# Stiff rod / tube - What material and diameter

I need a long arm which will transmit motion from a motor to a pen. It needs to be about 500mm long.

What I want to try and do is reduce the bend (i.e make it stiff) but also make it fairly light.

Question A) In this case, am I better off going for a tube rather than a solid rod?

Question B) Our of the materials available to me (which is carbon fibre, aluminium, steel or wood) - is there any of this which would have an obvious benefit?

• What load is applied? How is that load applied? ie what direction? – Solar Mike Jan 4 '19 at 13:37
• @SolarMike I'm not certain of the load applied, but not significant (small stepper motor), but the load is applied to the tube end (along the long edge) – Jimmy Jan 4 '19 at 14:38

When maximizing the shaft lateral stiffness vs. weight, one generally wants to optimize both the geometry and material, as you yourself have wisely decoupled in your question.

It's very common to optimize a system by constructing a ratio (termed a figure of merit) of geometric or material parameters.

I'm going to assume that the rod is loaded as a cantilever beam:

A tube is stiffer than a solid rod per weight, so it always benefits you (in terms of stiffness vs. weight alone) to use a tube, and in fact as wide a tube as possible. You can verify this by taking the ratio of the polar area moment of inertia (which mediates the stiffness) to the area (which mediates the weight): $$\frac{I}{A}\propto\frac{r_2^4-r_1^4}{r_2^2-r_1^2}=r_2^2+r_1^2.$$

The limit is the strength of the tube; thin-walled tubes easily buckle under a lateral load, unlike solid rods. And of course space limitations may constrain the use of a very wide tube.

For selecting the material, a useful technique is to use an Ashby chart:

Here, in terms of material properties, the stiffness is mediated by the elastic modulus, and the weight is mediated by the density. You're looking for materials that lie to the left and to the top. (Here, we're implicitly taking the ratio of the elastic modulus to the density.) If low weight is very important (at the expense of not maximizing stiffness), then wood is appropriate. If stiffness is very important (at the expense of not minimizing weight, then a ceramic would be suitable.

Note the constraints that come into play: neither wood nor a ceramic are as easy to machine into a thin-walled tube as metal, and metal can generally resist weathering and high temperature better than wood and can generally resist brittle fracture better than a ceramic. Consequently, geometry and material selection is often going to involve tradeoffs. The figure-of-merit approach allows to quantitatively identify the best solution by suitably weighting the parameters.

A compromise between efficiency and practicality:

Use a 1/4" type L copper tubing with brass fitting adaptor (1/4 to 1/2) or similar as base connection.

Then solder 2 gauge 16 electrical copper wires from the sides of brass fitting to the tip of tubing at the 50cm end so it forms a rigid horizontal long triangle.

Readily available materials, total cost under \$30.00 including the soldering gun, stiff enough vertically to support its own weight. And reinforced laterally by the two wires to take the load of pen.