TLDR; It’s theoretically possible to increase tube stiffness by using fiberglass packing tape.

# Intro

A common trick from R/C planes is to stiffen wings with strapping/filament tape. I got curious if this method would make a difference for the steel tubes used in a Lowrider.

The tape application would leave space for the bearings, so it would have no effect on the rolling surface. The calculations suggest that standard tape would be enough to increase stiffness by a bit. With an optimized tape the result might even be significant enough to warrant the effort.

# Theory

Absolute deflection of a fixed beam is \frac{P*x^2*(3L-4x)}{48EI}. In our case we have a beam composed of two different materials so we need to take that into account.

Nicely, this part of solid mechanics is linear, so in this case overall composite deflection of two materials is \frac{P*x^2*(3L-4x)}{48(E_1I_1+E_2I_2)}. In other words, if we want to look at relative changes in deflection, we can compare E_1I_1 to E_1I_1+E_2I_2. Thus, the percent increase in stiffness is simply:

\psi = \frac{E_2I_2}{E_1I_1}

# Worked example

Let’s apply a few strips of 1" wide strapping tape to a 1" steel tube and see what happens.

## Strapping tape mechanical properties

For the sake of the example, let’s use https://www.weasyadhesives.com/product/heavy-duty-strapping-filament-tape-232h/, which is with 0.17mm in thickness and has a breaking strength of 1450N/25mm wide.

### Area moment of inertia

We can calculate I_2 by using area moment equations. I’ll leave those off here, but because tape application is relatively uniform and symmetric let’s just calculate the area moment as the percentage of a tube:

I_{tape} = \frac{\pi}{4} (r_{o.d.}^4-r_{i.d.}^4) \frac{3 w_{tape}}{\pi r_{i.d.}}

where the r_{i.d.} is the o.d. of the steel tube, the r_{o.d.} is the i.d. + the tape’s thickness, and w_{tape} is the tape’s width.

I_{tape} = \pi/4((0.5*.0254+.00017)^4-(0.5*.0254)^4) * (3*1)/(\pi*1) = 1.07*10^{-9}\bf{m^4}

### Modulus of Elasticity

We can calculate I_{tape} from the above, but we need to know the tape’s Modulus of Elasticity (MoE). This isn’t given, but we can get a reasonable guess by assuming that the tape uses standard E-glass. We get there by trying to calculate what percentage of the tape is fiberglass (while assuming the rest is backing and adhesive).

- The area of 25mm of tape is 0.17*25 = 4.25mm^2.
- The breaking strength of this piece of tape is 1450N.
- This gives 1450N/4.25mm^@ = 341N/mm^2 = 341MPa.
- The ultimate yield strength for E-glass is 2GPa. (https://www.azom.com/properties.aspx?ArticleID=764)
- We calculate the percent strength .341/2 = 17%

(17% is a surprisingly reasonable percent fill. Standard vacuum-infused layups get around 40%. The highest you’ll see is in pulltruded rods, the best of which come close to 70%.)

Therefore our MoE is 17% of pure glass (~75GPa), which gives us:

E_{tape} = 13\bf{GPa}

## Steel tube mechanical properties

Assume a standard 1" x 0.049" steel tube, and published mechanical properties.

E_{steel} = 200\bf{GPa}

I_{steel} = \frac{\pi}{4} (r_{o.d.}^4-r_{i.d.}^4) = \frac{\pi}{4}(1^4-(1-0.049*2)^4)\frac{.0254^4}{2^4} = 6.91*10^{-9}\bf{m^4}

## Conclusion

\psi = \frac{E_{tape}I_{tape}}{E_{steel}I_{steel}} = \frac{13\rm{e}9 * 1.07\rm{e}-9}{200\rm{e}9 * 6.91\rm{e}-9} = 1\%

## Caveats

1%, just with a few pennies in tape?!! Well, there are a couple things to keep in mind:

### Adhesive effects

I didn’t account at all for the loss of shear strength due to the adhesive. This will probably be substantial, but certainly not overwhelming.

### Construction

I don’t think it would be easy to get 3 strips of 1" wide tape onto a 1" tube. Sure, it’s possible, but it’s gonna be hard. Better would be a 1/2" or 3/4" tape.

### Tube wall

I choose a pretty thin wall. A 0.065" wall would be naturally 25% stiffer.

# Afterward

I’ve not tested this. This is pure theory with some educated guesses, so the reality is going to be somewhat different, probably with less impact than predicted. Still, the theory is sound and I would be shocked to see a dramatic departure from the results.

I also don’t know how long the tape would last. It could be days, weeks, months, or years. YMMV.

More and thicker tape would increase the benefits. If there were some kind of carbon fiber strapping tape (does that exist?) then there would be an almost automatic 6x jump in the additional stiffness. Pultruded rods, adhered with double-stick tape, might have a tremendous bang for the buck.

BTW, if you’d like to buy the above tape, it’s so similar to mcmaster’s product, https://www.mcmaster.com/7686A13/, that it’s a good guess they’re the same part number and manufacturer.