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Getting a Stiffy – wall thickness vs. tubing diameter

Here we go, I promised myself I’d dig some numbers up to prove this…

Bigger or thicker?

One of the common things people do to increase the stiffness of a frame is to increase the wall thickness of the tubing, typically going up from 0.9mm to 1.2mm.

Now most commercial frames that I’ve seen never feel the need to go thicker than 0.9mm and here’s why:

It’s actually better to increase the overall diameter of the tube rather than the wall thickness – you end up with a tube that’s not only stiffer but also lighter!

Let’s see if we can prove that with some numbers in a very simple example. Take a tube ( pretend it doesn’t weigh anything!) stick one end to a wall, then hang a weight on the end:

Et Viola, a cantilever tubular beam.

Et Viola, a cantilever tubular beam.

We can calculate the elastic deflection \delta (we’re not going to snap it) using

c32ad9c31ad56dbb27215044fcaaba64

where

F = Force acting on the tip of the beam

L = Length of the beam (span)

E = Modulus of elasticity

I = Area moment of inertia

For a tube the Area moment of inertia is

473c4e0b819edaff5aec00ab856e80a7

where

r1 = internal radius of the tube

r2 = external radius of the tube

And for Grade 9 (3/2.5) Titanium the Modulus of elasticity is around 107Gpa or 107,000,000,000N/m2

Nearly there!

Triton-Bikes-February-2012_125

How bendy?

Right let’s take our first tube, a typical downtube on a frame, which is 500mm long and 38mm overall diameter we’re going to try and make it stiffer than stock with a wall thickness of 1.2mm. Hang a 20kg weight off the end of it and the end will flex by…

…umm…

…takes of shoes and socks to start counting on toes…

…carry the one and take away the number you first thought of…

 3.2mm deflection.

Now let’s compare that with a 500mm tube with a 45mm overall diameter with a wall thickness of 0.9mm. Banging the numbers through we get a figure of

Ooo, look, 2.5mm deflection!

Nice, so we know it’s stiffer but it must be heavier, right? Let’s have a look…

How heavy?

Mass = density x volume which in our case is

mass = D( π(r22 – r12)L )

For grade 9 Titanium the density is around 4480 kg/m³

For our first 38mm x 1.2mm tube that’s 310 grams

and for our second 45mm x 0.9mm tube it’s 279 grams!

QED – we can make a tube stiffer, with the smallest increase in weight, by making the overall diameter larger instead of increasing the wall thickness.

When do we stop?

So you could just carry on making the tube larger and larger while making the wall thickness thinner and thinner? Err, no.

As you might imagine there’s a practical limit to how thin you can make the wall thickness of a tube before it starts acting like a coke can and becomes easy to crush!

I’ve seen titanium bike tubing quoted as thin as 0.6mm in the center of double butted sections (it’s still 0.9mm on the ends) but 0.7mm is the more common thickness used. If common is the right word, of course, double butted tubes are still quite rare because of the extra costs associated with making them…

Wow, well done for making it this far! Do let me know if my sums look wrong, but I think I’m there – in the mean time stay tuned for some  more Gallery posts of people’s fine rides…