ROYAL A~"'- ~, ~ SL.tSit i~J£Nr ~ c:.,, ''-" = i~i~. 0 i ; ~ D . . . . . . . .

M I N I S T R Y O F S U P P L Y

R . & M . N o . 3 0 7 3 (15,610)

A.R.O. Technical Report

A E R O N A U T I C A L RESEARCH C O U N C I L

REPORTS A N D M E M O R A N D A

On the Post,Buckling Behaviour of Stiffened Plane Sheet Under Shear

.By E. H. MANSFIELD, M.A.

© O'own copTig,kt 19 5 8

LONDON :

/

HER MAJESTY'S STATIONERY OFFICE

I958 P~Ic:~ 7s. 6d. NET

k

On the Post-Buckling Plane Sheet

Behaviour of Stiffened Under Shear

By E. H. MANSFIELD, M.A.

COMMUNICATED BY THE DIRECToR-GENERAL OF SCIENTIFIC RESEARCH (AIR),

MINISTRY OF SUPPLY

Reports and Memoranda No. 3 o 7 3 "

November, 19 5 2

Summary.~The post-buckling behaviom" of a flat plate reinforced by stringers and frames is considered theoretically,_ attention being concentrated on the case of pure shear loading. Formulae and graphs are presented for the rapid determination of the shear stiffness, shear strain and induced compressive stresses in stringers and frames.

The analysis is an extension of work by Kromm and Marguerre.

1. Introduction.--It is well' known that a reinforced sheet can carry shear loads considerably in excess of the initial buckling load. The behaviour of the sheet under very high loads has been fully explained by the tension-field theory of Wagner 1. The transition from initial buckling to a pure tension field is of particular importance in the aeronautical field and it has been considered theoretically by Kromm and Marguerre ", Kuhn ~ and LeggetP and experimentally by CrowtheP, Van der Neuff and others.

The equations derived by Kromm and Marguerre are complicated, and for pure shear loading the solutionswere confined to the four limiting combinations of zero and infinite direct stiffness of the stringers and frames. In the present report these equations are modified so that t h e post-buckling behaviour may be predicted quickly with moderate accuracy for any combination of stringer and frame stiffness. Formulae and graphs are presented for the post-buckled s ta te to determine the overall shear stiffness of the sheet, the overall shear strain of the sheet, t h e stringer stress, and the frame stress.

L1. Assumptions.--The following assumptions are made regarding the structure (see Fig. 1)::

(a) The material is elastic

(b) The frames are not attached to the skin

(c) The stringers have negligible torsional rigidity

(d) The frames and stringers do not bend. Assumptions made in the analytical solution are given in the text and in Appendix I.

* R.A.E. Report Struct. 136, received 8th February, 1953.

2. Statement of the Prob lem. - -The structure of the reinforced plate is specified, apart from ,overall scale factors, by the relative direct stiffnesses of the stringers (S) and t h e frames (F). The loading on the reinforced plate is specified by the shear stress ~ and the nominal direct stress a. T h e six unknown quanti t ies are:

7 the overall shear strain

as the average stringer compressive stress

ar the frame compressive stress

'~ the buckle wave-number

the buckle wave-angle.

zl the max imum plate deflection.

2.i . Use o f Non-dimensional Symbols . - -Considerable simplification in the analys is and in the p resen ta t ion of results is effected by us ing non-dimensional symbols. The symbols men t ioned above are non-dimensional because they are expressed in terms of their initial buckling values. Precise definitions are given in the List of Symbols. I t will be noted tha t at the onset of buckling under pure shear

= r = ~ = ~ = 2 ) (2)

a ~ a s ~ ~ y ~ A ~ 0 J ' . . . . . . . .

while at the onset of buckling under pure compression in the direction of the stringers

a = as = ~ = 1 )

J (2) " c = 7 = c ~ = A = 0 ' . . . . . . . .

provided F is zero.

The amounts of material in the stringers, frames and plate are in the ratio

S : F : I .

A value of A equal to uni ty corresponds to a deflection of about 1.7 t imes the skin thickness.

2.2. Equations for So lu t i on . - -The six equations containing the six unknowns 7, as, ar, ~, ~, d are, essentially, those obta ined by K r o m m and Margnerre * who used a strain-energy me thod of solut ion. Brief details of the analysis leading up to the equations are given in Appendix I. T h e equations are:

cot2A 2 (1 + v)( r - - . , ) -- 2 + ~2, • . . . . . . . . . . . (3)

(1 + F ) a ~ - - ~{Sa~ - - a(1 + S ) } = 1 + 2 T ~ J d2 ,. . . . . . . . . . (4)

2;t" A as - - ~ F a F + { S a s - - a (2 + S~) - - 2 + ~2 , • . . . . . . . . . . . (S)

;¢ - - 2 2A ~ Fay -- 4 (2 + e2),~, . . . . . . . . (6)

2v 22 -4- 3 2.d ~ - - - - F a y = - - , . . . . 2 ~ ( 2 - / ~ 2 ) ~ . . . . (7)

~2 ~ 2 I ~2 I o~ -- {Sas -- a(2 ,-}- S)} -- q- 1 ,t 2

2 + (2 ~ ~ ) + ~2(2 + ~ ) "

2

(s)

Equation (3) comes from a consideration of the average shear strain in the plate. Equations (4) and (5) come from a consideration of equilibrium of direct loads in stringers, Iram. es and plate and from compatibility of direct strains. Equations (6) to (8) are derived from mxmm~smg the total strain energy with respect to ~, ~ and A.

in solving equations (3) to (8), Kromm and Marguerre considered only the Special cases in which F was zero or infinite. In the present report no such restriction is made but attention .is concentrated on the case of pure shear loading, i.e., we take ~ = 0. The solutions so ob ta ined may be applied indirectly to all cases in which a is not zero, because S occurs only in the combination {Sas -- a(1 + S)}£ A solution obtained for

S = S~ ] (9) , , , . . . . . , . . ° • ,

a : O ] will be a solution for

S = &

d / . . . . . . . . . . . (10 2.3. Solution of the Equations with ~ = 0 - i n any particular problem S and F are known and

We require the behaviour oLy, d~/dy ( theshear stiffness), as and ~ as ~ increases. Two distinct methods of solution are presented here. The first solution is by elimination and is obtained by regarding), as the independent variable instead of ~. This involves considerable computation and is unsatisfactory for finding d,/dy. The method is summarised in section 3. The second method of solution is indirect and is based on the exact behaviour of d~/dy, ),, as, ~ immediately after buckling (, = 1) and under very high shear loads (~---> oo), but makes use of the known general behaviour of the functions for predicting the. behaviour in t h e range 1 < ~ < oo. The method, though approximate, admits of considerable generality and can be relied upon to be accurate to within 2 to 3 per cent. The method is discussed in detail in section 4.

3. Direct Parametric So-lution.--By regarding Z as the independent variable, equations (3) to (8) can be manipulated to give"

[cd(1 + S)(). 4 + 2). 2 + 5) -- 2(;~ 2 + 1)(2(! + S) -t z vS(2 2 , 1)}] ×

× [F~).~(). 2 + 1) + a~F{2).~(). ~ + 2) -- v(). ~ + ).~ + 2)} + 2).~(1 + 3 F -- v).~f)]

= [2(~ ~ + 1)((1 + F)().~ - - 1) + 2 ~ F } - - ~ F ( ) . ' -+: 2~ ~ + 5)] ×

× [~() . 0 + ).~ + 2 ~- 8(8). ' + ) . ~ + 2)) + 2).' + 2s).~(8). ~ - - ~ ) ]

as an equation for determining ~, and

8)1~ = ~2(2 + c~2)2[a2(1 ~- S)(). ~ -~- 2). 2 @- 5) - - 2 ( , t ~ + 1){2(1 + . S) + vS(). ~ - - 1)}] .-. a~().4 + ).2 + 2 + S(3). ~ + ).~ + 2)} + 2~ ~ + 2).~S(3). 2 - - v)

(11)

(12)

%/1! . . . . . . . . (131 = g ().~ + 2). ~ + 5) - ).~(2 + . . . . . . " "

Once )., c~, A, ~ are known the functions y, as and a / m a y be found directly from equations (3), (8) and (6) respectively. For values of ~ lying in the range 1 < v < 5, the app.roximate range for ). is 1 < 2 < 1.5. A numerical example based on these equations is given m Appendix II. The corresponding equations in which G is non-zero are given in Appendix III .

- 4 : Indirect Solutio~.--Before this method can be applied expressions are required for

r / ~ ] d d~ {dash, ( d % ( d ~ 4 , / d % [ d ~ i (da 4 /da 4

and these are derived in Appendix IV.

A* (71667}

The application of the method to the particular case of the shear stiffness will now be considered in detail. The other cases considered (shear strain, stringer stress and frame stress) are similar and the results only will be given.

4.1. Post-Buckling Behaviour of the Shear Stiffmss.--Prior t o buckling the shear stiffness d~/d), is unity. Immediately after buckling the shear stiffness drops to a value given by

d, _-- 1 + ( 1 + v){1 + 2 S + S F + S F ( 6 + 4 v _ , , 2 ) 1 J " " " "

which has been plotted in Fig. 2.

Similarly, the rate of change of the shear stiffness with shear load-J~_ (d]2)~ is a known L~~ \ ~ Y / A 1

function. The shear stiffness for large values of ~, plotted ill Fig. 3, is given by

t 03= 1 + ~ - + S ( 3 + ~) + ( 1 + S ) . ~ 2 . . . . . . . . . (15) where

~/12(1 + 3S)(1 + F ) - 2SF~,~ I ~ 2 = F ( 1 + S ) . . . . . . . . . . . ( 1 6 )

Up to the present the analysis is exact in so far as it represents the solution of equations (3) to (8). The stiffness for values of T in the range 1 < , < oo is represented approximately by an equation.of the form

d-7 = (~)03 + I ( ~ ) 1 - (17) dv]031 ' " . . . . . . . . .

which is correct at ~ = 1 and oo and should give a good representation of the known general

behaviour at intermediate values. The constant K is chosen so that [@_(d.] 1 win be correct, whence " ~ \~Y/J

dr

K ~ . . . . . .

and is plotted in Fig. 4. Figs. 5 and 6 have been prepared from equation (17) and demonstrate the change in shear stiffness due to changes in S and F. A detailed example is given in section 5.

4.1.1. 02btimum value of S for maximum post-buckled shear stiffmss.--For a given amount of stiffening material W(-- S + F) there is an optimum value of S for which the stiffness im- mediately after buckling is a maximum. This value, ob ta inedby differentiating equation (14) reduces to

s = w(1 + 2j - 1

3(1 -+- ~) . . . . . . . . . . . . . . (19)

Equation (19) will not, or course, be valid unless it yields a value of S sufficiently large to warrant the original assumption of negligible stringer bending.

4.2. Post-Buckling Behaviour of the Shear Strai~.--The shear strain could be obtained by integrating equation (17) but this would have to be done graphically and it is simpler to obtain

from the approximate relation

- - 1 - - + - - ~-~/~ . . . . . . . . . . . ( 2 0 ) 03 1 03

4

4.3. Post-Buckling Behaviour of the Stringer S tress . - -At the onset of buckl ing the str inger compressive stress is de termined by

s~ _ 2 + F(2 + 4v) . . . . . (21) ] 1 - 1 + 2 8 + 5 F + S F ( 6 + 4~ - - ~ ) ' " "

while for very high values of the shear load

(d~q 2(~2 + ,) . . . . . . . . (22) & 1+; - (1 + 3 s ) s . " . . . . . . .

and the approximate re la t ion for determining the stringer stress is

~+ _ ( a ~ q _ t { a ~ s ~ _ { a ~ q t+-,,, . . . . . . (23) T -- 1 \dT]o~ t \ d T ] + \ & ] l ) " . . . .

The functions (das/&)l, (dS~s/d~)l, (d~s/&)+, ~' are p lo t ted in Figs. 71 8, 9 and 10.

4.4. Post-Buckling Behaviour of the Frame S t ress . - -A t the onset of buckling the frame compressive stress is de te rmined by

(d~ 6 4 + s (4 + 2~) . . . . . . (24) dT ] l = 1 + 28 + 5F + S F ( 6 + 4~, -- +,~) ' ""

while for very high values of the shear load

(e~ 4 _ 2 . . . . . . (25) dT! ~ F~® " . . . . . . . . . . .

and the approximate relation for determining the frame stress is

~ ( e ~ i _ t ( a ~ _ ( e ~ t~-~,, . . . . . . (26) - - 1 - - \ t i t 1+ t \ & 1+ \ & 11t ' ""

The functions (d~ddT)+, (dF~f l&)l , (d~fldT)®, K" are p lo t ted in Figs. 11, 12, 13, and 14.

5. Numerical Example . - -Consider a s tructure in which

t = 0. 048 in.

b = 4"-8 in.

a = 20 in.

Section area of stringer = 0. 115 sq in.

Section area of frame = 0. 192 sq in.

E = 10 × 106 lb/sq in.

v = 0 . 3 .

We require the shear stiffness, shear strain, stringer stress and frame stress at shear stresses of 5,000, 10,000 and 15,000 lb/sq in.

Wi th these dimensions 0.115

S = = 0 . 5 4-8 × 0.048

O. 192 F = = 0 . 2

20 × 0.048

~2Et ~ ~* -- 3(1 - - v')b " = 3,600 lb/sq in.

= 1 corresponds to a buckl ing stress of

~/(2)~* = 5,100 lb/sq in.

5 (716671 A*2

This value may be compared with the known exact value, for simply supported edges, of 4,850 lb/sq in. The conditions along the edges are those of ' con t inu i ty ' rather than simple support so that the error suggested by these figures (6 per cent) will not in fact be so great (the error arises, of course, from the assumed mode of buckling). For convenience we can assume initial buckling stress of 5,000 lb/sq in.

At 5,000 lb/sq in. (~ = 1) the relative shear stiffness drops to a value given by equation (14) or Fig. 2"

(d~)l = 0 . 7 3

From Figs. 3 ,4 , 7, 9, 10, 11, 13 and 14 we have

~ = 0 " 3 4 , '~---1"66, \d~:] =0.71, \ d T ] ,

( h : ] l = 1"70, \ d ~ ] ~ 0 = 4 " 7 3 ' K " = 0 . 3 7 .

= 1"80, K' 0"62,

We are now in a position to consider the state of tile structure at 10,000 lb/sq in. (, = 2) and at 15,000 lb/sq in. (~ -- 3). Thus, from equation (17)

d~ = o . 34 + 0. 39/ 1.oo ,

From equation (20)

whence

whence

From equation (23)

whence

- - -- 0"34 + 0"39/~ °'88,

(?'),=2 = 2 .79 ,

(r)~=~ = 5" 02.

as = 1 " 8 0 - 1.09/z °'"" T - - 1

(as).=2 = 1"09 ,

corresponding to an actual stress of 1.09~* = 3,920 lb/sq in. Similarly,

(Zs)~=3 = 2.49, (a~),=2 = 2.39, (~p),=~ = 5.42.

These results have been plotted in Fig. 15 and compared with values obtained by the lengthier direct solution considered numerically in Appendix II.

6. Co~clusions.--The post-buckling behaviour under shear of plane sheet reinforced by stringers and frames has been considered theoretically. An approximate method of analysis, based on work by Kromm and Marguerre 2, has been developed. Formulae and graphs have been presented from which it is possible to determine the shear stiffness, shear strain, stringer stress and frame stress in the post-buckled state.

6

b

6g

S

F

E

G

f fs

f fF

(7

LIST OF SYMBOLS

Skin thickness

Stringer pitch ~

Frame pitch

Section area of stringer divided by bt

Section area of frame divided by at

Young's modulus

Shear modulus

Poisson's ratio

. Buckling stress of plate under pure compression

a~Et ~ 3 ( 1 - ~)b ~

Compressive stress in stringer divided by ~*

Compressive stress in frame divided by a*

Nominal appIied, compressive stress divided by ~* stringer and one panel ,= (1 -t- S)b¢ ~ * )

(load applied to one

Nominal shear stress divided by ~/(2)~*, so t h a t , = 1 at initial buckling under pure shear

wm~x Maximum plate deflection = ½(crest to trough deflection)

4 t

~0 Angle between the nodal lines and stringers (see Fig. 1)

= ~/2 cot % so that e = 1 at onset of buckling under pure shear

---- b divided by distance between nodal lines (see Fig. 1)

7 = G multiplied by shear strain divided.by ~/(2)~*, so that prior to buckling . , 7 = 7 r .

W ---- S + F

/~,K,K' " Decay factors

Suffices ~ a n d . refer to conditions when ~ 1 and. as ~ ~ co.

L ,

7

No. Author

1 H. Wagner . . . . . .

2 A. Kromm and K. Marguerre ..

3 Paul Kuhn . . . . . .

4 D.M.A. Leggett . . . . . .

5 F. Crowther and N. Sanderson..

6 A. Van der Neut and W. K. G. Floor

7 E.H. Mansfield . . . . . .

R E F E R E N C E S

Title, etc.

Ebene Blechwandtr~iger mit sehr dfinnen Stegblechen. Z. t~lugtech. Motorluflseh. Vol. 20. p. 200. 1929.

Verhalten eines von Schub und Druckkraften beanspruchten Plattenstreifens oberhalb der Beulgrenze. L.F.F. Vol. 14. p. 627. 1937.

Investigations on the incompletely developed plane diagonal-tension field. N.A.C.A. Report 697. 1940.

The stresses in a flat panel under shear when the buckling load has been exceeded. R. & M. 2430. September, 1940.

Experimental investigation into plate web spars under shear. R.A.E. Reports Struct. 7 and 8. A.R.C. 11,570. February, 1948, and A.R.C. 11,573. March, 1948.

Experimental investigation of the post-buckling behaviour of flat plates loaded in shear and compression. N.L.L. Report S.341. 1949.

Some identities on structural flexibility after buckling. R.A.E. Report Struct. 137. 1952.

A P P E N D I X I

Derivat ion o f the EquaEions Obtained by K r o m m and Marguerre

Add i t i ona l No ta t ion

Ox, OF Cartes ian co-ordinates , Ox parallel to s t r ingers (see Fig. 1)

$ Airy ' s stress func t ion

w Deflect ion n o r m a l to p lane of p la te

1 W a v e l e n g t h of buckles

G Shear m o d u l u s

D F lexura l r ig id i ty of p la te = Et3/12(1 - - v ~)

p~, py Average va lues of t he compress ive stresses in the p la te

q Average shear s t ress . in t he p la te

e,,, eyy Average compress ive s t ra ins in the s t r ingers and f rames

~y Average shear s t ra in in the p l a t e

V St ra in energy

Suffix ~ or y af ter ~ or w indicates different iat ion.

The equa t ions govern ing the pos t -buck l ing behav iou r of t he p la te are

V~de = E{ (w,~,) 2 - - w,~w,~,} . . . . . . . . . . . . . . . (27)

Ov'w = - - + . . . . . . . . . . . . . ( 2 8 )

Owing to the ex t r eme diff iculty of ob ta in ing an exact so lu t ion of these equat ions , a p p r o x i m a t e m e t h o d s m u s t be employed . A form for w is chosen which includes var ious p a r a m e t e r s and which gives a good r ep resen ta t ion of t he t y p e of d i s tor t ion observed in practice. An a p p r o x i m a t e so lu t ion of equa t ion (27) is t h e n possible b y s imple in tegra t ion. The var ious p a r a m e t e r s are found f rom a m i n i m u m - e n e r g y t h e o r e m which is used in preference to equa t ion (28).

8

The assumed form for the deflection is

~Y ~ (x cot ~) ] W=W~a~COSuCOS7 - - y

= ZUma x COS T COS b

(29)

Subst i tu t ing this expression in equat ion (27) and integrat ing gives, for the par t icular integral,

E W m a x 2 '

For the complementary function only terms which give rise to average values of the stresses are considered ;

. . ( 3 1 ) ¢(°) = - - ½P,x ~ - - q x y - - ~ p , y , . . . . . . . . . . and we have

¢ = ¢(p)+ ¢(~). .. . . . . . . . . . . . . . . . (32)

Restr ict ing ¢ in this way means t ha t is is not possible to satisfy, the bounda ry conditions exactly, but average values for the overall shear and compressive stratus may be found"

e,y = q + u ~ w ~ 2 c o t W . . . . (33) 4P ' " . . . . . . . . .

i b y - - v p . ~ w .... 2{(/)~ } (34) %Y-- E + 8l ~ + c ° t 2 ~ ' " . . . . . . .

. . . . . . ( a s ) p , - + . . . . . e.~ = - E 8 P . . . .

Equa t ions (33), (34), (35) correspond to equations .(3), (4), (5) respectively of the main text .

I t is convenient to regard the strains e,,, %y and e,y as fixed and then express the fact t ha t the strain energy of th6' plate is a min imum with respect to l, ~, Wm~ (the static analogue of Kelvin 's min imum energy theorem).

From well-established theory the s t rain energy in a strip of plate of length l is

V -- - {(¢~, + ¢,,)~ -- 2(1 + v)(¢,,¢,, -- 5,,~)} • ; - ~ 1 2 J - b ~ 2

+ D{(w,,~ + wyy) ~ - 2(1 -- v ) ( w = w y y - w,y~)) 1 d x d y . . . .

and if the values of ¢ and w, given by equations (32) and (29) are subst i tu ted we find

V~4Wm~x~(1 + Jl ~) sin' V Yg'tt2Wmax 2 ~2)2 V = E t b l L " - 2 ~ + 96b~(1 - - V~ {(1 + + 4X ~ cos ~ ~o}

1 + _ p { ½ p 2 + ½ p 2 vlb,p, + (1 + v)q2}l . . . . . . . . . . (37)

and the average strain energy per uni t length is

? = V/Z . . . . . . . . . . . . . . . . . . . . . (38)

9

The conditions of minimum strain energy are now

aW o r ~ = O, . . . . . . . . . . (39)

- - _ _ . , . ~0 or ~ = 0 , . . . . . . . (40)

~Wma~ or ~ = 0 . . . . . . . . . . . (41)

subject to equations (33) to (35) being satisfied.

On introducing the non-dimensional parameters used in the main text, and equating e**, ayy to the strains in the stringers and frames, equations (39), (40), (41) may be written, respectively:

2~2A 2 2 + cd + 2{Sas - - a(1 + S)} -[- ~2(FaF + 1) - - 4~ , + ½(1 + 22)(2 + ~ ) = 0 , . . (42)

(1 + (2 + 7 ) 2 ~ + { S ~ s - - ~(1 + S)} - - F ~ + (2 - - ~ ) , / ~ . - - 1 = 0 . . . . . . . (43)

(1 + ( (2+~2) 2 2 + { S ~ s - - a ( 1 + S ) } - t - F a F 2-t-~2 2 ~ -t- - -

- - 2 ~ + ½~2 + (1 + ~2)2(2 + ~ ) = 0 (44) 8~2 . . . . . . . . . . .

Equations (6) to (8) of the main t ex t are derived by suitable combination of equations (42) to (44).

A P P E N D I X n

Example Based on Direct Parametric Solution

The example considered here is the same as that considered in section 5, for which S = 0.5, F = 0 " 2 , and v = 0.3.

W e n o w choose an a r b i t r a r y v a l u e for ~ a n d s u b s t i t u t e in e q u a t i o n (11). L e t us t ake , for c o n v e n i e n c e , 2 = %/2 so t h a t ~ = 2. , E q u a t i o n (11) is n o w

(19"5~" - - 18" 90) (1" 2~ ' + 2 .72~ ~ + 5 .92) - - (7 .92 - - 0.78~2)(16~" + 19.4) = Z = 0 . ( 4 5 )

This e q u a t i o n m a y be so lved b y t r ia l a n d error.

W i t h ~2 = 2, Z = - - 2 . 0 9 ;~ w i t h ~" =- 2 .1 , Z = + 40 .2 , so t ha t , on in te rpo la t ing , we can t a k e ~2 = 2 .005 .

F r o m e q u a t i o n (12) we n o w find A 2 = 1 .573 a n d f rom e q u a t i o n (13) ~ = 2 .10 , whence , f r om e q u a t i o n s (3), (8) and (6), y = 2 . 9 5 , as = 1 .18 a n d aF = 2"77.

Va lues of 22 equa l to 1 .1 a n d 2 . 5 h a v e also b e e n t aken . T he n u m e r i c a l resul t s of all t he ca lcu la t ions are p r e s e n t e d in t a b u l a r f o rm below.

22

1.0 1.1 2.0 2.5

~ 2

1.00 "~ 1.102 2.005 2.377

1 . 0 0 1.050 1 . 4 1 6 1.542

. . . . . L I ~ ,-g

0 1.00 0 . 1 1 5 6 I 1 . 0 8 1.573 2.10 2.530 2.85

1.00 1.11 2.95 4.56

i f 8

0 0.06 1.16 2.14

0 0.14 2.77 5.25

! 0

A P P E N D I X 11I

For the case of combined shear and compression it is convenient to obtain the solution for t h e case in which the rat io of applied compressive load/appl ied shear load remains cons tant during loading. If we denote this ratio b y ~ we have

= o(1 + S ) I , . . . . . . . . . . . . . (46)

and the equations corresponding t o those of para. 3 are

[{e'(1 + S). + ~ } ( Z ' @ 24' + 5) -- 2(2' +: 1){2(1 + S) + vS(~' -- 1)}] x

x [~'F,t2(~ ~ + 1) + ~2F{2Z~(,t ~ + 2) - - v(;L ~ + X~ + 2)} + 2;~(1 + 3 F -- vFX~)]

= [2(,~ ~ + 1){(1 + F)(,V -- 1) + 2 v F } - - vF~(~. ~ + 2~ ~ + 5)] X "

× [~'{~ + ~= + . 2 + S(3~ ~ + ~ -+: 2)} + ~ ( X ~ + 1) + 2-~ ' + 2S~=(3~ ' -- v)], . . (47)

8A ~ = ,1'(2 + ~)~[{~'(I + S) + ~}(~ + 2,t ~ + 5) -- 2(1 ~ + 1){2(1 + S) + ~S(a ~- I)}] (48) ¢~{;¢ + ~2 + 2 + S(3;¢ + 2~ -47 2)} + ~fl(;t ~ + 1) + 2;¢-}- 2;,~S(3;t ~ -- v) -'

+ 1) [} ( , V + 2;~ ~ + 5 ) - ~ ( 2 + ~)~ ' " . . . . . . . . . . . . . (49)

o(i + s ) = . . . . . . . . . . . . . (5o) Knowing ,~, ~, A, 3, o the functions y, ~s and ~ m a y be found direct ly from equations (3), (8)

and (6) respectively. _.

,,, -- .- - A P P E N D I x -IV .... - " "

[' ('1] . Derivat ion of the Differentials (d'c/@)l, ~ ~ / , etc.

In determining the differentials (&/d),)~, ~

in which the f lex ib i l i ty of a s t ructure is related to the flexibili ty of a s t ructure constrained to buckle in certain fixed modes.. Identi t ies (C) and (G) of Ref. 7 s ta te t ha t the flexibil i ty d),/dT:' and cross-flexibilities, e.g., dos/d'~, at the onse t of buckling do not depend on tile rate of change of mode shape. Thus we m a y take ,t = ~ = 1 in equations (3), (4), (5) and (44), i.e., (41), which on wri t ing F for (1/3)A 2 reduce s imply to

( 1 + F ) a ~ - - : ~ S ~ s = 4 r ] . . . . . . (51) (1 + S) s ± ,vo = 2 r . . . . .

. . . . . ~ - - 1 - - F o r F ~- ½ S a s = I '

The flexibilities on this basis do not va ry w i t h , so t ha t by simple el iminat ion we can deduce equat ions (14), (21), (24) of the m a i n t e x t . - '

T o c a l c u l a t e l d { ~]dr__

-

a n d we shal l Calculate (d'y/d*=), from iden{ity (E) of Ref. 2.

~-11

d~ 2 )1 and \ d~ ~ )1

2will be calculated from ident i ty (H).

Consider now tha t 2 is the independent variable and consider the s tate of the structure when ,~. has increased to (1 + d2). F rom equat ion (11) by subst i tu t ion and neglecting second order quant i t ies it will be found t ha t ~ has increased to (! + oc'd,~), where

, . . . ( s 2 / = g_ + S + 7F + SF(7 + 3 v - - ~,~)] . . . .

Fur thermor e , from equat ion (12)'

dZ ] ~ = 2 I + S + 7F + SF(7 + 3 v - - ~) ' ""

so t ha t from equat ion (13) , , = 1 + ~ dZ, where

3 [1_ + 2S + 5F + SF(6 + 4~, -- ~,~)] (54)

~ = 2 \ 1 + S + 7F + SF(7 + 3~ -- ~) ] . . . . .

Now let the structure be constrained to buckle in this mode alone (Z = 1 + d2, ~ = 1 + ~'d~). The stiffness of such a constrained structure may be de te rmined from a set of equations similar to equat ion (51), namely,

(1 + , ) (r - ~)

(1 + F ) ~ - ~S~s

(1 + S)~s - ~F~F

-- 1 -- F ~ { 1 -- (3/2) dZ} -- {Sos (1 -- ~' da)

= rE1 + {2 + (1/3)~'} dzj

= 4PE1 + {½ + (1/3)~'} dZ] *

= 2/ ' [ ! + {2 -- (2/3),~'} d;t]

= r { 1 - ( s / 3 ) ~ ' d ~ }

. . ( s s )

If these equations are solved by simple el imination they give expressions for the flexibilities in the form

from equat ion (54).

And similarly

#

Now since

+ +

we have by equat ing coef-ficients in equations (56) and (58)

whence, from ident i ty (E) of Ref. 7"

(d~r~ 3A1 ~]~- 2~ ""

12

& /,,~ = \ & )l + -ff

& J,~ = \ & )1 +

I 0 @ •

. . . . . . ( s s )

. . . . . . (57)

. . . . . . ( s 8 )

. . . . . . ( s 9 )

Simi la r ly we h a v e f rom i d e n t i t y (H)"

-ida-- 1i (dT:/@) ~o, etc., a re f o u n d b y s t a n d a r d m e t h o d s if we observe t h a t t --+ oo a s , --+ oo.

(60)

13

G i 6

/ ~ / /

/ / /

/ / / / /

/ ~ # / / / 4 ~ /

I / o . / / / /"

/ / ~ -/ " /= / /

/

/

/ /

/_ __J

FIG. 1. Figure showing notation.

/

%-

6

1.0

0 . ~ t

f - 0"~, J - 1 " - - ' - - - - " " - - - - ' - -

f J

f o.'~ j

i I I i t

I , l J ,

l I J

I I

I l I -0 ~..0 3"0 o0

1 1 i f I [

i 0.6 j - ,

0 , 4 .

F I G . '2.

0,5 0 0.~

l J

0 . 6 0.~, S

S h e a r s t i f fness at o n s e t of buck l ing .

0"7

o.e ~ I ~

I ~ ~ - . I I I

f

o-~ I I I i

o, t ~i

I I

0 0.2. o-4- 0 , 6 0 . 8 1.0 2_,0 3 -0 o o

FIG. 3. shear stiffness Under high.-shear loads.

is

I' -

2,4

z.a ..... i i _. - _ .

2.1

"] .-

I

i

i i ''

I I

"°I o ~ .... o ~ .... o'0 o!~ ,/0 ~~"J F

FIG. 4. The decay factor K.

0.8

0.6

0.4

0-2

0 0

\

"-----2 S=O

5 , 1 . 0

~ 5 : e Q

. 'FIGi 5.

a 3 4 T

Variaf ioff of stiffne~s ~vith z and S. --

lS

0 - ~

,a.z.

0 . 4 -

0.~_

0 0

FIG. 6.

P_ 1- 3

Variation of stiffness with z and F.

F" = oc'

F : O ' 5

- - F , O . I

~ - - - F = 0

4

I.O

I I

/ I / /

0.8

/ /

\~-~ ,), / / /

I I I I t I

O 0 0.~- 0 - 4 0 . 6 0 . ~ I -0 ~-.0 ~ . 0 oo

S

FIG. 8. Rate of increase of stringer load at- onset' o f buckling.

I I

/ F ' = o- I I

/ ~ /Z~° ' a I / ~ ' = I,O I

/ / / F ~ co

0 0

I I I

. . . . . i

I

I I I I I I I I I I

0.::" O. ~ O. ~, ~ 0 - 8 -I.0 ~ ,0 .~.0 oo

FIG. 9. Rate of increase of stringer stress under high shear loads.

1 8

O. o

O - E

0 ' 5

K)

0.4-

S= ~.0 S : 1.0 ~ : 0 , 5

~ S = O

"0<

0 0 . ~ 0 . 4 - O-to O'E5 I - 0 O0

FIG. 10. The decay factor n ' .

2 . 5

P.,O

I - 5

I-C

O-~

I

I

I

I

i ] - - . . i

1

I

I

I

I

I I I I

O 0 O-~

FiG. 11.

I I

0 - 4 0 . ~ 0 .~ , I ' 0 ?...0 S

Rate of increase of frame stress at onset of buckling.

I I

~ . 0

1 9 (71667) A**

1.0

0 ' 8

0 . 6

0 0

FIG. 12.

I t I I I I i

I

~ ~ ~ ' - - " ' - ~ ~.o /

I I I - - ~ - ~ " ° ' a 1

~ - I I I o , ] I I I I

O.P- O .a - 0 " 6 0.1~ 1.0 ~ ' -0 .~.0 O0 s

Rate of increase of frame load at onset of buckling.

G

4-

~ 5 = 0 ' 5

~S=oO

I

I I I I I L . . . . . . .

I I I I I

I I

I 1 r f I

I I

I o I

0 0'~_ 0.~- 0 . 6 0.~, 1.0 1"5 . 2. oo

F

FIC. 18. Rate of increase of frame stress under high shear loads.

20

0 . 6

0 " 5 f

J

o . ~

y,',

0 - ~

o. .~ //~

/ /

I/ i / i /

O.I i[ I

o 0 0"~_

/ f

I / / t

I

~ - - - - - - I . . . . . . _ _ ~ " "

I

I

I I

I

I I I I I

0 , 4 O - G 0 - ~

I

I I I

I - 0

/ . . . - - -

/ J

~ = 0 -?_

S : 0 - 5

S : I - 0

5 , P..-O

OO

FIG. 14. The decay factor K".

0 . 8

0 - 6

0-4

O.?.

/

I

/ FIG. 15.

/ j /

/ ~ 6r

I

,t_v a,

a- 5 22

Example considered in section 5 and Appendix II.

: R A P I D INCURECT

~O! -L IT ION.

e : Olla.ECT S O L U T I O N

21 (7t667) \Vt. 52/8943 K7 5/58 Hw, PRINTED IN GREAT BRITAIN

Publications of the Aeronautical Research Council

R. & M. No . 30J

A N N U A L T E C H N I C A L REPORTS OF THE A E R O N A U T I C A L R E S E A R C H COUNCIL ( B O U N D V O L U M E S )

I939 Vol. I. Aerodynamics General, Performance, Airscrews, Engines. 5os. (52s.). Vol. II. Stability and Control, Flutter and Vibration, Instruments, Structures, Seaplanes, etc.

:63s. (65s.)

194o Aero and Hydrodynamics, Aerofoils, Airscrews, Engines, Flutter, Icing, stability and Control, Structures, and a miscellaneous section. Sos. (52s.)

1941 Aero and Hydrodynamics, Aerofoils, Airscrews, Engines, Flutter, stability and Control, Structures. 63s. (65s.)

1942 Vol. I. Aero and Hydrodynamics, Aerofoils, Airscrews, Engines. 75s. (77s.) Vol. II. Noise, Parachutes, Stability and Control, Structures, Vibration, Wind Tunnels.

47s. 6d. (49 s. 6d.)

1943 Vol. L Aerodynamics, Aerofoils, Airserews. 8os. (82s.) Vol. II. Engines, Flutter, Materials, Parachutes, Performance, Stability and Control, Structures.

9os. (92s. 9 £) 1944 Vol. I. Aero and Hydrodynamics, Aerofoils, Aircraft, Airserews, Controls. 84.r. (86s. 6d.)

Vol. II. Flutter and Vibration, Materials, Miscellaneous, Navigation, Parachutes, Performance, Plates and Panels, Stability, Structures, Test Equipment, Wind Tunnels. 84s. (86s. 6d.)

i945 Vol. I. Aero and Hydrodynamics, Aerofoils. I3os. (I32s. 9d.) Vol. II. Aircraft, Airscrews, Controls. 13os. (132s. 9d.) Vol. III. Flutter and Vibration, Instruments, Miscellaneous, Parachutes, Plates and Panels,

Propulsion. I3OS. (132s. 6d.) Vol. IV. Stability, Structures, Wind Tunnels, Wind Tunnel Technique. 13os. (132s. 6d.)

Annual Reports of the Aeronautical Research Council-- 1937 2s. (2s. 2d.) 1938 IS. 6d. (Is. 8d.) 1939-48 3 s. (3s. 5d.)

Index to all Reports and Memoranda published in the Annual Technical Reports, and separately--

April, 195o - - R. & M. 2600 2s. 6d. (2s. xod.)

Author Index to all Reports and Memoranda of the Aeronautical Research Council--

I9O9--January, 1954 R. & M. No. 2570 !5 s. (15s. 8d.)

Indexes to the Technical Reports of the Aeronautical Research Council--

December I, I936--June 3 o, I939 July I, I939--June 3 o, 1945 July I, I945--June 3 o, 1946 July I, I946--December 31, 1946 January I, I947--June 3 o, 1947

Published Reports and Memoranda Council--

Between Nos. 2251-2349 Between Nos. 2351-2449 Between Nos. 245 I-2549 Between Nos. 2551-2649 Between Nos. 2651-2749

R. & M. No. 185o Is. 3,/. (is. 5d.) R. & M. No. 195o is. (IS. 2d.) R. & M. N6. 2o5o IS. (rs. 2d.) R. & M. No. 215o is. 3d. (is. 5d.) R. & M. No. 2250 IS. 3d. (is. 5d.)

of the Aeronautical Research

R. & M. No. 2350 R. & M. No. 2450 R. & M. No. 2550 R. & M. No. 265o R. & M. No. 2750

is. 9 d. (is. I id .) 2s. (2s. 2a'.) 2s. 6d. (2s. Iod.) 2s. 6d. (2s. lod.) 2s. 6d. (2s. Iod.)

Prices in brackets inNude postage

HER MAJESTY'S STATIONERY OFFICE York House, Kingsway, London W.C.z ; 423 Oxford Street, London W.I ; i3a Castle Street, Edinburgh 2 ; . 39 King Street, Manchester 2 ; 2 Edmund Street, Birmingham 3 ; t°9 St. Mary Street, Cardiffl Tower Lane, Bristol t

8o Chichester Street, Belfast, or through any bookseller.

S.O. Code ,No. 23-3073

R. & M. No . 30:i