I appreciate your comments and couldn’t agree more: all good, and the intent is to increase the knowledge base and disspell myths and misinformation when it comes to good wheel design and construction. Let the information overload continue
Thanks for the clarification on spoke contribution to lateral wheel stiffness being proportional to the square of bracing angle (and the semantics are important: it’s spoke contribution at issue). One small note I’d make is that if you go through the math, using Gavin’s spoke modulus of 206 GPa (appropriate for stainless steel), 100 kg-f/mm stiffness corresponds to a 1.3 mm diameter spoke. More realistically, for the 1.83 mm diameter Gavin uses (odd, but slightly larger than 15G or 1.8 mm) the spoke stiffness is very close to 200 kg-f/mm. This only helps your argument, though: the lateral component per spoke is effectively double what you calculated in your example.
Where I’m having an issue is the sentence, “You stated that rim depth is a more important factor than hub bracing angle…for the typical ranges encountered. But that last part is key…” I’d say that’s just the point, but turn it around: if the purpose of discussing and listing bracing angles in the hub review and hub specs is so readers can make informed decisions on hub selection, then isn’t it more appropriate to compare front hubs and rear hubs separately?
As an example, neglecting the outlier Soul-Kozak front hub, the range of hub bracing angles is 7.1 (C-K R45) to 7.9 (Alchemy ELF) degrees. Squaring the ratio (7.9/7.1)^2 gives a 24% increase in stiffness due (only) to the spoke bracing angle. A 24% increase in lateral wheel stiffness due to the rim can be achieved by increasing the rim depth by (1.24)^3 or 91% increase in rim depth. The thing is, one has a wide range of choices for rims; not so for hubs: a front hub cannot be substituted for a rear to get a substantial increase in spoke bracing angle, but one can easily choose a stiffer rim to more than make up for it. Fairwheel sells KinLin rims in section depths of 19, 22, 27, 30, and 38 mm, for example. If I felt a customer asking for, say, a XR-200 22 mm deep rim (most likely for the weight savings) but weighed 200 lbs., I’d probably steer them towards the XR-300 30 mm deep rim (and 65 grams heavier) for overall strength as well as lateral stiffness. The XR-300 is 254% stiffer radially than the XR-200, meaning building a front wheel with the widest flange spacing in the review on the shallower rim would be less stiff laterally than the narrowest hub in the review (outside of the Soul-Kozak, as stated previously) using the XR-300 rim, all else being equal.
The conclusion is that, for a given wheel–front vs. rear, MTB vs. road, etc., etc.–the most important practical considerations for wheel stiffness is, in order of importance: (1) rim; (2) number of spokes; (3) hub bracing angle, neglecting the odd hub outliers; (4) spoke gauge (Rinard says it’s a small effect; 11% increase for 2.0 straight gauge vs. 2.0/1.45/2.0 butted with 32 spokes in his test). Interesting.
Your comment on the FIR Rialto rim at “700 grams” (it’s more like 650 g–still substantial, but with many Al rims at 400-500 grams it’s not a huge stretch, particularly for a tandem build) only matters to a few people, and I agree not many would choose that rim…although I’d readily point out that Fairwheel offers the KinLin XR-380 38 mm deep rim, which at 550 g might indicate there’s some market for such things. A more relevant response is one word: carbon. There’s plenty of light weight carbon offerings in a wide variety of rim depths, so for those wheel customers who are concerned about lateral wheel stiffness, an easy solution is to steer them towards a deeper-section carbon (assuming they’re in the market for one in the first place) rim. Note that ENVE, as a good example, has refined their Smart System 6.7 wheels by using a 60 mm section depth for the front rim and a 70 mm section depth for the rear rim–an interesting and relevant point overlooked by the reviewer in the Fairwheel product introduction athttp://fairwheelbikes.com/cycling-blog/products/enve-smart-system-6-7-rims.html. Doesn’t sound like much, but that 10 mm difference front to rear is worth (70/60)^3=60% increase in radial stiffness on the rear, which helps make up for the inherently lower lateral rear stiffness due to the hub bracing angle.
We haven’t even talked about rim width, which obviously also makes a big difference to those who are concerned about lateral wheel stiffness. I’ve always thought the crazy super-wide DH rims with shallow section depth (e.g., Sun Rhino and the like) are a poor design as the wide rim might be OK laterally, but not without enough section depth to keep the wheel stiff and strong radially…and add to the lateral rigidity.
The last issue I’d like to address is wheel strength, not stiffness. As Rinard, Jobst Brandt, and others have said, the strength of the wheel is only maintained until certain spokes lose tension. In a severely biased rear wheel, that happens sooner, but that must be balanced against the engineering advantages of a wider flange spacing. As the original review stated, one way to achieve that is by using lighter-gauge spokes on the NDS vs. DS–something I’ve done in all my wheel builds since 1991. To the extent that all of the rear hubs in the review end up with a NDS:DS spoke tension ratio of around 50%, that argues strongly in favor of the 2:1 triplet rear lacing pattern, since the spoke strain can be equalized using the same-gauge spokes on both sides, thus making the wheel as strong as possible but also as stiff as possible (vs. using differentially-guaged spokes NDS:DS, assuming conventional materials where you’re limited in the choices of available spoke gauges). That also indicates an optimal left-side flange distance/diameter so that the tension ratio is exactly 50% if used with triplet lacing.
It’s too bad Gavin’s analyses didn’t include similar graphs for lateral stiffness vs. J (rim torsional moment) and I_rr (rim bending moment about the radial axis), as that’s what you really need to look at if you’re obsessed with lateral wheel stiffness. Maybe someone out there can add to the knowledge base with this info? There’s always another layer on the onion to be peeled.
All the best,