anirudh aircraft design ii

8/12/2019 Anirudh Aircraft Design II

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GROUP C2

Aircraft Design II Report

Unconventional Medium Commuter A/C

Ajinkya Desai, Anirudh Gupta, Ronak Karia4/20/2013


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ContentsList of Figures ......................................................................................................................................... 4

List of Tables .......................................................................................................................................... 4

Chapter 1: Statistical Study of Structural Design Features of Existing Aircraft ..................................... 6

1.1 Introduction ................................................................................................................................... 6

1.2 Salient Structural Design Features of Existing Aircraft ................................................................ 7

1.2.1 ATR 72[Ref.1] ....................................................................................................................... 7

1.2.2 British Aerospace ATP [Ref. 2] ............................................................................................. 8

1.2.3 Ilyushin Il-114 [Ref. 3] .......................................................................................................... 8

1.2.4 SAAB 2000 ............................................................................................................................ 8

Chapter 2: V-n Diagram and Span-wise Lift Distribution ...................................................................... 9

2.1 V-n Diagram ................................................................................................................................. 9

Procedure and Calculations ............................................................................................................. 9

Formulae: ........................................................................................................................................ 9

Speeds for critical loads: ............................................................................................................... 12

Gust loads: .................................................................................................................................... 12

Final V-n diagram ......................................................................................................................... 13

2.2 Distribution of Aerodynamic Loads on the Wing ....................................................................... 14

Introduction ................................................................................................................................... 14

Procedure and Calculation ............................................................................................................ 14

Graphs and Variations ................................................................................................................... 15

2.3 Summary ..................................................................................................................................... 16

Chapter 3: Distribution of shear forces and moments ........................................................................... 16

3.1 Wing Discretization .................................................................................................................... 16

3.2 Center of gravity & Aerodynamic Centre locations. .................................................................. 17

3.3 Formulation ................................................................................................................................. 17

3.4 Graphs and Variations ................................................................................................................. 18

Chapter 4: Idealization of Wing Section ............................................................................................... 21

4.1 Material selection and properties ................................................................................................ 21

4.2 Actual Structure of Wing section ................................................................................................ 214.3 Idealized wing section properties ................................................................................................ 21

Procedure and Calculation ............................................................................................................ 21

Formulae ....................................................................................................................................... 21

Sample Calculation for root section .............................................................................................. 22


Chapter 5: Determination of Axial and Shear Stresses ......................................................................... 24

5.1 Bending Moments and Stresses .................................................................................................. 24

Procedure and Calculations: .......................................................................................................... 24

Formula: ........................................................................................................................................ 24


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5.2 Shear Stress in Beams ................................................................................................................. 25

5.2.1 Shear Stress arising from Shear Forces Vx, Vy........................................................................ 25


Formulae: ...................................................................................................................................... 25

5.2.2 Shear Stress arising from Torsion ............................................................................................ 26


Formulae: ...................................................................................................................................... 26


5.3 Principal Stresses ........................................................................................................................ 27

Procedure and Calculations ........................................................................................................... 27


5.4 Factor of Safety ........................................................................................................................... 28

Formulae ....................................................................................................................................... 28

Calculation .................................................................................................................................... 28

Chapter 6: Aero-elastic Parameters....................................................................................................... 29

6.1 Shear Centre and Elastic Axis ..................................................................................................... 29

Formula ......................................................................................................................................... 29


6.2 Critical Divergence Speed .......................................................................................................... 30

Procedure ...................................................................................................................................... 30

Formula ......................................................................................................................................... 30

Sample Calculation at the Tip ....................................................................................................... 30

Conclusion .................................................................................................................................... 30

Chapter 7: Wing Weight Calculation and Comparison with Empirical Estimate ................................. 30

Assumptions .................................................................................................................................. 30

Calculations ................................................................................................................................... 31

Conclusion .................................................................................................................................... 32

Chapter 8: Buckling and Crushing Loads ............................................................................................. 32

8.1 Crushing Loads on Spars and Ribs ............................................................................................. 32Procedure ...................................................................................................................................... 32

Formula ......................................................................................................................................... 32


8.2 Buckling of Columns and Plates ................................................................................................. 34

Formulae ....................................................................................................................................... 34

Assumptions .................................................................................................................................. 34


Alternative approach ..................................................................................................................... 35

Conclusions ........................................................................................................................................... 35


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References ............................................................................................................................................. 36

List of Figures

Figure 1. Two spar wing structure of ATR ............................................................................................. 7Figure 2. Structural features of the fuselage of ATR-72 ......................................................................... 7Figure 3. Material decomposition of the parts of the ATR-72 aircraft ................................................... 8

Figure 4. v-n diagram ............................................................................................................................ 13Figure 5. Spanwise distribution of lift for different load factors and critical speeds ............................ 15Figure 6.Spanwise distribution of drag for different load factors and critical speeds........................... 15Figure 7. Spanwise distribution of pitching moment for different load factors and critical speeds .... 16Figure 8. Schmatic of Section Spacing ................................................................................................. 17Figure 9. Plot of Vx against spanwise station from root ....................................................................... 18Figure 10. . Plot of Vy against spanwise station from root ................................................................... 19

Figure 11. . Plot of Mx against spanwise station from root .................................................................. 19

Figure 12. . Plot of My against spanwise station from root .................................................................. 20Figure 13. . Plot of Mt against spanwise station from root ................................................................... 20Figure 14. Spanwise variation of area moment of inertia ..................................................................... 22Figure 15. Area moment of inertia inclusive of ribs along the span ..................................................... 23

Figure 16. Position of the centre of gravity, inclusive of ribs, against spanwise location from root .... 23Figure 17. Area of section [inclusive of ribs] along span ..................................................................... 24

Figure 18. Variation of the bending normal stress with spanwise location from the root for cruiseconditions .............................................................................................................................................. 24Figure 19. Variation of the bending normal stress with spanwise location from the root for given

critical condition ................................................................................................................................... 25Figure 20. Variation of tau_xz with spanwise station ........................................................................... 26

Figure 21. Variation of tau_yz with spanwise station ........................................................................... 26Figure 22. Variation of the larger principal stress with spanwise location ........................................... 27Figure 23. Position of shear centre along the chordline, variation with spanwise location .................. 29Figure 24. Crushing loads for front and rear spar ................................................................................. 33Figure 25. Front and rear spar loads on ribs, variation along spanwise location .................................. 34Figure 26. Plot of buckling stress with the longeron number ............................................................... 35

List of TablesTable 1. Existing Turboprop Aircraft of the given kind ......................................................................... 6Table 2. Existing Turboprop Aircraft of the given kind ......................................................................... 6

Table 3. Design features of aircraft under study ..................................................................................... 6

Table 4. Calculation of maximum and minimum Cza ............................................................................ 9Table 5. Variation of the Cza with the angle of attack.......................................................................... 10

Table 6. : Coordinates for parabolas OA and OB (V-n diagram) ......................................................... 12Table 7. Gust load Factors .................................................................................................................... 13

Table 8. Calculation of lift coefficient and lift for different load factors and critical speeds ............... 14Table 9. Calculation of the moment coefficient from local angle of attack of untwisted wing ............ 16

Table 10. Material properties of structural elements ............................................................................ 21Table 11. Wing design attributes .......................................................................................................... 21

Table 12. Area moment of inertia for root section per longeron........................................................... 22

Table 13. Area moment of inertia at the root section ............................................................................ 22Table 14. Non-trivial solutions of the Eigen Value problem, variation with span-wise location ......... 27

Table 15. Wing geometric and aerodynamic parameters ...................................................................... 30

Table 16. Mass of each longeron .......................................................................................................... 31Table 17. Mass of ribs ........................................................................................................................... 31


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Table 18. Front and rear spar loads on ribs, variation with spanwise location ..................................... 32

Table 19. Buckling force and stress on longerons of idealized section ................................................ 34


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Chapter 1: Statistical Study of Structural Design Features of Existing

Aircraft

1.1 IntroductionThe aircraft presented is an unconventional design of a medium commuter aircraft. It is a five engine

turboprop aircraft, with a low wing and canard configuration. The engines are small in size. Theconventional aircraft is twin turboprop (powerful engines) with aft-tail configuration. One example ofthe conventional aircraft is the ATR-72 aircraft. A comparison is drawn between this conventionalaircraft and our design, in table 1, to justify the emulation of this aircraft.

Table 1. Existing Turboprop Aircraft of the given kind

Requirements/Features Medium Commuter Aircraft ATR-72 Series 500

Cruise Speed 600 kmh-1

for best range, at an

altitude of 7 km at MTOW

459 kmh-1

economical cruise

speed at 95% MTOW

Balanced Field Length 1100 m 1223 m

Service Ceiling 10 km 7.62 km

Payload 70 pax + 3 crew + 20 kg baggage

each

68 pax + 3crew + 20 kg

baggage each

Range 1200 km with max. Payload and IFR

reserve

1200 km

Cabin Height 1.75m -

Seating Economy class with galley and

lavatory

Economy class with galley and

lavatory

Other existing aircraft considered are twin turboprop aircraft, presented in table 2.

Table 2. Existing Turboprop Aircraft of the given kind

Turboprop Aircraft We(kg) MTOW (kg) Passengers and Crew

ATR 72500 12,950 22,500 68 + 2

British Aerospace ATP 13,595 22,930 4+64

Ilyushin Il-114 13,500 23,500 64+2

IPTN N-250 13,665 22,000 50-70

YS-11A-200 14,600 23,500 64+2

Xian MA60 13,700 23,500 60 + 2Saab 2000 13,800 22,800 58 + 2

Fokker 50 12,250 20820 58

The design features of the unconventional aircraft presented are summarized in Table 3 for reference.

Table 3. Design features of aircraft under study

Parameters Unconventional (C2)

Range 1387.25 km

Aspect ratio 15

Wing area 63.94 mCruise Speed 600 km/hr


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L/D 21.43

Weight 21988 kgf

Balanced Field Length 965 metres

1.2 Salient Structural Design Features of Existing Aircraft

1.2.1 ATR 72[Ref.1]

Wings Two-spar, fail-safe wings. Materials: Mainly of aluminium alloy, with leading- edges of Kevlar/Nomex sandwich. The

outer wing-box structure is made of composite materials including carbon monolithic

structure, carbon/Nomex sandwich and Kevlar/Nomex sandwich. The wing top skin panels aftof rear spar are of Kevlar/Nomex with carbon reinforcement. The flaps and ailerons havealuminium ribs and spars, with skins of carbonfibre/Nomex and carbon/epoxy respectively.

Fuselage Semi-monocoque fail-safe fuselage: Uses a substructure to which the airplanes skin is

attached. The substructure, which consists of bulkheads and/or formers of various sizes andstringers, reinforces the stressed skin by taking some of the bending stress from the fuselage.

Materials: Fail-safe stressed skin, mainly of light alloy except for Kevlar/Nomex sandwich.The engine cowlings are of CFRP/Nomex and Kevlar/Nomex sandwich, reinforced withCFRP in nose and underside. The propeller blades have metal spars and GFRP/polyurethaneskins. The structure of the ATR-72 is generally as for ATR 42, but new wings outboard of

engine nacelles have CFRP front and rear spars. The self-stiffening CFRP skin panels andlight alloy rib result in a weight saving of 120 kg (265 lb).

Tail The horizontal stabilizer is made stronger using co-cured multispars and the vertical fin is

strengthened using panels and co-bonded stringers. Materials: The vertical tail and horizontal tail are made mainly of aluminium alloy. There are

CFRP/Nomex sandwich rudder and elevators.

Figure 1. Two spar wing structure of ATR

Figure 2. Structural features of the fuselage of ATR-72


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Advantages of CompositesThe Q400 has bigger (and consequently heavier) engines, and uses little or no composites in itsaircraft structure, unlike the ATR72 which extensively uses proven lightweight composites in thewing, and tail plane. Composite materials make up 19% of the total weight of the structure. ATR 72ssecondary structures are extensively made of composite material, which are not subject to corrosion.In addition, the ATR 72 innovates by the use of carbon fiber for its outer wings and tail, thus reducing

weight further.

Figure 3. Material decomposition of the parts of the ATR-72 aircraft

The in-service advantages of composites are numerous, like Immunity to corrosion and fatigue Reduction of inspection Payload gain and fuel savings.

1.2.2 British Aerospace ATP [Ref.2]Cantilever low-wing monoplane.

The all-metal fuselageis circular in cross section and is of semi-monocoque fail safe design.The airframe is exceptionally strong and with durability and maintainability in high cycle,short sector operations.

Materials: The primary load-bearing structure is constructed from advanced alloys.Lightweight composites are used selectively on non-critical secondary structures.

1.2.3 Ilyushin Il-114 [Ref.3]

Conventional low-wing monoplane Two-spar wingswith a removable leading-edge on outer panels. Circular-section, semi-monocoque fuselageis built as five sub-assemblies. Materials: Approximately 10 per cent of the airframe by weight is made of composites. It uses

The fuselage is made of aluminium alloy The tail unit is metallic in nature.

1.2.4 SAAB 2000 Two-spar wings, fin and tailplane. Materials: Wing and fuselage primary structures are made of metal/metal bonded aluminium

alloy, with honeycomb sandwich fin. They use composites for ailerons (CFRP/Nomex), flaps(CFRP skins), wing/body fairings (Kevlar/Nomex), nosecone (GFRP/Nomex), rudder andelevators (GFRP leading-edges and CFRP skins), propeller blades, and cabin floor (carbon

fibre sandwich).


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Chapter 2: V-n Diagram and Span-wise Lift DistributionA study of the v-n diagram and span-wise lift distribution at the critical loads obtained from the v-ndiagram has been presented. The results are for a medium commuter, 5-turboprop engine aircraft, with

an unconventional canard configuration. All calculations are for the clean configuration.

2.1 V-n DiagramThe basic strength and flight performance limits are specified in the form of a v-n diagram. Aircraftload factor (n) is the quantity estimated which expresses the maneuvering of the aircraft as a multipleof the standard acceleration due to gravity. The air speed is indicative of the dynamic pressure. Thelimit load factor is a function of airspeed. The variation is shown using V v/s n diagram for boththe values of n and for all values of V till the maximum attainable velocity (flight envelope).

Procedure and Calculations The normal aerodynamic force coefficient, CZa, is evaluated as a function of angle of attack in

Table 1.1.

The limit load factors- (n+ and n-) are observed from literature and the stall speeds formaximum and minimum CZaare calculated.

Drag divergence speed is observed from previous report, Ref. 4. Gust load factors at drag divergence speed are evaluatedThe following is the relation used to compute n

The smallest speed (VA) corresponding to the positive limit load factor is computed. The graph isconstant for all speeds further till the maximum speed corresponding to which the aircraft experiencesthe maximum dynamic pressure (q) i.e. at the drag divergence speed (VC) . The point representing

maximum q and maximum load factor is clearly important for structural design.

Similar calculations are done for the negative limit load factor and again the smallest speed iscomputed (VB). It is evident from the above equations that CZacalculation is necessary. The data usedto obtain the V-n diagram is procured as follows as per the procedure explained above.

Formulae:

The given are the angles of attack obtained from existing literature on the NLF-0414 airfoil, whichwe have used for our wing. The 2-D lift, drag and moment coefficients were also available fromRef.[4]from which the wing aerodynamic coefficients had been calculated in Ref. [5], i.e. theprevious report, and have been tabulated as shown below.

Table 4. Calculation of maximum and minimum Cza

CL CD Cz= CLcos + CDsin CMa CMa*CW/Lt Cza

-9.25 -0.844 0.031 -0.838 -0.058 -0.023 -0.861

15 1.309 0.052 1.278 -0.029 -0.011 1.267


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Table 5. Variation of the Cza with the angle of attack

Cl Cd Cm CL CD CZ=

CLcos + CDsin

CMa CMaC/Lt Cza

-9.25 -0.6567 0.03161 -0.0661 -0.844 0.031 -0.838 -0.0583 -0.023 -0.861

-9 -0.6404 0.02947 -0.0657 -0.821 0.030 -0.816 -0.0580 -0.023 -0.838

-8.75 -0.6225 0.02734 -0.0654 -0.798 0.029 -0.793 -0.0577 -0.022 -0.816

-8.5 -0.6031 0.02525 -0.0652 -0.775 0.028 -0.771 -0.0575 -0.022 -0.793

-8.25 -0.5816 0.02342 -0.065 -0.752 0.028 -0.748 -0.0573 -0.022 -0.770

-8 -0.5587 0.02177 -0.0649 -0.729 0.027 -0.725 -0.0573 -0.022 -0.748

-7.75 -0.5347 0.02032 -0.0649 -0.706 0.026 -0.703 -0.0573 -0.022 -0.725

-7.5 -0.5102 0.01903 -0.0648 -0.682 0.025 -0.680 -0.0572 -0.022 -0.702

-7.25 -0.4853 0.01794 -0.0648 -0.659 0.025 -0.657 -0.0572 -0.022 -0.679

-7 -0.46 0.01708 -0.0647 -0.636 0.024 -0.634 -0.0571 -0.022 -0.657

-6.75 -0.4342 0.01645 -0.0648 -0.613 0.024 -0.612 -0.0572 -0.022 -0.634

-6.5 -0.4076 0.01602 -0.065 -0.590 0.023 -0.589 -0.0573 -0.022 -0.611

-6.25 -0.3809 0.01558 -0.0652 -0.567 0.022 -0.566 -0.0575 -0.022 -0.589-6 -0.3545 0.01511 -0.0653 -0.544 0.022 -0.543 -0.0576 -0.022 -0.566

-5.75 -0.3279 0.01467 -0.0655 -0.521 0.021 -0.521 -0.0578 -0.023 -0.543

-5.5 -0.301 0.0143 -0.0657 -0.498 0.021 -0.498 -0.0580 -0.023 -0.520

-5.25 -0.2738 0.01403 -0.0659 -0.475 0.020 -0.475 -0.0581 -0.023 -0.498

-5 -0.2478 0.01359 -0.066 -0.452 0.020 -0.452 -0.0582 -0.023 -0.475

-4.75 -0.2211 0.01348 -0.0662 -0.430 0.020 -0.430 -0.0584 -0.023 -0.452

-4.5 -0.193 0.01324 -0.0667 -0.407 0.019 -0.407 -0.0588 -0.023 -0.430

-4.25 -0.1646 0.013 -0.0671 -0.384 0.019 -0.384 -0.0592 -0.023 -0.407

-4 -0.136 0.01278 -0.0676 -0.361 0.018 -0.361 -0.0596 -0.023 -0.385

-3.75 -0.1078 0.01251 -0.068 -0.338 0.018 -0.339 -0.0600 -0.023 -0.362

-3.5 -0.0791 0.01236 -0.0685 -0.316 0.018 -0.316 -0.0604 -0.024 -0.340-3.25 -0.0504 0.01217 -0.069 -0.293 0.017 -0.293 -0.0609 -0.024 -0.317

-3 -0.0219 0.01195 -0.0695 -0.270 0.017 -0.271 -0.0613 -0.024 -0.295

-2.75 0.0061 0.01162 -0.0701 -0.247 0.017 -0.248 -0.0618 -0.024 -0.272

-2.5 0.0332 0.01103 -0.0708 -0.225 0.017 -0.225 -0.0625 -0.024 -0.250

-2.25 0.0595 0.00997 -0.0718 -0.202 0.016 -0.203 -0.0633 -0.025 -0.227

-2 0.0854 0.00922 -0.0721 -0.180 0.016 -0.180 -0.0636 -0.025 -0.205

-1.75 0.1152 0.0095 -0.0725 -0.157 0.016 -0.158 -0.0640 -0.025 -0.182

-1.5 0.1445 0.00984 -0.0727 -0.135 0.016 -0.135 -0.0641 -0.025 -0.160

-1.25 0.1739 0.01009 -0.0729 -0.112 0.016 -0.112 -0.0643 -0.025 -0.137

-1 0.2017 0.01064 -0.0724 -0.090 0.016 -0.090 -0.0639 -0.025 -0.115

-0.75 0.2285 0.01093 -0.0718 -0.067 0.016 -0.067 -0.0633 -0.025 -0.092

-0.5 0.2572 0.0111 -0.0718 -0.045 0.016 -0.045 -0.0633 -0.025 -0.070

-0.25 0.287 0.0112 -0.0723 -0.022 0.016 -0.022 -0.0638 -0.025 -0.047

0 0.3157 0.0111 -0.0726 0.000 0.016 0.000 -0.0641 -0.025 -0.025

0.25 0.3443 0.01107 -0.0728 0.022 0.016 0.022 -0.0642 -0.025 -0.003

0.5 0.3724 0.01117 -0.0729 0.045 0.016 0.045 -0.0643 -0.025 0.020

0.75 0.4012 0.01121 -0.0733 0.067 0.016 0.067 -0.0647 -0.025 0.042

1 0.4306 0.01114 -0.0737 0.089 0.016 0.089 -0.0650 -0.025 0.064

1.25 0.4602 0.01102 -0.0742 0.111 0.016 0.112 -0.0655 -0.026 0.086

1.5 0.4904 0.0107 -0.0745 0.134 0.016 0.134 -0.0657 -0.026 0.108

1.75 0.521 0.01018 -0.0747 0.156 0.016 0.156 -0.0659 -0.026 0.1312 0.5517 0.00966 -0.075 0.178 0.016 0.178 -0.0662 -0.026 0.153


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2.25 0.5819 0.00937 -0.0753 0.200 0.016 0.201 -0.0664 -0.026 0.175

2.5 0.6123 0.00905 -0.0759 0.222 0.017 0.223 -0.0670 -0.026 0.197

2.75 0.6431 0.00858 -0.0763 0.244 0.017 0.245 -0.0673 -0.026 0.218

3 0.6737 0.00819 -0.0767 0.266 0.017 0.267 -0.0677 -0.026 0.240

3.25 0.7034 0.00789 -0.0773 0.288 0.017 0.289 -0.0682 -0.027 0.262

3.5 0.7333 0.00764 -0.0778 0.310 0.018 0.311 -0.0686 -0.027 0.2843.75 0.7632 0.00746 -0.0783 0.332 0.018 0.333 -0.0691 -0.027 0.306

4 0.789 0.00747 -0.078 0.354 0.018 0.354 -0.0688 -0.027 0.328

4.25 0.8055 0.00871 -0.0768 0.376 0.019 0.376 -0.0678 -0.026 0.350

4.5 0.8198 0.01 -0.0756 0.39808 0.0189 0.39834 -0.0667 -0.026 0.372

4.75 0.8331 0.01116 -0.0742 0.420 0.019 0.420 -0.0655 -0.026 0.395

5 0.8468 0.01216 -0.0728 0.442 0.020 0.442 -0.0642 -0.025 0.417

5.25 0.8626 0.013 -0.0717 0.464 0.020 0.464 -0.0633 -0.025 0.439

5.5 0.878 0.01363 -0.0704 0.486 0.021 0.486 -0.0621 -0.024 0.461

5.75 0.8965 0.01424 -0.0696 0.508 0.021 0.507 -0.0614 -0.024 0.483

6 0.9168 0.0148 -0.069 0.530 0.022 0.529 -0.0609 -0.024 0.505

6.25 0.9363 0.01539 -0.0683 0.552 0.022 0.551 -0.0603 -0.023 0.527

6.5 0.9564 0.01593 -0.0677 0.573 0.023 0.572 -0.0597 -0.023 0.549

6.75 0.9766 0.01644 -0.067 0.595 0.023 0.594 -0.0591 -0.023 0.571

7 0.9965 0.01697 -0.0663 0.617 0.024 0.615 -0.0585 -0.023 0.592

7.25 1.0175 0.01741 -0.0658 0.639 0.024 0.637 -0.0581 -0.023 0.614

7.5 1.0365 0.01799 -0.0649 0.660 0.025 0.658 -0.0573 -0.022 0.636

7.75 1.0578 0.0184 -0.0644 0.682 0.025 0.679 -0.0568 -0.022 0.657

8 1.0779 0.01889 -0.0637 0.704 0.026 0.701 -0.0562 -0.022 0.679

8.25 1.0956 0.01954 -0.0627 0.726 0.027 0.722 -0.0553 -0.022 0.700

8.5 1.1155 0.02004 -0.062 0.747 0.027 0.743 -0.0547 -0.021 0.722

8.75 1.1353 0.02054 -0.0613 0.769 0.028 0.764 -0.0541 -0.021 0.7439 1.1548 0.02106 -0.0606 0.791 0.029 0.785 -0.0535 -0.021 0.765

9.25 1.1731 0.02167 -0.0597 0.812 0.030 0.806 -0.0527 -0.021 0.786

9.5 1.1867 0.02258 -0.0582 0.834 0.030 0.828 -0.0513 -0.020 0.808

9.75 1.2059 0.02312 -0.0575 0.856 0.031 0.849 -0.0507 -0.020 0.829

10 1.2241 0.02372 -0.0566 0.877 0.032 0.869 -0.0499 -0.019 0.850

10.25 1.2417 0.02437 -0.0557 0.899 0.033 0.890 -0.0491 -0.019 0.871

10.5 1.2588 0.02503 -0.0548 0.920 0.034 0.911 -0.0484 -0.019 0.892

10.75 1.2757 0.02573 -0.0538 0.942 0.034 0.932 -0.0475 -0.019 0.913

11 1.2911 0.02653 -0.0527 0.964 0.035 0.953 -0.0465 -0.018 0.935

11.25 1.303 0.02757 -0.0512 0.985 0.036 0.973 -0.0452 -0.018 0.956

11.5 1.3128 0.02878 -0.0495 1.007 0.037 0.994 -0.0437 -0.017 0.97711.75 1.3297 0.02948 -0.0487 1.028 0.038 1.015 -0.0430 -0.017 0.998

12 1.3451 0.03029 -0.0477 1.050 0.039 1.035 -0.0421 -0.016 1.019

12.25 1.3591 0.03119 -0.0465 1.072 0.040 1.056 -0.0410 -0.016 1.040

12.5 1.3724 0.03215 -0.0454 1.093 0.041 1.076 -0.0401 -0.016 1.060

12.75 1.3857 0.03313 -0.0442 1.115 0.042 1.096 -0.0390 -0.015 1.081

13 1.3988 0.03413 -0.0431 1.136 0.043 1.117 -0.0380 -0.015 1.102

13.25 1.4106 0.03522 -0.0419 1.158 0.044 1.137 -0.0370 -0.014 1.123

13.5 1.4217 0.03639 -0.0407 1.179 0.045 1.157 -0.0359 -0.014 1.143

13.75 1.4318 0.03768 -0.0395 1.201 0.046 1.178 -0.0349 -0.014 1.164

14 1.4389 0.03924 -0.038 1.223 0.047 1.198 -0.0335 -0.013 1.185

14.25 1.4398 0.04136 -0.036 1.244 0.048 1.218 -0.0318 -0.012 1.206

14.5 1.4472 0.04297 -0.0346 1.266 0.050 1.238 -0.0305 -0.012 1.226


12/36

14.75 1.4568 0.0444 -0.0336 1.288 0.051 1.258 -0.0296 -0.012 1.247

15 1.4656 0.04594 -0.0327 1.309 0.052 1.278 -0.0289 -0.011 1.267

Speeds for critical loads:

The positive and negative limit load factors- (n

+

, n

-

) have been taken as +4 and -2 respectively (Ref.3,4).

For = 15o, CZa= 1.267, n = +4

This gives the value of VA= 117.94 m/s.

For = -9.25o, CZa= -0.861, n =

This gives the value of VB= 101.17 m/s

At drag divergence, VC = 222.22 m/s giving us the maximum speed for n+. This is the indicated

airspeed as evaluated from Ref. 4., i.e. the previous report.

This is the same as the dive speed, VD, since it is a commuter aircraft. Hence VD= VC= 222.22 m/s.

At cruise, the speed as per our aircraft requisites is VE= 166.66 m/s.

Table 6. : Coordinates for parabolas OA and OB (V-n diagram)

V (m/s) Upper Parabola (n+) Lower Parabola (n-)

0 0 0

10 0.0288 -0.0195

20 0.1152 -0.078

30 0.2592 -0.1755

40 0.4608 -0.312

50 0.72 -0.4875

60 1.0368 -0.702

70 1.4112 -0.9555

80 1.8432 -1.248

90 2.3328 -1.5795

100 2.88 -1.95

110 3.4848 -2.3595

101.17 2.947786243 -1.995896936

117.94 4.006034957

Gust loads:


13/36

where;

a= 0.0877 per degree = 5.0279 per radian

where; 7.098 ft; g = 32.2 ft/s2, = 0.002378 slug/ft3;

The gust velocity is taken to be + 50 ft/s as explained in Ref. 1.

Table 7. Gust load Factors

U (ft/s) KU (effective gust velocity) V (knots) ng

50 39.9 VD= 431.96 3.47, -1.47

Final V-n diagram

Figure 4. v-n diagram


14/36

2.2 Distribution of Aerodynamic Loads on the Wing

IntroductionThe span-wise wing loading is extremely essential to be determined for the bending moment and structuralanalysis of the wing. The aircraft design requires detailed inertia and aerodynamic load distribution toallow one to perform detailed sizing for various structural elements and their geometric locations.Schrenks approximation method has been used to obtain the wing loading. Results have been presented fora wing without twist, as was done in the previous course. In this method, the load distribution is determined

both using the elliptical assumption of wing as well as trapezoidal assumption. The actual loading is thus,approximately the average of the two loadings.

Procedure and Calculation

For the four critical load conditions of the set (VA, n+), (VB, n-), (VC, n+), (VE, n-), obtain the CZa. From CZa, we can obtain the lift coefficient of the wing for each case as in Table 8. Calculate the average chord from the Schrenks approximation for sectional lift. Obtain the spanwise lift, drag and moment coefficients from the relations mentioned in class. Obtain the spanwise lift, drag and moment distribution from the above aerodynamic coefficients.

The following relations are used to obtain the average chord:

where;

For a 3-D wing, with the approximation of an elliptic load distribution, we have the following relations:

Ours is an untwisted wing, so the effects of twist on the local lift coefficient are ignored. Hence:

The local drag coefficient is calculated using the following relation.

Table 8. Calculation of lift coefficient and lift for different load factors and critical speeds

Velocity (m/s) n CL

VA= 117.94 4 1.267 15 1.309 11152.39VB= 101.17 -2 -0.861 -9.25

-0.844 5291.17

Vc= 222.22 4 0.446 5.2 0.462 13973.79VE= 166.66 -2 0.398 4.5 0.398 6770.99


15/36

Graphs and VariationsThe spanwise variation for the lift distribution and drag distribution has been presented in Figure 2.1and Figure 2.2 respectively. It must be noted that although the load factors are the same for two sets ofv-n. The angles of attack and hence the coefficient of lift for each will be different leading todifference in the spanwise loading. The Schrenks approximation has been used to evaluate the liftcoefficient as a function of the spanwise location.

Figure 5. Spanwise distribution of lift for different load factors and critical speeds

Figure 6.Spanwise distribution of drag for different load factors and critical speeds

-40000.0

-30000.0

-20000.0

-10000.0

0.0

10000.0

20000.0

30000.0

40000.0

0 5 10 15 20L(y)

[N/m]

y (m)

L(y), VA, 4

L(y), VB, -2

L(y), VC, 4

L(y), VE, -2

0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20

D(y)

[N/m]

y (m)

D(y), VA, 4

D(y), VB, -2

D(y), VC, 4

D(y), VE, -2


16/36

Table 9. Calculation of the moment coefficient from local angle of attack of untwisted wing

Velocity (m/s) CLw CMaVA= 117.94 1.309 - 2.1 -0.0635VB= 101.17 -0.844 1.37

-0.0656

VC= 222.22 0.446 -0.72 -0.0633VE= 166.66 - 0.398 0.65 -0.0645

Figure 7. Spanwise distribution of pitching moment for different load factors and critical speeds

2.3 SummaryThe v-n diagram for a medium commuter, unconventional commuter aircraft with canardconfiguration has been presented. Four sets of V and n, corresponding to different angles of attack,have been plotted in part I. Gust loads have been taken into account for VC, and the gust velocity isassumed 50ft/s as mentioned in class. In part II, the spanwise lift, drag and moment distribution havebeen plotted, starting from the CZavariation with angle of attack. The procedures for each of the partshave been explained.

Chapter 3: Distribution of shear forces and momentsHaving calculating the aerodynamics loads with the span-wise position, the next task is to determine

the shear force, bending moment and torsional moment as a function of the span. These would be used

to calculate the shear flow and the stresses in the wing.

3.1 Wing DiscretizationTo calculate these as a function of span, we start from the boundary condition at the wing tip, whereall the forces and moments are zero, and march towards the root. We discretize the wing, for

numerical purposes, into many span-wise sections. The said forces and moments are calculated foreach section.

To account for the greater variation near the tip, the sections are smaller and hence more frequent nearthe tip as compared to the root. We used an AP series discretization to achieve this.

-14000

-12000

-10000

-8000

-6000

-4000

-2000

0

0 5 10 15 20

M(y)

[Nm/m]

y (m)

M(y), VA, +4

M(y), VB, -2

M(y), VC, +4

M(y), VE, -2


17/36

Figure 8. Schmatic of Section Spacing

A total of 60 stationswere taken.

Our span length is 30.969m. Thus, L = 0.008461m.

The coordinate axis chosen:

Xalong the chord (LE to TE)

Yupward

Zalong the span

3.2 Center of gravity & Aerodynamic Centre locations.

Before proceeding to the shear forces and the moment calculations, we first need to calculate x cgandycg (the location of the center of gravity with span-wise position). This is done using CAD, for astation of unit chord and then scaled for different span wise positions using a scaling factor c.

Xcg = 0.265443*cYcg = -0.002445*c

The aerodynamic centres were taken to have an offset of 0.25cfrom the leading edge, along the X-direction.

3.3 Formulation

For the actual calculations, the alternate formulation given in the notebook was used. Since the wingdoes not have any twist, no further corrections were necessary. The set of equations used are

mentioned below-


18/36

Where, ; and . is the mass of the discrete element and n is the load factor.3.4 Graphs and Variations

The figures 9-13 present the variations of the shear forces and moments for different load factors and

critical speeds against the spanwise locations. An obvious observation is that the loads and momentsare maximum at the root and decrease along the span towards the tip where the loads and momentsare zero, it being a free end. The maximum loads are observed to be for the combination of n = 4 andv = 222.22, which is maximum load factor along with drag divergence speed. This was as anticipated.

Figure 9. Plot of Vx against spanwise station from root


19/36

Figure 10. . Plot of Vy against spanwise station from root

Figure 11. . Plot of Mx against spanwise station from root


20/36

Figure 12. . Plot of My against spanwise station from root

Figure 13. . Plot of Mt against spanwise station from root


21/36

Chapter 4: Idealization of Wing Section

4.1 Material selection and properties

Table 10. Material properties of structural elements

Structure Material YoungsModulus

E (GPa)

ShearModulus

-G (GPa)

Density(g/cc)

Yieldstrength

(GPa)

Utimatestrength

(GPa)

Poissonsratio ()

Spar

webs

7075-T6 Al 71.7 26.9 2.81 0.503 0.572 0.33

Skin 7075-T6 Al 71.7 26.9 2.81 0.503 0.572 0.33

Longitudi

nal

7075-T6 Al 71.7 26.9 2.81 0.503 0.572 0.33

4.2 Actual Structure of Wing section

Table 11. Wing design attributes

Airfoil AR Root chord

(Cr)

Taper ratio () Quarter chord

sweep (c/4)

Number of stations

NLF-0414 15 2.8477 m 0.45 0o 30

The root section of the wing with properties as mentioned in Table xyz, is modelled in CATIA. Theactual structure comprises two wing spars and six longitudinal. The dimensions of each of these havebeen marked in Figure xyz. Since majority of the trailing edge is used for flaps (appx. 20%), nostiffeners have been used for the same.

Spars- These are I shaped and two in number. They are placed at 30% chord from the LE and 70%chord from the LE respectively, as mentioned in Ref. Xyz.

Longitudinal- These are six in number and have a rectangular cross section.

4.3 Idealized wing section properties

Procedure and Calculation1. Construct a CAD model schematic of the wing section with airfoil skin, spars and

longitudinal.2. Design the idealized wing section by substituting the webs with two booms each, longitudinal

with individual booms and by modelling the skin with panels in turn replaced by booms.3. Fit a coordinate system in the plane of the section, with x axis along the geometric chord andorigin at quarter chord (since ).

4. Calculate for each of these sections5. Calculate total area and coordinates of the centroid for each section and plot them as a

function of span-wise location

Formulae


22/36

Sample Calculation for root section

Table 12. Area moment of inertia for root section per longeron

Area x(m) y(m) i_xx (x 10-4

kgm2)

i_xy (x 10-4

kgm2)

i_yy (x 10-4

kgm2)

0.0012566 -0.53 0.105 0.1385 -0.6992 3.5288

0.00166 -0.292 0.188 0.5867 -0.911 1.4147

0.0012566 -0.04 0.232 0.6764 -0.1164 0.02

0.00378 0.142 0.169 1.0796 0.9076 0.763

0.005533333 0.382 0.262 3.7983 5.5391 8.0776

0.001809 0.545 0.267 1.2896 2.6327 5.3747

0.001809 0.885 0.263 1.2513 4.2109 14.1709

0.00166 1.041 0.25 1.0375 4.3205 17.9917

0.003026333 1.281 0.152 0.6992 5.893 49.6668

0.003026333 1.281 -0.023 0.016 -0.8917 49.6668

0.00166 1.041 -0.181 0.5438 -3.128 17.99170.001809 0.885 -0.13 0.3057 -2.0814 14.1709

0.001809 0.545 -0.137 0.3395 -1.3509 5.3747

0.00166 0.382 -0.135 0.3025 -0.8562 2.4233

0.001134 0.142 -0.05 0.0284 -0.0806 0.2289

0.0012566 -0.04 -0.123 0.1901 0.0617 0.02

0.00166 -0.292 -0.104 0.1795 0.504 1.4147

0.0012566 -0.53 -0.057 0.0408 0.3796 3.5288

Summing up the entries in the fourth, fifth and sixth columns respectively, we obtain Table 13.

Table 13. Area moment of inertia at the root section

Root Section Ixx(x 10-4m4) Root section Ixy (x 10-4m4) Root Section Iyy (x 10-4m4)

12.5036 14.3335 195.8279

Graphs and Variations

Figure 14. Spanwise variation of area moment of inertia

0

50

100

150

200

250

0 5 10 15 20

Mom.ofInertia(x1E-04kgsqm.)

Spanwise location from the root

ixx

ixy

iyy


23/36

Figure 14 describes the variation of the areal moment of inertia with the span-wise distance from theroot chord, for half the wing. The data points represent the stations. As expected, the areal moment ofinertia about the y axis is larger for all the stations as compared to that about the x axis.

Figure 15. Area moment of inertia inclusive of ribs along the span

Figure 15 describes the variation of the same, but inclusive of the ribs. Wherever there is a sudden

jump in the physical quantity, be it area moment of inertia or centre of gravity or area of the wingsection, there are ribs inserted. The ribs help in picking up additional load, hence reducing the stress

of the longerons at the different stations. These ribs help increase the factor of safety. The density ofthe ribs is higher towards the root section. Figures 16 and 17 present the latter two quantities.

Figure 16. Position of the centre of gravity, inclusive of ribs, against spanwise location from root

-1000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 5 10 15 20 25 30 35Area

momentofinertia(x1E-04kgmetr

e

sq.)


Ixx

Ixy

Iyy

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25 30 35

Positionofc.g.

foreachstation(m)[ribs

inclusive]

Spanwise location from root

cgx

cgy


24/36

Figure 17. Area of section [inclusive of ribs] along span

Chapter 5: Determination of Axial and Shear Stresses

5.1 Bending Moments and Stresses

Procedure and Calculations:1. For a wing section of isotropic material, the axial stress or bending stress can be calculated.2.

These are calculated using the coordinates of the booms as obtained by an idealization of thewing section.

3. The bending stress is plotted against the span-wise location for the boom with maximumstress in figure 18.

Formula: Graphs and Variations

Figure 18. Variation of the bending normal stress with spanwise location from the root for cruise conditions

-0.10000

0.00000

0.10000

0.20000

0.30000

0.40000

0.50000

0.60000

0 5 10 15 20 25 30 35

Area(sqmetre)[ribsinclusive]

Spanwise location from root

-60000

-50000

-40000

-30000

-20000

-10000

0

10000

0 10 20 30 40

AxialLoad(kPa)


sigma_zz (cruise)


25/36

Figure 19. Variation of the bending normal stress with spanwise location from the root for given critical condition

5.2 Shear Stress in Beams

5.2.1 Shear Stress arising from Shear Forces Vx, Vy

Procedure and Calculations:

1. The shear flow is first resolved by treating each section as an open section. The increment inshear flow across a boom is calculated using the formula.

2. The equilibrium condition on moments and shear flows is applied.3. The next set of equations is obtained by equating the rates of twist for both the sections.4. Torsional equations are solved and the shear flow from torsion is also added.5.

The shear stress (x,s) is obtained from the shear flow6. x,sis componentized into xzand yzand the maximum value of each of these is plottedagainst the span-wise stations in figures 20, 21.

Formulae:The change in shear flow across a boom is given by

where Aris the area of the r

thboom.

The equilibrium conditions are used as follows

The rate of twist for the Rthsection is given as Where q0,Ris the constant shear flow in the R

thsection, if the section were closed and AR is the swept

area by the line joining a boom to the centroid along the segment joining two consecutive booms.

-300000

-250000

-200000

-150000

-100000

-50000

0

50000

0 10 20 30 40

AxialLoad(kPa)


sigma_zz ( v = 222, n = 4)


26/36

5.2.2 Shear Stress arising from Torsion

Procedure and Calculations:1. qIand qIIare assumed to be the constant shear flow in the Ithand IIthsections respectively.2. Calculate the lengths of each individual segment of each section.3. Obtain the rate of twist for each section using the formula. These will give two sets of

coupled equations.4. Equate the rates of twist. This gives one equation in qIand qII.5. The second equation is obtained by twisting-moment balance.

Formulae:The rate of twist for the R

thsection at a station is given by

The moment balance equation is as follows


Figure 20. Variation of tau_xz with spanwise station

Figure 21. Variation of tau_yz with spanwise station

-0.02

-0.01

0

0.010.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 5 10 15 20 25 30 35

She

arstress(kPa)


tau_xz

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0 5 10 15 20 25 30 35

Shearstress(kPa)


tau_yz


27/36

5.3 Principal Stresses

Procedure and Calculations

The principal stresses are calculated from the Eigen values of the following stress tensor.

Hence we need the solutions to the following equation.

This gives The roots of this equation are given by 3= 0 and The Eigen values are the three principal stresses. These are tabulated in table 14.


Figure 22. Variation of the larger principal stress with spanwise location

Table 14. Non-trivial solutions of the Eigen Value problem, variation with span-wise location

Station 1(x 1E+05 kPa) 2(x 1E+05 kPa)

30 -0.0313554 0.0313554

29 -0.027778866 0.027747603

28 -0.024508662 0.024206075

27 -0.021817311 0.02055296626 -0.020143046 0.016531549

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 5 10 15 20 25 30 35

PricncipalStress(x1E+05kPa)


lambda_1


28/36

25 -0.020265835 0.012000762

24 -0.023712937 0.007377906

23 -0.011590698 0.01037117

22 -0.049538624 0.001764196

21 -0.074618777 0.000819957

20 -0.008656527 0.00426386719 -0.153114841 0.000196682

18 -0.009456617 0.001619444

17 -0.276129348 5.22986E-05

16 -0.013253209 0.000439956

15 -0.452113839 9.81793E-06

14 -0.019763626 9.41048E-05

13 -0.687645377 1.16094E-08

12 -0.02880346 8.94932E-06

11 -0.041438925 1.67707E-05

10 -1.151243946 7.04208E-06

9 -0.043207493 2.13077E-05

8 -0.045983185 5.04088E-05

7 -0.048933896 9.21353E-05

6 -0.052064571 0.000143759

5 -0.055374391 0.000199086

4 -0.058853101 0.000249041

3 -2.618018852 0.000181137

2 -0.076908415 0.000315634

1 -0.081081441 0.000300167

5.4 Factor of Safety

FormulaeThe Von-Mises criterion for yield failure is used. The Von-Mises yield stress is given by-

CalculationFrom table .... the set of largest principal stresses are taken to evaluate the failure stress. Hence we

have 1= -2.618018852 x 105kPa, 2= 0.000181137 x 10

5kPa and 3= 0 kPa.


29/36

Hence factor of safety using the material properties of the alloy is found to be 1.92. The design is failsafe and at first sight may seem like it is over designed. The factor of safety is higher than the generalvalue of 1.5 for most aircraft.

Chapter 6: Aero-elastic Parameters6.1 Shear Centre and Elastic Axis

FormulaThe position of the shear centre is given by

The locus of the shear centre is the elastic axis. The distance between the shear centre and the

aerodynamic centre is given by e = EAAC = exa.c. = ex, since aerodynamic centre is taken as the

origin. The results are plotted in Figure 23.


Figure 23. Position of shear centre along the chordline, variation with spanwise location

If the c.g. is ahead of the e.a. then the aircraft remains free of flutter at all times. If the e.a. lies

between the c.g. and the a.c. then there is a possibility of flutter. This is an interesting case. If we takethe locus of the shear centres of only the stations without ribs, we find that the c.g. is ahead of the

shear centre at all times. This can be observed in Figure 23. At the position where there are ribs, theelastic axis is just ahead of it. This means that there is a possibility of flutter.

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25 30 35

Positionalongthechord

line(m)withAC=(0,0

)

Spanwise stations from the root

ex (m)

cgx(m)


30/36

6.2 Critical Divergence Speed

Procedure1. The elastic axis is obtained by taking the locus of the shear centres at each station.2. The areal moment of inertia at each station can be obtained either from the CAD drawing or

by calculating the angular deflection of each cell at every station and using the formula given.3. The stiffness constant- kis obtained from the formula for each station and the average value

over the span is taken to be the final value used for calculating divergence speed.

Formula Where k= GJ/L and J = Mt/G

Sample Calculation at the TipThis calculation is presented for the tip since the critical divergence speed is lowest at the tip of thewing. This is because the shear centre is closest to the aerodynamic centre at the tip.

From an average of all the local kat each station, the resultant value is k= 2.547 x 107S.I. unit. The

other physical quantities used are tabulated in Table 15.

Table 15. Wing geometric and aerodynamic parameters

Parameter Value

Wing Area for reference (S) 63.94 m2

CL 6.07 per radian

Location of shear centre at tip (e) 0.340 m from the A.C.

Hence we get ConclusionThe critical divergence speed is the maximum at the root section and minimum at the tip. Theminimum value, i.e. the value at the tip is compared to the drag divergence speed and found to besmaller. Hence, the critical divergence speed for flutter lies outside the flight envelope and is neverreached.

Chapter 7: Wing Weight Calculation and Comparison with Empirical

Estimate

The statistical estimate of the wing weight was done in Ref. The wing weight was then evaluated to beWw= 2207.194 kgf.

Assumptions1. The longerons are not tapering all the span of the wing. They are of constant cross section.

This would help stiffening at the tips as well. The skin lengths are negotiatedcorrespondingly.

2. Thickness of the ribs = 3 cm (thicker ribs make them less susceptible to crushing)3. Since idealization of the wing section conserves the area, the mass calculations are done using

the idealized sections.


31/36

CalculationsTable 16. Mass of each longeron

Longeron ID Area (sq. m) Mass (kg)

1 0.0012566 54.18390087

2 0.00166 71.578287

3 0.0012566 54.18390087

4 0.00378 162.991521

5 0.005533333 238.59429

6 0.001809 78.00308505

7 0.001809 78.00308505

8 0.00166 71.578287

9 0.003026333 130.4938289

10 0.003026333 130.4938289

11 0.00166 71.578287

12 0.001809 78.00308505

13 0.001809 78.0030850514 0.00166 71.578287

15 0.001134 48.8974563

16 0.0012566 54.18390087

17 0.00166 71.578287

18 0.0012566 54.18390087

Total Mass of Longitudinal Stringers 1598.110304

Table 17. Mass of ribs

Section Area of rib (sq. m) Mass of rib (kg)

30 0 0

29 0 0

28 0 0

27 0 0

26 0 0

25 0 0

24 0 0

23 0.11103 9.359829

22 0 0

21 0 0

20 0.1302 10.9758619 0 0

18 0.147341 12.4208463

17 0 0

16 0.168552 14.2089336

15 0 0

14 0.194423 16.3898589

13 0 0

12 0.225625 19.0201875

11 0.243458 20.5235094

10 0 0

9 0.284094 23.94912428 0.307117 25.8899631


32/36

7 0.332096 27.9956928

6 0.359155 30.2767665

5 0.388421 32.7438903

4 0.420025 35.4081075

3 0 0

2 0.490803 41.37469291 0.530276 44.7022668

Total mass of the ribs 365.2395288

Conclusion

Total Mass of the Wing = Mass of the longerons (idealized sections) + Mass of ribs

= 1598.110 + 365.239 = 1963.349 kg

Marginal Mass = 2207.1941963.349 = 243.845 kg.

Chapter 8: Buckling and Crushing Loads

The ribs and spars are susceptible to crushing in the direction transverse to the span. The stringers aresusceptible to buckling in the direction along the span. It is important to take these into considerationsince it is possible that a structural member buckles before yielding.

8.1 Crushing Loads on Spars and Ribs

Procedure1. Evaluate the stress topat the upper cap of the spars, for each of the stations and tabulate it.2. The height of the spars is already known at each location. The equivalent thickness of the

spars is calculated.3. The crushing force is calculated and plotted against the span-wise location in Figure 24.4. Since the crushing load varies from front to rear, both the loads are plotted for comparison.

FormulaThese loads are evaluated for wing spars using the following formula

Where teis the equivalent thickness of the top of the spar and h is the depth of the spar.


Table 18. Front and rear spar loads on ribs, variation with spanwise location

Station Front Spar (N) Rear Spar (N)

30 0 0

29 1.77502E-05 1.76884E-08

28 0.001647071 1.23505E-06

27 0.028412723 1.37937E-05

26 0.168481094 4.09885E-05

25 1.034299802 0.000132379

24 3.201444384 5.5303E-05

23 0.03472572 0.005831748


33/36

22 26.12236742 0.005853426

21 94.89577643 0.10952591

20 0.641410004 0.173363078

19 421.7361361 15.49031129

18 3.07591959 1.412741005

17 2190.279777 264.669992516 6.664945531 4.561289729

15 6149.217799 819.8993809

14 15.19136079 7.721833854

13 7675.752264 167.758745

12 19.49580188 6.14615573

11 4.857560685 1.142434779

10 5411.921213 44.72936641

9 16.80921612 4.30730888

8 19.06458413 5.277919972

7 21.78299803 6.594449587

6 26.60798408 9.418839732

5 32.79341467 14.12950108

4 49.74289357 35.30462457

3 37406.39364 3423.632932

2 77.13475563 65.45767228

1 82.45884919 76.16178469

Figure 24. Crushing loads for front and rear spar

The figure 24 shows the variation in Pcrushfor both, the front and the rear spars. It is evident that thecrushing load varies from the front to the rear spar. These variations are without the ribs. The crushingloads for the rib locations are presented in Table for reference. These are not plotted, since the ribs arenot really divided into two spars. Rather, the ribs can be thought of as simply supported between thefront and the rear spar. For this reason, W frontand Wrearare plotted separately in figure 25 for the ribs.The ribs are laterally unsupported and hence their crushing load is low.

-5000

0

5000

10000

15000

20000

25000

30000

35000

40000

0 5 10 15 20 25 30 35

Pcrushforeachspar(new

ton)

Spanwise station from the root

Front Spar (N)

Rear Spar (N)


34/36

Figure 25. Front and rear spar loads on ribs, variation along spanwise location

8.2 Buckling of Columns and Plates

FormulaeThe critical load for buckling of columns(longerons) in the first mode is given by-

The effective length is the one calculated from the case for one end fixed and the other end free forhalf span of the wing.

AssumptionsThe buckling loads have been calculated for the longerons of the idealized section. Hence, all analysisis done using the equations for column buckling.


Table 19. Buckling force and stress on longerons of idealized section

Longeron

ID

Pcr1 (x 1E+05) Pcr2 (x 1E+05) P (x 1E+05) Buckling Stress (GPa)

1 0.104115022 2.651939503 2.756054525 0.219326319

2 0.440921754 1.063134976 1.50405673 0.090605827

3 0.508289063 0.015053055 0.523342118 0.041647471

4 0.811340115 0.573409481 1.384749596 0.036633587

5 2.854480997 6.07046301 8.924944007 0.161294169

6 0.969167473 4.039139326 5.008306799 0.276854992

7 0.940346265 10.64967746 11.59002372 0.640686773

8 0.779696968 13.52103271 14.30072968 0.86148974

9 0.525462439 37.32532636 37.8507888 1.250714466

10 0.012031234 37.32532636 37.33735759 1.233749012

11 0.408698438 13.52103271 13.92973115 0.839140431

12 0.229753963 10.64967746 10.87943142 0.601405827

0

10

20

30

40

50

60

70

80

90

0 5 10 15 20 25

WrearandWfro

nt(newton)


Wfront (N)

Wrear (N)


35/36

13 0.255162848 4.039139326 4.294302174 0.237385416

14 0.227359636 1.821138903 2.048498539 0.123403526

15 0.021305455 0.172022844 0.193328299 0.017048351

16 0.142871307 0.015053055 0.157924362 0.012567592

17 0.134931239 1.063134976 1.198066214 0.072172663

18 0.030682059 2.651939503 2.682621562 0.213482537

From figure 26 it can be observed that at longeron number 11, which endures the maximum axialstress/ normal stress, the bucking stress is approximately 1 GPa, which is almost double the yieldstress of 0.503 GPa. Hence, it can be concluded that the longerons will prefer yielding to buckling.Since it has already been proved that the structure is fail safe, there is no issue.

Figure 26. Plot of buckling stress with the longeron number

Alternative approachThe portion of skin between the stiffenersmay buckle as a plate simply supported on 4 sides.

Where tskis the thickness of the skin and bskis the width of the skin between two stiffeners. Since thisis the critical stress, the width of the least wide panel should be taken for the calculation.

ConclusionsThe structural design features of a medium commuter aircraft of unconventional configuration have

been presented. All aerodynamic loads and moment are calculated for all critical speeds and loadfactors and taken into account for structural safety.

The focus is mainly on wing design. It is a two-spar wing, just like its conventional counterpart, theATR-72. The material used is a very strong and durable alloy of aluminium. This alloys has a highdensity and high yield strength of 0.503 GPa. The cross section of the wing is idealized using 18longerons. Longeron 11 carries the largest stress. Each of the longerons are theoretically tested foryielding and buckling. The wing design is fail safe and has a factor of safety of 1.9, which is abovethe average value of 1.5. This was achieved by proper placement of ribs along the span, so as to

reduce the stresses taken by the longerons. The stringers and skin, hence, do not yield and area also

0

0.2

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0.6

0.8

1

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0 2 4 6 8 10 12 14 16 18 20

Bucklingstress

(GPa)

Longeron Number


36/36

resistant to buckling. Buckling loads are higher than yield stresses. This ensures complete safety ofthe wing structure. Aeroelastic forces are also taken into account and tackled.

The major trade-off of any design calculation is the limitation on weight. In this case, there is amargin of weight between to weight estimated from empirical calculations and the weight calculatedfrom the structural elements. Hence, the wing design is quite competitive.

References1. A. Amendola, G. Iannuzzo, P. Cerreta, R. Pinto, 2011, Future aero-structure for the next

generation green civil aircraft,Aerodays 2011, Alenia Aeronautica,2. Jane's All the World's Aircraft, 1995-96, Page 155-1583. Jane's All the World's Aircraft, 1988-19894. www.airfoiltools.com, NASA/LANGLEY NLF 0414F AIRFOIL (nlf414f-il) Xfoil prediction

polar at RE=1,000,000, last visited at 13thFebruary, 2013

5. Desai A., Karia R., Final Report for Design of Medium Commuter (Unconventional) Aircraft,2013, Department of Aerospace Engineering, IIT Kanpur.

6. Raymer D. P., Aircraft Design: A Conceptual Approach 4th Edition. AIAA, Reston, VA,1999.

anirudh aircraft design ii

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