Jim, your calculations look spot-on to me, and I would only add that rim design (i.e. a channel section rim vs. a box-section design) can effectively change the air volume independent of outside rim diameter or tire profile and width, but I certainly agree with your postulation that for a fixed pressure, casing tension is proportional to tire section (where section is a function of area as determined by both width and profile).
Jim, a great observation as always, wrt the rider's body, the bike, and the luggage acting as shock absorbers. Anecdotally, I know I arrive at the bottom of a rocky, uneven downhill faster and less fatigued when I post instead of remaining seated. It is just so much easier than being rattled to death, and removing my weight from the saddle and pivoting through the pedals, cranks, and BB make life much easier for the bike as well. I have a couple earlier editions of _Bicycling Science_ and return to them again and again to marvel over the authors' approach. There is so much to ponder in the book that it is hard to absorb it in one go and so I'll read, then ponder, and then re-read some time later with a new perspective. I love books and treatises that make me think like that.
Tying this further to the thread at hand...
Also much-beloved is my 1974 copy of Fred De Long's _Guide to Bicycles and Bicycling_. He presented some remarkable engineering work on the science of bicycling, and approached it with an engineering background (in the 1930s, he was a manufacturing engineer for SKS bearings). He was best known as an avid club cyclist who worked to better rationalize and apply CPSC (Consumer Product Safety Commission) regulations in the States and was a tireless advocate for cycling. In my early years of riding, I learned to really think about bicycling in scientific terms from his publications. His section on the "Human Engine" was a nice collection of contemporary thinking on cyclists' kinesiology, circa 1974, and even included data that would later be known as VO2MAX uptake. In his chapter on rims and tires, he devoted considerable space to what he called "Serviceability and Rolling Resistance". Among his more interesting observations is this one:
"The 26x1-3/8-inch blackwall tire can be expected to have a rolling resistance 40% greater than that of a good 27x1-1/4-inch tire. Eight percent of this is due to size difference, but the balance is due to the heavier walls and treads used to give more service [on the 26" tire]. A balloon tire with a lower inflation pressure may require that the cyclist expend from 2-1/2 to 3 times more energy. ...the tire design varies with the intended service." [pg. 167]
I believe this -- and the era in which he was writing -- contributed to the prevailing philosophy that tires with large section width, run at lower pressures are inefficient compared to more sporting, narrower tires of larger diameter. I believe the real truth was obscured by the relative differences in construction materials. Large-section balloon tires of the era were simply horrid from an enthusiast's perspective. They were made of poor-quality materials, were extraordinarily heavy, and the sidewalls were so stiff that I recall myself being surprised when a tire that appeared full when parked simply collapsed under my weight, flat of all air; the stiff sidewalls alone maintained the shape of the carcass until overloaded when I mounted the bike. By contrast, even the poorest quality 27" tire of the day was a lightweight marvel in comparison and employed much higher quality materials in its construction. Tubulars -- especially Mrs. Pye's handmade silk sewups from New Jersey used on board tracks by Six-Day riders back in the '30s and '40s and the much later cotton training sewups and Clement Criterium Setas and heavier Roubaix road tires pretty much blew them away, but were the dedicated province of the hardcore club rider or racer here in the States at that time, not the general recreational rider or rare tourist. For some years -- too many in my opinion -- bicycles in the States were considered sidewalk toys, intended largely for children in the eyes of the general public. This has improved greatly, but still holds to an extent in some areas, where they are instead viewed as expensive recreational toys, much like skis, and similarly intended for use in relatively restricted venues such as parks and bike paths (Why cartop a bike? Why not ride from your front door?). I ran into this on my 2010 Great Basin tour, where I was told by elderly ranchers that bicycles belonged on sidewalks and not amidst desert sagebrush. They simply did not think it possible, and rated as nil my chances for a successful transit.
Sorry for the digression, but it does put into perspective some of Fred's views at the time he wrote his _Guide_ and places them in the larger context of American cycling at the time, some 37 years ago. Among his further observations, he notes, "...The actual [recommended] pressure [in a tire] will depend on load. Note that rolling friction decreases as pressure is raised, but the shock and vibration also increase."
In more contemporary treatises, Tony Hadland has spent considerable space on the subject, starting about a dozen years ago. I had the privilege of corresponding with him briefly some years ago regarding the origins of a Folder of unknown provenance in his possession. His root page is here:
http://www.hadland.me.uk/Relevant portions of his website are here:
http://www.hadland.me.uk/lafford.htm , where he presents his observations on tires most suitable for recumbents and small-wheelers, and here:
http://www.hadland.me.uk/page15.html , where he talks about small wheels for adult bicycles.
Though Tony's work is geared toward smaller wheels than we might consider and use here on the Thorn Forum, his general observations and recommendations are still valid, and especially interesting (to me, anyway) is his recounting of the theories of Frenchman Paul de Vivie ("Velocio", see
http://en.wikipedia.org/wiki/Paul_de_Vivie for background) from the 1920s, in which he advocated balloon tyres (sic) of up to 2.25" (57mm) cross-section on small rims. Hadland editorializes, "The idea of a reduction of tyre diameter being matched by a corresponding increase in cross-sectional area certainly has merit. The volume of air and pressure remains the same as in the conventional tyre, while the wider cross-section compensates for (and can even improve on) the otherwise harsher ride of the small wheel." Most relevant to our discussions in the earlier posts on this thread, Hadland goes on to observe...
"...As for rolling resistance, a reasonable prima facie indicator is the length of the tyre print (under a known weight) divided by the inflated tyre radius.10 For a given tyre pressure and load, the contact patch area is approximately constant, regardless of tyre diameter. (For example, a tyre inflated to 50 psi and carrying a load of 100 lbs has a contact patch with an area of approximately 100/50 square inches, ie. 2 square inches, whatever format the tyre may be.11) However, with the Vélocio approach to small wheels the patch is wider but shorter. Thus compensation is obtained for the otherwise higher rolling resistance.
However, this compensation depends on superior lightweight tyre carcass construction. This is difficult to achieve because, the larger the cross-section, the stronger the carcass must be to hold a given pressure. For economy of manufacture, the strength of wide section tyres often comes from thicker, heavier and less flexible materials, and results in a higher rolling resistance," (in this latter section, Hadland references the work of Rob Van der Plas, 'Rolling Resistance', _Bicycle_, UK, February 1984).
And there we have it, in my opinion. As evident in Schwalbe's tests and charts, primary factors in rolling resistance (at whatever level) are not the area of the contact patch, but the shape and how it influences deformation of the carcass, coupled with the quality of materials used in constructing a tire. From this we can infer that poorly designed tires using inferior materials will be slower regardless of the pressure used, and the inverse -- carefully designed tires made with high-quality materials will not only roll more easily and comfortably from the get-go, they will allow for greater exploitation of those characteristics through proper inflation -- *and* will tolerate lower pressures and increase handling and ride comfort *without* incurring the same penalties in rolling resistance suffered by their poorly-constructed kin. Fatter-section tires run at lower pressures (provided they are of quality materials and construction) will have a wider contact patch that causes less deformation and therefore lower rolling resistance than a narrower tire, which is characterized by a more longitudinal contact patch more subject to deformation and, therefore, relatively higher rolling resistance, other factors being held equal.
The fact that not all of us have had equal success in experiencing low pressure, wide tires that ride fast may be due to the idea that not all tires are created equal or have design parameters and materials optimized for the purpose. Certainly, I've ridden some tires that were bog slow *regardless* of pressure (see my earlier post in this thread referencing the 700x35C Michelin City Pilots) and they were irredeemable regardless of my adjustments and the tubes and rims selected. When it all goes right -- and that seems to depend on a number of variables and in some cases, purpose-matched components like rims and friendly frame clearances -- it is a postulate that holds true in spades and is a genuine joy to experience.
Andre, I took the opportunity to peruse your off-list site, quickly became lost in it, and thoroughly enjoyed your writings and photography. Your incomparable Utopia Kranich is about as close as I can imagine to a true systems approach to maximize the superb qualities of large-section tires operated at low pressures. It has to be _the_ optimum design for this philosophy and is an incredible machine. I've enjoyed your observations and insights to this thread, and especially the reference to Lotus Engineering and Colin Chapman's designs ("Simplicate, then add lightness" was one of his engineering battle cries). Carrying the comparison a bit further, I was struck by how much like a bicycle my French friend's Citroen 2CV appeared when we examined it in his barn. Basic, simple, lightweight, and with incredible versatility and economy -- all in a vehicle originally intended to help farmers get their pigs and produce to market in the most economical, reliable manner possible.
Lets all work on getting that extended research grant on the general topic; the reference to an "..agreeable warm place with good cycling" sold me on the idea!
And...I'm delighted with my Duremes run at appropriate pressures and can't wait to try them "in anger" so to speak, under load and in a proper expedition setting.
Best,
Dan.
