The evaluation of matrix elements for the analysis of swept-hack wing structures by the method of obligue coordinates

^ REPORT No. 44 TECHNISCHE HOGESCHOOL S VLIEGTUIGBOUWKUNDE Kanaalstraat lO - DELFT O

S '^^FT 2 6 Mei 1951

THE COLLEGE OF AERONAUTICS

CRANFIELD

THE EVALUATION OF MATRIX ELEMENTS FOR THE

ANALYSIS OF SWEPT-BACK WING STRUCTURES

BY THE METHOD OF OBLIQUE CO-ORDINATES

by

S. R. LEWIS, B.Sc.

Department of Aircraft Design

This Report must not be reproduced without the permission of the Principal of the College of Aeronautics.

TECHNISCHE HOGESCHOOL VLIEGTUIGBOUWKUNDE Koaaalattaat 10 - DELFT R e p o r t No. l+JLj. A p r i l ,

1951-26

Me* ^^^^

THE COLLEGE OF ABRONAUTICS

C R A N F I E L D

The evaluation of matrix elements for the analysis of swept-hack wing structures

by the method of obligue coordinates

by

-S. R. Lewis, B.Sc.

oOo

SUMMARY

This report is an addition to the College

f 1)

of Aeronautics Report No. 31 • The purpose of the report is to enable r^ne to obtain the matrix elements • used in the analysis of swept-back wing structures oy obliq.ue coordinates in a very rapid manner.

Part I gives the numerical evaluation of the matrix element A.. together with tables and graphs. The parameters chosen are those common to aircraft design.

Part II gives the numerical evaluation of a.. and a^ ^ together with tables and graphs.

Part III gives approximations for certain section constants, and the methods of extrapolation and interpolation of the constants from the curves of figures 3 - 1i)- These simplified expressions give a reasonable degree of accuracy and are satisfactory for preliminary investigations.

a„ a. _R E o -2-NOTATION

= section area of the stringer. = section area of the rib boom.

= -stringer pitch measured parallel to the ribs. = rib pitch measured parallel to the stringers. = thickness of the top and bottom skins.

('7c/2-a) = angle of sweepback of the mid-chord point line (see fig, 1 ).

s Modulus of Elasticity.

= Poissons ratio. Value taken as 0.3 for all calcT-.lations.

A.., a. , (i = 1,2,3J D = 1s2,3) = Matrix elements for '' "'•'' the determination of the stresses and

strains.

Ox, Oy, Oz = axes of reference.

i 1 t?

c

i b ^ ^ -\ 1 1 , 1 b 1> ,, ^ Figure 1 Figure 2

-3-Part I.

Values of the matrix elements A..

o _ . A. A- ^^ for varying values of a^ and __Ë_ , --, .

a g t a ^ t AR TABLE a 30 35

k5

55 60 90 a g t ajjt = .5 A^^Et A^^Et A22Et =.2 'A23 E t A33Et 0 . 9 3 0 0 . 5 1 6 1.50 1. 18 1 . 7 5 3 . 5 0 0 . 8 8 7 0.1+11 1 . 3 5 l . l i ; 1.61 3 . 3 1 O . 8 I 5 0 . 2 2 9 1.09 1 . 0 6 1 . 3 5 3 . 0 3 0. 757 .0779 .8i+9 0 . 9 7 6 1 . 0 7 2.814. 0.732 .01i;9 .729 .938 0.930 2.77 0 . 6 7 9 -0.121 0 . 3 6 7 . 8 i ; 7 0.1+77 2 . 6 U 0 . 6 6 0 - 0 . 1 6 8 0 0 . 8 1 2 0 2.71+ ^S* 2R_ aj^t =.5 =.6 A^^Et A^2Et A^3Et A22Et A23Et A,,Et 0 . 8 5 8 0 . 3 5 0 1 . 2 5 0 , 8 0 2 1.19 2 . 6 6 0 . 8 i ; 1 0 . 2 8 3 1. 17 0 . 7 8 3 1,11 2 . 6 0 0 . 8 0 1 0 . 1 8 9 1.01 0-71+3 0.91+7 2 . 5 2 0 . 7 5 5 . 0 5 6 0 . 8 2 5 . 7 0 2 0 . 7 7 1 2 . 5 1 0 . 7 3 2 . 0 1 0 8 . 7 2 5 . 6 8 2 0 . 6 7 6 2 . 5 2 0.671+ - 0 . 0 9 0 1 0.381+ . 6 3 2 0 . 3 5 7 2 . 5 7 0.652 -0.126 0 0.613 0 2.71 agt ^R aj^t

=.5

=i.q A^^Et A^2Et A^3Et A22Et A23Et A,,Et 0 . 8 2 1 0 . 2 6 5 1. 12 0 . 6 0 7 0 . 9 0 2 2.23 O . 8 I 7 0 . 2 1 5 1.06 0 . 5 9 6 0.81+6 2 . 2 2 0 . 7 9 3 .121+ . 9 6 0 . 5 7 3 0.730 2.25 i 0.751+ .01+37 . 8 1 1 .51+8 0 . 6 0 2 2 . 3 3 0.732 .0085 .722 . 5 3 6 0 . 5 3 1 2 . 3 8 0 . 6 7 2 - 0 . 0 7 2 0.391+ . 5 0 5 0 . 2 8 5 2 . 5 3 0.61+6 -0. 102 0 0.1+92 0 2 . 6 9 / T a b l e I I

-1+-TABLE I I a * H . = 2

1

«E*

j ' ' E * - '

[ a g t AR i ^R^ A^^Et • ^12^"^ A^ 3Et A22Et A23Et A33Et A^^Et A^2Et A^3Et A22Et A23Et ' A33Et 1 A^^Et i A^gEt • A^3Et ^ A22Et ^ 2 3 ^ *

A,,Et

1 33 i 30 1.1+8 0 . 8 2 2 2 . 3 8 1.35 2 . 2 5 1+.92 1.31 0.531+ 1.90 0 . 8 7 6 1.1+6 3 . 6 1 1.22 0 . 3 9 5 1.67 0.61+9 1.08 2 . 9 8 35 1.38 0 . 6 3 8 2. 10 1 . 2 5 1.96

k'k5

1.27 0.1+26 1.76 0 . 8 3 1 1.31 3.1+2 1.21 0 . 3 2 0 1 . 6 0 0.621+ 0 . 9 8 3 2 . 9 1 1+5 1.21 0 . 3 3 9 1 . 6 2 1.09 1.50 3 . 7 ^ . . 1 . 1 1. 18 0 . 2 3 6 1.1+8 0 . 7 5 9 1.01+ 3 . 1 2 1.16 0 . 1 8 1 1.1+1 0 . 5 8 2 0 . 8 0 0 2 . 7 9 i 55 1.08 0 . 1 1 2 1.22 0 . 9 8 0 1. 11 3 . 2 6 1.08 0 . 0 8 0 3 1.18 0.701+ 0 . 7 9 8 2 . 9 0 1.08 0 . 0 6 2 6 1.16 0.51+9 0 . 6 2 2 i 2 . 7 0 ï 60 ! 1.01+ 0 . 0 2 1 0 1 . 0 3 0 . 9 3 8 0 . 9 3 6 3 . 0 7 1 . 0 3 0 . 0 1 5 3 1 . 0 2 0 . 6 8 2 0 . 6 8 0 2 . 8 2 1 . 0 3 0 . 0 1 2 1.02 0 . 5 3 6 0 . 5 3 5 2 . 6 7 ; 75 i 0 . 9 3 2 - 0 . 1 6 6 0.501+ . 8 5 5 0.1+53 2 . 7 1 0 . 9 2 3 - 0 . 1 2 3 0 . 5 2 6 . 6 3 7 0 . 3 3 8 2 . 6 5 0 . 9 1 9 -0.0981+ 0 . 5 3 9 . 5 0 8 0 . 2 6 9 2 . 6 2 i

90 1

0 . 8 9 7 - 0 . 2 2 8 0 1. 10

0 1

2 . 8 0 1 0 . 8 8 1 1 - 0 . 1 7 1 0 0 . 8 2 9

0 1

2 . 7 5

0.872 1

- 0 , 1 3 7 0 0.661+ 0 2 . 7 2 TABLE I I I

1

Aq a g t

h -2

1 Aq \ a g t = - 0 5

K - 6

a g t Ap

14-^-°

A^^Et A^ 2Et A^3Et AggEt A23Et A33Et A^^Et A^2Et A^3Et ^ 2 2 ^ * A23Et A33Et A^^Et A^2Et A, , E t 13 A22Et A23Et A33Et 30 1.60 0 . 8 8 7 2 . 5 7 1.39 2 . 3 5 5 . 2 3 1.1+0 0 . 5 7 1 2.01+ 0 . 8 9 2 1.512 3 . 8 0 1. 30 0.1+21 1.78 0 . 6 5 7 1.11 3 . 1 3 35 1.1+8 0 . 6 8 5 2 . 2 5 1.27 2 . 0 3 1+.69 1.35 0.1+55 1.88 0.81+1 1.35 3 . 5 9 1.29 0.31+0 1.70 0 . 6 2 9 1.01 3.01+

1+5 1 55

1.29 0 . 3 6 1 1 . 7 3 1.10 1 . 5 3 3 . 8 8 1 . 2 5 0 . 2 5 1 1 . 5 7 0 . 7 6 2 1.06 3.21+ 1.15 0 . 1 1 8 1.29 0 . 9 8 1 1.12 3 . 3 3 1. 11+ 0.081+9 1 . 2 5 0.701+ 0 . 8 0 3 ; 2 . 9 7 1 . 2 3 j 1.11+ 1 0 . 1 9 2 ] 0 . 0 6 6 2 1.1+9 1 . 2 3 O.58I+ 0.51+9 0 . 8 1 3 0 . 6 2 6

2.89 1 2.77

60 1.09 0 . 0 2 2 2 1.09 0 . 9 3 8 0 . 9 3 7 3 . 1 3 1.09 0 . 0 1 6 1 1,08 0 . 6 8 2 0 . 6 8 1 2 . 8 7 1.09 0 . 0 1 2 7 1.08 0 . 5 3 6 0 . 5 3 5 2 . 7 3 75 0 . 9 7 7 -0.171+ 0 . 5 2 8 . 8 5 6 0.1+1+9 2 . 7 3 0 . 9 6 8 - 0 . 1 2 9 0 . 5 5 2 . 6 3 8 0.331+ 2 . 6 7 0 . 9 6 3 - 0 . 1 0 3 0 . 5 6 5 . 5 0 8 0 . 2 6 6 2 . 6 3

90 1

0 . 9 3 9 ] - 0 . 2 3 8 0 1 . 1 6 0

2.80 1

0.922 1

- 0 . 1 7 9 0 0 . 8 6 7 0 1 2 . 7 5 j 0 . 9 1 1 -0.11+3 0 0.691+ f 0 2 . 7 2

COLLEGE OF AERONAUTICS REPORT No 4 4 . FIG 3 A , | X STt: 3 C 4 - 0 S O 6 0 7 0 8 0 S O

£

COLLEGE OF AERONAUTICS REPORT No 4 4 .

FIG 4

Agjg^x E t

COLLEGE OF AEPONAUTICS PEPORT No 4 4 .

FIG 5

COLLEGE OF AERONAUTICS REPORT N o 4 4 .

FIG 6

A , j >C C t

COLLEGE OF AERONAUTICS REPORT No 4 4 FIG

7

A , 2 ^

^ ^

Aai

K E t S O

COLLEGE OF AERONAUTICS REPORT No 4 4 FIG 8

A « x E t

_»I5

A » i >c Cb

3 0 4 0

_{s o}

6 0 T O 8 0

_ao

COLLEGE OF AERONAUTICS REPORT No 4 4

FIG

9

A « X E t

COLLEGE OF AERONAUTICS REPORT No 4 4 . FIG lO asb Aft aot" - . 5 ' S

©

ast A^ Opt =.|

®

A R

%-=os

®"

2-5 2'0 A a ' K X i S t _^23" 1-5 A s 2 X C b i*o

®^'*\^

\ , ^ •

ks

^ \ \ . \

N,

3 0 4"© So 6 0 7 C 8 0 9 0

COLLEGE OF AERONAUTICS REPORT No 4 4 . FIG

lO

asfe Aft =.-5

®

Qst _{A H} =.i =-a

®

Aa ^ = • 0 5

0"

2 ' S 2-0 A ^ ' K X . E t _^23 1-5 A 3 2 X £ b 1 0

\\®"

©^'^v^

. \ , ^

V

^ • ^

s.

3 0 4'®

_So

6 0 T O 8 0 9 o

COLLEGE OF AERONAUTICS REPORT No 4 4 2 0 I'S i-e FIG I I 1 - ^ 1*2 l-O

^ 2 3

X £ t

A 3 2

X E t 8 ^ \ >

X

3 C ^ O 5 0 GO

cC

T O

_ao

9 0

5

-Part II

U i l

(Et)^

1 ^ Cl a g t - 5 ' a ^ t - ^ ^ i A q Ap

|agV.3^a^--0

U s

AR

^

U s

AR | a g t - ^ ' a j ^ t - ^

h ^ • 1 ^^^ 1 0

a g t - 0 5 . a ^ t - 2

! Aq Ap i Aq Ap 30 8 . 8 5 1 3 . 0 3 1 7 . 2 0 5 . 5 6 8 . 5 6 1 1 . 5 5 5.11+ 8 . 0 0 1 0 . 8 5

35

5.U3

-1 7 . 9 1 1 0 . 3 9 3 . 5 0 5 . 2 5 6 . 9 9 3 . 2 6 i+.92

6.57

t

1+5 i 55

2.59 j 1.51+

3.68 1 2.1i;

i+.78 1.7i+ 2 . 5 0 3 . 2 6 1 . 6 4 2 . 3 6 3 . 08 2.7i^ 1 . 0 7 1.i+9 1.91 1.01 1.1+1 1.81 60 1.26 1.71+ 2 . 2 1 . 8 9 1.22 1.56 .8i+7 1.17 1.i+8 75 . 8 6 1 . 1 5 1.1+1+ . 6 2 . 8 1 ^ 1 . 0 5 . 60 . 8 0 1 . 0 0 90

.76

1.00 1 . 2 5

.56

.71+ . 9 3 . 5 3

.71 1

. 8 9 V a l u e s of (a. .) •^ J

a°

1^11

E t

K

JEt

h i 3 .

l E t

1^22

E t ^ 2 3 . E t j E t 30 8 . 7 9 7 . 2 5 - 7 . 6 1 32 i 35 7 . 3 8 5 . 9 3 - 6 . 2 6 8 . 7 9 ; 7 . 3 8 5 . 8 2 1+.U8 -1+.77 5 . 8 2

-7.61 -6.26 1-1+. 77

1 i

7 . 3 6 1 6.01+1 U . 5 8 1 ; 1+3 3.1+6 2.31+ - 2 . 5 3 3.1+6 - 2 . 5 3 1^5 ! 50 3.1 ! 2,kk j

2.02I 1.44

- 2 . 2 h 1 . 5 7 3 . 1 - 2 ? 2 . i ; 2 2 . 1 0 2.1+U 55 ] 60 2 . 0 [ 1 . 6 9 I . 0 6 I . 8 0 - 1 . 1 5 2 . 0 - . 8 5 1.69

-1.571-1.15I-.85

1.51 j l . 1 3 i . 8 7 7 5 ' 1 . 2 2 .1+2 - . 3 1 6 1.22 - . 3 1 6 .1+8 90 j I . I O I . 3 3 j 1 0 1. 10 0 . 3 8 ^12 E t

S 3 .

E t « 2 3 = ^21 E t - 5 1 E t ^32 Et ~ Et •

COLLEGE OF AERONAUTICS REPORT No 4 4

FIG 12 2 0

COLLEGE OF AERONAUTICS REPORT N « . 4 4

FIG 13

COLLEGE OF AERONALTICS REPORT No 4 4 FIG 14 Et. -o-o - T ' O - 6 * 0 - 5 - 0 3 "^•O - 3 ' 0 -a*o - I ' O

o

\ t \ \ \ \ \ \ \ \ \ \ \

A

\ \

VKVUUCS o r (<i-u)fa POfi? vfowïaNriisio V5«VL.UES o r A ^ i s i o x e

Et

Et.

E t Et

V \ \ \ \ \ \ \ '^s ^^"^^ •• _{»«» ^^__} 3 0 ^€> S O 6 0 _-r©

_{s o}

_{s o}

Part III

Formulae giving approximate values of the matrix-elements with maximum percentage variation from the actual

calculated values.

The following expressions (1 and 2) hold for angles of sweepback between 30° and 60°.

/ A \ A A^^Et = j - .015 ~j + .012] (60 - a) - .78 -^-^ + 1.12

\ S / S (1) The maximum percentage error for the range of parameters chosen in figure 3 is ± 1 0 percent, and this occurs at a = 30°. For larger values of a the error will be less.

A^^Et = (- .0105 - ^ + .0126) (60 - a) - . L^88 -^:r +1.01

2 2 \ aj^t / a^^t

(2) The maximum percentage error for the range of parameters chosen in figure L|. is i 12 percent, and tnis occurs at a = 30°. For larger values of a the error v;ill be less. Greater accuracy can be obtained by direct extrapolation or interpolation (see example (5) below).

Simple linear expressions for the other matrix components A,-,, A,,, A,^ etc. would lead to large

in-33 13 3"^

accuracies and the best method of obtaining these values is by extrapolation and interpolation of the curves, figures 9-15.

The examples below indicate the method of obtaining the values of A,.,, A. ^ etc.

33 "3 Example 1 A^-^Et _{^ ^}

a = 1+3°

A -

= •

288

^ = .

05

;

it is satisfactory to assume a straight line variation between two parameters chosen in figure 5 for 5 ^ with

Ap ^ ^ Ao Ao S

A - constant. Similarly, for - ^ with - £ ^ constant ajjt a ^ a g r a straight line variation is satisfactory.

Ag A ^

^—1^ = . 1 and ^—^ = . 05 would g i v e S R

A33Et = 3^88_^-3^l8 X . 1 5 + 3.88

-7-A,

_S

_{" K}

-—r

= .5 and —-r = .05 would give

agt a^^t

A-._Bt = ^-^^

7 ^'^

X .15 + 3.09

33 ->'' .1+

= 3.31 (1+)

^s

for the value of

—+

of .288 by interpolation of the

S

values (3) and (1;) above, we have

A^^Et = U. 11+2 - ljjL^^2 - 3.31 X .188 = 3.76.

The actual calculated value of A,^Et is 3.81, which shows

a percentage error of 1 . 3 % .

Example 2 A,„Et

The assumptions are as for Example 1, (see figs. 6 and 7 )

a = 1+3° A = • 288 X = . 05

_{agt aj^t}

S R

—-r = . 1 and

~rr =

. 05 would give

agt a^^t

A^^Et = ' ^ ^ ^ ^ ' ^ ^ X . 1 5 + .391+

= .kk (5)

Aq Ap

- ~ r = . 5 and -rrr = . 0 5 would g i v e _{a g t a^^t}

A,„Et =

_{12 . [4.}

'^^

"1 '^^"^ X .15 + .26

= .287 (6)

^s

for the value of ~ V = .288 by interpolation of the values

a„z

(5) and (6) above, we have

A^gEt

= .kk -

'^^ : '^^"^ X .188 = .368.

The actual calculated value of A.p^t is .381, which shows

a percentage error of 3-k % •

Example 3 A.,Et

The assumptions are as for Example 1, (see figs. 8 and 9) a = 43° ^ = . 288 A = . 05 _{agt aj^t}

Ag A„

-—T- = ,1 and -—r = .05 would give agt aj^t A Et = 1-688 - 1.53^ X ^^5 ^ ^,688 13^" ~ .1+ = 1.71+5 (7) A, cs A p - ^ = . 5 and •—: = . 05 would g i v e a g t a^^t A^ Et = i i i M - Z _ i i ^ X . 1 5 + 1.11+8 = 1.222 (8)

-8-^S

for the value of -=T- = .288 by interpolation of the aqt

values (7) and (8) above, we have

A^ ^Et = 1.222 + ''•7^^ - 1-222 ^ ^ ^ gg ^ 1. i;67.

The actual calculated value of A. ,Et is l.lj.72, which shows a percentage error of 0.3 % .

Example U Ao,Et

The assumptions are as for Example 1, (see figs.10 and 11) A.

a = 1+3° ^ = . 2 8 8 ^ = . 0 5 a, t a ^ t -^

A _{S i>}

~—r = . 1 and T-rr = . 05 would g i v e _{a g t a^^t} Ag^Et = li^23-^-L^ll X . 1 5 + 1.575

= 1.755 (9)

A, _{'S ti}

-—T- = . 5 and -~r = . 05 would give agt a^^t

A23Et = ^-^^ " -^"^^ X .15 + 1.1+1

= 1.573 (10) ^S

for the values of - ^ = .288 by interpolation of the aqt

values (9) and (10) above, we have

Ao,Et = 1.573 + 1-75^ - 1-523 X .212 = 1.67. The actual calculated value of A^^Et is 1,77 which shows a percentage error of 5 . 6 % .

Example 5

The values of AppEt can be obtained by direct interpolation or extrapolation of the curves of figure ij..

Aq

For large variations in — ^ there is very little change agt Ar,

in value of A^joEt for fixed value of _£_ '^'^ A A a„t

A A R a = U3° - ^ = .288 ~ \ = .05. _{agt aj^t}

The value can be obtained by extrapolation of the curves A

b e t w e e n v a l u e s of _B_ = . 2 and . 6 , r e s p e c t i v e l y . apt

Taking mean values of the curves Ap - ^ = . 2 A„„Et = 1 . 1 0 ajjt 22 ^ = ,6 AppEt = . 7 6 aj^t 22 T h e r e f o r e f o r - ^ = . 0 5 , A,„Et = 1.10 + ^-^^-'1^ x . 1 5 ^ = 1 . 2 3 . The actual calculated value of AppEt is 1.31» which shows

0/ a percentage error of 6 9o •

-9-CONCLUSIONS

(1) Variations in the values of A.. Et for specified

Aq 11 Ap values of -S_. are very small for large variations in

-—r-aqt ^p throughout the range of a.

This term occurs in the expression for direct stress and shear stress and indicates that the stresses are not critical for guite large variations in the

parameters.

(2) Variations in the values of App Et for specified values of _£— are very small for large variations in Aq apt

- ^ , for values of 30 < a < 7 5 • Beyond this the varia-aot

tion is large.

(3) Variations in the values of A,, Et for specified

^ A q ^^ A p v a l u e s o f _ £ _ a r e l a r g e f o r small v a r i a t i o n s i n -JL- . a ^ aj^t (Ij.) V a r i a t i o n s i n the v a l u e s o f A , p E t or A p . / E t f o r s p e c i f i e d v a l u e s o f - § — a r e r e a s o n a b l y l a r g e f o r A-n ^S'*' variations in SL— . a^

(5) Variations in the values of A. ,/Et or A.^./Et for specified values of - ~ are reasonably large fo..-»

Ap ^S* variations in -—. ,

aj^t

(6) Variations in the values of A^^-Et or A^r,/Et

Ap '^'> -"^

for specified values of -J-r are reasonably small for Aq ^R^

large variations in - ^ . agt

ooOoo

I am indebted to Dr. Kirkby and the Computing Section of the Aerodynamics Department for the valuable aid given in the computation of the numerical examples.

Reference 1

College of Aeronautics Report No. 31, "On the application of oblique coordinates to problems of plane elasticity and swept-back wings", by W. S. Hemp.