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Effects of geometric imperfections and large amplitudes on vibrations of an initially stressed mindlin plate
Applied Acoustics 35 (1992) 265-282
~,~:.\~.~,-~
Effects of Geometric Imperfections and Large Amplitudes on Vibrations of an Initially Stressed Mindlin Plate Lien-Wen Chen* & Chuan-Cheng Lin Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan 70104
(Received 19 February 1990; revised manuscript received 15 August 1991; accepted 19 September 1991)
ABSTRACT Non-linear equations of motion for a transversely isotropic plate having geometric imperfections and in a general state of non-uniform initial stresses are derived. The effects of both transverse shear deformation and rotatory inertia are included. The modal equations of a simply supported plate subjected to in-plane extensional stress and pure bending stress are obtained by performing the Galerkin procedure and then solved by the Runge-Kutta integration technique. The effects of geometric imperfections on large amplitude vibration behaviour are investigated. It is found that the vibration behaviour of an initially stressed plate is very much dependent on the order of initial amplitudes as well as geometric imperfections. The effects of various parameters on the non-linear vibration frequencies are studied.
NOTATION a,b
D D* E G G* h
Plateform dimensions of plate Extensional modulus [Eh/(1- v2)] Flexural modulus [Eh3/12(1- v2)] Young's modulus Shear modulus Transverse shear rigidity Thickness of plates
* To whom correspondence should be addressed. 265
Applied Acoustics 0003-682X/92/$05.00 © 1992 Elsevier Science Publishers Ltd, England. Printed in Great Britain
266
Lien-Wen Chen, Chuan-Cheng Lin
Initial extensional stress [k = 12Nxb2/(rt2h2D)] k K (alh)lzt Mx, M~.,Mxr Bending moment resultants ni Components of the unit normal with respect to the spatial frame mx, u,,,Ux, Stress resultants r a/b u> i) In-plane displacement of middle surface Components of initial and disturbed displacement Uio, Ui W Lateral displacement w/h Initial amplitude of free flexural vibration Xi Coordinates of a point x, Body force
fll qs P P tTij, ~ ij T
Angles of rotation of a normal to the middle surface in the Xl-X 3 and x2-x 3 planes, respectively The ratio of bending stress to extensional stress [fll = am/a,] Kronecker delta Components of initial imperfection Dimensionless geometric imperfection Poisson's ratio Density Initial and perturbing stress Dimensionless time [ = t[D/(pha2)] 1/2]
1 INTRODUCTION The non-linear analysis of elastic plates has received considerable attention and continues to remain of major concern because of its severe operational conditions in many mechanical, aerospace and hydrospace structural applications. Past investigators who had studied the effects of initial stresses on vibration problems have mainly concentrated on the perfect fiat plate. However, it is possible that certain deviations (the so-called geometric imperfections) between the manufactured plate and designed shape may exist. The studies on non-linear vibrations of an initially stressed, geometric imperfect plate have received relatively little attention although the statical and dynamical behaviour are very much dependent on the existence of geometric imperfections. A comprehensive review of the various developments in the geometrically non-linear behaviour of thin plates has been given by Chia. 1 Celep studied the effect of initial imperfection on the free vibration of rectangular plates 2
Vibrations of an initially stressed mindlin plate
267
and circular plates 3 by using the classical thin plate model. The influences of transverse shear deformation and rotatory inertia on the large amplitude vibration of imperfect plates are also studied by the above author. 4 Yamaki e t al. 5"6 investigated the effect of geometric imperfections and initial edge inplane displacements on the postbuckling and free vibrations of a rectangular plate. Recently, the effects of geometric imperfections on thenon-linear vibration of circular plates, 7 rectangular plates with hysteresis damping 8 and laminated composite plates 9 were conducted by Hui. In many practical applications, plate structures are subjected tO static loads causing internal stress fields. The presence of such stresses affects the vibration characteristics significantly. Herrmann & Armanakas 1° used a variational procedure to derive the equations of motion for an initially stressed thick plate. Brunelle & Robertson I ~ used the average stress method and the variational formulation to derive the governing equations for a thick plate in an arbitrary state of initial stress and then studied the linear vibration behaviour of a thick plate in a state of in-plane compression and bending load. ~2 Chen and Doong extended the theory of Brunelle ~ to derive the non-linear equations of motion for a thick plate in a general state of non-uniform initial stress to study the large amplitude vibrations of initially stressed rectangular 13 and circular plates. 14 Finite element solutions on large amplitude vibrations of plates under in-plane compression were presented by Mei ~5 and Yang & Han. ~6 The effects of geometric imperfections on the linear vibrations of simply supported plates under in-plane biaxial or uniaxial compression were examined by Hui ~7' 18 and Ilanko & Dickinson. ~902° The presence of small imperfections may significantly raise the frequencies and the sensitivity to imperfections increases with the amount of preload. Ilanko and Dickinson also observed experimentally and obtained close agreement with theoretical results. Furthermore, the finite element model for the non-linear vibrations of an imperfect plate was developed by Kapania & Yang. 2~ The arbitrary geometry of imperfections can be formulated by using a variable order polynomial. In the current study, the non-linear equilibrium equation in terms of a second Piola-Kirchhoff stress component 22 is generalised to include the effect of geometric imperfections. The equation is perturbed by an incremental deformation. 23 The products of incremental stresses with lateral (incremental and initial) displacements must be preserved here. The non-linear equations of motion, based on the large defection theory of von Karman's assumptions, for an initially stressed imperfect plate including the effects of transverse shear deformation and rotatory inertia are derived by the average stress method. For the sake of mathematical simplicity, the initial imperfections of the example problem studied here is assumed to be
268
Lien-Wen Chen, Chuan-Cheng Lin
sinusoidal and symmetric with respect to the plate centre. A simply supported plate under the effects of in-plane compressive (tensile) plus bending stress is presented to study the influences of geometric imperfections on the large amplitude vibration behaviour. On the basis of an assumed mode satisfying the geometric boundary conditions, the modal equations of motion are obtained by performing the Galerkin procedures. The previbration state (initial displacement field due to initial stress) of plates must be determined in the first phase of solution. The initial displacements are then superimposed on the non-linear dynamic equations to examine the large amplitude vibration behaviour by performing the method of Runge-Kutta. The non-linear periods for initially imperfect plates are close to those of Singh et al. 24 for a doubly curved shell without initial stress. The present studies will show that the effects of initial stresses on vibrations are dependent on both the order of initial amplitudes and geometric imperfections. It is remarkable that the initial stress can significantly affect the vibration behaviour for some special initial amplitude of vibration. Therefore, the large amplitude vibrations of an initially stressed, imperfect plate are worth investigating although the linear vibration behaviour is studied.17- 20
2 NONLINEAR EQUATION The nonlinear theory of curved plates due to geometric/imperfection is much more complicated than that of perfect plates. By introducing the imperfection component r/S, the nonlinear equation of equilibrium in terms of a second Piola-Kirchhoff stress component 22 can be expressed as ~xl I-(~i~S+
~u2/~xj + ~ns/~x~)cr*] + ,x'* = 0
(1)
where the xl are material (Lagrangian) coordinates that originally coincided with a spatial Cartesian coordinate system. The us and r/S are Cartesian components of the displacement and imperfection vector given with respect to the spatial coordinate system. The a* are Kirchhoff stress components taken with respect to undeformed areas. The X* are body force components referred to initial volume. The 6js symbol is the Kronecker delta. For notational convenience, a comma preceding an index implying partial differentiation with respect to the appropriate material coordinate will be used in the following text. The Lagrangian strain components ei~are defined as F,ij -~- ~(Ul, j "~ Uj. i -~- Uk.iUk, j "JI- Uk,fflk. i "JI- Uk,iYlk,j)
(2)
If relative extensions and shears are small, the values oftr* and X* are as a
Vibrations of an initially stressed mindlin plate
269
first order approximation equal to the actual stresses aii and the actual body force X s. These approximations will be adopted in the present work. The problem of interest here is a body in a state of non-uniform initial stresses, which is in static equilibrium and is subjected to a time varying incremental deformation. Following a technique described by Bolotin 23 the following quantities are introduced.
fi =
Uso + Us
,~'s = Xs + AX~ + )?~ - pu]
(3)
~ij =eij + g~j where, for example, ks, uso and ~7~represent the final displacement, initial displacement and incremental displacement, respectively. The term -p~'~ is the inertia force due to perturbation and the superior double dot denotes the second partial derivative with respect to time. Using the fact that both the initial and the final states satisfy the nonlinear eqn (1) gives the following non-linear equation for the incremental stresses and displacements.
[((~js "~- Us,j -~ as,j "~- ?~s.j)~ij],i "~- (aijas,j),i 21- AX$ ~1_Xs = P~i's
(4)
Similarly, the incremental Lagrangian strain components are related to incremental displacement, initial displacement and geometric imperfection by
eij = ½[~i,j -~ ~j,i "~- Uk,ilgk,j "~- Uk.j(l'lko.i "JI-nk,i) "~- Uk,i(glko.j "~- l~k,j) ]
(5)
The previbration state of static equilibrium must be determined first by using the non-linear eqn (1) and then the large amplitude vibrations of an initially stressed plate can be solved by the dynamic eqn (4). The form ofeqn (4) is the same as that of eqn (1) by dropping (trifis.j).i and (6uusa). i. For the sake of brevity, only the dynamic eqn (4) is used here to derive the equations of motion.
3 INITIALLY STRESSED I M P E R F E C T PLATE The incremental displacement fields for a point off the middle surface of a rectangular plate are assumed to be of the following form: UI(XI,X2,X3,
t)
=
U(Xl,x2, t) + Xa~(Xx,X2, t)
~2(xl, x 2, x 3, t) = v(x l, x2, t) + x3fl(x 1, x 2, t) /~3(Xl, X2, X3, t)= W(X1, X2, t)
(6)
Lien-Wen Chen, Chuan-Cheng Lin
270
where u, v, w, represent the displacements of the middle surface and, ~t and fl stand for the bending slope in the x, and x 2 directions. In the current study, only the vertical imperfection deviated from the plate surface is considered. Thus the field of geometric imperfection can be assumed as )71 =/~2 = 0
/13 = r/(x,,x2)
(7)
To simplify the non-linear equations, von Karman's assumptions for large deflection theory of plates are adopted here. By using eqns (5) and (6), the stress-displacement relations for a transversely isotropic material are a l l = (E/I - v2)(el x + ve22)
a22 = (El1 -- v2)(e22 + re11) a,2 =Ge,2
(8)
trl3 = ~c2G*(u'.I + a)
a23 = x2G*("'.2 +/~) where /311 = U. 1 "~ w.21/2+ w.,(Wo.1 + r/,,) + x3a.1 C22 = V.2 + W22/2 + W,2(Wo,2 + r/,2) + x3fl,2
(9)
F"12 ~"//.2 "[- V,1 + W, 1W,2 "1- W,I(Wo.2 "q- ?],2) -1- W,2(Wo. 1 "1- ?].1) "[- X3(~.2 -1- fl,')
G* takes into account the effects of transverse shear isotropy. Wois the initial displacement in the thickness direction due to initial stresses. Furthermore, the products of displacement gradient and stress in eqn (4) are so small they can be dropped except for t~nff3.~ and 17i2/t3o 2. In order to clarify the derivation procedure, it is useful to partially write out eqn (4):
(aij~l,j).l -[-#i,.i"[-Xl
"{- A X 1 ~--p/~"
(10)
(aUff2.j)., + #i2a +,~2 + AX2 =pf"
(11)
(tTijff3.j),i-[-ffi3.i--b [6il(ffs,j--J- l~,o.j--J-n.j)] i-J- x3 + mx3-~- p~;
(12)
For subsequent use in the equations of motion, the following initial stress resultants and material parameters are defined: (N~, Ny, Nxy, a~, Qy) = ~(at ,, a22, a, 2, a, 3, a23) dxs (Mx, Mr, Mxy, Q*,Q*)=S(tr,,,tr22,a,2,a13 ,a23)xadx a * = S (a , l, az2, cr 12).v 2 d.v 3 (M~* , M;* , M~r) D = E h / ( 1 --v 2) D*=Eh3/12(1 - v 2)
(13)
Here all the integrals are through the thickness of the plate from - h i 2 to + h/2.
Vibrations o/"an initially stressed mindlin plate
271
I ntegrating eqns (10)-(12) through the thickness o f the plate and using the stress-displacement relations (8) will result in an x : e x t e n s i o n , an x 2extension and an xa-shear equation. Furthermore, multiplying eqns (10) and (11) by x 3, an x~-moment and an x 2 - m o m e n t equation can be obtained by performing the same procedures in the derivation of the extensional equations. The x~-extension equation is D[u., + w2.1/2 + w.xD 1 + v(v.2 + w2.2/2 + w.2D2)]. 1 + Gh[u.2 + v l + w iw2 + w.ID2 + w 2D1].2 + (NxU. l + Mx~.l + Nxyu,2 + MxycX.2 + ~Qx).I + (Nru,2 + M:t.2 + Nxr u, l + Mx:t. t + ctQr),/+f~ = phil
(14)
where D x = r/.1 + Wo,1 and D 2 = q.2 + wo.2. The xz-extension equation is Gh[u,2 + I),1 + W IW,2 + w, ID 2 + w , 2 D 1 ] , l + D[v2 -t- w22/2 + w.2D 2 + v(u,1 + w21/2 + w, 1DI)],2
+ (N:.I + m 3., + N ,v.2 + m 3.2 + + (Nrv,2 + M, fl,2+ mxyv, l + Mxyfl, l + flQ,),2+fy = phi:
(15)
The x3-shear equation is hh'2G*[(ot.t + w l), 1 "l'l"(~.2 "l" W 2),2] + { G h ( w + q + wo).2(u.2 + v.t + w.aw2 + w.lD 2 + w 2 D 0 + D ( w + r/+ Wo).I[/2.1 "l" 1t'21/2 + w.,D1 + v(v.2 + wZa/2 + w.2D2)]}. , + [D(w+ r/+ Wo).2[v.2 + w22/2, + w,2D 2 + v(u, I + w2.1+ wADI)] + G h ( w + q + wo).l(u.a + v A + w A w 2 + W lD 2 + w.2Dt)}. 2 + (mxw.l + mxyw.2).l + (mxywl + Nyw,2).2 + q = phf~;
(16)
The x l - m o m e n t equation is
D*(~X.l + Yfl,2).l "{- (D*/2)(1 -- v)(~x,2+ fl.l),2 -- hxZG*( ~ + w.l) + (MxU.l + M ~*. I + M,,yu,2 + M*y~ 2 ctQ:)A* + (mru.2 + M,*ct.2 + M~rU.l + M*y~. 1 + ~Q*).2 - (Qxu.t + Q*~.I + Qyu.2 + Q%t.2) + m x = pha~i/12
(17)
The x 2 - m o m e n t equation is (D*/2)(I -- v)(c¢,2 + fl,1),l.l.D*(~I + vfl,2),2-- hx2G*(fl + w,2 )
+(M~v , + M*fl., + M~,v.2 + M*,fl.2 + flQ:,).l + (M.~.v.2 + M*fl.2 + Mx, v,, + M*,fl., + flQ*),2 -(Q:v,, + Qxfl.1 * + Qyv,2 + Q*fl.2) + mr = ph3]]/12
(18)
Lien-Wen Chen, Chuan-Cheng Lin
272
4 EXAMPLE PROBLEM Consider a simply supported plate in a state of initial stress which is taken as (19)
trl 1 = a,~ + 2x3trm/h
with all other initial stresses assumed to be zero. am and tr. are constant. It is comprised of a tensile (compressive) stress plus a bending load (see Fig. 1). The only non-zero initial stress resultants are Nx = ha.
M:, = fllhZa./6
M * = h3aJ12
(20)
where fll = am/a.. Suppose that the edges are all simply supported. The boundary conditions ave
v=w=fl=Nx=M~=0
at
x 1=0
and
a
(21)
u=w=ct=Ny=Mr=0
at
x 2=0
and
b
(22)
The geometric imperfections are assumed to be r/(x, y) = hp sin (nx,/a) sin (nx2/b)
(23)
where/a is a non-dimensionalized constant with respect to plate thickness, i.e. the maximum imperfection at the middle surface of the plate. Lateral loads and body forces are taken to be zero. Jx,fr, q, rex, mr = 0
(24)
The following one term mode shapes which are the same as the mode shapes of C h e n & D o o n g ~3 and satisfy the geometric boundary conditions of eqns (21) and (22) are assumed. w = h ~ t ) sin (rtxl/a) sin ( z x 2 / b )
(25)
{u, ~} = {ha(t), 8(0} cos (xxl/a) sin (lzx2/b)
(26)
{v, fl} = {h~(t), fl(t)} sin (rtxl/a) cos (xx2/b)
(27)
Then, by substituting the assumed vibration modes (25)-(27) and the assumed imperfection shape (23) into the equations of motion (14)--(18), and employing the Galerkin procedures, one obtains [C,j] {X} = [M,j] { 2 }
(28)
where {X} x = {~(t), f'(t), ~(t), ~(t), ~(t)}. X3
t =-
X1
hl
+S o n
Fig. !.
The state of initial stress combination.
Om
Vibrations of an initially stressed raindlin plate
273
T h e f o l l o w i n g coefficients a n d n o n - d i m e n s i o n a l p a r a m e t e r s are used: C t l = - - 6 [ 2 + r2(1 - v)] - - k r 2 / K C12 = - 6 r ( 1 + v) C~3 = - 32[4 + (1 -- 3v)r2](w + 21a + 2Wo)/(3Kn 2)
C14 = - f l l k r 2 / ( 6 K) Cls=O C21 = C12 C22 = - 6 1 2 r 2 + (1 - v)] - kr2/K C23 = - 3 2 1 4 r2 + (1 - 3v)](w + 2~u + 2Wo)/(3Krr 2) C24 = 0"0
C2s = - fltkr2/(6K) C31 = 64[2 - (1 - 3v)r2](w + # + Wo)/(3Kn 2) C32 = 64rE2r 2 - (1 - 3v)](w + # + Wo)/(3Kn 2) C3a = - 3 [ 2 r 2 + 9( r4 + 1)](w + # + wo)(w + 2/~ + 2Wo)/(8K) -- 12G'(1 + r 2) -- k r 2 / K C34 = - 12G'/K
C35 = -- 12G'r/K C42 = 0 C,1 = C14 C,4 = -- [1 + r2(1 -- v)/2 + C,5 = --r(1 + v)/2
C43 = C34
1 2 G ' K 2] -- kr2/(12K)
C53 = C35 C5,, = C45 C52 = C52 = [(1 v)/2 + r 2 + 1 2 G ' K 2] - kr2/(1210 C55 Mt~ = M22 = M33 = l
Maa = M 5 5 = 1/12 M,i = O f o r i ~ j where
r = a/b K = (a/h)/n z = t[D/(pha2)] 1/2 k = 12b2NJ(n2Dh 2)
5 NUMERICAL
RESULTS
In this study, n u m e r i c a l w o r k has b e e n d o n e for a t r a n s v e r s e l y i s o t r o p i c s q u a r e p l a t e (a/b = 1, h/b = 0.1) w i t h all edges simply s u p p o r t e d , b y the m e t h o d o f R u n g e - K u t t a . T h e s h e a r c o r r e c t i o n f a c t o r is a s s u m e d as 5/6. T h e initial c o n d i t i o n s f o r free flexural v i b r a t i o n are t a k e n to be ~ (z --- 0) = 0 a n d ~ z = O) = W/h, w h e r e W/h r e p r e s e n t s the initial a m p l i t u d e o f the plate. T h e
Lien-Wen Chen, Chuan-Cheng Lin
274 40.00
33.00
~O
26. O0
14 ®
19.00
12.00 ........
r ................................
5.00
0.10
'
'
0.28
0.46
C
' 0.64
' 0-82
1.00
W/h
Fig. 2.
Comparisons on non-linear periods for isotropic square plate, - - - - , Ref. 24, - present result; A, h/b=0"25; B, h/b=O'5; C, h/b=O'l.
minus sign of the initial stress (k = 12Nxb2/(x2Dh2)) represents compression and vice versa. The calculated periods are transformed into frequencies which are defined as ~2=
(29)
12pha20~2/(~z2D)
For verifying the accuracy of the current calculations, Fig. 2 indicates that the non-linear periods for imperfect plates are in good agreement with those presented in Ref. 24 for doubly curved shells. It is reasonable that the current non-linear periods are somewhat lower than those of Ref. 24 since Singh et al. 24 neglected the in-plane inertia terms and fourth and higher order derivatives with respect to time. I .30
k=O
U =0.0-U =0.1-
1.20
ij = 0 . 2 lJ = 0 . 3 p =0.4=0.5-
I.I0
fl 1 . 0 0
O. 90 0.80
0.70 0.60 -1.00
, -0.60
, -0.20
, 0.20
I 0.60
1.00
W/h
Fig. 3.
Effects o f i n i t i a l a m p l i t u d e o n n o n - l i n e a r f r e q u e n c i e s f o r t h i c k s q u a r e plate.
Vibrations o f an initially stressed mindlin plate
275
The effects of initial amplitude on non-linear frequency for an unstressed thick plate with various orders of geometric imperfections are shown in Fig. 3. When the plate is perfectly fiat (/~ = 0), the curve is symmetric with respect to W/h = 0 and the non-linear frequencies increase with the increase in initial amplitude. The vibration behaviour is the so-called hard spring type. For an imperfect plate, however, the curve is no longer symmetric. The minimum frequency, corresponding to linear vibrations, increases with the increasing imperfection amplitude owing to a curvature effect and finally becomes a local maximum surrounded by two adjacent local minima. That is to say, a significant rise in natural frequencies may occur in the presence of geometric imperfections. Furthermore, the variations in frequency with various orders of geometric imperfections are not uniform when different initial amplitudes are given. A similar phenomenon also occurs when the initial stress comes into play. Thel:efore, the effects of large amplitudes on the vibrations of an initially stressed imperfect plate are worth investigating although the linear vibrations of an imperfect plate under in-plane load have already been studied.' 7- 20 Figures 4a-4c illustrate the effects of initial amplitude on non-linear frequency for an initially stressed plate with different orders of geometric imperfections (/1 = 0.25, 0.5 and 0.75). The result with a perfectly flat plate without initial stress is re-plotted to show how the vibration behaviour changes. The trends are similar to those of an unstressed plate (Fig. 3) having geometric imperfections because the initial displacement (due to initial stress) and the geometric imperfections have a similar effect in the dynamic eqn (12). Figure 4a shows the results for small geometric imperfection /a = 0.25. The vibration behaviour is similar to that of perfect plate. On the 2.00
I .60
I .2O
0.80
"'-
0.40
Fig. 4a. Frequencies
I -1 . 0 0
k=O
.,~.-"
~//
J i i -0.50 0.00 0.50 I .00 W/h v e r s u s i n i t i a l a m p l i t u d e f o r a s q u a r e p l a t e : . . . . , k = O, It = O: - -
-2.00
I -1.50
.
It = 0"25.
"Lien-Wen Chen, Chuan-Cheng Lin
276 2.00
,, ,,,,,,
I .70
"",
k=3
~'0l8.01"1~0 0.50
.-°';'" .""'" I -I .50
-2.00
i -1 .00
I -0.50
I 0.00
i 0.50
I .00
W/h F i g . 4b.
Frequencies versus initial amplitude for a square plate; . . . . , k = 0 , / ~ = 0; - 1~ = 0-5. 2.00
1 • 70
i
,,,,,
k=3
1.40
-
",, "',,,,,,,,,,,,,,
1.10 k=O
"'-,
,'
0.80
0.50 -2.00
i -1 . 5 0
i -I .00
i -0.50
i 0.00
t 0.50
1 .00
W/h F i g . 4e.
Frequencies versus initial amplitude for a square plate; . . . . , k = 0 , / ~ = 0; - /~ = 0-75.
right side of Fig. 4b and Fig. 4c, the non-linear frequencies of an initially stressed plate having a large imperfection amplitude are greater than those of an unstressed plate even if the initial stresses are compression. On the other side, the non-linear frequencies of plates with a large negative initial amplitude decrease under the influence of initial stresses. It is remarkable that the tensile stress can raise the frequencies of plates and vice versa in most cases but the trends are reversed for small amplitude vibrations of plates having large geometric imperfections. Further, the fact that the effect of initial stresses on vibration behaviour is less influential for some special initial amplitudes such as W/h = - 0 . 8 and 0.4 in Fig. 4c is also interesting.
Vibrations of an initially stressed mindlin plate
277
Thus the vibration behaviour of an initially stressed plate is very much dependent on not only geometric imperfection but also initial amplitude. The plots of frequencies versus geometric imperfection for an initially stressed plate are shown in Figs 5a-5c for different initial amplitudes. Figure 5a is the case of a large amplitude vibration with W/h = - 1 and 1. The frequencies of positive initial amplitude are larger than those of negative initial amplitude. The trend of the curves for negative initial amplitude decreases and then increases with the increasing initial imperfection whereas the tendency for positive initial amplitude increases monotonously. The differences vanish in the case of perfectly fiat plate vibrations. Figure 5b
.40
..~:~/j
,
I 30 •
1 .20 I .10 I .00
%. 90 0.80 O.TO 0.60 0.00
=
0.20
l
,
0.40
I
0.60
0.80
! .0
1.1
Fig. 5a.
Frequencies versus geometric imperfections for an initially stressed plate; . . . . , W/h= i; - - - , W/h= - ! ; A, k = 3 ; B, k - - 0 ; C, k - - - 1 - 5 . I • 50
I
.30
l.TO A
° ° °- o" °"""
''"
,/
0.90
0.70
0.50 0.00
I.... O, 20
I O.
I
40
0.60
"'"'''""
[''"'" 0.80
1 .00
P
Fig. 5b.
Frequencies versus geometric imperfections: . . . . , W/h = 0"5; k = 3 ; B, k = 0 ; C, k = - 1 - 5 .
, W/h = --0.5; A,
Lien-Wen Chen, Chuan-Cheng Lin
278
l .80
1 .60
I .40
A B
1.20
I .00 0.80 0.60 0.40 0.00
i 0.20
I O. 4 0
a 0.60
J 0.80
I .00
F i g . 5c. L i n e a r f r e q u e n c i e s v e r s u s g e o m e t r i c i m p e r f e c t i o n s : A, k = 3; B, k = 0; C, k = - 1.5.
shows the case with W/h= +0"5. The frequencies of positive initial amplitude are no longer greater than those of negative initial amplitude. The non-linear frequencies for positive and negative initial amplitudes are equal for some orders of geometric imperfections. When the initial amplitude becomes smaller and can be neglected (Fig. 5c) the differences between positive and negative initial amplitude vanish. The frequencies, corresponding to linear vibration, increase monotonously with the increase of geometric imperfections. That is to say, the presence of geometric imperfections can raise the natural frequencies of an initially stressed plate. It is noticeable that the variations of geometric imperfections are less influential in tensile stressed plates (A curve) than in compressed plates (C curve) and the compressive stress is less important in plates having a large order of geometric imperfections because the plates are stiffened by the curvature effect. The effects of in-plane initial stresses on large amplitude vibrations of a plate having geometric imperfections p = 0.3 and 0"75 are shown in Figs 6a and 6b respectively. The initial amplitudes considered here are W/h = 0.5 and - 0 . 5 . In the case o f p = 0.3, the compressive in-plane stresses can reduce the stiffness of plates in most cases. The bending stress is unimportant in the case of small compressive loads. The non-linear frequencies of W/h = 0-5 are larger than those of W/h = - 0 - 5 . However, the non-linear frequencies of W/h = 0-5 are smaller than those of W/h = - 0 . 5 when/~ = 0.75. This result can also be observed in Fig. 5b. Notice that the vibration frequencies are raised under the influence of compressive stress when W/h=-0"5. The effects of bending stress are not uniform in W/h = 0.5. The small amplitude vibration behaviour of an initially stressed plate having geometric
Vibrations of an initially stressed mindlin plate
279
imperfection amplitudes 0.3 and 0-75 are also demonstrated in Figs 6c and 6d respectively. The compressive initial displacement can raise the vibration frequencies of plates due to the bending effect. The vibration frequencies decay in the presence of compressive stress when the order of geometric imperfection amplitude is small or zero. However, the effect of initial displacement is more influential than that of compressive stress when the imperfection amplitude is equal to or larger than about a quarter of the plate thickness. In dynamic equation (12), it is obvious that the compressive stress can reduce the stiffness of plates whereas the initial displacement can raise that of plates. Thus the non-linear vibration behaviour of an initially imperfect plate is very much dependent on the magnitude of the geometric imperfections and initial stress. 0.84 0,80
iJ=O. 3
A
0,76
~
O. 72
B
/--B==- 20-.X 13,= 0
0.68 0.64 0.60 -3.00
I -2.50
, -2.00
I -I .50 k
t -I.00
I -0-50
0.00
Fig. 6a. Effects of in-plane initial stress on frequencies: A, W/h = 0"5: B, W/h = - 0 ' 5 . I .30
1.20 l .10 I • O0
O. 90 0.80 0.70
FBj= 2 0
~=0
0.60 O. 50 -3.00
I
-2.50
I
-2.00
I
- I .50
I
- I .00
I
-0.50
0,00
k
Fig. 6b. Effects o f in-plane initial stress on frequencies: A, W/h=--0-5: B, W/h =0.5.
Lien-Wen Chen, Chuan-Cheng Lin
280 0.90
/--e,]- 2 0
0.80 B
0.70
0.60
0.50
0.40
0.30 -3.00
Fig. 6¢.
-2.50
-2.00
-I .50 k
-I .00
-0.50
0.00
Effects of in-plane initial stress on linear frequencies: A, p = 0; B, p = 0"3. 1 .45 I .41 ! .3T
t.33 fl 1 .29 I .25
1,21 1.17 -:3.00
Fig. 6(I.
-2.50
-2.00
-I .50 k
-I .00
-0.50
0.00
Effects of in-plane initial stress on linear frequencies:p = 0.75.
6 CONCLUDING
REMARKS
Non-linear equations of motion for an initiallystressed transversely isotropic plate having geometric imperfections are derived. The effects of transverse shear deformation and rotatory inertia are included. The equations derived here can be used to study non-conservative as well as conservative stability and dynamic problems for sundry states of initial stress. The preliminary results indicate the following: (1)
The present studies are in close agreement with those of Singh el
aL 2+
Vibrations of an initially stressed mindlin plate
(2)
(3) (4)
281
The effects of geometric imperfections on the vibrations of unstressed plates are similar to the effects of initial stresses on the vibrations of imperfect plates. A significant increase in n a t u r a l frequencies may occur for an initially imperfect plate even in the absence of initial stresses. The vibration behaviour is very much dependent on initial amplitude as well as geometric imperfection.
The results presented do not cover all the possible cases of this problem. They do indicate, though, some of the many interesting effects that can be studied with the present equations. Large amplitude vibration problems involving various boundary conditions and various initial stress distributions of thick plates are still to be investigated. Particularly interesting problems are the effect of time varying initial stresses on the large amplitude vibrations, which will be studied in the future.
REFERENCES 1. Chuen-Yuan Chia, Nonlinear Analysis of Plates. McGraw-hill, New York, 1980. 2. Celep, Z., Free flexural vibration of initially imperfect thin plates with large elastic amplitudes. ZAMM, 56 (1976) 423-8. 3. Celep, Z., An analogy between free vibration of a plate and of a particle of mass. Journal of Sound and Vibration, 53 (1977) 323-31. 4. Celep, Z., Shear and rotatory inertia effects on the large amplitude vibration of the initially imperfect plates. Journal of Applied Mechanics, 47 (1980) 662-6. 5. Yamaki, N., Otomo, K. & Chiba, M., Nonlinear vibrations of a clamped rectangular plate with initial deflection and initial edge displacement. Part 1: Theory. Thin Wall Structures, I(1) (1983) 3-29. 6. Yamaki, N., Otomo, K. & Chiba, M., Nonlinear vibrations of a clamped rectangular plate with initial deflection and initial edge displacement, Part 2: Experiment. Thin Wall Structures, 1(1) (1983) 101,19. 7. Hui, D., Large amplitude axisymmetric vibrations of geometrically imperfect circular plates. Journal of Sound and Vibration, 91(2) (1983) 239-46. 8. Hui, D., Effects of geometric imperfections on large amplitude vibrations of rectangular plates with hysteresis damping. Journal of Applied Mechanics, 51 (1984) 216--20. 9. Hui, D., Soft-spring nonlinear vibrations of antisymmetrically laminated rectangular plates. International Journal of Mechanical Science, 27(6) (1985) 397-408. 10. Herrmann, G. & Armanakas, A. E., Vibrations and stability of plates under initial stress. Journal of Engineering Mechanics Division, 127, 458-87. !1. Brunelle, E. J. & Robertson, S. R., Initially stressed Mindlin plates. AIAA Journal, 12(8) (1974) 1036-45. 12. Brunelle, E. J. & Robertson, Vibrations of an initially stressed thick plate. Journal of Sound and Vibration, 45(3) (1976) 405-16.
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13. Chen, L. W. & Doong, J. L., Large amplitude vibration of an initially stressed moderately thick plate. Journal of Sound and Vibration, 89(4) (1983) 499-508. 14. Chen, L. W. & Doong, J. L., Large amplitude vibration of an initially stressed thick circular plate. AIAA Journal, 21 (1983) 1317-24. 15. Mei, C., Large amplitude vibration of plate with initial stresses. Journal of Sound and Vibration, 60(3) (1978) 461-4. 16. Yang, T. Y. & Han, A. D., Buckled plate vibrations and large amplitude vibrations using high-order triangular elements. AIAA Journal, 21(5) (1983) 758-66. 17. Hui, D. & Leissa, A. W., Effects of geometric imperfections on vibrations of biaxially compressed rectangular flat plates. Journal of Applied Mechanics, 50, (1983) 750-6. 18. Hui, D., Effects of geometric imperfections of frequency-load interaction of biaxially compressed antisymmetric angle ply rectangular plates. Journal of Applied Mechanics, 52 (1985) 155-62. 19. llanko, S. & Dickinson, S. M., The vibration and postbuckling of geometrically imperfect, simply supported, rectangular plates under uni-axial loading, Part 1: theoretical approach. Journal of Sound and Vibration, 118(2) (1987) 313-16. 20. Ilanko, S. & Dickinson, S. M., The vibration and postbuckling of geometrically imperfect, simply supported, rectangular plates under uni-axial loading, Part 2: experimental investigation. Journal of Sound and Vibration, 118(2)(1987) 313-16. 21. Kapania, R. K. & Yang, T. Y., Buckling, postbuckling, and nonlinear vibrations of imperfect plates. AIAA Journal, 25(10) (1987) 1338-46. 22. Fung, Y, C., Foundations of Solid Mechanics. Prentice-Hall, Englewood Cliffs, NJ, 1965. 23. Boiotin, V. V., Nonconservative Problem of the Theory of Elastic Stability. Macmillan, New York, 1963. 24. Singh, P. N., Sundararajan, V. & Das, Y. C., Large amplitude vibration of some moderately thick structural elements. Journal of Sound and Vibration, 36(3) (1974) 375-87.
~,~:.\~.~,-~
Effects of Geometric Imperfections and Large Amplitudes on Vibrations of an Initially Stressed Mindlin Plate Lien-Wen Chen* & Chuan-Cheng Lin Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan 70104
(Received 19 February 1990; revised manuscript received 15 August 1991; accepted 19 September 1991)
ABSTRACT Non-linear equations of motion for a transversely isotropic plate having geometric imperfections and in a general state of non-uniform initial stresses are derived. The effects of both transverse shear deformation and rotatory inertia are included. The modal equations of a simply supported plate subjected to in-plane extensional stress and pure bending stress are obtained by performing the Galerkin procedure and then solved by the Runge-Kutta integration technique. The effects of geometric imperfections on large amplitude vibration behaviour are investigated. It is found that the vibration behaviour of an initially stressed plate is very much dependent on the order of initial amplitudes as well as geometric imperfections. The effects of various parameters on the non-linear vibration frequencies are studied.
NOTATION a,b
D D* E G G* h
Plateform dimensions of plate Extensional modulus [Eh/(1- v2)] Flexural modulus [Eh3/12(1- v2)] Young's modulus Shear modulus Transverse shear rigidity Thickness of plates
* To whom correspondence should be addressed. 265
Applied Acoustics 0003-682X/92/$05.00 © 1992 Elsevier Science Publishers Ltd, England. Printed in Great Britain
266
Lien-Wen Chen, Chuan-Cheng Lin
Initial extensional stress [k = 12Nxb2/(rt2h2D)] k K (alh)lzt Mx, M~.,Mxr Bending moment resultants ni Components of the unit normal with respect to the spatial frame mx, u,,,Ux, Stress resultants r a/b u> i) In-plane displacement of middle surface Components of initial and disturbed displacement Uio, Ui W Lateral displacement w/h Initial amplitude of free flexural vibration Xi Coordinates of a point x, Body force
fll qs P P tTij, ~ ij T
Angles of rotation of a normal to the middle surface in the Xl-X 3 and x2-x 3 planes, respectively The ratio of bending stress to extensional stress [fll = am/a,] Kronecker delta Components of initial imperfection Dimensionless geometric imperfection Poisson's ratio Density Initial and perturbing stress Dimensionless time [ = t[D/(pha2)] 1/2]
1 INTRODUCTION The non-linear analysis of elastic plates has received considerable attention and continues to remain of major concern because of its severe operational conditions in many mechanical, aerospace and hydrospace structural applications. Past investigators who had studied the effects of initial stresses on vibration problems have mainly concentrated on the perfect fiat plate. However, it is possible that certain deviations (the so-called geometric imperfections) between the manufactured plate and designed shape may exist. The studies on non-linear vibrations of an initially stressed, geometric imperfect plate have received relatively little attention although the statical and dynamical behaviour are very much dependent on the existence of geometric imperfections. A comprehensive review of the various developments in the geometrically non-linear behaviour of thin plates has been given by Chia. 1 Celep studied the effect of initial imperfection on the free vibration of rectangular plates 2
Vibrations of an initially stressed mindlin plate
267
and circular plates 3 by using the classical thin plate model. The influences of transverse shear deformation and rotatory inertia on the large amplitude vibration of imperfect plates are also studied by the above author. 4 Yamaki e t al. 5"6 investigated the effect of geometric imperfections and initial edge inplane displacements on the postbuckling and free vibrations of a rectangular plate. Recently, the effects of geometric imperfections on thenon-linear vibration of circular plates, 7 rectangular plates with hysteresis damping 8 and laminated composite plates 9 were conducted by Hui. In many practical applications, plate structures are subjected tO static loads causing internal stress fields. The presence of such stresses affects the vibration characteristics significantly. Herrmann & Armanakas 1° used a variational procedure to derive the equations of motion for an initially stressed thick plate. Brunelle & Robertson I ~ used the average stress method and the variational formulation to derive the governing equations for a thick plate in an arbitrary state of initial stress and then studied the linear vibration behaviour of a thick plate in a state of in-plane compression and bending load. ~2 Chen and Doong extended the theory of Brunelle ~ to derive the non-linear equations of motion for a thick plate in a general state of non-uniform initial stress to study the large amplitude vibrations of initially stressed rectangular 13 and circular plates. 14 Finite element solutions on large amplitude vibrations of plates under in-plane compression were presented by Mei ~5 and Yang & Han. ~6 The effects of geometric imperfections on the linear vibrations of simply supported plates under in-plane biaxial or uniaxial compression were examined by Hui ~7' 18 and Ilanko & Dickinson. ~902° The presence of small imperfections may significantly raise the frequencies and the sensitivity to imperfections increases with the amount of preload. Ilanko and Dickinson also observed experimentally and obtained close agreement with theoretical results. Furthermore, the finite element model for the non-linear vibrations of an imperfect plate was developed by Kapania & Yang. 2~ The arbitrary geometry of imperfections can be formulated by using a variable order polynomial. In the current study, the non-linear equilibrium equation in terms of a second Piola-Kirchhoff stress component 22 is generalised to include the effect of geometric imperfections. The equation is perturbed by an incremental deformation. 23 The products of incremental stresses with lateral (incremental and initial) displacements must be preserved here. The non-linear equations of motion, based on the large defection theory of von Karman's assumptions, for an initially stressed imperfect plate including the effects of transverse shear deformation and rotatory inertia are derived by the average stress method. For the sake of mathematical simplicity, the initial imperfections of the example problem studied here is assumed to be
268
Lien-Wen Chen, Chuan-Cheng Lin
sinusoidal and symmetric with respect to the plate centre. A simply supported plate under the effects of in-plane compressive (tensile) plus bending stress is presented to study the influences of geometric imperfections on the large amplitude vibration behaviour. On the basis of an assumed mode satisfying the geometric boundary conditions, the modal equations of motion are obtained by performing the Galerkin procedures. The previbration state (initial displacement field due to initial stress) of plates must be determined in the first phase of solution. The initial displacements are then superimposed on the non-linear dynamic equations to examine the large amplitude vibration behaviour by performing the method of Runge-Kutta. The non-linear periods for initially imperfect plates are close to those of Singh et al. 24 for a doubly curved shell without initial stress. The present studies will show that the effects of initial stresses on vibrations are dependent on both the order of initial amplitudes and geometric imperfections. It is remarkable that the initial stress can significantly affect the vibration behaviour for some special initial amplitude of vibration. Therefore, the large amplitude vibrations of an initially stressed, imperfect plate are worth investigating although the linear vibration behaviour is studied.17- 20
2 NONLINEAR EQUATION The nonlinear theory of curved plates due to geometric/imperfection is much more complicated than that of perfect plates. By introducing the imperfection component r/S, the nonlinear equation of equilibrium in terms of a second Piola-Kirchhoff stress component 22 can be expressed as ~xl I-(~i~S+
~u2/~xj + ~ns/~x~)cr*] + ,x'* = 0
(1)
where the xl are material (Lagrangian) coordinates that originally coincided with a spatial Cartesian coordinate system. The us and r/S are Cartesian components of the displacement and imperfection vector given with respect to the spatial coordinate system. The a* are Kirchhoff stress components taken with respect to undeformed areas. The X* are body force components referred to initial volume. The 6js symbol is the Kronecker delta. For notational convenience, a comma preceding an index implying partial differentiation with respect to the appropriate material coordinate will be used in the following text. The Lagrangian strain components ei~are defined as F,ij -~- ~(Ul, j "~ Uj. i -~- Uk.iUk, j "JI- Uk,fflk. i "JI- Uk,iYlk,j)
(2)
If relative extensions and shears are small, the values oftr* and X* are as a
Vibrations of an initially stressed mindlin plate
269
first order approximation equal to the actual stresses aii and the actual body force X s. These approximations will be adopted in the present work. The problem of interest here is a body in a state of non-uniform initial stresses, which is in static equilibrium and is subjected to a time varying incremental deformation. Following a technique described by Bolotin 23 the following quantities are introduced.
fi =
Uso + Us
,~'s = Xs + AX~ + )?~ - pu]
(3)
~ij =eij + g~j where, for example, ks, uso and ~7~represent the final displacement, initial displacement and incremental displacement, respectively. The term -p~'~ is the inertia force due to perturbation and the superior double dot denotes the second partial derivative with respect to time. Using the fact that both the initial and the final states satisfy the nonlinear eqn (1) gives the following non-linear equation for the incremental stresses and displacements.
[((~js "~- Us,j -~ as,j "~- ?~s.j)~ij],i "~- (aijas,j),i 21- AX$ ~1_Xs = P~i's
(4)
Similarly, the incremental Lagrangian strain components are related to incremental displacement, initial displacement and geometric imperfection by
eij = ½[~i,j -~ ~j,i "~- Uk,ilgk,j "~- Uk.j(l'lko.i "JI-nk,i) "~- Uk,i(glko.j "~- l~k,j) ]
(5)
The previbration state of static equilibrium must be determined first by using the non-linear eqn (1) and then the large amplitude vibrations of an initially stressed plate can be solved by the dynamic eqn (4). The form ofeqn (4) is the same as that of eqn (1) by dropping (trifis.j).i and (6uusa). i. For the sake of brevity, only the dynamic eqn (4) is used here to derive the equations of motion.
3 INITIALLY STRESSED I M P E R F E C T PLATE The incremental displacement fields for a point off the middle surface of a rectangular plate are assumed to be of the following form: UI(XI,X2,X3,
t)
=
U(Xl,x2, t) + Xa~(Xx,X2, t)
~2(xl, x 2, x 3, t) = v(x l, x2, t) + x3fl(x 1, x 2, t) /~3(Xl, X2, X3, t)= W(X1, X2, t)
(6)
Lien-Wen Chen, Chuan-Cheng Lin
270
where u, v, w, represent the displacements of the middle surface and, ~t and fl stand for the bending slope in the x, and x 2 directions. In the current study, only the vertical imperfection deviated from the plate surface is considered. Thus the field of geometric imperfection can be assumed as )71 =/~2 = 0
/13 = r/(x,,x2)
(7)
To simplify the non-linear equations, von Karman's assumptions for large deflection theory of plates are adopted here. By using eqns (5) and (6), the stress-displacement relations for a transversely isotropic material are a l l = (E/I - v2)(el x + ve22)
a22 = (El1 -- v2)(e22 + re11) a,2 =Ge,2
(8)
trl3 = ~c2G*(u'.I + a)
a23 = x2G*("'.2 +/~) where /311 = U. 1 "~ w.21/2+ w.,(Wo.1 + r/,,) + x3a.1 C22 = V.2 + W22/2 + W,2(Wo,2 + r/,2) + x3fl,2
(9)
F"12 ~"//.2 "[- V,1 + W, 1W,2 "1- W,I(Wo.2 "q- ?],2) -1- W,2(Wo. 1 "1- ?].1) "[- X3(~.2 -1- fl,')
G* takes into account the effects of transverse shear isotropy. Wois the initial displacement in the thickness direction due to initial stresses. Furthermore, the products of displacement gradient and stress in eqn (4) are so small they can be dropped except for t~nff3.~ and 17i2/t3o 2. In order to clarify the derivation procedure, it is useful to partially write out eqn (4):
(aij~l,j).l -[-#i,.i"[-Xl
"{- A X 1 ~--p/~"
(10)
(aUff2.j)., + #i2a +,~2 + AX2 =pf"
(11)
(tTijff3.j),i-[-ffi3.i--b [6il(ffs,j--J- l~,o.j--J-n.j)] i-J- x3 + mx3-~- p~;
(12)
For subsequent use in the equations of motion, the following initial stress resultants and material parameters are defined: (N~, Ny, Nxy, a~, Qy) = ~(at ,, a22, a, 2, a, 3, a23) dxs (Mx, Mr, Mxy, Q*,Q*)=S(tr,,,tr22,a,2,a13 ,a23)xadx a * = S (a , l, az2, cr 12).v 2 d.v 3 (M~* , M;* , M~r) D = E h / ( 1 --v 2) D*=Eh3/12(1 - v 2)
(13)
Here all the integrals are through the thickness of the plate from - h i 2 to + h/2.
Vibrations o/"an initially stressed mindlin plate
271
I ntegrating eqns (10)-(12) through the thickness o f the plate and using the stress-displacement relations (8) will result in an x : e x t e n s i o n , an x 2extension and an xa-shear equation. Furthermore, multiplying eqns (10) and (11) by x 3, an x~-moment and an x 2 - m o m e n t equation can be obtained by performing the same procedures in the derivation of the extensional equations. The x~-extension equation is D[u., + w2.1/2 + w.xD 1 + v(v.2 + w2.2/2 + w.2D2)]. 1 + Gh[u.2 + v l + w iw2 + w.ID2 + w 2D1].2 + (NxU. l + Mx~.l + Nxyu,2 + MxycX.2 + ~Qx).I + (Nru,2 + M:t.2 + Nxr u, l + Mx:t. t + ctQr),/+f~ = phil
(14)
where D x = r/.1 + Wo,1 and D 2 = q.2 + wo.2. The xz-extension equation is Gh[u,2 + I),1 + W IW,2 + w, ID 2 + w , 2 D 1 ] , l + D[v2 -t- w22/2 + w.2D 2 + v(u,1 + w21/2 + w, 1DI)],2
+ (N:.I + m 3., + N ,v.2 + m 3.2 + + (Nrv,2 + M, fl,2+ mxyv, l + Mxyfl, l + flQ,),2+fy = phi:
(15)
The x3-shear equation is hh'2G*[(ot.t + w l), 1 "l'l"(~.2 "l" W 2),2] + { G h ( w + q + wo).2(u.2 + v.t + w.aw2 + w.lD 2 + w 2 D 0 + D ( w + r/+ Wo).I[/2.1 "l" 1t'21/2 + w.,D1 + v(v.2 + wZa/2 + w.2D2)]}. , + [D(w+ r/+ Wo).2[v.2 + w22/2, + w,2D 2 + v(u, I + w2.1+ wADI)] + G h ( w + q + wo).l(u.a + v A + w A w 2 + W lD 2 + w.2Dt)}. 2 + (mxw.l + mxyw.2).l + (mxywl + Nyw,2).2 + q = phf~;
(16)
The x l - m o m e n t equation is
D*(~X.l + Yfl,2).l "{- (D*/2)(1 -- v)(~x,2+ fl.l),2 -- hxZG*( ~ + w.l) + (MxU.l + M ~*. I + M,,yu,2 + M*y~ 2 ctQ:)A* + (mru.2 + M,*ct.2 + M~rU.l + M*y~. 1 + ~Q*).2 - (Qxu.t + Q*~.I + Qyu.2 + Q%t.2) + m x = pha~i/12
(17)
The x 2 - m o m e n t equation is (D*/2)(I -- v)(c¢,2 + fl,1),l.l.D*(~I + vfl,2),2-- hx2G*(fl + w,2 )
+(M~v , + M*fl., + M~,v.2 + M*,fl.2 + flQ:,).l + (M.~.v.2 + M*fl.2 + Mx, v,, + M*,fl., + flQ*),2 -(Q:v,, + Qxfl.1 * + Qyv,2 + Q*fl.2) + mr = ph3]]/12
(18)
Lien-Wen Chen, Chuan-Cheng Lin
272
4 EXAMPLE PROBLEM Consider a simply supported plate in a state of initial stress which is taken as (19)
trl 1 = a,~ + 2x3trm/h
with all other initial stresses assumed to be zero. am and tr. are constant. It is comprised of a tensile (compressive) stress plus a bending load (see Fig. 1). The only non-zero initial stress resultants are Nx = ha.
M:, = fllhZa./6
M * = h3aJ12
(20)
where fll = am/a.. Suppose that the edges are all simply supported. The boundary conditions ave
v=w=fl=Nx=M~=0
at
x 1=0
and
a
(21)
u=w=ct=Ny=Mr=0
at
x 2=0
and
b
(22)
The geometric imperfections are assumed to be r/(x, y) = hp sin (nx,/a) sin (nx2/b)
(23)
where/a is a non-dimensionalized constant with respect to plate thickness, i.e. the maximum imperfection at the middle surface of the plate. Lateral loads and body forces are taken to be zero. Jx,fr, q, rex, mr = 0
(24)
The following one term mode shapes which are the same as the mode shapes of C h e n & D o o n g ~3 and satisfy the geometric boundary conditions of eqns (21) and (22) are assumed. w = h ~ t ) sin (rtxl/a) sin ( z x 2 / b )
(25)
{u, ~} = {ha(t), 8(0} cos (xxl/a) sin (lzx2/b)
(26)
{v, fl} = {h~(t), fl(t)} sin (rtxl/a) cos (xx2/b)
(27)
Then, by substituting the assumed vibration modes (25)-(27) and the assumed imperfection shape (23) into the equations of motion (14)--(18), and employing the Galerkin procedures, one obtains [C,j] {X} = [M,j] { 2 }
(28)
where {X} x = {~(t), f'(t), ~(t), ~(t), ~(t)}. X3
t =-
X1
hl
+S o n
Fig. !.
The state of initial stress combination.
Om
Vibrations of an initially stressed raindlin plate
273
T h e f o l l o w i n g coefficients a n d n o n - d i m e n s i o n a l p a r a m e t e r s are used: C t l = - - 6 [ 2 + r2(1 - v)] - - k r 2 / K C12 = - 6 r ( 1 + v) C~3 = - 32[4 + (1 -- 3v)r2](w + 21a + 2Wo)/(3Kn 2)
C14 = - f l l k r 2 / ( 6 K) Cls=O C21 = C12 C22 = - 6 1 2 r 2 + (1 - v)] - kr2/K C23 = - 3 2 1 4 r2 + (1 - 3v)](w + 2~u + 2Wo)/(3Krr 2) C24 = 0"0
C2s = - fltkr2/(6K) C31 = 64[2 - (1 - 3v)r2](w + # + Wo)/(3Kn 2) C32 = 64rE2r 2 - (1 - 3v)](w + # + Wo)/(3Kn 2) C3a = - 3 [ 2 r 2 + 9( r4 + 1)](w + # + wo)(w + 2/~ + 2Wo)/(8K) -- 12G'(1 + r 2) -- k r 2 / K C34 = - 12G'/K
C35 = -- 12G'r/K C42 = 0 C,1 = C14 C,4 = -- [1 + r2(1 -- v)/2 + C,5 = --r(1 + v)/2
C43 = C34
1 2 G ' K 2] -- kr2/(12K)
C53 = C35 C5,, = C45 C52 = C52 = [(1 v)/2 + r 2 + 1 2 G ' K 2] - kr2/(1210 C55 Mt~ = M22 = M33 = l
Maa = M 5 5 = 1/12 M,i = O f o r i ~ j where
r = a/b K = (a/h)/n z = t[D/(pha2)] 1/2 k = 12b2NJ(n2Dh 2)
5 NUMERICAL
RESULTS
In this study, n u m e r i c a l w o r k has b e e n d o n e for a t r a n s v e r s e l y i s o t r o p i c s q u a r e p l a t e (a/b = 1, h/b = 0.1) w i t h all edges simply s u p p o r t e d , b y the m e t h o d o f R u n g e - K u t t a . T h e s h e a r c o r r e c t i o n f a c t o r is a s s u m e d as 5/6. T h e initial c o n d i t i o n s f o r free flexural v i b r a t i o n are t a k e n to be ~ (z --- 0) = 0 a n d ~ z = O) = W/h, w h e r e W/h r e p r e s e n t s the initial a m p l i t u d e o f the plate. T h e
Lien-Wen Chen, Chuan-Cheng Lin
274 40.00
33.00
~O
26. O0
14 ®
19.00
12.00 ........
r ................................
5.00
0.10
'
'
0.28
0.46
C
' 0.64
' 0-82
1.00
W/h
Fig. 2.
Comparisons on non-linear periods for isotropic square plate, - - - - , Ref. 24, - present result; A, h/b=0"25; B, h/b=O'5; C, h/b=O'l.
minus sign of the initial stress (k = 12Nxb2/(x2Dh2)) represents compression and vice versa. The calculated periods are transformed into frequencies which are defined as ~2=
(29)
12pha20~2/(~z2D)
For verifying the accuracy of the current calculations, Fig. 2 indicates that the non-linear periods for imperfect plates are in good agreement with those presented in Ref. 24 for doubly curved shells. It is reasonable that the current non-linear periods are somewhat lower than those of Ref. 24 since Singh et al. 24 neglected the in-plane inertia terms and fourth and higher order derivatives with respect to time. I .30
k=O
U =0.0-U =0.1-
1.20
ij = 0 . 2 lJ = 0 . 3 p =0.4=0.5-
I.I0
fl 1 . 0 0
O. 90 0.80
0.70 0.60 -1.00
, -0.60
, -0.20
, 0.20
I 0.60
1.00
W/h
Fig. 3.
Effects o f i n i t i a l a m p l i t u d e o n n o n - l i n e a r f r e q u e n c i e s f o r t h i c k s q u a r e plate.
Vibrations o f an initially stressed mindlin plate
275
The effects of initial amplitude on non-linear frequency for an unstressed thick plate with various orders of geometric imperfections are shown in Fig. 3. When the plate is perfectly fiat (/~ = 0), the curve is symmetric with respect to W/h = 0 and the non-linear frequencies increase with the increase in initial amplitude. The vibration behaviour is the so-called hard spring type. For an imperfect plate, however, the curve is no longer symmetric. The minimum frequency, corresponding to linear vibrations, increases with the increasing imperfection amplitude owing to a curvature effect and finally becomes a local maximum surrounded by two adjacent local minima. That is to say, a significant rise in natural frequencies may occur in the presence of geometric imperfections. Furthermore, the variations in frequency with various orders of geometric imperfections are not uniform when different initial amplitudes are given. A similar phenomenon also occurs when the initial stress comes into play. Thel:efore, the effects of large amplitudes on the vibrations of an initially stressed imperfect plate are worth investigating although the linear vibrations of an imperfect plate under in-plane load have already been studied.' 7- 20 Figures 4a-4c illustrate the effects of initial amplitude on non-linear frequency for an initially stressed plate with different orders of geometric imperfections (/1 = 0.25, 0.5 and 0.75). The result with a perfectly flat plate without initial stress is re-plotted to show how the vibration behaviour changes. The trends are similar to those of an unstressed plate (Fig. 3) having geometric imperfections because the initial displacement (due to initial stress) and the geometric imperfections have a similar effect in the dynamic eqn (12). Figure 4a shows the results for small geometric imperfection /a = 0.25. The vibration behaviour is similar to that of perfect plate. On the 2.00
I .60
I .2O
0.80
"'-
0.40
Fig. 4a. Frequencies
I -1 . 0 0
k=O
.,~.-"
~//
J i i -0.50 0.00 0.50 I .00 W/h v e r s u s i n i t i a l a m p l i t u d e f o r a s q u a r e p l a t e : . . . . , k = O, It = O: - -
-2.00
I -1.50
.
It = 0"25.
"Lien-Wen Chen, Chuan-Cheng Lin
276 2.00
,, ,,,,,,
I .70
"",
k=3
~'0l8.01"1~0 0.50
.-°';'" .""'" I -I .50
-2.00
i -1 .00
I -0.50
I 0.00
i 0.50
I .00
W/h F i g . 4b.
Frequencies versus initial amplitude for a square plate; . . . . , k = 0 , / ~ = 0; - 1~ = 0-5. 2.00
1 • 70
i
,,,,,
k=3
1.40
-
",, "',,,,,,,,,,,,,,
1.10 k=O
"'-,
,'
0.80
0.50 -2.00
i -1 . 5 0
i -I .00
i -0.50
i 0.00
t 0.50
1 .00
W/h F i g . 4e.
Frequencies versus initial amplitude for a square plate; . . . . , k = 0 , / ~ = 0; - /~ = 0-75.
right side of Fig. 4b and Fig. 4c, the non-linear frequencies of an initially stressed plate having a large imperfection amplitude are greater than those of an unstressed plate even if the initial stresses are compression. On the other side, the non-linear frequencies of plates with a large negative initial amplitude decrease under the influence of initial stresses. It is remarkable that the tensile stress can raise the frequencies of plates and vice versa in most cases but the trends are reversed for small amplitude vibrations of plates having large geometric imperfections. Further, the fact that the effect of initial stresses on vibration behaviour is less influential for some special initial amplitudes such as W/h = - 0 . 8 and 0.4 in Fig. 4c is also interesting.
Vibrations of an initially stressed mindlin plate
277
Thus the vibration behaviour of an initially stressed plate is very much dependent on not only geometric imperfection but also initial amplitude. The plots of frequencies versus geometric imperfection for an initially stressed plate are shown in Figs 5a-5c for different initial amplitudes. Figure 5a is the case of a large amplitude vibration with W/h = - 1 and 1. The frequencies of positive initial amplitude are larger than those of negative initial amplitude. The trend of the curves for negative initial amplitude decreases and then increases with the increasing initial imperfection whereas the tendency for positive initial amplitude increases monotonously. The differences vanish in the case of perfectly fiat plate vibrations. Figure 5b
.40
..~:~/j
,
I 30 •
1 .20 I .10 I .00
%. 90 0.80 O.TO 0.60 0.00
=
0.20
l
,
0.40
I
0.60
0.80
! .0
1.1
Fig. 5a.
Frequencies versus geometric imperfections for an initially stressed plate; . . . . , W/h= i; - - - , W/h= - ! ; A, k = 3 ; B, k - - 0 ; C, k - - - 1 - 5 . I • 50
I
.30
l.TO A
° ° °- o" °"""
''"
,/
0.90
0.70
0.50 0.00
I.... O, 20
I O.
I
40
0.60
"'"'''""
[''"'" 0.80
1 .00
P
Fig. 5b.
Frequencies versus geometric imperfections: . . . . , W/h = 0"5; k = 3 ; B, k = 0 ; C, k = - 1 - 5 .
, W/h = --0.5; A,
Lien-Wen Chen, Chuan-Cheng Lin
278
l .80
1 .60
I .40
A B
1.20
I .00 0.80 0.60 0.40 0.00
i 0.20
I O. 4 0
a 0.60
J 0.80
I .00
F i g . 5c. L i n e a r f r e q u e n c i e s v e r s u s g e o m e t r i c i m p e r f e c t i o n s : A, k = 3; B, k = 0; C, k = - 1.5.
shows the case with W/h= +0"5. The frequencies of positive initial amplitude are no longer greater than those of negative initial amplitude. The non-linear frequencies for positive and negative initial amplitudes are equal for some orders of geometric imperfections. When the initial amplitude becomes smaller and can be neglected (Fig. 5c) the differences between positive and negative initial amplitude vanish. The frequencies, corresponding to linear vibration, increase monotonously with the increase of geometric imperfections. That is to say, the presence of geometric imperfections can raise the natural frequencies of an initially stressed plate. It is noticeable that the variations of geometric imperfections are less influential in tensile stressed plates (A curve) than in compressed plates (C curve) and the compressive stress is less important in plates having a large order of geometric imperfections because the plates are stiffened by the curvature effect. The effects of in-plane initial stresses on large amplitude vibrations of a plate having geometric imperfections p = 0.3 and 0"75 are shown in Figs 6a and 6b respectively. The initial amplitudes considered here are W/h = 0.5 and - 0 . 5 . In the case o f p = 0.3, the compressive in-plane stresses can reduce the stiffness of plates in most cases. The bending stress is unimportant in the case of small compressive loads. The non-linear frequencies of W/h = 0-5 are larger than those of W/h = - 0 - 5 . However, the non-linear frequencies of W/h = 0-5 are smaller than those of W/h = - 0 . 5 when/~ = 0.75. This result can also be observed in Fig. 5b. Notice that the vibration frequencies are raised under the influence of compressive stress when W/h=-0"5. The effects of bending stress are not uniform in W/h = 0.5. The small amplitude vibration behaviour of an initially stressed plate having geometric
Vibrations of an initially stressed mindlin plate
279
imperfection amplitudes 0.3 and 0-75 are also demonstrated in Figs 6c and 6d respectively. The compressive initial displacement can raise the vibration frequencies of plates due to the bending effect. The vibration frequencies decay in the presence of compressive stress when the order of geometric imperfection amplitude is small or zero. However, the effect of initial displacement is more influential than that of compressive stress when the imperfection amplitude is equal to or larger than about a quarter of the plate thickness. In dynamic equation (12), it is obvious that the compressive stress can reduce the stiffness of plates whereas the initial displacement can raise that of plates. Thus the non-linear vibration behaviour of an initially imperfect plate is very much dependent on the magnitude of the geometric imperfections and initial stress. 0.84 0,80
iJ=O. 3
A
0,76
~
O. 72
B
/--B==- 20-.X 13,= 0
0.68 0.64 0.60 -3.00
I -2.50
, -2.00
I -I .50 k
t -I.00
I -0-50
0.00
Fig. 6a. Effects of in-plane initial stress on frequencies: A, W/h = 0"5: B, W/h = - 0 ' 5 . I .30
1.20 l .10 I • O0
O. 90 0.80 0.70
FBj= 2 0
~=0
0.60 O. 50 -3.00
I
-2.50
I
-2.00
I
- I .50
I
- I .00
I
-0.50
0,00
k
Fig. 6b. Effects o f in-plane initial stress on frequencies: A, W/h=--0-5: B, W/h =0.5.
Lien-Wen Chen, Chuan-Cheng Lin
280 0.90
/--e,]- 2 0
0.80 B
0.70
0.60
0.50
0.40
0.30 -3.00
Fig. 6¢.
-2.50
-2.00
-I .50 k
-I .00
-0.50
0.00
Effects of in-plane initial stress on linear frequencies: A, p = 0; B, p = 0"3. 1 .45 I .41 ! .3T
t.33 fl 1 .29 I .25
1,21 1.17 -:3.00
Fig. 6(I.
-2.50
-2.00
-I .50 k
-I .00
-0.50
0.00
Effects of in-plane initial stress on linear frequencies:p = 0.75.
6 CONCLUDING
REMARKS
Non-linear equations of motion for an initiallystressed transversely isotropic plate having geometric imperfections are derived. The effects of transverse shear deformation and rotatory inertia are included. The equations derived here can be used to study non-conservative as well as conservative stability and dynamic problems for sundry states of initial stress. The preliminary results indicate the following: (1)
The present studies are in close agreement with those of Singh el
aL 2+
Vibrations of an initially stressed mindlin plate
(2)
(3) (4)
281
The effects of geometric imperfections on the vibrations of unstressed plates are similar to the effects of initial stresses on the vibrations of imperfect plates. A significant increase in n a t u r a l frequencies may occur for an initially imperfect plate even in the absence of initial stresses. The vibration behaviour is very much dependent on initial amplitude as well as geometric imperfection.
The results presented do not cover all the possible cases of this problem. They do indicate, though, some of the many interesting effects that can be studied with the present equations. Large amplitude vibration problems involving various boundary conditions and various initial stress distributions of thick plates are still to be investigated. Particularly interesting problems are the effect of time varying initial stresses on the large amplitude vibrations, which will be studied in the future.
REFERENCES 1. Chuen-Yuan Chia, Nonlinear Analysis of Plates. McGraw-hill, New York, 1980. 2. Celep, Z., Free flexural vibration of initially imperfect thin plates with large elastic amplitudes. ZAMM, 56 (1976) 423-8. 3. Celep, Z., An analogy between free vibration of a plate and of a particle of mass. Journal of Sound and Vibration, 53 (1977) 323-31. 4. Celep, Z., Shear and rotatory inertia effects on the large amplitude vibration of the initially imperfect plates. Journal of Applied Mechanics, 47 (1980) 662-6. 5. Yamaki, N., Otomo, K. & Chiba, M., Nonlinear vibrations of a clamped rectangular plate with initial deflection and initial edge displacement. Part 1: Theory. Thin Wall Structures, I(1) (1983) 3-29. 6. Yamaki, N., Otomo, K. & Chiba, M., Nonlinear vibrations of a clamped rectangular plate with initial deflection and initial edge displacement, Part 2: Experiment. Thin Wall Structures, 1(1) (1983) 101,19. 7. Hui, D., Large amplitude axisymmetric vibrations of geometrically imperfect circular plates. Journal of Sound and Vibration, 91(2) (1983) 239-46. 8. Hui, D., Effects of geometric imperfections on large amplitude vibrations of rectangular plates with hysteresis damping. Journal of Applied Mechanics, 51 (1984) 216--20. 9. Hui, D., Soft-spring nonlinear vibrations of antisymmetrically laminated rectangular plates. International Journal of Mechanical Science, 27(6) (1985) 397-408. 10. Herrmann, G. & Armanakas, A. E., Vibrations and stability of plates under initial stress. Journal of Engineering Mechanics Division, 127, 458-87. !1. Brunelle, E. J. & Robertson, S. R., Initially stressed Mindlin plates. AIAA Journal, 12(8) (1974) 1036-45. 12. Brunelle, E. J. & Robertson, Vibrations of an initially stressed thick plate. Journal of Sound and Vibration, 45(3) (1976) 405-16.
282
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13. Chen, L. W. & Doong, J. L., Large amplitude vibration of an initially stressed moderately thick plate. Journal of Sound and Vibration, 89(4) (1983) 499-508. 14. Chen, L. W. & Doong, J. L., Large amplitude vibration of an initially stressed thick circular plate. AIAA Journal, 21 (1983) 1317-24. 15. Mei, C., Large amplitude vibration of plate with initial stresses. Journal of Sound and Vibration, 60(3) (1978) 461-4. 16. Yang, T. Y. & Han, A. D., Buckled plate vibrations and large amplitude vibrations using high-order triangular elements. AIAA Journal, 21(5) (1983) 758-66. 17. Hui, D. & Leissa, A. W., Effects of geometric imperfections on vibrations of biaxially compressed rectangular flat plates. Journal of Applied Mechanics, 50, (1983) 750-6. 18. Hui, D., Effects of geometric imperfections of frequency-load interaction of biaxially compressed antisymmetric angle ply rectangular plates. Journal of Applied Mechanics, 52 (1985) 155-62. 19. llanko, S. & Dickinson, S. M., The vibration and postbuckling of geometrically imperfect, simply supported, rectangular plates under uni-axial loading, Part 1: theoretical approach. Journal of Sound and Vibration, 118(2) (1987) 313-16. 20. Ilanko, S. & Dickinson, S. M., The vibration and postbuckling of geometrically imperfect, simply supported, rectangular plates under uni-axial loading, Part 2: experimental investigation. Journal of Sound and Vibration, 118(2)(1987) 313-16. 21. Kapania, R. K. & Yang, T. Y., Buckling, postbuckling, and nonlinear vibrations of imperfect plates. AIAA Journal, 25(10) (1987) 1338-46. 22. Fung, Y, C., Foundations of Solid Mechanics. Prentice-Hall, Englewood Cliffs, NJ, 1965. 23. Boiotin, V. V., Nonconservative Problem of the Theory of Elastic Stability. Macmillan, New York, 1963. 24. Singh, P. N., Sundararajan, V. & Das, Y. C., Large amplitude vibration of some moderately thick structural elements. Journal of Sound and Vibration, 36(3) (1974) 375-87.