University Project

1. 1
FEA LAB REPORT ME3602
Student Number: 1635505
Abstract
The product under investigation is the designof an aluminium bicycle crankarmthat will be attached to
light weight, high specification bicycles. The distribution of stress from the force applied decides the
amount of potential deformationand yielding. Thisis key to determine the maximum weighta rider can
be for the operationofthecrankarm. Afactor of safety is appliedof 2 whichgivesa totalforce of 2000N
and a rider weight of 100kg, much above the average weight of a regular rider.
The problemunderinvestigationisanoverallvolume/massreductionofmaterialby atleast50%. Astatic
FEA (FiniteElementAnalysis)willbedoneusingAnsysWorkbench 19.1. TheFEAiskey intheoptimization
duringthedesignphasetodevelopaviableproductwithouthavingtomanufactureandtestmanyvariants.
The original will be modelled in Solidworks, have a mesh derived from convergence & optimization, and
thenitsstructuralpropertieswillbesimulated.Fromthisare-designwillbeputthroughthesameprocess.
The bike pedal will be omitted for the sake of simplicity as the forces are already at an above average
amount compared to the normal operating force it would be subjected to.
The re-designed optimized bicycle crank arm fulfilled the requirements of mass removal while staying
under the maximum yield stress of the material.
1.0 Introduction
1.1 FEA History
FEA hasbeenaneffectiveengineeringtoolsincebeforethemoderncomputer.It’sutilitygoesback tothe
1940’s, withthefirstdevelopmentinthefieldbeingaccreditedtoA.Hrennikoff[1]in 1941 andR.Courant
in 1943[2]. Bothoftheseprofessorsoftheirfield approachedthemethodthroughmeshdiscretizationof
a continuousdomainintoasetof discrete sub-domains(elements). Thisrootofthismethodcomesfrom
matrix methods and finite difference methods of analysis. Both of these methods are considered FEM
(Finite Element Method). Any mathematical or physical problem that fulfil the requirementsof the prior
mentioned techniques can then be solved to the level of accuracy they provide.
1.2 FEA Theory
Thetheory ofacontinuousstructureallowsforaninfinitenumberofproblemsdependingonthesituation
the structure is put under. The role of FEA is to simulate a situation that makes approximations of the
behavioural qualities. One of these behavioural qualities for example are the potential stresses prone to
occur once the boundary conditionsof the componentare defined. Another is corresponding values for
targetedareasoffailurethatcanhighlightwheredesignchangescouldbemade. Accuracyoftheresultsis
reflected in the user’s ability to apply a mesh made up of elements that matches the structural reality
through mesh optimization.

2. 2
2.0 The Problem Under Investigation
2.1 Overview of Problem
Figure 1 CAD Drawing of Original Crankarm
Inthisstudy, thestateofthe problemisa staticstructuralcomponent. FEM isappliedtothiscomponent
toproducevisualisationsofitsstructuralpropertiesunderaforce:TotalDeformation,EquivalentStresses,
Structural Errors, etc. The analysis of these visualizationscan be used to determine design changes to
reducevolumeofmaterial, thusweight&cost. Thecrankarmwillbeanalysedat0o
,45o
, &90o
tosimulate
the1stquarterofmotionfromariderthatgenerates mostoftheforcefor motion. With90o
beingwhere
the maximum force is being applied & 0o
being the minimum. The torsional effects of a pedal are
disregarded in this model for the sake of simplicity as a pedal is not specified in the brief.
2.2 Boundary Conditions
Figure 2: Left to Right: Fixed Supports, Cylindrical Supports, Frictionless Supports
Fixedsupportswillbeusedonwherethe boltfacewouldinteractwiththecrank armasthesespots
wouldberigidly heldby theforceof thebolt.
CylindricalSupportsareusedontheinteriorboltholesastheseare3-Dinshapeandfulfilthe
requirements.Itispreferredoverafixedsupportasitbettermatchesthereality of thegeometry.
A frictionlesssupportisusedforthemainholeasitwouldn’tbemoving/deforminginthenormal
direction. Theholewouldhaveabearinginitallowingforitsrotationasitconvertsthepedallingintothe
movementofthesprocket&chain tomovethewheels.

3. 3
Figure 3: Left to Right: 0o
, 45o
, 90o
Thebearingforceis chosen by assumingwherethepedalwillbe exertingitsforceonthecrankarm. Inside
that hole, the bearing would be sitting, pushing its weight on the material downwards from the riders
applied force. A shaft is not used in this model as the pedal & interconnectionparts are not specified
thereforeitis assumedtheforceof2000Nisdistributedontheholeitself. Becauseofthistheforceisonly
acting on the bottom half of the circle facing the ground for the specified position (0O
, 45O
, 90O
). Thisis
because that bottom half is the only part of the hole the force is directly acting on.
2.3 HandCalculations Verifyingthe Model
Toconfirmtheresultsfromthemodeltotheoreticalcalculationshandcalculationsweredoneintheform
of transformationofthecrank geometrytoacantileverbeam.Theareasweresimplifiedtorectanglesto
matchtheiry-valuedescentfromoneendtotheother.By breakingitintodifferentrectanglesitallowed
for matrixcalculationtobedoneforgreateraccuracy.
Usingthematrixcalculation: ( 𝐹) = ( 𝑢)[ 𝑘], whereFisforce, uis displacement, andk isspringconstant.
Withtheboundaryconditionsu1 being0 duetoit beingaconstrainedsupport,samewithR1, withthe
force being2000N asspecified.
Thisgavethefinaltotaldeformationu6 tobe2.62x10-4 mwhichiswithin2% oftheFEA model[Appendix].
Usingtheequationforstress fromdisplacementmatrices
𝜎 = 𝐸
(𝑢 𝑖+1 + 𝑢 𝑖)
𝑙
=
2.62𝑥10−4 + 1.90𝑥10−4
0.17
= 1.89𝑥108 𝑃𝑎
ConsideringthatthemaximumstressobtainedfromFEAis1.78x108
Pathatgivesan8% difference,
confirmingthattheFEAmodelisacceptableconsideringassumptionsmadeforbeamsimplification.
Figure 4 Configuration of transformation into a matrix cantilever beam

4. 4
3.0 Part 1 – Finite Element Analysis of the Crankarm
3.1 Convergence toFindOptimal Mesh
Before any analysiscan be done there needsto be an optimal meshfoundthrough the method of mesh
optimization.Thiscanbedonebychoosingareasonablestartingpointforacoursemeshbasedoffthesize
of thesample. Fortheseelementtypesastarting elementsizeof10mmwhichgeneratesasmallnumber
(>3000) of nodes was generated to find the solutions of maximumdisplacement. The element size was
then reduced multiple times until the near maximum (around 200k) nodes had been generated for this
available product license. Four mesh types were chosen: Automatic, Tetrahedron (Quad), Sweep
(Quad/Tri), and Hex-Dominant. The mesh was chosen as course.
Figure 5: Left to Right: Automatic, Tetrahedron (Quad), Sweep (Quad/Tri), Hex-Dominant
Maximum displacementwas chosen as the convergence quality for the y-axis due to its easy tointerpret
values. For instance, the convergence value was consistently 2.67x10-4 m. Hex-Dominant, Tetrahedron
(Quad), andAutomaticallhadtoget toelementsize2.5mmtobeabletoreach theconvergencevalueof
2.67x10-4 m however Sweep (Quad/Tri) reached that value at 10mm.
Figure 6: Convergence plot of 3D Mesh
2.20E-04
2.30E-04
2.40E-04
2.50E-04
2.60E-04
2.70E-04
2.80E-04
0 50000 100000 150000 200000 250000
MaximumDeformation(m)
Number of Nodes
Number of Nodes vs Maximum Deformation (m)
Auto Tetrahedron (Quad) Sweep - Quad/Tri Hex-Dominant

5. 5
3.2 Analysisof Structural Properties
Sweep (Quad/Tri)2.5mmwaschosenwithfinemeshforthe besteffectivenessof capturingtheaccuracy
of the values gained for the structural properties in a reasonable time frame.
Force Angle (Degrees)
90 45 0
Skewness 0.479 0.45 0.453
Total Deformation Maximum m 2.67E-04 1.35E-04 9.74E-06
Equivalent Stress Pa 1.74E+08 9.38E+07 2.18E+07
Normal Stress Maximum Pa 1.03E+08 4.58E+07 2.16E+07
Normal Stress Minimum Pa -1.01E+08 -5.77E+07 -1.40E+07
Figure 7 Table of Structural Properties, Yellow (highest) to Red (Lowest)
Skewnessistheangularmeasureofelementquality withrespecttotheanglesofideal elementtypes[3].
The lower skewness is, the more accurate the generation of the triangles in the meshis. If the skewness
was 1 it wouldbecompletely unacceptable, withanything below0.6asacceptableandbelow0.4asideal
in this 3D mesh. The skewness is in an acceptable range for the orientations.
Total Deformation
Figure7 showsthatthe90o
positionofthecrank armhasthemaximumforceexerted fromtherider due
to the amount of total deformation. Whereas 45o
comes in second with 0o
as the least deformed. This
couldbefromhowthevectorsoftheforcetravelthroughthearm. At 0o
theforceistravellingequallyalong
either side of the crank arm which can be seen in Figure 16 (Appendix). This causes the least amountof
deformation as there is more volume of the material the force needs to travel through thus the
compressiondownwards is lessened. In comparison, at the 90o
position the force is exerted along the
lengthofthecrank arminsteadofdownwardsintoit. Theforcealongthelengthcreatesmomentumwhich
aids the force in causing maximum deformation in the crank arm.
Equivalent Stress
Thestress concentrationsoccurringinthebarwerein differentlocationsfor0o
whichoccurredinthe
pedalbearingholecomparedto the45o
&90o
whichoccurredintheboltholes.Inthe0o
themaximum
occurredinthemiddleofthe innerwallsastheforce wouldpushthematerialoutwardanddownwards
towardsthecrank bearinghole.Asmentioned thisforceoccursonthebottomhalfofthedirectionthe
pedalbearingisactingtowards.Thus,inthebottomredlineiswherethe maximumcompressionis
occurringandinthetoplineiswhere themaximumtensionisoccurring.In the45o
, themaximumis in
thebottomboltholeastheforceisactingdownwardsandinwardstowardsit. Whereasthe90o
iscausing
equalmaximumsonthetopandbottom, whereequalmaximumcompressionandtensionis occurring.

6. 6
Figure 8: Left to Right: 0, 45, 90 degrees (Maximum Locations)
Normal Stress
A normalstressoccurswhenamember(inthiscase, theelement)isloadedby anaxialforce. Anaxial
force beingaforce thatdirectly actsonthecentreaxisofan object. Forthe0o
, thelogicwouldbethat
themaximumaxialcompressionisoccurringdirectlyunderthepedalbearingholeandthemaximum
axial tensionisoccurring attheboundaryofthelowerhalfof thecircle thatmeetsthetophalf. This
highlightswherethemaximumcompressionandtensionisoccurringasthey arenearequalinmagnitude
butoppositeinsign. Wherethebluewouldbecompressionandtheredwouldbetension. Thiscanthen
be appliedtothe45o
and90o
wheretensionwouldbeoccurringatthetopholeasit ispulleddownwards
while thebottomholewouldbecompressedby thematerialabove.
Figure 9: Left to Right: 0, 45, 90 degrees (Min/Max Symmetry Locations)
Structural Error
Structuralerrorisastatisticin Mechanicaltodeterminewheretherecouldbeimprovementsinthe
mesh. ThisusuallyoccursinareasofhighstressconcentrationascanbeseenfromFigure10 compared
toFigure8. Thespotsofmaximumstressesarewherethestructuralerrorsareoccurring, specificallyin
thespotsoftension. Anotherthingtonoteisthatthestructuralerrorincreasesasitgoesfrom0o
to90o
(Figure7)where thecrankarmpositionofmaximumstresshasthehigheststructuralerror.Thiscanbe
reducedby optimizingthemeshthroughsizing&spheresofinfluence.

7. 7
Figure 10: Left to Right: 0, 45, 90 degrees (Maximum Locations)
4.0 Part 2 – Redesign and Mesh Optimisation
4.1 Design1: Basic Manufacturable Shape, I-Beam
Thefirst designsfocuswasonkeepingitashapethathadvery
low manufacturablecostsby reducingshapecomplexity. Material
wasremovedfromaroundthe3 smallboltsholesastherewasno
stressonthemduringmaximumforcethereforethemasswas
deadweight. Thelengthofthebarbetween thepedalbearing
andcrankshaftbearingcouldbereducedin volumeequally from
bothsidesforsymmetricaldistributionofforce.
4.2 Design2: 3-D Printable Shape, Honeycomb (Cross Section)
Hexagonshavebeenin interestintermsoftheefficiency ofsizing
plusstrengthcomparedtoothershapes.Ingraphene, aoneatom
thick carbonlayeristhestrongestmaterialevermeasuredand
takestheformofa hexagonallattice[4]. Beesforinstance, usea
hexagonalshapetoholdthemaximumamountofhoney inthe
smallestamountofspaceandthusneedmaximumstructural
strength[5]. Forthesereasonsahexagonalpattern wasused.
4.3 Design3: Interior Truss Bar
Trussbarsof10mmwidthwereusedhere, brokenintoeachhalf
of gradientoflength. Thiswastoseeif by hollowingoutthe
lengthofthebarthere couldbethegreatestreductioninvolume
while thickeningthebarstoprovideresistanceagainstbending.

8. 8
4.4 Optimal Design: Basic Manufacturable Shape, I-Beam
Design1 waschosenasithadthegreatestreductioninmasswith59.53%whileprovidingthelowest
valuesinstructuralproperties(NormalStressAverage&Un-Averaged,StructuralError, ShearStress,
DirectionalDeformation, TotalDeformation, &EquivalentStress)comparedtotheothertwodesigns.The
meshchosen forroughdesigncomparison wasHex-Dominant2.5mmduetoitbeingthesecondoptimal
meshintheoriginalcrankarmandthatSweep couldn’tbeusedinthere-designsduetotheir
asymmetricalshapeinallaxis. Alongwiththisithadtheeasiesttomanufactureshapemakingitthebest
choice.
Furthermeshoptimizationwasdonetoimprovetheresultsthroughfacesizingaroundtheinnerholes
andedge/facesizingaroundtheexteriorlengthsofthebar. Spheresofinfluence wereplacedaroundthe
centreof thetwoboltholeswheremaximumstressesoccurandatthepedalbearingholehoweverusing
specificedge andface sizingprovedtocreateacleanermesh andreducedtrianglecount.
Skewnesswasreportedas0.504indicatingsatisfactorymesh.
Figure 11: Before Optimization (Left), After Optimization (right)
Figure 12: Convergence plot of 3D Mesh
6.53E-04
6.54E-04
6.55E-04
6.56E-04
6.57E-04
6.58E-04
6.59E-04
6.60E-04
0 50000 100000 150000 200000
TotalDeformation(m)
Number of Nodes
Number of Nodes vs Total Deformation (m)
Auto Tetrahedron (Quad) Hex-Dominant

9. 9
Force Angle (Degrees)
Property Unit 90 45 0
Total Deformation Maximum m 6.59E-04 3.32E-04 2.10E-05
Equivalent Stress Pa 1.21E+08 6.77E+07 2.15E+07
Normal Stress Maximum Pa 1.08E+08 5.70E+07 1.68E+07
Normal Stress Minimum Pa -9.51E+07 -5.01E+07 -2.09E+07
Figure 13: Table of Structural Properties, Yellow (Highest) to Red (Lowest)
Total Deformation
Itcanbe seenfromFigure17 thattheshapeoftotaldeformationremainedthesameexceptforinthe0o
positionwherethedeformationoccurredunderneaththepedalbearingholeinsteadofaboveit. This
couldbeduetothe removalofmaterialintoanI-shapeunderneaththatledtothischange. Theamount
of deformationwasgreaterthantheoriginalcrank howeverinalldesignsitwashigher indicatingadirect
correlation. Thiscouldbebecausetheremovalof50% massnomatterinwhat geometry configuration
leadstostructuralweakeningthatincreasestotaldeformation.
Equivalent Stress
Theequivalentstressesthatoccurreddiffergreatly fromtheoriginaldesign.Forthe0o
thestressesare
morespreadoutaroundthepedalbearing, withless concentrationaroundtheholeandmoretowards
thedip intotheI-shapebeam.Forthe45o
and90o
thestresseshavebeenre-locatedfromaroundthe
outerboltholeintotheinnercentreofthehole. Overalltherehasbeenanoticeabledecreasein
maximumstressby0.53E+08.
Figure 14: Left to Right: 0, 45, 90 degrees (Maximum Locations)
Normal Stress
Theshapere-designhasspreadoutandre-locatedthetensionandcompressionforcesactingonthe
crank arm. ItcanbeseenfromFigure9 & 15 thattheseforcesaren’tasconcentrated. Inthe0o
thecolour
aroundtheholehaschangedfromapaleblue(compression)toayellow(tension)fromthematerial
beingpulledinwardstowardsthecentreoftheI-beam.Inthe45o
and90o
thetension&compressionis

10. 10
nolongerconcentratedaroundtheoutsideofthe boltholebutspreadoutalongthebeamfromthe
innercentre. Thenormalmaxandmins valuesarevery similartotheoriginalcrankarm.
Figure 15: Left to Right: 0, 45, 90 degrees (Max/min Locations)
5.0 Conclusions
Ithasbeenfoundthattheanalyzationoftheoriginalcrank armforvariousstructuralqualitiessuchastotal
deformation, directional deformation, equivalent stresses, normal stresses, and structural error had
paintedabasicimagetomakedesigndecisionson. Withthefocusbeingthatthestressesaroundthetwo
bolt holes needingto have extra care takento them. The areasaroundthe other 3 bolt holes hadnearly
nostress/deformationoccurring thus allowingthatmaterialtobecompletelyremovedasitwasservingno
purpose.
The materials choice of aluminium allows for easier production as it is a face-centred cubic metal. This
meansthatitis a tightly packedandbecauseofthatithasa highernumber ofslip systemsresultinginits
atomic structure being soft & malleable [6]. This softness of metal allows for the use of many forms of
machining that would cut costscompared to carbonfibre. For manufacturing, bothof the circular ends
couldbemadeonalathewiththelargerendhavingtheadditionofmachining toremovetheextramaterial
aroundtheboltholes.TheI-beamwouldbemanufacturedusingeithercold/hotrollingorthroughwelding
3 flat beamstogether.Afterthese3 shapeshavebeenmanufacturedthey wouldbeweldedtogether. The
other option would be cast-moulding for large-scale manufacturing.
The removal of 50% volume was exceeded in the chosen design by 9.53% with along with a 43.78%
reduction in maximum equivalent stressduring maximumforce orientation. However, total deformation
increasedfromthe amountofmassremoved. Thedistributionofcompressionandtensionforcesshould
be preferredtoconcentrationaroundtheouterboltholeasconcentrationscanleadto greaterchanceof
fail in that specific area. In conclusion, the objective was met with a trade-off between mass and total
deformationhoweveritis 230% belowyieldstressatmaximumforceorientationwithafactorsafety of2
thusitwillnotreachthepermanentelasticdeformationofaluminiuminevenanabnormaloperationforce.

11. 11
References
[1] Hrennikoff, A. (1941). “Solution of problems of elasticity by the framework method”. Journal of
applied mechanics 8.4: 169–175.
[2] Courant,R. (1943).“Variationalmethodsforthesolutionofproblems of equilibrium and vibrations”.
Bulletin of the American Mathematical Society 49: 1–23.
[3] Engmorph.com. (2019). [online] Available at: https://www.engmorph.com/skewness-finite-elemnt#!
[Accessed 19 Nov. 2019].
[4]ColumbiaEngineering.(2019).EvenwithDefects,GrapheneIsStrongestMaterialintheWorld.[online]
Available at: https://engineering.columbia.edu/news/even-defects-graphene-strongest-material-world
[Accessed 19 Nov. 2019].
[5] Microscopy-uk.org.uk. (2019). Micscape Microscopy and Microscope Magazine. [online] Available at:
http://www.microscopy-uk.org.uk/mag/indexmag.html?http://www.microscopy-
uk.org.uk/mag/artsep98/hexagon.html [Accessed 19 Nov. 2019].
[6] Depts.washington.edu. (2019). MetalsStructure. [online] Available at:
https://depts.washington.edu/matseed/mse_resources/Webpage/Metals/metalstructure.htm [Accessed
4 Dec. 2019].

12. 12
Appendix
Figure 16: Left to Right: 0, 45, 90 degrees (Overview of Deformation)
Figure 17: Left to Right: 0, 45, 90 degrees (Overview of Deformation)

13. 13
Area:
𝐴1 = 𝑤1 𝑡 = 0.078 ∗ 0.01 = 7.8𝑥10−4
𝑚2
𝐴2 = ( 𝑤1 + (
𝑤2 − 𝑤1
𝐿
) 𝑦) ∗ 𝑡 = (0.078 + (
0.03 − 0.078
0.17
) 0.012) ∗ 0.01 = 7.46𝑥10−4
𝑚2
𝐴3 = (0.078 + (
0.03 − 0.078
0.17
)0.024) ∗ 0.01 = 7.12𝑥10−4
𝑚2
𝐴4 = (0.078 + (
0.03 − 0.078
0.17
)0.036) ∗ 0.01 = 6.78𝑥10−4
𝑚2
𝐴5 = (0.078 + (
0.03 − 0.078
0.17
)0.048) ∗ 0.01 = 6.45𝑥10−4
𝑚2
𝐴6 = 𝑤2 𝑡 = 0.03 ∗ 0.01 = 3𝑥10−4
𝑚2
Spring constants:
𝑘 𝑒𝑞 =
(𝐴𝑖+1 + 𝐴𝑖 )𝐸
2𝑙
𝑘1 =
(7.46𝑥10−4
𝑚 + 7.8𝑥10−4
𝑚)71𝑥109
𝑃𝑎
2 ∗ 0.012𝑚
= 4.52𝑥109
𝑁𝑚−1
𝑘2 =
(7.12𝑥10−4
𝑚 + 7.4610−4
𝑚)71𝑥109
𝑃𝑎
2 ∗ 0.012𝑚
= 4.31𝑥109
𝑁𝑚−1
𝑘3 =
(6.78𝑥10−4
𝑚 + 7.12𝑥10−4
𝑚)71𝑥109
𝑃𝑎
2 ∗ 0.012𝑚
= 4.11𝑥109
𝑁𝑚−1
𝑘4 =
(6.45𝑥10−4
𝑚 + 6.78𝑥10−4
𝑚)71𝑥109
𝑃𝑎
2 ∗ 0.012𝑚
= 3.91𝑥109
𝑁𝑚−1
𝑘5 =
(3𝑥10−4
𝑚 + 6.45𝑥10−4
𝑚)71𝑥109
𝑃𝑎
2 ∗ 0.012𝑚
= 2.80𝑥109
𝑁𝑚−1
Global Stress Matrix * Displacement Matrix = Force Matrix:
[
𝑘1 −𝑘1 0 0 0 0
−𝑘1 𝑘1 + 𝑘2 −𝑘2 0 0 0
0 −𝑘2 𝑘2 + 𝑘3 −𝑘4 0 0
0 0 −𝑘3 𝑘3 + 𝑘4 −𝑘4 0
0 0 0 −𝑘4 𝑘4 + 𝑘5 −𝑘5
0 0 0 0 𝑘5 𝑘5 ]
109
∗
(
𝑢1
𝑢2
𝑢3
𝑢4
𝑢5
𝑢6
)
=
(
𝑅1
0
0
0
0
𝐹 )
Matrix After Values Inputted with Boundary Conditions (u1 = 0, R1= 0, F = 2000):
[
1 0 0 0 0 0
−4.52 8.83 −4.31 0 0 0
0 −4.31 8.42 −4.11 0 0
0 0 −4.11 8.02 −3.91 0
0 0 0 −3.91 6.71 −2.8
0 0 0 0 −2.8 2.8 ]
109
∗
(
0
𝑢2
𝑢3
𝑢4
𝑢5
𝑢6)
=
(
0
0
0
0
0
2000)
After putting this matrix into a Gaussian Elimination calculator, and solving for linear equations via substitution:
u1 u2 u3 u4 u5 u6
0 m 0.44x10-4 m 0.91x10-4 m 1.39x10-4 m 1.90x10-4 m 2.62x10-4 m