joejack951

110 kgf is the tension on the spoke. You are not assuming anything by calculating just the stretch in the spoke given that tension. If you wanted to get the total deflection of the system with a 110 kgf tension, then you would need to factor in the rim, nipple, spoke, and hub which gets a little more complicated.

DannoXYZ 

I'm not trying to get total deflection of the entire system, but rather just the spoke-length. This is a load-in-series problem and we can solve it similar to a chain-links or springs-in-series type of configuration. Does the links above and below link C in any way change its load significantly? Let's assume we size a chain with strength such it reaches yield at 200kg so that the weight of the links themselves are immaterial relative to the actual load (each link weighs a couple of grams):



What is the load on link A, B, C, D ??? Does it matter what position link C is in? If we move it up or down one link, does its load change significantly?

I'm comparing nipple, hub, and perhaps even rim deflection to the weight and position of the other links in the chain. I'm saying that the majority of the total system deflection is in the spokes. Mike has given us the rim-deflection before & after inflation. Let's calculate the spoke values. I don't think the amount the hub or nipple stretches will affect the spoke-deflection by a significant amount (or total deflection for that matter).

Mike Mills

I'm not sure about that assumption. I think the inner wall of the rim may deflect inwards once under tension. Most of them are flat and thin relative to the direction of spoke tension. And being made of soft aluminum doesn't help either. It wouldn't surprise me at all if inner rim wall deflection was equal to or greater than spoke strain once put under tension. Once you doubt an assumption, you must do the analysis. You can analyze them independantly just to get a feel for the strain each might see. In order to understand the "real" situation, you need to analyze them together. It isn't that hard, it's like two series/stacked springs. I just haven't yet done it.

joejack951

I believe you are over complicating things. You KNOW the tension on the spoke. That's what the tension gauge measures. The tension gauge is not telling you the force pulling in on the rim or the force at the hub. You ought to be able to use that force and the cross section, length, and tensile strength of the spoke to determine the deflection.

As to your chain link illustration, every link should be experiencing the same tension in that scenario. They will not all experience the same deflection though assuming different geometry/materials for each link.

As for Mike's comment, I agree that the inner wall of the rim will deflect inwards under tension. It must as there's no such thing as a completely rigid material. The deflection might be so tiny as to be inconsequential but it will deflect some amount. I don't think the rim deflection is inconsequential when looking at the full wheel system though. But it's immaterial to the question of spoke deflection. It would only matter if we were trying to determine the exact dimensions of a fully built wheel with rim, spokes, nipples, and hub.

Mike Mills

Perhaps I am mistaken, I thought the issue was deflection of the rim with/without internal pressure in the tire - inflated versus not inflated. That is why I was looking at internal stress in the rim with pressure.

If this is the question, then perhaps another thing to look at is the pressurized tube bearing on the spoke nipple and the consequential relaxation of spoke tension. This is an easy calculation, in itself. This should get you in the ball park:

Change in spoke tension = tube pressure * spoke nipple head area

If the change is a large fraction of the initial tension (change / preload > 0.1), then one has to look further into rim deflection. If it is 0.01 (1%) or less, I think it would be easy to dismiss it as inconsequential.

Question - Is the rim/spoke/hub a linear system? Does 1/4 turn of the nipple always cause the same increase in spoke tension? Does spoke tension vary non-linearly with nipple tightening?

DannoXYZ 

Yes that was the goal. My initial assertion was that the rim compresses under pressure, shortens the distance between the rim and hub, thus reducing spoke-tension. You've already calculated the rim-deflection due to the pressure-change. All we need now is the initial spoke-stretch due to a known tension. Then calculate the new tension when the rim deflects inwards from pumping up the tube.

Since the spoke is operating in the middle of its stress/strain curve at about 1/2 yield, and the change in tension is so minor, I think we can assume that the it is linear under the operating parameters we are seeing. However the spoke+nipple+rim may not be a linear system. Depending upon the rim-construction, single-wall clincher, double-wall, box-section vs. aero, bare hole or eyelets, we may see rim deflection that is less or more than spoke-stretch. And the way the hole puckers out with increased tension may lead to non-linear behaviors. However, we're not spinning any nipples here, just pumping up the tube. And for the minor amount of deflection you've calculated, let's assuming it's linear.