On a theory of sandwich construction

TECHNISCHE HOGESCHOOL VLIEGTUIGBCUV/KUMDE TECHNISCHE ÜKSVEeSETEST DELFT

LUCHTVAART- iiTTEeHUHEK ]2 l i j f i 1 9 5 0 .- K l u y v e r w e g t . - 2 6 2 9 H S D E L F T REPORT MO. 1 5 M a r c h . 1 9 4 8 . T H E C O L L E G E O F A E R O N A U T I C S G R A N F I E L D On a T h e o r y of S a n d w i c h C o n s t r u c t i o n - h y .

Yl.S. Hemp, M.A.

of the Department of Aircraft Design

SUMMARY

• . • s

The theory of sandwich construction developed in this paper proceeds from the simple assumption that the filling has only transverse direct and shear stiffnesses, corresponding to its functional requirements (§1). This supposition permits integration of the equilibrium equations for the filling (§2). The resulting integrals are used to study the compression

buckling of a flat sandwich plate (§3). The formulae obtained are complex, but may be simplified in practical cases (§4). A second approach to sandwich problems is made in §5, where a

theory of "bending" of plates is outlined. This generalises the usual theory, making allowance for flexibility in shear. This

approach is applied to overall compression buckling of a plate in §6, and agreement with the previous calculations is found. This suggests the possibility of calculating buckling loads for curved sandwich shells. A siaiple example, the symmetrical

buckling of a circular cylinder in compression is worked out in 87. The theory developed would seem applicable to all cases of buckling of not too short a wave length (§8).

1 Assumptions

The construction of a plate built according to the principles of Sandwich Construction is shown in Fig.l.

Metal or plywood faces are glued to the surface of a low density filling. The faces are the principal load carrying agent. The function of the filling is to stabilise the faces against lateral buckling and to provide a shear connection between the faces without which the plate could not transmit bending actions. The filling may contribute to the load carrying capacity of the plate, but it is not essential that it should do so. The advantage of Sandwich Construction lies in the great flexural and torsional rigidity of plates con-structed by this method. This rigidity arises from the stiffness of the faces in their planes combined with their relatively large separation.

A Z . .^ ,t

('.'.-,:,/'.'•:'-. V'" '*"' .''

"~* .'•/;i'-.-; 'A

h ^ x

liEii

no other part with a thickne E and extend between 2h large The theory of Sandwich Construction developed in thi paper proceeds from an ideal model in which the component' parts fulfil their essential functions but play

at all. The faces are idealised as thin plates t of isotropic material having Young's Modulus Poisson's ratio cr. The filling is assumed to the middle surfaces of the faces with thickness

C9mpared with t. It will be assumed homogeneous, but

anistropic, with direct stiffness at right angles to the faces and shear stiffness in planes at right angles to the faces^ Other kinds of stiffness of the filling will be taken as zero. If Cartesian axes are taken with Ox and Oy in the middle surface of the filling and Oz at right angles to the faces, the stress-strain relations for the filling can be

written:-ss Xx Y„ = 0, = Le Y y = 0 = Ce

_zz>

) JZf _^x = Le _{zx> X,.} = 0 ) (1)

The notation for stress and strain components is that of Love's Treatise (Ref. 1 ) . C is Young's Modulus in the

Oz direction while L is the shear modulus in the Oyz and Ozx planes.

^ 2 , The Displacement

The displacement in the filling can be calculated from equations (1) and the stress equations of equilibrium which can be written remembering (1)

as:-^ Z x _ as:-^ ciYz

è 2 = o

azx

It follows that Zx and Yg and that

are functions of x and y alone,

rd^x

ax

= - ( ^ + ^ | ) - Zzo (3)

20 is (Z^) „ , a function of x and y. Using where Z

formulae expressing the strain components in terms of the displacement (u, v, w ) , the &z 2;z «»w <Stv ^y "^ dz _ ^lU. ^ w _ "" ^yz * az '*' dx " ®zx (4)

we obtain by substitution from (1) and simple integration the

formulae:-i.

u = w =

z£

_è_ /a£x

+

aYz.

ec

aix

ax d y

z£ _d_

,aZx ^

dY^ 6Ü ^y ^-J^ +• " 5 ^ _zf

,dZx

20

^alE

+ ÖY _dy zf 'aZzo + 2 ( ^ -^lo) + Uo L dx 2C gx ) - z^ 2ÏÏ zZ zo & Z 2 0 ay

.

+ Wr

z(Iz

L _{• J y '} +- Tf

where (UQ, V Q , W Q ) is the displacement of the plane z = 0. Equation (5) expresses the displacement in terms of six

arbitrary functions of x and y, namely vo §3. and Wf Buckling in Compression -•x' 'ZO' u 0» -00 < X •< +

A sandwich plate, occupying the region ^ , 0 ^ y :S b, - h:é z-^ + h, is compressed in

the X direction by a uniform' load P per unit length. The edges y = o, b ere simply supported. The plate will become unstable e.t a certain critical value of P. To find this v a l u e , a small displacement (u, v, w ) is imposed upon the uniform compression and the examination of the

possibility of equilibrium in this buckled form is carried out in the u^aal way. The displacement (u, v, w ) is given by ( 5 ) . This satisfies equilibrium conditions in the filling. The six unknown functions involved are determixied by the

boundary conditions at the faces.

The calculations are simplified somewhat by introducing the arcal dilatation A of the faces» Thi: I S

related to the applied forces per unit area

by the equation

-•x and Y z

V ^ ^

(1

-a"

Et ( Ö Z _'X

ax

aYz

ay

(6)

where in this, as in subsequent equations, the upper sign

-refers to z = h and the lower to z = -h. From (5) it

follows that

/^

- ^ 2 / h £ ( d ^ ^ 5Y,) ^ h2 )

- " \i 6C( ^lE + -5y) - -gc ^20 + ^^0^

ax ay' ax "ay'

Substituting from (7) into (6) and adding and subtracting the

resulting

equations:-( 6G ^ + X ^ - Et ) equations:-(-alE -ay)

= h V ^ Q .. (8)

ryS^auo dVo __

Y£

^ 4

^ ^ ?>x ^ 3 y ) =^2C V^^zo ^-^^

The remaining condition of equilibrium at the faces

is that of balance of normal forces. Here the effects of the

initial compression P must be introduced as well as the

external force Zg. The resulting equations

are:-'"''*•' i é '("'^'ih i <2,)^^^i^ = 0 .. (10)

S u b s t i t u t i n g from (3) and (5) i n t o (10) and again adding and

s u b t r a c t i n g the r e s u l t i n g e q u a t i o n s :

-{

«>»''•{ i i " j ( ^ . = { . . .

("^*+

IM^^o

•• •• ••

(12)

< " ^ * " l é * H ' Zzo = 0 •• <13)

Equations (8), (9), (12) and (13) involve only the four

unknowns

^ ^ + ^ 7 2>Uo

^

3vo

ax "^ dy ' ^20 . -gx + - ^ and

WQ

This relative simplicity is due to the use of A

K-The calculation of critical loads is unaffected by this

artifice. The equations fall into two sets. Equations (8)

and ( 1 2 ) i n v o l v e o n l y ^ ^ x è^z and WQ, w h i l e e q u a t i o n s

ax ay

( 9 ) and ( 1 3 ) i n v o l v e Z^„ and ^ U Q dVo .

^° ax "^ "ay

There are thus two distinct types of

buckling:-(a) Symmetric. Here i ^ + ^ ^ ^ o and so w is

ax

sJ

°

an odd function of z. The critical loads follow from (13).

(b) Anti-symmetric. Here Z„o = ^-ii2 + ^J° = 0 and w

^° 5x Ty

is an even function of s. The critical loads follow from (8) and (12) which yield when

az

_2c , dYz ÖX "ay is eliminated:-h^D r-78 h J P d^ ^ 4 hD T>76 (^2 ( l - C r 2 ) D ) _ 7 4

3C V + ^ C aS2 V -

T ;

V + (^ ^ Et i V

hP $

"SI ^x

V^

+ ( l - c r ^ ) P B' _{2 E t} öx« Wf = 0 ( 1 4 )

The critical values of P follow from (13) and (14) by assuming that w and hence Z20 a^^ ^o» vary as sin /7x sin TTy , where A is the, as yet unknown,

J^ " b

half-wave length. The formulae are:- .

.ype (a) P = ^ ( | . A ^ 2 , | b £ C ( A ^ 2

.. (15)

Type (b) •^Eth^

r2 i>,2

p = 2 TT^Eth^ ,b

,

A)2 l ^ ^ i i ? " ^ "LÏÏP^^"A^^"5Cb^ ^^-^A^^ i

"^^^^^"^^ '^ ^ ^ i^^sth . .^ b2^ rr^Eth^ ,, b2 p ) (^ ^Lb2(i::^«)(i+;^)+3Gb4(i-cr2)(i+A^) )

.. .. (16)

§4. Discussion of the Buckling Formulae

The value of the smallest critical load follows from (15) and (16) by choslng A to'make P a minimum. This is easy in the case of symmetrical buckling and yield!

V,

In practice = (1 ^

^^^A4

b^C /JT^D) (17) /T^hD -^ > 1 and so hD

A =

nrh)

(18)

w h i c h shows t h a t s y m m e t r i c b u c k l i n g o c c u r s i n s h o r t , w a v e -l e n g t h s of t h e o r d e r of t h e s a n d w i c h t h i c k n e s s 2h'''^. The c o r r e s p o n d i n g c r i t i c a l l o a d i s g i v e n b y :

-^ c r l t = 4 ( H ^ ) ^ ^15)

The formula (16) for anti-symmetrical buckling is much more difficult to interpret. If the filling is so rigid that the effects of C and L can be disregarded, the problem reduces to that of an ordinary plate and so for minimum P, which Y/ill be written Pg, the condition is X = t>. This gives

p -. 8/7gEth^ (20)

^ (l-^^jP^

Now so long as A is of the same order as b, inspection of (15) shows that of the various terms of the correcting fraction only the unities and the term involving L in the denominator need be retained. Under these conditions equation (16) can be written

% E = *(T-^-)V ^ 1 ^ 3 ( 1 . ^ , { .. (21,

Where, Pg = 2hL (22) The minimum value of P occurs when

^4 = fizl^' (33,

and this yields for Pcrit "^^^ formula

i + -è- + - ^ — (24)

ï'crlt "I's 2Ps i6P^2

The formula (24) governs the overall buckling of a sandwich panel, as opposed to the short wave wrinkling v/hich is

governed by (19). Its range of accuracy is revealed by (23), which shows that it is certainly valid for Pg ^ 3Pg. Comparison may be made with the formula for a strut with lov/ shear stiffness which is

P — r r = TT + ^ .. .. • (25) • ^ c r i t P E P s w h e r e P-g i s now t h e E u l e r l o a d p e r u n i t l e n g t h . :i-'.,< ^ /]j i s p r o p o r t i o n a l t o ^ ^ ^ ' ^ / c h ' ^ ) " 'which i n p r a c t i c e i s of t h e o r d e r of u n i t y .

The relation (16) gives a further minimum value of P when A/t) ^ < 1. Expansion in powers of A/^ up to

A /h^ gives a formula with a minimum at

A

7r(^)*

" ^3C'

(26)

The corresponding critical value of P

is:-^crlt = ^ ^ ( f ) *

(27)

Comparison with equation (19) shows that the critical load for anti-symmetrical wrinkling is larger than that for the

symmetrical variety.

S5. Bending

The problem of the overall buckling of a sandwich panel may be approached via a theory of bending of sandwich plates. This may be developed from the displacement formulae

(5) by taking that part of the displacement which is anti-symmetric about z = 0. The displacement at the face z = h, written (u', v', w ' ) , is then given by

u' I -w'

lif J L f ^ Z

-h2 2C X Ö Z 6C ay ^ax X

f

^Zx ax 3Y^ ay SY, ) + •) + h ( ^ h ( ^ w, Ö W Q N d W p

By

(28)

The stress resultants T^S Tg' s.^'^ S' in the face z = h are given by Ti' = Et (1-0-2) ^ S » , <5>u' Sv' ^ (•as- +^-—r-) By T2' = Et (l-<72)^By

/9v' , _ au'.

I - Et •2(i+cr)

d

a V' X + 8 x d u'

ay

(29)

Neglecting the contribution from the bending of the faces, the formulae for the normal stress resultants N, and Nj_ and the stress couples G,, Qp and H for the sandwich plate as a whole can be

written:-N , = ZhZjr_ N. = 2hY„ (30)

-G, = 2hT,' , Gg = 2hTp' , H = -2hS' .. .. (31) The sign convention for the quantities Tj', Tg', S', N, , Ng, G|, Gp and H is given in Pig,2.

T'^^'

I

GO

^

^--S'

H

L^x

^ S '

J-^T.'

F a c e z = h

^i

y

F i g . 2

Sandwich as a whole

The quantities Z^- and Y„ may be eliminated using (30). Relations between G,, Gp, and H and the normal displacement

of the middle surface W Q can be obtained by substituting from (28) into (29) and thence into (31). The result may be written:

G, =

-D,(K,

+a-Kp - 1 ( ^ -.cr^Is) .. h (afp ,ais\

' [ ^ 2 n L l a x - y y ) Ï2Ü(ï5c^ dy2)j Gg = -D.^Kg +crK, H = D, (1 - c r ) 2 h L ^ a y 3 x J 12G($y2 8 X ^ ) j

i^(3Ng aN,

4hLlax 3 y ) 12C 3x3y h S^_£ where D, = 2Eth' ₍₁ (32) (33) Ö X"^ 2 3y2 Y = ^ ^ 0

Sxay

(34)

and p is the transverse load per unit area of the plate, which is given by equation (35) below. Equations (32)

generalise the usual bending moment - curvature relations to allow for flexibility of the filling in shear and transverse tension and compression. In practice the terms in p are usually small and may therefore be omitted.

The theory of the bending of sandwich plates is completed by the usual equilibrium

equations:-dN, dNg

"Jx ^

bJ

ÖG, d H è X " a y

a X ^ a- " P = '

- N, = 0

^ 1 + ^ - Ng = 0

(35)

S6. Alternative Calculation of Overall Compression Buckling A calculation of the buckling load for compression buckling with half-wave length X of the order of b can be based upon the bending theory of §5, Allowance for the initial compression p is made by writing

Ü = -P ^ ^ ^ 0

5

x*^ (36)

The equations (32) and (35) are solved by writing

Wo = W sin ^^ sin 'Iz

A b N, - n,cos ''ii^sin ?I2:

A ^

G , = g, sin ^-^sin ii^

H = h cos -2 c o s ^

I A ^

X

Ng = ng sin '-i^ cos ^iZ

^ „ . TTx TTj

2 "^ ^2 s m -- s m -^

A

(37)

where "7,, n,, ng, g , gg, and h, are constants.

Substitution from (37) and the elimination of these constants yields the following formula for

P:-P =

~T2

(A "O)

1 + ïï""^,

2hLb E

1 - ^ !

_X'

(38)

It is to be remarked that the terms in p in (32) have been omitted.- Inspection of (33) and (20) shovvs that equation (38) is identical with equation (21). The approach via the

bending theory of §5 yields the same result for overall

buckling as the more exact calculations of §3. This suggests the possible application of the formulae (32) to more

difficult problems, --/aoh as those of the buckling of curved shells.

S7. Symmetrical Buckling of a Circular Cylinaer in Compression

The application of the formulae (32) to problems of curved shells may be exemplified by the simple case of the buckling of a circular cylinder in a symmetric mode. The assumed cross-sectional deformation is shown in Fig. 3.

"w" the radial displacement is a function of x the distance along the axis of the cylinder. The hoop tensile strain £ g

-^' ^ ' " • .^ is w/r . Assuming no change in direct stress parallel to the

that the axis, it follows x-wise value strain - C w . has the Fig. 3 8

-The hoop tension Tg is then given by 2Et _w ^2 = ( i ^ : ; 2 ) ( ^ 2 - ^ ^ ^ i ) = 2Et.^ (39) The e q u a t i o n s of e q u i l i b r i u m a r e : -SN, Tg

- 5 3 E " T ^ P - 0

S G - N . ( 4 0 )

where N, and G, are the shear and bending moment. The pressure p arises from the initial compression P and is given

byt-9 2vy

P = -P -:rr^ (41)

ax^

Finally the bending moment - curvature relation'follows from

(32):-( a w 1 aN, )

G, = -D, (sx2 - 2hL -a5^) (42)

Elimination of Tp, N,, p and G, from (39), (40), (41) and (42)

yields:-1 -

D, d^

2hL dx2/

P f e ^ ^ w ) . . , | l l = 0 . . (43)

The critical load is obtained from (43) by assuming w proportional to sin " x A . This yields the

result:--2 I _ \, A E S 2 . A S2 i P E

/PS - ^ — ) W ^ ) - 3 p^^^^^ ^ ^ ^ ^ ^ ^ ^

Ps A

A

'J^'

(44)

where, I^E "^ '^ (2EtD, )

'E

AK =

7r(£i£!,^-2Et

(45)

Pg and A -^ are the buckling load and half-wave length for the case where shear flexibility of the filling is small. pQ is given by (22), The minimum value of P in equation

(^4) occurs when

AE

1 - i IE ₍₄₆₎

T h i s g i v e s f o r P c r i t "^^® f o r m u l a :

The hoop tension Tg is then given by

"£= ( i r k ( ^ 2 - ^ « i ' = 2Etf

(39, The equations of equilibrium

are:-dN, Tg

ax

^ G j + p = 0 H, = 0 (40)

where N, and G, are the shear and bending moment. The pressure p arises from the Initial compression P and is given

by:--n ^ w , .-, s

F i n a l l y the bending moment - c u r v a t u r e r e l a t i o n follows from ( 3 2 ) :

-(-^ _J^ ©N, )

~ "" 2hL "a^) •• (42)

G, - -D, (sx2

Elimination of Tp, N,, p and G, from (39), (40), (41) and (42)

yields:-^ "iïïï L2Ayields:-^ÏÏ2yields:-^"7éyields:-^Jyields:-^ D.

S w . 2Et ^4

= 0 .. (43)

The critical load is obtained from (43) by assuming w proportional to sin " x A . This yields the

result:-p/p.= (iMf . ( A ) ' - * a

'J^' _{^S\fk + £ E ( A E ) 2 / .. (44)}

i^s

A

where, ^ E "^ '^ (2EtD, )'

AE =

JTi^)^

(45)

Pg and A -^ are the buckling load and half-wave length for tne case where shear flexibility of the filling is small. Pc is given by (22), The minimum value of P in equation

(Ï4) occurs when

AE Pc (46)

This gives for P c r i t "^^^

formula:-c r l t = PHI (1

-PE

Equation (47) is valid so long as (46) yields a wave length .sufficiently long to justify the use of the bending theory

of §5. For practical application P E >^ 3 Pg would seem quite a reasonable limitation. 2

§8. The Field of Application of the Theory

The type of construction to which the theory of this paper is directly applicable is that class of sandwich in which the filling consists of resin impregnated paper honeycomb. In this case the stiffnesses of the filling

conform almost exactly to the equations (1), The formulae for the buckling loads (19), (24) and (47) would seem then to be appropriate to this type of construction.

The formula (19) which covers the case of wrinkling will certainly not apply to other kinds of filling^ whose

comprehensive elastic properties induce a dying away of surface waves at points distant from the surface and so lead to a

buckling formula which is approximately independent of the filling thickness *'''. On the other hand the formulae (24) and

(47) v/hich apply to overall buckling 'in wave lengths large

compared v/ith the plate thickness may reasonably be expected to apply to all practical fillings.

References

1. "The Mathematical Theory of Elasticity". Love. 2. "Theory of Elastic Stability". Timoshenko.

3. "Instability of Sandwich Struts and Beams", H.L. Cox. (R & M.2125)