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o. LU ai [/"•" VUEGtUIGBOUWKUr^DE BIBLIOTHEEKTHE COLLEGE OF AERONAUTICS
CRANFIELD ,
Klu\A/9rweQ iS DELFT
,THE BUCKLING IN COMPRESSION OF PANELS
WITH SQUARE TOP-HAT SECTION STRINGERS
by
W . S. HEMP, M.A. and K. H. GRIFFIN. B.Sc. of the Department of Aircraft Design.
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
-hy-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 betv/een these plates leads to a relation betv/een the buckling stress and the variables of the
panel geometry.
The results thus obtained are 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.
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Table of Principal Symbols Used
L = Stringer height b = Stringer pitch
t = Skin thickness t = Stringer thickness t-u = Buckling stress f^ = 3.62Et^/b^
f„ = Average stress in a skin panel a °
f = Edge stress in a skin panel H = Amplitude of buckle
H = Amplitude of initial irregularity X = Wavelength of buckled form
Other subsidiary symbols are defined where they occur in the report.
PART 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 Fig.1. The stringers are square, of
side h, and are uniformly spaced to a pitch b. The skin and stringers have thibknesses t, t
resocctively. ^
,<
h
><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.
{k) The buckling is sinusoidal in the
direction of compression, of wavelength 2 ^ , By (1)s A rnay vary continuously, and its
magnitude is determined by a minimum ürocess (§1.2)
(5) Perpendicular to the direction of compression, the buckling is symmetrical with respect to each stringer, in a mode as
shown in Pig. 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. A. e»
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Fig. 2, 1.2, Method of TreatmentWe 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 v/e need only consider one of them. One boundary condition, that of equilibrium, is
that the sum of the moments at each junction Rhou].d 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 Fig. 3.
Using assumption (7), we may analyse the deflections of plate IV into
IVa: A syiTimetrical buckling, by moments
^(M^ + Mg +- M7) to edge angles - ^{Q ^ + ^2) IVb: An antisymmetric buckling, by moments
^(Mg + M^ - M^) to edge angles i{^ ^ - ^2) Thus we deal with the deflections of the
five plates listed below.
Plate I II III IVa IVb Width h h b-h h h Thick -nesf?i + ^s t t *s '^s Moments | Edge-Angle ; M, M2
M3
-HM.+M2+M3)
i(M2+M3-M^)
Ö102
02
-^(81+82)
HO^-
^2)
Mode Symmetric j" 1
" i
tr 1 1 AntisymmetricApplying 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 A • 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 X • This gives the value of f for v/hich a buckled mode of wavelength 2 A is "iDossible. If v/e 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'TT' and width '-C , 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. U.
^ \ I 1 i I
Fig.
h.
From assumption (1+) the normal deflection must be of the form
W =
^r^(y)sin{ttx/X)
Since the deflections are proportional to sin(»Tx/>i), the rotation angles 0 at y = ±-^/2 will vary
likev/ise, as will the couples M acting on these edges.
Let & = eoSin(7rx/;L), M = MQSin(trx//), where Ö Q » M are constants. Then it can be shovm
that for equilibrium under the action of f,
v/e must have , , ,3 — (symmetric buckling) 1
^K(i/b)(r/t)2/-^/^^^^/,(f/b,r/t)
M/e = M ^ ^ o
(antisymmetric buckling5 v/here ( 2 ) and / > I/ÖC] / y / . X p - l ^ f o r o C ^ V1//3 j= V |-287(irb/A)(t/t5/(f/fo).i(rrb/A)y(. ^..^ ^;
K i s a f u n c t i o n of f/fQ-,A.A o n l y ; and fg = 3.62E(t/b)'=^ i s t h e b u c k l i n g s t r e s s of a l o n g p l a t e of w i d t h b , t h i c k n e s s t , h a v i n g simply s u p p o r t e d e d g e s .
I.U. Application to the
Problem;-Por each of the plates I - IVb we v/rite down equation
(1):-I M^= K[(hA)(tg/t)2F(h/b,tg/t)/^^ /
II U^^ 4Sh/h)Fih/b,^)J62 ( (3)
III M3= K[(1-(hA) F(^-{h/b),^)] & 2 J
IVa -(M^+M2+M3)= K[(h/b) (tg/t)2p(h/b , t g / t | [ (^^+^^2) ]
IVb - M^+M2+M3 = K £ ( h / ' b ) ( t g / t ) 2 G ( h / b , t g / t ) ] ( a ^ - ^ ) J
S u b s t i t u t i n g for M.. ,M2,M., from (3) i n (U), and then
e l i m i n a t i n g <9^, 65 and v/riting h = b / n ; P ( r / n , t 7 t ) = P ^ ( t / t ) , v/e o b t a i n
(5)
(3P?(t3/t)+G^(t^/t)^(p?(0-f(n-1)P^_^(1)) I
+ ( t 3 / t ) 2 p ^ ( t g / t ) [p^(tg/t)+3G^(tg/t)^ = 0 {'
(5) is an implicit equation in the non-dimensional parameters f/f{^,//b, t /t,h/b(=''/n). Thus, given t /t,h/b (the geometry of the panel), we can
determine the ratio f^/fQ as outlined in § 1.2.
1.5» Calculation of Results
For a given value of b/^. , the left-hand-side of (5) was tabulated at intervals for f/f^ of 0.1. The value of f-i3/fQ(b/JiJ v/as then obtained by
graphical inverse interpolation to zero. This was done for several values of b/,^; %/f(j v/as then plotted against b/i, and its minimum value found graphically. The results of these calculations are shovim in Table I.
Special
Cases;-("l) 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 iDlates of width (b-h), thickness
t.(h<V2)
i.e. f^ = 6.3lE(t/(b-h)^^
V ^ o = 1.7y(l-(h/b)j2
(2) s/t small;- When ^/t is small, (5) approximates to
(3Pf(tg/t)+ G^tg/t))^Ff(1)+(n-1)p;;_^(1)j = 0 (7)
Thus its solution is the smallest value of /T£ which makes either factor of the left-hand side of
(7) vanish.
NOW P?(1)+(n-1)FjJ_.,(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 a-s in Pig. 5.
^'^^-. As H/b i n c r e a s e s , t h e s t r i n g e r s buckle f i r s t , and 3 F ? ( t s / t ) + G ^ ( t g / t ) = 0 (9) i s t h e b u c k l i n g equation f o r t h e s t r i n g e r s clamped by t h e unbuckled skin, a s i n P i g . 6. Fig. 6. (9) has t h e e x p l i c i t s o l u t i o n
fb/fo = 1.17(t/t)V(hA)^
where b / ^ = 1.125 b / h /Thus . . . (10)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 term of (5) are small, the
first term is no longer negligible compared with them, and its effect is to "round off the corner" of the intersection.
The final result of these calculations is shown in Fig.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 v/as done with a radius of h/10 for the cases b/h = 5, ^Q/^ = 1>1.33 and also the case t = 0.
In none of these was the drop in f-^/^o greater than "5%. Also, it will be seen that this is necessarily
an over-estimate of the actual effect, as there is assximed 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 and 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 III, 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 U.
The stringers were of height 1", with ^" flanges, and were riveted to the panels at a rivet pitch of I". The stringers were Rpaced to pitches of 3,U,5,6,7,8 inches giving values of l^/b between 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. Tv/o 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 tv/o methods of estimation for the same panel, and between the various panels. In some of the estimations (mostly by the (amplitude)^-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,
1
2.2, N.A.C.A. Test Programme
Quite an extensive programme of panel
testing v/as carried out by Messrs. ?^A. Hickman and N.P. Dow at the Langley Memorial Laboratory during
I9U6-8 (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 were tested having four values of *s/t, namely 0.39,0.63,1.00, I.25. The stringer
thickness in each case was O.OU". The stringer flange widths b^ were respectively 0.85", 0.75", 0.65", 0.55". For 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 panol 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/* = 1.00, 1.25 the cross-sectional area of the stringer flanges was large compared with that of the skin to v/hich they were 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 Fig. 9.
Again, the scatter is noticeable. But the experimental points for ^s/t = 0.39 agree remarkably v/eil v;ith the theoretical curve.
Por ^^/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 F E R E N C E S (1) Hickman, y\A. , and Dow,N.P. (2) HU, P.C.,Lundquist, E.E, and Batdorf, S.B. Compressive strength of 21+G - T aluminium-alloy flat panels Vi^ith longitudinal formed hot-section stiffeners having four ratios of stiffener thickness to skin thickness. N.A.C.A. Technical Note 1553, March l9i4-8.
Effect of small deviations from flatness on effective width and buckling of plates in compression N.A.C.A. Technical Note 1121+, January 19U7.
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 LO allow for the effect of initial vmving 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 ^averap-e (the average compressive stress carried by the plate] and ^edge
(the compressive stress at the edge of the plate), and between ^edge ^^^ H (the amplitude of the "true" buckle), can then be calculated for a series of values of Hj-^/t; using approximate methods and large deflection plate theory. The results of such a calculation are shown in Pigs. 10,11.
Thus, for three values of
H/t;-H^/t
Drop in the initial value of Young's Modulus Drop in ^-5(^3-^5 method) Drop in f^(H^-fg method) 0,05 2^% 3i% 1-2i^(?) 0.10
3%
7%
1-3fo(?) 0.205%
11-15?S 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. Hovvever, as the (amplitude)2 _ strain curve departs from the straight-line form, in the neighbourhood of f„/f-u = 1 , even with small Ho/t, it is impossible to use this method v/hen failure follov/s closely after buckling.
As these results apply only to initial
buckles having the same wavelength as the final system, they are of little use in correcting experimental
observations for these effects. Hov/ever, they do
show hov/ 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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COMPARISON OF THE THEORETICAL RESULTS
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