Non-linear vibration and postbuckling of isotrophic thin circular plates on elastic foundations

Journal ofSound

and Vibration (1986) 107(2), 253-263

NON-LINEAR

VIBRATION

OF ISOTROPIC ON

THIN

ELASTIC

AND

POSTBUCKLING

CIRCULAR

PLATES

FOUNDATIONS

P. C. DUMIR Department of Applied Mechanics, Indian Institute of Technology, New Delhi-l 10016, India

(Received 29 March 1985, and in revised form 6 July 1985)

An approximate solution for the large deflection axisymmetric responses of isotropic thin circular plates resting on Winkler, Pastemak and non-linear Winkler foundations is presented. Plates with edges elastically restrained against rotation and in-plane displacement are considered. Von K&man type equatitis in terms of transverse deflection and stress function are employed. A one term mode shape is used to approximate the transverse deflection and Galerkin’s method is used to obtain an equation for the central deflection which has the form of a Duffing’s equation. Non-linear frequencies, postbuckling response to radial load at the edge and the maximum transient response to transverse step load have been obtained. It is shown that sufficiently accurate results are obtained by this method. Numerical results are presented to illustrate the effect of various parameters.

1. INTRODUCTION The large amplitude axisymmetric vibrations of isotropic uniform thin circular plates have been discussed by several investigators. “Assumed-space-mode” solutions for simply supported and clamped plates with movable and immovable edges have been reported [ 1,2] in which the von Kgrm&n equations were used. Similar studies have been conducted [3,4] for Berger’s equations. An “assumed-time-mode” approach for the von K&-mPn equations, have also been presented [5,6], in which finite differences and finite elements were used, respectively. Multimode non-linear vibration analysis has been reported [7], for the von Kgrmgn equations. There have been few studies of the non-linear response of plates on elastic foundations [8]. The static response of plates on Winkler foundations under uniformly distributed transverse load has been studied by Sinha [9], using Berger’s equations. Bolton [lo] employed the Galerkin method with three terms to analyze the static response of circular plates ‘on Winkler foundations. Gajendra [ 1 l] and Datta [ 121 used Berger’s equations and a one term Galerkin method to analyze non-linear free vibration of immovable clamped and simply supported circular plates on Winkler foundations. Dumir et al. [ 131 have presented an “assumed-time-mode” analysis of the non-linear vibration of circular plates on Winkler, Pasternak and non-linear Winkler foundations. They used the von Kgrrnin equations and the orthogonal point collocation method for spatial discretization. The non-linear transient response of immovable clamped and simply supported circular plates on Winkler-Pastemak foundations, subjected to uniformly distributed step loads, has been investigated by Nath [14], using Chebyshev series and the Houbolt technique. Axisymmetric postbuckling analyses of isotropic thin circular plates have been carried out by several investigators [15-191 using various techniques. Raju and Rao [20] have 253 002240X/86/1

10253+ 11 $03.00/O

@ 1986 Academic Press Inc. (London) Limited

254

P. C. DUMIR

used finite elements to study the postbuckling response of circular plates, with edges elastically restrained against rotation, resting on Winkler foundations. This paper presents an approximate one term “space-mode-solution” of the governing von K&man type equations for isotropic uniform thin plates resting on Winkler, Pasternak and non-linear Winkler foundations. Perfect bonding between the plate and the foundation has been assumed. Plates with edges elastically restrained against rotation and in-plane displacement are considered. The non-linear free vibration response, the postbuckling response to a radial load at the edge and the maximum transient response to a uniformly distributed transverse step load are obtained. The effects of various parameters are investigated.

2. MATHEMATICAL

FORMULATION

k, the non-linear The foundation is modelled in terms of the Winkler parameter parameter k, and the shear parameter g of the Pasternak foundation [8] (a list of nomenclature is given in the Appendix). For the axisymmetric case, there is a distributed force on the circular plate given by -[ kw* - k, w*3 - gL( w*)],

(1)

where w*( r, t) is the transverse deflection of the plate and L( ) = [ r( ),,],r/ r. Including the effect of the foundation [8], the governing equations for the moderately large axisymmetric deflection of a circular plate as given by Chia [21] in terms of w* and stress function F” become DL*(w*)-(h/r)(F;wlr,),,=q-y*hw;,-

kw*+k,w*‘+gL(w*),

L2( F*) = -(E/r) N = h(F?/r),

w;w;,,

(2)

D = Eh3/ 12( 1 - v’),

No = hF:r, H

y*h=yh+ I0

yoke

dz.

(3)

The mass densities of the plate and the foundation are y and yO. The effective mass density [ 181 y* 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) = w*(r, t)n(z).

(4)

The radius and thickness of the plate are a and h, and E and v are Young’s modulus and Poisson’s ratio. The uniformly distributed transverse load is q. Upon introducing the dimensionless parameters w=w*/h, K = ka4/ Eh3, the governing

equations

F = F*/ Eh2, K, = k,a4/ Eh, (2) reduce

P = r/a,

T = [D/ y*ha4]“*t,

G = ga2/ Eh3,

to the dimensionless

[1/12(1-v2)](~+V4~)+K~-K,~3-GV2~-(F’~’)’/~=Q, V4F = -w’w”Ip,

Q = qa4/ Eh4,

(5)

forms (6a) (6b)

where ( )’ and ( *) are derivatives with respect to p and T and V’( ) = [p( )‘]‘/p. For a plate with an elastically restrained outer edge, with rotational and in-plane stiffnesses kt and k:, subjected to an applied in-plane radial force resultant N* at the outer edge, the

PLATES

ON

ELASTIC

255

FOUNDATIONS

boundary conditions are r=a:

N I =N*-K*u* I

M,=kXw:,

(7)

.

In terms of the dimensionless parameters Kb, Ki and N, Kb = 12Kza/Eh3,

Ki= K?a/Eh,

N = (N*/Eh)(a/h)*,

(8)

the boundary conditions in dimensionless form are p= 1:

K,(F”-

vF’)+ F’= N,

[(l-v~)K,+Y]w’+w”=O,

w = 0. (9,lO)

An approximate solution is obtained by assuming the non-linear free vibrations to have the same spatial shape, i.e., W(P, T) = HT)(l+

C,p2+ C,P4),

(11)

and equations (10) then yield C _ _6+2~+2(1v2)Kb 15+v+(l-v2)Kb ’

c = l+v+(l-v2)Kb 2

(12)

5+v+(l-v2)Kb’

Substituting expression (11) in equation (6b), integrating the resulting equation making use of boundary condition (9) yields F=-~Z[C,p2+(C:/16)~4+(C,C2/18)p6+(C:/48)p8]+C4Np2,

and (13)

c =_3C:[(3-~)Ki+1]+4C,C2[(5-~)Ki+1]+2C:[(7-~)Ki+l] 3 24[1+(1_v)Ki]

>

1

(14)

c4=[2{l+(l-V)Ki}]’

Substituting w(p, T) and F from equations (11) and (13) into the equation of motion (6a) and applying the Galerkin procedure (multiplying both sides by pw(p, T) and integrating from p = 0 to p = l), one obtains Duffing’s equation A,c~+(A~+A,N)~#J+A,+~=A~Q,

(15)

Ai=[1/12(l-~~)](Z~+ClZ2+C2Z4), A2=[16C2/3(1-v2)]Zo+K(Zo+C~Z2+C2Z4)-4G(C,Z~+4C,Z,), A3 = -8C4(C,Z,,+4C2Z2),

A4=-Kr[Zo+3CrZ2+3(C:+C2)Z4+(C:+6CrC2)Z6 +3(c:+c:c,)z*+3c,c~z,,+c:z,,] +2[4C3CrZ~+(l6C3C~+C~)Z2+5C~C2Z4+~CiC~Z6+~C~Z~], Ij=

I

A~=ZO,

~p’~~+C,p’+~,p~~~~=~~/~j+z)1+~~~/~~+~~1+/~~+~~1.

(16)

The responses for different loadings are obtained as follows. (a) The postbuckling response is obtained from equation (15) by setting 4 = Q = 0, to yield n = N/N,, = 1+ E,+~,. (17) where the buckling load N,, and the postbuckling parameter e1 are given by NC,= -Az/A3,

E, = A4/A2.

(18)

256

P. C. DUMIR

Equation

(15) can be written

load n as

in terms of the in-plane A,&+A2(1-n)4+A,&=A,Q.

(b) The static response

under

uniformly

(19)

distributed

load Q is given by

Q=a,(l-n)++cY,+’

(20)

with a,=Az,/A5.

ay1= A,lA,, (c)

Free non-linear

vibrations

are governed

(21)

by

4 + w;< 4 + &@) = 0, where the dimensionless

linear

frequency

w. is related

(22) to the dimensional

frequency

o$

by wo= ~$[y*ha~/D]r’~=

[(AJA,)(l-

n)]“’

(23)

TABLE 1

Comparison of frequencies

( Y = 0.3, n = 0)

Present

Collocation \

r

Tl To K

K,

G

Kb

0

0

0

0 0 0 0

4 10 4 4 4 4 4 10 4 4 4 4 4 10 4 4 4 4

0 0 0 0 -1 -2 0 0 0 0 -2 2 0 0 0 0 -2 2

0 0

0.5 1.5 0 0 0 0 0.5 1.5 0 0 0 0 0.5 1.5 0 0

2 5 CJ 03 2 0 0 0 0 0 0 0 0 0 0 0 0 co co co 03 az co

Ki

w.

0

4.9470 4.9470 4.9470 4.9470 6.7006 7.8699 10.3280 IO.3280 6.7006 8.2557 11.5619 9.9960 12.7848 8.2557 8.2557 8.2557 11.5619 9.9961 12.7848 8.2557 8.2557 12.2616 14.6924 13.6655 16.1105 12.2616 12.2616

0.2 2 co 00 co XI 0 2 0 0 0 0 0 0 co co 03 co co co co co co 00 co co

-

C=l 0.9145 0.8640 0.7305 0.6518 0.7673 0.8155 0.8608 0.9495 0.8270 0.9665 0.9825 0.9767 0.9856 0.9368 0.9097 0.8188 0.8937 0.8650 0.9105 0.7839 0.8590 0.8949 0.9231 0.9127 0.9348 0.8743 0.9171

c=2 0.7509 0.6538 0.4741 0.3968 0.5161 0.5788 0.6484 0.8353 0.5953 0.8837 0.9352 0.9161 0.9460 0.8025 0.7405 0.5834 0.7080 0.6556 0.7422 0.5366 0.6453 0.7103 0*7700 0.7470 0.7976 O-6718 0.7566

Tl To WO

4.9350 4.9350 4.9350 4.9350 6.6674 7.8139 10.2158 10.2158 6.6674 8,2484 11.5566 9-9846 12.7656 8.2484 8.2484 8.2484 11e5566 9.9846 12.7656 8.2484 8.2484 12.1673 14.6138 13.6237 16.1070 12.1673 12.1673

C=l 0.9119 0.8614 0.7278 0.6493 0.7639 0.8107 0.8530 0.9424 0.8239 0.9656 0.9820 0.9759 0.9848 0.9356 0.9080 0.8189 0.8944 0.8662 0.9123 0.7816 0.8614 0.8895 0.9194 0.9100 0.9344 0.8684 0.9120

c=2 0.7475 0.6448 0.4645 0.3897 0.5031 0.5638 0.6335 0.8242 0.5827 0.8829 0.9349 0.9147 0.9442 0.7974 0.7310 0.5774 0.7074 0.6515 0.7409 0.5245 06407 0.6981 0.7605 0.7393 0.7948 0.6568 0.7470

PLATES ON ELASTIC

and the non-linearity

257

FOUNDATIONS

parameter E is given by E = AJ[A2(

1 - n)].

(24)

For the case of no applied in-plane edge force (n = 0), the non-linearity parameter E of non-linear oscillation is the same as the postbuckling parameter sl. The non-linear period for amplitude C can be expressed in terms of the complete elliptic integral K of the first kind: T/ To = 2K( p)/[ P( 1 + EC’)“~],

p=[2(1+&c2)/(&c2)]-“2.

where

(d) The maximum deflection response to a uniformly obtained by integrating equation (19) to yield

distributed

(25)

step load Q0 is

A,~2/2+A2(1-n)~2/2+A,+4/4-A,Q,,+=constant=0. Thus the maximum

deflection

(26)

&,.,,, is given by

Qo= (%/2)(I

- n)&l,X+

3. RESULTS

(27)

(%/4)&w

AND DISCUSSION

As a check on the extent of the accuracy achieved by the approximate method, the present results for linear and non-linear frequencies, buckling and postbuckling loads, static deflection response under uniformly distributed load, and maximum deflection response to a uniformly distributed step load are compared with the numerical results obtained by the method of orthogonal point collocation used by the author [13,19,22].

TABLE

Comparison

of postbuckling

2

loads ( Ki = 0, v = 0.3) -N

I

\

Present

, K

K,

G

Kb

4=0

0

0

0

0

0.3852 0.5709 0.9006 1.1337

1

4 9 CCJ 4

0

0

0 1 4 9 a,

4

0

1

0 1 4 9 co

4 4 4 4

1 -1 1 -1

0 0 0 0

0 0 co co

14652 1.0728 1.2622 1.5862 1.8021 2.0652 2.0728 2.2622 2.5862 2.8081 3.0652 1.0728 1.0728 2.0652 2.0652

l#J=l

9=2

0.4865 0.6833 1.0379 1.2944 1.6795

0.7901

1.1740 1.3746 1.7235 1.9627 2.2795 2.1740 2.3746 2.7235 2.9627 3.2795 1.0745 1.2736 2.1962 2.3628

1.0203 14497 1.7763 2.3223 1.4777 1.7116 2.1353 24446 2.9223 2.4177 2.7116 3.1353 34446 3.9223 l-0795 1.8759 2.5890 3.2557

, d=O 0.3844 0.5661 0.8751 1 a0745 1.3445 1.0728 1.2581 1.5413 1.6982 1.8794 2.0728 2.2581 2.5413 2.6982 2.8794 1.0728 1.0728 1.8794 1.8794

Collocation $J=l 0.4865 0.6835 1.0302 1.2673 1.6093 1.1721 1.3750 1.7070 1.9090 2.1641 2.1721 2.3750 2.7070 2.9090 3.1641 1.0718 1.2749 2.0905 2.2377

-r +=2 O-7769

1.0092 1.4507 1.7868 2.3344 1.4564 1.6976 2.1408 2.4576 2.9260 2.4565 2.6976 3.1408 3.4577 3.9260 1.0488 1.9250 2.5998 3.2559

258

P. C. DUMIR TABLE

Comparison

qf maximum

3

transient response to step load

(v = 0.25, K, = 0, n = 0) d max

Kb

K,

00

K

co

cc

15

0

cc

10

co

ffi

12

0 5 10 5 10 0 5 10 5 10 0 5 10 5

G

Present

0 0 0 2 4 0 0 0 2 4 0 0 0 2

2.202 1.993 1.794 1.454 0.989 2.380 2.064 1.767 1.378 0.851 2.524 1.996 1.723 1.300

Collocation dynamic analysis 2.332 2.082 1.851 I.520 I.031 2.372 2.039 1.731 1.384 0.846 2.713 2.221 1.829 1.372

Figure 1. Comparison of static response of plates on Winkler foundations. v = 0.3, K, = G = 0. (a) K, = a; Kb =a; - - -,Kb =O. 1,K =O; 2, K = 9.157.Coilocation results: x, Kb =O, K =O;O, K,=O, (b) K,=O.--, K=9.157;A, Kb=qK=O;m,Kh=co, K=9.157.

PLATES ON ELASTIC

FOUNDATIONS

259

The excellent accuracy of the collocation method has been demonstrated [13,19]. The results for non-linear free vibration are compared in Table 1. It is observed that over a wide range of foundation parameters for Winkler, Pasternak and non-linear Winkler foundations good accuracy is achieved by the present method since the maximum difference for the linear frequency o,, is 1.2% and for the period ratio T/ T0 at C = 2 is 2.5%. The buckling and postbuckling loads for movable plates are compared in Table 2, for five sets of foundation parameters and five values of rotational stiffness of the edge. It is seen that engineering accuracy is achieved for the buckling and postbuckling loads by the approximate method. The buckling loads obtained for the simply supported case are very accurate and the postbuckling loads differ by a maximum of 3%. The buckling loads for the clamped case are less accurate with a maximum difference of lo%, but the postbuckling loads for 4 = 2 are very accurate. The non-linear static deflection response of plates on Winkler foundations subjected to uniformly distributed load is compared in Figure 1. It is noted that the maximum difference in the response is about 8%.

TABLE

4

Response of plates on linear Winkler foundations (v = O-3, K1 = G = 0, n = 0)

Tl To I

K 0

Kb

Ki

0

0 0.2 2 co az 00 co

0 0 0 2 5 cc 2 0.2 CT) 4

10

0 0 0 0 2 5 co 2 0.2 03

2 0.2 0 0 0.2 2 co co 00 co

0 0 0 0 2 5 o;,

2 0.2 0 0 0.2 2 00 co co co

2 0.2 co

2 0.2 0

a1 1.4376 1.4376 1.4376 1.4376 2.5682 3.4808 5.8608 2.5682 1.5844 5.8608 4.0030 4.0030 4.0030 4.0030 5.0668 5.9356 8.2608 5.0668 4.1404 8.2608 7.8511 7.8511 7.8511 7.8511 8.8146 9.6179 Il.8608 8.8146 7.9745 Il.8608

a3 0.3776 0.6580 1.7093 2.6604 2.4344 2.3709 2.7619 1.6038 0.6590 0.8571 0.3776 0.6580 I.7093 2.6604 2.4344 2.3709 2.7619 1.6038 0.6591 0.8571 0.3776 0.6580 1.7093 2.6604 2.4344 2.3709 2.7619 1.6038 0.6591 0.8571

-

N,

0.3853 0.4392 0.9247 1.7048 0.4867 1.4652 1.0728 1.2230 2.5748 3.3633 1.2719 2.0652 2.1042 2.3988 5.0500 5.8510 2.4498 2.9652

E =

E,

0.2627 0.4577 1.1890 1.8507 0.9479 0.6812 0.4713 0.6245 0.4160 0.1463 0.0943 0.1644 0.4270 0.6646 0.4805 0.3994 0.3343 0.3165 0.1592 0.1038 0.048 1 0.0838 0.2177 0.3389 0.2762 0.2465 0.2328 0.1820 0.0826 0.0723

00

4.9474 4.9474 4.9474 4.9474 6.7006 7.8699 10.3280 6.7006 5.2035 10.3280 8.2557 8.2557 8.2557 8.2557 9.4116 10.277 12.2616 9.4116 8.4117 12.2616 11.5619 11.5619 11.5619 11.5619 12.4136 13.0819 14.6924 12.4136 11.6737 14.6924

C=l 0.9145 0.8640 0.7305 0.6518 0.7673 0.8155 0.8608 0.8270 0.8741 0.9495 0.9665 0.9438 0.8714 0.8188 0.8587 0.8782 O-8949 0.8997 0.9454 0.9634 0.9825 0.9701 0.9276 0.8937 0.9107 0.9192 0.923 1 0.9383 0.9705 0.9740

c=2 0.7509 0.6538 0.4741 0.3968 0.5161 0.5788 0.6484 0.5953 0.6713 0.8353 0.8837 0.8203 0.6666 0.5835 0.6447 0.6787 0.7103 0.7198 0.8245 0.8743 0.9352 0.8947 0.7803 0.7080 0.7427 0.7611 0.7700 0.8064 0.8959 0.9072

P. C. DUMIR

260

The maximum deflection response to uniformly distributed step loads is compared in Table 3. The maximum error in the maximum response is about 5%, 2%, and 10% for the immovable clamped, immovable simply supported and movable clamped cases respectively. It may be concluded from these comparisons that reasonably good engineering accuracy is achieved by the approximate method for deflection response under a variety of loadings. The results for plates with elastically restrained edges, resting on linear Winkler (Kr = G = 0) foundations, without an applied edge load are given in Table 4. Similar results for Pasternak (K, = 0) and non-linear Winkler (G = 0) foundations are presented in Table 5. Based on these results, some observations can be made. The buckling load

TABLE

5

Response of plates on Pasternak and non-linear Winkler foundations (v = 0.3, n = 0)

K

K,

G

K,

K,

4

0

0

0 4 co co

0 0 0 co

4

0

0.5

0 0 4 00 00

00 0 0 0 00

4

0

1

0 0 4 00 00

00 0 0 0 co

0 0 4 Co co

co 0 0 0 co

0 0 4 co

03 0 0 0

0

00

4-l

4 4 4 4 4

0

0.5 0.5 0.5 1 1

0 0 0 0~~ 0

a, 4.0030 5.7040 8.2608 8.2608 4.0030 5.8686 7.5019 10.2608 10.2608 5.8686 7.7342 9.2999 12.2608 12.2608 7.7342 4.0030 5.7040 8.2608 8.2608 4.0030 4.0030 5.7040 8.2608 8.2608 4.0030

n3

-NC,

F = E,

*o

0.3776 0.4936 0.8571 2.7619 2.6604 0.3776 0.4936 0.8571 2.7619 2.6604 0.3776 0.4936 0.8571 2.7619 2.6604 0.7491 0.8405 1.1905 3.0952 3.0318 0.1919 0.3201 0.6905 2.4290 2.2890

1.0728 1.5862 2.0652 -

0.0943 0.0865 0.1038 0.3343 0.6646 0.0643 0.0658 0.0835 0.2692 0.4533 0.0488 0.053 1 0.0699 0.2253 0.3440 0.1871 0.1474 0.1441 0.3747 0.7574 0.0479 0.0561 0.0836 0.2940 0.5718

8.2557 10.0527 12.2616 12.2616 8.2557 9.9961 11.5287 13.6655 13.6655 9.996 1 11.4754 12.8361 14.9381 14.9381 11.4754 8.2557 10.0527 12.2616 12.2616 8.2557 8.2557 10.0527 12.2616 12.2616 8.2557

1.5728 2.0862 2.5652 2.0728 2.5862 3.0652 1.0728 1.5862 2.0652 1.0728 1.5862 2.0652 -

C=l 0.9665 0.9692 0.9634 0.8949 0.8188 0.9767 0.9762 0.9702 0.9127 0.8650 0.9822 0.9807 0.9748 0.9254 0.8924 0.9368 0.9491 0.9502 0.8844 0.8008 0.9825 0.9796 0.9702 0.9058 0.8382

c=2 0.8837 0.8918 0.8743 0.7103 0.5835 0.9161 0.9144 0.8950 0.7470 0.6556 0.9343 0.9292 0.9098 0.7751 0.7054 0.8025 0.8344 0.8372 0.6902 0.5586 0.9354 0.9256 0.8949 0.7323 0.6120

NC, increases with the foundation parameters K, G and support stiffness parameters Ki, Kb. The linear deflection parameter (or and the linear frequency w0 increase with K, G and rotational stiffness Kb of the support. The non-linearity parameter E (also the postbuckling parameter .sr) increases with in-plane stiffness Ki for simply supported plates ( Kb = 0) and decreases with rotational stiffness Kb for immovable plates ( Ki = 00). The effect of Ki on the non-linearity parameter is greater than that of Kb. The parameter (Ye, reflecting the non-linear stiffening of the plate, is independent of K and G. But the non-linearity parameter E decreases with K and G because the linear parameter a,

PLATES ON ELASTIC

FOUNDATIONS

261

increases with K and G. Thus the effect of geometric non-linearity decreases with K and G. In all cases, the effect of foundation shear parameter G on the response parameters is more pronounced than that of the Winkler parameter K. The relative effect of K and G decreases with Kb since the stiffness of the plate without foundations increases with &. For non-linear Winkler foundations, the non-linearity parameter E is significantly affected by the non-linear foundation parameter K, . It decreases with softening foundations (K, > 0) and increases with hardening foundations (K, < 0). The effect of in-plane radial force n on the response of clamped and simply supported plates with movable edges is given in Table 6. The linear deflection parameter (Y, and TABLE

E$ect

6

ofin-plane radial force n on the response (Ki = 0, v = 0.3) I

Kb

0

00

n

0.8 0.7 0.6 0.5 0.3 0.0 -0.3 -0.8 0.8 0.7 0.6 0.5 0.3 0.0 -0.3 -0.8

a1

‘0.2875 0.4313 0.5750 0.7188 1.0063 1.4376 1.8688 2.5876 1.1722 1.7582 2.3443 2.9304 4.1026 5.8608 7.6191 10.5495

ff3

E

0.3776 0.3776 0.3776 0.3776 0.3776 0.3776 0.3776 0.3776 0.8571 0.8571 0.8571 0.8571 0.8571 0.8571 0.8571 0.8571

1.313 0.8756 0.6567 0.5254 0.3753 0.2627 0.2021 0.1459 0.73 13 0.4875 0.3656 0.2925 0.2089 0.1463 0.1125 0.0813

00

2.2125 2.7098 3.1290 3.4983 4.1393 4.9474 5.6409 6.6376 4.619 5.657 6.532 7.303 8.641 10.328 11.776 13.856

T/ T, C=l

o-7135 0.7795 0.8204 0.8484 0.8843 0.9145 0.9322 0.9496 0.8058 0.8570 0.8868 0.9062 0.9302 0.9495 0.9605 0.9709

c=2 0.4561 0.53 10 0.5857 0.6280 0.6899 0.7509 0.7914 0.8356 0.5653 0.6420 0.6946 0.7332 0.7865 0.8353 0.8658 0.8974

the linear frequency o0 are significantly influenced by the presence of an in-plane force. The effect of a compressive force (n > 0) is more pronounced than that of a tensile force (n CO). The non-linear deflection parameter a3 is of course independent of n. The non-linearity parameter E increases with n. 4. CONCLUSIONS

A unified approximate approach to determine linear and non-linear frequencies, buckling and postbuckling loads, static response under transverse load, and maximum response to uniformly distributed step load, has been presented in this paper. The von K&man equations were used. Isotropic circular plates with edges elastically restrained for rotation and in-plane displacement, resting on Winkler, Pasternak and non-linear Winkler foundations have been considered. Reasonably accurate results for engineering applications are obtained by this method. The effect of various parameters has been investigated. The buckling load N,,, the linear deflection parameter CY~ and the linear frequency o0 increase with the foundation parameters K, G and the rotational stiffness Kb of the edge support. The non-linearity parameter E (or the postbuckling parameter E,) increases with the

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in-plane support stiffness Ki, and decreases with the rotational support stiffness Kh and the foundation parameters K and G. The effect of in-plane stiffness K, is greater than that of rotational stiffness Kt,. Also, the effect of the shear parameter G of the foundation K. The linear parameter (Y, and the linear is greater than that of the Winkler parameter frequency decrease with the in-plane force n, but the non-linearity parameter F increases with it. The effect of a compressive force is more pronounced than that of a tensile force.

REFERENCES 1. N. YAMAKI 1961 Zeitschriftfiir angewandte Mathematik und Mechanik 41, 501-510. Influence of large amplitude on flexural vibration of elastic plates. 2. J. L. NOWINSKI 1962 hoceedings of the Fourth United States National Congress of Applied Mechanics, 325-334. Nonlinear transverse vibration of circular plates built-in at the boundary. 3. W. A. NASH and J. R. MODEER 1959Proceedings of Symposium on the Theory of Thin Elastic Shells, International Union of Theoretical and Applied Mechanics, North Holland. Certain approximate analysis of the nonlinear behaviour of plates and shallow shells. 4. T. WAH 1963Proceedings of the American Society of Civil Engineers, Journal of the Engineering Mechanics Division 89, l-5. Vibration of circular plates at large amplitudes. 5. C. L. HUANG and B. E. SANDMAN 1971 International Journal of Non-linear Mechanics 6, 451-468. Large amplitude vibrations of rigidly clamped circular plates. 6. J. N. REDDY, C. L. HUANG and I. R. SINGH 1981International Joumalfor Numerical Methods in Engineering 17, 527-541. Large deflections and large amplitude vibrations of axisymmetric circular plates. 1983 Proceedings of the American Society of Civil 7. M. SATHYAMOORTHY and M. E. PRASAD Engineers, Journal of the Engineering Mechanics Division 109, 1114-l 122. Multimode nonlinear analysis of circular plates. 8. V. Z. VLASOV and U. N. LEONTIEV 1966 Beams, Plates and Shells on Elastic Foundations. Israel Program for Scientific Translations, Jerusalem (translated from Russian). 9. S. N. SINHA 1963 Proceedings of the American Society of Civil Engineers, Journal of the Engineering Mechanics Division 89, l-24. Large deflection of plates on elastic foundation. 10. R. BOLTON 1972Proceedings American Society of Civil Engineers, Journal of the Engineering Mechanics Division 98, 629-640. Stresses in circular plates on elastic foundations. 11. N. GAJENDRA 1967International Journal of Non-linear Mechanics 2, 163-172. Large amplitude vibration of plates on elastic foundations. 12. S. DA-ITA 1976 International Journal of Non-linear Mechanics 11, 337-345. Large amplitude free vibration of irregular plates placed on elastic foundation. 13. P. C. DUMIR, Ch. R. KUMAR and M. L. GANDHI 1986 Journal of Sound and Vibration. Non-linear axisymmetric vibration of orthotropic thin circular plates on elastic foundations. 14. Y. NATH 1982 International Journal of Non-linear Mechanics 17, 285-296. Large amplitude response of circular plates on elastic foundations. 15. K. 0. FRIEDRICHS and J. J. STOKER 1942Journal of Applied Mechanics 9, A7-A14. Buckling of the circular plate beyond the critical thrust. 16. S. R. BODNER 1955Quarterly of Applied Mathematics 12,397-401. The postbuckling behaviour of a clamped circular plate. 17. M. YANOWITCH 1956Communications in Pure and Applied Mathematics 9,661-672. Non-linear buckling of circular elastic plates. 18. H. B. KELLER and E. L. REISS 1958 Proceedings of the 3rd International Congress Applied Mechanics, Prouidence, 375-385. Nonlinear bending and buckling of circular plates. 19. P. C. DUMIR and K. N. KHATRI 1984Fibre Science and Technology 21,233-245. Axisymmetric postbuckling of orthotropic thin tapered circular plates. 20. K. K. RAJU and G. V. RAO 1984 Computers and Structures 18, 1183-1187. Postbuckling of cylindrically orthotropic circular plates on elastic foundations with edges elastically restrained against rotation. 21. C. Y. CHIA 1980Nonlinear Analysis of Plates. New York: McGraw-Hill. 22. P. C. DUMIR, M. L. GANDHI and Y. NATH 1983Journal of Composite Materials 17,478-491. Nonlinear static and transient analysis of orthotropic thin circular plates with elastically restrained edge under central load.

PLATES

ON

APPENDIX:

a, h C D E, v F*, F g, k G, K H k, >K Kg, K: Kb, Ki M N*, N NC, L q,

NO

Q QO r, P

6 7 u* w*, w T To a19 a3 Y9

Yo

Y* E ; Alax 2, 0 WO

ELASTIC

FOUNDATIONS

NOMENCLATURE

radius and thickness of the plate dimensionless amplitude of vibration of centre flexural rigidity, Eh3/ 12( 1 - v2) Young’s modulus, Poisson’s ratio stress function, F = F*/ Eh2 Pastemak foundation parameters ga2/ Eh3, ka4/ Eh3 dimensionless foundation parameters: depth of single layer foundation non-linear foundation parameter, K, = k,a4/ Eh rotational and in-plane stiffness of the edge support dimensionless stiffnesses: 12Kga/Eh3, Kta/ Eh radial moment resultant applied radial in-plane force at edge, N = N*a2/Eh3 buckling load Nl NC, in-plane force resultants uniformly distributed transverse load, Q = qa4/ Eh4 uniformly distributed step load radius, dimensionless radius r/a time, dimensionless time r = [D/( y*ha4)]‘12t radial displacement at midplane transverse displacement, w = w*/ h period for amplitude C, linear period linear and non-linear parameters in static response mass densities of plate and foundation effective mass density non-linearity parameter of non-linear vibrations postbuckling parameter central deflection, maximum central deflection frequency, dimensionless frequency o = w*[ yha4/ Dll/’ linear frequency

( 1,( 1’ 4 )/aT, a )ldP

263