Lightweight Beam

Structure Optimisation

Structural Design Project

ABSTRACT This report aims to detail the design and optimisation of a beam which must withstand given strength tests. The chosen cross-sectional shape for our beam was an I-I-beam due to its superior lightweight structural strength. Through detailed analysis calculations, and computer model simulations, it is found that our design will sufficiently pass the strength tests. Reiny Brown, Pietro Casabianca, Tom Blake

Overall Weight: 620 grams Overall Cost: 23.80 GBP

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Table of Contents

Nomenclature 1.0 – Nomenclature ....................................................................................................................................... 3

2.0 – Introduction .......................................................................................................................................... 3

2.1 – Project Overview ............................................................................................................................... 3

2.2 – Testing Procedure and Constraints ................................................................................................... 4

3.0 – Beam Selection ..................................................................................................................................... 5

3.1 – Cross-section Selection ..................................................................................................................... 5

3.1.1 – Single C-Section ......................................................................................................................... 5

3.1.2 – Triangular-Section ...................................................................................................................... 5

3.1.3 – Double C-Section ....................................................................................................................... 6

3.1.4 – I-I-beam ...................................................................................................................................... 6

3.2 – Final Cross-Section Decision and Optimization ................................................................................ 7

3.2.1 – Beam Length .............................................................................................................................. 7

3.2.2 – Decision and Optimization ......................................................................................................... 7

4.0 – Testing Requirements and Calculations .............................................................................................. 9

4.1 – Calculating 𝐼𝑐 ..................................................................................................................................... 9

4.2 – Rivet Pitch ....................................................................................................................................... 10

4.3 – Plate Bearing Failure ....................................................................................................................... 10

4.4 – Stiffness Requirements ................................................................................................................... 11

4.4.1 – Three-point Bending Stress ..................................................................................................... 11

4.4.2 – Deflection Estimate.................................................................................................................. 11

4.5 – Compression Requirements ............................................................................................................ 12

4.5.1 – Global Buckling ........................................................................................................................ 12

4.5.2 – Critical Inter-Rivet Buckling Stress ........................................................................................... 13

4.5.3 – Critical Flange and Web Buckling Stresses .............................................................................. 13

5.0 – Cost and Weight Estimates ................................................................................................................ 14

5.1 – Cost Estimate .................................................................................................................................. 14

5.2 – Weight Estimate .............................................................................................................................. 14

6.0 – Prediction of Failure ........................................................................................................................... 14

6.1 – Three-point Bending Failure Load .................................................................................................. 14

6.2 – Compression Test Failure Load ....................................................................................................... 14

7.0 – Future Optimisation ........................................................................................................................... 15

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7.1 – Material Choice ............................................................................................................................... 15

7.1.1 – Mass Index of Stiffness ............................................................................................................ 15

7.2 – Flange Lips ....................................................................................................................................... 16

8.0 – Conclusion ............................................................................................................................... 16

Appendix A: Derivations ................................................................................................................... 17

Appendix B: Beam Section Selection ................................................................................................. 18

Appendix C: Beam Optimization ....................................................................................................... 23

Appendix D: Moment of Inertia and Rivet Calculations ..................................................................... 26

Appendix E: Beam Test Diagrams ..................................................................................................... 28

Appendix F: Table of Equations .................................................................................................................. 29

Bibliography .................................................................................................................................... 30

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1.0 - Nomenclature:

𝒕 Material thickness 𝒚 Intermediate distance

𝝈𝒃 Bearing strength 𝒃 Breadth

𝝈𝟎.𝟏% 0.1% Proof Stress 𝒅 Depth

𝝉 Shear Strength �̅� Centroidal Coordinates

𝝉𝒓𝒊𝒗𝒆𝒕 Rivet shear strength 𝒑 Rivet Pitch

𝑬 Modulus of Elasticity 𝑸 First moment of area

𝝆 Density 𝒒 Shear flow

𝑴 Bending Moment 𝝈𝒃𝒆𝒏𝒅𝒊𝒏𝒈 Bending stress

𝜹, 𝒗 Deflection 𝝈𝒑 Inter-rivet buckling stress

𝑷 Load 𝝈𝒄𝒓𝒊𝒕−𝒇𝒍𝒂𝒏𝒈𝒆 Critical flange buckling stress

𝑳 Length 𝝈𝒄𝒓𝒊𝒕−𝒘𝒆𝒃 Critical web buckling stress

𝑰𝑪 Moment of inertia for beam 𝝂 Poisson Ratio

𝑰𝒐 Moment of inertia for individual

component 𝑨 Area

𝒎 Mass 𝒓 Internal bend radius

2.0 - Introduction: 2.1 - Project Overview: “The objective of this project is to design and manufacture a beam which will withstand given

strength tests, while keeping mass and cost minimum” (Watson, 2019). From an aeronautical

point of view, the importance for a lightweight design is due to its ability to reduce fuel costs and

increase passenger capacity. In addition, it would improve aircraft performance, sustainability

and range. However, it is as important that the materials used for the beam are cost effective for

the required strength characteristics. The beam must also be easily manufactured meaning a

simplistic, yet effective, design is important. A simple beam, along with clear and detailed

engineering drawings, will allow for a reduction in: manufacturing time, material waste and, most

importantly, the possibility of manufacturing errors which could lead to weaker structures. By

eradicating manufacturing errors, the beam’s actual strength will be closer to the predicted

strength and unintended beam characteristics will be prevented in testing.

The aim of this report is to outline the testing procedure, provide a clear description and

explanation of the beam selection and final optimised design and produce a detailed analysis of

the beam. A variety of calculations will be used to predict the stiffness, strength and buckling of

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the beam, along with the forces exerted on the rivets, to prove that the beam will be able to

withstand the forces undertaken during the tests. Lastly, potential changes to the design will be

proposed for future optimisation.

2.2 - Testing Procedure and Constraints The material that will be used is Aluminium Alloy L156, with a plate thickness of 1.220mm. The

beam’s stiffness will be examined under a three-point bending test and the compressive

buckling strength will be tested in a compressive test. These tests will be outlined in further

detail later on in the report. Additionally, there is a choice between using two different sized

Domed Head Carbon Steel Pop Rivet. These are TSBD 44 BS and TSBD 54 BS weighing in at 0.5g

and 1g respectively. TSBD 44 BS requires a hole diameter of 3.3mm and has a tensile strength of

1550N and shear strength of 1150N. TSBD 54 BS requires a hole diameter of 4.1mm and has a

tensile strength of 2500N and shear strength of 1730N (Watson A. , 2019).

The geometric constraints for the beam are, the structure’s cross-section must be no larger

than a 100mm square for the beam’s outer dimensions. While for the inner dimensions, it must

be possible to fit a 50mm diameter cylinder within the beam, i.e. must be no thinner than

50mm. The length of the beam will be 500mm, however this will be explained further on

(Watson A. , 2019). Autodesk Fusion 360, a 3D CAD software which “offers professional-grade

3D mechanical design, documentation and product simulation tools” (Autodesk, 2019), will be

used to simulate an accurate digital model of our beam under the given tests stated above. This

will demonstrate potential weaknesses and strengths in the designed beam before it is even

manufactured. In doing so, a predicted failure load can also be found.

Table 1: Important material properties of Aluminium Alloy L156

PROPERTY VALUE

𝑷𝒍𝒂𝒕𝒆 𝑻𝒉𝒊𝒄𝒌𝒏𝒆𝒔𝒔, 𝒕 1.220 mm 𝑩𝒆𝒂𝒓𝒊𝒏𝒈 𝑺𝒕𝒓𝒆𝒏𝒈𝒕𝒉 , 𝑩 550 MPa 𝟎. 𝟏% 𝑷𝒓𝒐𝒐𝒇 𝑺𝒕𝒓𝒆𝒔𝒔, 𝝈𝟎.𝟏% 210 MPa 𝑺𝒉𝒆𝒂𝒓 𝑺𝒕𝒓𝒆𝒏𝒈𝒕𝒉, 𝝉 210 MPa

𝑷𝒐𝒊𝒔𝒔𝒐𝒏’𝒔 𝑹𝒂𝒕𝒊𝒐, 𝝂 0.33 𝑴𝒐𝒅𝒖𝒍𝒖𝒔 𝒐𝒇 𝑬𝒍𝒂𝒔𝒕𝒊𝒄𝒊𝒕𝒚, 𝑬 69 GPa

𝑫𝒆𝒏𝒔𝒊𝒕𝒚, 𝝆 2.8 × 103 kg/m3

5

3.0 - Beam Selection and Optimisation 3.1 - Cross-Section Selection

To initiate the optimisation procedure, four popular beam cross-sections were modelled in the Autodesk Fusion 360 simulation environment in order to utilise the cloud-computing facility for finite element analysis (FEA) for non-linear static stress loads. The main advantage of using FEA is that it can split the solution into many smaller iterations, allowing for the location of failure to be more easily identified (Hardcastle, 2014). Despite non-linear analysis requiring greater computing power, it provides a far greater degree of accuracy due to the non-linearities existing in this force application. Examples of these errors can be seen in Appendix B: Figure 1(a) when compared to Appendix B: Figure 1(b). The majority of differences occur between linear and non-linear static stress tests when the stress felt by the object exceeds the yield strength (0.1% proof stress in this application). However, the trade-off between these solving methods are in the form of time, a non-linear solve will take a considerably longer time to finalize. (DASI Solutions, 2012).

For visual aid, exaggerated results (as referenced below) display the maximum deformation at

2.5% of the actual model size.

3.1.1 – Single C-Section

Out of all the sections looked at in this report, a C-section beam is the lightest and simplest

design. However, a light and inexpensive design is useless unless it can withstand the required

loads. The non-linear FEA results are shown in Appendix B: Figure 2, where (b) is the

exaggerated deflection of (a). The test simulation results return a maximum stress value of

433.5MPa, of which far exceeds the tensile stress of Aluminium L156, at 365MPa.

3.1.2 – Triangular-Section

Having a triangular cross-section permits a simpler design, while still being able to accept the

internal pipe. The shape means an extra face can be removed, compared to I- and boxed-

beams, leading to reduced mass and overall cost. There are however a number of

disadvantages which outweigh the benefits.

The first issue is with manufacturing. Although it would make sense to think that the

manufacturing of this beam would be easier than others, consisting of only three pieces, the

bending machine would be unable to create the required angles of this shape.

Secondly, to allow for the cylinder to fit inside the triangular beam, each length side will have to

be larger than other beam designs. Not only would this increase the weight of the structure,

thus reducing the weight advantage, but it would also decrease the structure’s strength. Due to

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the triangular shape of the beam, it is more likely that a dent could form during the three-point

bending test, introducing a new stress concentration on the loaded face of the beam. This

would lead to a greatly reduced critical buckling stress, making catastrophic failure likely.

The non-linear FEA results of the triangular cross-section are shown in Appendix B: Figure 3,

where (b) is the exaggerated deflection of (a). The test simulation results return a maximum

stress value of 309MPa, and although this does not exceed the tensile stress of Aluminium

L156, at 365MPa, it does however exceed the 0.1% proof stress of 216MPa. This means the

beam will be permanently deformed after the load is applied, an event that should be avoided.

3.1.3 - Double C-Section

Also known as a boxed-section, this beam is a potential choice due its ability to withstand both

high torsional and biaxial bending due to a large moment of inertia on both axes compared to

the triangular and singular C-sections. This is due to the majority of the mass being distributed

far from the centre of mass. This holds true through the non-linear FEA simulation, see

Appendix B: Figure 4, where it achieves a maximum stress value of 306.3MPa, a value far less

than the previous two cross sections, yet still large enough to suffer from permanent

deformation as it exceeds the 0.1% proof stress of 216MPa.

3.1.4 – I-I-Beam

In theory, by making area moments larger, the rigidity of a structure is increased. This can be achieved most efficiently by having the cross-sectional area concentrated away from the beam centre (Bailey, Bull, & Lawrence, 2013). It can withstand large amounts of bending loads with minimal structure weight required. For this reason, it is one of the most efficient and commonly used structural items in engineering (Doshi, 2016). As the maximum compression and tension occurs at furthers points away from the neutral axis, the wide outer flanges on the top and bottom surfaces of the beam are able to distribute the loads more effectively. With considerable proportion of its mass far away from the neutral axis, it has a comparatively large moment of inertia, which is one of the main reasons for its impressive strength to weight ratio (Zhou, 2018).

Potential downsides to the use of an I-beam are that it is less resistant to shear stress forces,

due to the reduced material thickness along the web section. However, as long as the shear

stresses experienced by the beam are less than the shear strength of the material, failure of the

beam should not occur. Another potential disadvantage to using an I beam is its relatively weak

strength to torsional loads. This could especially be a problem during the compressive test as it

is more prone to buckling compared to a box section beam. In order to prevent this buckling,

and increase its torsional load resistance, the web height must not be too large. Therefore, a

fine balance between buckling strength and bending strength must be made (Zhou, 2018).

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The FEA analysis confirms the theory, returning a maximum stress value of 72.28MPa, see

Appendix B: Figure 5. This is far smaller value than in the previous cross-sections, resulting in a

beam that behaves only in the elastic region of it stress-strain diagram.

3.2 - Final Cross-Section Decision and Optimisation:

3.2.1 - Beam Length The beam length was decided as 500mm simply to cater the requirements of both tests. For the three-point bending test, the beam must be no shorter than 450mm to enable it to be held by both supporting pins. While with the compressive test, its loading pads are exactly 500mm apart, so a beam of 450mm length would fall short. 3.2.2 – Decision and Optimisation

The I-I-beam cross section was chosen due to various reasons. First of all, observing the figures

in Appendix B, it is clear that the I-I-beam suffered from minimal deflection and buckling

compared to the C- and the triangular sections. Although the double C-section had reduced

deflection I-I-beam, its buckling was significantly worse. Additionally, as previously mentioned,

the maximum stress value experienced in the I-I-beam was lower than any of the other designs.

The equation for deflection in three-point bending

𝛿 =𝑃𝐿3

48𝐸𝐼𝐶

From this equation, it is clear to see that a beam with a greater moment of inertia will have a

smaller overall deflection, hence reducing the risk of permanent deformation. Due to having

the mass farther from the centre, I-I-beams have a larger moment of inertia.

Proceeding, optimisation of cross-sectional dimensions is necessary, this requires dimensions to

be minimised to reduce mass per unit length whilst also maintaining adequate structural

Beam

Cross-

section

Number

of folds

Number

of

guillotine

cuts

Number

of rivets

Cost for

folds

(GBP)

Cost for

guillotine

cuts

(GBP)

Cost for

rivets

(GBP)

Total

Cost

(GBP)

Cost

percentage

(%)

Triangular 3 8 3 1.095 1.480 1.710 4.285 82

Double C-

section 4 8 4 1.460 1.480 2.280 5.220 100

I-I-beam 4 8 4 1.460 1.480 2.280 5.220 100

Table 1: Comparative costs of various different cross-sectional shapes. It is briefly worth mentioning

that the cost per rivet includes the cost per drilled hole for rivets

(1)

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integrity to withstand the test loads with minimal deformation, prioritising an elastic

deformation only. The optimization process developed from a single fully constrained

parametric model of a generic I-I-section of length 500mm. From Autodesk Fusion 360, a

parametric dimension interface can be accessed to input a series of dimensions to size the

cross-section accordingly. 15 different cross sections were chosen to be analysed, each varying

between 5m and 10mm in each dimension of depth, breadth and web width. These dimensions

where later converted into 15 individual beams, each subjected to the same three point

bending environment, see Appendix C: Figure 1, note the fully-fixed constraints as well as the

semi-fixed constraints. The bottom supports utilize a x,y,z constraint, whereas the top loading

platen is only constrained in the x and z direction, allowing vertical movement to apply a load.

Contacts made between the beam bodies, supports and loading platen can be seen as the

separation contacts in Appendix C: Figure 2. Note the friction coefficient of 0.61 (Beardmore,

2013) is used between these bodies to accurately represent the behaviour of the bodies

relative to one another. An intelligent auto-mesh has also been applied to generate an accurate

representation of bodies as a collection of triangles; mesh becomes more densely populated

around areas of curves, compared to flat surfaces where fewer elements are required, see

Appendix C: Figure 3.

For simplicity these beams were modelled without rivets or holes, and bonded body contacts

are set; this will give an estimation accurate enough to decipher the cross-section optimization

dimensions as it is assumed enough rivets will be used to prevent failure. The final 15th beam

during the analysis failed due to a singularity being detected, this could not be resolved and

therefore has been omitted from the optimization stage.

The results of these simulations can be seen in Appendix C: Table 1. The table can be

interpreted by red signalling high values of stress and deformation and green signalling more

favourable values. It can be seen that the most favourable beams from these simulations are

beams 3, 6 & 14. Combining these stress and displacement results with Appendix C: Table 2 it is

intuitive to select beam 14 as it has the lowset mass, and therefore the lowest unit cost out of

the tested beam.

To conclude, from the FEA analysis and optimisation the cross section of beam number 14

(dimensioned 55mm depth, 90mm breadth and 55mm web width) will be selected for further

investigation. See Appendix C: Figure 4. It was decided not to simulate buckling as the results

would poorly reflect the behaviour of the beam in a buckling environment, due to the

simplification of the rivets via a bonded contact.

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It was planned to carry out a final, fully represented, model, including rivets, FEA simulation however the solver being used did not have a sufficient element storage allocation. Resulting in an error message upon commencing a solve.

4.0 - Testing Requirements and Calculations:

4.1 - Calculating Ic

The moment of inertia for the entire beam, Ic, can be calculated using the parallel-axis theorem.

It was first considered that the individual moment of inertia value, 𝐼𝑜 , for the origin of the fillets

could be calculated using the following equation, where r is the radius of the fillet (Fenner &

Watson, 2012):

𝐼𝑜 = 𝑟4 [(1 −5𝜋

16) − (1 −

𝜋

4) (

(10 − 3𝜋)2

(12 − 3𝜋)2)] = 0.007545116𝑟4

However, as this equation was derived for an I-beam, with the web placed directly through the centre line, this equation would give negligible values which would not change the final 𝐼𝑐 value by much. Therefore, the moment of inertia can be roughly calculated by ignoring the fillet curves, and assuming right-angle joints (as shown in Appendix D, Figure 1). This also allows for each section to be modelled by a rectangle, whose individual moment of inertia (𝐼𝑜) is:

𝐼𝑜 =𝑏𝑑3

12

�̅� =∑ 𝑦𝐴

∑ 𝐴=

11750.76

427.30≈ 27.5

ℎ = �̅� − 𝑦

𝑰𝒄 = 𝐼𝑜−𝑡𝑜𝑡𝑎𝑙 + ℎ2𝐴𝑡𝑜𝑡𝑎𝑙 = 𝟐𝟒𝟎𝟔𝟗𝟖. 𝟖𝟐 𝒎𝒎𝟐

Part Area (mm2) y (mm) yA (mm3) Io (mm4) h (mm) h2A (mm4)

1 109.80 54.39 5972.02 13.62 -26.89 79393.32

2 109.80 0.61 66.98 13.62 26.89 79393.32 3 61.15 27.50 1681.63 12800.05 0 0

4 61.15 27.50 1681.63 12800.05 0 0

5 21.35 53.17 1135.18 2.65 -25.67 14068.56 6 21.35 53.17 1135.18 2.65 -25.67 14068.56

7 21.35 1.83 39.07 2.65 25.67 14068.56

8 21.35 1.83 39.07 2.65 25.67 14068.56

Total 427.30 - 11750.76 25637.94 - 215060.88

Table 2: Second Moment of Inertia calculations

(2)

(3)

(4)

10

4.2 - Rivet Pitch Before rivet pitch may be calculated, shear flow (q) must be found from the equation. Note that in Appendix D Figure 2, the shear flow diagram for a regular I beam, shows a small amount of shear in the flanges, compared to large shear in the web.

𝑞 =𝑉 ∙ 𝑄

𝐼𝑐

Where V is equal to 𝑃

2 in a three-point bending setup. However, we multiplied this by an error

factor of 2 to overestimate the shear flow and, thus, consolidate that the beam won’t fail. Q is given as:

𝑄 = ∫ 𝑦𝑑𝐴 =

𝐴

�̅� ∙ 𝑏 ∙ 𝑡

Therefore, the shear flow is found to be

∴ 𝑞 =𝑃 ∙ �̅� ∙ 𝑏 ∙ 𝑡

𝐼𝑐=

(2400)(27.5 × 10−3)(90 × 10−3)(1.22 × 10−3)

240698.82 × 10−12

≈ 30107.33497 𝑁𝑚−1 Rivet pitch is now found, allow the minimum number of rivets to be calculated too.

𝑝 =𝜏𝑟𝑖𝑣𝑒𝑡

𝑞=

1730

30107.33497≈ 0.05746108 𝑚

𝑴𝒊𝒏𝒊𝒎𝒖𝒎 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒓𝒊𝒗𝒆𝒕𝒔 =𝐿

𝑝=

0.5

0.05746108≈ 𝟗 𝒑𝒆𝒓 𝒔𝒊𝒅𝒆

Keeping the minimum number of rivets required in mind, it was decided to go for 12 rivets instead. This is due to two important reasons. Firstly, both plates have two rows of rivets. This means an even number of rivets is required per plate, so that an even number of rivets can go into each row. Therefore, the number of rivets can only be any even number starting from 10 onwards. However, if 10 were to be chosen, 5 rivets would go in each row. This would be problematic as it means there would be two rivets in the middle section which would interfere with the top loading platen. Thus, 12 rivets were decided as the best choice. 4.3 – Plate Bearing Failure Plate bruising is the phenomena when the bearing pressure between the rivets and aluminium plates becomes excessive, weakening the overall structure strength. This is demonstrated by

(5)

(6)

(7)

(8)

(9)

11

Appendix D Figure 3, and to make sure bruising won’t occur the maximum force each hole can experience before bruising will be calculated using Equation 10.

𝑃𝑚𝑎𝑥 =𝜎𝑏𝑡𝑑𝑟𝑖𝑣𝑒𝑡−ℎ𝑜𝑙𝑒

𝑝𝑎𝑐𝑡𝑢𝑎𝑙=

(550 × 106)(1.22 × 10−3)(4.1 × 10−3)

(92 × 10−3)≈ 29.9 𝑘𝑁

Since this value is much larger than any force the beam will experience, plate bruising is not a concern. Another critical phenomenon is plate tearing, which is the complete failure of the plate’s rivet hole. As it has been shown that plate bruising won’t be an issue, neither will plate tearing. 4.4 - Stiffness Requirements: To determine the stiffness of the beam a three-point bending test will be used. Appendix E Figure 1 briefly illustrates the test set up. The distance between the two supporting pins will be 450mm and a force of 2.4kN will be applied to the beam exactly midway down between the pins. Also, as the loading pin is circular, it can be assumed that the force applied will act like a point load. The maximum acceptable deflection of the beam is 9mm. Appendix E Figure 2 clearly shows what type of deflection should be expected from the three-point bending test. The maximum deflection will occur mid-span, directly underneath the applied load. At the supporting pins (x=0m and x=0.45m) there will be no deflection and no bending moment. These are known as the boundary conditions.

4.4.1 - Three-point Bending Stress: The bending moment for three-point bending is given by:

𝑀 =𝑃𝐿

4=

(2400)(0.5)

4≈ 300 𝑁𝑚

By solving for M, the bending stress can then be calculated using:

𝜎𝑏𝑒𝑛𝑑𝑖𝑛𝑔 =𝑀 ∙ 𝑦

𝐼𝑐=

(300)(27.5 × 10−3)

240698.82 × 10−12≈ 𝟑𝟒. 𝟐𝟖 𝑴𝑷𝒂

This is well below the proof stress shown in Table 1, therefore there will be no permanent deformation, successfully completing the test requirements. 4.4.2 - Deflection Estimate: The deflection equation for a three-point bending beam is derived in Appendix A. From this derivation the following equation for beam deflection is acquired:

𝛿 =𝑃𝐿3

48𝐸𝐼𝐶

(11)

(12)

(10)

12

This is a rough method to hand calculate the deflection of the beam. For this test, 𝑃 is 2400N, L is 0.5m, and E is 69GPa. Substituting these values into the equation:

𝛿 =(2400) × (0.5)3

(48)(69 × 109)(240698.82 × 10−12)≈ 𝟎. 𝟑𝟕𝟔 𝒎𝒎

It is clear to see that this is a rather small amount of deflection, which is not realistic and not expected to occur during the actual test. This is due to the formula being used to calculate the deflection, 𝛿, is considered for solid beam in three-point bending. Due to the small thickness of the aluminium sheet and the mechanics of thin walled structures, a correction factor, 𝑘𝑓 must

be applied to acquire rough estimates. The correction factor for this type of test is usually 6 ≤ 𝑘𝑓 ≤ 10. In this report, a 𝑘𝑓 of 10 will be used, therefore the final equation for calculating

the deflection is:

𝛿𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑 = 𝑘𝑓𝛿 = (10)(0.376) ≈ 𝟑. 𝟕𝟔 𝒎𝒎

As this is less than 9mm, the beam deflection is within limits to pass the three-point test. 4.5 - Compression Requirements: Appendix E Figure 3 shows the testing set up for the compressive test. Note that a 6kN force is applied to the cross section surface. To determine if the compressive strength of the beam is satisfactory, a compressive test load of 6kN will be applied to the beam (shown by Figure 3), and the structure must not fail. The overall normal compressive stress on a beam with cross-sectional area (A), under a specific load (P) is calculated by using Equation 14 Where the load is given as 6kN and the area was found earlier in Section 4.1.

𝜎𝑚𝑎𝑥 =𝑃

𝐴=

6000

427.3 × 10−6≈ 𝟏𝟒. 𝟎𝟒 𝑴𝑷𝒂

Compressive stress is negative, however for ease of calculation it has been omitted. Therefore, the maximum compressive stress felt by the beam can now be compared to the maximum stresses the components of the beam can withstand. If these following values are greater than 𝜎𝑚𝑎𝑥, the beam will pass the test. In addition, 𝜎𝑚𝑎𝑥 is also well below the proof stress shown in Table 1. 4.5.1 – Global Buckling Global buckling for the I-I-beam is found as:

𝜎𝑐𝑟𝑖𝑡−𝑔𝑙𝑜𝑏𝑎𝑙 = (𝜋2𝐸

𝐿2) (

𝐼𝑐

𝐴) ≈ (

𝜋2(69 × 109)

0.52) (

240698.82 × 10−12

427.30 × 10−6)

≈ 𝟏𝟓𝟑𝟒. 𝟒𝟓 𝑴𝑷𝒂

(13)

(14)

(15)

13

As this is well above the maximum compressive stress (𝜎𝑚𝑎𝑥), the beam will withstand global buckling. 4.5.2 – Critical Inter-Rivet Buckling Stress The inter-rivet buckling stress is found by using Equation 16:

𝜎𝑐𝑟𝑖𝑡−𝑝 = 2.47 ∙ 𝐸 (𝑡

𝑝)

2

= 2.47(69 × 109) (1.22 × 10−3

0.05746108 )

2

≈ 𝟕𝟔. 𝟖𝟑 𝑴𝑷𝒂 With a value much greater than the maximum stress the beam will experience; it is safe to say that the rivets will hold during the compressive test. 4.5.3 – Critical Flange and Web Buckling Stresses The equations for calculating critical buckling stresses are shown below. Note that d is not depth in this instance but is the flange breadth. Whereas h is the breadth of the flat plate bounded by two joints.

𝜎𝑐𝑟𝑖𝑡−𝑓𝑙𝑎𝑛𝑔𝑒 =0.43 ∙ 𝜋2𝐸

12(1 − 𝜐2)(

𝑡

𝑑)

2

=0.43(𝜋)2(69 × 109)

12(1 − 0.332)(

2 × 1.22 × 10−3

14.53 × 10−3)

2

≈ 𝟕𝟕𝟐. 𝟐𝟓 𝑴𝑷𝒂

𝜎𝑐𝑟𝑖𝑡−𝑤𝑒𝑏 =4 ∙ 𝜋2𝐸

12(1 − 𝜐2)(

𝑡

ℎ)

2

=4(𝜋)2(69 × 109)

12(1 − 0.332)(

1.22 × 10−3

44.15 × 10−3)

2

≈ 𝟐𝟎. 𝟗𝟏 𝑴𝑷𝒂 It can therefore be concluded that the beam will successfully withstand the compression test, since all the component’s critical stresses are greater than the maximum compressive stress the beam will experience.

(16)

(17)

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14

5.0 – Cost and Weight Estimates 5.1 - Cost Estimate:

5.2 – Mass Estimate: The overall mass of the beam is calculated through the following equation:

𝑚𝑡𝑜𝑡𝑎𝑙 = 𝑚𝑝𝑙𝑎𝑡𝑒−𝑡𝑜𝑡𝑎𝑙 + 24 ∙ 𝑚𝑟𝑖𝑣𝑒𝑡 − 24 ∙ 𝑚ℎ𝑜𝑙𝑒

𝑚𝑡𝑜𝑡𝑎𝑙 = (∑ 𝐴) 𝐿𝜌 + 24 ∙ 𝑚𝑟𝑖𝑣𝑒𝑡 − 2 ∙ 24 ∙ 𝜋𝑟2𝑡𝜌

𝑚𝑡𝑜𝑡𝑎𝑙 = (427.3 × 10−6) × 0.5 × 2.8 × 103 + (24 × 0.001)

− (48 × 𝜋 ×(4.1 × 10−3)2

4× 1.22 × 10−3 × 2.8 × 103)

∴ 𝑚𝑡𝑜𝑡𝑎𝑙 ≈ 0.59822 + 0.024 − 0.002164 ≈ 0.620𝑘𝑔 ≈ 620𝑔

6.0 – Prediction of Failure 6.1 – Three-point Bending Failure Load As stated in Section 4.3, the maximum acceptable deflection of the beam is 9 mm. Therefore, by rearranging Equation (1) for P, and letting 𝛿𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑 = 9 𝑚𝑚, the failure force can be calculated.

𝑃 =48𝛿𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑𝐸𝐼𝐶

𝐿3=

48(9 × 10−3)(6 × 109)(240698.82 × 10−12)

(0.5)3≈ 𝟒𝟗𝟗𝟏 𝑵

Hence, for a force of 5kN or above, the beam in three-point bending will fail. 6.2 – Compression Test Failure Load To find the failure load for the compression test, the component with the lowest critical stress is used as it would fail first. From Section 4.4, the web has the lowest buckling stress. By rearranging Equation (14) to solve for force.

𝑃 = 𝜎𝑚𝑎𝑥𝐴 = (20.91 × 106)(427.3 × 10−6) ≈ 𝟖𝟗𝟑𝟓 𝑵 Therefore, any force of 9kN or above would cause beam failure in the compression test.

Procedure/Part Quantity Cost / GBP Total Part Cost / GBP

Guillotine cuts (2 per rectangular piece)

8 0.185 1.480

Folds 4 0.365 1.460

Drilled hole for rivets 24 0.400 9.600

Rivets 24 0.170 4.080

Aluminium per 100g 598g 1.200 7.176 Total Beam Cost = 23.796 ≈ 23.80 GBP

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15

7.0 – Future Optimisation

7.1 – Material Choice The material used for this project, Aluminium Alloy L156, was perfectly adequate for this project, with superior strength to weight ratio compared with other materials such as Mild Steel. One of the major disadvantages however with using aluminium is that it has no standard fatigue limit making its failure unpredictable. This can lead to early retirement of the structure causing excess waste, or to possible failure, depending on how conservative the user of the structure is. Mild steel is a possible replacement material for Aluminium Alloy L156, despite its reduced strength properties. The use of Mild steel would reduce the cost of manufacturing in various ways. First of all, the cost of Mild steel per volume compared to the L156 is far cheaper and easier to obtain. Also, Mild Steel is much simpler to work on, with folds and cuts being easier to produce, allowing for a reduction in manufacturing time and cost. 7.1.1 – Mass Index of Stiffness Through a simplified analysis of a beam in a three-point bending configuration, a relationship between mass and mass index is found. Stiffness is given by load (P) divided by deflection (), which can be equated by rearranging Equation (1).

𝑃

𝛿=

48𝐸𝐼

𝐿3

Given the beam has a cross-sectional breadth b and depth d, I is given as:

𝐼 =𝑏𝑑3

12

Thus,

𝑃

𝛿=

4𝐸𝑏𝑑3

𝐿3

And mass is equal to density times volume:

𝑚 = 𝜌 ∙ 𝑉 = 𝜌 ∙ 𝑏𝑑𝐿 Rearranging for b and subbing the new equation into Equation (20) gives:

𝑃

𝛿=

4𝐸 (𝑚

𝜌𝑑𝐿) 𝑑3

𝐿3=

4𝐸𝑚𝑑2

𝜌𝐿4

Lastly, the equation can be rearranged so that m is given as:

(20)

16

𝑚 = (𝑃

𝛿) (

𝐿4

4𝑑2) (

𝜌

𝐸)

Assuming that all other parameters remain constant, the beam with the lowest mass can be found by searching for a material with the lowest mass index for stiffness. This is shown by the third bracketed term in Equation (21).

7.2 – Flange Lips Beam buckling is significantly improved, and failure loads are increase when flange lips are added. These improvements increase at an increasing rate as the flange lip height is increased. However, above a certain height may not add any extra benefit. Although these flange lips would increase the strength of the beam, it was not implemented in our design for a few major reasons. Mainly, manufacturing difficulty and time would increase, as for the flange lips to be beneficial, measurements and bending must be far more accurate. This would also lead to an increase in overall price and weight. For a beam that is already able to withstand the required tests without flange lips, it would have been against the optimisation process to include them. They would, however, be an optimisation aspect if the loads the beam must withstand increase greatly (Ragheb, 2007).

8.0 – Conclusion

Through the optimisation process we were able to determine the ideal cross-sectional shape for our beam. This process was heavily improved by incorporating computer model simulations through Autodesk Fusion 360 and allowed for a detailed analytical look into the beam behaviour. The detailed analysis section showed that our beam will be able to withstand the forces subjected to it in both the stiffness and compressive tests. Also, the internal stresses within the beam will be within tolerance, allowing for only minimal buckling. Observing both the weight and cost of the beam, the beam structure is reasonably cost effective and adequately lightweight. From this report it is clear, that this beam has achieved all the objectives set out for it, and will satisfy the testing requirements, proving to be a viable lightweight structure for load purposes.

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17

Appendix A: Derivations

1. Derivation of three-point bending beam deflection equation The bending moment equation and governing equation for deflection v, is shown as follows (Zhou, 2018):

𝑀 =𝑃𝑥

2 𝐸𝐼𝑣′′ = −𝑀

where 𝑃 is the load, 𝑥 is the distance along the beam, 𝑀 is the bending moment, 𝐸 is the modulus of elasticity, 𝐼 is the second moment of inertia and 𝑣 is the deflection. Therefore,

𝐸𝐼𝑣′′ = −𝑃𝑥

2

𝐸𝐼𝑣′ = −𝑃𝑥2

2+ 𝐶1

𝑎𝑡 𝑥 =𝐿

2, 𝑣′ = 0 ∴ 𝐶1 =

𝑃𝐿2

16

𝐸𝐼𝑣 = −𝑃𝑥3

2+

𝑃𝐿2

16𝑥 + 𝐶2

𝑎𝑡 𝑥 = 0, 𝑣 = 0 ∴ 𝐶2 = 0 Thus

𝐸𝐼𝑣 = −𝑃𝑥3

2+

𝑃𝐿2

16𝑥

𝑣 =𝑃𝑥

48𝐸𝐼(3𝐿2 − 4𝑥2)

Replacing 𝑣 with 𝛿 and calculating for the beam centre where 𝑥 =𝐿

2 :

𝛿 =𝑃𝐿3

48𝐸𝐼

18

Appendix B: Beam Section Selection

Figure 1. (b) Non-linear static stress test for a cantilever beam. SOURCE: (DASI Solutions, 2012)

Figure 1. (a) Linear static stress test for a cantilever beam. SOURCE: (DASI Solutions, 2012)

19

Figure 2: (b) Non-linear static stress FEA three-point bending results – C-section ADJUSTED

Figure 2: (a) Non-linear static stress FEA three-point bending results – C-section ACTUAL

20

Figure 3: (a) Non-linear static stress FEA three-point bending results – Triangular-section ACTUAL

Figure 3: (b) Non-linear static stress FEA three-point bending results – Triangular-section ADJUSTED

21

Figure 4: (a) Non-linear static stress FEA three-point bending results – Double C-section ACTUAL

Figure 4: (b) Non-linear static stress FEA three-point bending results – Double C-section ADJUSTED

22

Figure 5: (a) Non-linear static stress FEA three-point bending results – I-I-section ACTUAL

Figure 5: (b) Non-linear static stress FEA three-point bending results – I-I-section ADJUSTED

23

Appendix C: Beam Optimisation

Figure 1: FEA loading Environment Fusion 360

Figure 2: FEA loading environment contact configuration

24

Table 1: I-I Beam non-linear static stress FEA three-point bending results

Cuts Folds Hole Rivets Area(mm^4)

1 90 90 60 8 4 24 24 500.49 0.701 0.723 25.03

2 90 90 55 8 4 24 24 512.69 0.718 0.740 25.23

3 90 85 55 8 4 24 24 488.29 0.684 0.705 24.82

4 80 90 60 8 4 24 24 476.09 0.667 0.688 24.62

5 80 90 55 8 4 24 24 488.29 0.684 0.705 24.82

6 80 85 55 8 4 24 24 463.89 0.649 0.671 24.41

7 70 90 60 8 4 24 24 451.69 0.632 0.654 24.21

8 70 90 55 8 4 24 24 463.89 0.649 0.671 24.41

9 70 85 55 8 4 24 24 439.49 0.615 0.637 24.00

10 60 90 60 8 4 24 24 427.29 0.598 0.620 23.80

11 60 90 55 8 4 24 24 439.49 0.615 0.637 24.00

12 60 85 55 8 4 24 24 415.09 0.581 0.603 23.59

13 55 90 60 8 4 24 24 415.09 0.581 0.603 23.59

14 55 90 55 8 4 24 24 427.29 0.598 0.620 23.80

Beam

DesignBeamMass

(kg)

Overall

Mass(kg) Cost(GBP)

PlateDepth

(mm)

Breadth

(mm)

WebWidth

(mm)

Table 2: I-I Beam overall weight and cost analysis

25

Figure 4: The selected cross-section design Figure 3: Meshing of a curved surface

26

Appendix D: Moment of Inertia and Rivet Calculations

Moment of Inertia Diagram:

Shear Flow Diagram:

Figure 1: Simplified diagram of the beam cross-section for the calculation of the beam’s moment

of Inertia. It is worth noting the replacement of the smooth fillet curves with right angles to allow

for the beam to modelled by a combination of rectangles

Figure 2: Shear flow distribution of a regular I-beam. Note that

although not the exact same distribution of an I-I beam, it is fairly

similar SOURCE: (Sharma, N.d.)

27

Plate Failure

Figure 3: Bearing strength of plate diagram

SOURCE: (Watson A. , 2018)

28

Appendix E: Beam Test Diagrams Three-point bending:

Compressive Test:

Figure 1: Three-point bending test set up with a 2.4kN

test load SOURCE: (Ćurković , Bakić, Kodvanj, & Haramina,

2010)

Figure 2: The type of deflection that will occur during the

test SOURCE: (Zhou, 2018)

Figure 3: The compressive test set up, where a

6kN load will be applied to the beam SOURCE:

(Watson A. , 2019)

29

Appendix F: Table of Equations 1 – Three-point Bending Deflection ....................................................................................................... 7/10

2 – Moment of Inertia Component Value for a Fillet .................................................................................. 9

3 – Centroidal Coordinate ............................................................................................................................. 9

4 – Moment of Inertia for Entire Beam ........................................................................................................ 9

5 – Shear Flow ............................................................................................................................................. 10

6 – First Moment of Area ............................................................................................................................ 10

7 – Shear Flow Solution .............................................................................................................................. 10

8 – Rivet Pitch .............................................................................................................................................. 10

9 – Minimum Number of Rivets ................................................................................................................. 10

10 – Plate Bruising ....................................................................................................................................... 11

11 – Three-point Bending Moment ............................................................................................................ 11

12 – Three-point Bending Stress ................................................................................................................. 11

13 – Three-point Bending Deflection Correction ....................................................................................... 12

14 – Normal Compressive Stress ................................................................................................................ 12

15 – Global Buckling Critical Stress ............................................................................................................. 12

16 – Critical Inter-Rivet Buckling Stress ...................................................................................................... 13

17 – Critical Flange Buckling Stress............................................................................................................. 13

18 – Critical Web Buckling Stress ................................................................................................................ 13

19 – Overall Mass Estimate......................................................................................................................... 14

20 – Three-point Bending Deflection Rearranged ..................................................................................... 15

21 – Mass Index for Stiffness Proof ............................................................................................................ 16

30

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