Shear effects on vibration of plates

Journal of Sound and Vibration (1978) 60(2), 308-311

SHEAR EFFECTS ON VIBRATION OF PLATES 1. INTRODUCTION Large amplitude vibration of moderately thick isotropic, orthotropic as well as composite rectangular plates have been studied in the past by Mindlin (using linear theory) and others [1-6]. Recently the study has been extended to cover a rather widely used plate geometry, i.e., skew plates [7, 8]. The Berger approximation has been incorporated in some of these papers and the results thus obtained have been shown to be reasonably good for practical engineering purposes. The author has suggested an approximation similar to Berger's for tile non-linear static and dynamic analysis of skew plates and the applicability of this approximation has been supported by numerical results [9, 10]. This approximation is the basis of the study of the effects of transverse shear deformation and rotatory inertia on the large amplitude vibration of isotropic skew plates in reference [7]. The governing equations for the large amplitude vibration of plates wherein the effects of transverse shear deformation and rotatory inertia are included have been presented in different forms by the different authors of the papers cited previously. However, all of them have assumed the in-plane displacement components tt or v at a distance z away from the median surface of the plate to consist of u~ or v~ (median surface displacements) and a quantity equal to z times the rotation along the x or y direction represented by cz or ft. It is obvious that ce equals - w ~ and fl equals - w y when the effect of transverse shear deformation is not considered in the formulation. When one makes an attempt to introduce these simplifications into the final governing equations presented by the earlier investigators [2-8, I I] the resulting equations do not appear to be suitable for application to the linear and non-linear vibration of plates where the transverse shear and rotatory inertia effects are not considered. Also, in some cases the final governing equations themselves are of such a nature that these simplifications cannot be easily introduced [3, 4]. For example, equations (30), (31), (32) and (33) of reference [2] do not simplify to give the required linear or non-linear equations when the transverse shear effect is neglected. However, such simplified linear or non-linear equations can be derived by referring back to the set of equations (19)-(28) of reference [2]. This obviously involves an additional effort. In the light of these observations, it is the purpose of this note to show that a system of dynamic equations can be derived to take into effect the transverse shear and the rotatory inertia in a manner which can be simplified without any difficulty when one does not want to "see" these effects in the formulation. Tracing constants, as has been suggested in references [2] and [3], have been used to make the reduction simpler. An orthotropic skew plate is considered for illustration as these equations are more general and cover the plate geometries dealt with in this note. 2. EQUATIONS OF MOTION

The following 13 equations of motion, derived by means of Hamilton's principle and the use of variational calculus, involve the stress resultants N~, Ny and Nxy, the moment resultants Mx, My and M~, the shearing forces Vx and Vr, the rotations ccand p, and the displacement components tt ~ v ~ and w:

M x . x + M ~ y . , - V ~ , - a , R , ct.,t=O,

M,.,+M,,.,-V,-atR,

p.,,=O,

(1,2)

n.x -- (12/h3)(cxt Mx + c~2 My + ct4 M:,,) = O, (3) 308 0022--460X/78/180308 + 04 $02.00/0 9 t 978 AcademicPress Inc. (London) Limited

309

LETTERS TO TIlE EDITOR

ft., - ( 1 2 / h ' ) (c,z M,, + c22 My -4- c=,, M=,) = O, cq, + B.x -- (121h 3) (c,, Mx + c24 M, + c,, M=,) = O, ot + w.,, - ( 6 T d 5 h ) ( c s s

V, + cs, V,) = O,

fl + w., - (6T,/5h) (c~6 V= + c66 V,) = O, t , - - ( I / h ) ( c t , N = + c,2

N,

+ c,,t N x , ) =

0,

c,z -- (l/h) (c,, N= + c,2 N, + c2, N~,) = 0, ? - (l/h) (c,, N= + c2, N, + c , N=,) = O,

N=.= + N,,., = pt hu],,

N,., + N~,.= = pz hv~

Vx, x + Z,.,-b Nxw.,, + N,w.,,-I- 2Nx, w.,, + ll = 0 ,

(4) (5) (6) (7) (8) (9) (10) (11, 12)

(13)

where It = c { q . ( x , y ) - pthw.,,}, q.(x,y) is the lateral load per unit area of the plate, h is the thickness of the plate, pt is the mass per unit volume of the plate, as = (cp~h3]12) and e = cosO, 0 being the skew angle. Rl and T. are the tracing constants which represent the influences of rotatory inertia and transverse shear deformation, respectively. Also

{au} = [a,A {eu}

or

{~u} = [c,A {au}"

The coefficients aij and ctj are not defined hcre but can be found in reference [7] or [8]. If an approximation similar to Berger's is introduced for skew plates [9, I0] equations (1)-(7) remain unchanged whereas all the other six equations simplify to e = e~ + 25 ez + (22 + 23)? = ~2h2C(t)[12,

(14)

an he{w== + 2~ w.,, + 2(22 + 23) w.=,} + Vx.= + V,., + 1~ = 0,

(15)

where el = cuO~ + svO.,, + 89 ~, = v ~ + 89

)q = (a22/anW z,

s=

sin O,

~ = cu ~ + sv~ + v~ + w.~ w . ,

22 = (--2alJaH) u2,

3.3 = (--2a2Jau) 'n.

Equations (1)-(7), (14) and (15) are then the nine equations in this approximation involving the nine quantities M=, M , M=,., V=, V, u, p, e and w. One can now simplify equations (I)-(13) in a manner which is easy for use at a later stage. The procedure is as follows. Equations (1)-(5) are solved for V= and Vy in terms of a and ft. Substituting for V~ and Vy in equations (6) and (7) gives

D, fl,~,, + D5 fl,=y + D6fl...) + R,T.(Dvct.,, + Ds P..) = 0,

Ot + W,x -- Ts(D10t.Xx + D , ct,,y + D 3 ot,yy +

(16)

1ff+ W., -- Ts(D9ot.xx + Dto~.xr + D z : o~.,,+ Dl2fl.xx "l-Dt'~ff.:~, + D,, ft.,,) + R, T,(D,s ~.,, + D,6 ft.,,) = 0.

(17)

With the in-plane inertia terms neglected, the rcmaining six equations, namely equations (8)-(13), are either simplified in terms of u ~ v~ and w or written in terms of a stress function F , where N= = hF.,,, N, = h F ~ , N,, = -hF=,. Not forgetting the compatibility condition (to be obtained from equations (8)-(10)) and writing equations (8)-(13) in terms o f F , one obtains c ~ F . . . . - 2c~, ~ x , , , + (2el. + c . ) F,,,,,, - 2c,, F,,,,,,, + e,, F . , . , = w?,,, - w.,,,, w,., I, + I2 + I 3 = 0 ,

(I 8)

(19)

310

LETTERS TO TIlE EDITOR

where I, = h(F.,, w . ~ + F.xx w.,, - 2F.~, w.~,),

/3 = DlTct .... W D I s t:t,zxy --b D l g c t . x y y

-I-

D2o~.y~y

+ D2t fl .... + D22 fl.~x, + D2s fl.~,, + Dz( ft.,,, + R, o2scct.xt, + ft.,it). Equations (16)-(19) are then the four equations governing the large amplitude flexural vibration of orthotropic skew plates wherein the effects of transverse shear and rotatory inertia are taken into account. Coefficients D~ to D2s are not defined here for the sake of saving space. Similar coefficients that are defined for isotropic skew plates can be found in references [7] and [8]. If the transverse shear effect is not to be included in the formulation, one sets T , - - 0 in equations (16)-(19). Likewise, if the rotatory inertia effect is to be eliminated R, is set equal to zero. 3. NO TRANSVERSE SHEAR EFFECT

When the effect oftransverse shear deformation is omitted, Ts = 0 and hence from equations (16) and (17) it follows that, O~ ~ - - W x ,

f l ~ - - W 1 ~.

Substituting for ct and fl in equation (19) gives

+ (D2o + D23) w.,,,, + D2+ w.m,} - R, D2s(a2/at 2) (v 2 w) = o.

(20)

Equations (I 8) and (20) are the required equations. In addition, if R, = 0 then equations (18) and (20) agree with those in relerence [10] where these effects are not considered. It is worth mentioning here that such a straightforward simplification is not possible with the equations presented by the earlier investigators [3, 4]. 4. EQUATIONS IN THE BERGER TYPE APPROXIMATION

Since equations (1)-(7) hold good in this case, equations (16) and (17) which are derived from equations (1)-(7) are also applicable to this case. Substituting for Vx and Vy as outlined earlier puts equation 05) in the form

11 + a~l he{w.x~ + 21 w.,, + 2(3.2 + ).a) w.x,} + 13 = 0.

(21)

Equations (14), (16), (17) and (21) are the required equations. When the effects of transverse shear and rotatory inertia are both neglected these equations readily simplify to those in reference [10]. In conclusion, it is to be mentioned that equations (16)-(19) or equations (14), (16) (17) and (21) in the Berger type approximation are the two sets of dynamic equations (each consisting of four equations) applicable for orthotropic skew plates. These two sets are simple enough for reduction when the transverse shear or rotatory inertia effect is to be eliminated from them. ct and fl could further be eliminated from these two sets of equations by writing each set to consist of only two equations either in terms of w and F or in terms of w and e (corresponding to a Berger type approximation). This procedure is very similar to the one outlined in reference [8] for isotropic skew plates and hence is omitted here. The main advantage of this approach is the simplicity with which the governing equations may be applied for plates with or without the shear and rotatory inertia effects. With a p p r o priate simplifications, the same set ofgoverning equations can be used to study the individual

LETTERS TO TIlE EDITOR

311

effect of the transverse shear deformation or the rotatory inertia. Finally, the numerical work performed with the equations presented here indicates that the results remain absolutely unchanged whether one considers the equations presented here or those presented earlier [7, 8, t 1] as long as the effects of transverse shear and rotatory inertia are included. Department of Civil Enghleering, The University of Calgary, Calgary, Alberta, Canada T2N 1N4

M . SATttYAMOORTHY

(Received9 May 1978)

REFERENCES

1. R. D. MINDLIN1951 Journal of Applied Mechanics 18, 31-38. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. 2. C. I. Wo and J. R. VINSON1969 Journal of Composite Materials 3, 548-561. On the nonlinear oscillations of plates composed of composite materials. 3. C. I. Wo and J. R. VINSON1969 Journal of Applied Mechanics 36, 254-260. Influences of large amplitude, transverse shear deformation and rotatory inertia on lateral vibrations of transversely isotropic plates. 4. P. N. SINGH,Y. C. DAS and V. SONDARARAJAN1971 Journal of Sound and Vibration 17, 235-240. Large-amplitude vibration of rectangular plates. 5. C. I. Wo and J. R. VINSON1971 Journal of the Acoustical Society of America 49, 1561-1567. Nonlinear oscillations of laminated specially orthotropic plates with clamped and simply supported edges. 6. J. RAMACHANDRAN1975 Journal of the Franklin Institute 299, 359-362. Vibration of variable thickness plates at large amplitudes. 7. M. SATHYAMOORTHY1977 Journal of Sound and Vibration 52, 155-163. Shear and rotatory inertia effects on large amplitude vibration of skew plates. 8. M. SATHYAMOORTHY1978 International Journal of Solids and Structures. Vibration of skew plates at large amplitudes including shear and rotatory inertia effects. 9. M. SATHYAMOORTHYand K. A. V. PANDALAI1973 American Institute of Aeronautics and Astronautics Journal 11, 1279-1282. Large amplitude flexural vibration of simply supported skew plates. 10. M. SATHYAMOORTHYand K. A. V. PANDALAI1973 Journalofthe Franklin Institute 296, 359-369. Nonlinear vibration of elastic skew plates exhibiting rectilinear orthotropy. 1I. M. SATHYAMOORTHY1978 American Institute of Aeronautics and Astronautics Journal 16, 285286. Vibration of plates considering shear and rotatory inertia.