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Stress Analysis of a Bicycle ME3213 Project
David Lopez, Jovan Mayfield, & Pierre Marc Paras
12/9/2010
Stress Analysis of a Bicycle
December 9, 2010
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Contents 1 Abstract ................................................................................................................................................. 3
2 Introduction .......................................................................................................................................... 3
3 Design .................................................................................................................................................... 4
3.1 Bicycle Chain ................................................................................................................................. 4
3.2 Chain Wheel .................................................................................................................................. 4
3.3 Pedal/Crank Assembly .................................................................................................................. 5
3.4 Bicycle Frame ................................................................................................................................ 6
3.5 Handlebars .................................................................................................................................... 7
4 Assumptions .......................................................................................................................................... 8
5 Analysis ................................................................................................................................................. 8
5.1 Bicycle Chain ................................................................................................................................. 9
5.2 Chain Wheel .................................................................................................................................. 9
5.3 Pedal/Crank Assembly ................................................................................................................ 10
5.4 Bicycle Frame .............................................................................................................................. 10
5.5 Handle Bars ................................................................................................................................. 11
6 Test Design .......................................................................................................................................... 11
7 Summary ............................................................................................................................................. 12
8 Bibliography ........................................................................................................................................ 12
Figure 1: Bicycle chain ................................................................................................................................... 4
Figure 2: Chain Wheel ................................................................................................................................... 4
Figure 3: Pedal/Crank assembly .................................................................................................................... 5
Figure 4: Bicycle Frame ................................................................................................................................. 6
Figure 5: Handlebar ....................................................................................................................................... 7
Table 1: Maximum Stress Found ................................................................................................................. 11
Stress Analysis of a Bicycle
December 9, 2010
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1 Abstract The design project allows us to get a fundamental understanding of how materials and the mechan-
ics of those materials are used in real world situations, with real world application. We were required to
design a two-wheel bicycle that can safely withstand 250 lbs and an acceleration of 15 m/s2. This design
was done first, on the individual parts to ensure that each portion of the bicycle was capable of support-
ing the stresses induced by the forces and reactions acting on them. After making educated assump-
tions, analysis was performed on the individual parts, and finally on the total design for the final testing.
The bicycle performed as expected from the assumptions and calculations.
2 Introduction In order to perform more accurate analysis the bicycle was broken down to familiar shapes and
members such as prismatic bars, hollow tubes, and bars with holes through the face. The designing of
the bicycle also required us to divide the bike into five portions: the bicycle chain, the chain/wheel con-
nection, the pedal/crank assembly, the bicycle frame, and the handlebars. Treating the portions as free-
body diagram like structures, the forces acting on each portion were determined, and the stresses
whether axial, shear, torsion, or bending moment were determined and calculated. After solving how
the bicycle would act in the separate portions, they needed to be assembled to gain a full analysis to
ensure that the bicycle would act as expected after analyzing the parts. Also the parts were directly as-
sociated with each other, such as the bicycle chain to chain/wheel connection, so it was essential that
the numbers were in agreement with each other as they shared the tension in the chain.
After determining what forces and stresses the bicycle would experience the next step was to
choose materials able to securely withstand these effects. After choosing a factor of safety (FS) of 2, the
material for each component was chosen, which could withstand the maximum stresses of each portion
of the bicycle with the FS enforced.
Stress Analysis of a Bicycle
December 9, 2010
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3 Design
3.1 Bicycle Chain
Figure 1: Bicycle chain
3.2 Chain Wheel
Figure 2: Chain Wheel
Stress Analysis of a Bicycle
December 9, 2010
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3.3 Pedal/Crank Assembly
Figure 3: Pedal/Crank assembly
Stress Analysis of a Bicycle
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3.4 Bicycle Frame
Figure 4: Bicycle Frame
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3.5 Handlebars
Figure 5: Handlebar
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4 Assumptions A bike chain is constructed with many small links. These small links are like bars but with two holes
drilled into them. It can be assumed that there is constant tension all throughout the entire chain. Also,
there are normal stresses acting upon the outer and inner links within the chain. The tension of the
chain is in relation to the radius of the chain wheel and the force applied on the pedals. Along with nor-
mal stresses, there are shear and bearing stresses acting in the chain. All the shear and bearing stresses
are placed upon the pins holding the links together.
In the chain wheel, we assume that there is little or no friction. We also as assume that the radius is
even and the same throughout the entire wheel.
In the pedal and crank assembly, it is assumed that the biker presses down with a force of 60 lbs on
each pedal. Upon looking at all the forces in a pedal crank, it is expected to break or split the crank into
two different and simple bars. The forces acting upon the crank is an assumed combination of bending
moment and torsion. It can be assumed all the torsion and bending moment comes from the force ap-
plied upon the pedal. In this project we are assuming no friction between parts and for the force on the
pedal to be constant.
In the bicycle frame, we assume that all of the riderβs weight is placed on the seat. We assume that
all the reaction forces are equal to the weight.
5 Analysis Before we begin our analysis of various components, it first necessary to find the required force
applied on the pedal, such that the bicycle will accelerate at a rate of 15 ft/s2. To do this first the inertial
force is found
πΌπ = ππ =π
ππ =
250 πππ
32.2 ππ‘π 2
15 ππ‘
π 2 = 116.46 πππ
Since this is the resisting force of the motion of the bicycle, this is also the force resisting the mo-
tion of the gear, or the tension in the chain. Now the required force applied on the pedal is found
ππΆ = 0 = πΉπΏπΆ β ππ
πΉ =ππ
πΏπΆ= 116.46 πππ 3.5 ππ
7 ππ= 58.23 πππ
Stress Analysis of a Bicycle
December 9, 2010
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5.1 Bicycle Chain The tensile force in the chain, which is equal to inertial force of the bicycle, was found to be
π = 116.46 πππ
Since the links contain holes in the stress in the link is not distributed evenly, but reaches a maxi-
mum near the hole. The maximum stress acting on the outer links in the chain is found by
ππππ₯ = πΎππππ = πΎπΉ
π΄
Where K, the stress-concentration factor, was found to be 2.4, from Fig. 2-63, on page 167 of the
textbook.
ππππ₯ = 2.4 116.46 πππ 2
. 0625 ππ . 25 ππ = 3726.72 ππ π
The shear stress acting on the pin holding the links together is given by
π =πΉ
π΄=
π
2ππ2= 116.46 πππ 2
π . 0625 ππ 2= 4745.01 ππ π
π1,2 = Β±ππ₯π¦ = Β±4745.01 ππ π
5.2 Chain Wheel The spokes on the chain wheel which pull on the chain links, experience a shear stress on the area
between the base of the spoke and the chain wheel. The area on the base of each spoke is roughly 1/8
in by 3/8 in.
ππ₯π¦ =πΉ
π΄=
116.46 πππ
. 125 ππ . 375 ππ = 2484.48 ππ π
π1,2 = Β±ππ₯π¦ = Β±2484.48 ππ π
Stress Analysis of a Bicycle
December 9, 2010
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5.3 Pedal/Crank Assembly To analyze the maximum state of stress acting on the crank, we choose a point on the surface of
the crank, for maximum shear stress, and a point at the furthest distance away from the pedal, for max-
imum bending moment stress. Since the stresses acting on the crank are a combination of the torsion
and bending moment, the total state of stress is given by
ππ₯ = 0
ππ¦ =π
π=
πΉπΏπΆππ3 32
= 58.23 πππ 7 ππ
π . 5 ππ 3 32 = 33215.1 ππ π
π =ππ
πΌπ=
16πΉπΏπππ3
=16 58.23 πππ 4 ππ
π . 5 ππ 3= 9490.0 ππ π
With the state of stress found at this point on the crank, the principle stresses and maximum
shear stress are calculated as shown below
π1,2 =ππ₯ + ππ¦
2Β±
ππ₯ β ππ¦
2
2
+ ππ₯π¦2 =
33215.1 ππ π
2Β±
β33215.1 ππ π
2
2
+ 9490.0 2
π1 = 35735.3 ππ π
π2 = β2520.2 ππ π
ππππ₯ = ππ₯ β ππ¦
2
2
+ ππ₯π¦2 =
β34224.7 ππ π
2
2
+ 9778.48 2 = 19127.7 ππ π
5.4 Bicycle Frame The point chosen on the bicycle frame is on the top surface of the tube segment labeled BC. This
point is also located very near point B in figure 4.
ππ₯ =π
π=
ππΏπ΅πΆπ4 π2
4 β π14 π2
= 250 πππ 24 ππ
π4 . 625 ππ 4 β . 5 ππ 4 . 625
= 52999.9 ππ π
ππ¦ = 0
π = 0
ππππ₯ =ππ₯2
=52999.9 ππ π
2= 26499.9 ππ π
Stress Analysis of a Bicycle
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5.5 Handle Bars The point chosen on the handle bars it located on the surface of the tube, 45Λ from the top, at
point A.
ππ₯ =π
π=
πΏπ»πΉπ4 π2
4 β π14 π2
= 7.25 ππ 20 πππ
π4 . 625 ππ 4 β . 5 ππ 4 . 625
= 1280.83 ππ π
ππ¦ = 0
ππ₯π¦ = ππ
πΌπ=
ππ2π2 π2
4 β π14
= 5 ππ 20 πππ . 625 ππ π2 . 625 ππ 4 β . 5 4
= 441.666 ππ π
π1,2 = ππ₯ + ππ¦
2Β± (
ππ₯ + ππ¦
2)2 + ππ₯π¦
2 = 1280.83 ππ π
2Β±
1280.83 ππ π
2
2
+ 441.666 ππ π2
π1 = 1418.36 ππ π
π2 = β137.528 ππ π
ππππ₯ = ππ₯ β ππ¦
2
2
+ ππ₯π¦2 =
1280.83 ππ π
2
2
+ 441.666 ππ π2 = 777.943 ππ π
6 Test Design Tabulated below are the results for the maximum stresses experiences by various components on
the bicycle. With a factor of safety of two, the maximum stresses are doubled.
Table 1: Maximum Stress Found
Component Maximum Stress Max Stress w/
Factor of Safety
Chain Link 3.72672 ksi 7.45344 ksi
Chain Pin 4.74501 ksi 9.49002 ksi
Chain Wheel 2.48448 ksi 4.96896 ksi
Pedal Crank 35.7353 ksi 71.4706 ksi
Bicycle Frame 52.9999 ksi 105.9998 ksi
Handle Bars 1.41836 ksi 2.83672 ksi
Using Table H-3: Mechanical Properties, materials with a minimum yield stress that meet the crite-
ria are chosen. For the chain link, pin, and wheel, an aluminum alloy can be used. Table H-3 lists alumi-
num alloys as having yield stresses in the range of 5-70 ksi, so one with at least 10 ksi can be used. The
bicycle frame material was chosen to be high-strength steel, which is listed as having yield stress in the
range of 50-150, so one with a yield stress of at least 110 ksi would be used.
Stress Analysis of a Bicycle
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7 Summary We found that breaking the bicycle into smaller portions of beams with stresses and moments to
be the most effective way in constructing and designing the bicycle as a whole. All of the calculations
were aimed to observe all the stresses that could potentially affect the bicycle design when supporting a
load of 250 lbs as well as an acceleration of 15 m/s2. Though the bicycle could clearly be divided into
smaller members much more, the five designs in this project cover enough if not most of the needed
parts when looking at and constructing a bicycle. Also, in this project friction was not factored in. Upon
designing this project, we found the importance of yield, maximum, principle, and shear stresses. These
maximum are necessary especially when provided with a factor safety in ensuring a low risk of failure. It
is essential to understand that all parts are closely associated with each other. We found that the type of
material chosen makes a massive impact on the design as a whole.
8 Bibliography Gere, J. M., & Goodno, B. J. (2009). Mechanics of Materials. Cengage Learning.
