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Vibration Analysis of a Rectangular Plate
Journal of The Franklin Institute DEVOTED
Volume
296,
.Number
TO SCIENCE AND THE MECHANIC
November
5
Vibration Analysis of a Rectangular by M. CENGiZ
ARTS
and BRUNO
DijKMECit
A.
1973
Plate*
BOLEY~
Department of Engineering, Cornell University Thurston Hall, Ithaca, New York Presented
ABSTRACT:
stressed
herein
rectangular plates
plate is stressed
either thermally
the other two edge8 are taken are restrained
i8 a detailed
in both pre-
to remain
modes
superimposed
coupling
among
these modes,
given. Some numerical
or by the u&axial
to be either freely
straight.
vibration
analysis
Both
upon are
of the small, free
and post-buckling
rangea. The
in-plane
movable
studied. A criterion
simply
and for
of pre-
supported
loads on two opposite
or immovable,
the symmetrical
both primary-
oscillations
and
anti-aymnzetricul
secondary-buckling, the existence
edges,
and all the edges small and
of coupling
the i8
results are then presented.
Notation Throughout this paper, the Cartesian convected coordinates are used. The subscripts x and y, together with f, F, w and W, stand for partial differentiation. Dots and primes are used to designate partial differentiations with respect to time t and x or y, respectively. All quantities are deflned when they first arise. The list of notation follows. w, w v, v AT
lateral component of displacement vector in-plane components of displacement vector temperature difference between the plate and its support aspect ratio of plate, = a/b f,r? Airy stress function h thickness of plate E ,v Young’s modulus, Poisson’s ratio D bending rigidity of plate, = Eh8/12( 1 -vt) q,Q lateral load on plate average tractions in the x- and y-direction Pz, P,; PW P” oJ, oJ0 frequency and lowest natural frequency of plate 01 coefficient of linear thermal expansion 6 Kronecker delta v”: Laplace operator, = as/ax* + @/ay2 u,
u;
qYfY 9)
=
fzs SW+fw ssz - 2fGY92”
* This work was supported by the U.S. Office of Naval Research. t On leave from The Technical University of Istanbul. $ Present address : Dean of Engineering, The Technological Institute, University, Evanston, Illinois 60201.
305
Northwestern
M. Cengiz D%meci
and Bruno A. Boley sin nz(7rz/a), co8 m(7rx/a) sin n(q/b), co9 n(q/b) +l ifp = T, -1 ifp+r = 0, Ootherwise 40, (32D/Eh)*, DW, critical mess per unit area of plate = AT/AT, time
I. Introduction
This paper is concerned with a problem in the theory of small vibrations of prestressed plates. Such vibrations have been studied by several authors (l-4), but all previous researches have been based on a one-term approximation for the static lateral deflection solution and on a few terms for the description of the superimposed small oscillations. Certain conclusions were drawn, concerning the type of coupling which will arise among the several vibration modes, but the validity of those conclusions was of necessity limited by the approximations implied in the small number of terms maintained in the solution. It is the purpose of the present paper to develop the solution in greater generality, so as to obtain the precise rules governing the coupling of vibration modes. In order to reach the desired results just outlined, it is unfortunately necessary to develop in detail the solutions of the equations governing both the static post-buckling behavior of simply supported rectangular plates and the superimposed oscillations. Following a brief review of the von Kkman equations for the large deflection of plates in Sect. II this is done in Sects. III and IV. The linearized versions of the general results are presented in Sect. V, and then some numerical results in Sect. VI. The last section is devoted to conclusions. ZZ. Formulation
of the Problem
The behavior of a homogeneous isotropic rectangular plate (Fig. 1) under the action of an in-plane load is described, under well-known assumptions, by von K&man’s equations [e.g. Ref. (5)]. In dimensionless form, they can be expressed by V4F = 404(W,
-
W),
\
(14
v4w=-37+Q+404(F,W)
1
with F = f/f,,,
{U, 8, W> = f
f,, =
W; = 32D,Eh:
40,
{wv, w>>
Q=;9
q,, = DW,,
D = Eh3/12(1 -3).
(lb) i
Here, h is the thickness of the plate, E Young’s modulus, v Poisson’s ratio, D the bending rigidity of the plate, w the lateral component and u, v the
306
Journal
of The Franklin
Institute
Vibration Analysis of a Rectangular Plate
FIG. 1.
Simply supported rectangular plate.
in-plane components of the displacement Airy stress function. In order to complete the formulation introduce suitable boundary conditions. be simply supported, that is W(0, y, t) = W(a, y, t) = W(x, W,,(O,
Y, t) = W,,(a,
vector, q the lateral load and f the of the problem, it is necessa.ry to The plate edges are here taken to
0, t) =
Y, t) = W,,(x,
W(x,
b, t) = 0,
0, t) = W,,(x,
b, t) = 0
I
(2)
and will be restrained to remain straight, U(a,y,t)-U(O,y,t)
V(x,b,t)-
a(FYy-~FZZ-4W9&, s0
= U, = 7
V(x,O,t)
= v, = 7
b(F,,-J$y-4W;)dy s0
(3) I
which can be replaced by
u(a, y,t) - u(O,y, 0 = V(z, b, t) -
-ibAT), 0
V(x, 0, t) = v,
(4)
if a uniform temperature increment AT exists between the plate and its supports. In Eq. (4), 01denotes the coefficient of linear thermal expansion. The shear stresses are equal to zero at all edges, so that F&=0 The average tractions
atx/a=O,l,
y/b=O,l.
(5)
are
(6) Vol.396,N0.5,November1973
307
M. Cengiz Dci’lcmeci and Bruno A. Boley It is necessary to prescribe either the elongations from each of the pairs (U&p,) and (V,,pV). 111. Displacement
and Stress
The displacement
Functions,
function
U, and V, or one quantity
and Boundary
Conditions
will be taken in the usual form: (7a)
W(x, Y, t) = W(x, y) -I-K,(x, Y, t)
which must satisfy the equations of motion (1) and the boundary conditions (2)-( 6). Here, W, stands for the static solution of the plate, while W, indicates the small (compared to W,) oscillatory motions superimposed upon the buckling configuration of the plate. The terms W, and W, are chosen in the form of the trigonometric series: w,(XTY) = 5 5 w,,-LY,T mn
Ub)
ABwz@,
?.I, t)
=
22 c a
with Xn, = sinmz2,
Kj3
-Gyj
B
Y, = sinngy,
*,
=
cosnzy. b
I
(7c)
Here, the integer pairs (m, n) and (ar,p) assume values dependent on the type of oscillatory motions (symmetric or antisymmetric) and on the buckling configurations (primary or secondary) being considered in the analysis. The actual plate motions will, of course, require infinite series in these equations, i.e. M = N = A = B = CO;the possibility of using finite numbers of terms is, nevertheless, maintained here for purposes of later discussions. Displacements in the form (7b) clearly satisfy all the initial and boundary conditions for simply supported rectangular plates except for those stipulating that the edges remain straight. It can, however, be shown that the latter conditions are also satisfied. This will be done following the determination of the Airy stress function below. In the case of periodic oscillations with frequency w, the vibration amplitudes can be expressed by V&(t) = faB exp iw(t - to).
(8)
The lateral load representing
the inertia forces then becomes: .. q = -pzi; = -pwowz.
Thus, Q =$
= -$+&X,yg.
w
(gb)
Substituting the displacement function (7) into the first equation of Eqs. (la), one obtains the plate equations of motion for the dynamical Airy
308
Journal of The Franklin Institute
Vibration Analysis of a Rectangular Plate stress function as
which can be written, symmetric form : VPF = 4
2 2 55
mm7
after carrying
8
out the indicated
operations,
in the
W,, W,,[Smnrsq5,,z,h,4,. c+hS - (m2 s2 + n2 r2) X, Y, X,.E]
MNAB
The solution of Eq. (ll),
after considerable
simplifications,
MNAB
takes the form:
--
+ 2 x c c mncu$
z
WL
Tq9 -Lncup + $
iJ $
f
&!7 WY?/Lq+
Wa)
where L pQr8= - (ps-
qrJ2(K,+,,+,
&+r +,+s + %r,p--s &p-r &I,-,)
+ ( PS + qrJ2(Kp+r,q-s &+r v&9-,+ K&,p+s &p-r &I,,,,
(12b)
with ifp=q,
+l
0,
otherwise.
(1533)
The constants PI and P2 stand for the mean membrane stresses in the x- and y-direction, respectively, and they can either be prescribed or be evaluated from the given boundary conditions. They are related to the edge displacements by the equations
(13) and
MN mna$
Vol. 296, No. 5, November
1973
A
B I
.
(14)
309
M. Cengix DMmeci and Bruno A. Boley When a uniform temperature difference AT exists between the plate and the supports, these relations become
MN
-25
A
5
I3
I
z ~m2%n,~,~Wm,~a~
and P2 = 0 in the case of freely movable immovable edges, we read:
unloaded
(15)
edges. For the case of
(164 and
Returning integrals :
310
to the remaining
boundary
conditions,
we first carry out the
Journal
of The Franklin
Institute
Vibration Analysis of a Rectangular Plate
ABAB
where Eqs. (7) and (12), and the relationships “L pqrs,yydx = s0
i
2a@
+J(l -
6,) #~-s -
&+J9
0 (18)
bLpqw,‘s,rl dy = s0
;
2bq2%.J(l- &,) &p-v -
&+rl
0
are used. Substituting
these results into Eqs. (3), we finally obtain U, and V,:
MNAB
a,nd
MN -2C
A
B
IiS I3 Zn2L,~n~Kn~$3+v4 win (I $
I
(19b)
which show that the edges remain straight in the static as well as dynamic range.
IV.
Equations
of Motion
To obtain the equations of motion for the free vibrations of rectangular plates in the post-buckling range, the displacement function (7), the lateral load (9) and the stress function (12) are substituted into Eq. (1) with the
Vol. 206,No. 5, November1973
311
M. Cengiz Diilcmeci and Bruno A. Boley
Ii. (20)
MNAB
1n:the case of equilibrium, i.e. for the calculation of the static large-deflection solution, this equation reduces to
in which PF’ and P,(8) denote the membrane stresses in the static case in the x- and y-direction, respectively. Using this equation, various methods of calculating the deflection amplitudes have been studied, for instance, in Refs. (6-10). Further, in Ref. (6), a method of iteration has been developed in the same spirit as that of Boley (lo), and its results have been evaluated on the basis of the effective width of plates [see, e.g. Ref. (ll)]. The results of Ref. (6) will be used in this paper. By the use of Eq. (21), Eq. (20) can be simplified to the form:
ABAB_
_ =
0.
Here, the notation: pia’
312
=
p1 _ p(s)1 ,
P;d’=P,-P$’
(23)
JoumaI
of The Franklin
Institute
Vibration Analysis of a Rectangular Plate is introduced.
With the help of Eqs. (15) and (16), we obtain
mn (24) P$’ = pia’
=
0
in the case of freely movable unloaded edges, and 1
P:“’ = -(“!“)2(~AT,-~~~+-2)W~~), 1-G
WE ?r
MNAB + 2 C lZ C mnap
E (vm2 + n2 h2) L,
hp
K,,
RF
(25)
in the case of immovable unloaded edges. It is clear that any of the pairs (PI, P2), (P,,AT) or (P2, AT) may be prescribed. Now, we wish to single out the equation governing the dynamic displacement term VP,,. To do so, we multiply Eq. (22) by XPYC and integrate over the plate area. The following integrals are useful: QXP-yq,
X,Y,dxdy
-;(;)2PIy2-;@2P..ix2] =
-7T 4X2ab(l*2P1+(r2P2) WpgSpll&, 0a
!
where 4&
= (P
- C7t)2 (sS+t,ll8p+U.cr + “$4 a&J + (P
%&kl
+ CJQ2 (s,+l,, E&U+ $4 sq+U,J,
= - (ss - hr)2 (Kg+l;h+&r,h+s,k,r + &h--S + (gs + hr)2 (Kg+l;h--sLgw+r,h--s,k,l
Vol. 296,
No. 5, November
1973
+
Kg-r,h+s
(27a) Qs,~--s,k,J L&,h+s,k,,)
313
M. Cengiz Dtiknteci and Bruno A. Boley and ifp = q, - 1, ifj9-tq = 0, 0, otherwise.
+I,
EC= The derived result is finally
I
Wb)
(28) This equation together with (24) and (25) represent the equation of motion for a particular pair of vibration modes (p, u) in the case of a simply supported rectangular plate stressed either thermally or by in-plane loads, while the other two edges are either freely movable or completely restrained laterally. With the aid of Eqs. (24) and (25), Eq. (28) can also be rewritten in terms of the temperature difference AT rather than of PI and P2; the results need not be stated explicitly here [see Ref. (S)]. In the prebuckling range, the static solution WI is equal to zero, and the equation of motion (28) becomes
which, by means of Eqs. (15) and (16), might be expressed in terms of AT as well. V. Linearized
free
Vibrations
The preceding analysis is now restricted to small amplitude vibrations, Accordingly, the higher order terms of v$ are omitted, and then the linearized versions of the general results are given in this section. The linearized form of the stress function (12) is
MNAB
(30)
314
Journal
of The Franklin
Institute
Vibration Analysis of a Rectangular Plate The relationships increment AT are
between the edge stresses and the uniform temperature
(31a) P2 = 0 and
MN
(aAT)-;sm2
@lb) Wkn
in the case of freely movable edges, and
W-W
and P:d) = _ 1 P:“’ = -(“(~)“(LxAT)-;;~+~~~)W&], 1-G w; 7r
2(~AT)-;
&vm2+n2)12) m It
(=‘b) W;,
vm2 + h2n2) a,, S,, iF$ W,, in the case of immovable edges. The linearized equations of motion are
- 4h2$5 ($ Py mn
+ a2 P$d”‘)s,, a,, w,,
(33) as a special case of Eq. (28). Using Eqs. (31), this equation is expressed in
Vol.
296, No.
5, November
1973
315
M. Cengix DSmeci and Bruno A. Boley terms of AT, for later use, by
-
~2wKnrsap + 2Jc$nn,s%m K Ej3= 0
(34)
in the case of freely movable edges. In the prebuckling range, Eqs. (33) and (34) reduce to 5
(35)
and
= O.
~(~)?F-~~+[(p~+~~u~~2-~(~~,.+AT+Q, The mode shapes and frequencies the equations of motion just given.
(36)
of the plate can be found by solving
VI. Examples With the help of the general results obtained in the preceding articles, some numerical results are now presented. The rectangular plate treated is simply supported and initially stressed under a constant temperature increment AT. While the edges x = 0 and x = a are immovable and subjected to the temperature increment, the edges y = 0 and y = b are freely movable. The displacement function is taken in the form: WI&y) = AX,Y,+BX,,Y,+CX,Y,, with
Kh,y,t)
= Gf,y,+%xTy,+~~“y,
1
(374
The first term of WI represents the first mode of the primary buckling shape, whereas its second and third terms arise as a result of the method of iteration (6), (10) employed in calculating the statical deflection amplitudes of the large buckled plate. Using Eqs. (21) and (24), these amplitudes are readily found to be @=-&1+2A2, cr
B = C = &A3,
Pa)
where aAT,+
316
f 0
2
.
t 38b)
Journal of The Franklin
Institute
Vibration Analysis of a Rectangular Plate Now, we rewrite Eqs. (34) and (35) for the pair (B, 1) in the explicit form:
with O
/3=01,u,v.
(40b )
Here, w,, is the lowest natural frequency of an unstressed simply supported rectangular plate. With the help of Eqs. (l), we obtain this frequency as: wo” = -f
04(l+A2)2* i
By the use of Eqs. (8), (37)-(40) and (41),the frequencies are obtained for the following cases.
(41) of the plate
Case 1. Square plate, h = 1; a = 2, CT= 4, v = 0; cog= 4(D/p) (rr/a)4, Fig. 2. The vibration modes 5 and 4 are coupled in the post-buckling range. A general criterion for the coupling among the vibration modes is given in the next section. Case 2. Rectangular plate, X = 3; 01= 1, cr = 3, v = 5; W: = lOO(D/p)/(r/a)J, Fig. 3. The modes 5 and r) are coupled, while the mode E is uncoupled with the modes 5 and 7. The mode t, however, would be coupled with the mode r9r, if this mode was included in Eqs. (37).The coupling evidently depends on the terms employed in Eqs. (37) only [cf. Refs. 1,2)].In Fig. 3, the experimental results of Ref. (1)are also shown. The inclusion of further terms in the static solution improves the agreement with the experimental results. Nevertheless, the effect of the first buckling mode is predominant.
Vol. 296, No. 5, November
1973
317
M. Cengiz D6kmeci and Bruno A. Boley
Second opprox.
554 45-
I?%.
2. Vibration frequencies of buckled rectanguler plate, h = a/b = 1.
5
Mode I 0 0
I 2
I 3
I 4
I 5
I 6
I 7
I 9
I 9
IO
Fm. 3. Vibration frequencies of buckled rectangular plate, X = 3.
Case 3. Rectangular Fig. 4. The vibration post-buckling ranges.
plate, x = 4; 01= 2, a = 4, v = 0; W; = 289(D/p) (r/a)4, modes 5 and t are uncoupled in both the pre- and
Case 4. Rectangular plate, x = 5; CII = 1, u = 3, v = 5; W: = 676(0/p) (n/a)4, Fig. 5. All the vibration modes are uncoupled in both of the ranges. If the mode r,l were included, the modes 5 and 7 would be uncoupled, whereas the mode 5 would be coupled with the new mode E,‘;l. Further details regarding the numerical results given above can be found in Ref. (6).
318
Journal of The Franklin
Institute
Vibration Analysis of a Rectangular Plate
x=4 6-
nrst approximation Second opprox.
FIG. 4. Vibration frequencies of buckled rectangular plate, A = 4.
60
30
Fir*+ opproximatlon Second opprox.
2 -0
00
00
IO
20
30
40
50
60
70
6.0
90
10 0
0
FIG. 5. Vibration frequencies of buckled rectangular plate, A = 5.
VIZ.
Concluding
Remarks
A general analysis of the free vibrations of the initially buckled rectangular plates has been presented. The plate is assumed to be simply supported at all edges ; the two edges are stressed by either temperature increment or in-plane loads, and the unloaded edges are either immovable or freely movable, but are considered to remain straight during the deformation. Both symmetrical and antisymmetrical vibration modes and both primary and secondary-buckling are treated in the analysis. The equations of motion have been studied in general, and the linearized version of which has been given.
Vol. 296, No. 5, November 1973
319
M. Cengiz
Ddcmeci and Bruno A. Boley
From the analysis presented the following conclusions may be drawn: The series of functions (7)used for displacements satisfy all the boundary conditions, regardless of the number of terms involved. In the prebuckling range, namely 0 < 1,
W,, = 0
for all m and n,
all vibration modes are uncoupled, if the small oscillations are only considered, as is indeed well known. In the post-buckling range, no uncoupled harmonic modes exist, even in the case of small oscillatory motions. This conclusion is overlooked in the previous studies, since they were based on a finite number of harmonics only. The symmetrical and antisymmetrical small vibration modes are uncoupled as a result of the character of the terms arising in the static solution of primary buckling treated. The exact nature of the coupling among the vibration modes depends on the static solution on which they are superimposed, be that the one corresponding to a primary or a secondary buckling mode. Inspection of the general equations of motion (34) reveals that if and only if one of the relations : is satisfied for any pair of integers (m, n), (r, s) and (01,j3) which respectively refer to any harmonic present in the static and dynamic parts of the displacement function, with any choices of the plus and minus signs, then the vibration mode (p, u) is uncoupled. It is often mentioned [see, e.g. Ref. (l)] that the vibration in a mode whose shape is identical to the buckling shape is uncoupled from all other harmonics. This is not so, as may be seen from the example already given (cf. A = 3 and h = 4). The above criteria may be employed to determine the terms which it is reasonable to include in a calculation involving a finite number of harmonics. For example, for a plate with aspect ratio h = 3, Table I may be set up, in which the terms qPC are listed in the order of increasing frequency o at the buckling load. A cross in the table indicates coupling and a zero indicates absence of coupling; the table assumes that only one term (i.e. A = W,,) has been used in the static large-deflection solution. The table shows that coupling with wai will not occur unless at least 5 terms are used, i.e. those arising in the q9rth vibration mode at buckling. Of course, the conclusions are different if more terms are employed in the static solution. For instance, if W,, and W,, are used then coupling with WI’,,occurs. The amplitude W,, of the first mode of the buckling shape has been found numerically to have a significant effect on the dynamic behavior of plate in the post-buckling range. The agreement with the experimental results (1)is materially improved by the inclusion of further terms in the static solution, [see Fig. 31. In the analysis, the equations of oscillatory motions are directly obtained from Eq. (1) in a straightforward manner. It is of interest to note that these
320
Journal of The Franklin Institute
Vibration Analysis of a Rectangular Plate equations can also be derived Refs. (1,2)].
by the use of the Lagrange
equation
[cf.
TABLE I
References
(1) (2) (3) (4)
(5) (6) (71 (8) (9) (19) (11)
R. L. Bisplinghoff and T. H. H. Pian, “On the vibrations of thermally buckled bars and plates”, Proc. Ninth IUTAM Congr., Brussels, Vol. VII, p. 307, 1957. Y. Shulman, “On the vibration of thermally stressed plates”, in “Development in Mechanics”, Univ. of Texas Press, p. 233, 1959. Y. W. Chang and E. F. Masur, “Vibrations and stability of buckled panels”, ASCE, EM, Vol. 91, p. 1, 1965. S. A. Ambartsumian, G. E. Bagdasarian, S. M. Durgarian and V. Ts. Gnuny, “Some problems of vibration and stability of shells and plates”, Int. J. Solid Struct., Vol. 2, p. 59, 1966. B. A. Boley and J. H. Weiner, “Theory of Thermal Stresses”, Wiley, New York, 1960. M. C. Dokmeci, “Statics and dynamics of plates with large deflections”, Thesis, Cornell University, 1972. M. Stem, “Behavior of buckled rectangular plates”, ASCE, EM, Vol. 84, p. 59, 1960. E. F. Masur, “On the analysis of buckled plates”, Proc. Third U.S. National Congr. Appl. Mechx., p. 411, 1958. A. C. Walker, “The postbuckling behavior of simply supported square plates”, Aero. Quart., Vol. 20, p. 203, 1970. S. R. Boley, “A procedure for the approximate analysis of buckled plates”, J. Aero. Sci., Vol. 22, p. 337, 1955. B. A. Boley, “The shearing rigidity of buckled sheet panels”, J. Aero. Sk., Vol. 17, p. 356, 1950.
vol. 296, No. 5, November1973
321
Volume
296,
.Number
TO SCIENCE AND THE MECHANIC
November
5
Vibration Analysis of a Rectangular by M. CENGiZ
ARTS
and BRUNO
DijKMECit
A.
1973
Plate*
BOLEY~
Department of Engineering, Cornell University Thurston Hall, Ithaca, New York Presented
ABSTRACT:
stressed
herein
rectangular plates
plate is stressed
either thermally
the other two edge8 are taken are restrained
i8 a detailed
in both pre-
to remain
modes
superimposed
coupling
among
these modes,
given. Some numerical
or by the u&axial
to be either freely
straight.
vibration
analysis
Both
upon are
of the small, free
and post-buckling
rangea. The
in-plane
movable
studied. A criterion
simply
and for
of pre-
supported
loads on two opposite
or immovable,
the symmetrical
both primary-
oscillations
and
anti-aymnzetricul
secondary-buckling, the existence
edges,
and all the edges small and
of coupling
the i8
results are then presented.
Notation Throughout this paper, the Cartesian convected coordinates are used. The subscripts x and y, together with f, F, w and W, stand for partial differentiation. Dots and primes are used to designate partial differentiations with respect to time t and x or y, respectively. All quantities are deflned when they first arise. The list of notation follows. w, w v, v AT
lateral component of displacement vector in-plane components of displacement vector temperature difference between the plate and its support aspect ratio of plate, = a/b f,r? Airy stress function h thickness of plate E ,v Young’s modulus, Poisson’s ratio D bending rigidity of plate, = Eh8/12( 1 -vt) q,Q lateral load on plate average tractions in the x- and y-direction Pz, P,; PW P” oJ, oJ0 frequency and lowest natural frequency of plate 01 coefficient of linear thermal expansion 6 Kronecker delta v”: Laplace operator, = as/ax* + @/ay2 u,
u;
qYfY 9)
=
fzs SW+fw ssz - 2fGY92”
* This work was supported by the U.S. Office of Naval Research. t On leave from The Technical University of Istanbul. $ Present address : Dean of Engineering, The Technological Institute, University, Evanston, Illinois 60201.
305
Northwestern
M. Cengiz D%meci
and Bruno A. Boley sin nz(7rz/a), co8 m(7rx/a) sin n(q/b), co9 n(q/b) +l ifp = T, -1 ifp+r = 0, Ootherwise 40, (32D/Eh)*, DW, critical mess per unit area of plate = AT/AT, time
I. Introduction
This paper is concerned with a problem in the theory of small vibrations of prestressed plates. Such vibrations have been studied by several authors (l-4), but all previous researches have been based on a one-term approximation for the static lateral deflection solution and on a few terms for the description of the superimposed small oscillations. Certain conclusions were drawn, concerning the type of coupling which will arise among the several vibration modes, but the validity of those conclusions was of necessity limited by the approximations implied in the small number of terms maintained in the solution. It is the purpose of the present paper to develop the solution in greater generality, so as to obtain the precise rules governing the coupling of vibration modes. In order to reach the desired results just outlined, it is unfortunately necessary to develop in detail the solutions of the equations governing both the static post-buckling behavior of simply supported rectangular plates and the superimposed oscillations. Following a brief review of the von Kkman equations for the large deflection of plates in Sect. II this is done in Sects. III and IV. The linearized versions of the general results are presented in Sect. V, and then some numerical results in Sect. VI. The last section is devoted to conclusions. ZZ. Formulation
of the Problem
The behavior of a homogeneous isotropic rectangular plate (Fig. 1) under the action of an in-plane load is described, under well-known assumptions, by von K&man’s equations [e.g. Ref. (5)]. In dimensionless form, they can be expressed by V4F = 404(W,
-
W),
\
(14
v4w=-37+Q+404(F,W)
1
with F = f/f,,,
{U, 8, W> = f
f,, =
W; = 32D,Eh:
40,
{wv, w>>
Q=;9
q,, = DW,,
D = Eh3/12(1 -3).
(lb) i
Here, h is the thickness of the plate, E Young’s modulus, v Poisson’s ratio, D the bending rigidity of the plate, w the lateral component and u, v the
306
Journal
of The Franklin
Institute
Vibration Analysis of a Rectangular Plate
FIG. 1.
Simply supported rectangular plate.
in-plane components of the displacement Airy stress function. In order to complete the formulation introduce suitable boundary conditions. be simply supported, that is W(0, y, t) = W(a, y, t) = W(x, W,,(O,
Y, t) = W,,(a,
vector, q the lateral load and f the of the problem, it is necessa.ry to The plate edges are here taken to
0, t) =
Y, t) = W,,(x,
W(x,
b, t) = 0,
0, t) = W,,(x,
b, t) = 0
I
(2)
and will be restrained to remain straight, U(a,y,t)-U(O,y,t)
V(x,b,t)-
a(FYy-~FZZ-4W9&, s0
= U, = 7
V(x,O,t)
= v, = 7
b(F,,-J$y-4W;)dy s0
(3) I
which can be replaced by
u(a, y,t) - u(O,y, 0 = V(z, b, t) -
-ibAT), 0
V(x, 0, t) = v,
(4)
if a uniform temperature increment AT exists between the plate and its supports. In Eq. (4), 01denotes the coefficient of linear thermal expansion. The shear stresses are equal to zero at all edges, so that F&=0 The average tractions
atx/a=O,l,
y/b=O,l.
(5)
are
(6) Vol.396,N0.5,November1973
307
M. Cengiz Dci’lcmeci and Bruno A. Boley It is necessary to prescribe either the elongations from each of the pairs (U&p,) and (V,,pV). 111. Displacement
and Stress
The displacement
Functions,
function
U, and V, or one quantity
and Boundary
Conditions
will be taken in the usual form: (7a)
W(x, Y, t) = W(x, y) -I-K,(x, Y, t)
which must satisfy the equations of motion (1) and the boundary conditions (2)-( 6). Here, W, stands for the static solution of the plate, while W, indicates the small (compared to W,) oscillatory motions superimposed upon the buckling configuration of the plate. The terms W, and W, are chosen in the form of the trigonometric series: w,(XTY) = 5 5 w,,-LY,T mn
Ub)
ABwz@,
?.I, t)
=
22 c a
with Xn, = sinmz2,
Kj3
-Gyj
B
Y, = sinngy,
*,
=
cosnzy. b
I
(7c)
Here, the integer pairs (m, n) and (ar,p) assume values dependent on the type of oscillatory motions (symmetric or antisymmetric) and on the buckling configurations (primary or secondary) being considered in the analysis. The actual plate motions will, of course, require infinite series in these equations, i.e. M = N = A = B = CO;the possibility of using finite numbers of terms is, nevertheless, maintained here for purposes of later discussions. Displacements in the form (7b) clearly satisfy all the initial and boundary conditions for simply supported rectangular plates except for those stipulating that the edges remain straight. It can, however, be shown that the latter conditions are also satisfied. This will be done following the determination of the Airy stress function below. In the case of periodic oscillations with frequency w, the vibration amplitudes can be expressed by V&(t) = faB exp iw(t - to).
(8)
The lateral load representing
the inertia forces then becomes: .. q = -pzi; = -pwowz.
Thus, Q =$
= -$+&X,yg.
w
(gb)
Substituting the displacement function (7) into the first equation of Eqs. (la), one obtains the plate equations of motion for the dynamical Airy
308
Journal of The Franklin Institute
Vibration Analysis of a Rectangular Plate stress function as
which can be written, symmetric form : VPF = 4
2 2 55
mm7
after carrying
8
out the indicated
operations,
in the
W,, W,,[Smnrsq5,,z,h,4,. c+hS - (m2 s2 + n2 r2) X, Y, X,.E]
MNAB
The solution of Eq. (ll),
after considerable
simplifications,
MNAB
takes the form:
--
+ 2 x c c mncu$
z
WL
Tq9 -Lncup + $
iJ $
f
&!7 WY?/Lq+
Wa)
where L pQr8= - (ps-
qrJ2(K,+,,+,
&+r +,+s + %r,p--s &p-r &I,-,)
+ ( PS + qrJ2(Kp+r,q-s &+r v&9-,+ K&,p+s &p-r &I,,,,
(12b)
with ifp=q,
+l
0,
otherwise.
(1533)
The constants PI and P2 stand for the mean membrane stresses in the x- and y-direction, respectively, and they can either be prescribed or be evaluated from the given boundary conditions. They are related to the edge displacements by the equations
(13) and
MN mna$
Vol. 296, No. 5, November
1973
A
B I
.
(14)
309
M. Cengix DMmeci and Bruno A. Boley When a uniform temperature difference AT exists between the plate and the supports, these relations become
MN
-25
A
5
I3
I
z ~m2%n,~,~Wm,~a~
and P2 = 0 in the case of freely movable immovable edges, we read:
unloaded
(15)
edges. For the case of
(164 and
Returning integrals :
310
to the remaining
boundary
conditions,
we first carry out the
Journal
of The Franklin
Institute
Vibration Analysis of a Rectangular Plate
ABAB
where Eqs. (7) and (12), and the relationships “L pqrs,yydx = s0
i
2a@
+J(l -
6,) #~-s -
&+J9
0 (18)
bLpqw,‘s,rl dy = s0
;
2bq2%.J(l- &,) &p-v -
&+rl
0
are used. Substituting
these results into Eqs. (3), we finally obtain U, and V,:
MNAB
a,nd
MN -2C
A
B
IiS I3 Zn2L,~n~Kn~$3+v4 win (I $
I
(19b)
which show that the edges remain straight in the static as well as dynamic range.
IV.
Equations
of Motion
To obtain the equations of motion for the free vibrations of rectangular plates in the post-buckling range, the displacement function (7), the lateral load (9) and the stress function (12) are substituted into Eq. (1) with the
Vol. 206,No. 5, November1973
311
M. Cengiz Diilcmeci and Bruno A. Boley
Ii. (20)
MNAB
1n:the case of equilibrium, i.e. for the calculation of the static large-deflection solution, this equation reduces to
in which PF’ and P,(8) denote the membrane stresses in the static case in the x- and y-direction, respectively. Using this equation, various methods of calculating the deflection amplitudes have been studied, for instance, in Refs. (6-10). Further, in Ref. (6), a method of iteration has been developed in the same spirit as that of Boley (lo), and its results have been evaluated on the basis of the effective width of plates [see, e.g. Ref. (ll)]. The results of Ref. (6) will be used in this paper. By the use of Eq. (21), Eq. (20) can be simplified to the form:
ABAB_
_ =
0.
Here, the notation: pia’
312
=
p1 _ p(s)1 ,
P;d’=P,-P$’
(23)
JoumaI
of The Franklin
Institute
Vibration Analysis of a Rectangular Plate is introduced.
With the help of Eqs. (15) and (16), we obtain
mn (24) P$’ = pia’
=
0
in the case of freely movable unloaded edges, and 1
P:“’ = -(“!“)2(~AT,-~~~+-2)W~~), 1-G
WE ?r
MNAB + 2 C lZ C mnap
E (vm2 + n2 h2) L,
hp
K,,
RF
(25)
in the case of immovable unloaded edges. It is clear that any of the pairs (PI, P2), (P,,AT) or (P2, AT) may be prescribed. Now, we wish to single out the equation governing the dynamic displacement term VP,,. To do so, we multiply Eq. (22) by XPYC and integrate over the plate area. The following integrals are useful: QXP-yq,
X,Y,dxdy
-;(;)2PIy2-;@2P..ix2] =
-7T 4X2ab(l*2P1+(r2P2) WpgSpll&, 0a
!
where 4&
= (P
- C7t)2 (sS+t,ll8p+U.cr + “$4 a&J + (P
%&kl
+ CJQ2 (s,+l,, E&U+ $4 sq+U,J,
= - (ss - hr)2 (Kg+l;h+&r,h+s,k,r + &h--S + (gs + hr)2 (Kg+l;h--sLgw+r,h--s,k,l
Vol. 296,
No. 5, November
1973
+
Kg-r,h+s
(27a) Qs,~--s,k,J L&,h+s,k,,)
313
M. Cengiz Dtiknteci and Bruno A. Boley and ifp = q, - 1, ifj9-tq = 0, 0, otherwise.
+I,
EC= The derived result is finally
I
Wb)
(28) This equation together with (24) and (25) represent the equation of motion for a particular pair of vibration modes (p, u) in the case of a simply supported rectangular plate stressed either thermally or by in-plane loads, while the other two edges are either freely movable or completely restrained laterally. With the aid of Eqs. (24) and (25), Eq. (28) can also be rewritten in terms of the temperature difference AT rather than of PI and P2; the results need not be stated explicitly here [see Ref. (S)]. In the prebuckling range, the static solution WI is equal to zero, and the equation of motion (28) becomes
which, by means of Eqs. (15) and (16), might be expressed in terms of AT as well. V. Linearized
free
Vibrations
The preceding analysis is now restricted to small amplitude vibrations, Accordingly, the higher order terms of v$ are omitted, and then the linearized versions of the general results are given in this section. The linearized form of the stress function (12) is
MNAB
(30)
314
Journal
of The Franklin
Institute
Vibration Analysis of a Rectangular Plate The relationships increment AT are
between the edge stresses and the uniform temperature
(31a) P2 = 0 and
MN
(aAT)-;sm2
@lb) Wkn
in the case of freely movable edges, and
W-W
and P:d) = _ 1 P:“’ = -(“(~)“(LxAT)-;;~+~~~)W&], 1-G w; 7r
2(~AT)-;
&vm2+n2)12) m It
(=‘b) W;,
vm2 + h2n2) a,, S,, iF$ W,, in the case of immovable edges. The linearized equations of motion are
- 4h2$5 ($ Py mn
+ a2 P$d”‘)s,, a,, w,,
(33) as a special case of Eq. (28). Using Eqs. (31), this equation is expressed in
Vol.
296, No.
5, November
1973
315
M. Cengix DSmeci and Bruno A. Boley terms of AT, for later use, by
-
~2wKnrsap + 2Jc$nn,s%m K Ej3= 0
(34)
in the case of freely movable edges. In the prebuckling range, Eqs. (33) and (34) reduce to 5
(35)
and
= O.
~(~)?F-~~+[(p~+~~u~~2-~(~~,.+AT+Q, The mode shapes and frequencies the equations of motion just given.
(36)
of the plate can be found by solving
VI. Examples With the help of the general results obtained in the preceding articles, some numerical results are now presented. The rectangular plate treated is simply supported and initially stressed under a constant temperature increment AT. While the edges x = 0 and x = a are immovable and subjected to the temperature increment, the edges y = 0 and y = b are freely movable. The displacement function is taken in the form: WI&y) = AX,Y,+BX,,Y,+CX,Y,, with
Kh,y,t)
= Gf,y,+%xTy,+~~“y,
1
(374
The first term of WI represents the first mode of the primary buckling shape, whereas its second and third terms arise as a result of the method of iteration (6), (10) employed in calculating the statical deflection amplitudes of the large buckled plate. Using Eqs. (21) and (24), these amplitudes are readily found to be @=-&1+2A2, cr
B = C = &A3,
Pa)
where aAT,+
316
f 0
2
.
t 38b)
Journal of The Franklin
Institute
Vibration Analysis of a Rectangular Plate Now, we rewrite Eqs. (34) and (35) for the pair (B, 1) in the explicit form:
with O
/3=01,u,v.
(40b )
Here, w,, is the lowest natural frequency of an unstressed simply supported rectangular plate. With the help of Eqs. (l), we obtain this frequency as: wo” = -f
04(l+A2)2* i
By the use of Eqs. (8), (37)-(40) and (41),the frequencies are obtained for the following cases.
(41) of the plate
Case 1. Square plate, h = 1; a = 2, CT= 4, v = 0; cog= 4(D/p) (rr/a)4, Fig. 2. The vibration modes 5 and 4 are coupled in the post-buckling range. A general criterion for the coupling among the vibration modes is given in the next section. Case 2. Rectangular plate, X = 3; 01= 1, cr = 3, v = 5; W: = lOO(D/p)/(r/a)J, Fig. 3. The modes 5 and r) are coupled, while the mode E is uncoupled with the modes 5 and 7. The mode t, however, would be coupled with the mode r9r, if this mode was included in Eqs. (37).The coupling evidently depends on the terms employed in Eqs. (37) only [cf. Refs. 1,2)].In Fig. 3, the experimental results of Ref. (1)are also shown. The inclusion of further terms in the static solution improves the agreement with the experimental results. Nevertheless, the effect of the first buckling mode is predominant.
Vol. 296, No. 5, November
1973
317
M. Cengiz D6kmeci and Bruno A. Boley
Second opprox.
554 45-
I?%.
2. Vibration frequencies of buckled rectanguler plate, h = a/b = 1.
5
Mode I 0 0
I 2
I 3
I 4
I 5
I 6
I 7
I 9
I 9
IO
Fm. 3. Vibration frequencies of buckled rectangular plate, X = 3.
Case 3. Rectangular Fig. 4. The vibration post-buckling ranges.
plate, x = 4; 01= 2, a = 4, v = 0; W; = 289(D/p) (r/a)4, modes 5 and t are uncoupled in both the pre- and
Case 4. Rectangular plate, x = 5; CII = 1, u = 3, v = 5; W: = 676(0/p) (n/a)4, Fig. 5. All the vibration modes are uncoupled in both of the ranges. If the mode r,l were included, the modes 5 and 7 would be uncoupled, whereas the mode 5 would be coupled with the new mode E,‘;l. Further details regarding the numerical results given above can be found in Ref. (6).
318
Journal of The Franklin
Institute
Vibration Analysis of a Rectangular Plate
x=4 6-
nrst approximation Second opprox.
FIG. 4. Vibration frequencies of buckled rectangular plate, A = 4.
60
30
Fir*+ opproximatlon Second opprox.
2 -0
00
00
IO
20
30
40
50
60
70
6.0
90
10 0
0
FIG. 5. Vibration frequencies of buckled rectangular plate, A = 5.
VIZ.
Concluding
Remarks
A general analysis of the free vibrations of the initially buckled rectangular plates has been presented. The plate is assumed to be simply supported at all edges ; the two edges are stressed by either temperature increment or in-plane loads, and the unloaded edges are either immovable or freely movable, but are considered to remain straight during the deformation. Both symmetrical and antisymmetrical vibration modes and both primary and secondary-buckling are treated in the analysis. The equations of motion have been studied in general, and the linearized version of which has been given.
Vol. 296, No. 5, November 1973
319
M. Cengiz
Ddcmeci and Bruno A. Boley
From the analysis presented the following conclusions may be drawn: The series of functions (7)used for displacements satisfy all the boundary conditions, regardless of the number of terms involved. In the prebuckling range, namely 0 < 1,
W,, = 0
for all m and n,
all vibration modes are uncoupled, if the small oscillations are only considered, as is indeed well known. In the post-buckling range, no uncoupled harmonic modes exist, even in the case of small oscillatory motions. This conclusion is overlooked in the previous studies, since they were based on a finite number of harmonics only. The symmetrical and antisymmetrical small vibration modes are uncoupled as a result of the character of the terms arising in the static solution of primary buckling treated. The exact nature of the coupling among the vibration modes depends on the static solution on which they are superimposed, be that the one corresponding to a primary or a secondary buckling mode. Inspection of the general equations of motion (34) reveals that if and only if one of the relations : is satisfied for any pair of integers (m, n), (r, s) and (01,j3) which respectively refer to any harmonic present in the static and dynamic parts of the displacement function, with any choices of the plus and minus signs, then the vibration mode (p, u) is uncoupled. It is often mentioned [see, e.g. Ref. (l)] that the vibration in a mode whose shape is identical to the buckling shape is uncoupled from all other harmonics. This is not so, as may be seen from the example already given (cf. A = 3 and h = 4). The above criteria may be employed to determine the terms which it is reasonable to include in a calculation involving a finite number of harmonics. For example, for a plate with aspect ratio h = 3, Table I may be set up, in which the terms qPC are listed in the order of increasing frequency o at the buckling load. A cross in the table indicates coupling and a zero indicates absence of coupling; the table assumes that only one term (i.e. A = W,,) has been used in the static large-deflection solution. The table shows that coupling with wai will not occur unless at least 5 terms are used, i.e. those arising in the q9rth vibration mode at buckling. Of course, the conclusions are different if more terms are employed in the static solution. For instance, if W,, and W,, are used then coupling with WI’,,occurs. The amplitude W,, of the first mode of the buckling shape has been found numerically to have a significant effect on the dynamic behavior of plate in the post-buckling range. The agreement with the experimental results (1)is materially improved by the inclusion of further terms in the static solution, [see Fig. 31. In the analysis, the equations of oscillatory motions are directly obtained from Eq. (1) in a straightforward manner. It is of interest to note that these
320
Journal of The Franklin Institute
Vibration Analysis of a Rectangular Plate equations can also be derived Refs. (1,2)].
by the use of the Lagrange
equation
[cf.
TABLE I
References
(1) (2) (3) (4)
(5) (6) (71 (8) (9) (19) (11)
R. L. Bisplinghoff and T. H. H. Pian, “On the vibrations of thermally buckled bars and plates”, Proc. Ninth IUTAM Congr., Brussels, Vol. VII, p. 307, 1957. Y. Shulman, “On the vibration of thermally stressed plates”, in “Development in Mechanics”, Univ. of Texas Press, p. 233, 1959. Y. W. Chang and E. F. Masur, “Vibrations and stability of buckled panels”, ASCE, EM, Vol. 91, p. 1, 1965. S. A. Ambartsumian, G. E. Bagdasarian, S. M. Durgarian and V. Ts. Gnuny, “Some problems of vibration and stability of shells and plates”, Int. J. Solid Struct., Vol. 2, p. 59, 1966. B. A. Boley and J. H. Weiner, “Theory of Thermal Stresses”, Wiley, New York, 1960. M. C. Dokmeci, “Statics and dynamics of plates with large deflections”, Thesis, Cornell University, 1972. M. Stem, “Behavior of buckled rectangular plates”, ASCE, EM, Vol. 84, p. 59, 1960. E. F. Masur, “On the analysis of buckled plates”, Proc. Third U.S. National Congr. Appl. Mechx., p. 411, 1958. A. C. Walker, “The postbuckling behavior of simply supported square plates”, Aero. Quart., Vol. 20, p. 203, 1970. S. R. Boley, “A procedure for the approximate analysis of buckled plates”, J. Aero. Sci., Vol. 22, p. 337, 1955. B. A. Boley, “The shearing rigidity of buckled sheet panels”, J. Aero. Sk., Vol. 17, p. 356, 1950.
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