I promised math some time ago. Now it is time to bust out the engineering paper!

After my first post Trent brought up the issue of how the wood planks will wear with constant sliding. I thought this was a valid concern. He proposed lining the channels with a friction reducing material. But before opting for this more expensive option I wanted to do a little analysis to see if it would be an issue.

To see if wear will be an problem I have done my best (a bit of googling) to research wood wear in asperity contact with steel. I don’t have access to an academic database and this is not a subject within tribology that seems to have seen a lot of work. As a result I haven’t found a lot of information to estimate how much the planks will wear. But that hasn’t stopped me before, so I will charge forward blindly with some analysis using little information, less expertise, and a series of wild assumptions. (Now this is all pretty quick and dirty so please feel free to check my math and question my assumptions)

The first of these assumptions is that the wear will be similar to that in wooden journal bearings, operating under asperity conditions. It seems reasonable to me, they both have wood sliding on metal and the wood sliding in a channel is basically a journal bearing of infinite diameter. So all the data I will be using to estimate wear rates will be pulled from papers dealing with plain bearings.

The first task involved is to estimate the contact forces between the planks and the channel. This part is a simple bit of static analysis, based on the assumptions that the maximum loading will be roughly the weight of a 250lb climber (split 150lb per end of plank), it will be applied parallel to the plank, and that it will be applied at the tip of a 4″ tall hold. As shown below. (note: I forgot a force in my free body diagram, we will assume the chain is applying a force straight up on the back surface of the plank.)

This geometry shows a 4″ tall hold on a 2×8 plank. The contact forces F, will be a function of the length of the wear patch (Lw) (where the wood is actually touching steel), and the the climbers weight (Load).

F=Load x 5.5in / (7.5in-Lw)

If we assume that wood will wear into a roughly 1 inch square contact patch relatively quickly we will have around 7.4kg/cm^2 of pressure in the contact patch. We can use the traditional PV ratings of plain bearings to assert that we should get a reasonable lifespan out of the planks, following the initial wear-in. This is based on the PV limits presented by (Wilcock and Booser 1957, Anon. 1977, Steuernagle 2001) shown below for wooden bearings. Our operating conditions will fall roughly withing the red circle, depending on speed and actual loading.

This is probably all the analysis I should bother to do since I have a whole lack of information but I promised Mark some math. So I am going to charge on even more blindly to estimate the amount of wear we can expect to occur.

A means of estimating wear in wood journal bearings is presented by Weir and Morogoro 1976. They propose that wear depth (W ) can be estimated from sliding Distance (D), contact pressure (Pc) and a wear coefficient (Cw) specific to wood type and lubrication.

W=Cw x D x Pc

However, it is clear from later research (Sathre 2005) that speed and pressure have very different, non-linear effects on wear. In this work they don’t explicitly provide a model for wear so I have created one by applying a curve fit to their data: the equation below. (it matches all 7 available data points weee) Here V is sliding velocity in m/min and the wear coefficient Cw=3.28 x 10^(-7) matches the wear they observed for maple. W is the wear in mm, provided a contact pressure in kgf/cm^2 and sliding distance D in km. (sorry about all the mixed units that are coming up but I am just too lazy to convert my units to SI)

This relationship will fall apart past a certain velocity since thin film lubrication will start to reduce wear as velocity increases. But our assumed operating conditions are pretty similar to those used by Sathre so this should provide a reasonable estimate. In fact it should be conservative since, it has been suggested, that increasing wear with speed is due to heat build up in bearings. Heat will not be an issue here since our thin channels will dissipate heat readily.

W=Cw x D x Pc^2.11 x V^.645

Making the wild conjecture that pine, having 1/3 the Janka hardness of the lubricated maple used by Sathre, will wear 3x faster we will estimate how fast cheap pine planks will wear out using Cw=9.84 x 10^(-7).

We can see that as the plank wears the length of the contact patch Lw will grow and contact pressure will decrease as a function of the amount of wear that has occurred. The relationship between Lw and W is defined as below, where c represents the initial clearance between the plank and the channel and a is the distance from the center of the plank to the corner. I leave it to the reader to derive the relationship as a trigonometric exercise.

a=((0.75in)^2+(3.75in)^2)^0.5

Lw=2*W/sin(2*asin((0.75in+c/2+W)/a)-2*asin(0.75in/a))

Contact pressure can then be defined for a 1inch deep channel as:

Pc=F/(Lw x 1in)

There is no way I am going to attempt to find the resulting integral of all this, so I solved it numerically using Octave to get the plot below. Assuming c=0.06in initial clearance and a wall speed of V=15m/min (pretty damn fast). I would consider a reasonable lifespan for the wall to consist of 30min of use 4 times a week for 5 years will resulting in 3000min. of use if a plank is being stood on ~1/10 of the time. Under these conditions the plank will wear about .006″ per side so about 1/64″ total. I would consider this a very reasonable wear rate….. If any of my assumptions can be counted on.

So as a result of this analysis I am not going to worry about my wood choice or lining the channels with a low friction material. I will, however, lubricate the wood with a very viscous lubricant, preferably beeswax or the wear will likely be unacceptable.

Most of the assumptions made here are conservative. There is no realistic way a climber will be applying all their weight to a single plank 1/10th of the time. But I also have no reason to believe that pine won’t have 10x the wear coefficient of maple. So this seems like something we will just have to try out and see what happens.