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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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