Table Of ContentANALYSIS AND DESIGN
OF STRUCTURAL
SANDWICH PANELS
by
H O W A RD G. A L L EN
Senior Lecturer, Dept. of Civil Engineering, The University,
Southampton
P E R G A M ON P R E SS
OX FORD -LONDON-EDINBURGH - N EW YORK
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Copyright © 1969 Howard G. Allen
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First edition 1969
Reprinted 1969
Library of Congress Catalog Card No. 68-24060
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PREFACE
THIS book is intended to appeal to those engineers who are
faced for the first time with the arduous task of finding a way
through the tangled literature on sandwich theory.
It is not, however, an encyclopaedia of sandwich construction.
The aim has been not to mention every paper on the topic which
has appeared in print, but to provide a simple guide to the prin-
cipal aspects of the theory of sandwich construction and to the
assumptions on which it is based. This has entailed a very selec-
tive interpretation of the literature and a number of advanced
topics have been omitted in order to do justice to the essentials.
The opening chapters deal with the relatively simple problems
of sandwich beams and struts; these chapters (and especially
Chapter 4) serve as an introduction to the later ones on panels.
The treatment throughout is in engineering terms. For example,
the discussion of the bending of sandwich beams in Chapter 2
grows naturally from the ordinary theory of bending; it does
not start (as it might) with a general consideration of the equili-
brium of elements of the core and the faces and with the compa-
tibility of core and face deformations. Mathematical rigour has
been sacrificed to a clear interpretation of the physical realities
of sandwich behaviour. For instance, emphasis is given to the
real significance of the assumptions made in sandwich theory
concerning the thicknesses and relative stiffnesses of the faces
and the core.
The bending and buckling of sandwich panels is introduced
in Chapter 5. It is hoped that the simplified presentation will
make the material intelligible even to readers who hitherto have
÷ PREFACE
had little acquaintance with the theory of bending of plates and
with the use of strain energy for such problems. The more
thorough panel analysis in Chapter 6 is merely an extension of
the preliminary work of Chapter 5.
The panel analyses in Chapters 5 and 6 (based on the Ritz
method) and in Chapter 7 (based on the derivation of differential
equations for a sandwich plate) are in many ways equivalent and
the designer may adopt the technique which pleases him best. The
equivalence of the two methods, and of others, is demonstrated
in Chapter 9.
Within the limits set by its introductory nature, the book has
been arranged to provide information of immediate practical value.
This is especially true of Chapter 10, in which are summarized the
results derived in other chapters and which may be used without
detailed reference to the rest of the book. Furthermore, the con-
tents list contains an indication of those parts of the book which
are of an introductory or elementary character and which are
suitable for a first reading.
Most of the literature on sandwich construction is concerned
with aerospace applications but there is also a considerable inter-
est among the building industry in the possibilities of structural
and semi-structural panels. Unlike those for aircraft, panels de-
signed for use in buildings and for other semi-structural purposes
often tend to be designed with rather thick faces and weak cores.
In such circumstances the assumption that the faces are thin
(often made in aircraft papers) may not be valid. The validity of
the assumption and the consequences of avoiding it are explored
at greater length than usual. For this reason the book should be
of interest not only to aeronautical engineers but also to anyone
concerned with the design of sandwich panels in the building,
plastics and boat-building industries.
The information on which this book is based is taken almost
entirely from readily available British or American publications.
This is purely a matter of convenience and it should be noted that
PREFACE xi
almost parallel information exists in other languages, particularly
Russian and Dutch. Although great initial impetus was given to
the development of sandwich theory by the wartime investiga-
tions of British workers, anyone who writes a book on sandwich
construction must find it impossible to avoid repeated reference
to the very extensive post-war studies of sandwich construction
performed at the U.S. Forest Products Laboratory, which must
surely come first in any list of acknowledgements. In particular
I am grateful to FPL for permission to use fig. 16 of Report
1810 as the basis of my Fig. 8.11.
The many plate-buckling graphs which have appeared in FPL
reports and elsewhere have served as models for the equivalent
graphs in this book. Nevertheless, all of my own graphs for the
bending and buckling of sandwich plates are based on original
calculations. I owe a great debt to Mr. G. M. Folie, who prepared
the computer programmes for this work.
Finally I wish to express my thanks to the Royal Aeronautical
Society (for permission to use figs. 5, 13 and 14 of ref. 10.1 as
the basis of Figs. 8.5 and 8.6), to the American Institute of Aero-
nautics and Astronautics (for permission to use fig. 4 of ref. 9.3
in the preparation of Figs. 8.13 and 8.14) and to the McGraw-
Hill Book Company (for permission to copy fig. 2.10 of ref. 35.15
in the outlines of Fig. 9.4).
LISTOF PRINCIPAL SYMBOLS
AND DEFINITIONS
ELASTICITY
x, y,z Rectangular coordinates, æ is normal to the faces,
positive downwards. In a beam or strut, χ lies
along the axis of the member.
u, v, w Displacements in the x-, y-, z-directions.
Tensile strains
Tensile stresses \ In x-, y- z-directions.
9
EX9 Ey, Ez Moduli of elasticity1
yxy* yyz Shear strains
In zx-, xy-,
Shear stresses
>>z-planes.
Gzx> xGy> y G Moduli of rigidity (shear moduli) 1 J
Poisson's ratios in the xy-plane. When a is the
x
only non-zero stress, v— —e/e.
x y x
g= 1
BEAMS AND STRUTS
b Width of beam
t Face thickness
d Distance between centre-lines of opposite faces Figs.
= c + t 2.1, 2.17.
h Overall thickness of sandwich
L Span
/ Second moment of area, especially total value of
Eqn. (2.19).
faces about centroid of sandwich beam
t Suffixes are omitted where the material is isotropic and where there is
no ambiguity. It is usually obvious which symbols refer to the properties
of the faces and which to the properties of the core. Where it is not, the
suffixes / and c are used.
xii
LIST OF PRINCIPAL SYMBOLS AND DEFINITIONS xiii
If Sum of second moments of area of faces
Eqn.(2.19).
about their own separate centroidal axes
q Distributed transverse load on beam, per unit length
Q Shear force Fig. 2.2.
M Bending moment
Ä Displacement at a particular point.
W Point load.
D Flexural rigidity of beam or strut (see Table on p. xv).
Ρ Axial load in strut.
MISCELLANEOUS
Suffixes 1 and 2 (i) Upper and lower faces respectively.
(ii) Primary and secondary transverse
displacements (Section 2.5).
A = b d2/c
PANELS
a, b Length and width of panel, in χ and y
directions.
M, M, M , (M ) Bending and twisting moments
x y xy yx Per unit
Tensile and shearing membrane
width
forces
Fig. 5.2.
Qx>Qy Shear forces
Transverse load per unit area.
»y> Dy Bending and twisting stiffnesses Per unit
X
Shear stiffnesses width
Definitions,
Section 7.2.
BQx-> BQy Modified shear stiffnesses Formulae,
{B = b*D ; B = b*D ) Section 7.6.
Qx Qx Qy Qy
xiv LIST OF PRINCIPAL SYMBOLS AND DEFINITIONS
v v "Poisson's ratios" defined in terms of
x> y
curvature. Equations (7.1), (7.2).
g = \-vv
x y
m, η Positive integer suffixes denoting mode of
deformation, usually with m half-waves in
^-direction, η half-waves in j>-direction.
λ If a line is drawn through the core of an
undeformed sandwich, normal to the
middle plane, then, as the sandwich de-
forms, the line rotates in the zx-plane
μ μ(θ\νΙàγ)
through an angle X(dw/dx) (Fig. 4.1).
Similarly, is the rotation in the
j>z-plane.
Ρ Compressive edge load per unit width.
Ê Non-dimensional plate buckling coeffi-
cient. Suffixes 1, 2 and 3 merely dis-
tinguish the values used in the compar-
able equations (5.29), (6.36) and (7.57).
WRINKLING
/ Half-wavelength.
2?i Non-dimensional buckling coefficient in the basic wrinkling
formula, α =ΒΕ} Ε* (equation (8.12)).
λ
Non-dimensional buckling coefficient in the alternative wrink-
ling formula, a = B\(E EG)* (equation (10.16)).
Fc c
^ = ^[2(1 + ^)]*.
B Non-dimensional coefficient in formula for face failure asso-
2
ciated with initial irregularities,
=
G BE / E (equation (8.34)).
2 C
ρ Non-dimensional coefficient which defines character -of .sand
wich for purpose of wrinkling analysis,
- T { * }*
LIST OF PRINCIPAL SYMBOLS AND DEFINITIONS XV
ISOTROPIC PANELS, BEAMS, STRUTS
Beam or strut Panel
Flexural rigidity D D
Flexural rigidity, neglect- D = Ebtd2/2 D = Etd2l2(\-Vf)
x f 2 f
ing local bending stiff- (cylindrical bending)
ness of faces D = D = Eft d212
x v
(anticlastic bending)
n2 Ù = ——+n2
a2
Nominal ratio of flexural £ ð2 Ef ct Q 2(n% \-vs* E ftc ) G, b2
rigidity to core shear ~ ¾ Ί2
stiffness
~G_ ð72 Eftd2 c
b2 2(1 - v2f) Ggd2
_ ð2 Ejbtd2 c
Special case of above when £ L_2 ð2 2E f GdtJ bd2 ð2 Ef dt
faces are very thin ;c=d. Q 2(1-ν2.) G b2
See Chapter 5. ~"T~G U c
e
Factor for sandwiches with
thick faces, Chapters 4 [À+îη2 3d2\ ζ []+QÜ^3d2]
and 6
DEFINITIONS
Antiplane core One in which σ = o = x = 0. As a
χ y xy
consequence, the shear stresses x , x
zx yz
are independent of z.
Core shear stiffness AG = Gbd2/c (for beam of width b).
Very thin face One with negligible stiffness in bending
about its own axis, and sufficiently thin
for d to be equated with c.
xvi LIST OF PRINCIPAL SYMBOLS AND DEFINITIONS
Thin face One with negligible stiffness in bending
about its own axis, but not so thin that
d and c may be equated.
Thick face One with significant stiffness in bending
about its own axis, and too thick for d
and c to be equated. See Section 10.1 for
quantitative assessments of face thick-
nesses.
