Calhoun: The NPS Institutional Archive

Theses and Dissertations Thesis Collection

1967-12

Comparison of theoretical and experimental stresses

in an F8U-3 airplane wing loaded in torsion

Messerschmidt, Donald Bruce

Monterey, California. U.S. Naval Postgraduate School

http://hdl.handle.net/10945/11810

NPS ARCHIVE1967MESSERSCHMIDT, D.

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HCOMPARISON OF THEORETICAL AND

STRESSES IN AN FSU-3 AfRPLANILOADED fN TORSION

DONALD BRUCE MESSERSCHMIDT

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COMPARISON OF THEORETICAL AND EXPERIMENTAL

STRESSES IN AN F8U-3 AIRPLANE WING LOADED

IN TORSION

by

Donald Bruce Messerschmidt

Major, United States Marine Corps

B. S., U. S. Naval Academy, 1959

Submitted in partial fulfillment of the

requirements for the degree of

MASTER OF SCIENCE IN AERONAUTICAL ENGINEERING

from the

NAVAL POSTGRADUATE SCHOOL

December 1967

ABSTRACT

The stresses near the root of an F8U-3 airplane wing loaded in

torsion were found using the matrix-force method of analysis. The wing

was represented by a simplified, idealized model consisting of axial-

load carrying flange bars and shear carrying cover plates. The theoret-

ical stresses in the idealized model were found by using the IBM 360/67

computer to solve for the redundant loads. These stresses show favor-

able comparison with the experimental stresses determined from strain

gages mounted on the cover skin of the wing. This study shows that the

matrix-force method is applicable to thick-skinned wings.

1'

NAVAL POSTGRADUATE SCHOOLMONTEREY CA 93943-5101

TABLE OF CONTENTS

List of Tables

List of Illustrations

Table of Symbols

Introduction

Theoretical Analysis

Experimental Results

Discussion of Results

Conclusions

References

Appendix A: Program Energy

Appendix B: Program Recrose

PAGE

5

7

9

11

16

37

42

45

59

60

65

LIST OF TABLES

TABLES PAGE

I Geometry of Idealized Model 47

II Geometry of Idealized Model 48

III The Edge Shear Forces on the Panels 49

IV Matrix A 50

V Matrix B 51

VI • Matrix F 52

VII Matrix P 53

VIII Experimental Results at Y = 34.5 54

IX Comparison of Results at Y = 35.5 55

X Comparison of Theoretical Results at Y = 34.5 56

XI Matrix of all Internal Loads, vith 57

Center Section Included

XII Matrix of all Internal Loads, with Center Section 58

not Included

LIST OF ILLUSTRATIONS

FIGURE PAGE

1. F8U-3 Wing Center Section 12

2. F8U-3 Wing Center Section 14

Planform Geometry and Member Location

3. Geometry of Cut at W.S. 25.095 19

4. Geometry of Cut at W.S. 81.98 20

5. Geometry of Cut at W.S. 21

6. Exploded View of Outer Section 22

7. Exploded View of End Rib 23

8. Determinate Beam 18

9. Cross Section of Determinate System 24

10. Flange Bar 27

11. Force Diagram of Flange Bar 29

12. Force Diagram of Beam Web 30

13. Force Diagram of Cover Panel 31

14. Sweep Coupling of Cover Panel 32

15. Wing Mounted on Loading Jig 38

16. Wing with Cover Skin Removed 39

17. Junction Panel and Budd/Datran 41Digital Strain Indicator

TABLE OF SYMBOLS

Matrix of coefficients of non-redundant loadswA Inverse of A matrix

a Redundant load (pounds)r

A.I.B. Aft intermediate beam

a Non-redundant load (pounds)nr r

{AT/ Matrix of resultant loads

B Matrix of coefficients of redundant and applied loads

C Matrix of deflection influence coefficients

C.I.B. Center intermediate beam

fcjj J matrix designated for computer program

E Modulus of elasticity, (psi)

F Matrix of flexibility influence coefficients

F.B. Front Beam

F.I.B. Forward intermediate beam

FR.I.B. Front Intermediate beam

IgJ IjI matrix augmented by trivial relations

G Shear modulus, (psi)

IhI Strain energy matrix

J Matrix formed by product |A "IB

L Spanwise length, (inches)

P Applied force, (pounds)

\J?f Column matrix of applied loads

q Generalized end loads, (pounds)

R.B. Rear beam

R.I.B. Rear intermediate beam

Stress coefficient matrixW

T Torque, (inch - pounds)

t Thickness (inches)

U Internal strain energy (inch - pounds)

UC C matrix designated for computer program

W Panel edge length (inches)

W.S. Wing Station

X X-axis coordinate of wing, (inches)w

Y Y-axis coordinate of wing, (inches)

C Strain, (micro - inches per inch)

/-<. Poisson's ratio

Normal stress (psi)

T Shear stress (Psi)

Angle of principal axis, measured from load reference axis,

the C.I.B.

Angle of sweep

Rectangular matrix

[ J Row matrix

{ J Column matrix

A

[)

10

INTRODUCTION

The purpose of this study is to predict the stresses in an F8U-3

airplane wing loaded in torsion using the matrix-force method of analysis.

The stresses determined analytically are compared with the stresses ob-

tained experimentally in order to determine the accuracy and consistency

of the results.

The primary structure of the F8U-3 wing center section consists of

a simple multicell torque box, as shown in Figure 1. Spanwise bending

is carried by the relatively thick skins which are fabricated from

7079-T6 aluminum alloy plate. The skins are machined to varying thick-

nesses both spanwise and chordwise in order to obtain an optimum distri-

bution of material. The skins are continuous from the wing fold rib

inboard to the wing centerline and from the front beam to the rear beam.

It is this area, between wing fold ribs, port and starboard, and Front

and Rear Beams, fore and aft, which defines the wing center section,

the area of interest in this study. The sweep of the wing is 42° at

the quarter chord. The wing thickness varies from 47« at the tip to

5.017o at the root. It is this combination of a high degree of sweep

and low thickness ratio which raises problems with structural analysis

of the wing. Due to bending-torsion interaction, or sweep-coupling,

structural analysis by engineering beam theory methods runs into diffi-

culty when analyzing stresses near the wing root.

This thesis describes the matrix-force method of analysis for

determining the stress distribution in such a wing. Among the advan-

tages of this method are:

(1) It results in a consistent set of internal stresses and external

deflections

.

11

Actual wing

WS 81.98

Idealized wing

FIG. I

F8U-3 WING CENTER SECTION

(2) The structure can be altered and re-analyzed with only minor

changes in the computer data cards.

(3) The "Bookkeeping" can be simplified and organized in a logical

manner

.

The matrix-force method of analysis uses forces as the unknowns and

treats the structure as an assembly of elastic components. The method,

as described in Reference 1, is divided into three parts:

A. Idealize the structure

1. Replace the actual structure with a mathematical model.

2. Make the model statically determinate by cutting redundantmembers

.

3. Write the equilibrium equations that determine the load distribu-tion in the structure

B. Calculate and tabulate the flexibilities of the individual members

comprising the structure.

C. Perform matrix operations.

In this report, the wing is "cut" outboard of the intermediate

rib (Y = 81.98), and it is assumed that the stress distribution herew

can be accurately calculated using the engineering theory of torsion,

T — ^2.Aq . The dihedral of the wing is neglected, and the wing is

assumed to be supported on four rigid supports located at A, B, C,

and D in Figure 2. The aft supports are pins which can support fore

and aft and vertical loads. The forward supports can support vertical

loads only. The only loading condition considered in this study is

symmetrical wing torsion. From symmetry, the reactions at all four

support points are determined by equilibrium as functions of the applied

loads

.

13

1U

This study was conducted at the Naval Postraduate School, Monterey,

California, during 1967. The author is indebted to Professor C. H. Kahr

who conceived the need for the investigation and who rendered valuable

assistance and encouragement throughout the project. The author expresses

gratitude to Aeronautical Laboratory Technicians R. A. Besel and T. B.

Dunton whose assistance and cooperation were essential to the completion

of the study.

15

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hen

The model selected to represent the wing structure consisted of

flat plates which carry only shear and cylindrical bars which carry

only axial load. This model was selected for compatibility with the

previous work done in Reference 2. Analysis of the model can be as

accurate as the matrix-force method and arithmetic permit; however,

the correlation between theoretical and experimental stresses will

depend upon how closely the model represents the actual structure.

This study is not an attempt to define the best model for the analysis,

but to calculate stresses in what was tacitly assumed to be an adequate

model. Most of the examples of similar studies currently available in

the literature deal with thin-skinned shells. There is little evidence

on whether or not such models are suitable for the thick-skinned wing

plates used in this study. An important factor in constructing the

idealized model is the change of the wing geometry to model geometry.

The model consists of bars and flat shear carrying plates. It is ex-

tremely important to insure that these plates are, in fact, mathemat-

ically flat. For example, the depths of the ribs at WS 25.095 and WS

81.98 were such that the four points AFGH in Figure 1 did not lie in a

plane. Therefore, panel AFGH is warped and has panel edge shears which

have components perpendicular to plane AFGH. These shears are not

accounted for in the equilibrium equations written for a flat panel

and will produce errors in the solution. Reference 3 shows that for

a similar wing such discrepancies yielded determinate member loads

which were out of equilibrium as much as 5 to 10%. After altering

the beam depths to make the panels flat, this trouble disappeared. The

alteration required a change of less than 1% of each dimension.

17

After determining model geometry from the adjusted dimensions of

the wing, the cross sectional area of the bars and the thickness of the

shear panels were calculated. Each bar area of the model consists of

the area of the beam flange plus one half the skin area between adjacent

beams. The beam web areas were not included, using the methods of

Reference 4. The effect of combining the axial load carrying skin with

the beam flange to form the model's cylindrical bar is to remove Poisson's

effect from the analysis. This means that any chordwise contraction of

the cover skins is neglected. The wing skin thickness varies in the

chordwise direction in section ABCD and in both spanwise and chordwise

directions in section AJBEF. An average skin thickness between beams

was determined and used as the shear plate thickness. Using constant

thickness shear plates in place of tapered skin simplifies the analysis

but introduces possible sources of error. Model geometry and areas are

listed in Tables I and II. Diagrams of cross sectional geometry are shown

in Figure 3, 4, and 5. An exploded view of the outer section of the

idealized model from Y = 25.095 to Y =81.98 is shown in Figure 6.WWAn exploded view of the intermediate rib at Y =81.98 is shown in

w

Figure 7.

2. Make the model statically determinate by cutting redundant members,

The most complex beam that is truly determinate, so that the flange

and shear loads can be determined from equilibrium equations, is the

3- flange, 3 web beam shown in Figure 8.

02Fig 8.

DETERMINATE BEAM

18

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23

In this beam, loads can be obtained from equilibrium only, without con-

sidering flange areas, material stiffness, or deformation of the structure.

In order to make the idealized model of this study a determinate structure,

it was necessary to cut the 5 interior webs and 11 of the flanges, re-

sulting in the structure shown in Figure 9.

Flange 14

Fig. 9

CROSS SECTION OF DETERMINATE SYSTEM

The number of redundancies arising from the geometry of a box structure

has been given by Reference 4 and others as B + N - 4 where B is the

number of longitudinal effective flanges continuous across each junction

and N is the number of closed cells stiffened by ribs at tip end of bay.

For the outer section of this model, B = 14 and N = 6, which results in

16 degrees of redundancy. This agrees with the 16 members which were

cut to provide the determinate structure of Figure 7.

3. Write the equilibrium equations that determine the load distri-

bution in the structure.

Considering the outer wing section from station 25.095 to 81.98,

there are the following 49 unknown member loads.

12 skin panels14 beam flanges7 beam webs

10 rib flange loads at station 81.986 rib webs at station 81.98

49 total unknowns

24

The redundant forces come from the following members

1 F.B. , , upper flange2 FR.I.B. , both flanges2 F.I.B. both flanges2 C.I.B. both flanges2 A.I.B. both flanges2 R.I.B. both flanges5 webs 8 •• 12

16 redundant forces

Thus, there are 49 - 16 or 33 equations of equilibrium required to

characterize the model. These equations are:

a) Six equations of equilibrium of free body of bay between stations

Y = 25.095 and 81.98w

(1) P =x

(2) P =y

(3) p = oz

(4) M =0xx

(5) M =0yy

(6) M =0zz

b) Eleven equations of equilibrium of the redundant beam flanges, elements

16, 18, 20, 22, 24, 29, 31, 33, 35, 37, and 39.

c) Ten equations of forces and moments, F = , M =0 of thex yy

intermediate beam segments of the rib at Y =81.98w

d) Six equations of equilibrium of the rib posts on the beam ends at

Y =81.98, elements 53 through 58.w

The addition of the center section Y = to 25.095 can be thoughtw

of as an extention of the horizontal flanges. The forces in the X and Z

directions are reacted by the horizontal and vertical supports at Y =

25.095, but the Y forces carry through to the centerline, Y = 0.w

25

Although the center section is 16° redundant (the same as the out-

board section) , these are not new redundancies but meraly components

of the forces in the outer section. Further investigation shows that for

symmetrical wing loading, the bending meoment, M , is constant from

Y = 25.095 to Y = 25.095 and that torsional and vertical shear arew w

zero. The axial load on each flange at Y = 25.095 is the same as the° w

load on each flange at Y = 25.095. Hence there is no shear on anyw

panel or web in the center section. (It was later shown that due to

a slightly unsymmetrical load, there are, in fact, shearing stresses

in the center section panels.) Since the addition of the center section

results in no new redundancies and no new external loads, there can be

no more than the 33 independent equilibrium equations already proposed

for the outer section. Since there are more than 33 structural elements

involved, there are several possible combinations of 33 independent

equations. Those chosen were considered to be those most likely to

have the larger stress concentrations.

B. Calculate and tabulate the flexibilities of the individual members

comprising the structure.

As described in Section C, the internal strain energy of the wing

may be written as:

Z£U - L«. a. • o% J [%,4J «11 (1)

The elements in the F matrix define the elasticity of the structure,

If skin thicknesses or flange areas are changed, only the elements of

the F matrix will be affected. If the geometry of the model is changed,

the equations of equilibrium will also be affected. Multiplying out

equation 1,

26

2EU = *.f„ + <*z /L * •• a, Fhn +2q,qx Fiz 1- 2a,a Fl3 + (2)

Each term of Equation 2 can be regarded as a small increment of strain

2energy ^ U associated with the load a. or a. . a.. Since the load a.

l i j l

may be applied to more than one structural element, the corresponding

F.. term may be made up of contributions from several structural elementsnThe F matrix must be a square symmetric matrix where F. . = F...

1. Flange Bars.

The calculation of the F matrix elements for the flanges was done

following the example of Reference 4. The energy in a uniform bar under

a linearly varying axial force is

u =/

2AEdx

(3)

where o and q - are end forces as shown in Figure 10.

\\\\\\\\\\\\\\\\\

?- D -R£

Fig. 10

FIANGE BAR

Rewriting Equation 2 for this single element

2 * 22EU =%,£,+ ?%, %* fm + % z £ (4)

therefore

f" Lho 3

;

2LL dx (5)

T" 3A **

(6)

27

"M.

-MLand

(7)

f = f =''* " 'zi 6A

In the case of linearly tapered members, the flexibilities may be

obtained as functions of the end area ratios. Reference 1 shows the

flexibility influence coefficients modified by a function, Q, of the

end areas which was obtained from a graph of Q versus A. /A., the end

area ratio.

£ . -i- d> f =— <& f = -^ 6 ( 8 )

For most of the flange elements, the generalized end loads, q.

and q. were linear functions of the unkown member loads, or:

? . = <«, *<*, ^ " •<"4i, (9)

?j " *' * .ft «e ' ' ' '

ft ^ (10)

where C&* = the coefficient of the i'th member load applied to one end

of the bar,

and Q. = the coefficient of the i'th member load applied to the other

end of the bar.

Now taking each force acting on the ends of the flange individually,

that is:

f ,= °t> a < (ii)

?* * P> *>(12)

and substituting into Equation 4, we find that

beu = a #Y<£ +eu.p,fa y (3/4 )d3)

28

and from Equation 1, for a single force, a

2EU = a? F„

Comparing Equations 13 and 14,

Similarly, for the other forces acting on the same flange,

22.

and for the cross product term,

(14)

(15)

(16)

(17)

The forces on element 14 are shown in Figure 11.

For example, the flexibility influence coefficients for element 14 were

determined as follows:

% *-! -

A, = /. 3 -t -.._/— TO "? «_ Az --/.3— i. /y. j —»

Fig. 11

FORCE DIAGRAM OF FLANGE BAR

*» - h, A = *«^ - ^y«=Z7 &.

= /0 - /67

?/ = -3.23/ a^ +ax/ -/o.3S3a.

?*- -^e **>**„

2 2,

/\ =*0. 233 (-/0. 9*3) * * * /o./il^/o. 5si)(o) +ZO, 333 (o) = 2439. 433

2

29

For element 14, there are 4 more coefficients to be found; F

F21, 21'

F2, 21'

and F20, 21'

The F2, 2

20, 20'

coefficient for element 14 is

added to other F ~ coefficients for elements 7, 39, 101, and 126 to

make up the total F9

_ flexibility influence coefficent found in the F

matrix.

2. Symmetrical Shear Panels

Reference 3 gives the energy of the symmetrical trapezoidal

panel in terms of the shear force on the shorter parallel side, as:

2E

so that

vH^^lH^'W a

may be written for F - of element 1

'-' (w.33S(/2,46o)

S/£ (.2oo)g (.OOG78) + /32

The forces on element 1 are shown in Figure 12

W,

a =.8Z7a,c 1

Fig. 12

FORCE DIAGRAM OF BEAM WEB

(18)

(19)

Other F1 1

coefficients appear from elements 40 and 51 and add to pro-* >

*

duce the final F found in the F Matrix.

30

The shear panel edge loads were determined by assigning a unit

shear force, a., to the outboard edge of all panels and the forward

edge of the rib webs. From the equilibrium of the free body, forces

a, , a , and a may be found as functions of the unit force a..

% - ai h '

Wc

a = a W. / Wc 11 c

a, = a. 1 / Wd id c

(20)

The geometry of the panels and the load ratios are shown in Table III.

3. Non-symmetrical shear panels.

a) The cover skin panels have roughly the shape of a parallelogram,

with the inclined sides skewed slightly. The maximum difference in

inclination, 0-8= 3° 34', occurs at the forward panels, elements 38

and 15. These panels were treated in accordance with Reference 6

which shows that the energy of the swept panel is given as

££CT = r/^-^K^37fe))]where T = Cot ~@ + cot 8 cot @ + cot *

6

The forces on element 15 are shown in Figure 13.

(21)

(22)

a.G/ement /&

K&

ex..

w.

a4

so that

' nsr\

may be written for F

Fig. 13

FORCE TJIAGRAM OF COVER PANEL

= E2

Za 4z

20, 20of element 15.

2o,2o ' 3*(3/'£9

) .S7*(/+ A ZCe) (3. S9Q) = SO. S2S

31

b) The cover skin panels introduce additional forces due to interaction

between torsion and bending. Twist of a swept wing produces bending and

vice-versa. This interaction was introduced by considering the stress

distribution in the rhomboidal cover skin panels, as shown in Figure 12.

These panels were assumed to be parallelograms in order to simplify the

analysis. Consider the plate shown, loaded with a uniform shear force,

a , and axial load, f .

a

- a f = 2a tan A

Fig. 14

SWEEP COUPLING OF COVER PANEL

The axial force, f , was assumed concentrated at the flanges which were

made up mostly of skin material, so that the cover panels carry only

shear. However, it is seen in Figure 12 that the shear forces along

the sides of the panel produce additional tensile stresses, f ', which

do work on the displacements produced by the primary axial stress, f .

For this panel, the strain energy can be written.

2 2(23)

The third term in Equation 22 is due to the interaction between the

axial and shearing forces and does not appear in an equation, written

for a rectangular panel. This term causes the stress distribution in

this model to show a root stress concentration. Reference 3 states that

"Studies made on simple one cell swept box models with streamwise ribs,

utilizing parallelogram shaped cover sheets, show that unless this cross-

product term is introduced, there will be nothing in the analysis to

32

differentiate between a straight and swept wing, and there will be no

3root stress concentration factor" . For the purpose of calculating the

energies in the flanges adjacent to each panel, this axial load, f

'

t >

was aplit equally and added to the end of each flange. These coupling

loads are shown as dashed vectors in Figure 6.

C. Perform matrix operations.

The matrix-force method, as described in References 1 and 3,

expresses the energy of the entire structure as the successive product

of three matrices.

3£U - \a\ [f\ {a} < 2«

where a, a are the internal loads acting on the members. TheInF matrix is symmetrical and represents the elasticity of the structure.

The internal loads were found by writing the equilibrium equations for

the determinate structure and for each redundant member. The number

of equations must equal the total number of unknown internal loads less

the number of redundants. The equations are written so that all the

independent external loads and redundants are collected on one side and

non-redundant loads on the other side of the equality;

such as

M («*]=

[£

] {P

\

ar} HP

where A and B are matrices of the coefficients of the loads.

Solving for the non-redundant loads, a , as functions of the appliednr

loads, P , and the redundants, a , we get:

{anr}= ^[B]{P\a

r } = [j}{p\a r}9»

33

Then, by adding the trivial relation between the redundants

and expanding J to include Equation 4 results in

which may be transposed, yielding Equation 6

{«} - l«j - ki

aA Wr

Now, substituting into Equation 1,

2EU= [p\ar \6

TFG {P\CLr }

(27)

(28)

(29)

(30)

H is a symmetric matrix. Its members are the coefficients of the

strain energy terms when the energy is written as a function of applied

and redundant loads. Partitioning H into four sub-matrices,

[«]

"pup '£ pxr

H.2/ r |

H22 rxr

yP - appl/ed loads - 33

r - redundant /oads - /6

Rewriting Equation 29.

2£V* P H„ +2Par Hl£+ar H& (31)

Minimizing the strain energy with respect to the redundant, a,yields

12 r 2Z

Rearranging M- -

en] {p}

(32)

(33)

34

The matrix of all the unknown internal loads is given by

total(a)

m -TOTAL

P*f>

£A/.

Wr J

n (34)

S is the unit load distribution matrix for the indeterminate structure.

Each S. term is the value of the i'th member load when the structure1m

is subjected to a unit value of the m'th applied load.

Using the notation of Reference 5, the strain energy can be

written:

U = ±[f\[c]{p] (35)

where C is the matrix of flexibility influence coefficients,

stituting from Equation 11 into Equation 23,

Sub.

2EPJ[#]MH

therefore

Citr r

F

- / I \EN GTG

= _L EN H EN

and substituting EN = — H1-1

22H21

from Equation 10,

c = ^lk HIZ

H.12.

-IH

Note that Equation 34 may be expanded to read

a. = s., P. +s.. P, ... s. P1 111 122 1mm

a2 = S

21Pl+ s

22P2

s9

P2m m

(36)

(37)

(38)

(39)

35

I

Therefore, each internal load, a. , may be found by substituting a

particular set of applied loads, ^Pj . For any other loading condition,

only the matrix of applied loads needs to be changed to obtain the new

internal loads for that loading (flight) condition. This enables a

rapid evaluation of the maximum load on any element over a range of

applied loads.

A computer program, originally set up by Reference 2 for the CDC 1604

computer was rewritten in Fortran IV to fit the IBM 360/67 computer and

appears in Appendix A. This program performs the complete matrix oper-

ation outlined in Section C with input A and B from the flexibility matrix

and the applied loads, P. Matrices A, B, F, and P appear in Tables IV,

V, VI, and VII.

36

EXPERIMENTAL RESULTS

In order to ascertain whether or not the results obtained from the

strain energy method were accurate for this type of thick skinned wing

construction, it was necessary to have experimental data with which to

compare the results of the idealized model. The wing is mounted in a

jig which closely approximates the actual method of joining the wing to

the aircraft fuselage, as shown in Figure 15. Since the wing is a

continuous structure, the root fixity coefficient is not changed as when

a single span wing is mounted on a completely rigid bulkhead. The

loading condition investigated was a torque of 336,000 in. lbs. This

load was considered to be well within the elastic range, but still of

sufficient magnitude to give adequate strain levels throughout the

structure for repeatable strain measurements. Some strain gages were

installed on the wing when received from Ling-Temco-Vought Aerospace

Corp., and others were installed at the Naval Postgraduate School.

Figure 16 shows the wing with one cover skin removed for installation

of strain gages. Wing section Y = 34.5 was particularly well instru-

mented, with strain gage rosettes placed on both sides of top and bottom

skins to check for differential bending effects. Strain gage data were

reduced to principal stresses and directions through computer program

"Recrose", written in Fortran IV for the IBM 360/67 computer, as shown

in Appendix B. The torque loads were applied with four hydraulic cylin-

ders through a heavy fitting attached to the wing fold lugs. The applied

torque was designed to be perpendicular to the elastic axis. The load

was applied at both wing fold axes; and to ensure a symmetrical load,

the four hydraulic cylinders were supplied through a common manifold.

37

•~l

'

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31

Load magnitude was determined by a manifold pressure gage in the

hydraulic pressure system and checked by four Dillon dynamometers

attached to each cylinder. The strain readings were taken with a

Budd/Datran digital strain indicator (Model TC22) . All gages were

wired to a common junction panel as shown in Figure 17. All gages

were balanced at a zero reading before applying the load. Eighteen

gages were read at each loading in about one minute. This minimized

the chance for drift in gage reading. All gages were rechecked for

their zero reading after the load was released and sufficient time

had elapsed for hysteresis effects to be minimized. All testing was

done at night to eliminate the thermal effects of sunlight shining on

the wing. Reference 2 contains all strain gage coordinates, gage fac-

tors and resistances. It also shows that the strains are in the linear

region by comparing the gage readings at a loading of 8000 psi (336,000

in. lbs. of torque) with those taken at 7000 psi (294,000 in. lbs. of

torque) and extrapolated to 8000 psi.

Strain readings for the rosettes on the cover panels appear in

Table VIII. The readings were averaged for the back- to-back gages

and entered in program "Recrose" to compute the stresses. The normal

and shearing stresses and the principal axes for the skin panels are

also shown in Table VIII.

40

T

DISCUSSION OF RESULTS

There are two areas of comparison in this study. The first is the

comparison of the theoretical results with the experimental results, and

the second is the comparison of the theoretical analysis of the wing

with and without including the center section.

In the first area, the stresses as found in Table IX show fairly

close agreement. In the panels where the best stress agreement occurs,

such as panel 38, the strain readings on both sides of the skin are

nearly equal. The strain readings of back-to-back gages were averaged

and used to find the experimental stresses at each station. Where the

corresponding interior and exterior gages showed considerable variance

in their strain readings, the experimental and theoretical stresses also

had greater variance. The difference in strain readings is due to

differential bending in the cover skin and possible torsional resistance

of the pivot rib at the rear fitting, both of which were neglected in the

theoretical analysis. The differential bending was neglected because

the model used for the flexibility coefficient of the cover skin was a

flat panel which carries only shear. The torsional resistance of the

pivot rib was neglected because the effect of an unknown amount of slack

movement about the wing hinge could not be incorporated into the

theoretical analysis. The strain gages themselves are believed to be

accurate to within 5 micro-inches of the true strain values. Strain

readings found by Holgren and Comfort in 1963 were very close to those

found in this study. The substitution of aluminum pads between the

jig and the loading floor for the original plywood pads, as recommended

in Reference 2, does not seem to have made much difference in the strain

42

values. The original gages were carefully installed and waterproofed

which seems to account for their longevity. Most of the gages have

been in place for over four years. The normal expected life-span of

this type gage in industrial use is one or two years. The first factor

mentioned, that of differential bending and torsional resistance of the

pivot rib, seems most likely to be the cause of the difference between

theoretical and experimental results. The differences do not seem to

be too great since some of the references mention that errors of 100%

are sometimes found in the case of thin skinned structures. More con-

clusive comparisons could be made if the torque loading were increased

and larger strain readings were available. However, it was decided that

a significant increase in the torque loading beyond the 336,000 in. lbs.

used in this study might permanently warp the wing and render it useless

for further study.

The second area of comparison is between the theoretical results

found in this study, and those of Reference 2. Including the center

section, from Y = to Y = 25.095 resulted in changing 25 flexibilityWWinfluence coefficients and did not change the equilibrium equations. In

effect, it moved the wing root from the pivot rib at Y = 25.095 to thew

center rib at Y =0. Most of the coefficients changed by 3 or 4%, and

none of them changed by more than 137 ; yet the matrix of resultant loads,

j*AT/ , shows some considerable changes. In comparing (at| of Reference

2 with the /AT] matrix found in this study, the former loads will be

designated as aj (Reference 2). The greatest change was in the load

on the rear intermediate beam, R.I.B. a. (Reference 2) = 1010.4; and

a,„ = 10,263.5. This load is applied to element 24 and will influence

the stresses on panels 23 and 25. The flexibility coefficient associated

43

with this force has a much smaller change, F,„, ,_ (Ref. 2) = 13.3832

and F,„ ,L = 14.3406. The purpose of mentioning these changes is to

show that the cumulative effect of minor changes in 25 out of 129 non-

zero flexibility coefficients may be great. This means that small errors

in deciding upon the true area and length of the elements of the ideal-

ized model may be significant. When the actual structure to be idealized

has irregular cut-outs and joints, and reduction in skin, web, and

flange areas occurs in steps rather than linearly, considerable effort

will be involved in just constructing an idealized model. By examining

Table X, it is seen that inclusion of the center section relieved the

stresses at the leading edge of the wing which was the area of greatest

discrepancy in Reference 2. Both normal and shear stresses are much

closer to the experimental values. The stresses in the rear panels are

also improved, but it is felt that the pins at point B and C in Figure 2

fit snugly enough to offer some torsional resistance of the pivot rib at

the rear fitting, which was not accounted for in the theoretical analysis.

Tables XI and XII list the resultant loads, a. through a,Q

for the wing with

and without center section included. The agreement between theoretical

and experimental stresses throughout the wing seems to bear out the val-

idity of the matrix-force method, and the approach as described in this

study in the prediction of stresses in a wing prior to its construction.

For greater accuracy, it would be necessary to use more elements which

would require the calculation of additional equilibrium equations and hence

expansion of the matrices.

44

CONCLUSIONS

The matrix-force method of analysis using the strain energy theory,

as proposed in Reference 1, is acceptable for a thick-skinned wing.

It can be expected to predict the stresses in a complex, highly redun-

dant structure, loaded in torsion, with a reasonable degree of accuracy.

For increased accuracy, the long slender panels representing the cover

skin should be broken up, and smaller elements used which would reduce

the size of the structural grid but increase the number of unknowns.

Reference 7 mentions a method of accomplishing this by dividing each

panel into a number of narrow longitudinal strips which could be treated

as rectangles with constant shear flow in each one. The difference in

shear forces of adjacent strips would be taken by assuming concentrated

stringer areas in the spanwise direction. This method would still

neglect Poisson's effect. From the results of this study, it appears

that a better panel would be one whose length and width were of similar

size.

In summation, it was found that the accuracy of the matrix-force

method was improved considerably (compared with the results found in

Reference 2) by the inclusion of 14 additional members which resulted

in changes to 25 flexibility coefficients. Generally speaking, it must

be possible to attain a very high degree of accuracy by fundamentally

the same method as has been used in this study, by dividing the structure

into a much larger number of elements. This involves a larger number of

redundancies and much more numerical work. This could be programmed to

be accomplished by the computer. An experienced stress analyst, who

knows the stress distribution of most of the elements of the structure

45

on which he is working, would be able to concentrate the redundancies

to the unknown points, thus reducing the computational work considerably.

The experimental analysis would be improved by the addition of more strain

gages on the inside of the wind cover skins.

Because of the number of man - hours involved in removing and

replacing the cover skins, however, it would not be practical to install

additional gages.

46

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48

TABLE III

THE EDGE SHEAR FORCES ON THE PANELS

Piece wiwc lb Id ab/a i = ac/a i

= ad/ai=No. in. in. in. in. xb/wc w

i/wc , Vw

c

1 5.64 6.82 14.3408 14.3287 2.1027 .8269 2.10092 6.82 7.44 13.5077 13.5041 1.8155 .9166 1.81503 7.44 7.66 13.0307 13.0303 1.7011 .9712 1.70104 7.66 7.54 12.5756 12.57 54 1.6678 1.0159 1.66785 7.54 7.06 12.1370 12.1296 1.7191 1.0679 1.71806 7.06 6.30 11.8046 11.7984 1.8737 1.1206 1.87 277 5.64 7.24 79.3021 79.3021 10.9533 .7790 10.95338 6.82 8.94 74.9206 74.9206 8.3803 .7628 8.38039 7.44 9.66 71.3846 71.3846 7.3897 .7701 7.3897

10 7.66 9.70 68.5890 68.5890 7.0710 .7896 7.071011 7.54 9.20 66.4567 66.4567 7.2235 .8195 7.223512 7.06 8.32 64.9038 64.9038 7.8009 .8485 7.800913 6.30 7.20 63.8339 63.8339 8.8658 .8750 8.865815 14.3287 20.8705 79.3021 74.9206 3.7997 .6865 3.589717 13.5041 19.0991 74.9206 71.3846 3.9227 .7070 3.737519 13.0303 17.7 87 9 71.3846 68.5890 4.0130 .7325 3.855921 12.5754 16.5114 68.5890 66.4567 4.1540 .7616 4.024923 12.1296 15.2393 66.4567 64.9038 4.3608 .7959 4.258925 11.7984 14.1418 64.9038 63.8339 4.5894 .8342 4.513828 11.8046 14.1748 64.9038 63.8339 4.5788 .8327 4.503330 12.1320 15.2456 66.4567 64.9038 4.3590 .7957 4.257132 12.5756 16.5132 68.5890 66.4567 4.1535 .7615 4.024434 13.0307 17.7879 71.3846 68.5890 4.0130 .7325 3.855936 13.5077 19.1022 74.9206 71.3846 3.9220 .7071 3.736938 14.3408 20.8878 79.3021 74.9206 3.7965 .6865 3.5868

49

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50

TABLE VMATRIX B INPUT FORM

11 1.0 1343 3 -1.0 3355 5 -1.0 5377 7 -1.? 8 1

8 4 96.1099 8 58 8 7.4179 8 9812 1.8124 813816 3.0187 817820 2.5220 821824 3.3099 825828 3.2013 829832 2.8758 833836 -45.6107 837840 -3.5 384 841844 -2.5220 84S848 -3.3090 84^945 -1.0 10221123 1.0 11361237 7.2236 12481349 -1.0 14 71538 -7.8009 15431642 1 .0 17301829 -1.0 18351934 -8.3804 19392010 -0.99999 20112014 0.99915 20152018 0. 99980 201O2022 -0.60391 20232026 -0.45373 20272030 0.55916 20312039 -0.65009 2 0402043 -0.48235 20442047 0.55916 20482121 -0.75927 21222125 -0.87645 21262129 0.79688 21302133 0.89114 21392142 -0.85597 21432146 0.79688 214722 1 56.885 22 222 5 56.885 22 622 9 1.12519 2 2102213 -2.34025 22142217 0.27134 22182221 3.39403 22222225 3.64622 22262229 3.84895 22302233 3.2J813 22392242 -3.93750 22432246 -3.R4895 224723 8 56.8554 23 92312 56.8702 23132316 -56.8827 2317232^ 0.0 23212324 -63.5718 23252328 15.8462 23292332 78.4498 23332341 -47.9'j19 23422345 15.3462 23462349 78.4498 241227 9 7.5385 28 829 3 1.0 29 429 7 1.0 29 32911 -0.D0844 29122^15 -0.02295 29162919 0.03219 29202923 0.01487 29242927 0.01009 29292931 0.01249 29322935 -0.77119 29362939 -0. 01415 29402943 -0.00971 29442947 -0.01487 294*3018 -1.0 3m3216 -1.0 331?

O.J 220 1.0 244 -1.0 1

1.0 4 4 -1.0 436 1.0 1

1.0 6 6 -1.0 63 8 1.0 1

55.2466 8 2 69.5754 8 3 8 3.0796 1

108.6853 8 6 120.8150 8 7 132.6135 1

5.9191 810 4.2884 811 3.0187 1

0.5448 814 0.5448 815 1.8124 1

4.2884 818 5.9191 819 7.4172 1

3.2013 822 3.5384 823 3.5707 1

2.8758 826 2.3642 827 2.5220 1

3.5384 8 30 3.5707 831 3.3099 1

2.3642 834 -15.9213 835 -30.7843 1-60.8687 838 -75.9532 839 -3.2013 1

-3.5707 842 -3.3099 843 -2.8758 1

-3.2013 846 -3.5384 847 -3.5707 1

-2.8758 921 1.0 934 8.3804 1

1.0 1035 7.3897 1046 -1.0 1

7.0710 1147 -1.0 1224 1.0 1-1.0 1325 1.0 1338 7.8009 1

0.0 14 8 7.^563 1532 -1.0 1

1.0 1631 -1.0 1637 -7.2236 1

-1.0 1736 -7.0710 1741 1.0 1-7.3897 1840 1.0 1928 -1.0 1

l.n 20 8 -0.99948 20 9 -0.99980 1-0.99996 2012 -0.99974 2013 -0.99915 1

0.99974 2016 0.99996 2017 0.99999 1

0.99948 2020 -0.69666 2021 -0.65009 1

-0.55916 2024 -0.51788 2025 -0.48235 1

0.69666 2028 0.6 5009 2029 0.60391 10.51788 2032 0.48235 2033 0.45373 1

-0.60391 2041 -0.55916 2042 -0.51788 1

0.69666 2045 0.65009 2046 0.60391 1

0.51788 2049 0.48235 2120 -0.71732 1-0.79688 2123 -0.82936 2124 -0.85597 1-0.89114 2127 J. 71732 2128 0.75927 1

0.82936 2131 0.85597 2132 0.87645 1-0.75927 2140 -0.79688 2141 -0.82936 1

-0.87645 2144 0.71732 2145 0.75927 1

0.82936 2148 0.85597 2149 0.87645 1

56.885 22 3 56.885 22 4 56.885 1

56.885 22 7 56.885 22 8 1.83113 1

0.27134 2211 -0.4 8011 2212 -1.30551 1

-2.34025 2215 -1.30551 2216 -0.48011 1

1.12519 2219 1.83113 2220 2.59681 1

3.84895 2223 4.02233 2224 3.9375J 1

3.20813 2227 2.59681 2228 3.39403 1

4.02233 2231 3.93750 2232 3.64622 1-3.39403 22 40 -3.84895 2241 -4.02233 1

-3.64622 2244 -2.59681 2245 -3.39403 1

-4.^2233 2248 -3.93750 2249 -3.64622 1

56.8736 2 310 56.8844 2311 56.8827 1

56.8367 2314 -56.8367 2315 -56.8702 1-56.8844 2318 -56.8736 2319 -56.8554 1

-15.8462 2322 -31.8510 2323 -47.9019 1

-78.4498 2326 -92.3667 2327 0.0 1

31.8513 2330 47.9019 2331 63.5718 1

92.3667 2339 -15.8462 2340 -31.8510 1

-63.5718 2343 -78.4498 2344 0.0 131.8510 2347 47.9019 2348 63.5718 1

6.8182 2511 7.4397 2610 7.6599 1

1.0 29 1 1.0 29 2 1.0 1

1. J 29 5 1.0 29 6 1.0 1

0.03219 29 9 0.01978 2910 0. 00477 1-0.02295 2913 -0. 04114 2914 -0.04114 1

-0.00844 2917 0.00477 2918 0.01978 1

0.01009 2921 0.01415 2922 0.01555 1

0. )1249 2925 0.00971 2926 0.00705 1

0.01415 2929 0.01555 2930 0.01487 1

0.00971 2933 0.00705 2934 -0.76286 1-0.78969 2937 -0.81957 2938 -0.84856 1

-0. J1555 2941 -0.01487 2942 -0.01249 1

-0.01009 2945 -0.^1415 2946 -0.01555 1

-0. 31249 2949 -0.00971 30 9 1.0 1

1.0 3117 -1.0 3211 1.0 1

1.0 3315 -1.0 2

51

<r\J

org^coegremccvO ^-<—• ref^re>teg»-iONccvOo^o*-, CJ>r~-

m r\i re .-« rem •-<o 0* -J- co in re >fr .4- in co r- <f eg re -* h-O^ re ~*ore>^iPr^rexinin^re.^vC^-^Xr^<)inegegr\jo^NO»tinxOr'~vO

>£>m >4- ^-*o >o 0^ r- reo re r- eg r- r- s re eg c* —• re -«—«oo r» reoirrein<Njp-rec, r\iegCT> r\jror~-c^rOC0'-^v}" inccrg ocotn>0 mt-H | fVjrsj | ^f-l | .-I | r-lr-1 f\j I Oil

-< I III ^

Or- core

r-mr- •

• • «r\j

c;f\jir.

qo

rg r^^re^>j-rer^»^rec\i^>^.-o^^o~r\jegegr\j<^rr*^ >^^ro>4->i-.j-eg >*- ao cooO •—• <—' <\i re re -4" in >o >c r— oo ooo 1*eg in r—& —• *f in go coo >J* oo

^^^^,^^,-h,_i,_»,^^h,_i,—l,--^<\]r\Jegr\Jpgre^ererere^J->J-«T

aONOOOcocooore.-leg>a•^-<-^^t reacrgegvocorer- .-iegrgr->4-NCOeg—i.-iinoovO>oOvOcor^>j-ejrein<c>0'-'egr^cr coogoege-a^rein^-c^c>j- in r- r- eg gD ego go in ego in r-ooo eg s r— en —i .-i eg gon cr* sOo C- -tf

- oo(mnot go s go.-<ejr- r\j cc o-- >t eg >t—'«cct <ff>ir. >tcrcr-»c>ocr-rec

CO

O

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En

XMH

xo»-<CMnor\irvjo«-<a -j-ega-aoGCr^C . crcooc r-KMrvj^rvjr-m^-rogDre^Orea^gDccregogooreegininrecocoNO O >o >fr re—•.—

i

ir —tre I-< I <-• --ieg I .-knj i rj^- i-i i i

I I I I I ->

c^rer^eg*t<^gogorexac<gvCgo^vCX^«^Or^inr^rr—rein^xo^rer-eg re .—t >}- re—i>rm(Mr-ti—<< r^—t^l-^e^gegeg<\J^\J^er'^re>J-.J-re>}'.>J-

egro^-;Do> 00'--^egegre>i-Lr.g^gor-••aocoo o—i-4-vOC—•reinvOr-C^rer-^^.^^^^^^.—i.-H,—i,-*.—t,—trgegrgegegrererererere>a->}-

meg «j-re •j-f*- s >o oo s ino re •—iac>t<—'0s co>4">iegi—i^rereccs co re c,

> >3-o<^r^{M^>3-e>iOinrecrrsJO'-«orvjm>ca0reore ocgore•4-rerea- r-ineginCTN <\j^>ocoegaN ^—*.~ r^rersjacccr^-r-if-^mm—ivCvCcom j-\m»-4f\,r~-mvCi«~ m.rnr-re^'n^xcOi-oc^—ivj-r^^ininNtr^^sC^rr

rMOs r--cor^eoCoo>tro>3"r~-r\joorvjvOcoocNoreinej.^cor^r^egr-roincrNjrNjfNje-rvjrvjc^r^o^tnr*--r*-eg nJ-^h(\j i

.-<.—i |.—irere

i i |.—i—

i

J- I I I II

<r;oegegr-ac>frOt—i <Nm -4- r-i rvi

-H00 I I

I

egvJ-vCino c •minr\ir-fv->frinrAccccrsjvCCr cr-4-vOoCt-Hrem>nejafv'vO^j- ^^r^ 1^>^(^^H>tre'^>^>t"^i'-^vj-r<^c\j>J-r^r^)rvir\irere>3-rr,

i^->t vl" 4-

ejrvvCcoooc—<egegrn>j-iningop~r~-cocr oo>d-NCccceg>3-gor"-»eggD_i _i _4 _i ^4 ^^ ^_ _i _< ^h ^-4 ^i ,_< ^j ^_ cm ej eg e . re re re re^ rr ^- ,j-

O invfco>i- revocoo^ o >o •—' in re re^ofMcCr-H^-cot-fOvOvONO^^mej>Ou-ov >^C'ifav «Org>t<~^->oa;>^cs re-H>c^jo>tr\jOK >tvOinocrfN «rx^- Os

sor^^r^cocr>ue^HC^orNj <^rMg3>^'HsOO>r^orNjr>-r-t'^vCcorerecrv-rersiir..-OMr*iC/Cer-tscxmoor: •4-oa-'", >tr~sD—'vj->too^^(\j>t<3r>-oo

C* ^lt-. xrer^o^vOir. ^reu> o^occcf^Lnov >ooc-egfMro>40cocoxo^-^vjrer^a —ivocr^reif- xrergvC in re ojee re >jac(M re>DOm^-ieg^-t

r" rr,>J- (%j| ^| I HHf\| | nr<l I (^ h|C|I I I I I I

>-i^Hm%j-f>inLnegr--r~^,

C7v crxr>-f^-—im, cro^regjxc fMC sOr^rr<. —imcr

(nj f-i >j- rn^^>3- re-^s^>rro,re^H>rr<".-Hre,rer\jevJf\ir<*rere,>j-r^. >j- >J-sJ-xj-

-^egu'xav C^o«^'^egre(^>rin>cr^f^xcrao^' l^f<-oe>j>tinr-x<-<incr^h^ _i ^-( ,_, ^^ ^-i^ ^-i_(_ _< ^i _t c\j (\i rg eg re re re re r^ re «J- >r >T

52

TABLE' VII

MATRIX' F'

Input Form

1 1500.002 252,803 714.004 528.005 109.006 -210.507 -2782.008 -4523.569 -5212.17

10 -5568.7411 -5395.8612 -4593.5813 -4654.9414 -4654.9415 -4593.5816 -5395.8617 -5568.7418 -5212.1719 -4523.5620 2390.7321 4825.5822 5395.0223 6125.1724 6229.2525 5802.5226 2738.2727 2390.7328 4825.5829 5395.0230 6125.1731 6229.2532 5802.5233 2738.27

1

1

1

1

11

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

11

1

1

1

1

2

53

TABLE VIII

EXPERIMENTAL RESULTS AT Yw = 34.w 5

PANELNO.

GAGE NO.STRAIN

READINGS AVERAGESTRAINS

O" maxTinax

Inte-rior

Exte-rior

Inte-rior

Exte-rior CT min

15 486 61 61 1094.02485 118 118 1008.22484 -50 -50 -922.41

21 27 9 459 17 25 21 851.68280 458 95 100 97.5 726.87281 457 04 -14 -5 -602.06

25 367 468 16 -43 -13.5 678.15368 467 -47 -99 -7 3 627.44369 466 29 11 20 -576.74

28 28 474 -19 34 7.5 960.4729 473 137 86 111.5 827.8630 47 2 27 -8 9.5 -695.25

30 89 477 9 9 9 457.6690 47 6 55 63 59 512.2791 475 -12 -20 -16 -566.87

34 19 480 -10 -21 -15.5 599.4320 479 76 65 70.5 568.2221 47 8 29 10 19.5 -537.02

38 13 483 -80 -7 6 -7 8 495.3714 482 61 62 61.5 803.4915 481 35 42 38.5 -1111.62

54

Comj

T^BLE IX

at Station Yw = 34>arison of Results .5. Load ing

336, 000 in. lbs. of torque. Principal axis is measured

from load reference axis, the Center Intermediate Beam,

which has 33° 58' sweep back.

PANELNO.

MAX. or' MIN.normal; stress

MAXIMUM SHEARSTRESS PRINCIPAL AXIS

Experi- Theore- Experi- Theore- Experi- Theore-mental tical mental tical mental tical

15 1094.02 1605.47 1008.22 1333.88 31° 52' 30° 12'

17 1530.10 1092.17 28° 16'

19 1047.07 895.67 38° 32'

21 851.68 678.85 726.87 845.91 40° 52' 40° 43'

23 -7 92.63 819.44 48° 05'

25 -602.06 -970.55 627.44 1033.12 38° 48' 49° 20'

REAR BEAM

28 960.47 767.42 827.86 870.20 44° 44' 47° 36'

30 457.66 626.57 512.27 556.54 39° 20' 45° 24'

32 813.00 669.19 40° 12'

34 599.43 851.95 568.22 761.82 37° 26' 39° 55'

36 -7 67 . 60 901.15 35° 49'

38 -1111.62 -1052.83 803.49 1160.03 27° 11' 33° 56'

55

TABLE X

COMPARISON OF THEORETICAL RESULTS AT STATION Yw = 34.5

(1) WITHOUT INCLUDING CENTER SECTION, AS IN REF. 2.

(2) INCLUDING CENTER SECTION

MAXIMUM MINIMUM MAXIMUMPANELNO.

NORMAL STRESS NORMAL STRESS SHEAR STRESS

(1) (2) (1) (2) (1) (2)

15 2234 1605 -1280 -1062 1757 1334

17 2025 1530 -651 -654 1338 1092

19 1420 1047 -630 -744 1025 896

21 1009 67 9 -67 9 -1013 844 846

23 827 846 -619 -793 723 819

25 994 1095 -418 -970 706 1033

28 813 97 3 -373 -7 67 593 870

30 714 626 -478 -486 596 556

32 835 813 -505 -525 670 669

34 1068 852 -47 8 -67 2 773 762

36 1575 1034 -575 -7 68 107 5 901

38 1982 1267 -1004i

-1053 1493 1160

56

TABLE XI

MATRIX OF ALL INTERNAL LOADS{AT}- [S] {P}

FROM IDEALIZED MODEL WITH CENTER SECTION INCLUDED

1 174.64054260

2 1325.450S2773

3 462.C38330C8

4 474.22583008

5 675.57104492

6 1:66.64038086

7 -597.00024414

8 -3451.01147461

9 -34<38. 75415039

1C -4256.81250000

11 -5356.. 39062500

12 -5181.C39C6250

1? -4892.48437500

14 -73 50.367187 50

15 -7291.81250000

16 -5 829. 66 796 « 75

17 -5258.5 1953125

18 -4711.71093750

19 -4013.84619141

20 - 34:7. 25048828

21 397;. 23242188

22 2286.73657227

23 2093.25366211

24 -1755.84497370

2 5 -5 79.25122^70

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

4 9

266.98681641

1781.35253906

3944.2023925^

-8407.73046875

-7190.99218750

-5754.24218750

-4264.46093750

-2847. 58496094

650.^9536133

716.78784180

392.5336^141

-136. 54725647

1679.07592773

6767. 17968750

6785.34375000

5224.67187500

985.44262695

10263. 53515625

3625.59106445

61 34. 81640625

510 3.66015 625

3034. 79296875

4810. 60546875

6030.46484375

57

TABLE HI

MATRIX OF ALL INTERNAL LOADS{AT}«[S]{P}

FROM IDEALIZED MODEL WITH CENTER SECTION NOT INCLUDED

1

2

3

4

5

6

7

8

9

1C

11

12

13

14

15

16

17

18

IS

2C

21

22

23

2 4

25

-92.323eQ676

1*92.31518555

-6G. 23043823

-222.62962341

-253. n «P3483 c

-94.43901C62

442.254636 67

-2286.37548828

-3785.3C151367

-4119. C 9765625

-4763.9843750C

-4985. 19921875

-4969.66796675

-4492.2734375C

-435^.52343750

-4 8 86,99609 375

-5C1 ?.?195312 c

- 4 7 6 5 . 9 1 C 1 5 6 2 5

-4219.2539C625

-3867. ^4985352

5143.6R75QCCC

2CC. 63256836

-379.16625977

- 1720.40966797

-1354.33C56641

26

27

28

29

30

31

32

33

34

35

36

37

38

39

4C

41

42

43

44

45

46

47

48

49

-11CC. 95019531

-939.57446289

-859.61694336

-3524.74560547

-3361.27685 54 7

-3227.84252 C 3C

-2977.405C293O

-2714.C4394531

767.79956C55

904.96679687

592.9165C391

-17.30769348

-720.509C332C

8591.921875CC

8726. 140fc25CC

7870.3671675C

4969.57031250

1C1C .43847656

546C.7343750C

8681. 625 COCCC

R357.C3125CCC

798C.82C3125C

4503.7617167?

769.419A335<;

*8

REFERENCES

1. Wehle, L. B., Jr., and W. Lansing. "A method for Reducing the

Analysis of Complex Redundant Structures to a Routine Pro-

cedure," Journal of the Aeronautical Sciences, pp. 677-

684, October 1952

2. Holgren, M. A., and C. L. Comfort. "Comparison of Experimental

Stresses and Deflections with Those Predicted by a Strain

Energy Method for an F8U-3 Wing Loaded in Torsion." Unpub-

lished Master ' s thesis, Naval Postgraduate School, Monterey,

California, 1963.

3. Smith, C. F. "Stress and Deflection Analysis of Aircraft Struc-

tures Using Strain Energy Theory in Conjunction with Electronic

Digital Computers," Report 10917. Dallas, Texas: Chance

Vought Aircraft, Inc., November 1957.

4. Bruhn, E. F. , and A. F. Schmitt. Analysis and Design of Aircraft

Structures , Vol. 1, Chapter A7 . Cincinnati, Ohio: Tri-State

Offset Company, 1965.

5. Lang, A. L. , and R. L. Bisplinghof f . "Some Results of Swept -

back Wing Structural Studies," Journal of the Aeronautical

Sciences , Vol. 18, No. 11, pp. 705-717, November 1951.

6. Garvey, S. J. "The Quadrilaterial Shear Panel," Aircraft

Engineering , Vol. 23, pp. 134-144, May 1951.

7. Eggertz, S. "Calculation of Stresses in a Swept Multicell

Cantilever Box Beam with Ribs Perpendicular to the Spars

and Comparison of Results," Report 54. FLGTEKNISKA

FORSOKSANSTALTEN, the Aeronautical Research Institute of Sweden,

Stockholm, March 1954.

59

START

CLEAR ARRAYS

FORM A , B , UC

1

PATT P-ATTQC T RETURN

APPENDIX A

PROGRAM ENERGY

YES— PRINT MATRIXSINGULAR

INVERT A ,T^ITE AI

CALL MATMPYCJ = A"'B

FORM G FRCM CJ ^YADDING TRIVIAI RELATIONS

WRITE G

TRANSPOSE G

GAIT MATMPYH = G

rF G , WRITE H

FORM P

PARTITION H FORM H21KDFORM '^22

WRITE H22

CAI.L GAUSS 3 RETURN

YESH PRINT MATRIXSINGUI AR

' ' 2 T "?°, I RITE >P-?I

3 ) (U

CALL MATMPYENP = H22I x H21

EN= -ENP

UTIN — I/EN

CALL MATMPYS = G X UNITN

WRITE S

CALL MATMPYAR = EN X PWRITE AR

CALL MATMPYAT = S X P

WRITE AT

FORM C

C-l/E ( H11+H12 XEN )

WRITE C

END

60

//ENERGY JOB ASZ 5,0 1FP , 5, 5 , MESSER SCHM IDT ,D.B . ,MSGLEVEL 1

// FXEC FORTCLG//FURT.SYSIN DD •

C COEFFICIENT MATRICES At B. AND UC ARE BASIC STRUCTURAL DATAODIMENSION A(50, 50). AI (50,50), B (50,50), CJ (50, 50).UC (50,50), x1G(50,50).H(50, 5O) t GT(5O,5O),H22(5O,5O),H12(5O,50) , UCG (50,50),2H? 1(50, 50) ,H22 1(50, 51), EN (16,33 ) , AR (16) , UNI TN(50, 50) ,S (50,50) ,

3AT(50),H12N(5O,SO) ,C (50, 50) .LIST (50),P(3 5) ,ENP(16,33 )

OEQUIVALENCE (A, C J, UCG, H22 , H21 , S) , (B,G,H12,C),(UC,H),HAI.GT.H2 2I ,UNITN,H12N)

C CLEAR ARRAYS TO RECEIVE INPUTDO 10 1=1,5000 10 J-l.50A(I ,J-) = 0.0B(I ,J ) - 0.0UC(I, J) = 0.0

10 A I (I, J ) = 0.0C READ NON REDUNDANT MATRIX A

31 CALL INPUT(50,A)GO TO (31, 32), NEXT

C READ LOAD AND REDUNDANT MATRIX B32 CALL INPUT(5 J, B)

GO TO (32, 33), NEXTC FORM ENEKGY COEFFICIENT MATRIX UC

33 CALL INPUT(50,UC)GO TO (33,35), NEXT

35 DO 36 I = 1,49DO 36 J = 1,49

36 UC(J,I) = UC(T ,J)C INVFRT THE A MATRIX

CALL GAUSS3(33,1.F-U ,A,AI,KER)GO T0(365 ,364),KER

364 PRINT 151151 F0RM4T(24H MATRIX A IS SINGULAR)36 5 PRINT 2 0037 PRINT 3838QF0RMAT(64H A MATRIX INVERSE BY GAUSS3, 9 COLUMNS, IE 1'9, 10 1

8

1PER PAGE//PRINT 100, (fAH T, J), J- It 9) , 1-1,33)PRINT 20)PRINT 100, ((AI (I, J), J = 10, 18), I = 1,33)PRINT 200PRINT 1 00 i ((AI (I, J), J =19, 2 7), 1=1,33)PRINT 200 ,.PRINT 300,C(AI ( I, J), J=2 8,33), 1=1,33)

DC FORMAT(/9F10.6 )

150 FORMAT(/11F10.6)200 FDRMAT(lHl)

C POST MULTIPLY AI WITH B MATRIXCALL MATMPY(33,33,49,AI ,B,CJ)

C FORM G MATRIX BY ADDITION OF TRIVIAL RELATIONS TO CJDO 40 1=34,49

40 C J ( I , I ) = 1 . )DO 41 1=1 ,49DO 41 J=l,49

41 G(I, J) = CJClt J)PRINT 200

42 PRINT 4343 F0RMAT(50H G MATRIX , FORMED FROM CJ WITH TRIVIAL RELATIONS//)

PRINT 100,((G(I,J ),J = 1,9),I=1,49)PRINT 200PRINT 100,((G(I,J),J=10,18), 1=1,49)PRINT 200PRINT 100 tCCGCIt J)t J=19,?7), 1=1,49)PRINT 20

J

PRINT 100, (CG (I ,J), J = 28,36) , 1 = 1,49)PRINT 200PR I NT 100, (CG(I ,J), J = 37, 45) , 1 = 1,49 )PRINT 200PRINT 4 00, ((G (I , J) , J =46,4 9), 1 = 1,49)

300 FORMAT ( / 6F13.7 )

400 FORMAT (/ 4F13 .7 )

C TRANSPOSE GDO 44 1=1,49DO 44 J=l,49

44 GT(I,J) = G( J, I)C FORM H MATRIX BY TRIPLE PRODUCT (GT)(UC)(G)

CALL MATMPY (49, 49,49, UC,G, UCG)CALL MATMPY (49, 49,49, GT, UCG, H)PRINT 200PRINT 45

61

45

50d6j0

51057

581

54

52152?

524

32 5

53

5M540

5 6

Oq

62

63

nor

818

59

FORMATPRINTPR INTPRINTPP INTPRINTPRINTPP INTPRINTPR INTPRINTPRINTFORMATFORMATRFAD LDO 510P 11) =

RE AD (5FORMATGO TOPARTITDO 5 4DO 54I I = I

JJ = JH22CI I

PRINTPR INTFORMATPR INTPRINTI N V E R TCALL GGO TOCPR INTPP INTPR INTPRINTDO 5 3DO 5^KK = I

H21CKKFORM RDO 54000 540SUM =DO 541SUM -

ENPCIDO 56TO 56ON ( I , JFORM y

DO 94DO 44UMITNDO 62UNIT'MDO 63DU 63K K =UN IT nIC

FORM SCALL IM-

PRINTPP INTFORMATPRINTPP INTPR INT1PR PITPR INT 1

R FOUNDun 8^SUM =DO 3 1

SUM =AH CI )

DO 59LISTCIPR INT

IX H FORMED BY TRIPLE,49),J) , J=l,9), I-

iJ) , J-10,18) ,

,J),J* 19,27),

,J ) ,J>28,36),

,J ), J=37,45; t

,J ),J=46,49),)

4)X P

P(I)) .NEXTF12.4, II )

NEXTTR IX

(44H MATR50?, re hc i

20?500, ff H(I200500, ff H(I20u500, CCHd20050), CC Hfl20060J, ff H( I

( /9F13.4( / 4F13.

OAD MATR

I

I = 1,35.

,58) I, (

( 12 ,

f 57,581) ,

ION H MA1-34,49J = 34, 49

-33-33, JJ ) = H(200522, f(H?2C/8F12.4 )

2V0522 t ((H22

H2? MATAUSS 3 (

1

525,524) ,

15 j

20 i

522, (Y H22522, (f H2?I = 34,49J-1,33

-33, J ") H ( I

EOUNOAMT1=1,16K=l, 33

. uJ - 1 , 1

6

SUM + H22IM ,J)»H21f J,K )

K) = SUM1=1,16J =1,33

) = -1.ATRIX UNI

PRODUCT, GTXUCXG//)

-1,49)

-1,49)

= 1,49)

= 1,49 )

-1,49 )

I, J)

fit J), J - 1,8), I - 1,16 )

f I, J) , J = °,16), I = 1, 16 )

R I X6,1.E-10, H22, H22I.KFR)KER

IM,J),J = !,8),I = 1,16)I (I , J ), J=9,16) , 1-1, 16]

,J)LOAD COEFFICIENT MATRIX, EN,

(I

= 1,33= 1,33

, J) = 1

» ENP(I, J

)

ITN BY PLACING EN BELOW UNIT MATRIX

(I= 1,33

, I) = 1 .

1=34,44J-1,33

-33I , J ) = FNMATRIX .

A T M P Y (4920.'80?f43H STR

15", (fS(I20j

US(i ,50,20.50,ANT

usu ,

LOAD= 1,16

33

I

r

J = 1

SUM + EM= SUM1=1,49

) = ] M? ">

!

1

CKK,J)S = (G)CUNITN)

,49,33, G, UNITN, S

)

ESS COEFFICIENT MATRIX,J), J=l,ll ) , I = 1,49)

J) , J=12,2?) ,1 = 1,49 )

J ),J»23,33), 1=1,49 )

MATRIX, AR

= (G) (UNITN) // )

( I, J)*Pf J)

*2

PRINT 60600FORMAT (50H REDUNDANT LOAD MATRIX. AR - H22I H21 P //

1 16H AR( 1-16)= Af ))

PRINT 700,(LIST(J> ,AR(J) ,J=1,16)700 FORMAT ( I 13X, I 2 , F20. 8 )

TOTAL UNKNOWN INTERNAL LOADS. AT- (S)(P)DO 640 1=1,49SUM- 0.0DO 641 J=l,33

641 SUM = SUM + S(I,J)«P(J)64^ AT( I) = SUM

PRINT 200PRINT 900

, .

9000FORMAT ( 45H MATRIX OF ALL INTERNAL LOADS. AT » (S)(P) //1 16H AT (I) - A C ) )

PRINT 1000, (L 1ST CI ),ATM) t 1 = 1,49)1000 FORMAT ( /13X,I2»F20*8)

DO 52 1=1,33DO 52 J=34,49

52 H12CI.J-33) = H(I f J )

E - 10300000.0 .•

INFLUENCE COEFFICIENT MATRIX. C - ( Hll + (H12) (EN)) / EDO 90 I - 1,33DO 90 K = 1,33SUM - 0.3DO 95 J = 1,16

95 SUM = SUM t H12(I

,

J)*EN(J,K)90 H12Nd,K •) = SUM

SCALE - 10000.DO 70 1-1,33DO 70 J-1,33

70 C CI , J ^ = CCH(ItJ) + H12N(I,J ))/E)*SCALEPRINT 200PRINT 73

733FORMAT (45 H MATRIX OF DFFLECTION INFLUENCE COEFFICIENTS /135H C =C(H11 +CH12HN))/ E)*(SCALE))PRINT 150, ((C( I, J ) ,J= 1,11) , 1=1,33 )

PRINT 200PRINT 150, ((C(I, J), J=12,?2), 1=1,33^PRINT 200

,

PRINT 153* CCCCIfJ) ,J=2 3,3 3) ,1=1,33)RETURNEND

,.

SUBROUTINE GAUSS3(N,EP,AS,XS,KER)IMPLICIT REAL»8(A-H),REAL*8(0-P )

REAL»4 AS(5D,53),XSC50,5 n )*EPDIMENSION (A 5 , 50 ) , X (50 , 50 )

DO 1 I=1,NDO 1 J=1,NA(I,J)=AS (I ,J">

1 X ( I , J ) = .

DO 2 K=1,N2 XCK,K)=1.C

1^ DO 34 L=l ,NKP=0Z=0.0DO 12 K=L,NIF (Z-DABS(A(K, L))) 11,12, 12

11 Z=DABS A K,LK P= K

12 CONTINUEIF(L-KP) 13,2>, 20

13 DO 14 J=L,NZ=A(L,J

)

A(L,J)»A(KP,J)1^ AfKP,J)=Z

DO 15 J-l.NZ=X(L,J)X ( L , J ) = X ( K P , J •)

15 X(KP,J)=Z20 IF(DABS(A(L,D) -PP)5 ),50, 3030 IF(L-Nm,34,3431 LP1=L+1

DO 36 K=LP1,NIF( A(K,D) 32,36,32

32 RATIO=-A(K,L)/A(L,L)DO 33 J=LP1,N

33 ACK, J)= A(K, J)-RATIO»A(L, J)DO 35 J=1,N

35 X(K,J)=X(K, J")-RATIO*X(L, J )

36 CONTINUE

63

3 440

41

4243

51

- r

i

21

ln

1•'

'.

1 . \

CUNTOP AI I-Nun 4

lf( I

I IP1DC 4s=s+XCI I

KER =UO 5Dfl 5XS(IkLTUKER^RETUENnSURRr i meon l

no l

SUMDC ?SUMen,^FTUendSUBRI'IMEruui( K( SREADFUR*"U() 1

IF CNT

INUE^ 1= 1

*1-I3 J = l1

i-ri)4-ii + i

2 K=IA CI I,, J) =(1

1 1 = 1

1 J=lt J ) =

RN2RN

HUT inNSI CM" 1= 1

. K=l= >. J

? J == SUMK ) =

,N

1,43,43

I P 1 , NK)»X(K, i

)

x( ii, j)-s)/a( ii,

m

,NX( I, J)

F WATVPY ( L , M, N, A, R, C )

AO'>,5 0, R(50,50), C (50, 5^1, L

iN

1 , M+ A ( I , J ) •

SUNer j,k)

CCNIF (

IF (

IF (

IF (

CO i

h.FTUPR PIrruMSTOPEND

RN

OUT INMS I ONVALL'N) , 12)

(r> , 2

\T ( 4'i 1 = 1

K( I )

INUEI l.NFI2.NlI3.NFI 4 . N E

n ciiRNt inAT ('

F INPUT (N, *

)

A ( N , N ) , K C 6 )

r.E(Kfl),Jl),fKC2),J2),(Kf3),J3),(K('4),Il) f

, (Kf*). 13)1) I1,J1,A1, I?,J2,A2, I3,J3,A3, I4,J4,A4, NEXT(212, F12.6), I 1 )

• 6.r,T. p; ) C-,0 ro l

r 'j

.*•

) a c 1 1 , j n - a i

. 0) A(I2,J2) = A2

. •' ) A CI3, J3 ) = A3

. .' ) A (14, JM = A42), NEXT

, K , II , J 1 , A 1 , I 2 , J 2 , A 2 , I 3 , J 3 , A 3 , I 4 , J4 , A4K = ',FS/ IX, 4(-M2,E12.M)

6) i

APPENDIX BPROGRAM RECROSE

START

1 = , L =

1 = 1 + 1

READ N,E,U

READ GAGE N0.,€;,^,e3

CORRECT FOR GAGE FACTOR

A= e^e3

B- €, -«.

c 2e£-k

0=1.0 - II

H=1.0 - U

R = SQRT (B*+C*)

%U - (V2H)R

°^v - 5/2 (A/G+R/H)

or = E/2 (A/G-R/H)

PHIPRI=l/2 ARCTAN 2 (C/B)

NO— PHIPRI = PHIPRI + 1.5707

PHIPRI = PHIPRI/.017U5

D = T (C0S2#PHIPRI)

cr = (cr +(T )/2+D/

v /nax mm'

I

CF m fa +<r )/2-D3 V m« T mm '/

7! = - T *SIN (2.0 PHIPRI)o max

NO— PRINT TITI.E

YES

PRINT DATA

Kl) YES-* GO TO 1

NO

END

©66

//////ccccc

RECROSE JUB ASZ5,01FPEXEC FORTCLG

FORT.SYSIN DO *

RECTANGULAR STRAINOF PRINCIPAL AXES,AND STRESSES ALONG

MESSERSCHMinT,D.B. ,MSGLEVEL 1

RUSETTE DATA REDUCTION FOR THE DET ERM I N AT I CNPRINCIPAL STRESSES, MAXIMUM SHEARING STRESSES,THE ROSETTE AXES 1 AND 3.

4iOOi n1021 )3

///

RLI

I

REEF.

CEEEr,

HABCRTSspI

ppDSSTI

wH

9SLI

IN

L

I

FF

FF

3'

3F33S3TR

C

EAD-0'0*I+1EAD1-E12-F23 = E3ORRE1-E12 = F23=E3= 1. i

-1.9-El*= E1-= C2.= SQRAUMAIGMAIGMIHI PRF CBHIPRHI PR= TAUIGMAIGMAAUO =

F flRITEKITEIGMA= L + 1F CLR ITF= 1

F CIORMAOR MAORMAHRMAS IGM14.71 PSI

' ,/,13, '

FTURNO

(5, 100") N,E,U

5,10* 1. JE* 1 . E* 1 . J FCTION* 1 .

r> 3

1 IGAGE,E1,E2, E3-06-06-06FOR A GAGE FACTOR = 1.95

J303*1

-uu

E3E30*E2)TfCB«X=(E/X=(E/N =(E /K= 'J .

r>

.GE.'iR=PH I

U=PH I

MAX»Ci = r<rs3 = rcs-TAUM.GT. 1

C6, 1

C6, 1

3, PHI1

. L T . 6(6, 1

-A•2) +(2.->2. .')

2.0)• CAT.0)PRR +PRK/0SC2IGMAIGMAAX*S) GO') 2

)

^3)PRO,

CC*»2))*H)) *R• CCA/G) *(R/H))CCA/G)- CR/H))AN2 fC,PO)GO TO 21.5707963.1745329E- r,

l

. 1*PHIPRR")X+SIGMIN)/2. +X+S IGMIN)/2.0) -DINC2.0*PHIPRR)TO 3

IGAGE,F1,SIGMAX,E2,SIGMIN,F3,TAUMAX,SIGMA1,PHI^RR,TAUO

6) GO32)

TO 4

.LT.N) GO 'TO 1

T C IH,E17.7,F10.5)T CI 8.3F12.5;T < 1M1 >

.

T CT6,'R0SFTTE NUMBER' , I B , // , T 16 , El= ,F14.7, IM/IN ,T45,AX»',F14.7, ' PSI' ,/,T16, 'E2- '«F14.7, ' IN/ I N '

, T45 , 'S IG V I N-' ,

, ' PSI ',/ , T16, -:3=' ,F14.7, ' IN/IN ' ,T45, 'TAUMAX= '.F14.7,',//, Til, 'SIGMA U',F14.7, ' P S I' , T45 ,

' PH I PR=',F14.7,' RADIANT11,'SIGMA 3=',F14.7, ' PS I' , T45 , ' <>H I PR.' ,F14.7, 'DEGREES',/,TAU o- ',F 14.7, 'PSI ' ,//

O.SYSIN CD *

H1521252830\h38

484

'. 1 ,3-3j»HH

/«

3621•13

7q

157861

'8

11897

-73,1 1159,70,61

118,

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5

5G55n

-13. , 5-5.2 0. \ J9.,5

-16..019.,538.,5

-50.,0

66

INITIAL DISTRIBUTION LIST

NO. COPIES

1. Defense Documentation Center 20

Cameron StationAlexandria, Virginia 22314

2. Library 2

Naval Postgraduate SchoolMonterey, California

3. Commandant of the Marine Corps 1

Headquarters, U.S. Marine CorpsWashington, D. C. 22214

4. Chairman, Department of Aeronautics 2

Naval Postgraduate SchoolMonterey, California

5. Professor Charles H. Kahr 3

Department of AeronauticsNaval Postgraduate School

6. Professor Ulrich Haupt 1

Department of AeronauticsNaval Postgraduate SchoolMonterey, California

7. Major Donald B. Messerschmidt 2

381-A Bergin DriveMonterey, California

8. Dr. E. S. Lamar 1

Chief ScientistNaval Air Systems CommandDepartment of NavyWashington, D. C. 20360

9. Mr. G. L. Desmond 1

Aerodynamics and Structures Technology AdminstratorDepartment of the NavyNaval Air Systems CommandWashington, D. C. 20360

10. Commander 1

Naval Air Systems CommandDeaprtment of the NavyWashington, D. C. 20360

11. Prof. A. E. Fuhs 1

Department of AeronauticsNaval Postgraduate SchoolMonterey, Calif. 93940

67

Security Clarification

DOCUMENT CONTROL DATA • RAD(Security claatitication ol tltlt, body of mbmtrmct and indexing annotation muat ba antatad whan tf» ormrall taaott la clamaltlad)

1. ORIGINATING ACTIVITY (Corporal* author)

Naval Postgraduate School

Monterey, California

2a. REPORT SECURITY CLASSIFICATION

Unclassified2fc CROUP

3 REPORT TITLE

COMPARISON OF THEORETICAL AND EXPERIMENTAL STRESSES IN AN F8U-3 AIRPLANE WINGLOADED IN TORSION

4. DESCRIPTIVE NOTES (Typa ol rmport and tncluaira dataa)

AUTHORfS) (Lmat nam*, Unit nmma, initial)

Messerschmidt, Donald B.

« REPORT OATE

December 1967

7a. TOTAL NO. OF PACES

66

7b. NO. OF REFS

• S. CONTRACT OR SRANT NO.

•. PROJECT NO.

• a. ORIGINATOR'S REPORT NUMEBRfSJ

• 6 OTHER REPORT NOfSJ (Any otharmia tapori)

that may

10. AVAILABILITY/LIMITATION NOTICES

11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

Naval Postgraduate School

Monterey ,. , CaL ifprnia

IS- ABSTRACT

The stresses near the root of an F8U-3 airplane wing loaded in torsionwere found using the matrix-force method of analysis. The wing was represen-ted by a simplified, idealized model consisting of axial-load carrying flangebars and shear carrying cover plates. The theoretical stresses in the ideal-ized model were found by using the IBM 360/67 computer to solve for the redun-dant loads. These stresses show favorable comparison with the experimentalstresses determined from strain gages mounted on the cover skin of the wing.This study shows that the matrix-force method is applicable to thick-skinnedwings

.

DD FORM1 JAN 84 1473

Security Classification

69

UNCLASSIFIEDSecurity Classification

KEY WO R DS

Matrix-force method

Strain energy analysis

Swept-wing structural analysis

DD ,?o":..1473 '«*«, UNCLASSIFIEDS/N 0101-807-6821 Security Classification

thesM558

3 2768 00416481 4

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