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# Tension Ring

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## Tension Ring

(OP)
I've been looking at a turret design and trying to determine if I could run a strap around the tails of the rafters so that I could eliminate the internal bracing:

I'm trying to figure out how to calculate the tension in the strap given a uniform (snow load) on the turret roof. The steeper the pitch the less the tension in the strap. This reminds me of some of my calculus problems from my college days, but it is probably much simpler than that.

A larger overhang would counter the flattening effect of the load on the roof and reduce the tension in the strap.

If you take the tributary load at the center of the roof as half the distance from the center to the outside wall the point load at the center would be given as P = (S + D) * pi * (d/4)^2.

The horizontal force exerted by each rafter would then be: Fh = P/tan(Theta) * (1/n) where "n" equals the number of rafters.

The tension in the strap from basic statics would be T = (Fh/2) / cos(omega) where omega = 90 * (1 - 1/n).

Anyone care to check my math and see if it adds up?

I'm assuming an overhang of zero or at least conservatively ignoring the countering affect of the overhang.

A confused student is a good student.
Nathaniel P. Wilkerson, PE
www.medeek.com

### RE: Tension Ring

this is a puzzle to me:
haven't worked through your formulae but it looks like you're assuming rafter thrust is the same for both the 'non-continuous' rafters as well as the rafters that are continuous to the ridge.
the framing system is something i might use in a pyramid shaped roof, which I've modeled in the past as intersecting 3 hinged arches, where the horizontal thrust of the 4 ridge beams is supported by a 'tension ring' with 4 'straight sides' each connection resolvable by 'basic statics'. the non-continuous rafters i would model as simply supported beams with no lateral thrust.
in the present case, if i looked at it as 4 intersecting 3 hinged arches, 8 ridge beams, but with tension ring consisting of 8 'curved sides'...the tendency to 'straighten that curve' will induce a thrust into the non-continuous rafters. (and yes, a diaphragm can be installed which will 'cover a multitude of sins', but to me the puzzle is to see how the system truly works)
as the model is extended further, at some point, the tension ring is perpendicular to the thrusting rafter and one is dividing by cos90-ish, as in your equation, and this is no longer basic, at least my kind of basic, statics.
Seems like a hoop stress condition, which I've seen but never worked through the math on....

### RE: Tension Ring

(OP)
Yes, for simplicity I'm assuming the same rafter thrust for all rafters.

A confused student is a good student.
Nathaniel P. Wilkerson, PE
www.medeek.com

### RE: Tension Ring

then i suspect hoop stress is your model and the simple beam supporting the upper end of your non-continuous rafters needs to be designed for some lateral bending.

### RE: Tension Ring

Reminds me of Frank Lloyd Wright's studio near Chicago where he used a great big chain looped around to counteract thrust. not that his designed were necessarily what you call "Engineered".

### RE: Tension Ring

(OP)
Great minds think alike.

A confused student is a good student.
Nathaniel P. Wilkerson, PE
www.medeek.com

### RE: Tension Ring

Except for the overhang, the structure should act like a conical shell. If the total weight is W, the vertical reaction is also W. The vertical reaction per unit length of circumference is W/(2*pi*r). If the roof slopes at angle θ, the radial reaction is W/(2*pi*r*tanθ) per unit length of circumference and the tension in the strap is W/(2*pi*tanθ).

The overhang does not contribute to the strap tension but adds to the vertical reaction.

BA

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