Posted by: Ian | July 18, 2010

Difficulty

In my last post I referred once more to the excellent book 100 Greatest Cycling Climbs, which describes a selection of the most renowned British hill-climb courses. This is already for British cyclists what the book of The Munros is to Scottish hill walkers, and I expect that in time a second volume will be out to compare with the The Corbetts. Almost as much as the book, I admired the author for the inspiration to write it, although my admiration has been tempered a little since discovering that there is a pre-existing US equivalent – The Complete Guide to Climbing (by bike), which includes a list of the 100 most difficult climbs in the States.

I was curious to see how difficulty is measured. In the British book each hill is given a 1 to 10 rating, with 1 meaning “hard” and 10 meaning “it’s all you can do to keep you bike moving”. There is no attempt at a formula and the author simply asserts that “the rating is an amalgamation of gradient, length, the likely hostility of the riding conditions and the condition of the surface”. The author of the US book gives a half-hearted attempt at a formula:

Difficulty = sqrt(average grade) * elevation gain * altitude adjustment * surface adjustment * grade variability.

The first two terms are quite straightforward (though I’m not sure whether he uses net elevation gain or, as he seems to say, total elevation gain). The other three terms are fudges. The altitude adjustment is an unspecified correction for riding at high altitude; all he discloses is that it equates to a 1% increase in difficulty at 2,000 feet. The surface adjustment boosts the difficulty of any unpaved sections of the climb by 25%. The grade variability hikes the difficulty of each section of the climb that is more than 5% steeper than the mean overall grade by 0.025. This reflects the reasonable view that a climb with a mix of steeper and less steep gradients is harder than one that ascends on a steady slope, but the author does not specify exactly what constitutes a section, so it’s not really possible to check the rating given for yourself.

A more committed attempt to specify difficulty in a formula is given on the website climbbybike. I’m not giving a link to it because it’s such a technically poor site that is painfully slow to run; when I was on it yesterday it kept crapping out with script time-outs. It’s a great shame because if it worked the site would be very useful, or at least fascinating. The formula that they specify for difficulty is:

Difficulty = 400 * H/D + H^2/D + D/1000 + max(T-1000,0)/100

where H is altitude gain, D is distance and T is mountain height, all in metres.

H/D is, of course, the average grade, and the final term is akin to the altitude adjustment of the first formula.

The site also references a couple of alternative formulae but this is the one that it uses. It produces a number that doesn’t make much intrinsic sense so, more helpfully, it gives a difficulty ranking. This is exactly what we want but again the implementation seems highly flawed. Yesterday, I couldn’t find any climbs on the site that it ranked in its global top five. The hardest in England was Hard Knott Pass, which it placed at somewhere below 1,000 on its global ranking. To be fair, the author of the British 100 climbs book says that geographically we can’t compete with climbs like Mont Ventoux and Haleakala. I looked both of these up on climbbybike and found that Mont Ventoux only manages a ranking of 131. Haleakala did better, being 17 globally and 2nd in the US.

As you can see, it gives these nice gradient maps, with colour coding for severity of slope. Mont Ventoux has several kilometres with a gradient of 7-9%, whereas Haleakala is generally sloping only at around 6% but goes on for over twice the distance. For anyone actually planning to do it though, be aware that the last kilometre is not as benign as this chart indicates, since the final section up to the telescopes kicks in cruelly at 12%.

The US climbs book also ranks Haleakala second in difficulty, but puts a climb in Washington ahead of it, whereas climbbybike puts another Hawaiian mountain first and ranks the Washington climb third.

For comparison, our own Dunkery Beacon has three straight red 0.5 km sections flanked by three yellows but the climb is judged to run for only 5 km. Here’s a snap of it from the Tour of Wessex:

Today the climbbybike site is running much faster but the rankings have undergone some interesting changes. Mott Street in Essex has been promoted to the global number two. I’ve cycled up it with Stu and Emily – it’s quite near Stu’s – and while it’s a great ride even the local cycling club wouldn’t make any such claim for it. In the British 100 climbs book it’s rated 3/10.

Of course, there are factors other than those in the formulae that affect how hard a ride feels to us. Weather is one, which is alluded to in the non-formula formulation of the British book but not included in the others. For example, I’ve been up Dunkery Beacon five times, three of them straight up past the beacon itself and twice on the route to the West that goes through the cobbled brook. Far and away the most difficult time was on last year’s Exmoor Beast, when riders ahead of me were literally getting blown off their bikes. Then, the route bore no relationship to how it is in good weather, as on the photo above.

Also, the more you do a route the easier it gets: familiarity is your friend.

Even more mundane matters can dramatically affect the difficulty of a ride. On Thursday Gavin, Emily and I met just before 7 am at West Hampstead and cycled down to Regent’s Park for loops. Since it occurred to me a few weeks ago that it’s doable to complete each one mile split in under 3 minutes that has been my aim. The first mile, which is an easy one and cycled with fresh legs, took 2:58. The next, which is even easier, took 2:56. The one after that took 3:00. I wondered what was happening as this was far slower than my recent times, while I wasn’t any more tired and I’d had the same breakfast. When the fourth mile took me 3:01 with much more effort – standing on the pedals to try to generate momentum rather than just cycling at high cadence from the saddle – I reconciled myself to doing what I could and hoping to keep the time as low as I could manage. Just before the fifth mile ended I saw the problem: my rear tyre was flat.

I pulled over to change it and as I did so the rain, which had grown from a drizzle to a shower, intensified sharply. Emily caught up and helped me but I still made a meal of it as I couldn’t conclusively find where the tyre had punctured. I did locate the puncture in the tube but for once I’d removed the tube from the tyre and so couldn’t track it back.  While I was faffing around Gav lapped me. It’s not hard to spot the point on this chart of my heart rate where it happened (the graph starts inside the park):

Once I got the Tempo back on the road we left the park and tootled across town. We were late enough that the Rapha cafe was just about to open as we came to it; we stopped there for a flat white, which was excellent.

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Responses

  1. […] few months ago I wrote about different formulae that have been proposed for quantifying the difficulty of a ride and I’ve incorporated the formula used by ClimByBike into my sheet. A nice feature of this […]

  2. Ant’s tip for the day. Line up the logo on your tyre with the valve on your inner tube. That way when you flat and find the hole in the tube you know where to look in the tyre.


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