The buckling in compression of panels with square top-hat section stringers

^ 12 Iuli1950

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Kluyverweg 1 - 2029 HS DELF

THE COLLEGE OF AERONAUTICS

CRANFIELD

THE BUCKLING IN COMPRESSION OF PANELS

WITH SQUARE TOP-HAT SECTION STRINGERS

by

W. S. HEMP, M.A. and K. H. GRIFFIN, B.Sc.

TECHNISCHE HOGESCHOOL VLIEGTUIGBOUWKUNDE REPORT No. 29 J u n e . 19U9 T H E C O L L E G E OF A E R O N A U T I C S . C R A N F I E L D

"The Buckling in Compression of Panels v/ith Square Top-Hat Section Stringers

-by-W.S. Hemp, M.A., and K.H. Griffin, B.Sb.. of the Department of Aircraft Design.

— o O o —

SUMMARY

A simplified panel model is described, together virith a number of assumptions about the

mode of its buckling. The approach to the calculation of the buckling stress is by splitting the panel

into a number of flat olates and treating these by the ordinary plate theory. Use of the boundary conditions between these plates leads to a relation between the buckling stress and the variables of the panel geometry.

The results thus obtained aro compared with two sets of recent experimental v/ork; and an appendix is included to show the effect of initial panel irregularities on the experimental determination of buckling stresses.

Table of Principal Symbols Used = Stringer height = Stringer pitch = Skin thickness = Stringer thickness = Buckling stress = 3.62Et^/b^

= Average stress in a skin panel = Edge stress in a skin panel = Amplitude of buckle

= Amplitude of initial irregularity = Wavelength of buckled form

Other subsidiary symbols are defined where they occur in the report.

L

t

a

H

Ho

X

-2-PAJIT THEORETICAL CONSIDERATIONS

1.1. Statement of the Problem

We investigate the local buckling of stringer-skin combinations v;ith a cross-section perpendicular to the direction of the stringers as shown in Pig.1. The stringers are square, of side h, and are uniformly spaced to a pitch b. The skin and stringers have thiblcnesses t, t

respectively, ® •/ i<:

h

-V J _V....^ .-iv ">.

„, » L

t

Fig. 1 Physical

Assumptions;-(1) The panels are large enough in both directions for edge-effects to be ignored. (2) The stringers have square corners and

no flanges, their sides being attached directly to the skin.

(3) The panels buckle under the action of a compressive stress f, parallel to the direction of the stringers.

(U) The buckling is sinusoidal in the

direction of compression, of wavelength 2 ^ .

By (1)5 A may vary continuously, and its

magnitude is determined by a minimum process (§1.2)

(5) Perpendicular to the direction of compression, the buckling is symmetrical with respect to each stringer, in a mode as

shown in Fig. 2,

(6) During buckling, the stringer corners A,B, C,D (Fig.2) remain fixed in space, and the right angles at these corners are conserved.

(7) Deflections are small enough for the principle of Superposition to hold. ^ - . _ J \ \ )

J

Fig. 2. 1.2, Method of Treatment

We regard the panel shown in Pig. 1 , as if it were built up of separate flat plates, each buckling under the action of the compressive stress f and moments distributed along the edges parallel to f. These moments are caused by interaction with the other plate(s) attached to this edge.

Since the buckling is symmetrical about the stringers, the two stringer sides will have similar modes of

buckling, and we need only consider one of them. One boundary condition, that of equilibrium, is

that the sum of the moments at each junction should be zero. The other, arising from assumption (6),

is that the angles of rotation at the edges of adjoining plates should be equal.

The four basic plates are shown in Pig. 3»

-h-Using assumption (7), we may analyse the deflections of plate IV into

IVa: A symmetrical buckling, by moments

i(M^ + Mg +- M^) to edge angles - ^{Q ., + ^2)

IVb: An antisymmetric buckling, by moments

^(Mg + M, - M.,) to edge angles i(Q ^ - Ö2)

Thus we deal with the deflections of the five plates listed below.

Plate I II III IVa IVb Width h |h jb-h h :h Thick -ness *s t

t

*s *s Moments M, M2 M3 iiU^+U^+Mj) iiU^+MyUj^) Edge-Angle Ö1

1 02

1 ^2

|-i(Öi+02)

\

He^-

^2)

Mode Symmetric

" 1

II II 1 Antisymmetric

Applying the formulae of plate theory to each of these five "basic plates", we obtain relations between the unknown moments and rotation-angles at each edge (§ 1.3). These relations contain the stress f and the half-wavelength X • There are five of these relations, containing three independent moments and two independent angles. We may eliminate the moments and angles and obtain an equation between f and A • This gives the value of f for which a buckled mode of wavelength 2 A is iDOssible. If we find the value of /\ which gives the minimum value of f, we shall have found the least stress for which a buckled form can exist. This, then, is the buckling stress f, .

1.3. General Mathematical Results

, Consider a long plate of thickness IT' and

width >< , under the action of a compressive stress f in the direction of its length. Take axes in the plane of the plate as shown in Pig. 1+.

Fig.

k.

Prom assumption (k) the normal deflection must be

of the form

W =

W^iy)sin{rTx/X)

Since the deflections are proportional to sin(irx/A), the rotation angles 9 at y = +-f/2 vv'ill vary

likewise, as will the couples M acting on these edges.

Let Q = eoSin(7rx/A), M = MQSin(tTx//),

where Ö Q » M are constants. Then it can be shown

that for equilibrium under the action of f,

we must have , , „ -. (symmetric b\iGkling)

jK(-^/b)(r/t)^K/.. .^^(f/b,r/t)

°'^° (K(^)(r/t)2(;,/,^^^/,(/A,T/t)

(antisyraraetric bTickling

where

( 1 )

(2)

and

1/oc] /f' , > p-lA-foro(^V

K is a function of f/fo.,A./b only; and fo = 3.62E(t/b)'^ is the buckling stress of a long plate of width b, thickness t, having simply 'supported edges.

^ ,k» Application to the

Problem;-For each of the plates I - IVb we v/rite down equation

(1);-I M.,=

Kl{hA>){t^/tfF(h/b,t^/t)J^^

II M2= K[_(h/b)p(h/b,1)jÖ2

(

(3)

III M3= 4(1-(h/b)

F(^-{h/h),^)] d

IVa -(M.|+M2+M,)= K IVb - M.,+M2+M3 = K[_(h/b)(tg/t)2G(h/b,tg/t)j(a.,-^)

Substituting for M.| ,M2,M, from (3) in {U-), and then

eliminating 0^, $2 and writing h = b/n;F(r/n,t7t)=F^('?:/t),

2 ^ ^

{h/io)(t^/t)hih/h,t^/t)j{e^+B^)

Uu)

we obtain

(5)

(3F^(t3/t)+G^(t/t)^('p?(1) + (n-1)P^_.,(1)) 1 +(t3/t)2p^(tg/t) [p?(t^/t)+3G?(tg/t)^ = 0 (

(5) is an implicit equation in the non-dimensional parameters f/f^,AA,"t^/t,h/b(=1/n). Thus, given t /t,h/b (the geometry of the panel), we can

determine the ratio i^^/^Q as outlined in § 1.2.

1.5» Calculation of Results

For a given value of b/^( , the left-hand-side of (5) v/as tabulated at intervals for f/f,-, of 0.1.

The value of t-^/f^{'b/j^ v/as then obtained by

graphical inverse interpolation to zero. This was done for several values of b/^; fb/fo was then plotted against b/i, and its minimum value found graphically. The results of these calculations are shown in Table I.

Special

Cases;-(1) s/t large;- If t—:^ 0 while t^ remains finite, then the skin alone will buckle,the stringers exerting a clamping effect. So the buckling stress is that for clamped-edge -olates of width (b-h), thickness

t.(h<V2)

i.e. fx, = 6.3lE('t/(b-h))^

^ ^ (6)

V ^ o = 1.7y(l-(hA)|'

(2) s/t small;- When ^/t is small, (5) approximates to

(3P?(t3/t)+ G^t^/t)^[p^(1)+(n-1)pJJ_^(1)^ = 0 (7)

Thus its solution is the smallest value of /f^ which makes either factor of the left-hand side of

(7) vanish.

Now P?(1) + (n-1)P^..,(1) = 0 (8) is the buckling equation for the skin held at points

alternately h,(b-h) apart. i.e. this is the case where the stringers exert no rotatory influence on the skin, but are merely stiff enough in their own plane to

prevent normal displacement of the skin at their corners. Thus the buckling mode is as in Pig.- 5»

Pig. 5.

As H/b increases, the stringers buckle f i r s t , and

3 P f ( t s / t ) + Gf(tg/t) = 0 (9)

i s the buckling equation for the stringers clamped

by the unbuckled skin, as in Pig. 6.

<: ,>, ; :r.izzxa:xx:xi''T.fXroj^rji„:i.'i,,Txrri':rr'i"?Ti:,.'> t,<! . ' - T I naxiT:rr-:x:.c:3» P i g . 6 .

(9) has the explicit solution

f^/fo = 1.17(t / t ) V ( h A ) ^

S'

where b/^, = 1.125 "b/h

(10)

-8-Thus for a given (small) t /t the solution of (5) is initially the curve which is the solution of (8), and after their intersection, by the cubical hyperbola whose equation is (10). Near the intersection, when both factors of the second terra of (5) are small, the first term is no longer negligible ocuiparecl v;ith them, and its effect is to "round off the corner" of the intersection.

The final result of these calculations is shown in Pig.7. 1.6. Variations from Assumptions

It will be noted that up to the present no account has been taken of the fact that "practical" stringers have rounded corners. An attempt has been made to allov/ for this in the following manner.

The plates forming the stringers are unaltered, but the condition of conservation of angles is replaced. Using the theory of cylindrical shells, a change in angle at the corners is inserted, compatible with the bending of a cylindrical quadrant of the required radius and thickness under the given moments. This leads to new terms in the buckling equation (5), which is then solved as before.

This was done with a radius of h/10 for the cases

b/h = 5, '^s/'^ ~ 1j1»33 and also the case t = 0.

In none of these was the drop in f-j^/fo greater than

3%- Also, it v/ill be seen that this is necessarily

an over-estimate of the actual effect, as there is assumed to be no reduction in the dimensions of the

stringer plates. In fact, the rounding of the corners reduces the width of the stringer top ana sides,

increasing their stiffness , and partly counter-balancing the effect of the increased flexibility

introduced.

It was, therefore, decided that the effect of the curved stringer corners could safely be neglected. It is difficult to allow for the stringer flanges, as their effect will vary with the method of con-struction employed. It is clear, hov/ever, that the effect will be to increase the stiffness of panel m , thus increasing the buckling stress. However, for small ts/t,h/b, the effect should only be small.

PART II : COMPARISON WITH EXPERIMENT

2.1. Work of A.J. Monk

In preparing his thesis for the Diploma of the College of Aeronautics, Lt.(E) A.J. Monk

tested 36 panels of construction similar to that considered in part I. These panels had four skin sections supported by five stringers, and were square, giving a ratio (panel length/stringer pitch) of

approximately k»

The stringers were of height 1", with ^" flanges, and were riveted to the panels at a rivet pitch of f". The stringers were spaced to pitches

of 3,U,5,6,7,8 inches giving values of h/]-, betvi^een 0.125 and 0.333. The ratio tg/t covered the values 0.35, 0.U5, 0.60, 0.75, 1.00, 1.33.

The ends of the panels were cast in a low melting-point alloy, and machined flat and parallel before testing. Two methods of estimating the buckling-stress were used. The first was to find

the point where the load-strain curve changed direction. Secondly, the square of the buckle-amplitude was plotted against the strain. This should give a straight line intersecting the strain axis at the buckling-strain.

The results of these tests showed appreciable scatter, both between the two methods of estimation for the same panel, and between the various panels. In some of the estimations (mostly by the (amplitude)2. strain method), no reliable answer could be obtained, as the curves bore little resemblance to the expected form. In less doubtful cases an estimate has been made, and a (?)appended to the entry in the diagram. These results, plotted against the appropriate

theoretical curves, are shovm in Pig. 8.

The scatter and the small number of panels tested make it difficult to obtain either confirmation or refutation of the theory, and indicate the need for a large-scale test programme with statistical analysis of the results. However, the experimental values show quite fair agreement with the theory.

-10-2.2, N.A.C.A. Test Programme

Quite an extensive programme of panel

testing was carried out by Messrs. W.A. Hickman and

N.F. Dow at the Langley Memorial Laboratory during

1914.6-8 (Ref.1). These tests were carried out primarily to obtain failing-stress values, but buckling-stress measurements were also taken. These were done by the "strain-reversal" method- (Ref.2),

Panels v/ere tested having four values of *s/t, namely 0.39,0.63,1.00, 1.25. The stringer

thickness in each case was O.OU". The stringer flange widths b^ were respectively 0.85", 0.75", O.65", 0.55". Por each value of tg/t, h/tg and (b-h-b^)/t took the values, 20,30,UO,60 and 25,35,50,75 respectively. Pour panels, of varying lengths, were tested for each variant of panel geometry, so 256 panels were tested

in all. Some of these v/ere for values of ^/h

outside the range we are discussing; others had

buckling stresses too great for accurate determination of the tangent modulus (required for calculation of fo) • "For "'^s-/t = 1.00, 1.25 the cross-sectional area of the stringer flanges was large compared with that of the skin to v/hich they wore attached, and so the test specimens bore little relation to the

theoretical model considered. However, there remains a large number of test results, and these are shown in Pig. 9.

Again, the scatter is noticeable. But the experimental points for ^s/t = 0.39 agree remarkably well v;ith the theoretical curve.

For "^s/t = 0.63, however, the tendency is for the experimental points to be some 15^ higher than the theoretical, v/hile maintaining the general trend of variation. The reason for this is not yet understood,

R E P E R E N C E S (1) Hickman, F.A. , and Dow,N.F. (2) HU, P.O.,Lundquist, E.E, and Batdorf, S.B. Compressive strength of 21+ G - T aluminium-alloy flat panels with longitudinal formed hot-section stiffeners having four ratios of stiffener thickness to skin thickness. N.A.C.A. Technical Note 1553, March 1948.

Effect of small deviations from flatness on effective width and buckling of plates in compression N.A.C.A. Technical Note 11 2l+,

-12-APPENDIX

The Effect of Initial Irregularities on the Estimation of Buckling Stress

For the special case of panels with

simply supported edges, it is possible to allow for the effect of initial waving in the following

manner. Assume a wave of amplitude H , and of the same wavelength as the final buckle system, to be present in the plate initially. The theoretical relations between ^average ("the average compressive stress carried by the plate) and ^edge

(the compressive stress at the edge of the plate), and between ^edge and H (the amplitude of the "true" buckle), can then be calculated for a series

of values of EQ/I; using approximate methods and

large deflection plate theory. The results of such a calculation are shown in Figs. 10,11.

Thus, for three values of

H/t;-H ^ t

Drop in the initial value of Young's Modulus

Drop in ^i^^-^Q method)

Drop in f]-,(H -f^ method) 0.05 2^^ 3^^ 1-2ifo(?) 0.10 3% 1% 1-3fo(?) 0.20 ll-l5fo 1-5fo(?)

It will be seen that the (amplitude) - strain method of estimating buckling stress is less susceptible to error due to initial irregularities than is the

load-strain method. HoY/ever, as the (amplitude)2 _ strain curve departs from the straight-line form, in the neighbourhood of f„/fv, = 1 , even with 'small

e D

Hp/t, it is impossible to use this method when failure folloviTs closely after buckling.

As these results apply only to initial

buckles having the same v/aveiength as the final system, they are of little use in correcting experimental

observations for these effects. However, they do

show how underestimates of f^ can arise in experimental work, and underline the need for the greatest possible accuracy in the manufacture of test panels used for the checking of theoretical work.

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THE BUCKLING STRESS IN COMPRESSION OF PANELS

WITH SQUARE TOP-HAT SECTION STRINGERS

Ï

t - Skin Thickness. ts" Stringer Thickness. b - Stringer Pitch. h - Stringer Height. E - Young's Modulus. fb~ Buckling Stress. fo-3.62 E(Vb)*

FIG.

7

h, ₇

4

* + •t%. = 0 3 5 .

4;

3 -2 1

L - - —

o o 1 ^ _ ^ ^ ' ^ < \ \ t s , = o 45. O 2 0-3 Y^'^ PA

COMPARISON OF THE THEORETICAL RESULTS

OF FIG. 7 WITH THE EXPERIMENTAL

COMPARISON WITH AMERICAN RESULTS (N ACA TN.I553)

h

Vt -

°-3=-\

k

i

o-i 0 £ O S

+

^'^ +

+ + 0 4 . + +

4-+

+

+

, +

+

+

+

4-+

+

+

4- '4-t j / s O ©3. o i 0 - 2 _{o 3} % 0 4 .

Fir;

o

C O F A .

R E P . N O .

29

H'-fe CURVES FOR VARIOUS H<

5 o

4 o

3 o

2 o