BICYCLE WHEEL SPOKE LACING

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2 Background BICYCLE WHEEL SPOKE LACING Finite Element Analysis of Spoke Lacing Patterns 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 1 Trailing Spokes, Drive Side Shown Figure 2 Leading Spokes, Drive Side Shown 2

3 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 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 Figure 3 Fully Assembled Model 3

4 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 CX- Ray spokes is given by Sapim as 1600 N/mm 2. 4

5 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/mm 2, and the non-drive side spokes are tensioned to 390 N/mm 2. 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/mm 2 on a drive side trailing spoke, and the minimum tension was found to be 248 N/mm 2 Figure 4a Pretension on a non-drive side leading spoke. The maximum tension is less than half of the 1600 N/mm 2 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/mm 2, although significantly reduced from the original tension of 390 N/mm 2, 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/mm 2 (maximum) and 285 N/mm 2 (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 5

6 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/mm 2 and 183 N/mm 2, 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 Spoke Lacing Table 1 - Spoke Tension Values Maximum Tension Minimum Tension Max Tension Range Force (N) Location Force (N) Location Force (N) Location 3x / 3x 747 DS Trailing 248 NDS Leading 142 NDS Leading 2x / 2x 763 DS Trailing 285 NDS Leading 192 DS Leading 3x / Radial 800 DS Trailing 305 NDS Radial 186 DS Leading Radial / 3x 680 DS Radial 183 NDS Leading 207 NDS Leading find that the tension is less than half of the tensile strength of the Sapim CX-Ray spokes. The wheel with radial spokes on the drive side has the lowest 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. 6

7 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 7

8 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. 8

9 Table 2 Lateral Deflection Spoke Lacing 3x / 3x 2x / 2x 3x / Radial Radial / 3x Lateral Deflection 0.23 mm 0.23 mm 0.26 mm 0.18 mm 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 show in the next section, this wheel is the least stiff overall. It appears that placing the radial spokes on the drive side provides some balance to this wheel. The 3x/Radial wheel has the greatest lateral deflection, and thus the greatest tendency to rub the 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. 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 to make their designs very stiff (at least on the racing end of things). However, no matter Figure 10 Complete Wheel Stress and Deflection 28 Spokes 3x DS / Radial NDS Torque plus Rider Weight Deflection Magnified 100x 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 9

10 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 1.20 energy is calculated by the finite element 1.00 analysis. Power Loss (Watts) Conclusion 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. 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. 10