ELASTIC I3UCICILING Of A SIMPLY
SUPPORTED !RECTANGULAR SANDWICH
PANEL SUBJECTED TO COMBINED
EDGEWISE BENDING AND COMPRESSION
No. 1857
`Sepiember-1-946.
""T4 REVIEWED
Ai:FIRMED
1962
Please return to:
Wood Engineering Research
Forest Products Laboratory
Madison, Wisconsin 53705
This Report is One of a Series
issued in Cooperation with the
ANC-23 PANEL ON SANDWICH CONSTRUCTION
of the Departments of the
AIR FORCE, NAVY, AND COMMERCE
UNITED STATES DEPARTMENT OF AGRICULTURE
FOREST PRODUCTS LABORATORY
MADISON 5 , WISCONSIN FOREST SERVICE
I n Cooperation with the University of Wisconsin
40*
TABLE OF CONTENTS
Page
I.
SUMMARY AND CONCLUSIONS
1
II.
INTRODUCTION 2
III.
NOTATION IV.
MATHEMATICAL ANALYSIS 12
1. Determination of Displacement Functions which
Satisfy Equilibrium of the Core 12
2. Core Strains 18
3. Facing Strains Associated with Deformations of the
Middle Surfaces of the Facings (Membrane Strains). 19
8
a. Upper facing 20
b. Lower facing 21
4. Facing Strains Associated with Bending of the Facings
About Their Own Middle Surfaces 23
a. Upper facing 23
24
26
b. Lower facing 5. Elastic Energy of the Core 6. Elastic Energy of the Facings Associated with
Membrane Strains 29
a. Upper facing 29
b. Lower facing 34
7. Elastic Energy of the Facings Associated with Bending
of the Facings About Their Own Middle Surfaces 41
a. Upper facing 41
b. Lower facing 45
Page
8. Total Elastic Energy of the Simply Supported
Sandwich 9. Potential Energy of the Edge Loads 10. Equations Which Define Instability of the Sandwich 48
51
V. NUMERICAL COMPUTATIONS 55
62
1. General Case 62
2. Case a = 0, E Finite c
63
3. Case a.= 0, E = cc c
63
4. Case a = 2, E = co c
67
5. Case a General, E = 00 c
71
74
VI. USE OF DESIGN CURVES VII. DISCUSSION OF RESULTS 75
1. General Case 75
2. Case a = 0, Otherwise General 77
3. Case a = 0, E = 00, G
=G
= G, t' =t c
yz xz — —
77
4. Case a = 2, E =00 , G
= G, t' =t =G
c
yzxz — —
VIII. BIBLIOGRAPHY IX. APPENDICES 84
87
89
1. Appendix A -- Algebraic Details 89
2. Appendix B -- Integration Formulas 90
3. Appendix C -- Examples of Integrations 4. Appendix D
5. Appendix E
Determinant Elements for
Case a = 0, E = 00 6. Appendix F -- Determinant Elements for
Case a = 2, E = 00 c
X. FIGURES 91
95
100
Determinant Elements for
Case a = 0, E Finite c
103
121
ELASTIC BUCKLING OF A SIMPLY SUPPORTED
RECTANGULAR SANDWICH PANEL SUBJECTED TO
COMBINED EDGEWISE BENDING AND COMPRESSION*
By
W. R. KIMEL, Engineer
Forest Products Laboratory, Forest Service
U. S. Department of Agriculture
I. SUMMARY AND CONCLUSIONS
A theoretical analysis is made of the problem of the elastic
buckling of simply supported rectangular sandwich panels acted upon
by any combination of edgewise bending and compression on opposite
This report is one of a series (ANC-23, Item 56-5) prepared and distributed by the Forest Products Laboratory under U. S. Navy,
Bureau of Aeronautics Nos .NAer 01684 and NAer 01593 and U. S.
Air Force No. DO 33(616)-56-9. Results here reported are preliminary and may be revised as additional data become available.
Maintained at Madison, Wis. , in cooperation with the University of
Wisconsin.
Report No. 1857
1.
edges. The solution is based on the assumption that the sandwich
panel is composed of isotropic plate facings of unequal thickness and
an orthotropic core subjected only to anti-plane stress. The mathematical solution of the problem is based upon a Rayleigh-Ritz energy
method using a double Fourier series with configuration parameters
which are constants of integration obtained from solution of the core
equilibrium equations. The specific method of approach is thought
not to have been previously applied in sandwich analyses. The solution is in the form of a characteristic determinant of order infinity,
except in the special case of pure edgewise compression, in which
case the determinant is of order six. Evaluation of an order eighteen
principal minor from the determinant of order infinity is made to obtain data for design curves.
Design curves based on the additional assumptions of membrane facings and infinite transverse modulus of elasticity of the core
are compiled for the case of pure edgewise compression and for the
case of pure edgewise bending. These design curves are believed
sufficiently accurate for use in the design of a great many panels of
modern design.
2.
Equations are presented from which the critical load on sandwich
panels composed of plate facings and orthotropic cores with finite
transverse moduli of elasticity can be obtained. Unfortunately, however,
considerable computational labor will be involved in the determination
of critical load for this more general case. Design curves can be constructed from these more general equations, if the time and expense involved in their preparation can be justified.
II. INTRODUCTION
An elastic sandwich is a structural component consisting of two
relatively thin external members called facings separated by and bonded
to a relatively thick internal member called the core. The facings are
commonly a material with comparatively high strength and stiffness,
whereas the core is commonly a material of lighter density and relatively low strength and stiffness. The resulting layered-type structure
is characterized 1..,y an extremely high strength-weight ratio as compared
to that obtainable with the use of a single homogeneous material. For
this reason, its primary field of application has been in guided missile
and airframe assemblies, for example, wings, wall panels, webs of
beams, and so forth. The thin facings of the sandwich, if not bonded to
the core, are incapable of resisting reasonable design loads in their
own plane because of their inability to resist becoming elastically unstable. A primary function of the core is thus seen to be to maintain
3.
the stability of the sandwich. A typical elastic sandwich panel is composed of aluminum facings and aluminum honeycomb core. The aluminum facings are commonly bonded to the core by epoxy or vinyl phenolic
resins. Recently improved methods of fabrication and the development
of improved bonding agents have made practical the use of sandwich construction in an increasing number of different fields of application.
Continually increasing applications of the elastic sandwich as a
structural component have made necessary the analytical development
of equations defining the stability criteria for different geometrical configurations of panels subjected to various combinations of loading. Because of the many variables which enter the problem, a study of experimental data alone cannot be expected to yield all the information which
rational design procedures require. A panel of a committee composed
of representatives of the Department of the Air Force, Department of
the Navy, and Department of Commerce has been organized to unify,
interpret, and present known rational approaches and data for sandwich
structure design. It is known as the ANC-23 Panel. 1 The problem
chosen for analysis in this thesis is one that has been clearly indicated
by the ANC-23 Panel as being unsolved, and one which is encountered
in the design of structures using sandwich plates in their construction.
The purpose of this thesis is, therefore, to present a rigorous
derivation of the stability criteria for the simply supported rectangular
1
–See reference 1.
4.
sandwich panel when it is subjected to combinations of edgewise bending
and compression loadings.
—may be defined as that state of stress that exAnti-plane stress?
hibits stress components which are zero in a state of plane stress. Conversely, a state of plane stress exhibits stress components which are
zero for anti-plane stress. Because sandwich cores have such low load
carrying capacities in the direction of the plane of the panel as compared
to the relatively stiff facings, the normal stresses and shear stresses in
the core in the direction of the plane of the panel are assumed to be
negligible. Thus, the sandwich core in this analysis is assumed to be
subjected to a state of anti-plane stress. This assumption has been
used in many previous analyses and is known to represent actual sandwich construction very well. The facings are treated by isotropic thin
plate theory, that is, plane sections initially perpendicular to the
median plane of the plate remain plane during deformation in accordance
with the Bernoulli-Navier hypothesis.
The specific method of approach used in the solution of this problem is believed not to have been previously applied to sandwich analysis,
but an analogous method has been commonly applied to nonlayered
systems.
Equations for the three rectangular components of core displacement are found which satisfy the core equilibrium equations and the
..,See reference 5 and figure 2.
5.
boundary conditions of the simply supported panel. By evaluating the
displacements of the core at the interfaces (the junctions of the core and
the facings) and equating these core displacements to displacements of
the facings at these interfaces, displacements at any point in the facings
may be found. From these displacement equations, strains and subsequently elastic energy of both the facings and the core may be expressed.
It is noteworthy that the displacement functions written from a solution
of the core equilibrium equations are, in this particular analysis, equal
to zero until the sandwich starts to buckle. The edge loads are applied
to the facings in the conventional manner as in the related ordinary
3
plate problem presented by Timoshenko.Now, let the elastic energy of a general elastic system with respect to an undeflected configuration of that system be V and the potential energy of the external loads with respect to their undeflected positions immediately prior to buckling be T. Then the total energy of the
system, U, with respect to the undeflected state may be expressed as
U = V - T.
For a condition of instability, U must have a stationary value with re spect to any arbitrary change in configuration of the system.
The configuration of the system is defined in this thesis by
parameters A
mn
, B
mn , •
.,G
—See reference 14, page 351.
mn which are actually constants of
6.
integration obtained from solutions of the core equilibrium equations.
Thus,
dA + aU dB +
mn a Bnin mn
a Arrin
+ au dG
= o,
mn
8Grnn
from which it is clearly evident that the buckling criteria are:
8T
_ av aAn,in a Arnn a Arrin
au
au
aB
_
mn
au
aG
_
mn
av DT
aB
aB
mn
-0
=o
mn
(1)
av
aT
a Grnn a Grnn
- o.
This solution is seen to be a Rayleigh-Ritz method, but it differs from
the Rayleigh-Ritz procedure as applied to the conventional stability
analyses of ordinary plate problems in that certain of the differential
equations of equilibrium associated with the problem (the core equilibrium equations) are herein satisfied exactly, while the facings are
mathematically attached to the core by requiring displacement continuity between core and facings at the interfaces. The facings are then
treated by the conventional energy method for plates.
This method of approach to sandwich stability analysis was
prompted by, and is believed to be complementary to, a sandwich
7.
stability analysis by Raville4 who solved all the differential equations of
equilibrium in the facings as well as in the core. It is believed that the
method of analysis presented in this thesis, together with the method
presented by Raville, exhibit fundamental methods of approach whereby
it may be expected that if a problem has been solved in the thin plate or
shell literature,its counterpart in the sandwich panel can be solved.
In this thesis, literal solutions are derived within the scope of
the aforementioned assumptions; however, the reduction of these equations for use in preparing usable design curves seemed prohibitive in
view of the many parameters involved. Therefore, in the preparation
5
of design curves, the additional assumptions of membrane facings —
and infinite transverse modulus of elasticity of the core are made. The
transverse modulus of elasticity of the core refers to the modulus of
elasticity of the core in a direction perpendicular to the facings. These
assumptions have been previously used-6- and are known to represent
actual sandwich construction very well. It is to be emphasized that
literal equations are presented from which the numerical calculation of
4
—See reference 12.
5
In using the design curves, the actual flexural rigidity of the spaced
facings, D, may be used as a good approximation. See Use of
Design Curves and Discussion of Results.
6
—See references 1, 2, 3, 8, 11, 13, and 16. It is shown in this thesis that
the "tilting" method involves this assumption. See Discussion of
Results.
8.
critical load for any one particular sandwich may be computed without
the latter two assumptions. It is believed that design curves can be
prepared from these more exact literal equations if the time and expense involved in their preparation can be justified. Such design curves
would be of limited usefulness, however, in view of the many complex
parameters that would of necessity be involved in this case.
III. NOTATION
x, y, z
rectangular coordinates (fig. 1)
a
length of sandwich in direction of loading
b
width of sandwich in direction perpendicular to loading
c
thickness of core
t
thickness of upper facing
t'
thickness of lower facing
E
modulus of elasticity of facings
Poisson' s ratio of facings
Ec
xz
G yz
modulus of elasticity of core in z direction -- transverse
modulus of elasticity of core
modulus of rigidity of core in xz plane
modulus of rigidity of core in yz plane
N bmaximum value of loading (load per unit width, b, of panel)
due to pure edgewise bending (fig. 4)
N cvalue of loading (load per unit width, b, of panel) due to
pure edgewise compression (fig. 4)
9.
No
Nb +
Lc (fig. 4)
Nxvalue of N o at any location on loaded edge (fig. 4)
value of No at buckling
N or
2N
b
N0
a
normal strain in core in z direction
E
ZC
Y xzc
shear strains in core
Yyzc
x , E y y xy
x
E
E' ,
x y
normal and shear strains in upper facing
normal and shear strains in lower facing
xy
membrane strains in upper facing
y
xM yM , xyM
EE
E
4)
i
i
y
xM, EyM xyM
t
membrane strains in lower facing
bending strains in upper facing
EE
xB yB xyB
E , E 1
yl
xB yB xyB
bending strains in lower facing
T
U, V, W
displacements of upper facing in x , y , and z directions, respectively
u' v'w'
displacement of lower facing in x , y , and z directions, respectively
U , V , W
C
C
C
displacements of core in x, y, and z directions,
respectively
I
m, n, p, q,
A
, B
mn
mn,
D
T
mn ,
T
E
j
C
mn ,
xzc, yzc
integers
mn
F
,
mn , G mn
configuration parameters
shear stresses in core in xz and yz planes,
respectively
10.
normal stress in core in z direction
zc
0xM' °yM' TxyM
,
, a-'
a. '
xM yM xyM
°xB' °yB' TxyB
membrane stresses in upper facing
membrane stresses in lower facing
bending stresses in upper facing
Cr ,
bending stresses in lower facing
z'
distance from middle surface of upper facing
xB yB TyB
x
distance from middle surface of lower facing
V celastic energy of core
V mF ,
MF,
elastic energy of upper and lower facing, respectively, associated with membrane strains
IF
V BF' V B
elastic energy of upper and lower facing, respectively, associated with bending strains
V
total elastic energy of sandwich
E rr2
5
c(1 -
1
Omn
0 mn
)
2
2
yz a n
2 2
G
xz m b
G
2 n2
+ —
2
b2
m
a
2 a2
2
m +n
2
b
kma
critical load factor corresponding to loading defined by a
1 1.
I
moment of inertia of spaced plate facings,
(c + 20 3 c3
for t = t'
12
IM
moment of inertia of spaced membrane facings,
2
c t
2
I BM
fictitious moment of inertia of spaced plate facings,
t(c + t)2
2
for t = t'
flexural rigidity of spaced plate facings, El
D
1- p.
2
D Mflexural rigidity of spaced membrane facings,
EI
M
1- 11
D
BM
2
fictitious flexural rigidity of spaced plate facings,
EI
BM
12
S
IT
Ect
2
2
2Ga (1 - 11 )
2
Tr
Ect
2
2
for t = t'
for t = t'
2Gb (1 - p. )
(1.2
mn
Pmn
x
1
+ S1)mn
m2 a2 1
ocr
H , J , K
13,01
13,(1 P,q
(1-
2 D
Tr
M
elements of determinants
12.
IV. MATHEMATICAL ANALYSIS
The sandwich panel and its relation to the coordinate system are
shown in figure 1. Figure 3 illustrates different combinations of loading
which are defined by different values of
a.
Specifically, as shown in
figure 4,
2N
2 N,
b
b + N o N o(2)
N
a -
where Nb is the maximum loading on the sandwich (pounds per inch) due
to pure edgewise bending and N c is the loading on the sandwich (pounds
per inch) due to pure edgewise compression. This scheme makes possible a general solution for determination of the critical load of a simply
supported rectangular sandwich panel subjected to any combination of
edgewise bending and compression.
1. Determination of Displacement Functions
which Satisfy Equilibrium of the Core
In accordance with the assumptions outlined in the Introduction,
o- , cr , and T xyc
XC yC
are assumed equal to zero. The core is therefore
in a state of anti-plane stress. ? A differential element of the core is
shown in figure 2. Summations of forces in the x , y , and z directions,
respectively, give the following equations:
7
—Anti-plane stress is defined in the Introduction.
1 3.
aT
xzc
=0
az
aT
(3)
yzc
(4)
az
and
aT
aT
xze
ax
au
yzc
zc -0.
ay
(5)
az
Relations between stress and strain are applicable, that is:
o- zc = E E
c zc
T xzc
=G
(6)
xz Yxzc
(7)
y
yzc = G yz yzc.
(8)
and
In addition, the strains and displacements are related by the following
equations:
aW
E
zc
c
(9)
az
au
=
xzc
c
-az
aw
c
ax
(10)
and
av
Y yzc
c
aw
c
az + ay
Equations (6) through (11) allow the equilibrium equations, (3), (4), and
(5), to be written as follows:
14.
a 2 u c a 2 wc
az2
2
a vc
az 2
o
(12)
2
a wC
ayaz - 0
(13)
axaz
2
2
a 2 w
a v c a wc
c
G
+
c) + G (
+
)+E
xz(c
axaz
2
Yz ayaz
2
a y
ax
2
a 2 u
a wc
az
2 = 0.
(14)
In order to satisfy the boundary conditions of the simply supported panel,
as well as the core equilibrium equations, the core displacements are
assumed to be of the forms
co
\
(1) (z) cos m7TX sin nTrY
UC =
fmn
a
m= n=1
(15)
>°° f.(2) (z) sin m irx
a cos
mn
m=1 n=
(16)
co
f(3) (z) sin m rx sin
wec sin
mn
a
m=1 n=1
(17)
and
00
where f(1)
in
` n1' (z) denote separate functions of z alone.
mn (z), and f(3
mn (z), f(2)
These functions of z alone will be determined by the requirement that
equations (15), (16), and (17) satisfy the core equilibrium equations (12),
(13), and (14). It is to be noted that equations (15), (16), and (17) satisfy
the boundary conditions,
15.
u
0,
=
c
Y=°
y=b
M
x
=
0,
v
c
=
0,
we
=
x=0
x=a
and
M
y=0
y=b
0,
x=0, y=0
x=a, y=b
y
=
0,
x=0
x=a
where M and M signify edge moments on the panel about axes parallel
x
to the x and y axes respectively. Substitution of equations (15), (16), and
(17) into equations (12), (13), and (14) and requiring that the resulting
equations be valid for all values of x and y as well as for all values of
m and n results in the equations
2 ()
(3)
d f
(z) (1n
mTrdfmn (z)
_ 0
2
1- a
dz
dz
2 (2)
d f
(z)
mn
2
+ b
dz
(18)
df(3)
mn (z)
dz
(19)
and
-GXZ
mTrdf(1)
mn (z)
a
dz
2 2
n Tr (3)
2 fmn (z)
b
The functions f
2 2
m Tr (3)
f
mn •(z)
•
a 2
- G yz
(2) (z)
nTrdfmn
b
dz
d 2 f(m
3 ) (z)
+ Ec dz
2
- O.
(3)
(2)
(1)
(z) will now be found which
(z), f
(z), and f
mn
mn
mn
satisfy equations (18), (19), and (20). Differentiation of equation (20)
with respect to z gives
(20)
16.
Ins
2 (1)
(3) (z)
2 dfmn
mir d f mn (z)
m2Tr
-G xz a
dz
2
dz 2a
G
yz
d 2 f (2)
(z)
mn
dz 2
4-.
3)
n 2Tr2 cif mn (z)
b2
d
dz
d3 (3)
f
+ Ec
mn
dz
(z)
- O.
(21)
Substitution of equations (18) and (19) into (21) yields
d3f(m3) (z)
n
- 0
dz 3
(22)
which can be integrated directly to the form
2
)
(z)
=
A
f (3
mn
mn 2 + B mn z + Cmn'
(23)
Substitution of equation (23) into equation (19) with subsequent integration
results in
z3
nTr
z 2 + Dmn z + E
Amn
b
+
Brim
6-mn '
fm
(2n) (z) =
(24)
Substitution of equation (23) into equation (18) gives an equation which is
integrable to
(1)
f m) (z) =
z3
z2
a [A mn —6- + mn -T + F mn z + Gmni.
Mir
(25)
A mn
,Bmn
,Cmn,Dmn,E
mn,F
mn , and G mn in equations (23), (24), and
(25) are constants of integration which are not all independent. To determine relations between these integration constants, equations (23), (24),
and (25) are substituted into equations (18), (19), and (20). Equations (18)
17.
and (19) are seen to be satisfied identically by this substitution, but equation (20) yields the relation
m 2 Tr 2
2
a
Gxz (Fmn - C mn ) + n2 2 Gyz (Dmn - Cmn)
+ Ec A
mn = 0.
(26)
By substituting equations (23), (24), and (25) into equations (15), (16),
and (17) there results the equations
oo
c=-
U
3
mir[A —
—
z +B
a mn 6
m=1
+ Gmn
00
V
00
mni
+ F mn z
(27)
nir [A
z3 + B
z2 + D
mn z
b mn 6 - mn 2
m=1 n=1
+E
mn z2
yrx sin —
nlrY
cosm_
a
-
c
2
sin miTx cos nTrY
a
(28)
and
oo
W
c
=
A
m=1 n=1
2
z + B
z+ C
mn
mn 2
mn
mirx
a
s i n — sin
niry
(29)
The above equations for core displacements satisfy the equations of
equilibrium of the core if the constants A
and F
C
D
are
mn
mn'
mn
18.
related by equation (26). Thus, it is seen that there are only six independent constants of integration in the expressions for the core
displacements.
2. Core Strains
now obtained by substiC
The core strains,E zc' y xzc andyyzare
tuting equations (27), (28), and (29) into equations (30), (31), and (32)
which follow directly.
aw
E
zC
c
(30)
az
=
Y xzc
aw c au c
ax
az
(31)
and
aw
y yzc
1
c
= .••n••••
av c
(32)
—
ay
az
This substitution gives
co
E zC
co
=,
m=1 n=1
00 00
xz C
=>
m=1 n=1
(33)
b
A mn + B mn ) sin rmrx sin "Y
a
nry
mrx
(Cmn - F mn ) "Mr cos -- sin
a
a
(34)
and
yzC
.
G mn - Dmn ) b sin
/Y1 rx
a
c os nry.
.
(35)
19 .
The core shear strains are thus seen to remain constant with variations
in z alone. This was evident, of course, from the original core equilibrium equations (3) and (4). Again it is emphasized that A =, mn
C
,
D
mn
, and F
mn
are not independent.
3 Facing Strains Associated with
Deformations of the Middle Surfaces
of the Facings (Membrane Strains)
The expressions for the displacement components of points in
either facing may be obtained by requiring displacement continuity be tween the core and facings at their bonding surfaces (interfaces), that is,
the interface displacements of the facings must be equal to the interface
displacements of the core. The middle surface displacements of the fac ings may be expressed in terms of these interface displacements by
assuming:
1.
w and w', the z -direction displacements of the upper and
lower facings respectively, are constant through the facing thicknesses.
In analyses, where displacements associated with incipient buckling are
under consideration (as in this paper), this assumption seems particularly
valid.
2.
u and u', the x-direction displacements in the upper and lower
facings respectively, vary linearly through the facing thicknesses.
3.
v and v' , the y-direction displacements in the upper and lower
facings respectively, vary linearly through the facing thicknesses.
20.
The facing strains are arbitrarily divided into two distinct parts.
One part is that associated with strains in the middle surfaces of the
facings, that is, the membrane strains. The other part is that associated with strains caused by bending of the facings about their own middle
surfaces.
a. Upper facing. --The displacements at the middle surface of
the upper facing may therefore be evaluated as follows:
u
aw
c
ax
t
t
z=- —
2
ucl
V
z =- —
2
2
z=0
c
iz =0
+
t
2
Ow
ay
(3 6)
z=0
c
(37)
z=0
and
z= -
t
w
(38)
c
z=0
2
The membrane strains in the upper facing may be written as follows:
au
z=
E
xM
ax
z= -
Y M aY
t
a zw I
c
au l
c
ax
al
e
2
z =0
av c
- ay z =0
2
ax
(3 9)
2
z=0
2w I
+—
2
(40)
ay -
z=0
21.
and
av
z
z = - —
Y
2
= -
—2
ax
xyM ay
(41)
au ca 2 w c
ay
t axay
z=0
av c
ax= 0
z=0
By substituting equations (27), (28), and (29) into equations (39), (40),
and (41), the following relations are obtained for the membrane strains
in the upper facing:
E
(G
x
—mn
C )
mn- 2
22
n
sin
2
a
22
t c ) n
E - —
E
YM
mn 2 mn b2
M TrX
sin—
nny
s
(42)
m nx 4
sm. — sm. nny
—
a
(43)
a
and
Y xyM
G +E
+ E mn
mn
"'
M 11
- t C_ )
mn a
- nr
M ?DC
nary
b cos —
a cos
(44)
b. Lower facing. --The displacements in the middle surface of the
lower facing may be evaluated as follows:
ti =uc
l z = c + —
2
z=c
t' awe
2 ax
(45)
z=c
22.
v'
=V
z = c + 7
t'
c
aw
c
2 ay
z=
(46)
Z=C
and
w'l
(47)
t' = we
z=c
z=c+ —
2
•
The membrane strains in the lower facing may be written as follows:
8u'
z =
e
xM
'M
t'
au
ax
av,
e
c+
=
c
ax
z = c + t'
av
-
ay
Z =C
t'
c
ay
2
t' a wc
2
2
ax
2
a w
( 48
Z = C
c
( 49 )
ay2
Z =C
)
Z=C
and
au ,ay,
z = c + —i
2
2
xyM
t'
z=c+—
8x
ay
(50)
au
ay
c
av
a 2w
I
- t z=c
c
8x8y
c
ax
Z=C
7 = C
By substituting equations (27), (28), and (29) into equations (48), (49), and
(50), the following relations are obtained for the membrane strains in the
lower facing:
23.
3 t' c 2
2 t' c
C mn t
c
c
Amn
(—6 + 4 + B
(-- 2
+ 2 + 2
mn
xM
m 22
+ F mn c + G mn
co
E Y M
= >
m=1 n=1
a2
a in nr
sinm—
—y
a
(51)
c 3t+' c 2
c: 2 t., c. Cmn t'
Amn
(—
+B
(—2+2-- +
6
mn
4
2 2
+ Dmn c + Emni n Tr
b 2
MITX
sin — sin
nry
a
(52)
and
co
xyM
= -
T
c° m=1 n=1
[Amn t c
3
3
2
t' c ) + Brain (c 2 + c) + Cmnt'
2
+(D mn + Fmn) c + Emn+ Gmn mr nr
— cos MTDC cos nry (53)
a b
a
b•
4. Facing Strains Associated with Bending of the
Facings About Their Own Middle Surfaces
(a) Upper facing. --The facing strains due to bending of the facings
about their own middle surfaces may be determined from a knowledge of
the slopes and curvatures of each facing in the xz and yz planes. These
slopes and curvatures may be evaluated by differentiations of equation(38).
24.
Thus, it is seen that the facing strains due to bending of the upper facing
about its own middle surface may be derived from equation (38), and
hence from equation (29), as follows:
22
m
- - z'
mn
a
m= n=1
z = -
E xB = Z '
a2wi
2
1Tsin
2
ax
. nry
mTr xsin
a
(54)
2
a w
e
yB
= z' ay 2
z=
2 2
2-
n Tr
z'
-
mn
•
sin
M 1TX sin lurY
b2
a
b
(55)
and
N
aw
t
awi
Z
a
xyB ay
ax
t
Z=
4_a
ax
co
ay
M Tr
mn
Z= -2
axay
I
co
C
- 2z'
2
I
= 2z' >
m71 n=
2
a w
a
nil.
c os
M 1TX
a
cos niry
(56)
where z' is measured from the center plane of the upper facing so that
< t
t <
= z = — .
-—
2
2
(b) Lower
facing. --Similarly, the facing strains due to bending of
the lower facing about its own middle surface may be derived from equation (38), and hence from equation (29), as follows:
25.
co
tri
0
(I)
0
0
N
N
1
N
(N3
at
N
E
kI
NXI
E I aj
g
.—.0
0
E
E
E
U
"4
U
U
+
-.5
+
+
U
0
U
U
0
E
111
-I-
+
I
sk
SlAril
0
,--I
I I
g
Alg
_,
II
s
A
_
.,...,4
A
-
:
N
,1-1
0
0
0
P4
rd
0
N
-N
I
"0
•H
II
I
II
4:J I NI
I I
74:4 IN
0
:15)
+
N
-+--) I
U
II
U
+
U
II
I I
N
N
N
0
;_,
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a)
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r..)
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a)
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8
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al
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on
E
g
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PC)
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c.)
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r.)
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co
rtt
CO
N
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II
II
II
cn
,-i
-
-
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ccl
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V II
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-
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r
d)
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I; J IN
26.
5. Elastic Energy of the Core
The elastic energy of the core due to the anti-plane stress components may be expressed as
a b c
Vc = —
2
E
ZC ZC
+T
yzc Yyzc
+T
XZC YXZ C )
dx dy dz. (60)
000
An energy expression due to elastic energy of the core may be expressed
in terms of core strains by substituting equations (6), (7), and (8) into
equation (60). This substitution gives
a b c
2
2
y ) dx dy dz.
y
+G
( E c E2 c + G
yz yz
2
xz xz
V=1
c 2
(61)
0
If equations (9), (10), and (11) are substituted into equation (61), there
results
55
a b c
V
c
21
aw 2
[E c
aw 2
CI
c
Gxz az ax
au
0 0 0
av
+G
yz
awc
c
+
oz
ay
2
dx dy dz
(62)
From equations (27), (28), and (29) the following equations may be written:
au
8z
c
m=1 n=
2
Tr A
z
maxsi n nay
cos z+F
—+B
mn
a mn 2 mn
b
a
(63)
27.
av c
2
m
z
mn z + Dmn sin
illn —
2 +B
a
nTr
az
b
.
TO
c
mry
os —
(64)
co
awc
ax
co
m=1 n=1
A mn
z
2
+ B mn z + Cmn
2
M IT
a
cos 1-r - Trx sin 11117.
a
(65)
8w
2
c
A
ay
mn + B mn z + C mn
2
b
r12Trx cos n
nom'
a
(66)
and
awc
i nmrrx ;s mry
a
az
(67)
It is convenient to write equation (62) as
V =V
+V
+V c
c1
c2
c3
(68)
where
vc
1
a b
aw 2
E cdx dy dz
az
(69)
o o 0
a b c
v
„
au c
Gxz (
000
az
aw
c 2
+
ax
)
(ix dy dz
(70)
28.
'31
N
tr)
N
cs4
N
N
-d
N
N
4
N
I c0
0
C
N
•=4
81A
ct,
.•••••n
Cn
•••0
8
A E
U)
cd
rl I
w QIN
0
r•Cl
0
N
N
1
N1
0
I
I
II
.-4
0
• r-I
+.5
(/)
8
U
cd
cd
c.c)
0
• ri
1
I
,.0
•
cd
8
.0
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4-1
0
I
U
0
• ri
—I IN
sv\rA
-4
'34
en
0
NI
0
"0
0
cd
29.
8
which, upon integration —may be expressed respectively as
c 1
=
E abc
c 24 (A 2
n=
mn
c2 + 3A
mn
B
mn
c + 3B
(75)
mn )
VcL, -
22
abc rn TT
2
2 (C mn - mn )
xz 8 m= n=1 a
(76)
V
abc
yz 8
(77)
and
c3
=G
A complete expression for the elastic energy of the core which is subjected to anti-plane stress may now be written by substituting equations
(75), (76), and (77) into equation (68). Thus, it is seen that
v c
abc c
2
2
— (A
8
3
m.n c + 3A mn B mn c + 3B mn )
m
=l n=
+G
22
m TT
xz
a
2 (C mn
Fmn)
2
+ Gyz n2 2
(C mn
- D m n)
]
.
(78)
6. Elastic Energy of the Facings
Associated with Membrane Strains
(a) Upper facing. --The elastic energy of the upper facing associated with strains in its middle surface (membrane strains) may be written
as follows:
B ee Appendix C for further details.
30.
a b t
2
VMF
( cr
f-
yM + TxyM
xM xM yM E
xy
m)
dx dy dz .
0 0 0
(79)
This energy expression may be written in terms of strains by substituting
xm -
E ,
2
1-p,
(80)
yM)
xM
E
yM
66
1-p.
(81)
2
and
T
xyM
E
2(1+p,) YxyM
(82)
into equation (79) to give
a b t
V MF
E
2
1
(E
2
)0
0
+E
5
0 1-1'12 xM
1
2(1+0 xyM
2
+
YM
dx dy dz,
E
xM yM
)
(83)
It is convenient to write equation (83) as
V MF V MF 1 VMF2
+ V MF3 VMF4
(84)
where
V
MF1
E
2
2(1-p. )
2
xM
0 0 0
dx dy dz
(85)
31.
•••••••n•
N
co
N
N
• -0
"0
'•0
"0
4
4
0q•.°
N
00
0
• r--1
ti)
0
• rl
u)
0 1 cd
co
In
F., I a!
0
•.-I
cn
•1
cn
W
d
0
Cd
a)
0
N
"ti
rcl
N
" "g
"ti
X
X
"ci
rt
1
l
N X
?"
X
cd \-----sp
N
4.1
,–,
14
s-C\i'
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u)
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cti
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+
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a)
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cti
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en
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.......
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0
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N
15
N
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r1-1
›'
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0
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32.
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cri
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-11
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ai
O
U
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rd
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-
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O
cn
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01
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41
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O
U)
od
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U
8
.:14
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—I
II
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t'
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II
8
g1
1,---]
r\c---..
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ro
zi.,
c0
rd
N.--.
41
r4.8
cd
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II
1 1
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N
r.4
4.1
33.
vMF 3
and
co CO
2 2 n
22
abtp,E
m
(Ginn - 2
— C )(E
2 /
nin rnn- 2Cmn)
40-p. ) m=1 n=1 a 2
b2
(95)
y
2
abtE \-°° ) 00 m 2 Tr 2 n 2 IT 2
rnn
t
C
mn
)
. (96)
VMF4 16(1+p,) /
2 (Gmn+ E
a 2b
m=1 n=1
A complete expression for the elastic energy of the upper facing due to
the membrane strains in the upper facing may now be written by substituting equations (93), (94), (95), and (96) into equation (84). Thus, it is
seen that
00 o0[ 4 4
M
Tft
2 n 4 Tr 4
V
MF
4 (Gmn 2 C mn )
2 >
4 (Emn
8(1-p. ) m=1 n=1 b
a
abtE
2222
2
- —2 C mn ) + 2p, m2n 2 (G mn 2 Cmn
) (Emn
b
a
Cmn)
2 2
m Tr n 2 Tr 2
21
2
(G
+
Emn
t
C
)
.
mn
2
2
a
b
inn
(97)
As shown in Appendix A, equation (97) may be algebraically simplified
to the form
V
MF
abtE
8(1 -µ2)
22
2
t
n2Tr
2 (Gmn - 2 Cmn) + 2 (Emn
111. Tr n
t
1-p. m 2 Tr 2 n 2 7r 2
2
+
-—
C
)
2
Emn
Gmn
2 mn
2
2
a
b
(98)
34.
OD) Lower facing. --The elastic energy of the lower facing associated with strains in the middle surface of the lower facing (membrane
strains) may be written as follows:
a b c+t'
V'
MF
2
) dx dy dz
y'
+ T I
El
cr '
El
(cr xM xM yM yM xyM xyM
J
0 0 c
(99)
As was shown in the case of the upper facing, this energy expression
may be written in terms of membrane strains alone, as follows:
a b c+t'
1
)
[E'
xM
1-p, 2
1
Vm
' F E
2
0 0 c
1
(y1
2(1 +p,) xy
2
2
+ (e'yM) 2 + 2p. e'
El
xM yM]
Y
(100)
dx dy dz
It is convenient to write equation (100) as
+ V'
+ V'
V'
V'
+ V'
MF MF 1 MF2 MF3 MF4
(101)
where
a b c+t'
V'
MF 1
E
2(1 -p.
2)
(e'xm)
2
dx dy dz
(102)
0 0 c
a b c+it'
E
MF2
2(1-p.
2
5)
0 c
dy dz
( ElYM ) 2 dx(103)
35.
a b c +t'
V'
MF3
1.J.E
2
1
00
x.
e'
dx dy dz
YM
(104)
and
a b c+t'
E
V'
MF4 4(1+4)
2
00c
xyM
dx dy dz
(105)
36.
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CD
a)
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g
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41.
7. Elastic Energy of the Facings
Associated with Bending of the Facings
About Their Own Middle Surfaces
(a) Upper facing. --The elastic energy of the upper facing associated with strains caused by bending of the upper facing about its own
middle surface may be written as follows:
a b
V
BF
=1
2
-2-
(oxB
55
5
00
E
xB
+
o-
y B iB
+T
xyB NxyB
) dx dy dz'
(11 6 )
where z' is measured from the center plane of the upper facing. This
energy expression may be written in terms of strains by substituting
o-
xB
=
1-p,
2
( E
xB
+
cryB 1-42 (EyB +
p. E
yB
xB
)
(117)
)
(118)
and
T
—
xyB
E
(119)
2 (l+p,) NxyB
into equation (116) to give
+t
a b
2
V
BF
=
E
2
11 5
o o
t
1
-p.
2
2
2
(e xB
yB
1
2
+dx dy dz'
2( 1 41 ) xyB
+ E
+ 2µE
E
xB yB
)
(120)
42.
It is convenient to write equation (120) as
VBF
V BF 1
+V
BF2
+ V BF3 + V BF4
(121)
where
v
BF1
a b +22
E s 5
e xB dx dy dz'
2
2(1-p. )
t
0 0
2
(122)
t
ab+
V
51 5
E
BF2
2(1-p.
o o
dx dy dz'
(123)
yB dx dy dz'
(124)
2
Y xyB dx dy dz' ,
(125)
yB
t
-7
ab
_ p.E
v BF3--
2
1-1-1' 0 0
and
+
2
je xB
t
_2
E
+t
ab z
VBF4 = 4(1+p.)
50 05
t
2
43.
co
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U)
a)
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Le)
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0
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01,
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0
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0
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a)
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I XI
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III
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11
45.
A complete expression for the elastic energy of the upper facing associated with strains caused by bending of the upper facing about its own
middle surface may now be written by substituting equations (130), (131),
(132), and (133) into equation (121). Thus, it is seen that
V BF
-
3
abt E
96( 1 - [J. 2 )
+ 2(1-p.)
oo
4 4
4 4
2222
+ n w + 2p. m w n w
b4
a2 b2
m 7r
n=
a4
m27r2 2
2
n Tr C
mn
2
2
a b
(134)
and this equation can be algebraically simplified to the form
co
3
abt
E
>m 2 ir 2 n 2 7r 2 \ 2 2
V
Cmn
BF 96(1-11 2 ) m= n= \
b2
a 2
(135)
(b) Lower facing. --The elastic energy of the lower facing associated with strains caused by bending of the lower facing about its own
middle surface may be written as
t'
ab+2
1
((T xB
T
BF =2
0 0
t'
2
t
y' ) dx dy dz"
El
+ T 1
xB+yB
yB
xyB
xyB
E (136)
where z" is measured from the middle plane of the lower facing. As
was shown in the case of the upper facing, this energy expression, equation (136), may be written in terms of bending strains alone, as follows:
46.
t'
a b +
7
1
E
=
V'
2
BF
15
0 0
1-4 2
5
t'
) 2 + (el yB )2 + 2p.c'
[c'
(
xB
c'
xB yB
-—
(N1
)
2(11+4) xyB
2
(137)
dx dy dz"
It is convenient to write equation (137) as
V'
= V'
+ V'
+ V'
+ V'
BF BF 1 BF2 BF3 BF4
where
(138)
t'
a b + 2
E
BF1
2(1-p. 2 )
15
xB
) 2 dx dy dz"
(139)
0 0 t'
- -2-
E
(e , )2
dx dy dz"
yB
2
BF2 2(1-11 )
j5
t'
0 0
t'
2
(140)
a b
V'
BF-3
I-LE2
E'
e'
dx dy dz"
xB yB
(141)
and
0
V
' - t
a b 2
5
E
BF4 4(1+p.)
0
( .y'xyB
t'
2
dx dy dz"
(142)
47.
Substitution of equations (57), (58), and (59) into equations (139), (140),
(141), and (142) yields, respectively,
4 4
2
2
v,
ab(t 1 )3 E >
Tr
m
(A
---4—
mn
c
7
+
Brnn
c
+
C
mn
)
(143)
BF 1 96(1-11 2 ) m= n=1
a
oo
00
4 4
22
ab(t
'
)3E
>
n Tr
c
V'
4 (Arnn 7 + Bmn c + C mn )
BF2 96(1-p. 2
) m=1 n=1
00
(144)
00
2 2 2 2
2
v ,
_ ab(t 1 ) 3 p.E
m Tr n IT
c 2
BF3(A
—
+
B
c
mn 2 +
mn C mn )
48(1-11, 2 ) m=1 n=1
b
a2
(145)
and
00
00
2 2 2 2
2
2
111 Tr n Tr ,
kArnn + B mn c + Cmn ) .
48(1+11) m= n=1 a 2
b2
_ ab(t ' )3E
BF4
(146)
A complete expression for the elastic energy of the lower facing associated with strains caused by bending of the lower facing about its own
middle surface may now be written by substituting equations (143), (144),
(145), and (146) into equation (138). Thus, it is seen that
00
3
v ,
_ abt1)
(
E
BF 96
(1-p. 2 ) m=
00
(me2r2 n2_2)2
2
2
c
+ "
(A
+B
c
+
C
)
mn
mn
mn
a 2
b2
(147)
48.
00
N
^
r-N-T
-i
9,
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.
m
n
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0
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(1)
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cn
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49.
It will be recalled that the parameters A m
n , C mn , D , and F
mn
in
equation (149) are not all independent. The relation between these
parameters is given in equation (26), which when solved for F mn yields
1 a
Fmn Cmn P mn G
z
E
c
Amn + G
D mn )
2( 2
xz m TT
Yz b2
(150)
where
G
0,
—r1
1+
2 n2
Yz a
G
2 2
xz m b
(151)
•
2 n 2
m
0 mn
2+ 2
a
b
(152)
and
5=
En
2
(153)
c(1-1.1.2)
and substitution of equation (150) into equation (149) permits the writing
of an expression for the total elastic energy of the simply supported sandwich in which all of the parameters, A
G
mn are now independent. Thus,
mn
, B
mn
, C
, D
, Emn, and
mn
mn
50.
I
0
is)
,--(
0
-....,
o
41
41
No
C-1
INA
.c
+
I
0 1 0
0
N
N.—.
0
IN
1
rd
g I
Sk
)
ICC
1.
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g
1 1
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g
cr)
A
NI
(..)
NI d NI M
n]
N n N,.0
g
N
›,
r\l 41
The.----,
1
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rd CN3
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+
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at
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ca.
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+
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+
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a.
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+
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en
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+
51.
9. Potential Energy of the Edge Loads
The potential energy of the edge loads with respect to the unde fleeted configuration of the panel, immediately prior to buckling, is denoted by T, and may be derived in exactly the same manner as in the
analogous plate analysis. —9 The edge loads are defined in terms of Nx
pounds per inch of sandwich acting on edges perpendicular to the
x axis as shown in figure 4. Since the core was assumed initially to be
incapable of carrying loads in directions parallel to the facings, all the
edge loads must be mathematically applied to the facings. It is of interest to note that in actual sandwich applications, the edges must be designed so that these same edge loads are actually applied to the facings.
It is recalled that the core displacement functions given in equations (27),
(28), and (29) are compatible with zero edge moments about the edges of
the sandwich panel. This necessitates that the strains in both facings
be the same in the direction of loading. This means that the stresses in
both facings will be the same for facings of like materials. Thus,
ab
T = j
N x
t +
aw 2
[t. (— ) + t' ( aw' )
ax
ax
dx dy
(155)
00
aw
,
where — and aw
— refer to the slopes in the xz plane of the upper and
ax
ax
—
lower facings, respectively. Because of the nature of the application of
the edge loads shown in figure 4,
-See reference 14, page 351.
52.
ay
Nx = N o (1 - -fo-) .
(156)
Substitution of equation (156) into equation (155) gives
aw 2
t (—) + t . ( a
ax
ax
dx dy (157)
which may be written as
a b
N
T o
2(t + t')
S1
00
N
t
ax
aw. dx dy
a 2 + t' (—)
ax
ab
oa
2(t + t' )b
awl 2
aw 2
y t (—) + t' (—) dx dy. ax
ax
(158)
00
Because of displacement continuity between the core and facings at the
interfaces and with reference to equation (65), it is seen that
aw
nTry
mTrx
mur
—
cos
—
sin
a
mn a
aw
ax ax
(1 59)
and
aw,
ax
aw
ax
2
- 2- Amn 2 + B mn c
c
c
m=1 n=1
Z= C
+ cm &
a
nTry
mTrx
cos — sin —
a
It follows thatIl
10
.-=See
Appendix C for further details.
(160)
53.
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r•.1
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0
±
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+
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±
+
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+
0 ......1,
4>
u
I
--il.
. ri I
rd
0
N
'1
-.I-,
±
+
N u I mi
g
E
N
"
U
4,
k.___...v.----,
N
e \I
N
N UI N
o
E
4
t_.____.1
X
±
CO
I
H
0
rd
rci
-d
0
7-J
+
• r-I
rd
U
-0
-0
0
7.,
4->
,.0
+
cd
N
'--.,
N
-0
U
PC1
+
8,--1
0
+.,
•ri
O
W
rd
N
0
±
0O
• r-I
4-,
rd
0
0-1
0
U
+
i_t_i
r0
4.,
al
(1)
....,
U g
CA
+
Pq
........
0
m
--1
_
+
•',
--....
0
Nur
In
.0
.-4
P:1
U
+
N
U
U
It
C.)
.-n
U
4,
•,-1
+
H
C.)
.1-1
.--.. NU I r\1
0
r-I
MI
0
1
a)
..-)
I/\1
I
+
N
..-40
a.)
-0
+
0
Id
• H
'',
U
+
4-,
0
U
O
cd
PC1
-
0
0
N
-d
g
r_...____,
• ri
I
L......:1_/
-
....-...
N
N
U
+
0
a)
-0
cn
1....:/.) ......i
ii
8
cd
iti
d":I-)
rn
g
g
.--..,
-0
0
Z
+
._..)
4 .„
co
AI
+I
.v-1 I
a)
a)
H
ccor
+
55.
10. Equations Which Define
Instability of the Sandwich
As indicated in the introduction of this paper, the total energy of
the system, (V - T), must possess a stationary value at instability, that
is,
av aA rriri
aT
-0
aAmn
(167)
av
as
aT = n
aB
mn
(168)
av aT - 0
ac
aac
mn
mn
(169)
av aD
mn
aT - 0
aD
mn
(170)
av OT
aE mn
(171)
aEmn
avaT
- o .
aGmn aGmn
(172)
0
r--4
U
e.) I r\I
E
E 1 cd
+
..N.......
• r"--7-1̀
0
1
0
M
U
---....
N
41
7-)
0
..-.,
0X
O
7-4 f r\I
0
ni
-F
...>
--'0
u 1
0
E
O
O
NI
id
Cr'
Ln
.
.......
ch
O
o
.,--n
1-71
cr'
U
0
w
U 0
+
+
U
0
E
ICD_......1
N
W
,..0
NI 0 I N-0
+
r .7 u
N
U
0
,..-.-0
E
E
0
.......
NI
g
't
g
I")
H
c' M
±
1N
d
. . X
0 0
0
E
PC1
al
,—I
.......•
_Lo
.+J
+
.1-1
'c
N
U
N
N rq
NT N
,—s
..--...
a
NI
L--"1._i
+
0
+
E
u
0
E
u
,-,
41
-1-.)
2
g
cr)
0
-IN Ion
o
1-j,
0
0
1N
E
0 INPCI
U
0
C.
) 1 N
+
<4
-F
41
r>
E
..d
N
U
NU
Clq
0 I rq
4-1
• 1-1
u
E
A
enVI.0
N
cti N
...--..•,-4
+
Eht
±
o
+
A
° i
!
.---..
.0
,,-1
. .c)
71..
NI
N
Cll
cd
.....,
N
n
++
.......
0
O
E
*(
----,
vNI-71,
U1
N
l"-.......•
O
'14
MI 0 I N4
+
+
I
cd
41
o
cr)E
o
......,
E
E
N I rq
N
CD
.......,
. .ci-1p
..1
0
0
I
Q41
N
1--
1
TN
u
U
1
N'-'
±
'-ci
----
+
al
o
E
4
r
00
cn I-4,
+
O
#a
o
03
e------4,-----)
N4-1:I,
U
+
N uI1d''
N
ed
c)
ii
1
fxl
E
(.0
....1
'-,J
Z ° N.1
4.-1--'
0
CD
'gip
bi)
0
1 t,
...../
7i1J
u
0
ci)
en
N
ca
E
E
Ni
I
U1
N
eq
0
NI
56 .
N
filU I >I
N
0
+
N
N
k
0 (NA
,
E
g
-.14..
• 1
"0
N1I10
<4
--_____I
1.---L–__.
I-1
-0
•-0
0
X
C13
0I
-H
• r-1
a)
;..i
a)
4
57.
N
o o
U
NI
\ Ira
U
NUIN
IN
N
UI
>4
Ncd
U
O
II
o
U
N I NI
t
U
74--)
1
U
N
U
4-)
CD
..••••••n
Cr4
N
CID
NU
Lr)
••••••••
.•••••n
ct)
Nu
0
O
()IN
0
c0
'dI
0
4c.d4
cr'
a)
1
'ho
71-1
IN
N E I rqrd
d
co
I NI
cd
0
•H
58.
I
U
0
0
I
NI 0 IN
O
•H
rd
Li-i
. .
-0
0
•,-1O
u,
•H
-d
ti
O
0
N I N
moo
+
0
on
7-)
U
---. o
NE
ca
N
r-I
1-7---1
,
t..]
NI t
(..)
........
o
N
0
+
rd
U
E
U
N
1.4
X
+
._____4_...
..1
L__1_
- _1
0
E
l______Ne---/
+
N
0
k
LC
E
U
0
r-4
"--4-)
.......
0
4-)
ni
0
U
en
cd
0E
;-4
• r-/
I
0N-1
-4
0 +
1_:...71
N
Pa
4.. f)t,.
U
rq0 I Ng
±
No IN
,-.) 1 ,,,
r--_I
E I d
0
01
----0
cDE
E
2- I N
E
0
cv
Lo
E
E
E
U
•C
74-,
cd41
+
+
NI
4-) I
0
E
0--
.0
.........
-1-o)
.0
.0
...._...
rd
..--..
N
N
d
E
INS
\10
O I 4
r‘1 cd I
I
1 0
"ft;-) 1 N
E
0
E
N N
E
—0
E
0
E
N
n
i
ct
CA
NuIN
tn.
U
r..-,....,--7,
O
•HI
4-,
d
co
O
,-,--1
O
0
o
•H
a.
NN
I N -0
N
0
(..)
+
0
E
0
(DE
+
,
-i
8"1 E
+
U
7.1 Ni
0
E
m
1
c0 I N
+
+
0
E
diN
o
E
.
I
I-7-1
E
4.1
IN0
NI r,-0
t I r\1
++
N
O
M
E
+
o
E
t
0
.,..
N (NI
I
NN
LO
in
,--i
........
E
0
0
0
c.)
C.)
+
1N
•r,
E
0
+
. 0
1-----1
+
41
W
tdr
N INS
0
u
o
.,-1
cr'
<i)
O
U
.
<4
O
.--...
N•H
••••n••
0
E
<4
rp
±
.•-•.0
r-I
(\14-' 1 't
+
tu)
N
0 rl
0 q-
±
,ft-I
......
H
E
ill
in
r-
c)
g
n
0
0
N
N cd NI
±
E
0
N irq
E I cd
+
0
41
E
E
N
X
OI
.H I
u
ti
0
E
PC)
±
ca
rd
'0
0
.,--4r)
0I
-H
•'-1
•H
(1)
co
..0
a)
4
ci)
59 .
U)
a)
U
0
•H
NJ
b.0
INJ
cd
Cd
I
U
r•I
rd
0
U
N
U
0
•H
U)
•H
Nig U
•H
7:1
NE
cd
C_
O
N
U
H
I
r\14
0
•H
I0
ort
r\IdNd
crl
a)
0
,z)
cd
ir)
U)
0
U
•H
ca
Ca)
0
0
• HI
•H
N
60.
N
N
cno
i
w
N
1‘4)
cc)
E
NO r\l.0
.,
to
Q
t._______Ne.-)
7t,
1
cti
I
Lo
N
0
x
(.,
W
+
---0 7
4-)
u
cd
E
A
7.4
;.,
O
E
0
I
N
U
N
U
---.0
E
0
a.
C.)
o
L___
u ..j
E
N
INA
e\10
U
N IN
IN
vd
E
+
21 eq
0
IN
c
r-0-1
PA
o
1N
,0
r-----g-1
+
0
E
0
W
+
E
(.)
Ni
+
II
,...._"........._,
N
IN
0
E
w
E
+
.-4-'
+
E
I,,,
l
E
cd
0
+
i
NU I eq
g
a> E
+
E
o
•r-I
al
N
1
rs, S
IN
E
CU
N IN
g
N
O
•1-1
•r-I
4-)
U)
ilk
k
LO
n
N.--.. N
E
(...4-4-' ,t4
N
U
1
–I–
u
dX
0 0
1
N
N
k
ct N
,....••
U
0
E
N
u
, ..,
No 1 N.-0
±
±
6 1.
cn
co
I.o
U
U
N
111
U
U
U
U
N NI
El cd
O
U
- +)
IN
0
CJ
\I0 N4
0
U
r
NE I NI rd
0
NIN
U
O
N
U
+
0
0 0
U
r \I 0 I Nr0
•0
O
•H
(.1)
O
to
N
62.
Equations (173), (174), (175), (176), (177), and (178) constitute a linear,
homogeneous set which will be satisfied by a value of N o equal to Nocr
,
, D
, C
, B
the buckling load. If all of the parameters, A mn
mn mn mn
E mn , and G mn are identically equal to zero, this set of equations is
satisfied, but this is a trivial solution associated with the nonbuckled
state of the sandwich. The solution of interest is that which satisfies
the set of equations (173) through (178) when at least one of the parameters
A mn through G mn assumes a value other than zero. Such a solution can
be obtained by equating to zero the determinant of the coefficients of the
in the set of equa and G
E
parameters A, B , C,
D
mn
mn mn mn mn mn
tions (173), (174), (175), (176), (177), and (178). Since there is an infinite
number of these parameters in the infinite number of equations which
constitute this set, the resulting determinant is of order infinity unless
= 0, in which case the determinant is of order six. Fortunately, however, for a k 0 a relatively small part of the determinant of order infinity will yield a satisfactory approximation to the critical load.
V. NUMERICAL COMPUTATIONS
1. General Case
A solution of equations (173) through (178) will provide the critical
load for the most general case of sandwich panel and loading. For a
specific panel, a numerical solution for critical load can be obtained.
63.
For further details of such a general case solution, see the section entitled Discussion of Results.
2. Case a = 0, E c Finite
The equation which defines the critical load for the case a = 0
(pure edgewise compression, see fig. 3), is found by equating to zero
the determinant of the coefficients of A , B
mn
mn •,
.,G
mn
in equations
(173) through (178). This equation is
H
H
1,1
2,1
H
H
1, 2
2,2
.
H
. H
1,6
2,6
0
H
6,1
H
6,2
where the elements
H p
.
(179)
H6,6
after some reduction, are given in Appendix D.
For the determination of the critical load of any specific panel loaded in
edgewise compression (a = 0), equation (179) may be solved numerically
for N ocr•
3. Case a = 0, E c = 00
The assumption of infinite transverse modulus of elasticity of the
core (E c = co) introduces welcome simplifications into equation (179).
This assumption is believed valid in analyses relating to modern sandwich
64.
11
construction. — In this case, equation (179) can be written as
J
1,1
J
2,1
J
J
.
J
...
J
1,2
2,2
1,6
2,6
(180)
'1 6,1
J
...
j6,2
where the elements J 6,6
are given in Appendix E. For this case, when
ID,c1
G = G = G and t' = t , the determinant in equation (180) may be
yz xZ — —
simplified by judicious additions and subtractions of multiples of rows
and columns.
Thus, equation (180) reduces to
2
t (c + t)
2
1 1
+1
S I,rnn
2
ocr
2
D t. rnn
2 2
m a
••••
(181)
vmm.
where
_
(c +
3
-c
3
(182)
12
1)
2 a2
2
= m + n —
2
mn
b
11
— See Discussion of Results.
(183)
65.
EI
D =
1-4
(184)
2
and
2
S
Tr
=
Ect
(185)
2
2
2Ga(1- )
Equation (181) can be written as
N
ir
=
ocr
2D
2
k'
m0
(186)
where
2
k'
-
m0
b
2
m a 2
cD
2
mn
1
2
t (c + t)
2I
1
+1
S rnn
(187)
Further simplifications result when the facings are sufficiently thin to
be considered membranes. In such cases t < c , so that terms involving
either the square or cube of t are small enough to be neglected. This
latter assumption is equivalent to the assumption of membrane facings,
and is known to represent many actual sandwich constructions very well.
This assumption of membrane facings permits equations (186) and (187)
to be written, respectively, as follows:
N
and
ocr
=
7r2D M
2 k m0
b
(188)
66.
k
m0
-
2
b2
mn
22 1 + S iD
m a
mn
(189)
12
where —
El
D
M
- 1- p.
(190)
2
and
I
tc2
M 2
(191)
Instead of the parameter, S, a related parameter, W, may be used in
equation (189). Thus, equation (189) may be expressed as
2
mn
2
k
m0
b
2 2
m a
1+
b
(192)
2
—2- W (D
mn
12
— The simplificatioi, from equation (187) to equations (189) and (192) requires a modification (in equation (186)) of D, the flexural rigidity
of the spaced plate facings. The assumption of membrane facings,
El
=
— will effect this simplification. A more accufor which D
2
M
1-p.
rate assumption which will produce this same simplification is to
EI
BM
This point is discussed further in Discussion
use D
=
2
BM
1-p.
of Results.
6 7.
where
2
W-
r Ect
(193)
2
2
2Gb (1-p. ) '
Equation (192) is presented graphically in figure 5.
4. Case a = 2, E c = co
For the determination of critical load in all cases where a is
12
other than zero, a determinant of order eighteen is solved.- Thus, the
equation defining the critical load of the sandwich panel for the case
a = 2 (pure edgewiise bending) is found by equating to zero the determinant of the coefficients of A
,B
, . . , G
in equations (173)
m
n
mn
mn
through (178). This equation is
K
K
1,1
2,1
K
K
l, 2
K.,
.
2,2
K
2,18
(194)
K 18, K 18, 2
where the elements K
G
xz
p, q
.
K
18, 18
are given in Appendix F. For the case where
= G yz = G and t = t' , the determinant in equation (194) may be
12
—See Discussion of Results for further details relating to choice of
order of the determinant used here.
68.
0
;-1
0
A
-C3
(1)
U
▪ dc 71,U
▪
cr)
0
or)
so
ON N
N
H:34
cc
•••••..../
0
•H
cll
tlk)
• g
H
LH
• H
1
--(I A
.--,
+ t‘l
N
—4,
N
• H
cd
k
0
.,-1
rq
r-1
0
Z
ei
N
cr)
4,
E
---I I A
r‘a
u
714
1
Hi
+
e••].••••-n
--1
1
•11
cd
k
E
I
0
Z
(NI
r"11 I as
e,
co
N
E
0, N
‘r) I '
4-)
.+J
0
•
Cd
r
•
ci)
CI)
,-0
U
(I)
c
,-0
--,
4-I
•H
0.•
E
•H
U)
H . . E
V
(l)
—4
1
r--1
(,)
•H
NI
,t,0 .........
0
• I-4
4-4
—4I A
—4
(.. )
-.....
-i-J
'0
N
7114
±
4-) I—,
+ rs4
a)
E
1
.--1
N--..
E
c.]
(t 't
NC
I
.
0
Z
r-il]
I Q`
,
C
69.
n••-.1
rd
7)54
O
44
I n
•--1
r:2, I
"0
0
•
d' t
4
CD
C/)
O
1:1
a)
cd
z
.0I
0
I
U
—1
CU
M
111
N
1'
tr-1
E
1n1
O
ON
r-1
I
I
• t-I
7:1
rd
I
I
0
• n-1
4-)
a)
U
0
r.)
••1
cd
0
cri
a)
0
CID
4a)
cf>
bz
0
•U
Z0
Z0
crI
•n) (7'
E
•.0 I In
CT, N
oz1
a)
0
a)
CD
a)
H
G P 0 827391-9
70.
Expansion of the determinant in equation (196) leads to the equation
Dmw
Nocr
2
m 1
1+S
4
ml
2
ml
2 2
2m a
2
m3
1+S
m3 )
2
m2 m
+ St t
t
(197)
2
16
48 2
(-2 ) 1 + S 1) ml +( 9 ) 1 + S t m3
which may be expressed as
Tr
Nocr
2
D
b2
k
(198)
m2
where
2
2
ir b
k = —
2 2
m2
m a
2
2
\ (2
m3
m2
m
l
1 + S m3
+S
1 + St
m2-)
ml
2
2
2 t
2
(D
c.)‘,) m3
ml
/96 \
1+S
9
‘251 1 +
m3
(199)
71.
or in terms of the parameter W as
2
ml
7
k
m2
=
m
2
b
2
b2
1 + W —
2
a
\
2
96
( )
25
2
2
c.
2
m3
2
m li
.I.2
ml
2
b
1+W2
a
.I.
1 + W b2
2 '1' rn3
a
, 2
2
m3
+ (32)
9
2
b
iI
1 + W 2(m3
ml
m3
a
1 + 111/41.
7
a
m2
(200)
Equation (199) is presented graphically in figure 7. Equation (200) is
presented graphically in figure 6.
5. Case a General (See Figure 4), E = 00
c
A reduced form of the determinant in the equation which defines
the critical load for the case with membrane facings, E
G yz = G
= 00 ,
c
= G, t = t' , anda is any value defined by equation (2), is
xz
72
O
dIN
N
Ncd N
N
O
U
z
0
dIN
•
---4
N
N
N
N al N 0
1 c)
cd 't
P
U
0
k
z
I
H
U
0
Z
N
N
d
HE)4
..0 l
,
ri)
,..]
e
z
U
0
d
00 WI
'7r
N
IN
+
,--4
d IN
IA
N
c N
N cd
N
U
z
0
O
73.
Expansion of equation (201) yields the following equation:
13m2 P m3 -
(13
▪
13 1112
6
+ ( 1- )
P m2
m1
X
131113)
3
-
13m3 P m2 13 m3 ) +
a.)
N
22
oc r ma
D M
4
X2(
(3rril
2
48 2
( 2S )
(P
m1
x)
TT
2
9
(
P m3
X)
=
0
(202)
where
mn -1 + S
2
mn
(203)
mn
and
2
= N
m a2
ocr
17-
2 D
1
M
(
(1
-
2
)
•
(204)
No design curves were made for values of a other than a = 0 and a = 2.
Equation (202) is presented with the thought that design curves for various values of a can be prepared if desired. Until such curves are compiled, the designer can numerically solve equation (202) for the critical
load of any particular panel.
74,
VI. USE OF DESIGN CURVES
In using either figure 5, 6, or 7 to obtain the critical load of a
particular sandwich panel, select the correct member of the family of
4
curves-1- by calculating the value of the parameter S (or W if using
either figure 5 or 6) from the physical properties of the particular sandwich under consideration. Then, read the lowest value of k ma correa of the panel. The integer m associated with the
sponding to the ratio —
particular curve from which k ma
is selected, indicates the number of
half sine waves into which the panel will buckle if its critical load is applied. The critical load of the panel, N OC r , can now be computed by substituting this value of k _ into either equation (188) or equation ((198),
mu
depending on whether the panel is to be subjected to pure edgewise compression (a = 0) or pure edgewise bending (a = 2). It should be noted that
figure 5 applies for a = 0, whereas figures 6 and 7 are both applicable
for a = 2.
In the curves shown in figures 5 and 6, the parameter W is used
to separate each member of the family of curves, whereas in figure 7,
the parameter S is used for this purpose. Curves 6 and 7 use parameters
W and S respectively, to define the k m2 for the same identical case of
loading (a = 2). It is believed that the designer will find the use of figures
14
—A family of curves is here defined as that set of curves corresponding
to a particular value of S (or W).
75.
6 and 7 together will aid in the interpolation to a correct value of k
m2
for any specific sandwich.
If the facings are very thin, then the assumption of membrane
facings is sufficiently accurate, and the designer may use a flexural
rigidity factor denoted by D M . For sandwich panels with facings of
thicknesses such that the assumption of membrane facings is deemed inaccurate, the designer is advised to use the flexural rigidity denoted by
D. A further discussion of the different approximations for flexural ri-
gidity is contained in the section entitled Discussion of Results.
VII. DISCUSSION OF RESULTS
1. General Case
The general case discussed here refers to a sandwich panel composed of elements with the following properties:
1. The core is orthotropic and is capable of resisting only antiplane stress. The transverse modulus of elasticity of the core ,
E c , is
finite.
2. The facings are isotropic and are of unequal thickness.
The loading on the panel for this general case is any combination of edgewise bending and compression on opposite edges of the panel (denoted by
a. from equation 2).
The solution for critical load in such a general case may be effected by equating to zero the determinant of the coefficients of A
mn
,
76 .
B
mn
,
, G mn in equations (173) through (178). This characteristic
determinant is of order infinity, except in the special case of pure edgewise compression (a = 0), in which case the determinant is of order six.
A close approximation to the critical load may be obtained by replacing
this determinant of order infinity by its first principal minor of order
eighteen. Specifically, this first principal minor of order eighteen is
composed of the coefficients of the configuration parameters A ml' Amt'
A
m3'
B m l' B m2' B 3 '
, G m 1' G m2' G m3 in equations (173)
through (178). The equation formed by setting this first principal minor
15
to zero is believed to yield a solution which is sufficiently accurate—
for design. For combinations of loading defined by values of a close to
zero, smaller principal minors will yield sufficient accuracy for de6
sign. 1- It is to be emphasized that this method of solution will involve
15
—For the analogous homogeneous plate problem with a = 2 (see refer-
ence 14, page 355), Timoshenko asserts, "the difference between the
third and fourth approximation is only about one-third of one percent."
These third and fourth approximations to which Timoshenko refers
are analogous to the principal minors of order eighteen and twentyfour, respectively, referred to in this thesis. Therefore, for a = 2,
it is believed that the principal minor of order eighteen will provide
accuracy within about one -third of one percent of that obtainable
from the principal minor of order twenty-four.
16
—The error in any given calculation for critical load may always be
estimated by again solving for critical load using the next larger
principal minor. However, the principal minors which must be considered are (in order of increasing accuracy) of order 6, 12, 18, 24,
30, ... , so that this error is not easily determined.
77.
7
a considerable expenditure of computational labor,- 1-"-- and is not recommended unless the designer believes the core-flattening effect which accompanies incipient buckling is appreciable. This flattening effect is
believed negligible for a majority of panels. Following this idea, equation (202) has been derived assuming that E = 00. Equation (202) is
c
applicable for any combination of edgewise bending and compression
(any value of a), for t' = t, and G G
xz.
yz =G.
2. Case a = 0, Otherwise General
For the case of pure edgewise compression (a = 0), equation (179)
can be solved numerically for the critical load of any particular panel.
This equation includes the effect of core-flattening on the critical load,
and is sufficiently general to accomodate sandwich panels with orthotropic cores and plate facings of unequal thickness. This solution has
not been reduced to design curves.
3. Case a = 0, E = 00, G
c
yz = Gxz = G t' = t
The case discussed here refers to a sandwich panel composed of
elements with the following properties:
1. The core is isotropic and is capable of resisting only antiplane stress. The transverse modulus of elasticity of the core, E c , is
infinite.
7 A numerical solution for the critical load of a particular panel involves
the solution of an order eighteen numerical determinant and is therefore possible. A general literal solution for N ocr seems impossible.
GPO 827391-B
78.
2. The facings are isotropic and are of equal thickness.
The loading on the panel for this case is pure edgewise compression.
The critical load for this case is defined by equation (186). Equation (186) may be reduced to equation (188) by a modification of the flexural rigidity factor, D, of the spaced plate facings. The design curves
shown in figure 5 were constructed from equation (188) and are therefore
applicable to this case subject to the aforementioned modification. The
modification here referred to is that involved in assuming
,2
t (c + t)
2I
(205)
For facings with negligible flexural rigidity, the assumption of membrane facings is valid, that is, t < c, so that all terms involving squares
or cubes of the facing thickness, t, may be assumed negligible and
1)). Therefore, with the assumption of
hence I = IM(see equation (191)).
membrane facings, equation (205) is seen to be identically satisfied. A
much closer approximation which will give this identical simplication
occurs if I in equation (205) is taken equal to a fictitious area moment
of inertia denoted by I BM' where
Z , + t) 2 .
—
I BM= 2
(206)
To further emphasize the fact that the use of I BM in place of I in equation (205) is a much better approximation than is the use of I in place
79.
of I, a comparison of these approximations to I follows:
,3
3
+ 2t) - c
12
I-
3
t
+—
BM 6
I
I
(207)
I
(208)
2 2 3
+ ct + t =
m
3
(209)
Associated with I, I, and I are the flexural rigidity factors D,
BM
D
BM
, and D M
of the spaced facings. For example,
D=
EI
2
•
I is not significantly different from I
therefore, the designer is adBM'
vised to use the exact value of D with figures 5, 6, and 7 in cases where
the facings are believed too thick to be treated as membranes. For
many applications, however, the assumption of membrane facings and,
therefore, the use of I M will be sufficiently accurate.
Equation (186),which gives the critical load for this case when the
facings are considered as plates, reduces to
2
lirn
G-11.-00
N
OC T
-
D
2
tt
2 2 mn
ma
TT
(210)
80.
when the modulus of rigidity of the core is infinite. Equation (210) is
identical to the result obtained from the analogous homogeneous plate
18
analysis, — where D is the flexural rigidity of the spaced plate facings.
When the facings are considered as membranes (see equation 188), this
critical load is again expressed by equation (210), except that D becomes
D M' the flexural rigidity of the spaced membrane facings.
Equation (186) reduces to
lim
G--•-0
71.2
Et 3 2
N ocr = 2 2 2 cl. mn
)
m a
(211)
when the modulus of rigidity of the core is zero. Equation (211) yields
18 for two simply-supported rectangular homogeneous
the critical load--plates of thickness t. When the facings are considered as membranes
(see equation (188), this critical load reduces to zero as would be
expected.
The critical load for a sandwich strip with plate facings in plane
strain may be obtained from equation (186). It is
lim N oc r
b—..co
ir22
m
2
a
—See reference 14, page 328.
(c + t)2
2I
1 -- 1
m 2S
+1
(212)
8 1.
Where the facings are considered as membranes (see equation 188), the
critical load of a sandwich strip is again expressed by equation (212),
except that D becomes D , the flexural rigidity of the spaced membrane
facings.
For infinite modulus of rigidity of the core, equation (212) reduces to the Euler column equation for two homogeneous spaced strips
in plane strain, that is,
lim N
ocr
b -n—• 00
11110
2 2
_wm
D
2
a
(213)
G---i► oo
With infinite modulus of rigidity of the core and with membrane facings,
the critical load of a sandwich strip may be expressed by equation (213),
wherein D is replaced by D
M
With G = 0, equation (212) yields the familiar Euler column
formula for two homogeneous strips in plane strain, that is,
N
ocr
-
7r
2
a
m
2
2
E t3
2 6
1-p,
•
(214)
An additional limiting case of interest is that which occurs when
either m is made infinite or a is made zero. In this case, the critical
load on a sandwich panel with plate facings (defined in equation (186)) is
infinite, that is,
82.
ocr
(215)
= oo,
or
However, this critical load for a sandwich panel with membrane facings
(see equation 188) is finite, that is „
N
lim
ocr
=cG.
(2 16)
00
or
The finite limiting value of critical load given in equation (216) is characteristic of sandwich analyses wherein the assumptions of E c = 00 and
of membrane facings are made. This limiting value of the critical load
—
may be attributed to the shear instability of the core, 19
a
for each ofthe
The value of — corresponding to the minimum k m0
b
individual curves in figure 5 is found by setting
8k
m0
a
a—
(217)
=0.
b
Thus,
k
m0
min
19
—See reference 1.
11
1-W
1+W
(218)
8 3.
Therefore, the equation of the horizontal tangent to each family of
20
curves — is
4
m0
min
(1 + W)
(W < 1)
2
.
(219)
Also, note that
k
1
m0
a=0
b
(220)
w
Equation (219) is of particular interest to the designer because it can be
used to compute the critical load factor, kof a particular panel
m0'
a
whenever — > 1, provided W< 1. For values of W = 1, equation (219) is
not valid. The designer may use the equation
k
m0
=
1
W
(w 1)
for the determination of critical load factor , k
(221)
a
for all 7
1
). ratios when
m0 '
the parameter W = 1.
The result from this limiting case analysis, that is, equation
(181), is identical with the result obtained from the so-called 'tilting"
21
method of analysis. — This comparison seems to reveal the fullda mental nature of the assumptions involved in the "tilting" method,
20
—A family of curves is here defined as the set of curves corresponding
to a particular value of W (or 5).
21
—See references 2, 3, 16.
84.
that is, the core is in anti-plane stress and the transverse modulus of
elasticity of the core, E c , is infinite.
4. Case a = 2, E c = co, Gyz Gxz G'
=t
The case discussed here refers to a sandwich panel composed
of elements with the following properties:
1. The core is isotropic and is capable of resisting only antiplane stress. The transverse modulus of elasticity of the core, E c , is
infinite.
2. The facings are isotropic and are of equal thickness.
The loading on the panel for this case is pure edgewise bending.
The critical load for this case is defined by equation (195). Equation (195) may be simplified to give equation (198) by a modification of
the flexural rigidity factor, D, of the spaced plate facings. This modification is based on the same assumption-given in the approximate equation(205). The design curves shown in figure 6 and in figure 7 give km2
a
versus — for the same case of sandwich panel and loading; however, the
parameter W is used to distinguish between families of curves in figure
6, whereas, the parameter S is used to distinguish between families of
curves in figure 7.
Equation (195), which gives the critical load for this case when
the facings are considered as plates, reduces to
85.
DTI-
4
2
2
2
cD
.4
m1
m2
m3
48 2 2
16 2 2
) 13 + (—)
25
m 1 9
m3
(1.
lim N ocr 2m 2 2
a
(222)
when the modulus of rigidity of the core is infinite. Equation (222) is
22
identical to the result obtained from the homogeneous plate analysis,—
where D, is the flexural rigidity of the spaced plate facings. When the
facings are considered as membranes (see equation (198)), this critical
load is again expressed by equation (222), except that D becomes D
M
,
the flexural rigidity of the spaced membrane facings.
Equation (195) reduces to
lim N
ocr
3
4
Tr
2
2 2
6(1-il ) 2m a
Et
2
2
2
(D
m2m3
ml
2
2
48 )+ (16)
.1.2
25
ml
9
m3
d)
(223)
when the modulus of rigidity of the core is zero. Equation (223) yields
22
the critical load — for two simply supported rectangular homogeneous
plates of thickness t subjected to pure edgewise bending. When the facings are considered as membranes (see equation (198)), this critical
load reduces to zero as would be expected.
22
—See reference 14, page 355.
GPO E127391-7
86.
When either m is taken equal to infinity or a is taken equal to
zero, the critical load on a sandwich panel with plate facings (see equation (195)) is infinite, that is,
lira
N
0 cr
= Do.
(224)
or
However, this critical load for a sandwich panel with membrane facings
(see equation (198)) is finite, that is,
lim
(225)
N 0 C T = 1.886 c G.
m—+00
or
The finite limiting value of critical load given in equation (225) is characteristic of sandwich analyses in which the assumptions of E c = 00 and
of membrane facings are made, As stated previously, this limiting
value of the critical load may be attributed to the shear instability of the
core.
A form of equation (225) which is useful for design is
2
ir D
lim
N
Oc r
b
2
M
1.886
2
a
2
Sb
—
2
(S -a=2- = 0.215) . (226)
87.
Equation (226) can be used to compute the critical load for sandwich
2
a
panels subjected to pure edgewise bending when S —>
= 0.215 . That
2
b
this is so is evident from a study of figure 7. In a like manner, study of
equation (225) together with figure 6 reveals that the critical load
2
ir D
li m.
♦
m— w.co
N ocr = b
2
M 1.886
(W
0.215)
(227)
is applicable for computing the critic \al load when W 0.215.
VIII. BIBLIOGRAPHY
(1) ANC-23 Panel
1955. Sandwich Construction for Aircraft, second edition, Part II.
(2) Ericksen, W. S.
1950. Deflection Under Uniform Load of Sandwich Panels Having
Facings of Unequal Thickness, Forest Products Laboratory
Report No. 1583-C.
(3) , and March, H. W.
1950. Compressive Buckling of Sandwich Panels Having Facings of
Unequal Thickness, Forest Products Laboratory Report
No. 1583-B.
(4) Eringen, A. Cemal
1952. Bending and Buckling of Rectangular Sandwich Plates, Proc.
First U. S. National Congress of Applied Mechanics, pp.
381-390.
(5) Goodier and Hsu
1954. Nonsinusoidal Buckling Modes of Sandwich Plates, Jour.
Aero. Sci. , Vol. 21, No. 8.
(6) Gough, G. S. , Elam, C. F. , and de Bruyne, N. A.
1940. The Stabilisation of a Thin Sheet by a Continuous Supporting
Medium, Jour. Royal Aero. Soc.
88.
(7) Kuenzi, Edward W.
1955. Mechanical Properties of Aluminum Honeycomb Cores,
Forest Products Lab oratory Report No. 1849.
(8)
, and Ericksen, W. S.
1951. Shear Stability of F lat Panels of Sandwich Construction,
Forest Products Laboratory Report No. 1560.
(9) Love, A. E. H.
1927. The Mathematic al Theory of Elasticity, Cambridge University
Press, Londo n, England, Fourth edition.
(10) Nash, W. A.
1954. Bibliography on Shells and Shell-like Structures, TMB
Report 863, U. S. Navy Dept. , David Taylor Model Basin,
Washington.
(11) Norris, Charles B.
1956. Compr essive Buckling Design Curves for Sandwich Panels
with Isotropic Facings and Orthotropic Cores, Forest
Pro ducts Laboratory Report No. 1854.
(12) Raville, M. E.
1955. Buckling of Sandwich Cylinders of Finite Length Under
Uniform External Lateral Pressure, Forest Products
Laboratory Report No. 1844-B.
(13)
1955.
Deflection and Stresses in a Uniformly Loaded Simply
Supported, Rectangular Sandwich Plate, Forest Products
Laboratory Report No. 1847.
(14) Timoshenko, S.
1936. Theory of Elastic Stability, First edition, N. Y.
(15)
, and Goodier, J.
1951. Theory of Elasticity, Second edition, N. Y.
(1 6) Williams, D. , Leggett, D. M. A. , and Hopkins, H. G.
1941. Flat Sandwich Panels Under Compressive End Loads,
Report No. A.D. 3174, Royal Aircraft Establishment.
(17) Zetlin, Lev
1955. Elastic Instability of Flat Plates Subjected to Partial Edge
Loads, Proc. Amer. Soc. of Civil Engineers, Vol. 81,
Paper No. 795.
89.
IX. APPENDICES
1 . Appendix A -- Algebraic Details
Equations (97) and (114) for the elastic energy due to the membrane strains in the upper and lower facing, respectively, may be written in the forms shown in equations (98) and (115) by identical algebraic
simplifications. Thus, for the upper facing, let
Rrnn = Grnn t mn (228)
t
mn 2 mn
(229)
Q
mn
=E
so that equation (97) may be written as
4 4
2
4 mn
7T
+ 2p,
44
n Tr
2
Q
4 mn
b
2 2 2 2
2 2 2 2
m Tr n Tr
1-p, m 1T n Tr
Q +
(R
2
2 R
mn mn 2
mn + Qmn)2
2
2
a b
a b
(230)
22
2 2
in Tr
Tr
a2 Rrnn + 2-Qmn
abtE 7c°
2
8(1-p, - ) rn7 n=1
[(
2 2 2
1-p, m Tr n Tr 22
R
- Q
)
2
2
mn
2
mn
a
b
(231)
90.
and this latter equation may be seen to be identical with equation (98).
A parallel simplification may be effected for the elastic energy due to
the membrane strains in the lower facing if it is noted that for the lower
facing
3 t c 22
c
R' = A (— +
) + Bmn
mn mn 6 4
2
t'c
—) +
2
t'
—
C mn
2
+
F mn c + G mn
(232)
and
3
,c t'c
Q' = A k--+
mn mn 6 4
2
2 t'c
'
c + E mn
+ B (c— + —) + Cmn L
2+D
mn
mn 2 2
(233)
2. Appendix B
Integration Formulas
Integration formulas used in this thesis include the following:
( i and j are integers)
b
TrY dy =
cos i---
cos
,
for i = j
(234)
j
(235)
for i = j
(236)
0
b
j Try
cos my
— cos
dy = 0
b
b
for i
b
sin
i y
sin
j Try
dy
91.
0
iTry
jTry
sin —
sin
b
b
y sin
iTry
sin
jTry
= 0
b
i
j
(237)
for i = j
(238)
for
,
2
dy = 7 ,
0
y
b
sin ilTY sin iTrY dy = 0
, for
i j and
i t j even
(239)
j and i t j odd,
(240)
2
iTry
jTry
4b
ij
y sin b
sin b dy = Tr22
22
( i
-J )
for i
3. Appendix C -- Examples of Integrations
Included in this appendix are details of integration of equation
(72) and of equation (163). These integrations are typical of many others
in this thesis.
Equation (72) may be written as
92.
a b c
V
c1
00
c
2
00
(A mn z + B mn ) sin M
m
=1 n=1
0 0 0
00
00
(A . z+B pi
p=1 i=1
sin
7p
rx
si.n
a
Try
lDC
a
sin
nay
b
dx dy dz.
(241)
Equation (241) requires the integration of the product of a double infinite
series by itself. The integration formulas (236) and (237) show that Vci
. However, when
in equation (241) is zero when m p and/or n
m = p and n = i , integration with respect to x and y in equation (241)
gives
E
c
V
=
c1
2
c
ab
4
5
0
.
m=1 n=1
(Amn z + Bmn)2 dz.
(242)
Integration of equation (242) gives
co
ab >
V = E
c -24 c 1
m=1 n=
(Amn c + B mn )
3
3
- B mn
Amn
which is identical with the integrated form given in equation (75).
(243)
93.
Equation (163) may be written as (see equation (159))
a b
2
ax
(T'
w ) dx dy =
0 0
a b
00
Y
00
MTr
C mn a
cos
mTrx
n Try
sin
a
00
p
C 21 2r- cos 7TX sin iTry
— dx dy .
p=1 i=1
P I a
a
(244)
b
Integration of equation (244) gives values other than zero only when
m =—
p. Thus, the integration of equation (244) with respect to x gives
—
5 (a-2- 1 )
ab
a
2
2 dx dy = —
ax
0
0
22
m 7r
C mn Y
2
m=1 n=1
a
sin
na
co
X
z
i =1
iC . sinYi - dy
b
mi
(245)
Now, when i = n, equation (245 integrates to a value I where (see inP
tegration formula (238))
I1 =
C
22
2 m 7r
mn
2
a
(246)
When i n and i t n is even, equation (245) integrates to a value 1
2
where (see integration formula (239))
1 = 0.
2
(247)
GPO B27391-6
94.
•••••••n
co
5E
d
• 0r-I
N
...._.
• -
• r-I
0
0
,
0
k
0
.m
,-1-1
0
O
•r-I
1--in I
"c:1
0
,-1
ct
-rd
;-1
b.0
a)
-1..N/ I
4a'
..
a)
a)
LH
NI n cnI
I
cd
E
0
C)
O
g
E
0
a)
-0
1
0
to
a)
81"1,--1
8 kr0ill
;54
•
1—n
I
C)
O
r Cii
N
r 4-
▪
• r-I
• r-I
>
at
O
4-4
I
g
r..1
0
---
u)
a)
at
+4
U
2:43
0
to
• -1
.--.
.--.
en
..-1
.......
Ln
N
rg
N
O
-)
d
0
.,
4-)
cd
0
0
0-'
a)
8-E
"Ci -d
0
00
/4,
to
.,-.1
r.),.0
-H
1 NI ,
't
•r-I I
cd IN
-0
O
cd
Q 1
N
O
4-4
(TS
cd
.......
--I -45.)
---..
.•—•••.
•14
N 00
7t41
N
O
o
.,-1
ird
4-1
O
cl
F-1
b.o
.........
(..)
N
8
kN
N
I c°
II
-0
4
N
c't I e>ti
—
I
a)
11
a)
...--•
ro ------Tho
--1
cd
cn
cd \ ...--------. \CD
a)
-0
N
H
-._.
0
0
• .-1
+.4
cd
0
co
a)
E
NI
a!
,--1
II
81/ \ 711
g
.0
0
at
..
---...
I-71,
N
N
0
1—.__.1
N
'-ci
44
d''
at
E
u)
.,-,
46
C>
NI
..0
-714
1—•el
•1-.1 I
N
N
IN n
r--4
I
g
I
c.)
O
a)
a)
0I
n
Cid
N
O
.44
......
O
4 IN
0
a.)
N
II
N
• 0
,--1
4c-d)
0
cr
8 1/\1 II
•,-1
't
N
,--1
‘.1-4
1-.1
,-i
--.
kE 0
8
a)
.
s-1
,.
•
NI
1
cr)
......
-0
NI
• ri
—I
g
cll
r.......--......-1
0
N
95.
4. Appendix D -- Determinant Elements
for Case a = 0, E c Finite
The elements of the determinant in equation (179) are as follows:
2
2 E
Ec
O mn c
C
c
a
H = -( +
11
2 12
m2 2 ) + (G
12
t ? Sir
xz
TT
n
2
E
2
c
2 2
2
b Tr G xz
2
a
2
c
O rrin c
H 1,4
+
c)(-
Ec
c
2
Gxz Tr
2
2
2
2
111 IT
2
0
c + ctlc
‘f mn
H1,2 = Omn (
2
A- 12c
H 1,3 = °Ilan
3
3
3
12
c
E 2)
G xz
E
cc
2)
Gxz
Tr
3
2
2
2
Ec
E
c G yz a n + n c 11 _ _o_y_L) t_ O mn c
c
2 G
12
2 2
2
2
67.xz m b
b
xzGxzir
E C
n2
2 2 G
b tr
xz
Ec c
H
2 2
1,5 G
xz W
-II 2
2 C pmn
96.
O rrin c
3
Ec
c
( 2
H1,6 =1
H 2 , 1 = 0 mn (
G
xz Tr
+ ct'
2
2)
o
mn c
2
E
_
- 12
2 2
E cc 0 mn(t ' ) 2
H, 2 12
e Er-
c0 2(t' )
mn
H2 ,3 -12
3
cc
G xz
2
(c + ct, )
[e
mn 2
i2
0 m2 c
(t + t') a 2 T
N
2
o
m
c2 + ct'
2
t'
+ 0 mn2 + c )(
2
) (t + t' ) a 2 8
_ yz
c 2 + ct') n 2
z c (1 H 2, 4 = O rrin (
2
xz
H
2
2, 5 = 0
2
c + ct'
H2, 6 = e mn
2
2
0 c3
c „ mn
H3, 1 = 27( - 12
Ec
c
G xz
2)
97.
E
3, 2
H3,
H
H
H
2
t'611. O mn
(C + ct'
+ Onan (2
CO mn t3
3, 3 = -
3, 4
2
c2
t'
c )(
3t'
-
c
m
+ N o ( t + tt
2
V a 2 60
mn
G
n 2
yz c 2
c
(1
---)(
- -2")
2
G
xz
3, 5 = 0
c2
3,6 = - 2H
+)
2
2
2
G yz a n
H
4, 1
2G
t Ixz m 2 b 2
E
ct
2
G
n
b2 c (1 -G z
2
E
n
c 111 2
c p rnn
2 2 G
b ir
xz 2
xz
nc
m12
3
E
cc
2)
G XZ 11.
98.
H
2
=n
G yz
(1 G
xz
c
2
4, 2
2
fc + ctl
mn k
2
1
2
t'
yz
H A , = — c ( 1- G ) mn ( 2 +
2
c)
xz
b
2G
yz
p mn +
H 4, 4 = —
2 v 6
b
n
2 ,
H4, 5
i
=
a
2 2 cp mn
2
n
c (1 H 4,6 = —
2
E
c
c
H5, 1 = t 2 G
XZ
H
5, 2
=0
xz
2
2
n2
b
c (1 -
yz
xz
2
m 2 n 1 11 2 2
c p
mn
2
z
a b2
99.
H 5, 3 = 0
H
5, 4
= tic°
2
2
m
H = b
2 "7
5, 5
n a
+ tl )
H 5,6 = 0
H6,1
_t O
c3
rnn ( - 12 )
O
H
6, 2 =
mn
c 2 + ct'
)
2
2
H6,3 =
H
mn
2
') - t ((t
)
= tic e
6,4
6
mn
2
+ t' c O
mn
1 00.
b
146 , 5
2
m
2
=2
n a
H 6,6 = (t + ti)
5. Appendix E -- Determinant Elements
for Case a = 0 , E c = co
The elements of the determinant in equation (180) may be found by
taking the limit of the determinant in Appendix D as E co . This is
1
done by multiplying row 1, row 2, and column 1 by the factor E
— and
then taking the limit as E —;co. Thus, the elements of the determinant
c
in equation (180) are as follows:
J 1,1 - G
J
1,2
=
1
xz m
2
a
2 2
71-
c
ctl
2
G
xz
t t Oc 2
2 2
G ir
xz
yz
b
2
a
2
2 m 2
1-h
2
O rrin (c +t')
2
a 2n
J 1,3 - - G
2
2
xz m b
G
+n
t
t 5c
G
xz
B m n 2
m2
c ( 2
P mn
a
n 2 1-4 \
2 2 1
b
101.
1, 4
4
J 1,
=
G
yz
2
a
G
xz m
n
2
b
2
2
2
J
=
0,
J 3,1
3, 2
J 3, 3
G
= H
= H
=
4, 1
J4,
J
a
G Yz
=
J
P mn )
G
G
3, 2
2
=
1,
n2
2
J
xz
2,4
= J
emn 7
J 3, 4
a
2
xz m
2
n2
b2
- t t
= H
n
b
2
c
G
xz
1
0
2)
V
Pmn c —2
+ —2 G
(
=
2
2
J 3, 5
2
2,6
,m n 1
b
a
3, 4'
2
= J
2,5
ti
tt 5c
G
b
J
2,3
(that is, no change from element
3, 3'
yz
2, 2
xz m
2
1-4
b. 2 2
n
V 5c m
G
xz a 2
6
2, 1
2
c
(1
yz
2 G.
- G2
b
xz
xz
t'Oc n 2 1+11
G
2
2
xz b
1, 5
J1,
2
t, 6 n
G
= H
yz
xz
H
3, 2
in Appendix D)
3, 5'
J
3,6
=H
3,6
I
-
2
Pmn)
G PO 827391-5
102.
J
4,2
J
5, 1
=H
4,2 ,
J
J
4,5 = H 4,5 ,
=
t t Sc n 2
.L2
G
xz b
5,5
4,6
J
=H
4,4
=H
4,4 ,
4,6
2
J
=H
4,3 ,
1+11
H5,2'
J
=H
J
J 5,2
4,3
5,5
,
5,3 = H 5, 3 ,
J
5,6
=H
J 5,4 = H
5,4 ,
5,6
2
J6, 1 =
J
6,2
=H
2
n 1-p,\
t'Oc ,m
a2 b Z 2
6,2 ,
J 6,5 = H6,5'
J 6,3 = H
6,3 ,
J6,6 H6,6
J
6,4
=H
6,4
,
103.
6. Appendix F -- Determinant Elements
for Case ..._
a = 2, E c = co
The elements of the determinant in equation (194) may be found
by a scheme similar to that used in Appendix E, but it is more convenient to deduce these elements from a study of the determinant in Appendix E together with equations (173) through (178). In any event, the elements of the determinant in equation (194) are listed in order as follows:
All unlisted elements, K
p, q
, have a value of zero.
2
2
2
1 a
t t 8c
1 a
1-11
+
K 1, 1 G
(1 + 7 -2- 2 )
2 2
2 2
b2
xz m Tr G TT
xz
c , ce8
K
— () mi. (c + t')
1,4 - 2— [ G
xz
G
K
1, 7
yz a
2
1
G
2 2
xz m b
m 2
i
1 1-p,
+7
)
2
2
a
b
t' 6c
1
0
L + p
G
ml 2 ml
xz
2
2
G
y z a
1
1 c f
yz
K
G
1, 10
2 2 - ti6 2 G
‘1 - G
xz m b
b
xz
xz
G
K, 13 _
1, 1D
t t 6c 1 14-p.
G
-2- 2
xz b
1-p,
2 Pml)
104.
2
t i 6c I m 4_ 1 1-4)
' 2 2 2
K 1,16 = G
b
xz a
2
2
2
1-p.,
a
4
t'Oc
a
(1 + —
)
K =
2 2
2
2
2
2 2
2,2 G
m
b
G ir
xz m ir
1 xz
K 2,5
K
2,8
=
=
c
cti 6
2
Gxz
0
(c + tt)
m2
Gyz a 2 4
4 L.:\
. 1 + p
_ t i bc [e
(---E +
m2 c 1112
Gxz m2 2
2 2
1
a
b2
xz na b
G
2
4 ,1 o 4 c
K
2 G xz
2 2
2,11 G
b
xz m b
G
2,14
K
2
yz 1 p.
Pm 2 )
2
G xz
t t 6c 4 1+p,
2 2
G
xz b
= - — —
2,17
K
yz a
3,3 G
2
t' 6c m 4 1-p.
+
b
—
(
2
2
G
a
xz
1 a
2
2 2
xz m Tr
2
9
(1 + b
2 2
2
G -tr
ttEic
xz
a
2
2
1-11
.
2/
105.
K
ctiö
=
- G
3, 6 2
0
xz
a
2
m3
(c + ti]
9 t'Oc [
= - Gyz
2 2 G
E3m3
K 3, 9
xz
xz m b
2
e 5c 9
G
2
xz b
2(
Gyz 1-4
c
1 -2 G
G
2 P m3 )
b
xz
xz
1+11
2
2
t'Oc m
( 2
G
xz a
K3, 18
9
b
1-p.
2
2
= 1,
K4,4 -1,K 5, 5
G
K
K
7, 1
= -
G
cEor
7, 4
12
yz
a
2
xz m
2
2
2 +_92 1 1.1.‘
2
+ Pm3 ( a2
9
9
3, 12yz a 2 2
xz m b2
K 3, 15 7--
m
e
1
b
K
6, 6
=1
t'Oc
2
G
xz
2
3
2
Ornl (e) + e Eor
ml 2
e mi (
+ p
,m
ml
c2 +
2
2
a2
)
e mi
1
b2
1-11
2 ]
t'
m
+
a
2
pml
106.
2
m
t'
16,
—
9 o t + t 2
a
X7, 5
K
(t') 3
22 2 {t3
+ 12
+ Sw
p
= —2- G
m13
yz ml
7, 7
2
2
m
m
2 P m1 c
a
a
2 1
2
b
,
+ t'
t'
rn1 2
2 1-p. 2
2 P 1
2 c
N a 2
16 o m
=K7
z
c
8
a
G
2
t' m2
2 [1
w
(
- c (1 -
= -2
Gyz pm1 + t' Sir
K
p mt c)
+
G ' •e ml
b 2
7, 10
a
xz
b
-
K
7, 13
=
m2 1 1 c 2 2
p
2 2 2
m1
a b
2
2
t
Ow
2
1
1
+ t' 6 .7 2
0
2 m1
2
1 1-p, c
m
Pml
2 2 2
b
a
2
+m
1 (0
2 ml 2 2 Pml
a
b
c)
107.
2
m
— 2 -0m l + t'On2
K7, 16 = 2
a2
m2
t' m
(e ml
a
2
Pml c)
a
1
a
K
K
K
K
K
8, 2
8, 4
=-
b
1
2 2 c Pml
2
yz a
4
7
G2
1
2
xz
= -
16
9
cEor
8, 5 -
2
e
8, 7
16
9
m2
pm2 c
+
4
1-4,
a
2
c 2 + ce
3
2
t'
m2
(t')
t
+
'Ow
0
(
)
(0
+
p
rn2
2
m2
m2
m2
a
2
t'
48
m
=-2 N a
5 o t
8, 6
t t
2
a
=
0
m2
N at t
2
o
m
t + t l2
a
2
12
t'Oc
Gxz
N a
o m2
—
c
2
a
108.
K
8, 8
=
)3
t: +
(t—
2
471- 2 n
+ 57r
112
2 yz P mt 61r e2m2
b
2
2
m 4 111, 2 2
m2
c) + 2 2 2 c Pm2]
mt
a b
a 2 Pm2
2
48 N o a m
2
K8, 9 = - 25
a
2
+ 61.2
411.
= —
K
2 G yz P mt
8,11
2
t'
4 c 1 - &"-r-z) (O m2
xz
+ a72 P mt c)
m 2 4 1- 11 2 2
2 b1
2 2 c P mt
a
2
2
— t' m
t 2 4 t tA, 2 {.4
K 8,14 = 2 57r 2 - m2
b 2 (0 m2 2 +a2 P m2 c)
bn
4_
m 2 4 1-4
- 2 2 2 c Pm2
a b
109.
K
= -
8,17
t
2
2
+m
2
a
K
K
K
9,3
9, 5
9, 6
=
=
-
_
2
2
, 2 m 2
2 m
t '
57r 2+
t* 5v
P m2 c
2 m2
2 ‘ Q m2
a
a
a
4 - p.
1 2 c Pm2
2
G
9
yz a
2 2
G
xz rn b
t' 5c
Gxz
t'
—+
c(
m3 2
Pm3
2
a
9
b
2
111)
2
t'
48
m2
- — N a,
25
o t
2
t'
a
c5-rr 22, , 3
+
12 0 m3
2
5v 0
c2 + ct,
2
m3
(
m2
t'
( 0 m3 2 + 2 p m 3 c )
a
48 N o a m 2
2
K 9,8 = - 25
c
a
GPO 827391-4.
110.
K
2
t'
[t 3(t I ) 32
+ u lT [ (3 m3 2
3 + 12
22
G p + 67r B.
b2
9, 9
Y z m3
-
9ir
2
m 2
a 2 Pm3 c)
9Tr2
K
9,12
=
m 2 9 1--p. c 2 2
Pm31
a2 b 2 2
,
2
2
9
(0
—2- c (1 -
+ t
G p
Y z m3
b 2b
xz
c)
p
-+
2 m3
m3 2
a
2
9 1 11 2 2
m
2 b 2 2 c Pm3
a
2
K
9,15
2
2 9
t
+ ti OTT
- — oir —
2
m3
2
b
m
2
9
2
tl
m
9
—+ — 0
—
2 ( m3 2
2
m3 c)
a
b
2 c pm3
a b
2
m2
2
2
t' m
2
2m
p rn3 c)
+
2
T
m3
(0
[—
T
+
t'Oir
= - — Errr
K
2
m3
2
9,18 .
a
a
a
m
2
9 111
+ 2 2 2 c Pm3
a b
=
G
K
10,1 G
yz a
2
1 2 2
xz m b
K
= t'Sir
10,4
&
. f
L
1
c2
( 1
2G
b
xz
2
b
xz
2
t' m
1 -2- + —2- p rn1 : 2
1
a
K10, 7 = b2 G y z pmt + t'6,1T
2
it
2
=+ t Ecir
K10,10
2 Gyz P mt
b
yz 1 }j.
2 P m 1)
xz
[L2 c (1 H
2
2
yz.
1 114
) - m
2 2 2
a
b
xz
G
2
G
1
7 c (1 b
xz
2
2
+m
1 1-p, _2 p2
2 2 2
ml
a b
m2
1
Gyz
K
= t'67r2
c (1 10,13
b2[ 12
Gxz l a 2 b 2 2 c Pml
c (1
112.
2 m2 1
--T 2
a b
K10,16 =
2
4
_ G yz a
2 2
K11, 2 - G
xz m b
11,5
2
= t
K11,8
4rr
b
0
m2
G
c (1
t
(
c
2
yz
G - 2 Pmt)
xz
2
yz 14 c I
‘ 1 G - 2 Prnz)
6 b 2 G
xz
xz
+
2
)
2
2
G
yz \
G xz
yz p mt
Tr 2
a
+ 5Tr
11,11
4ir
2
G
yz p mt
t'
2
m2
+ 2 P m2
61r
2
m 2 4 1-4 2 2
2 c pmz
2
a b
c
a
4 1- 4 c Z 2
P mt}
b 2 2
2
K
Gyz)
G
xz
[42 c (1
G
1 G yz
xz
12
4
2 c(1
113.
+ m 2 4 1-1.1. c
Pm2
a Z b 2 2
11,14 = 51r
m 2
2
K11,17 = t1r2
6
yz
1-p,
2
-
C (1 -
a b
xz
2
yz a 2 9 _ t io
K12, 3 G xz m b 2
b
p
m2
)
2
Gc (1 G yz 1-4
2 Pm3)
xz
xz
4.n
K12,6 = t'67r2 B m3 (
2
K 2
1-,
= -
b
-
yz)
Gxz
G
c2 + ctt
2
p
yz m3
m2 9
2 b2
a
9
2 c (1 xz
t
3 +
m2
a
111
2 c
2
2
Pm3
9
pm3 c] 1- c(1
b2
114.
2
971. 2
K12,12
b2
-
m
a
2
2
G
G yz P m3
9
b
1-1.1.
ti
1
2 c(
b
2 2
2 c Pm3}
2
G
t'6
K12,15 =
1 -
2
K12,18 =
tI51T. 2-m
T
9
t' 5c 1
z
G
xz b
2
G
xz
—2- c (1 - GYz
a b
xz
l+p,
(c2 + ct'
K13, 4 = tt E1T2 e
-6-Y )
xz
m1
2
1
b
2
2
9
+ 1222
2
a
b
1 -p,
2- P m 3 )
1-p,
c
p
m3i
115.
2 2
t 57r 1
u ml 2
K13, 7 - 2
b
m2
K
13,10
tt m 2
12
2 Pmt
a
ft
1 1-p.
b 2 2 c Pm1
Gyz 1
G
2
xz b
= t Sr
2 1
1 m 2 111
K
= Sir — (— +
) (t + )
2 2 2
13,13
2
b
b
a
2
2 m 1 1+p,
2 (t ++ t')
K13,16 = °Tr
a b
14, 2
=
G
be 4 l+µ
2 2
xz b
K
= t ! b.rr2 0
(
m2
14, 5
c 2 + ct i ,
2
4
2
b
2
m 1 1-µ
cp
2 2
2
a
b
1
t
116.
2 2
t 5Tr
4
0m2
2
Km, 8 -
m 2c) 4
t 2[
t
(m
°2 2- + a
Pm2 b2
2
m 4 111
2 2 2 c Pm2
a
b
,
K14,11
=
c (1 -
O2
t'ir
[b2
G yz \
G
xz
4
m2
7+
b
2
m 1 4
4
2 + 2 211) (t tt)
K14,14 - 57r2 2
a
b
Ki4,17 =
b
2 m 2 4 l+p,
OIT
(t + t')
b
i 5c 9 1+1.4,
2 2
K15, 3 = G
xz b
c 2 + ct I
2
2
K15, 6 = 57r e m3 (
9
b2
a
2
4
b
1 1.1.
2
Pm2-
t
117.
2
2 2
9
t m
9
2
2 0m3 7 + t'Sir (0 m3 -2- + 2 p m3 c) 7
K15, 9 —
a
t
2
9 1 -1.1
2 2
?
a b
m
15,12
= ti8Tr
Pm3
9 m2 9
G
2
(1 xz b
Ki5,15 = OTr z
m2 1-1
=
+—
2
a
1 (1
2
2
b
K15,18
c
b
671. 2 m
a
2
9
2
b
2
)
2
+
2 2 2
a b
cp
m3
(t + tt)
l+µ (t + tt)
2
2
2
1-1.1,‘
t5c i m 1
K16,1 = G
‘ 2 + 2 2
xz a
b
K
= t r OTr 2 0(
m 16, 41
c
2
+ ct I
2
m2
2
a
GPO 827391-3
118.
2
2
2
m
ti m
2
m
pm 1 c)
2- +
+ t Ern- 1(0
2
ml
a
a
Lm
2
2
t 5ir
- 2
K16, 7
m 2 1 1-p,
2 2 2 c Pmt
a
b
2
2 n2 1-4
1
m
1ZN
m
= tI6Tr2
2 - a 2 2 2 c Pml
[.b 2 c ‘ 1 - G
K16,10
b
xz a
K16,13 =
K
2
1
2 m
2
2
a
b
+µ
(t + tl)
2
2 m 2 t rn 2
J.
)
16,16 =
K17,2 =
K
17, 5
=
a2
b2
2
2
4 1-p.
t'Sc m
Gxz ( a2 + —
b2 —
2 )
2
bir 0
m2
(
c
2 +
2
)
m2
2
a
(t + t')
319.
2
m2 2
2+ t 6Tr2 / [0
2 511.
a
2
-
17 , 8
K
e
2
_
b
2
17,11
b
c
-
G
2
yzx
)
xz
2
m
2
m 4
-—
1-p.
2
2
2
a
a
2 4 1+1.1.
2 2 (t + t)
2
a b
m
=
17,14
4
= t67r
5.1r
c Prn2
2
2
a2
m2
2
K
1or m2 ( 2 + 2 2
)
17,17 = E2
b
a a
K18, 3
rn2
= _
2
t töc m
2
G
xz a
(
( c
2
= t6-rr 8
K
18, 6
m2
2 Pm2
a
1
m 4
K
t
2
t
m
4
9 1-p.)
22
b
2
+ ct m 2
2
)2
a
(t
+ t')
—
2
b
c0
' m2
m2
C]
a
120.
2
2
, m
-t ‘ m
+ t'Orr2 (0
P
m3 + 2 m3 c) a
m3 2
a
a
2
Z 2
t E rr
K
= - 2
18, 9
m 2 9 ill
+
a
2
b
2 c pm3]
2
2 9 c
2
t18,12 = EcTr
K
2
2
m 9 11.1.
_yz\ m
2 2 c Pm3
2
2
G
xz a a b
2
2 m 9 1+11
2 2 (t + tt)
-7—
=
67r
K18,15
a b
K
18,18
= 67r
2m
2
'7
a
2
m 9 1-p.
( 2 + b2 2 )
a
+ t.)
1. -129
Figure 1. --Isometric drawing of a sandwich panel.
Figure 2. --Differential element of the core.
Figure 3. --Top view of sandwich panel showing different combinations of
edgewise bending and compression as defined by a. N o is the load in
pounds per inch of sandwich.
Figure 4. --Pictorialized definition of a, N_, N,, N_, and
--11. —L1