Cmqnitm & Slnrctrups Vol. 43. No. 3, pp. 431-437.1992 Printedin Great Britain.
~~~9~ s5.00 + 0.00 Q 1992 Fwgamotl Press Ltd
FREE VIBRATION OF ISOTROPIC AND ORTHOTROPIC SQUARE PLATES WITH SQUARE CUTOUTS SUBJE~E~ TO IN-PLANE FORCES H. P. LEE and S. P. LIM Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511 (Received 2 April 1991)
Abstract-The analysis of a simply supported square plate with a square cutout subjected to in-plane forces is carried out by sub-dividing the plate into sub-domains. Compatibility between sub-domains is enforced through a point-matching technique at appropriate common points. The frequencies are then computed by the Rayleigh method. The forces are assumed to act in the plane of the undeformed middle surface of the plate. Numerical results are presented for isotropic and orthotropic square plates with central square cutouts.
subjected to uniform lows :
NOTATION rectangular Cartesian coordinates dimension of the plate dimension of the square cutout thickness of the plate in-plane forces per unit length transverse deflection mass density angular frequency non~im~sional frequency parameter potential energy kinetic energy Poisson’s ratio of isotropic plates Poisson’s ratio of orthotropic plates bending stiffness of isotropic plates bending stiffness of orthotropic plates Young’s modulus of isotropic plates shear modulus of isotropic plates shear moduli of orthotropic plates critical buckling load
forces is as fol-
+ 2N9, &+N$.
In the case of a specially orthotropic plate with the principal axes of orthotropy parallel to the plate edges, the equation becomes
reference tensile in-plane force 1.
The maximum potential energy and kinetic energy for an isotropic plate subjected to in-plane forces are given by
The transverse free vibration of isotropic rectangular plates subjected to in-plane uniaxial, biaxial or shear forces were reported in [l-8]. Numerical results for natural frequencies were obtained by either solving the analytical equations or by the Ritz method. In [!?-111 the studies were extended to orthotropic plates. All the reported works are confined to rectangular plates without any cutout. A simple numerical method based on the Ray leigh’s method is presented for predicting the natural frequencies of rectangular plates with rectangular cutouts subjected to in-plane forces. The forces are assumed to act in the ~defo~~ middle surface of the plate.
(3) 2. ANALYSIS The differential equation of motion expressed in rectangular coordinates for a rectangular plate
H. P. LEEand S. P. LIM
A similar expression for potential energy can be formulated for specially orthotropic plates
If the in-plane forces N, and N, are assumed to be constant with zero Nx, around the boundary, eqn (1) is simplified to the case of constant coefficients. Analytical solutions can be obtained for rectangular plates with two sides simply supported and the remaining sides clamped, free or simply supported. Approximate solutions can also be obtained by equating the maximum potential energy to the maximum kinetic energy based on an assumed series of deflection functions which satisfy the given boundary conditions. For a rectangular plate with a rectangular cutout as shown in Fig. 1, the in-plane forces are no longer uniform throughout the domain of the plate due to the presence of the internal free edges and stress concentration near the corners of the cutout. There is therefore no closed-form solution for the governing differential equation. In the present analysis, the rectangular plate is divided into smaller sub-domains based on the mode shapes and the location of the cutout. In the case of a central cutout, the first few mode shapes can be pre-determined due to the symmetry of the cutout about the geometrical axes. The sub-domains for these mode shapes namely X, - Y,, X, - YI, X, - Y,, and X2 - Y2 are shown in Fig. 2. The values if m and n in X,,, and n in X,,, and Y, represent the number of half-waves along the x and y axes, respectively.
for the mode consideration.
The boundary conditions for the assumed deflection function for each sub-domain are redefined according to the external simply supported as well as the internal free-edge conditions. Sub-domain 1,3, 5, and 7 of Fig. 2 are taken to be part of a complete plate with four edges simply supported. Each of the sub-domains 2, 4, 6, and 8 is however treated as part of a plate with three sides simply supported and the remaining side free. A deflection function W, which satisfies all the external boundary conditions and all, or part of the internal free edge conditions, is then assumed for each sub-domain. The assumed deflection function can also be simplified further as a product of beam functions for the given modified boundary conditions. The function W, is of the form
The adjacent sub-domains will have one of the two functions, X, or Y, in common. The relative magnitudes of coefficients A,, are determined by matching the maximum deflections at the corners of the cutout. For the Xl - Y, mode W,=
W,= W,= W,=A,sinFsiny
W,= W,= A,rsiny, rl
Fig. 1. A square plate with a square cutout subjected to in-plane forces.
A2 and A4 are determined by matching the maximum deflection at the comer with x = r, and y =s,.
Free vibration of square plates with cutouts For the _I’,- Y, mode WV,==W,=A,sinEsin$ al2
W;= W,= A,xsinE $1 aJ2’ W8= A,xsiny, r1
A, = A, sin?
For the X, - Y, mode
. RX . ny W, = W, = A, sm a sm b/2
z; z z z 2
; 2 d 2 2 2 S! 64 2 z 0
A2=A1s’nm W,= W*=A~~sin~,
NdNcr Fig. 3. Fundamental frequency parameters for a square plate with a square cutout subjected to biaxial compressive forces.
For the X, - Y, mode EX . ny W, = A, sinasmW
W;= A,xsinE a/2’ Sl
W8= A,zsinz b/2’ Ii
The natural frequency for the mode shape is determined by equating the maximum potential energy to the maximum kinetic energy. The frequency is expressed in terms of a non-dimensional parameter given by
The linear functions x/r and y/s are beam functions for simply supported-free end-conditions. The potential energy and kinetic energy of the plate are assumed to be the respective sums of the potential energy and kinetic energy of each sub-domain. The expressions for potential energy given in eqns (3) and (5) are derived for a rectangular plate with equal loading on opposite external edges of the plate. Such condition is not satisfied for sub-domains 2, 4, 6, and 8 of Fig. 2 due to the internal free edges of the cutout. The potential energy will therefore be overestimated if expressions in eqns (3) and (5) are used for its computation. To account for the stress free loading condition on one side of the sub-domains, the contributions of the integrals ’ dx dy
(11) The flexural rigidity for isotropic plate, D, is replaced by D,. for orthotropic rectangular plates.
are multiplied by correction factors a and 6, respectively. a is taken to be 0.5 for sub-domains 4 and 8 as one side of the sub-domains for loading in the x direction is free, and 1.0 for sub-domains 2 and 6 as loadings in the y direction are equal and opposite. By the same event, /? is taken to be 0.5 for sub-domains 2 and 6, and I .Ofor sub-domains 4 and 8.
u. / 0.0 0 1 0.2 0.3
Fig. 4. Fund~en~i frequency parameters for a square plate with a square cutout subjected to biaxial tensile forces.
H. P. LEEand S. P. LIM Table 1. Material properties for the orthotropic plates Case No.
YXY 0.25 0.25
(3) and (5) that a combination of N, and NYexists for reducing the computed natural frequency to zero for the given mode. The correspoinding values of N, and A; are the buckling loads for the mode shape under consideration. The buckling loads for rectangular plates with rectangular cutouts have been examined in . The predicted buckling loads are found to be in good agreement with the reported finite element results in [14, 1.51 for isotropic square plates with square cutouts. 3. 01
Nx/Ncr Fig. 5. Fundamental frequency parameters for a square plate with a square cutout subjected to uniaxial compressive forces. In the following sections, numerical results for natural frequencies are presented for simply supported isotropic and orthotropic square plates with central square cutouts. The plates are subjected to uniaxial tensile or compressive forces with N, equal to zero for the first case, and biaxial loadings with N, = N, for the second case study. For a rectangular plate subjected to uniaxial or biaxial compressive forces, it can be seen from eqns
The fundamental non-dimensional frequency parameters for a square plate subjected to uniformly distributed biaxial tensile or compressive forces are presented in Figs 3 and 4. The compressive forces have been normalized by the buckling load for the fundamental mode. The behaviour of rectangular plates under in-plane forces are the same as that of a complete plate without any cutout. The natural frequencies are found to increase with increased magnitude of tensile in-plane forces or decreased magnitude of compressive in-plane forces. The natural frequencies for the fundamental mode of square plates subjected to uniaxial tensile forces are the same as that of the biaxial forces of half
* x2-Yl -__1 * XL-Yb -_
Nx/i%t = 15
: : z’
Nx/Nt = 20 ___*__---“--
Fig. 6. Fundamental frequency parameters for a square plate with a square cutout subjected to uniaxial tensile forces with NJN, = 10.
Fig. 7. Fundamental frequency parameters for a square plate with a square cutout subjected to biaxial tensile forces with EJE,, = 40.
Free vibration of square plates with cutouts
Nx/Nt = 0
&J&t=10 Nr/Nt =
l;I ::‘! :
,a’ / ,
__.&$/ I , /- ,* /
Fig. 8. Fundamental frequency parameters for a square plate with a square cutout subjected to biaxial tensile forces with EJE,= 3.
Fig. IO. Fundamental frequency parameters for a square plate with a square cutout subjected to uniaxial forces with EJE,= 3.
the magnitude. The predicted frequency curves for the uniaxial and biaxial loadings are identical if the in-plane compressed forces are normalized by the respective buckling loads. The higher modes of a square plate subjected to uniaxial loading having a central square cutout of size c/a = 0.5 are presented in Figs 5 and 6. For a square plate subjected to uniaxial tensile forces, there is a tendency
for the frequency of the Xi - Y, mode to be lower than the fundamental mode for large cutout sizes. It should be noted that the natural frequencies of a rectangular plate with a cutout is dependent on the Poisson’s ratio of the plate. The Poisson’s ratio is taken to be 0.3 in the foregoing numerical computations.
* Xl-Y1 \
* x2-YI -___
0 0 1
Nx/Ncr Fig. 9. Fundamental frequency parameters for a square plate with a square cutout subjected to uniaxial compressive forces with EJE,,= 40.
Nx/Ncr 1I. Fundamental frequency parameters for a square plate with a square cutout (c/a = 0.5) subjected to uniaxial compressive forces with EJE, = 40.
H. P. Lp: and S. P. LIM
0 I 0.2 0 3 0.4 0.5 0 6 0.7 0.8 0.9
2” 0’1 0’2
0 '3 0 '4 0 '5 0 '6 0 '7 0 B L/d
Fig. 12. Fundamental frequency parameters for a square plate with a square cutout (c/o = 0.5) subjected to uniaxial compressive forces with E,JE, = 3.
Fig. 14. Fundamental frequency parameters for a square plate with a square cutout (c/a = 0.5) subjected to uniaxial tensite forces with E,/E.” = 3 and ~~/N, = 10.
are shown in Figs 11-14. The curves for the uniaxial
The effect of orthotropy
is examined for square plates with material properties shown in Table 1. The variation of the fundamental frequencies with increased in-plane forces are the same as that of the isotropic plates. The results are presented in Figs 7-10. The results for the higher modes of a square plate having a central square cutout of size c/a = 0.5
and biaxial loadings are identical for normalized compressive forces. For in-plane tensile forces, the nautral frequencies for the uniaxial and biaxial loadings are equal only for the X, - Y, and the X, - Y, modes with the biaxial forces equal to half of the uniaxial forces. As in the case of the isotropic plates, there is a tendency for the X, - Y, mode to be lower than the fundamental mode for large cutout sizes. 4. CONCLUSION
A simple method for predicting the natural frequencies of simply supported rectangular plates with rectangular cutouts have been presented for both the uniaxial and biaxial in-plane forces. Numerical results for square plates with square cutouts have been presented for both the isotropic and orthotropic plates. Increased in-plane tensile forces increase the natural frequencies whereas increased compressive forces decrease the natural frequencies until the state of buckling.
* Xl-Yi l
G. Herrmann, The influence of initial stress on the dynamic behavior of elastic and viscoelastic plates.
Fig. 13. Fundamental frequency parameters for a square plate with a square cutout (c/u = 0.5) subjected to uniaxial tensile forces with E,/E, = 40 and NJN, = 10.
Pub. Inst. Ass. for Bridge and Structural Engng 16, 275-294 (1956). S. M. Dickinson, Lateral vibration of rectangular plates subject to in-plane forces. J. Sound Vibr. 16, 46.5-472 (1971). S. F. Bassily and S. M. Dickinson, Buckling and lateral
vibration of rectangular plates subject to in-plane loads-a Ritz approach. J. Sound Vjbr. 24, 219-239 (1972).
Free vibration of square plates with cutouts
4. S. F. Bassily and S. M. Dickinson, CorrigendumBuckling and lateral vibration of rectangular plates subject to in-plane load-a Ritz approach. J. Sound
orthotropic plates under in-plane forces. J. Appl. Mech. 38, 699670 (1971). 10. S. R. Soni and C. L. Amba Rao, Vibration of or-
Vibr. 29, 505-508 (1973). 5. S. F. Bassily and S. M. Dickinson, Vibrations of plates
thotropic rectangular plates under in-plane forces. Compur. Strucf. 4, 1105-l 115 (1974). P. A. A. Laura and L. E. Luison, Vibration of orthotropic rectangular plates with edges possessing different rotational flexibility and subjected to in-plane forces. Comput. Strucl. 9, 527-532 ((978). _ A. W. Leissa. Vibration ofalates. NASA SP-160 (1978). H. P. Lee, S: P. Lim and 8. T. Chow, Elastic buckling of rectangular plates with rectangular cutouts. To be published. A. B. Sabir and F. Y. Chow, Elastic buckling of flat panels containing circular and square holes In Instability and Plastic Collapse of Steel Swucture (Edited by J. L. Morris), pp. 31 l-321. Granada, London (1983). R. Narayanan and F. Y. Chow, Ultimate capacity of uniaxially compressed perforated plates. Thin- Walled
subject to arbitrary in-plane loads-a perturbation approach. J. Appl. h4ech. 40, 1023-1028 (1973). 6. P. A. A. Laura and E. Romanelli, Vibrations of rectangular plates elastically restrained against rotation along all edges and subjected to a biaxial state of stress.
J. Sound Vibr. 37, 367-377 (1974). 7. C. E. Gianetti, L. Diez and P. A. A. Laura, Transverse
vibrations of rectangular plates with elastically restrained edges and subject to in-plane shear forces. J. Sound Vibr. 54, 409-417
8. L. Diez, C. E. Gianetti and P. A. A. Laura, A note on
transverse vibrations of rectangular plates subject to in-plane normal and shear forces. J. Sound Vibr. 59, 503-509 (1978). 9. S. M. Dickinson, The flexural vibration of rectangular
2, 241-264 (1984).