The tension-spoked bicycle wheel has endured for over 140 years due to its simplicity and elegance. In the days of the "ordinary" or "high-wheel" the bicycle was practically just one huge wheel with a slender frame and saddle. Although materials and manufacturing techniques have continued to envolve, the central design of the wheel has not significantly changed since the 1870s.

Despite its age, the bicycle wheel still poses many unanswered questions: what makes a wheel strong? How and why do wheels fail? What is the ideal strength of the wheel? I aim to answer these questions through the lens of engineering mechanics and mathematical modeling.

## Optimizing spoke tension

If the spokes of a bicycle wheel tensioned beyond a safe limit, the wheel can spontaneously collapse into a taco shape without applying any external forces. Exploiting an analogy with the classical beam-on-elastic-foundation problem, we've developed a mathematical model which predicts the critical spoke tension.

The key parameter controlling the maximum safe tension is the ratio between rim bending stiffness and twisting stiffness. The lower of the two stiffnesses controls the overall strength and stiffness of the wheel. Thus a single-wall rim (which is very flexible in twisting) will buckle much more easily than a double-wall rim (which has more equal bending and twisting stiffness).

## Strength of the bicycle wheel

Bicycle wheels can buckle (or taco) during use when the applied radial load (weight of the bike and rider) exceeds the wheel strength. The failure load is generally only reached after the spokes at the bottom of the wheel have completely lost tension and gone slack under the applied load. Therefore, the behavior of the wheel becomes nonlinear prior to failure, making an analytical solution next-to-impossible.

However, we have derived a simplified model for the strength of the wheel by considering the competition between two failure modes: (1) spoke buckling and (2) rim buckling. Increasing the spoke tension increases the spoke buckling load (because it has to lose more tension before it goes slack), but decreases the rim buckling load (because higher spoke tension leads to a lower lateral wheel stiffness).

In most cases, the model accurately predicts the average strength of the wheel, as calculated from nonlinear finite-element analysis (FEA). The horizontal spread of the data in the figure above reflects the fact that the wheel strength depends slightly on spoke tension. Additionally, since the *post*buckling behavior of wheels is unstable, the peak load is somewhat sensitive to imperfections.

## Finite-element simulations for wheels

I wrote bikewheelcalc in Python to simulate the mechanical response of a bicycle wheel to different loads. This program can be used to calculate the stiffness (along different directions), rim stresses, and change in spoke tensions for wheels with custom geometries, materials, or lacing patterns. I hope it can be useful to wheelbuilders and mechanics in designing wheels or selecting components, or to students learning about finite-element simulation and stress analysis.

See my previous post for an example of calculating wheel stiffness using **bikewheelcalc**

## Publications & presentations

**M. Ford** and O. Balogun, "Analytical Model for the Radial Strength and Collapse of the Bicycle Wheel", International Cycling Safety Conference 2017, Davis, CA. (Presentation Slides)

**M. Ford**, J.M. Papadopoulos, and O. Balogun, "Buckling of the bicycle wheel", Proceedings of the 2016 Bicycle and Motorcycle Dynamics Conference.

## Media mentions

Bicycling Magazine - This Professorâ€™s Bike Experiments Might Change the Way You Ride