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On the non-linear vibrations of rectangular plates
Journal of Soundand Vibration (1978) 56(4), 521-530
ON THE NON-LINEAR
VIBRATIONS
OF RECTANGULAR
PLATES G. PRATHAPAND T. K. VARADAN Departmentof Aeronautics, Indian Institute of Technology, Madras 600036, Zndia (Received 21 June 1977, and in revisedform 12 October 1977)
A solution, based on a one-term mode shape, for the large amplitude vibrations of a rectangular orthotropic plate, simply supported on all edges or clamped on all edges for movable and immovable in-plane conditions, is found by using an averaging technique that helps to satisfy the in-plane boundary conditions. This averaging technique for satisfying the immovable in-plane conditions can be used to resolve many anisotropic and skew plate problems where otherwise, when a stress function is used, the integration of the u and v equations becomes difficult, if not impossible. The results obtained herein are compared with those available in the literature for the isotropic case and excellent agreement is found. Results available for the one-term mode shape solutions of these problems are compared and the non-linear effect is presented as functions of aspect ratio and of the orthotropic elastic constants of the plate. The results are further compared with those based on the Berger method and the detailed comparative studies show that the use of the Berger approximation for large deflection static and dynamic problems and its extension to anisotropic plates, skew plates, etc., can lead to quite inaccurate results.
1. INTRODUCTION
There has been a considerable amount of work done on large amplitude vibration of rectangular plates. The earliest was that of Chu and Herrmann [l], in which the dynamic equivalents of the von Karman equations were used to study the problem of an isotropic plate simply supported on all edges with immovable in-plane conditions. Subsequently, Yamaki [2] studied the isotropic plate for both simply supported and clamped plates for three different cases of in-plane boundary conditions. Other notable contributions include those of Srinivasan [3], Eisley [4,5], Bennett [6], Ramachandran [7], and the extension of the Berger approximation to rectangular plates by Nash and Modeer [8], Wah [9] and Mayberry and Bert [lo]. The case of an orthotropic plate was considered by Sathyamoorthy and Pandalai [ 111 for movable in-plane boundary conditions. The purpose of this paper is to collect and compare the data for the one-term mode shape solutions of the problems of isotropic and orthotropic plates for boundary conditions which are either simply supported on all edges or clamped on all edges and for in-plane conditions which are either movable on all edges or immovable on all edges. The effect of non-linearity is presented as functions of the geometrical and elastic properties of the orthotropic plate. An averaging technique, used to satisfy the in-plane boundary conditions, is found helpful in resolving anisotropic plate problems when using the stress function approach. The similarity to a simplification suggested, but used incorrectly, by Bert [12] is pointed out and the correct approach presented. A comparative study of the Berger approximation indicates that there are many doubts regarding the applicability of the Berger equations, even to the isotropic rectangular plate, and therefore extensions to anisotropic or skew plate problems may have little meaning. 521
G. PRATHAP AND
522
T. K. VARADAN
2. THEORY
2.1.
EQUATIONS OF MOTION
The von Karman equations applicable to the dynamic problem of an orthotropic plate are (a list of symbols is given in Appendix II) D, w.xxxx+ D,, w. xx,.y+ D, w,yyyy= F, yy~‘9xx + F. xxw, yy- 2F.,, w, xy - PW,tfr + (2K12 + &,I F,,,,, + Ki 1F. YYYY
K22
F,
xxxx
=
W.zxv -
W,,,
W.,,,
(1) (2)
where the stress-strain relations are taken in the form ox = E,, s, +
&2&y,
E, 1 = M El2
=
D,, = E, = E,, h3/12,
0,
=
42
1 - vxy vy,>v
J% vxy/(l
-
vxy v,J
cc +
E22
=
E,
=
E22
4/U
E,,
-
vyx/(l
-
~xy =
vw
(3)
‘3x,.
VA
(4)
ad
vxy v,A
2D,, + DC = D,, = (2E,, + 4G) h3/12,
Dz2 = D, = E22 h3/12, (5)
hK, I= 1/Em
hKn = -v,,lEx
= -vxylEyr
kKJ3 = l/G,
hK22 = l/E,,
(6)
where h is the thickness of the plate. 2.2.
BOUNDARY
CONDITIONS
The boundary conditions for w may be identified without any difficulty, for a plate bounded by x = x1,x2 and y = y,,y, as (i) on edges x = x1, x2 w=O W ,x
=
or 0
NX,w,,+NXYw,Y+ V,=O, RW,,,
or
+
D12
W,,,
=
(7)
0,
(8)
(ii) on edges y = y1,y2 w=O W ,y=O 69
on
comers
w=O
or
N,,w,,+
or
4.W,yy+
bbvl),
or
V,=O,
(9)
42W,xx=0,
(10)
(x29~2), (x2,yl), hy2)
wVXY=O,
(11)
where, as usual, N,, = F, yy, I’, = 0, w, xxx+
(42
+
N,, = Nxy = -F, xy, 406) w,xw
(12)
N,, = F, xx,
v, = D, w, wp +
(42
+
40,)
w, xx,..
(13)
The in-plane boundary conditions however merit some detailed attention as they are often misunderstood and they play an important role in non-linear problems. Since the non-linear formulation introduces two in-plane displacement quantities, u and a, and the expression for the total potential energy contains quadratic terms of the first derivatives of u and a, one expects that there will be two in-plane boundary conditions in u and u at each point of an edge. The only unimpeachable method of identifying the in-plane boundary conditions is through the variational principle. On the edges x = x1, x2, the conditions for u are Y1 [SuN,J;; dy = 0. I Yl
523
NON-LINEARVIBRATIONSOFPLATES
Since in the variational approach 6u can be any arbitrary function, the above equation is satisfied in any one of the following three simple instances (although theoretically, at any rate, an infinite number of ways exist in which 6u and N,, can be prescribed along the edge) : (i) 24= 0 on the edges x = x1,x2, called the immovable condition; (ii) N,, = 0 on the edges x = x1,&, called the stress free condition, there being no restriction on the edge movement in the x direction; (iii) j$ N,,dy = 0 and 6u = constant on edges x = x1,x2, called the movable condition; in other words, the edge is maintained by a distribution of forces in such a way that the displacement 6u is a constant [2]. The argument may be extended to all the edges for both u and Dso that these three simple types of in-plane conditions can be tabulated as shown in Table 1. Theoretically, therefore, TABLET In-plane boundary conditions
/ Immovable x=x1,x2
u=o
Y=Yl,Yz
v=o
Y =Yl,Y?
U= 0
x=x1,x2
lJ=0
Classification A Stress free
\
Movable P,= Jz N,,dy=O P,=J:;N,,dx=O P~,=J~;N,,dx=O P,,= 5::N,,dy = 0
N,,=O NSV=o N,,=O
N,,=O
one has as many as 3’ possible combinations of these simple boundary conditions alone. However, in the literature, the following four cases of in-plane restraint are defined [2] : case (a) : N,,= N,,= 0 on x=x1,x1; N,,=N,,=O on Y=Y,,Y,; (14) case (b) : u=Nx,,=0 on x=x1,x2; v=N,, =0 on Y=Y,,Y~; (15) case(c):
Px=Nxv=O,
24= const.
on
x=x1,x2;
Py=Nxy=O,
D = const.
on
Y=Y,,Y~;
(16)
case(d): u=v=O
on
x=x1,x2;
u=v=O
on
y=yI,y2.
(17)
2.3. SOLUTIONSOF SPECIFIC PROBLEMS
2.3.1. Isotropicplates Chu and Herrmann [l] solved the large amplitude vibrations of an isotropic rectangular plate for the simply supported case, using the mode shape w =f(t) sin (xx/a) sin @y/b),
(18)
where x1 = 0, x2 = a, y1 = 0, y2 = b, and choosing the in-plane conditions as u = 0 on x = 0, a and u = 0 on y = 0, b. Note that the boundary conditions in v and u on edges x = 0,a and y = 0, b, respectively, are not prescribed at all. The in-plane equations in u and o were solved, yielding u = (nf’/l6u){cos (2ny/b) - 1 + (d/P) v}sin (2Xx/a), o = (rr~*/16b){cos(2Xx/a) - 1 + (b2/aZ) v} sin (2ny/b).
(1%
524
G. PRATHAP
AND T. K. VARADAN
The equation in w was then solved to yield the modal equation of the Duffing type: viz., d2f/dt2 + af+ Bf3= 0, (20) where a > 0 and b > 0. Yamaki [2] solved this same problem, but took care to prescribe the in-plane boundary conditions more completely as u = N,, = 0 on x = 0, a and u = N,, = 0 on y = 0, b. Solving the equations in stress function form by using the Galerkin technique, he arrived at exactly the same result as Chu and Hermann. This is not surprising as the mode shapes obtained for u and o (equation (19)) automatically reduce to the additional implied boundary condition N,, = 0 on all the four edges. Yamaki solved both the simply supported plate and the clamped plate for the three boundary conditions denoted as cases (a), (b) and (c) in equations (14)-(16). It is interesting to note that the solution for case (d) in equation (17), which represents the actual immovable edge, rarely appears in the literature. In this paper, this case is attempted by an averaging procedure for an orthotropic plate, as shown below.
2.3.2. Orthotropic plates The mode shapes for the clamped plate defined by the edges x = -a, a and y = -b, b and for the simply supported plate whose boundaries are x = 0, a and y = 0, b are chosen as w1=
+f(t){1 + cos (m/a)}{ 1 + cos (ny/b)},
w2 = f(t) sin (m/a) sin (?ry/b),
(21)
respectively. These mode shapes are substituted into equation (2) and the differential equation in F is solved to obtain the particular integral F,, as F,,, =
b,, cos 7
+ b,, cos 7
+ beI cos g,os
T
F,, =
+ b31 ,os$
+ b,, cos T
+ b,l cos ~COS~
+ b,, cos F,osF
f’(t),
(22)
{b12cosWa) + b2,~0s CW/Wf2(t),
(23)
where the second subscripts 1 and 2 denote the clamped and simply supported cases, respectively, and the b’s are given in Appendix I in their explicit forms. The complete solution is F = FD + F, where the complementary solution, F,, in both cases is taken in the form [ll], F, = 1’7~y2/2 + n,, x2/2 - Rx, xy.
(24)
one recognizes immediately that the constants lij,,N,, and RX,,contribute directly to the in-plane stresses N,,, N,, and N,,, respectively, and these constants will be determined by
using the boundary conditions. Solutions are attempted here for two boundary condition cases which can be enforced with very little difficulty for the choice of the complementary solution used here. For the specially movable case, in both cases, RX = N, = N,., = 0 and the in-plane stresses are due only to the particular integral. The immovable boundary conditions u = 0, o = 0 on each edge are enforced in the following average manner as the actual integration and satisfaction of the equations for u and u on all edges is very difficult, if not impossible, for some cases: e.g., anisotropic plates and skew plates. They are, **au X2av 4X2,Y)
-
4Xl.Y)
=
Q.x=o,
s
@2,Y)
-
4-%,Y)
=
s
-jpx=o,
x1
Xl
Y2
‘* 4-T
Y2) -
4&Y,)
=
s
YI
au aydY=O,
hY2)
-
4XvY,)
=
s
II
;dy=O.
(25)
NON-LINEAR
VIBRATIONS OF PLATES
525
With use being made of these conditions, the terms IV,, = F, ,,,,,IV,.,,= F, xx and NX,,= -F, xy can be integrated over the region of the plate, to yield the constants
for a clamped plate, and
for a simply supported plate. Note that on all edges NXy= 0, which suggests that the immovable edge condition satisfied as in equations (25) for this problem and for this particular mode shape is insensitive to the u conditions on the y = yl,yz edges and the v conditions on the x = x1,x2 edges and behaves exactly in the same way as assumed by Chu and Herrmann [I] and Yamaki [2]. Equation (2) is then solved by the Galerkin technique, leading to a modal equation of Duffing’s type, for all cases considered, which can be written as d2fjdr2 + af+ j?f” = 0, withf=f//r,
(28)
T = o, t, o $ = rr4&/pa2 b2, p being the mass per unit area of the plate, and
(29)
(30) where 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. The results for the immovable case of the isotropic plate are in excellent agreement with those of Chu and Herrmann [l] and Yamaki [2]. Since it does not appear to be possible to establish rigorously the mathematical validity of the averaging technique used here, the only proof of its usefulness is its ability to reproduce the results obtained earlier for an isotropic case [l, 21. It is expected that equally acceptable results would obtain in the orthotropic case. 2.4. SOLUTION TO THE MODAL EQUATION With the initial conditionsf(0) = f.and dfkdr(0) = 0, the solution to the modal equation is f(7) =f0’Cn(wt, K,), where K: =fi%w + rfi), o/w,= (1 + yfy,
T/T, = 2 F(K,)/n m;,
G. PRATHAP
526
AND T. K. VARADAN
y = /l/a, coo = &, Cn is the elliptic cosine integral and F(K,) the complete elliptic integral of the first kind. Note that the ratio of the non-linear to linear frequencies, or the ratio of the time periods T/T,, and its dependence on the amplitude ratiof, is governed entirely by the
0.6
/,-0.:
--
__-_-.
-7 \
Immovable
o-4
Y
o-3 k-
_-----.
-Movable
0.2
0-I
0
2
4
6
8 Aspect
Figure 1. Rectangular clamped plate. -,
1
IO
51
ratio
Isotropic; ----,
,
2.5 i/
\ e ~---------
type 1; -+,
orthotropy
, -
orthotropy
type 2.
, -
-
Immovable
_._-_-_ --__ __ Immovable
Berger
Movable
Aspectratio Figure 2. Simply supported plate. -,
Isotropic; ----,
orthotropy
type 1; -.-,
orthotropy
type 2.
magnitude of y. In this paper therefore, the non-linear behaviour of plates is interpreted in terms of the value of y and this is represented as the dependent variable in graphical form, in Figures 1 and 2. Figure 3 shows an actual plot of the time period with respect to amplitude for a few specific examples.
521
NON-LINEAR VIBRATIONSOF PLATES I-0 I-
0.9
0.8
F
0.7
xi 2
O-6
0.5
0.4
0.3
O
I
0.4
I
0.8
Amplitude,
I
I
I
I.2
I.6
2.c
i
Figure 3. Time period ratio vs. amplitudef.
3. BERGER
----,
Berger.
APPROXIMATION
In 1955, Berger [13] presented a new approach to the analysis of isotropic plates with large deflections, in which the elastic strain energy due to the second strain invariant is disregarded. Physically, this amounts to neglecting the influence of membrane shear stresses and adding a part of the product term involving membrane normal stresses. A result of this formulation is that the sum of the membrane stresses is a constant throughout the plate. Berger’s approach yields field equations that are not coupled, in contrast to the von Karman equations, and has therefore been used widely in the literature and even modified to include large deflection and large amplitude vibration problems of anisotropic plates and shallow shells [14-161. In 1972 Nowinski and Ohnabe [17] showed that the Berger approach for circular plates may lead to grave inaccuracies and that for a movable edge it becomes meaningless. Other inconsistencies have been pointed out: e.g., Schmidt [18] suggests that fortuity is a possible explanation of the efficacy of the Berger method, and even then, this is true only of the magnitude of the finite deflection at the centre of the plate. We have noticed inaccuracies in the application of the Berger approximation to rectangular plates, and especially of the inability of the Berger approach to reproduce the qualitative nature of the non-linear behaviour with respect to aspect ratio of the plate, or orthotropicity of material properties. This argument hinges upon the quantity y = /3/a appearing in the modal equation d2f/dr2 + af + fif 3 = pg. Here p is a constant and q is the uniformly distributed load per unit area. It is easy to see that yf2 is a measure of the ratio of stretching and bending energies. Thus y determines the nonlinear behaviour of the load-deflection curve in the static case and the frequency amplitude relationship in the dynamic case, and is a measure also of the departure of these relationships from their linear values. The larger the value of y, the greater becomes the hardening effect. The accuracy of a one-term solution in a non-linear problem has its limitations. Even so, it should be interpreted in terms of the accuracy obtained in determining y. The Berger method involves dropping out the energy term corresponding to the second invariant of the strains in the middle plane and this would therefore be reflected in the value of y obtained in the Berger approximation. The value of y thus obtained, when compared with that obtained from the
528
G. PRATHAP AND T. K. VARADAN
von Karman equations, will be a measure of the stretching energy that is over- or underestimated in the Berger approximation. The Berger approximation, for purposes of comparison with equations (30), yields, for an orthotropic plate with immovable in-plane conditions,
Note that the quantity El, does not appear in the above expressions. Nowinski and Ohnabe [17] showed that the Berger approximation yielded inaccurate results for circular plates with movable in-plane conditions. This has been explained by Vendhan [ 19] in an investigation of the variationally derived in-plane boundary conditions which showed that the Berger equations implied zero non-linearity in the case of plates with movable edges. An extensive numerical study, reported in reference [ 191,indicated that on the basis of a single mode expansion, the Berger equations lead to an entirely different in-plane behaviour from that suggested by the von Karman equations. Although it would appear from the results of reference [ 191and Figure 3 that the natural frequencies do not differ very much and that these overall results appear to be of acceptable accuracy, a more appropriate criteria for accuracy will be the comparison of y as done here, in which case the error in y may be as large as 40 % (clamped isotropic at large aspect ratio, Figure 1).
4. DISCUSSIONS AND CONCLUSIONS
Figures 1 and 2 show plots of y as a function of aspect ratio for different types of orthotropicity. Note that the error of the Berger approximation increases with increasing aspect ratio. The results indicate that the use of equations (25) to satisfy the immovable boundary conditions is equivalent to the solutions of Chu and Herrmann [l] and Yamaki [2]. In the latter, the U, u equations were obtained from the stress functions. However, in the case of anisotropic rectangular plates or skew plates, which reflect an equivalent anisotropic behaviour, this integration to obtain u, u from the stress function becomes difficult and the proposed method could solve the problem more easily. Another interesting feature to be noted is the nature of the non-linear part of the solutions. The solution of the in-plane equations or the compatibility equation can be thought of as being divided into the particular integral and the complementary solution. The complementary solution is obtained by imposing the boundary conditions, as in equations (25). Thus the non-linear behaviour due to in-plane stresses can be thought of as of two kinds, one due to the interaction of I(, u, w quantities as reflected in the in-plane differential equations and compatibility conditions, and the other from the in-plane stresses due to the nature of edge restraint. In this regard, it is interesting to note that Bert [12] used a simpli6cation similar to that proposed in this paper to satisfy the boundary conditions, but in the process of integrating to obtain the in-plane stresses he lost that part of the non-linearity arising from the particular integral, so that only the complementary solution part that arises from the satisfaction of the immovable edge condition (i.e., R,,lo, in the present analysis) has remained, thus underestimating the total non-linear effect by a considerable extent. Bert’s values for this problem thus will be (compare with equations (30))
NON-LINEARVIBRATIONS OF PLATES
529
Thus in the present analysis, the in-plane stress contributions iv,, iv, due to the complementary solution alone are constant, whereas according to Bert the total in-plane stresses N,, NY remain a constant in the region of the plate. It may also be noted that Vendhan [20] differentiates the non-linear behaviour into two parts, a contribution from a coupling of u, D and w terms and a contribution from w,,, wz,,terms, and arrives at the erroneous conclusion that Bert’s results [12] are due to the neglection of u-u stretching. There is considerable uncertainty still as to the applicability and range of validity of the Berger approximation. Very recently, Jones and Mazumdar [21] cited Vendhan [ 19] as having established the efficacy of the Berger approximation. However, the present analysis suggests that the neglecting of stretching strain energy due to the second strain invariant is not justifiable at all as it leads to a considerable loss of accuracy. The extension of Berger’s procedure to orthotropic and anisotropic plates [ 151and to the effect of transverse shear deformation and rotatory inertia on non-linear vibrations [22] may have little meaning in reflecting correctly the non-linear behaviour due to the nature of the inherent inaccuracy of the Berger approximation as seen from Figures 1 and 2. It would seem useful and justifiable to carry out a thorough examination of the errors involved in Berger applications to such problems on a similar basis. ACKNOWLEDGMENT The authors are deeply indebted to Professor K. A. V. Pandalai, Department of Aeronautical Engineering, Indian Institute of Technology, Madras for his valuable advice and encouragement during the course of this investigation. REFERENCES 1. H. CHU and G. HERRMANN1956 Journal of Applied Mechanics 23, 532-540. Influence of large
amplitudes on free flexural vibrations of rectangular elastic plates. Itiuence of large amplitudes on flexural vibrations of elastic plates. 1965 American Institute of Aeronautics and Astronautics Journal 3,1951-1953. A. V. SRINIVASAN Lar8e1amplitude free oscillations of beams and plates. J. G. EISJXY1964American Institute of Aeronautics and Astronautics Journal 2,2207-2209. Large amplitude vibration of buckled beams and m&angular plates. J. G. EISLEX1964 Zeitschrift fur angewandte Mathematik und Physik 15, 167-175. Non-linear vibration of beams and rectangular plates. A. Bs~~rrrr 1972American Institute of Aeronautics and Astronautics Journal 10,1145-l 146.Some approximations in the nonlinear vibrations of unsymmetrically laminated plates.
2. N. YAMAKI 1961 Zeitschrift fur angewandte Mathematik und Mechanik 41,501-540.
3. 4. 5. 6.
7. J. RAMACHANDRAN 1974 Nuclear Engineering Design 30, 40247. Nonlinear vibrations of elastically restrained rectangular orthotropic plates. 8. W. A. NASH and J. R. MODEER1960 in Proceedings Symposium on 27teory of Thin Elastic Shells, (De& August 1959). New York : Interscience Publishers, Inc. Certain approximate analyses of the nonlinear behaviour of plates and shallow shells. 9. T. WAH 1963 International Journal of Mechanical Sciences 5,425438. Large amplitude flexural vibration of rectangular plates. 10. B. L. MAYBERRYand C. W. BERT1969 Shock and Vibration Bulletin 39, 191-199. Experimental
investigation of non-linear vibrations of laminated anisotropic panels. 11. M. SATH~AMOORTHY and K. A. V. PANDALAI1970 Journal of the Aeronautical Society of India 22,264266. Nonlinear flexural vibrations of orthotropic rectangular plates. 12. C. W. BERT1973 Journal of Applied Mechanics 40,452458. Non-linear vibration of a rectangular
plate arbitrarily laminated of anisotropic material. 13. H. M. BERGER 1955 Journal of Applied Mechanics 22,465472. A new approach to the analysis of large de&&ions of plates. 14. M. SATH~AMOORTHY and K. A. V. PANDALAI1972Journal of Aeronautical Society of India 24, 409414;
25, l-10. Large amplitude vibrations of certain deformable bodies, Part I and II.
G. PRATHAP AND T. K. VARADAN
530 15.
16. 17. 18. 19. 20. 21.
T. IWINSKI and J. L. NOWINSKI 1957 Arch. Me& Stor. 9, 593403. Orthotropic plates with large deflection. J. L. NOWINSKI and I. A. ISMAIL 1964 Zeitschrift fur angewandte Mathematik und Physik 15, 449456. Certain approximate analysis of large deflections of cylindrical shells. J. L. NOWINSKI and H. OHNABE 1972 International Journal of Mechanical Sciences 14, 165-170. On certain inconsistencies in Berger equations for large deflections of plastic plates. R. SCHMIDT 1974 Journal of Applied Mechanics 41, 521-523. On Berger’s method in the nonlinear theory of plates. C. P. VENDHAN 1975 International Journal of Mechanical Sciences 17,461468. A study of the Berger equations applied to the nonlinear vibration of elastic plates. C. P. VENDHAN 1975 American Institute of Aeronautics and Astronautics Journal 13, 1092-1094. Modal equations for the nonlinear flexural vibrations of plates. J. MAZUMDAR and R. JONES 1977 Journal of Sound and Vibration 50, 389-397. A simplified approach to the large amplitude vibration of plates and membranes.
22. C.-I. WV and J. R. VINSONJournal of Applied Mechanics Paper No. 69-APM-10. Influences of large amplitudes, transverse shear deformation and rotary inertia on lateral vibrations of transversely isotropic plates. APPENDIX
b,l =-A
1 (i
x1$ b2
+w12
+&3)
+K"$
I
7
hz =
II
APPENDIX
1
32K,,G’
II: LIST OF SYMBOLS
dimensions of plate constants in stress function expression elastic rigidities of plate elastic rigidities of plate stress function amplitude of vibration shear modulus elastic constant thickness of plate elastic constants in-plane stress resultants constants in stress function in-plane and transverse displacements of plate constants from modal equation strains in plate stresses in plate
a2
b,,=Lff.
32K,, a2
ON THE NON-LINEAR
VIBRATIONS
OF RECTANGULAR
PLATES G. PRATHAPAND T. K. VARADAN Departmentof Aeronautics, Indian Institute of Technology, Madras 600036, Zndia (Received 21 June 1977, and in revisedform 12 October 1977)
A solution, based on a one-term mode shape, for the large amplitude vibrations of a rectangular orthotropic plate, simply supported on all edges or clamped on all edges for movable and immovable in-plane conditions, is found by using an averaging technique that helps to satisfy the in-plane boundary conditions. This averaging technique for satisfying the immovable in-plane conditions can be used to resolve many anisotropic and skew plate problems where otherwise, when a stress function is used, the integration of the u and v equations becomes difficult, if not impossible. The results obtained herein are compared with those available in the literature for the isotropic case and excellent agreement is found. Results available for the one-term mode shape solutions of these problems are compared and the non-linear effect is presented as functions of aspect ratio and of the orthotropic elastic constants of the plate. The results are further compared with those based on the Berger method and the detailed comparative studies show that the use of the Berger approximation for large deflection static and dynamic problems and its extension to anisotropic plates, skew plates, etc., can lead to quite inaccurate results.
1. INTRODUCTION
There has been a considerable amount of work done on large amplitude vibration of rectangular plates. The earliest was that of Chu and Herrmann [l], in which the dynamic equivalents of the von Karman equations were used to study the problem of an isotropic plate simply supported on all edges with immovable in-plane conditions. Subsequently, Yamaki [2] studied the isotropic plate for both simply supported and clamped plates for three different cases of in-plane boundary conditions. Other notable contributions include those of Srinivasan [3], Eisley [4,5], Bennett [6], Ramachandran [7], and the extension of the Berger approximation to rectangular plates by Nash and Modeer [8], Wah [9] and Mayberry and Bert [lo]. The case of an orthotropic plate was considered by Sathyamoorthy and Pandalai [ 111 for movable in-plane boundary conditions. The purpose of this paper is to collect and compare the data for the one-term mode shape solutions of the problems of isotropic and orthotropic plates for boundary conditions which are either simply supported on all edges or clamped on all edges and for in-plane conditions which are either movable on all edges or immovable on all edges. The effect of non-linearity is presented as functions of the geometrical and elastic properties of the orthotropic plate. An averaging technique, used to satisfy the in-plane boundary conditions, is found helpful in resolving anisotropic plate problems when using the stress function approach. The similarity to a simplification suggested, but used incorrectly, by Bert [12] is pointed out and the correct approach presented. A comparative study of the Berger approximation indicates that there are many doubts regarding the applicability of the Berger equations, even to the isotropic rectangular plate, and therefore extensions to anisotropic or skew plate problems may have little meaning. 521
G. PRATHAP AND
522
T. K. VARADAN
2. THEORY
2.1.
EQUATIONS OF MOTION
The von Karman equations applicable to the dynamic problem of an orthotropic plate are (a list of symbols is given in Appendix II) D, w.xxxx+ D,, w. xx,.y+ D, w,yyyy= F, yy~‘9xx + F. xxw, yy- 2F.,, w, xy - PW,tfr + (2K12 + &,I F,,,,, + Ki 1F. YYYY
K22
F,
xxxx
=
W.zxv -
W,,,
W.,,,
(1) (2)
where the stress-strain relations are taken in the form ox = E,, s, +
&2&y,
E, 1 = M El2
=
D,, = E, = E,, h3/12,
0,
=
42
1 - vxy vy,>v
J% vxy/(l
-
vxy v,J
cc +
E22
=
E,
=
E22
4/U
E,,
-
vyx/(l
-
~xy =
vw
(3)
‘3x,.
VA
(4)
ad
vxy v,A
2D,, + DC = D,, = (2E,, + 4G) h3/12,
Dz2 = D, = E22 h3/12, (5)
hK, I= 1/Em
hKn = -v,,lEx
= -vxylEyr
kKJ3 = l/G,
hK22 = l/E,,
(6)
where h is the thickness of the plate. 2.2.
BOUNDARY
CONDITIONS
The boundary conditions for w may be identified without any difficulty, for a plate bounded by x = x1,x2 and y = y,,y, as (i) on edges x = x1, x2 w=O W ,x
=
or 0
NX,w,,+NXYw,Y+ V,=O, RW,,,
or
+
D12
W,,,
=
(7)
0,
(8)
(ii) on edges y = y1,y2 w=O W ,y=O 69
on
comers
w=O
or
N,,w,,+
or
4.W,yy+
bbvl),
or
V,=O,
(9)
42W,xx=0,
(10)
(x29~2), (x2,yl), hy2)
wVXY=O,
(11)
where, as usual, N,, = F, yy, I’, = 0, w, xxx+
(42
+
N,, = Nxy = -F, xy, 406) w,xw
(12)
N,, = F, xx,
v, = D, w, wp +
(42
+
40,)
w, xx,..
(13)
The in-plane boundary conditions however merit some detailed attention as they are often misunderstood and they play an important role in non-linear problems. Since the non-linear formulation introduces two in-plane displacement quantities, u and a, and the expression for the total potential energy contains quadratic terms of the first derivatives of u and a, one expects that there will be two in-plane boundary conditions in u and u at each point of an edge. The only unimpeachable method of identifying the in-plane boundary conditions is through the variational principle. On the edges x = x1, x2, the conditions for u are Y1 [SuN,J;; dy = 0. I Yl
523
NON-LINEARVIBRATIONSOFPLATES
Since in the variational approach 6u can be any arbitrary function, the above equation is satisfied in any one of the following three simple instances (although theoretically, at any rate, an infinite number of ways exist in which 6u and N,, can be prescribed along the edge) : (i) 24= 0 on the edges x = x1,x2, called the immovable condition; (ii) N,, = 0 on the edges x = x1,&, called the stress free condition, there being no restriction on the edge movement in the x direction; (iii) j$ N,,dy = 0 and 6u = constant on edges x = x1,x2, called the movable condition; in other words, the edge is maintained by a distribution of forces in such a way that the displacement 6u is a constant [2]. The argument may be extended to all the edges for both u and Dso that these three simple types of in-plane conditions can be tabulated as shown in Table 1. Theoretically, therefore, TABLET In-plane boundary conditions
/ Immovable x=x1,x2
u=o
Y=Yl,Yz
v=o
Y =Yl,Y?
U= 0
x=x1,x2
lJ=0
Classification A Stress free
\
Movable P,= Jz N,,dy=O P,=J:;N,,dx=O P~,=J~;N,,dx=O P,,= 5::N,,dy = 0
N,,=O NSV=o N,,=O
N,,=O
one has as many as 3’ possible combinations of these simple boundary conditions alone. However, in the literature, the following four cases of in-plane restraint are defined [2] : case (a) : N,,= N,,= 0 on x=x1,x1; N,,=N,,=O on Y=Y,,Y,; (14) case (b) : u=Nx,,=0 on x=x1,x2; v=N,, =0 on Y=Y,,Y~; (15) case(c):
Px=Nxv=O,
24= const.
on
x=x1,x2;
Py=Nxy=O,
D = const.
on
Y=Y,,Y~;
(16)
case(d): u=v=O
on
x=x1,x2;
u=v=O
on
y=yI,y2.
(17)
2.3. SOLUTIONSOF SPECIFIC PROBLEMS
2.3.1. Isotropicplates Chu and Herrmann [l] solved the large amplitude vibrations of an isotropic rectangular plate for the simply supported case, using the mode shape w =f(t) sin (xx/a) sin @y/b),
(18)
where x1 = 0, x2 = a, y1 = 0, y2 = b, and choosing the in-plane conditions as u = 0 on x = 0, a and u = 0 on y = 0, b. Note that the boundary conditions in v and u on edges x = 0,a and y = 0, b, respectively, are not prescribed at all. The in-plane equations in u and o were solved, yielding u = (nf’/l6u){cos (2ny/b) - 1 + (d/P) v}sin (2Xx/a), o = (rr~*/16b){cos(2Xx/a) - 1 + (b2/aZ) v} sin (2ny/b).
(1%
524
G. PRATHAP
AND T. K. VARADAN
The equation in w was then solved to yield the modal equation of the Duffing type: viz., d2f/dt2 + af+ Bf3= 0, (20) where a > 0 and b > 0. Yamaki [2] solved this same problem, but took care to prescribe the in-plane boundary conditions more completely as u = N,, = 0 on x = 0, a and u = N,, = 0 on y = 0, b. Solving the equations in stress function form by using the Galerkin technique, he arrived at exactly the same result as Chu and Hermann. This is not surprising as the mode shapes obtained for u and o (equation (19)) automatically reduce to the additional implied boundary condition N,, = 0 on all the four edges. Yamaki solved both the simply supported plate and the clamped plate for the three boundary conditions denoted as cases (a), (b) and (c) in equations (14)-(16). It is interesting to note that the solution for case (d) in equation (17), which represents the actual immovable edge, rarely appears in the literature. In this paper, this case is attempted by an averaging procedure for an orthotropic plate, as shown below.
2.3.2. Orthotropic plates The mode shapes for the clamped plate defined by the edges x = -a, a and y = -b, b and for the simply supported plate whose boundaries are x = 0, a and y = 0, b are chosen as w1=
+f(t){1 + cos (m/a)}{ 1 + cos (ny/b)},
w2 = f(t) sin (m/a) sin (?ry/b),
(21)
respectively. These mode shapes are substituted into equation (2) and the differential equation in F is solved to obtain the particular integral F,, as F,,, =
b,, cos 7
+ b,, cos 7
+ beI cos g,os
T
F,, =
+ b31 ,os$
+ b,, cos T
+ b,l cos ~COS~
+ b,, cos F,osF
f’(t),
(22)
{b12cosWa) + b2,~0s CW/Wf2(t),
(23)
where the second subscripts 1 and 2 denote the clamped and simply supported cases, respectively, and the b’s are given in Appendix I in their explicit forms. The complete solution is F = FD + F, where the complementary solution, F,, in both cases is taken in the form [ll], F, = 1’7~y2/2 + n,, x2/2 - Rx, xy.
(24)
one recognizes immediately that the constants lij,,N,, and RX,,contribute directly to the in-plane stresses N,,, N,, and N,,, respectively, and these constants will be determined by
using the boundary conditions. Solutions are attempted here for two boundary condition cases which can be enforced with very little difficulty for the choice of the complementary solution used here. For the specially movable case, in both cases, RX = N, = N,., = 0 and the in-plane stresses are due only to the particular integral. The immovable boundary conditions u = 0, o = 0 on each edge are enforced in the following average manner as the actual integration and satisfaction of the equations for u and u on all edges is very difficult, if not impossible, for some cases: e.g., anisotropic plates and skew plates. They are, **au X2av 4X2,Y)
-
4Xl.Y)
=
Q.x=o,
s
@2,Y)
-
4-%,Y)
=
s
-jpx=o,
x1
Xl
Y2
‘* 4-T
Y2) -
4&Y,)
=
s
YI
au aydY=O,
hY2)
-
4XvY,)
=
s
II
;dy=O.
(25)
NON-LINEAR
VIBRATIONS OF PLATES
525
With use being made of these conditions, the terms IV,, = F, ,,,,,IV,.,,= F, xx and NX,,= -F, xy can be integrated over the region of the plate, to yield the constants
for a clamped plate, and
for a simply supported plate. Note that on all edges NXy= 0, which suggests that the immovable edge condition satisfied as in equations (25) for this problem and for this particular mode shape is insensitive to the u conditions on the y = yl,yz edges and the v conditions on the x = x1,x2 edges and behaves exactly in the same way as assumed by Chu and Herrmann [I] and Yamaki [2]. Equation (2) is then solved by the Galerkin technique, leading to a modal equation of Duffing’s type, for all cases considered, which can be written as d2fjdr2 + af+ j?f” = 0, withf=f//r,
(28)
T = o, t, o $ = rr4&/pa2 b2, p being the mass per unit area of the plate, and
(29)
(30) where 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. The results for the immovable case of the isotropic plate are in excellent agreement with those of Chu and Herrmann [l] and Yamaki [2]. Since it does not appear to be possible to establish rigorously the mathematical validity of the averaging technique used here, the only proof of its usefulness is its ability to reproduce the results obtained earlier for an isotropic case [l, 21. It is expected that equally acceptable results would obtain in the orthotropic case. 2.4. SOLUTION TO THE MODAL EQUATION With the initial conditionsf(0) = f.and dfkdr(0) = 0, the solution to the modal equation is f(7) =f0’Cn(wt, K,), where K: =fi%w + rfi), o/w,= (1 + yfy,
T/T, = 2 F(K,)/n m;,
G. PRATHAP
526
AND T. K. VARADAN
y = /l/a, coo = &, Cn is the elliptic cosine integral and F(K,) the complete elliptic integral of the first kind. Note that the ratio of the non-linear to linear frequencies, or the ratio of the time periods T/T,, and its dependence on the amplitude ratiof, is governed entirely by the
0.6
/,-0.:
--
__-_-.
-7 \
Immovable
o-4
Y
o-3 k-
_-----.
-Movable
0.2
0-I
0
2
4
6
8 Aspect
Figure 1. Rectangular clamped plate. -,
1
IO
51
ratio
Isotropic; ----,
,
2.5 i/
\ e ~---------
type 1; -+,
orthotropy
, -
orthotropy
type 2.
, -
-
Immovable
_._-_-_ --__ __ Immovable
Berger
Movable
Aspectratio Figure 2. Simply supported plate. -,
Isotropic; ----,
orthotropy
type 1; -.-,
orthotropy
type 2.
magnitude of y. In this paper therefore, the non-linear behaviour of plates is interpreted in terms of the value of y and this is represented as the dependent variable in graphical form, in Figures 1 and 2. Figure 3 shows an actual plot of the time period with respect to amplitude for a few specific examples.
521
NON-LINEAR VIBRATIONSOF PLATES I-0 I-
0.9
0.8
F
0.7
xi 2
O-6
0.5
0.4
0.3
O
I
0.4
I
0.8
Amplitude,
I
I
I
I.2
I.6
2.c
i
Figure 3. Time period ratio vs. amplitudef.
3. BERGER
----,
Berger.
APPROXIMATION
In 1955, Berger [13] presented a new approach to the analysis of isotropic plates with large deflections, in which the elastic strain energy due to the second strain invariant is disregarded. Physically, this amounts to neglecting the influence of membrane shear stresses and adding a part of the product term involving membrane normal stresses. A result of this formulation is that the sum of the membrane stresses is a constant throughout the plate. Berger’s approach yields field equations that are not coupled, in contrast to the von Karman equations, and has therefore been used widely in the literature and even modified to include large deflection and large amplitude vibration problems of anisotropic plates and shallow shells [14-161. In 1972 Nowinski and Ohnabe [17] showed that the Berger approach for circular plates may lead to grave inaccuracies and that for a movable edge it becomes meaningless. Other inconsistencies have been pointed out: e.g., Schmidt [18] suggests that fortuity is a possible explanation of the efficacy of the Berger method, and even then, this is true only of the magnitude of the finite deflection at the centre of the plate. We have noticed inaccuracies in the application of the Berger approximation to rectangular plates, and especially of the inability of the Berger approach to reproduce the qualitative nature of the non-linear behaviour with respect to aspect ratio of the plate, or orthotropicity of material properties. This argument hinges upon the quantity y = /3/a appearing in the modal equation d2f/dr2 + af + fif 3 = pg. Here p is a constant and q is the uniformly distributed load per unit area. It is easy to see that yf2 is a measure of the ratio of stretching and bending energies. Thus y determines the nonlinear behaviour of the load-deflection curve in the static case and the frequency amplitude relationship in the dynamic case, and is a measure also of the departure of these relationships from their linear values. The larger the value of y, the greater becomes the hardening effect. The accuracy of a one-term solution in a non-linear problem has its limitations. Even so, it should be interpreted in terms of the accuracy obtained in determining y. The Berger method involves dropping out the energy term corresponding to the second invariant of the strains in the middle plane and this would therefore be reflected in the value of y obtained in the Berger approximation. The value of y thus obtained, when compared with that obtained from the
528
G. PRATHAP AND T. K. VARADAN
von Karman equations, will be a measure of the stretching energy that is over- or underestimated in the Berger approximation. The Berger approximation, for purposes of comparison with equations (30), yields, for an orthotropic plate with immovable in-plane conditions,
Note that the quantity El, does not appear in the above expressions. Nowinski and Ohnabe [17] showed that the Berger approximation yielded inaccurate results for circular plates with movable in-plane conditions. This has been explained by Vendhan [ 19] in an investigation of the variationally derived in-plane boundary conditions which showed that the Berger equations implied zero non-linearity in the case of plates with movable edges. An extensive numerical study, reported in reference [ 191,indicated that on the basis of a single mode expansion, the Berger equations lead to an entirely different in-plane behaviour from that suggested by the von Karman equations. Although it would appear from the results of reference [ 191and Figure 3 that the natural frequencies do not differ very much and that these overall results appear to be of acceptable accuracy, a more appropriate criteria for accuracy will be the comparison of y as done here, in which case the error in y may be as large as 40 % (clamped isotropic at large aspect ratio, Figure 1).
4. DISCUSSIONS AND CONCLUSIONS
Figures 1 and 2 show plots of y as a function of aspect ratio for different types of orthotropicity. Note that the error of the Berger approximation increases with increasing aspect ratio. The results indicate that the use of equations (25) to satisfy the immovable boundary conditions is equivalent to the solutions of Chu and Herrmann [l] and Yamaki [2]. In the latter, the U, u equations were obtained from the stress functions. However, in the case of anisotropic rectangular plates or skew plates, which reflect an equivalent anisotropic behaviour, this integration to obtain u, u from the stress function becomes difficult and the proposed method could solve the problem more easily. Another interesting feature to be noted is the nature of the non-linear part of the solutions. The solution of the in-plane equations or the compatibility equation can be thought of as being divided into the particular integral and the complementary solution. The complementary solution is obtained by imposing the boundary conditions, as in equations (25). Thus the non-linear behaviour due to in-plane stresses can be thought of as of two kinds, one due to the interaction of I(, u, w quantities as reflected in the in-plane differential equations and compatibility conditions, and the other from the in-plane stresses due to the nature of edge restraint. In this regard, it is interesting to note that Bert [12] used a simpli6cation similar to that proposed in this paper to satisfy the boundary conditions, but in the process of integrating to obtain the in-plane stresses he lost that part of the non-linearity arising from the particular integral, so that only the complementary solution part that arises from the satisfaction of the immovable edge condition (i.e., R,,lo, in the present analysis) has remained, thus underestimating the total non-linear effect by a considerable extent. Bert’s values for this problem thus will be (compare with equations (30))
NON-LINEARVIBRATIONS OF PLATES
529
Thus in the present analysis, the in-plane stress contributions iv,, iv, due to the complementary solution alone are constant, whereas according to Bert the total in-plane stresses N,, NY remain a constant in the region of the plate. It may also be noted that Vendhan [20] differentiates the non-linear behaviour into two parts, a contribution from a coupling of u, D and w terms and a contribution from w,,, wz,,terms, and arrives at the erroneous conclusion that Bert’s results [12] are due to the neglection of u-u stretching. There is considerable uncertainty still as to the applicability and range of validity of the Berger approximation. Very recently, Jones and Mazumdar [21] cited Vendhan [ 19] as having established the efficacy of the Berger approximation. However, the present analysis suggests that the neglecting of stretching strain energy due to the second strain invariant is not justifiable at all as it leads to a considerable loss of accuracy. The extension of Berger’s procedure to orthotropic and anisotropic plates [ 151and to the effect of transverse shear deformation and rotatory inertia on non-linear vibrations [22] may have little meaning in reflecting correctly the non-linear behaviour due to the nature of the inherent inaccuracy of the Berger approximation as seen from Figures 1 and 2. It would seem useful and justifiable to carry out a thorough examination of the errors involved in Berger applications to such problems on a similar basis. ACKNOWLEDGMENT The authors are deeply indebted to Professor K. A. V. Pandalai, Department of Aeronautical Engineering, Indian Institute of Technology, Madras for his valuable advice and encouragement during the course of this investigation. REFERENCES 1. H. CHU and G. HERRMANN1956 Journal of Applied Mechanics 23, 532-540. Influence of large
amplitudes on free flexural vibrations of rectangular elastic plates. Itiuence of large amplitudes on flexural vibrations of elastic plates. 1965 American Institute of Aeronautics and Astronautics Journal 3,1951-1953. A. V. SRINIVASAN Lar8e1amplitude free oscillations of beams and plates. J. G. EISJXY1964American Institute of Aeronautics and Astronautics Journal 2,2207-2209. Large amplitude vibration of buckled beams and m&angular plates. J. G. EISLEX1964 Zeitschrift fur angewandte Mathematik und Physik 15, 167-175. Non-linear vibration of beams and rectangular plates. A. Bs~~rrrr 1972American Institute of Aeronautics and Astronautics Journal 10,1145-l 146.Some approximations in the nonlinear vibrations of unsymmetrically laminated plates.
2. N. YAMAKI 1961 Zeitschrift fur angewandte Mathematik und Mechanik 41,501-540.
3. 4. 5. 6.
7. J. RAMACHANDRAN 1974 Nuclear Engineering Design 30, 40247. Nonlinear vibrations of elastically restrained rectangular orthotropic plates. 8. W. A. NASH and J. R. MODEER1960 in Proceedings Symposium on 27teory of Thin Elastic Shells, (De& August 1959). New York : Interscience Publishers, Inc. Certain approximate analyses of the nonlinear behaviour of plates and shallow shells. 9. T. WAH 1963 International Journal of Mechanical Sciences 5,425438. Large amplitude flexural vibration of rectangular plates. 10. B. L. MAYBERRYand C. W. BERT1969 Shock and Vibration Bulletin 39, 191-199. Experimental
investigation of non-linear vibrations of laminated anisotropic panels. 11. M. SATH~AMOORTHY and K. A. V. PANDALAI1970 Journal of the Aeronautical Society of India 22,264266. Nonlinear flexural vibrations of orthotropic rectangular plates. 12. C. W. BERT1973 Journal of Applied Mechanics 40,452458. Non-linear vibration of a rectangular
plate arbitrarily laminated of anisotropic material. 13. H. M. BERGER 1955 Journal of Applied Mechanics 22,465472. A new approach to the analysis of large de&&ions of plates. 14. M. SATH~AMOORTHY and K. A. V. PANDALAI1972Journal of Aeronautical Society of India 24, 409414;
25, l-10. Large amplitude vibrations of certain deformable bodies, Part I and II.
G. PRATHAP AND T. K. VARADAN
530 15.
16. 17. 18. 19. 20. 21.
T. IWINSKI and J. L. NOWINSKI 1957 Arch. Me& Stor. 9, 593403. Orthotropic plates with large deflection. J. L. NOWINSKI and I. A. ISMAIL 1964 Zeitschrift fur angewandte Mathematik und Physik 15, 449456. Certain approximate analysis of large deflections of cylindrical shells. J. L. NOWINSKI and H. OHNABE 1972 International Journal of Mechanical Sciences 14, 165-170. On certain inconsistencies in Berger equations for large deflections of plastic plates. R. SCHMIDT 1974 Journal of Applied Mechanics 41, 521-523. On Berger’s method in the nonlinear theory of plates. C. P. VENDHAN 1975 International Journal of Mechanical Sciences 17,461468. A study of the Berger equations applied to the nonlinear vibration of elastic plates. C. P. VENDHAN 1975 American Institute of Aeronautics and Astronautics Journal 13, 1092-1094. Modal equations for the nonlinear flexural vibrations of plates. J. MAZUMDAR and R. JONES 1977 Journal of Sound and Vibration 50, 389-397. A simplified approach to the large amplitude vibration of plates and membranes.
22. C.-I. WV and J. R. VINSONJournal of Applied Mechanics Paper No. 69-APM-10. Influences of large amplitudes, transverse shear deformation and rotary inertia on lateral vibrations of transversely isotropic plates. APPENDIX
b,l =-A
1 (i
x1$ b2
+w12
+&3)
+K"$
I
7
hz =
II
APPENDIX
1
32K,,G’
II: LIST OF SYMBOLS
dimensions of plate constants in stress function expression elastic rigidities of plate elastic rigidities of plate stress function amplitude of vibration shear modulus elastic constant thickness of plate elastic constants in-plane stress resultants constants in stress function in-plane and transverse displacements of plate constants from modal equation strains in plate stresses in plate
a2
b,,=Lff.
32K,, a2