Effect of Cell Geometry on the Shear Modulus and on Density of Sandwich Panel Cores1
A simple analytic theory for the effect of cell geometry on both the shear modulus and the density of sandwich panel core is presented. The core shear modulus in different directions is analyzed to include the effects of the angle a and the aspect ratio b/a of the cell. It is also found that the minimum cell weight of the sandwich core depends both on the cell angle a and the cell aspect ratio b/a. The theory compares fairly well with some available experiments. The cell geometry chosen is so general that the regular hexagonal and square cells of commercial sandwich cores are special cases.
Introduction ONE of the principal elements in sandwich structure
is the core. For bending, it transmits the shear between the tension stress of one face to the compression stress of the other. It carries the transverse compression resulting from loading nor-mal to the panel. It is, therefore, very essential to the sandwich structure.
For efficient sandwich structure, the core must be lightweight and strong in carrying the loadings, particularly transverse shear. Since the early development of sandwich structures in the Glenn L. Martin Company [l],2 it has been found that the efficient core is made of thin metal foil formed into the geometrical shape of cells such as honeycomb, square cell, etc. The present investi-gation is a simple analysis to calculate the core elastic properties from the geometrical shapes of the cells. As far as budding and plastic effects of core cells are concerned, some later report will be given.
The present simple theory seems to agree fairly well with the experiments of Kuenzi [2], although the quality of the testing samples is not entirely satisfactory.
Assumptions:
1 The cell walls of the core behave elastieally. 1 This research was supported by the United States Air Force
through the Air Force Office of Scientific Research of the Air Research and Development Command, under Contract Number AF 18(603)-112. Reproduction in whole or in part is permitted for any purpose of the United States Government.
1 Numbers in brackets designate References at end of paper. Contributed by the Metals Engineering Division and presented at
the Winter Annual Meeting, New York, N. Y., November 27-De-c e m b e r 2 , 1 9 6 0 , o f T H E AMERICAN SOCIETY OF MECHANICAL E N G I -NEERS. Manuscript received i.t ASME Headquarters, June 29, 1959. Paper No. 60—WA-52.
2 No buckling of cell walls occurs. 3 The geometrical shape of the cell remains the same in each
plane parallel to the sandwich faces, although displaced. 4 The bond between the two cell walls is assumed perfect. 5 The direction of applied shear force is the same as the
direction of shear displacement (some small deviation is neglected).
6 The thickness of cell wall (t) is much smaller than the characteristic dimension (a). The bond width (b) is also much larger than (<)• The thickness of the core (I) should also be much larger than (<).
Analysis Let the upper face of a typical cell be given an arbitrary shear
displacement I in a direction at an angle 6 from the z-axis as shown in Fig. 1(6) and calculate the shear strains of the individual faces.
The shear strain 7! of the two parallel faces 322 "3" and 566'5* are equal and can be shown to be
7 , - | cos 6
The corresponding shear force is 71 Gbt, and the component of this force in the ^-direction is
P, = 7, Gbt cos 6 = ' - y cos2 d (1)
From the geometry of the cell and the displacement, it is easy to show that the shear strain 72 of another set of parallel faces 211 "2" and 544'5" is
72 = 7 cos (a — 6)
-Nomenclature-A = area of core cell in x-y plane as
shown in Fig. 1
a, b = core cell dimensions as shown in Fig. 1
G = shear modulus of core foil Gg = equivalent shear modulus of the
core in ^-direction
<78 _ ^(fl+r/2) r a y 0 0f orthogonal shear Ge
moduli P = shearing load t = thickness of core foil
tb = thickness of bond between core cells as shown in Fig. 1
I =
Pb Pc
Pt
core cell dimension as shown in Fig. 1
core cell angle shear displacement angle as shown
in Fig. 1 density of bonding material equivalent density of core density of core foil material
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and the shear strain 73 of the last set of parallel faces 433"4" and 611 "6" is
73 = | cos (a + 6)
In a similar manner, it is found that the corresponding force components are
Fig. l(o) Core call geometry
Fig. 1(b) Core cell under arbitrary shear displacement
1 1 p—I—1 r (I «.o*
P2 = — cos2 (a - 6)
IGat Pz Y cos2 ( a +
(2)
(3)
The average shear stress in the plane of the panel face can now be evaluated from Eqs. (1), (2), and (3), assuming that the shear stress direction is the same as that of the displacement,
re = P_ A
2Pi + T\ + P,
re _ 2IGbl (
IA \ cos2 6 + - [cos2 (a - 6 ) + cos2 (a + 0)]> (4)
where P = total shearing force and A = area of cell hexagon. In setting P = 2Pi + P2 + Pz it can be seen from Fig. 1(a) how the cell faces are distributed such that each cell consists of the correct portion of core foil.
The hexagon area A can be seen to be
1.4
12
1.0 3 ^>0.8 s o ~ 0.6
0.4
02
jJoc-0* 1
_ _ 1 \ 1 \ 1 1 _ 1 1 b/o-0 oc.90* b/o-0
. b/a"l/5
^V^b/o'oo
0' 10* 20* 30* 40* 50* e
60* 70* 80* 90*
Fig. 2(a) The full lines are the plot of the ratio of equivalent shear moduli (a/tXGfl/G) versus shear displacement angle 6 for different values of cell aspect ratio b/a at a = 45 deg. The dotted line is the locus of the common intersection point for different values of a .
Fig. 2(b) The full lines are the plot of the ratio of equivalent shear moduli (a/r)(G#/G) versus shear displacement angle 6 for different values of cell aspect ratio of b/a at a = 60 deg. The dotted line is the locus of the common intersection point for different values of a .
90* Fig. 2(c) The full lines are the plot of the ratio of equivalent shear moduli (a/f)(G^/G) versus shear displacement angle 8 for different values of cell aspect ratio b/a at a = 90 deg. The dotted line is the locus of the common intersection point for different values of a .
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( a \ A = 2ab sin a I 1 + — cos a J (5)
Substituting Eq. (5) into Eq. (4) gives
( a \ . I 1 + — cos a I sin a
( a \ . 11 + — cos a I sin a
(7 <
^ + + + 1 ^ - 2 sin2 a J cos 20
2 sin (h a. \ a I 1- cos a I (7)
This will be referred to as the ratio of equivalent shear moduli and is an even function with respect to 6. From the choice of co-ordinates, the cell is oriented symmetrically with respect to the i/-axis. Therefore it is sufficient to consider Eq. (7) for 0 S 9 £ •JT/2. Eq. (7) is plotted versus 6 for three values of a at different values of b/a. Figs. 2 (a, b, and c) are, respectively, for a = 45 deg, a = 60 deg, and a = 90 deg. All the curves of each family always intersect at a particular common point for each given value of a. The locus of this intersecting point in parametric equations of a can be found:
(h G
a sin a
cos 20 =
t 2 sin2 a 4- cos oc — 1
1 — cos a 2 sin2 a + cos a — 1
(8)
The locus of these intersecting points is shown in Figs. 2 (a, b, and c) by dotted lines. This curve approaches 0 = 35 deg 16 min asymptotically for a = 0.
The equivalent shear modulus in the direction of 0 + ir/2 is given by the following equation
cos2 0 + — [cos2 (a - 0) + cos2 (a + 0)] IGl 26
re = — 7 \ (6) at
The over-all shear strain y of the cell itself, as given by the initial arbitrary displacement I, is y = l/l• This can be used, together with Eq. (6), to give the equivalent shear modulus in the 0-direction,
, i r l i cos2 0 + £ [cos2 (a - 0) + cos2 (a + 0)] re I IGt 26
Go = — = 7 — 7 y I at
G ( 9 + x / 2 ) a_
G ' t (a + 1 ) ~ [ ( I + 1) " 2
• fb \ 2 sin a I h cos a I \a )
i 26
(9)
If the ratio of orthogonal shear moduli is now defined as <70 = G(e+ir/2)/Gg, Eqs. (7) and (9) give the result
If both sides of the above equation are divided by Gl/a, the fol-lowing is obtained
Fig. 3 Ratio of orthogonal shear moduli gg versus shear displacement angle 0 for different values of cell aspect ratio b/a and core cell angle a as shown in Table 1
_ . . - . . . sin2 a Table 1 Important special cases corresponding to curves for constant values of
b 1 + -
in Fig. 3
Curve 1 2 3 4 5 6 7 General
x 6 a & — a
sin2 a 0 1
6 1 4
1 3
3 8
5 12
1 2
General x 6
a & — a l + » a
0 1 6
1 4
1 3
3 8
5 12
1 2
Important special cases
a CO ; oo • co 2 ; | ; 5 1 ; 2 ; 3 i - A - 2 2 ' 4
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1 + -
/ 2 sin2 a\ 1 + 11 — \ cos 20
(10)
a ,
sin2 a sin2 a '
+ 1
— = 2 sin2 a - 1 a
Setting 0 = 0 and 0 = 7r/2 in Eq. (7) gives the following
i G ' i
+ cos2 a.
• f b a. \ sin a I f- cos a ) \a )
(13)
b/a
9* = a.
+ cos a
sin2 a G * b ,
— + cos2 a a
(15)
which is plotted in Fig. 3 for different values of the parameter sin2 a/( 1 + b/a). One interesting property should be men-tioned. It can be seen from Eq. (10) that, when 0 = 45 deg, go = 1 and is independent of a and b/a. This corresponds to the common intersecting point in Fig. 3. It can also be shown from Eq. (10) that two curves symmetrical with respect to 6 — ir/4 always exist provided that a and b/a for each curve is related to the other as follows
(ID
Equation (13) has been plotted in Fig. 4 for several values of the core cell angle a and similarly equation (15) is shown in Fig. 5. Both families of curves are useful for practical sandwich design.
Effects of Cell Geometry on Core Density Parameters Introducing the core density parameters pfl pcl and pb as de-
fined in the notation, the weight of one cell may be written as
pc Al = (2bt + 2at)p,l + blttPt.
Substituting (5) for cell area A and solving for the equivalent density ratio (pc/p /)(a/<), the following is obtained
These pans of symmetrical curves are shown in Fig. 3. It is interesting to note from Fig. 3 that the cells with the
geometry (b/a = 0, a = t/4), (b/a = 1/2, a = tt/3), and (b/a = 1, a = 7t/2) are isotropic and thus have the same properties for any value of 0. In general, if the core is imposed to be isotropic then gg = 1 for all values of 6. This condition yields the following simple relation
A Pf
1 + A ( 1 + i . A . A ) a \ 2 p, t )
• ( b o. ^ sin a I 1- cos a: I (16)
(12)
2.0
Fig. 5 Ratio of principal orthogonal shear moduli gx versus cell aspect ratio b/a for different values of core cell angle a
b/a
Fig. 6 Plot of equivalent density ratio (pc/p/)(a/t) versus cell aspect ratio Fig. 4 Plot of the ratio of equivalent shear moduli (G^/GXa/t) ver- b/a for different values of core cell angle a and bond weight parameter sus cell aspect ratio b/a for several values of core cell angle a (Pb/p/Xtb/t)
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Fig. 7 Plot of core cell angle a versus. ce ll aspect raUo &/a for mini~ mum. density cor.
,6Q
140 ~ CORRUGATED CORE , , EXPANDED CORE
,20
100
:; 1" 80 z ~ :Ii 1< 60 ~ "-x ~ ~.
.0
20
20 40 60 80 100 120 G .. (THE ORETICAL )
Fig. 8 Comparison of th eory and experimenl
This equation bas been plotted in Fig. 6 for two cases. First, the bond density is neglected and, secondly, (Pb/P,)(tb/t) is assumed to be litO. I nspection of honeycomb core indicates that the bond is very thin in comparison with the core foil , thuB (pdp,) (t,/t) c: 0.1 is considered a reasonable upper limit. Fig. 6 shows that the effect of bond weight docs not amount to a substantial port ion of the corc density, especially when b/a is small.
Minimum Density Core Another interesting problem is to find the minimum density of
the core with respect to a for a given P, and alt. From the density expression, Eq. (16), neglecting bond weight, a relation for minimum core density is obtained by differentiation of (Pe/PI) (a/t) with respect to IX. T his yields
Thus a can be expressed as a function of b/a for a minimum density core. T his rela tionship is shown in Fig. 7. I n the actual
Journal of Basic Engineering
sandwich core cell b/a ¢ O. However, in the figure, curves corresponding to b/a = 0 are plotted to show the limiting case.
Comparison of the Theory and Experiments A general comparison of the theory with experimental values
[2 ] is shown in Fig. 8. It is seen that the t heory generally pre· dicts a higher value of Gz than obtained experimentally. The discrepancy may be due to the fact that the prescnt theory does not take into account buckling of cell walls, nor is any bond fa ilure considered .
Acknowledgments The authors wish to thank Mrs. Helen Woodward and Mr .
James J. Baltes for their assistance throughout t he investigation.
References 1 Glenn L. Martin Company, M artin Slar, June, 1952, p. 2. 2 E . W. Kuenzi, "Mechanical Properties of Aluminum Honey
