1 Introduction

Plates are important structural components that have many applications in civil, aerospace, automobile and marine structures, etc. Depending on the purpose, these plates may be isotropic or orthotropic, with the latter becoming increasingly adopted due to its numerous advantages. Both static and dynamic analyses of these elements require that certain theory or, more precisely, plate theory be used to arrive at the governing equations subject to some boundary conditions. Starting from the classical plate theory (CPT), several works have been reported and quite a number of other higher-order theories and their variations exist today.

The CPT or the Kirchhoff plate theory [1] is very appealing in terms of simplicity and ease of solution. It has been successfully applied to free vibration analysis of orthotropic plates in [2–7]. However, the fact that it does not take into account the effect of transverse shear stresses limits its application to thin plates. It is also worth mentioning that the effect of shear stresses is more pronounced in orthotropic plates. As a result, natural frequencies are overestimated by the CPT in case of thick/orthotropic plates. If a more reliable solution is needed, other plate theories that incorporate shear deformation effects need to be sought. A number of such shear deformation theories are available in the literature, including the first-order shear deformation theory (FSDT) [8–10] and the higher-order shear deformation theories (HSDT) [11–20]. Typical instances where free vibration analyses of orthotropic plates were carried out based on these theories can be found in [21–24] based on the FSDT and in [25–28] based on the HSDT. The FSDT assumes a constant distribution of shear stress across the plate’s thickness, thereby violating zero-stress condition on the plate’s top and bottom surfaces. In order to take care of this inconsistency, use of a correction factor becomes necessary in the FSDT. On the other hand, the HSDT do not require the use of such correction factor and they, still, satisfy the zero-traction boundary condition at the top and bottom surfaces of the plate. But, this happens at the expense of more complex formulation, involving several unknowns, that becomes inconvenient to handle. Such set-backs encountered in form of lesser accuracy in the CPT and the inconveniencies in formulating the higher-order theories can be well taken care of by the adoption of the two-variable refined plate theory (RPT) of Shimpi [29], which was later extended by Shimpi and Patel [30] to orthotropic plates. The RPT is very similar to the CPT, and it still accounts for shear stress distribution that satisfies zero-traction boundaries without the need for any correction factor. RPT expresses the governing equations in terms of only two unknown functions. Zig-zag and layer-wise theories of plates [31] are other useful plate theories whose recent applications to laminate glass and solar panels prove successful. Detailed analysis of free vibrations of functionally graded plates has been performed using several variants of plate models in [32].

Each of the aforementioned theories yields a set of governing differential equations, some of which require advanced methods of solution. Consequently, several studies reported the application of various methods to analyze free vibration of orthotropic plates. Some of these include the works reported in [33, 34] based on the Navier’s method, [35] based on the Levy’s method, [36–38] based on the finite element method, [36, 39–41] based on the Rayleigh–Ritz method, [42–44] based on the state–space method, [36, 45] based on the differential quadrature method, [46–48] based on the meshless method including the RBF, [49, 50] based on the Galerkin method, [51, 52] based on the discrete singular convolution (DSC) method, [53] based on the extended Kantorovich method, [54] based on the mixed variational formulations and [7, 55, 56] based on the exact solutions. A more comprehensive review of recent literature on free vibration analysis of general composite plates can be found in the work of Sayyad and Ghugal [57].

As evident in the above and other numerous studies on the use of various solution tools and theories for free vibration of orthotropic plates, it can be noticed that the application of differential transform method (DTM) and Taylor collocation method (TCM) for free vibration of orthotropic plates is not popular, if any. Works such as those reported in [58, 59] utilized the DTM only for isotropic and not for orthotropic plates. As for the TCM, most of the published research on its application addresses other problems but not free vibration of orthotropic plates.

This paper has dual objectives, one of which is to present a study based on a very efficient and relatively simple theory, the two-variable refined plate theory (RPT) [30], which outperforms the CPT. In addition, the inconveniencies in formulating the first-order and higher-order shear deformation theories can be avoided by the use of the refined plate theory. The theory allows quadratic shear stress distribution that satisfies zero-traction boundaries without the need for any correction factor. Additionally, as the name implies, the two-variable RPT expresses the governing equations in terms of only two unknown functions, contrary to the need for several functions in many of the higher-order theories. The similarity, in some ways, of the refined theory with the CPT speaks enough about its simplicity. The second objective of the paper is to present two solution schemes, the DTM and TCM, having some similarities with respect to their origin and to illustrate how they can, conveniently, be used in the analysis of free vibrations of the orthotropic plates based on the RPT. The work reported in [30] presents some results for free vibration of simply supported square orthotropic plate based on this theory. Later, Thai and Kim [35] extended the analysis to consider other boundary conditions. However, aside from the simply supported square plate, results of the work reported in [35] has been presented without further verification with other methods. Hence, the present work will have a secondary benefit of serving as a verification of some of the results reported in [35]. But, in addition, some cases whose analytical solutions are not readily available, particularly based on the two-variable refined plate theory, are also analyzed. In order to verify the results for those cases with no analytical solution, formulation of three more plate theories, the CPT, the FSDT [10, 60, 61] and the HSDT [62], was implemented and solved using both the DTM and TCM. The robustness and competitiveness of DTM and TCM in providing reliable results make them good alternatives to the analytical solutions.

2 Two-variable refined plate theory (RPT) for orthotropic plate [30]

The two-variable RPT of Shimpi and Patel [30] has many attractive features that make it competitive with other well-established and robust plate theories. First, the theory is similar to the CPT and, hence, it is simple. In addition, there is no need for shear correction factor as required in some shear deformation theories, and it still takes into account the effect of shear stress distribution that satisfies zero-traction boundaries. This way, the RPT takes care of both the lesser accuracy in the CPT and the complexities in formulating the first-order and higher-order shear deformation theories. Detailed derivation of RPT for orthotropic plates can be found in [30], but its overview is given as follows.

2.1 Coordinate system and orthotropic plate configuration

The plate under consideration has dimensions of \(a\times b\times h\) and is oriented with respect to a right-handed Cartesian coordinate system as shown in Fig. 1. The displacement components U, V and W corresponding to the x, y and z coordinate directions, respectively, are shown in bracket. For free vibration, the top and bottom surfaces are free. On the other hand, convenient boundary conditions can be applied at the edges \(x=0,x=a,y=0\) and \(y=b\).

Fig. 1
figure 1

Geometry and orientation of the plate considered

Full size image

2.2 Displacements, constitutive equations and stress resultants

According to CPT, the displacement components are given by Eq. (1).

$$\begin{aligned} U=-z\frac{\partial W}{\partial x};V=-z\frac{\partial W}{\partial y}; W=W_b \end{aligned}$$
(1)

where \(W_b \) is the transverse displacement due to bending effect alone.

It is clear from Eq. (1) that, by default, any subsequent formulation in terms of U, V and W based on CPT neglects the effect of transverse shear. On the contrary, the RPT assumes that both U, V and W consists of bending components (\(U_b ,V_b \,\mathrm{and} \,W_b \), respectively) and shear components (\(U_s ,V_s \,\mathrm{and} \,W_s \), respectively) thus.

$$\begin{aligned} U=U_b +U_s ;V=V_b +V_s ;W=W_b +W_s \end{aligned}$$
(2)

where

$$\begin{aligned} U_b =-z\frac{\partial W_b }{\partial x};U_s =h\left( {\frac{1}{4}\left( {\frac{z}{h}} \right) -\frac{5}{3}\left( {\frac{z}{h}} \right) ^{3}} \right) \frac{\partial W_s }{\partial x};V_b =-z\frac{\partial W_b }{\partial y};V_s =h\left( {\frac{1}{4}\left( {\frac{z}{h}} \right) -\frac{5}{3}\left( {\frac{z}{h}} \right) ^{3}} \right) \frac{\partial W_s }{\partial y} \end{aligned}$$

The six independent strain components are obtained by using the displacement expressions given by Eq. (2) in the following strain-displacement relations.

$$\begin{aligned} \varepsilon _x =\frac{\partial U}{\partial x};\varepsilon _y =\frac{\partial V}{\partial y};\varepsilon _z =\frac{\partial W}{\partial z};\gamma _{xy} =\frac{\partial V}{\partial x}+\frac{\partial U}{\partial y};\gamma _{yz} =\frac{\partial W}{\partial y}+\frac{\partial V}{\partial z};\gamma _{zx} =\frac{\partial U}{\partial z}+\frac{\partial W}{\partial x} \end{aligned}$$
(3)

Equation (3) can be substituted in the constitutive equations to obtain the corresponding stresses given by Eq. (4).

$$\begin{aligned} {\varvec{\sigma }} ={\overline{{\varvec{Q}}}} \,{\varvec{\varepsilon }} \end{aligned}$$
(4)

where \(\sigma =\left[ {\sigma _x ,\sigma _y ,\tau _{xy} ,\tau _{yz} ,\tau _{zx} } \right] ^{T};\varepsilon =\left[ {\varepsilon _x ,\varepsilon _y ,\gamma _{xy} ,\gamma _{yz} ,\gamma _{zx} } \right] ^{T}\) and \({\overline{{\varvec{Q}}}}\) is a \(5\times 5\) matrix whose elements are

$$\begin{aligned} {\overline{Q}}_{11} =Q_{11} =\frac{E_1 }{1-\nu _{12} \nu _{21} };\quad {\overline{Q}}_{12} ={\overline{Q}}_{21} =Q_{12} =\frac{\nu _{12} E_1 }{1-\nu _{12} \nu _{21} };\quad {\overline{Q}}_{22} =Q_{22} =\frac{E_2 }{1-\nu _{12} \nu _{21} } \end{aligned}$$

\({\overline{Q}}_{33} =Q_{66} =G_{12} ;{\overline{Q}}_{44} =Q_{44} =G_{23} ;{\overline{Q}}_{55} =Q_{55} =G_{31} \) and \({\overline{Q}}_{ij} =0\) for other combinations of i and j.

\( E_i \left( {i=1,2} \right) ,G_{ij} \left( {i,j=1,2,3\, \& \, i\ne j} \right) \) and \( \nu _{ij} ( {i,j=1,2\, \& \, i\ne j} )\) are the Young’s modulus, shear modulus and Poisson’s ratio, respectively. The subscripts 1, 2 and 3 correspond to the three coordinate directions, x, y and z, respectively.

Expressions for the stress resultants (moments and shear forces) can be obtained from Eq. (5a) to arrive at Eq. (5b).

$$\begin{aligned} \left\{ {{\begin{array}{l} {{\begin{array}{l} {M_x^b } \\ {M_y^b } \\ {M_{xy}^b } \\ \end{array} }} \\ {M_x^s } \\ {M_y^s } \\ {M_{xy}^s } \\ {Q_x } \\ {Q_y } \\ \end{array} }} \right\}= & {} {\mathop {\int }\limits _{z=-h/2}^{z=h/2}} \left\{ {{\begin{array}{l} {{\begin{array}{l} {\sigma _x z} \\ {\sigma _y z} \\ {\tau _{xy} z} \\ \end{array} }} \\ {\sigma _x \left( {-\frac{1}{4}z+\frac{5}{3}z\left( {\frac{z}{h}} \right) ^{2}} \right) } \\ {\sigma _y \left( {-\frac{1}{4}z+\frac{5}{3}z\left( {\frac{z}{h}} \right) ^{2}} \right) } \\ {\tau _{xy} \left( {-\frac{1}{4}z+\frac{5}{3}z\left( {\frac{z}{h}} \right) ^{2}} \right) } \\ {\tau _{zx} } \\ {\tau _{yz} } \\ \end{array} }} \right\} \mathrm{d}z \end{aligned}$$
(5a)
$$\begin{aligned} \left\{ {\begin{array}{{l}} {M_{x}^{b} } \\ {M_{y}^{b} } \\ {M_{{xy}}^{b} } \\ {M_{x}^{s} } \\ {M_{y}^{s} } \\ {M_{{xy}}^{s} } \\ {Q_{x} } \\ {Q_{y} } \\ \end{array} } \right\}= & {} \left[ {\begin{array}{llllllll} - D_{{11}} &{} - D_{{12}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ - D_{{12}} &{} - D_{{22}} &{} 0 &{} 0 &{} 0&{} 0&{}0&{}0\\ 0&{}0&{} - D_{{66}} &{}0&{}0&{}0&{}0&{}0 \\ 0&{}0&{}0&{} - \frac{1}{{84}}D_{{11}} &{}- \frac{1}{{84}}D_{{12}} &{}0&{}0&{}0 \\ 0&{}0&{}0&{} - \frac{1}{{84}}D_{{12}} &{} - \frac{1}{{84}}D_{{22}} &{}0&{}0&{}0 \\ 0&{}0&{}0&{}0&{}0&{} - \frac{1}{{84}}D_{{66}} &{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}A_{{55}} &{}0 \\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}A_{{44}}\\ \end{array}} \right] \left\{ {\begin{array}{{l}} {\frac{{\partial ^{2} W_{b} }}{{\partial x^{2} }}} \\ {\frac{{\partial ^{2} W_{b} }}{{\partial y^{2} }}} \\ {2\frac{{\partial ^{2} W_{b} }}{{\partial x\partial y}}} \\ {\frac{{\partial ^{2} W_{s} }}{{\partial x^{2} }}} \\ {\frac{{\partial ^{2} W_{s} }}{{\partial y^{2} }}} \\ {2\frac{{\partial ^{2} W_{s} }}{{\partial x\partial y}}} \\ {\frac{{\partial W_{s} }}{{\partial x}}} \\ {\frac{{\partial W_{s} }}{{\partial y}}} \\ \end{array} } \right\} \end{aligned}$$
(5b)

The material parameters are

$$\begin{aligned} D_{11} =\frac{Q_{11} h^{3}}{12};D_{22} =\frac{Q_{22} h^{3}}{12};D_{12} =\frac{Q_{12} h^{3}}{12};D_{66} =\frac{Q_{66} h^{3}}{12};A_{44} =\frac{5Q_{44} h}{6};A_{55} =\frac{5Q_{55} h}{6} \end{aligned}$$

2.3 Governing equations of motion

In deriving the equations of motion based on RPT, expressions for the kinetic and potential energies are, firstly, written as in Eqs. (6) and (7), respectively [30, 35].

$$\begin{aligned} E_k= & {} \frac{1}{2}\mathop \int \limits _{-\frac{h}{2}}^{\frac{h}{2}} \mathop \int \limits _0^b \mathop \int \limits _0^a \rho \left[ {\left( {\frac{\partial U_b }{\partial t}+\frac{\partial U_s }{\partial t}} \right) ^{2}+\left( {\frac{\partial V_b }{\partial t}+\frac{\partial V_s }{\partial t}} \right) ^{2}+\left( {\frac{\partial W_b }{\partial t}+\frac{\partial W_s }{\partial t}} \right) ^{2}} \right] \mathrm{d}x \,\mathrm{d}y\, \mathrm{d}z \end{aligned}$$
(6)
$$\begin{aligned} U= & {} \frac{1}{2}\mathop \int \limits _{-\frac{h}{2}}^{\frac{h}{2}} \mathop \int \limits _0^b \mathop \int \limits _0^a \rho \left[ {\sigma _x \varepsilon _x +\sigma _y \varepsilon _y +\tau _{xy} \gamma _{xy} +\tau _{yz} \gamma _{yz} +\tau _{zx} \gamma _{zx} } \right] \mathrm{d}x\, \mathrm{d}y\, \mathrm{d}z \end{aligned}$$
(7)

Equations (6) and (7) are substituted in Eq. (8), written using Hamilton’s principle. Integrating the resulting equation by parts, and due to the independence of \(\delta W_b \) and \(\delta W_s \), their coefficients can be collected separately which give the governing equations of motion, given by Eq. (9), for free vibration of orthotropic plates.

$$\begin{aligned} \mathop \smallint \nolimits _0^t \delta \left( {E_k -U} \right) \mathrm{d}t=0 \end{aligned}$$
(8)
$$\begin{aligned}&L_1 \left[ {W_b } \right] +I_0 \left( {\frac{\partial ^{2}W_b }{\partial t^{2}}+\frac{\partial ^{2}W_s }{\partial t^{2}}} \right) =q \end{aligned}$$
(9a)
$$\begin{aligned}&L_1 \left[ {W_s } \right] +84\left( {L_2 \left[ {W_s } \right] +I_0 \left( {\frac{\partial ^{2}W_b }{\partial t^{2}}+\frac{\partial ^{2}W_s }{\partial t^{2}}} \right) } \right) =84q \end{aligned}$$
(9b)

where the linear differential operators \(L_1 \) and \(L_2 \) are defined thus

$$\begin{aligned} L_1 \left[ \cdots \right]= & {} D_{11} \frac{\partial ^{4}\left( \cdots \right) }{\partial x^{4}}+2\left( {D_{12} +2D_{66} } \right) \frac{\partial ^{4}\left( \cdots \right) }{\partial x^{2}\partial y^{2}}+D_{22} \frac{\partial ^{4}\left( \cdots \right) }{\partial y^{4}}-I_2 \frac{\partial ^{2}}{\partial t^{2}}(\nabla ^{2}\left( \cdots \right) )\\ L_2 \left[ \cdots \right]= & {} -\left( {A_{55} \frac{\partial ^{2}\left( \cdots \right) }{\partial x^{2}}+A_{44} \frac{\partial ^{2}\left( \cdots \right) }{\partial y^{2}}} \right) \end{aligned}$$

\(I_0 \) and \(I_2 \) are inertias given by \(\rho h\) and \(\rho h^{3}/12,\) respectively.

For free vibration, the external transverse load \(q=0\).

It is important, at this point, to discuss the concept of consistency which arises when deciding the conformity of the various plate theories/models with linear three-dimensional theory of elasticity. This is of concern because there are several plate theories, including the RPT presented in this work, that are based on some priori assumptions which may result in system of equations that violate the elasticity theory. Interestingly, however, Shimpi [29] has already reported that the RPT is variationally consistent. Details of the consistent approach are well addressed by Kienzler [63], where he derived some consistent plate theories from the basic equations of three-dimensional linear theory of elasticity by applying the uniform-approximation technique which, depending on the order of the approximation chosen, yields a set of governing partial differential equations without invoking any priori assumptions or shear correction factors. Some recent works where similar idea was used include [64] where the concept is applied to anisotropic materials and [65] where comparison of various linear plate theories in light of consistent second-order approximation is made.

2.4 Free vibration equations for the orthotropic plate under consideration

The present study considers rectangular orthotropic plates with the two opposite boundaries, parallel to the x-axis (i.e., \(y=0\) and \(y=b)\), simply supported (i.e., Levy plates). The first part of the analysis considers a case of three combinations of clamped (C), simply supported (S) and free (F) edge conditions on the two remaining boundaries (i.e., at \(x=0\) and \(x=a\)), keeping one of them to be simply supported. Consequently, plates with one end simply supported and the other free, one end simply supported and the other clamped, and both ends simply supported are referenced as SF, SC and SS plates, respectively. Next, multi-span plates and plates with stepped thickness and end rotational springs, whose analytical solutions are not readily available, particularly based on the two-variable refined plate theory are also analyzed in this work.

Assuming a solution, given by Eq. (10), for the transverse displacement components \(W_b \) and \(W_s \), it can easily be seen that the boundary conditions of zero displacement and zero moment at \(y=0\) and \(y=b\) are automatically satisfied [35].

$$\begin{aligned} W_b \left( {x,y,t} \right)= & {} \mathop \sum \limits _{n=1}^\infty w_b \left( x \right) e^{i\omega _n t}\sin \left( {\beta y} \right) \end{aligned}$$
(10a)
$$\begin{aligned} W_s \left( {x,y,t} \right)= & {} \mathop \sum \limits _{n=1}^\infty w_s \left( x \right) e^{i\omega _n t}\sin \left( {\beta y} \right) \end{aligned}$$
(10b)

where the unknown functions \(w_b \left( x \right) \) and \(w_s \left( x \right) \) need to be determined. \(\beta =n\pi /b\) and the natural frequency of the nth mode is denoted by \(\omega _n \).

Substituting Eq. (10) in the two governing equations given by Eq. (9) yields the system of two coupled equations given by Eq. (11). Similarly, the boundary conditions can be written as given in Table 1.

$$\begin{aligned}&D_{11} w_b^{{\prime }{\prime }{\prime }{\prime }} -\left[ {2\beta ^{2}\left( {D_{12} +2D_{66} } \right) -\omega _n^2 I_2 } \right] w_b^{{\prime }{\prime }} -\left[ {\omega _n^2 I_0 +\omega _n^2 I_2 \beta ^{2}-D_{22} \beta ^{4}} \right] w_b -\omega _n^2 I_0 w_s =0 \end{aligned}$$
(11a)
$$\begin{aligned}&D_{11} w_s^{{\prime }{\prime }{\prime }{\prime }} -\left[ {84A_{55} +2\beta ^{2}\left( {D_{12} +2D_{66} } \right) -\omega _n^2 I_2 } \right] w_s^{{\prime }{\prime }} -\left[ {84\omega _n^2 I_0 +\beta ^{2}\left( {\omega _n^2 I_2 -D_{22} \beta ^{2}-84A_{44} } \right) } \right] w_s\nonumber \\&\quad -\,84\omega _n^2 I_0 w_b =0 \end{aligned}$$
(11b)

where \(\left( \cdots \right) ^{{\prime }}=\frac{\mathrm{d}\left( \cdots \right) }{\mathrm{d}x},\left( \cdots \right) ^{{\prime }{\prime }}=\frac{\mathrm{d}^{2}\left( \cdots \right) }{\mathrm{d}x^{2}}\) and so on.

In the subsequent analyses, use is made of a non-dimensional space variable \({\bar{x}}=x/a\), where a is the x-directional dimension of the plate shown in Fig. 1. Hence, \(0\le {\bar{x}} \le 1\).

Table 1 Boundary conditions for the SS, SC and SF plates
Full size table

3 Differential transform method (DTM)

Originally proposed by Zhou [66], the DTM is a semi-analytical method that, in recent decades, gained recognition for solving problems governed by differential equations. Its concept stems from the Taylor series expansion, and solution of the original problem is approximated in form of polynomials. However, unlike in the Taylor series method, symbolic evaluation of the derivatives is avoided in DTM. Instead, they are obtained by some recursive relations obtainable from the transformed governing equations.

3.1 Fundamentals of DTM

Consider an analytic function w(x) that is sufficiently differentiable within the domain of interest. By definition, the differential transformation of w(x) near the point \(x=x_0 \) is \(W\left( k \right) \) given by Eq. (12). Conversely, the differential inverse transform of \(W\left( k \right) \) is given by Eq. (13). Due to the consequence of Eq. (12), Eq. (13) can be rewritten in the form given by Eq. (14); the Taylor series expansion.

$$\begin{aligned} W\left( k \right)= & {} \frac{1}{k!}\left( {\frac{\mathrm{d}^{k}w\left( x \right) }{\mathrm{d}x^{k}}} \right) _{x=x_0 } \end{aligned}$$
(12)
$$\begin{aligned} w\left( x \right)= & {} \mathop \sum \limits _{k=0}^\infty \left( {x-x_0 } \right) ^{k}W\left( k \right) \end{aligned}$$
(13)
$$\begin{aligned} w\left( x \right)= & {} \mathop \sum \limits _{k=0}^\infty \frac{\left( {x-x_0 } \right) ^{k}}{k!}\left( {\frac{\mathrm{d}^{k}w\left( x \right) }{\mathrm{d}x^{k}}} \right) _{x=x_0 } \end{aligned}$$
(14)

For an integer \({\bar{N}}\) large enough to result in a negligibly small value of the summation \(\mathop \sum \nolimits _{k={\bar{ N}} +1}^\infty \left( {x-x_0 } \right) ^{k}W\left( k \right) \), the approximation of the original function (in our case, the solution) given by Eq. (13) can be truncated and rewritten as shown in Eq. (15). This way, \({\bar{N}} \) represents the number of terms sufficient enough for the series convergence.

$$\begin{aligned} w( x )=\mathop \sum \limits _{k=0}^{{\bar{N}} } \left( {x-x_0 } \right) ^{k}W( k ) \end{aligned}$$
(15)

Since the aim of DTM is to transform the original differential equation(s) into recursive formulas needed to evaluate the terms \(W\left( k \right) \left\{ {k,0,{\bar{N}} } \right\} \), some basic mathematical operations required in the transformation are given by Eqs. (16) to (23).

$$\begin{aligned}&\displaystyle w( x)&{\xrightarrow {{\mathrm{transforms}} {\mathrm{to}}}}&\quad W( k ) \end{aligned}$$
(16)
$$\begin{aligned}&\displaystyle cw( x)&\quad cW(k) \end{aligned}$$
(17)
$$\begin{aligned}&\displaystyle \alpha _1 w_1 (x)+\alpha _2 w_2 ( x )+\cdots +\alpha _n w_n ( x )&\quad \alpha _1 W_1 \left( k \right) +\alpha _2 W_2 \left( k \right) +\cdots +\alpha _n W_n \left( k \right) \end{aligned}$$
(18)
$$\begin{aligned}&\displaystyle \frac{\mathrm{d}w( x )}{dx}&\quad \left( {k+1} \right) W\left( {k+1} \right) \end{aligned}$$
(19)
$$\begin{aligned}&\displaystyle \frac{\mathrm{d}^{2}w( x )}{\mathrm{d}x^{2}}&\quad \left( {k+1} \right) \left( {k+2} \right) W\left( {k+2} \right) \end{aligned}$$
(20)
$$\begin{aligned}&\displaystyle \frac{\mathrm{d}^{3}w( x )}{\mathrm{d}x^{3}}&\quad \left( {k+1} \right) \left( {k+2} \right) \left( {k+3} \right) W\left( {k+3} \right) \end{aligned}$$
(21)
$$\begin{aligned}&\displaystyle \frac{\mathrm{d}^{4}w\left( x \right) }{\mathrm{d}x^{4}}&\quad \left( {k+1} \right) \left( {k+2} \right) \left( {k+3} \right) \left( {k+4} \right) W\left( {k+4} \right) \end{aligned}$$
(22)
$$\begin{aligned}&\qquad \vdots&\qquad \quad \vdots \nonumber \\&\qquad \vdots&\qquad \quad \vdots \nonumber \\&\displaystyle \frac{\mathrm{d}^{n}w\left( x \right) }{\mathrm{d}x^{n}}&\quad \frac{\left( {k+n} \right) !}{k!}W\left( {k+n} \right) \end{aligned}$$
(23)

3.2 Free vibration analysis based on DTM

This section presents the application of DTM for the free vibration analysis of orthotropic plates based on the two governing equations given by Eq. (11). For that purpose, the bending and shear displacement components of the transverse displacement are approximated around \({\bar{x}}={\bar{x}}_0 \) as follows.

$$\begin{aligned} w_b \left( {{\bar{x}}} \right) =\mathop \sum \limits _{k=0}^{{\bar{N}} } ({\bar{x}}-{\bar{x}}_0 )^{k}W_b \left( k \right) ;\quad w_s \left( {{\bar{x}}} \right) =\mathop \sum \limits _{k=0}^{{\bar{N}} } ({\bar{x}}-{\bar{x}}_0 )^{k}W_s \left( k \right) \end{aligned}$$
(24)

It should be noted that \(W_b \left( k \right) \) and \(W_s \left( k \right) \) in the present section are constants and are, therefore, different from \(W_b \left( x \right) \) and \(W_s \left( x \right) \) appearing in the previous sections (which are continuous functions of the space variable x).

Substituting the transform values of Eqs. (16) to (23) in Eq. (11) results in the following equations.

$$\begin{aligned} W_b \left( {k+4} \right)= & {} \frac{c_1 \left( {k+1} \right) \left( {k+2} \right) W_b \left( {k+2} \right) +c_2 W_b \left( k \right) +c_3 W_s \left( k \right) }{\left( {k+1} \right) \left( {k+2} \right) \left( {k+3} \right) \left( {k+4} \right) } \end{aligned}$$
(25a)
$$\begin{aligned} W_s \left( {k+4} \right)= & {} \frac{c_4 \left( {k+1} \right) \left( {k+2} \right) W_s \left( {k+2} \right) +c_5 W_s \left( k \right) +c_6 W_b \left( k \right) }{\left( {k+1} \right) \left( {k+2} \right) \left( {k+3} \right) \left( {k+4} \right) } \end{aligned}$$
(25b)

where

$$\begin{aligned} c_1= & {} \frac{\left( {2\beta ^{2}\left( {D_{12} +2D_{66} } \right) -\omega _n^2 I_2 } \right) }{D_{11} },c_2 =\frac{\left( {\omega _n^2 I_0 +\omega _n^2 I_2 \beta ^{2}-D_{22} \beta ^{4}} \right) }{D_{11} }, \end{aligned}$$
(26a)
$$\begin{aligned} c_3= & {} \frac{\omega _n^2 I_0 }{D_{11} },c_4 =\frac{\left( {84A_{55} +2\beta ^{2}\left( {D_{12} +2D_{66} } \right) -\omega _n^2 I_2 } \right) }{D_{11} }, \end{aligned}$$
(26b)
$$\begin{aligned} c_5= & {} \frac{\left( {84\omega _n^2 I_0 +\beta ^{2}\left( {\omega _n^2 I_2 -D_{22} \beta ^{2}-84A_{44} } \right) } \right) }{D_{11} },\quad c_6 =\frac{84\omega _n^2 I_0 }{D_{11} } \end{aligned}$$
(26c)

Since the three cases considered in the first part of the work are the SF, SS and SC plates, the boundary conditions at \({\bar{x}}=0\) (i.e., the S-type boundary) are, therefore, common to all the three plate cases. Consequently, the displacement and moment expressions from Table 1 for this boundary are written using Eq. (24) near the point \({\bar{x}}_0 =0\) to arrive at Eq. (27).

$$\begin{aligned} W_b \left( 0 \right) =0,\quad W_s \left( 0 \right) =0,\quad W_b \left( 2 \right) =0,\quad W_s \left( 2 \right) =0 \end{aligned}$$
(27)

At \({\bar{x}}=1,\) different boundary conditions are applicable to the SF, SS and SC plates. These are obtained from expressions in Table 1 written, again, using Eq. (24) near the point \({\bar{x}}_0 =0\) to arrive at the following.

C-type boundary :

$$\begin{aligned} \mathop \sum \limits _{k=0}^{{\bar{N}} } W_b \left( k \right) =0,\quad \mathop \sum \limits _{k=0}^{{\bar{N}} } W_s \left( k \right) =0,\quad \mathop \sum \limits _{k=1}^{{\bar{N}} } kW_b \left( k \right) =0,\quad \mathop \sum \limits _{k=1}^{{\bar{N}} } kW_s \left( k \right) =0 \end{aligned}$$
(28)

S-type boundary :

$$\begin{aligned}&\mathop \sum \limits _{k=2}^{{\bar{N}} } W_b \left( k \right) =0,\quad \mathop \sum \limits _{k=2}^{{\bar{N}} } W_s \left( k \right) =0,\quad -D_{11} \mathop \sum \limits _{k=3}^{{\bar{N}} } k\left( {k-1} \right) W_b \left( k \right) +D_{12} \beta ^{2}\mathop \sum \limits _{k=1}^{{\bar{N}} } W_b \left( k \right) =0, \end{aligned}$$
(29a)
$$\begin{aligned}&-\left( {\frac{1}{84}} \right) \left( {D_{11} \mathop \sum \limits _{k=3}^{{\bar{N}} } k\left( {k-1} \right) W_s \left( k \right) -D_{12} \beta ^{2}\mathop \sum \limits _{k=1}^{{\bar{N}} } W_s \left( k \right) } \right) =0, \end{aligned}$$
(29b)

F-type boundary :

$$\begin{aligned}&-D_{11} \mathop \sum \limits _{k=2}^{{\bar{N}} } k\left( {k-1} \right) W_b \left( k \right) +D_{12} \beta ^{2}\mathop \sum \limits _{k=1}^{{\bar{N}} } W_b \left( k \right) =0, \end{aligned}$$
(30a)
$$\begin{aligned}&-\left( {\frac{1}{84}} \right) \left( {D_{11} \mathop \sum \limits _{k=2}^{{\bar{N}} } k\left( {k-1} \right) W_s \left( k \right) -D_{12} \beta ^{2}\mathop \sum \limits _{k=1}^{{\bar{N}} } W_s \left( k \right) } \right) =0,\end{aligned}$$
(30b)
$$\begin{aligned}&-D_{11} \mathop \sum \limits _{k=3}^{{\bar{N}} } k\left( {k-1} \right) \left( {k-2} \right) W_b \left( k \right) +\left( {\left( {D_{12} +4D_{66} } \right) \beta ^{2}-I_2 \omega _n^2 } \right) \mathop \sum \limits _{k=1}^{{\bar{N}} } kW_b \left( k \right) =0\end{aligned}$$
(30c)
$$\begin{aligned}&-D_{11} \mathop \sum \limits _{k=3}^{{\bar{N}} } k\left( {k-1} \right) \left( {k-2} \right) W_s \left( k \right) +\left( {\left( {D_{12} +4D_{66} } \right) \beta ^{2}-I_2 \omega _n^2 +84A_{55} } \right) \mathop \sum \limits _{k=1}^{{\bar{N}} } kW_s \left( k \right) =0 \end{aligned}$$
(30d)

Since the values of the transformation terms \(W_b \left( k \right) \) and \(W_s \left( k \right) \) are known for \(k=0\hbox { and }2\) as given by Eq. (27), one can write the terms for \(k=1\,\hbox {and}\,3\) in terms of some unknown constants \(\zeta \) and \(\eta \) as follows.

$$\begin{aligned} W_b \left( 1 \right) =\zeta _{b,} W_b \left( 3 \right) =\eta _b , W_s \left( 1 \right) =\zeta _s \hbox { and } W_s \left( 3 \right) =\eta _s . \end{aligned}$$
(31)

where the subscripts b and s stand for the bending and shear, respectively.

Using the recursive formulas given by Eq. (25) and values of the terms \(W_b \left( k \right) \) and \(W_s \left( k \right) \) for \(k=0, 1, 2, 3\) given by Eqs. (27) and (31), one can find that, \(W_b \left( 4 \right) =0\) and \(W_s \left( 4 \right) =0\). Similarly, \(W_b \left( 5 \right) \) and \(W_s \left( 5 \right) \) are obtained as follows.

$$\begin{aligned} W_b \left( 5 \right) =\left( {c_2 \zeta _b +c_3 \zeta _s +6c_1 \eta _b } \right) /120,\qquad \quad W_s \left( 5 \right) =\left( {c_6 \zeta _b +c_5 \zeta _s +6c_4 \eta _s } \right) /120 \end{aligned}$$
(32)

Continuing in similar manner, the subsequent terms are obtained in terms of \(\zeta _{b,} \eta _b ,\zeta _s \) and \(\eta _s \) as shown in Table 2. These were evaluated up to the \({\bar{N}} th\) terms (i.e., \(W_b \left( {{\bar{N}} } \right) \) and \(W_s \left( {{\bar{N}} } \right) )\).

Table 2 Values of the terms \(W_b \left( k \right) \) and \(W_s \left( k \right) \) in the DTM formulation
Full size table

Substituting the terms from \(W_b(0)\) and \(W_s(0)\) to \(W_b({\bar{N}})\) and \(W_s({\bar{N}})\) in Eqs. (28), (29) and (30) for the SF, SS and SC plates respectively, the following system of algebraic equations is obtained.

$$\begin{aligned}&\left[ {\begin{array}{l} a_{11} \mathop \sum \limits _{k=3}^{{\bar{N}}} \mathrm{Coeff.}\left[ {W_b \left( k \right) ,\zeta _b } \right] \,\,a_{12} \mathop \sum \limits _{k=3}^{{\bar{N}}} \mathrm{Coeff.}\left[ {W_b \left( k \right) ,\eta _b } \right] \,\,a_{13} \mathop \sum \limits _{k=3}^{{\bar{N}}} \mathrm{Coeff.}\left[ {W_b \left( k \right) ,\zeta _s } \right] \,\,a_{14} \mathop \sum \limits _{k=3}^{\mathop {{\bar{N}}}\limits } \mathrm{Coeff.}\left[ {W_b \left( k \right) ,\eta _s } \right] \\ a_{21} \mathop \sum \limits _{k=3}^{{\bar{N}}} \mathrm{Coeff.}\left[ {W_s \left( k \right) ,\zeta _b } \right] \,\,a_{22} \mathop \sum \limits _{k=3}^{{\bar{N}}} \mathrm{Coeff.}\left[ {W_s \left( k \right) ,\eta _b } \right] \,\,a_{23} \mathop \sum \limits _{k=3}^{{\bar{N}}} \mathrm{Coeff.}\left[ {W_s \left( k \right) ,\zeta _s } \right] \,\,a_{24} \mathop \sum \limits _{k=3}^{{\bar{N}}} \mathrm{Coeff.}\left[ {W_s \left( k \right) ,\eta _s } \right] \\ a_{31} \mathop \sum \limits _{k=3}^{{\bar{N}}} \mathrm{Coeff.}\left[ {W_b \left( k \right) ,\zeta _b } \right] \,\,a_{32} \mathop \sum \limits _{k=3}^{{\bar{N}}} \mathrm{Coeff.}\left[ {W_b \left( k \right) ,\eta _b } \right] \,\,a_{33} \mathop \sum \limits _{k=3}^{{\bar{N}}} \mathrm{Coeff.}\left[ {W_b \left( k \right) ,\zeta _s } \right] \,\,a_{34} \mathop \sum \limits _{k=3}^{{\bar{N}}} \mathrm{Coeff.}\left[ {W_b \left( k \right) ,\eta _s } \right] \\ a_{41} \mathop \sum \limits _{k=3}^{{\bar{N}}} \mathrm{Coeff.}\left[ {W_s \left( k \right) ,\zeta _b } \right] \,\,a_{42} \mathop \sum \limits _{k=3}^{{\bar{N}}} \mathrm{Coeff.}\left[ {W_s \left( k \right) ,\eta _b } \right] \,\,a_{43} \mathop \sum \limits _{k=3}^{{\bar{N}}} \mathrm{Coeff.}\left[ {W_s \left( k \right) ,\zeta _s } \right] \,\,a_{44} \mathop \sum \limits _{k=3}^{{\bar{N}}} \mathrm{Coeff.}\left[ {W_s \left( k \right) ,\eta _s } \right] \\ \end{array}} \right] \left[ {\begin{array}{l} \zeta _b \\ \eta _b \\ \zeta _s \\ \eta _s \\ \end{array}} \right] \nonumber \\&\quad =\left[ {\begin{array}{l} 0 \\ 0 \\ 0 \\ 0 \\ \end{array}} \right] \end{aligned}$$
(33)

The summation symbol \(\mathop \sum \nolimits _{k=3}^{{\bar{N}} } \mathrm{Coeff.}\left[ {W_b \left( k \right) ,\zeta _b } \right] \) used, signifies the sum of coefficients of \(\zeta _b \) in the terms \(W_b \left( k \right) \left\{ {k=3,4,\ldots ,{\bar{N}} } \right\} \). Definition of other elements in the matrix of Eq. (33) follows similar convention. Values of the terms \(a_{ij} ,\left\{ {i,j=1,2,3,4} \right\} \) depend on the boundary condition of the plate and are given in the appendix.

Determinant of the \(4\times 4\) coefficient matrix of Eq. (33) is obtained in each case of the plate with different boundary conditions and equated to zero in order to avoid trivial solution. Solving the resulting equation(s) yields the natural frequency \(\omega _n \).

4 Taylor collocation method (TCM)

Collocation method refers to a family of various numerical solution techniques of differential equations based on selecting a candidate solution that, discretely, satisfies the governing equation and boundary conditions at certain number of domain and boundary points (collocation points), respectively. What differs between such methods is the choice of candidate solutions. The radial basis function (RBF) collocation method, for example, uses some suitable RBF centered at a given number of points. The TCM, on the other hand, makes use of Taylor polynomials derived from the Taylor series expansion of the candidate solution.

Taylor series was discovered to be useful in solving integral equations encountered in physical sciences and engineering by past researches such as [67]. The formalized approach has been presented by Kanwal and Liu [68]. Later, extension of the method to solve different kinds of differential equations and their systems has been carried out by different researchers, with the work reported in [69] as one of the early ones. Analysis of plates using this method is not very popular. However, its recent application to axisymmetric plates and shells have proved successful [70]. The concept of TCM is similar to that of DTM with the exception that the former is fully numerical while the latter is semi-analytical. In other words, contrary to the evaluation of coefficients of the Taylor series expansion by some recursive formulas in the DTM, in TCM the domain of interest needs a physical discretization followed by a collocation procedure to generate a number of algebraic equations equal to the number of the Taylor series coefficients that are finally determined by solving the system of equations so generated.

4.1 TCM formulation

The basic principle of TCM and its formulation are explained in terms of a general governing equation (of a dependent variable \(w\left( x \right) )\), over the domain \({\Omega }\), described by a domain operator L subject to some generic boundary conditions, on the boundary \({\Gamma }\left( {\hbox {a}_0 ,\hbox {a}_1 } \right) \), described by a boundary operator B as given by Eq. (34). f and g are some continuous functions of the position variable.

(34a)
(34b)

To use the TCM, we seek a solution of Eq. (34) in the form given by Eq. (35), a Taylor series expansion of w around point c. As a priori, the function w is assumed to have nth derivatives in the interval of expansion.

$$\begin{aligned} w\left( x \right) =\mathop \sum \limits _{n=0}^\infty \frac{w^{\left( n \right) }\left( c \right) }{n!}\left( {x-c} \right) ^{n},\qquad \hbox {a}_0 \le x, c\le \hbox {a}_1 , \end{aligned}$$
(35)

The above infinite series can be truncated at \(n={\bar{N}} \) (where \({\bar{N}} \) is chosen large enough to ensure the series convergence) as shown in Eq. (36).

$$\begin{aligned} w\left( x \right) =\mathop \sum \limits _{n=0}^{{\bar{N}} } \frac{w^{\left( n \right) }\left( c \right) }{n!}\left( {x-c} \right) ^{n},\qquad \hbox {a}_0 \le x,c\le \hbox {a}_1 , \end{aligned}$$
(36)

To find the \({\bar{N}} +1\) unknowns (\(w^{\left( n \right) }\left( c \right) )\) in Eq. (36), the geometry of the problem is modeled by \({\bar{N}} +1\) randomly distributed nodes such that \({\bar{N}} +1=N_d +N_b \), where \(N_d \) and \(N_b \) represent the domain and boundary nodes, respectively. Applying Eqs. (34a) and (34b) at the \(N_d \) discrete nodes in the domain and \(N_b \) nodes on the boundary, respectively, results in \({\bar{N}} +1\) algebraic equations as shown below.

(37a)
(37b)

Solving the above system of equations yields the values of the coefficient \(w^{\left( n \right) }\left( c \right) \) for \(n=0,1,2,\ldots {\bar{N}} \). The formal procedure for the free vibration analysis based on TCM is given in the following section.

4.2 Free vibration analysis based on TCM

It can be recognized that the two fourth-order governing equations for free vibration of orthotropic plates, given by Eq. (11), can be written in the general form given by Eq. (38).

$$\begin{aligned} \mathop \sum \limits _{k=0}^4 \left( {P_{kj} w_b ^{\left( k \right) }\left( {{\bar{x}}} \right) +Q_{kj} w_s ^{\left( k \right) }\left( {{\bar{x}}} \right) } \right) =0,\qquad j=1,\hbox {}2, \quad 0\le {\bar{x}}\le 1 \end{aligned}$$
(38)

where the coefficients P and Q are given, in terms of \(c_i \left\{ {i,1,6} \right\} \) (Eq. 26), thus

\(P_{01} =-c_2 ;P_{21} =-c_1 ;Q_{01} =-c_3 ;Q_{02} =-c_5 ;Q_{22} =-c_4 ;P_{02} =-c_6 ;P_{41} =Q_{42} =1\) and

\(P_{kj} =Q_{kj} =0\) for combinations of k and j otherwise.

Similarly, the general form of the eight boundary conditions (Table 1) can be written as follows.

$$\begin{aligned} \mathop \sum \limits _{j=0}^3 \left( {\alpha _{ij} w_b ^{\left( j \right) }\left( 0 \right) +\beta _{ij} w_b ^{\left( j \right) }\left( 1 \right) +\gamma _{ij} w_s ^{\left( j \right) }\left( 0 \right) +\kappa _{ij} w_s ^{\left( j \right) }\left( 1 \right) } \right) =0,\quad \hbox {}i=0,\hbox {}1,\hbox {}2,\ldots ,7\hbox {} \end{aligned}$$
(39)

The coefficients \(\alpha ,\beta ,\gamma \) and \(\kappa \) in Eq. (39) depend on the actual boundary condition thus.

  1. i.

    At \({\bar{x}}=0\), \(\beta _{ij} =\kappa _{ij} =0\) for all i and j while \(\alpha \) and \(\gamma \) are common for SC, SS and SF plate types as given below.

    $$\begin{aligned} \alpha _{00} =\gamma _{10} =1;\quad \alpha _{22} =-D_{11} ,\alpha _{20} =D_{12} \beta ^{2};\quad \gamma _{32} =-\left( {\frac{1}{84}} \right) \left( {D_{11} } \right) ;\quad \hbox {}\gamma _{30} =\left( {1/84} \right) D_{12} \beta ^{2}\hbox {} \end{aligned}$$

    \(\alpha _{ij} =\gamma _{ij} =0,\) for combinations of i and j otherwise.

  2. ii.

    At \({\bar{x}}=1, \quad \alpha _{ij} =\gamma _{ij} =0\) for all i and j, while \(\beta \) and \(\kappa \) are given, depending on the fixity condition, as follows.

C-type boundary :

$$\begin{aligned} \beta _{40} =\kappa _{50} =\beta _{61} =\kappa _{71} =1 \end{aligned}$$

\(\beta _{ij} =\kappa _{ij} =0,\) for combinations of i and j otherwise.

S-type boundary :

$$\begin{aligned} \beta _{40} =\kappa _{50} =1;\quad \beta _{62} =-D_{11} ;\quad \beta _{60} =D_{12} \beta ^{2};\quad \kappa _{72} =-\left( {\frac{1}{84}} \right) \left( {D_{11} } \right) ;\quad \kappa _{70} =\left( {1/84} \right) D_{12} \beta ^{2} \end{aligned}$$

\(\beta _{ij} =\kappa _{ij} =0,\) for combinations of i and j otherwise.

F-type boundary :

$$\begin{aligned} \beta _{42}= & {} \beta _{63} =\kappa _{73} =-D_{11} ;\quad \beta _{40} =D_{12} \beta ^{2};\quad \kappa _{52} =-\left( {\frac{1}{84}} \right) \left( {D_{11} } \right) ;\quad \kappa _{50} =\left( {\frac{1}{84}} \right) D_{12} \beta ^{2};\\ \beta _{61}= & {} \left( {\left( {D_{12} +4D_{66} } \right) \beta ^{2}-I_2 \omega _n^2 } \right) ;\quad \kappa _{71} =\left( {\left( {D_{12} +4D_{66} } \right) \beta ^{2}-I_2 \omega _n^2 +84A_{55} } \right) \\ \beta _{ij}= & {} \kappa _{ij} =0, \hbox { for combinations } of \,i \hbox { and } j \hbox { otherwise}. \end{aligned}$$

Equation (40) gives the approximate solutions for the bending and shear components of the transverse displacement, written as two truncated Taylor series expansions around \({\bar{c}} \) in each case. \({\bar{N}} \) is selected large enough to ensure the series convergence.

$$\begin{aligned} w_b \left( x \right)= & {} \mathop \sum \limits _{n=0}^{{\bar{N}} } \frac{w_b ^{\left( n \right) }\left( {{\bar{c}} } \right) }{n!} \left( {{\bar{x}}-{\bar{c}} } \right) ^{n},\qquad \qquad 0\le {\bar{x}},{\bar{c}} \le 1,\quad N \ge 3 \end{aligned}$$
(40a)
$$\begin{aligned} w_s \left( x \right)= & {} \mathop \sum \limits _{n=0}^{{\bar{N}} } \frac{w_s ^{\left( n \right) }\left( {{\bar{c}} } \right) }{n!}\left( {{\bar{x}}-{\bar{c}} } \right) ^{n},\qquad \qquad 0\le {\bar{x}},{\bar{c}} \le 1,\quad N\ge 3 \end{aligned}$$
(40b)

A set of \({\bar{x}}_i ,\left( {i=0,1,2,\ldots ,{\bar{N}} } \right) \) collocation points at some specified intervals within the problem domain is used in order to find the unknown Taylor coefficients \(w_b ^{\left( n \right) }\left( {{\bar{c}} } \right) \) and \(w_s ^{\left( n \right) }\left( {{\bar{c}} } \right) \quad \left\{ {n=0,1,2,\ldots {\bar{N}} } \right\} \). However, because there are two boundary conditions at each of the two boundary nodes, this will result in two extra equations more than the number of unknowns. Therefore, the present work chooses to create two more virtual nodes outside the domain, one adjacent to each of the boundary nodes at a distance \(d{\bar{x}}\). Consequently, the following range for \({\bar{x}}_i \) results.

$$\begin{aligned} -d{\bar{x}}={\bar{x}}_0<{\bar{x}}_1<\cdots {\bar{x}}_{N-1} <{\bar{x}}_N =1+d{\bar{x}} \end{aligned}$$

where, for a uniform interval, the value of \({\bar{x}}_i \) becomes

$$\begin{aligned} {\bar{x}}_i =-\mathrm{d}{\bar{x}}+i\frac{1+2\mathrm{d}{\bar{x}}}{{\bar{N}} },i=0,1,2,\ldots {\bar{N}} \end{aligned}$$
(41)

Writing Eq. (40) in terms of matrices, one obtains.

$$\begin{aligned} w_b \left( {{\bar{x}}} \right)= & {} {\bar{X}} M_0 A_b \end{aligned}$$
(42a)
$$\begin{aligned} w_s \left( {{\bar{x}}} \right)= & {} {\bar{X}} M_0 A_s \end{aligned}$$
(42b)

where

$$\begin{aligned} {\bar{X}}= & {} \left[ {1\,\left( {{\bar{x}}-{\bar{c}} } \right) \,\left( {{\bar{x}}-{\bar{c}} } \right) ^{2}\,\cdots \,\left( {{\bar{x}}-{\bar{c}} } \right) ^{{\bar{N}} }} \right] \\ M_0= & {} \left[ {{\begin{array}{ccccc} \frac{1}{0!} &{}0 &{}0 &{}\cdots &{}0 \\ 0&{}\frac{1}{1!}&{}0&{}\cdots &{}0 \\ 0&{}0&{}\frac{1}{2!}&{}\cdots &{} 0 \\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \\ 0&{}0&{}0&{}\cdots &{}\frac{1}{{\bar{N}} !} \\ \end{array} }} \right] \\ A_b= & {} \left[ {W_b ^{\left( 0 \right) }\left( {{\bar{c}} } \right) \,W_b ^{\left( 1 \right) }\left( {{\bar{c}} } \right) \,W_b ^{\left( 2 \right) }\left( {{\bar{c}} } \right) \,\cdots \,W_b ^{\left( {{\bar{N}} } \right) }\left( {{\bar{c}} } \right) } \right] ^{T}\\ A_s= & {} \left[ {W_s ^{\left( 0 \right) }\left( {{\bar{c}} } \right) \,W_s ^{\left( 1 \right) }\left( {{\bar{c}} } \right) \,W_s ^{\left( 2 \right) }\left( {{\bar{c}} } \right) \,\cdots \,W_s ^{\left( {{\bar{N}} } \right) }\left( {{\bar{c}} } \right) } \right] ^{T} \end{aligned}$$

Applying Eq. (42) at \({\bar{x}}_i \) collocation points yields

$$\begin{aligned} w_b \left( {{\bar{x}}_i } \right)= & {} {\bar{X}} _i M_0 A_b\qquad i=0,1,2,\ldots ,{\bar{N}}\end{aligned}$$
(43a)
$$\begin{aligned} w_s \left( {{\bar{x}}_i } \right)= & {} {\bar{X}} _i M_0 A_s \qquad i=0,1,2,\ldots ,{\bar{N}} \end{aligned}$$
(43b)

where

$$\begin{aligned} {\bar{X}} _i =\left[ {1\,\left( {{\bar{x}}_i -{\bar{c}} } \right) \,\left( {{\bar{x}}_i -{\bar{c}} } \right) ^{2}\,\cdots \,\left( {{\bar{x}}_i -{\bar{c}} } \right) ^{{\bar{N}} }} \right] \end{aligned}$$

The vector of zero derivatives of the solution can be denoted as follows

$$\begin{aligned} w_b ^{\left( 0 \right) }= & {} \left[ {w_b \left( {{\bar{x}}_0 } \right) w_b \left( {{\bar{x}}_1 } \right) \,w_b \left( {{\bar{x}}_2 } \right) \cdots w_b \left( {{\bar{x}}_{{\bar{N}} } } \right) } \right] ^{T},\\ w_s ^{\left( 0 \right) }= & {} \left[ {w_s \left( {{\bar{x}}_0 } \right) w_s \left( {{\bar{x}}_1 } \right) \,w_s \left( {{\bar{x}}_2 } \right) \cdots w_s \left( {{\bar{x}}_{{\bar{N}} } } \right) } \right] ^{T} \end{aligned}$$

Hence, Eq. (43) can be written in the form shown below.

$$\begin{aligned} w_b ^{\left( 0 \right) }= & {} CM_0 A_b \end{aligned}$$
(44a)
$$\begin{aligned} w_s ^{\left( 0 \right) }= & {} CM_0 A_s \end{aligned}$$
(44b)

where

$$\begin{aligned} C = \left[ {\bar{X}_{0} \,\bar{X}_{1} \, \cdots \,\bar{X}_{{\bar{N}}} } \right] ^{T} = \left[ {\begin{array}{ccccc} 1&{}\left( {\bar{x}_{0} - \bar{c}} \right) &{}\left( {\bar{x}_{0} - \bar{c}} \right) ^{2} &{} \cdots &{}\left( {\bar{x}_{0} - \bar{c}} \right) ^{{\bar{N}}} \\ 1&{}\left( {\bar{x}_{1} - \bar{c}} \right) &{}\left( {\bar{x}_{1} - \bar{c}} \right) ^{2} &{} \cdots &{}\left( {\bar{x}_{1} - \bar{c}} \right) ^{{\bar{N}}}\\ \vdots &{} \vdots &{} \vdots &{} \cdots &{} \vdots \\ 1&{}\left( {\bar{x}_{{\bar{N}}} - \bar{c}} \right) &{}\left( {\bar{x}_{{\bar{N}}} - \bar{c}} \right) ^{2} &{} \cdots &{}\left( {\bar{x}_{{\bar{N}}} - \bar{c}} \right) ^{{\bar{N}}} \\ \end{array}} \right] \end{aligned}$$

Similarly, the vector of the derivatives becomes

$$\begin{aligned} w_b ^{\left( k \right) }= & {} CM_k A_b \end{aligned}$$
(45a)
$$\begin{aligned} w_s ^{\left( k \right) }= & {} CM_k A_s \end{aligned}$$
(45b)

where

$$\begin{aligned} w_b ^{\left( k \right) }= & {} \left[ {w_b ^{\left( k \right) }\left( {{\bar{x}}_0 } \right) \,w_b ^{\left( k \right) }\left( {{\bar{x}}_1 } \right) \,w_b ^{\left( k \right) }\left( {{\bar{x}}_2 } \right) \,\cdots \,w_b ^{\left( k \right) }\left( {{\bar{x}}_{{\bar{N}} } } \right) } \right] ^{T},\\ w_s ^{\left( k \right) }= & {} \left[ {w_s ^{\left( k \right) }\left( {{\bar{x}}_0 } \right) \,w_s ^{\left( k \right) }\left( {{\bar{x}}_1 } \right) \,w_s ^{\left( k \right) }\left( {{\bar{x}}_2 } \right) \,\cdots \,w_s ^{\left( k \right) }\left( {{\bar{x}}_{{\bar{N}} } } \right) } \right] ^{T} \end{aligned}$$

Applying Eq. (38) at the discrete interior points (\(i=1,2,\ldots ,{\bar{N}} -1)\) yields

$$\begin{aligned} \mathop \sum \limits _{k=0}^4 \left( {P_{kj} w_b ^{\left( k \right) }\left( {{\bar{x}}_i } \right) +Q_{kj} w_s ^{\left( k \right) }\left( {{\bar{x}}_i } \right) } \right) =0,\quad j=1,\hbox {}2, \end{aligned}$$
(46)

or, equivalently

$$\begin{aligned} \mathop \sum \limits _{k=0}^4 \left( {P_{kj} w_b ^{\left( k \right) }+Q_{kj} w_s ^{\left( k \right) }} \right) =0,\quad j=1,\hbox {}2, \end{aligned}$$
(47)

Substituting Eq. (45) in Eq. (47) yields

$$\begin{aligned} \left[ {\mathop \sum \limits _{k=0}^4 P_{kj} CM_k } \right] A_b +\left[ {\mathop \sum \limits _{k=0}^4 Q_{kj} CM_k } \right] A_s =\left[ 0 \right] ,\qquad j=1, 2,\quad 0\le {\bar{x}}\le 1 \end{aligned}$$
(48)

Since the system of equations given by Eq. (48) is a consequence of collocating over the interior nodes to satisfy Eq. (38), it therefore represents a set of \(\left( {{\bar{N}} -i} \right) \) equations involving \(\left( {{\bar{N}} +1} \right) \) unknowns. Hence, the remaining \(\left( {i+1} \right) \) equations are generated as given in Eq. [49] by considering the boundary conditions given by Eq. (39).

$$\begin{aligned} \left[ {\mathop \sum \limits _{j=0}^3 \left( {\alpha _{ij} X_0 M_j +\beta _{ij} X_{{\bar{N}} } M_j } \right) } \right] A_b +\left[ {\mathop \sum \limits _{j=0}^3 \left( {\gamma _{ij} X_0 M_j +\kappa _{ij} X_{{\bar{N}} } M_j } \right) } \right] A_s =\left[ 0 \right] \end{aligned}$$
(49)

The overall set of the system of equations required to solve for the vector \(\hbox {A}=\left[ {A_b ,A_s } \right] ^{T}\) of the unknown coefficients is given by Eq. (50) as a combination of domain and boundary equations.

$$\begin{aligned} \left[ {{\begin{array}{ll} {\left[ {\mathop \sum \limits _{k=0}^4 P_{kj} CM_k } \right] }&{} {\left[ {\mathop \sum \limits _{k=0}^4 Q_{kj} CM_k } \right] } \\ {\left[ {\mathop \sum \limits _{j=0}^3 \left( {\alpha _{ij} X_0 M_j +\beta _{ij} X_{{\bar{N}} } M_j } \right) } \right] }&{} {\left[ {\mathop \sum \limits _{j=0}^3 \left( {\gamma _{ij} X_0 M_j +\kappa _{ij} X_{{\bar{N}} } M_j } \right) } \right] } \\ \end{array} }} \right] \left[ {{\begin{array}{ll} \\ {A_b } \\ \\ \\ \\ {A_s } \\ \\ \end{array} }} \right] =\left[ {{\begin{array}{l} \\ 0 \\ \\ \\ \\ 0 \\ \\ \end{array} }} \right] , \end{aligned}$$
(50)

The natural frequency parameters \({\bar{\omega }} _n \) are obtained by equating determinant of the \(2\left( {{\bar{N}} +1} \right) \times 2\left( {{\bar{N}} +1} \right) \) coefficient matrix of Eq. (50) to zero and solving the resulting equation.

5 Results and discussion

By implementing the DTM and TCM formulations presented in the previous sections, results for free vibration analysis of orthotropic rectangular plate, based on the present theory (the RPT), with boundary conditions as given in Table 1 are presented in the first part of the present section.

Comparison of the predicted results with those of other theories is, first, illustrated in Sect. 5.1 for orthotropic plates whose exact solution is available in the literature. Section 5.2 presents an extension of the problem to cover SF, SC and SS rectangular plates. Convergence study for both DTM and TCM is given in Sect. 5.2.1, followed by a parametric study, in Sect. 5.2.2, to illustrate the effect of different values of the thickness ratio (a / h), the aspect ratio (a / b) and the modular ratio (\(E_1 /E_2 )\). For the sake of verification, analytical results reported in [35] are given in each case.

Results for some cases whose analytical solutions are not readily available, particularly based on the two-variable refined plate theory, are presented in Sect. 5.2.3. Formulations of three more plate theories, the CPT, the FSDT [10, 60, 61] and the HSDT [62], were implemented and solved using both the DTM and TCM for verification purposes. Similar to the case of RPT presented in the present work, the high-order theory in [62] considers both the displacements in x and y directions, u and v, respectively, to be composed of shear and bending components. However, while the bending components in [62] are similar to that of CPT, the shear components are assumed to be exponential in nature with respect to the thickness coordinate. For that reason, their theory is also referred to as the exponential shear deformation theory.

5.1 RPT-based DTM and TCM versus other plate theories

In order to demonstrate the performance of DTM and TCM based on the RPT, a simply supported square orthotropic plate made from same material as that reported in the reference of Reddy [17] and, later, adopted in [30, 35], is considered. The material properties are \(E_1 =20,830ksi,E_2 =10,940ksi,G_{12} =6,100ksi,G_{13} =3,710ksi,\upsilon _{12} =0.44\) and \(\upsilon _{21} =0.23\) .

Three more plate theories, namely CPT, FSDT [10, 60, 61] and HSDT [62], were also formulated, implemented and solved using both the DTM and TCM in order to demonstrate differences between models and advantages of the proposed analysis. Details of the governing equations of these theories can be found in the respective mentioned references.

As shown in Table 3, natural frequency parameters \({\bar{\omega }} _n =\omega _n h\sqrt{\rho /Q_{11} }\) predicted by DTM and TCM based on the present theory compare favorably with the exact solution [71] based on 3D elasticity theory and the HSDT [62] with a maximum deviation from the exact solution of 0.67 % (on the conservative side). However, it can be noticed that, while the CPT overestimates the natural frequencies, the FSDT results are a little less conservative compared to the present theory, at least for the orthotropic plate considered here. Although so far there is no available plate theory that is both as simple as the CPT and as accurate (in all cases) as the other more complex higher-order theories, the RPT attempts to satisfy such qualities to some extents. For instance, its formulation resembles that of the CPT and, once the shear terms are dropped, the governing equations (Eq. 11) of the RPT turn, exactly, to the equation of CPT. Second, the elimination for the need of shear correction as required in some shear deformation theories gives the present theory an additional advantage because it still satisfies the zero-traction boundary condition at the top and bottom surfaces of the plate. In addition, the resulting system of equations are easier to solve than those of other higher-order theories.

With regard to the method of solution, the results shown in Table 3 based on DTM and TCM agree perfectly with those reported in Refs. [30, 35]. However, while both the DTM and TCM require lower \(\bar{N}\) value to achieve convergence for all the four modes based on RPT, the \(\bar{N}\) value needed in the FSDT and HSDT solution increases with increase in the mode and becomes much higher than those needed in RPT. This is not surprising considering the complexity of the governing equations of FSDT and HSDT (available in the respective references) compared to those of RPT (Eq. 11) and CPT.

By extending the analysis to cover additional plate types in terms of boundary conditions, geometric dimensions and modular ratios, numerical results in the subsequent subsections are used to elaborate more on the capability of the two methods of solution (DTM and TCM) proposed in this research.

Table 3 Natural frequency parameters \({\bar{\omega }} _n =\omega _n h\sqrt{\rho /Q_{11} }\) of simply supported square plate with \(\frac{a}{h}=10\)
Full size table

5.2 Extension to general rectangular plates with different boundary conditions

In this section, consideration is, initially, given to SF, SS and SC orthotropic plates having generic rectangular shapes with various thickness ratios. Non-dimensionalized material properties [35, 72] are assumed thus: \(G_{12} /E_2 =G_{13} /E_2 =0.5,G_{23} /E_2 =0.2,\upsilon _{12} =0.25\) and \(E_1 /E_2 \) is varied. First, convergence studies for the DTM and TCM are carried out, followed by parametric studies. Next, multi-span plates and plates with stepped thickness and end rotational springs are considered.

5.2.1 Convergence studies

In order to demonstrate the convergence behavior of DTM and TCM applied to SF, SS and SC plates, nine cases of rectangular orthotropic plates are considered for each of SF, SS and SC plates, resulting in a total of 27 cases. The convergence results for the fundamental natural frequency parameters \({\bar{\omega }} =\omega \frac{a^{2}}{h}\sqrt{\rho /E_2 }\) illustrating effects of the aspect ratio, thickness ratio and modular ratio are, respectively, presented in Tables 4, 5 and 6. Results predicted based on CPT and the analytical solution reported in [35] are also shown. Good agreement exists between the DTM, TCM and analytical results, with the TCM resulting in lower percentage errors almost consistently. However, it should be noted that, based on the CPT, both DTM and TCM provide solutions having the same accuracy. In terms of computational demand, more computer memory is needed in TCM since a system matrix of size \(2\left( {{\bar{N}} +1} \right) \times 2\left( {{\bar{N}} +1} \right) \) needs to be handled, while DTM formulation results in only a \(4\times 4\) matrix size and, hence, it is less resource-consuming.

The purpose of using different values of a / h is to demonstrate the effect of plate thickness on the result. It is evident, from Table 5, that the magnitude of the error in CPT is higher for lower thickness ratios \(a/h=5\,\mathrm{and}\,20\) (i.e., thick plates). This is expected due to the fact that CPT formulation neglects the effect of shear which, on the other hand, is taken care of in RPT. Shear effects are only negligible in case of thin plates, and consequently, the error in the plates with higher values of a / h (such as 100) becomes negligible in the predictions by CPT.

Increase in the modular ratio also causes increase in the shear effect, particularly for the SS and SC plates. That is why, as given in Table 6, error in the solution based on CPT also increases with increase in \(E_1 /E_2 \). Such effect is, however, not observed in the SF plate because the free end relaxes the stiffness of the plate which, in turn, causes less shear effect.

Table 4 Convergence of fundamental frequency parameters \({\bar{\omega }} =\omega \frac{a^{2}}{h}\sqrt{\rho /E_2 }\) for plates with \(\frac{a}{h}=50\) and \(\frac{E_1 }{E_2 }=10\)
Full size table
Table 5 Convergence of fundamental frequency parameters \({\bar{\omega }} =\omega \frac{a^{2}}{h}\sqrt{\rho /E_2 }\) for plates with \(\frac{a}{b}=2.0\) and \(\frac{E_1 }{E_2 }=10\)
Full size table
Table 6 Convergence of fundamental frequency parameters \({\bar{\omega }} =\omega \frac{a^{2}}{h}\sqrt{\rho /E_2 }\) for plates with \(\frac{a}{b}=0.5\) and \(\frac{a}{h}=100\)
Full size table

5.2.2 Parametric studies

Apart from the 27 cases analyzed in Sect. 5.2.1 above, a comprehensive parametric study consisting of 45 cases with variations in the material modular ratio and geometric ratios is carried out for each of the SF, SS and SC plate types. Hence, a total of 135 cases with the DTM and TCM applied to obtain solutions are presented. The end results of the fundamental natural frequency parameters \({\bar{ \omega }} =\omega \frac{a^{2}}{h}\sqrt{\rho /E_2 }\) are presented in Tables 7, 8 and 9 for the SF, SS and SC plates, respectively. Variables targeted for the parametric study comprise of different values of the thickness ratio (a / h), the aspect ratio (a / b) and the modular ratio (\(E_1 /E_2 )\). These results cover other ranges of the variables not covered in Sect. 5.2.1, and they are presented in a more definite pattern to enhance understanding the obvious roles played by each of the variables. Value of \({\bar{N}}\) used in each method is indicated as well as the error computed with respect to the analytical results available in [35].

It can be noticed that, throughout the results presented in Tables 7, 8 and 9, the effect of increasing the aspect ratio and thickness ratio is to cause a corresponding increase in the fundamental natural frequency parameter, hence, increase in the fundamental natural frequency for all the plate types studied. Similarly, the effect of increasing the modular ratio causes an increase in the fundamental natural frequency parameter for the SS and SC plates. On the other hand, no such effect is observed in the case of SF plate because of its free end which lowers the plate’s stiffness. Denoting the fundamental natural frequency parameters for the SF, SS and SC plates as \({\bar{\omega }} _{SF} \), \({\bar{\omega }} _{SS} \) and \({\bar{\omega }} _{SC} \), respectively, it can be seen that effect of end fixity suggests that, under the same condition (same thickness ratio, aspect ratio and modular ratio), values of the fundamental natural frequency parameters have magnitudes in the order \({\bar{\omega }} _{SC}>{\bar{\omega }} _{SS} >{\bar{\omega }} _{SF} \). This is also confirmed by a general analysis performed for arbitrary geometry and for elastic stiffeners along the part of the boundary in [73]. It is logical to have such behavior due to the fact that increase in fixity results in higher reactive effects, in form of increase in the stiffness, thereby amplifying the natural frequency.

It is worth mentioning that the both the two numerical methods, DTM and TCM, adopted in obtaining these solutions do not require iterations as is the case with the analytical results reported in [35]. Additionally, unlike other element-based numerical techniques, the DTM and TCM can be used to present solutions in symbolic form which warrants the possibility of obtaining other secondary variables by differentiation, integration, etc. Also, the obvious advantage of eliminating the need for method calibration as required in other meshless methods, such as the Kansa method and method of fundamental solutions, adds to the appealing nature of the presented methods.

Table 7 Fundamental frequency parameters \({\bar{\omega }} =\omega \frac{a^{2}}{h}\sqrt{\rho /E_2 }\) for SF plate
Full size table
Table 8 Fundamental frequency parameters \({\bar{\omega }} =\omega \frac{a^{2}}{h}\sqrt{\rho /E_2 }\) for SS plate
Full size table
Table 9 Fundamental frequency parameters \({\bar{\omega }} =\omega \frac{a^{2}}{h}\sqrt{\rho /E_2 }\) for SC plate
Full size table

5.2.3 Multi-span plates and plates with stepped thickness and end rotational springs

Since the problems presented in Sects. 5.2.1 and 5.2.2 are for single-span plates analyzed by the proposed methods and verified using an existing analytical solution, the present section aims at presenting results for some cases whose analytical solution is not readily available particularly based on the two-variable refined plate theory. For the sake of verification, formulation of three more plate theories, the CPT, the FSDT [10, 60, 61] and the HSDT [62], was implemented and solved using both the DTM and TCM. These are the same plate theories against which comparison of the results in Sect. 5.2.1 was made.

The analysis starts by, firstly, considering a multi-span Levy plate, consisting of (\(n-1\)) internal supports. Various engineering applications of such plates exist, such as in bridge decks, floor slabs, aeroplane parts, etc. Such problems can be, conveniently, modeled as plates with internal line supports are shown in Fig. 2. The problem now goes beyond just satisfying the governing equations in a span and the external boundary conditions as is the case in the analysis carried out in the previous sections. In this case, the governing equations must be satisfied in each of the “n” spans in turn. Also, in addition to the boundary conditions given in Table 1, the compatibility conditions given by Eqs. (51) to (58) must hold at the interface between the ith and (\(i+1\))th span in order to ensure continuity at the location of the internal supports. Consequently, the system of equations that needs to be handled becomes different and much more involving than the case of single-span plates earlier analyzed.

$$\begin{aligned} \left( {w_b } \right) _i= & {} 0 \end{aligned}$$
(51)
$$\begin{aligned} \left( {w_s } \right) _i= & {} 0 \end{aligned}$$
(52)
$$\begin{aligned} \left( {w_b } \right) _{i+1}= & {} 0 \end{aligned}$$
(53)
$$\begin{aligned} \left( {w_s } \right) _{i+1}= & {} 0 \end{aligned}$$
(54)
$$\begin{aligned} \left( {w_b^{\prime } } \right) _i= & {} \left( {w_b^{\prime } } \right) _{i+1} \end{aligned}$$
(55)
$$\begin{aligned} \left( {w_s^{\prime } } \right) _i= & {} \left( {w_s^{\prime } } \right) _{i+1} \end{aligned}$$
(56)
$$\begin{aligned} \left( {-D_{11} w_b^{{\prime }{\prime }} +D_{12} \beta ^{2}\hbox {}w_b } \right) _i= & {} \left( {-D_{11} w_b^{{\prime }{\prime }} +D_{12} \beta ^{2}\hbox {}w_b } \right) _{i+1} \end{aligned}$$
(57)
$$\begin{aligned} \left( {-\left( {1/84} \right) \left( {D_{11} w_s^{{\prime }{\prime }} -D_{12} \beta ^{2}w_s } \right) } \right) _i= & {} \left( {-\left( {1/84} \right) \left( {D_{11} w_s^{{\prime }{\prime }} -D_{12} \beta ^{2}w_s } \right) } \right) _{i+1} \end{aligned}$$
(58)

The two proposed methods, DTM and TCM, have been implemented based on RPT to solve a three-span plate problem with each span having a square shape. The same problem was also solved based on three more plate theories (CPT, FSDT and HSDT) using the two techniques (DTM and TCM), and the results are shown in Table 10.

In order to generalize the case of multi-span Levy plates, plates with end rotational springs at the two external boundaries (at \(x=0\) and \(x=a\) or, in non-dimensionalized form, \({\bar{x}} =0\) and \({\bar{x}} =1)\) are considered. The springs can represent location of continuity with adjacent plate members whose analysis is not included. However, more interestingly, elastically supported plates represent the most general scenario of boundary condition. The springs with non-dimensional stiffness \({\bar{k}}_s =k_s /\left( {a^{2}E_2 } \right) \) are used to model the two-span plate shown in Fig. 3. \(k_s \) is the actual stiffness constant of the spring. Results achieved using the presented techniques in this work for \(\delta =0.3\) are shown in Table 11 based on both RPT and other plate theories.

Fig. 2
figure 2

Multi-span plate (thickness not shown) consisting of (\(n-\)1) internal supports

Full size image
Table 10 Frequency \({\bar{\omega }} =\omega \frac{a^{2}}{h}\sqrt{\rho /E_2 }\) for a three-span plate with \(\frac{a}{b}=3.0\), \(\frac{a}{h}=10\) and \(\frac{E_1 }{E_2 }=5\)
Full size table
Fig. 3
figure 3

A two-span plate (thickness not shown) with end rotational springs of stiffness \({\bar{k}}_s \)

Full size image
Table 11 Frequency \({\bar{\omega }} =\omega \frac{a^{2}}{h}\sqrt{\rho /E_2 }\) for a two-span plate and plate with stepped thickness and end rotational springs: \(\frac{a}{b}=3.0\), \(\frac{a}{h}=10\) and \(\frac{E_1 }{E_2 }=5\)
Full size table

The approach of formulating the multi-span problem above can be extended to deal with Levy plates with stepped thickness which can have any number of thickness segments “n.” Again, for the sake of generalization of the edge fixity conditions, end rotational springs are incorporated in the model as shown in Fig. 4. Although the problem consists of a single span, the presence of the thickness steps results in the need to ensure continuity conditions at the interface between the segments by satisfying the additional equations given by Eqs. (59) to (66). These conditions are different from those (Eqs. 51–58) satisfied in the case of multi-span plates. Results for a plate with two thickness steps of equal length in x-direction are shown in Table 11. It should be noted that, unlike the previous cases treated for SS, SC and SF plates, and unlike the multi-span plates without springs, the moment boundary conditions at the two edges (i.e., at \(x=0\) and \(x=a)\) in the case of plates with rotational springs needs to be modified. These modifications are given by Eqs. (67) and (68) and are applied to both the two-span plate with end springs and the plate with stepped thickness and end springs above.

It is clear in all the three problems presented in this section that the RPT-based DTM and TCM predicts the frequency factors that agree closely with the HSDT with marginal differences on the conservative side, whereas the CPT overestimates these factors. It is interesting to note that since TCM is fully numerical, it turns out to be more flexible than DTM which is semi-analytical in nature. As a result, the former can be applied (but with different formulation from the one presented in this work) to handle non-classic (non-canonic) plate shapes, while the applicability of the latter for such cases needs to be established by further research.

Fig. 4
figure 4

Plate with stepped thickness (“n” segments) and end rotational springs of stiffness \({\bar{k}}_s \)

Full size image
$$\begin{aligned} \left( {w_b } \right) _i= & {} \left( {w_b } \right) _{i+1} \end{aligned}$$
(59)
$$\begin{aligned} \left( {w_s } \right) _i= & {} \left( {w_s } \right) _{i+1} \end{aligned}$$
(60)
$$\begin{aligned} \left( {w_b^{\prime } } \right) _i= & {} \left( {w_b^{\prime } } \right) _{i+1} \end{aligned}$$
(61)
$$\begin{aligned} \left( {w_s^{\prime } } \right) _i= & {} \left( {w_s^{\prime } } \right) _{i+1} \end{aligned}$$
(62)
$$\begin{aligned} \left( {-D_{11} w_b^{{\prime }{\prime }} +D_{12} \beta ^{2}\hbox {}w_b } \right) _i= & {} \left( {-D_{11} w_b^{{\prime }{\prime }} +D_{12} \beta ^{2}\hbox {}w_b } \right) _{i+1} \end{aligned}$$
(63)
$$\begin{aligned} \left( {-\left( {1/84} \right) \left( {D_{11} w_s^{{\prime }{\prime }} -D_{12} \beta ^{2}w_s } \right) } \right) _i= & {} \left( {-\left( {1/84} \right) \left( {D_{11} w_s^{{\prime }{\prime }} -D_{12} \beta ^{2}w_s } \right) } \right) _{i+1} \end{aligned}$$
(64)
$$\begin{aligned} \left( {-D_{11} w_b^{{\prime }{\prime }{\prime }} +\left( {\left( {D_{12} +4D_{66} } \right) \beta ^{2}-I_2 \omega _n^2 } \right) w_b^{\prime } } \right) _i= & {} \left( {-D_{11} w_b^{{\prime }{\prime }{\prime }} +\left( {\left( {D_{12} +4D_{66} } \right) \beta ^{2}-I_2 \omega _n^2 } \right) w_b^{\prime } } \right) _{i+1} \nonumber \\ \end{aligned}$$
(65)
$$\begin{aligned} \left( {-D_{11} w_s^{{\prime }{\prime }{\prime }} +\left( {\left( {D_{12} +4D_{66} } \right) \beta ^{2}-I_2 \omega _n^2 +84A_{55} } \right) w_s^{\prime } } \right) _i= & {} \left( -D_{11} w_s^{{\prime }{\prime }{\prime }} +\left( \left( {D_{12} +4D_{66} } \right) \beta ^{2}\right. \right. \nonumber \\&\left. \left. -I_2 \omega _n^2 +84A_{55} \right) w_s^{\prime } \right) _{i+1} \end{aligned}$$
(66)
$$\begin{aligned} -D_{11} w_b^{{\prime }{\prime }} +D_{12} \beta ^{2}\hbox {}w_b= & {} {\bar{ k}}_s w_b^{\prime } \end{aligned}$$
(67)
$$\begin{aligned} -\left( {1/84} \right) \left( {D_{11} w_s^{{\prime }{\prime }} -D_{12} \beta ^{2}w_s } \right)= & {} {\bar{k}}_s w_s^{\prime } \end{aligned}$$
(68)

6 Conclusions

In this paper, free vibration analysis of orthotropic plates has been carried out using the differential transform method (DTM) and Taylor collocation method (TCM) based on the two-variable refined plate theory. Detailed formulations of the methods have been given, followed by their implementation on plates having different end conditions. The results have been verified using the analytical solution in cases of SS, SC and SF plates based on the same theory, and consequently, various factors ranging from geometric to material parameters have been studied. Cases of multi-span plates and plates with stepped thickness and end rotational springs were also studied in order to show the capability of the proposed methods and the plate theory, the RPT, in solving problems whose analytical solutions are not readily available. Comparison have also been made with the results based on other plate theories, such as the classical plate theory, the first-order shear deformation theory and the high-order shear deformation theory. In light of the outcome of the analyses, the following conclusions can be drawn.

  1. 1.

    Although the classical plate theory is the simplest and most appealing compared to other shear deformation theories, and since it cannot be relied upon in all situations, the two-variable refined plate theory serves as a good alternative since its formulation is relatively simple and resembles that of the classical plate theory, and yet it takes into account the effect of shear deformation. In addition, the resulting system of equations are easier to solve than those of other higher order theories.

  2. 2.

    Contrary to the case of some shear deformation theories, the present theory does not require any shear correction factor and it results in quadratic distribution of the shear stress satisfying zero-traction boundary condition at the top and bottom surfaces of the plate.

  3. 3.

    Compared to the present theory, the classical plate theory overestimates the natural frequencies.

  4. 4.

    Capability of DTM and TCM to analyze orthotropic plates has been proven, and good agreement exists between the results achieved and the analytical ones for SS, SC and SF plates on the one hand, and the FSDT and HSDT for multi-span plates and plates with stepped thickness and end rotational springs on the other.

  5. 5.

    In cases of SS, SC and SF plates based on CPT, accuracy level of both DTM and TCM is the same. However, based on RPT, results obtained from TCM are, in few cases, better (though not significantly different from the DTM). On the other hand, the DTM requires less computer memory in its solution because the final size of the system matrix is much smaller than that needed to be handled in the TCM. For example, while solution for SF, SS and SC plates using the DTM requires dealing with just a \(4\times 4\) matrix, the TCM requires one to deal with a \(2\left( {{\bar{N}}+1} \right) \times 2\left( {{\bar{N}}+1} \right) \) matrix for the same plates.

  6. 6.

    The effect of increasing the aspect ratio and thickness ratio is to cause a corresponding increase in the fundamental natural frequency parameter, hence, increase in the fundamental natural frequency for all the plate types studied.

  7. 7.

    The effect of increasing the modular ratio causes an increase in the fundamental natural frequency parameter for the SS and SC plates. On the other hand, no such effect is observed in the case of SF plate because of its free end which lowers the plate’s stiffness.

  8. 8.

    Denoting the fundamental natural frequency parameters for the SF, SS and SC plates as \({\bar{\omega }} _{SF} \), \({\bar{\omega }} _{SS} \) and \({\bar{\omega }} _{SC} \), it can be seen that under the same condition (same thickness ratio, aspect ratio and modular ratio), values of the fundamental natural frequency parameters have magnitudes in the order \({\bar{\omega }} _{SC}>{\bar{\omega }} _{SS} >{\bar{\omega }} _{SF} \). Similar finding is confirmed in [73].

  9. 9.

    Both the two numerical methods, DTM and TCM, adopted in obtaining these solutions do not require iterations as is the case with the analytical results reported in [35].

  10. 10.

    Unlike other element-based numerical techniques, the DTM and TCM can be used to present solutions in symbolic forms which warrants the possibility of obtaining other secondary variables by differentiation, integration, etc.

  11. 11.

    The obvious advantage of eliminating the need for method calibration as required in other meshless methods, such as the Kansa method and method of fundamental solutions, adds to the appealing nature of the presented methods.