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Free vibration of composite plates using the finite difference method
Thin-Walled Structures 42 (2004) 399–414 www.elsevier.com/locate/tws
Free vibration of composite plates using the finite difference method Karim S. Numayr, Rami H. Haddad , Madhar A. Haddad Civil Engineering Department, Jordan University of Science and Technology, P.O. Box 3030, Irbid, Jordan Received 10 August 2002; received in revised form 30 July 2003; accepted 30 July 2003
Abstract The finite difference method was used to solve differential equations of motion of free vibration of composite plates with different boundary conditions. The effects of shear deformation and rotary inertia on the natural frequencies of laminated composite plates are investigated in this paper. Four cases are studied: neglecting both shear deformation and rotary inertia, considering only rotary inertia, considering only shear deformation, and considering both. Solutions were obtained for symmetric and angle-ply laminated plates. The factors that affect natural frequencies of different composite plates, such as span-to-depth ratio, aspect ratio, angle-ply, and lamination sequence were also investigated. Results were found to agree well with exact and approximate solutions reported in literature. Shear deformation showed a considerable effect on the natural frequencies for composite plates, whereas the rotary inertia effect was found to be negligible. # 2003 Elsevier Ltd. All rights reserved. Keywords: Plates; Laminates; Composite materials; Free vibration; Natural frequency; Shear deformation; Rotary inertia
1. Introduction Laminated and fibrous composites are increasingly used in various areas, especially in the aerospace industry where they are subjected to dynamic loads. Laminated structural elements exhibit properties, which are more favorable than those of single-layer and isotropic ones. Also, fiber reinforced composite materials have high strength to density ratio, and relatively low cost. Fibers can be oriented
Corresponding author. Tel.: +962-2-709-5111; fax: +962-2-709-5018. E-mail address: [email protected] (R.H. Haddad).
0263-8231/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2003.07.001
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to provide minimum structure weight for a given structure geometry and a given system of loads. The ratio of in-plane Young’s modulus to transverse shear modulus is relatively high for composite plates because of the great difference between elastic properties of fiber filament and matrix materials. This leads to using the classical plate theory, which neglects transverse shear deformation, is invalid for most composite plates, even those which are geometrically thin, including shear deformation and/or rotary inertia yields a mathematical complexity of the dynamic problem of composite plates. Therefore, a numerical solution such as the finite element method or the finite difference method can be used to solve this class of problem. Different studies have been conducted to determine the dynamic properties of composite laminates [1–13]. Whitney [1] considers a higher order theory, which includes the effect of transverse shear and rotary inertia deformations. His approach is an extension of theories developed by Reissner [2] and Mindlin [3] for homogeneous isotropic plates. An exact solution was attained for the simply supported case, only. Wu and Vinson [4] studied the effect of shear deformations on the fundamental natural frequency of composite plates with different boundary conditions: clamped, simply supported, and combined clamped and simply supported edges. This solution was based on Galerkin’s method and assumed functions analogous to those of Warburton [5]. The author found that for a plate composed of highly anisotropic composite layers with either clamped or simply supported edges, the effect of transverse shear deformations is significant for both small and large amplitude vibration. Using another approach, Hearmon [6] applied Rayley–Ritz method to especial orthotropic plates, with clamped and simply supported boundary conditions, neglecting transverse shear, and rotary inertia deformations. Bert [7] presented the effect of shear deformation on vibrations of antisymmetric angle-ply laminated rectangular plates. The displacement formulation of heterogeneous shear deformation plate theory oriented by Yang, Norris, and Stavsky [8] was used. Numerical results are presented showing the parametric effect of aspect ratio, length-to-thickness ratio, number of layers and lamination angle. It was concluded that: (a) the effect of relative transverse shear deformation on the fundamental frequency is greater for antisymmetric angle-ply plates than for homogeneous isotropic plates of the same dimensions, and (b) the effect of plate aspect ratio (a/b) on the fundamental frequency is more pronounced in thicker plates (low a/h ratio) than that of thin plates (high a/h ratio). Bhimaraddi and Stevens [9] presented a higher order theory for free vibration of orthotropic, homogeneous, and laminated rectangular plates. The theory accounts for in-plane inertia, rotary inertia, and shear deformation effects. The proposed method used Hamilton’s principle and assumed parabolic variations for transverse shear strains across the thickness of plate. The main conclusions were: (a) the thickness of plate has a more pronounced effect on the behavior of composite plates than that of the behavior of isotropic plates; (b) the transverse shear deformation effect are more pronounced in thin composite laminated plates; and (c) the frequencies predicted by the present analysis are closer to exact values than those predicted by shear deformation theory.
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Reddy [10] presented a layerwise theory for the analysis of free vibration of laminated plates. This theory is the most current and sophisticated theory in which full account is given to various three-dimensional effects. The classical laminated plate theory, the first order shear deformation laminated plate theory, and the thirdorder shear deformation plate theory were considered. The elasticity equations were solved by utilizing the state-space variables and the transfer matrix. Results were also obtained for symmetric and antisymmetric laminates. Mirza and Li [11] presented an analytical approach based on the reciprocal theorem for the free vibration of sandwich plates. Whereas, Marco and Ugo [12] prev v v v sented a higher order plate element formulation for symmetrically [0 /90 /0 /90 / v 0 ] square plate for the case of simply supported boundary conditions. The transverse shear stress distribution across the thickness of thick multilayered anisotropic plates was obtained using the higher order finite element procedure. Whitney [13] presented solutions for flexural vibration frequency of symmetric and nonsymmetric laminates by adopting the bending theory for anistropic laminated plates developed by Yang, Norris, and Stevsky [8]. The latter theory considered shear deformation and rotary inertia in the same manner as that in Mindlin’s theory for isotropic homogenous plates. The results indicated that shear deformation can be quite significant for composites with a span-to-depth ratio as high as 20, and insignificant for homogeneous isotropic plates with the same ratio. Based on the previously outlined literature, it is clear that the behavior of composites plates subjected to free vibration was not well established since none of the approximate or the exact solutions obtained covered all possible boundary conditions, geometric factors, and different types of composite materials. In addition, individual or combined effects of shear deformation, and/or rotary inertia on the natural frequency of composite plates was not fully understood. Hence, this paper aims at utilizing the finite difference method in order to: 1. Perform dynamic analysis of composite plates in order to have a better understanding of the dynamic behavior of different plates in lieu with different parameters associated with the problem. 2. Find the natural frequency of symmetrically laminated and fiber-reinforced composite plates of various boundary conditions and comparing obtained results with those in the literature. 3. Study the effect of including and excluding shear and rotary inertia deformations, that is studying the following four cases: (a) neglecting both shear and rotary inertia deformations; (b) considering only rotary inertia deformation; (c) considering only shear deformation; (d) considering both shear and rotary inertia deformations. 4. Study the effect of fiber orientation on the natural frequency. 2. Problem formulation In order to have a better insight into the effects of shear deformation and rotary inertia on the solution of the dynamic problem of composite plates with laminates,
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cases (a) to (d) of objective (3) were considered in the formulation of the differential equations of motion. In the derivation process, the following basic assumptions were made: 1. The plate is constructed of an arbitrary number of orthotropic layers bonded together. The principal axes of material of an individual layer need not to coincide with geometric axes of the plate. 2. The plate is thin and has a constant thickness; i.e. the thickness h is much smaller than other dimensions. 3. The inplane displacements u, v in x and y directions, respectively, and the transverse displacement w in the z direction are all small compared to the plate thickness. 4. Inplane strains ex ; ey , and cxy are small compared to unity. 5. Each ply obeys Hooke’s law; linear elastic behavior. 2.1. Case (a): Neglecting both shear deformation and rotary inertia The dynamic equilibrium of the infinitesimal element shown in Fig. 1 yields the following partial differential equation of motion [1]:
D11
@4w @4w @4w @2w þ 2 ð D þ 2D Þ þ D ¼ q 12 66 22 @x4 @x2 @y2 @y4 @t2
ð1Þ
where D11, D12, D22, and D66 are rigidities in the principle materials direction, and q is the average mass density of all laminates.
Fig. 1. Nomenclature for moments and transverse force in plate element subjected to free vibration.
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2.2. Case (b): Considering rotary inertia deformation, only For this case, the translational partial differential equation of motion becomes:
D11
@4w @4w @4w @4w @4w @2w þ 2 ð D þ 2D Þ þ D I I ¼ q 12 66 22 @x4 @x2 @y2 @y4 @x2 @t2 @y2 @t2 @t2
where, I ¼
Ð h=2
h=2 q z
2
ð2Þ
dz is the mass moment of inertia per unit area.
2.3. Case (c): Considering shear deformation only For this case of including shear deformation, the inter-laminar shear strains are: @w @x @w cyz ¼ wy þ @y
cxz ¼ wx þ
where wx, wy are rotations in the x and y directions, respectively. Application of rotational and translational dynamic equilibrium of the infinitesimal element yields the following three partial differential equations of motion;
D11
@ 2 wy @ 2 wx @ 2 wx 5 @w A þ D þ ð D þ D Þ w þ ¼0 12 66 66 55 x 6 @x @x2 @x@y @y2
ð3Þ
@w wy þ ¼0 @y
ð4Þ
@ 2 wy @ 2 wy 5 @ 2 wx þ D66 ðD12 þ D66 Þ þ D A44 22 6 @x@y @x2 @y2
@wy @ 2 w 5 @wx @ 2 w 5 @2w A55 þ 2 þ A44 þ 2 ¼q 2 6 @x 6 @y @t @x @y
ð5Þ
where A44, A55 are the shear rigidities. These equations cannot be reduced to one equation as in the previous two cases, even in the absence of rotary inertia terms. 2.4. Case (d): Considering both shear and rotary inertia deformations Differential equations of motion for this case is similar to that of the previous 2 @2w case except that the inertia moments I @@tw2x and I @t2y appear on the right hand sides of Eqs. (3) and (4), respectively. The general form of partial differential equations of motion for angle-ply laminates, including shear deformation and rotary inertia,
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are as follows: @ 2 wy @ 2 wy @ 2 wx @ 2 wx @ 2 wx þ D66 þ 2D16 þ D16 þ ðD12 þ D16 Þ 2 2 2 @x @x@y @y @x @x@y @ 2 wy 5 @w @w A55 wx þ þ D26 þ A45 wy þ 4 @x @y @y2 @ 2 wx ¼I @t2
D11
@ 2 wy @ 2 wy @ 2 wx @ 2 wx @ 2 wx þ D þ ð D þ D Þ þ D þ 2D 12 66 26 66 26 @x2 @x@y @y2 @x2 @x@y 2 @ wy @w @w k A45 wx þ þ D22 þ A44 wy þ @x @y @y2 @ 2 wy ¼I @t2 5 @wx @ 2 w @wx @wy @2w A55 þ 2 þ A45 þ þ2 6 @x @x@y @x @y @x 2 2 @wy @ w @ w þ 2 þA44 ¼q 2 @y @t @y
ð6Þ
D16
ð7Þ
ð8Þ
in which coupling rigidity terms D26 and A45 are included in the formulation.
3. Method of solution The following separation of variables is used: wðx; y; tÞ ¼ wðx; yÞeiXt
ð9Þ
For the transverse displacement function and wx ðx; y; tÞ ¼ wx ðx; yÞeiXt
ð10Þ
iXt
ð11Þ
wy ðx; y; tÞ ¼ wy ðx; yÞe
for the rotation functions, where t is the time and X is the natural frequency. The resulted partial differential equations (PDE) in terms of mid-plane variables (x,y) are then solved using the finite difference method by substituting the central difference derivatives of displacements and rotations in the PDE. Several boundary conditions are considered for each one of the four cases. For instance, for a simply supported x-edges (y ¼ 0 or y ¼ b), the boundary conditions in discrete form would be: wðm; nÞ ¼ 0; wðm; n þ 1Þ ¼ wðm; n 1Þ; wy ðm; n þ 1Þ ¼ wy ðm; n 1Þ; and wx ðm; nÞ ¼ 0; where (m,n) is a point on the x-edge as shown in Fig. 2. Whereas, for a clamped y-edge (x ¼ 0 or x ¼ a), the boundary conditions would be: wðm; nÞ ¼ 0; wðm þ 1; nÞ ¼ wðm 1; nÞ; wx ðm; nÞ ¼ 0; wy ðm; nÞ ¼ 0; where (m,n) is a point on the y-edge, Fig. 2.
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Fig. 2. A schematic showing the plate mesh numbering.
The UNIX system was used to solve the eigen value problem of the different formulated matrices to find the natural frequencies. Because of the large size of stiffness and mass matrices, it was difficult to enter data by hand; therefore mesh generation programs were made using Fortran language. The computation time for these programs was small even though their sizes reached up to (50 50). These matrices were either symmetric or nonsymmetric depending on the case studied and on the boundary conditions. JACOBI and DETERMINANT programs were used to find the natural frequencies for symmetric and nonsymmetric matrices, respectively.
4. Analysis and discussion of results In this section, the results obtained in this paper were validated by comparing them with those reported in the literature for plates of varying dimensions and boundary conditions considering cases (a) to (d). In addition, the convergence rate, as the number of mesh divisions increased, was discussed. Finally, using the present technique, the natural frequencies for composite plates with varying width to depth ratios, aspect ratios, angle-ply values and lamination sequences were obtained, while considering different composite materials. The materials used and their properties are listed in Table 1. 4.1. Validation and convergence of present results The results obtained using the finite difference method have been compared with approximate and exact values reported in literature for plates with varying boundary conditions, geometric parameters, and materials properties, for the cases of including or excluding shear deformations and/or rotary inertia. Nondimensional
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Table 1 Engineering properties of composite materials Property
Material
E1 E2 m12 m12 G12 G12 G12 q
B/E [14]
G/E [15] (AS/3501-6)
Glass/E [16]
Steel
224.649 12.753 0.256 0.0146 4.434 4.434 2.497 2440
144.800 9.650 0.300 0.020 4.140 4.140 3.450 1389
9.710 3.250 0.290 0.097 0.903 0.903 0.236 1347
207 N.A. 0.270 N.A. 83 N.A. N.A. 7849
B, boron; E, epoxy; G, graphite. v
v
v
v
fundamental frequencies ðx11 Þ for a [0 /90 /90 /0 ] boron/epoxy simple-fixed supported square plate of different span-to-depth ratios (a/h) as calculated by the present technique, Vinson and Chou [14] and Hearman [6] considering case (a) are listed in Table 2. Tables 3 and 4 show nondimensional fundamental frequencies for a fixed boron/epoxy square plate, computed in the present study, compared to exact and approximate solutions [4] for cases (a) and (c), respectively. Good agreement between the present results and those reported by Wu and Vinson [4] can be Table 2 Nondimensional fundamental frequeny of a fixed-simple cross-ply boron/epoxy square plate for different span depth ratios, Case (a) (h ¼ 2:5 cm) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a/h x11 ¼ x11 a2 q=ET h3
10 20 30 40 50
Present
Hearmon [6]
Vinson and Chou [14]
0.624823 0.631608 0.631388 0.632112 0.632438
0.632463 0.6324631 0.6324631 0.6324631 0.6324631
0.643435 0.643435 0.643435 0.643435 0.643435
Table 3 Nondimensional fundamental frequencies of a fixed cross ply boron/epoxy square plate, Case (a) (a ¼ b ¼ 75 cm; h ¼ 2:5 cm) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mode number xmn ¼ xmn a2 q=ET h3
1 2 3 4
m
n
Present
Hearmon [6]
Vinson and Chou [14]
1 1 1 1
1 2 3 4
0.9001671 1.3150953 2.1395865 3.3076284
0.9049738 1.3239921 2.1667976 3.3888332
0.9072362 1.3733217 2.2264539 3.4474835
K.S. Numayr et al. / Thin-Walled Structures 42 (2004) 399–414 Table 4 Nondimensional fundamental frequencies (a ¼ b ¼ 75 cm; h ¼ 2:5 cm) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mode number xmn ¼ xmn a2 q=ET h3
1 2 3 4
of
a
fixed
boron/epoxy
square
m
n
Present
Vinson and Chou [14]
1 1 1 1
1 2 3 4
0.8323117 1.2198056 1.8062964 2.6693444
0.8384779 1.262727 1.8566729 2.748363
407
plate,
case
(c)
noticed. Yet, it should be indicated that as the mode number increased, the present results showed some divergence from those found by Hearmon [6] and Vinson and Chou [14]. Figs. 3 and 4 show convergence rate of the nondimensional fundamental frequency ðx11 Þ as the number of divisions is increased for a cross-ply boron/epoxy rectangular plate and for cases (b) and (d), respectively. As can be seen, the rate of convergence depends on the initial guess; it increases if the exact value turned out to be larger than the initial guess and vice versa. 4.2. Effect of shear deformation and/or rotary inertia To have better insight about the effect of including or excluding shear deformation and/or rotary inertia on the dynamic properties of composite plates with different boundary conditions, Tables 5 and 6 are presented herein. As can be noticed, including shear deformation had significantly affected the natural
Fig. 3. Convergence rate of the nondimensional fundamental frequency for a simply supported cross-ply boron/epoxy rectangular plate (a ¼ 75 cm; b ¼ 50 cm; h ¼ 2:5 cm).
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Fig. 4. Convergence rate of the nondimensional fundamental frequency for a fixed supported cross-ply boron/epoxy rectangular plate (a ¼ 75 cm; b ¼ 50 cm; h ¼ 2:5 cm).
frequency values, contrary to that when including rotary inertia. Moreover, it should be indicated that the percentage change in the nondimensional fundamental frequencies due to the inclusion of shear deformation is dependent on the spanTable 5 Present nondimensional fundamental frequeny of a fixed-simple cross-ply boron/epoxy square plate for different span depth ratios (h ¼ 2:5 cm) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a/h xmn ¼ xmn a2 q=ET h3
10 20 30 40 50
Case (a)
Case (b)
Case (c)
Case (d)
0.624823 0.631608 0.631388 0.632112 0.632438
0.62264 0.62618 0.62687 0.62773 0.62799
0.471879 0.579435 0.619187 0.62276 0.629389
0.471879 0.5794345 0.6191887 0.6227567 0.6293886
Table 6 Present nondimensional fundamental frequencies of a fixed boron/epoxy cross-ply square plate (a ¼ b ¼ 75 cm; h ¼ 2:5 cm) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mode xmn ¼ xmn a2 q=ET h3 number m n Case (a) Case (b) Case (c) Case (d) 1 2 3 4
1 1 1 1
1 2 3 4
0.9001671 1.3150953 2.1395865 3.3076284
0.8999772 1.3107414 2.1296714 3.2831078
0.8323117 1.2198056 1.8062964 2.6693444
0.8321170 1.2193327 1.7963615 2.6448522
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to-depth ratio, as well as the mode number as indicated by Tables 5 and 6, respectively. The results showed less effect of including shear deformations at higher span depth ratios, and at a lower mode number. 4.3. Effect of plate geometry and boundary conditions on natural frequency 4.3.1. Span depth ratio Fig. 5 shows the effect of span-to-depth ratio (a/h) on the fundamental frequency of a simply supported square cross-ply boron/epoxy plate for cases (a) and (c). The span-to-depth ratio has almost a negligible effect on the results when shear deformation is excluded, case (a), and a significant effect when shear deformation is included, case (c), especially at low ratios. At higher span-to-depth ratios greater than 30, the influence of including shear deformation on nondimensional frequencies is minor; although values remain smaller than those obtained for case (a). Analysis of composite plates with different boundary conditions was carried out; fixed (fixed at four edges), fixed-simple (fixed from two adjacent edges and simple at the other two edges), and simple (simply supported at four edges). The results obtained show, as expected, that natural frequencies are more for the fixed boundary conditions than those for the fixed-simple and less for the simple boundary conditions. This is attributed to the stiffness which is the highest in fixed plates and the lowest in simply supported ones. The above results showed that the effect of shear deformation on a fixed plate is more than that on a simply supported one: the effective length for a fixed plate is lower hence a/h ratio (Fig. 6). Fig. 7 shows the effect of span depth ratio on the nondimensional fundamental frequency, considering case (c), for a plate fixed from two adjacent edges and simple from the other two edges. The plate is made of either graphite, epoxy glass,
Fig. 5. Effect of span depth ratio on the nondimensional fundamental frequency for a simply supported cross-ply boron/epoxy square plate (h ¼ 2:5 cm).
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boron/epoxy, or steel. The span–depth ratio had a considerable effect on the fundamental frequency of plates made with composite materials, as the curves showed an increasing trend especially at lower a/h ratios. On the other hand, curves corre-
Fig. 6. Comparison of the nondimensional fundamental frequency for cross-ply boron/epoxy square plate under different boundary conditions (h ¼ 2:5 cm).
Fig. 7. Comparison of the nondimensional fundamental frequency for fixed-simple cross-ply square plate composed of different materials (h ¼ 2:5 cm).
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sponding to isotropic materials remained almost constant, regardless of the span– depth ratio. This is attributed to the difference in the elastic properties between fiber filament and matrix materials which leads to a high ratio of in-plane Young’s modulus to transverse shear modulus for the composite plates. 4.3.2. Aspect ratio Figs. 8 and 9 show the effect of aspect ratio on the nondimensional fundamental frequency for a simply supported and a fixed boron/epoxy rectangular plates (b ¼ 50 cm; h ¼ 2:5 cm), respectively. As the aspect ratio increased, the nondimensional fundamental frequency increased because the ratio of area to perimeter decreased, thus the stiffness increased. The results of Fig. 9 suggests that when shear and rotary inertia deformations are included, the rate of change of natural frequency to aspect ratio decreases because the area subjected to shear increases. 4.4. Effect of composite material properties Figs. 10 and 11 show the effect of both fiber orientation and lamination sequence on the fundamental frequencies for boron/epoxy square and rectangular plates, v respectively. For the square plate, there is symmetry for the orientation angle 45 , thus a maximum fundamental natural frequency is obtained; the two maximum components are equal in each direction of the plane of the plate. Whereas for the v rectangular plate the maximum fundamental frequency happens to be between 60 v and 75 , because the dimensions are not equal and the two components resulting from the fiber in the short and long direction are different: the component in the long direction was smaller than that in the short direction. Moreover, it is clear
Fig. 8. Effect of aspect ratio on the nondimensional fundamental frequency for a simply supported cross-ply boron/epoxy plate (b ¼ 50 cm; h ¼ 2:5 cm).
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that natural frequency is increased when adopting a lamination sequence in the v v v v manner h=0 =0 =h rather than 0 =h=h=0 , because the distance between the neutral surface and the fiber components is increased.
Fig. 9. Effect of aspect ratio on the nondimensional fundamental frequency for a fixed supported crossply boron/epoxy plate (b ¼ 50 cm; h ¼ 2:5 cm).
Fig. 10. Effect of fiber orientation and lamination sequence on the nondimensional fundamental frequency for a simply supported boron/epoxy square plate (a ¼ b ¼ 50 cm; h ¼ 2:5 cm).
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Fig. 11. Effect of fiber orientation and lamination sequence on the nondimensional fundamental frequency for a simply supported boron/epoxy rectangular plate (a ¼ 50 cm; b ¼ 75 cm; h ¼ 2:5 cm).
5. Conclusions Based on the results discussed before, the following conclusions can be drawn: 1. Including transverse shear deformation in the dynamic analysis of composite plates resulted in a significant change in the natural frequency of such plates. 2. Rotary inertia terms have a minor influence on the values of natural frequency and therefore could be neglected for the materials used in this study. 3. The effect of shear deformation on the natural frequencies corresponding to higher modes is more pronounced than that on those corresponding to lower modes. 4. The effect of including transverse shear deformation is greater for fully clamped plates than for a simply supported ones. 5. The effect of shear deformation is more pronounced for a plate with low spanto-depth ratio than that for a plate with higher span-to-depth ratio. 6. Fiber orientation (h) has a great effect on the determination of natural frequencies of angle-ply composite plates. For a square plate, maximum fundav mental frequency is obtained when fibbers were oriented at an angle of 45 . v v 7. Composite plates with lamination sequence h=0 =0 =h have higher nondimenv v sional fundamental frequencies than those having 0 =h=h=0 sequence for all values of (h). References [1] Whitney JM. Structural analysis of laminated an isotropic plates, 1st ed. Western Hemisphere. Technical Publishing Company; 1987, p. 263–95.
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[2] Reisner E. The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics 1945;12:69–77. [3] Mindlin RD. Influence of rotary inertia and shear on flexural motion of isotropic, elastic plates. Journal of Applied Mechanics 1951;18:336–43. [4] Wu CI, Vinson JR. Nonlinear oscillations of laminated specially orthotropic plates with clamped and simply supported edges. Journal of Acoustical Society of America 1971;49(5):1561–8. [5] Warburton G. The vibration of rectangular plates. Proceeding of the Institute of Mechanical Engineering 1954;168:371–82. [6] Hearmon R. The frequency of flexural vibrations of rectangular orthotropic plates with clamped or simply supported edges. Journal of Applied Mechanics 1959;26:537–42. [7] Bert CW, Chen TC. Effect of shear deformation on vibration of antisymmetric angle-ply laminated rectangular plate. Institute Journal of Solid Structures 1977;14:265–473. [8] Yang PC, Norris GH, Stavsky Y. Elastic wave propagation in hetrogeneous plates. International Journal of Solid and Structures 1996;2:1665–84. [9] Bhimaraddi A, Stevens LK. A higher order theory for free vibration of orthotropic, homogeneous, and laminated rectangular plates. Journal of Applied Mechanics 1984;51:95–8. [10] Reddy JN. A simple higher-order theory for laminated composite plates. Journal of Applied Mechanics 1984;51(84):745–52. [11] Mirza S, Li N. Analytical approach to free vibration of sandwich plates. AIAA Journal 1995;33(12):1988–90. [12] Marco DS, Ugo IC. Analysis of thick multilayered anisotropic plates by a higher order plate element. AIAA Journal 1995;33:2435–7. [13] Whitney JM, Pagano NJ. Shear deformation in hetrogeneous anisotropic plate. Journal of Applied Mechanics 1970;37:1031–6. [14] Vinson JR, Chou TW. Composite materials and their use in structures. Essex, UK: Applied Science Publishers Ltd; 1975, p. 247–309. [15] Abramovich H. Shear deformation and rotary inertia effects on vibration of composite beams. Composite Structures 1992;20:165–73. [16] Jones RM. Mechanics of composite materials. Washington, DC, USA: McGraw-Hill; 1975.
Free vibration of composite plates using the finite difference method Karim S. Numayr, Rami H. Haddad , Madhar A. Haddad Civil Engineering Department, Jordan University of Science and Technology, P.O. Box 3030, Irbid, Jordan Received 10 August 2002; received in revised form 30 July 2003; accepted 30 July 2003
Abstract The finite difference method was used to solve differential equations of motion of free vibration of composite plates with different boundary conditions. The effects of shear deformation and rotary inertia on the natural frequencies of laminated composite plates are investigated in this paper. Four cases are studied: neglecting both shear deformation and rotary inertia, considering only rotary inertia, considering only shear deformation, and considering both. Solutions were obtained for symmetric and angle-ply laminated plates. The factors that affect natural frequencies of different composite plates, such as span-to-depth ratio, aspect ratio, angle-ply, and lamination sequence were also investigated. Results were found to agree well with exact and approximate solutions reported in literature. Shear deformation showed a considerable effect on the natural frequencies for composite plates, whereas the rotary inertia effect was found to be negligible. # 2003 Elsevier Ltd. All rights reserved. Keywords: Plates; Laminates; Composite materials; Free vibration; Natural frequency; Shear deformation; Rotary inertia
1. Introduction Laminated and fibrous composites are increasingly used in various areas, especially in the aerospace industry where they are subjected to dynamic loads. Laminated structural elements exhibit properties, which are more favorable than those of single-layer and isotropic ones. Also, fiber reinforced composite materials have high strength to density ratio, and relatively low cost. Fibers can be oriented
Corresponding author. Tel.: +962-2-709-5111; fax: +962-2-709-5018. E-mail address: [email protected] (R.H. Haddad).
0263-8231/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2003.07.001
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to provide minimum structure weight for a given structure geometry and a given system of loads. The ratio of in-plane Young’s modulus to transverse shear modulus is relatively high for composite plates because of the great difference between elastic properties of fiber filament and matrix materials. This leads to using the classical plate theory, which neglects transverse shear deformation, is invalid for most composite plates, even those which are geometrically thin, including shear deformation and/or rotary inertia yields a mathematical complexity of the dynamic problem of composite plates. Therefore, a numerical solution such as the finite element method or the finite difference method can be used to solve this class of problem. Different studies have been conducted to determine the dynamic properties of composite laminates [1–13]. Whitney [1] considers a higher order theory, which includes the effect of transverse shear and rotary inertia deformations. His approach is an extension of theories developed by Reissner [2] and Mindlin [3] for homogeneous isotropic plates. An exact solution was attained for the simply supported case, only. Wu and Vinson [4] studied the effect of shear deformations on the fundamental natural frequency of composite plates with different boundary conditions: clamped, simply supported, and combined clamped and simply supported edges. This solution was based on Galerkin’s method and assumed functions analogous to those of Warburton [5]. The author found that for a plate composed of highly anisotropic composite layers with either clamped or simply supported edges, the effect of transverse shear deformations is significant for both small and large amplitude vibration. Using another approach, Hearmon [6] applied Rayley–Ritz method to especial orthotropic plates, with clamped and simply supported boundary conditions, neglecting transverse shear, and rotary inertia deformations. Bert [7] presented the effect of shear deformation on vibrations of antisymmetric angle-ply laminated rectangular plates. The displacement formulation of heterogeneous shear deformation plate theory oriented by Yang, Norris, and Stavsky [8] was used. Numerical results are presented showing the parametric effect of aspect ratio, length-to-thickness ratio, number of layers and lamination angle. It was concluded that: (a) the effect of relative transverse shear deformation on the fundamental frequency is greater for antisymmetric angle-ply plates than for homogeneous isotropic plates of the same dimensions, and (b) the effect of plate aspect ratio (a/b) on the fundamental frequency is more pronounced in thicker plates (low a/h ratio) than that of thin plates (high a/h ratio). Bhimaraddi and Stevens [9] presented a higher order theory for free vibration of orthotropic, homogeneous, and laminated rectangular plates. The theory accounts for in-plane inertia, rotary inertia, and shear deformation effects. The proposed method used Hamilton’s principle and assumed parabolic variations for transverse shear strains across the thickness of plate. The main conclusions were: (a) the thickness of plate has a more pronounced effect on the behavior of composite plates than that of the behavior of isotropic plates; (b) the transverse shear deformation effect are more pronounced in thin composite laminated plates; and (c) the frequencies predicted by the present analysis are closer to exact values than those predicted by shear deformation theory.
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Reddy [10] presented a layerwise theory for the analysis of free vibration of laminated plates. This theory is the most current and sophisticated theory in which full account is given to various three-dimensional effects. The classical laminated plate theory, the first order shear deformation laminated plate theory, and the thirdorder shear deformation plate theory were considered. The elasticity equations were solved by utilizing the state-space variables and the transfer matrix. Results were also obtained for symmetric and antisymmetric laminates. Mirza and Li [11] presented an analytical approach based on the reciprocal theorem for the free vibration of sandwich plates. Whereas, Marco and Ugo [12] prev v v v sented a higher order plate element formulation for symmetrically [0 /90 /0 /90 / v 0 ] square plate for the case of simply supported boundary conditions. The transverse shear stress distribution across the thickness of thick multilayered anisotropic plates was obtained using the higher order finite element procedure. Whitney [13] presented solutions for flexural vibration frequency of symmetric and nonsymmetric laminates by adopting the bending theory for anistropic laminated plates developed by Yang, Norris, and Stevsky [8]. The latter theory considered shear deformation and rotary inertia in the same manner as that in Mindlin’s theory for isotropic homogenous plates. The results indicated that shear deformation can be quite significant for composites with a span-to-depth ratio as high as 20, and insignificant for homogeneous isotropic plates with the same ratio. Based on the previously outlined literature, it is clear that the behavior of composites plates subjected to free vibration was not well established since none of the approximate or the exact solutions obtained covered all possible boundary conditions, geometric factors, and different types of composite materials. In addition, individual or combined effects of shear deformation, and/or rotary inertia on the natural frequency of composite plates was not fully understood. Hence, this paper aims at utilizing the finite difference method in order to: 1. Perform dynamic analysis of composite plates in order to have a better understanding of the dynamic behavior of different plates in lieu with different parameters associated with the problem. 2. Find the natural frequency of symmetrically laminated and fiber-reinforced composite plates of various boundary conditions and comparing obtained results with those in the literature. 3. Study the effect of including and excluding shear and rotary inertia deformations, that is studying the following four cases: (a) neglecting both shear and rotary inertia deformations; (b) considering only rotary inertia deformation; (c) considering only shear deformation; (d) considering both shear and rotary inertia deformations. 4. Study the effect of fiber orientation on the natural frequency. 2. Problem formulation In order to have a better insight into the effects of shear deformation and rotary inertia on the solution of the dynamic problem of composite plates with laminates,
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cases (a) to (d) of objective (3) were considered in the formulation of the differential equations of motion. In the derivation process, the following basic assumptions were made: 1. The plate is constructed of an arbitrary number of orthotropic layers bonded together. The principal axes of material of an individual layer need not to coincide with geometric axes of the plate. 2. The plate is thin and has a constant thickness; i.e. the thickness h is much smaller than other dimensions. 3. The inplane displacements u, v in x and y directions, respectively, and the transverse displacement w in the z direction are all small compared to the plate thickness. 4. Inplane strains ex ; ey , and cxy are small compared to unity. 5. Each ply obeys Hooke’s law; linear elastic behavior. 2.1. Case (a): Neglecting both shear deformation and rotary inertia The dynamic equilibrium of the infinitesimal element shown in Fig. 1 yields the following partial differential equation of motion [1]:
D11
@4w @4w @4w @2w þ 2 ð D þ 2D Þ þ D ¼ q 12 66 22 @x4 @x2 @y2 @y4 @t2
ð1Þ
where D11, D12, D22, and D66 are rigidities in the principle materials direction, and q is the average mass density of all laminates.
Fig. 1. Nomenclature for moments and transverse force in plate element subjected to free vibration.
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2.2. Case (b): Considering rotary inertia deformation, only For this case, the translational partial differential equation of motion becomes:
D11
@4w @4w @4w @4w @4w @2w þ 2 ð D þ 2D Þ þ D I I ¼ q 12 66 22 @x4 @x2 @y2 @y4 @x2 @t2 @y2 @t2 @t2
where, I ¼
Ð h=2
h=2 q z
2
ð2Þ
dz is the mass moment of inertia per unit area.
2.3. Case (c): Considering shear deformation only For this case of including shear deformation, the inter-laminar shear strains are: @w @x @w cyz ¼ wy þ @y
cxz ¼ wx þ
where wx, wy are rotations in the x and y directions, respectively. Application of rotational and translational dynamic equilibrium of the infinitesimal element yields the following three partial differential equations of motion;
D11
@ 2 wy @ 2 wx @ 2 wx 5 @w A þ D þ ð D þ D Þ w þ ¼0 12 66 66 55 x 6 @x @x2 @x@y @y2
ð3Þ
@w wy þ ¼0 @y
ð4Þ
@ 2 wy @ 2 wy 5 @ 2 wx þ D66 ðD12 þ D66 Þ þ D A44 22 6 @x@y @x2 @y2
@wy @ 2 w 5 @wx @ 2 w 5 @2w A55 þ 2 þ A44 þ 2 ¼q 2 6 @x 6 @y @t @x @y
ð5Þ
where A44, A55 are the shear rigidities. These equations cannot be reduced to one equation as in the previous two cases, even in the absence of rotary inertia terms. 2.4. Case (d): Considering both shear and rotary inertia deformations Differential equations of motion for this case is similar to that of the previous 2 @2w case except that the inertia moments I @@tw2x and I @t2y appear on the right hand sides of Eqs. (3) and (4), respectively. The general form of partial differential equations of motion for angle-ply laminates, including shear deformation and rotary inertia,
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are as follows: @ 2 wy @ 2 wy @ 2 wx @ 2 wx @ 2 wx þ D66 þ 2D16 þ D16 þ ðD12 þ D16 Þ 2 2 2 @x @x@y @y @x @x@y @ 2 wy 5 @w @w A55 wx þ þ D26 þ A45 wy þ 4 @x @y @y2 @ 2 wx ¼I @t2
D11
@ 2 wy @ 2 wy @ 2 wx @ 2 wx @ 2 wx þ D þ ð D þ D Þ þ D þ 2D 12 66 26 66 26 @x2 @x@y @y2 @x2 @x@y 2 @ wy @w @w k A45 wx þ þ D22 þ A44 wy þ @x @y @y2 @ 2 wy ¼I @t2 5 @wx @ 2 w @wx @wy @2w A55 þ 2 þ A45 þ þ2 6 @x @x@y @x @y @x 2 2 @wy @ w @ w þ 2 þA44 ¼q 2 @y @t @y
ð6Þ
D16
ð7Þ
ð8Þ
in which coupling rigidity terms D26 and A45 are included in the formulation.
3. Method of solution The following separation of variables is used: wðx; y; tÞ ¼ wðx; yÞeiXt
ð9Þ
For the transverse displacement function and wx ðx; y; tÞ ¼ wx ðx; yÞeiXt
ð10Þ
iXt
ð11Þ
wy ðx; y; tÞ ¼ wy ðx; yÞe
for the rotation functions, where t is the time and X is the natural frequency. The resulted partial differential equations (PDE) in terms of mid-plane variables (x,y) are then solved using the finite difference method by substituting the central difference derivatives of displacements and rotations in the PDE. Several boundary conditions are considered for each one of the four cases. For instance, for a simply supported x-edges (y ¼ 0 or y ¼ b), the boundary conditions in discrete form would be: wðm; nÞ ¼ 0; wðm; n þ 1Þ ¼ wðm; n 1Þ; wy ðm; n þ 1Þ ¼ wy ðm; n 1Þ; and wx ðm; nÞ ¼ 0; where (m,n) is a point on the x-edge as shown in Fig. 2. Whereas, for a clamped y-edge (x ¼ 0 or x ¼ a), the boundary conditions would be: wðm; nÞ ¼ 0; wðm þ 1; nÞ ¼ wðm 1; nÞ; wx ðm; nÞ ¼ 0; wy ðm; nÞ ¼ 0; where (m,n) is a point on the y-edge, Fig. 2.
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Fig. 2. A schematic showing the plate mesh numbering.
The UNIX system was used to solve the eigen value problem of the different formulated matrices to find the natural frequencies. Because of the large size of stiffness and mass matrices, it was difficult to enter data by hand; therefore mesh generation programs were made using Fortran language. The computation time for these programs was small even though their sizes reached up to (50 50). These matrices were either symmetric or nonsymmetric depending on the case studied and on the boundary conditions. JACOBI and DETERMINANT programs were used to find the natural frequencies for symmetric and nonsymmetric matrices, respectively.
4. Analysis and discussion of results In this section, the results obtained in this paper were validated by comparing them with those reported in the literature for plates of varying dimensions and boundary conditions considering cases (a) to (d). In addition, the convergence rate, as the number of mesh divisions increased, was discussed. Finally, using the present technique, the natural frequencies for composite plates with varying width to depth ratios, aspect ratios, angle-ply values and lamination sequences were obtained, while considering different composite materials. The materials used and their properties are listed in Table 1. 4.1. Validation and convergence of present results The results obtained using the finite difference method have been compared with approximate and exact values reported in literature for plates with varying boundary conditions, geometric parameters, and materials properties, for the cases of including or excluding shear deformations and/or rotary inertia. Nondimensional
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Table 1 Engineering properties of composite materials Property
Material
E1 E2 m12 m12 G12 G12 G12 q
B/E [14]
G/E [15] (AS/3501-6)
Glass/E [16]
Steel
224.649 12.753 0.256 0.0146 4.434 4.434 2.497 2440
144.800 9.650 0.300 0.020 4.140 4.140 3.450 1389
9.710 3.250 0.290 0.097 0.903 0.903 0.236 1347
207 N.A. 0.270 N.A. 83 N.A. N.A. 7849
B, boron; E, epoxy; G, graphite. v
v
v
v
fundamental frequencies ðx11 Þ for a [0 /90 /90 /0 ] boron/epoxy simple-fixed supported square plate of different span-to-depth ratios (a/h) as calculated by the present technique, Vinson and Chou [14] and Hearman [6] considering case (a) are listed in Table 2. Tables 3 and 4 show nondimensional fundamental frequencies for a fixed boron/epoxy square plate, computed in the present study, compared to exact and approximate solutions [4] for cases (a) and (c), respectively. Good agreement between the present results and those reported by Wu and Vinson [4] can be Table 2 Nondimensional fundamental frequeny of a fixed-simple cross-ply boron/epoxy square plate for different span depth ratios, Case (a) (h ¼ 2:5 cm) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a/h x11 ¼ x11 a2 q=ET h3
10 20 30 40 50
Present
Hearmon [6]
Vinson and Chou [14]
0.624823 0.631608 0.631388 0.632112 0.632438
0.632463 0.6324631 0.6324631 0.6324631 0.6324631
0.643435 0.643435 0.643435 0.643435 0.643435
Table 3 Nondimensional fundamental frequencies of a fixed cross ply boron/epoxy square plate, Case (a) (a ¼ b ¼ 75 cm; h ¼ 2:5 cm) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mode number xmn ¼ xmn a2 q=ET h3
1 2 3 4
m
n
Present
Hearmon [6]
Vinson and Chou [14]
1 1 1 1
1 2 3 4
0.9001671 1.3150953 2.1395865 3.3076284
0.9049738 1.3239921 2.1667976 3.3888332
0.9072362 1.3733217 2.2264539 3.4474835
K.S. Numayr et al. / Thin-Walled Structures 42 (2004) 399–414 Table 4 Nondimensional fundamental frequencies (a ¼ b ¼ 75 cm; h ¼ 2:5 cm) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mode number xmn ¼ xmn a2 q=ET h3
1 2 3 4
of
a
fixed
boron/epoxy
square
m
n
Present
Vinson and Chou [14]
1 1 1 1
1 2 3 4
0.8323117 1.2198056 1.8062964 2.6693444
0.8384779 1.262727 1.8566729 2.748363
407
plate,
case
(c)
noticed. Yet, it should be indicated that as the mode number increased, the present results showed some divergence from those found by Hearmon [6] and Vinson and Chou [14]. Figs. 3 and 4 show convergence rate of the nondimensional fundamental frequency ðx11 Þ as the number of divisions is increased for a cross-ply boron/epoxy rectangular plate and for cases (b) and (d), respectively. As can be seen, the rate of convergence depends on the initial guess; it increases if the exact value turned out to be larger than the initial guess and vice versa. 4.2. Effect of shear deformation and/or rotary inertia To have better insight about the effect of including or excluding shear deformation and/or rotary inertia on the dynamic properties of composite plates with different boundary conditions, Tables 5 and 6 are presented herein. As can be noticed, including shear deformation had significantly affected the natural
Fig. 3. Convergence rate of the nondimensional fundamental frequency for a simply supported cross-ply boron/epoxy rectangular plate (a ¼ 75 cm; b ¼ 50 cm; h ¼ 2:5 cm).
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Fig. 4. Convergence rate of the nondimensional fundamental frequency for a fixed supported cross-ply boron/epoxy rectangular plate (a ¼ 75 cm; b ¼ 50 cm; h ¼ 2:5 cm).
frequency values, contrary to that when including rotary inertia. Moreover, it should be indicated that the percentage change in the nondimensional fundamental frequencies due to the inclusion of shear deformation is dependent on the spanTable 5 Present nondimensional fundamental frequeny of a fixed-simple cross-ply boron/epoxy square plate for different span depth ratios (h ¼ 2:5 cm) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a/h xmn ¼ xmn a2 q=ET h3
10 20 30 40 50
Case (a)
Case (b)
Case (c)
Case (d)
0.624823 0.631608 0.631388 0.632112 0.632438
0.62264 0.62618 0.62687 0.62773 0.62799
0.471879 0.579435 0.619187 0.62276 0.629389
0.471879 0.5794345 0.6191887 0.6227567 0.6293886
Table 6 Present nondimensional fundamental frequencies of a fixed boron/epoxy cross-ply square plate (a ¼ b ¼ 75 cm; h ¼ 2:5 cm) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mode xmn ¼ xmn a2 q=ET h3 number m n Case (a) Case (b) Case (c) Case (d) 1 2 3 4
1 1 1 1
1 2 3 4
0.9001671 1.3150953 2.1395865 3.3076284
0.8999772 1.3107414 2.1296714 3.2831078
0.8323117 1.2198056 1.8062964 2.6693444
0.8321170 1.2193327 1.7963615 2.6448522
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to-depth ratio, as well as the mode number as indicated by Tables 5 and 6, respectively. The results showed less effect of including shear deformations at higher span depth ratios, and at a lower mode number. 4.3. Effect of plate geometry and boundary conditions on natural frequency 4.3.1. Span depth ratio Fig. 5 shows the effect of span-to-depth ratio (a/h) on the fundamental frequency of a simply supported square cross-ply boron/epoxy plate for cases (a) and (c). The span-to-depth ratio has almost a negligible effect on the results when shear deformation is excluded, case (a), and a significant effect when shear deformation is included, case (c), especially at low ratios. At higher span-to-depth ratios greater than 30, the influence of including shear deformation on nondimensional frequencies is minor; although values remain smaller than those obtained for case (a). Analysis of composite plates with different boundary conditions was carried out; fixed (fixed at four edges), fixed-simple (fixed from two adjacent edges and simple at the other two edges), and simple (simply supported at four edges). The results obtained show, as expected, that natural frequencies are more for the fixed boundary conditions than those for the fixed-simple and less for the simple boundary conditions. This is attributed to the stiffness which is the highest in fixed plates and the lowest in simply supported ones. The above results showed that the effect of shear deformation on a fixed plate is more than that on a simply supported one: the effective length for a fixed plate is lower hence a/h ratio (Fig. 6). Fig. 7 shows the effect of span depth ratio on the nondimensional fundamental frequency, considering case (c), for a plate fixed from two adjacent edges and simple from the other two edges. The plate is made of either graphite, epoxy glass,
Fig. 5. Effect of span depth ratio on the nondimensional fundamental frequency for a simply supported cross-ply boron/epoxy square plate (h ¼ 2:5 cm).
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boron/epoxy, or steel. The span–depth ratio had a considerable effect on the fundamental frequency of plates made with composite materials, as the curves showed an increasing trend especially at lower a/h ratios. On the other hand, curves corre-
Fig. 6. Comparison of the nondimensional fundamental frequency for cross-ply boron/epoxy square plate under different boundary conditions (h ¼ 2:5 cm).
Fig. 7. Comparison of the nondimensional fundamental frequency for fixed-simple cross-ply square plate composed of different materials (h ¼ 2:5 cm).
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sponding to isotropic materials remained almost constant, regardless of the span– depth ratio. This is attributed to the difference in the elastic properties between fiber filament and matrix materials which leads to a high ratio of in-plane Young’s modulus to transverse shear modulus for the composite plates. 4.3.2. Aspect ratio Figs. 8 and 9 show the effect of aspect ratio on the nondimensional fundamental frequency for a simply supported and a fixed boron/epoxy rectangular plates (b ¼ 50 cm; h ¼ 2:5 cm), respectively. As the aspect ratio increased, the nondimensional fundamental frequency increased because the ratio of area to perimeter decreased, thus the stiffness increased. The results of Fig. 9 suggests that when shear and rotary inertia deformations are included, the rate of change of natural frequency to aspect ratio decreases because the area subjected to shear increases. 4.4. Effect of composite material properties Figs. 10 and 11 show the effect of both fiber orientation and lamination sequence on the fundamental frequencies for boron/epoxy square and rectangular plates, v respectively. For the square plate, there is symmetry for the orientation angle 45 , thus a maximum fundamental natural frequency is obtained; the two maximum components are equal in each direction of the plane of the plate. Whereas for the v rectangular plate the maximum fundamental frequency happens to be between 60 v and 75 , because the dimensions are not equal and the two components resulting from the fiber in the short and long direction are different: the component in the long direction was smaller than that in the short direction. Moreover, it is clear
Fig. 8. Effect of aspect ratio on the nondimensional fundamental frequency for a simply supported cross-ply boron/epoxy plate (b ¼ 50 cm; h ¼ 2:5 cm).
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that natural frequency is increased when adopting a lamination sequence in the v v v v manner h=0 =0 =h rather than 0 =h=h=0 , because the distance between the neutral surface and the fiber components is increased.
Fig. 9. Effect of aspect ratio on the nondimensional fundamental frequency for a fixed supported crossply boron/epoxy plate (b ¼ 50 cm; h ¼ 2:5 cm).
Fig. 10. Effect of fiber orientation and lamination sequence on the nondimensional fundamental frequency for a simply supported boron/epoxy square plate (a ¼ b ¼ 50 cm; h ¼ 2:5 cm).
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Fig. 11. Effect of fiber orientation and lamination sequence on the nondimensional fundamental frequency for a simply supported boron/epoxy rectangular plate (a ¼ 50 cm; b ¼ 75 cm; h ¼ 2:5 cm).
5. Conclusions Based on the results discussed before, the following conclusions can be drawn: 1. Including transverse shear deformation in the dynamic analysis of composite plates resulted in a significant change in the natural frequency of such plates. 2. Rotary inertia terms have a minor influence on the values of natural frequency and therefore could be neglected for the materials used in this study. 3. The effect of shear deformation on the natural frequencies corresponding to higher modes is more pronounced than that on those corresponding to lower modes. 4. The effect of including transverse shear deformation is greater for fully clamped plates than for a simply supported ones. 5. The effect of shear deformation is more pronounced for a plate with low spanto-depth ratio than that for a plate with higher span-to-depth ratio. 6. Fiber orientation (h) has a great effect on the determination of natural frequencies of angle-ply composite plates. For a square plate, maximum fundav mental frequency is obtained when fibbers were oriented at an angle of 45 . v v 7. Composite plates with lamination sequence h=0 =0 =h have higher nondimenv v sional fundamental frequencies than those having 0 =h=h=0 sequence for all values of (h). References [1] Whitney JM. Structural analysis of laminated an isotropic plates, 1st ed. Western Hemisphere. Technical Publishing Company; 1987, p. 263–95.
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[2] Reisner E. The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics 1945;12:69–77. [3] Mindlin RD. Influence of rotary inertia and shear on flexural motion of isotropic, elastic plates. Journal of Applied Mechanics 1951;18:336–43. [4] Wu CI, Vinson JR. Nonlinear oscillations of laminated specially orthotropic plates with clamped and simply supported edges. Journal of Acoustical Society of America 1971;49(5):1561–8. [5] Warburton G. The vibration of rectangular plates. Proceeding of the Institute of Mechanical Engineering 1954;168:371–82. [6] Hearmon R. The frequency of flexural vibrations of rectangular orthotropic plates with clamped or simply supported edges. Journal of Applied Mechanics 1959;26:537–42. [7] Bert CW, Chen TC. Effect of shear deformation on vibration of antisymmetric angle-ply laminated rectangular plate. Institute Journal of Solid Structures 1977;14:265–473. [8] Yang PC, Norris GH, Stavsky Y. Elastic wave propagation in hetrogeneous plates. International Journal of Solid and Structures 1996;2:1665–84. [9] Bhimaraddi A, Stevens LK. A higher order theory for free vibration of orthotropic, homogeneous, and laminated rectangular plates. Journal of Applied Mechanics 1984;51:95–8. [10] Reddy JN. A simple higher-order theory for laminated composite plates. Journal of Applied Mechanics 1984;51(84):745–52. [11] Mirza S, Li N. Analytical approach to free vibration of sandwich plates. AIAA Journal 1995;33(12):1988–90. [12] Marco DS, Ugo IC. Analysis of thick multilayered anisotropic plates by a higher order plate element. AIAA Journal 1995;33:2435–7. [13] Whitney JM, Pagano NJ. Shear deformation in hetrogeneous anisotropic plate. Journal of Applied Mechanics 1970;37:1031–6. [14] Vinson JR, Chou TW. Composite materials and their use in structures. Essex, UK: Applied Science Publishers Ltd; 1975, p. 247–309. [15] Abramovich H. Shear deformation and rotary inertia effects on vibration of composite beams. Composite Structures 1992;20:165–73. [16] Jones RM. Mechanics of composite materials. Washington, DC, USA: McGraw-Hill; 1975.