Due: 23/9/13

AbstractIn the analysis of the beam, plate and brick models for the original cross bar design under both point loads and UDLs, minimal discrepancies were encountered across the displacements and the stresses. The only real anomaly rose in the stress of the plate model under a point load, which was unexpectedly low. This was most likely attributed to the plate model’s overlapping geometries, which skew areas, masses and second moment of inertias, in turn distorting the measured stress.

Point Force UDL

Max Displacement (mm) Max Stress (MPa)Max Displacement (mm) Max Stress (MPa)

Beam 1.202 18.887 0.694 6.304Plate 1.275 17.377 0.745 6.058Brick 1.264 19.041 0.730 6.354

The optimisation recommended by this report is very similar to the original design, only much thinner in sections and with fillets smoothing out the step from the flanges to the thinner middle walls. This adding of fillets provided enough strength to the system so as to allow substantial mass to be taken out of cross bar. The optimised system was found to allow just under the maximum 2mm deflection, while being about 600g (40% lighter than the original design).

It is recommended that this structure be extruded, as extrusion is well suited for the production of aluminium. It should be noted that the filleted edges smooth out some sharp edges of the original model. This will lower the stress placed on the die, meaning less energy is needed for extrusion and less wear associated with the die. As an aside, the filleted edges are also an added safety feature, as they limit the risk or cuts caused by small sharp edges.

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Design OriginalMass (kg) 0.601 1.021

Centre of Mass x,y,z coordinates (mm) 20.5, 19.9986, 540 20.5, 20.451, 540Max Vertical Displacement (mm) -1.992 -1.264

Maximum Bottom Side Stress (MPa) 27.130 16.849

ContentsAbstract......................................................................................................................................2

Introduction................................................................................................................................4

Part A– Analysis........................................................................................................................5

Assumptions...........................................................................................................................5

Beam Model...........................................................................................................................5

Point Force..........................................................................................................................6

Uniformly Distributed Load...............................................................................................6

Plate Model.............................................................................................................................7

Point Force..........................................................................................................................8

Uniformly Distributed Load...............................................................................................9

Brick Model..........................................................................................................................10

Point Force........................................................................................................................11

Uniformly Distributed Load.............................................................................................11

Comparison...........................................................................................................................11

Part B - Design.........................................................................................................................13

Assumptions.........................................................................................................................13

Considerations and Objectives.............................................................................................13

Design premise.....................................................................................................................14

Final Design Recommendation................................................................................................15

Section..................................................................................................................................15

Component Characteristics...................................................................................................16

Conclusion................................................................................................................................17

Bibliography.............................................................................................................................17

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IntroductionRoof rack systems for cars are widespread in their use and versatile in their applications. As such, the design of roof racks is dependent to a large extent on the type of use it will likely see, which effectively translates to the loading scenario it is expected to experience. The roof rack system considered in this report is marketed as being suitable for heavy duty use, and is rated to 80kg of mass, or 40kg on each rail. In order to test the claim made by the manufacturer, and to optimise the mass of the cross bar, an appropriate computational model is required to adequately simulate the mechanics of the system.

Finite Element Analysis (FEA) is a computational method of analysing mechanics of modelled structures, and their response to applied loads. The software Strand7 is one such instance FEA software, and offers the use of multiple element definitions to construct models. Each element is different in how it processes applied loads and the associated deflections, meaning that a model made of one element can differ substantially from a model made with another type of element. It is evident that Strand7 is suitable in modelling the roof rack system, and can be done so with multiple model types to compare results.

Optimising a roof rack system will obviously focus on meeting two goals; minimising mass and maximising strength (or resistance to deflection). Stricter constraints are concerned with maintaining key geometries of the cross bar, so as to ensure all mounts and slotted areas necessary for the roof rack are still able to be utilised. FEA models will be used to give an accurate approximation of the optimised systems performance compared to that of the original.

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Part A– AnalysisThe cross bar is made of aluminium alloy and is assumed to have a yield strength of 200MPa, with a cross section given in Figure 1.

Figure 1: Given Cross Section of the Roof rack cross bar

This report aims to analyse the stresses and the maximum deflections of this cross bar for the prescribed loading conditions using three types of Finite Element models in Strand7: a beam model, a plate model and a brick model.

AssumptionsThe following assumptions are made in the analysis of the crossbar

Young’s Modulus is 69 GPa (aluminium alloy) The distance between end pin supports is 1080mm Each cross bar sees 400N of force (which can be idealised as a point force or

uniformly distributed load (UDL)). Total fibre stress is an adequate indicator of combined stress for a beam, and von

mises stress is adequate for plate and brick models. The supports of the model are rigid in all degrees of freedom, except rotation allowing

bending along the beam; this achieves a pinned structure that does not transfer moments.

Beam ModelThe beam model was constructed in the following way:

1. A model was made of the cross section given using plates, and then saved as a geometry template.

2. A single beam of length 1080mm was then built in a separate Strand7 file. 3. Both end nodes were restrained as per the model assumptions above4. The beam was then subdivided in order to refine results. 14 beams were produced, as

further subdivision would be met with diminishing improvement in accuracy.

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5. Nodes were added in the middle of the top face, and on either end of the bottom face. 6. Rigid links between these offset nodes and their centroidal counterparts were made to

closely simulate where the structure would be loaded and constrained (Figure 2).

Figure 2: Beam model, with visible cross section and rigid linking

The resulting data for model properties and results from the load cases below are summarised in Table 1.

Point ForceA load case was created wherein a 400N point force acts at a node exactly halfway along the length of the structure. The figure below depicts the structure with an exaggerated vertical displacement scale of 10%.

Figure 3: Results from Strand7 showing variation total fibre stress throughout the beam model

It is of interest to note that the highest stress occurs at the top face of the beam: 29.53 MPa in compression, almost double that of the bottom face, which was 18.89MPa. Figure 3 is particularly useful as it simultaneously shows distribution of stress, magnitude of stress, and the shape of deformation.

Uniformly Distributed LoadA separate load case was constructed, this time consisting of a uniformly distributed load equivalent to 400N. As there is no point force resulting in high stresses, this load case results in a slightly smoother stress transition.

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Figure 4: Stress concentration with a 10% displacement scale for the UDL

For the UDL above, the vertical displacement profile is not noticeably different to the point load case. However, the magnitude of the maximum vertical displacement is halved in the instance of the distributed load. Stress is similarly smaller, this time by a factor of roughly three. These results are largely intuitive, as the force is more uniformly applied and therefore less concentrated in nature. As the beam scenario is strictly two dimensional, no outward flaring occurs in the cross section of the model.

Point load UDLMax Vertical Displacement (mm) 1.202 0.694Max Horizontal Displacement (mm) 0 0Maximum Bottom side Stress (MPa) 18.887 6.304Maximum Topside Stress (MPa) -29.532 -13.431

Table 1: Summary of displacements and stresses for the beam model

It is known that the stress in a beam in bending is given by σ=MyI

, where M is the applied

moment, I is the second moment of area and y is the distance from the neutral axis to the point in question. As the model is restrained at the bottom, the stress is expected therefore to be higher on the top surface, as it is furthest from the axis of bending. This is confirmed in the table above. Negative signs for stresses denote compression, which is expected for the top of the beam.

Plate ModelFor the plate model, a different approach is needed to construct a simulation:

1. The cross section previously used to generate the beam model is altered to have single beams running through the centroid of each plate.

2. These beams are subdivided into roughly equal lengths of 2.5mm3. The beams are then extruded at increments of 2.5mm over the full 1080mm length

(Figure 5)4. A single node is placed in between the bottom feet of the model at either end, and is

constrained to meet model assumptions above5. From here, rigid links can be used to link this ‘floating’ node to the nodes on the

bottom flanges at either end of the plate model, which will simulate the clamping of the cross bar (Figure 6).

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6. Care was taken to splay multiple rigid links to nodes adjacent to the corner nodes. This aimed to better distribute the stresses generated due to simulated clamping, avoiding spuriously high stresses at nodal restraints. This also avoids a singular stiffness matrix, as plate elements are unable to sustain rigid constraint along one edge alone.

Figure 5: Cross section of the plate model, with different coloured section showing different plate properties

Figure 6: A look at how rigid links were meshed across the sides of plates

Point ForceWith the application of the point force on the plate model, a node was placed halfway along the model, directly between the two top flanges. With rigid linking (again making sure to mesh the links to adjacent nodes to spread forces), a force could be applied to the top of the structure through the floating node.

Figure 7: Stress concentration caused by the application of a point force

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It is obvious from Figure 7 that applying a point force in this manner, although better than simply adding a nodal load, still results in misleadingly high stress concentrations not consistent with the real world scenario. This means that the results recorded for the stress in the top of the beam are questionable. Similar such stress concentrations occur at the end supports of the plate model (Figure 8). These high stresses occur because the model has been constrained not to transfer moments at the joints; in the real system, this stress would not occur as some moment would indeed be shared with the mounts.

Figure 8: Stress concentrations caused at end supports

It is noted that a plate model (and later, the brick model) is able to experience lateral flaring of the cross section. The bottom flanges obviously experience the largest out-of plane-movement, but it is clear that the maximum flaring displacement is comparatively minimal (being around 100 times less than the vertical displacement in Table 2) and so is treated as insignificant.

Uniformly Distributed LoadThe uniformly distributed load was applied through the use of a face pressure to the top flat sections of the model. An equivalent pressure p was needed from an equivalent load of 400N acting on the area. As such

p= 400

2∗(9.5−( 2.52 ))∗1080

p=0.02245 MPa

The plate model idealises the top flanges as being half their thickness (1.25mm ) shorter than they are in the actual structure, owing to the fact they are simply beams extruded to have thicknesses (see Figure 5).

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Figure 9: UDL applied to the plate model, with stress variance and a 10% displacement scale

A pressure applied to the top of the structure appears to emerge from the underside, but this is merely a graphical representation of the loading magnitude. The application of a uniform load does not necessitate additional rigid linking on the top surface, so a smoother transition of stress distribution occurs. The high peak stresses are still incident at the supports, but they can continue to be ignored.

Point load UDLMax Vertical Displacement (mm) 1.275 0.745Max Horizontal Displacement (mm) 0.0145 0.0065Maximum Bottom Side Stress (MPa) 17.377 6.058Maximum Top Side Stress (MPa) -22.497 -12.9218

Table 2: Summary of results for the plate model

It should be noted, the top side stresses recorded for the brick and plate model may be confounded by high stress concentrations from applied point loads. The extent to which this point force affects the behaviour of the structure is not clear, and it is difficult to say how different models will handle this loading type. Therefore, stresses will henceforth be compared along the bottom of the beam for the sake of continuity.

Brick ModelThe brick model was constructed in a method similar to that of the plate model; only this time bricks are extruded from a plate cross section:

1. The cross section created for the beam model template is subdivided to make roughly 2.5mm square plates.

2. The plates are then extruded in increments of 2.5mm to the full 1080mm length, creating bricks.

3. The same process for the plate model is used to create the end restraints from rigid links.

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Figure 10: Cross section of the brick model with rigid linking

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Point ForceIn the same manner as previously described for the plate model, a point force is distributed through rigid linking meshed in multiple directions to alleviate singular areas and spurious stress results at the point of application.

Figure 11: Meshing of the applied point force for the brick model

Uniformly Distributed LoadIt should be noted from earlier that the plate model idealises the top flanges as being less wide than they are in the actual structure, owing to the fact they are simply beams extruded to have thicknesses. The brick model more closely simulates the face for the distributed loading, allowing the full area of the top flanges to bear the force. Therefore

pbrick=400

2∗9.5∗1080

p=0.01949 MPa

As can be seen, the effective pressure seen in the brick model is closer to actuality, and is about 13% less than that of the plate model.

Point load UDLMax Vertical Displacement (mm) 1.264 0.729Max Horizontal Displacement (mm) 0.0154 0.0067Maximum Bottom Side Stress (MPa) 16.849 6.354Maximum Top Side Stress (MPa) -23.354 -13.276

Table 3: Summary of Brick model results

It is observable that again, the vertical displacements for a uniformly distributed load are half that of the point load case and the stresses are similarly lower. Flaring is again insubstantial.

ComparisonAll models use the same material, and thus have the same value for material density. As such, discrepancies in mass between models are solely owing to differences in volume. It is intuitive that the beam and the brick model are identical in mass; they only differ in the way they are defined as elements. However, Table 4 indicates that the plate model has more mass, most likely a consequence of the fact that there are overlapping geometries in the cross section, and thus, excess mass.

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Beam Plate BrickMass (kg) 1.02089 1.03912 1.02089

Centre of Mass (mm) (20.5, 20.549, 540) 20.5, 20.633, 540 20.5, 20.549, 540Table 4: Summary for all models’ mass properties

The centre of mass is given as a set of (x,y,z) coordinates in reference to Figure 12 below. It is obvious that all the models should have their ‘z’ component the same, as they are all extruded to the same length. The ‘y’ coordinate is the only discrepancy, being different in the plate model; this is most likely a result of overlapping masses creating spurious area concentrations in corners and joins between plates, shifting the centre of mass vertically upward. This would also mean the plate model would have a different ‘I’ value than the beam and brick models. As the overlapping masses are symmetric about the y axis, it makes sense that the plate’s ‘x’ coordinate be congruent with the other two ‘x’ components.

Figure 12: Schematic showing how coordinate systems are defined

In Table 5, max stresses are taken on the bottom side for reasons explained above. For the beam model, total fibre stress is used, and for plate and brick models, the analogous von mises stress is used.

Point Force UDL

Max Displacement (mm) Max Stress (MPa)Max Displacement (mm) Max Stress (MPa)

Beam 1.202 18.887 0.694 6.304Plate 1.275 17.377 0.745 6.058Brick 1.264 19.041 0.730 6.354

Table 5: Summary of all model data

The following things can be noted about the results: The Uniformly Distributed Loads consistently result in about half the displacement of

the point loads The stresses are always lower in the UDL, for reasons explained previously. Across all three models, the displacements for both the UDL and point force load

cases are all within at most 0.07mm of each other, a very reasonable cluster of results. Similarly, the max stresses in the UDL case are all very close (a reasonable 0.296

MPa difference at worst). All of these small discrepancies will be caused by how the different element types are defined, and how they deform under loading. The finer meshing and the ability to handle

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more complex degrees of freedoms make the brick and plate model theoretically more accurate, barring any peculiar geometric behaviour arising from their use.

The only real anomaly arises with the max stress of the plate model under a point load. Although experiencing a reasonable (if slightly larger than expected) deflection, the stress in the plate model is unexpectedly lower than the other two by a not insignificant 1.6 MPa, while the plate and beam agree within a much more acceptable 0.1 MPa. These anomalies are most likely owing to how the plate model is defined; beams extruded to plates acquire a thickness but are not geometrically offset to accommodate this. As such, areas and masses tend to overlap at the junction of sections. This would result in dubious values for the area, mass and second moment of inertia, all of which would in turn skew the recorded stress. The plate model is therefore acceptable as a rough indicator but not to be used as a final simulation of system performance.

Part B - Design

AssumptionsIt is assumed in the design process that:

The geometry of the bottom flanges and the overall width and height of the section cannot be changed

The clearances between the middle bar and lower flanges given in the original design are assumed as a minimum for clips and mounts to be able to slot into.

These clearances should be able to be increased without affecting the ability to make use of these clips, mounts or attachments

The general shape of the bottom flanges are assumed necessary for its function Von mises stress is an appropriate approximation of combined stress A suitable design cannot exceed 2mm deflection at any point in the structure Any aluminium alloy can be used, as they all have the same E value of 69GPa. Vertical deflection as a result of vertical loading is the only design parameter

trying to be met in this instance; torsional loading need not be considered.

Considerations and ObjectivesThe aim in the design process was to minimise the weight while staying below a maximum allowable deflection limit in the prescribed load case. This effectively meant that the section of the cross member had to be thinned down and optimised to maintain a level of strength when subjected to bending.

Through applying Castigliano’s theorem, the deflection at the midpoint of the structure is:

δ= P L3

48 EI

Where E is young’s modulus, P is an applied point load, L is length and I is the second moment of area. This implies that for constant material, geometric constraints and loading:

δ∝ 1I

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It is known that the deflections for the given models area all around 1.2 – 1.3 mm. To reach a maximum deflection of 2mm, I is able to be reduced.

Given

I=∑ (b d3

12+ A r2)

We note that thinning sections (reducing b, d and A parameters) and bringing outer flanges closer to the axis of bending (reducing r) are all options that will reduce I. As changing the height of the cross section or changing the distance between lower flanges and the middle bar could impinge on clearances required for any roof rack attachments, changing ‘r’ in any substantial sense is not an option.

Other considerations in design include Production methods; aluminium is almost always extruded owing to the ease and

low cost involved (Trakhtenberg, Wiemerslage, 2007). In extrusion of aluminium, it is generally more optimal to design close to uniform

wall thickness, as this decreases stress on the extrusion die and increases die lifespan. (Drozda, Wick, Bakerjian, Veilleux, Petro, 1984)

1mm is generally the limiting wall thickness of extruded aluminium. (Sapa Profiles UK Ltd, 2013)

Aluminium alloy can be extruded to 0.1 mm tolerance (Sapa Group UK Ltd, 2013)

Design premiseAlthough the plate model produces dubious results, the ability to easily offset plates and change thicknesses of individual sections make the model uniquely suited to rough initial testing. As such, the design was tested in the easily constructed plate model, giving an undershoot of stress. Once an appropriate model was found, the final draft was completed with a brick model for better accuracy.

Recall, we aim to maximise I=∑ (b d3

12+ A r2)

An effort was made to move mass away from the neutral axis and more toward the outer regions, thus increasing ‘A’ and ‘r’ in the above equation. This was effectively removing a great deal of mass through thinning the members, then putting a fraction of that mass in a more strategic point to strengthen the structure. Fillets were placed at the junction between the thinner middle members and the much thicker upper and lower flanges. The added strength the fillets gave to the section meant a great deal of mass could be shaved from the members in the middle of the section (from 2.5 mm to 1.1mm, which accounts for a considerable amount of mass given the size of the middle sections).

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Figure 13: Use of fillets in the optimised design

Final Design Recommendation

SectionThe optimised design recommends the following changes:

All walls previously 2.5mm are to be thinned down to 1.6mm The side walls should be thinned from 1.9mm to 1.1mm The section running across the middle is to be thinned from 2.5mm to 1mm The edge created from the stepped thickness at the side walls is to be smoothed out by

the addition of fillets

Figure 14: Final schematic of the optimised cross section

In this manner, all important clearances (marked with red arrows on Figure 14) were maintained. These distances were never made smaller, but in most instances were made bigger as a result of thinning sections. This will ensure any clips or fittings designed to slot into the section will still be able fit into the sections. The overall height and width of the section has been maintained, as has the geometry of the bottom and top flanges.

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Component CharacteristicsA brick model was chosen to simulate the optimised design, as this type of model is generally the most accurate representation of the actual system. To get a more consistent picture of what changes have been effected from optimisation, Table 6 compares the brick model of the original system with the optimised one.

Table 6: Comparison of final optimisation to original structure

The above values result from the application of a point load, as it was observed that this load case represents the worst case scenario of the two loading schemes. As an aside, horizontal displacement is again noted to be minimal and therefore largely inconsequential, and the difference in the centres of mass is only minimal, as mass was removed and redistributed fairly equally. Also, the use of triangular elements is justified, as they are sufficiently far away from peak stress areas and therefore do not skew results.

Evidently the mass of the component was able to be reduced to approximately 600g, a reduction of roughly 40%. This was all done while observing the specified deflection limit of 2mm. In terms of stresses, the structure does not approach the assumed yield strength of 200MPa; if the stress is taken to be at the bottom of the beam

Factor of safety=σ yield

σ

Factor of safety= 20027.13

=7.372

In fact, even the largest (and obviously spurious) stresses that occur at the supports in are only around 90MPa, which is still not close to yielding.

Figure 15: Stress Distribution of final optimisation.

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Design OriginalMass (kg) 0.601 1.021

Centre of Mass x,y,z coordinates (mm) 20.5, 19.9986, 540 20.5, 20.451, 540Max Horizontal Displacement (mm) 0.083 0.015

Max Vertical Displacement (mm) -1.992 -1.264Maximum Bottom Side Stress (MPa) 27.130 16.849

Maximum Top Side Stress (MPa) 35.570 23.354

ConclusionThe final design called for a cross bar very similar to the original design, only much thinner in sections and with fillets smoothing out the step from the flanges to the thinner middle walls. This addition of fillets provided enough additional strength to the system so as to allow substantial mass to be taken out of cross bar. The optimised system was found to be about 600g, which is 40% lighter than the original design.

It is recommended that this structure be extruded, as extrusion is well suited for the production of aluminium. It should be noted that the filleted edges smooth out some sharp edges of the original model. This will lower the stress placed on the die, meaning less energy is needed for extrusion and less wear associated with the die. As an aside, the filleted edges are also an added safety feature, as they limit the risk or cuts caused by small sharp edges.

Bibliography

Drozda, Tom; Wick, Charles; Bakerjian, Ramon; Veilleux, Raymond F.; Petro, Louis (1984), Tool and manufacturing engineers handbook: Forming, SME

Sapa Group UK Ltd. (2013). Standards for Aluminium Mill Products. Retrieved 9 18, 2013, from Sapa Group: http://www.sapagroup.com/pages/522554/Standards%20for%20Alum.%20Products%2012-07.pdf

Sapa Profiles UK Ltd. (2013). Extrusion Design Guide. Retrieved 9 18, 2013, from Aluminium Design: http://www.aluminiumdesign.net/design-support/extrusion-design-guide/

Trakhtenberg, E., & Wiemerslage, G. (2007). A study of minimum wall thicknesse for extruded aluminum. Albuquerque: A05 Synchotron Radiation Facility.

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