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Static and transient response of rectangular plates
Thin-Walled Structures 5 (1987) 125-143
Static and Transient Response of Rectangular Plates
A. Bhimaraddi* Department of Civil Engineering, University of Melbourne, Parkville, Victoria 3052, Australia (Received 20 April 1985; revised version accepted 27 August 1986)
ABSTRACT Static and transient analysis of composite plates is presented using a recently proposed shear deformation theory. The dynamic response is obtained by employing the numerical time integration scheme due to Newmark. The results obtained by using the classical plate theory (CPT) and the Mindlintype shear deformation theory (SDT) are compared with those obtained by using the proposed theory. The comparison studies reveal that the linear stress distribution, as assumed in CPT and SDT, differs considerably from the predicted nonlinear distribution of the proposed theory.
NOTATION
a,b E~, Ey, Ez G~y, Gxz, G yz h hl, h2, h3 m,n Nx, Ny, Nxy ] Mx, My, Mxy ], M', My, M', J
Length and width of the plate. Localized load dimensions (see Fig. 1). Young's moduli. Shear moduli. Total thickness of the plate. Thicknesses of the individual layers. N u m b e r of half waves in the x- and y-directions. Stress-resultants,
* Present address: Department of Civil Engineering, University of Canterbury, Christchurch l, New Zealand. 125
Thin-Walled Structures 0263/8231/87/$03.50 O Elsevier Applied Science Publishers Ltd, England, 1987. Printed in Great Britain
A. Bhimaraddi
126
q:o, q zi t tp U, V, W UO~ l-gh V()~ VI~ W 0
x,y,z
Applied loads on the outer and the inner surface of the plate in the z-direction. Time coordinate. Fundamental natural period (= 2rr/l~). Displacement components in the x-, y- and z-directions. Generalized displacement quantities. Cartesian coordinate system. Relative thickness ratios in laminated plates (fl, = h 2 / h b r 3 = h2/h3, fl = f l l = f13).
Shear strains in cartesian coordinates. Time step-size used in numerical time integration. Normal strains in the x-, y- and z-directions. Poisson's ratio. Angular frequency. Mass density of the material of the plate. Mass densities of individual layers. Normal stresses in the x-, y- and z-directions. Shear stresses in cartesian coordinates.
Txy, 3/ xz, "f yz
At •x, Ey, E z
J[~xy 1/ P Pl, P2, P3 O-x, O'y, dr z
Txy~ J'xz~ "l'yz
(T, AT, Tp) =
(t, At, tp)
Ex
(a, v, ~ ) = - - (u, v, w) aqo 1
( ~ x , ~ y , Txz, r yz, Txy) = - - (O'x, O'y, Txz, ryz, Txy) qo
l INTRODUCTION Laminates consisting of multiple layers of fiber-reinforced composite materials have come into extensive use in technology. Thus, there is an increasing need for accurate mathematical modelling of the behaviour of composite structures. It is possible to treat structural elements constructed of laminates by applying three-dimensional elasto-dynamics to each individual layer in conjunction with appropriate continuity and boundary conditions. However, the resulting analysis is usually so complicated that it is not practical to carry out. To avoid this complexity, it is the usual engineering practice to propose certain hypotheses regarding the kinematics of deformation and then to work with the stiffnesses which are integrated
Static and transient response of rectangular plates
127
through the thickness. For example, in classical plate theory (CPT)1-5 it is assumed that normals to the midplane before deformation remain straight and normal to the midplane after deformation; the implication is that the thickness shear deformations are negligible. It has long been known that due to their relatively low transverse shear modulus, plates made of filamentary composite materials exhibit much larger thickness shear effects than those made of homogeneous isotropic materials. Hence, a mathematical model to analyse such structures, in order to be closer to reality, must incorporate the effects of transverse shear deformation. There exist many plate theories that account for transverse shear strains. 6-~3Mindlin-type shear deformation theories (SDT) are based on the assumption that the normals to the midplane before deformation remain straight but no longer remain normal to the midplane after deformation; the implication is that the midplane displacement components are assumed to be linear functions of the normal coordinate (z) and hence warping of cross-sections is not allowed. Although the above mentioned shear deformation theories yield acceptable solutions, they do not adequately predict response characteristics of plates in cases such as: when accurate higher order frequencies are required; when there exists an appreciable thermal gradient through the thickness. Also, questions relating to interlaminar stresses, edge effects, and delamination in composites can be addressed only when the higher order theories which account for such effects as: transverse shear deformation, transverse normal strain, and warping of cross-sections are employed. Usually, higher order theories are based on the assumed form for displacement components--taken as power series expressions in the normal coordinate (z). Various such formulations can be found in refs 14-23. It may be said here that most of these higher order theories employ too many u n k n o w n generalized displacements in order to achieve parabolic variation for transverse shear strains. In addition, most of these theories 14-~8do not satisfy shear-free conditions on the top and bottom surfaces of the plate. Hence, to account for this deficiency, shear correction factors have been introduced 14.~5in a manner similar to the Mindlin-type theories. 7-. Further discussion on the applicability and the accuracy of such formulations can be found in ref. 16. The higher order plate theories in which in-plane displacement components are assumed as cubic polynomials in z, which results in parabolic variation for transverse shear strain distribution, can be found in refs 24 and 25. These theories employ fewer unknown generalized displacements (equal to those used in the Mindlin-type theories) and also they satisfy shear-free surface conditions. However, the displacement forms assumed in ref. 25 are different from those assumed in ref. 24. Also, the equilibrium
128
A. Bhirnaraddt
equations presented in ref. 24 are variationally inconsistent with the assumed displacement forms. Recently, variationally consistent equilibrium equations have been derived, using Hamilton's principle, by Reddy 26'27for the displacement forms proposed by Levinson. 24 Although the proposed theories in refs 24-27 satisfy shear-free conditions and employ the same number of unknown generalized displacements to obtain cubic variation for in-plane displacement components, it is to be noted that CPT equations can be obtained as a special case from ref. 25 but not from refs 24, 26 and 27. Also, almost any function ~:(z) (see eqn (2) below), whose first derivative vanishes at +_h/2 and is non-zero elsewhere, can be used in ref. 25 resulting in a corresponding higher order theory. Such various alternative forms for sO(z) and the procedure for systematic reduction of equilibrium equations for plates and shells in terms of stressresultants using the three-dimensional stress equilibrium equations have been given in ref. 28. The superiority of the proposed plate theory (PPT) has been demonstrated in ref. 25, by comparing the results obtained by PPT, SDT and CPT with those obtained by using the three-dimensional elasticity 29-3~theory, in the context of free vibration analysis. In this paper further comparison of PPT, SDT and CPT has been dealt with in the context of static and forced vibration analysis of composite plates. A solution to forced vibration response has been obtained by numerical integration of the resulting system of dynamic equilibrium equations using Newmark's method. 32,33
2 G O V E R N I N G E Q U A T I O N S OF MOTION The components of displacements are assumed as follows (refer to the Notation). u ( x , y , z , t) = uo(x, y , t ) + ( u ~ ( x , y , t ) - zwo..~ v(x,y,z,t)
= v o ( x , y , t ) + ~ v l ( x , y , t ) - zWo,y
w(x,y,z,t)
= Wo(x,y,t)
(I)
where sc = z
4Z2~
1 - -fh-~]
(2)
In these equations, a comma indicates differentiation with respect to the letter(s) followed by it. It can be easily verified, by writing out the expressions for strains z5'28'34using the above displacement components, that the transverse shear strains, and hence stresses, vanish on the top and bottom surfaces of the plate and that they vary parabolically across the thickness of
Static and transient response of rectangular plates
129
the plate. Hence, there arises no necessity to introduce shear correction factors, as is usually done in the Mindlin-type shear deformation theories. It may be observed that the form of in-plane displacement components in eqn (1) is not the only choice. There could be other possible alternatives to choose for ~:(z) to accomplish the above mentioned purpose. The other alternative forms for ~:(z) and a discussion of the accuracy of the associated equations may be found in refs 28 and 35. Finally, if one wants to have a greater number of terms in the w-expression, corresponding terms should be added to the u- and v-expressions to have the shear-free boundary conditions satisfied. This will, of course, result in a more accurate but more complicated theory involving the consideration of thickness normal strain effects, and a greater number of u n k n o w n generalized displacements. Hence, to limit the complexity of the problem to a reasonable degree, the w-expression has been chosen to be a constant across the thickness of plate. The constitutive relations corresponding to a monoclinic type of material (or for any layer in the case of a laminate) are of the form, Ory = C12E x-I- C22Ey q- C26~/xy
Orx = CllEx-{- C12Ey + Cl6"Yxy
(3) rxy :
C16E x d- C26Ey "~- C66)l xy
7XZ = 644~x z -~ C45~/y z
Tyz = Ca5~/xz -~- C55,)/yz
We have the following definitions for stress-resultants appropriate to the present theory 25'28
[NxNyNx] s['] Mx My Mxy M" My M'y
=
-Z
(O'x,Ory,rxy)dZ
(4)
and the equilibrium equations in terms of the stress-resultants are written as, 28 Nx, x+Nxy, y = - f
pu..dz
Nxy'x+Ny'Y = - I
Pv'ndz
M',x+ M'xy,y - ! I
r~z~,dz = - f
M'y,.+ M~,y- f
"/'yz~:ldZ
=
-
M .... +2Mxy,.y+My, yy= q+ f
p~u,,dz
f p,v,ttdz p(zu,..+ZV, y.+W,~)dz
(5)
A. Bhimaraddi
130 where
4z: (6)
~1 = ( , z = 1 - h2
and q = qzo - qz~. Here, qzo and q:~ are the applied loads in the z-direction on the top and bottom surfaces of the plate respectively. Substitution of stressresultants, expressed in terms of the generalized displacements of the problem, in eqns (5) results in the following set of dynamic equilibrium equations, L8 = p
(7)
where
6 T = {Uo, Vo, UbVbWo}
p T = { 0 , 0 , 0 , 0 , q}
and the operators L ijare symmetric and have the following forms: B "LI1 ~
~"A
i11
m
-A~-
A 10 Bl6
L13
A 66
L22
A 22
A 26
( ),~+2
_
( ),x,,+
=
L24
B66
B26
B22
L33
D~I
D16
D66
L44_
-- D66_
__ D 2 6 _
_ 022_
~
n
m
m
£
0
N
0
£
+
0
(),.-
()
N
0
R -
_/~55_
(),.,
Static and transient response of rectangular plates -L
12-
FA 16-
i
=
B66+ B,2 ( ),xy +
L23
BI6
B66+ B12
-L34 -
-O16-
-/~66 + D12-
-L25]
=
-r
0
B26
()x~+
ILl51:_ IL35]
0
326-
-A66 + A12-
B16
L 14
131
()
( ),yy-
0
B26
- ~z~45_
- 026 -
2B66+B12 /)11] ( ) .... 3
016
D26I ( )'~YYY-n16]
( ),xxy--
( )' xt,
_ [ 2B66+ n12 () ....
t522
)~xyy
2/966+ O12 ]
1
B26 ( ) , xxy--3
J
( )' Y ' -
]
( ) , xxy
[-026
( )'''
L55 = D l l ( ) ..... +4D16( ),x~xy+2(2D66+D12)( ),xxyy+4D26( ),xyyy
+D2E( ),yyyy+P( ),xx.+P(
),yy.-£(
),.
where ( ) .... etc., refer to derivatives of the column matrix of generalized displacements shown in eqns (7) and,
B,j
Bo
Bo
Di/
Do
Di/
[~ M RN] Q
=
=
Co
z z~
~
dz
the
(8)
z 2 z~ ~2
f
O
[lz~:] z 2 z~ ~2
dz
(9)
N o t e that A,j, . . . , R are to be computed by evaluating the indicated integration over the thickness of plate between the limits -h/2 to +hi2. The
132
A, Bhimaraddi
b o u n d a r y conditions along the edge of the plate require that either one m e m b e r of each of the following six pairs, or six linearly independent combinations of them, must be specified. 2~ Along the x = constant: Nxuo ; N~yvo ; M ' u l
; M'xyVl
; Qxwo ; M~wo,~
Along the y = constant: N~yuo ; Nyvo ; MiryUl ; M'vvl ; Qywo ; Mvwo, y
where Qx = M~, ~ + 2M~y, ~+ Muo, ,, + Q u i, , , - Pwo, xtt
(lo)
Qy = 2M~y, x+ My, v+ Mvo, t,+ Q v l , . - Pwo, y.
The governing equations of the classical plate theory can be obtained by neglecting the third and fourth rows and columns, which correspond to u~ and v ~displacement parameters, from the matrices of eqn (7). Similarly all the other relations such as, stress-resultants, boundary conditions, etc., can be obtained by ignoring the terms containing u~ and v,.
3 S O L U T I O N OF T H E E Q U I L I B R I U M E Q U A T I O N S The closed-form solutions to plate dynamic equilibrium equations, (7), with generally orthotropic (monoclinic) material are difficult to obtain. However, specially orthotropic, simply supported plates render possible closed-form solutions in the form of a double Fourier series. Since the solution is in series form and each term in it is explicitly defined by a set of simultaneous equations it is easy to sum the series to any desired degree of accuracy. The following simply supported boundary conditions are assumed Nx = V o = M ' =
(alongx = 0,a)
Vl= Wo= Mx=O
u0 = Ny = ul = My = w0 = My = 0
(alongy = 0,b)
and correspondingly, the solutions for eqns (7), with C~6 = C26 = C~5 -- 0, are assumed as
Uo(x,y,t) = ~ ~ ~0m.(t)a'lm.(x,y); v0(x,y,t)= ~ Y~Vo,.n(t)a'2r..(x,y) n
m
n
m
(11)
Static and transient response of rectangularplates u~(x,y,t) = Y" ~ U~,~.(t)~lm.(x,y) ; v~(x,y,t)= n
Wo(x,y,t) =
m
~
~ n
133
~ V~,..(t)cb2.,n(x,y) m
~ Wo,,.(t)~3,.,(x,y)
n
rtl
where
do1.,. = cos(mrrx/a)sin(nrry/b) ; ~2.,. = sin(m~rx/a)cos(nrry/b) ~3,.. = sin (mTrx/a) sin (n~ry/b)
(12)
T h e spatial variation of the loading, to be consistent with the solution of the p r o b l e m , has been assumed in the form of a double Fourier series as,
q(x,y,t) = ~
~ qm.(t)sin(m~rx/a)sin(mry/b)
n
(13)
rtz
w h e r e the terms q.,.(t) in the above series representation of the load may be d e t e r m i n e d as 1
q~.(t)
=
"~
o
o
q (x, y, t) sin (mlrx/a) sin (ncry/b) dx dy
(14)
For a uniform load of intensity qo(t), q.,.(t) becomes
qm.(t) = 4qo( t) t l - cosrmr) (1 - cosmr)
(15)
mnT.r 2
For a uniform load of intensity qo(t) over the rectangular area (~ × b) with the centre at (x0, y0), as shown in Fig. 1, q,..(t) is q m.(t) = ~ s i n rnnrr
(m¢rxo/a) sin (nrryo/b) sin (rmr'd/2a) sin (nrr-b/2b)
(16)
For a sinusoidally distributed load q(x, y, t) = qosin(rrx/a) sin(try/b), q m.(t) is given as q u ( t ) = qo(t)
m = n = 1
qll(t) = 0
m-2
(17)
and
n>-2
O t h e r loading distributions can be analysed by inserting the appropriate relations into eqn (14). Substituting eqns (11) and (13) in eqns (7), the following system of second order ordinary differential equations, variable in time, for each set of (m, n) values, can be obtained, Mi+K~
= q
(18)
134
A. Bhimaraddi
a I--
F133
I
~-- xo-- ~ X
L ir~
U
T
Fig. 1. Description of patch loading.
where K and M are the symmetric stiffness and mass matrices of the order 5 x 5 and we have the following definitions for ~, A and q,
= {t#,,m.. v0.,o, t31=°, v,=.. m :
{Uomn, Vomn, Ulmn , Vlmn" Womn!}T
q = {O,O,O,O, qm,(t)} T
Note that the superposed dots indicate differentiation with respect to time. For a given set of (rn, n) values the A vector can be obtained at various desired values of time by solving eqns (18) using time integration scheme due to Newmark. 32.33By summing the series over all (m, n) values up to the desired degree of accuracy, displacement and stress distribution in space and time can be obtained for a specified variation of the external loading in space and time. However, for a static problem the A vector is computed by solving the set of algebraic equations which are obtained from eqns (18) by ignoring the time dependence of displacements and loads. Finally, it should be noted here that the transverse shear stress distribution across the thickness, in the case of SDT and CPT, has been obtained by first computing the values of shear forces and then distributing them according to the parabolic law. However, the actual shear stress distribution is constant according to SDT and is zero according to CPT.
Static and transient response of rectangular plates
135
4 DISCUSSION OF NUMERICAL RESULTS In all of the numerical results presented herein, zero initial conditions for displacements and velocities have been assumed. All of the computations have been carried out in double precision on a VAX/VMS computer. The numerical results for transient response have been obtained by using a time step-size (AT) equal to 1% of the predominant natural period of the system. The accuracy and the correctness of the algebra and the computer coding developed have been ensured by comparing the results with those available in the literature. 12 The SDT results in the present study have been obtained by using a shear factor value of zr2/12. There are different shear factor values available in the literature for homogeneous and laminated plates which are often obtained for a particular problem under consideration. Hence, it becomes extremely difficult to choose these values for the problems dealt with in this paper. Usually, a value of rr2/12 is used in the absence of a rational explanation as to the use of other values than ~.2/12.12,13Further discussion on the use of shear factor values may be found in ref. 35. Detailed studies on the static and dynamic response of plates with various geometric and material parameters have been undertaken. 35However, only a few results have been given here for the sake of brevity. Unless otherwise stated, the spatial location of the points to which the values of various displacements and stress correspond to is as given below,
(~, O'x,o-y) at
x y z 1 a b h 2
(~, V,¥xy) at
x y . . a b
.
4 'h
1
2
(¥x~,¥yz) at
x y . . a b
1 z . . 4 'h
0
.
z
1
The influence of material orthotropy has been considered using E,,/Ey values of 1, 25 and 40 while keeping Gxy = Gxz = Gy~ = ½Ey, and ~xy = ¼, p l = p2 = p3 = p. Figures 2-5 show comparisons of through-the-thickness variations of stresses and displacements obtained using various plate theories. The loading considered corresponds to the centrally applied static patch load (eqn 16) on the top surface of the plate. The different geometric and material parameters considered have been indicated in the diagrams. Figure 2 shows the comparison of results with the elasticity solution 29for
A. Bhimaraddi
136
0 ,25 -q.
PP'[
/ ~ SDT. CPT
I
h ._ 0 1 Q"- " '
"~]
O
N
0 -1 -5- .~>
-o26
i
"~\
~ " /
- 0.50
I
I
I
-1.2
I
-0,8
I
I
-0.4
J 1.2
i
0
0.4
0.8
O'x= CTy 0 .50~.~T, 0.25
CPT
PPT / ~
.
Elasticity
~
L=
.
~- = 1__
o
b
h
8
= 0.1
a =1 • -5
o
-0.25 - 0,50
I
I
-24
I
I
-1.6
I
I
-O.8
0.8
O
1.6
2.z,
0-~:% 0 50
~-~
PPT
/Elasticity•
a
" ~ . . ~ P T SDT 0 .25
6
1
k=&1 o
£=i b
o
-0 25 - 0 5O
I
0.036
I
I
0024
I
I
O.O12
I
I
0
t
-O. O12
I
I l"""~J
-0024
-0.036
Fig. 2. Comparison of through-the-thickness variation of stresses and displacements obtained using various plate theories for a square isotropic plat_e (/~ = 0.25) subjected to a centrally applied static patch load (5 × b).
an isotropic square plate. It is evident that as the patch load dimensions decrease the nature of the stress distribution across the thickness of the plate becomes more and more nonlinear. The stress distribution predicted by PPT is closer to the elasticity solution than CPT and SDT, in terms of both the prediction of the nonlinear nature and the magnitude. Though the differences in CPT and SDT are not seen on the present scale, it has been observed that CPT predicts higher values (i.e. closer to the elasticity solution) than SDT. The in-plane displacements predicted by all three theories are in very good agreement with the elasticity results. The elasticity results for composite plates are not available in the literature and hence further comparison of various plate theories has been
Static and transient response of rectangular plates __
0.50
137
CPT
I
E 5
a
I
b
I
0.25
-0.25 -0.5C
~SOT, CPT ppT- - j I
I
I -
I
I ~I"~IPPTI 0 I0
I
-10
20
I 20
I
I
I
I
I
-8
I
-4
I
0
i I
I 4
I
0.50 0,25
.Actual CPT
.Actu(11 CPT r" ,, 50T~
JE N
"
,,
_
0/90
CPT
SOT "~
Q/b
0
-0.25
Pl(1te = 1
h2/hl = 1
~ P T
E = --6 = ~-
~
- 0.50
I
-0.1
-0.2 "Cxz =
hle " 6 - -6- ~-
SOT PPT I
I
I
-0.2
-0.3
-0.6
= 0.1
E x / E y = 25 , b (is shown
I
-I.0 X 10 -~
Z x z = '~yz
",','Cy z
Fig. 3. Comparison of through-the-thickness variation of ~'x and ¥= obtained using various plate theories for two-layered (0/90) square plates subjected to a centrally applied static patch load.
0.25
~N
CPT
"
h = 0.15 (1
h = 0.05 (:1
~
o -0.25 -0.50
i
I
I
-20
I
I
-10
0
10
20
I
-4
I
0
-2
2
4
0.50 0/90/0
SOT 0.25 ~Actu(11 CPT r" .. SOT _
{
IActuol CPT Actu(1I S O T
-0.25 -0.50 - I
Cp••PT SOT
oh--= 0.15
h =0.05 ,
, -0.1
I
I
-0.2
"~XZ
-0.02
¢
h2 _ h2 _ "6"~1- - ~ 3 - p =1 Ex/Ey
i
I
Pt(1t
o/b = 1
-0.06 -0.04 rE"XZ
i
i
= 25
~- = = h/(1 (is shown
-0.08
Fig. 4. Comparison of ~ and ¥= obtained using various plate theories for three-layered (0/90/0) square plates, with different thickness values, subjected to a centrally applied static patch load.
138
A. Bhimaraddi
0.50 Ex//Ey : SPT, CPT / PPT
0
N
Ex/Ey= ~0
PPT C p T ~ T ~ SDT ~.'~. "
10
0.25
-0.25 -
0.50
I
_,~
I
I
-2
I
0
2
4
8
-~
0 50 0•90•0 Plate
SOT
0.25 .Actuot cPr
/.Actual
~~T
.. SDT
-0.25
-0.50
~Xy=10
~,,./SOT
CPT SOT
~'~k,~ p PT ~CPT
ta/b =
I
h2_ h2 6 = ~-fi = -~1 ~-
E._Ex= 40
h/a = 01
Ex/Ey -0.04
-0108
Z'×z
i
i
-0.04
as shown
-0.08 "~xz
Fig. 5. Comparison of ~'x and -~xz obtained using various plate theories for three-layered (0/90/0) square plates, with different Ex/Ey values subjected to a centrally applied static patch load.
made taking PPT as the standard solution, due to its superior performance in the case of homogeneous (isotropic and orthotropic) plates when compared with elasticity results. 28,35Further, the agreement between PPT and SDT, in the prediction of ~, has been observed to be excellent (SDT overestimates by 2 %), whereas, CPT underestimates the same by more than - 5 0 % in some thick and highly orthotropic plates considered. Also, since there was a better agreement in the prediction of in-plane displacements than in the prediction of stresses by various plate theories, the comparison studies have been made only for stresses (Figs 3-5). It may be observed from these figures that for small patch load sizes, for higher thickness values, and for higher EJ Ey ratios the nature of the normal stress distribution becomes more and more nonlinear. Hence, there is a considerable difference in the linear stress distributions predicted by CPT and SDT. In all of the cases considered CPT and SDT underestimate the normal stress values. The smooth variation of the shear stress distribution, in the present case of laminated plates, can be attributed to the fact that the shear moduli values (Gxz = Gyz = ½Ey; Ey corresponding to the 0 ° layer) are the same for all the layers. Figures 6--8 show the variation of normal stress with time for a suddenly applied load, constant in time and varying sinusoidaUy in space (eqn 17).
Static and transient response of rectangularplates
-20
q
- 50
o
Orthotrcl:)ic
-IE
.//
\~
//
-12
,t;
\k
~
q [
\ o[
.o~og=~o~=./ Plate
~=o.,s
139
'I / \ \ . / ' - - x \
-40
AT=0.2 Isotropi¢ Plate i o/b:, ./
i
/ /
\
-20
-8
/I
Y -4
~
T
6
\\\AT:0.2 /
t'li; \ \~
-30
!
2
L
z
\..
/--PPT
i
\X~!
-10
4
8
B
12
T
16
CD)
(o)
Fig. 6. Variation of the normal stress (~x) as a function of time (T), based on various plate theories, for homogeneous square plates under sinusoidally distributed loading.
%
-25o
sot /
-,~o
___
,,\o/~o ~,o,=
'b~
o/b : -I00
.... .....
PPT SDT CPT
qo h :0.10 Q
- -
-6C
,/i . . . . ,
\\
o/~o ~,o,,
/ -4C
-50
-
I
I
10
I
I
20
I
T (a)
I
30
I
i
40
20
6
12
18
24
T (b)
Fig. 7. Variation of the normal stress (~-x) as a function of time (7"), based on various plate theories, for two-layered (0/90) square plates under sinusoidally distributed loading (h/ a = 0.05 and0.1).
A. Bhimaraddi
140 - 12'0
/,,
_,oo
-80
//
'\ 't
il
AT:0,2
-,o
i
't 't o1 o/o
-60
- 2o
6
12
18
///
'~
24
20
'~\
"~
///
1
PpT
4
8
T (a)
/
'\
\
// ....
~,=02
~/o/9o/o P,o,,/
~
T (b)
12
\
/
16
20
Fig. 8. Variation of the normal stress (~x) as a function of time (T), based on various plate theories, for three-layered (0/90/0) square plates under sinusoidallydistributed loading.
Such plots for other stresses and displacements have not been given since the variation was similar. The selection of this limited type of loading has been m a d e to limit the consideration of an excessively large number of terms in the series while adequately demonstrating the solution procedure. It may be observed from these figures that as the thickness of the plate increases the differences in SDT and CPT, when compared with PPT, increase. The slightly undulating nature of the curves for 0/90 (Fig. 7(b)) is attributed to the contribution of the in-plane and transverse shear modes to a greater extent than in 0/90/0 or homogeneous plates. Figure 6 reveals that for isotropic plates all the three theories agree fairly well (even for a -- 0.15); whereas, for an orthotropic plate of the same thickness, there is a considerable difference in the prediction of dynamic response by different theoriesl The maximum difference in SDT, even for such a smoothly varying load in space, is about - 1 0 % (0/90/0 plate with = 0.15), and that in CPT is about - 3 5 % .
h~
h/a
5 CONCLUSIONS In conclusion, we note that the stress and displacement distribution across the thickness of the plate is nonlinear. This nonlinearity increases with the increase in thickness, with the increase in the material orthotropy, and with
Static and transientresponseof rectangularplates
141
the decrease in the localized load dimensions. The assumption of linear distribution of stresses and displacements by the classical plate theory and the Mindlin-type theories differs considerably from the nonlinear distribution predicted by the proposed plate theory. More refined higher order theories could be developed in which the effects of normal strains can be taken into account, besides satisfying the shear-free surface conditions, using the following series expressions for assumed displacement forms U=
N M zm+l E ~nun-- E Wm, x n=0 m=0m+l
V
N ~ zm+l E ~nVn-Wm'y n=0 m=0m+l M
W:
E Zo Wm m=0
N = M = 0 corresponds to the classical plate theory and N = 1, M = 0 corresponds to the present plate theory.
ACKNOWLEDGEMENTS Financial assistance, in the form of a writing award, was provided by the Faculty of Engineering, University of Melbourne, Australia. In addition the author thanks Professor L. K. Stevens and the referee for their comments and suggestions.
REFERENCES 1. Timoshenko, S. and Woisnowsky-Krieger, S., Theory of Plates and Shells, 2nd edn, McGraw-Hill Book Company, New York, 1959. 2. Reissner, E. and Stavsky, Y., Bending and stretching of certain types of heterogeneous aeolotropic elastic plates, Trans. ASME, J. Appl. Mech., 28 (1961) 402--8. 3. Dong, S. B., Pister, K. S. and Taylor, R. L., On the theory of laminated anisotropic shells and plates, J. Aerospace Sci., 29 (1962) 969-75. 4. Whitney, J. M. and Leissa, A. W., Analysis of heterogeneous anisotropic plates, Trans. ASME, J. Appl. Mech., 36 (1969) 261--6. 5. Ashton, J. E. and Whitney, J. M., Theory of Laminated Plates, Technomic Publishing, Stamford, Conn., 1970. 6. Reissner, E., The effect of transverse shear deformation on the bending of elastic plates, Trans. ASME, J. Appl. Mech., 12 (1945) A-69.
142
A. Bhimaraddi
7. Mindlin, R. D., Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates, Trans. ASME, J. Appl. Mech., 18 (1951) 31-8. 8. Huang, T. C., Application of variational methods to the vibration of plates including rotary inertia and shear, Proc. of the 7th Midwestern Mech. Conf., Held at Michigan State Univ., Sept. 6-8, 1961, Developments in Mechanics, Vol. 1, Edited byJ. E. Lay and L. E. Malvern, pp. 61-72. 9. Panc, V., Theories of" Elastic Plates, Noordhoff International Publishing, Leyden, 1975. 10. Yang, P. C., Norris, C. H. and Stavsky, Y., Elastic wave propagation in heterogeneous plates, Int. J. Solids and Structures, 2 (1966) 1542-50. 11. Whitney, J. M. and Pagano, N. J., Shear deformation in heterogeneous anisotropic plates, Trans. ASME, J. Appl. Mech., 37 (1970) 1031--6. 12. Reddy, J. N., On the solution to forced motions of rectangular composite plates, Trans. ASME, J. Appl. Mech., 49 (1982) 403-8. 13. Reddy, J. N., Dynamic (transient) analysis of layered anisotropic composite material plates, Int. J. Numer. Methods in Engng, 19 (1983) 237-55. 14. Whitney, J. M. and Sun, C. T., A higher-order theory for extensional motions of laminated composites, J. Sound and Vib., 30 (1973) 85-97. 15. Nelson, R. B. and Lorch, D. R., A refined theory for laminated orthotropic plates, Trans. ASME, J. Appl. Mech., 41 (1974) 177-83. 16. Lo, K. H., Christensen, R. M. and Wu, E. M., A high-order theory of plate deformation, Pt-l: homogeneous plates, Trans. ASME, J. Appl. Mech., 44 (1977) 663-8. 17. Lo, K. H., Christensen, R. M. and Wu, E. M., A high-order theory of plate deformation, Pt-2: laminated plates, Trans. ASME, J. Appl. Mech., 44 (1977) 669-76. 18. Kant, T., Numerical analysis of thick plates, Computer Methods in Appl. Mech. Engng, 31 (1982) 1-18. 19. Reissner, E., On transverse bending of plates, including the effects of transverse shear deformation, Int. J. Solids and Structures, 11 (1975) 569-73. 20. Reissner, E., A twelfth order theory of transverse bending of transversely isotropic plates, ZAMM, 63 (1983) 285-9. 21. Srinivas, S., A refined analysis of thick anisotropic laminates, J. Sound and Vib., 30 (1973) 495-507. 22. Pagano, N. J., Stress fields in composite laminates, Int. J. Solids and Structures, 14 (1978) 385-400. 23. Pagano, N. J. and Soni, S. R., Global-local laminate variational model, Int. J. Solids and Structures, 19 (1983) 207-28. 24. Levinson, M., An accurate, simple theory of the statics and dynamics of elastic plates, Mech. Res. Cornmun., 7 (1980) 343--50. 25. Bhimaraddi, A. and Stevens, L. K., A higher order theory for free vibration of orthotropic, homogeneous, and laminated rectangular plates, Trans. ASME, J. Appl. Mech., 51 (1984) 195-8. 26. Reddy, J. N., A refined nonlinear theory of plates with transverse shear deformation, Int. J. Solids and Structures, 20 (1984) 881-96. 27. Reddy, J. N., A simple higher-order theory for laminated composite plates, Trans. ASME, J. Appl. Mech., 51 (1984) 745-52. 28. Bhimaraddi, A. and Stevens, L. K., On the higher order theories in plates and shells, lnt. J. Struct., 6 (1986) 35-50.
Static and transient response of rectangularplates
143
29. Srinivas, S., Rao, A. K. and Joga Rao, C. V., Flexure of simply supported thick homogeneous and laminated rectangular plates, ZA MM, 49 (1969) 44958. 30. Srinivas, S. and Rao, A. K., Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates, Int. J. Solids and Structures, 6 (1970) 1463-81. 31. Srinivas, S., Joga Rao, C. V. and Rao, A. K., An exact analysis for vibration of simply supported homogeneous and laminated rectangular plates, J. Sound and Vib., 12 (1970) 187-99. 32. Newmark, N. M., A method of computation for structural dynamics, Proc. ASCE, J. Engng Mech. Div., 85 (1959) 67-94. 33. Bathe, K.-J. and Wilson, E. L., Numerical Methods in Finite Element Analysis, Prentice-Hall of India Pty Ltd, New Delhi, 1978. 34. Timoshenko, S. and Goodier, J. N., Theory of Elasticity, 2nd edn, McGrawHill, New York, 1951. 35. Bhimaraddi, A., Static and dynamic response of plates and shells, PhD Thesis, Department of Civil Engineering, University of Melbourne, February 1985.
Static and Transient Response of Rectangular Plates
A. Bhimaraddi* Department of Civil Engineering, University of Melbourne, Parkville, Victoria 3052, Australia (Received 20 April 1985; revised version accepted 27 August 1986)
ABSTRACT Static and transient analysis of composite plates is presented using a recently proposed shear deformation theory. The dynamic response is obtained by employing the numerical time integration scheme due to Newmark. The results obtained by using the classical plate theory (CPT) and the Mindlintype shear deformation theory (SDT) are compared with those obtained by using the proposed theory. The comparison studies reveal that the linear stress distribution, as assumed in CPT and SDT, differs considerably from the predicted nonlinear distribution of the proposed theory.
NOTATION
a,b E~, Ey, Ez G~y, Gxz, G yz h hl, h2, h3 m,n Nx, Ny, Nxy ] Mx, My, Mxy ], M', My, M', J
Length and width of the plate. Localized load dimensions (see Fig. 1). Young's moduli. Shear moduli. Total thickness of the plate. Thicknesses of the individual layers. N u m b e r of half waves in the x- and y-directions. Stress-resultants,
* Present address: Department of Civil Engineering, University of Canterbury, Christchurch l, New Zealand. 125
Thin-Walled Structures 0263/8231/87/$03.50 O Elsevier Applied Science Publishers Ltd, England, 1987. Printed in Great Britain
A. Bhimaraddi
126
q:o, q zi t tp U, V, W UO~ l-gh V()~ VI~ W 0
x,y,z
Applied loads on the outer and the inner surface of the plate in the z-direction. Time coordinate. Fundamental natural period (= 2rr/l~). Displacement components in the x-, y- and z-directions. Generalized displacement quantities. Cartesian coordinate system. Relative thickness ratios in laminated plates (fl, = h 2 / h b r 3 = h2/h3, fl = f l l = f13).
Shear strains in cartesian coordinates. Time step-size used in numerical time integration. Normal strains in the x-, y- and z-directions. Poisson's ratio. Angular frequency. Mass density of the material of the plate. Mass densities of individual layers. Normal stresses in the x-, y- and z-directions. Shear stresses in cartesian coordinates.
Txy, 3/ xz, "f yz
At •x, Ey, E z
J[~xy 1/ P Pl, P2, P3 O-x, O'y, dr z
Txy~ J'xz~ "l'yz
(T, AT, Tp) =
(t, At, tp)
Ex
(a, v, ~ ) = - - (u, v, w) aqo 1
( ~ x , ~ y , Txz, r yz, Txy) = - - (O'x, O'y, Txz, ryz, Txy) qo
l INTRODUCTION Laminates consisting of multiple layers of fiber-reinforced composite materials have come into extensive use in technology. Thus, there is an increasing need for accurate mathematical modelling of the behaviour of composite structures. It is possible to treat structural elements constructed of laminates by applying three-dimensional elasto-dynamics to each individual layer in conjunction with appropriate continuity and boundary conditions. However, the resulting analysis is usually so complicated that it is not practical to carry out. To avoid this complexity, it is the usual engineering practice to propose certain hypotheses regarding the kinematics of deformation and then to work with the stiffnesses which are integrated
Static and transient response of rectangular plates
127
through the thickness. For example, in classical plate theory (CPT)1-5 it is assumed that normals to the midplane before deformation remain straight and normal to the midplane after deformation; the implication is that the thickness shear deformations are negligible. It has long been known that due to their relatively low transverse shear modulus, plates made of filamentary composite materials exhibit much larger thickness shear effects than those made of homogeneous isotropic materials. Hence, a mathematical model to analyse such structures, in order to be closer to reality, must incorporate the effects of transverse shear deformation. There exist many plate theories that account for transverse shear strains. 6-~3Mindlin-type shear deformation theories (SDT) are based on the assumption that the normals to the midplane before deformation remain straight but no longer remain normal to the midplane after deformation; the implication is that the midplane displacement components are assumed to be linear functions of the normal coordinate (z) and hence warping of cross-sections is not allowed. Although the above mentioned shear deformation theories yield acceptable solutions, they do not adequately predict response characteristics of plates in cases such as: when accurate higher order frequencies are required; when there exists an appreciable thermal gradient through the thickness. Also, questions relating to interlaminar stresses, edge effects, and delamination in composites can be addressed only when the higher order theories which account for such effects as: transverse shear deformation, transverse normal strain, and warping of cross-sections are employed. Usually, higher order theories are based on the assumed form for displacement components--taken as power series expressions in the normal coordinate (z). Various such formulations can be found in refs 14-23. It may be said here that most of these higher order theories employ too many u n k n o w n generalized displacements in order to achieve parabolic variation for transverse shear strains. In addition, most of these theories 14-~8do not satisfy shear-free conditions on the top and bottom surfaces of the plate. Hence, to account for this deficiency, shear correction factors have been introduced 14.~5in a manner similar to the Mindlin-type theories. 7-. Further discussion on the applicability and the accuracy of such formulations can be found in ref. 16. The higher order plate theories in which in-plane displacement components are assumed as cubic polynomials in z, which results in parabolic variation for transverse shear strain distribution, can be found in refs 24 and 25. These theories employ fewer unknown generalized displacements (equal to those used in the Mindlin-type theories) and also they satisfy shear-free surface conditions. However, the displacement forms assumed in ref. 25 are different from those assumed in ref. 24. Also, the equilibrium
128
A. Bhirnaraddt
equations presented in ref. 24 are variationally inconsistent with the assumed displacement forms. Recently, variationally consistent equilibrium equations have been derived, using Hamilton's principle, by Reddy 26'27for the displacement forms proposed by Levinson. 24 Although the proposed theories in refs 24-27 satisfy shear-free conditions and employ the same number of unknown generalized displacements to obtain cubic variation for in-plane displacement components, it is to be noted that CPT equations can be obtained as a special case from ref. 25 but not from refs 24, 26 and 27. Also, almost any function ~:(z) (see eqn (2) below), whose first derivative vanishes at +_h/2 and is non-zero elsewhere, can be used in ref. 25 resulting in a corresponding higher order theory. Such various alternative forms for sO(z) and the procedure for systematic reduction of equilibrium equations for plates and shells in terms of stressresultants using the three-dimensional stress equilibrium equations have been given in ref. 28. The superiority of the proposed plate theory (PPT) has been demonstrated in ref. 25, by comparing the results obtained by PPT, SDT and CPT with those obtained by using the three-dimensional elasticity 29-3~theory, in the context of free vibration analysis. In this paper further comparison of PPT, SDT and CPT has been dealt with in the context of static and forced vibration analysis of composite plates. A solution to forced vibration response has been obtained by numerical integration of the resulting system of dynamic equilibrium equations using Newmark's method. 32,33
2 G O V E R N I N G E Q U A T I O N S OF MOTION The components of displacements are assumed as follows (refer to the Notation). u ( x , y , z , t) = uo(x, y , t ) + ( u ~ ( x , y , t ) - zwo..~ v(x,y,z,t)
= v o ( x , y , t ) + ~ v l ( x , y , t ) - zWo,y
w(x,y,z,t)
= Wo(x,y,t)
(I)
where sc = z
4Z2~
1 - -fh-~]
(2)
In these equations, a comma indicates differentiation with respect to the letter(s) followed by it. It can be easily verified, by writing out the expressions for strains z5'28'34using the above displacement components, that the transverse shear strains, and hence stresses, vanish on the top and bottom surfaces of the plate and that they vary parabolically across the thickness of
Static and transient response of rectangular plates
129
the plate. Hence, there arises no necessity to introduce shear correction factors, as is usually done in the Mindlin-type shear deformation theories. It may be observed that the form of in-plane displacement components in eqn (1) is not the only choice. There could be other possible alternatives to choose for ~:(z) to accomplish the above mentioned purpose. The other alternative forms for ~:(z) and a discussion of the accuracy of the associated equations may be found in refs 28 and 35. Finally, if one wants to have a greater number of terms in the w-expression, corresponding terms should be added to the u- and v-expressions to have the shear-free boundary conditions satisfied. This will, of course, result in a more accurate but more complicated theory involving the consideration of thickness normal strain effects, and a greater number of u n k n o w n generalized displacements. Hence, to limit the complexity of the problem to a reasonable degree, the w-expression has been chosen to be a constant across the thickness of plate. The constitutive relations corresponding to a monoclinic type of material (or for any layer in the case of a laminate) are of the form, Ory = C12E x-I- C22Ey q- C26~/xy
Orx = CllEx-{- C12Ey + Cl6"Yxy
(3) rxy :
C16E x d- C26Ey "~- C66)l xy
7XZ = 644~x z -~ C45~/y z
Tyz = Ca5~/xz -~- C55,)/yz
We have the following definitions for stress-resultants appropriate to the present theory 25'28
[NxNyNx] s['] Mx My Mxy M" My M'y
=
-Z
(O'x,Ory,rxy)dZ
(4)
and the equilibrium equations in terms of the stress-resultants are written as, 28 Nx, x+Nxy, y = - f
pu..dz
Nxy'x+Ny'Y = - I
Pv'ndz
M',x+ M'xy,y - ! I
r~z~,dz = - f
M'y,.+ M~,y- f
"/'yz~:ldZ
=
-
M .... +2Mxy,.y+My, yy= q+ f
p~u,,dz
f p,v,ttdz p(zu,..+ZV, y.+W,~)dz
(5)
A. Bhimaraddi
130 where
4z: (6)
~1 = ( , z = 1 - h2
and q = qzo - qz~. Here, qzo and q:~ are the applied loads in the z-direction on the top and bottom surfaces of the plate respectively. Substitution of stressresultants, expressed in terms of the generalized displacements of the problem, in eqns (5) results in the following set of dynamic equilibrium equations, L8 = p
(7)
where
6 T = {Uo, Vo, UbVbWo}
p T = { 0 , 0 , 0 , 0 , q}
and the operators L ijare symmetric and have the following forms: B "LI1 ~
~"A
i11
m
-A~-
A 10 Bl6
L13
A 66
L22
A 22
A 26
( ),~+2
_
( ),x,,+
=
L24
B66
B26
B22
L33
D~I
D16
D66
L44_
-- D66_
__ D 2 6 _
_ 022_
~
n
m
m
£
0
N
0
£
+
0
(),.-
()
N
0
R -
_/~55_
(),.,
Static and transient response of rectangular plates -L
12-
FA 16-
i
=
B66+ B,2 ( ),xy +
L23
BI6
B66+ B12
-L34 -
-O16-
-/~66 + D12-
-L25]
=
-r
0
B26
()x~+
ILl51:_ IL35]
0
326-
-A66 + A12-
B16
L 14
131
()
( ),yy-
0
B26
- ~z~45_
- 026 -
2B66+B12 /)11] ( ) .... 3
016
D26I ( )'~YYY-n16]
( ),xxy--
( )' xt,
_ [ 2B66+ n12 () ....
t522
)~xyy
2/966+ O12 ]
1
B26 ( ) , xxy--3
J
( )' Y ' -
]
( ) , xxy
[-026
( )'''
L55 = D l l ( ) ..... +4D16( ),x~xy+2(2D66+D12)( ),xxyy+4D26( ),xyyy
+D2E( ),yyyy+P( ),xx.+P(
),yy.-£(
),.
where ( ) .... etc., refer to derivatives of the column matrix of generalized displacements shown in eqns (7) and,
B,j
Bo
Bo
Di/
Do
Di/
[~ M RN] Q
=
=
Co
z z~
~
dz
the
(8)
z 2 z~ ~2
f
O
[lz~:] z 2 z~ ~2
dz
(9)
N o t e that A,j, . . . , R are to be computed by evaluating the indicated integration over the thickness of plate between the limits -h/2 to +hi2. The
132
A, Bhimaraddi
b o u n d a r y conditions along the edge of the plate require that either one m e m b e r of each of the following six pairs, or six linearly independent combinations of them, must be specified. 2~ Along the x = constant: Nxuo ; N~yvo ; M ' u l
; M'xyVl
; Qxwo ; M~wo,~
Along the y = constant: N~yuo ; Nyvo ; MiryUl ; M'vvl ; Qywo ; Mvwo, y
where Qx = M~, ~ + 2M~y, ~+ Muo, ,, + Q u i, , , - Pwo, xtt
(lo)
Qy = 2M~y, x+ My, v+ Mvo, t,+ Q v l , . - Pwo, y.
The governing equations of the classical plate theory can be obtained by neglecting the third and fourth rows and columns, which correspond to u~ and v ~displacement parameters, from the matrices of eqn (7). Similarly all the other relations such as, stress-resultants, boundary conditions, etc., can be obtained by ignoring the terms containing u~ and v,.
3 S O L U T I O N OF T H E E Q U I L I B R I U M E Q U A T I O N S The closed-form solutions to plate dynamic equilibrium equations, (7), with generally orthotropic (monoclinic) material are difficult to obtain. However, specially orthotropic, simply supported plates render possible closed-form solutions in the form of a double Fourier series. Since the solution is in series form and each term in it is explicitly defined by a set of simultaneous equations it is easy to sum the series to any desired degree of accuracy. The following simply supported boundary conditions are assumed Nx = V o = M ' =
(alongx = 0,a)
Vl= Wo= Mx=O
u0 = Ny = ul = My = w0 = My = 0
(alongy = 0,b)
and correspondingly, the solutions for eqns (7), with C~6 = C26 = C~5 -- 0, are assumed as
Uo(x,y,t) = ~ ~ ~0m.(t)a'lm.(x,y); v0(x,y,t)= ~ Y~Vo,.n(t)a'2r..(x,y) n
m
n
m
(11)
Static and transient response of rectangularplates u~(x,y,t) = Y" ~ U~,~.(t)~lm.(x,y) ; v~(x,y,t)= n
Wo(x,y,t) =
m
~
~ n
133
~ V~,..(t)cb2.,n(x,y) m
~ Wo,,.(t)~3,.,(x,y)
n
rtl
where
do1.,. = cos(mrrx/a)sin(nrry/b) ; ~2.,. = sin(m~rx/a)cos(nrry/b) ~3,.. = sin (mTrx/a) sin (n~ry/b)
(12)
T h e spatial variation of the loading, to be consistent with the solution of the p r o b l e m , has been assumed in the form of a double Fourier series as,
q(x,y,t) = ~
~ qm.(t)sin(m~rx/a)sin(mry/b)
n
(13)
rtz
w h e r e the terms q.,.(t) in the above series representation of the load may be d e t e r m i n e d as 1
q~.(t)
=
"~
o
o
q (x, y, t) sin (mlrx/a) sin (ncry/b) dx dy
(14)
For a uniform load of intensity qo(t), q.,.(t) becomes
qm.(t) = 4qo( t) t l - cosrmr) (1 - cosmr)
(15)
mnT.r 2
For a uniform load of intensity qo(t) over the rectangular area (~ × b) with the centre at (x0, y0), as shown in Fig. 1, q,..(t) is q m.(t) = ~ s i n rnnrr
(m¢rxo/a) sin (nrryo/b) sin (rmr'd/2a) sin (nrr-b/2b)
(16)
For a sinusoidally distributed load q(x, y, t) = qosin(rrx/a) sin(try/b), q m.(t) is given as q u ( t ) = qo(t)
m = n = 1
qll(t) = 0
m-2
(17)
and
n>-2
O t h e r loading distributions can be analysed by inserting the appropriate relations into eqn (14). Substituting eqns (11) and (13) in eqns (7), the following system of second order ordinary differential equations, variable in time, for each set of (m, n) values, can be obtained, Mi+K~
= q
(18)
134
A. Bhimaraddi
a I--
F133
I
~-- xo-- ~ X
L ir~
U
T
Fig. 1. Description of patch loading.
where K and M are the symmetric stiffness and mass matrices of the order 5 x 5 and we have the following definitions for ~, A and q,
= {t#,,m.. v0.,o, t31=°, v,=.. m :
{Uomn, Vomn, Ulmn , Vlmn" Womn!}T
q = {O,O,O,O, qm,(t)} T
Note that the superposed dots indicate differentiation with respect to time. For a given set of (rn, n) values the A vector can be obtained at various desired values of time by solving eqns (18) using time integration scheme due to Newmark. 32.33By summing the series over all (m, n) values up to the desired degree of accuracy, displacement and stress distribution in space and time can be obtained for a specified variation of the external loading in space and time. However, for a static problem the A vector is computed by solving the set of algebraic equations which are obtained from eqns (18) by ignoring the time dependence of displacements and loads. Finally, it should be noted here that the transverse shear stress distribution across the thickness, in the case of SDT and CPT, has been obtained by first computing the values of shear forces and then distributing them according to the parabolic law. However, the actual shear stress distribution is constant according to SDT and is zero according to CPT.
Static and transient response of rectangular plates
135
4 DISCUSSION OF NUMERICAL RESULTS In all of the numerical results presented herein, zero initial conditions for displacements and velocities have been assumed. All of the computations have been carried out in double precision on a VAX/VMS computer. The numerical results for transient response have been obtained by using a time step-size (AT) equal to 1% of the predominant natural period of the system. The accuracy and the correctness of the algebra and the computer coding developed have been ensured by comparing the results with those available in the literature. 12 The SDT results in the present study have been obtained by using a shear factor value of zr2/12. There are different shear factor values available in the literature for homogeneous and laminated plates which are often obtained for a particular problem under consideration. Hence, it becomes extremely difficult to choose these values for the problems dealt with in this paper. Usually, a value of rr2/12 is used in the absence of a rational explanation as to the use of other values than ~.2/12.12,13Further discussion on the use of shear factor values may be found in ref. 35. Detailed studies on the static and dynamic response of plates with various geometric and material parameters have been undertaken. 35However, only a few results have been given here for the sake of brevity. Unless otherwise stated, the spatial location of the points to which the values of various displacements and stress correspond to is as given below,
(~, O'x,o-y) at
x y z 1 a b h 2
(~, V,¥xy) at
x y . . a b
.
4 'h
1
2
(¥x~,¥yz) at
x y . . a b
1 z . . 4 'h
0
.
z
1
The influence of material orthotropy has been considered using E,,/Ey values of 1, 25 and 40 while keeping Gxy = Gxz = Gy~ = ½Ey, and ~xy = ¼, p l = p2 = p3 = p. Figures 2-5 show comparisons of through-the-thickness variations of stresses and displacements obtained using various plate theories. The loading considered corresponds to the centrally applied static patch load (eqn 16) on the top surface of the plate. The different geometric and material parameters considered have been indicated in the diagrams. Figure 2 shows the comparison of results with the elasticity solution 29for
A. Bhimaraddi
136
0 ,25 -q.
PP'[
/ ~ SDT. CPT
I
h ._ 0 1 Q"- " '
"~]
O
N
0 -1 -5- .~>
-o26
i
"~\
~ " /
- 0.50
I
I
I
-1.2
I
-0,8
I
I
-0.4
J 1.2
i
0
0.4
0.8
O'x= CTy 0 .50~.~T, 0.25
CPT
PPT / ~
.
Elasticity
~
L=
.
~- = 1__
o
b
h
8
= 0.1
a =1 • -5
o
-0.25 - 0,50
I
I
-24
I
I
-1.6
I
I
-O.8
0.8
O
1.6
2.z,
0-~:% 0 50
~-~
PPT
/Elasticity•
a
" ~ . . ~ P T SDT 0 .25
6
1
k=&1 o
£=i b
o
-0 25 - 0 5O
I
0.036
I
I
0024
I
I
O.O12
I
I
0
t
-O. O12
I
I l"""~J
-0024
-0.036
Fig. 2. Comparison of through-the-thickness variation of stresses and displacements obtained using various plate theories for a square isotropic plat_e (/~ = 0.25) subjected to a centrally applied static patch load (5 × b).
an isotropic square plate. It is evident that as the patch load dimensions decrease the nature of the stress distribution across the thickness of the plate becomes more and more nonlinear. The stress distribution predicted by PPT is closer to the elasticity solution than CPT and SDT, in terms of both the prediction of the nonlinear nature and the magnitude. Though the differences in CPT and SDT are not seen on the present scale, it has been observed that CPT predicts higher values (i.e. closer to the elasticity solution) than SDT. The in-plane displacements predicted by all three theories are in very good agreement with the elasticity results. The elasticity results for composite plates are not available in the literature and hence further comparison of various plate theories has been
Static and transient response of rectangular plates __
0.50
137
CPT
I
E 5
a
I
b
I
0.25
-0.25 -0.5C
~SOT, CPT ppT- - j I
I
I -
I
I ~I"~IPPTI 0 I0
I
-10
20
I 20
I
I
I
I
I
-8
I
-4
I
0
i I
I 4
I
0.50 0,25
.Actual CPT
.Actu(11 CPT r" ,, 50T~
JE N
"
,,
_
0/90
CPT
SOT "~
Q/b
0
-0.25
Pl(1te = 1
h2/hl = 1
~ P T
E = --6 = ~-
~
- 0.50
I
-0.1
-0.2 "Cxz =
hle " 6 - -6- ~-
SOT PPT I
I
I
-0.2
-0.3
-0.6
= 0.1
E x / E y = 25 , b (is shown
I
-I.0 X 10 -~
Z x z = '~yz
",','Cy z
Fig. 3. Comparison of through-the-thickness variation of ~'x and ¥= obtained using various plate theories for two-layered (0/90) square plates subjected to a centrally applied static patch load.
0.25
~N
CPT
"
h = 0.15 (1
h = 0.05 (:1
~
o -0.25 -0.50
i
I
I
-20
I
I
-10
0
10
20
I
-4
I
0
-2
2
4
0.50 0/90/0
SOT 0.25 ~Actu(11 CPT r" .. SOT _
{
IActuol CPT Actu(1I S O T
-0.25 -0.50 - I
Cp••PT SOT
oh--= 0.15
h =0.05 ,
, -0.1
I
I
-0.2
"~XZ
-0.02
¢
h2 _ h2 _ "6"~1- - ~ 3 - p =1 Ex/Ey
i
I
Pt(1t
o/b = 1
-0.06 -0.04 rE"XZ
i
i
= 25
~- = = h/(1 (is shown
-0.08
Fig. 4. Comparison of ~ and ¥= obtained using various plate theories for three-layered (0/90/0) square plates, with different thickness values, subjected to a centrally applied static patch load.
138
A. Bhimaraddi
0.50 Ex//Ey : SPT, CPT / PPT
0
N
Ex/Ey= ~0
PPT C p T ~ T ~ SDT ~.'~. "
10
0.25
-0.25 -
0.50
I
_,~
I
I
-2
I
0
2
4
8
-~
0 50 0•90•0 Plate
SOT
0.25 .Actuot cPr
/.Actual
~~T
.. SDT
-0.25
-0.50
~Xy=10
~,,./SOT
CPT SOT
~'~k,~ p PT ~CPT
ta/b =
I
h2_ h2 6 = ~-fi = -~1 ~-
E._Ex= 40
h/a = 01
Ex/Ey -0.04
-0108
Z'×z
i
i
-0.04
as shown
-0.08 "~xz
Fig. 5. Comparison of ~'x and -~xz obtained using various plate theories for three-layered (0/90/0) square plates, with different Ex/Ey values subjected to a centrally applied static patch load.
made taking PPT as the standard solution, due to its superior performance in the case of homogeneous (isotropic and orthotropic) plates when compared with elasticity results. 28,35Further, the agreement between PPT and SDT, in the prediction of ~, has been observed to be excellent (SDT overestimates by 2 %), whereas, CPT underestimates the same by more than - 5 0 % in some thick and highly orthotropic plates considered. Also, since there was a better agreement in the prediction of in-plane displacements than in the prediction of stresses by various plate theories, the comparison studies have been made only for stresses (Figs 3-5). It may be observed from these figures that for small patch load sizes, for higher thickness values, and for higher EJ Ey ratios the nature of the normal stress distribution becomes more and more nonlinear. Hence, there is a considerable difference in the linear stress distributions predicted by CPT and SDT. In all of the cases considered CPT and SDT underestimate the normal stress values. The smooth variation of the shear stress distribution, in the present case of laminated plates, can be attributed to the fact that the shear moduli values (Gxz = Gyz = ½Ey; Ey corresponding to the 0 ° layer) are the same for all the layers. Figures 6--8 show the variation of normal stress with time for a suddenly applied load, constant in time and varying sinusoidaUy in space (eqn 17).
Static and transient response of rectangularplates
-20
q
- 50
o
Orthotrcl:)ic
-IE
.//
\~
//
-12
,t;
\k
~
q [
\ o[
.o~og=~o~=./ Plate
~=o.,s
139
'I / \ \ . / ' - - x \
-40
AT=0.2 Isotropi¢ Plate i o/b:, ./
i
/ /
\
-20
-8
/I
Y -4
~
T
6
\\\AT:0.2 /
t'li; \ \~
-30
!
2
L
z
\..
/--PPT
i
\X~!
-10
4
8
B
12
T
16
CD)
(o)
Fig. 6. Variation of the normal stress (~x) as a function of time (T), based on various plate theories, for homogeneous square plates under sinusoidally distributed loading.
%
-25o
sot /
-,~o
___
,,\o/~o ~,o,=
'b~
o/b : -I00
.... .....
PPT SDT CPT
qo h :0.10 Q
- -
-6C
,/i . . . . ,
\\
o/~o ~,o,,
/ -4C
-50
-
I
I
10
I
I
20
I
T (a)
I
30
I
i
40
20
6
12
18
24
T (b)
Fig. 7. Variation of the normal stress (~-x) as a function of time (7"), based on various plate theories, for two-layered (0/90) square plates under sinusoidally distributed loading (h/ a = 0.05 and0.1).
A. Bhimaraddi
140 - 12'0
/,,
_,oo
-80
//
'\ 't
il
AT:0,2
-,o
i
't 't o1 o/o
-60
- 2o
6
12
18
///
'~
24
20
'~\
"~
///
1
PpT
4
8
T (a)
/
'\
\
// ....
~,=02
~/o/9o/o P,o,,/
~
T (b)
12
\
/
16
20
Fig. 8. Variation of the normal stress (~x) as a function of time (T), based on various plate theories, for three-layered (0/90/0) square plates under sinusoidallydistributed loading.
Such plots for other stresses and displacements have not been given since the variation was similar. The selection of this limited type of loading has been m a d e to limit the consideration of an excessively large number of terms in the series while adequately demonstrating the solution procedure. It may be observed from these figures that as the thickness of the plate increases the differences in SDT and CPT, when compared with PPT, increase. The slightly undulating nature of the curves for 0/90 (Fig. 7(b)) is attributed to the contribution of the in-plane and transverse shear modes to a greater extent than in 0/90/0 or homogeneous plates. Figure 6 reveals that for isotropic plates all the three theories agree fairly well (even for a -- 0.15); whereas, for an orthotropic plate of the same thickness, there is a considerable difference in the prediction of dynamic response by different theoriesl The maximum difference in SDT, even for such a smoothly varying load in space, is about - 1 0 % (0/90/0 plate with = 0.15), and that in CPT is about - 3 5 % .
h~
h/a
5 CONCLUSIONS In conclusion, we note that the stress and displacement distribution across the thickness of the plate is nonlinear. This nonlinearity increases with the increase in thickness, with the increase in the material orthotropy, and with
Static and transientresponseof rectangularplates
141
the decrease in the localized load dimensions. The assumption of linear distribution of stresses and displacements by the classical plate theory and the Mindlin-type theories differs considerably from the nonlinear distribution predicted by the proposed plate theory. More refined higher order theories could be developed in which the effects of normal strains can be taken into account, besides satisfying the shear-free surface conditions, using the following series expressions for assumed displacement forms U=
N M zm+l E ~nun-- E Wm, x n=0 m=0m+l
V
N ~ zm+l E ~nVn-Wm'y n=0 m=0m+l M
W:
E Zo Wm m=0
N = M = 0 corresponds to the classical plate theory and N = 1, M = 0 corresponds to the present plate theory.
ACKNOWLEDGEMENTS Financial assistance, in the form of a writing award, was provided by the Faculty of Engineering, University of Melbourne, Australia. In addition the author thanks Professor L. K. Stevens and the referee for their comments and suggestions.
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142
A. Bhimaraddi
7. Mindlin, R. D., Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates, Trans. ASME, J. Appl. Mech., 18 (1951) 31-8. 8. Huang, T. C., Application of variational methods to the vibration of plates including rotary inertia and shear, Proc. of the 7th Midwestern Mech. Conf., Held at Michigan State Univ., Sept. 6-8, 1961, Developments in Mechanics, Vol. 1, Edited byJ. E. Lay and L. E. Malvern, pp. 61-72. 9. Panc, V., Theories of" Elastic Plates, Noordhoff International Publishing, Leyden, 1975. 10. Yang, P. C., Norris, C. H. and Stavsky, Y., Elastic wave propagation in heterogeneous plates, Int. J. Solids and Structures, 2 (1966) 1542-50. 11. Whitney, J. M. and Pagano, N. J., Shear deformation in heterogeneous anisotropic plates, Trans. ASME, J. Appl. Mech., 37 (1970) 1031--6. 12. Reddy, J. N., On the solution to forced motions of rectangular composite plates, Trans. ASME, J. Appl. Mech., 49 (1982) 403-8. 13. Reddy, J. N., Dynamic (transient) analysis of layered anisotropic composite material plates, Int. J. Numer. Methods in Engng, 19 (1983) 237-55. 14. Whitney, J. M. and Sun, C. T., A higher-order theory for extensional motions of laminated composites, J. Sound and Vib., 30 (1973) 85-97. 15. Nelson, R. B. and Lorch, D. R., A refined theory for laminated orthotropic plates, Trans. ASME, J. Appl. Mech., 41 (1974) 177-83. 16. Lo, K. H., Christensen, R. M. and Wu, E. M., A high-order theory of plate deformation, Pt-l: homogeneous plates, Trans. ASME, J. Appl. Mech., 44 (1977) 663-8. 17. Lo, K. H., Christensen, R. M. and Wu, E. M., A high-order theory of plate deformation, Pt-2: laminated plates, Trans. ASME, J. Appl. Mech., 44 (1977) 669-76. 18. Kant, T., Numerical analysis of thick plates, Computer Methods in Appl. Mech. Engng, 31 (1982) 1-18. 19. Reissner, E., On transverse bending of plates, including the effects of transverse shear deformation, Int. J. Solids and Structures, 11 (1975) 569-73. 20. Reissner, E., A twelfth order theory of transverse bending of transversely isotropic plates, ZAMM, 63 (1983) 285-9. 21. Srinivas, S., A refined analysis of thick anisotropic laminates, J. Sound and Vib., 30 (1973) 495-507. 22. Pagano, N. J., Stress fields in composite laminates, Int. J. Solids and Structures, 14 (1978) 385-400. 23. Pagano, N. J. and Soni, S. R., Global-local laminate variational model, Int. J. Solids and Structures, 19 (1983) 207-28. 24. Levinson, M., An accurate, simple theory of the statics and dynamics of elastic plates, Mech. Res. Cornmun., 7 (1980) 343--50. 25. Bhimaraddi, A. and Stevens, L. K., A higher order theory for free vibration of orthotropic, homogeneous, and laminated rectangular plates, Trans. ASME, J. Appl. Mech., 51 (1984) 195-8. 26. Reddy, J. N., A refined nonlinear theory of plates with transverse shear deformation, Int. J. Solids and Structures, 20 (1984) 881-96. 27. Reddy, J. N., A simple higher-order theory for laminated composite plates, Trans. ASME, J. Appl. Mech., 51 (1984) 745-52. 28. Bhimaraddi, A. and Stevens, L. K., On the higher order theories in plates and shells, lnt. J. Struct., 6 (1986) 35-50.
Static and transient response of rectangularplates
143
29. Srinivas, S., Rao, A. K. and Joga Rao, C. V., Flexure of simply supported thick homogeneous and laminated rectangular plates, ZA MM, 49 (1969) 44958. 30. Srinivas, S. and Rao, A. K., Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates, Int. J. Solids and Structures, 6 (1970) 1463-81. 31. Srinivas, S., Joga Rao, C. V. and Rao, A. K., An exact analysis for vibration of simply supported homogeneous and laminated rectangular plates, J. Sound and Vib., 12 (1970) 187-99. 32. Newmark, N. M., A method of computation for structural dynamics, Proc. ASCE, J. Engng Mech. Div., 85 (1959) 67-94. 33. Bathe, K.-J. and Wilson, E. L., Numerical Methods in Finite Element Analysis, Prentice-Hall of India Pty Ltd, New Delhi, 1978. 34. Timoshenko, S. and Goodier, J. N., Theory of Elasticity, 2nd edn, McGrawHill, New York, 1951. 35. Bhimaraddi, A., Static and dynamic response of plates and shells, PhD Thesis, Department of Civil Engineering, University of Melbourne, February 1985.