Vibrations of an Initially Stressed Thick Plate

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Journal of Sound and Vibration (1976) 45(3), 405-416

VIBRATIONS OF AN INITIALLY STRESSED THICK PLATEE. J. BRUNELLE~

Department of Mechanics, Mechanical and Aeronautical Engineering,Rensselaer Polytechnic Institute, Troy, New York, U.S.A.

ANDS. R. ROBERTSONSGTE Laboratories, Waltham, Massachusetts 02154, U.S.A.

(Received 14 April 1975,and in revisedform 27 September 1975)By using previously derived equations for a thick p late in an arbitrary state of non-uniforminitial stress the vibrational behavior of a thick, simply supported rectangular plate subjectedto initial stress is investigated . The initial stress is taken to be a combination of pure b endingplus extensional stress in the plane of the plate. The vibrational behavior is correlated to thebuckling behavior in order to explain how , in certain cases, higher modes can vibrate atfrequencies lower than those of the lower order m odes.

1. INTRODUCTIONPast investigations of the effects that initial stresses have on the vibrations of beams and plateshave been mainly concerned with in-plane com pressive or tensile stresses. Very little has beendone w ith respect to the formulation and solution of equations for beams and plates in anarbitrary state of initial stress. This is particularly true for the cases of thick beams an d plateswhere rotatory inertia and transverse shear effects are impo rtant. The first attempt to deriveequations of motion for a thick plate in a state of initial stress was made by Herrmann andArmenakas [ 11. They used a variational procedure to derive the governing equations. Theirformulation had certain inconsistencies in that they used two different definitions of gener-alized stress, the boundary and lateral loading terms were not clearly defined and, based onunclear argu ments, certain terms were dropped from the final equations. Brunelle [2] derivedthe governing equations for a thick beam in an arbitrary state of initial stress. It was foundthat the equations derived in reference [l] would not reduce down to the equations for thethick beam. In addition, it was found that the terms dropped in reference [l], from the finalequations, do contribute to the behavior of the thick beam in a significant way. Sun [3]derived the equations for a thick beam having an initial, in-plane com pressive stress by usingboth the Trefftz and Biot formulations. Wh en the equations in reference [2] are specializedto the case of an initial, in-plane compressive stress, they agree with Suns equations, derivedby using the Trefftz formulation. Brunelle and Robe rtson [4] derived the equations for athick plate in an arbitrary state of initial stress by two different methods. In the derivationsthe Trefftz generalized stress was used throughout and no terms were dropped from the finalequations. The equations were used to study the static buckling behavior of a simply sup-ported, thick plate under combined initial compressive stress and bending stress acting inthe plane of the plate. The p urpose of this paper is to extend that work to the considerationof the effects of the initial, in-plane comp ressive (tensile) plus bending stress on the vibratorybehavior of a simply-supported, thick plate.

7 C onsultant o Watervliet Arsenal, W atervliet, N ew York 1218 9, U.S.A.$ Formerly w ith The Ballistic Research Laboratory, Aberdeen Proving Ground, Maryland, U.S.A.405


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40 6 E. J. BRUNELLE AND S. R. ROBERTS ONThough the governing equations for an initially stressed, thick plate are rather cumb er-some, they are given here in order to show how they simplify down for the problem discussedhere. An attempt is made to relate some of the unusual effects the initial stress field has onthe vibratory behavior of the plate to its buckling behavior.

2. THE GOVERNING EQUATIONSThe problem of interest here is that of a thick plate that is in a state of non-uniform initialstress, which is in static equilibrium and subjected to a time varying incremental deformation.

In accordance with a technique described by Bolotin [5] the following quantities are intro-duced :

zi, = u + ii,, (1)

8, j = b j + Oij, (2)

@,=P,+AP,+& (3 )J?s=Xs+AXs+IP,-pB,, (4 )where, for example, li,, u, and zi, represent the final displacement, initial displacement andthe perturbing displacement, respectively. The terms Ap, and AX, represent changes in theinitial surface traction and body force due to the perturba tion. The term -pif, is the inertiaforce due to the perturbation where the superior double dot denotes the second partialderivative with respect to time.In deriving the thick plate equations the following displacement field was assum ed inorder to account for transverse shear and in-plane extensional deformation:

r&(x,, x2,x3,2) = u,(x,, x2, t) + x3 $,(x1, x1. r), (5)&(x1. x2, x 3 , t ) = U 2 ( X l r x 2 , t ) + x 3 J / * (x , * x 2 > t h (6 )a , @ , , x 2 , t > = 4 % x 2 , t ). (7)

8, and ti2 are the in-plane displacements and ~ 2~s the lateral deflection of the neutral surface.ui, u2 and w are displacements of the neutral surface. Ic/1 nd I,$? re the angular changes oflines initially normal to the neutral surface. Note that the incremental displacements definedin equations (5)-(7) do not have superior bars. They should not be confused with the initialdisplacements of equation (1). However, since the initial deformation will appear only interms of stress resultants, this should cause no difficulty. The above assum ption for the dis-placement field is analogous to that used in the derivation of the Mindlin plate equationsWI .The incremental stress-displacement relations are taken to be those of linear elasticity:

5 -11 - &A-,.1 ~,.z),a,, = +2.2 + &,l),

a 2 = G(4.z + %.A (10)CT - K z G * (4 . 3 + a 3 . 1 ) >3 - (1 1 )6 2 3 = K 2 G *@2 . 3 + 0 3 . 2 )~ ( 1 2 )

where G* accoun ts for the effect of transverse isotropy. If G* = G the mate rial is isotropic.K is M indlins shear correction factor [6].


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40 8where

E. J. BRUNELLE AND S. R. ROBERTSON

hl29= J (X,+dX,)d. Y3 + w ,,(a,+, - a,) + w .,(o,: - a,) + a;3 - a,.

-hi2

The x,-moment equation is

h+ 2 1//1.*(42 - G2) + $,(d3 + 633) + 5:3 + cr ; 1 U8*)

The x,-moment equation is

-Q:$2.2 -Nz~2-r2G*h(~2+w~2)+m,=~~2. (19)where

+ Yj (cJ3: - 032) $2.2 + (d3 + 033) $2 + 52: + 5, 1 (19*)The boundary conditions, in terms of normal and tangential co-ordinates, on the plateare given below. First, the following traction resultants are defined :

AFnn = [ AP,, dx,, Fnn = j Is, dx,t

AF,, = Apt dxj,s F,, = 1 I% dxj,AF,, = Ap, dx,,1 F.3 = 1 A dx,,

AM., = j AP, x3 dx3, &z, A j P. x3 dx,,

AM,, = j Apt x3 dx,, A?,,, = j & x3 dx,, (20)


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VIBRATIONS OF A STRESSED THICK PLATE 40 9where all integrals are from -h/2 to +h/2 and

Bn + & I = ~#I,Ll + a, a,., + t 7 , 3 C , , + C , (2 1 )i 4 + AP, = gnn ~t ,n + 0 . t Ut . r + ~ 3 4 . 3 + @, a , ( 2 2 )B 3 + A P , = c n n ~ 3 , . + g n c u 3 . r + g n 3 ~ 3 . 3 + 5 V n 3 , (23)

where the subscripts n and t denote the normal and tangential directions of the plates edge,respectively. The terms contain ingthe initial stresses account for the change in the initialboundary conditions due to the incremental deformation.The five boundary traction conditions are4, + AF,, = N u,., + M . L + N,,, u,,r + M,, $,,, + Q,, ti. + D(u,,, + VU & (24)R,, + AF,, = N, ~r,n + M. Ic/,,, + N,,, u,,, + M ,, $,., + Q. $t + GW,,, + u,,.), (25)

F n 3 + A J n 3 = N . w . n + N n t w , , + K z G * W , t + w , J , (2 6 )

l l ; i , + A M , , = M % + M f I C / n . n + M , % I . 1 + M , I I / , , , + Q ,* $. + w ,., + v& t>, (27)m, + AM,, = M, nf,, + MZ $,., + Mm,~t,f + M,: tit,, + QE $, + ; (1 - v) (A,, + $,,,>. (28)Alternative displacement boundary conditions are

u, = %n, u, = %, w = W 3 , 4 + = G . , * t = * . t , (2 9 )where the quantities on the right sides are prescribed.If a rectangular plate were being considered the boundary conditions would be rephrasedin x1, x2 co-ordinates. For example, on the edge where x2 is constant the unit normal hascomponents (0, I, 0) and the subscript IZ orresponds to the x,-direction while the subscript tcorresponds to the x,-direction. Thus, for instance, equation (28) would become

These equations are rather cumbersome. However, for many particular problems theycan be simplified to something manageab le as well as interesting.3. SIMPLY SUPPORTED RECTANG ULAR PLATE

Consider a simply supported thick rectangular plate in a state of initial stress. The state ofinitial stress is cl1 = a, +2x30,/h, (30)

f

Figure 1. The rectangular plate and the initial stress field.


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410 E. J. BRUNELLE AND S. R. ROBERTSO Nwith all other initial stresses assumed to be zero. bN and c,+, are taken to be constants so thatthe initial stress field is uniform. It is comprised of a tensile (compressive) plus a bendingstress: see Figure 1. From equations (13) the only non-zero initial stress resultants are

Nx = ha,, M, = h2 cr,+,/6, M: = h3a,/12.Lateral loads and body forces are taken to be zero:

f x , . f y , 4 , mx , m,0 .The equations of motion (15)-( 19) simplify toMX IC/ .11 + Nxu,,,, + WC,, + vu& + GQ,,,, + ~2.12) = @A ,Mx$ 2.l, + Nxu,,,, + GN U,,,, + ~2.11) + Wu,.,, + vu,.,J = phii,,N, w.11 + KG* h($,,, + w ,,l) + KG* h($z.L + w .~J = phi i,

(31)

(32)

M: $,,I, + Mxu ,,~, + gtJ(II/u,+ v$z.zJ+ g,, 1.22 + $2.12) - K2 G* h(lC/, + w ,,) = $i,,A.Gh3

M,*IC/ .11 + Mx4.1, + ,,(h,21 + ti2.11) + g($ 2.22 + ~41.12) - K2 G* h($2 + w,J = $32.(33)

The boundary conditions (24)-(29) are, for the simply supp orted plate, on the x1 = constantedges,

w = 0, *2 = 0, u2 = 0,~~~+~~~,=~,I(/,,,+N,~,.,+~(u,.,+vu,,,)=O,

~~,+~Mr~=Mx*~~,~+M,u~.,+~(~,,,+v~,.,)=o, (34)and, on the x2 = constant edges,w = 0, *I = 0, ur =o,

F22 + AF22 = M, $2.2 + N, ~2.2 + W42.2 + u,.,) = 0,

a22 + AM,: = M,* $2.2 + Myu2.2 = W$,., + ~$1.1) = 0. (35)Displacements of the following form will satisfy equations (33) and boundary conditions

(34) and (35):

w= W~~sin(~)sin(~)exp(iwr),Ic/1= r,..cos~~)sin(~jerp(i,r),

ti2 = Y,..,i,(~)cos(~jenp(icur),

u1 = U,,cos(y)sin( y)exp(iwr),

u2 = U,..sin( y)cos(y)erp(ior).Brunelle and Robertson [4] have used the above solution to study the static buckling behaviorof the simply-supported thick plate. The characteristic equation found for that case may


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V I B R A T I O N S O F A S T R E S S E D T H I C K P L AT E 411also be used to study the free vibration problem. The characteristic equation is found bysubstituting the assum ed displacem ent field of equations (36) into the equations of motion(33). Since the equations are homogeneous, the determinant of the coefficients m ust vanish inorder for a solution to exist. This procedure gives the following characteristic equation forthe determination of the buckling loads and the natural frequencies. The characteristicequation in non-dimensional form is

whereb,, =m 2+n2 7 (u/b)2 + m (a/b)k lK - a2/K,

b 12 = ( ) m nu /b2 , bm = 0, b,, = m 2(u/b) (k/K ) jY/(L jK ),

bn =O, 21 --j---= ( + ) m n a / b 3

b22 = (nu /b)2 + l-v2 + (a/b)2 k /K m 2 - a2/K,1 b23 = 0,b,, = 0, b2, = (m a/b )2 (k/K ) 8/(3KY 2, b,, = 0, b,, = 0,bJ3 = (a/b)2 (m + k Sm 2 + (na/b)2 - SR 2, bs4 = m (alb)2,bj5 = n(4b)3, b,, = SjI (a/b)2 m 2 k /(3K)12, b,, = 0,

b43 m(alb)2, b,, = S m 2[(u/b)2 k /K + I ] + T (a/b)2 n2 + (a/b)2/S - R 21K ,b,, = Sm n(a/b) (1 + v)/2, bs = 0, b,, = Sj ?(a/b)2 m 2 k /(3K)li2,

bs3 = n@/b), b, , = Sm n(a/b) (1 + v)/2,bs5 = S{m2[(a/b)2 k /K + (1 - v )/2] + (na/b )2 + (a/b)2/S - SJ2/K},

and where the following non-dimensional parameters have been used :B = Q A 4 l ~ M Q = K2G*h/9 =12K2G*/hD, M = ph3/ 129 = phi D,S = Eh2/G* b2( 1 - v) = Dh/b2 G* (transverse isotropy parameter),

S, = NJ9 = 12N Jh2D , k = (b/n)2 S, = 12b2N&r2 h 2 D (buckling coefficient),K = (U/h)2 /K 2, Q 2 = pa402/hz2 ~~ D (non-dimensional frequency).

Taking K = 7r2/12 [6] gives Q = n2G */hD = S(7c/b)2, (3K)12 = 6a/z ch, k /K= (b/a)2Nx /D,SJ2/K = ph a2c02/x2 D, k S = NJK~ G* h = 12N,..7t2G* h. f12 has been non-dimensionalizedwith respect to the first Bernoulli-Euler beam frequency. v is taken to be O-30.


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412 E . J . B R U N E L L E A N D S . R . R O B E R T S O N

I: I 2 3 5J/L1

Figure 2. Buckling curve envelopes for various values of 8.There are so many parameters that can be varied that it would be difficult to present

results for all cases. From the numerous problems solved, only a few typical cases will beselected for discussion. These cases will illustrate the salient features of the way that the thickplate considered here behaves. The initial in-phase compressive (tensile) stress is containedin the buckling coefficien t, k. If k is positive the stress is tensile. The initial in-plane bendingstress is contained in /?. When p = 0 and/or k = 0, there is no initial bending stress . For theproblem considered here a/h = 10 and S = 0.05. For this case, Brunelle and Robertson[4] have comp uted the static buckling loads for /I = 0,lO. In these cases n = I. A composite

1 htchness -shear

40

i \r Extensmal20 -

Flexural/ 1 /-3 -2 -I 0 / 2

h

i

:

Figure 3. Frequency uersus non-dimensional extensional load for the five plate modes when m = 2, n = 1,a/b = 2 and b = 5.


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VIBRATIONS OF A STRESSED THICK PLATETABLE 1

N at ural f requencies for k = 0, a/b = 2, m = 2, n = 1 and 4 = 0.

41 3

a Mode type7.42741 Flexural

18.4509931.18787 Extensional

100.33580166.22782 i Thickness-shear

plot of the buckling load envelopes for j? = 0, 5, 10 is given along with the classical result fora simply su pported thin plate with only an in-plane compressive stress acting in Figure 2.

The envelope for a = 5 is new. W hat these envelopes reveal is the mod e shape a plate w illhave when the compressive load exceeds that o f the envelope for a given width to lengthratio, a/b. These curves are calculated by determining the roots of the characteristic equationwhen Q = 0. Physically, this means that a compressive stress is sought which completelynegates the rigidity of one of the flexural mod es of the plate so that it will oscillate at zerofrequency with a finite am plitude, in the absence of a lateral force. Th ere will be five naturalfrequencies associated with each mode (i.e., for each pair o fm and n ). If k = 0 (i.e., no in-planeinitial stress) the equations of motion separate into a pair of coupled extensional equationsand the flexural equation coupled to the two thickness shear equations. Since k = 0 impliesgN = 0 this m eans that oM = 0 regardless of the value of /?. The natural frequencies for thiscase, when m = 2, n = 1 and a/b = 2 are given in Table 1. Figure 3 show s the behavior ofeach mod e as the in-plane load go es from tensile to compressive when p = 5. Notice how thelowest natural frequency (flexure) is noticeably affected by the variation in k . Also, noticehow it drops to zero when k reaches the buckling load for this case (see Figure 2). The changein k affects the extensional frequencies to a much lesser degree and has a very small affecton the thickness shear modes. For the remainder of the discussion only the flexural mod ewill be considered.

Figure 4 is a very interesting plot. It is a plot of natural frequen cy versus initial tensile andcompressive stress for various m when a/b = 2, n = 1 and p = 5. It can be seen that the fre-quency curve for m = 1 crosses those for the higher mod es. Also, fro m Figure 2 it can beseen that this plate w ill buckle in the second mode m = 2, n = 1. It can be seen that the fre-quency curve form = 2 goes to zero at the buckling load for this case. The interesting thing tonote is that prior to buckling, the frequency curve for m = 1 is above that for m = 2. Thissays that the lowest mode m = 1, n = 1, vibrates at a higher frequency than the second mod em = 2, n = 1. It can also be seen that all the higher mod es are more affected than the lowestmod e by increased compressive stress. In general, frequency curves for mod es of lowerorder than the mod e in which buckling occurs will cross above the curve for the buckling

a

hFigure 4. Frequency versus initial extensional stress for various values of m when n = 1 , a/b = 2 and B = 5.


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41 4 E. J. BRUNELLE AND S. R. ROBERTSON

Figure 5. Frequency versus initial extensional stress for various values of m when n = 1, a/b = 2 and /3 = 0.

mode prior to buckling. It is interesting that the stiffness? of a higher mode can be madeless than that of a lower mode by the proper variation of the compressive load. M odes higherthan the buckling mode cannot have their frequency curves brought below that of the bucklingmode. Thus, if the aspect ratio a /b is such that the plate buckles in the first mode (see Figure2), the frequency curves will not cross. Another explanation of this behavior can be madewith the aid of the buckling curves of Figure 2. Consider the curve for /? = 0 when m = 1.It is apparent that as the aspect ratio increases from zero, the buckling load for that m odedecreases until it reaches a minimu m, after which it continues to increase. As it increases,it crosses above the curves for the higher modes. Thus, the first mode will become progressivelystiffer u i s -a& the higher modes as a /b increases. Thus, when a/b = 2, for example, theload necessary to cause buckling in the first mode is greater than the load necessary to causebuckling in the second mode. This implies that the first mode is stiffer than the secondmode in the neighborhood of the buckling load for this case. Therefore, as one examines thefrequency behavior of the various modes a s the compressive load increases, one would expecta transition to take place where those modes lower than the buckling mode become stifferthan the buckling mode. This has been borne out in the discussion of Figure 4. This behavioris also observed when /? = 0, 10 .Figure 5 illustrates the sam e effects as Figure 4 except for the fact that /? = 0. Figure 6illustrates these effects for /3 = 10. Altho ugh the figure does not show it clearly, the curveform = 2 passes above the curve form = 3 just prior to reaching the buckling load. Examina-tion of Figure 2 indicates that buckling occurs very close to the cusp near a/b = 2 on thefl = 10 curve. The computer output indicates that buckling indeed occurs when m = 3 for avalue of k = -3.1265. The value of k = -3.1473 when m = 2 is also found which indicates

t For a linear vibration problem, each mode can be represented by an equivalent spring m ass. For a fixedmass, as the frequency increases the spring constant increases with its square. Thus, if the first mode has ahigher frequency than the second mode it may be thought to be stiffer since its associated spring constant islarger.


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VIBRATIONS OF A STRESSED THICK PLATE 415

-6 -5 -4 -3 -2 -I 0 I 2 3 4 5 6k

Figure 6. Frequency uersus initial extensional stress for various values of m when n = 1, a/b = 2 and B = 10.

kFigure 7. Frequency oersus initial extensional stress for various values of m when n = 1, a/b = 4 andj?=5.

20 I I-5

n IO- 432m =l

I I0 5 IO

PFigure 8. Frequency versus the ratio of initial bending to extensional stress, for various values of m whenn=l,a/b=2andk=-2.


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416 E.J. RUNELLEAND S.R.ROBERTSONthat the curve for m = 2, as well as that for m = I, passes above the curve for nz= 3 prior tobuckling. Mo reover, the curve form = 1 is above that for m = 2.

Figure 7 illustrates these same results when p = 5 and n = 1 when ajb = 4. In this casebuckling occurs when m = 5 and all the curves m = I to 4 pass above it just prior to buckling.Interestingly enough, the frequency for m = 1 is higher than for m = 2 which is higher thanthat for m = 3, etc., just prior to buckling. Thus, for mode s b elow the buckling mod e, thefrequency prior to buckling undergoes a transition after which the frequency increases asthe order of the mode decreases.

Figure 8 show s the variation of the natural frequency as /? varies w hen a/b = 2, n = 1 andk = -2. No te that the increase in initial bending stress, ,!I, has little effect on the flexuralmod es shown excepting the slight softening of the higher modes. Of course, in the neighbor-hood of the buckling load, p has a dramatic effect as seen in Figure 2.

4. CONCLUSIONAlthough the results presented do not cover all the possible cases for this problem they

typify its behavior and point out some interesting consequences of an initial stress field.There is a lot of room for the future study of the effects that other initial stress fields wouldhave on this and other geometries.

REFERENCES1. G. HERRMANN and A. E. ARMENAKAS 1962Transactions of the American Society of Civil Engineers

127, 458. Vibratio ns and stability of plates under initial stress.2. E. J. BRUNELLE 1972 ournal of Applied Mechanics 39, 819-821. Stability an d vibrations of trans-versely isotropic beams under initial stress.3. C. T. SUN 1972 JournaI of Applied Mechanics 39, 282-285. On the equations for a Timoshenkobeam under initial stress.4. E. J. BRUNELLE and S. R. ROBERTSON 1974American Institute of Aeronautics and Astronautics

Journal 12, 1036-1045. Initially s tresse d Mindlin p lates.5. V. V. BOLOTIN 1963Non-Conservative Problems of the Theory of Elastic Stability. The MacMillanCompany.6. R. D. MINDLIN 19.51ournal of Applied Mechanics 18 , 31-38. Influence of rotary inertia andshear on flexural motions of isotropic elastic plates.

Vibrations of an Initially Stressed Thick Plate

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