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Coil spring stress equation for tubular (vs. solid) wire?

Coil spring stress equation for tubular (vs. solid) wire?

Coil spring stress equation for tubular (vs. solid) wire?

Looked through my Wahl's Mechanical Springs text on how to compute the shear stress in a coil spring when the "wire" is actually a coiled tube.  I finally decided to take the "J" (torsional area moment) in the standard spring equation, and replace it with a (Do^4-Di^4) term.  More specifically, the "standard" coil spring shear stress equation is

Tau = 16PR/(pi D3) {(4c-1)/(4c-4)}

And a revised equation for a hollow section would be

Tau = 16PR Do/[pi(Do3-Di3)] {(4c-1)/(4c-4)}

Any comments regarding the substitution?  Does anybody have a directly solved reference for such a beast (coiled tube spring)?  Better, does anyone with access to the SMI software know if it can analyze springs wound from tube?

Thanks for any and all help.

RE: Coil spring stress equation for tubular (vs. solid) wire?

I guess my first question is, why tubular rather than solid? Reduction in weight?

It looks as if the substitution makes sense. I've searched around and can't find anything having to do with tubular springs.


RE: Coil spring stress equation for tubular (vs. solid) wire?


Thanks for the reply.  The reason (don't laugh) is to have a flexible hydraulic tube, in a place where a hose would be abraded and/or possibly subjected to collapse from external pressure.  Thus, we need a rigid tube, but flexible enough to extend in the axial direction.  The tube spring works, but I am trying to analyze for stresses and fatigue life, given different windings, preset conditions, etc.

RE: Coil spring stress equation for tubular (vs. solid) wire?


Believe me, I don't laugh at anything (besides homework problems) smile. Your application makes sense to me.

My inclination would be to say that as long as you're very conservative as far as your axial extension/compression vs. the working length of the "spring", I can't see why they cannot be solved in the same manner.

I see it as being akin to bending a tube vs. bending a solid  rod. In the linear elastic portion, they are very similar.

Maybe build in a higher safety factor for fatigue, just for that warm and fuzzy.


RE: Coil spring stress equation for tubular (vs. solid) wire?

I found the following reference from 1919 on the subject:


There have been some more recent studies on hollow springs fabricated from reinforced composites.  The following are some links:




The Meccanica article uses the following formula for maximum torsional stress:

τmax = Cf · [16Mt/πdo3(1 − a4)]

where a = ri/ro, do = 2ro, and Cf = [4c-(1+a2)] / 4c-1.  C = R/ro and Mt is the torsion moment due to spring compression.

RE: Coil spring stress equation for tubular (vs. solid) wire?

Thanks very much TVP, a star for the book reference, and I'd give you a dozen more for the equation.  The articles I found also, but the only free source had a derivation that generated the equation I listed; I had trouble verifying their derivation, and thus posted here.   

RE: Coil spring stress equation for tubular (vs. solid) wire?

Good link, TVP.


RE: Coil spring stress equation for tubular (vs. solid) wire?

I have this one, I think is the same as you have, but transformed for easiness of use:

Tmax = 16·Mt/[pi·(D^3)(1-((d^4)/(D^4))]

Where D is the outer diameter and d inner diameter.

Using the next formula you can calculate the tube section equally resistant in torsion to a solid section:

M^3 = D^3[1-(d^4)/(d^4)]

Where M is the diameter of the solid section and D & d have the same meaning as in the previous formula. So you can obtain Tmax for a determined solid section and then, "translate" it to a tube, with the thickness of your choice.

It is interesting to point that, in pure torsion, you always require less area with a hollow section vs. solid section (which translates into lighter components). The problem is that, in turn, higher outer diameters are required.

Localized failure (theoretically) can also be a problem when designing a hollow section spring (crippling of the wall), specially if the forming operation of the spring is not carefully carried out (deformed walls, roundness defects, non-concentricites...), which could not be so uncommon, as bending tube is somewhat more difficult than bending wire or bar.

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