# Properties of an hexagonal shape vs round tube shape

In a bike frame build, are there any differences between hexagonal cross-section tube instead of the typical round tubes?

Assuming that both are made from the same aluminium alloy and have the same wall thickness, are there differences in weight, strength, or durability? Does the hexagonal shape have any particular advantages over round tube?

• By thickness, do you mean tube wall thickness? As opposed to tube diameter? – wwarriner Apr 29 '16 at 22:04
• Tube wall thickness, yes. – Rfraile Apr 29 '16 at 22:07
• Would the hexagon inscribe (like this) or circuscribe (like this) the circle? – Wasabi Apr 30 '16 at 1:29

An additional factor is that any section which has defined corners will tend to concentrate stress at the corners rather than it being evenly distributed throughout the section.

With tubing this can be a double effect that you potentially have work hardening from the manufacturing process concentrated at the corners as well. If the section has a uniform wall thickness then it certainly means that some of the material is effectively wasted as the corners will encounter yield before the flats and at worst it can lead to crack propagation points and fatigue.

It is a fairly good rule of thumb in structures that any sort of discontinuity represents at best an inefficiency and at worst a potential failure point. An ideal structure will have a smoothly varying section with section size proportional to the stresses on it, in fact bones are an excellent example of this. Although this is, in most cases, impractical for fabricated frame structures.

Of course there are other pragmatic considerations to be taken into account for example square or rectangular tube is much easier to join than round as square tube can just be mitred to the required angle with a saw whereas round tube needs to be notched with a specialist machine or painstakingly hand fitted.

It seems likely to me that hex tube would be a fairly poor compromise between square and round/elliptical section tube.

I can't think of much advantage to using a hexagonal tube, except perhaps cosmetic - but this is subjective. Hexagonal tube is likely to be much more expensive as it is a non-standard shape.

If it is assumed that both the round and hexagonal tubes are extruded (which is likely for a bike frame) we can ignore any cold-forming or welding issues. For them to be the same strength (and weight) they would have to have the same cross-sectional area. The axial stiffness is $EA$, where $E$ is Young's modulus which depends on the material (assumed to be the same in this case) and $A$ which is the cross sectional area.

A hexagon with an equal cross-sectional area to a circle of radius $r$ and (thin) wall thickness $t$ must have side lengths of: $a=\frac{1}{3} \left(\pi r+\sqrt{3} t\right)$ if the wall thickness is assumed to be the same.

While bending is not a major factor in bike frames we can compare the bending stiffness of the two shapes anyway. Bending stiffness is $EI$ where $I$ is the second moment of area of the cross-section.

$I$ of a (thin) circle is: $I_c = \pi r^3 t$

$I$ of a hexagon (equal area and wall thickness as circle) is: $I_h =\frac{5}{18} t \left(9 a^3-9 \sqrt{3} a^2 t+12 a t^2-2 \sqrt{3} t^3\right) = \frac{5}{54} t \left(\pi ^3 r^3+3 \pi r t^2\right)$ after substituting $a$ from above.

We can immediately see that the hexagon has a lower second moment of area in comparison to the circle. In fact the second moment of area of the hexagon is about 91% of the circle.

We can also do a simple buckling comparison by considering the Euler buckling load: $\frac{\pi^2 EI}{L^2}$, where $L$ is the length of the tube. Again, the circle is more resistant to buckling since its $I$ is larger.

So this basic analysis shows that a hexagon of equal area and wall thickness would be weaker under bending and have a lower buckling load.

Some factors which I have not considered:

• localised buckling of tubes under bending (not a straightforward analysis - especially for the hexagon). For example the Brazier effect.
• Impact resistance. The hexagon may be more resistant to localised impact, particularly if that impact occurs at a corner. This is probably offset by the fact that a circle would deflect projectiles better.

In general a large diameter circular tube is the most efficient shape under axial loading.