# Spoke Patterns

## Crossings v. Stiffness

One of the issues of lore regarding bike wheels is the relative
merits of different cross patterns. You can have anything from no
crossings (more commonly termed *radial* spoking) up to
generally around 3 or 4 cross. The limit is a function of the ratio
between the rim diameter and the hub diameter. For the geometry as
set out on the main
analysis page, the limit is four cross.

This page examines the effect on the effective stiffness of the
wheel under vertical loads of different spoking patterns.

## The analyses

The basic model is as per the main
analysis page, with the
difference being only that there are variable number of crossings to the
spoking. Section properties, spoke numbers and applied loads are all
identical, and match those in the main page.

## Results

As discussed on the main page, you get most feel for the analysis
results by looking at the deformed shapes, so here they are:

0 cross (radial):

1 cross:

2 cross:

3 cross:

4 cross (almost tangential):

Already, it should be apparent that the answer is that different
spoking makes very little difference to the stiffness. All the plots
have deflection scaled up 100 times, and you'd be struggling to see
any difference.

Numerically, I can extract the vertical deflection at the middle of
the contact patch:

model | deflection mm per
1000N | stiffness N per mm |

0 cross | 0.1628mm | 6143 |

1 cross | 0.1642mm | 6090 |

2 cross | 0.1654mm | 6046 |

3 cross | 0.1675mm | 5970 |

4 cross | 0.1703mm | 5872 |

## Conclusions

So, a radially spoked wheel is about 4.6% stiffer than a tangentially
spoked one. Alternatively, if you apply 1000N (about 100kg, 220lb) to
each of the wheels, the tangential (four-cross) spoked one deflects 0.0075mm
(0.0003 inch) more than the radial spoked. Since the tyre is likely
to deflect several millimetres
at least (if 3mm, that's 400 times more deflection) I conclude the
spoking is unlikely to make a discernible difference to the vertical
stiffness of the wheel.

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