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Nonlinear axisymmetric transient analysis of orthotropic thin annular plates
Fibre Science and Technology 21 (1984) 23--40
Nonlinear Axisymmetric Transient Analysis of Orthotropic Thin Annular Plates P. C. D u m i r , M. L. G a n d h i a n d Y o g e n d r a N a t h Department of Applied Mechanics, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi--110016 (India)
SUMMARY This investigation deals with the geometrically nonlinear, axisymmetric, transient elastic response for stresses and deflection of cylindrically orthotropic thin annular plates subjected to uniformly distributed and ring loads. The dynamic analogues of yon Kdrmdn equations in terms of normal displacement w and stress function ~ have been employed. The displacement w and stress function ~bare expanded infinite power series. The orthogonal point collocation method in the space domain and the Newmark-fl scheme in the time domain have been used. Three types of uniformly distributed dynamic loadings, namely, step function, sinusoidal pulse and exponentially decaying load, and a case of step function ring load at the free edge have been considered. Detailed results are presented for a clamped and simply supported annular plate with a free hole, and an annulus supported at the inner boundary and free at the outer edge, for different values of orthotropic parameter and annular ratio. The effect of elastic rotational and inplane edge restraints has also been studied.
NOMENCLATURE Outer radius, inner radius and thickness of the plate. Coefficients of power series expansion of w and ~. al, bi Coefficients in quadratic extrapolation. A1, A2, A3 D Eoha/12(fl - vg). Elastic moduli; Poisson's ratios. E o, E,, vo, v, K~,K~' Rotational and inplane stiffness of support. 23 Fibre Science and Technology 0015-0568/84/$03.00© ElsevierAppliedSciencePublishers Ltd, England, 1984. Printed in Great Britain a,b,h
24
P. C. Dumir, M. L. Gandhi, Yogendra Nath
Kb, Ki
Az
K~ a/D, K* a/hEo, dimensionless stiffnesses. Radial and circumferential bending moments. Number and radii of collocation points. Inplane forces, transverse shear. Intensity of uniformly distributed load, total ring load at the edge. qa4/ E,h 4, paZ/ Erh4, non-dimensional loads. Radius, non-dimensional radius. Time, non-dimensional time. Transverse and inplane displacements; stress function. Non-dimensional displacements and stress function. Parameters of the Newmark-fl scheme. Eo/E,, orthotropic parameter. Mass density. Time step.
a*,
am', a b" Radial and circumferential stresses; membrane and
g,,Mo N, Pi N,,No; Q, q,P Q,P r, p l, "C W*, U*, I~¢* W, U, I]/
¢7r,0 ()',(') Subscript J Subscripts p, i Movable Immovable
b/(a - b).
bending stresses. (a/h)2(~,.~/E,), dimensionless membrane stresses. (a/h)Z[ab,i~(h/2)/E,], dimensionless bending stresses. Differentiation with respect to p and z. Step of marching. Predicted value, value at the ith collocation point. g i = 0 (2 i ~-- - 1).
Ki = ~ (2,. = 1).
INTRODUCTION Fibre reinforced composites are increasingly being introduced into a variety of structures in aeronautical, mechanical and marine engineering applications where high ratios of stiffness and strength to weight are of paramount importance. Annular thin plate structures of composite materials have wide applications in these areas. The nonlinear response of these structures is of interest in their economical design. The nonlinear frequency-amplitude relation of orthotropic annular plates has been obtained by Nowinski1 using the Galerkin technique. The nonlinear transient analysis of orthotropic annular plates has received little attention. Alwar and Reddy2 studied the transient analysis of orthotropic
Nonlinear transient analysis of orthotropic annulus
25
annular plates subjected to a uniformly distributed step load using the Chebyshev series in space and the Houbolt scheme in time. To the best of the authors' knowledge, there are no results available for other transient distributed and edge loads for annular plates with or without flexible supports. The object of the present investigation is to study the transient response of orthotropic annular plates under three types of uniformly distributed dynamic loadings, namely, step function, sinusoidal pulse and exponentially decaying load, and under one case of step function ring load at the free edge. Detailed results are presented for a clamped and simply supported annular plate with a free hole, and an annulus supported at the inner boundary and free at the outer edge, for different values of the orthotropic parameter fl and the annular ratio. The effect of elastic rotational and inplane edge restraint has also been studied. The nonlinear differential equations for a cylindrically orthotropic plate, in terms of normal displacement w and stress function ~, have been employed. The deflection w and stress function ¢, are expanded in a finite power series and the differential equations have been discretised spacewise by collocating at the zeros of a Legendre polynomial. The Newmark-fl scheme is used for time-marching. An iterative scheme has been used to solve the nonlinear equations by predicting one of the variables in the nonfinear terms as a mean of its values at the two previous iterations. The predictions at the first iteration are taken as the quadratically extrapolated values from the preceding three steps. Time and space-wise convergence studies have revealed that five collocation points and A¢ = 0.004 yield accurate results for plates supported at the outer edge, whereas five collocation points and A¢ = 0.008 yield accurate results for plates clamped at the inner edge. An independent check on the accuracy of results has been incorporated in the form of a balance of energy. The sum of the kinetic energy and the strain energy of the plate at each time step is calculated and compared with the sum of the initial kinetic energy and the external work done up to that instant. This energy balance has been found to have been maintained very well at every time step. MATHEMATICAL FORMULATION The governing differential equations for the moderately large axisymmetric static deflection of a cylindrically orthotropic thin circular
26
P. C. Dumir, M. L. Gandhi, Yogendra Nath Kb
P
]
I" b4,----- ° -----,~
Ki
P
Kb
f,o~
I }..,. t
L o~1--
Ki
CASE I
CASE II
PLATE WITH FLEXIBLE SUPPORTS
PLATE CLAMPED AT INNER EDGE
1
IMMOVABLE
STEP
CLAMPED
IMMOVABLE SIMPLY SUPPORTED
EXPERIMENTAL
SINUSOIDAL
STEP
UNIFORMLY DISTRIBUTED LOADS
EDGE LOAD
Fig. 1. Edge conditions and dynamic loadings.
plate with the orthotropy axis coinciding with the axis of the plate have been given by Nowinski. 3 Neglecting damping and inplane inertia and including the transverse inertia term, these equations are modified as" D
rw*
+w*
--w*
-w,*d/ =
(1)
(q-Thw*,)rdr-Qr(r°)
1 fl hEo ~b*, + - r ~k,*----r2 ~k*+
,,2
(',,,,j
(2)
=0
where N, -
~'* r
N o = ~,*
Q, =
Eo h3
D = 12(fl-- v2) U~ ~O = - -
r
m*
G0 =----
Eo
E0
E,
~r = U *
~}/-h/2 v,
fl = ~
V0
if, =
m* O"r
Vr - -
C'/h/2 tr*=d z
'"
"~
lt,
,~2
2~"v'r"
m*
O'rm*
--
Er
V8
Gr0
-~0
(3)
Nonlinear transient analysis of orthotropic annulus
27
Introduce the following dimensionless parameters: w
.
w* . h
.
( a - b) ~b* ~b D qa4 .
b a-b
~-
r- b p=-a-b F D ]1/2
pa2
Q=e,h' e=E,h"
d
0
t
(4)
where a and b are the outer and inner radius of the plate and h is its thickness. The uniformly distributed load is q, andp is the ring load at the edge. The geometry, loading and edge conditions are depicted in Fig. 1. The governing equations reduce to the following dimensionless form: (p + 0 2w'' + (p + ~)w" - #w' - (p + OOw'
F6(fl_-_ v2) 1 = (P + ~J) L rcfl ~(I + zj) e -t (1+~),, 1 ~ { 1 2 ( ~ - v 2 ) Q - ~ t_ (P + ' ) d p ] =(P+~)F
(Case I)
6(fl--vg) 1 nfl (1 + ~)~ P
-~(1+~)4
8
Q - ~ (p + O d p
(CaseII)
(p + ~)20" + (p + 0 0 ' - 8 0 + 6(8 - vo~)(p + O(w') 2 = 0
(5) (6)
For an elastically restrained outer edge with rotational and inplane stiffnesses Kb* and K*: M,(1) = K~'w*(1) N,(1) = -/C~ u*(1) (7) Introduce dimensionless stiffnesses, K b and Ki: 1 + 2b
1 -~- ~i
a
Kb = 1 - 2-----~= K* ~
K~
I - 2i
K*a hE o
(8)
For the two cases shown in Fig. 1, the boundary conditions take the following dimensionless form: Case I p = 0:
v8
w"(0) + ~- w'(0) = 0
~k(0) = 0
p = 1: w(1) = 0 [(1 + 2b) + v0(l - 2b)]W'(1) + (1 -- 2b)(1 + ¢)W'(1) = 0 [(1 - 2 i ) - Vo(1 + ;q)]~O(1) + (1 + 2i)(1 + ¢)~k'(1) = 0
(9)
28
P. C. Dumir, M. L. Gandhi, Yogendra Nath
Case II p=O:
w(0) = 0
p=l:
w'(0) = 0 vo
w"(1) + 1 - - ~ w'(1) = 0
q/(o) - vd/(o)/¢ = o
~O(1) = 0
(I0)
The initial conditions are assumed as: w(p, O) = ~:(p, O) = O, and differential eqn (5) yields the initial accelerations: #(p, O) = 12(//- v2)
fl
Q(p, O)
The dimensionless stresses are:
,r" =
& = 12(/~- vg) (p + ¢)
(~)1 o~o*
fl(l+O 2
¢~(h/2) = ( h ) 2 a~*(h/2) =E, a}(h/2) = ( h ) 2 ab'(h/2) =E,
2 ( f l - v~) w"+p+¢,]
t~(1+¢)~ ( ' w,+ yaw") 2(fl-- vao) ~
(11)
METHOD OF SOLUTION The time is incremented in small steps Az. Nonlinear eqns (5) and (6) are solved iteratively at step J by linearising them for each iteration by writing the nonlinear terms as: (~w')s = ~,spw~
(w') 2 = w'spw'j
(12)
where the predicted term fj, is taken as the mean of the previous two iterations. For the first iteration, the predicted value is extrapolated quadratically from the values at three previous steps:
fsp = a t ( f , - , ) + a2(fs-2) + a a ( f , - 3)
(13)
where A 1, A 2, A 3 for the different stages are: 1, 0, 0 ( J = 1); 2, - 1, 0 ( J = 2 ) ; and 3, - 3 , 1 ( J > 3).
Nonlinear transient analysis of orthotropic annulus
29
In the present study the orthogonal point collocation method has been used to solve ¢qns (5) and (6) with the zeros of a Lcgendre polynomial as collocation points. For N collocation points, the deflection and stress function are expanded as polynomials in non-dimensional radius, a:
N+3 w(p) =
N+2 p'-
0 < p _< 1
=
m=l
(14)
n=l
The differential eqns (5) and (6) are collocated at the zeros of an Nth order Legendrc polynomial in the range 0 to 1. The inertia term, fi~,in cqn (5) is discretised using the Newmark-fl s c h e m e :
fOs=
.
ws2a(A~)
,,
t
a(A¢)I
(0:)
(I 5)
- 1 wJ- 1
with Wj = 14:j_I "~"[(I -- ~)Wj_ 1 + (~ti)j](A'~) Wj_ I(A'c) + [(0"5 -- ~)WJ- 1 + {XwJ](A'~)2
Wj = w j _ 1 +
The collocationequationsfor differentialcqns (5) and (6) are: N+3 X [(m--1){(Piq-~)2(m - 2)(m - 3)p7'-* m=l + (Pi + ~)(m - 2)p~'-3 - (fl + (Pl + ¢)(~z~)i)P~ '-2 }
+(Pi+ :_-+l ------T g(A'0 ¢) (1 +I~)4/m ~ + £p7,+i)i m as F 6 ( f l - v2)
=(pi+~)L ~fl x
"
-
1 1 ( I + ~ ) 2 P J + (I+~)4
{21+ ~(A~)2 (Case I)
(16a)
P. c. Dumir, M. L. Gandhi, Yogendra Nath
30 N+3
~I(m--1){(pi+~)2(m--2)(m--3)pm-'* m=l
+ (Pi + ~)(m - 2)p7'- 3 (fl + (Pi + ~)(~'/Jp)i)pim-2 } _
+(pi+~) 1 (p~'+'- 1 p.7'-l)la, " ~(Az) 2 ( 1 + ~ ) " \ m - + l 4-~ m [ - 6 ( f l - v2) 1 1 = (Pi + ~) L ~-fl (1 + ~)2 PJ + (1 + ~)~ x
f'l' {12(fl; v2)
Ws-,
ws-~
Qj 4 ot(Az)2 Voc(Az)
+(0~___5_5-1 ) ~ s - x } ( P + ~)dP 1 (Case II)
(16b)
N+3
-~ [6(fl - vg)(w'sQi(m - l)(pl + ~)pT'- 2la,, m=l N+2
+ )'
[(Pi +~)2( n - 1 ) ( n - 2)p~'-3
n=l
+(pi+~)(n- 1)p~-2-flp~-l]b.=O
i= 1, ..., N
(17)
The five equations for boundary conditions are solved for al, a 2, a3 and b~, b 2 in terms of the remaining N a's and b's, respectively. Equations (16) and (17) are the 2N discretised equations for these a's and b's. These are solved by Gaussian elimination with pivoting. The iterations are continued until w(0) or w(1), ~O'(0)and ~b'(0) satisfy a relative convergence criterion within 0.1 ~o accuracy. After the converged solutions for the a's and b's at step J are obtained, the procedure is repeated for the (J + 1)th step. RESULTS AND DISCUSSION In the present investigation, a nonlinear transient analysis of orthotropic annular plates has been carried out using the orthogonal point
l=
=
oo
..,= °
"T
'7
(0) M
I I I I I 1 ~ N ¢y d
+ .,= "~
1
°
A+ 0
I 1
I
I
i eO I I (0) M
J I
I l l i i + l l
o
(0) M
0
e,i [,.T.
P. C. Dumir, M. L. Gandhi, Yogendra Nath
32
TABLE 1 Convergence Studies: Values at Maximum Deflection in First Cycle Case I. Immovable clamped plate, b/a=0.25, /~=10, uniformly distributed sinusoidal pulse load with amplitude Q = 10 and period z=0'4
N
Az
z
w(O)
trb,(1)
am(l)
1.308 4 1-308 8 1.309 3 1.307 0
11.536 5 11 '539 2 11.538 2 11.528 0
1.493 9 1'495 3 1.496 4 1.493 6
1.3114 1.308 8 1.308 4
11.5136 11'539 2 11 '525 4
1.4710 1'495 3 1-494 5
Time convergence ( N = 5) 0.005 0-004 0"003 0.001
0"340 0-340 0.339 0.339
Space convergence (Az = 0.004) 4 5 6
0.340 0-340 0' 340
Case II. Plate clamped at inner edge, b/a = 0.25, fl = 5, uniformly distributed step load Q = 2 N
Az
z
w(1)
trb,(O)
a~,(O)
0.7324 0.7326 0.733 0 0.732 7
5-6364 5.7120 5.720 4 5.717 1
0-392 0.393 0-392 0.390
0-735 5 0.7326 0.732 3
5.425 2 5.7120 5.753 3
0.396 5 0.393 1 0'393 3
Time convergence ( N = 5) 0.010 0"008 0'006 0'002
0.320 0.320 0.318 0.314
1 1 7 8
Space convergence (Az = 0.008) 4 5 6
0.320 0-320 0-320
collocation method collocating at the zeros of Legendre polynomials for spatial discretisation, and the Newmark-fl method (~ = 0.25, 6 = 0.5) for time-marching. The larger Poisson's ratio has been taken as 0.25, The influence of the orthotropic parameter and annular ratio on the large amplitude deflection and stress responses has been studied. Detailed time and space-wise convergence studies have been carried out and it is revealed that five collocation points and Az =0.004 yield results of engineering accuracy for plates supported at the outer edge whereas five collocation points and Az = 0.008 yield good results for annular plates
Nonlinear transient analysis of orthotropic annulus
2l
bla =0.25
T
.
I
.
I
~
I
I ~4b~-a--,.F
2
I
33
~=I
I
J
I -~b~
I B=I 2
0
0.0 Fig. 4.
0.2
O.t,
"(:
0.6
0.8
Deflection response to step load at the inner edge.
clamped at the inner edge. Typical results for convergence studies for clamped orthotropic plates under uniformly distributed sinusoidal load are given in Table 1. Convergence results for an orthotropic annular plate clamped at the inner edge under uniformly distributed step load are also given in Table 1. The deflection response to a uniformly distributed step function load is shown in Fig. 2 along with the results obtained by Alwar and Reddy. 2 There is good agreement in the results. The phase shift noticed in the results may be because of the Houbolt scheme used by Alwar and Reddy.2 The deflection response to a uniformly distributed sinusoidal pulse for a clamped plate is plotted in Fig. 3 for two values of the orthotropic parameter, fl = 1 and 2, and two values of the annular ratio, b/a = 0.25 and 0-5. The deflection response to step load at the inner edge for fl = 1 and 2 for clamped and simply supported plates is shown in Fig. 4. It can
P. C. Dumir, M. L. Gandhi, Yogendra Nath
34
3
~
f
i
i
r
i Q
....
I
~ ~
~ ~
I
I
I
I 0.2
I
I
2~
'
~
i r i ! I,,-b
o=t
i
I
b/o:
0.25 O.SO
)
!
i
) i
[ I I
:ol 1 2
2
0
I
I
I I I I
I
L
I
I
t
~ I 0.5
I I I I 1
J 2
I
I
I 5
I I I I
w(0)m
x for
3
2 E o v
0.1
Fig.
5.
Effect
of ~ on
an
immovable
clamped
10
plate.
be noted that the maximum response and 'period' decrease with/~ and b/a. The presence of modes other than the fundamental is also evident. Considering three types of uniformly distributed loadings, namely, step function, exponential decay and sinusoidal pulse, and a case of step function ring load at the edge, the influence of the orthotropic parameter /~ on the maximum deflection and stresses has been studied for two values of annular ratio and the results are shown in Figs 5-8. The maximum deflection in the first cycle and the maximum stresses near the instant of
E
~
.
~
~
0-2
~
--,-,
0.50
o.1o
2
,
P, ~~ -
--//_2
/I
~r ~ ~, , ~ - - - - - - - - ~ ~
I
0-5
I
¢
I
I
I
1.0 /3
;
2-0
I
3
"Y
5.0
I~'--P-'I--'I'-I
t
10.0
"~-';'~--~'~
# on ~r(l)max for an immovable clamped plate.
I
i,, - - ' ~ , ~ ' r - - ' ~ ' ~ P ~ ' " ~ ' ~ " P ~ " ~ - - - - - " ~
Fig. 6. Effect of
0-1
5
, ,~
1
.--~----"~-----~---'~-----'----~~_=:~---~-=-
,~
O' . . . .
S
10
Fig. 7.
0-1
•
I~
i
i
i
i
0.5
I
i
l
I
i
I
I
I
i
I
1.0 tl
I
I
,
2
,ol
rr,
2
i
I
o.1o
o.so
'. ::
i
I
I
I
1
I
S
I
I
._o__
I
.....
i
'
i
i
,
10
1
Effect of ~ on w(O)m.xfor an immovable simply supl~rted plate.
0.2
I
-,,,~.._
~',,,~,~.
' ,~.:.'~.;.v~'
0
0.2
0.I
I
1
I
L
0.5
I
I
I
I
1.0 /!
I
J
t~-'~
I I I
0.50
/..i%
z
1 b/a: 0.25 /
2.0
I
I
~
I
~
;
I
I
I
o
1.6
3.2
4.0
. o
I
I
i
O.Z
~
4
/~
/
I
i
i
\
0.4
I
ORTHOTROPIC:
,
i
i
"C
~=
1
0.6
I
2
,/,
i
1 k b = ki =-1 2 -,I,
I
I
/
i
f
J
0.8
i
F i g . 9. Effect o f flexibility (•i = 2b) o n t h e d e f l e c t i o n r e s p o n s e to a u m f o r m l y d i s t r i b u t e d s t e p l o a d ( Q = 10, b/a = 0.5).
-0.810.0
0.0
0.8
1.6
2.4
-0.81
0.0
0.8
I
b/a = 0.5 ISOTROPIC:O = 1
2_~ I
1
~ z .
"
~"T"~r~ 5.0 10.0
I
~ - . ~ .
.
~
"~'
Fig. 8. Effect of/3 on a0(0)m~ for an i m m o v a b l e simply supported plate.
J
•
~- a~.l
P
I
~,o-~
0
18
o12 b~
E
12
,~5~,o~! ' ~ , o , - ~,o,
Nonlinear transient analysis of orthotropic annulus
37
these maximum deflections are considered. The effect of the orthotropic parameter/J on w(0)m ~ and on a~(1)max and o~,(1)=~, is shown in Figs 5 and 6, respectively for immovable clamped plates. It can be noted that the effect of/~ is less for larger values of b/a, Both the deflection and stresses are greater for smaller values of b/a. The effect of/~ on w(0),~, and on a~(0)=~1 and o~0(0)m~ is shown in Figs 7 and 8, respectively, for an immovable, simply supported plate. The effect of inplane and rotational flexibilities (~,i= 2e) on the deflection response to a uniformly distributed step function load is shown
/"J ~: 0.5 ~ 1
~ 2
2
~ 3
1
8
I::
-
(1)~ rO~(1)
---
L
L
E
0
OI, -1
I
t
t 0 ~b
I
I
~CIIP" 1 -1
1 0
t
I 1
'~b
Fig. 10. Effect of flexibility (2 i = 2b) of the support on maximum deflection and stresses (b/a = 0.5).
in Fig. 9 for b/a = 0.5 and fl = 1 and 2. It can be concluded from Fig. 9 that the maximum deflection response and time 'period' decrease with 2 and ft. The effect ofinplane and rotational flexibilities on w(0)max,a~(1)ma, and o'm(1)m~of orthotropic plates subjected to a uniformly distributed step load is shown in Fig. 10. The maximum central deflection increases sharply near 2 = - 1 whereas the radial bending stress increases with 2.
A
l
~r
Fig. 11.
-2-4 0.0
-1.6
-0.8
0.0
0.8
1.6
J
I
0.6 T,,
I
~1~
I
2
I
I~
I
--o-
I
I
t
G9
I
I
-- 0.25
bla • 0.10
r
t
I
I
I
I
J
I
I
1.2
I
t
I
l
D e f l e c t i o n response o f a
plate claml~cl at the inner edge under a uniformly distributed sinusoidal pulse.
0.3
I
2
[
0.0
0
o J<
I
0.2
~,/o--o.,o-
r
f
,
XP
I
0.4
I
J
I
"C
,
I
\,
\
0.6
~
r
I
0.8
"~'-- "~
\
i
t
,
T
Fig. 12. Deflection response of an annular plate clamped at the inner edge under step function uniformly distributed load and ring load.
_
I
OI 0.1
a 0.2
I
0.5
I
I
I
I
I
I
I
I
I
I
I J I J 1
-db6-
~o-~
I
I
I 2
o.25
2
I
0.10
1 b/a:
0-8
I
J
I
I
J
I
I
I 5
I
I
f
I
I
i
I
t ] J 10
i
I
Fig. 13. Effect of fl on w(l)m,~ for an annular plate clamped at the mner edge.
E
E
I
. 0-2
.
.
~/.2
.
. 0-5
.
Pt ~
'
.
O--I't
'"
1
41
2t ---L'
2
---o-
2
'
....
5
0.25
'
10
/
/
Fig. 14. Effeetofflon e,(0)m~, for an annular plate clamped at the inner edge.
Ol~---- . 0.1
51
~oI
251
01
10l
15
20
'0'
40
P. C. Dumir, M. L. Gandhi, Yogendra Nath
The results for annular plates clamped at the inner edge and free at the outer edge are given in Figs 11-14. The deflection response to a uniformly distributed sinusoidal pulse load is plotted in Fig. 11 for two values of the orthotropic parameter, fl = 1 and 2, and two values of the annular ratio, b/a = 0.10 and 0.25. Similar results for a uniformly distributed step load and a step function ring load at the outer edge are given in Fig. 12. The maximum deflection and 'period' decrease with fl and annular ratio b/a. The influence offl on W(1)maxand arb(0)max and tTm(0)maxhas been plotted in Figs 13 and 14, respectively, for two values of the annular ratio, b/a = O. 1 and 0.25. Three types of uniformly distributed loadings, namely, step function, exponential decay and sinusoidal pulse, and a case of step function ring load at the outer edge have been considered. It is noted that the predominant bending stresses decrease with b/a for all cases while these decrease with fl except for the case of sinusoidal pulse. Several new results reported in the present investigation may be of interest in the design of composite annular plate structures.
REFERENCES 1. J. L. Nowinski, Nonlinear vibrations of elastic circular plates exhibiting rectilinear orthotropy, ZAMP, 14 (1963) pp. 112-24. 2. R. S. Alwar and B. Sekhar Reddy, Large deflection static and dynamic analysis of isotropic and orthotropic annular plates, Int. J. Non-Linear Mechanics, 14 (1979) pp. 347-59. 3. J. L. Nowinski, Cylindrically orthotropic plates, ZAMP, 11 (1960) pp. 218-28. 4. J. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1976.
Nonlinear Axisymmetric Transient Analysis of Orthotropic Thin Annular Plates P. C. D u m i r , M. L. G a n d h i a n d Y o g e n d r a N a t h Department of Applied Mechanics, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi--110016 (India)
SUMMARY This investigation deals with the geometrically nonlinear, axisymmetric, transient elastic response for stresses and deflection of cylindrically orthotropic thin annular plates subjected to uniformly distributed and ring loads. The dynamic analogues of yon Kdrmdn equations in terms of normal displacement w and stress function ~ have been employed. The displacement w and stress function ~bare expanded infinite power series. The orthogonal point collocation method in the space domain and the Newmark-fl scheme in the time domain have been used. Three types of uniformly distributed dynamic loadings, namely, step function, sinusoidal pulse and exponentially decaying load, and a case of step function ring load at the free edge have been considered. Detailed results are presented for a clamped and simply supported annular plate with a free hole, and an annulus supported at the inner boundary and free at the outer edge, for different values of orthotropic parameter and annular ratio. The effect of elastic rotational and inplane edge restraints has also been studied.
NOMENCLATURE Outer radius, inner radius and thickness of the plate. Coefficients of power series expansion of w and ~. al, bi Coefficients in quadratic extrapolation. A1, A2, A3 D Eoha/12(fl - vg). Elastic moduli; Poisson's ratios. E o, E,, vo, v, K~,K~' Rotational and inplane stiffness of support. 23 Fibre Science and Technology 0015-0568/84/$03.00© ElsevierAppliedSciencePublishers Ltd, England, 1984. Printed in Great Britain a,b,h
24
P. C. Dumir, M. L. Gandhi, Yogendra Nath
Kb, Ki
Az
K~ a/D, K* a/hEo, dimensionless stiffnesses. Radial and circumferential bending moments. Number and radii of collocation points. Inplane forces, transverse shear. Intensity of uniformly distributed load, total ring load at the edge. qa4/ E,h 4, paZ/ Erh4, non-dimensional loads. Radius, non-dimensional radius. Time, non-dimensional time. Transverse and inplane displacements; stress function. Non-dimensional displacements and stress function. Parameters of the Newmark-fl scheme. Eo/E,, orthotropic parameter. Mass density. Time step.
a*,
am', a b" Radial and circumferential stresses; membrane and
g,,Mo N, Pi N,,No; Q, q,P Q,P r, p l, "C W*, U*, I~¢* W, U, I]/
¢7r,0 ()',(') Subscript J Subscripts p, i Movable Immovable
b/(a - b).
bending stresses. (a/h)2(~,.~/E,), dimensionless membrane stresses. (a/h)Z[ab,i~(h/2)/E,], dimensionless bending stresses. Differentiation with respect to p and z. Step of marching. Predicted value, value at the ith collocation point. g i = 0 (2 i ~-- - 1).
Ki = ~ (2,. = 1).
INTRODUCTION Fibre reinforced composites are increasingly being introduced into a variety of structures in aeronautical, mechanical and marine engineering applications where high ratios of stiffness and strength to weight are of paramount importance. Annular thin plate structures of composite materials have wide applications in these areas. The nonlinear response of these structures is of interest in their economical design. The nonlinear frequency-amplitude relation of orthotropic annular plates has been obtained by Nowinski1 using the Galerkin technique. The nonlinear transient analysis of orthotropic annular plates has received little attention. Alwar and Reddy2 studied the transient analysis of orthotropic
Nonlinear transient analysis of orthotropic annulus
25
annular plates subjected to a uniformly distributed step load using the Chebyshev series in space and the Houbolt scheme in time. To the best of the authors' knowledge, there are no results available for other transient distributed and edge loads for annular plates with or without flexible supports. The object of the present investigation is to study the transient response of orthotropic annular plates under three types of uniformly distributed dynamic loadings, namely, step function, sinusoidal pulse and exponentially decaying load, and under one case of step function ring load at the free edge. Detailed results are presented for a clamped and simply supported annular plate with a free hole, and an annulus supported at the inner boundary and free at the outer edge, for different values of the orthotropic parameter fl and the annular ratio. The effect of elastic rotational and inplane edge restraint has also been studied. The nonlinear differential equations for a cylindrically orthotropic plate, in terms of normal displacement w and stress function ~, have been employed. The deflection w and stress function ¢, are expanded in a finite power series and the differential equations have been discretised spacewise by collocating at the zeros of a Legendre polynomial. The Newmark-fl scheme is used for time-marching. An iterative scheme has been used to solve the nonlinear equations by predicting one of the variables in the nonfinear terms as a mean of its values at the two previous iterations. The predictions at the first iteration are taken as the quadratically extrapolated values from the preceding three steps. Time and space-wise convergence studies have revealed that five collocation points and A¢ = 0.004 yield accurate results for plates supported at the outer edge, whereas five collocation points and A¢ = 0.008 yield accurate results for plates clamped at the inner edge. An independent check on the accuracy of results has been incorporated in the form of a balance of energy. The sum of the kinetic energy and the strain energy of the plate at each time step is calculated and compared with the sum of the initial kinetic energy and the external work done up to that instant. This energy balance has been found to have been maintained very well at every time step. MATHEMATICAL FORMULATION The governing differential equations for the moderately large axisymmetric static deflection of a cylindrically orthotropic thin circular
26
P. C. Dumir, M. L. Gandhi, Yogendra Nath Kb
P
]
I" b4,----- ° -----,~
Ki
P
Kb
f,o~
I }..,. t
L o~1--
Ki
CASE I
CASE II
PLATE WITH FLEXIBLE SUPPORTS
PLATE CLAMPED AT INNER EDGE
1
IMMOVABLE
STEP
CLAMPED
IMMOVABLE SIMPLY SUPPORTED
EXPERIMENTAL
SINUSOIDAL
STEP
UNIFORMLY DISTRIBUTED LOADS
EDGE LOAD
Fig. 1. Edge conditions and dynamic loadings.
plate with the orthotropy axis coinciding with the axis of the plate have been given by Nowinski. 3 Neglecting damping and inplane inertia and including the transverse inertia term, these equations are modified as" D
rw*
+w*
--w*
-w,*d/ =
(1)
(q-Thw*,)rdr-Qr(r°)
1 fl hEo ~b*, + - r ~k,*----r2 ~k*+
,,2
(',,,,j
(2)
=0
where N, -
~'* r
N o = ~,*
Q, =
Eo h3
D = 12(fl-- v2) U~ ~O = - -
r
m*
G0 =----
Eo
E0
E,
~r = U *
~}/-h/2 v,
fl = ~
V0
if, =
m* O"r
Vr - -
C'/h/2 tr*=d z
'"
"~
lt,
,~2
2~"v'r"
m*
O'rm*
--
Er
V8
Gr0
-~0
(3)
Nonlinear transient analysis of orthotropic annulus
27
Introduce the following dimensionless parameters: w
.
w* . h
.
( a - b) ~b* ~b D qa4 .
b a-b
~-
r- b p=-a-b F D ]1/2
pa2
Q=e,h' e=E,h"
d
0
t
(4)
where a and b are the outer and inner radius of the plate and h is its thickness. The uniformly distributed load is q, andp is the ring load at the edge. The geometry, loading and edge conditions are depicted in Fig. 1. The governing equations reduce to the following dimensionless form: (p + 0 2w'' + (p + ~)w" - #w' - (p + OOw'
F6(fl_-_ v2) 1 = (P + ~J) L rcfl ~(I + zj) e -t (1+~),, 1 ~ { 1 2 ( ~ - v 2 ) Q - ~ t_ (P + ' ) d p ] =(P+~)F
(Case I)
6(fl--vg) 1 nfl (1 + ~)~ P
-~(1+~)4
8
Q - ~ (p + O d p
(CaseII)
(p + ~)20" + (p + 0 0 ' - 8 0 + 6(8 - vo~)(p + O(w') 2 = 0
(5) (6)
For an elastically restrained outer edge with rotational and inplane stiffnesses Kb* and K*: M,(1) = K~'w*(1) N,(1) = -/C~ u*(1) (7) Introduce dimensionless stiffnesses, K b and Ki: 1 + 2b
1 -~- ~i
a
Kb = 1 - 2-----~= K* ~
K~
I - 2i
K*a hE o
(8)
For the two cases shown in Fig. 1, the boundary conditions take the following dimensionless form: Case I p = 0:
v8
w"(0) + ~- w'(0) = 0
~k(0) = 0
p = 1: w(1) = 0 [(1 + 2b) + v0(l - 2b)]W'(1) + (1 -- 2b)(1 + ¢)W'(1) = 0 [(1 - 2 i ) - Vo(1 + ;q)]~O(1) + (1 + 2i)(1 + ¢)~k'(1) = 0
(9)
28
P. C. Dumir, M. L. Gandhi, Yogendra Nath
Case II p=O:
w(0) = 0
p=l:
w'(0) = 0 vo
w"(1) + 1 - - ~ w'(1) = 0
q/(o) - vd/(o)/¢ = o
~O(1) = 0
(I0)
The initial conditions are assumed as: w(p, O) = ~:(p, O) = O, and differential eqn (5) yields the initial accelerations: #(p, O) = 12(//- v2)
fl
Q(p, O)
The dimensionless stresses are:
,r" =
& = 12(/~- vg) (p + ¢)
(~)1 o~o*
fl(l+O 2
¢~(h/2) = ( h ) 2 a~*(h/2) =E, a}(h/2) = ( h ) 2 ab'(h/2) =E,
2 ( f l - v~) w"+p+¢,]
t~(1+¢)~ ( ' w,+ yaw") 2(fl-- vao) ~
(11)
METHOD OF SOLUTION The time is incremented in small steps Az. Nonlinear eqns (5) and (6) are solved iteratively at step J by linearising them for each iteration by writing the nonlinear terms as: (~w')s = ~,spw~
(w') 2 = w'spw'j
(12)
where the predicted term fj, is taken as the mean of the previous two iterations. For the first iteration, the predicted value is extrapolated quadratically from the values at three previous steps:
fsp = a t ( f , - , ) + a2(fs-2) + a a ( f , - 3)
(13)
where A 1, A 2, A 3 for the different stages are: 1, 0, 0 ( J = 1); 2, - 1, 0 ( J = 2 ) ; and 3, - 3 , 1 ( J > 3).
Nonlinear transient analysis of orthotropic annulus
29
In the present study the orthogonal point collocation method has been used to solve ¢qns (5) and (6) with the zeros of a Lcgendre polynomial as collocation points. For N collocation points, the deflection and stress function are expanded as polynomials in non-dimensional radius, a:
N+3 w(p) =
N+2 p'-
0 < p _< 1
=
m=l
(14)
n=l
The differential eqns (5) and (6) are collocated at the zeros of an Nth order Legendrc polynomial in the range 0 to 1. The inertia term, fi~,in cqn (5) is discretised using the Newmark-fl s c h e m e :
fOs=
.
ws2a(A~)
,,
t
a(A¢)I
(0:)
(I 5)
- 1 wJ- 1
with Wj = 14:j_I "~"[(I -- ~)Wj_ 1 + (~ti)j](A'~) Wj_ I(A'c) + [(0"5 -- ~)WJ- 1 + {XwJ](A'~)2
Wj = w j _ 1 +
The collocationequationsfor differentialcqns (5) and (6) are: N+3 X [(m--1){(Piq-~)2(m - 2)(m - 3)p7'-* m=l + (Pi + ~)(m - 2)p~'-3 - (fl + (Pl + ¢)(~z~)i)P~ '-2 }
+(Pi+ :_-+l ------T g(A'0 ¢) (1 +I~)4/m ~ + £p7,+i)i m as F 6 ( f l - v2)
=(pi+~)L ~fl x
"
-
1 1 ( I + ~ ) 2 P J + (I+~)4
{21+ ~(A~)2 (Case I)
(16a)
P. c. Dumir, M. L. Gandhi, Yogendra Nath
30 N+3
~I(m--1){(pi+~)2(m--2)(m--3)pm-'* m=l
+ (Pi + ~)(m - 2)p7'- 3 (fl + (Pi + ~)(~'/Jp)i)pim-2 } _
+(pi+~) 1 (p~'+'- 1 p.7'-l)la, " ~(Az) 2 ( 1 + ~ ) " \ m - + l 4-~ m [ - 6 ( f l - v2) 1 1 = (Pi + ~) L ~-fl (1 + ~)2 PJ + (1 + ~)~ x
f'l' {12(fl; v2)
Ws-,
ws-~
Qj 4 ot(Az)2 Voc(Az)
+(0~___5_5-1 ) ~ s - x } ( P + ~)dP 1 (Case II)
(16b)
N+3
-~ [6(fl - vg)(w'sQi(m - l)(pl + ~)pT'- 2la,, m=l N+2
+ )'
[(Pi +~)2( n - 1 ) ( n - 2)p~'-3
n=l
+(pi+~)(n- 1)p~-2-flp~-l]b.=O
i= 1, ..., N
(17)
The five equations for boundary conditions are solved for al, a 2, a3 and b~, b 2 in terms of the remaining N a's and b's, respectively. Equations (16) and (17) are the 2N discretised equations for these a's and b's. These are solved by Gaussian elimination with pivoting. The iterations are continued until w(0) or w(1), ~O'(0)and ~b'(0) satisfy a relative convergence criterion within 0.1 ~o accuracy. After the converged solutions for the a's and b's at step J are obtained, the procedure is repeated for the (J + 1)th step. RESULTS AND DISCUSSION In the present investigation, a nonlinear transient analysis of orthotropic annular plates has been carried out using the orthogonal point
l=
=
oo
..,= °
"T
'7
(0) M
I I I I I 1 ~ N ¢y d
+ .,= "~
1
°
A+ 0
I 1
I
I
i eO I I (0) M
J I
I l l i i + l l
o
(0) M
0
e,i [,.T.
P. C. Dumir, M. L. Gandhi, Yogendra Nath
32
TABLE 1 Convergence Studies: Values at Maximum Deflection in First Cycle Case I. Immovable clamped plate, b/a=0.25, /~=10, uniformly distributed sinusoidal pulse load with amplitude Q = 10 and period z=0'4
N
Az
z
w(O)
trb,(1)
am(l)
1.308 4 1-308 8 1.309 3 1.307 0
11.536 5 11 '539 2 11.538 2 11.528 0
1.493 9 1'495 3 1.496 4 1.493 6
1.3114 1.308 8 1.308 4
11.5136 11'539 2 11 '525 4
1.4710 1'495 3 1-494 5
Time convergence ( N = 5) 0.005 0-004 0"003 0.001
0"340 0-340 0.339 0.339
Space convergence (Az = 0.004) 4 5 6
0.340 0-340 0' 340
Case II. Plate clamped at inner edge, b/a = 0.25, fl = 5, uniformly distributed step load Q = 2 N
Az
z
w(1)
trb,(O)
a~,(O)
0.7324 0.7326 0.733 0 0.732 7
5-6364 5.7120 5.720 4 5.717 1
0-392 0.393 0-392 0.390
0-735 5 0.7326 0.732 3
5.425 2 5.7120 5.753 3
0.396 5 0.393 1 0'393 3
Time convergence ( N = 5) 0.010 0"008 0'006 0'002
0.320 0.320 0.318 0.314
1 1 7 8
Space convergence (Az = 0.008) 4 5 6
0.320 0-320 0-320
collocation method collocating at the zeros of Legendre polynomials for spatial discretisation, and the Newmark-fl method (~ = 0.25, 6 = 0.5) for time-marching. The larger Poisson's ratio has been taken as 0.25, The influence of the orthotropic parameter and annular ratio on the large amplitude deflection and stress responses has been studied. Detailed time and space-wise convergence studies have been carried out and it is revealed that five collocation points and Az =0.004 yield results of engineering accuracy for plates supported at the outer edge whereas five collocation points and Az = 0.008 yield good results for annular plates
Nonlinear transient analysis of orthotropic annulus
2l
bla =0.25
T
.
I
.
I
~
I
I ~4b~-a--,.F
2
I
33
~=I
I
J
I -~b~
I B=I 2
0
0.0 Fig. 4.
0.2
O.t,
"(:
0.6
0.8
Deflection response to step load at the inner edge.
clamped at the inner edge. Typical results for convergence studies for clamped orthotropic plates under uniformly distributed sinusoidal load are given in Table 1. Convergence results for an orthotropic annular plate clamped at the inner edge under uniformly distributed step load are also given in Table 1. The deflection response to a uniformly distributed step function load is shown in Fig. 2 along with the results obtained by Alwar and Reddy. 2 There is good agreement in the results. The phase shift noticed in the results may be because of the Houbolt scheme used by Alwar and Reddy.2 The deflection response to a uniformly distributed sinusoidal pulse for a clamped plate is plotted in Fig. 3 for two values of the orthotropic parameter, fl = 1 and 2, and two values of the annular ratio, b/a = 0.25 and 0-5. The deflection response to step load at the inner edge for fl = 1 and 2 for clamped and simply supported plates is shown in Fig. 4. It can
P. C. Dumir, M. L. Gandhi, Yogendra Nath
34
3
~
f
i
i
r
i Q
....
I
~ ~
~ ~
I
I
I
I 0.2
I
I
2~
'
~
i r i ! I,,-b
o=t
i
I
b/o:
0.25 O.SO
)
!
i
) i
[ I I
:ol 1 2
2
0
I
I
I I I I
I
L
I
I
t
~ I 0.5
I I I I 1
J 2
I
I
I 5
I I I I
w(0)m
x for
3
2 E o v
0.1
Fig.
5.
Effect
of ~ on
an
immovable
clamped
10
plate.
be noted that the maximum response and 'period' decrease with/~ and b/a. The presence of modes other than the fundamental is also evident. Considering three types of uniformly distributed loadings, namely, step function, exponential decay and sinusoidal pulse, and a case of step function ring load at the edge, the influence of the orthotropic parameter /~ on the maximum deflection and stresses has been studied for two values of annular ratio and the results are shown in Figs 5-8. The maximum deflection in the first cycle and the maximum stresses near the instant of
E
~
.
~
~
0-2
~
--,-,
0.50
o.1o
2
,
P, ~~ -
--//_2
/I
~r ~ ~, , ~ - - - - - - - - ~ ~
I
0-5
I
¢
I
I
I
1.0 /3
;
2-0
I
3
"Y
5.0
I~'--P-'I--'I'-I
t
10.0
"~-';'~--~'~
# on ~r(l)max for an immovable clamped plate.
I
i,, - - ' ~ , ~ ' r - - ' ~ ' ~ P ~ ' " ~ ' ~ " P ~ " ~ - - - - - " ~
Fig. 6. Effect of
0-1
5
, ,~
1
.--~----"~-----~---'~-----'----~~_=:~---~-=-
,~
O' . . . .
S
10
Fig. 7.
0-1
•
I~
i
i
i
i
0.5
I
i
l
I
i
I
I
I
i
I
1.0 tl
I
I
,
2
,ol
rr,
2
i
I
o.1o
o.so
'. ::
i
I
I
I
1
I
S
I
I
._o__
I
.....
i
'
i
i
,
10
1
Effect of ~ on w(O)m.xfor an immovable simply supl~rted plate.
0.2
I
-,,,~.._
~',,,~,~.
' ,~.:.'~.;.v~'
0
0.2
0.I
I
1
I
L
0.5
I
I
I
I
1.0 /!
I
J
t~-'~
I I I
0.50
/..i%
z
1 b/a: 0.25 /
2.0
I
I
~
I
~
;
I
I
I
o
1.6
3.2
4.0
. o
I
I
i
O.Z
~
4
/~
/
I
i
i
\
0.4
I
ORTHOTROPIC:
,
i
i
"C
~=
1
0.6
I
2
,/,
i
1 k b = ki =-1 2 -,I,
I
I
/
i
f
J
0.8
i
F i g . 9. Effect o f flexibility (•i = 2b) o n t h e d e f l e c t i o n r e s p o n s e to a u m f o r m l y d i s t r i b u t e d s t e p l o a d ( Q = 10, b/a = 0.5).
-0.810.0
0.0
0.8
1.6
2.4
-0.81
0.0
0.8
I
b/a = 0.5 ISOTROPIC:O = 1
2_~ I
1
~ z .
"
~"T"~r~ 5.0 10.0
I
~ - . ~ .
.
~
"~'
Fig. 8. Effect of/3 on a0(0)m~ for an i m m o v a b l e simply supported plate.
J
•
~- a~.l
P
I
~,o-~
0
18
o12 b~
E
12
,~5~,o~! ' ~ , o , - ~,o,
Nonlinear transient analysis of orthotropic annulus
37
these maximum deflections are considered. The effect of the orthotropic parameter/J on w(0)m ~ and on a~(1)max and o~,(1)=~, is shown in Figs 5 and 6, respectively for immovable clamped plates. It can be noted that the effect of/~ is less for larger values of b/a, Both the deflection and stresses are greater for smaller values of b/a. The effect of/~ on w(0),~, and on a~(0)=~1 and o~0(0)m~ is shown in Figs 7 and 8, respectively, for an immovable, simply supported plate. The effect of inplane and rotational flexibilities (~,i= 2e) on the deflection response to a uniformly distributed step function load is shown
/"J ~: 0.5 ~ 1
~ 2
2
~ 3
1
8
I::
-
(1)~ rO~(1)
---
L
L
E
0
OI, -1
I
t
t 0 ~b
I
I
~CIIP" 1 -1
1 0
t
I 1
'~b
Fig. 10. Effect of flexibility (2 i = 2b) of the support on maximum deflection and stresses (b/a = 0.5).
in Fig. 9 for b/a = 0.5 and fl = 1 and 2. It can be concluded from Fig. 9 that the maximum deflection response and time 'period' decrease with 2 and ft. The effect ofinplane and rotational flexibilities on w(0)max,a~(1)ma, and o'm(1)m~of orthotropic plates subjected to a uniformly distributed step load is shown in Fig. 10. The maximum central deflection increases sharply near 2 = - 1 whereas the radial bending stress increases with 2.
A
l
~r
Fig. 11.
-2-4 0.0
-1.6
-0.8
0.0
0.8
1.6
J
I
0.6 T,,
I
~1~
I
2
I
I~
I
--o-
I
I
t
G9
I
I
-- 0.25
bla • 0.10
r
t
I
I
I
I
J
I
I
1.2
I
t
I
l
D e f l e c t i o n response o f a
plate claml~cl at the inner edge under a uniformly distributed sinusoidal pulse.
0.3
I
2
[
0.0
0
o J<
I
0.2
~,/o--o.,o-
r
f
,
XP
I
0.4
I
J
I
"C
,
I
\,
\
0.6
~
r
I
0.8
"~'-- "~
\
i
t
,
T
Fig. 12. Deflection response of an annular plate clamped at the inner edge under step function uniformly distributed load and ring load.
_
I
OI 0.1
a 0.2
I
0.5
I
I
I
I
I
I
I
I
I
I
I J I J 1
-db6-
~o-~
I
I
I 2
o.25
2
I
0.10
1 b/a:
0-8
I
J
I
I
J
I
I
I 5
I
I
f
I
I
i
I
t ] J 10
i
I
Fig. 13. Effect of fl on w(l)m,~ for an annular plate clamped at the mner edge.
E
E
I
. 0-2
.
.
~/.2
.
. 0-5
.
Pt ~
'
.
O--I't
'"
1
41
2t ---L'
2
---o-
2
'
....
5
0.25
'
10
/
/
Fig. 14. Effeetofflon e,(0)m~, for an annular plate clamped at the inner edge.
Ol~---- . 0.1
51
~oI
251
01
10l
15
20
'0'
40
P. C. Dumir, M. L. Gandhi, Yogendra Nath
The results for annular plates clamped at the inner edge and free at the outer edge are given in Figs 11-14. The deflection response to a uniformly distributed sinusoidal pulse load is plotted in Fig. 11 for two values of the orthotropic parameter, fl = 1 and 2, and two values of the annular ratio, b/a = 0.10 and 0.25. Similar results for a uniformly distributed step load and a step function ring load at the outer edge are given in Fig. 12. The maximum deflection and 'period' decrease with fl and annular ratio b/a. The influence offl on W(1)maxand arb(0)max and tTm(0)maxhas been plotted in Figs 13 and 14, respectively, for two values of the annular ratio, b/a = O. 1 and 0.25. Three types of uniformly distributed loadings, namely, step function, exponential decay and sinusoidal pulse, and a case of step function ring load at the outer edge have been considered. It is noted that the predominant bending stresses decrease with b/a for all cases while these decrease with fl except for the case of sinusoidal pulse. Several new results reported in the present investigation may be of interest in the design of composite annular plate structures.
REFERENCES 1. J. L. Nowinski, Nonlinear vibrations of elastic circular plates exhibiting rectilinear orthotropy, ZAMP, 14 (1963) pp. 112-24. 2. R. S. Alwar and B. Sekhar Reddy, Large deflection static and dynamic analysis of isotropic and orthotropic annular plates, Int. J. Non-Linear Mechanics, 14 (1979) pp. 347-59. 3. J. L. Nowinski, Cylindrically orthotropic plates, ZAMP, 11 (1960) pp. 218-28. 4. J. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1976.