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On the flexural behaviour of orthotropic sandwich plates
Build. Sci. Vol. 8, pp. 127-136. Pergamon Press 1973. Printed in Great Britain
I
I
I
l
I
On the Flexural Behaviour of Orthotropic Sandwich Plates D. P. RAY*
P. K. SINHAt In this paper, the orthotropic sandwich plates having orthotropic faces and core, unequalface thicknesses and dissimilarities in face materials (including different Poisson ratios for facings) are considered. The stress-strain relations and differential equations are derived for bending of such plates by applying the variational principle of complementary energy. Various formulae and a few numerical results are obtained for bending of a simply supported plate under action of uniaxial compression and uniformly distributed transverse load. The validity of Vianello approximation is also studied and the results are discussed.
Mx, My Mx~.
NOMENCLATURE
The notations used in the present paper have been arranged alphabetically as given below: a b
h hx, h2
N~,Ny N~,,.
length of plate parallel to x-axis. width of plate parallel to y-axis.
P Px Q;,, Qy
2 2 core thickness. distances of neutral plane of the plates from the mid-planes of top and bottom facings respectively. 4P k
=
sx, sy
U V
F~
k,,
uniaxial buckling parameter defined by equation (36). m, n number of longitudinal and lateral half waves for plate respectively. q transverse loading per unit area of plate. qo uniformly distributed loading on plate. qm, constant defined in equation (27). t~, l2 thickness of top and bottom faces respectively. w deflection of plate parallel to z-axis. x, y, z coordinate axes. D~, D , D~y stiffnesses defined by equations (15), (16), (17) respectively. [Ext, Eye] , ] principal moduli of elasticity of orthotropic [Ex2, Eye] J faces, top and bottom respectively.
W
O'x2 , O'y 2
l'x~, 1, l'xy 2
Txzc, T vzc P
4n2Dy Fy--
bending moments in plate per unit run. twisting moment in plate per unit run. normal forces in plate per unit run. shear forces in plate parallel to xy-plane per unit run. compressive load in plate per unit run. compressive buckling load per unit run. transverse shear forces per unit run of plate parallel to xz-plane and yz-plane respectively. transverse shear stiffnesses per unit run of orthotropic sandwich plate, in xz-plane and yz-plane respectively. strain energy of plate. work done by inplane forces in course of lateral deflection of plate. work done by surface forces over the portion of the surface where displacements are prescribed. normal stresses per unit area of top face parallel to x- and y-axes respectively. normal stresses per unit area of bottom face parallel to x- and y-axes respectively. shear stresses per unit area in xy-plane of top and bottom faces respectively. transverse shear stresses per unit area of core in xz-plane and yz-plane respectively. Poisson ratio of isotropic face.
Vxl, V~l
Poisson ratios of orthotropic top face.
Vx2, VF2
Poisson ratios of orthotropic bottom face.
Vx, Vy
defined by equation (20).
b2
p
Dx Dy S~
G1, G2
moduli of rigidity in xy-plane for top and bottom faces respectively. G~:~, G~,:~ moduli of rigidity of orthotropic core in xz-plane and yz-plane respectively. 1 complementary energy of plate. I~ modified complementary energy of plate.
Sy rl = a/b 4s, Dxy
* Professor of Civil Engineering, Indian Institute of Technology, Kharagpur, India. ~"Senior Research Fellow, Department of Civil Engineering, Indian Institute of Technology, Kharagpur, India.
(1 - vxvy)D~ w, fl~, Yl 127
generalised boundary displacements. Lagrangian multipliers.
128
D. P. Ray and P. K. Sinha
i. INTRODUCTION IN SOME elastic bodies consisting of several different materials, where the continuity of displacement components on the interlace between the various materials is ensured without any slipping or tearing, the principle of minimum complementary energy establishes an alternative but convenient method for formulation of the problem. Particularly, when the equilibrium equations are known in terms of stresses, the approximate stress-strain relations, and subsequently the governing differential equations and the boundary conditions can be conveniently derived by applying the above variational principle. The principle was first applied by Reissner[l] to obtain stress-strain relations for the linear bending ofisotropic sandwich plates. Wang[2] used the same technique in solving thin homogeneous and sandwich plates with large deflections. Cheng[3], and Cheng et al.[4] extended it further for bending and buckling of orthotropic sandwich plates having isotropic faces and orthotropic core. In subsequent papers, Ueng and Lint5] and Ueng[6] utilised the same principle in bending of orthotropic (.both faces and core) sandwich plates with similar face materials and unequal face thicknesses. In the present study, the problem is further generalised to take into account the effects of orthotropy of face and core materials, unequal face thicknesses and dissimilarities in face materials (including different face Poisson ratios). Besides the works cited above, none of the otherworks on orthotropic sandwich construction,viz: that of Liboveand Batdorf[7], Plantema[8], Allen[9], Chang et al.[1012], Eringen [13], Norris[l 4-16] and others[ 17-20], considered all these effects together. Though Libove and Batdorf took into account the general case of orthotropy and presented a small deflection theory to describe the approximate elastic behaviour of orthotropic sandwich plates, the stiffnesses defined in the paper failed to interpret the physical significance in terms of material properties and geometrical proportions of the faces and the core. The present paper has been aimed at fulfilling this analytical interstice. Besides this, the expressions for deflections in combined bending of simply supported orthotropic sandwich plates are also derived. Finally, the flexural behaviour of orthotropic sandwich plates for a particular case under action of uniformly distributed transverse load and uniaxial compression is discussed. 2. B A S I C
ASSUMPTIONS
The basic assumptions made in the analysis are briefly outlined in the following: (a) Both the faces are orthotropic but may be
different in material and thickness. The Poisson's ratios of the faces are also different. The faces are thin and have negligible bending rigidities. The forces in the plane of the sandwich plate are taken entirely by two facings. (b) The core, being weak, behaves like a homogeneous orthotropic (in transverse shear rigidity) elastic continuum. However, by weak-core it is meant that the core is weak in carrying faceparallel stresses, but strong enough in resisting transverse shear. So the core carries the main part of the transverse shear loading, but undergoes considerable shear deformation. As a consequence of this antiplane stress condition of the core (figure l), it can be shown that r.~:,, and ry:c are
Y
dx
_
/
//--
V ~Tyzc
! / t
I
II
.
/
:t
I/
," ,Or~zc . :-~.~c + T o y
.
.
.
.
.
/
I
4- -'--J
/ l _ / , . , + ° > , °,z
/Z---ff~/ ~'/J ~"
! i - Ozc+~dz ' -O r z c dz "rxzc + x dz
Y
3z
Fig. 1. Stresses in a core element.
constant through the thickness of the core. This is obvious from the equilibrium equations for the core as given by O,Zx:c
c~z
-
0;
~ry=c
&
-
0
(1)
(c) The deflections of the plate are small, and are constant through the plate thickness. (d) The inplane forces act on the neutral plane of the plate. (e) Failure of the plate does not occur at the interfaces due to failure of bonds. Assumptions (a) and (b) are based on the relative stiffness of the core and the facings, but they are on the conservative side for the so-called weak-core sandwich structure. In fact, the faces may be considered thin, if the condition (100 > c/t > 5.77) is fulfilled[9]. Assumption (c) implies that the transverse shortening of the core is neglected in the analysis. It, therefore, ensures safety against wrinkling failure. However, for most of practical sandwich construction, the above assumptions may be considered justified.
On the Flexural Behaviour of Orthotropic Sandwich Plates
Inserting the relations (3) and (4) in (2) the strain energy for bending and shear can be rewritten as a b I'M x My vxl
3. STRESS-STRAIN RELATIONS AND DIFFERENTIAL EQUATIONS Let figure 2 represent a rectangular sandwich plate element which consists of an orthotropic core of thickness h, and two orthotropic membrane upper and lower facings of thickness t 1 and t 2 respectively. The xy-plane denotes the neutral plane of the plate with z-axis along the normal to this plane. The axes of orthotropy are assumed to be parallel to the x and y directions. Based on the above assumptions, the strain energy for bending and shear of the sandwich plate of length a and width b is given by
a b
I;[
~-2~
0 0
1 [M2x M 2
vx2 - 2 -
+
+
h
2
,
:
M~y
Q2
\Oxzc %=)J Next, the equilibrium of the sandwich plate element gives rise to the following equations:
-, Ox~Oyl)
~m~ 0N~
"Ox +--~y =o
0 0 tl 2 +-~11 r x y l + t 2
ax2 ~x2
..[
0"Y2 2Vx2 Ex2 ax2tTy2
ON, aN~, ....ay +"~-x = 0
Ey2
a b
ff(
(6a)
am, aM~y Qx = O ax ay am~ aM.,
t2 "c2y21 dx dy
+½
129
Oy
(2)
rx=~ TT~ dx dy
h ~ G.c:
Ox
Qy = 0
---{- ~+q+N
O0
Ox
x 02w N ~2w ~2W ~y2 + y ~y2 + 2Nxy Ox Oy = 0
where the relation vxEy = vyE~ is used.
y.~Ux --
T
(6b) lq
'
....
~r/7/://d////~,~9~////S/'/////~/~'~ h'T
/ My,/
hi h 2 i ~ - - - - ~ - - - ~ %
~M~,
~Y--~-
V'--Neulral plane I ~?~"'-[--"-N. }1
"-{~
' t3;~
*, h, ='2 h2
Fig. 2. Stress resultants acting on a sandwich element of unit area.
Since the stresses are uniform across the thickness, the stress resultants are as follows:
Mx = ~ x l t l C
= ~x2t2 c
M r = ~ % 1 t l c = ~tTy2t2e Mxy = ~-ZxyltlC = ~Zxy2t2C
(3)
where
t I t2 e=h+5+ T
The inplane forces Nx, Ny and Nxy being constant during lateral deflection, the equations (6a) are automatically satisfied. So the other three equilibrium equations (6b) constitute the auxiliary conditions or constraints which restrict the class of admissible stress variations in the application of the principle of minimum complementary energy. The work done by the inplane forces in the course of the lateral deflection of the plate is as follows a b
:cr .P:
V=½oJo JL \~)+ ,~)
and Qx
=
¢'Cxzc
2N law\ [aw\l
(4)
D. P. Rm' and P. K. Sinha
130
The work done by the surface forces over that portion of the surface where the displacements are prescribed is given by =
(?W
Sr
~' I'
14)
where
b W
py
/
"
8w
dw
E x l ~ x 2 l l l 2 C2
Dx
+9(2Mxy+9(sMx]dy
151
( 1-- vxvy)(t 1E~.1+ t2E,2 )
0
Dy
a
O~y -
8w
3w
16)
I +t2Ey2)
(1--VxVy)(tlEy
Ax = o x=,.i
2G~G2t~t2c 2 Gltl +G2t~_
(17)
C2
(18)
Sx =
o
+o~sMxr+~6Mr]
dx
(8)
s,=
Jy= 0
y=b
where ~ , ~2 . . . . % are the generalised boundary displacements. The application of the variational theorem of complementary energy to the bending problem, now, involves minimising the functional I - [(U+V)-W] with the constraint conditions as the equilibrium equations (6b). Introducing Lagrangian multipliers w, //~ and ~ ~ the modified complementary energy takes the form as follows: a b
I~= U + V - W +
w~--~x +~,.1, + q 0 o
(~2W
(SMx
+ YJ \ 0y
and Vxlt2Exz + Vx2tlExl ~"x
tlE~l+t2E~z (2O)
Vy
vylt2Ey2 + Vy2tlEyl tlEvl +t2Eyz
D~ and Dy can be defined as effective bending stiffnesses, D~y as the effective twisting stiffness and Vx, vy as effective Poisson's ratios of the orthotropic sandwich plate element as a whole. In addition to Euler equations, the following boundary conditions are also obtained:
(~2W w =
+ 2N~,r~x -fy
(?M~,y
8x
)
Qr
dx dy
(9)
Now carrying out the variation 6I,. = 0, integrating by parts and transforming the appropriate
surface integral to line integral by applying Green's Theorem, the variational equations (Euler's equations), which are equivalent to the required stress-strain relations, are given by M~ = D~ 88~ +v~Dy ~)71 ~
(lO)
My = n y ~ + v y D ~ Off1 8) 8x
(ll)
Dxy (O71 .F63fl l ~
MXY=-- 2 \ 8 x
@]
9(1 o r
8w
Q~ 8w . . . s~ 8x
8w]
6 Qx+Nx-~x+ N xyt~l--:,] = 0
]21 ~
9(2
or
8Mxy
fl I ~
9(3
or
6Mx = 0
=
0
(21)
and at v = 0, b W ~
9(4
or
6 Q,,+Nyp,--:+Nxysx /
fl l ~
9(5
or
6Mxr = 0
~i(22)
)' 1 =
9(6
or
6My = 0
J
0 i
!
Next applying the stress-strain relations in the equilibrium equations and eliminating /~ and "/1 the following three governing differential equations are obtained:
1
+.~__ ~ ) _ _ 11 0 x
+LgtT+Vx
,,
o,,
(12)
_FD; ' a3 //, .
(19)
at x = 0, a
(~2 W
+ NX ~xz + Ny ~
c~ T~ G,..c
(13)
L
a3 q
w =0 (23)
On the Flexural Behaviour of Orthotropic Sandwich Plates
131
The lateral load q can be expressed as q : ~ ~ q,,, sin m
-
Dy-%-v-v3+(Oxr+v~,Dx)
mlZX
a
n
w = 0
Qx
The third differential equation is obtained from the last of equations (6b) and it is given by
• ~ Qxmn cos m~xa sin n~y -b-
:
in
n
Qy ~- ~ ~ Qym,f sin mzrx m n a W -~-
w+q = 0
(25)
By changing notations, it can be observed that the differential equations (23) through (25) are identical to those developed by Libove and Batdorf[7]. But in the present analysis the seven physical constants, namely, D~, Dy, D~y, S~, Sy, v~ and vy, of an orthotropic sandwich plate can be given some theoretical interpretations in terms of constituent material properties as well as face and core geometry. Further, the relations (15), (16), (17) and (20) with suitable simplifications for a particular sandwich construction can be reduced to those obtained by Cheng[3], Ueng[6], Allen[9] and the other authors. However, the shear stiffnesses S~ and Sy have been defined differently by the different authors considering whether the faces are thin (c 4: h) or very thin (c = h).
4. SOLUTION FOR COMBINED BENDING
For all practical sandwich plate construction, having simply supported edges, the supports are applied over the entire thickness. Usually, the displacements of the faces along the lines of support are prevented by presence of such support conditions. Then the boundary conditions, for such simply supported edges, are given by
cos
nzrY7
(28)
13
•m n Win. sin mlrxa sin -nny7
where m and n are integers. The expressions for w, M x and My can be derived as follows: b4
t Amnqmn
o0 oo
w = x4Dr ~ ~ (4/Fr)(Nx[m2/q2] + Nyn2)A. + D:. x sin
Mx = - n2
m n
mnx nny sin - a b
(29)
Bm.qm. (4/Fy)(Nx[m2/q 2] + Nyn2)A'n + D ' . x sin
mztx mty sin a b
b2~
(30)
C'.q,.n
My = -~ m . (4/Fr)(N~,[m2/q 2] + NynZ)A'. + D ' . × sin
a t x = 0, a
(27)
b
where m and n are integers. The solution of the governing differential equations (23)-(25) has been assumed to be of the following form (24)
+2Nxy
nlzy
sin-
mTrx nny sin m a b
(31)
where A m pn
=
-e
1-
½/~
"-~
+(/~-~,v~)
q2
(32)
w=01
B;,. = t ~ + v ~ n ~ + M~ = 0 I
7
1-
/~¢ ,)-z-
(26a)
QY - 0 Sy a n d y = 0, b
w=0]
M~=0 ~Qx- 0
C ' . = vx ~-5-+n + 2
1-
vx0 ~/--~-
(26b)
q2 \ e
e /
e n4J
(34)
132
D. P. Ray and P. K. Sinha
fi ] V V
m2n2
l
: 4~
'74 (m4
2m2n2
.,/
,
)+t
(35)
Similar expressions can also be obtained for the shears Qx and Qy. Now let us consider a sandwich plate which is under action of any transverse load q and uniaxial compression Px (i.e. Nx = - Px and Ny = 0). From the equation (29), the instability condition gives k,..
4P~ Fy
D~,.
(m2/rl2)A;nn
b4 ~ ~ mnx mny w = r?D----~~ ~, (w),,, sin - - a sin ---if-- (37) where
A'.,.q,.. (38)
Again, in absence of compressive force, P, the deflection becomes b 4 ~ GO mnx. nrty Wo = rr4Dy ~.~ ~ (Wo)mn s i n - a sm--ff-
(39)
where A'.q.,. (w0),.. = ~
(40)
Substituting the relation (36) into the equations (38) and (40), and rearranging, it can be shown that (W)m . --
-
(Wo)mn -
1 -k/km.
Wo w
-
-
-
1-k/kjl
(42t
gives reasonable approximate values of the deflection.
(36)
where Px = - N x and Ny = 0. Let P be any compressive load which is less than the critical load Px. Let k = (4P/Fy). Then from the equation (29), the following expression is obtained for deflection:
(W)mn = -- k(mZfrlZ)A;nn + D~,.
Again, for each term of the series, the equation (41t is valid. But it is to be noted that for each such component deflection (for particular values of m and n), the corresponding buckling load parameter is different, and hence the magnification factor (l-k/k,..)-1 is also different. Here the total deflection is also obtained by summation of all the components. The process being too complicated, in some cases the well-known Vianello's simplified formula (kll corresponds to m = I. n = I) given by
(41)
Similar expressions can also be obtained for moments and shears. From the equation (41), it is seen that the deflection under action of only transverse load is magnified in presence of compressive edge loads, and also, when the compressive edge load approaches critical load, the deflection tends to become infinity. The expression (40) suggests that (Wo),,. is composed of a number of components depending on particular integer values of m and n, and the total deflection is obtained by the series summation.
5.
FLEXURAL
BEHAVIOUR
Here the results are obtained for sandwich plates acted upon by uniformly distributed transverse load qo and uniaxial compression P. In the case of uniform transverse loading, all the formulae given above generally yield satisfactory results, when only a few terms m = 1, 3 . . . . and n = 1 of the infinite series are used in computation. Hence for most o f the cases, when r/is approximately unity or less, it is expected that simplified Vianello formula (42) will give good approximations of the actual behaviour. However, when r/assumes values higher than unity it is interesting to study the validity of Vianello approximations. The deflections have been computed along the longitudinal centre line (y = b/2) for different cases of orthotropy and shear stiffness values. The computations have been carried out by taking only the first two terms of the doubly infinite series, namely, m = 1, n = 1; m = 3, n = 1 for three different cases: k = 0, k/k,., = 0.4 and k/k,., = 0.8. Similar results are also computed with the Vianello formula (42). These are then plotted as shown in figures 3-5. While the dotted lines indicate Vianello approximations, the firm lines represent the results obtained by taking the first two terms of the series. Figure 3 shows that when r/ = 1"0 the Vianello formula gives good results for k = 0, k/k,,, = 0.4 and k/km, = 0"8 even when the core is orthotropic. However, though only one value of ~b, i.e. q5 = 0-2, has been considered, it is expected that similar good approximations will be obtained for other values of 4'. Figure 4 gives such deflections for plates having q = 2-0 and ~b = 0.2. It is seen that when k/k,,, = 0.4, i.e. the edge load is 40 ~o of the critical load, theVianello formula gives good approximations. But when k/k,,. = 0.8, there is considerable deviation between the corresponding firm and dotted curves.
On the Flexural Behaviour of Orthotropic Sandwich Plates 2"01(0)
0"4[O ( )(
=0'4
~ = 0'4
I ~=,.o
,8= I ' 0
~ %
~5F
133
o.31-- ,# =0.8
~= 0 . 8 u: 0"25
u =0.25 o 3= Q
.....
//
0.5
~
k
=
O
0.2
0.1
I
/ f I
0
20
40 rr x / a ,
2.0
60
(b) Isotropic plate = I-0
__
80
k- 2 276
,=,.o
..//-
./k.,,~
I
60
0.4 (b) Isotropic plate
1.5
N
I
40
7r x / a , degrees
degrees
# = i o
%
I
20
80
= 2"286 _
% %.
0- ~ -
~,-"
V'=08
]
-I
-~f/.//" K~..13~8 /
u =0.25
I'0 (:3
_< ~k
£0
0"1 ~
0
20
40
60
80
20
0'4l(c)'
,~ 2 5
7rx/a, degrees
2'0
_
I
k:2-51
~:O.e
% %
u = 0-25
I.C
I
I
40
I
60
80
~'x/o, degrees
(c) • = 2.5 ,8:1.0
1 0.31--
#--12o
~e~,'-
~v=0.8 u =0.25
~ .f'~"
....
~k=2"51
-
/
0.5
k
20
40 rrx/a
.Q %,
I
0
80
0
( = 5.0
I
/9= 1.0 ~=0.8 0=0.25
~"
I
20
1
I
40
60
1
80
"tr x / a ~degree s
, degrees 0.4
2°l(d)J
,.5F
60
I(a) ,~= 5.0
~- ....
,e=,.o
~=0'8 u:0.25
0.3
~ ./;'/"
~"4k=2. 6
o
I.o
/"t"
2.6
Q
0-2 -
/
-
£0
_
k
~
~k=O
0-1
0
20
40
60
x/a, degrees
3. [(a)-(d)]Deflection of sandwich plates (~/= 1"0, ~b= 0"2)along longitudinal centre line (y = b/2).
Fig.
r
80
i
20
i
40
rr x/o, degrees
4. [(a)-(d)]Deflection of sandwich (b=0"2) along longitudinal centre
Fig.
Vianello approx.
i
60
i
80
plates (tl = 2.0, line (y = b/2). , Two terms of series
D. P. R a t a n d P. A'. S i n h a
134 0 16 (a)
(c )
] [
~:os
.... o
4' =0
I
k=3-2
012 ~C3
o
o 08
012
~o
008
,E"-
004
C'
I
I
20
40
I
04
I
60
r
80
v-x/c] , degrees
20
40 7rx/o
016
(b) ¢ : 0 2 _. ~
~A _ k =
2'23
60 ,
8'0
degrees
_ 0
20
=
s_ . . . . . . .
'
012 aao
008 ¥ 004 50~
20
40
",'rx/a,
60
~
80
C04
degrees 40
?C
x /a,
60
80
degrees
Fig. 5. [(a) (d)] De[tection o f isotropic sandwich plates (v = 0'25, l/ = 3'0) along longitudinal centre line (y = b;2). Viane/lo appro.v. , Two terms o f series ~- - . Tabh, 1. Deflection w o f uniJbrndy loaded sandwich plates (0 = 2.0; ~b = 0-2) along longitudinal centre line (y - b.C). Isotropic fitces and orthotropie core. Values o f (zr6/16) (Dw/qoaZb -') ~x a degree
c = 0"4 Calculation 1*
~: = 1.0
Calculation 2+
Calculation 1
~: = 2'5
C a l c ul a t i on 2 k/k ....
0 20 40 60 80 90
0 0"112 0"188 0"219 0"224 0'224
0 0'105 0'182 0"219 0"231 0'232
0 0'107 0'185 0'224 0"236 0"240
~:
5.(1
C a l c ul a t i on I
Calculation 2
C a l c ul a t i on 1
Calculation 2
0 0"103 0"181 0'224 0-240 0"241
0 O'lO0 0"178 0'224 0-241 0"243
0 0"101 0"179 0"223 0-240 0'242
0 0"100 0.178 0-223 0"242 0"244
0 0-168 0"282 0"330 0"336 0"336
0 0"148 0"262 0-330 0"356 0"359
0 0"162 0"278 0"336 0"352 0'354
0 0.150 0.267 0.376 0.364 0.367
0.4
0 0'103 0"181 0"224 0'240 0'244
k/k,,,, = 0-8 0 20 40 60 80 90
0 0"178 0"273 0"276 0'245 0.239
0 0"132 0-228 0"276 0"291 0"291
0 0.182 0.292 0.318 0.305 0.304
0 0-146 0.256 0"318 0.341 0.347
* Calculation 1 refers to those c o m p u t e d from the e qua t i ons (37) and (41) t a k i n g only the first two terms of the doubly infinite series. + Calcu lation 2 is based on the Vianello f o r m u l a (42).
On the Flexural Behaviour o f Orthotropic Sandwich Plates
135
Table 2. Deflection w of uniformly loaded isotropic sandwich plates ( t / = 3"0) along longitudinal centre line (y = b/2). Values of 0r6/16)(Dw/qoa2b 2) ~rx a
degree
~b = 0 Calculation 1*
~b = 0'2
Calculation 2#
Calculation 1
~b = 0'5
Calculation 2
~b = 1'0
Calculation 1
Calculation 2
Calculation 1
Calculation 2
0 0.0780 0.1250 0.1349 0.1283 0-1270
0 0.0710 0.1180 0.1349 0.1357 0'1350
0 0.1050 0'1670 0.1782 0.1683 0-1660
0 0.0967 0.1574 0.1782 0.1770 0.1760
0 0'1150 0.1678 0.1515 0.1170 0'1110
0 0.0799 0.1325 0.1515 0.1525 0"1519
0 0.1368 0-2018 0'1942 0.1610 0.1548
0 0.1053 0.1715 0.1942 0'1925 0.1914
k/km, = 0'4 0 20 40 60 80 90
0 0.0494 0-0809 0.0910 0.0901 0.0896
0 0.0453 0'0769 0.0910 0.0941 0.0942
0 0'0614 0.992 0.1087 0.1052 0.1042
0 0.0554 0'0935 0.1087 0'1108 0.1108 k/k.,. = 0"8
0 20 40 60 80 90
0 0.0832 0.1250 0.1095 0.0845 0.0802
0 0.0544 0.0925 0.1095 0.1131 0.1132
0 0.1017 0'1456 0.1263 0.0920 0.0862
0 0.0645 0.1087 0.1263 0.1277 0.1277
* Calculation 1 refers to those computed from the equations (37) and (41) taking only the first two terms of the doubly infinite series. t Calculation 2 is based on Vianello approximations.
T h e deviations tend to increase as e decreases. Thus the central deflection calculated b y the Vianello f o r m u l a is only 3.7 % higher at e = 5.0, while that at e = 0.4 is a b o u t 22 % higher (Table 1). Figure 5 gives results for isotropic sandwich plates, for ~/ = 3.0 a n d for different values o f ~b. Here the deviation is m u c h m o r e p r e d o m i n a n t , particularly when k/k,,, = 0.8. The d e v i a t i o n in the central deflection varies f r o m a b o u t 41 to 23.8 %, when ~b varies from 0.0 to 1-0 respectively (Table 2). Hence, when the compressive edge l o a d is m o r e t h a n 40 % o f the buckling load, the Vianello f o r m u l a is always seen to predict results being widely deviated f r o m those o b t a i n e d f r o m the equations (37) a n d (41), a n d the m a x i m u m deflection is always unconservative. M o r e o v e r , while the results from the Vianello f o r m u l a suggests that the m a x i m u m deflection should occur at the centre o f the plate, the actual m a x i m u m deflection is f o u n d to be somewhere a l o n g the y = b/2 line other t h a n the centre. It is expected t h a t the m o m e n t s a n d shears being higher derivatives o f deflections, errors involved in such c o m p u t a t i o n s will be even m o r e p r o m i n e n t .
is because o f the fact that in this analysis m u c h greater flexibility in selection o f materials a n d thicknesses for the facings has been ensured. This permits a wider choice o f materials a n d p r o p o r t i o n s in regard to e c o n o m i c design o f sandwich plates, particularly e m p l o y i n g a d v a n c e d c o m p o s i t e materials for facings. F u r t h e r , it is observed that, in some o f the cases, because o f the simplified expression and easy accessibility to desk c o m p u t a t i o n , the Vianeilo f o r m u l a (42) can be suggested as a possible substitution for the expression (41), which involves lengthier c o m p u t a t i o n s t h r o u g h the series summation, w i t h o u t any serious loss o f accuracy. Thus, it is n o t e d that the Vianello f o r m u l a (42) gives g o o d a p p r o x i m a t i o n s o f the actual b e h a v i o u r o f o r t h o tropic sandwich plates u n d e r c o m b i n e d b e n d i n g with u n i f o r m l y distributed transverse l o a d a n d uniaxial c o m p r e s s i o n so long as the aspect ratio q is unity o r less, and, for all practical purposes, r e a s o n a b l y g o o d results can also be expected for other cases, when r/ = 2 or even 3, a n d the c o m pressive edge l o a d is limited to 4 0 % o f the critical load.
6. C O N C L U S I O N S It is f o u n d t h a t the theoretical d e v e l o p m e n t carried o u t in the present s t u d y is m o r e general t h a n those d o n e b y the previous investigators. This
Acknowledgements--The authors would like to express their appreciation to the Research and Development Organisation, Ministry of Defence, New Delhi for the financial support to carry out research on sandwich structures.
136
D. P. Ray amt P. K. Shtha
REFERENCES I. E. REISSNER,On bending of elastic plates, Quarterly ~fApplied Mathematics 5, I, April (1947). 2. C.T. WANG, Principle and application of complementary energy method for thin homogeneous and sandwich plates and shells with finite deflections, NACA TN 2620, 1952. 3. S. CHENG, On the theory of bending of sandwich plates, Proc. Fourth U.S. Nationa/ Congress of Applied Mechanics, Vol. I (1962). 4. S. CHENGand N. AL-RUBAYI,Elastic stability of orthotropic sandwich plates, Developments in Mechanics, Vol. 5, Proc. of the llth Midwestern Mechanics Conf, Sept. (1969). 5. C. E. S. UENG and Y. J. LIN, On bending of orthotropic sandwich plates, AIAA Journal 4, Dec. 1966. 6. C. E. S. UENG, Some remarks on the deflection of orthotropic sandwich plates, J. Composite Matls. 2, 1968. 7. C. LIBOVEand S. B. BATDORE,A general small deflection theory for flat sandwich plates, NACA Report No. 899, 1948. 8. F.J. PLANTEMA,Sandwich Construction--The bending and buckling of sandwich beams, plates, and shells, Vol. l, p. 132. Wiley, New York (1966). 9. H.G. ALLEN, Analysis and Design of Structural Sandwich Panels, Vol. l, p. 87. Pergamon Press, 1969. 10. C. C. CHANG and 1. K. EBCIOGLU,Elastic instability of rectangular sandwich panel of orthotropic core with different face thickness and materials, AFOSR TN 58-221, ASTIA AD 154-122, University of Minnesota, March 1958. 11. C.C. CHANG, B. T. FANG and I. K. EBCIOGLU,Elastic theory of a weak-core sandwich panel initially warped, simply supported and subjected to combined loadings, Proc. Third U.S. National Congr. Appl. Mech. (1958). 12. C. C, CHANGand B. T. FANG, Initially warped sandwich panel under combined loadings, J. Aero Space Science 27, 10, Oct. (1960). 13. A. C. ERINGEN, Bending and buckling of rectangular sandwich plates, Proc. First U.S. National Congr. Appl. Mech. (1951). 14. C. B. NORRIS, Compressive buckling design curves for sandwich panels with isotropic facings and orthotropic cores, FPL Report No. 1854 (1956). (Revised Jan. 1958.) 15. C. B. NORRIS, Compressive buckling curves for simply supported sandwich panels with glass-fabric-laminate facings and honeycomb core, FPL Report No. 1867 (1958). 16. C. B. NORRIS, Compressive buckling curves for flat sandwich panels with dissimilar facings, FPL Report No. 1875, 1960. 17. W. S. EPaCKSENand H. W. MARCH, Compressive buckling of sandwich panels having dissimilar facings of unequal thickness, FPL Report No. 1583-B (1950). (Revised Nov. 1958.) 18. W. S. ERICKSEN,Effects of shear deformation in the core of a flat rectangular sandwich panel-deflection under uniform load of sandwich panels having facings of unequal thickness, FPL Report No. 1583-C (1950). 19. W. S. ERICKSEN,Supplement to: Effects of shear deformation in the core of a flat rectangular sandwich panel-deflection under uniform load of sandwich panels having facings of moderate thickness, FPL Report No. 1583-D (1951). 20. H. W. MARCH, Effects of shear deformation in the core of a flat rectangular sandwich panel, FPL Report No. 1583 (1948). Dans cet ouvrage, on consid6re les plaques sandwich orthotropiques ayant des faces et un noyau orthotropiques, des 6paisseurs diff6rentes de faces et des diff6rences de mat6riau de faces (y compris des rapports de Poisson diff6rents des surfaces). Les relations contrainte-d6formation et les 6quations diff~rentielles sont d6riv~es pour le fl6chissement de telles plaques en appliquant le principe variationel d'6nergie compl6mentaire. Diverses formules et quelques r6sultats num~riques sont obtenus pour le fl6chissement d'une plaque simplement support6e sous l'action d'une compression uniaxiale et d'une charge transversale uniform6ment distribu6e. On 6tudie 6galement la validit6 de l'approximation de Vianello et les r6sultats sont discut6s. In diesem Bericht werden orthotrope Verbundplatten mit orthotropen Aussenseiten und Kernen, ungleichen Dicken der Aussenseite und Un~hnlichkeiten in Aussenseitenmaterialien (einschliesslich verschiedenen Poisson Verh~iltnissen fiir Verkleidungen) behandelt. Die Spannungs-Dehnungsbeziehungen und Differentialgleichungen fiir Biegung solcher Platten werden durch Anwendung des Variationsprinzips der Komplement/~renergie abgeleitet. Es werden verschiedene Formeln und ein paar zahlenm/issige Ergebnisse ffir Biegung einer einfach gestiJtzten Platte unter Aussetzung einachsigen Drucks und gleichf~rmig verteilter Querbelastung erhalten. Auch wird die Giiltigkeit der Vianello Ann/iherung untersucht und die Ergebnisse werden besprochen.
I
I
I
l
I
On the Flexural Behaviour of Orthotropic Sandwich Plates D. P. RAY*
P. K. SINHAt In this paper, the orthotropic sandwich plates having orthotropic faces and core, unequalface thicknesses and dissimilarities in face materials (including different Poisson ratios for facings) are considered. The stress-strain relations and differential equations are derived for bending of such plates by applying the variational principle of complementary energy. Various formulae and a few numerical results are obtained for bending of a simply supported plate under action of uniaxial compression and uniformly distributed transverse load. The validity of Vianello approximation is also studied and the results are discussed.
Mx, My Mx~.
NOMENCLATURE
The notations used in the present paper have been arranged alphabetically as given below: a b
h hx, h2
N~,Ny N~,,.
length of plate parallel to x-axis. width of plate parallel to y-axis.
P Px Q;,, Qy
2 2 core thickness. distances of neutral plane of the plates from the mid-planes of top and bottom facings respectively. 4P k
=
sx, sy
U V
F~
k,,
uniaxial buckling parameter defined by equation (36). m, n number of longitudinal and lateral half waves for plate respectively. q transverse loading per unit area of plate. qo uniformly distributed loading on plate. qm, constant defined in equation (27). t~, l2 thickness of top and bottom faces respectively. w deflection of plate parallel to z-axis. x, y, z coordinate axes. D~, D , D~y stiffnesses defined by equations (15), (16), (17) respectively. [Ext, Eye] , ] principal moduli of elasticity of orthotropic [Ex2, Eye] J faces, top and bottom respectively.
W
O'x2 , O'y 2
l'x~, 1, l'xy 2
Txzc, T vzc P
4n2Dy Fy--
bending moments in plate per unit run. twisting moment in plate per unit run. normal forces in plate per unit run. shear forces in plate parallel to xy-plane per unit run. compressive load in plate per unit run. compressive buckling load per unit run. transverse shear forces per unit run of plate parallel to xz-plane and yz-plane respectively. transverse shear stiffnesses per unit run of orthotropic sandwich plate, in xz-plane and yz-plane respectively. strain energy of plate. work done by inplane forces in course of lateral deflection of plate. work done by surface forces over the portion of the surface where displacements are prescribed. normal stresses per unit area of top face parallel to x- and y-axes respectively. normal stresses per unit area of bottom face parallel to x- and y-axes respectively. shear stresses per unit area in xy-plane of top and bottom faces respectively. transverse shear stresses per unit area of core in xz-plane and yz-plane respectively. Poisson ratio of isotropic face.
Vxl, V~l
Poisson ratios of orthotropic top face.
Vx2, VF2
Poisson ratios of orthotropic bottom face.
Vx, Vy
defined by equation (20).
b2
p
Dx Dy S~
G1, G2
moduli of rigidity in xy-plane for top and bottom faces respectively. G~:~, G~,:~ moduli of rigidity of orthotropic core in xz-plane and yz-plane respectively. 1 complementary energy of plate. I~ modified complementary energy of plate.
Sy rl = a/b 4s, Dxy
* Professor of Civil Engineering, Indian Institute of Technology, Kharagpur, India. ~"Senior Research Fellow, Department of Civil Engineering, Indian Institute of Technology, Kharagpur, India.
(1 - vxvy)D~ w, fl~, Yl 127
generalised boundary displacements. Lagrangian multipliers.
128
D. P. Ray and P. K. Sinha
i. INTRODUCTION IN SOME elastic bodies consisting of several different materials, where the continuity of displacement components on the interlace between the various materials is ensured without any slipping or tearing, the principle of minimum complementary energy establishes an alternative but convenient method for formulation of the problem. Particularly, when the equilibrium equations are known in terms of stresses, the approximate stress-strain relations, and subsequently the governing differential equations and the boundary conditions can be conveniently derived by applying the above variational principle. The principle was first applied by Reissner[l] to obtain stress-strain relations for the linear bending ofisotropic sandwich plates. Wang[2] used the same technique in solving thin homogeneous and sandwich plates with large deflections. Cheng[3], and Cheng et al.[4] extended it further for bending and buckling of orthotropic sandwich plates having isotropic faces and orthotropic core. In subsequent papers, Ueng and Lint5] and Ueng[6] utilised the same principle in bending of orthotropic (.both faces and core) sandwich plates with similar face materials and unequal face thicknesses. In the present study, the problem is further generalised to take into account the effects of orthotropy of face and core materials, unequal face thicknesses and dissimilarities in face materials (including different face Poisson ratios). Besides the works cited above, none of the otherworks on orthotropic sandwich construction,viz: that of Liboveand Batdorf[7], Plantema[8], Allen[9], Chang et al.[1012], Eringen [13], Norris[l 4-16] and others[ 17-20], considered all these effects together. Though Libove and Batdorf took into account the general case of orthotropy and presented a small deflection theory to describe the approximate elastic behaviour of orthotropic sandwich plates, the stiffnesses defined in the paper failed to interpret the physical significance in terms of material properties and geometrical proportions of the faces and the core. The present paper has been aimed at fulfilling this analytical interstice. Besides this, the expressions for deflections in combined bending of simply supported orthotropic sandwich plates are also derived. Finally, the flexural behaviour of orthotropic sandwich plates for a particular case under action of uniformly distributed transverse load and uniaxial compression is discussed. 2. B A S I C
ASSUMPTIONS
The basic assumptions made in the analysis are briefly outlined in the following: (a) Both the faces are orthotropic but may be
different in material and thickness. The Poisson's ratios of the faces are also different. The faces are thin and have negligible bending rigidities. The forces in the plane of the sandwich plate are taken entirely by two facings. (b) The core, being weak, behaves like a homogeneous orthotropic (in transverse shear rigidity) elastic continuum. However, by weak-core it is meant that the core is weak in carrying faceparallel stresses, but strong enough in resisting transverse shear. So the core carries the main part of the transverse shear loading, but undergoes considerable shear deformation. As a consequence of this antiplane stress condition of the core (figure l), it can be shown that r.~:,, and ry:c are
Y
dx
_
/
//--
V ~Tyzc
! / t
I
II
.
/
:t
I/
," ,Or~zc . :-~.~c + T o y
.
.
.
.
.
/
I
4- -'--J
/ l _ / , . , + ° > , °,z
/Z---ff~/ ~'/J ~"
! i - Ozc+~dz ' -O r z c dz "rxzc + x dz
Y
3z
Fig. 1. Stresses in a core element.
constant through the thickness of the core. This is obvious from the equilibrium equations for the core as given by O,Zx:c
c~z
-
0;
~ry=c
&
-
0
(1)
(c) The deflections of the plate are small, and are constant through the plate thickness. (d) The inplane forces act on the neutral plane of the plate. (e) Failure of the plate does not occur at the interfaces due to failure of bonds. Assumptions (a) and (b) are based on the relative stiffness of the core and the facings, but they are on the conservative side for the so-called weak-core sandwich structure. In fact, the faces may be considered thin, if the condition (100 > c/t > 5.77) is fulfilled[9]. Assumption (c) implies that the transverse shortening of the core is neglected in the analysis. It, therefore, ensures safety against wrinkling failure. However, for most of practical sandwich construction, the above assumptions may be considered justified.
On the Flexural Behaviour of Orthotropic Sandwich Plates
Inserting the relations (3) and (4) in (2) the strain energy for bending and shear can be rewritten as a b I'M x My vxl
3. STRESS-STRAIN RELATIONS AND DIFFERENTIAL EQUATIONS Let figure 2 represent a rectangular sandwich plate element which consists of an orthotropic core of thickness h, and two orthotropic membrane upper and lower facings of thickness t 1 and t 2 respectively. The xy-plane denotes the neutral plane of the plate with z-axis along the normal to this plane. The axes of orthotropy are assumed to be parallel to the x and y directions. Based on the above assumptions, the strain energy for bending and shear of the sandwich plate of length a and width b is given by
a b
I;[
~-2~
0 0
1 [M2x M 2
vx2 - 2 -
+
+
h
2
,
:
M~y
Q2
\Oxzc %=)J Next, the equilibrium of the sandwich plate element gives rise to the following equations:
-, Ox~Oyl)
~m~ 0N~
"Ox +--~y =o
0 0 tl 2 +-~11 r x y l + t 2
ax2 ~x2
..[
0"Y2 2Vx2 Ex2 ax2tTy2
ON, aN~, ....ay +"~-x = 0
Ey2
a b
ff(
(6a)
am, aM~y Qx = O ax ay am~ aM.,
t2 "c2y21 dx dy
+½
129
Oy
(2)
rx=~ TT~ dx dy
h ~ G.c:
Ox
Qy = 0
---{- ~+q+N
O0
Ox
x 02w N ~2w ~2W ~y2 + y ~y2 + 2Nxy Ox Oy = 0
where the relation vxEy = vyE~ is used.
y.~Ux --
T
(6b) lq
'
....
~r/7/://d////~,~9~////S/'/////~/~'~ h'T
/ My,/
hi h 2 i ~ - - - - ~ - - - ~ %
~M~,
~Y--~-
V'--Neulral plane I ~?~"'-[--"-N. }1
"-{~
' t3;~
*, h, ='2 h2
Fig. 2. Stress resultants acting on a sandwich element of unit area.
Since the stresses are uniform across the thickness, the stress resultants are as follows:
Mx = ~ x l t l C
= ~x2t2 c
M r = ~ % 1 t l c = ~tTy2t2e Mxy = ~-ZxyltlC = ~Zxy2t2C
(3)
where
t I t2 e=h+5+ T
The inplane forces Nx, Ny and Nxy being constant during lateral deflection, the equations (6a) are automatically satisfied. So the other three equilibrium equations (6b) constitute the auxiliary conditions or constraints which restrict the class of admissible stress variations in the application of the principle of minimum complementary energy. The work done by the inplane forces in the course of the lateral deflection of the plate is as follows a b
:cr .P:
V=½oJo JL \~)+ ,~)
and Qx
=
¢'Cxzc
2N law\ [aw\l
(4)
D. P. Rm' and P. K. Sinha
130
The work done by the surface forces over that portion of the surface where the displacements are prescribed is given by =
(?W
Sr
~' I'
14)
where
b W
py
/
"
8w
dw
E x l ~ x 2 l l l 2 C2
Dx
+9(2Mxy+9(sMx]dy
151
( 1-- vxvy)(t 1E~.1+ t2E,2 )
0
Dy
a
O~y -
8w
3w
16)
I +t2Ey2)
(1--VxVy)(tlEy
Ax = o x=,.i
2G~G2t~t2c 2 Gltl +G2t~_
(17)
C2
(18)
Sx =
o
+o~sMxr+~6Mr]
dx
(8)
s,=
Jy= 0
y=b
where ~ , ~2 . . . . % are the generalised boundary displacements. The application of the variational theorem of complementary energy to the bending problem, now, involves minimising the functional I - [(U+V)-W] with the constraint conditions as the equilibrium equations (6b). Introducing Lagrangian multipliers w, //~ and ~ ~ the modified complementary energy takes the form as follows: a b
I~= U + V - W +
w~--~x +~,.1, + q 0 o
(~2W
(SMx
+ YJ \ 0y
and Vxlt2Exz + Vx2tlExl ~"x
tlE~l+t2E~z (2O)
Vy
vylt2Ey2 + Vy2tlEyl tlEvl +t2Eyz
D~ and Dy can be defined as effective bending stiffnesses, D~y as the effective twisting stiffness and Vx, vy as effective Poisson's ratios of the orthotropic sandwich plate element as a whole. In addition to Euler equations, the following boundary conditions are also obtained:
(~2W w =
+ 2N~,r~x -fy
(?M~,y
8x
)
Qr
dx dy
(9)
Now carrying out the variation 6I,. = 0, integrating by parts and transforming the appropriate
surface integral to line integral by applying Green's Theorem, the variational equations (Euler's equations), which are equivalent to the required stress-strain relations, are given by M~ = D~ 88~ +v~Dy ~)71 ~
(lO)
My = n y ~ + v y D ~ Off1 8) 8x
(ll)
Dxy (O71 .F63fl l ~
MXY=-- 2 \ 8 x
@]
9(1 o r
8w
Q~ 8w . . . s~ 8x
8w]
6 Qx+Nx-~x+ N xyt~l--:,] = 0
]21 ~
9(2
or
8Mxy
fl I ~
9(3
or
6Mx = 0
=
0
(21)
and at v = 0, b W ~
9(4
or
6 Q,,+Nyp,--:+Nxysx /
fl l ~
9(5
or
6Mxr = 0
~i(22)
)' 1 =
9(6
or
6My = 0
J
0 i
!
Next applying the stress-strain relations in the equilibrium equations and eliminating /~ and "/1 the following three governing differential equations are obtained:
1
+.~__ ~ ) _ _ 11 0 x
+LgtT+Vx
,,
o,,
(12)
_FD; ' a3 //, .
(19)
at x = 0, a
(~2 W
+ NX ~xz + Ny ~
c~ T~ G,..c
(13)
L
a3 q
w =0 (23)
On the Flexural Behaviour of Orthotropic Sandwich Plates
131
The lateral load q can be expressed as q : ~ ~ q,,, sin m
-
Dy-%-v-v3+(Oxr+v~,Dx)
mlZX
a
n
w = 0
Qx
The third differential equation is obtained from the last of equations (6b) and it is given by
• ~ Qxmn cos m~xa sin n~y -b-
:
in
n
Qy ~- ~ ~ Qym,f sin mzrx m n a W -~-
w+q = 0
(25)
By changing notations, it can be observed that the differential equations (23) through (25) are identical to those developed by Libove and Batdorf[7]. But in the present analysis the seven physical constants, namely, D~, Dy, D~y, S~, Sy, v~ and vy, of an orthotropic sandwich plate can be given some theoretical interpretations in terms of constituent material properties as well as face and core geometry. Further, the relations (15), (16), (17) and (20) with suitable simplifications for a particular sandwich construction can be reduced to those obtained by Cheng[3], Ueng[6], Allen[9] and the other authors. However, the shear stiffnesses S~ and Sy have been defined differently by the different authors considering whether the faces are thin (c 4: h) or very thin (c = h).
4. SOLUTION FOR COMBINED BENDING
For all practical sandwich plate construction, having simply supported edges, the supports are applied over the entire thickness. Usually, the displacements of the faces along the lines of support are prevented by presence of such support conditions. Then the boundary conditions, for such simply supported edges, are given by
cos
nzrY7
(28)
13
•m n Win. sin mlrxa sin -nny7
where m and n are integers. The expressions for w, M x and My can be derived as follows: b4
t Amnqmn
o0 oo
w = x4Dr ~ ~ (4/Fr)(Nx[m2/q2] + Nyn2)A. + D:. x sin
Mx = - n2
m n
mnx nny sin - a b
(29)
Bm.qm. (4/Fy)(Nx[m2/q 2] + Nyn2)A'n + D ' . x sin
mztx mty sin a b
b2~
(30)
C'.q,.n
My = -~ m . (4/Fr)(N~,[m2/q 2] + NynZ)A'. + D ' . × sin
a t x = 0, a
(27)
b
where m and n are integers. The solution of the governing differential equations (23)-(25) has been assumed to be of the following form (24)
+2Nxy
nlzy
sin-
mTrx nny sin m a b
(31)
where A m pn
=
-e
1-
½/~
"-~
+(/~-~,v~)
q2
(32)
w=01
B;,. = t ~ + v ~ n ~ + M~ = 0 I
7
1-
/~¢ ,)-z-
(26a)
QY - 0 Sy a n d y = 0, b
w=0]
M~=0 ~Qx- 0
C ' . = vx ~-5-+n + 2
1-
vx0 ~/--~-
(26b)
q2 \ e
e /
e n4J
(34)
132
D. P. Ray and P. K. Sinha
fi ] V V
m2n2
l
: 4~
'74 (m4
2m2n2
.,/
,
)+t
(35)
Similar expressions can also be obtained for the shears Qx and Qy. Now let us consider a sandwich plate which is under action of any transverse load q and uniaxial compression Px (i.e. Nx = - Px and Ny = 0). From the equation (29), the instability condition gives k,..
4P~ Fy
D~,.
(m2/rl2)A;nn
b4 ~ ~ mnx mny w = r?D----~~ ~, (w),,, sin - - a sin ---if-- (37) where
A'.,.q,.. (38)
Again, in absence of compressive force, P, the deflection becomes b 4 ~ GO mnx. nrty Wo = rr4Dy ~.~ ~ (Wo)mn s i n - a sm--ff-
(39)
where A'.q.,. (w0),.. = ~
(40)
Substituting the relation (36) into the equations (38) and (40), and rearranging, it can be shown that (W)m . --
-
(Wo)mn -
1 -k/km.
Wo w
-
-
-
1-k/kjl
(42t
gives reasonable approximate values of the deflection.
(36)
where Px = - N x and Ny = 0. Let P be any compressive load which is less than the critical load Px. Let k = (4P/Fy). Then from the equation (29), the following expression is obtained for deflection:
(W)mn = -- k(mZfrlZ)A;nn + D~,.
Again, for each term of the series, the equation (41t is valid. But it is to be noted that for each such component deflection (for particular values of m and n), the corresponding buckling load parameter is different, and hence the magnification factor (l-k/k,..)-1 is also different. Here the total deflection is also obtained by summation of all the components. The process being too complicated, in some cases the well-known Vianello's simplified formula (kll corresponds to m = I. n = I) given by
(41)
Similar expressions can also be obtained for moments and shears. From the equation (41), it is seen that the deflection under action of only transverse load is magnified in presence of compressive edge loads, and also, when the compressive edge load approaches critical load, the deflection tends to become infinity. The expression (40) suggests that (Wo),,. is composed of a number of components depending on particular integer values of m and n, and the total deflection is obtained by the series summation.
5.
FLEXURAL
BEHAVIOUR
Here the results are obtained for sandwich plates acted upon by uniformly distributed transverse load qo and uniaxial compression P. In the case of uniform transverse loading, all the formulae given above generally yield satisfactory results, when only a few terms m = 1, 3 . . . . and n = 1 of the infinite series are used in computation. Hence for most o f the cases, when r/is approximately unity or less, it is expected that simplified Vianello formula (42) will give good approximations of the actual behaviour. However, when r/assumes values higher than unity it is interesting to study the validity of Vianello approximations. The deflections have been computed along the longitudinal centre line (y = b/2) for different cases of orthotropy and shear stiffness values. The computations have been carried out by taking only the first two terms of the doubly infinite series, namely, m = 1, n = 1; m = 3, n = 1 for three different cases: k = 0, k/k,., = 0.4 and k/k,., = 0.8. Similar results are also computed with the Vianello formula (42). These are then plotted as shown in figures 3-5. While the dotted lines indicate Vianello approximations, the firm lines represent the results obtained by taking the first two terms of the series. Figure 3 shows that when r/ = 1"0 the Vianello formula gives good results for k = 0, k/k,,, = 0.4 and k/km, = 0"8 even when the core is orthotropic. However, though only one value of ~b, i.e. q5 = 0-2, has been considered, it is expected that similar good approximations will be obtained for other values of 4'. Figure 4 gives such deflections for plates having q = 2-0 and ~b = 0.2. It is seen that when k/k,,, = 0.4, i.e. the edge load is 40 ~o of the critical load, theVianello formula gives good approximations. But when k/k,,. = 0.8, there is considerable deviation between the corresponding firm and dotted curves.
On the Flexural Behaviour of Orthotropic Sandwich Plates 2"01(0)
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Fig. 5. [(a) (d)] De[tection o f isotropic sandwich plates (v = 0'25, l/ = 3'0) along longitudinal centre line (y = b;2). Viane/lo appro.v. , Two terms o f series ~- - . Tabh, 1. Deflection w o f uniJbrndy loaded sandwich plates (0 = 2.0; ~b = 0-2) along longitudinal centre line (y - b.C). Isotropic fitces and orthotropie core. Values o f (zr6/16) (Dw/qoaZb -') ~x a degree
c = 0"4 Calculation 1*
~: = 1.0
Calculation 2+
Calculation 1
~: = 2'5
C a l c ul a t i on 2 k/k ....
0 20 40 60 80 90
0 0"112 0"188 0"219 0"224 0'224
0 0'105 0'182 0"219 0"231 0'232
0 0'107 0'185 0'224 0"236 0"240
~:
5.(1
C a l c ul a t i on I
Calculation 2
C a l c ul a t i on 1
Calculation 2
0 0"103 0"181 0'224 0-240 0"241
0 O'lO0 0"178 0'224 0-241 0"243
0 0"101 0"179 0"223 0-240 0'242
0 0"100 0.178 0-223 0"242 0"244
0 0-168 0"282 0"330 0"336 0"336
0 0"148 0"262 0-330 0"356 0"359
0 0"162 0"278 0"336 0"352 0'354
0 0.150 0.267 0.376 0.364 0.367
0.4
0 0'103 0"181 0"224 0'240 0'244
k/k,,,, = 0-8 0 20 40 60 80 90
0 0"178 0"273 0"276 0'245 0.239
0 0"132 0-228 0"276 0"291 0"291
0 0.182 0.292 0.318 0.305 0.304
0 0-146 0.256 0"318 0.341 0.347
* Calculation 1 refers to those c o m p u t e d from the e qua t i ons (37) and (41) t a k i n g only the first two terms of the doubly infinite series. + Calcu lation 2 is based on the Vianello f o r m u l a (42).
On the Flexural Behaviour o f Orthotropic Sandwich Plates
135
Table 2. Deflection w of uniformly loaded isotropic sandwich plates ( t / = 3"0) along longitudinal centre line (y = b/2). Values of 0r6/16)(Dw/qoa2b 2) ~rx a
degree
~b = 0 Calculation 1*
~b = 0'2
Calculation 2#
Calculation 1
~b = 0'5
Calculation 2
~b = 1'0
Calculation 1
Calculation 2
Calculation 1
Calculation 2
0 0.0780 0.1250 0.1349 0.1283 0-1270
0 0.0710 0.1180 0.1349 0.1357 0'1350
0 0.1050 0'1670 0.1782 0.1683 0-1660
0 0.0967 0.1574 0.1782 0.1770 0.1760
0 0'1150 0.1678 0.1515 0.1170 0'1110
0 0.0799 0.1325 0.1515 0.1525 0"1519
0 0.1368 0-2018 0'1942 0.1610 0.1548
0 0.1053 0.1715 0.1942 0'1925 0.1914
k/km, = 0'4 0 20 40 60 80 90
0 0.0494 0-0809 0.0910 0.0901 0.0896
0 0.0453 0'0769 0.0910 0.0941 0.0942
0 0'0614 0.992 0.1087 0.1052 0.1042
0 0.0554 0'0935 0.1087 0'1108 0.1108 k/k.,. = 0"8
0 20 40 60 80 90
0 0.0832 0.1250 0.1095 0.0845 0.0802
0 0.0544 0.0925 0.1095 0.1131 0.1132
0 0.1017 0'1456 0.1263 0.0920 0.0862
0 0.0645 0.1087 0.1263 0.1277 0.1277
* Calculation 1 refers to those computed from the equations (37) and (41) taking only the first two terms of the doubly infinite series. t Calculation 2 is based on Vianello approximations.
T h e deviations tend to increase as e decreases. Thus the central deflection calculated b y the Vianello f o r m u l a is only 3.7 % higher at e = 5.0, while that at e = 0.4 is a b o u t 22 % higher (Table 1). Figure 5 gives results for isotropic sandwich plates, for ~/ = 3.0 a n d for different values o f ~b. Here the deviation is m u c h m o r e p r e d o m i n a n t , particularly when k/k,,, = 0.8. The d e v i a t i o n in the central deflection varies f r o m a b o u t 41 to 23.8 %, when ~b varies from 0.0 to 1-0 respectively (Table 2). Hence, when the compressive edge l o a d is m o r e t h a n 40 % o f the buckling load, the Vianello f o r m u l a is always seen to predict results being widely deviated f r o m those o b t a i n e d f r o m the equations (37) a n d (41), a n d the m a x i m u m deflection is always unconservative. M o r e o v e r , while the results from the Vianello f o r m u l a suggests that the m a x i m u m deflection should occur at the centre o f the plate, the actual m a x i m u m deflection is f o u n d to be somewhere a l o n g the y = b/2 line other t h a n the centre. It is expected t h a t the m o m e n t s a n d shears being higher derivatives o f deflections, errors involved in such c o m p u t a t i o n s will be even m o r e p r o m i n e n t .
is because o f the fact that in this analysis m u c h greater flexibility in selection o f materials a n d thicknesses for the facings has been ensured. This permits a wider choice o f materials a n d p r o p o r t i o n s in regard to e c o n o m i c design o f sandwich plates, particularly e m p l o y i n g a d v a n c e d c o m p o s i t e materials for facings. F u r t h e r , it is observed that, in some o f the cases, because o f the simplified expression and easy accessibility to desk c o m p u t a t i o n , the Vianeilo f o r m u l a (42) can be suggested as a possible substitution for the expression (41), which involves lengthier c o m p u t a t i o n s t h r o u g h the series summation, w i t h o u t any serious loss o f accuracy. Thus, it is n o t e d that the Vianello f o r m u l a (42) gives g o o d a p p r o x i m a t i o n s o f the actual b e h a v i o u r o f o r t h o tropic sandwich plates u n d e r c o m b i n e d b e n d i n g with u n i f o r m l y distributed transverse l o a d a n d uniaxial c o m p r e s s i o n so long as the aspect ratio q is unity o r less, and, for all practical purposes, r e a s o n a b l y g o o d results can also be expected for other cases, when r/ = 2 or even 3, a n d the c o m pressive edge l o a d is limited to 4 0 % o f the critical load.
6. C O N C L U S I O N S It is f o u n d t h a t the theoretical d e v e l o p m e n t carried o u t in the present s t u d y is m o r e general t h a n those d o n e b y the previous investigators. This
Acknowledgements--The authors would like to express their appreciation to the Research and Development Organisation, Ministry of Defence, New Delhi for the financial support to carry out research on sandwich structures.
136
D. P. Ray amt P. K. Shtha
REFERENCES I. E. REISSNER,On bending of elastic plates, Quarterly ~fApplied Mathematics 5, I, April (1947). 2. C.T. WANG, Principle and application of complementary energy method for thin homogeneous and sandwich plates and shells with finite deflections, NACA TN 2620, 1952. 3. S. CHENG, On the theory of bending of sandwich plates, Proc. Fourth U.S. Nationa/ Congress of Applied Mechanics, Vol. I (1962). 4. S. CHENGand N. AL-RUBAYI,Elastic stability of orthotropic sandwich plates, Developments in Mechanics, Vol. 5, Proc. of the llth Midwestern Mechanics Conf, Sept. (1969). 5. C. E. S. UENG and Y. J. LIN, On bending of orthotropic sandwich plates, AIAA Journal 4, Dec. 1966. 6. C. E. S. UENG, Some remarks on the deflection of orthotropic sandwich plates, J. Composite Matls. 2, 1968. 7. C. LIBOVEand S. B. BATDORE,A general small deflection theory for flat sandwich plates, NACA Report No. 899, 1948. 8. F.J. PLANTEMA,Sandwich Construction--The bending and buckling of sandwich beams, plates, and shells, Vol. l, p. 132. Wiley, New York (1966). 9. H.G. ALLEN, Analysis and Design of Structural Sandwich Panels, Vol. l, p. 87. Pergamon Press, 1969. 10. C. C. CHANG and 1. K. EBCIOGLU,Elastic instability of rectangular sandwich panel of orthotropic core with different face thickness and materials, AFOSR TN 58-221, ASTIA AD 154-122, University of Minnesota, March 1958. 11. C.C. CHANG, B. T. FANG and I. K. EBCIOGLU,Elastic theory of a weak-core sandwich panel initially warped, simply supported and subjected to combined loadings, Proc. Third U.S. National Congr. Appl. Mech. (1958). 12. C. C, CHANGand B. T. FANG, Initially warped sandwich panel under combined loadings, J. Aero Space Science 27, 10, Oct. (1960). 13. A. C. ERINGEN, Bending and buckling of rectangular sandwich plates, Proc. First U.S. National Congr. Appl. Mech. (1951). 14. C. B. NORRIS, Compressive buckling design curves for sandwich panels with isotropic facings and orthotropic cores, FPL Report No. 1854 (1956). (Revised Jan. 1958.) 15. C. B. NORRIS, Compressive buckling curves for simply supported sandwich panels with glass-fabric-laminate facings and honeycomb core, FPL Report No. 1867 (1958). 16. C. B. NORRIS, Compressive buckling curves for flat sandwich panels with dissimilar facings, FPL Report No. 1875, 1960. 17. W. S. EPaCKSENand H. W. MARCH, Compressive buckling of sandwich panels having dissimilar facings of unequal thickness, FPL Report No. 1583-B (1950). (Revised Nov. 1958.) 18. W. S. ERICKSEN,Effects of shear deformation in the core of a flat rectangular sandwich panel-deflection under uniform load of sandwich panels having facings of unequal thickness, FPL Report No. 1583-C (1950). 19. W. S. ERICKSEN,Supplement to: Effects of shear deformation in the core of a flat rectangular sandwich panel-deflection under uniform load of sandwich panels having facings of moderate thickness, FPL Report No. 1583-D (1951). 20. H. W. MARCH, Effects of shear deformation in the core of a flat rectangular sandwich panel, FPL Report No. 1583 (1948). Dans cet ouvrage, on consid6re les plaques sandwich orthotropiques ayant des faces et un noyau orthotropiques, des 6paisseurs diff6rentes de faces et des diff6rences de mat6riau de faces (y compris des rapports de Poisson diff6rents des surfaces). Les relations contrainte-d6formation et les 6quations diff~rentielles sont d6riv~es pour le fl6chissement de telles plaques en appliquant le principe variationel d'6nergie compl6mentaire. Diverses formules et quelques r6sultats num~riques sont obtenus pour le fl6chissement d'une plaque simplement support6e sous l'action d'une compression uniaxiale et d'une charge transversale uniform6ment distribu6e. On 6tudie 6galement la validit6 de l'approximation de Vianello et les r6sultats sont discut6s. In diesem Bericht werden orthotrope Verbundplatten mit orthotropen Aussenseiten und Kernen, ungleichen Dicken der Aussenseite und Un~hnlichkeiten in Aussenseitenmaterialien (einschliesslich verschiedenen Poisson Verh~iltnissen fiir Verkleidungen) behandelt. Die Spannungs-Dehnungsbeziehungen und Differentialgleichungen fiir Biegung solcher Platten werden durch Anwendung des Variationsprinzips der Komplement/~renergie abgeleitet. Es werden verschiedene Formeln und ein paar zahlenm/issige Ergebnisse ffir Biegung einer einfach gestiJtzten Platte unter Aussetzung einachsigen Drucks und gleichf~rmig verteilter Querbelastung erhalten. Auch wird die Giiltigkeit der Vianello Ann/iherung untersucht und die Ergebnisse werden besprochen.