DEFLECTION AND STRESSES IN A
UNIFORMLY LOADED, SIMPLY SUPPORTED,
41.1
RECTANGULAR SANDWICIl PLATE
Original report dated December 1955
Information Reviewed and Reaffirmed
September 1962
LOAN COPY
No. 1847
•
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
1111111,
FOREST PRODUCTS LABORATORY
MADISON •5, WISCONSIN UNITED STATES DEPARTMENT OF AGRICULTURE
FOREST SERVICE
In C ooperaton
with
the University of Wisconsin
DEFLECTION AND STRESSES IN A UNIFORMLY LOADED,
SIMPLY SUPPORTED, RECTANGULAR SANDWICH PLATE!
By
MILTON E. RAVILLE, Engineer
Forest Products Laboratory, 2
. Forest Service
U. S. Department of Agriculture
Summary
A theoretical solution is presented for the deflection and stresses in
a uniformly loaded, simply supported, rectangular sandwich plate.
The solution is applicable to sandwich plates having an orthotropic
core of arbitrary thickness and isotropic facings. The facings may
be of equal or unequal thickness. Numerical results and curves are
included.
Introduction
The purpose of this report is to obtain formulas from which the deflection and stresses in a uniformly loaded, simply supported, rectangular sandwich plate may be computed. The sandwich plate is assumed to consist of isotropic facings separated by and bonded to an
!This report is one of a series (ANC-23, Item A-7) prepared and distributed by the Forest Products Laboratory under U. S. Navy
Bureau of Aeronautics Order No. NAer 01684 and U. S. Air Force
No. AF-18(600)-102, Amendment A6(55-286). Results here reported are preliminary and may be revised as additional data become available. Original report dated December 1955.
—Maintained at Madison, Wis. , in cooperation with the University of
Wisconsin.
Report No. 1847
orthotropic core. The core is considered to have such a small loadcarrying capacity in the plane of the plate as compared to that of the
facings that the normal stresses in the core in the plane of the plate
and the shear stresses in the core on planes perpendicular to the facings and in directions parallel to the facings may be neglected. The
analysis of the facings is based on the usual small deflection theory
of laterally loaded plates.
Notation
x, y, z
rectangular coordinates (fig. 1)
a
width of sandwich plate
b
length of sandwich plate
a
c
thickness of core
t 1thickness of upper facing
t2
thickness of lower facing
E
modulus of elasticity of facings
v
Poisson's ratio of facings
Ec
modulus of elasticity of core in z direction
Gxz
modulus of rigidity of core in xz plane
yz
modulus of rigidity of core in yz plane
q
intensity of uniform external lateral loading
cr znormal stress in core in z direction
.r xz
xz , yz
Report No. 1847
shear stresses in core
-2-
E
normal strain in core in z direction
shear strains in core
Yrs
uC , v C, w C
displacements of core in x, y, and z directions
Nx
, Ny
, N xy
normal forces and shear force per unit length of
upper facing
N'
N' N'
x y
normal forces and shear force per unit length of
lower facing
M M , M
x
y xy
bending moments and twisting moment per unit
length of upper facing
xy
Mx, MY,
y
transverse shear forces per unit length of upper
facing
Q ,,Qy
10'
y
bending moments and twisting moment per unit
length of lower facing
transverse shear forces per unit length of lower
facing
u, v, w
, v', w'
E x, E y , • y xy
displacements of upper facing in x,
tions, respectively
and z direc-
displacements of lower facing in x, E, and z directions, respectively
normal strains and shear strains in upper facing
normal strains and shear strains in lower facing
m, n
integers
Arnn' B mn Crnn 1
'mn' I mn' Kmn Lmn
Report No. 1847
-3-
constants
k
16qa 4 (1-v2)
ir 6 E/
ti t 2 )(
(1+2)
Sx
t1 + tz)
2
c+
w z Ect t
1 2
Gxz a 2 (1-v 2 ) (t + t )
1
2
z
Sy
Ect
1
t2
G yz a 2 (1-v 2 ) (t + t )
2
1
Theoretical Analysis
The dimensions of the sandwich plate and the coordinate system used
in the analysis are illustrated in figure 1. The method of analysis
consists of determining expressions for the core displacements that
satisfy the core equilibrium equations and the boundary conditions.
The arbitrary constants that appear in these expressions for the core
displacements are then evaluated from consideration of the equilibrium of the facings in conjunction with the requirement that the displacements of the core and facings be equal at their mutual interfaces.
Equilibrium of the Core
A differential element of the core is shown in figure 2. In accordance
with the assumptions outlined in the Introduction, a. , cr and r
in the core are assumed to be zero. From the Summation of forces in
the x, y, and z directions, respectively, the following three equilibrium equations of the core are obtained:
Report No. 1847
-4-
(1)
6T
yz = 0
(2)
Z
and
6cr
31'
z
xz ♦ yz
'...-•••••••••
6y
6x
6z
(3)
= 0
On the basis of Hooke's law, the following stress-strain equations are
applicable:
(4)
cr z = Ecez
T
xz
G
xzyxz
(5)
G y
(6)
and
T =
yz
yz yz
Also, the strains and displacements are related as follows:
6w
e
z
6u 6w
6w c
Y
xz =
6x
and
6vc 6 wc
=—+
Yz 6z
6y
„,.
Report No. 1847
-5-
Equations (4) through (9) enable the equilibrium equations of the core,
equations (1), (2), and (3), to be expressed as follows:
w a2 u
• = 0
(10)
cv
• = 0
a yaz
(11)
a xaz az2
62 w
and
E
c
a z w c G
az'
az
_ 62 u c) G
a w 3 2 v c) . 0
33caz
Yz 6y2
a y az
ax2
xz
(12)
The expressions for the core displacements are assumed to be of the
following form:
uc =
1
(z) cos mirx sin 213
- rY
a
cos
2 (z) sin rn"
a
(13)
(14)
and
3 (z) sin
mint
a
sin
nor
(15)
It is noted that the above expressions satisfy th
= e0 b
b, oundary conditions
that we = 0 at all boundaries and that (M
(u. c )m = 0, and (v )
c x=0 = 0. The three functions of z in equations (13),
x=a
(14), and (15) are determined, as follows, from the requirement that
Report No. 1847
-6-
these equations satisfy equilibrium equations (10), (11), and (12). If
equations (13), (14), and (15) are substituted into equations (10), (11),
and (12) and it is specified that the resulting equations be valid for all
values of x and y, the following equations are obtained:
f(z)
1
+
3 (z) = 0
a
(16)
nir
2 (z) + —
b f 3 (z) = 0
11
(17)
and
Ec f 33
2 2
Gxz
-
Yz
[11
2 2
f
b
MIT
f3 (z) +
a f 1 iz%
11
'
f2 (z )
3 (z)
=
0
(18)
where the primes denote derivatives with respect to z. From equations
(16) and (17)
f1 (z) = - 1-111r f3(z) + Amn
a 3
(19)
f (z) + B mn
2 (z) = - mr
b3
(20)
and
where Amn and B mn are constants of integration. The substitution of
the above values of f (z) and f (z) into equation (18) yields the follow1
2
ing differential equation:
f 3 (z)
Gxz
G yz nn
E cAmn
mn + E c — mn
Bmn
Integration of the above equation yields:
Report No. 1847 -7-
f3 (z) =
ff
2-
G
Gx
z
Amn
÷c
( -Ec
-E B mn
z2 + C mn + F mn c (21)
The functions f 1 (z) and f (z) can now be determined by substituting the
2
above value of f 3 (z) into equations (19) and (20) and performing the indicated integrations. The results are:
f1 (z)=Amn
-
z-
Tr 2 mn
—z -B
—6 a 2 E cmn 6 ab
w 2 m2 G xz 3
m C
2 a mn
- mc
a
z3
E
c
n z Hmn c
and
f2(z) = - A mn
1. 2
[
mn Gxz z 3 + B
mn z
ab E c
n
- 7 I-3- Cum z 2 -
n 2 ri Z
Gyz
6 b2 Ec
nc
-F F mn z + L mn c
The functions of z that appear in equations (13), (14), and (15) having
been determined, it is possible by redefining the arbitrary constants
to express the core displacements as follows:
u =
>2"
m=1
2
marrc %A
3 + B mn 4 z% tnn 3
• 2 +Cmn--)
•
n=1
c
+ F mn -c + Hrnn] cos
Report
No. 1847
c
In" sin mil'
a
-8-
(24)
V =
C
nitct •
b
2
z 3z
„6„,
mn 2
3 rnn 3
_4 A-
C
inn c
Di
cos -a
n • + Lmn sin niing
+ Km
a
w
[4 Amn•7 + Bum
=
c
(25)
(26)
—T1
+ C mn sin nl " sin 111
a
The above expressions for the core displacements satisfy the equilibrium equations of the core if
G
A
Fmn
+ Gyznp Km -8a
c mn
n
xz
where p = t.
Thus it is seen that there are actually only six arbitrary constants
present in the expressions for the core displacements.
Since, from equations (4) through (9),
z
T
=E
xz
6v1 c
C 6z
G
Xz
Report No. 1847
alt
f
C+
6VV
ci
6 z6x
-9-
(2 7)
and
6Ar
C 4. --2)
31.
T
yz= G
yz (
the core stresses may be expressed as follows:
cr = Ec
c
z
G
T
XZ
=
(8 AArun
xz
+
mn cos
Brnn)
=TX
a
sin
sin
•
M 'RX
Ea
a
sin
(28)
(29)
and
03
T
Yz
Kmn
>
sin
M1TX
a
c m= f n=1
ril
cos –a
(30)
In the analysis which follows it is shown that the displacements and
stresses in the facings may be expressed in terms of these same arbitrary constants and that these constants can be evaluated from consideration of the eqttilibrium of the facings.
Equilibrium of the Facings
A differential element of the upper facing of the plate is shown in figure 3; the forces in the plane of the facing are shown in figure 3(a),
and the remainder of the forces and the moments are shown in figure
3(b). The summation of forces in the x, y, and z directions yields the
following three equations, respectively:
aNx M•x 6x
yy
Report No. 1847
=-T
(31)
xz
-10-
3N dNxy
ay
ax
_ T
(32)
yz
and
aQ
)3c
3Q Y = q
6y
(33)
cr) z = -
Also, the summation of moments about the x and y axes, respectively,
yields
2,/s/L
dM xy
and, since M
I1 %
Qy = - T yz 2 /
(34)
= M
xy,
Yx
61vIx 6Mx Q
x
t
(T)
= - T xz
(35)
If equations (34) and (35) are solved for Q and Qx and these values are
substituted into equation (33), the result is:
2
d Mx
a
2
Mxy
ax a y
-
t , dT xz
2 ax
2 m
6
Y=_
2
3y
q (o`z)z
2
(36)
+ -/
6y
The equilibrium equations of the upper facing are thus reduced to equations (31), (32), and (36).
The equilibrium equations that apply to the lower facing are obtained in
a similar manner on the basis of figure 4. The summation of forces in
the x, y, and z directions yields
Report No. 1847
-11-
6N'x
6,1t
61‘1'
6N'
_/
6N'
=
6y
(37)
X Z
=
(38)
y
and
aca;c
6x
U1'
+
6y
=
z
(39)
c
2
The summation of moments around the x and / axes yields
am.' am'
_
6y
_
6x
1 = T
y
and, since Me
yx -
t2
yz 2
(40)
y ,
WVI;c 6111,4
6x
(41)
QX1 = TXZ ("124
6y
The substitution of the values of Q ' and
obtainedd from equations
(40) and (41) into equation (39) results in:
62 MX
z
6x
M;cv 62
2
6x6y
t2
)Txz
= a.)
aT yz
(42)
?)r
Report No. 1847
c
z = 2-
6y2
-12-
The forces and moments per Unit length of the facings are related to
the displacements of their respective middle surfaces by the following
equations:
_ Et 2( au +v
Nx
v
Ny
N =
xY
v au%
ax
N 1 =
Y
Et 1 I au , av%
2(1+0 ay ax
Et 3
1 (
2
ax
Eti 3
-7
o
v o
Y )
2
2
+ v aui)
2 , ( w'r
2 (
+
2(14-0 ay
Et 3
2
W )
ay
2
6x
2 ( 6
w+v
2
12(1-10 )
w + v 6 w)
ay 2ax
12(1-v2)
,
1-v 2 ay•
Y
2
M - Myv -
Et
a .w +v
12(1-v2)
Et 1 3
aui
Et
Et
1 av
2 dy
1-v
Et2
M
I
-
2
Mr
=
xy
6y2
2
(1-M- + v
Et23
12(1+v)
)
2
12(1-v2) 2
a w)
12(1+v) axay
Et 3
2
ay ax2
2
w 1)
(43)
axay
When the foregoing expressions for the forces and moments in the facings are substituted into the equilibrium equations of the facings, the
equilibrium equations of the upper facing become
Et
l
1 -2
v
a
2
u
ox
Report No. 1847
1-v 6 2 u
2 ) 6
y
2
( 2)
v]
ax6y.
-13-
_T
xz
(44)
Et
1
[6 2 2
1-v
1
1-v
%
2
Z
6 v / 1+v%
u
i)x 6y
6Y
(45)
= - T
YZ
and
Et 1
3
4
V w) = q
12(1-v 2 )
(Tz ) z
t1 6T xz 6T
yz )
= - .G. + 2 ( 6x + a
2
(46)
and the equilibrium equations of the lower facing become
Et 2
62u
1-v 2
ax z
Et 2
aZvi
1.-10
yZ
,1
,
Zu"
2v,
1+v ) 1
(-
33cy
6y
.)2
1+v
Z
l/
6x2,
Z 13.
ax.y
=T
(47)
,Lz
= T yz
(48)
and
3
2 4
7 (V w ) =
12(1-v-)
Et
((rz )z
2
+
t 6T xz
-2- (Th;c
6T
yz
(4 9 )
Matching Displacements at the Boundaries
Between the Core and the Facings
The equilibrium equations of the facings, equations (44) through (49),
may be expressed in terms of the core displacements by equating the
interface displacements of the facings to the corresponding'interface
displacements of the core and then expressing the middle surface dieplacements of the facings in terms of the interface displacements. In
so doing, it is assumed that w and w' are constant through the facing
thicknesses and that__
u, u', v, and ; T-vary linearly through the thicknesses. Thus,
Report No. 1847
-14-
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Report No. 1847
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..--..
I-1 U
4.1 IN
.—o
N
N
n••••
..4
ty
..•
s
......
4
,go o
cd
1N
I
44 q11
0
co
tri
..+
44
O
01%1
j
......
♦
1
4
i
IIN
Z:
....
gay IN
S'
N
CI.
P.i
N
0.,
N
Ng+
1
0
•-• el
....•
1r
4.1
0
p.4
W
N
N
i I N
N
+
N
1__1
id
cr 41
N .0
cr, o
#.1:I
0
li
wg
.......
+
)
.
0
4.,N311nni
..4 I N
nt4...i
N
U
I
N...-...
>i
rnI
......
U
en
...i N
.......
....1
tn1 44
,_,
FLI
t4
r.---1
44...
+
0
..4
IN
I___.;:-1
4-u
...N Irti
1 1
N
.....
44
I
+
1
.
•
W0
m
II
.N 3
f—s
›.• M. + .
v
I=
0
.-4114
W
Wu II+
+
g
0
N
0
0
.-4
01
0
1
gt.1
+
•••n
1
u
.--i
.......
N
1%1
k 0
0
0
+
..—.,
N
a
0
+
cs1
0
l
cdel
+
....P
...4
4.1
E
ai
A
a
4
N
N
't
nn
f n1
m
I
g
+
...I
%moo
ill
a
P.4 1.0
U
g
+
I nti
+
a
1:11
+
^
.,_,N I g.I
+
—1 1 so
g
44
IN
a.
a
.t N
..I
1 AP 1
,------.
Ni
a.
.-74
• I
1
•.14
gr
N
gl
+
4
I
0
N+
O.
N
N
0
O
+
NEg IW
a
Report No. 1847
IN
%MO
N
ir
0
4.b
14
re'o
.7
.4
0 IN
W
44 VI
4-.
0
471
..
II
.....
......
a
N
.1-a
a
a.
No
+
Ns
0I
Ili 10
0
-18-
11
+
2
+ n 2 p 2 ) (A
mn + B mn + C ain))
6 Gxz a
+ 2 Brnn ) + 6 G a
3
12 E a 4 (1-v2)
c
(4
Amn
4
ir E t 3 c
2
3 (1-y 2 )
m F
E t2 2 c
mn
3 (1-v 2 )
yz
Tr3 E t
c
nP Kmn = °
(62)
A literal solution for the constants Amn , B mn , Cmn , F inn, Kmn , Hmn,
and L
mn obtained on the basis of equation 27)and equations (57) through
(62) is very lengthy and contains too many parameters to be of practical
value for design purposes. These equations can be simplified enough to
render a practical solution possible if certain additional assumptions
are made. The amount of error introduced by making further simplifying assumptions can be determined in any particular case by obtaining
a numerical solution based on the foregoing general system of equations.
To obtain , the aforementioned simplification. it is assumed that the flexural stiffnesses of the individual facings are negligible and that the modtaus of elasticity of the core in the z direction (E) is infinite. This additional assumption in regard to the core results in a core analysis that
is identical with that obtained on the basis of the so-called
method commonly used in sandwich analysis. The neglect of the flexural stiffnesses of the facings is known to be justifiable for most practical sandwich constructions. As a result of these assumptions, the
system of equations, equations (27) and (57) through (62), reduces to
the following:
G m
xz
8a
+
np K = —
E
mn yz
mn irc c A mn
Report No. 1847
-19-
(271)
Om%
ti
tn
IM
N
N
N
..—..
• N N
@
i4
IIN
t.,/
J
1
44
u
••n••
I
+
+
N
a.
N
1u
Z°
q1/4/
1
......
N
•••••nsi
.
.1.1
1,n..j
+
.....,
N
j
E
I
MA IN
+
0
4.1
.....,
.-4
-,—,
n-iiNI
m
'II
+
I
0 >.'
11
•Ila
N
N
#
0
4.
1
N
N
›,
II
„Th
^
eSji
cd
-4
g
E
,--.
rilF
N
IN
,--1
Q.
N
0
I
a
+
414
+
Tr
Ne
Cd
Report No. 1847 -20-
.....
a
.0
--....
;LI
.....
JJ
...0
A-3
E
pa
U
r4
V
li
03
d
.....0
+
0I
E
rf
+
..a
0
r'.1
a
Ei
Wqr4
II
i
X
•
0
N
N
0..4
0.
......
a.
II
N
N
N
N
).
U
0 I
I
A
."
,4
-
..+•n
......
+
-....
saNi
X
4.863
NI
N
W
N
IN
Ct.
41
(0
rd
1:1
N
0
••=r,
h
N
N N h
t4
r4)4
+
..
N
t IN
to
0
II
II
0
....
+
44
t.)0 I
N
t
SI
.) 0 I NI
L............–.-1
NN
cl.
N
13
.t I 1.4
W
I
0.
No
.74
c.)
o I co
0
Report No. 1847
0
'`=4
re
8
H
•
in
E
I
t 0 oi
g (D
04
I N
I:
+
a
.4
Tr
+
+
NEN
.....
rci
ii
1
ta
U. 1
til cl
a
U 4.,N
41
N
-21-
i
mo
cr
a, co
a 4)
..4
.. 44
•011) 4944
l$ RI
L7
‘t M
:3
..0
.0
nCI
f---------4_----s
U)
1
,
u
N
N
N
N
N
a.
a
NE
N
IN
N
L:L
TV
I
—t
N
N
41"
N
U
4.i
5
U
N
u
I
ra
••••I
+
-
V
N
N
E
44
Report No. 1847
-22-
E
0'
NIN
"14.1N
NC.
41 41
N
.4 .1
E
+
N.....
CI
CI
t IN
+
1
.. 0
S
1....
t..)
N
...,41
+
i
LI
,
.....
t IN
I•er
a.
I ti
...NI
I Tr
+4'4
4.ri
4 N 0
..-s
4
ILI
I 41
N
0
ril
N 5
N
0 NI0
E 44 +
,-I 1
+
5
.....
NT..
N
N
N0
41
.
64
4
.11
4+
I NI;
4 NU
I
+
it
..-4
.I .4
4.)
N
a.
rd
trs
4J
+
..
.4.0
.0
N
r.
fib..
rei
.....4.)
.1.n
EnK
1.4 ...
i
1-4
N
N
No.
0
+ N
N
O.
N
W
N
+
N N
40
I +
...4
4.n
...1
44
v
.0..
.1
1
....0.-
II
II
0
.0
-23-
N
5
E
co
+
NT%
a.
0NI
0
1.4
44
II
Report No. 1847
41
0
N
H
where
16 qa 4 (1-v2)
k
EI
t
t
-
1 t 2
H
t + t
c
t 2
l
1
2
z
w Ec t 1 t2
Sy =
— G
xza 2 (1-v 2 ) (t1 + t 2)
S
w2 E c t t
l 2
+t )
2
Ga
2
yz 2 (1-v )(t 1
Lateral Deflection
Since, under the assumptions used in obtaining equations (63) through
(69), Amn and B mn are equal to zero, the expression for the deflection
given by equaticTria6) becomes
co
wc = w = w' =>
m=1
(70)
Cmn sin m x sin r114
a
m and n are odd
The maximum deflection occurs at the center of the plate, and, with the
substitution of the expression for C mn given by equation (65), it may be
expressed as
Report No. 1847
-24-
m +
0.n
2
max = k m=1 n=1
7. (-1)
2 1
mz
mrx (m2 + n2 p 2) 2 [1
s + ( 1-v ) (m2 + 7 7 2 S S
x Y
Y
2
_2 p 2 sx)]
.
+ n2 p 2]
+(1-v)(m2S
2
+ ( 1-2v ) n2 p 21 sx
1
(7 1 )
m and n are odd
If the moduli of rigidity of the core are set equal to infinity, Sxand S
are zero, and the above solution reduces to the classical Navier solution for the deflection of a homogeneous plate provided that the moment
of inertia is taken to be that of the spaced facings of the sandwich plate.
Core Stresses
The expressions for the core stresses are obtained by substituting the
values of the constants given by equations (63), (64), (66), and (67) into equations (28), (29), and (30). These expressions are:
1
.
inn sin
rn
TT
x
11 a sin P.
(72)
m and n are odd
Report No. 1847
-25-
••n
N
rtiftT-ft
if
1:1
V
0
0
a)
a)
14
14
Id
0 I
ni
CI I
b
0
cd
cd
El l
>,...
NE
V/
r.*
......
.......
0
?
I IN
..--.
0i
›..
V)
N
---..
N
I C•1
N
n.
rq
N
0
+
,..
♦
M 4
0
Ns
.
14
NO 1::-°4I
co
0
u
.5
co
g
0
4.1
ra
at
to
5
0
1J
.,...-------.1.-----Th.
V)
a.
.--.
Eel
a.
+
+
CI
0
trl
.6
rd
17.71
..
TA
..--,
r.1
N
N
0
—\
0
W
N
0
1,..
.5
to
BI
r1
N
e7
81A17
to
0
4.1
r%3
a. +
N
R.
N
td
a° 4.71
.-.0
÷
.--;
u
rn
M
0"
(i)cu
N
O.
•
N
0
0
+
±
-
to
cu
ri
N
I-
Report No. 1847
...
-26-
t4
4-.1
I
A-
NI
N
N
4.J
N
–1
AJ
/-r
+
-I-
t,
u
u
CI)
til
Xi
4•J
II
......----_____
0
'------.--n
IAN
v
i
II
to
b
V
0
•,-,
bl
t4H
0
a)
Id
0
0
5
a)
ci
Id
.0
E°
5
rd
rd
bD
COI
• H
4.1
• H
Sd
N
„.9
ca
o
-1
0
to
al
al
1.1
$.1
cts
&L
a)
II
X
7.-J
t-
I-1
.5
r--1
0
in $
• H
C/7
N
N
^
N
0
Ni
.T.
I
.1
H..,
0.1
+
4
El
a
.--1
g
rd
El
43
RI
Cr. c_c_i14
Cd
N
N
5
E
g
--.
rr
—i
.......
...i
II
B
II
'6'
N
8
A
il
ci
4)
C1)
1-i
8
A 5
V\ g
In
U
R)
U
0
v.i
4J
^
N
.1.4
›-•
r-1
sd
tr)
5
N
a:
N
0
N
a_
Q.
. fir
N
0
ci
5
RI
Report No. 1847
-27-
N
0
+
Q.
N
0
+
Facing Stresses
The expressions for the forces and moments per unit length of the facings are given by equations (43). Under the present assumption that
the flexural stiffnesses of the individual facings are zero, the six moment expressions are zero. The forces per unit length of the facings
may be evaluated by first expressing them in terms of the core displacements by means of equations (50) and then substituting in the
values of the core displacements given by equations (24), (25), and (26),
remembering that mn and B mn are zero. The results of these substitutions may be expressed as
A
Nx
Et 1Et2
=
2 (e x + ve
142: =
(E;c + y e )
1-v 2
1-v
y)
Et l
t
,
Ny= - E
(ve x + e y )
N =
tve
+ e ' )
2
2
1 v
1-v
N
-
xY 2
Et,
(y )
(1+ V ) XY
Report No. 1847
N'Y=
4 X
-28-
Et7
(y1 )
1n7VT XY
(77)
0
4)
Id,
at
Cd
01
a
ao
O
III
r-
71
_0
M
CO
§
14
I---1
t I ri
Z
0
r--1
4
0
M
i
P 1 rd
I Cd
riki
1
..-0
5
g
sru
• r1
1
•9
*d
j1
K
44
j
0
I rc4d
g
rai
♦
gZI3
44 g
1 r%1
SIN
Mkid1
1
a
0
U
1
a
0
C..)
t E
4r1
♦
U
Mb
r41.0
U
N
----.
N
4.1
-IN
LI
..-....--.
N1
N
N
sl$
+
M
11
......
......
E
N
Ng
Ncl
3
N
0.
cd
N
E
LL
....Li
..-I
4 .1
1 N
+
U
N
N
to
0
c.)
.
o
.1.._...I
a.
11:1
E
ol
0
U
--II
I__I
4
Elk Pi
41
I
$4
ID
1
II
0
8 v `0
91X-4
11
8k8
El
-.
81"I A
gkil
$3
it
.",a,
.....,
II
tr
A El
/\ 5
8
ri
A
II
II
024
Report No. 1847
II
wo>.‘
It
It
'
vo
N
mo
-29-
›.
>14
C
to
O
.,-1
PO
TI
0
1:1
PO
0
14
a)
w
O
tr
0.1
14
i
;
Pel
PO
0
w
cii
A
I
4,N1
vtd
td
0i
0
d
+
4,
1
+
4401N
A
E l
E l
El
0 to
.c'
VO tg
k
aI
il I
PCI
t'd
1:1
5
0
k
b
id
PO
0
of
rd
-0
1
k
rd
:II
>-• w
Pli
al ck;
4 .5
-0
0 "
RI
i'4
1
14
- 113
.0to
P 1
a*
., tu
rd
jdP
w 1,-,
-1:11
. to
.
1
>.
5 I rd
0 I
4
to
NIA
0
.14
0)
g
0 Id
.0
to
ag
0
44 1°
lad
j
8."1°-41i
^
tIJ
al 0
k
iv\rt
0
-4
il
1..) 114.s0
0
to .0
0
4.1
B ki.II
C.)
Oft.
O
co
014 4.1
E
N
N
44
131
o)
44
4—.
0 •dg
o
+ N
0 Pd
N
+
u
04
sua
O
ID
u
4.1
N
ID .
♦
oi
4 g
N
.4
4.1
to
ii
U
0
+
.0
O
2N
......
m al
4.1
O
O
0
..—.
to u)
4.6N
O
0
.—..
..— N
+
O
I
J.
N
Er
•ri
N
II
rd
ekr
II
0
I
+
.4
II
8
/ \ 0
II
>•n
1.4
..
P.
Report No. 1847
rn
en
k
0.1 ("-
NO
n . •••n••
R. .0
O
az
c^) 0
k
Lc'II
a)
44
4..
NI +
► 1 .dri
N N
rd
A
1
IV
-30-
♦1N
+
u
......
.....
42N3
r4
N
01
N
.0
44
4ir,
4.
n••4
N ot
II
N
ilki 7
0
.....
N
4.b..4
1
II
11.14
n0
81"17cl
4.a
NO rid
A
II
"vi>4
rd
o
0
0
2
fo
r'
Cd
CO
14 1
r4 1
1I
ij
Ts
C
1
Cd
0I
El
5l
M
I 4
tt)
0
t.)
01
pia
11 Id
co
0
0
)4
N
0.
N
N
O
8IA
0
13
11
SIN
8
0,
N
__)4
A
N
4.14
+
N
I N0.
♦
+ N
E
I
N
P*1
N
4.1
N
I
•••mr,
♦
4.4
p41
N
N
41
•n••••
N
a.
N
N
0
IN N
4.1
N
0.
N
5
4.n
NINed
N
a.
N
0
+.
N
ttl
N
II
),*
Report No. 1847 n•1
-31-
0
11
and
2 + (m 2 - vn 2 p 2 ) Sx + (- vm 2 + n 2 p 2 ) S
mn
2
(m 2 + n 2 p 2 )
y
/1--v (m2
S + n 2 p 2 Sx)]
2 k
1
As would be expected, the absolute values of N x and N__', N and N ,
x Y
Y
Nxy
and N 'xy
are equal, respectively, as shown by equations (77) and
the•subsequent expressions for the strains. The maximum values of
1\13c andoccur at the center of the plate, and these values may be
NY
obtained from the following:
Et
(Nx)
max 1-v 2
Et
(N
)
y
l
l
max 1-v 2
(Ex
+ ve
max
(78)
)
'max
+
(ve
xmax
(79)
'max
where
t
- -
E
max
a2
m+n
t+t
(
2 ) (c +
t 1 + t2
12 2
1
-1) 2
4jmn
m and n are odd
Ymax
= - k
2 2
a 2
t 2
t
+ t
1
(c +
2
(tt- + t 2
m+n
> T°3
1
m=1 n=1 (-1)
Il mn
m and n are odd
x y is zero at the center of the plate and reaches a maximum at the
corners. No attempt was made to calculate values of N since such
xy
values would be of little importance for design purposes.
Report No. 1847
-32-
Numerical Computations
Calculations were made for the maximum deflection, the maximum
shear stresses in the core, and the maximum normal forces per unit
length of the facings. For purposes of calculation, equations (71),
(78), (79), (75), and (76) were expressed as follows:
(w)
=kC
=k
max 1
(Nx)
(80)
= k i (C 2 + vC )
max
(N )
Y
(T
XZ
=k
ma
ax
)
=
+ vC )
2
1
k
max
2 C4
and
( TyZ )
=k
max
(84)
2 C5
where
k-
16 qa 4 (1-v 2 )
Tr
Tr
6 EI
2
(
a
k
a )
E t
2
t
t
1 2
(c
(1-v ) (t + t )
1
2
T
Report No. 1847
-33-
l
+t
2
2)
16 qa2
w4
(c +
tl
t2
2
)
and
k
2 -
16 cla
t +t
2
1
2 )
n 3 (c +
The coefficients C 1 through C 5 represent the corresponding double infinite series in equations (717 (78), (79), (75), and (76). Since the
series represented by C 1 , C 2 , and C 3 in equations (80), (81), and (82)
are alternating in sign when summed over either the Ws or the mt s,
the values of these coefficients can be obtained with sufficient accuracy
by summing a finite number of terms and using Euler' s transformation
on the last few terms in cases where convergence is slow. In obtaining the values of C 1 , C 2 , and C 3 given in table 1, the first 21 terms of
the double infinite series were used. The double infinite series that
appear in the expressions for the core shear stresses, represented by
C 4 and C 5, are more difficult to sum because C 4 alternates only when
summed over then' El and C 5 alternates only when summed over the m l s.
The nonalternating part of these series was summed by the method
suggested by Gumowskil and the resulting partial sums could then be
summed using Euler's transformation. In all of the numerical work,
the value of Poisson's ratio of the facings was taken as 0. 3.
It is of interest to note that the values for S x= S = 0 in table 1 represent
the deflection, moment, and shear coefficients for a uniformly loaded
homogeneous plate with a moment of inertia equal to that of the spaced
facings of the sandwich plate. Thus, if the proper conversion factor
is used in each case, these values may be shown to agree with those
given by Timoshenko for the homogeneous plate problem. ±
3
—Gumowski, Igor. Summation of Slowly Converging Series. Journal
of Applied Physics Vol. 24, No. 8, p. 1068. 1953.
4
—Timoshenko, S. Theory of Plates and Shells. p. 133. New York.
1940.
Report No. 1847
-34-
T•
Conclusions
A general solution for the deflections and stresses in a uniformly
loaded, simply supported, rectangular sandwich plate is contained in
this report. This solution, based on the assumptions outlined in the
Introduction, consists of expressions for the deflections and stresses
in the form of double Fourier series in which the coefficients must be
obtained from equations (27) and (57) through (62).
In order to reduce the amount of numerical work necessary for the
preparation of design curves, certain additional simplifying assumptions are made. On the basis of these additional assumptions the necessary Fourier coefficients may be expressed as shown in equations
(63) through (69). The solution for the deflections and stresses is then
represented by equations (70), (72 1 ), (73), (74), and (77); and the expressions for the maximum deflection, the maximum shear stresses
in the core, and the maximum forces per unit length in the facings are
given by equations (71), (75), (76), (78), and (79). Numerical results
based on equations (71), (75), (76), (78), and (79) are given in table 1,
and design curves based on these values are shown in figures 5 through
16.
Report No. 1847
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