Torque
and Spoke Tension

## Torque and spokes

Our other articles about torque describe how it works for the bicycle frame, fork and drivetrain. We'll look here at the effect of torque on spoke tension.

This is quite different. Spokes do not twist or bend, and so do not transmit torque as such: they transmit it through changes in their tension. The amount of these changes depends on the spoking pattern and the dimensions of the hub and rim.

We'll look at front-wheel braking, and then at the rear wheel, where the spokes may have to resist torque in both directions.

## Torque and spoke tension in front-wheel braking

A rim brake transmits all torque to the frame or fork directly -- none through the spokes: the hub is free to turn, except for the tiny amount of torque due to rolling resistance of the hub's bearings. For this reason, a wheel with a rim brake can be spoked radially, as long as the hub's flanges can withstand the direct outward pull of the spokes. .

With a hub brake -- drum, disc, coaster -- on the other hand, all of the torque from braking is transmitted through the spokes, and so the spokes must be laced in a cross pattern.

Let's have an example.

The strongest braking force which a skillful cyclist can safely achieve equals about half the weight of the bicycle and rider. Here as in the main torque article, we'll assume a 200-pound bicycle and rider. Then the force of deceleration is 100 pounds. We'll also assume that braking is only with the front wheel, with a radius of 13.38 inches, or 1.11 feet -- so the torque from braking is 100 pounds times 1.11 feet, or 111 pound-feet.

Changes in spoke tension due to braking are easiest to calculate at the hub, rather than the rim. If we assume that the spokes leave the hub tangent to the hub flange (directly in line with the rotation of the hub), then the total change in spoke tension is the torque divided by the radius of the hub's spoke hole circle.

For a typical small-flange hub as used with a disc brake, the radius is about an inch, 1/12 foot. So, the total change in tension of the spokes in our example with 111 pound-feet of torque is 111/(1/12) pounds, or 1332 pounds -- more than 1/2 ton. That sounds dangerous!

However, this load is distributed among all the spokes. Assuming that the change in tension due to braking is distributed evenly among the spokes (and geometry says that it will be), we have

Spoke
count
Tension
change
in pounds
48 28
36 37
32 42
28 48
24 56
20 67
16 83

Half of the spokes in the cross pattern are leading spokes, and the other half, trailing spokes. Applying a hub brake as in our example will increase tension by the amount shown for the trailing spokes and decrease it by the same amount for the leading spokes.

The cross-sectional area of a 2mm spoke shaft is 3.14mm2 or about .005 in2. At an optimal pre-tensioning of a steel spoke, 50,000 pounds per square inch, the tension on each spoke with no load on the wheel is about 250 pounds. The tension change due to braking is only a fraction of that, and so it looks as though we should be safe, even with a 16-spoke wheel. No spokes will be grossly overtensioned, or go completely slack.

But wait, it isn't that simple! The drawing below shows the wheel turning clockwise and the hub brake trying to retard it counterclockwise, as indicated by white arrows. The drawing shows only one leading spoke and one trailing spoke, for the sake of clarity.

A hub brake would increase the tension of the blue, dashed leading spoke and decrease the tension of the white, trailing spoke. The effective radius for tension change is the radius at which an extension of the spoke across the hub flange would come closest to the axle, and is at 90 degrees from a line extended from the center of the wheel. If the spoke actually extended to this point, it would be pulling directly against torque. As this radius becomes smaller, the tension change of the spokes becomes greater, because the spokes are no longer pulling or pushing as nearly directly against torque.

Two other angles are shown in the drawing: an 8-degree nipple angle and a 56-degree hub angle.

The nipple angle is the angle between the spoke, as seen from the side, and a line extending directly toward the center of the wheel. This angle defines the ability of the spoke to resist torque at the rim. A radial spoke has a zero nipple angle and cannot resist torque. On the other hand, if the nipple angle is larger than about 10 degrees, the spoke nipples bind, making them hard to adjust, bending and weakening the spokes bend where they exit the nipples.

The nipple angle increases with the effective spoking radius and decreases with the size of the rim. But, conveniently, the load which torque places on the spokes is the same for wheels with the same rim entry angle and number of spokes, regardless of wheel size: the effective spoking radius and the torque increase at the same rate.

The hub angle is the angle at which the spokes rise above the hub flange. If this angle gets too low, each spoke covers up the next one over, making spoke installation difficult.

The table below gives examples of effective spoking radius and nipple angle for different cross numbers, with the same size of rim and hub. We have a simple Microsoft Excel spreadsheet into which you may insert other numbers, and we are working on upgrading Damon Rinard's Spocalc so it will also show them. The table below shows some columns from our simple spreadsheet.

 hub diam., mm 50 50 50 50 50 hub radius, mm 25 25 25 25 25 rim diam., mm 600 600 600 600 600 rim radius, mm 300 300 300 300 300 flange spacing from CL 35 35 35 35 35 spoke hole diam., mm 2 2 2 2 2 spoke count 36 36 36 36 36 cross 0 1 2 3 4 cross angle 0.0° 20.0° 40.0° 60.0° 80.0° hub entry angle 90.0° 68.2° 46.7° 25.7° 5.2° rim entry angle 0.0° 1.8° 3.3° 4.3° 4.8° spoke length 276.2 277.8 282.5 289.4 297.7 bracing angle 7.2° 7.2° 7.1° 6.9° 6.7° eff. spoking radius, mm 0.0 9.3 17.1 22.5 24.9 Clear-ance, mm 8.6 8.5 7.3 5.1 2.3 hub flange dishing 7.2° 7.7° 9.7° 15.9° 73.3°

• Weight loading reduces the tension of spokes at the bottom of the wheel. Typically, the weight loading is distributed over 3 or 4 spokes at the bottom. When using the front brake hard, the front wheel carries most of the weight of the bicycle and rider. For our 200-pound rider, weight loading may reduce the tension of the spoke which is directly over the tire contact patch by as much as 100 pounds during hard braking -- assuming that that the road suface is smooth. Loading is greater as the wheel climbs over a bump.
• Lateral loading loosens the spokes at the bottom on one side of the wheel and tightens the spokes on the other side. Sharp steering motions and leaning the bicycle under the rider pruduce these loads, Lateral loading is much heavier on a tricycle, which does not lean into turns.
• Spokes with 2mm shafts are the thickest in common use on bicycles. Some spokes are swaged, with shafts as thin as 1.5 mm, and the optimal tension of these in an unloaded wheel is only 140 pounds.
• The braking load and weight on the wheel may be greater than in our example -- with a heavy rider, baggage, a tandem, a front-loader cargo bike.
• As rims are not perfect, spoke tension must be adjusted to true the wheel, and so the spokes are not tensioned perfectly evenly. Uneven spoke tension makes a wheel more prone to failure.
• If the hub barrel is not stiff in torsion, the spokes on the same side of the wheel as the brake will take most of the braking load. I read somewhere years ago that a hub with a thin barrel transmits only about 12% of the torque to the opposite flange -- and that is for a rear hub with a short barrel!
• The wheel may be dished, and so the spokes on one side are looser. These are the spokes on the side opposite A disk brake, and if the hub barrel ans spokes are stiff, these spokeswill go slack before the spokes on the near side.

How can these problems be addressed?

• Larger hub flanges reduce the change in spoke tension due to torque. Doubling the radius of the circle of spoke holes reduces the change in spoke tension by half. For this reason, spoke tension change is not a problem with a drum brake where the spokes attach at the outside of the drum. There are special problems and solutions though with very large hubs, and we have an article about that..
• A larger number of spokes is better.
• A high cross number helps, so the spokes leave the hub at a larger effective radius.
• A stiff -- large-diameter -- hub barrel sends more of the torque to the spokes opposite the brake disk, and this is advantageous as long as their tension is high enough so they remain under tension in hard brakin..
• Thicker spokes are better, as long as the rim and hub can withstand the tension. When a wheel is dished, thicker spokes on the dished-in side and thinner ones on the dished-out side manage stresses better -- see our article about checking spoke tension for an explanation. It may also make sense to lace the spokes on the dished-out side radially, as described in our article on wheelbuilding.
• A hub with more widely-spaced flanges and less dishing is better.
• A smaller wheel is stronger for the same number of spokes, and braking torque is lower. This is one reason that 20-inch wheels make good sense on "tadpole" (two wheels in front) tricycles.
• Good wheelbuilding practice with a true rim and optimal spoke tensioning avoid loose spokes which go slack.
• On a tricycle, the two wheels on the same axle need to be strongest against lateral force pushing from the outsidel in a turn, the wheel on the outside is carrying most of the weight and and withstanding centrifugal force. The extra strength can be achieved by tilting the wheel so its top is inboard of its bottom.
• Good advice to spare the wheel is to unweight it and release the brake for big bumps, and to avoid leaning the bicycle under the rider when braking.

## Torque and rear-wheel spoking

As noted in our main torque article, skidding limits rear-wheel braking with a conventional bicycle to approximately 30% the weight of the rider and machine except with extreme abuse such as locking the brake when slamming the wheel down onto the ground. So, a hub brake in the rear wheel generally does not impose as heavy a torque load as one in the front wheel. A lower or longer machine can put more torque loading on the rear wheel in braking.

A rear wheel also transmits driving torque from pedaling. Other than with shock loading from a sudden pedal thrust, this torque is limited by lifting ofthe front wheel, when the driving force is about half the bicyclist's weight with a conventional bicycle.

With a rear wheel, as torque loosens one and tightens the other spoke of a laced pair -- interference between spokes and the rear derailer may occur. With the trailing spoke on the outside, pedaling pulls the spoke pair to the inside. But note, if the wheel also has a hub brake, that will pull the spoke pair to the outside, and the only way to avoid problems it to provide enough clearance so interference cannot occur.

The large the effective spoking radius, the less the tension on the spokes will vary, and so the less this effect will occur.