Hoop stress and cross-bending stiffness for the bicycle rim

I have been revising the structural model of the bicycle wheel.  The revised bicycle rim model has added the effects of hoop stress and cross-bending stiffness to the rim structural model.  The hoop stress is the stress compressing rim along its circumference in the azimuthal direction due to spoke tension.  Cross bending is the phenomenon that couples a bending force in one direction with a displacement normal to it.  The term is computed by using an off-diagonal coefficient in the stiffness matrix. This coefficient is called cross-bending stiffness.   A bending moment directed in the radial direction causes bending not only in the radial direction but also causes bending in the axial direction and vice versa.   The cross bending stiffness is the proportionality constant in the stiffness matrix that relates the axial bending moment to the radial curvature.  Both hoop stress and cross bending stiffness were previously neglected as small (and inconveniently tedious to include in the derivation).

The motivation to revise the model was the poor comparison we observed between some measured data for the rim model and the model predictions.  This revision has been underway since I discovered the poor comparison last April, eight months ago, more or less.  I suspected that hoop stress was part of the problem because the wheel that showed the poorest comparison was a cyclocross wheel with high tension and 32 spokes.  The number of spokes and the high tension combine to give a much higher hoop stress than other wheels I had tested that compared better.  Also the shape of the measured wheel with a single perturbed spoke (one spoke is tightened one full turn) was deformed in the familiar taco shape of a damaged wheel.  The taco shape is widely described as being a consequence of buckling of the rim.  Bucking is an elastic instability associated with the rim in compression.  Hence, I supposed that the physics related to the buckling was also related to the shape observed in the perturbation experiments.

The changes have been added one at a time.  First, only the hoop stress was added to the axial bending equation.  This term helped the axial displacement show the expected taco shape but with a smaller amptitude than the experiment.  Better comparison was obtained by various parameter changes that resulted in less stiff rim or more stiff spokes.  I expected similar improvement for the radial displacement.  When I added hoop compression terms to that equation, it hardly affected the radial displacement at all.  I sought out some additional physical phenomenon that had been neglected and found that potentially the cross-bending coupling might act in the way needed to improve the comparison.  I started making this addition in early December 2014 and had time during Christmas travels to work on it.   The solution came together while we were in California.  I finished the derivation in January and programmed it shortly afterward.  I also have written a separate document in which the equations are derived.  This document improves on the original derivation not only in adding the hoop stress and cross-bending terms but also because the equilibrium force balances are derived with new illustrations that aid in understanding the directions of force vectors in the differential element of the circular rim.    

The derivation is about 40 pages long at present count.  The programming of the new model is only about ten lines of coding.  It is an amazing compression of the time and thought, the physics and mathematics of the derivation into a compact algorithm.  The compression is aided by the tools for matrix algebra provided in Scilab but even a Fortran program would probably only be 20 or 30 lines.  The compactness of the final answer is beneficial in many ways, not the least of which is that there are not so many signs and matching parentheses to go wrong.  The symmetry of the solutions for the radial and axial equations is reassuring if not scientifically sound evidence of its correctness.  

The revised algorithm for the coefficient matrices is quite a bit shorter than the previous wheel models.  This is not because the model is simpler, it is not.  It is because I had some new insight into the matrix equations.  I learned that I could construct the coefficient matrices because they possessed the special symmetry of Toeplitz matrices.  This form allows the matrices to be defined by the elements of the first row and column.  I also was able to reduce the computations to high level matrix expressions, eliminating some intermediate variables and some complicated term-by-term computations involving individual matrix elements contained within for loops.   The end result is that the model programming and the equations in the derivation document are very close to the same.

The results are less successful in matching the experimental results than hoped.   The following three figures show the orginal model, the revised model with cross-bending stiffness, and the experimental results.

The tension is a pretty good match which suggests that something is right about the model.  The axial shape has the right general shape but the wrong magnitude.  The radial shape seems totally wrong.  It is very interesting that the experimental radial displacement echoes the two hump (inverted dromedary?) shape of the axial displacement but the computed displacement does not show any evidence of the taco shape.  The stability of the spoke model in tension totally eliminates the instability of rim bending in the radial direction (the instability that would be present in the bare rim unsupported by spokes).  The plots shown are for nominal dimensions and material properties, but I experimented with a range of parameter values for stiffness of the rim and spokes.  Nothing moves radial displacement into a taco shape.  

To explain the discrepancy, it occurs to me that I have plotted the displacement of the center of bending of the rim.  That is, Euler-Bernoulli approximations reduce the three dimensional rim to an equivalent one dimensional infinitesimal wire whose bending stiffness parameters are the same as the actual rim.  We are plotting the displacement of that infinitesimal wire.  This approximation neglects the fact that the actual rim rotates about the center of bending and the measurements taken on the surfaces of the actual rim would be affected by that rotation.  The effect on the illustrated results would be an enhancement of the displacement at the center of bending.  A rough calculation assuming the rim cross section is rigid (the effect of the Euler-Bernoulli equations) and using the angle of the rim displacement and the external dimensions of the rim to a displacement due to the rotation indicates that most of the observed discrepancy between measured and calculated would be corrected.

I am setting the bicycle wheel modeling work aside for a while as I work on getting our house in Oak Ridge ready to sell.  Please comment on the results if you are interested in helping sort out the deficiency or if you are just dying for me to do it to see how it turns out.