FINITE ELEMENT ANALYSIS AND DESIGN OPTIMIZATION OF AN EBIKE STRUCTURE USING CARBON FIBER COMPOSITESStephen Roper (0swkr@queensu.ca), Dr. Il Yong Kim*, Luke Ryan**MECH 461 | Undergraduate Research ProjectDepartment of Mechanical and Materials Engineering | Queen’s University, Kingston, Ontario, Canada

Submitted for review Friday April 15th, 2016

Abstract: Electric bicycles are becoming an increasingly popular method of alternative transportation and as demand grows, so does the need for higher performance concerning range, speed and durability. In order to meet consumer demand, electric bicycles must incorporate new advanced materials into their design to decrease mass while maintaining structural characteristics such as stiffness and deflection. This research project investigates the design and optimization of a carbon fiber composite electric bicycle frame considering ply angle, ply number and stacking sequence within the laminate. The final result is a readily manufacturable design that achieves a mass reduction of 31%, while increasing stiffness by an average of 4% over extreme operating conditions. The final design recommendation utilizes a monocoque geometry and thirty-one distinct T300/976 carbon/epoxy plies to achieve overall improved structural performance.

Key Words: carbon fiber reinforced polymer, composite, electric bicycle, design, optimization, finite element analysis, compliance, HyperWorks, HyperMesh, OptiStruct

NOMENCLATURECFRP Carbon fiber reinforced polymereBike Electric bicycle Prepreg Pre-impregnated epoxy resin

X , X ' Longitudinal strength tension and compression

Y , Y ' Transverse strength tension and compressionS12 Shear strength~X Allowable stress X , X '

~Y Allowable stress Y , Y '

~S Allowable stress S12

ε 1, ε 2 Normal strain longitudinal and transverse

E1, E2 Young’s modulus longitudinal and transverse

γ12 Shear strain

ν12 Poisson’s ratio

G12 Shear modulus

F12 Tsai-Wu stress interaction termPCOMPP Element card, composite propertyMAT8 Material card, compositeFSOTSZ Output, free size to sizeSZTOSH Output, size to shuffleCSTRESS Output, composite stressCSTRAIN Output, composite strainCFAILURE Output, composite failureSRCOMPS Output, composite strength index

*Project Supervisor, kimiy@queensu.ca, **Project Mentor, luke.ryan@queensu.ca, Department of Mechanical and

Materials Engineering, Queen’s University, Structural and Multidisciplinary Design Lab, Jackson Hall 213. 1 INTRODUCTIONCarbon fiber reinforced composites are becoming an increasingly popular material for industrial and commercial applications due to their lightweight properties and high strength characteristics. Designers across multiple disciplines are now leveraging CFRPs to reduce mass, improve stiffness and increase fatigue and corrosion resistance in structural applications. The automotive industry is increasingly incorporating composite materials into lightweight vehicle design to enhance fuel efficiency and range. For example, companies like Hyundai, Volkswagen, and General Motors are exploring composite automotive chassis and novel manufacturing methods for electric and hybrid vehicles, while the BMWi3 is currently the largest-volume production car to incorporate CFRPs [1] [2]. Lightweight composite applications in the aerospace industry are even more extensive, with aircraft incorporating advanced carbon-polymer components into structural airframe and propulsion design to reduce landing weight fees, increased range and payload capabilities, and ultimately decrease operating cost [3]. The Airbus A350 XWB incorporates an industry leading 52% composite ratio with fuselage panels, window frames, and doors made from CFRPs [4]. Specific to this case study, advanced carbon composites are currently used for high-performance racing bicycles to further decrease weight and increase rider efficiency, with companies like Shimano developing multiple CFRP-based components [5].

This significance is extending even further into electric bicycles, which have seen a dramatic increase in consumer demand due to their ease of use and environmental appeal. As this alternate method of transportation continues to grow,

Stephen Roper

further improvements in vehicle power, torque, speed, range and efficiency are also desired to drive innovation and competition. These critical performance metrics are directly proportional to system mass, which is currently limited by large electric motors, bulky support frames and heavy batteries [6]. To match growing vehicle standards CFRPs are now being implement as the primary support structure material to effectively minimize mass without compromising strength and stiffness.

1.1 Literature ReviewPrevious research studies have examined conventional bicycle frame design and the transition to carbon fiber monocoque styles away from tubular geometry made from metallic alloys. For example, one case study specifically examined a racing bicycle CFRP frame for torsional stiffness in the bottom bracket and head tubes along with in-plane vertical compliance. This analysis was conducted considering weight, fatigue and vibration damping properties of two thin monocoque designs, called the modified beam frame and single bridge frame. With the selected geometry different composite layups were established for finite element (FE) testing and consisted of 6-layer and 10-layer laminate sections composed of unidirectional T300/5208 graphite-epoxy plies in various 0⁰ and 90⁰ orientations/stack-ups [7]. With an appropriate model, three different FE models and tests were developed to compare the two carbon monocoque design to against tubular baselines. The maximum stress criterion was used was to predict laminate failure and compare the different deflections from each test and bicycle model [7].

A second case study investigated the structural behavior of a composite bicycle frame using different stacking sequences and fiber directions to manually optimize the design. The objective of this work was to determine the optimal 8-ply stacking sequence and fiber direction for the frame design and propose structural changes to eliminate weak areas. The FE setup used a traditional tubular frame geometry and accounted for stress concentrations at the joints. The composite layup consists of a multi-directional quasi-isotropic laminate with 0°, 90°, and ±45° plies each with a thickness of 0.3mm T300 carbon-epoxy material properties. The model uses the maximum stress theory as the failure criteria to maximize a defined stress-strength ratio [8].

1.2 ObjectivesThe previous research conducted in this field applies manual optimization techniques to compare the deflection and stiffness performance of different frame geometries and predetermined laminate lay-ups. While the overall results from each case study improved frame design they did not include a detailed holistic investigation into the composite laminate development. This

research focuses on a systematic approach for composite optimization that manages complex geometrical analysis, unconventional material properties, and manufacturing requirements. The design optimization addresses three main objectives:(a) Free size optimization, to determine the ideal ply

orientation thickness distribution throughout the structure.(b) Dimension optimization, to transform the ply distribution

concept into discrete, manufacturable lamina thicknesses and shapes.

(c) Shuffle optimization, to re-order the plies considering manufacturing constraints and typical ply-book rules.

The culmination of this work is a final design recommendation considering structural geometry and CFRP laminate lay-up. Also suggested are applications for future work and investigation as well as brief comments on manufacturability. All of the analysis was conducted using Altair HyperWorks and the OptiStruct, HyperView and HyperMesh modules. The eBike geometry used in this case study is derived from the Cube Hybrid Nutrail 500 Hardtail Mountain eBike.

2 COMPOSITES AND FE MODELING2.1 Laminate DesignBackground research was conducted to develop an understanding of laminate design and the material properties of carbon fiber reinforced polymers. First, various commercially available carbon fiber fabrics were investigated, offering insight into possible configurations and material properties for structural applications. Unidirectional pre-impregnated resin plies were selected from this analysis [9]. Stacking angles were then researched and revealed the typical fiber orientations (0°, 90°, and ±45°) used in composite design, as well as related manufacturing constraints such as ply balancing (±45°) and maximum common-angle stacking. Lastly, ply thicknesses were examined to identify their role in CFRP applications. This investigation ultimately revealed the high customizability of laminated structures and their ability to achieve different mechanical properties from the same base materials. Typical unidirectional composites are shown in Figure 1.

Figure 1: Typical laminate definitions with a unidirectional stack (left) and a 0°-90° laminate stack (right) [10].

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2.2 Composite Failure TheoryTo further understand the mechanical properties of laminate structures orthotropic failure theories were studied to characterize ply deterioration. The main orthotropic yield functions considered in this analysis were the Maximum Strain and Tsai-Wu theories. These are readily supported by HyperWorks, along with Hill and Hoffman yield functions. The Maximum Strain theory provides suitable laminate strength predictions with respect to applied loads and stresses (in-plane and bending) and can isolate specific plies to suggest corresponding failure modes such as debonding and fiber failure. This criteria is typically used for unsymmetrical laminates and assumes a linear-elastic response where failure is predicted when strain reaches its limiting values (determined by uniaxial tensile experiments) [11]. To apply this criterion lamina elastic properties (E1 , E2 , ν12 ,G12) are required as basic inputs as well as longitudinal, transverse, and shear strengths in tension and compression (X ,Y , X ' , Y ' , S12). See Equation 1.

F=max(|ε1~X|,|ε 2

~Y |,|γ12~Y |) (1)

Like the Maximum Strain criteria, the Tsai-Wu theory assumes plane stress with a linear elastic lamina response and is preferred for predicting the onset of laminate failure, but not the specific mode (transverse, shear). This yield criteria is invariant under coordinate system rotation, making it superior to alternatives like the Hoffman criteria, and provides independent interactions among stress components. The same material properties used for the Maximum Strain theory are required here, however the ultimate strength values are insufficient for determining coefficients such as the stress interaction term F12[11]. The Tsai-Wu failure theory is presented in Equation 2.

F1σ 1+F2 σ2+F6 σ6+F11σ12+F22σ2

2 …+2 F12 σ1 σ2+F66 σ62=1(2)

In HyperWorks, these failure criteria can be used to generate failure index contour plots (Equation 3), to indicate the location and magnitude of high stress regions. The output can also be adjusted to include a composite strength ratio, which indicates the safety factor distribution throughout the structure (Equation 4).

FI>1 , Composite Failure (3)

SR>1 ,Composite Safety Factor (4)

2.3 Composite OptimizationWith an understanding of laminate design and orthotropic failure criteria, an optimization strategy was developed to systematically address various constraints and generate feasible design solutions. The final optimization methodology was conducted in three main phases using: (a) an Initial Model, (b) a Verified Model, and (c) a High Fidelity Verified Model. With each subsequent phase, design elements such as geometric properties, material definition, and mesh quality were slightly adjusted to improve FE behavior and achieve the best balance between lightweighting and stiffness characteristics. Correspondingly, the mathematical problem statements of each modeling phase were updated to reflect these design changes, however the principle objective function, constraints, and loading scenarios remained the same throughout. Within each of these modeling phases a distinct series of steps was carried out which closely followed the HyperWorks composite optimization engine.

First, a baseline analysis was developed using 7000 series aluminum to provide a method of comparison to the subsequent CFRP model. Next, the CFRP laminate was created by introducing four distinct plies at 0°, 90°, and ±45°, and each one assigned an oversized thickness to allow for future material removal and redistribution. From this super-ply CFRP optimization was executed in three steps: (a) free size, (b) discrete size, and (c) laminate shuffle.

Initially, free size optimization is used to create the ideal fiber angle thickness distribution throughout the structure, cutting away support material from the super-ply model where it is not required. Once the design concept is generated, dimension optimization converts the continuous thickness distribution into discrete manufactural ply thickness and shapes; shape optimization is done manually and requires user input to transform unconventional shapes into more manufacturable entities. Lastly, the shuffle optimization rearranges the laminate stack considering typical manufacturing constraints and good practices, generating a final, feasible, manufacturable design.

2.4 FE Model Setup2.4.1 Mesh QualityWhen developing and improving the FE models for each optimization phase (Initial, Verified, and High Fidelity Verified) two primary mesh parameters were considered: overall quality (Jacobian) and composite fiber orientation.

Using a Jacobian index plot, the quality of each element was analysed to identify difficult-to-mesh regions typically responsible for generating computational modeling error. This highlighted elements that varied significantly from the ideal shape by considering element warpage, size deviation, and

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skewed internal angles. This analysis then prompted local mesh refinement in these areas.

Using the composite material function, element orientations were also reviewed to ensure consistent alignment and directional referencing throughout the structure. Typically, a mesh fails fiber alignment within difficult-to-mesh areas (as defined by the Jacobian analysis) and is especially common in tri-elements. By redefining element orientations to a common reference discontinuities in the CFRP laminate were eliminated.

2.4.2 MaterialPrior research and case studies in bicycle frame optimization consistently applied T300 graphite/epoxy as the lamina material. To match these case studies and reflect current industry trends a T300 15k/976 carbon fiber prepreg (pre-impregnated resin) tape was selected for this analysis. This tape was applied to the structure in a unidirectional form with critical material properties sourced to satisfy the yield criteria requirements. See Table 1.

2.4.3 Loading and ConstraintsMountain bikes (and eBikes) are usually subjected to forces greater than steady-pedalling on a smooth road and instead experience a combination of bending, shear, torsion and tension or compression. The application of front-wheel and rear-wheel suspension systems are typically used to reduce these resulting forces, however bicycle design must still take into account the

high forces expected when traveling over rocky and uneven terrain at high speeds. In these applications failure analysis has revealed that bicycles typically break as a result of overload conditions (collisions, impact, jumps), or fatigue (cyclic loading) of unusually high stresses, rather than the build-up of material damage from smooth-riding [12]. Regulatory and independent testing standards agree with these observations, and typically evaluate components under static (maximum/overload) conditions and low-cycle fatigue (LCF).

In electric bicycles large loads are often induced by different mechanisms, such as impact, jarring bumps, strenuous braking or pedalling. These activities can develop forces ranging from 850N – 4000N and in some cases reflecting up to five times the rider weight [12]. To develop a comprehensive evaluation of frame durability under these high forces various mechanical tests have been established by regulatory bodies, such as ASTM and EU standards [13] [14]. Independent test services such as EFBe Prüftechnik (EU) have also been established and incorporate their own criteria above the standard regulations, and include maximum and overload tests exceeding 4,000N [15].

For the given FE modeling and optimization analysis only static loading conditions were considered. By utilizing the multi-loading capability of HyperMesh three different loading scenarios were developed to simultaneously consider critical in-plane and out-of-plane forces; See Table 2.

Table 1: T300 15k/976 general laminate properties [10].

Units Nominal Minimum MaximumPly Thickness mm 0.130 0.124 0.135

ρ kg/m3 1620 1580 1650E1 GPa 133.07 125.48 139.27E2 GPa 9.24 8.83 9.58G12 GPa 6.27 5.79 6.62ν12 - 0.318 - -X MPa 1427.21 1316.90 1509.95X ' MPa 1503.06 1116.95 1709.90Y MPa 39.02 31.23 44.95Y ' MPa 206.84 184.09 219.94S12 MPa 76.53 75.84 78.60

Table 2: Multi load case definition, considering in-plane and out-of-plane forces.

Case 1 (L1): Factory Static Test Case 2 (L2): Maximum Pedaling Force Case 3 (L3): Maximum Crash Force

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F3=4,500N

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3 ANALYSIS 3.1 Initial ModelThe initial model was developed using simplified eBike geometry in order to achieve a preliminary understanding of the system responses and optimization behavior.

First, the initial baseline simulation was conducted by modeling the eBike frame geometry with Al7005 as the structural material, where Table 3 shows the key system behaviors of interest. The compliance and mass responses are particularly important since these characteristics are explicitly defined in the CFRP mathematical optimization problem statement. This table also highlights the significance of the crash test (L3) as the worst loading condition, which shows the maximum compliance and displacement response for this model. Table 3: Al7005 baseline model, showing critical system behaviours.

System Behaviour UnitsBaseline Al7005

L1 L2 L3

Compliance [Nm] 2.34 0.202 43.9

Max. Displacement [mm] 1.55 0.421 5.37

Mass [kg] 3.14

Once the baseline analysis was complete the model properties were redefined, now considering a carbon fiber composite laminate to conduct the CFRP optimization. The mathematical problem statement used in all CFRP optimization phases (free size, dimension, and shuffle) is defined below:

OBJECTIVE FUNCTION:

MinimizeCw=∑ W iC i=CL 1+1.5 CL 2+2.0 CL 3

SUBJECT TO:

Mass ≤2.5 kgLaminateThickness ≤ 4.5mmPly Thickness≤ 0.127 mm

Ply Balance=± 45 °In this problem statement the objective is to minimize weighted compliance, which defines the ratio of deformation to the applied load in each scenario (L1, L2, and L3). Mathematically, this represents the strain energy of the structure with the input forces causing a translation of the body by some distance; conceptually, this can be considered the reciprocal of stiffness. This objective function was then subject to various constraints, the most significant being the desired final mass target with others considering manufacturing requirements.

The optimization was successfully executed from super-ply to shuffle and achieved an overall mass reduction of approximately 21%. In each phase the laminate properties were defined using the PCOMPP shell elements and MAT8 material property cards. In free size and discrete size, the output cards were adjusted to FSTOSZ and SZTOSH outputs respectively to retrieve the correct solver decks for the next optimization step. Additional control cards were also activated, including CSTRESS, CSTRAIN, CFAILURE and SRCOMPS to output the desired composite analysis results, like the strength ratio index. The detailed progression of each optimization phase with respect to mass and weighted compliance response from this analysis can be shown in Figure 2.

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F1=4,500N

F2 = 2,300N

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Al7005 Super Ply Free Size Discrete Size

Shuffle0

102030405060708090

100

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0.5

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Initial Model Optimization Response

Weighted Compliance Mass

Optimization Phase

Wei

ghte

d Co

mpl

ianc

e [N

m]

Mas

s [kg

]Figure 2: Optimization progression of the Initial Model.

The above optimization trend indicated that replacing the existing aluminum material with a CFRP laminate results in a decrease in both mass and compliance by approximately 21% each. Correspondingly, the structural stiffness increased and reduced the maximum deflection observed in the structure from 5.37mm to 4.65mm. These results were also confirmed by examining the maximum displacement and strength ratio contour plots, considering the expected worst results in Load Case 3 (See Figure 3). By examining Figure 3 (right) the resulting safety factor does not decrease below 2.95 in the high stress regions, indicating no ply or laminate failure for the final optimized system.

Figure 3: Final analysis of the optimized CFRP frame: (left) maximum dis-placement contour plot caused by load case 3 (right) safety factor contour plot. Notice the sharp discontinuity in the displacement plot at the down tube.

While this model did yield positive results, additional mass reductions beyond achieved 21% were desired to create a more suitable stiffness-to-weight ratio. Furthermore, the analysis revealed discontinuities in the difficult to mesh regions, indicating poor element quality and orientation definitions.

Both of these concerns were addressed in the Verified Model, next.

3.2 Verified ModelThe verified model was developed using key findings from the initial analysis which revealed various areas for improvement, including element refinement and greater mass reduction.

First, the elements around the observed discontinuity in the initial model were investigated to determine their cause. As mentioned in Section 2.4.1, this included a Jacobian analysis and element orientation check, which did reveal broken material alignment around the head tube elements. By carefully reassigning these elements to the standard predefined global reference the mesh orientations were corrected for this model. See Figure 4.

Figure 4: Area of the observed discontinuity: (left) initial model with broken element orientation, (right) verified model with corrected element orientation.

Next, the objective was adjusted with a greater mass reduction target and a more constrained laminate thickness requirements. All other variables remained unchanged.

OBJECTIVE FUNCTION:

MinimizeCw=∑ W iC i=CL 1+1.5 CL 2+2.0 CL 3

SUBJECT TO:

Mass ≤2.2 kgLaminateThickness ≤ 4.0 mmPly Thickness≤ 0.127 mmPly Balance=± 45 °

After conducting the optimization sequence for the verified model, the final CFRP design achieved a mass reduction of 42% but this time at the expense of compliance. That is, when the optimization entered the discrete size and shuffle optimization engines compliance increased above the aluminum baseline model, as shown in Figure 5. This indicated a loss of stiffness in the structure and an increase in the maximum displacement.

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This sudden increase at the discrete and shuffle optimization stages can be explained by examining the resulting orientation thickness distribution contour plots (Figure 6). Here, Figure 6 (left) shows the ideal distribution of plies throughout the structure that generates the highest stiffness and greatest weight savings. As discrete size optimization is introduced, manufacturing constraints become more important and drive a change in plie shape and thickness to accommodate for fabrication, as in Figure 6 (right). As ply shapes are changed dramatically or removed for simplicity, the compliance of the system increases away from the ideal minimum.

Al7005 Super Ply Free Size Discrete Size

Shuffle0

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40

60

80

100

120

0.0

0.5

1.0

1.5

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Verified Model Optimization Response

Weighted Compliance Mass

Optimization Phase

Wei

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mpl

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m]

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s [ k

g]

Figure 5: Optimization progression of the Verified Model. Note the increase in compliance above the aluminum baseline.

Figure 6: Optimization contour plots: (left) final free size optimization, (right) final discrete size optimization.

The final results from the verified model show significant weight savings from 3.14kg to 1.8kg, but with an overall increase in compliance by an average of 16%. This was confirmed when analysing the maximum displacement of Load Case 3, which increased to 6.45mm from the baseline 5.35mm. Despite this decrease in stiffness the optimized design did not

approach failure, as indicated by the ply strength ratio plot which indicated a minimum safety factor of 2.8. Lastly, the results from this model showed no discontinuities in any of the analysis contour plots, verifying the corrective element actions for future applications. 3.3 High Fidelity Verified ModelThe preliminary analysis, consisting of the Initial and Verified Models, provided a fundamental understanding in the design, analysis, and optimization response of a CFRP bicycle frame. These lessons were then applied to a more exact eBike geometry in the High Fidelity Verified Model to ultimately provide reliable results, eliminate simplification error and make final recommendations considering more current CFRP design practices. The updated geometry can be seen in Figure 7, which utilize a monocoque, thin walled box design. This is typically how carbon fiber laminates are used in high performance bicycles and reflects the transition away from the traditional tubular form.

Figure 7: Comparison of eBike geometry used in the different analysis phases: (left) simplified eBike geometry used in the Initial and Verified Models, (right) exact eBike geometry used in the High Fidelity Verified Model.

Before the baseline and optimization sequence, a preliminary investigation into the monocoque geometry transition was conducted to understand how system responses were expected to change from the tubular form. This analysis examined the inherent changes in stiffness and bending moment as a result of the changing area moment of inertia in the down tube. Equation 5 shows the traditional bending moment equation for a beam, which can be broken into the flexural modulus of rigidity (EI ) and the beam curvature (d y2/d x2). By rearranging this equation with respect to curvature, it can be shown that an increased flexural modulus will effectively increase stiffness.

M=EI dy2

d x2 (5)

Figure 8 shows the newly designed eBike cross section at the downtube, where the applied moment about the x-axis is greatest (Load Case 3). In this case the area moment of inertia is defined by Equation 6, while conventional tubular geometry is described by Equation 7.

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I xBOX= 1

12A B3− 1

12C D3 (6)

I xTUBE= π

64( A4−B4 ) (7)

Using these equations,

This hand calculation shows an increase in are moment of inertia for the monocoque geometry, resulting in an increased flexural modulus. As a result, the total structural stiffness is expected to increase due to the geometry adjustment alone.

Figure 8: Monocoque eBike cross sectional geometry, incorporating a rectangu-lar box instead of traditional hollow tubes.

After conducting the baseline aluminum analysis, these calculations were confirmed which showed a significant decrease in compliance for all load cases, and specifically of interest, Load Case 3. It is also important to note that the baseline mass changed as a result of the monocoque design implementation. These results are shown in Table 4. Table 4: Al7005 baseline model, showing critical system behaviours.

System Behaviour UnitsBaseline Al7005

L1 L2 L3

Compliance [Nm] 1.41 0.127 12.3

Max. Displacement [mm] 1.02 0.540 1.60

Mass [kg] 3.71

Next, the objective function for the monocoque design was developed, using an appropriate mass reduction target (2.6kg) and maximum laminate thickness (4mm), while maintaining all other constraints and design variables from earlier preliminary models.

These results for the monocoque CFRP design optimization are found in Table 5, as well as the resulting ply distribution in Figure 9.Table 5: Optimization response comparison to the baseline model (Table 4).

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I xBOX=6.47 ×10−7[mm4]>1.40× 10−7[m m4]=I xTUBE

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System Behaviour UnitsOptimized CFRP

L1 L2 L3

Compliance [Nm] -2% -1% -10%

Max. Displacement [mm] -51% +77% -10%

Mass [kg] -30.46%

Figure 9: Optimization progression: (left) maximum ply orientation thicknesses; (middle) discrete ply distribution for 0° plies; (left) optimized stacking sequence.

These results showed good optimization behavior, with the free size phase generating the ideal angle thickness distribution throughout the structure. Of particular interest are the patterns observed in the seat tube, which closely match the shapes developed in the Initial and Verified Models. Next discrete optimization transformed the continuous distribution into thirty-one discrete manufacturable ply shapes with standard thicknesses. Lastly, the stacking sequence was adjusted taking into account ply balancing and maximum orientation stacking succession. The optimization progression can also be found in Figure 10, which shows the system response from each phase.

Al7005 Super Ply Free Size Discrete Size

Shuffle21

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High Fidelity Verified Optimization Response

Weighted Compliance Mass

Optimization Phase

Wei

ghte

d Co

mpl

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e [N

m]

Mas

s [ k

g]Figure 10: Optimization progression of the High Fidelity Verified Model.

This analysis yielded characteristic and expected results. First, the super-ply and free size optimizations showed dramatically reduced mass and increased stiffness as a result of incorporating CFRPs as the structural material. As additional manufacturing considerations were introduced in the discrete and shuffle phases, the system continued to optimize mass, however, at the expense of compliance. One again, this is due to the laminate simplification from the ideal concept.

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Stephen Roper

To confirm these results, the maximum displacement (considering Load Case 3) was analyzed and indicated a deflection reduction of 10%, to 1.44mm from 1.60mm. Lastly, the failure index plot was studied and revealed no ply failure throughout the structure; the lowest safety factor was 7 in the high stress region at the top surface of the head tube.

4 DISCUSSION AND CONCLUSIONS The results from each analysis (Initial, Verified, and High Fidelity Verified) are provided in Figure 11, and highlight the optimization response with respect to weighted compliance and mass.

1.6 2.1 2.6 3.1 3.60

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Optimization Trends (All Models)

Initial Verified HF Verified

Mass [kg]

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Figure 11: Optimization trends for each model: Initial, Verified, and High Fi-delity Verified.

The Initial Model provided a suitable starting point for design optimization, using simplified geometry and a coarse mesh refinement to obtain preliminary results. While this analysis successfully decreased mass and compliance by 21%, additional weight savings were desired. Furthermore, discontinuities were observed in the final contour plots, requiring finer mesh adjustments to eliminate the error and confirm the results. The Verified Model was subsequently introduced to eliminate these discontinuities with element orientation refinement, and to drive the optimized mass target below 2.6kg. This successfully resulted in a mass reduction of 43% to 1.8kg (from 3.14kg), however compliance increased by an average of 16% indicating a loss of stiffness in the structure. Lastly, the High Fidelity Verified Model was introduced to merge all of the previous modeling insight while using a more exact representation of the eBike geometry. This successfully achieved a mass reduction of 31% from the baseline aluminum version, and decreased compliance by an average of 4%.

The final design conclusions and recommendations for the electric mountain bike are:1. The monocoque geometry has highest stiffness (in this

application) compared to conventional tubular geometry. This is inherently due to the adapted cross section, which increases the area moment of inertia to resist bending. The material properties of the carbon fiber composite material also make this design superior compared to all aluminum baseline models analysed in the investigation.

2. The high fidelity verified monocoque optimization offered a good balance between design variables, and the final results shown in Figure 9 represent a final manufacturable solution with good system response. These are:

(a) Mass: -31%(b) Compliance (L3): -5%(c) Compliance (Ave.): - 4.3%(d) Maximum Deflection (L3): -10%

3. The monocoque design did not approach ply failure, and had a significantly high safety factor in the high stress regions compared to the tubular geometry under the same loading case. Again, this is due to the geometry of the structure which distributes the carbon fiber laminate more effectively throughout the structure.

4. Continuing optimization with the High Fidelity Verified Model (monocoque design) offers the greatest opportunity for additional weight savings and increased stiffness in comparison to the tubular geometry. Optimization should be attempted to bring the mass below the best response obtained in the Verified Model (CFRP, 1.8kg).

(a) Decrease mass target for the monocoque optimization, from ≤ 2.6 kg to ≤ 1.8 kg

5. Considering the low deflection in these extreme load cases, mass reductions can be considered more favourable than stiffness losses. That is, an increase in compliance should be considered acceptable, up to a maximum deflection of 6.45mm (L3 in the Verified Model). This allows the eBike to operate more efficiently under normal operating conditions.

6. Buckling and fatigue analysis should be included in design optimization, as linear static analysis alone is not sufficient for predicting these other common failure modes.

5 ACKNOWLEDGMENTSDr. Il Yong Kim, for managing this project, providing technical guidance; Luke Ryan, for answering my questions, checking over models, and helping me on a weekly basis.

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Failure Mechanics of Composites, Amsterdam, Elsevier Science Publishers, 1985, pp. 86-97.[12] D. Wilson, Bicycling Science: Third Edition, Cambridge: The MIT Press, 2004. [13] ASTM Committee on Sports Equipment, Playing Surfaces, and Facilities, "Standard Test Methods for Bicycle Frames

(F2711–08)," ASTM International, 2012.[14] British Standards, "Montain Bicycles: Safety Requirements and Test Methods (BS EN 14766)," Standards Policy and

Policy Committee, 2005.[15] EFBe Prüftechnik GmbH, "Mechanical Tests for EPAC Trekking/City Bikes," EFBe Test Services, Waltrop, 2011.

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