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. Altabey^{1}
^{1}International Institute for Urban Systems Engineering, Southeast University, Nanjing, 210096, China
^{1}Department of Mechanical Engineering, Faculty of Engineering, Alexandria University, Alexandria, 21544, Egypt
Journal of Vibroengineering, Vol. 19, Issue 4, 2017, p. 28732885.
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
JVE Conferences
This paper presents a semianalytical 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 $h\left(y\right)$ 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 (${K}_{T}$) 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 Levytype 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 (Levytype) solution for multispan rectangular mindlin plates with two opposite edges simply supported. Abrate and Foster [2] used RayleighRitz method to investigate the free vibrations of rectangular composite plates with arbitrary number of intermediate line supports. Cheung and Zhou [3] used RayleighRitz 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 pb2 RayleighRitz 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 $n$th 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 (semianalytical and spline method). Dozio [9] presented accurate upperbound solutions for free inplane vibrations of singlelayer and symmetrically laminated rectangular composite plates with an arbitrary combination of clamped and free boundary conditions. He used RayleighRitz method to calculate inplane 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 temperaturedependent properties using harmonic differential quadrature method. They studied plate vibration which was subjected to static inplane forces while outofplane loading was dynamic. AlTabey [11] presented the finite strip transition matrix technique (FSTM) and semianalytical method to obtain the natural frequencies and mode shapes of symmetric angleply Graphite/Epoxy laminated composite variable thickness rectangular plate with classical boundary conditions (SSFF). Thinh et al. [12] examined the bending and vibration analysis of multifolding 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 BlockLanczos 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.
Semianalytical methods are welcomed in the literature as an alternative to the exact solution. In this paper a semianalytical 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 (${K}_{T}$) 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, angleply laminated, variable thickness, rectangular plates under the assumption of the classical deformation theory in terms of the plate deflection ${w}_{o}(x,y,t)$ is given by [17]:
$={m}_{o}\frac{h\left(y\right)}{{h}_{o}}\frac{{\partial}^{2}{w}_{o}}{\partial {t}^{2}}.$
Or in contraction form:
$={m}_{o}\frac{h\left(y\right)}{{h}_{o}}{W}_{tt},$
where: ${m}_{o}=\rho {h}_{o}$, the flexural rigidities ${D}_{ij}$ of the plate are given by:
where ${h}_{ok}$ is the distance from the middleplane of the plate according to ${h}_{o}$ to the bottom of the ${h}_{oth}$ layer as shown in Fig. 1. And $\overline{{Q}_{ij}^{k}}$ 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 ${Q}_{ij}^{k}$ by proper coordinate relationships they can be expressed in terms of the engineering notations as:
where: ${E}_{11}$, ${E}_{22}$ are the longitudinal and transverse Young’s moduli parallel and perpendicular to the fiber orientation, respectively and ${G}_{12}$ is the plane shear modulus of elasticity, ${\upsilon}_{12}$ and ${\upsilon}_{21}$ are the Poisson coefficients.
Fig. 1. 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:
$+{D}_{16}\frac{{h}^{3}\left(y\right)}{{h}_{o}^{3}}{W}_{xxxy}+\left(\frac{4{D}_{26}}{{h}_{o}^{3}}\frac{{\partial}^{2}{h}^{3}\left(y\right)}{\partial {y}^{2}}\right){W}_{xy}+\frac{4{D}_{26}}{{h}_{o}^{3}}{h}^{3}\left(y\right){W}_{xyyy}+\frac{8{D}_{26}}{{h}_{o}^{3}}\frac{\partial {h}^{3}\left(y\right)}{\partial y}{W}_{xyy}$
$+\left(\frac{{D}_{22}}{{h}_{o}^{3}}\frac{{\partial}^{2}{h}^{3}\left(y\right)}{\partial {y}^{2}}\right){W}_{yy}+\frac{{D}_{22}}{{h}_{o}^{3}}{h}^{3}\left(y\right){W}_{yyyy}+\frac{2{D}_{22}}{{h}_{o}^{3}}\frac{\partial {h}^{3}\left(y\right)}{\partial y}{W}_{yyy}={m}_{o}\frac{h\left(y\right)}{{h}_{o}}{W}_{tt}.$
The equation of motion Eq. (5) can be normalized using the nonDimensional variables $\xi $ and $\eta $ as follows:
$+4{\psi}_{4}\frac{1}{a{b}^{3}}{W}_{\xi \eta \eta \eta}+\frac{1}{ab}\frac{4{\psi}_{4}}{{h}^{3}\left(\eta \right)}\frac{{\partial}^{2}{h}^{3}\left(\eta \right)}{\partial {\eta}^{2}}{W}_{\xi \eta}+\frac{8{\psi}_{4}}{{h}^{3}\left(\eta \right)}\frac{1}{a{b}^{2}}\frac{\partial {h}^{3}\left(\eta \right)}{\partial \eta}{W}_{\xi \eta \eta}$
$+\frac{1}{{b}^{2}}\frac{1}{{h}^{3}\left(\eta \right)}\frac{{\partial}^{2}{h}^{3}\left(\eta \right)}{\partial {\eta}^{2}}{W}_{\eta \eta}+\frac{1}{{b}^{4}}{W}_{\eta \eta \eta \eta}+\frac{2}{{h}^{3}\left(\eta \right)}\frac{1}{{b}^{3}}\frac{\partial {h}^{3}\left(\eta \right)}{\partial \eta}{W}_{\eta \eta \eta}=\frac{{m}_{o}}{{D}_{22}}\frac{{h}_{o}^{2}}{{h}^{2}\left(\eta \right)}{W}_{tt},$
where $\beta =a/b$ is the aspect ratio, and:
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
Simply supported edges:
Clamped supported edges:
Free edges:
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.:
${\left.=2{\psi}_{3}\frac{1}{{a}^{3}}\frac{{\partial}^{3}{w}_{o}}{\partial {\xi}^{3}}\frac{1}{{b}^{3}}\frac{{\partial}^{3}{w}_{o}}{\partial {\eta}^{3}}{\psi}_{5}\frac{1}{{a}^{2}b}\frac{{\partial}^{2}{w}_{o}}{\partial {\xi}^{2}\partial \eta}4{\psi}_{4}\frac{1}{a{b}^{2}}\frac{{\partial}^{3}{w}_{o}}{\partial \xi \partial {\eta}^{2}}\right}_{\beta ={1}^{+}/2},$
where: ${K}_{T}$ is the elastic restraint coefficient given by: ${K}_{T}={T}_{b/2}{b}^{3}/{D}_{22}$, $T$ is translational stiffness per unit length, ${\psi}_{5}=({D}_{12}+4{D}_{66})/{D}_{22}$.
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 $\xi $direction as shown in Fig. 3, the shape function $W(\xi ,\eta ,t)$ may be assumed in the form:
where: ${Y}_{i}\left(\eta \right)$ is unknown function to be determined and ${X}_{i}\left(\xi \right)$ is chosen a priori, the basic function in $\xi $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 $\xi =$ 0 and $\xi =$ 1. By substituting of Eq. (13) into Eq. (6), multiplying both sides by ${X}_{j}\left(x\right)$ and after some derivatives, we can find:
$+\left(\frac{2{\psi}_{2}{\beta}^{2}}{{f}_{3}\left(\eta \right)}\frac{{c}_{ij}}{{a}_{ij}}+8{\psi}_{4}{\beta}^{2}a\frac{{f}_{1}\left(\eta \right)}{{f}_{3}\left(\eta \right)}\frac{{b}_{ij}}{{a}_{ij}}+{\beta}^{2}{a}^{2}\frac{{f}_{2}\left(\eta \right)}{{f}_{3}\left(\eta \right)}\right){Y}_{i,\eta \eta}$
$+\left(2{\psi}_{2}\beta a\frac{{f}_{1}\left(\eta \right)}{{f}_{3}\left(\eta \right)}\frac{{c}_{ij}}{{a}_{ij}}+\frac{{\psi}_{3}\beta}{{f}_{3}\left(\eta \right)}\frac{{d}_{ij}}{{a}_{ij}}+4{\psi}_{4}\beta {a}^{2}\frac{{f}_{2}\left(\eta \right)}{{f}_{3}\left(\eta \right)}\frac{{b}_{ij}}{{a}_{ij}}+\frac{4{\psi}_{4}{\beta}^{3}}{{f}_{3}\left(\eta \right)}\frac{{b}_{ij}}{{a}_{ij}}\right){Y}_{i,\eta}$
$+\left(\frac{{\psi}_{1}}{{f}_{3}\left(\eta \right)}\frac{{e}_{ij}}{{a}_{ij}}{\mathrm{\Omega}}^{2}\right){Y}_{i}=0,$
where:
${f}_{3}\left(\eta \right)=\frac{{h}_{o}^{2}}{{h}^{2}\left(\eta \right)},{a}_{ij}=\underset{0}{\overset{1}{\int}}{X}_{i}{X}_{j}d\xi ,{b}_{ij}=\underset{0}{\overset{1}{\int}}{X}_{j}{X}_{i,\xi}d\xi ,$
${c}_{ij}=\underset{0}{\overset{1}{\int}}{X}_{j}{X}_{i,\xi \xi}d\xi ,{d}_{ij}=\underset{0}{\overset{1}{\int}}{X}_{j}{X}_{i,\xi \xi \xi}d\xi ,{e}_{ij}=\underset{0}{\overset{1}{\int}}{X}_{j}{X}_{i,\xi \xi \xi \xi}d\xi .$
Fig. 3. Finite strip simulation on plate
From the beam Eigen function orthogonality, ${a}_{ij}={e}_{ij}=0$ for $i\ne j$, this agree for all types of boundary conditions except for plates having free edges in the $\xi $direction. The governing differential Eq. (14) can be written in form:
where:
and $\left[{E}_{ij}\right]=i\times j$ unit matrix.
A system of coupled fourth order equations are obtained which can be reduced to a system of first order differential equation:
where: $k=$1, 2, 3,…,$N$, $i=$1, 2, 3,…,$N$, $j=$1, 2, 3,…,$M$, coefficients of the matrix ${\left[{A}_{i}\right]}_{k}$ in equation, in general, are functions of $\eta $ and the eigenvalue parameter $\mathrm{\Omega}$. The vector ${Y}_{k}$ is given by:
where:
The relation under which the continuity conditions between the striped plates are satisfied may be expressed as:
where: ${\left[{T}_{i}\right]}_{j}$ is called the transition matrix of the strip $i$ while ${\left\{{Y}_{i}\right\}}_{j}$ and ${\left\{{Y}_{i1}\right\}}_{j}$ are the nodal vectors of the boundaries $i$ and $i1$. The solution is found using 2$N$number of initial vectors $\left\{{Y}_{0}\right\}$ at $\eta =$0. The transition matrix, Eq. (19) is applied across the stripped plate until just before the intermediate support at $y=b/2$, $\eta =1/2$ is reached. Thus, 2$N$number of solutions ${S}_{i}$ can be obtained. The true solutions $\left[S\right]$ can be written as a linear combination of these solutions as:
where ${C}_{i}$ are arbitrary constants, these constants can be determined by satisfying 2$N$number of boundary conditions at $\eta =$ 1/2 in Eqs. (10) and (12) of the intermediate elastic line support. And the matrix $\left[S\right]$ 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 ${C}_{i}$, $i=$1, 2, 3,…, 2$N$.
3. Results and discussion
In this section, the finite strip transition matrix (FSTM) approach is employed to investigate the free vibration of symmetrically laminated, angleply, variable thickness rectangular plates with intermediate elastic support in one direction with different elastic restraint coefficient (${K}_{T}$). The basalt FRP laminate composite plate was manufactured using five symmetrically, angleply, laminates with the fiber orientations [45°/–45°/45°/–45°/45°] of basalt fiber and a polymer resin matrix. The corresponding elastic modulus values were ${E}_{1}=$96.74 GPa, ${E}_{2}={E}_{3}=$ 22.55 GPa, and the Shear modulus values were ${G}_{1}={G}_{3}=$10.64 GPa, ${G}_{2}=$8.73 GPa. Poisson coefficients were ${\upsilon}_{1}={\upsilon}_{3}=$ 0.3, ${\upsilon}_{2}=$0.6 and the density was 2700 kg/m^{3}.
The frequency parameter $\mathrm{\Omega}$ is evaluated in nondimensional form, expressed as:
The plate with linear variable thickness, $h\left(y\right)$ is used (see Appendix) in nondimensional form:
where: $\mathrm{\Delta}$ is the tapered ratio of plate given by $\mathrm{\Delta}=({h}_{b}{h}_{o})/{h}_{o}$, $\left({h}_{o}\right)$ is the thickness of the plate at $\eta =$0 and (${h}_{b}$) is the thickness of the plate at $\eta =$1.
3.1. Convergence study and accuracy
In this subsection, a convergence investigation is carried out for the proposed method, first six frequencies are calculated and compared with available results in literatures. Table 1 presents a convergence and comparison study for isotropic, square ($\beta =$1.0), uniform thickness ($\mathrm{\Delta}=$0) plates with a midline support in each direction, the plate material has mechanical properties of ${\upsilon}_{1}={\upsilon}_{2}=$ 0.3, ${D}_{11}={D}_{22}=D=E{h}^{3}/\left[12\left(1{\upsilon}^{2}\right)\right]$, ${D}_{66}=\left(1\upsilon \right)D/2$. In this study the nondimensional frequency parameter $\mathrm{\Omega}$ become $\mathrm{\Omega}={\left(\rho h{\omega}^{2}{a}^{4}/D\right)}^{1/2}$. 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, 1921]. A very close agreement is observed.
Table 2 presented a convergence and comparison study for fully simply supported (SSSS) and fully clamped (CCCC) square ($\beta =$1.0), uniform thickness ($\mathrm{\Delta}=$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 Eglass/ epoxy material with the following properties are ${\upsilon}_{1}={\upsilon}_{3}=$0.23, $D=E{h}^{3}/\left[12\left(1{\upsilon}^{2}\right)\right]$, ${D}_{66}=\left(1\upsilon \right)D/2$. In this study the nondimensional frequency parameter $\mathrm{\Omega}$ become $\mathrm{\Omega}={\left(\rho h{\omega}^{2}{a}^{4}/\pi D\right)}^{1/2}$ and foundation elastic restraint coefficient is given by ${K}_{T}={k}_{f}{a}^{4}/D$. 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 semianalytical finite strip transition matrix (FSTM) technique.
Table 1. Convergence study of the first six frequency parameters of the isotropic square plates with a midline support in each direction
$N$

${\mathrm{\Omega}}_{1}$

${\mathrm{\Omega}}_{2}$

${\mathrm{\Omega}}_{3}$

${\mathrm{\Omega}}_{4}$

${\mathrm{\Omega}}_{5}$

${\mathrm{\Omega}}_{6}$


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
$N$

${K}_{T}$

${\mathrm{\Omega}}_{1}$

${\mathrm{\Omega}}_{2}$

${\mathrm{\Omega}}_{3}$

${\mathrm{\Omega}}_{4}$


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, angleply, 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 $\beta =$0.5 and tapered ratio of the plate thickness is $\mathrm{\Delta}=\mathrm{}$0.5. Four type of classical boundary conditions (SSSS, CCCC, SSFF and CCFF) as shown in Fig. 2 and different elastic restraint coefficients ${K}_{T}$ 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 midline of the plate.
Fig. 4. Variation of nondimensional frequencies parameter ($\mathrm{\Omega}$) with elastic restraint coefficient (${K}_{T}$)
The effect of intermediate elastic support on the nondimensional 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 (${K}_{T}$) as shown in Fig. 4. Fig. 5 shows the vibration behaviour of the variable thickness rectangular plate under varying elastic restraint coefficient (${K}_{T}$). As shown in the Fig. 5, the increasing values of frequencies with small elastic restraint coefficient (${K}_{T}$) 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 ${K}_{T}$ increases from 50 onwards, the nondimensional frequencies parameter are fast raised till value of ${K}_{T}$ reached 10^{4} and after this value there is almost negligible change in value of Nondimensional frequencies parameter.
Fig. 5. Variation of nondimensional frequencies parameter ($\mathrm{\Omega}$) with different mode number and elastic restraint coefficient (${K}_{T}$)
Fig. 6. Variation of nondimensional frequencies parameter ($\mathrm{\Omega}$) with elastic restraint coefficient (${K}_{T}$) and boundary conditions
Fig. 7. Variation of nondimensional frequencies parameter ($\mathrm{\Omega}$) 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, angleply, 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 semisimply supported (SSFF) condition, respectively. The other two boundary conditions (SSSS and CCFF) are showing an intermediate value. As shown in the Fig. 6, the nondimensional frequencies increase with the increase of the elastic restraint coefficient (${K}_{T}$) for all kind of support conditions (SSSS, CCCC, SSFF and CCFF).
Table 3. The first six frequencies of symmetrically, angleply, laminated, variable thickness rectangular plate with intermediate elastic line support for different elastic restraint coefficients, $(\mathrm{\Delta}=$ 0.5$)$, $(\beta =$ 0.5)
${K}_{T}$

${\mathrm{\Omega}}_{1}$

${\mathrm{\Omega}}_{2}$

${\mathrm{\Omega}}_{3}$

${\mathrm{\Omega}}_{4}$

${\mathrm{\Omega}}_{5}$

${\mathrm{\Omega}}_{6}$


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 semianalytical 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 (${K}_{T}$) 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 (${K}_{T}$) 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 semisimply 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.
References
 Xiang Y., Wei G. W. Exact solutions for multispan rectangular Mindlin plates. Journal of Vibration and Acoustics, Vol. 124, 2002, p. 545551. [Publisher]
 Abrate S., Foster E. Vibration of composite plates with intermediate line supports. Journal of Sound and Vibration, Vol. 179, 1995, p. 793815. [Publisher]
 Cheung Y. K., Zhou D. Vibration of rectangular plates with elastic intermediate linesupports and edges constraints. Journal of Thinwalled Structures, Vol. 37, 2000, p. 305331. [Publisher]
 Liew K. M., Wang C. M. Vibration studies on skew plates: treatment of internal line supports. Journal of Computer and Structures, Vol. 49, Issue 6, 1993, p. 941951. [Publisher]
 Cheung Y. K., Zhou D. Vibration analysis of Symmetrically Laminated rectangular plates with intermediate line supports. Journal of Computer and Structures, Vol. 79, 2001, p. 3341. [Publisher]
 Xiang S., Jiang S., Bi Z., Jin Y., Yang M. A nthorder meshless generalization of Reddy’s third order shear deformation theory for the free vibration on laminated composite plates. Journal of Composite Structures, Vol. 93, 2011, p. 299307. [Publisher]
 Thai H., Kim S. Free vibration of laminated composite plates using two variable refined plate theory. International Journal of Mechanical Sciences, Vol. 52, 2010, p. 626633. [Publisher]
 Ovesy H. R., Fazilati J. Buckling and free vibration finite strip analysis of composite plates with cutout based on two different modeling approaches. Journal of Composite Structures, Vol. 94, 2012, p. 12501258. [Publisher]
 Dozio L. Inplane free vibrations of singlelayer and symmetrically laminated rectangular composite plates. Journal of Composite Structures, Vol. 93, 2011, p. 17871800. [Publisher]
 Asadi H., Aghdam M. M., Shakeri M. Vibration analysis of axially moving line supported functionally graded plates with temperaturedependent properties. Journal of Mechanical Engineering Science, Vol. 228, Issue 6, 2014, p. 953969. [Publisher]
 AlTabey W. A. Vibration Analysis of Laminated Composite Variable Thickness Plate Using Finite Strip Transition Matrix Technique and MATLAB Verifications, book MATLAB Applications for the Practical Engineer. Technology and Medicine Open Access Book Publisher, 2014, p. 283620. [Search CrossRef]
 Thinh T. I., Binh B. V., Tu T. M. Bending and vibration analysis of multifolding laminate composite plate using finite element method. Vietnam Journal of Mechanics, Vol. 34, Issue 3, 2012, p. 185202. [Publisher]
 Ducceschi M. Nonlinear Vibrations of Thin Rectangular Plates: A Numerical Investigation with Application to Wave Turbulence and Sound Synthesis. Ph.D. Thesis, ENSTAParis Tech, 2014. [Search CrossRef]
 Yadav D. P. S., Sharma A. K., Shivhare V. Free vibration analysis of isotropic plate with stiffeners using finite element method. Journal of Engineering Solid Mechanics, Vol. 3, 2015, p. 67176. [Publisher]
 Küçükrendeci I., Morgül Ö. K. The effects of elastic boundary conditions on the linear free vibrations. Journal of Scientific Research and Essays, Vol. 6, Issue 19, 2001, p. 39493958. [Search CrossRef]
 Bert W., Malik M. Free vibration analysis of tapered rectangular plates by differential quadrature method: a semianalysis approach. Journal of Sound and Vibration, Vol. 190, 1996, p. 4163. [Publisher]
 Chakraverty S. Vibration of Plates. CRC Press, Taylor and Francis Group, 2009. [Search CrossRef]
 Reddy J. N. Mechanics of Laminated Composite Plates and Shells Theory and Analysis. 2nd Ed., CRC Press, Boca Raton London New York Washington, D.C., 2006. [Search CrossRef]
 Wu C. I., Cheung Y. K. Frequency analysis of rectangular plates continuous in one or two directions. Journal of Earthquake Engineering Structure Dynamics, Vol. 3, 1974, p. 314. [Publisher]
 Kim C. S., Dickinson S. M. The flexural vibration of line supported rectangular plate systems. Journal of Sound Vibration, Vol. 114, 1987, p. 2942. [Publisher]
 Leissa A. W. The free vibration of rectangular plates. Journal of Sound and Vibration, Vol. 31, 1973, p. 25793. [Publisher]
 Zhou D., Cheung Y. K., Lo S. H., Au F. T. K. Three dimensional vibration analysis of rectangular thick plates on Pasternak foundation. International Journal of Numerical Methods in Engineering, Vol. 59, 2004, p. 13131334. [Publisher]
 Ferreira J. M., Roque C. M. C., Neves A. M. A., Jorge R. M. N., Soares, C. M. M. Analysis of plates on Pasternak foundations by radial basis functions. Journal of Computational Mechanics, Vol. 46, 2010, p. 791803. [Publisher]
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