Let’s talk numbers

The science of hardwood frames

I’ve put this on a seperate page as i’m guessing it won’t be of much interest to most people. However, for the engineers amongst us and anyone else that is intrigued about how you can actually get wood to perform as well as a production bike, here are some of the key numbers that demonstrate the suprising suitability of wood as a performance material.

Bikes are typically made from aluminium, steel or carbon fibre. Each material has its benefits and drawbacks and gives a frame a different ‘feel’
– Aluminium – Light but flexible and weak
– Steel – Stiff and strong but heavy
– Carbon – Stiff and light but brittle and often vulnerable due to its thin wall thickness

All these materials have fairly consistent properties but wood is difficult to catagorise in the same general way as each species has very varied properties.

Typical frame material properties:

Density
ρ (kg/m3)
Youngs Modulus
E (GPa)
Yield/ Failure Stress
σy (N/mm2)
Aluminium (6061) 2700 70 300
Steel (Chromoly) 7850 205 1000+
Carbon 1600 70-100 600+
Hardwood (typical range) 600-1000 10-20 50-150
(Parallel to grain)

As expected, the wood is much weaker (σ) and much more flexible (E) than any of the other materials but it is also much lighter (ρ).

But if it’s so weak and flexible how can you possibly make a decent frame out of it?

Not all wood is suitable for making bike frames. In fact very few woods are. We need the stiffest, strongest wood available which tends to go hand in hand with the densest, heaviest wood available. This leads us to slow grown exotic hardwoods that have taken years and years to slowly grow and as a result are denser and stronger. Quick growing softwoods as used for most cheap joinery is likely to be less than half the strength and stiffness of a suitable hardwood. The range of hardwood properties in the table above has already cut out all the hardwoods that are not suitable for frame building (too flexible/ weak). I’ve included a non exhaustive list of woods that could be suitable for frame building down at the bottom of this page. I’ve also included typical material properties for comparison although these should be treated as rough numbers as there tends to be a reasonable variation in any of these properties from piece to piece.

Let’s start with a quick explanation of the engineering theory behind the tubes that make up a bike frame. For most bikes, stiffness is much more important than strength. By the time you’ve made a bike stiff enough to ride properly you’ve generally put in enough material that the frame is plenty strong enough for the job in hand.

Stiffness is governed be several factors:

  • Axial stiffness is related to E (see table above) x A (area of section). This controls how much the tube extends and shortens under push/pull forces
  • Bending stiffness is related to E (see table above) x I (second moment of area of section). This controls how much the tube bends.
  • Also, the strength of the tube to tension/ compression forces is related to A (area) x σ (yield/ failure stress of material)
  • And the strength of the tube to bending forces is related to σ (yield/ failure stress of material) x I (second moment of area) / r (radius)

The sharp eyed might have spotted that I have conveniently ignored the torsional stiffness (twist of the tube along its length related to GJ). Calculating torsional deflections for wood is a much more complex affair as the wood is anisotropic so has different material properties parallel and perpendicular to the grain. Combined with this, laminating several sheets together as we are intending to do for a bike frame will have a significant effect on the torsional performance that I don’t know how to quantify. To muddy the waters further, the shear modulus of elasticity (G) for wood is a very poorly documented property. So, forgive me if we gloss over that for this rudimentary comparison.

Let’s run through the numbers on a quick and simple example to see how it works. It’s worth noting at this stage (before someone points it out!) that the stiffness equations are all related to length. The longer something is the more flexible it is, hence smaller frames are stiffer than larger frames using the same materials. We’re going to assume that the length of the tube is the same for each option and all we are going to change is the material and tube size. So, i’ve left length out of the equations to make it simpler.

The down tube of a typical aluminium bike is, say, a 45mm diameter tube that is 2mm thick all round (I know they’re all oval these days but it makes the maths a bit harder!)
So, the area of the tube A = π(ro2 – ri2) is 270mm²
r is radius and the o and i subscripts refer to outer and inner radii
The second moment of area I = π(ro4 – ri4)/4 is 62580mm4

Using the factors above with an E = 70GPa and ρ = 2700kg/m3 for aluminium:
EA = 18912
EI = 4380600
y = 81kN
σyI/r = 0.834kNm
And weight = Aρ = 0.73kg/m

Now, lets pick a hardwood like Padauk that has a good balance between stiffness, strength and weight (see the table at the bottom of the page)
Padauk: ρ = 790kg/m3, E = 15.9GPa, σy = 116N/mm2 (parallel to grain)

Obviously, if we do the same calculation as before, using the Padauk properties then we will get a much weaker, more flexible tube. So lets make the tube bigger. Say 60mm diameter and 5mm thick all round. Using the same equations as above:
A = 864mm2
I = 329376mm4
So,
EA = 13737
EI = 5237078
y = 100kN
σyI/r = 1.274kNm
And weight = Aρ = 0.68kg/m

Now, comparing the 2 sets of numbers above, and including another line for a typical steel frame. I’ve intentionally left out carbon fibre because i know very little about it’s structural properties and don’t want to say anything that’s incorrect.

Dimensions Section Properties Material Properties Stiffness Strength Weight
Diameter Thickness A I E s r EA EI sA sI/r
mm mm mm2 mm4 Gpa N/mm2 kg/m3 kN kNm2 kN kNm kg
Aluminium 45 2 270 62580 70 300 2700 18912 4.38 81 0.834 0.73
Steel 40 0.8 99 18932 205 1000 7850 20197 3.88 99 0.947 0.77
Padauk 60 5 864 329376 15.9 116 790 13737 5.24 100 1.274 0.68

Or, using Aluminium as our standard

Comparison Stiffness Strength Weight
EA EI sA sI/r
Aluminium 100% 100% 100% 100% 100%
Steel 107% 89% 122% 113% 106%
Padauk 73% 120% 124% 153% 94%

Lets start with the worst result. The Padauk tube is 27% more flexible along its length. Is this a problem? The deflection that you notice when riding a flexible bike is from the bottom bracket moving from side to side as you pedal hard. This is a result of bending in the seat tube and down tube and also torsion (twist) in the down tube. The extension/ compression along the length of the individual tubes is fractional and plays very little part in the overall stiffness. So, no we’re not that worried about the EA value being a bit lower for our wooden frame.

On to the more interesting bits, the Padauk tube is 20% stiffer in bending, over 25% stronger and 6% lighter than the aluminium tube. Even if we assume that we can expect a reasonable variation in material properties from the wood, we are still looking at a material that is at least comparable in performance and weight to an aluminium or steel frame.

Really? What’s the catch? Why aren’t we all riding wooden bikes then?

Yes, really! The secret to it performing so well is increasing the diameter of the tube. You can see there is an r4 in the stiffness calculation. This means that, everything else remaining equal, if you double the diameter the stiffness increases by sixteen times(2x2x2x2). That’s why when you look at high performance carbon fiber road bikes they all have massive oversized tubes. In the example above, increasing the tube size and thickness had a bigger effect than the reduction in material properties so we ended up better off overall. The problem for bike designers comes with balancing weight, strength and stiffness in a frame. You’ll notice you never see a steel frame with oversize tubes. Why? We’ve just seen that making the tubes bigger has massive benefits for stiffness, so surely we should make all the tubes bigger. However, bigger tubes are also heavier so, to reduce the weight, the wall thickness of the tube has to get thinner. A typical steel frame will have 0.8mm thick tubes. Unfortunately if they go much thinner than that then the frame becomes too fragile and you’d dent it far too easily. Hence why my beloved Condor Pista has a dent in one of the seat stays from some clumsy goon in the bike shed at work. So, aluminium, carbon and steel bike frames reach a limit of the maximum tube size you can get to before either weight or fragility become a problem and, unsuprisingly, most performance bikes are at this limit.

This neatly brings me on to a further benefit of wooden frames. The resistance of a frame to being dented is related to the strength of the material and thickness of the tube squared. In reality a smaller diameter tube is more resistant to a larger tube with the same wall thickness due to the additional curvature but lets ignore that for a minute.

Material Thickness (mm) σy (N/mm2) σyt2 %
Aluminium 2 300 1200 100%
Steel 0.8 1000 640 53%
Padauk 5 58 1450 121%

In the table above you’ll notice i’ve used half the maximum stress for Padauk that we used previously. This is a bit of a made up number! When looking at how the tube would fail due to being dented a significant proportion of the failure machanism would be due to the wood failing perpendicular to its grain. Using a solid piece of wood we might expect the failure stress to be around a third of the along grain stress but laminating several thinner sheets of wood we would expect slightly better results assuming the glue is at least as strong as the wood. Hence my made up factor of 1/2. We can make it 1/3 instead if you’re feeling sceptical and it will show that the wood comes out slightly weaker than the aluminium but still much stronger than the steel. Renovo have done some tests which show this quite nicely that can be seen here.

At the foot of this page i’ve included a table of dense hardwoods and their assiciated properties. I thought it would be interesting to do a straight strength to weight ratio comparison between the different materials. In the graphs below a steeper line means a better strength or stiffness to weight ratio. Carbon fibre wins in both of them but the hardwoods do pretty well and the densest woods like Greenheart come out higher than any of the metals even before we have considered the benefits that we can gain from larger tubes as we calculated above.

I know you’re looking for the downsides, and it would clearly be nonsence if this article said that hardwood was going to revolutionise the world of cycling. As I said at the start, the strength issue is a bit of a side issue for performance bikes. Bikes very rarely break and if they do it’s due to a crash or significant manufacturing flaw. Wood is probably more susceptible to flaws or manufacturing errors than metal frames due to the variability of the specimens (this can obviously be reduced by careful selection of materials) and also the increased workmanship that has to go into making a wooden frame. That aside, the calculations showing the added strength of a wooden frame were more to show the sceptics that it certainly won’t fall to pieces due to the fact that wood is weaker than metal!

The green credentials of a wooden frame are somewhat reduced by the types of hardwood that have to be selected. Whilst it may beat an electricity guzzling aluminium frame on the carbon footprint test, finding a suitably dense wood requires the use of old, slow grown trees that will take 10s if not 100s of years to replace. Bamboo can be produced quickly and can show very high strength to weight ratios. It is used for scaffolding and many other structural applications in developing countries and may well provide a good compromise for frame building. It is unlikely to ever match the more specialist woods though.

Another issue is the cost. Sourcing exotic hardwoods in the UK is not a cheap business and building a frame takes plenty of time and effort, even using automation. Even at a high turnover I doubt a wooden frame could compete with off the shelf bikes.

Below is a list of hardwoods with properties that may be suitable for frame building. There are plenty of other woods with similar or even better properties but most of these are likely to be rare, expensive and potentially endangered (eg Ebony or many Rosewoods). As stated above, the values below are approximate mean values there will be a range for all the wood types and some vary by up to +/- 20%. Most of the data came from http://tropix.cirad.fr/index_gb.html which has plenty of information on a huge range of wood species. It’s a french site so I had to do some googling to work out what the french name for a lot of the species were.

Density Modulus of Elasticity (E) Bending Strength (σb) Crushing Strength (σc) Origin Colour
kg/m3 N/mm2 N/mm2 N/mm2
Ash 680 12.9 113 51 Europe
Beech 710 15.3 111 57 Europe
Bubinga 920 20 137 76 Africa
Greenheart 970 30 217 98 S America
Hard Maple 740 13 100 55 America
Jarrah 850 20 101 81 Australia
Jatoba 940 23.5 160 97 S America
Mahogony 560 11.8 77 46 Africa
Meranti 680 13 92 52 Asia
Oak 740 13.3 105 58 Europe
Padauk 790 15.9 116 65 Africa
Purpleheart 870 21.3 141 80 S America
Sapele 690 14 102 62 Africa
Teak 660 13.7 98 56 Africa
Walnut 660 11.8 117 64 Europe/ America
Wenge 880 21 144 85 Africa
Zebrano 750 15.5 110 62 Africa

So now you know the score, what are you waiting for? Choose some wood and get building!

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