- Email: [email protected]
Vibration and stability of a thin elastic plate resting on a non-linear elastic foundation when the deformation is large
Journal of Sound and Vibration (1979) 67(2), 284-288
VIBRATION NON-LINEAR
AND STABILITY OF A THIN ELASTIC PLATE RESTING ON A ELASTIC FOUNDATION WHEN THE DEFORMATION IS LARGE 1. INTRODUCTION
The non-linear vibration and dynamic stability of plates have received considerable attention in recent years because of the great importance and interest attached to structures of low flexural rigidity [l-3]. These easily deformable structures vibrate at large amplitudes so that the classical bending theory becomes inadequate and it is necessary to allow for moderately large deflections. The present work deals with the free vibration of a rectangular plate resting on a non-linear elastic foundation and subjected to a uniaxial compressive load. The static stability and the sensitivity of the plate to an initial imperfection are also discussed. The influence of the various parameters entering into the problem are shown graphically as well as the equilibrium paths and the stability boundaries of an imperfect plate. The mathematical analysis is based on Bubnov-Galerkin method.
2.
PROBLEM
DEFINITION
Consider a rectangular plate (a x b) made of an isotropic linearly elastic material (see Figure 1). The plate is resting on a non-linear elastic foundation and it is also subjected to a uniaxial compressive load P,. The problems of vibration and static instability to be considered in what follows here include the possibility of large deformations. Attention is confined to the middle plane so that the displacement components u and u can be ignored.
Figure
1. Plate geometry
and loading.
Consequently the resultant stresses Nij(i,j = x, y) are expressed as [4] N,, = Eh/(l -v~)(+w:++vw;)
- P,,
N,,, = E/r/(1 -v~)(~w~~+$Jw~~), (la-c)
Nxy = (EM1 + WV,, w,,),
where E is the modulus of elasticity, h is the plate thickness and v is Poisson’s ratio. The membrane forces produce a resultant transverse load (2)
2 = NX, W,XX - 2~XYWJY+ % W,YY. In the analysis the influence of the stresses Zi=(i= x, y,z), rotational components of the inertia force are disregarded.
inertia and in-plane
284 0022460X/79/220284
+ 05 $02.00/O
0
1979 Academic
Press Inc. (London)
Limited
LETTERS
TO THE EDITOR
285
On the basis of these assumptions, one can describe the large amplitude vibration of the plate by the non-linear differential equation DV2V2w + my,, + k, w + k, w3 = Z,
(3)
where D = Eh3/12(1 -v’), ti is the mass per unit area of the plate, and k, and k, are foundation parameters. We are primarily interested in the roles played by the various parameters in the problems of the vibration and static instability of the plate.
3.
PROBLEM
SOLUTION
Equation (3), through some approximate method (e.g., the Bubnov-Galerkin method) can be reduced to a set of several ordinary differential equations. In the first approximation one equation of the following form can be obtained : j-,, + Af + By
= 0.
(4)
It describes the motion of a single-degree-of-freedom model. If the plate is simply supported along the entire contour, the relation between w(x, y: t) andf(t) is as follows: w(x, y; t) = f(t) sin(rcx/a) sin(7ry/b).
(5)
For other boundary conditions equation (5) can be taken as or
w(x, Y;t) = C&(x, y)f(t),
w(x, y; r) = CX(x) Y(y)f(r),
(6, 7)
where 4(x, y) is the solution of the corresponding linear problem and X(x), Y(y) are appropriate beam functions. For a simply supported plate the coefficients A and B in equation (4) are
As is well known, the solution of equation (4) can be given in elliptic functions [S]. For the initial conditions t = 0:f
=fo,
.fI* = 03
(9)
one obtains T = {4/(A + Bfo2)“‘} F(L, &c),
where T is the complete period of the plate, F(L,$) of the first kind and
(10)
is a complete Jacobi elliptic integral
i2 = B’$/2(A +Bf,2).
(11)
The frequency of the fundamental mode is then obtained as o = 2n/T = rc(A+Bf02)i”/2F(A, 7c/2).
(12)
When the element which characterizes dynamic process is eliminated from equation (4), one has Af + Ef2 = 0,
(13)
286
LETTERS TO THE EDITOR
which describes load. Thefplays Assume now small transverse
the post-critical behaviour of a plate subjected to a uniaxial compressive the role of a “displacement measure”. that the plate, in addition to the membrane forces, is also subjected to a load distribution q(x, y) = qo sin(nx/cc) sin(rcy/b).
(14)
Then equation (3), in the static case, becomes DVzV2w + k, w + k, w3 = Z + qo sin(rrx/a) sin(rcy/b). Applying the Bubnov-Galerkin
(15)
method to equation (15), and simplifying, one obtains 2s + @-3 - q,d7/4 = 0,
(16)
where Pi = Ap(l-?)/I%,
B = Bp(l-?)/Eh.
Equation (16) can be written as P = A + Bf 2 - i&/f
(17)
where is = ((1 - v2)/Eh2)(n/c~)~P,,
a=x-p,
40 = qobl4.
Equation (17) represents the equilibrium paths in the (PJ) plane of the system being discussed. The co-ordinates F* andf* of the stationary points are obtained from the condition dP/df = 0.
(18)
Hence f* = -(qo/2B3”3
and
p* = a + 1439jp3qy3.
(19,2(J)
Eliminating now Cjobetween the last two equations one finds that maxima and minima of the imperfect paths lie on the locus given by F* -A” - 3Bf2* = 0, or (P*/a) - 1 - 3(z?/A”)f2* = 0.
(21)
The critical load of the linear case corresponds to f = 0. Equation (21) respresents the stability boundary and the points of maxima and minima are the critical points at which the paths exchange their stabilities.
4.
NUMERICAL
RESULTS-DISCUSSION
From equation (12) one can obtain all the information related to the influence of the various parameters on the frequency CD.The frequency increases monotonically with increases in the initial amplitudef, and in the foundation parameters (k,, k, > 0). When all the parameters entering into the picture remain unchanged and only P, varies, one can see (from Figure 2) that the frequency decreases with increasing P, and may even vanish when P, satisfies the condition 1 = 1, or 2A + Bf,2 = 0. The values of the elliptic integrals used have been obtained from reference [IS].
(22)
LETTERS TO THE EDITOR
Figure 2. w as a function of P,.
287
h = 0.5:v = 0.30; p = 0.24; E = 21 x 10’: k, = 500; k, = 5O;j” = 1: a = 1
When the plate is subjected to a transverse load distribution one has the case of an imperfect plate. For this case the equilibrium paths as given by equation (17) are shown in Figure 3. The stability boundary and the perfect equilibrium paths of the plate are also drawn in this figure. One can see from Figure 3 that in this case two branches result, one of which suffers an exchange of stabilities while the other is always stable.
100 -
I I
)T
I$
I -0.5
I
1 - 0.3
I
I -0.1
I 0.1
I
I 0.3
I
I 05
I
Figure 3. Equilibrium paths of the plate. h = 0.5; v = 0.30; p = 0.24; E = 21 x 105; k, = 500; k, = 50; cx= 5; Stable paths; ----, unstable paths.
b=l. -,
Department of Mechanics. University of Ioannina, loannina, Greece (Received 10 July 1979)
C. MASSALAS K. SOLDATOS G. TZIVANIDIS
288
LETTERS TO THE EDITOR REFERENCES
1. B. M. KARMAKAR 1911 Journal of Sound and Vibration 54,265-271. Non-linear dynamic behaviour of plates on elastic foundations. 2. J. RAMACHANDRAN 1973 Journal of Applied Mechanics 40, 630-632. Non-linear vibrations of a rectangular plate carrying a concentrated mass. 3. S. KISLIAKOV 1976 International Journal Non-linear Mechanics 11, 219-228. On the non-linear dynamic stability problem for thin elastic plates. 4. B. K. SHIVAMOGGI 1977 Journalof Soundand Vibration 54,75582. Dynamic buckling of a thin elastic plate: non-linear theory. 5. D. BYRD and M. FRIEDMANN 1954 Handbook of Elliptic Integrals for Engineers and Physicists. Berlin: Springer-Verlag.
VIBRATION NON-LINEAR
AND STABILITY OF A THIN ELASTIC PLATE RESTING ON A ELASTIC FOUNDATION WHEN THE DEFORMATION IS LARGE 1. INTRODUCTION
The non-linear vibration and dynamic stability of plates have received considerable attention in recent years because of the great importance and interest attached to structures of low flexural rigidity [l-3]. These easily deformable structures vibrate at large amplitudes so that the classical bending theory becomes inadequate and it is necessary to allow for moderately large deflections. The present work deals with the free vibration of a rectangular plate resting on a non-linear elastic foundation and subjected to a uniaxial compressive load. The static stability and the sensitivity of the plate to an initial imperfection are also discussed. The influence of the various parameters entering into the problem are shown graphically as well as the equilibrium paths and the stability boundaries of an imperfect plate. The mathematical analysis is based on Bubnov-Galerkin method.
2.
PROBLEM
DEFINITION
Consider a rectangular plate (a x b) made of an isotropic linearly elastic material (see Figure 1). The plate is resting on a non-linear elastic foundation and it is also subjected to a uniaxial compressive load P,. The problems of vibration and static instability to be considered in what follows here include the possibility of large deformations. Attention is confined to the middle plane so that the displacement components u and u can be ignored.
Figure
1. Plate geometry
and loading.
Consequently the resultant stresses Nij(i,j = x, y) are expressed as [4] N,, = Eh/(l -v~)(+w:++vw;)
- P,,
N,,, = E/r/(1 -v~)(~w~~+$Jw~~), (la-c)
Nxy = (EM1 + WV,, w,,),
where E is the modulus of elasticity, h is the plate thickness and v is Poisson’s ratio. The membrane forces produce a resultant transverse load (2)
2 = NX, W,XX - 2~XYWJY+ % W,YY. In the analysis the influence of the stresses Zi=(i= x, y,z), rotational components of the inertia force are disregarded.
inertia and in-plane
284 0022460X/79/220284
+ 05 $02.00/O
0
1979 Academic
Press Inc. (London)
Limited
LETTERS
TO THE EDITOR
285
On the basis of these assumptions, one can describe the large amplitude vibration of the plate by the non-linear differential equation DV2V2w + my,, + k, w + k, w3 = Z,
(3)
where D = Eh3/12(1 -v’), ti is the mass per unit area of the plate, and k, and k, are foundation parameters. We are primarily interested in the roles played by the various parameters in the problems of the vibration and static instability of the plate.
3.
PROBLEM
SOLUTION
Equation (3), through some approximate method (e.g., the Bubnov-Galerkin method) can be reduced to a set of several ordinary differential equations. In the first approximation one equation of the following form can be obtained : j-,, + Af + By
= 0.
(4)
It describes the motion of a single-degree-of-freedom model. If the plate is simply supported along the entire contour, the relation between w(x, y: t) andf(t) is as follows: w(x, y; t) = f(t) sin(rcx/a) sin(7ry/b).
(5)
For other boundary conditions equation (5) can be taken as or
w(x, Y;t) = C&(x, y)f(t),
w(x, y; r) = CX(x) Y(y)f(r),
(6, 7)
where 4(x, y) is the solution of the corresponding linear problem and X(x), Y(y) are appropriate beam functions. For a simply supported plate the coefficients A and B in equation (4) are
As is well known, the solution of equation (4) can be given in elliptic functions [S]. For the initial conditions t = 0:f
=fo,
.fI* = 03
(9)
one obtains T = {4/(A + Bfo2)“‘} F(L, &c),
where T is the complete period of the plate, F(L,$) of the first kind and
(10)
is a complete Jacobi elliptic integral
i2 = B’$/2(A +Bf,2).
(11)
The frequency of the fundamental mode is then obtained as o = 2n/T = rc(A+Bf02)i”/2F(A, 7c/2).
(12)
When the element which characterizes dynamic process is eliminated from equation (4), one has Af + Ef2 = 0,
(13)
286
LETTERS TO THE EDITOR
which describes load. Thefplays Assume now small transverse
the post-critical behaviour of a plate subjected to a uniaxial compressive the role of a “displacement measure”. that the plate, in addition to the membrane forces, is also subjected to a load distribution q(x, y) = qo sin(nx/cc) sin(rcy/b).
(14)
Then equation (3), in the static case, becomes DVzV2w + k, w + k, w3 = Z + qo sin(rrx/a) sin(rcy/b). Applying the Bubnov-Galerkin
(15)
method to equation (15), and simplifying, one obtains 2s + @-3 - q,d7/4 = 0,
(16)
where Pi = Ap(l-?)/I%,
B = Bp(l-?)/Eh.
Equation (16) can be written as P = A + Bf 2 - i&/f
(17)
where is = ((1 - v2)/Eh2)(n/c~)~P,,
a=x-p,
40 = qobl4.
Equation (17) represents the equilibrium paths in the (PJ) plane of the system being discussed. The co-ordinates F* andf* of the stationary points are obtained from the condition dP/df = 0.
(18)
Hence f* = -(qo/2B3”3
and
p* = a + 1439jp3qy3.
(19,2(J)
Eliminating now Cjobetween the last two equations one finds that maxima and minima of the imperfect paths lie on the locus given by F* -A” - 3Bf2* = 0, or (P*/a) - 1 - 3(z?/A”)f2* = 0.
(21)
The critical load of the linear case corresponds to f = 0. Equation (21) respresents the stability boundary and the points of maxima and minima are the critical points at which the paths exchange their stabilities.
4.
NUMERICAL
RESULTS-DISCUSSION
From equation (12) one can obtain all the information related to the influence of the various parameters on the frequency CD.The frequency increases monotonically with increases in the initial amplitudef, and in the foundation parameters (k,, k, > 0). When all the parameters entering into the picture remain unchanged and only P, varies, one can see (from Figure 2) that the frequency decreases with increasing P, and may even vanish when P, satisfies the condition 1 = 1, or 2A + Bf,2 = 0. The values of the elliptic integrals used have been obtained from reference [IS].
(22)
LETTERS TO THE EDITOR
Figure 2. w as a function of P,.
287
h = 0.5:v = 0.30; p = 0.24; E = 21 x 10’: k, = 500; k, = 5O;j” = 1: a = 1
When the plate is subjected to a transverse load distribution one has the case of an imperfect plate. For this case the equilibrium paths as given by equation (17) are shown in Figure 3. The stability boundary and the perfect equilibrium paths of the plate are also drawn in this figure. One can see from Figure 3 that in this case two branches result, one of which suffers an exchange of stabilities while the other is always stable.
100 -
I I
)T
I$
I -0.5
I
1 - 0.3
I
I -0.1
I 0.1
I
I 0.3
I
I 05
I
Figure 3. Equilibrium paths of the plate. h = 0.5; v = 0.30; p = 0.24; E = 21 x 105; k, = 500; k, = 50; cx= 5; Stable paths; ----, unstable paths.
b=l. -,
Department of Mechanics. University of Ioannina, loannina, Greece (Received 10 July 1979)
C. MASSALAS K. SOLDATOS G. TZIVANIDIS
288
LETTERS TO THE EDITOR REFERENCES
1. B. M. KARMAKAR 1911 Journal of Sound and Vibration 54,265-271. Non-linear dynamic behaviour of plates on elastic foundations. 2. J. RAMACHANDRAN 1973 Journal of Applied Mechanics 40, 630-632. Non-linear vibrations of a rectangular plate carrying a concentrated mass. 3. S. KISLIAKOV 1976 International Journal Non-linear Mechanics 11, 219-228. On the non-linear dynamic stability problem for thin elastic plates. 4. B. K. SHIVAMOGGI 1977 Journalof Soundand Vibration 54,75582. Dynamic buckling of a thin elastic plate: non-linear theory. 5. D. BYRD and M. FRIEDMANN 1954 Handbook of Elliptic Integrals for Engineers and Physicists. Berlin: Springer-Verlag.