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Non-linear vibration and postbuckling of orthotropic thin circular plates on elastic foundations
Applied Acoustics 19 (1986) 401-419
Non-linear Vibration and Postbuckling of Orthotropic Thin Circular Plates on Elastic Foundations P. C. Dumir* Department of Applied Mechanics, Indian Institute of Technology, New Delhi, 110016 (India)
S UMbtA R Y This paper presents an approximate solution of the large deflection ax'wmmetric response of cylindrically orthotropic thin circular plates resting on Winkle& Pa.~ternakand non-linear Winklerfoundations. Plates with edges elastically restrained aga~nst rotation and inplane disphzcement are analyzed. Von K6rm~n-type equations are employed. The deflection is approximated by a one-term mode shape and Galerkin's method is used to obtain Duffing's equation for the central dd~ection. No,:-linear frequencies, postbuckling response and maximum response under step load have been obtained. The effect of various parmneters is studied and it is shown that satisfactory engineering accurao is achieved by the present method.
INTRODUCTION The problems of non-linear vibration and postbuckling response of plates have been extensively treated by Chia t and Nayfeh and Mook. 2 The case of fibre-reinforced cylindrically orthotropic circular plates resting on Pasternak elastic foundations has not received much attention. An approximate one-term space mode solution of the non-linear vibration of circular plates with rectilinear orthotropy has been presented by Nowinski 3 using yon Kfirmfin equations. Non-linear free vibrations of cylindrically orthotropic circular plates have been reported ~ using finite elements. Gajendra7 and Datta ~ employed Berger's equations and * Present address: 1-12, IV-4/A-4, liT Campus, Hauz Khas, New Delhi, 110016 (India). 401 Applied Acoustics 0003-682X/86/$03"50 © Elsevier Applied Science Publishers Ltd, England, 1986. Printed in Great Britain
402
P.C. ~ r
used the one-term Galerldn method to analyse the non-linear free vibration of immovable clamped and simply supported isotropic circular plates resting on linear Winkler foundations. Dumir et aL 9 have presented an 'assumed..time-mode' solution of von K~rm~n's equations using the orthogonal point collocation method to analyse the non-linear vibration of orthotropic circular plates on Winkler, Pasternak and non-linear Winlder foundations. Non-linear transient response of immovable clamped and simply supported isotropic circular plates on WinklerPasternak foundations subjected to uniformly distr;.buted step load has been analysed by Nath ~° using the Chebyshev series and the Houbolt technique. Axisymmetric postbuckling analysis of orthotropic circular plates has been presented by Dumir and Khatri ~ by the orthogonal point collocation method. Raju and Rao 12 have used finite elements to obtain the postbuckling response of orthotropic circular plates, with edges elastically restrained against rotation, resting on linear Winkler foundations. This paper presents an approximate one-term 'space-mode-solution' of the yon K~rm~n-type governing equations of o]thotropic uniform thin plates resting on Winkler, Pasternak and non-linear Winkler foundations. A perfect bonding between the plate and the foundation has been assumed. Plates with edges elastically restrained against rotation and inplane displacement are considered. Non-linear free vibratioil response, the maximum transient response to uniformly distributed transverse step load and postbuckling response to radial load at the edge have been obtained. It is shown that the approximate method gives sufficient accuracy for engineering applications. The effect of the orthotropic parameter is investigated.
GOVERNING EQUATIONS The foundation is modelled in terms of the Winlder parameter, k, the non-linear parameter, kl, and the shear parameter, g, of the Pasternak foundation. 13 For the axisymmetric case, it introduces a distributed force on the circular plate given by - ( k w * - k , w . 3 - g V ~ w *)
(1)
where w*(r, t) is the '.ransverse deflection. The governing equations for
Non-linear vibration and buckling of plates on foundations
40".';
the moderately large axisymmetric deflection of a circular plate given by Chia ~ are modified to include the effect of the foundations~3:
= q -- 7 hw.. - kw* + klw *z + gVZw* ~,*,,+-
1 , /3 Eo 2 -• = r ~'" r 2~b +h~r(W* ) 0
(2)
where N, = ¢*/r
O = Eeh3/12(fl -- v~)
No = 'k,*,
fl = ~E, = ~v,
7% = 7h + f f
7o,/2(z)dz
(3)
The mass densities of the plate and foundation are 7 and 70- The effective mass density, '3 7*, is based on the assumption that the transverse displacement, w*(r, z, t), in a single layer foundation of depth H is given by w*(r, z, t) = ~,*(r, t ) , ~ z )
(4)
The radius and the thickness of the plate are a and h. The Young's moduli and Poisson's ratios are E,,Eo and Vr, VO. The orthotropic parameter is B. The stress function is ¢r* and q is the uniformly distributed transverse load. Introducing the dimensionless parameters:
w= T
0=~
ka 4 K = E/,~
o = -a
k la 4 K , = E,h
~=
ga 2 G = E,h----S
t
= qa_~_ 4 Q E,h"
(5)
The gove~ing equations (2) reduce to the following dimensionless form: g, + w'"' + -2 w"' -- /3 w" + # w' + l ~ / ; - v.~)
x[Kw_Klw
3 - G(pw,), ] = 12(fl-v~)Q ~ + (~w')' ~
p2~,, + pC/_/~q, + 6 ( / ~ - ~ ) p w '~ = o
where ( )' and (') are partial derivatives with respect to p and ~.
(6)
(7)
404
P.C. Dunar
For a plate with an elastically restrained outer edge, with rotational and inplane stiffnesses K* and K*, subjected to applied inplane radial force resultant N* at the outer edge, the boundary conditions are: r = a:
M,
K b w,,
N , = N * - K ~u*
(8(a))
where u* is the radial displacement at midplane. Introduce dimensionless parameters Kb, K~ and N: 12K~_____a Kb =
Kj =
E,,h 3
N=
(8(b))
E,h
Z,,h \ h i
The boundary conditions take the following dimensionless form: K,(¢' - vo~) + / ~ / = 1 2 ( ~ - v2)N (9(a)) p=l: +.
0
(9(b))
. o
The conditions of zero slope, zero shear and zero radial displacement at the centre are: w*,(0)=0
rlim(rQr)-- ~i~oD( - r w * . , , r - w ~ + / ~ ) - o
•
r
1
(10(a))
,
- v°~'*(O) = 0
(lO(b))
hE0 These internal conditions take the following dimensionless form: W(O)= 0
(
lira - pw" - w" + ~
p-'*O
";I
=0
¢(0)=0
(11 (a)) (I l(b))
METHOD OF SOLUTION The linear deflection response of an orthotropic circular plate subjected to a uniformly distributed transverse load, Q, is given byt4: w(p) = A + Bp 1 +p + 3(/~- v2)Qp'/[2~(9 -/1)]
p= ~
(12)
Non-linear vibration and buckling of plates on foundations
405
where A and B are determined from the two boundary conditions for w at the outer edge. An approximate solution for non-linear response is thus based on the following assumed "=:patialmode shape: w(p, z) = 0(T)[1 + CtP t +" + C2P']
(13)
This mode shape satisfies both of the conditions (1 lfa)) at p = 0. The boundary conditions (9(b)) yield
C2=(1 + p ) {
{(p + v0) +
12--p(1 + p ) + ( 3 - - p ) [ % + ( ( ~ - v ~ ) / , 8 ) K ~ , ] }
(14) "
Ct = -1 -C 2 The deflection from eqn (13) is substituted in eqn (7) and the resulting equation is solved subject to the boundary conditions (9(a)) and (1 |(b)) to yield = $2(C3p2p+ t + Cd~p+4 + CspV + C6PZ,)+ CvNpl,
(15)
where
2, (P + 1) 2c ,
(74=-
48(fl- v~)(p + I)CtC 2
(p+4)'-~
96(/ - volC 2 22 C~ = 49 - p C6 = - [ ~ -
(16)
v~K,)(C3 + C , + C5) + K,{(2p + I)C3
+ (p + 4)C, + 7Cs}]/[,8 + C l = ,8 + K l ( p -
K,tp -
v,)]
%)
Substituting w(p, t) and ~ from eqns (13) and (15) into the equation of motion (eqn (6)), applying the Galerkin procedure (multipIying both sides by pw(p,O and integrating from p = 0 to p = l) and using integration by parts yields an equation for the central deflection in the form of the Dufling equation: Ate; + (A2 + A3N)dp + A,,~pa = AsQ
(17),
406
P.C. Dundr
where t
Ct
I
C2
A x - 2 1 2 + 3 + p v+a +-'g- le
12~- ,2.) A 2 --
4(9 - fl)C2l 2
fl
x G[(p + l)Ctle+ t + 4CzI4] -t
1 ~ - ,~.)KA, (IS)
~ 3 = --CT[(P + 1 ) C t l 2 e + 4C21e+3]
A 4 -- - ( 1 q- p)ClC31ap+ 1 - {(1 -I- p ) C I C 4 Jr 4C2C3}12p+4 -{(1 + p)C,Cs + 4c2c,}1,+7 - ( 1 + p)ClCel2p- 4C2110C s -4C2C61p+ 3
p
Kt
,
C3 3C 2 .
C23 , 3Cl
+ " ~ " !10 + ~ ' p
A5 -
6 ~ - ,~) fl
3CIC 2 II 1+p 13+p+ 4------~ 11
3C2C2
+
3C 2
+ g ¥Tn r, + +
6CTC2
1,+,
1
12
where
(1 + P)CI 4C2 lj --- | pJ[(1 + p ) C l p p + 4C2P 3] dp = ( p + j + I) 4-( ~ - ~ ) d The response for different loadings is obtained as follows. (a) The postbuckling respon~ is obtained from eqn (17) by setting = Q = 0, to yield n --. ~-.,~/Sc,
= 1 + ~1~ 2
(19)
where the buckling load, Arc,, and the postbuckling parameter, el, are given by
Nc, = - A 2 / A 3
e.1 = A 4 / A 2
(20)
Non-linear vibration and buckling of plates on foundations
407
Equation (17) can be written as A t ~ + A2(1 - n ) ~
+ A 4 ~ 3 = AsQ
(2t)
(b) The static response under uniformly distributed load, Q, is given by Q -0tt(1-
ri)~b + ~ 3 ¢ 3
(22)
with ~l = A2/A5
~3 = A,/A s
(23) *
(c) Free non-linear vibrations are governed by + COo~(~+ ~b3) = 0
(24)
where the dimensionless linear frequency, ~0, is related to the dimensional frequency, o~0, by ~ 0 -" °J~[7*ha4/D] 1!2 = [(1 - n)A 2/A I] 1/2 (25)
and the non-linearity parameter, ~, is given by (26)
~: -- A ~ A 2 ( I -- n)
For the case of no applied inplane edge force (n = 0), the non-linearity parameter, e, of non-linear oscillation is the same as the postbuckling parameter, ~. The non-lib ;at period for amplitude, C, can be expressed in terms of the complete elliptic integral, K, of the first kind: T~ TO= 2K(S)/[n(I + tC2) 1/2]
(27)
where S=[~,C2/{2(I + ~C2)}]1/2
(d) The maximum deflection response to uniformly distributed step load Qo is obtained by integrating eqn (21) to yield Atq~'/2 + az(1 - n)~2/2 + A4d~4/4- AsQodp = constant = 0
(28)
Thus, the maximum deflection under step load Q0 is given by O~1
Qo= T(I --
O~3
+Z ~m,, 3
(29)
RESULTS AND DISCUSSION As a check on the accuracy of the approximate method, the present results for linear and non-linear frequencies, buckling and postbuckling
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Non-linear vibrazion and buckling of plates on foundations
411
TABLE 3 Comparison of Maximum Transient Response to Step Load (re --- 0.25, K! = O,/~ = 3, n = O)
Kb
g,
Qo
K
~
¢,~ Present
Dynamicanalysis using collocation
oo
oo
15
0 5 10 5 10
0 0 0 2 4
1.944 1.732 1.539 1-283 0.893
1.867 1 716 1-542 1-270 0.879
{J
oo
10
0 5 10
0 0 0
1-931 1.672 1.433
O0
0
12
5
2
1.15~
10
4
0"730
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0 0 0 2
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1-890 1-564 !'359 1"080
loads and the maximum deflection response to uniformly distributed step load are compared with the numerical re ;ults obtained by the method of orthogonal point collocation used by the author. 9'1~:5 Nonlinear free vibration results for orthotropic pl;~tes are compared in Table 1. It is noticed that, over a wide range of values of the foundation parameters and the edge stiffnesses, the error in the linear frequency, coo, ~s less than 1 per cent and the error in the period ratio, T/To, for C = 2 is less than 2 per cent for most cases. The comparison of buckling and postbuckling loads for platez with movable edges is given in Table 2 f o r / / = 2. The error in the buckling and postbuckling loads is much less than 5 per cent in most cases. The maximum deflection response to uniformly distributed step loads is compared in Table 3 f o r / / = 3. The error in the maximum response is less than 8 per cent. It is concluded from these comparisons that good engineering a~uracy is achieved by the approximate method for ~he deflection response under different types of loadings. The results for plates with elastically restrained edges, resting on linear Winkler (K! = G = 0), Pasternak (KI = 0) ai~d non-linear Winkler
412
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(G = 0) foundations, axe presented in Tables 4, 5 and 6 for the orthotropic parameters fl = 3, 5 and 10, respectively. Some observations can be made from these tabulated results. The deflection parameters, a~ and a3, the 1Aneaxfrequency, tOo, and the buckling load, Nc,, increase with the ~rthotropic parameter fl for all values of the foundation parameters, K, KI, G, and the edge stiffnesses, Ks, Ki. However, the effect of fl on the non-lineadt:/parameter, ~, or the postbuclding parameter, e~, is more complex. The linear deflection parameter, a~, the linear frequency, tOo, and the buckling load, Nc,, inc~ase with K, G and Ks. The non-linearity parameter, e, increases with K~(Kb = 0) and decreases w i ~ Ks (Ki = oo). The effect of K~on ~ is more than that of Ks. The non-linearity parameter 8 decreases with K and G. The effect of the foundation shear parameter, G, on the response is greater than that of the Winkler parameter, K.
CONCLUSIONS A unified simple approximate solution for linear and non-linear frequencies, buckling and postbuckling loads, and static and transient deflection response has been presented in this study using yon K/Lrmfin equations. Cylindrically orthotropic circular plates with edges elastically restrained for rotation and inplane displacement, resting on Winkler, Pasternak and non-linear Winkler foundations, are considered. It is shown that the accuracy achieved by the approximate method is satisfactory for engineering applications. The effect of various parameters has been investigated. The buckling load, Nc,, the deflection parameters, at1,0c3, and the linear frequency, tOo, increase with the orthotropic parameter, ft. The non-linearity parameter, 8, increases with the inplane support stiffness, Ki, and decreases with the rotational support stiffness, Kb, and the foundation parameters, K and G. The e~'cts of the inplane stiffness, Ki, and the shear parameter, G, of the foundation are more pronounced than the effects of the rotational stiffness, Kb, and the Winkler parameter, K, respectively. REFERENCES I. C. Y. Chia, Nonlinear analysis of plates, McGraw-Hill, New York, 1980. 2. A. H, Nayfeh and D. T. Mook, Nonlinear oscillations, Wiley-hiterscience, New York, 1979.
Non-linear vibration and buckling of plates on foundations
419
3. J. L. Nowinski, Nonlinear vibrations of elastic circular plates exhibiting rectilinear orthotropy. Zeitschrift flit angewandte Mathematik und Physik, 14 (1963), pp. 113-24. 4. G. V. Rao, K. K. Raju and L S. Raju, Finite element formulation for the large amplitude free vibrations of beams and orthotropic circular plates. Computers and Structures, 6 (1976), pp. 169-72. 5. J. N. Reddy, C. L. Huang and I. R. Singh, Large deflections and large amplitude vibrations of axisymmetric circular plate. International Journal of Numerical Methods in Engineering, 17 (1981), pp. 527-41. 6. G. V. Rao and K. K. Raju, Large amplitude axisymmetric vibrations of orthotropic circular plates elastically restrained against rotation. Journal of Sound and Vibration, 69 (1980), pp. 175-80. 7. N. Gajendra, Large amplitude vibration of plates on elastic foundations. International Journal of Nonlinear Mechanics, 2 (1967), pp. 163-72. 8. S. Datta, Large amplitude free vibration of irregular plates placed on elastic foundation. International Journal of Nonlinear Mechanics, I1 (1976), pp. 337-45. 9. P. C. Dumir, Ch. R. Kumar and M. L. Gandhi, Nonlinear axisymmetric vibration oforthotropic thin circular plates on elastic foundations. Journal of Sound and Vibration, 103 (1985), pp. 273-85. 10. Y. Nath, Large amplitude response of circular plates on elastic foundation-. International Journal of Nonlinear Mechanics, 17 (1982), pp. 285-96. ! 1. P.C. Dumir and K. N. Khatri, Axisymmetric postbuckling of ortho~ropic thin tapered circular plates. Fibre Science and Technology, 21 (I984), pp. 233-45. 12. K. K. Raju and G. V. Rao, Postbuckling of cylindrically orthotropic circular plates on elastic foundations with edge elastically restrained against rotation. Computers and Structures, 18 (1984), pp. ! 183-87. 13. V. Z. Klasov and U. N. Leontiev, Beams, plates and shells on plastic foundations. (Translated from the Russian.) Israel Program for Scientific Translations, Jerusalem, 1966. 14. S.G. Lekhnitski, Anisotropic plates, Gordon and Breach, New York, 1968. 15. P. C. Dumir, M. L. Gandhi and Y. Nath, Nonlinear static and transient analysis of orthotropic thin circular plates with elastically restrained edge under central load. Journal of Composite Materials, 17 (1983), pp. 478-91.
Non-linear Vibration and Postbuckling of Orthotropic Thin Circular Plates on Elastic Foundations P. C. Dumir* Department of Applied Mechanics, Indian Institute of Technology, New Delhi, 110016 (India)
S UMbtA R Y This paper presents an approximate solution of the large deflection ax'wmmetric response of cylindrically orthotropic thin circular plates resting on Winkle& Pa.~ternakand non-linear Winklerfoundations. Plates with edges elastically restrained aga~nst rotation and inplane disphzcement are analyzed. Von K6rm~n-type equations are employed. The deflection is approximated by a one-term mode shape and Galerkin's method is used to obtain Duffing's equation for the central dd~ection. No,:-linear frequencies, postbuckling response and maximum response under step load have been obtained. The effect of various parmneters is studied and it is shown that satisfactory engineering accurao is achieved by the present method.
INTRODUCTION The problems of non-linear vibration and postbuckling response of plates have been extensively treated by Chia t and Nayfeh and Mook. 2 The case of fibre-reinforced cylindrically orthotropic circular plates resting on Pasternak elastic foundations has not received much attention. An approximate one-term space mode solution of the non-linear vibration of circular plates with rectilinear orthotropy has been presented by Nowinski 3 using yon Kfirmfin equations. Non-linear free vibrations of cylindrically orthotropic circular plates have been reported ~ using finite elements. Gajendra7 and Datta ~ employed Berger's equations and * Present address: 1-12, IV-4/A-4, liT Campus, Hauz Khas, New Delhi, 110016 (India). 401 Applied Acoustics 0003-682X/86/$03"50 © Elsevier Applied Science Publishers Ltd, England, 1986. Printed in Great Britain
402
P.C. ~ r
used the one-term Galerldn method to analyse the non-linear free vibration of immovable clamped and simply supported isotropic circular plates resting on linear Winkler foundations. Dumir et aL 9 have presented an 'assumed..time-mode' solution of von K~rm~n's equations using the orthogonal point collocation method to analyse the non-linear vibration of orthotropic circular plates on Winkler, Pasternak and non-linear Winlder foundations. Non-linear transient response of immovable clamped and simply supported isotropic circular plates on WinklerPasternak foundations subjected to uniformly distr;.buted step load has been analysed by Nath ~° using the Chebyshev series and the Houbolt technique. Axisymmetric postbuckling analysis of orthotropic circular plates has been presented by Dumir and Khatri ~ by the orthogonal point collocation method. Raju and Rao 12 have used finite elements to obtain the postbuckling response of orthotropic circular plates, with edges elastically restrained against rotation, resting on linear Winkler foundations. This paper presents an approximate one-term 'space-mode-solution' of the yon K~rm~n-type governing equations of o]thotropic uniform thin plates resting on Winkler, Pasternak and non-linear Winkler foundations. A perfect bonding between the plate and the foundation has been assumed. Plates with edges elastically restrained against rotation and inplane displacement are considered. Non-linear free vibratioil response, the maximum transient response to uniformly distributed transverse step load and postbuckling response to radial load at the edge have been obtained. It is shown that the approximate method gives sufficient accuracy for engineering applications. The effect of the orthotropic parameter is investigated.
GOVERNING EQUATIONS The foundation is modelled in terms of the Winlder parameter, k, the non-linear parameter, kl, and the shear parameter, g, of the Pasternak foundation. 13 For the axisymmetric case, it introduces a distributed force on the circular plate given by - ( k w * - k , w . 3 - g V ~ w *)
(1)
where w*(r, t) is the '.ransverse deflection. The governing equations for
Non-linear vibration and buckling of plates on foundations
40".';
the moderately large axisymmetric deflection of a circular plate given by Chia ~ are modified to include the effect of the foundations~3:
= q -- 7 hw.. - kw* + klw *z + gVZw* ~,*,,+-
1 , /3 Eo 2 -• = r ~'" r 2~b +h~r(W* ) 0
(2)
where N, = ¢*/r
O = Eeh3/12(fl -- v~)
No = 'k,*,
fl = ~E, = ~v,
7% = 7h + f f
7o,/2(z)dz
(3)
The mass densities of the plate and foundation are 7 and 70- The effective mass density, '3 7*, is based on the assumption that the transverse displacement, w*(r, z, t), in a single layer foundation of depth H is given by w*(r, z, t) = ~,*(r, t ) , ~ z )
(4)
The radius and the thickness of the plate are a and h. The Young's moduli and Poisson's ratios are E,,Eo and Vr, VO. The orthotropic parameter is B. The stress function is ¢r* and q is the uniformly distributed transverse load. Introducing the dimensionless parameters:
w= T
0=~
ka 4 K = E/,~
o = -a
k la 4 K , = E,h
~=
ga 2 G = E,h----S
t
= qa_~_ 4 Q E,h"
(5)
The gove~ing equations (2) reduce to the following dimensionless form: g, + w'"' + -2 w"' -- /3 w" + # w' + l ~ / ; - v.~)
x[Kw_Klw
3 - G(pw,), ] = 12(fl-v~)Q ~ + (~w')' ~
p2~,, + pC/_/~q, + 6 ( / ~ - ~ ) p w '~ = o
where ( )' and (') are partial derivatives with respect to p and ~.
(6)
(7)
404
P.C. Dunar
For a plate with an elastically restrained outer edge, with rotational and inplane stiffnesses K* and K*, subjected to applied inplane radial force resultant N* at the outer edge, the boundary conditions are: r = a:
M,
K b w,,
N , = N * - K ~u*
(8(a))
where u* is the radial displacement at midplane. Introduce dimensionless parameters Kb, K~ and N: 12K~_____a Kb =
Kj =
E,,h 3
N=
(8(b))
E,h
Z,,h \ h i
The boundary conditions take the following dimensionless form: K,(¢' - vo~) + / ~ / = 1 2 ( ~ - v2)N (9(a)) p=l: +.
0
(9(b))
. o
The conditions of zero slope, zero shear and zero radial displacement at the centre are: w*,(0)=0
rlim(rQr)-- ~i~oD( - r w * . , , r - w ~ + / ~ ) - o
•
r
1
(10(a))
,
- v°~'*(O) = 0
(lO(b))
hE0 These internal conditions take the following dimensionless form: W(O)= 0
(
lira - pw" - w" + ~
p-'*O
";I
=0
¢(0)=0
(11 (a)) (I l(b))
METHOD OF SOLUTION The linear deflection response of an orthotropic circular plate subjected to a uniformly distributed transverse load, Q, is given byt4: w(p) = A + Bp 1 +p + 3(/~- v2)Qp'/[2~(9 -/1)]
p= ~
(12)
Non-linear vibration and buckling of plates on foundations
405
where A and B are determined from the two boundary conditions for w at the outer edge. An approximate solution for non-linear response is thus based on the following assumed "=:patialmode shape: w(p, z) = 0(T)[1 + CtP t +" + C2P']
(13)
This mode shape satisfies both of the conditions (1 lfa)) at p = 0. The boundary conditions (9(b)) yield
C2=(1 + p ) {
{(p + v0) +
12--p(1 + p ) + ( 3 - - p ) [ % + ( ( ~ - v ~ ) / , 8 ) K ~ , ] }
(14) "
Ct = -1 -C 2 The deflection from eqn (13) is substituted in eqn (7) and the resulting equation is solved subject to the boundary conditions (9(a)) and (1 |(b)) to yield = $2(C3p2p+ t + Cd~p+4 + CspV + C6PZ,)+ CvNpl,
(15)
where
2, (P + 1) 2c ,
(74=-
48(fl- v~)(p + I)CtC 2
(p+4)'-~
96(/ - volC 2 22 C~ = 49 - p C6 = - [ ~ -
(16)
v~K,)(C3 + C , + C5) + K,{(2p + I)C3
+ (p + 4)C, + 7Cs}]/[,8 + C l = ,8 + K l ( p -
K,tp -
v,)]
%)
Substituting w(p, t) and ~ from eqns (13) and (15) into the equation of motion (eqn (6)), applying the Galerkin procedure (multipIying both sides by pw(p,O and integrating from p = 0 to p = l) and using integration by parts yields an equation for the central deflection in the form of the Dufling equation: Ate; + (A2 + A3N)dp + A,,~pa = AsQ
(17),
406
P.C. Dundr
where t
Ct
I
C2
A x - 2 1 2 + 3 + p v+a +-'g- le
12~- ,2.) A 2 --
4(9 - fl)C2l 2
fl
x G[(p + l)Ctle+ t + 4CzI4] -t
1 ~ - ,~.)KA, (IS)
~ 3 = --CT[(P + 1 ) C t l 2 e + 4C21e+3]
A 4 -- - ( 1 q- p)ClC31ap+ 1 - {(1 -I- p ) C I C 4 Jr 4C2C3}12p+4 -{(1 + p)C,Cs + 4c2c,}1,+7 - ( 1 + p)ClCel2p- 4C2110C s -4C2C61p+ 3
p
Kt
,
C3 3C 2 .
C23 , 3Cl
+ " ~ " !10 + ~ ' p
A5 -
6 ~ - ,~) fl
3CIC 2 II 1+p 13+p+ 4------~ 11
3C2C2
+
3C 2
+ g ¥Tn r, + +
6CTC2
1,+,
1
12
where
(1 + P)CI 4C2 lj --- | pJ[(1 + p ) C l p p + 4C2P 3] dp = ( p + j + I) 4-( ~ - ~ ) d The response for different loadings is obtained as follows. (a) The postbuckling respon~ is obtained from eqn (17) by setting = Q = 0, to yield n --. ~-.,~/Sc,
= 1 + ~1~ 2
(19)
where the buckling load, Arc,, and the postbuckling parameter, el, are given by
Nc, = - A 2 / A 3
e.1 = A 4 / A 2
(20)
Non-linear vibration and buckling of plates on foundations
407
Equation (17) can be written as A t ~ + A2(1 - n ) ~
+ A 4 ~ 3 = AsQ
(2t)
(b) The static response under uniformly distributed load, Q, is given by Q -0tt(1-
ri)~b + ~ 3 ¢ 3
(22)
with ~l = A2/A5
~3 = A,/A s
(23) *
(c) Free non-linear vibrations are governed by + COo~(~+ ~b3) = 0
(24)
where the dimensionless linear frequency, ~0, is related to the dimensional frequency, o~0, by ~ 0 -" °J~[7*ha4/D] 1!2 = [(1 - n)A 2/A I] 1/2 (25)
and the non-linearity parameter, ~, is given by (26)
~: -- A ~ A 2 ( I -- n)
For the case of no applied inplane edge force (n = 0), the non-linearity parameter, e, of non-linear oscillation is the same as the postbuckling parameter, ~. The non-lib ;at period for amplitude, C, can be expressed in terms of the complete elliptic integral, K, of the first kind: T~ TO= 2K(S)/[n(I + tC2) 1/2]
(27)
where S=[~,C2/{2(I + ~C2)}]1/2
(d) The maximum deflection response to uniformly distributed step load Qo is obtained by integrating eqn (21) to yield Atq~'/2 + az(1 - n)~2/2 + A4d~4/4- AsQodp = constant = 0
(28)
Thus, the maximum deflection under step load Q0 is given by O~1
Qo= T(I --
O~3
+Z ~m,, 3
(29)
RESULTS AND DISCUSSION As a check on the accuracy of the approximate method, the present results for linear and non-linear frequencies, buckling and postbuckling
408
P. ¢. Dumb"
(.~
0
0
0
0
0
0
0
0
0
0
0
!
11
! ii 0 "[:
i i ~ i ~ 1~~ ~~~_ _ .
*
*
°
°
*
*
°
~
p
*
*
m
0
oo~
~ ~ ~ ~o~oo
0
0 0
0
Oqt"
Non.l~near vibration and buckling of plates on foundation~
~
°
°
.
°
0
.
0
°
0
,
0
°
0
~
0
~
0
~
0
0
0
0
~
0
0
0
~
0
0
0
0
0
0
0
0
°
~
°
o
.
.
0
0
~
Q
O
0
~
II
.
0
~
.
0
0
0
~
0
0
~
0
~
~
0
~
0
~
0
~
0
0
,
~ ~ ~ ~ ~
O
~
O
0
0
0
I m
0
.
0
oooooooooooo~
O
,
0
0
~
I g
~
410
P . C . Dumb"
II 6 II
II II .
o
°
°
°
.
°
.
,
.
°
.
.
.
°
°
°
°
°
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*
.
o
°
°
°
.
.
0 o
-[:: G.
8
0
il .
I
°
I
I
Non-linear vibrazion and buckling of plates on foundations
411
TABLE 3 Comparison of Maximum Transient Response to Step Load (re --- 0.25, K! = O,/~ = 3, n = O)
Kb
g,
Qo
K
~
¢,~ Present
Dynamicanalysis using collocation
oo
oo
15
0 5 10 5 10
0 0 0 2 4
1.944 1.732 1.539 1-283 0.893
1.867 1 716 1-542 1-270 0.879
{J
oo
10
0 5 10
0 0 0
1-931 1.672 1.433
O0
0
12
5
2
1.15~
10
4
0"730
2-083 1-791 ! .527 1"203 0.764
0 5 10 5
0 0 0 2
! .987 1.669 !.413 1"114
1-890 1-564 !'359 1"080
loads and the maximum deflection response to uniformly distributed step load are compared with the numerical re ;ults obtained by the method of orthogonal point collocation used by the author. 9'1~:5 Nonlinear free vibration results for orthotropic pl;~tes are compared in Table 1. It is noticed that, over a wide range of values of the foundation parameters and the edge stiffnesses, the error in the linear frequency, coo, ~s less than 1 per cent and the error in the period ratio, T/To, for C = 2 is less than 2 per cent for most cases. The comparison of buckling and postbuckling loads for platez with movable edges is given in Table 2 f o r / / = 2. The error in the buckling and postbuckling loads is much less than 5 per cent in most cases. The maximum deflection response to uniformly distributed step loads is compared in Table 3 f o r / / = 3. The error in the maximum response is less than 8 per cent. It is concluded from these comparisons that good engineering a~uracy is achieved by the approximate method for ~he deflection response under different types of loadings. The results for plates with elastically restrained edges, resting on linear Winkler (K! = G = 0), Pasternak (KI = 0) ai~d non-linear Winkler
412
P . C . Dum~"
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.
°
.
.
o
.
~
° m
o
o
.
.
° N
I
f
0
J o o o o ~
0
9"*
9
o o o o ~
0
0
o
,r
o M
~
Non-linear vibration a~l ~ucklmg o~'plates on foundations
0 ~
0 ~ 0 0 0
•
•
°
0 0 0
. . . .
°
°
~
~
.o I I
•
•
~
C~ ~
~
I I ~.-
•
°
•
•
0
~
I I
~
•
•
0
0
I I
•
•
(D
! I
°
t ~
oo
0o088
00088
c, o o 8 8
88
88
'*8
o'~88o
o',t
o',t
8~
8o
0
~
Q
--'
--'
0
"q"
'q"
'q"
'q"
m
88o
8 8~
I
413
414
?. C.
i G, 0
~1" q " q "
~1" ~D IP- ..-. I ~ m
m6~gg
L~
0
ggom
.--
I~
m
I~
I~
I " . (10 0 i~l
o¢~,,~g g g g
0
~r m
Non-linear vibration and buckling of plates on foundations
.
~ . ,
.
.
.
.
~
•
•
.
.
.
.
~
*I*j
v,~
0
¢~
.
.
.
•
.
~
~
.
~
¢~
•
~
.
°
.
~
*"
I
"--
.
415
416
P . C . D~m/r
!
| .
0
*
.
.
°
0
0
*
°
°
0
°
°
0
*
*
.
.
0
*
°
0
*
0
0
~
0
t
0
0
*
*
0
°
0
*
*
II
II
~
" ~
~r~-
r~
"
"
m
a. o
& ~J
m
oo~
r~
0
8888oo
oo~8888
0
0
*
0
Non-linear vibration and Buckling of plates on foundations
V1
fq
t~
~
~
,~
~
~0
r~
.m
m
~
-.
o o o o
.
,.,.,~,~,t',,lV
• ~
~
•
~
"
"o
~
F
~
.
.
" ~
~
~
oo
o o o o o
~,
,-.,,,-,,-~r,~'q"
•
~
r~
•
t~
o
('4
~
~
•
.
.
F , ~ . ~ F .
~ r ~ r : . .
,;~r'..
,~'~
•
;.:~r,~,,~
~_~,'~
~ . < r ~
oo
00o88
o0088
ooo88
88
88
o~
o,~
o,~
8o
8o
8
880
880
88o
0
O
¢>
'~
"
0
~
q"
'q"
~1"
i,m
.
417
418
P.c. Dum/r
(G = 0) foundations, axe presented in Tables 4, 5 and 6 for the orthotropic parameters fl = 3, 5 and 10, respectively. Some observations can be made from these tabulated results. The deflection parameters, a~ and a3, the 1Aneaxfrequency, tOo, and the buckling load, Nc,, increase with the ~rthotropic parameter fl for all values of the foundation parameters, K, KI, G, and the edge stiffnesses, Ks, Ki. However, the effect of fl on the non-lineadt:/parameter, ~, or the postbuclding parameter, e~, is more complex. The linear deflection parameter, a~, the linear frequency, tOo, and the buckling load, Nc,, inc~ase with K, G and Ks. The non-linearity parameter, e, increases with K~(Kb = 0) and decreases w i ~ Ks (Ki = oo). The effect of K~on ~ is more than that of Ks. The non-linearity parameter 8 decreases with K and G. The effect of the foundation shear parameter, G, on the response is greater than that of the Winkler parameter, K.
CONCLUSIONS A unified simple approximate solution for linear and non-linear frequencies, buckling and postbuckling loads, and static and transient deflection response has been presented in this study using yon K/Lrmfin equations. Cylindrically orthotropic circular plates with edges elastically restrained for rotation and inplane displacement, resting on Winkler, Pasternak and non-linear Winkler foundations, are considered. It is shown that the accuracy achieved by the approximate method is satisfactory for engineering applications. The effect of various parameters has been investigated. The buckling load, Nc,, the deflection parameters, at1,0c3, and the linear frequency, tOo, increase with the orthotropic parameter, ft. The non-linearity parameter, 8, increases with the inplane support stiffness, Ki, and decreases with the rotational support stiffness, Kb, and the foundation parameters, K and G. The e~'cts of the inplane stiffness, Ki, and the shear parameter, G, of the foundation are more pronounced than the effects of the rotational stiffness, Kb, and the Winkler parameter, K, respectively. REFERENCES I. C. Y. Chia, Nonlinear analysis of plates, McGraw-Hill, New York, 1980. 2. A. H, Nayfeh and D. T. Mook, Nonlinear oscillations, Wiley-hiterscience, New York, 1979.
Non-linear vibration and buckling of plates on foundations
419
3. J. L. Nowinski, Nonlinear vibrations of elastic circular plates exhibiting rectilinear orthotropy. Zeitschrift flit angewandte Mathematik und Physik, 14 (1963), pp. 113-24. 4. G. V. Rao, K. K. Raju and L S. Raju, Finite element formulation for the large amplitude free vibrations of beams and orthotropic circular plates. Computers and Structures, 6 (1976), pp. 169-72. 5. J. N. Reddy, C. L. Huang and I. R. Singh, Large deflections and large amplitude vibrations of axisymmetric circular plate. International Journal of Numerical Methods in Engineering, 17 (1981), pp. 527-41. 6. G. V. Rao and K. K. Raju, Large amplitude axisymmetric vibrations of orthotropic circular plates elastically restrained against rotation. Journal of Sound and Vibration, 69 (1980), pp. 175-80. 7. N. Gajendra, Large amplitude vibration of plates on elastic foundations. International Journal of Nonlinear Mechanics, 2 (1967), pp. 163-72. 8. S. Datta, Large amplitude free vibration of irregular plates placed on elastic foundation. International Journal of Nonlinear Mechanics, I1 (1976), pp. 337-45. 9. P. C. Dumir, Ch. R. Kumar and M. L. Gandhi, Nonlinear axisymmetric vibration oforthotropic thin circular plates on elastic foundations. Journal of Sound and Vibration, 103 (1985), pp. 273-85. 10. Y. Nath, Large amplitude response of circular plates on elastic foundation-. International Journal of Nonlinear Mechanics, 17 (1982), pp. 285-96. ! 1. P.C. Dumir and K. N. Khatri, Axisymmetric postbuckling of ortho~ropic thin tapered circular plates. Fibre Science and Technology, 21 (I984), pp. 233-45. 12. K. K. Raju and G. V. Rao, Postbuckling of cylindrically orthotropic circular plates on elastic foundations with edge elastically restrained against rotation. Computers and Structures, 18 (1984), pp. ! 183-87. 13. V. Z. Klasov and U. N. Leontiev, Beams, plates and shells on plastic foundations. (Translated from the Russian.) Israel Program for Scientific Translations, Jerusalem, 1966. 14. S.G. Lekhnitski, Anisotropic plates, Gordon and Breach, New York, 1968. 15. P. C. Dumir, M. L. Gandhi and Y. Nath, Nonlinear static and transient analysis of orthotropic thin circular plates with elastically restrained edge under central load. Journal of Composite Materials, 17 (1983), pp. 478-91.