Free vibration of basalt fiber reinforced polymer (FRP) laminated variable thickness plates with intermediate elastic support using finite strip transition matrix (FSTM) method

Wael A. Altabey1

1International Institute for Urban Systems Engineering, Southeast University, Nanjing, 210096, China

1Department of Mechanical Engineering, Faculty of Engineering, Alexandria University, Alexandria, 21544, Egypt

Journal of Vibroengineering, Vol. 19, Issue 4, 2017, p. 2873-2885. https://doi.org/10.21595/jve.2017.18154
Received 5 January 2017; received in revised form 29 January 2017; accepted 12 March 2017; published 30 June 2017

Copyright © 2017 JVE International Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Abstract.

This paper presents a semi-analytical method to investigate the effect of intermediate elastic support on the natural frequencies of basalt fiber reinforced polymer (FRP) laminated, variable thickness plates based on the finite strip transition matrix (FSTM) method. The plate has a uniform thickness in x direction and varying thickness hy in y direction. A singular value decomposition algorithm is employed at the intermediate support to eliminate the dependence of the solution of the first span on another span. By a new treatment of the intermediate line support, the dimension of the final matrix of the general solution will be the same as that of plates without intermediate support. Numerical results for different combinations of classical boundary conditions at the plate edges with different elastic restraint coefficients (KT) for intermediate elastic support are presented to obtain the first six frequency parameters. The illustrated results are in excellent agreement with solutions available in the literature, thus validating the accuracy and reliability of the proposed technique.

Keywords: free vibration, finite strip transition matrix, variable thickness plate, basalt FRP.

1. Introduction

Continuous plates and plates with intermediate stiffeners are very common in many engineering fields such as aerospace industries, civil engineering and marine engineering. Exact solutions of such plates are available only for some boundary conditions. For example, if two opposite sided are simply supported and the other sides may be any combinations of elastic, clamped and free, a Levy-type solution can be obtained for rigid stiffners [1].

In general, a numerical approach or an approximate method must be employed to find the natural frequencies and the mode shapes for different combinations of the boundary conditions. The vibration of plates with intermediate support attracts many researchers.

Xiang and Liew [1] presented an exact (Levy-type) solution for multi-span rectangular mindlin plates with two opposite edges simply supported. Abrate and Foster [2] used Rayleigh-Ritz method to investigate the free vibrations of rectangular composite plates with arbitrary number of intermediate line supports. Cheung and Zhou [3] used Rayleigh-Ritz method to study vibrations of symmetric laminated rectangular plates with intermediate supports. Liew and Wang [4] studied vibration of skew plates with internal line support using the pb-2 Rayleigh-Ritz method. Cheung and Zhou [5] used a set of static beam functions to analyze the vibration of orthotropic rectangular plates with intermediate elastic support. Xiang et al. [6] reported free vibration behavior of laminated seven composite plates based on the nth order shear deformation theory and this theory satisfies the zero transverse shear stress boundary conditions. Thai and Kim [7] examined the free vibration responses of laminated composite plates using two variables refined plate theory. Ovesy and Fazilati [8] employed the third order shear deformation theory for buckling and free vibration finite strip analysis of composite plates with cutout based on two different modeling approaches (semi-analytical and spline method). Dozio [9] presented accurate upper-bound solutions for free in-plane vibrations of single-layer and symmetrically laminated rectangular composite plates with an arbitrary combination of clamped and free boundary conditions. He used Rayleigh-Ritz method to calculate in-plane natural frequencies and modes shapes with a simple, stable and computationally efficient set of trigonometric functions. Asadi et al. [10] investigated the vibration analysis of axially moving functionally graded plates with internal line supports and temperature-dependent properties using harmonic differential quadrature method. They studied plate vibration which was subjected to static in-plane forces while out-of-plane loading was dynamic. Al-Tabey [11] presented the finite strip transition matrix technique (FSTM) and semi-analytical method to obtain the natural frequencies and mode shapes of symmetric angle-ply Graphite/Epoxy laminated composite variable thickness rectangular plate with classical boundary conditions (SSFF). Thinh et al. [12] examined the bending and vibration analysis of multi-folding laminate composite plate using finite element method based on the first order shear deformation theory (FSDT). They investigated the effect of folding angle on deflections, natural frequencies and transient displacement response for different boundary conditions of the plate. Ducceschi [13] studied the nonlinear vibrations of thin rectangular plates by developing of a numerical code able to simulate without restrictions. He described the large spectrum of dynamical features by the von Kármán equations. Yadav et al. [14] presented the free vibration analysis of stiffened isotropic plate by means of finite element method. They studied the effect of different boundary conditions, stiffeners location, thickness ratio, stiffener thickness to plate thickness and aspect ratio on the vibration analysis of stiffened isotropic plate, and calculated natural frequencies using Block-Lanczos algorithm. Küçükrendeci and Morgül [15] investigated the effects of elastic boundary conditions on the linear free vibrations. They found that frequency parameters increase when boron/epoxy used.

Semi-analytical methods are welcomed in the literature as an alternative to the exact solution. In this paper a semi-analytical method, the finite strips transition matrix (FSTM) method [16] has been employed to investigate the free vibration of basalt fiber reinforced polymer (FRP) laminated variable thickness rectangular plates with intermediate elastic support as shown in Fig. 1. A new treatment of the elastic intermediate boundary conditions using a singular values decomposition algorithm is introduced in this paper. Four different classical boundary conditions are considered in the analysis with different elastic restraint coefficients (KT) for intermediate elastic support to obtain the first six frequency parameters, some new data which can serve as the benchmark for further research are presented in this work.

2. Theory and formulation

2.1. Governing equations

The partial differential equation governing the vibration of symmetrically, angle-ply laminated, variable thickness, rectangular plates under the assumption of the classical deformation theory in terms of the plate deflection wo(x,y,t) is given by [17]:

(1)
D 11 4 w o x 4 + 4 D 16 4 w o x 3 y + 2 ( D 12 + 2 D 66 ) 4 w o x 2 y 2 + 4 D 26 4 w o x y 3 + D 22 4 w o y 4
              = - m o h y h o 2 w o t 2 .

Or in contraction form:

(2)
D 11 W x x x x + 4 D 16 W x x x y + 2 ( D 12 + 2 D 66 ) W x x y y + 4 D 26 W x y y y + D 22 W y y y y
              = - m o h y h o W t t ,

where: mo=ρho, the flexural rigidities Dij of the plate are given by:

(3)
D i j = 1 3 h 3 ( y ) h o 3 k = 1 n ( Q ¯ i j ) k ( h o k 3 - h o k - 1 3 ) ,               i , j = 1 ,   2 ,   3 .

where hok is the distance from the middle-plane of the plate according to ho to the bottom of the hoth layer as shown in Fig. 1. And Qijk¯ are the plane stress transformed reduced stiffness coefficients of the lamina in the laminate cartesian coordinate system. They are related to reduce stiffness coefficients of the lamina in the material axes of lamina Qijk by proper coordinate relationships they can be expressed in terms of the engineering notations as:

(4)
Q i j = Q 11 Q 12 Q 13 Q 12 Q 22 Q 23 Q 13 Q 23 Q 66 = E 11 1 - υ 12 υ 21 υ 21 E 11 1 - υ 12 υ 21 0 υ 21 E 11 1 - υ 12 υ 21 E 22 1 - υ 21 υ 12 0 0 0 G 12 ,

where: E11, E22 are the longitudinal and transverse Young’s moduli parallel and perpendicular to the fiber orientation, respectively and G12 is the plane shear modulus of elasticity, υ12 and υ21 are the Poisson coefficients.

Fig. 1. The geometrical model of Basalt FRP laminated variable thickness rectangular plate with intermediate elastic support

 The geometrical model of Basalt FRP laminated variable thickness rectangular plate  with intermediate elastic support

The substitution of Eq. (3) into Eq. (2) and after some derivation steps [18], the governing Partial differential equation can be written in form:

(5)
D 11 h 3 ( y ) h o 3 W x x x x + 2 ( D 12 + 2 D 66 ) h o 3 h 3 ( y ) y W x x y + 2 ( D 12 + 2 D 66 ) h o 3 h 3 ( y ) W x x y y
              + D 16 h 3 ( y ) h o 3 W x x x y + 4 D 26 h o 3 2 h 3 ( y ) y 2 W x y + 4 D 26 h o 3 h 3 ( y ) W x y y y + 8 D 26 h o 3 h 3 ( y ) y W x y y
              + D 22 h o 3 2 h 3 ( y ) y 2 W y y + D 22 h o 3 h 3 y W y y y y + 2 D 22 h o 3 h 3 y y W y y y = - m o h y h o W t t .

The equation of motion Eq. (5) can be normalized using the non-Dimensional variables ξ and η as follows:

(6)
ψ 1 1 a 4 W ξ ξ ξ ξ + 2 ψ 2 h 3 ( η ) 1 a 2 b h 3 ( η ) η W ξ ξ η + 2 ψ 2 1 a 2 b 2 W ξ ξ η η + ψ 3 1 a 3 b W ξ ξ ξ η
              + 4 ψ 4 1 a b 3 W ξ η η η + 1 a b 4 ψ 4 h 3 ( η ) 2 h 3 ( η ) η 2 W ξ η + 8 ψ 4 h 3 ( η ) 1 a b 2 h 3 ( η ) η W ξ η η
              + 1 b 2 1 h 3 ( η ) 2 h 3 ( η ) η 2 W η η + 1 b 4 W η η η η + 2 h 3 ( η ) 1 b 3 h 3 ( η ) η W η η η = - m o D 22 h o 2 h 2 η W t t ,

where β=a/b is the aspect ratio, and:

ξ = x a ,           η = y b ,           ψ 1 = D 11 D 22 ,           ψ 2 = ( D 12 + 2 D 66 ) D 22 ,           ψ 3 = D 16 D 22 ,           ψ 4 = D 26 D 22 .

2.2. Boundary conditions

In this paper, the boundary conditions along the x-direction and y-direction are considered by any combinations of the classical boundary conditions such as simply supported, clamped, or free. For the purpose of clarity, the symbol SFSC for example, means a plate having simply supported, free, simply supported and clamped edges at the boundaries, x= 0, y=b, x=a, and y= 0, respectively (start anticlockwise from the left edge of the plate). In the numerical computations, four different classical boundary conditions are considered in the analysis SSSS, CCCC, SSFF and CCFF as shown in Fig. 2.

Fig. 2. Representation of different support condition for the analysis

 Representation of different support condition for the analysis

Simply supported edges:

(7)
w o ξ = 0,1 = 0 ,           1 a 2 2 w o ξ 2 ξ = 0,1 = 0 ,               w o η = 0,1 = 0 ,             1 b 2 2 w o η 2 η = 0,1 = 0 .

Clamped supported edges:

(8)
w o ξ = 0,1 = 0 ,           1 a w o ξ ξ = 0,1 = 0 ,             w o η = 0,1 = 0 ,           1 b w o η y = 0,1 = 0 .

Free edges:

(9)
1 a 3 3 w o ξ 3 ξ = 0,1 = 0 ,           1 a 2 2 w o ξ 2 ξ = 0,1 = 0 ,             1 b 3 3 w o η 3 η = 0,1 = 0 ,           1 b 2 2 w o η 2 η = 0,1 = 0 .

2.2.1. Intermediate elastic line support

Since the treatment of the intermediate elastic line support conditions are the main objective of this paper we presented it in more details. At the intermediate elastic line support, y=b/2, the displacement must vanish and the moment must be continuous, i.e.:

(10)
K T w o = - 2 ψ 3 1 a 3 3 w o ξ 3 - 1 b 3 3 w o η 3 - ψ 5 1 a 2 b 2 w o ξ 2 η - 4 ψ 4 1 a b 2 3 w o ξ η 2 ,
(11)
1 b w o η   η = 1 - / 2 = 1 b w o η   η = 1 + / 2 ,
(12)
- 2 ψ 3 1 a 3 3 w o ξ 3 - 1 b 3 3 w o η 3 - ψ 5 1 a 2 b 2 w o ξ 2 η - 4 ψ 4 1 a b 2 3 w o ξ η 2 β = 1 - / 2
              = - 2 ψ 3 1 a 3 3 w o ξ 3 - 1 b 3 3 w o η 3 - ψ 5 1 a 2 b 2 w o ξ 2 η - 4 ψ 4 1 a b 2 3 w o ξ η 2 β = 1 + / 2 ,

where: KT is the elastic restraint coefficient given by: KT=Tb/2b3/D22, T is translational stiffness per unit length, ψ5=(D12+4D66)/D22.

2.3. Finite strip transition matrix (FSTM) method

The method is made when such a shape function is not conveniently obtained in case of discussing the plate problems by series. The plate may be divided into N discrete longitudinal strips spanning between supports as shown in Fig. 3. Simple basic displacement interpolation functions may then be used to represent displacement field within and between individual strips.

For a plate striped in the ξ-direction as shown in Fig. 3, the shape function W(ξ,η,t) may be assumed in the form:

(13)
W ξ , η , t = i = 0 N X i ξ Y i η e i ω t ,

where: Yi(η) is unknown function to be determined and Xi(ξ) is chosen a priori, the basic function in ξ-direction. The most commonly used is the Eigen function obtained from the solution of the differential equation of a beam vibration under the prescribed conditions of the stripe at ξ= 0 and ξ= 1. By substituting of Eq. (13) into Eq. (6), multiplying both sides by Xj(x) and after some derivatives, we can find:

(14)
i = 0 N j = 0 M β 4 f 3 ( η ) Y i , η η η η + 2 β 3 a f 1 ( η ) f 3 ( η ) Y i , η η η
              + 2 ψ 2 β 2 f 3 η c i j a i j + 8 ψ 4 β 2 a f 1 η f 3 η b i j a i j + β 2 a 2 f 2 η f 3 η Y i , η η
              + 2 ψ 2 β a f 1 η f 3 η c i j a i j + ψ 3 β f 3 η d i j a i j + 4 ψ 4 β a 2 f 2 η f 3 η b i j a i j + 4 ψ 4 β 3 f 3 η b i j a i j Y i , η
              + ψ 1 f 3 η e i j a i j - Ω 2 Y i = 0 ,

where:

Ω 2 = m o h η ω 2 a 4 h o D 22 ,           f 1 η = 1 h 3 η h 3 η η ,           f 2 η = 1 h 3 η 2 h 3 η η 2 ,
f 3 η = h o 2 h 2 η ,           a i j = 0 1 X i X j d ξ ,           b i j = 0 1 X j X i , ξ d ξ ,
c i j = 0 1 X j X i , ξ ξ d ξ ,           d i j = 0 1 X j X i , ξ ξ ξ d ξ ,           e i j = 0 1 X j X i , ξ ξ ξ ξ d ξ .

Fig. 3. Finite strip simulation on plate

 Finite strip simulation on plate

From the beam Eigen function orthogonality, aij=eij=0 for ij, this agree for all types of boundary conditions except for plates having free edges in the ξ-direction. The governing differential Eq. (14) can be written in form:

(15)
i = 0 N j = 0 M E i j Y i ' ' ' ' + U 1 i j U 0 i j Y i ' ' ' + U 2 i j U 0 i j Y i ' ' + U 3 i j U 0 i j Y i ' + U 4 i j - λ 2 U 0 i j Y i = 0 ,

where:

U 0 i j = β 4 t 1 η E i j ,             U 1 i j = 2 β 3 a t 2 η E i j ,
U 2 i j = 2 ψ 2 β 2 t 1 η c i j a i j + 8 ψ 4 β 2 a t 2 η b i j a i j + β 2 a 2 t 3 η ,
U 3 i j = 2 ψ 2 β a t 2 η c i j a i j + ψ 3 β t 1 η d i j a i j + 4 ψ 4 β a 2 t 3 η b i j a i j + 4 ψ 4 β 3 t 1 η b i j a i j ,
U 4 i j = ψ 1 t 1 η e i j a i j ,

and [Eij]=i×j unit matrix.

A system of coupled fourth order equations are obtained which can be reduced to a system of first order differential equation:

(16)
d d η Y k i j = A i k Y k i j ,

where: k= 1, 2, 3,…, N, i= 1, 2, 3,…, N, j= 1, 2, 3,…, M, coefficients of the matrix Aik in equation, in general, are functions of η and the eigenvalue parameter Ω. The vector Yk is given by:

(17)
Y k = Y ¯ 1 Y ¯ 2 Y ¯ i Y ¯ N ,

where:

(18)
Y ¯ i = Y i Y i ' Y i ' ' Y i ' ' ' .

The relation under which the continuity conditions between the striped plates are satisfied may be expressed as:

(19)
Y i j = T i j Y i - 1 j ,

where: Tij is called the transition matrix of the strip i while Yij and Yi-1j are the nodal vectors of the boundaries i and i-1. The solution is found using 2N-number of initial vectors Y0 at η= 0. The transition matrix, Eq. (19) is applied across the stripped plate until just before the intermediate support at y=b/2, η=1/2 is reached. Thus, 2N-number of solutions Si can be obtained. The true solutions S can be written as a linear combination of these solutions as:

(20)
S = i = 1 2 N C i S i ,

where Ci are arbitrary constants, these constants can be determined by satisfying 2N-number of boundary conditions at η= 1/2 in Eqs. (10) and (12) of the intermediate elastic line support. And the matrix S forms a standard eigenvalue problem. The natural frequencies of the system can be obtained from the conditions that the detainment of the S must vanish. An iteration algorithm is implemented to compute the natural frequency of the system and hence the constants Ci, i= 1, 2, 3,…, 2N.

3. Results and discussion

In this section, the finite strip transition matrix (FSTM) approach is employed to investigate the free vibration of symmetrically laminated, angle-ply, variable thickness rectangular plates with intermediate elastic support in one direction with different elastic restraint coefficient (KT). The basalt FRP laminate composite plate was manufactured using five symmetrically, angle-ply, laminates with the fiber orientations [45°/–45°/45°/–45°/45°] of basalt fiber and a polymer resin matrix. The corresponding elastic modulus values were E1= 96.74 GPa, E2=E3= 22.55 GPa, and the Shear modulus values were G1=G3= 10.64 GPa, G2= 8.73 GPa. Poisson coefficients were υ1=υ3= 0.3, υ2= 0.6 and the density was 2700 kg/m3.

The frequency parameter Ω is evaluated in non-dimensional form, expressed as:

(21)
Ω = m o h ( η ) ω 2 a 4 h o D 22 .

The plate with linear variable thickness, hy is used (see Appendix) in non-dimensional form:

(22)
h η = 1 + Δ η ,

where: Δ is the tapered ratio of plate given by Δ=(hb-ho)/ho, ho is the thickness of the plate at η= 0 and (hb) is the thickness of the plate at η= 1.

3.1. Convergence study and accuracy

In this subsection, a convergence investigation is carried out for the proposed method, first six frequencies are calculated and compared with available results in literatures. Table 1 presents a convergence and comparison study for isotropic, square (β= 1.0), uniform thickness (Δ= 0) plates with a mid-line support in each direction, the plate material has mechanical properties of υ1=υ2= 0.3, D11=D22=D=Eh3/121-υ2, D66=1-υD/2. In this study the non-dimensional frequency parameter Ω become Ω=ρhω2a4/D1/2. Two different classical boundary conditions are considered in the computational SSSS and CCCC. The computational results which are compared with values available from literatures [5, 19-21]. A very close agreement is observed.

Table 2 presented a convergence and comparison study for fully simply supported (SSSS) and fully clamped (CCCC) square (β= 1.0), uniform thickness (Δ= 0) plates with elastic foundation support. The elastic coefficient is taken equal to 500, 1390.2 for SSSS and CCCC respectively. The plates are manufactured from E-glass/ epoxy material with the following properties are υ1=υ3=0.23, D=Eh3/121-υ2, D66=1-υD/2. In this study the non-dimensional frequency parameter Ω become Ω=ρhω2a4/πD1/2 and foundation elastic restraint coefficient is given by KT=kfa4/D. From Table 2 it can be observed that the computational results are in an excellent agreement with exact frequency parameters presented in References [22, 23] and stable and fast convergence can be achieved with only a few terms of series solution (N= 3 to 7). This validates the precision of the semi-analytical finite strip transition matrix (FSTM) technique.

Table 1. Convergence study of the first six frequency parameters of the isotropic square plates with a mid-line support in each direction

N
Ω 1
Ω 2
Ω 3
Ω 4
Ω 5
Ω 6
SSSS
1
78.866
94.506
94.506
108.125
197.311
197.311
2
78.887
94.529
94.529
108.159
197.324
197.324
4
78.910
94.546
94.546
108.184
197.350
197.350
6
78.928
94.568
94.568
108.211
197.369
197.369
Ref [5]
78.957
94.590
94.590
108.240
197.392
197.392
Ref [19]
78.96
94.68
94.72
108.44
197.40
198.96
Ref [20]
78.958
94.826
94.826
108.41
197.50
197.50
Ref [21]
78.957
94.585
94.585
108.22
197.39
197.33
CCCC
2
108.222
127.346
127.346
144.026
242.386
242.386
4
108.243
127.365
127.365
144.048
242.758
242.758
5
108.259
127.382
127.382
144.071
242.773
242.773
7
108.282
127.398
127.398
144.099
242.801
242.801
Ref [5]
108.299
127.417
127.417
144.109
242.818
243.778

Table 2. Convergence study of the first four frequency parameters of the isotropic square plates with elastic foundation

N
K T
Ω 1
Ω 2
Ω 3
Ω 4
SSSS
2
500
3.0210
5.4828
5.4828
8.3017
3
500
3.0211
5.4836
5.4836
8.3019
4
500
3.0212
5.4842
5.4842
8.3023
7
500
3.0213
5.4847
5.4847
8.3029
Ref [22]
500
3.0214
5.4850
5.4850
8.3035
Ref [23]
500
3.0216
5.4846
5.4846
8.3051
CCCC
2
1390.2
5.2515
8.3785
8.3785
11.506
3
1390.2
5.2538
8.3811
8.3811
11.528
6
1390.2
5.2554
8.3843
8.3843
11.553
7
1390.2
5.2573
8.3879
8.3879
11.568
Ref [22]
1390.2
5.2588
8.4322
8.4322
11.674
Ref [23]
1390.2
5.2438
8.3129
8.3129
11.546

3.2. Laminated variable thickness plate with intermediate elastic line support

The results from the numerical computations using FSTM approach will be discussed here. Table 3 presents the first six frequencies of a symmetrically, angle-ply, laminated, variable thickness rectangular plate with intermediate elastic line support in one direction as shown in Fig. 1. The aspect ratio of the plate is β= 0.5 and tapered ratio of the plate thickness is Δ= 0.5. Four type of classical boundary conditions (SSSS, CCCC, SSFF and CCFF) as shown in Fig. 2 and different elastic restraint coefficients KT of intermediate elastic line support are considered in the computations to study the effect of intermediate elastic support on the natural frequencies of basalt (FRP) laminated variable thickness rectangular plate. The locations of the intermediate elastic line support is at mid-line of the plate.

Fig. 4. Variation of non-dimensional frequencies parameter (Ω) with elastic restraint coefficient (KT)

 Variation of non-dimensional frequencies parameter (Ω) with elastic restraint coefficient (KT)

The effect of intermediate elastic support on the non-dimensional frequencies of laminated variable thickness rectangular plate is computed and plotted in Figs. 4 and 5. From this figures, it is observed that the first six frequencies increase with the increasing of the value of elastic restraint coefficient (KT) as shown in Fig. 4. Fig. 5 shows the vibration behaviour of the variable thickness rectangular plate under varying elastic restraint coefficient (KT). As shown in the Fig. 5, the increasing values of frequencies with small elastic restraint coefficient (KT) are higher than the increasing values of frequencies with highest one, and the frequencies at high values of elastic restraint coefficient are almost constant.

After the value of KT increases from 50 onwards, the non-dimensional frequencies parameter are fast raised till value of KT reached 104 and after this value there is almost negligible change in value of Non-dimensional frequencies parameter.

Fig. 5. Variation of non-dimensional frequencies parameter (Ω) with different mode number and elastic restraint coefficient (KT)

 Variation of non-dimensional frequencies parameter (Ω) with different mode number  and elastic restraint coefficient (KT)

Fig. 6. Variation of non-dimensional frequencies parameter (Ω) with elastic restraint coefficient (KT) and boundary conditions

 Variation of non-dimensional frequencies parameter (Ω)  with elastic restraint coefficient (KT) and boundary conditions

Fig. 7. Variation of non-dimensional frequencies parameter (Ω) with different mode number and boundary conditions

 Variation of non-dimensional frequencies parameter (Ω)  with different mode number and boundary conditions

Influence of four different support conditions (SSSS, CCCC, SSFF and CCFF) on the vibration behavior of a symmetrically, angle-ply, laminated, variable thickness rectangular plate is computed and plotted in Figs. 6 and 7, From this figures, it can be seen that the frequencies are showing higher and lower value at fully clamped (CCCC) and semi-simply supported (SSFF) condition, respectively. The other two boundary conditions (SSSS and CCFF) are showing an intermediate value. As shown in the Fig. 6, the non-dimensional frequencies increase with the increase of the elastic restraint coefficient (KT) for all kind of support conditions (SSSS, CCCC, SSFF and CCFF).

Table 3. The first six frequencies of symmetrically, angle-ply, laminated, variable thickness rectangular plate with intermediate elastic line support for different elastic restraint coefficients, (Δ= 0.5), (β= 0.5)

K T
Ω 1
Ω 2
Ω 3
Ω 4
Ω 5
Ω 6
SSSS
50
22.1450
36.2210
53.5870
78.2360
105.5870
138.6970
150
34.6580
48.7340
66.1000
90.7490
118.1000
151.2100
400
45.8453
59.9213
77.2873
101.9363
129.2873
162.3973
750
55.0692
69.1452
86.5112
111.1602
138.5112
171.6212
1500
62.4709
76.5469
93.9129
118.5619
145.9129
179.0229
2500
67.5270
81.6030
98.9690
123.6180
150.9690
184.0790
5000
70.7832
84.8592
102.2252
126.8742
154.2252
187.3352
10000
72.7065
86.7825
104.1485
128.7975
156.1485
189.2585
1E+06
73.1195
87.1955
104.5615
129.2105
156.5615
189.6715
CCCC
50
28.4310
46.5025
68.7979
100.4437
135.5584
178.0668
150
40.9440
59.0155
81.3109
112.9567
148.0714
190.5798
400
52.1313
70.2028
92.4983
124.1440
159.2587
201.7672
750
61.3551
79.4267
101.7221
133.3678
168.4826
210.9910
1500
68.7569
86.8284
109.1239
140.7696
175.8843
218.3928
2500
73.8130
91.8845
114.1800
145.8257
180.9404
223.4489
5000
77.0692
95.1407
117.4361
149.0819
184.1966
226.7050
10000
78.9925
97.0640
119.3594
151.0052
186.1199
228.6283
1E+06
79.4055
97.4770
119.7724
151.4182
186.5329
229.0413
SSFF
50
12.2750
20.0773
29.7033
43.3663
58.5270
76.8799
150
24.7880
32.5903
42.2163
55.8793
71.0400
89.3929
400
35.9753
43.7777
53.4036
67.0666
82.2273
100.5802
750
45.1992
53.0015
62.6275
76.2905
91.4511
109.8041
1500
52.6009
60.4033
70.0293
83.6922
98.8529
117.2058
2500
57.6570
65.4594
75.0853
88.7483
103.9090
122.2619
5000
60.9132
68.7155
78.3415
92.0045
107.1652
125.5181
10000
62.8365
70.6388
80.2648
93.9278
109.0885
127.4414
1E+06
63.2495
71.0518
80.6778
94.3408
109.5015
127.8544
CCFF
50
19.5928
32.0466
47.4112
69.2194
93.4182
122.7123
150
32.1058
44.5596
59.9242
81.7324
105.9312
135.2253
400
43.2931
55.7469
71.1115
92.9197
117.1185
146.4127
750
52.5170
64.9707
80.3353
102.1436
126.3424
155.6365
1500
59.9187
72.3725
87.7371
109.5453
133.7442
163.0383
2500
64.9748
77.4286
92.7932
114.6014
138.8002
168.0944
5000
68.2310
80.6848
96.0494
117.8576
142.0564
171.3505
10000
70.1543
82.6081
97.9727
119.7809
143.9797
173.2738
1E+06
70.5673
83.0211
98.3857
120.1939
144.3927
173.6868

4. Conclusions

The work reported in this paper employs an efficient semi-analytical method for analysing the free vibration of thin basalt fiber reinforced polymer (FRP) laminated variable thickness rectangular plates with intermediate elastic support. A singular value decomposition algorithm has been employed to treat the intermediate support and reduce the dependence of the solutions at the intermediate elastic support. It is observed that the first six frequencies increase with increasing values of elastic restraint coefficient (KT) of intermediate elastic support, and the rate of increasing is different. It was found that the increasing rates of frequencies with a small elastic restraint coefficient (KT) are higher than the increasing rates of frequencies with highest one, and the frequencies at high values of elastic restraint coefficient are almost constant. On other hand, it observed that the frequencies values were influenced with change of the plate edges support between four different support conditions, for all first six frequencies are showing higher and lower value at fully clamped (CCCC) and semi-simply supported (SSFF) condition, respectively, the other two boundary conditions (SSSS and CCFF) are showing an intermediate value. Accuracy and convergence of solution was examined by comparing the numerical results obtained by the present method with those previously published. The results are in excellent agreement with results from the literature.

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