Analysis of sandwich panels: the significance of shear deformation

Analysis of sandwich panels: the significance of shear deformation H. G. ALLEN*

The implications of core shear deformations in the calculation of stresses, deflections and buckling loads are discussed in elementary terms. Simple criteria are given for the classification of panels according to the stiffness of the cores. Desirable attributes of good core materials are described

In structural analysis a sandwich panel is a thick sheet of lightweight material with a thin sheet of much stronger, stiffer material bonded securely to each side (Fig 1). If any of the essential features (in italics) are not provided, the structure is not strictly a sandwich panel and special methods of analysis must be employed. For example, ordinary sandwich theory cannot be used without modification for laminates such as plywood, in which the various layers are of comparable thickness and have stiffnesses of the same order of magnitude. It is commonplace to say that the faces of a sandwich resist the bending moments and the forces in the plane of the panel, while the core resists the transverse shear forces associated with loads perpendicular to the panel. However, this statement can only be valid if the core fulfils its proper functions. The core must be stiff enough in shear to prevent the faces from sliding over each other. Otherwise, defections will be increased and, in extreme cases, if the core is very flexible the faces act as two independent thin panels. In such cases the efficiency of the sandwich action is destroyed. A very flexible adhesive, which permits a face to slide relative to the adjacent core, has the same undesirable effect as a core which is too flexible in shear. The core must also be sufficiently rigid in the transverse direction to keep the faces the correct distance apart. Otherwise, resistance to local transverse loads is reduced, curved panels will show increased deflections and there may be local instability of the faces. In every case it is necessary to know whether the core fulfils the conditions laid down above, and to be able to quantify the effects of core shear flexibility. * Senior Lecturer, Dept of Civil Engineering, Southampton University, Southampton, Hampshire, England

Consider the panel shown in Fig 1 and suppose that the edges AB, CD are simply supported and AD, BC are free. The diagram then represents a simple beam of width a and span b,

CORE RIGID IN SHEAR If the shear stiffness of the core is large, shear deformations are negligible and it is possible to employ the ordinary bending theory. That is, plane sections remain plane and the strain varies linearly through the thickness of the panel. For example, the flexural rigidity is given by +

2

6

1

c ac 3

12

where E~ = E}/(1 vy), 2 "E'c=Ed(1-v~ z) EI, Ec and v;, vc are moduli of elasticity and Poisson's ratios of the face and core materials respectively. a. d, t, e are defined in Fig 1. -

/ Y

FIG 1

Notation

COMPOSITES June 1970

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ordinary small-deflection formulae normally used in such problems.

CORE SLIGHTL Y FLEXIBLE IN SHEAR This is the commonest practical case. Equations 3, 6 and 7 (for the evaluation of core shear stress, flexural rigidity and face stresses) are still valid. However, the core shear stress z is now associated with a shear strain 7 in the core r

_ Q

~= Gc

__~

AGe

8

De

where Gc is the shear modulus of the core. The quantity DQ may be defined as the shear stiffness (analogous with the flexural rigidity D) DQ = Goad 2 /c FIG 2 Shear stress distribution across thickness o f sandwich provided condition 2 is satisfied

It can be shown that the third term on the RHS is less than 1% of the first term, which is the dominant one, if

9

In the case of simply supported sandwich beams, the core shear deflection merely results in extra (shear) deflection over and above the bending deflection. The shear deflection diagram is similar to the bending moment diagram, factored by 1]DQ. Stresses in the faces and the core are not affected by small shear deformations. In a pin-ended sandwich strut the existence of shear deflections reduces the total stiffness of the strut and, in

,oo

-E'[ c \ c ]

>'-if-

2

Neglect of this last term is equivalent to assuming that the shear stress across the core is uniform, as shown in Fig 2. The shear stress in the core is 14=q~3

r=--~ Q A

where Q is the shear force and A is a notional core area given by A =

14,qO"2

3

ad 2 ..

%

12

!12

4

c

Evidently, when the faces are thin, A = ad = ac. The second term on the RHS of Equation 1 represents the local bending stiffnesses of the faces. It is less than 1% of the dominant first term if

_a >./q-O0 t

V 3

IO

I0

h2

8

= 5"77

5

Most sandwiches satisfy this condition. To summarise, provided conditions 2 and 5 are satisfied, the core shear stress is given by Equation 3 and the flexural rigidity of the sandwich is D = E) adt 2/2

6

The stresses in the faces due to a bending moment M are given approximately by o = M/adt

7

Deflections and buckling loads can be found for beams and panels by using the flexural rigidity, (6) in any of the

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COMPOSITES June 1970

I 2

o_ b

I 3

4 4

FIG 3 Coefficients for the maximum deflection o f a simply supported 'isotropic' panel with uniform transverse pressure (Equation 11}

consequence, the critical load is also reduced. The critical load, Pc,, is

6O

PE

5-5

P.

=

_ _

l

10 5.C

De

where, for a strut of length b, Pe is the Euler load

11

4...,

(1r2D/b 2). 1 In a rectangular panel, (Fig 1) simply supported on all four sides and subjected to a uniform transverse pressure q, two cases arise. If the panel is 'isotropic' in the sense that the faces have the same properties in all directions in the xy plane, and the core has the same shear moduli in the planes zx and yz, then the stresses and the bending deflections are the same as those in a similar panel without core shear deformations. However, there are additional deformations associated with the shear strains in the core. The maximum total deflection is given by

wm"x

- qb4 hi + qbZ h2 D DQ

11

The coefficients hi and h2 are defined in Fig 3 in terms of the aspect ratio of the panel, a/bJ The maximum bending moments can be found from any standard textbook on plates (eg z) and the stresses in the faces can be found from Equation 7. If the faces have different moduli of elasticity in the x and y directions (as in GRP with fabric reinforcement) or if the core has different shear moduli in the zx a n d y z planes, the panel is no longer 'isotropic' and is loosely described as 'orthotropic'. In this case the shear deflections interact with the bending deflections and a simple solution is no longer possible. The deflections, bending moments, twisting moments and shear forces are complicated functions of the proportions of the panel and the stiffnesses of the component materials. Fortunately design charts are avail:;,4, 5 able for the analysis of many common special cases. For example, a corrugated core with corrugations running in the x direction may be taken to have infinite shear stiffness in the zx plane, and finite but non-zero shear stiffness in the yz plane. The shear stiffness of a honeycomb core in the ribbon direction may be (typically) 2.5 times greater than the shear stiffness in the other direction. Ref 6 gives detailed formulae for the various stiffnesses of a panel with a corrugated core. If the orthotropy is not too extreme, a rough solution may be obtained by averaging the properties of the panel in the different directions (possibly weighting the average in favour of the properties in the short-span direction) and treating the panel as isotropic. When a sandwich panel is subjected to in-plane compressive edge forces, buckling will occur at some critical value of the edge force. The core shear strains contribute to the buckling deformation and they reduce the stiffness of the panel in consequence. The reduction in stiffness implies a reduction in the critical load. A common problem is that of a panel simply supported on four sides and subjected to a compressive force (P per unit length) in the x-direction,

4"..

T'2o

3'C

2.c 0

I 05

I I-0

I 1"5

I 2"0 ..g. b

I 2"5

I 30

I 3"5

I 4-0

FIG 4 Coefficients for the buckling load of a simply supported 'isotropic'panel with uniform compressive forces along two opposite edges (Equation 12)

along the edges x -- 0, x = a (Fig 1). The critical load is

n2D e. =--p-K

12

where K is given in Fig 4 in terms of the aspect ratio of the panel, a/b, and the ratio of the flexural and shear stiffnesses, D/b2DQ. The curve for D/b2DQ = 0 corresponds to the common solution for panels with no shear deformation.J l As the shear stiffness DQ of the core is reduced, K falls and the panel tends to buckle into more half-waves in the x-direction. Similar charts for more complicated load cases, for different boundary conditions and for orthotropic panels may be found in the literature. 3, 7 s, 9 1o

CORE VERY F L E X I B L E IN SHEAR This case is not likely to arise in highly efficient structures such as aluminium honeycomb panels, but it is relevant to certain panels with stiff faces and low-density expanded plastic cores. Furthermore, an understanding of this case throws light on the general behaviour of sandwich panels and provides some rough rules to classify them according to the stiffnesses of their cores. Consider, for example, the simply supported beam of span b, with a central point load W, shown in Fig ha. Fig 5b shows the ordinary bending deflection, the maximum value of which is

Wb 3 A1- 48D

13

The cross-sections ac and bd remain plane and perpendicular to the centre line.

COMPOSITES June 1970

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~w

deflections in Figs 5b and 5d and it can be written A = rA1

15

where r is obtained from Fig 6.1 Consider, for example, the curve for c/t = 3 (an extremely low value). At A(D/b2DQ = 0.01) the total deflection is only 12% greater than A~ (ie the ordinary bending deflection calculated by taking Gc = '~). The shear deformation of the core is small and the stresses are similar to those predicted by ordinary bending theory. At B(D/b2DQ = 0.1) the total deflection is 110% greater than A1. Core shear deformations are substantial and it can be shown that the local bending stresses in the faces, where they attempt to follow the kink in the shear deflection diagram, approach the values given by ordinary bending theory. From B to C the situation deteriorates further. From C to D the deviation of the curve from the line EF indicates that much of the load is being supported by the faces acting as independent beams. In the limit, beyond D, the core is completely ineffective (as in Fig 5e) and the stresses in the faces are many times greater than the simple predictions based on Gc = =. The deflection when Gc = 0 is

a

D Wb 3 D Df 48D Df AI

Wb 3

A= 48Ds

16

It is easy to show that, when c/t = 3, D/D¢ = 48. The condition A = 48A1 is represented in Fig 6 by the line GD, which is obviously an asymptote to CD.

[~ e I-

CLASS/F/CA TION OF SANDWICH PANELS

b

FIG 5 Deflection patterns for a sandwich beam with a central point load

Fig 6 can be used to place sandwich beams in welldefined categories. These same categories can be shown to be valid also for struts and panels, provided the dimension b

¢

T Fig 5c shows the shear deflections associated with the core shear strain 3' (= W/2DQ). The maximum value is

Wb A2 - 4D(2

14

The cross-sections ac and bd remain vertical. The distances ab and cd are unchanged by the shear deformation and the average direct stresses in the faces are therefore also unchanged. In reality, the faces must bend locally (about their own separate centre lines) in order to follow the pattern of the shear deformation. In Fig 5c the faces are required to bend to an infinite curvature at the points X, which they cannot do. The local bending stiffness of the faces therefore smooths out the sharp discontinuities in the shear deflection curve, as shown in Fig 5d. The shear deflection is reduced at the expense of introducing new local bending stresses in the faces. The total deflection is the sum of the

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C O M P O S I T E S June 1970

F, 2OO

100012

9O .~0

IOOC - -

I0 6 3

G

I

IA_.O.Ol

FIG 6

I 0"1

I

IO

Total deflection o f beam in Fig 5

K)O

t000

is correctly interpreted. In a transversely loaded panel, b represents the shorter of the two spans. In a strut, b represents the length. In a panel buckled by compressive edge forces along two opposite edges, b represents the width of the panel, measured across the direction of the edge forces. The categories are as follows

A

D/bZDQ<

0-01 (any practical

c/t)

Shear deflections small, stresses in faces predicted by ordinary bending theory (G~ = oo); buckling loads may be significantly reduced by core shoar flexibility in the range 0"001
0.01 <

D/bZDo <

c/t)

O"1 (any practical

Shear deflections comparable with bending deflections; stresses in faces given approximately by ordinary bending theory, except under concentrated loads; buckling loads substantially reduced C

0"1


10 (depending on

c/t)

Shear deflections dominant; stresses in faces no longer accurately represented by ordinary bending theory; buckling loads very low, heavily influenced by flexibility of core D

10 <

D/bZDQ(depending

on

c/t)

Core ceases to connect the faces together effectively; faces act as separate beams, panels or struts. Table 1 shows values of c/t and D/b2DQ for a few representative types of construction. Also listed are the categories into which the various types fall. Evidently there should be no problems with types 1 and 2, both of which can be expected to have fairly small shear deformations. Types 3 - 5 will show the effects of shear deformations quite markedly. In types 6 - 9 the core is not really stiff enough to connect the faces properly. The results in Table 1 are merely illustrative and should not be taken as definitive statements about the merits of

Table 1

1 2 3 4 5 6 7 8 9

S O M E G E N E R A L O B S E R VA T I O N S

In many ways, end-grain balsa (one of the original core materials) still serves as a perfect model for other core materials. Its structure consists of many fine tubes aligned perpendicular to the faces, providing considerable transverse stiffness. The connections between the tubes provide quite good shear strength and stiffness. The small size of the tubes permits good bond to be achieved between the faces and the core. Not much material is wasted in providing stiffness in planes parallel with the faces (stiffness in such planes is more useful in the faces themselves). Honeycomb materials may be imagined as attempts at artificial balsa wood. They have good strength and stiffness in relation to their weight but the size of the tubes (the cell size) cannot be made very small. This means a limitation on the amount of contact between the core and the faces, necessitating careful bonding. Expanded plastics tend to be weak in relation to their densities. This is partly due to the inherently low stiffness of the basic materials and partly because the properties of the foams are not efficiently orientated, as they are in balsa. Much has been written about the possibility of in-plane compressive stresses causing short-wavelength buckling of the faces. The time-honoured formula for this wrinkling stress (Equation 17) is still the best quick approximation.

Ow = 0"5 (Ey Ec Gc )1/3

17

However, theory indicates that this type of instability is only possible for certain types of sandwich. 1 Moreover, the stress at which wrinkling failure actually occurs depends on the extent of initial irregularities of the faces and on the

A comparison of various types of sandwiches FACES

Ref

the various materials. The precise values of D/b2DQ are considerably influenced by variations in c, t, b and in the quality and density of the foam, wood, honeycomb or other material employed in the core.

Material

AI AI GRP (cloth) GRP (mat) AI AI Steel AC AC

AI : A l u m i n i u m alloy AC : Asbestos cement

CORE

Ef

t

106 Ibf/in 2

in

10 10 2 1 10 10 30 3 3

0-028 0"022 0"060 0"125 0"036 0"036 0-036 0-375 0"375

Material

AI HC Balsa HDPF HDPF HDPF LDPF L D PF LDPF LDPF

Gc

c

b

Ibf/in 2

in

in

25 000 15 000 5 000 5 000 5 000 500 500 500 500

0"75 0-5 0"75 1"0 1-0 1"0 1"0 1"0 1"0

30 20 20 20 20 20 20 20 10

G R P : Glass reinforced plastics AI HC : A l u m i n i u m honeycomb

c/t

D/b2 D2

Category

26"8 22"7 12"5 8 27-8 27"8 27"8 2"67 2"67

0"0047 0-00315 0"0225 0"0312 0-09 0"9 2"7 2-8 11"3

1 1 2 2 2 3 3 3 4

HDPF : High density plastic foam LDPF : Low density plastic foam

COMPOSITESJune 1970 2 ] 9

3

Lood

'Structural sandwich composites', US Military Handbook 23A, Dept of Defense (1968) Williams, D. G., Chapman, J. C., 'Effect of shear deformation on uniformly loaded rectangular orthotropic plates', Proc ICE, Supp paper 7236 (1969)

t

Core

Allen, H. G., 'Graphs for the analysis of simply supported rectangular sandwich plates under uniform transverse pressure', Southampton Univ Dept Civil Eng Report CE/21/68 (1968) Libove, C., Hubka, R. E., 'Elastic constants for corrugated core sandwich plates', NACA TN 2289 (1951)

I

FIG 7

strength of the adhesive between the faces and the core. No completely general solution to this problem has been found. Where the possibility of wrinkling failure is suspected, it is essential to carry out compressive tests on the proposed form of construction. 12 It remains only to emphasise the importance of stiffeners to resist concentrated loads and reactions. Fig 7 illustrates the mode of action of an edge-stiffener of a type which might be provided all round the boundary of a panel. If the stiffener is omitted, the whole of the reaction R will be transmitted to the panel through shear in the lower face, resulting in large local deflections and premature failure. REFERENCES

1

2

Allen, H. G., 'Analysis and design of structural sandwich panels',Pergamon, Oxford (1969) Timoshenko, S. P., Woinowsky-Krieger, S., 'Theory of plates and shells', McGraw Hill, N Y (1969)

220

Plantema, F. J., 'Sandwich construction', Wiley, N Y (1966)

Action o f an edge stiffener

COMPOSt TES June 1970

Kuenzi, E. W., Morris, C. B., and Jenkinson, P. M., 'Buckling coefficients for simply supported and clamped flat rectangular sandwich panels under edgewise compression', US Forest Service Research Note FPL 070, Dec (1964) Jenkinson, P. M., Kuenzi, E. W., 'Buckling coefficients for flat rectangular sandwich panels with corrugated cores under edgewise compression', US Forest Service Research Paper FPL 25 (May 1965) 10

'Buckling loads in compression of flat sandwich panels',Roy Aero Soc Data Sheets 07.01.01, 02, 03, 04, also 07. 03. 01, Structures 1V (1956)

11

Timoshenko, S. P., Gere, J. M., 'Theory of elastic stability', McGraw Hill, N Y (1961)

12

'Compressive strength, edgewise, of flat sandwich constructions', ASTM Standard C364, (See also C365, C393, D1781, C273, C297 for other relevant tests)