aircraft structures - complementary energy analysis

8/7/2019 Aircraft Structures - Complementary Energy Analysis

http://slidepdf.com/reader/full/aircraft-structures-complementary-energy-analysis 1/28

1

Abstract

The problem required the calculation of the internal loads and moments in a fuselage frame with an

elliptical cross sectional shape and uniform depth. Complementary energy methods were used in

order to solve the problem, however the mathematics became tedious when considering an

analytical solution and as such much of the problem was solved numerically. This was simplified with

the use of Microsoft Excel, where certain integration problems were solved using the Trapezoidal

Rule over small increments (where the method is considered suitably accurate). The resultant

internal moment at the top of the frame was calculated as 60275 Nmm, the resultant internal axial

force at the same location was calculated as -160 N, and the rotation angle due to the applied

moment was calculated as 0.196 degrees. Finally the maximum stress at the top of the frame was

calculated as 8.559MPa.



2

Declaration



3

Contents

Abstract ............................................................................................................................................ 1

Declaration ........................................................................................................................................ 2

Nomenclature ................................................................................................................................... 4

1. Geometric Analysis .................................................................................................................... 5

2. Internal Loads ............................................................................................................................ 6

3. Complementary Energy Technique ............................................................................................ 7

4. Rotation Angle ......................................................................................................................... 10

5. Bending Stress ......................................................................................................................... 11

6. Final Results............................................................................................................................. 12

7. References ............................................................................................................................... 13

8. Appendix ................................................................................................................................. 14



5

1. Geometric Analysis

The following figure shows the right-hand half of the structure under analysis. Symmetry is exploited

in this case because the angular displacement in the vertical plane at the top and bottom of the

frame is equal to zero. Notably the effects of shear are ignored as the effects of bending momentand internal axial load are far greater in magnitude.

Figure 1: Analysed Section of the fuselage frame

Analysing a cross section with two elliptical defining perimeters required an expression relating the

radius of the ellipse with angle of rotation around the ellipse. For the geometry in the problem this

expression could be represented as:

() = ( sin ) + ( cos)

Hence for the thickness of the frame at any point, the difference between the outer and inner radii:

() = ( sin ) + ( cos) −

( sin ) + ( cos)

This calculation was required to calculate the moments of area for each infinitesimal sector of the

frame, required for determining the internal load distribution.



6

2. Internal Loads

Due to the structure being opened along the vertical axis and the wing moments situated along the

horizontal axis the loading was different for the top and bottom halves of the analysed section. Thus

for:

0 < ≤ 2 :

= + (1− cos)

Hence,

= 1

= (1−)

2 < < :

= + (1−) +

Hence,

= 1

= (1−)



7

3. Complementary Energy Technique

The principle of complementary energy states if: [1]

=

Then

= = 0

By definition,

=

=

And,

1 =

=

And in this case,

=

So,

= .1 = 0

Considering the internal moment:

= ( + (1−)) I + () I /

= 0

Separating into individual integrals and extracting terms not dependent on α,

0 = 1

I + 1

I

− I

+ 1

I /



8

From the above equation, the integrals can be represented by variables and a linear equation

established. Hence,

=

1

I

= I

= 1

I /

0 = − + +

Now considering the axial load and the complementary energy theorem,

0 = I (1 − cos)

= ((1−) + (1 −)) I + ( −) I /

= 0

Extracting non-dependent variables and separating integrals,

0 = 1

I − I

+ (1− )

I

+ 1

I /

− I

/

The new integrals can also be represented by new variables,

= (1− )I =

I /

Hence the linear equation becomes,

0 = − + + −



9

Without going into tedious arithmetic, the moment and axial load can be rearranged as the subjects

of the formulae to make numerical calculations easier:

=(−) −

= () − +

()() + Trapezoidal Rule used for evaluating integrals: [2]

= 1

2( − )( () + ())

Also for evaluating the infinitesimal moments of area, consider the moment of area of a rectangle

[3],

Figure 2: Cross section view of the frame, required to calculate the second moment of area

=1

12ℎ

In the case of the fuselage frame, the value for b is a constant (given as 10mm) and the value for h is

actually the local value for the frame thickness (difference of outer and inner radii). The value for

frame thickness was calculated for each degree of rotation and thus for each degree of rotation the

value for moment of area could be calculated. This local value for moment of area was required for

the numerical calculation of the integrals P, Q, R, S and T.



10

4. Rotation Angle

To find the angle of rotation consider the complementary energy theorem: [1]

=

− ∅= 0

Hence,

= ∅

Substituting for geometric properties as before yields:

∅=

For 0 < ≤ 2 :

= 0

For 2 < < :

= 1

Hence for the first segment all the terms fall away, leaving

∅ = + (1 −) +

.1

Which expands to

∅ = 1

I + 1

I ⁄ –

I ⁄

⁄ + 1

I

And in terms of the defined numerical integrals

∅ = + – +

Which is solved numerically in Microsoft Excel as Φ = 0.1959°.



11

5. Bending Stress

Considering the flexure equation, [3]

=

Where y is half the local frame thickness (if it is assumed that the neutral axis lies along the mean

radius line),

=1

2()

And

= 112.()

Substituting,

=3

5(())

With the aid of Microsoft Excel the moment was found for each radial node (iterated every degree

of rotation), and the above formula used to calculate the Bending stress (dominant stress) at every

node. The stress at the top of the frame was found as:

= .



12

6. Final Results

Internal moment at A 60275 Nmm

Internal axial load at A - 160 N

Structure rotation angle due to applied load 0.1959°

Maximum stress 8.559MPa



13

7. References

1. T.H.G. Megson, “Aircraft Structures for Engineering Students”, Third Edition, Butterworth-

Heinemann, 1999, pages 94, 95.

2.

T. Oliphant, “APPMCO3 Course and Lecture Notes”, School of Computational and AppliedMathematics, University of the Witwatersrand, 2010, Chapter 8 – Trapezoidal Rule.

3. Dr. F. Kienhofer, “MECN 3010 Course and Lecture Notes”, School of Mechanical Engineering,

University of the Witwatersrand, 2010.



14

8. Appendix

Microsoft Excel program

Constant inputs:

Mean radius r [mm] 3500

Structure thickness d 10

Frame thickness side t1 130

Frame thickness top t2 65

Applied moment Mapp[Nmm] 1000000

Elastic Modulus E [Mpa] 110000

Outer ellipse x axis ao 3565

Outer ellipse y axis bo 3532.5

Inner ellipse x axis ai 3435

Inner ellipse y axis bi 3467.5

Calculated data:

α[radians]

Radius

outer

Radius

inner h [mm] Ixx[mm^4] 1/Ixx Cos(α) (1-Cos(α))^2

0 3532.5 3467.5 65 228854.17

4.4E-

06 1 0

0.0174533 3532.50976 3467.48996 65.0198 229063.41

4.4E-

06 0.9998 2.31968E-08

0.0349066 3532.53905 3467.45985 65.0792 229691.65

4.4E-

06 0.9994 3.71092E-07

0.0523599 3532.58781 3467.40972 65.1781 230740.42

4.3E-

06 0.9986 1.87817E-06

0.0698132 3532.656 3467.33962 65.3164 232212.25

4.3E-

06 0.9976 5.93385E-06

0.0872665 3532.74353 3467.24964 65.4939 234110.67

4.3E-

06 0.9962 1.44803E-05

0.1047198 3532.85031 3467.1399 65.7104 236440.22

4.2E-

06 0.9945 3.00096E-05

0.122173 3532.97621 3467.01054 65.9657 239206.374.2E-

06 0.9925 5.55599E-05

0.1396263 3533.12108 3466.86172 66.2594 242415.57

4.1E-

06 0.9903 9.47105E-05

0.1570796 3533.28475 3466.69363 66.5911 246075.16

4.1E-

06 0.9877 0.000151577

0.1745329 3533.46703 3466.50648 66.9606 250193.38 4E-06 0.9848 0.000230804

0.1919862 3533.66771 3466.30051 67.3672 254779.31

3.9E-

06 0.9816 0.00033756

0.2094395 3533.88655 3466.07599 67.8106 259842.83

3.8E-

06 0.9781 0.000477527

0.2268928 3534.12329 3465.83319 68.2901 265394.583.8E-

06 0.9744 0.000656894



15

0.2443461 3534.37767 3465.57243 68.8052 271445.9

3.7E-

06 0.9703 0.000882344

0.2617994 3534.64937 3465.29402 69.3553 278008.78

3.6E-

06 0.9659 0.001161049

0.2792527 3534.93808 3464.99834 69.9397 285095.78

3.5E-

06 0.9613 0.001500656

0.296706 3535.24346 3464.68574 70.5577 292719.97

3.4E-

06 0.9563 0.001909274

0.3141593 3535.56515 3464.35662 71.2085 300894.85

3.3E-

06 0.9511 0.002395465

0.3316126 3535.90277 3464.0114 71.8914 309634.27

3.2E-

06 0.9455 0.002968226

0.3490659 3536.25593 3463.65052 72.6054 318952.31

3.1E-

06 0.9397 0.00363698

0.3665191 3536.6242 3463.27442 73.3498 328863.25 3E-06 0.9336 0.00441156

0.3839724 3537.00716 3462.88358 74.1236 339381.39

2.9E-

06 0.9272 0.005302191

0.4014257 3537.40435 3462.47849 74.9259 350521

2.9E-

06 0.9205 0.006319478

0.418879 3537.8153 3462.05966 75.7556 362296.17

2.8E-

06 0.9135 0.007474388

0.4363323 3538.23952 3461.62761 76.6119 374720.71

2.7E-

06 0.9063 0.008778231

0.4537856 3538.67652 3461.18289 77.4936 387808.01

2.6E-

06 0.8988 0.010242645

0.4712389 3539.12576 3460.72605 78.3997 401570.91

2.5E-

06 0.891 0.011879578

0.4886922 3539.58673 3460.25766 79.3291 416021.592.4E-

06 0.8829 0.013701266

0.5061455 3540.05886 3459.7783 80.2806 431171.38

2.3E-

06 0.8746 0.015720218

0.5235988 3540.54161 3459.28859 81.253 447030.67

2.2E-

06 0.866 0.017949192

0.5410521 3541.03438 3458.78911 82.2453 463608.73

2.2E-

06 0.8572 0.02040118

0.5585054 3541.5366 3458.2805 83.2561 480913.57

2.1E-

06 0.848 0.023089381

0.5759587 3542.04767 3457.76338 84.2843 498951.79 2E-06 0.8387 0.026027186

0.5934119 3542.56697 3457.2384 85.3286 517728.431.9E-

06 0.829 0.029228152

0.6108652 3543.09387 3456.70621 86.3877 537246.84

1.9E-

06 0.8192 0.032705983

0.6283185 3543.62776 3456.16747 87.4603 557508.49

1.8E-

06 0.809 0.036474508

0.6457718 3544.16798 3455.62283 88.5451 578512.87

1.7E-

06 0.7986 0.040547658

0.6632251 3544.71388 3455.07298 89.6409 600257.32

1.7E-

06 0.788 0.044939441

0.6806784 3545.26482 3454.51858 90.7462 622736.91

1.6E-

06 0.7771 0.0496639220.6981317 3545.82011 3453.96032 91.8598 645944.29 1.5E- 0.766 0.054735203



16

06

0.715585 3546.3791 3453.3989 92.9802 669869.58

1.5E-

06 0.7547 0.06016739

0.7330383 3546.94111 3452.83498 94.1061 694500.28

1.4E-

06 0.7431 0.065974581

0.7504916 3547.50545 3452.26927 95.2362 719821.11.4E-

06 0.7314 0.072170834

0.7679449 3548.07144 3451.70246 96.369 745813.94

1.3E-

06 0.7193 0.078770148

0.7853982 3548.63839 3451.13523 97.5032 772457.75

1.3E-

06 0.7071 0.085786438

0.8028515 3549.20561 3450.56829 98.6373 799728.49

1.3E-

06 0.6947 0.093233511

0.8203047 3549.77241 3450.00231 99.7701 827599.07

1.2E-

06 0.682 0.101125043

0.837758 3550.33811 3449.43799 100.9 856039.3

1.2E-

06 0.6691 0.109474556

0.8552113 3550.90199 3448.87602 102.026 885015.88

1.1E-

06 0.6561 0.118295392

0.8726646 3551.46339 3448.31706 103.146 914492.35

1.1E-

06 0.6428 0.127600692

0.8901179 3552.02161 3447.76181 104.26 944429.17

1.1E-

06 0.6293 0.137403372

0.9075712 3552.57596 3447.21092 105.365 974783.69 1E-06 0.6157 0.147716102

0.9250245 3553.12576 3446.66507 106.461 1005510.2

9.9E-

07 0.6018 0.158551276

0.9424778 3553.67033 3446.12492 107.545 1036560

9.6E-

07 0.5878 0.169920998

0.9599311 3554.20901 3445.5911 108.618 1067881.5

9.4E-

07 0.5736 0.181837056

0.9773844 3554.74113 3445.06427 109.677 1099420.4

9.1E-

07 0.5592 0.194310896

0.9948377 3555.26603 3444.54505 110.721 1131119.5

8.8E-

07 0.5446 0.207353608

1.012291 3555.78306 3444.03407 111.749 1162919.3

8.6E-

07 0.5299 0.220975898

1.0297443 3556.29159 3443.53194 112.76 1194757.7

8.4E-

07 0.515 0.235188069

1.0471976 3556.79097 3443.03926 113.752 1226570.58.2E-

07 0.5 0.25

1.0646508 3557.28059 3442.55662 114.724 1258291.7

7.9E-

07 0.4848 0.265421127

1.0821041 3557.75984 3442.08458 115.675 1289853.1

7.8E-

07 0.4695 0.281460423

1.0995574 3558.22813 3441.62372 116.604 1321185.2

7.6E-

07 0.454 0.298126374

1.1170107 3558.68486 3441.17458 117.51 1352217

7.4E-

07 0.4384 0.315426969

1.134464 3559.12948 3440.73769 118.392 1382876.8

7.2E-

07 0.4226 0.3333696721.1519173 3559.56142 3440.31358 119.248 1413091.6 7.1E- 0.4067 0.351961411



17

07

1.1693706 3559.98014 3439.90274 120.077 1442788.2

6.9E-

07 0.3907 0.371208558

1.1868239 3560.38512 3439.50566 120.879 1471893

6.8E-

07 0.3746 0.391116913

1.2042772 3560.77584 3439.12281 121.653 1500332.86.7E-

07 0.3584 0.411691688

1.2217305 3561.15183 3438.75464 122.397 1528034.4

6.5E-

07 0.342 0.432937492

1.2391838 3561.51261 3438.40159 123.111 1554925.4

6.4E-

07 0.3256 0.454858314

1.2566371 3561.85772 3438.06407 123.794 1580934.5

6.3E-

07 0.309 0.477457514

1.2740904 3562.18674 3437.74248 124.444 1605991.7

6.2E-

07 0.2924 0.500737804

1.2915436 3562.49923 3437.4372 125.062 1630028.5

6.1E-

07 0.2756 0.524701241.3089969 3562.79482 3437.14859 125.646 1652978.4 6E-07 0.2588 0.549349208

1.3264502 3563.07313 3436.87698 126.196 1674777.4 6E-07 0.2419 0.574682412

1.3439035 3563.33381 3436.6227 126.711 1695363.5

5.9E-

07 0.225 0.600700868

1.3613568 3563.57652 3436.38604 127.19 1714678.2

5.8E-

07 0.2079 0.62740389

1.3788101 3563.80096 3436.16727 127.634 1732665.5

5.8E-

07 0.1908 0.654790082

1.3962634 3564.00685 3435.96667 128.04 1749273.2

5.7E-

07 0.1736 0.682857334

1.4137167 3564.19393 3435.78445 128.409 1764452.65.7E-

07 0.1564 0.711602812

1.43117 3564.36196 3435.62084 128.741 1778158.9

5.6E-

07 0.1392 0.74102295

1.4486233 3564.51072 3435.47603 129.035 1790351.3

5.6E-

07 0.1219 0.77111345

1.4660766 3564.64004 3435.35018 129.29 1800993.5

5.6E-

07 0.1045 0.801869273

1.4835299 3564.74973 3435.24344 129.506 1810053.4

5.5E-

07 0.0872 0.833284638

1.5009832 3564.83968 3435.15594 129.684 1817503.8

5.5E-

07 0.0698 0.865353018

1.5184364 3564.90975 3435.08777 129.822 1823322.2

5.5E-

07 0.0523 0.89806714

1.5358897 3564.95987 3435.03903 129.921 1827490.8

5.5E-

07 0.0349 0.931418981

1.553343 3564.98996 3435.00976 129.98 1829997.1

5.5E-

07 0.0175 0.965399774

1.5707963 3565 3435 130 1830833.3

5.5E-

07 -2E-15 1

1.5882496 3564.98996 3435.00976 129.98 1829997.1

5.5E-

07 -0.017 1.035209399

1.6057029 3564.95987 3435.03903 129.921 1827490.85.5E-

07 -0.035 1.071016968



18

1.6231562 3564.90975 3435.08777 129.822 1823322.2

5.5E-

07 -0.052 1.107410965

1.6406095 3564.83968 3435.15594 129.684 1817503.8

5.5E-

07 -0.07 1.144378913

1.6580628 3564.74973 3435.24344 129.506 1810053.4

5.5E-

07 -0.087 1.181907609

1.6755161 3564.64004 3435.35018 129.29 1800993.5

5.6E-

07 -0.105 1.219983126

1.6929694 3564.51072 3435.47603 129.035 1790351.3

5.6E-

07 -0.122 1.258590824

1.7104227 3564.36196 3435.62084 128.741 1778158.9

5.6E-

07 -0.139 1.297715354

1.727876 3564.19393 3435.78445 128.409 1764452.6

5.7E-

07 -0.156 1.337340672

1.7453293 3564.00685 3435.96667 128.04 1749273.2

5.7E-

07 -0.174 1.377450045

1.7627825 3563.80096 3436.16727 127.634 1732665.55.8E-

07 -0.191 1.418026063

1.7802358 3563.57652 3436.38604 127.19 1714678.2

5.8E-

07 -0.208 1.459050653

1.7976891 3563.33381 3436.6227 126.711 1695363.5

5.9E-

07 -0.225 1.500505086

1.8151424 3563.07313 3436.87698 126.196 1674777.4 6E-07 -0.242 1.542369995

1.8325957 3562.79482 3437.14859 125.646 1652978.4 6E-07 -0.259 1.584625388

1.850049 3562.49923 3437.4372 125.062 1630028.5

6.1E-

07 -0.276 1.627250664

1.8675023 3562.18674 3437.74248 124.444 1605991.7

6.2E-

07 -0.292 1.670224623

1.8849556 3561.85772 3438.06407 123.794 1580934.5

6.3E-

07 -0.309 1.713525492

1.9024089 3561.51261 3438.40159 123.111 1554925.4

6.4E-

07 -0.326 1.757130932

1.9198622 3561.15183 3438.75464 122.397 1528034.4

6.5E-

07 -0.342 1.801018065

1.9373155 3560.77584 3439.12281 121.653 1500332.8

6.7E-

07 -0.358 1.845163486

1.9547688 3560.38512 3439.50566 120.879 1471893

6.8E-

07 -0.375 1.889543287

1.9722221 3559.98014 3439.90274 120.077 1442788.26.9E-

07 -0.391 1.934133072

1.9896753 3559.56142 3440.31358 119.248 1413091.6

7.1E-

07 -0.407 1.978907983

2.0071286 3559.12948 3440.73769 118.392 1382876.8

7.2E-

07 -0.423 2.023842719

2.0245819 3558.68486 3441.17458 117.51 1352217

7.4E-

07 -0.438 2.068911556

2.0420352 3558.22813 3441.62372 116.604 1321185.2

7.6E-

07 -0.454 2.114088373

2.0594885 3557.75984 3442.08458 115.675 1289853.1

7.8E-

07 -0.469 2.1593466742.0769418 3557.28059 3442.55662 114.724 1258291.7 7.9E- -0.485 2.204659608



19

07

2.0943951 3556.79097 3443.03926 113.752 1226570.5

8.2E-

07 -0.5 2.25

2.1118484 3556.29159 3443.53194 112.76 1194757.7

8.4E-

07 -0.515 2.295340368

2.1293017 3555.78306 3444.03407 111.749 1162919.38.6E-

07 -0.53 2.340652955

2.146755 3555.26603 3444.54505 110.721 1131119.5

8.8E-

07 -0.545 2.385909748

2.1642083 3554.74113 3445.06427 109.677 1099420.4

9.1E-

07 -0.559 2.43108251

2.1816616 3554.20901 3445.5911 108.618 1067881.5

9.4E-

07 -0.574 2.476142801

2.1991149 3553.67033 3446.12492 107.545 1036560

9.6E-

07 -0.588 2.521062007

2.2165682 3553.12576 3446.66507 106.461 1005510.2

9.9E-

07 -0.602 2.5658113682.2340214 3552.57596 3447.21092 105.365 974783.69 1E-06 -0.616 2.610362003

2.2514747 3552.02161 3447.76181 104.26 944429.17

1.1E-

06 -0.629 2.654684937

2.268928 3551.46339 3448.31706 103.146 914492.35

1.1E-

06 -0.643 2.698751131

2.2863813 3550.90199 3448.87602 102.026 885015.88

1.1E-

06 -0.656 2.742531508

2.3038346 3550.33811 3449.43799 100.9 856039.3

1.2E-

06 -0.669 2.785996981

2.3212879 3549.77241 3450.00231 99.7701 827599.07

1.2E-

06 -0.682 2.829118483

2.3387412 3549.20561 3450.56829 98.6373 799728.49

1.3E-

06 -0.695 2.871866993

2.3561945 3548.63839 3451.13523 97.5032 772457.75

1.3E-

06 -0.707 2.914213562

2.3736478 3548.07144 3451.70246 96.369 745813.94

1.3E-

06 -0.719 2.956129349

2.3911011 3547.50545 3452.26927 95.2362 719821.1

1.4E-

06 -0.731 2.99758564

2.4085544 3546.94111 3452.83498 94.1061 694500.28

1.4E-

06 -0.743 3.038553883

2.4260077 3546.3791 3453.3989 92.9802 669869.581.5E-

06 -0.755 3.079005711

2.443461 3545.82011 3453.96032 91.8598 645944.29

1.5E-

06 -0.766 3.118912975

2.4609142 3545.26482 3454.51858 90.7462 622736.91

1.6E-

06 -0.777 3.158247768

2.4783675 3544.71388 3455.07298 89.6409 600257.32

1.7E-

06 -0.788 3.196982455

2.4958208 3544.16798 3455.62283 88.5451 578512.87

1.7E-

06 -0.799 3.235089698

2.5132741 3543.62776 3456.16747 87.4603 557508.49

1.8E-

06 -0.809 3.2725424862.5307274 3543.09387 3456.70621 86.3877 537246.84 1.9E- -0.819 3.30931416



20

06

2.5481807 3542.56697 3457.2384 85.3286 517728.43

1.9E-

06 -0.829 3.345378442

2.565634 3542.04767 3457.76338 84.2843 498951.79 2E-06 -0.839 3.380709457

2.5830873 3541.5366 3458.2805 83.2561 480913.57

2.1E-

06 -0.848 3.415281766

2.6005406 3541.03438 3458.78911 82.2453 463608.73

2.2E-

06 -0.857 3.449070383

2.6179939 3540.54161 3459.28859 81.253 447030.67

2.2E-

06 -0.866 3.482050808

2.6354472 3540.05886 3459.7783 80.2806 431171.38

2.3E-

06 -0.875 3.514199046

2.6529005 3539.58673 3460.25766 79.3291 416021.59

2.4E-

06 -0.883 3.545491637

2.6703538 3539.12576 3460.72605 78.3997 401570.91

2.5E-

06 -0.891 3.575905675

2.687807 3538.67652 3461.18289 77.4936 387808.012.6E-

06 -0.899 3.60541883

2.7052603 3538.23952 3461.62761 76.6119 374720.71

2.7E-

06 -0.906 3.634009379

2.7227136 3537.8153 3462.05966 75.7556 362296.17

2.8E-

06 -0.914 3.661656218

2.7401669 3537.40435 3462.47849 74.9259 350521

2.9E-

06 -0.921 3.688338892

2.7576202 3537.00716 3462.88358 74.1236 339381.39

2.9E-

06 -0.927 3.714037609

2.7750735 3536.6242 3463.27442 73.3498 328863.25 3E-06 -0.934 3.738733266

2.7925268 3536.25593 3463.65052 72.6054 318952.313.1E-

06 -0.94 3.762407463

2.8099801 3535.90277 3464.0114 71.8914 309634.27

3.2E-

06 -0.946 3.785042528

2.8274334 3535.56515 3464.35662 71.2085 300894.85

3.3E-

06 -0.951 3.80662153

2.8448867 3535.24346 3464.68574 70.5577 292719.97

3.4E-

06 -0.956 3.827128298

2.86234 3534.93808 3464.99834 69.9397 285095.78

3.5E-

06 -0.961 3.84654744

2.8797933 3534.64937 3465.29402 69.3553 278008.78

3.6E-

06 -0.966 3.864864354

2.8972466 3534.37767 3465.57243 68.8052 271445.9

3.7E-

06 -0.97 3.882065249

2.9146999 3534.12329 3465.83319 68.2901 265394.58

3.8E-

06 -0.974 3.898137153

2.9321531 3533.88655 3466.07599 67.8106 259842.83

3.8E-

06 -0.978 3.91306793

2.9496064 3533.66771 3466.30051 67.3672 254779.31

3.9E-

06 -0.982 3.926846294

2.9670597 3533.46703 3466.50648 66.9606 250193.38 4E-06 -0.985 3.939461816

2.984513 3533.28475 3466.69363 66.5911 246075.16

4.1E-

06 -0.988 3.9509049393.0019663 3533.12108 3466.86172 66.2594 242415.57 4.1E- -0.99 3.961166985



21

06

3.0194196 3532.97621 3467.01054 65.9657 239206.37

4.2E-

06 -0.993 3.970240166

3.0368729 3532.85031 3467.1399 65.7104 236440.22

4.2E-

06 -0.995 3.978117591

3.0543262 3532.74353 3467.24964 65.4939 234110.674.3E-

06 -0.996 3.984793273

3.0717795 3532.656 3467.33962 65.3164 232212.25

4.3E-

06 -0.998 3.990262135

3.0892328 3532.58781 3467.40972 65.1781 230740.42

4.3E-

06 -0.999 3.994520017

3.1066861 3532.53905 3467.45985 65.0792 229691.65

4.4E-

06 -0.999 3.997563679

3.1241394 3532.50976 3467.48996 65.0198 229063.41

4.4E-

06 -1 3.999390804

3.1415927 3532.5 3467.5 65 228854.17

4.4E-

06 -1 4

Numerical integrations:

Integral P Integral Q Integral R Integral S Integral T Moment Bending Stress

7.6229E-

08

7.6223E-

08 8.837E-16 60274.5281 8.559696301

7.609E-08

7.6061E-

08 1.498E-14 60189.1956 8.542371927

7.5813E-08 7.5738E-08 8.513E-14 59933.224 8.49052575

7.5401E-

08

7.5257E-

08 2.94E-13 59506.6913 8.404536535

7.4856E-

08

7.4623E-

08 7.628E-13 58909.7275 8.285028707

7.4184E-

08 7.384E-08 1.647E-12 58142.5144 8.132862293

7.339E-08

7.2916E-

08 3.135E-12 57205.2856 7.949119208

7.248E-08

7.1858E-

08 5.436E-12 56098.3267 7.7350862937.1462E-

08

7.0675E-

08 8.785E-12 54821.9749 7.492235522

7.0343E-

08

6.9376E-

08 1.343E-11 53376.619 7.222201921

6.9131E-

08

6.7972E-

08 1.961E-11 51762.6991 6.926759712

6.7836E-

08

6.6473E-

08 2.76E-11 49980.707 6.607797299

6.6466E-

08

6.4889E-

08 3.764E-11 48031.1854 6.267291636

6.5031E-

08

6.3233E-

08 4.997E-11 45914.7283 5.907282541



22

6.3539E-

08

6.1514E-

08 6.481E-11 43631.9802 5.529847492

6.1999E-

08

5.9744E-

08 8.238E-11 41183.6365 5.13707735

6.0422E-

08

5.7933E-

08 1.029E-10 38570.4431 4.7310534465.8815E-

08

5.6092E-

08 1.264E-10 35793.1959 4.313826353

5.7186E-

08

5.4231E-

08 1.531E-10 32852.7409 3.887396633

5.5544E-

08

5.2359E-

08 1.832E-10 29749.9738 3.453697746

5.3896E-

08

5.0484E-

08 2.166E-10 26485.8397 3.014581251

5.2249E-

08

4.8614E-

08 2.534E-10 23061.3329 2.571804358

5.061E-084.6758E-

08 2.937E-10 19477.4966 2.127019829

4.8983E-

08

4.4922E-

08 3.374E-10 15735.4224 1.681768175

4.7375E-

08

4.3111E-

08 3.845E-10 11836.2502 1.237472037

4.5791E-

08

4.1332E-

08 4.349E-10 7781.16771 0.795432625

4.4234E-

08

3.9588E-

08 4.886E-10 3571.41017 0.356828036

4.2708E-

08

3.7884E-

08 5.456E-10 -791.74011 -0.0772867254.1216E-

08

3.6223E-

08 6.056E-10 -5306.9541 -0.505978237

3.9761E-

08

3.4608E-

08 6.686E-10 -9972.8563 -0.928431849

3.8345E-

08

3.3041E-

08 7.344E-10 -14788.026 -1.343947789

3.6969E-

08

3.1523E-

08 8.03E-10 -19750.995 -1.751936373

3.5636E-

08

3.0057E-

08 8.742E-10 -24860.253 -2.151912518

3.4346E-08 2.8642E-08 9.479E-10 -30114.243 -2.543489718

3.3099E-

08 2.728E-08 1.024E-09 -35511.365 -2.926373659

3.1896E-

08

2.5969E-

08 1.102E-09 -41049.975 -3.300355609

3.0738E-

08

2.4711E-

08 1.183E-09 -46728.385 -3.665305703

2.9623E-

08

2.3503E-

08 1.265E-09 -52544.866 -4.021166243

2.8552E-

08

2.2347E-

08 1.349E-09 -58497.647 -4.367945102

2.7523E- 2.124E-08 1.435E-09 -64584.913 -4.705709305



23

08

2.6537E-

08

2.0181E-

08 1.523E-09 -70804.811 -5.034578853

2.5593E-

08 1.917E-08 1.613E-09 -77155.446 -5.354720839

2.4689E-08

1.8204E-08 1.704E-09 -83634.884 -5.666343882

2.3824E-

08

1.7283E-

08 1.797E-09 -90241.15 -5.969692921

2.2998E-

08

1.6405E-

08 1.891E-09 -96972.233 -6.265044372

2.2209E-

08

1.5568E-

08 1.987E-09 -103826.08 -6.552701656

2.1457E-

08

1.4771E-

08 2.084E-09 -110800.61 -6.832991106

2.0739E-

08

1.4013E-

08 2.182E-09 -117893.69 -7.1062582442.0055E-

08 1.329E-08 2.282E-09 -125103.17 -7.372864415

1.9403E-

08

1.2603E-

08 2.384E-09 -132426.84 -7.633183774

1.8783E-

08

1.1949E-

08 2.487E-09 -139862.48 -7.887600602

1.8193E-

08

1.1327E-

08 2.592E-09 -147407.81 -8.136506939

1.7631E-

08

1.0735E-

08 2.698E-09 -155060.56 -8.380300511

1.7098E-08

1.0172E-08 2.807E-09 -162818.37 -8.61938294

1.6591E-

08

9.6357E-

09 2.916E-09 -170678.9 -8.854158201

1.6109E-

08

9.1258E-

09 3.028E-09 -178639.73 -9.085031329

1.5653E-

08

8.6405E-

09 3.142E-09 -186698.46 -9.312407336

1.5219E-

08

8.1785E-

09 3.258E-09 -194852.62 -9.536690329

1.4808E-

08

7.7385E-

09 3.376E-09 -203099.73 -9.7582828151.4419E-

08

7.3192E-

09 3.497E-09 -211437.29 -9.977585168

1.405E-08

6.9196E-

09 3.619E-09 -219862.74 -10.19499524

1.3701E-

08

6.5386E-

09 3.745E-09 -228373.52 -10.41090813

1.3371E-

08

6.1749E-

09 3.873E-09 -236967.04 -10.62571603

1.3059E-

08

5.8277E-

09 4.005E-09 -245640.69 -10.83980824

1.2764E-08 5.496E-09 4.139E-09 -254391.81 -11.0535712



24

1.2486E-

08

5.1788E-

09 4.277E-09 -263217.75 -11.26738873

1.2224E-

08

4.8752E-

09 4.419E-09 -272115.82 -11.48164218

1.1977E-

08

4.5843E-

09 4.564E-09 -281083.3 -11.696710831.1745E-

08

4.3054E-

09 4.713E-09 -290117.47 -11.9129722

1.1528E-

08

4.0377E-

09 4.867E-09 -299215.57 -12.13080252

1.1323E-

08

3.7805E-

09 5.025E-09 -308374.83 -12.35057717

1.1132E-

08

3.5329E-

09 5.188E-09 -317592.46 -12.57267124

1.0954E-

08

3.2944E-

09 5.356E-09 -326865.65 -12.79746006

1.0787E-08

3.0644E-09 5.53E-09 -336191.58 -13.02531979

1.0633E-

08

2.8421E-

09 5.709E-09 -345567.41 -13.25662808

1.049E-08 2.627E-09 5.895E-09 -354990.28 -13.49176469

1.0358E-

08

2.4185E-

09 6.086E-09 -364457.32 -13.7311122

1.0237E-

08 2.216E-09 6.285E-09 -373965.65 -13.97505669

1.0126E-

08

2.0192E-

09 6.491E-09 -383512.38 -14.22398847

1.0025E-08

1.8273E-09 6.704E-09 -393094.58 -14.47830287

9.9345E-

09 1.64E-09 6.926E-09 -402709.35 -14.73840094

9.8535E-

09

1.4567E-

09 7.156E-09 -412353.75 -15.00469033

9.782E-09 1.277E-09 7.395E-09 -422024.86 -15.27758601

9.7197E-

09

1.1005E-

09 7.644E-09 -431719.71 -15.55751116

9.6667E-

09

9.2668E-

10 7.903E-09 -441435.36 -15.844898

9.6227E-09

7.5513E-10 8.172E-09 -451168.86 -16.14018865

9.5876E-

09

5.8542E-

10 8.453E-09 -460917.22 -16.443836

9.5613E-

09

4.1714E-

10 8.746E-09 -470677.5 -16.75630467

9.5439E-

09

2.4988E-

10 9.051E-09 -480446.7 -17.07807186

9.5352E-

09

8.3225E-

11 9.37E-09 -490221.86 -17.40962831

9.5352E-

09

-8.322E-

11

9.5352E-

09 9.703E-09

-8.322E-

11 -500000 -17.751479299.5439E- -2.499E- 9.5439E- 1.005E-08 -2.499E- 490221.861 17.40962831



25

09 10 09 10

9.5613E-

09

-4.171E-

10

9.5613E-

09 1.041E-08

-4.171E-

10 480446.701 17.07807186

9.5876E-

09

-5.854E-

10

9.5876E-

09 1.079E-08

-5.854E-

10 470677.497 16.75630467

9.6227E-09

-7.551E-10

9.6227E-09 1.119E-08

-7.551E-10 460917.225 16.443836

9.6667E-

09

-9.267E-

10

9.6667E-

09 1.161E-08

-9.267E-

10 451168.857 16.14018865

9.7197E-

09

-1.101E-

09

9.7197E-

09 1.205E-08

-1.101E-

09 441435.365 15.844898

9.782E-09

-1.277E-

09 9.782E-09 1.25E-08

-1.277E-

09 431719.711 15.55751116

9.8535E-

09

-1.457E-

09

9.8535E-

09 1.298E-08

-1.457E-

09 422024.857 15.27758601

9.9345E-

09 -1.64E-09

9.9345E-

09 1.349E-08 -1.64E-09 412353.754 15.004690331.0025E-

08

-1.827E-

09

1.0025E-

08 1.401E-08

-1.827E-

09 402709.349 14.73840094

1.0126E-

08

-2.019E-

09

1.0126E-

08 1.457E-08

-2.019E-

09 393094.58 14.47830287

1.0237E-

08

-2.216E-

09

1.0237E-

08 1.515E-08

-2.216E-

09 383512.376 14.22398847

1.0358E-

08

-2.418E-

09

1.0358E-

08 1.576E-08

-2.418E-

09 373965.654 13.97505669

1.049E-08

-2.627E-

09 1.049E-08 1.64E-08

-2.627E-

09 364457.324 13.7311122

1.0633E-08

-2.842E-09

1.0633E-08 1.708E-08

-2.842E-09 354990.282 13.49176469

1.0787E-

08

-3.064E-

09

1.0787E-

08 1.779E-08

-3.064E-

09 345567.411 13.25662808

1.0954E-

08

-3.294E-

09

1.0954E-

08 1.853E-08

-3.294E-

09 336191.581 13.02531979

1.1132E-

08

-3.533E-

09

1.1132E-

08 1.932E-08

-3.533E-

09 326865.649 12.79746006

1.1323E-

08 -3.78E-09

1.1323E-

08 2.015E-08 -3.78E-09 317592.456 12.57267124

1.1528E-

08

-4.038E-

09

1.1528E-

08 2.102E-08

-4.038E-

09 308374.826 12.350577171.1745E-

08

-4.305E-

09

1.1745E-

08 2.194E-08

-4.305E-

09 299215.566 12.13080252

1.1977E-

08

-4.584E-

09

1.1977E-

08 2.29E-08

-4.584E-

09 290117.468 11.9129722

1.2224E-

08

-4.875E-

09

1.2224E-

08 2.392E-08

-4.875E-

09 281083.301 11.69671083

1.2486E-

08

-5.179E-

09

1.2486E-

08 2.499E-08

-5.179E-

09 272115.819 11.48164218

1.2764E-

08

-5.496E-

09

1.2764E-

08 2.612E-08

-5.496E-

09 263217.753 11.26738873

1.3059E-08 -5.828E-09 1.3059E-08 2.732E-08 -5.828E-09 254391.813 11.0535712



26

1.3371E-

08

-6.175E-

09

1.3371E-

08 2.857E-08

-6.175E-

09 245640.687 10.83980824

1.3701E-

08

-6.539E-

09

1.3701E-

08 2.99E-08

-6.539E-

09 236967.042 10.62571603

1.405E-08 -6.92E-09 1.405E-08 3.13E-08 -6.92E-09 228373.519 10.41090813

1.4419E-08

-7.319E-09

1.4419E-08 3.277E-08

-7.319E-09 219862.736 10.19499524

1.4808E-

08

-7.738E-

09

1.4808E-

08 3.433E-08

-7.738E-

09 211437.286 9.977585168

1.5219E-

08

-8.178E-

09

1.5219E-

08 3.597E-08

-8.178E-

09 203099.734 9.758282815

1.5653E-

08

-8.641E-

09

1.5653E-

08 3.77E-08

-8.641E-

09 194852.622 9.536690329

1.6109E-

08

-9.126E-

09

1.6109E-

08 3.953E-08

-9.126E-

09 186698.46 9.312407336

1.6591E-

08

-9.636E-

09

1.6591E-

08 4.146E-08

-9.636E-

09 178639.733 9.0850313291.7098E-

08

-1.017E-

08

1.7098E-

08 4.349E-08

-1.017E-

08 170678.895 8.854158201

1.7631E-

08

-1.073E-

08

1.7631E-

08 4.564E-08

-1.073E-

08 162818.372 8.61938294

1.8193E-

08

-1.133E-

08

1.8193E-

08 4.79E-08

-1.133E-

08 155060.557 8.380300511

1.8783E-

08

-1.195E-

08

1.8783E-

08 5.028E-08

-1.195E-

08 147407.815 8.136506939

1.9403E-

08 -1.26E-08

1.9403E-

08 5.28E-08 -1.26E-08 139862.475 7.887600602

2.0055E-08

-1.329E-08

2.0055E-08 5.544E-08

-1.329E-08 132426.837 7.633183774

2.0739E-

08

-1.401E-

08

2.0739E-

08 5.823E-08

-1.401E-

08 125103.165 7.372864415

2.1457E-

08

-1.477E-

08

2.1457E-

08 6.117E-08

-1.477E-

08 117893.691 7.106258244

2.2209E-

08

-1.557E-

08

2.2209E-

08 6.426E-08

-1.557E-

08 110800.609 6.832991106

2.2998E-

08

-1.641E-

08

2.2998E-

08 6.751E-08

-1.641E-

08 103826.082 6.552701656

2.3824E-

08

-1.728E-

08

2.3824E-

08 7.093E-08

-1.728E-

08 96972.2328 6.2650443722.4689E-

08 -1.82E-08

2.4689E-

08 7.452E-08 -1.82E-08 90241.1499 5.969692921

2.5593E-

08

-1.917E-

08

2.5593E-

08 7.829E-08

-1.917E-

08 83634.8836 5.666343882

2.6537E-

08

-2.018E-

08

2.6537E-

08 8.225E-08

-2.018E-

08 77155.4461 5.354720839

2.7523E-

08

-2.124E-

08

2.7523E-

08 8.639E-08

-2.124E-

08 70804.8111 5.034578853

2.8552E-

08

-2.235E-

08

2.8552E-

08 9.074E-08

-2.235E-

08 64584.9132 4.705709305

2.9623E-08 -2.35E-08 2.9623E-08 9.528E-08 -2.35E-08 58497.6469 4.367945102



27

3.0738E-

08

-2.471E-

08

3.0738E-

08 1E-07

-2.471E-

08 52544.8665 4.021166243

3.1896E-

08

-2.597E-

08

3.1896E-

08 1.05E-07

-2.597E-

08 46728.3852 3.665305703

3.3099E-

08

-2.728E-

08

3.3099E-

08 1.101E-07

-2.728E-

08 41049.9749 3.3003556093.4346E-

08

-2.864E-

08

3.4346E-

08 1.155E-07

-2.864E-

08 35511.3652 2.926373659

3.5636E-

08

-3.006E-

08

3.5636E-

08 1.211E-07

-3.006E-

08 30114.2433 2.543489718

3.6969E-

08

-3.152E-

08

3.6969E-

08 1.269E-07

-3.152E-

08 24860.2531 2.151912518

3.8345E-

08

-3.304E-

08

3.8345E-

08 1.329E-07

-3.304E-

08 19750.9951 1.751936373

3.9761E-

08

-3.461E-

08

3.9761E-

08 1.391E-07

-3.461E-

08 14788.0256 1.343947789

4.1216E-08

-3.622E-08

4.1216E-08 1.455E-07

-3.622E-08 9972.8563 0.928431849

4.2708E-

08

-3.788E-

08

4.2708E-

08 1.521E-07

-3.788E-

08 5306.95406 0.505978237

4.4234E-

08

-3.959E-

08

4.4234E-

08 1.588E-07

-3.959E-

08 791.74011 0.077286725

4.5791E-

08

-4.133E-

08

4.5791E-

08 1.658E-07

-4.133E-

08 -3571.4102 -0.356828036

4.7375E-

08

-4.311E-

08

4.7375E-

08 1.728E-07

-4.311E-

08 -7781.1677 -0.795432625

4.8983E-

08

-4.492E-

08

4.8983E-

08 1.8E-07

-4.492E-

08 -11836.25 -1.237472037

5.061E-08

-4.676E-

08 5.061E-08 1.873E-07

-4.676E-

08 -15735.422 -1.681768175

5.2249E-

08

-4.861E-

08

5.2249E-

08 1.947E-07

-4.861E-

08 -19477.497 -2.127019829

5.3896E-

08

-5.048E-

08

5.3896E-

08 2.022E-07

-5.048E-

08 -23061.333 -2.571804358

5.5544E-

08

-5.236E-

08

5.5544E-

08 2.096E-07

-5.236E-

08 -26485.84 -3.014581251

5.7186E-

08

-5.423E-

08

5.7186E-

08 2.171E-07

-5.423E-

08 -29749.974 -3.453697746

5.8815E-08 -5.609E-08 5.8815E-08 2.245E-07 -5.609E-08 -32852.741 -3.887396633

6.0422E-

08

-5.793E-

08

6.0422E-

08 2.318E-07

-5.793E-

08 -35793.196 -4.313826353

6.1999E-

08

-5.974E-

08

6.1999E-

08 2.391E-07

-5.974E-

08 -38570.443 -4.731053446

6.3539E-

08

-6.151E-

08

6.3539E-

08 2.461E-07

-6.151E-

08 -41183.637 -5.13707735

6.5031E-

08

-6.323E-

08

6.5031E-

08 2.53E-07

-6.323E-

08 -43631.98 -5.529847492

6.6466E-

08

-6.489E-

08

6.6466E-

08 2.596E-07

-6.489E-

08 -45914.728 -5.907282541

6.7836E- -6.647E- 6.7836E- 2.659E-07 -6.647E- -48031.185 -6.267291636



08 08 08 08

6.9131E-

08

-6.797E-

08

6.9131E-

08 2.719E-07

-6.797E-

08 -49980.707 -6.607797299

7.0343E-

08

-6.938E-

08

7.0343E-

08 2.775E-07

-6.938E-

08 -51762.699 -6.926759712

7.1462E-08

-7.068E-08

7.1462E-08 2.827E-07

-7.068E-08 -53376.619 -7.222201921

7.248E-08

-7.186E-

08 7.248E-08 2.874E-07

-7.186E-

08 -54821.975 -7.492235522

7.339E-08

-7.292E-

08 7.339E-08 2.917E-07

-7.292E-

08 -56098.327 -7.735086293

7.4184E-

08

-7.384E-

08

7.4184E-

08 2.954E-07

-7.384E-

08 -57205.286 -7.949119208

7.4856E-

08

-7.462E-

08

7.4856E-

08 2.985E-07

-7.462E-

08 -58142.514 -8.132862293

7.5401E-

08

-7.526E-

08

7.5401E-

08 3.01E-07

-7.526E-

08 -58909.728 -8.2850287077.5813E-

08

-7.574E-

08

7.5813E-

08 3.03E-07

-7.574E-

08 -59506.691 -8.404536535

7.609E-08

-7.606E-

08 7.609E-08 3.042E-07

-7.606E-

08 -59933.224 -8.49052575

7.6229E-

08

-7.622E-

08

7.6229E-

08 3.049E-07

-7.622E-

08 -60189.196 -8.542371927

-60274.528 -8.559696301

P Q R S T

5.7631E-

06

4.0353E-

20

2.8815E-

06 1.001E-05

-2.379E-

06

Results:

NA -160.07844 N

MA 60274.5281 Nmm

Φ 0.00342892 radians

aircraft structures - complementary energy analysis

Documents