Abstract
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.
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. 1The 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. 2Representation 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. 3Finite 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 1Convergence 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 2Convergence 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. 4Variation of nondimensional frequencies parameter (Ω) with elastic restraint coefficient (KT)
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. 5Variation of nondimensional frequencies parameter (Ω) with different mode number and elastic restraint coefficient (KT)
Fig. 6Variation of nondimensional frequencies parameter (Ω) with elastic restraint coefficient (KT) and boundary conditions
Fig. 7Variation of nondimensional 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, 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 3The first six frequencies of symmetrically, angleply, laminated, variable thickness rectangular plate with intermediate elastic line support for different elastic restraint coefficients, (Δ= 0.5), (β= 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.
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