aircrafts structures project

8/2/2019 Aircrafts Structures Project

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Aircrafts Structures Project:

Stress Analysis for a C-130 Center Wing Box Frame

MECH 536Aircraft Structures

Professor Pascal Hubert

McGill University

Paul Cebula - #260279934

Mukund Patel - #260279626

April 7, 2011


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Introduction

The Lockheed C-130 Hercules is a four engine turboprop military transport aircraft

designed and built by Lockheed Martin. It was originally designed to transport cargo and

military/medical personnel, however because of its versatile frame and popularity among pilots;

the basic airframe of the C-130A was soon adapted to numerous other tasks including: gunship,

science research support vehicle and even search and rescue operations aircraft. It is the main

tactical airlifter for many military forces, serving over 60 nations worldwide having over 40

different aircraft models (aerospace, 2011, p.1).

On February 14, 2005, the US Air Force grounded nearly 100 C-130E models because of

severe fatigue in the wings and the center wing box structure (Defense Industry Daily, 2007,

p.1). The purpose of this project is to conduct an idealized structure analysis of the center wing

box frame of a C-130 aircraft. The center wing box sits atop the fuselage and forms the

attachment point for both wings and all four engines, as shown in Figure 1. Once the stress

analysis is conducted, stiffeners and skins will then be sized for a safety factor of 1.5.

Figure 1: Picture of the C-130 center wing box frame under examination.


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Loading Case: Steady Flight with Maximum Payload.

The loading case to be examined will be steady cruising flight at 336 mph with a maximum

permissible takeoff weight of 155 000 lb. The specifications of the C-130 aircraft can be

obtained on the US Air Force website.

General Characteristics:

Length: 97 ft 9 in (29.8m)

Height: 38 ft 3 in (11.6 m)

Wingspan: 132 ft 7 in (40.4m)

Wing area: 1 745 ft2

(162.1 m2)

Max Takeoff Weight: 155 000 lb (70 300 kg)

Performance:

4 Allison T56-A-7 engines: 4,200 prop shaft horsepower/ engineCruise Speed: 336 mph (540 km/hr)

Range: 2 360 mi (3 800 km)

When conducting a stress analysis, the first step is to determine all of the external forces and

moments acting on the wing cross section by using a free body diagram. Because information on

military aircraft is difficult to obtain, several assumptions were made in order to simplify the

analysis being done.

Lift is evenly generated along both tail and middle wingspans Weight is uniformly distributed along the cargo containing fuselage section Drag is uniformly distributed along middle wing span section.

Engine Thrust:

Weight:

Assuming maximum permissible takeoff weight:


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Free Body Diagram

Figure 2: Top view of C-130 free body diagram

Figure 3: Front view of C-130 free body diagram


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Figure 4: Side view of C-130 free body diagram

Equilibrium Force Equation:


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Stress Analysis Approach

The next step is to take a cross section of the wing box under examination and calculate the wing

section properties by using equations 1-5. An initial value of 0.1ft2

will be the assumed cross

sectional area of the stringers in question. With this information, the axial force acting along the

stringers can then be calculated by using equation 6, and the shear flow can be found by using

equations 7-10.

Since the aircraft frame is made out of aluminum alloy 2014-T6, the maximum yield strength of

aluminum equals 58.01508 ksi. With a safety factor of 1.5:

If the axial stresses found in the stringers exceed 87 022.62 psi, then a second iteration ofcalculations with different stringer cross sectional areas must be performed. Continuous

iterations with different cross sectional areas will then be performed until the stringer axial

stresses are approximately 87 ksi. Once appropriate cross sectional areas are determined, the

thickness of the skins can be calculated by using equation 9, assuming max =43.511 ksi.An excel spreadsheet can be used to calculate the axial stresses and shear stresses for each

iteration of the calculations.

Section Properties

1. 2. 3. 4. 5. Stress Analysis

6.

[ ]7. [ ] 8. 9. 10. 11.


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Cross Sectional View of Wing Box

Assumptions

Material is linear elastic, same in tension and compression

Isotropic, homogeneous material Coordinate system located at centroid.

Figure 5: Cross section of the C-130 center wing box frame under examination.

Figure 6: Cross section of the C-130 center wing box with labeled areas and shear flows.


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Determine Forces acting at location (0, -7.5ft, 0)

(

)

Determine Moments acting at location (0, -7.5ft, 0)

( )

(

)( )


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Results

Initially, we assumed a cross sectional area value of 0.01ft2 for all of the stringers, however, this

resulted in a safety factor of close to 4. Since we wanted to optimize the design of the aircraft

structure, we performed several other iterations until the safety factor in all stringers were

approximately 1.5. After performing numerous iterations, the optimal cross sectional area for the

stringers are presented in Table 1:

Table 1: Optimal Cross Sectional Stringer Areas

Stringer An (ft^2)

1 0.003

2 0.003

3 0.003

4 0.004

5 0.0056 0.005

7 0.005

8 0.005

9 0.005

10 0.005

11 0.004

12 0.003

13 0.003

14 0.003

15 0.00316 0.003

17 0.003

18 0.003

19 0.004

20 0.005

21 0.005

22 0.005

23 0.005

24 0.004

25 0.003

26 0.003

27 0.003

28 0.003

SUM 0.108


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Finding Centroid

Using the bottom right hand corner of the wing box cross section (point 13 of Figure 5) as

reference, with all cross sectional area values equaling those presented in Table 1, the following

coordinates can be obtained:

Table 2: Stringer positions relative to reference point (point 13)

Stringer X' (ft) Z' (ft) AnZ' AnX'

1 -5.757 2.565 0.007695 -0.01727

2 -5.244 2.622 0.007866 -0.01573

3 -4.731 2.679 0.008037 -0.01419

4 -4.218 2.736 0.010944 -0.01687

5 -3.705 2.736 0.01368 -0.01853

6 -3.021 2.736 0.01368 -0.01511

7 -2.622 2.736 0.01368 -0.01311

8 -2.109 2.679 0.013395 -0.010559 -1.596 2.622 0.01311 -0.00798

10 -1.083 2.565 0.012825 -0.00542

11 -0.57 2.508 0.010032 -0.00228

12 0 2.28 0.00684 0

13 0 0 0 0

14 -0.456 -0.057 -0.00017 -0.00137

15 -0.912 -0.057 -0.00017 -0.00274

16 -1.368 -0.114 -0.00034 -0.0041

17 -1.824 -0.171 -0.00051 -0.00547

18 -2.28 -0.171 -0.00051 -0.00684

19 -2.736 -0.171 -0.00068 -0.01094

20 -3.192 -0.114 -0.00057 -0.01596

21 -3.648 -0.057 -0.00029 -0.01824

22 -4.104 0 0 -0.02052

23 -4.56 0.114 0.00057 -0.0228

24 -5.016 0.171 0.000684 -0.02006

25 -5.472 0.285 0.000855 -0.01642

26 -5.928 0.285 0.000855 -0.01778

27 -6.384 0.342 0.001026 -0.01915

28 -6.384 2.508 0.007524 -0.01915

SUM -88.92 34.257 0.140049 -0.33858


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Using these coordinates, the centroid can be determined using equations 1-2:

Similarly, the moment of inertias can be determined using equations 3-5. The table with moment

of inertia calculations can be found in appendix A:

With the section properties in hand, equations 6-7 yield:

The triangular area between each stringer can be calculated using Herons formula:

And from this:

The values calculated from the shear flow analysis can be viewed in Table 3.


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Table 3: Shear flow analysis

Stringer An,n+1 q (lb/ft) qn+1 (lb/ft) An,n+1*qn+1

1 0.400033 -1306.84 -1306.84 1707821

2 0.400033 -1322.47 -2629.31 3477193

3 0.400033 -1338.11 -3967.42 53088514 0.369168 -1805 -5772.42 10419208

5 0.492224 -2197.84 -7970.26 17517332

6 0.28713 -2119.96 -10090.2 21390824

7 0.383788 -2074.53 -12164.7 25236087

8 0.383788 -1931.64 -14096.4 27229201

9 0.383788 -1788.76 -15885.1 28414733

10 0.383788 -1645.88 -17531 28853929

11 0.637616 -1202.4 -18733.4 22524991

12 3.5739 -660.122 -19393.5 12802108

13 0.385006 1367.224 -18026.3 -2.5E+07

14 0.308655 1386.756 -16639.6 -2.3E+07

15 0.372011 1355.604 -15284 -2.1E+07

16 0.372011 1375.135 -13908.8 -1.9E+07

17 0.334647 1394.667 -12514.2 -1.7E+07

18 0.334647 1363.515 -11150.6 -1.5E+07

19 0.323276 1776.483 -9374.16 -1.7E+07

20 0.323276 2084.211 -7289.95 -1.5E+07

21 0.323276 1947.818 -5342.13 -1E+07

22 0.350892 1811.425 -3530.71 -6395609

23 0.31028 1590.559 -1940.15 -3085919

24 0.363888 1163.333 -776.814 -903693

25 0.230679 739.9803 -36.8337 -27256.2

26 0.310279 708.8281 671.9944 476328.6

27 3.518667 626.9923 1298.987 814454.7

28 0.472323 -1298.99 -2.8E-11 3.69E-08

SUM 16.7291 33284578

A summary of axial stress along the stringers, skin thickness and shear flow along the skins can

be viewed in Table 4.


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Table 4: Shear Flow and Thickness of Aircraft Skins and Axial Stress in Stringers.

Stringer qs (lb/ft) yy (psi) Thickness (ft)

1 7292.32 -67707.35 0.1894

2 7276.69 -72657.61 0.1889

3 7261.05 -77607.86 0.1885

4 6794.16 -82558.12 0.1764

5 6401.32 -84109.07 0.1662

6 6479.20 -86177.01 0.1682

7 6524.63 -87383.31 0.1694

8 6667.52 -85534.96 0.1731

9 6810.40 -83686.61 0.1768

10 6953.28 -81838.27 0.1806

11 7396.76 -79989.92 0.1921

12 7939.04 -68116.00 0.2061

13 9966.38 67856.05 0.2588

14 9985.92 72633.98 0.2593

15 9954.76 74012.60 0.2585

16 9974.30 78790.53 0.2590

17 9993.83 83568.46 0.2595

18 9962.68 84947.08 0.2587

19 10375.64 86325.71 0.2694

20 10683.37 84305.03 0.2774

21 10546.98 82284.36 0.2739

22 10410.59 80263.68 0.2703

23 10189.72 74843.71 0.2646

24 9762.49 72823.03 0.2535

25 9339.14 67403.05 0.2425

26 9307.99 68781.68 0.2417

27 9226.15 66761.00 0.2396

28 7300.17 -62412.44 0.1896


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Conclusion Size of Stiffeners and Skins

After conducting our analysis of the C-130 wingbox frame, we found that the stringers furthest

from the centroid had the largest axial stresses and those closest, had the least. Consequently, we

adjusted the values of the stringer areas accordingly such that those furthest away had a value of

0.005 ft2 and those closest had areas of 0.003 ft2. In terms of skin thickness, we found the

optimal thickness between each stringer such that a safety factor of 1.5 is preserved. The results

from our analysis are summarized in Table 5.

Table 5: Size of Stiffeners and Skins for C-130 wingbox frame.

Stringer An (ft2) Thickness (ft)

1 0.003 0.1894

2 0.003 0.1889

3 0.003 0.1885

4 0.004 0.17645 0.005 0.1662

6 0.005 0.1682

7 0.005 0.1694

8 0.005 0.1731

9 0.005 0.1768

10 0.005 0.1806

11 0.004 0.1921

12 0.003 0.2061

13 0.003 0.2588

14 0.003 0.259315 0.003 0.2585

16 0.003 0.2590

17 0.003 0.2595

18 0.003 0.2587

19 0.004 0.2694

20 0.005 0.2774

21 0.005 0.2739

22 0.005 0.2703

23 0.005 0.2646

24 0.004 0.2535

25 0.003 0.2425

26 0.003 0.2417

27 0.003 0.2396

28 0.003 0.1896


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Bibliography

Aerospaceweb.com. (2011, March). Lockheed C-130 Hercules Heavy Transport.

http://www.aerospaceweb.org/aircraft/transport-m/c130/

Defense Industry Daily. (2007, April). Keeping the C-130s Flying: Center Wing BoxReplacements. http://www.defenseindustrydaily.com/keeping-the-c130s-flying-center-

wing-box-replacements-03185/

U.S Air Force. (2009, October). C-130 Hercules Factsheet. http://www.af.mil/information

/factsheets/factsheet.asp?id=92


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Appendix A

Table A1: Stringer positions relative to Reference (point 13)

Stringer X' (ft) Z' (ft)

1 -5.757 2.5652 -5.244 2.622

3 -4.731 2.679

4 -4.218 2.736

5 -3.705 2.736

6 -3.021 2.736

7 -2.622 2.736

8 -2.109 2.679

9 -1.596 2.622

10 -1.083 2.565

11 -0.57 2.508

12 0 2.28

13 0 0

14 -0.456 -0.057

15 -0.912 -0.057

16 -1.368 -0.114

17 -1.824 -0.171

18 -2.28 -0.171

19 -2.736 -0.171

20 -3.192 -0.114

21 -3.648 -0.057

22 -4.104 0

23 -4.56 0.114

24 -5.016 0.171

25 -5.472 0.285

26 -5.928 0.285

27 -6.384 0.342

28 -6.384 2.508

SUM -88.92 34.257


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Table A2: Determining Moment of Inertia for wing cross section

Stringer Zn2 (ft^2) Xn2 (ft^2) AnZn2 (ft^4) AnXn2(ft^4) AnXnZn(ft^4)

1 1.608458 6.874884 0.004825374 0.02062465

-

0.009976055

2 1.756288 4.447881 0.005268863 0.01334364-

0.008384857

3 1.910615 2.547216 0.005731845 0.00764165

-

0.006618213

4 2.071441 1.172889 0.008285762 0.00469156

-

0.006234831

5 2.071441 0.3249 0.010357203 0.0016245

-

0.004101863

6 2.071441 0.012996 0.010357203 6.498E-05 0.000820372

7 2.071441 0.263169 0.010357203 0.00131584 0.003691676

8 1.910615 1.052676 0.009553075 0.00526338 0.007090942

9 1.756288 2.368521 0.008781438 0.01184261 0.010197799

10 1.608458 4.210704 0.00804229 0.02105352 0.013012245

11 1.467127 6.579225 0.005868506 0.0263169 0.012427425

12 0.966781 9.828225 0.002900342 0.02948468 0.009247466

13 1.681561 9.828225 0.005044682 0.02948468

-

0.012195934

14 1.832639 7.177041 0.005497917 0.02153112

-

0.010880089

15 1.832639 4.941729 0.005497917 0.01482519

-

0.009028159

16 1.990216 3.122289 0.005970647 0.00936687

-

0.007478386

17 2.15429 1.718721 0.00646287 0.00515616

-

0.005772661

18 2.15429 0.731025 0.00646287 0.00219307

-

0.003764779

19 2.15429 0.159201 0.00861716 0.0006368

-

0.002342529

20 1.990216 0.003249 0.009951078 1.6245E-05 0.000402064

21 1.832639 0.263169 0.009163195 0.00131585 0.003472369

22 1.681561 0.938961 0.008407803 0.00469481 0.006282754

23 1.398898 2.030625 0.006994488 0.01015313 0.00842709424 1.267313 3.538161 0.005069252 0.01415264 0.008470143

25 1.023638 5.461569 0.003070914 0.01638471 0.007093379

26 1.023638 7.800849 0.003070914 0.02340255 0.008477453

27 0.911548 10.556 0.002734643 0.031668 0.009305948

28 1.467127 10.556 0.00440138 0.031668

-

0.011806054

SUM 0.182345455 0.32824972 0.021640777


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