BICYCLE WHEEL SPOKE LACING
Finite Element Analysis of Spoke Lacing Patterns
Background
Most bicycles used spoked wheels because they are light, strong and perform well in a cross
wind. Front wheels (without disc brakes) are symmetric from right to left, and do not have to
transmit torque, therefore design of spoked front wheels is fairly straightforward. These
days, most front wheels are radially laced, meaning that the spokes radiate straight out from
the hub to the rim.
Rear wheels (and Front wheels with disc brakes) transfer torque from the hub to the
pavement. A radially laced wheel cannot do this; in order to transmit torque, the spokes
must come off of the hub flange at an angle. One group of spokes comes out at an angle
opposite to the wheel rotation such that they pull the rim when you apply power to the
pedals. These are called the trailing spokes. A second set of spokes, called the leading
spokes, emerges from the hub at an angle in the direction of the wheel rotation, seeming to
push the rim. The leading spokes are needed to balance the tension of the trailing spokes –
without them the wheel would collapse. Because the spokes come out at the hub flange at
alternating angles, they will cross each other one or more times. The number of crosses
(designated 1x, 2x, 3x, etc.) depends on the number of spokes and the angle that they come
out at.
Figure 2 – Leading Spokes, Drive
Side Shown
Figure 1 – Trailing Spokes, Drive
Side Shown
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Rear wheels are further complicated by the need to accommodate the gear cluster. The
freehub and cassette eat up space and force the spokes on the drive side to move closer to the
wheel centerline, resulting in a dished wheel. Just as the leading/trailing spokes need to
balance each other, so it is with the drive side and non-drive side spokes. Because the nondrive side spokes come in at a shallower angle, they must have reduced tension to balance
the drive side spokes.
The Problem
When a rear wheel is tensioned, all of the drive side spokes should be at approximately the
same high tension, while the non-drive side spokes should all be at some lesser tension.
When power is applied to the pedals, the tension in the trailing spokes increases, while the
tension in the leading spokes decreases. Because the non-drive side spokes already have a
lower tension, they run the risk of becoming unloaded. A spoke that becomes unloaded with
each wheel revolution will rapidly fail.
The quick fix to this problem is to lace the non-drive side spokes radially. This decouples the
non-drive side spokes from the pedal torque and eliminates the loss of tension on the nondrive side. Such half radial laced wheels are quite common. Other manufacturers stick with
the traditional fully crossed lacing, but make sure that the spokes have enough pretension to
ensure that the spokes always stay positively
tensioned. One major wheel manufacturer goes so far
as to lace the drive side spokes radially, with the
crossed spokes on the non-drive side. The
manufacturer claims that this pattern provides
optimum energy transfer.
The Study
In the face of many claims regarding spoke lacing
patterns, this study set out to scientifically determine
which lacing pattern builds the stiffest, most reliable
wheel that delivers the most wattage to the
pavement. To find out, we built a finite element
model of a bicycle wheel and put it through a
computer simulation to determine spoke tension,
lateral flex and energy absorption. The model
Figure 3 – Fully Assembled Model
consisted of generic rim, hub and spoke components.
The model began with an aluminum rim with a depth of 28 mm and width of 23 mm, typical
of the current trend toward wider rims (such as Williams System 28). This was paired with a
generic aluminum hub with dimensions based on the DT Swiss 240S hub. The wheels were
laced with steel spokes of round cross section. Although round in shape, the spokes are
modeled with a cross sectional area equivalent to the popular Sapim CX-Ray spokes. For the
purpose of this study, the model well represents the performance of these aero shaped
spokes. The spokes were laced to the rim in four patterns as follows:
1. 3x/3x – Three cross lacing on both sides
2. 2x/2x – Two cross lacing on both sides
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3. 3x/Radial – Three cross lacing on drive side / Radial on non-drive side
4. Radial/3x – Radial on drive side / Three cross lacing on non-drive side
All of the wheels were laced with 28 spokes. It is expected that the trends found in this
study will be applicable to wheels with greater or lesser numbers of spokes. Also, in a side
study, it was found that wheels with more spokes are stronger and stiffer, which comes as no
surprise.
The drive side spokes were pretensioned to 225 pounds (100 kg-f) which stresses them to
approximately 40 percent of the tensile strength of the Sapim CX-Ray spokes. The non-drive
side spokes were pretensioned to the lesser value of 135 pounds (60 kg-f) to balance the
wheel dish. This represents a highly tensioned wheel which can be expected from a quality
wheel builder using good quality components.
The wheels were loaded to simulate a straight ahead hard pedaling effort. It is assumed that
a force of 150 pounds is applied to the pedals which represents a light weight rider fully out
of the saddle standing on the pedal, or a heavier rider (such as the author) still seated in the
saddle but mashing very hard. This is typical of hard climbing effort and puts maximum
stress on the drive wheel. We have assumed that we are in the small ring (34T) of a compact
crankset, and have selected a 22 tooth gear in the rear cassette.
We used the state of the art ABAQUS CAE finite element modeling suite to build and
execute this finite element model. ABAQUS CAE is widely used in the defense, automotive
and aerospace industries, and is recognized as one of the leading mechanical simulation
programs on the market today.
The Results
When it comes to straight ahead power and performance, riders are concerned with lateral
flex and power transfer. Because a dished wheel is asymmetric, it will distort and twist
under load resulting in lateral deflections of the rim. These lateral deflections are not sensed
by the rider per se, but if they become large enough will result in brake rub which will spoil
any rider’s effort. In addition to the lateral deflection, there are also radial and torsional
(windup) deflections that contribute to the overall flexibility of the wheel. Each of these
deflection components rob the rider of precious energy and reduce the wattage delivered to
the pavement. Rather than get into the details of all of the deflection components, we used
total strain energy absorbed by the wheel as the metric to capture the net effect of all the
stiffness components. Once the strain energy absorption is known, the power loss (in watts)
is calculated by multiplying the strain energy per pedal stroke by the pedaling cadence.
Structural Adequacy and Fatigue Strength
Before delving into the lateral deflections and energy loss, we must check to make sure that
the wheel is structurally sound and will have a satisfactory fatigue life. The series of figures
that follow show the spoke tension in the wheels after pretension, under rider weight only,
and under rider weight plus pedaling effort. Spoke tension is represented in N/mm 2 which is
a measure of stress in the spoke material. For comparison, the tensile strength of the CXRay spokes is given by Sapim as 1600 N/mm2.
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The figures are shown for the 3x/3x laced
wheel. In the pretension case (Figure 4a), all
of the drive side spokes are evenly tensioned to
650 N/mm2, and the non-drive side spokes are
tensioned to 390 N/mm2. When rider weight is
added to the wheel (Figure 4b), the loads on
the spokes near the bottom of the wheel are
seen to decrease considerably, whereas the
tension in the remainder of the spokes
increases very slightly. Finally, when torque
is added, the tension in all of the trailing
spokes increases while the tension in the
leading spokes decreases. For the 3x/3x wheel,
the maximum tension was found to be 747
N/mm2 on a drive side trailing spoke, and the
minimum tension was found to be 248 N/mm2
Figure 4a –Pretension
on a non-drive side leading spoke. The
maximum tension is less than half of the 1600
N/mm2 tensile strength of the CX-Ray spoke, which can be said to have a Safety Factor of 2.1.
The minimum tension of 250 N/mm2, although significantly reduced from the original tension
of 390 N/mm2, still provides a positive tension on the wheel with a generous margin of safety.
The 3x/3x laced wheel is, therefore, considered structurally adequate and would be expected
to have a long life without concern of spoke breakage. Results for the 2x/2x laced wheel are
similar, but slightly higher (763 N/mm2 (maximum) and 285 N/mm2 (minimum)).
Figure 5 shows the full stress plot of the 3x/3x
wheel. This shows the stresses in the rim and
hub, as well as those in the spokes. As can be
seen, the stress in the hub and rim is much
lower than in the spokes. Figure 5
demonstrates that the spoke with the least
tension is a non-drive side leading spoke near
the bottom. This is because (1) the non-drive
side spokes have less tension to begin with; (2)
Rider weight causes the spokes at the bottom
to lose tension; and (3) Torque causes the
leading spokes to lose tension. The figure also
shows that the spokes with the highest tension
are trailing spokes on the drive side away from
the bottom. This is because of the trailing
spokes gain tension when torque is applied,
and because the drive side spokes have higher
tension to begin with. Note that rider weight
does not have much effect once you get away
from the bottom of the wheel.
Figure 4b – Add Rider Weight
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For the wheel with
radial lacing on the
non-drive side, the
reduction in tension of
the non-drive side
spokes under torque is
virtually eliminated.
However the drive
side spokes experience
a greater maximum
tension and a greater
range of tension as the
wheel revolves. It is
this range of tension
which results in
fatigue damage to the
spokes and nipples.
The wheel with radial
lacing on the drive
side showed maximum
and minimum tensions of 680 N/mm2 and 183 N/mm2, respectively. Although, there is still
plenty of tension on the non-drive side trailing spokes, this design provides the least margin
for losing tension. The leading spokes on this wheel also go through the greatest range of
tension.
Table 1 summarizes the spokes tensions in each of the wheels. The table provides maximum,
minimum and maximum range of tension for the spokes. For a wheel to be structurally
adequate, the maximum tension should be well under the tensile strength of the spokes. In
all four cases we
find that the
tension is less
than half of the
Maximum Tension
Minimum Tension
Max Tension Range
Spoke
tensile strength
Lacing
Force (N)
Location
Force (N)
Location
Force (N)
Location
of the Sapim
CX-Ray spokes.
3x / 3x
747
DS Trailing
248
NDS Leading
142
NDS Leading
The wheel with
2x / 2x
763
DS Trailing
285
NDS Leading
192
DS Leading
radial spokes on
the drive side
3x / Radial
800
DS Trailing
305
NDS Radial
186
DS Leading
has the lowest
Radial / 3x
680
DS Radial
183
NDS Leading
207
NDS Leading
maximum
tension.
Perhaps, this could be advantageous if a weaker spoke material such as aluminum is used,
but provides no real benefit when high strength steel spokes are used.
Table 1 - Spoke Tension Values
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It is also desired to ensure that all spokes remain under positive tension to avoid damage to
the spokes and nipples. Again all of these wheels maintain positive tension, however the
Radial/3x wheel has the least margin for error. Conversely, the 3x/Radial laced wheel
provides the greatest margin for error. With lower tensioned spokes, the radial lacing on the
non-drive side would definitely reduce the chances of losing tension, but should not necessary
in a well built, wheel with adequate pretension.
Finally, and most important, is the tension range. This is the range of tension that an
individual spoke will go through with each revolution of the wheel, and is the cause of metal
fatigue. The design with the lowest tension range will have the longest fatigue life, and will
provide the most reliable wheel. Table 1 shows that the 3x/3x wheel has the lowest tension
range and would be expected to have the longest life and greatest reliability.
Lateral Deflection and Brake Rub
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Powerful cyclists are often concerned about brake rub during all out climbing bursts and
sprints due to flexure of the wheel. Therefore, we used the finite element models to
determine the lateral deflection of the wheels under load. Figures 6 through 9 show the
lateral deflection of the rim for each lacing pattern. The deflections are in millimeters.
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Table 2 summarizes the maximum lateral deflection
of each wheel configuration. These results show that
the Radial/3x wheel has the least lateral deflection,
and therefore the least tendency to rub the brake
pads. This is somewhat surprising, since as we will
Lateral
Spoke Lacing
show in the next section, this wheel is the least stiff
Deflection
overall. It appears that placing the radial spokes on
3x / 3x
0.23 mm
the drive side provides some balance to this wheel.
2x / 2x
0.23 mm
The 3x/Radial wheel has the greatest lateral
3x / Radial
0.26 mm
deflection, and thus the greatest tendency to rub the
Radial / 3x
0.18 mm
brake pads. The two fully crossed wheels showed the
exact same lateral deflection and fell in between the
two half radial designs. All of these wheels show very small lateral deflection on the order of
¼ of millimeter. This is likely due to the fact that these wheels all have a high spoke count.
With lower spoke counts, the deflections will increase, but it is expected that the lacing
trends will continue to hold.
Table 2
Lateral Deflection
General Flexibility and Power Loss
A bicycle wheel, like anything else, has a characteristic springiness which will cause it to
absorb and store energy. When you push on the pedals some of your effort goes into flexing
the wheel rather than being delivered to the road as useful power. This energy that is
absorbed by the flexing wheel is
known as elastic strain energy.
It is the result of atoms in the
metal grains being stretched
apart. In theory, this strain
energy is recoverable, but the
human body does not have the
capacity to recover this energy.
Therefore, most of it is
dissipated as heat and is
wasted.
It is a principle of all springs
that the stiffer the spring the
less strain energy that it will
absorb under a given load. That
is why bicycle frame and
component manufacturers strive
Figure 10 – Complete Wheel Stress and Deflection
to make their designs very stiff
28 Spokes – 3x DS / Radial NDS
(at least on the racing end of
Torque plus Rider Weight – Deflection Magnified 100x
things). However, no matter
how stiffly they are made, all wheels will flex to some degree under load. Refer back to
Figure 5 to see the deflection of the 3x/3x wheel under torque load magnified 100 times.
Figure 10 shows the similar deflection for the 3x/Radial laced wheel. Note that the half
radial laced wheel deflects more than the 3x/3x wheel shown in Figure 5. It is interesting to
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see how the rims tends to flatten into a polygon with seven sides. The seven sides correspond
to each group of four spokes (DS Leading, DS Trailing, NDS Leading, NDS Trailing). The
deflection appears dramatic in the figures because of the 100x scale factor. When viewed at
1x scale factor, the deflection is not noticeable. Although it appears that the rim undergoes
the most deflection, it is actually the stretching of the spokes that absorbs the most elastic
strain energy.
Chart 1 below shows the power in watts absorbed by each of the wheels when power is
applied to the pedals. The power is calculated assuming that the wheel is flexed twice per
crank revolution (once for the left foot and once for the right foot). A pedaling cadence of 80
revolutions per minute is also assumed. The power is calculated as twice the strain energy
divided by the time it takes to make one complete crank revolution (80 rpm  0.75 seconds
per revolution). The
Chart 1 - Power Lost to Wheel Flex
value of the strain
energy is calculated
1.20
by the finite element
analysis.
1.00
0.80
Power
Loss 0.60
(Watts)
0.40
0.20
0.00
3x-3x
2x-2x
3x-Radial
Radial-3x
The chart shows that
the fully cross laced
wheels suffer
significantly less
power loss than the
half radial laced. In
fact the 3x/3x and
2x/2x wheels lose
half as much power
as the Radial/3x
wheel.
Conclusion
In this analysis the 3x/3x wheel provided the best combination of strength, stiffness, power
transfer and reliability. The 3x/3x wheel outperformed the two half radial wheels because all
of the spokes share the power. In the half radial wheels, only the crossed spokes transfer
power – the radial spokes only support rider weight. The 2x/2x wheel matched the 3x/3x
wheel in stiffness and power transfer, but the 3x/3x wheel won on fatigue strength. This is
because the spokes emerge from the hub at an angle nearly tangent to the hub flange. This
is true with a 28 spoke wheel; however, as the number of spokes in the wheel increases or
decreases, the number of spoke crossings would need to be adjusted to maintain the tangent
condition. Accordingly, 2x lacing should be used with 20 spokes or less, 3x lacing should be
used with 24 or 28 spokes, and 4x lacing should be used with 32 spokes or greater. So it is
concluded that the best rear wheel is cross laced on both sides with number of crossing
chosen to make the spokes as close to tangent to the hub flange as possible.
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