made it seem that the 'b' parameter was the actual deflection, and this was wrong. My apologies to all about this.

The V2 spreadsheet now has the 'a' parameter in feet: it is the HALF-width of the span. The 'b' parameter is now in inches of actual deflection and the Y column shows this in inches correctly. The graph in V2 has units of feet on the horizontal axis and inches of deflection on the vertical axis. The curve is still the same shape overall – a catenary.

The comment that there is little difference between the catenary curve and a circle at typical tarp deflections is correct of course. All the guy ropes and wind etc mean that mathematical perfection is completely irrelevant anyhow. Really, all you need is a SMOOTH curve.

Enjoy
Cheers
Roger

Is the curve applied to all edges of a shelter??? Top ridge, bottom edges?

Certainly it is useful when applied to the ridge line and to the bottom edges. Both are long and can flutter. I don't think it is as useful when applied to the ends: those edges are much shorter. But you can if you wish.

Cheers

Debra,

The curve is gradual enough and the fabric forgiving enough that there is zero pucker factor when sewing a flat felled seam in a curved ridge line.

I posted some pictures in an earlier thread on the subject.

The 9' ridge in the pictures has 3" of deflection (.33"/ft) in the middle. I don't think there would be any pucker factor with a larger deflection either.

-Lance

Roger-

I realize that in practical application it doesn't matter — especially, say, when a tarp is tensioned by multiple guylines and such…but for academic interest:

Does your spreadsheet calculate the Cat curve in the plane of the fabric? Or, does it calculate the curve in a vertical plane and then project it onto the fabric plane? If the latter, what angle between the fabric plane and vertical plane is assumed?

Cheers,

-Mike M

Thanks, Roger.

Still, the math is interesting. With an A-frame tarp made out of an ideal non-stretching material, supported on each apex by a guyline, and uniformly tensioned along two edges, I wonder what the optimal panel shape would be to most evenly distribute tension in the fabric.

[Sorry, my engineering geekiness is showing.] ;)

Mike,

Thanks for the thought provoking questions. Here's my two cents worth:

Does your spreadsheet calculate the Cat curve in the plane of the fabric? Or, does it calculate the curve in a vertical plane and then project it onto the fabric plane? If the latter, what angle between the fabric plane and vertical plane is assumed?”

Roger’s spreadsheet appears to present a curve in the same plane as the fabric. If you then project it from a vertical plane horizontally onto the angled plane of the fabric, the curve would no longer be a true catenary curve. For example, if the fabric was angled at 30deg from horizontal, the projected offset of each point along the new curve would be doubled (1/sin(30)). But that is not the same as a cat curve with twice the offset at mid span because you are mixing linier functions with hyperbolic functions. It’s kind of like mixing up the order of operation rules between addition, multiplication and exponents.

For practical purposes, it makes no difference. The test comparisons I’ve made were just a few thousandths of an inch different for a typical ridgeline and just a few tenths different using 10” and 20” of deflection. Roger’s spreadsheet makes a similar linier adjustment of .15346, but for our purposes here it doesn’t matter either. As Roger said, a smooth curve is what is most important.

Still, the math is interesting”. With an A-frame tarp made out of an ideal non-stretching material, supported on each apex by a guyline, and uniformly tensioned along two edges, I wonder what the optimal panel shape would be to most evenly distribute tension in the fabric.

I think that in the above scenario, a square panel would theoretically be the optimal shape. No curves are necessary because the theoretical fabric doesn’t stretch. The picture below helps illustrate the distribution of force throughout a panel. The stress throughout the square pannel appears more evenly distributed. Regardless of shape, the limiting factor is the concentration of force at the four corners.

Thanks for listening
-Lance

> I think that in the above scenario, a square panel would theoretically be the optimal shape.
> No curves are necessary because the theoretical fabric doesn’t stretch.

Hum … very theoretical. But even with Cuban fibre fabric there are other considerations. With a straight ridge line there is a very uneven distribution of tension in the fabric when the support is concentrated at just two points. At the middle of the ridge line the stress reaches a low. Assuming for the purpose of the exercise that the fabric is non-stretch, there remains the movement of the fabric due to stretch in the guy ropes. As soon as you permit any movement it is possible to create areas of zero stress in the fabric – and flutter can occur. Once you have flutter, you can have problems.
If you assume a uniform holding right along the edge of the tarp, then … hum … I suspect that the two cases are equivalent. However, an unlikely scenario :-)

Cheers

> answers in feet, inches and 8ths of an inch.
Sometimes, I pity America.
But then I realise it is all self-inflicted.

Cheers