# Buckling of the Bicycle Wheel

The spokes of a bicycle wheel brace and stiffen its slender rim, but only as long as the pre-tension is high enough to prevent them from going slack under external loads. However, if the spoke tension is too high, the rim will buckle into a non-planar shape. Despite its importance for wheel strength and safety, we know of no formula in the literature for the maximum tension a wheel can withstand before buckling. Here we derive such a formula. The non-dimensionalized buckling tension depends only on two parameters which are identified as the ratio of rim torsional stiffness to rim bending stiffness, and the ratio of lateral spoke bracing stiffness to rim bending stiffness. The buckled rim adopts a sinusoidal lateral profile where the number of positive peaks n >= 2 is determined by these two stiffness ratios. The lateral stiffness of the wheel may be represented by a system of equivalent springs: two series-connected springs for the rim bending and torsion stiffness, connected in parallel with a spring for the spoke system.