Large amplitude flexural vibration of stiffened plates

Journal of Sound and Vibration (1978) 57(4), 583-593

LARGE

AMPLITUDE

FLEXURAL

STIFFENED

VIBRATION

OF

PLATES

G. PRATHAP AND T. K. VARADAN Department of Aeronautics, Indian Institute of Technology. Madras-600036, India (Received 18 October 1977, arzdin rerisedfbrm

5 December 1977)

The large amplitude free flexural vibration of thin, elastic orthotropic stiffened plates is studied. The boundary conditions considered are either simply supported on all edges or clamped on all edges and the in-plane edge conditions are either immovable or movable. The governing dynamic equations are derived in terms of non-dimensional parameters describing the stiffening achieved, and the solutions are obtained on the basis of an assumed one-term vibration mode shape for various stiffener combinations. In all cases, the nonlinearity is found to be of the hardening type (i.e., the period of non-linear vibration decreases with increasing amplitude). Some interesting conclusions are drawn as to the effect of the stiffening parameters on the non-linear behaviour. A simple method of predicting the postbuckling and static large deflection behaviour from the results obtained in this analysis is included.

1. INTRODUCTION The non-linear or large amplitude flexural vibration of plates and shells has received considerable attention in recent years because of the great importance and interest attached to structures of low flexural rigidity. These easily deformable structures vibrate at large amplitudes and the classical bending theory becomes inadequate and it is necessary to allow for moderately large deflections. This paper deals with the large amplitude, free flexural vibrations of clamped and simply supported stiffened plates of two different in-plane edge conditions-movable and immovable. The governing equations are derived and applied to the case of a symmetrically stiffened plate with stiffeners in both x and y directions. These equations reduce to the well known von Karman equations [l] for an unstiffened isotropic plate and those in reference [ 21 for an unstiffened orthotropic plate. The solutions are obtained on the basis of a single term vibration mode shape by making use of Galerkin’s method. The non-linear behaviour for different geometrical and stiffener parameters is presented in the form of non-dimensional charts. In all cases, the non-linearity is found to be of the hardening type: i.e., the period of non-linear vibration decreases with increasing amplitude.

2. DERIVATION

OF GOVERNING

EQUATIONS

The geometry of the plate, stiffeners and co-ordinate system is shown in Figure 1. The principal directions of orthotropy are taken to coincide with the co-ordinate direction and the stiffeners run parallel to the edges of the plate. The neutral plane, for purposes of derivation, is assumed to be the middle plane of the plate, this assumption becoming exactly true only in the symmetrically stiffened case studied here. 583 0022-460X/78/0422-0583 $02.00/O 0 1978 Academic Press Inc. (London) Limited

584

G. PRATHAP AND T. K. VARADAN

Cc)

(b)

Figure 1. Plate and stiffener layout. (a) Geometry and co-ordinate notations for a repeating section; (c) smearing out of stiffeners.

system; (b) stiffener geometry and

The number of stiffeners is assumed to be large enough to consider them as a continuous distribution over the plate (smeared out approach). The torsional and lateral stiffnesses of the stiffeners as well as the effects of in-plane inertia are neglected in this analysis. Figure 1 shows a typical cross-section of the stiffeners running in the x direction. The stiffeners on either side of the plate have areas, spacing and eccentricities denoted by A,,, d,,, es1 and A,,, d,, esz, respectively. Similarly, for stiffeners running in the y direction, the notations, A,,, d,,, e,,, A,,, 4, and e,, are used (a list of symbols is given in Appendix II). The displacement pattern of the plate is described by U,v and w in the x, y and z directions, respectively. The middle surface (z = 0) strains are El

=

u,,

+

3w.L

E2 =

u*y +

3w3,

Y =

II,,

+

u,x

+

w,x

w .Y’

(1)

The strains at a point off the middle surface are s, = s1 - ~*,*,

e, =

E2 -

ZW,YY,

YXY =

Y -

2ZW,XY’

(2)

=

(3)

The stress-strain relations for an orthotropic plate are written as ax

=

&

8,

+

4512 &Y,

0,

=

&2

E,

+

E22

&yt

7xy

Wxy,

where

El1 = WU - v,yvyx), El2

=

J% vxy/U

-

vxy

vy3

=

E22

=

Ey

v,/U

Ey/(l

-

vxy

vxy

vy,),

vy,).

The total strain energy of the stiffened plate is the sum of the strain energies of the plate and the stiffeners. With the definitions C,, = E,,h, Cl2 = E12h, C,, = E,,h, C, = Gh, Dll =

585

LARGE VIBRATIONS OF STIFFENED PLATES

E,,h3/12,

Dzz = E,,h3/12,

and DG = Gh3/12, the strain energy of the un-

D12 = E,,h3/12

stiffened plate is Q=tJj

(C 11e: + c,, El + 2c12 E]E2+ c, y2 + D,, WfX, XY + D22 w.:Y + 2012 w,,, w,,, + 4Dci wfx,, d.x dy.

(4)

Following Varadan [3], the strain energy of the stiffeners along the .x and y direction may be written as US= US, i-

ur

us29

=

ur,

+

ur2,

where, for a typical case,

U,I = +%I ( i j[ (u,, + tw:J2$

- 2(u,, + 3w.Z~w,,, 2 (5)

+w,;XjA1 +;S1e’l)]dxdy), etc., and hence

(6)

ur=3 jj [Cu.,+ t’?#

a, + 2Cu.y + +w%, w,yyyr + w,2yy Al dx dy,

(7)

XY

where

+ &, A,M2, Y?= Er2Ar2e~2142 - E,1-41e,,/4,, Es2A2es,lds, - &,A,e,,l4 , A = EslK1 f A,le:l)141 + As2e~2Wsz~ I%= &,U,, + A1 e:,W, + Er,Kz+ h2 et2W2. a, = &I AslIds,+ Es,As21ds2, a, = Er, A&, L =

E,t(l,: +

The equilibrium equations and boundary conditions are derived by using the principle of minimum total potential, 6&J,+ u,+ u,-

V)=O

(8)

where V is the potential of the applied load. For a symmetrically stiffened plate (rS = yr = 01, the governing equations reduce to [4] DXYXXXX + D,, YXXYY + D, W,,,,, = F,,, W,,, + F,,, W,,, - 2F,,, W.,Yf 9 - & a:iat2, + WI2 K 1~,Y,YY

+

J&3)

FXXYY

+

K22

f:,,,,

F,,,

=

=

wb

-

WJX

W,YYP

where F,:,, = N,, = (CII + a,) &I+ -F,,,

C12

EZ>

Nyy = (C2, + a,) c2 + Cl2 6,

= Nx, = N,, = G Y,

K,,= (C2,+ 4lKG, + a,NCz2 + a,>- C&l,

KU = (GI + d/KG

+ a,)(Gz + 4 - Cf211

(9) (10)

586

G. PRATHAP AND T. K. VARADAN

+ a,)(C22+ a,) - Cf,l, D,, = 2D,, + 4Dc, Dx=D,,+Bs,

42 = -G/KG KS = l/C,,

D, = D,, + Pr,

A/PO = (1 + $3 ? = A, A&O d, + ps, A,,/Po 4, + PI, A,,/Po 4,+

or, -bllpo

4,

being the mass per unit area of the stiffened plate, p. the mass per unit area of the unstiffened plate, and ps,, p,, pr, and prz are the densities of the stiffener material.

pe

3. METHOD OF SOLUTION

First, the transverse displacement function w must be chosen to satisfy the boundary conditions on w. For a simply supported plate bounded by the lines x = 0, a and y = 0, b the mode shape is chosen as wI =At) sin (TX/~) sin (ny/b),

(11)

and for a clamped plate, bounded by the lines x = -a, CIand y = -b, b, the mode shape is chosen as

=;f(t)[l +cos(nx/a)][1 + cos(rcy/b)].

w1

(12)

These mode shapes are substituted into equation (10) and the differential equation in F is solved to obtain the particular integral F, as Fpl =

cosF+

b,, COST,

b,, cosz

27cY

+ b,, CO&OS b+

a

b,, cos%os-+ a

a

Fp, = blz cos z+ a

+ bbl cos?

271x a

b,,

b,, cos

f V),

(13)

(14)

where the second subscripts 1 and 2 denote the clamped and simply supported cases, respectively, and the bLj are given in their explicit forms in Appendix I. The complete solution is F= Fp + FC where the complementary solution F, in both cases is taken in the form [2] F, = B-&72)

+ &(x2/2) - g&y).

(15)

The constants RX,icr,and RX,,contribute directly to the in-plane stresses N,,, NY,and NXYand will be determined by using the boundary conditions. For a movable case, the boundary conditions are, on the edges Y YZ Nx, dy = 0, x = x1, x2, J”Nx, dv = 0, s Yl

Y1

and on the edges x2 Y = Yl,Y2,

J

Xl

NY, dx = 0,

x2 N,, dx = 0. s *1

LARGE VIBRATIONS

587

OF STIFFENED PLATES

For the immovable case, u = 21= 0 on all edges. This may be enforced in a simple averaged manner as

s x2all

0(x,,);) - U(Xl,Y)=

s y1

%dx=O,

x1

Xl )‘z

dx3 Y2) - v(x3 VI> =

s "au

axdx=O,

dx,, Y) - 4% Y) =

a”dy = 0, ay

“au u(x,)(2) - u(x,y,) =

s Y1 Sdy=O*

(16)

From these conditions, it follows that for a movable edge RX = fly = NX, = 0, and for an immovable edge

for a clamped plate, and

for a simply supported plate. For an unstiffened isotropic plate, these results agree exactly with those of Chu and Herrmann [l] and Yamaki [5], indicating that the use of equation (16) to satisfy the immovable conditions is exactly equivalent to the approaches of references [ 1] and [5]. Equation (9) is solved by the Galerkin technique, leading to a modal equation of Duffing’s type for all cases considered : i.e., d2f/dz2 + qf+ f3f” = 0, with .f=flh,

T = coot

and

(17)

0: = rr4D, Jpo a2 b2,

where i%= 12/%/E,1h3,

& = 12B,/~% h3, jjl

=

(i 0‘“;;;$ Or’+)g2 [0 01 A

+

;

;

l/24 16 ;

2

+

4(2~~ +

K33)

K22

b

+~,24,(~~+4~~~~33~+~6~~~~,

+1/6;+ [O

’

(2&z -t&3) K22 b K +??-a1101) 11

’

2

and

588

G. PRATHAP AND T. K. VARADAN

where

A = N + ~sWzz/E~~ + 4) - E~h%I/EzI&1 + 61, Cc, = a,lE, h. 6 = cc,lEl1 h, 1

The subscripts 1, 2 denote the clamped and simply supported cases, respectively, and 6 = 0 for the movable in-plane edge condition and 6 = 1 for the immovable case. With the initial conditionsno) =fO and (df/dT)(O) = 0, the solution to the modal equation is f(r) =fo Cn (~7, &I, where K: =fo”/2(1 + Y!& T/T, = 2F(&Y7rl/(l

o/o0 = (1 + $$)“*, + Y!:)

and y = /3/u, w0 = l/;jl, Cn is the elliptic cosine integral and F(K,) is the complete elliptic integral of the first kind. It is noted that the ratio of non-linear to linear frequencies, or the ratio of periods T/T,, and its dependence on amplitude ratiof, is governed entirely by the sign and magnitude of y. In this case, y is always a positive value, and the larger the value of y, the greater the hardening effect. In this paper, the non-linear behaviour of the plate is interpreted in terms of the value of y and its dependence on the various geometric and elastic parameters can be studied.

4. POSTBUCKLING AND STATIC LARGE DEFORMATION In two recent papers, Jones [6,7] presented without proof a simple method to determine an approximate relationship between the fundamental linear frequency and the static linear response of elastic systems using an inductively obtained proportionality constant C. Sundararajan [8] showed by formal derivation that such a relationship can be derived for any linear single degree of freedom system, or for any multiple degree of freedom system or continuous system by reducing them to an equivalent single degree of freedom system. The proportionality constant C is then obtained as a functional of the static deflection of the system under a uniform load and hence depends on the geometry and boundary conditions of the system. It is quite simple to extend this analogy to a non-linear continuous system such as the plate problem considered here. An assumption of a single mode shape reduces this continuous elastic system to a single degree of freedom system; in this problem, the corresponding single parameter is the deflection at the centre of the plate,j: Then the entire non-linear behaviour of the system is governed by the quantity y, so that once y is known for such a system, the non-linear free vibration response is governed by y as shown in section 3 above. When using the Galerkin method, it is equally simple to show that the static response of the system under a uniformly distributed load Q is given byf+ rf” = p Q, where y and p are constants determined

LARGE VIBRATIONS OF STIFFENED PLATES

589

by the Galerkin method and are therefore functionals of the mode shape used. Similarly, the postbuckling behaviour of the system is governed by the relationship N/N,, = 1 + #, where N is the buckling load required to produce deflectionfand N,, is the critical buckling load along any one edge. Thus, once y is known, the non-linear behaviour of the system in free vibration, static large deflection and postbuckling is predictable on the basis of a single mode shape solution. Jones [6,7] used proportionality constants C which were independent of the geometry and boundary conditions of the elastic system studied. Although C as derived by Sundararajan [8] depended on the geometry and boundary conditions, he was able to show that for clamped and simply supported plates, an approximate C value independent of the boundary conditions could be used. However, the same argument cannot be extended to the non-linear proportionality constant y in this paper. In fact, from Figures 3, 5, 6 and 7 of section 5 it is only too clear that y is very sensitive to the boundary conditions (both in-plane and w) and aspect ratio of plate.

5. DISCUSSIONS AND CONCLUSIONS The primary quantities entering into the problem are (i) the Young’s moduli and elastic constants Es,, Esz, E,,, E,,, Ell, E12, E,, and G, (ii) the stiffener quantities ArI, A,,, A,,, A,,, e II’ erzy esl, eszr d,,, d,,, ds, and d,,, and (iii) the plate dimensions a, b and h. From these quantities, for a symmetrically stiffened plate, the following non-dimensional parameters governing the problem can be extracted: E,,/E,,, E221&, G/E, 1, %, Es, i%, i$, 9 and the aspect ratio of the plate, a/b. Computations were carried out for an isotropic plate stiffened in one direction (El = 0, fir = 0) for a wide range of non-dimensional stiffener parameters &, /Is and aspect ratio. The computer program used is a very general one and can be used for any combination of the non-dimensional parameters listed above. In all the cases investigated here, the phenomenon of an increase in frequency with increasing amplitude of vibration (i.e., the hardening type of non-linearity) is observed. Figures 2 and 4 give the linear non-dimensional frequencies for a square plate, stiffened in one direction, for the clamped and simply supported cases, respectively. The linear frequencies increase with increase in ps but decrease with Ei,.This is to be expected because ps is a

Figure 2. Linear frequencies for clamped stiffened plate. 0; Unstiffened plate.

G. PRATHAP AND T. K. VARADAN

590

A: -

flexural

rigidity

flexural

of strlngen

ng!dlty

of plate

Figure 3. Hardening effect as function of stiffener ratios for square clamped plate. 0, Unstiffened plate; , immovable edge; ----, movable edge. IO

flexural a

=

16

12

8

4

0

rlgldlty

flexural

rigldlty

20

of stlffeWr5 of plate

Figure 4. Linear frequencies for simply supported stiffened plate. 0, Unstiffened plate.

I.8

0

4

I3 flexuml

P$=

flewrdl

16

12

20

ragldlty of sthroerr rigidity

of @ale

Figure 5. Hardening effect as function of stiffener ratios for square simply supported plate. c1, Unstiffened immovable edge; ----, movable edge. plate; -,

591

LARGE VIBRATIONS OF STIFFENED PLATES

c.01 / I

2

1

I

I

I

4

6

6

IO

Aspect

ratm

Figure 6. Hardening effect as function of aspect ratio for clamped plate. r7,= 0.5. -, movable; x, unstiffened (also [5]).

Immovable; ----,

measure of the effective bending rigidity provided by the stiffener and cl, is a measure of the volume of stiffener material as a fraction of the volume of the plate. Thus, for the same CI, frequencies increase the farther the stiffener centroids are from the middle plane. Similarly, for the same fis larger Zs indicates that larger volumes of stiffener material are used to obtain the same effective bending rigidity and this reduces the non-dimensional frequencies due to the increase in mass of the plate-stiffener combination. Figures 3 and 5 represent the non-linear behaviour of a square stiffened plate. It is observed that the relationship between non-dimensional frequency and amplitude for any stiffened plate is always of a less hardening nature than that of the corresponding unstiffened plate for most practical cases. The hardening effect is substantially larger for the immovable case than for the movable case. Further, increase in fls decreases the hardening effect, suggesting that increasing the effective bending rigidity of a plate decre,ases the non-linear behaviour. This appears to be true for both the immovable and the movable case. Conversely, increasing CI, without changing fis increases the hardening effect. Figures 6 and 7 depict the non-linear behaviour under varying aspect ratio. The hardening effect increases with aspect ratio as a

o.oK I

I 2

/ 4

/ 6 Aspect

/ 8

I IO

ratm

Figure 7. Hardeningeffect as function of aspect ratio for simply supported plate. ri, = 03. -, ----, movable; x, unstiffened (also [l, 51).

Immovable:

592

G. PRATHAP AND T. K. VARADAN

general rule. Moreover, at large aspect ratio, the influence of stiffener parameters on hardening becomes negligible. It is interesting to note that the separate effects of cl, and fl, on the linear frequencies and on y are directly opposite in nature. In fact, in most cases, a general conclusion may be drawn that systems which are stiffer for the case of linear behaviour exhibit less hardening behaviour when non-linear vibration is concerned. It is also interesting to observe that since the reduction of a continuous system to the equivalent single degree of freedom system by the use of a one-term mode shape and Galerkin technique is equivalent to the application of Rayleigh’s energy method, the linear part of the solution obtained will be an upper bound of the exact fundamental frequency. From this, one may argue by analogy that the non-linear behaviour as described by y obtained by such a method will be less hardening than would be obtained by more exact analysis. This has been found to be true from computational solutions to non-linear beam problems [9, IO] but the extension of a similar numerical solution to plate problems appears to be a formidable exercise.

ACKNOWLEDGMENT The authors thank Professor Indian Institute of Technology,

K. A. V. Pandalai, Department Madras, for his encouragement

of Aeronautical Engineering, and interest in the subject.

REFERENCES 1. H. CHU and G. HERRMANN 1956 Journal of Applied Mechanics 23, 532-W. amplitudes on free flexural vibrations of rectangular elastic plates.

Influence of large

2. M. SATHYAM~~RTHY and K. A. V. PANDALAI 1970 Journal of the Aeronautical Society of India 22,

264-266. Nonlinear flexural vibrations of orthotropic rectangular plates. 3. T. K. VARADAN 1975 Ph.D. Thesis, Zndian Znstitute of Technology, Madras. Nonlinear of tapered beams and stiffened plates and shells. 4. G. PRATHAP 1977 Ph.D. Thesis, Indian Institute of Technology, Madras.

5. 6. 7.

8. 9. 10.

problems

Nonlinear behaviour of flexible bars, anisotropic skew plates and stiffened plates. N. YAMAKI 1961 Zeitschriftfir angewandte Mathematik und Mechanik 41,501-540. Influence of large amplitudes on flexural vibrations of elastic plates. R. JONES 1975 Journal of Sound and Vibration 38, 503-504. An approximate expression for the fundamental frequency of vibration of elastic plates. R. JONES 1976 Journal of Sound and Vibration 44, 475-478. Approximate expressions for the fundamental frequency of vibration of several dynamic systems. C. SUNDARARAJAN 1977 Journal of Sound and Vibration 51, 493-499. Relationship between the fundamental frequency and the static response of elastic systems. G. PRATHAP and T. K. VARADAN American Institute of Aeronautics and Astronautics Journal 16, 88-90. Large amplitude free vibrations of tapered hinged beams. G. PRATHAP and T. K. VARADAN 1978 Journal of Sound and Vibration (in press). The large amplitude

vibrations

of tapered clamped

beams.

APPENDIX

I

b,, = -(1/32K,,)(az/bz),

b31 = -(1/512K&?/bz),

& = -(l/32~11)(~“la2),

b,, = -(1/512K1J(b2/a2),

&I = -1/32[16&1(a21~2)

+ 4(2K,, + K,& + K22(b’/a2)],

16K2,(b2/a2)1, h = -1/1WGI(a2/~2)+ WG2+ J&d + K22@2/a2)l, b,, = (1/32&)(b2/u2). b,, = W32Ku)(a2/b2),

b,, = -1/32[KI,(af/b2)

+ 4W12

+ &I

+

LARGE VJBRATIONS OF STIFFENED PLATES

APPENDIX

II: LIST OF SYMBOLS

Arl,.42,AS1,ASz cross-sectional area of stiffeners a, b dimensions of plate blJ constants in stress function expression Cll, Cz2, Clz, Cc elastic constants D,, Dxy, 4 Dll, Dz2, D,2, DG elastic rigidities &,G,G Eli, Ezz, E elastic constants Erl, E,,, E,,, Es, elastic moduli of stiffener material e,, , e,,, e,,, es2 stiffener eccentricities F, F,, F, stress function, particular integral, complementary solution f,f amplitude, non-dimensional amplitude of vibration h thickness of plate Z?,,Zr,,Zs,, Is2 moments of inertia of stiffener sections about their own centroids Kll, Kz2, Klz, KS3 elastic constants N,, N,, Nxy in-plane stress resultants constants in stress function % %, K, ZJ,us,, us,, G, Urz, Us, U, strain energy terms in plate and stiffeners U,u, w in-plane and transverse displacements x, y, z co-ordinate system used a, B, y constants from modal equation % Bs,6, BI stiffener constants c&,fls, B,,fl, non-dimensional stiffener constants a,, ay,c.ry strains in plate Q, c2,y midplane strains crx,o,, ‘sxy stresses in plate PO,Pe mass per unit area of unstiffened, stiffened plate psl, ps,, pI1, plz densities of stiffener material v,~, vyx Poisson’s ratios

593