#### Abstract

The paper presents the extension of an edge-based smoothed finite element method using three-node triangular elements for dynamic analysis of the functionally graded porous (FGP) plates subjected to moving loads resting on the elastic foundation taking into mass (EFTIM). In this study, the edge-based smoothed technique is integrated with the mixed interpolation of the tensorial component technique for the three-node triangular element (MITC3) to give so-called ES-MITC3, which helps improve significantly the accuracy for the standard MITC3 element. The EFTIM model is formed by adding a mass parameter of foundation into the Winkler–Pasternak foundation model. Two parameters of the FGP materials, the power-law index () and the maximum porosity distributions (), take forms of cosine functions. Some numerical results of the proposed method are compared with those of published works to verify the accuracy and reliability. Furthermore, the effects of geometric parameters and materials on forced vibration of the FGP plates resting on the EFTIM are also studied in detail.

#### 1. Introduction

The dynamic responses of the plates resting on the elastic foundation (EF) subjected to moving loads have been widely investigated by many scientists around the world. The results of these works may be applied for computation in many civil engineering problems such as vehicles running on roads and airplanes moving on runways. When investigating the response of structures on the EF, authors have mainly used the one-parameter Winkler model [1] or the two-parameter Winkler–Pasternak model [2]. For example, Xiang et al. [3] used an analytical method (AM) to study the free vibration of plates. Omurtag et al. [4] analyzed the free vibration of thin plates using the finite element method (FEM). Zhou et al. [5] used the Ritz method to consider the free vibration of rectangular plates. Ferreira et al. [6] studied the free vibration of the plates by radial basis functions. Duc et al. [7] studied the nonlinear thermal dynamic response of the shear deformable functionally graded (FG) plates. Mahmoudi et al. [8] developed a refined quasi-three-dimensional shear deformation theory for analyzing FG sandwich plates under thermomechanical load. Recently, Tran et al. [9] also investigated the free vibration of the FGP plates resting on the EFTIM.

Regarding the analysis of structures subjected to moving loads, some typical works can be mentioned here as follows. Wu et al. [10] studied the dynamic responses of nonuniform rectangular plates with various boundary conditions (BC) using the classical plate theory (CPT) and the FEM. Frýba [11] summarized the variety of problems about the dynamic response of the plates. Taheri and Ting [12] used Green’s functions to investigate the dynamic responses of plates with different boundary conditions. Hoang et al. [13] used the FEM to analyze dynamic responses of triple-layer composite plates with layers connected by shear connectors. Song et al. [14] considered dynamic responses of rectangular thin plates of arbitrary boundary conditions. Expanding the above studies for plates on the EF under moving loads, Zaman et al. [15] used the FEM based on the first-order shear deformation theory (FSDT) to analyze the airport pavements subjected to moving aircraft loads. In their work, the airport pavement and underlying soil medium were modeled as the thick plate, and a series of discrete linear springs and dashpots were acting at nodal points, respectively. Kim and Roesset [16] considered the dynamic responses of the plate resting on the EF under constant and harmonic moving loads. In their study, the double and triple Fourier transformations were used to approximate the moving loads, and the influences of the velocity parameters, load parameters, and the internal damping on the displacement were also examined. Huang and Thambiratnam [17, 18] investigated the dynamic responses of plates on the EF using the finite strip method (FSM). Seong-Min and McCullough [19] surveyed the dynamic responses for the plates resting on a viscous Winkler foundation. The influences of the varying amplitudes and a constant velocity on dynamic responses of the plates were considered in this paper. Also, Tran et al. [20] used the FSDT and FEM to study the dynamic responses of the sandwich composite plates with auxetic honeycomb core resting on the EF subjected to moving oscillator loads.

In a different aspect regarding the properties of materials, porosity can be appeared in the FG materials during the manufacturing process or intentionally created. Fundamentally, porosity reduces the stiffness of the structure; however, with excellent engineering properties like lightweight, good energy-absorbing capability, and strong thermal resistant properties, the FG porous (FGP) materials have still attracted the attention of many researchers. For example, Kim et al. [21] used an analytical method based on the modified couples stress to analyze the bending, free vibration, and buckling of the FGP microplates. Also, Coskun et al. [22] used a general third-order shear deformation theory (TSDT) to investigate the static, free vibration, and buckling of the FGP microplates. Rezaei and Saidi [23, 24] used an analytical method to study the free vibration of the rectangular and porous-cellular plates. Zhao et al. [25, 26] examined the free vibration and the dynamic responses of the FGP shallow shells by using an improved Fourier method. Regarding nonlinear problems, Li et al. [27] analyzed the nonlinear free vibration and dynamic buckling of the sandwich FGP plates with graphene platelet reinforcement (GPL) on the EF. Sahmani et al. [28] used the nonlocal theory to analyze the nonlinear large-amplitude vibrations of the FGP micro-/nanoplates with GPL reinforcement. Wu et al. [29] studied the dynamic of the FGP structures by using the FEM. In addition, Barati and Zenkour [30] used the FSDT and Galerkin’s method to analyze the free vibration of the FGP cylindrical shells reinforced by the GPL. Zenkour and Barati [31] considered the electro-thermoelastic free vibration of the FGP plates integrated with piezoelectric layers using an analytical method. Daikh and Zenkour [32] calculated the free vibration of the FG sandwich plates. Sobhy and Zenkour [33] considered the effects of the porosity distribution on the free vibration of the FG nanoplate using quasi-3D refined theory. Nguyen et al. developed the polygonal finite element method (PFEM) combined with HSDT to calculate the free vibration of FGP plates reinforced by the GPL [34] and active-controlled vibration of the FGP plate reinforced by the GPL [35]. Moreover, Tran et al. [36] investigated the free vibration of the FGP variable-thickness plates using an edge-based smoothed finite element method (ES-FEM). Lurlaro et al. [37] used the refine zigzag theory to analyze the free vibration of FG sandwich plates.

In another aspect regarding the development of numerical methods, the original MITC3 element [38] was integrated with an edge-based smoothed finite element method (ES-FEM) [39–44] to give so-called ES-MITC3 [45] for analysis of different structures. For example, Chau-Dinh et al. [45] proposed ES-MITC3 to study the free vibration for the isotropic plates. Nguyen et al. [46] extended ES-MITC3 to analyze the free vibration of the FG plates based on the TSDT. Pham et al. [9, 47–50] developed ES-MITC3 to investigate the static and vibration of structures. The numerical results in these works have almost shown that ES-MITC3 has the following advantageous properties: (1) ES-MITC3 can avoid the shear locking phenomenon even with the ratio of the thickness to the length of the structures reaching ; (2) ES-MITC3 has better accuracy than some available triangular elements in the literature such as MITC3 [38], DSG3 [51], and CS-DSG3 [52] and is a good competitor with the quadrilateral element MITC4 [53].

Based on the aforementioned literature review, the objective of this work is to further extend the ES-MITC3 element for investigating the dynamic responses of the FGP plates resting on the EFTIM subjected to moving loads. In this study, the forced vibration equation is based on the FSDT due to simplicity and high computational efficiency (which means that it can ensure a high accuracy with a lower computational cost). The accuracy and reliability of the proposed method are performed by comparing the present numerical results with those of other published works in the literature. Furthermore, the influences of geometric parameters, material properties, and loads on the dynamic responses of the FGP plates subjected to the moving loads are also examined in detail.

#### 2. Functionally Graded Porous Plates on the EF

Consider an FGP plate resting on the EFTIM subjected to a moving load as shown in Figure 1.

The FGP materials with the variation of two parameters and three distribution cases of porosity along with thickness are expressed as follows [21, 22]:where is the maximum porosity distribution value. The property of the FGP materials is obtained by formulation:where and are the typical material properties at the top and the bottom surfaces, respectively, and is the power-law index. The distributions of porosity through thickness of the plate are shown in Figure 2(a). The porosity distribution of case 1 is symmetric corresponding to the midplane-enhanced distribution, and those of case 2 and case 3 are bottom- and top surface-enhanced distributions, respectively. In addition, Figures 2(b)–2(d) show the distributions of the normalized typical property associated with three different cases of the porosity distributions with parameters , and .

**(a)**

**(b)**

**(c)**

**(d)**

The EFTIM model is built based on the Winkler–Pasternak foundation by adding a mass parameter of the foundation:where is the vertical displacement of the FGP plate; are the Winkler foundation stiffness and shear layer stiffness of the Pasternak foundation, respectively; in which the parameter is the influence level of the foundation mass on the free vibration response of the structures, which is determined by experiments; and is the density ratio of the foundation to the plate material . From equation (3), it can be seen that the investigation of the EFTIM model and Winkler–Pasternak foundation model on the static analysis is similar. However, these two models become complex in the dynamic response problems. If the influence of the foundation mass parameter is ignored, the EFTIM model will be equivalent to the Winkler–Pasternak foundation model, and hence, the EFTIM model will be close to the EF model in practice.

#### 3. The First-Order Shear Deformation Theory and Weak Form for FGP Plates

##### 3.1. The First-Order Shear Deformation Theory for the FGP Plates

The displacement at any point of the FGP plates in the present work based on the FSDT model can be expressed aswhere are five unknown displacements of the midsurface of the plate, and the bending strain field of the plate can be expressed asin which the membrane strain is defined as

The bending strain is defined asand the transverse shear strain is defined as

From Hooke’s law, the linear stress-strain relations of the FGP plates can be expressed aswherein which is effective Young’s modulus calculated by equation (2), and is Poisson’s ratio.

##### 3.2. Weak Form Equations

To obtain the equations of the motion of the FGP plate for the dynamic analysis, Hamilton’s principle is applied with the following form:where , and are the strain energy, the kinetic energy, and the work done by applied forces, respectively.

The strain energy is expressed aswhere the strain energy of the foundation is computed byand the strain energy of the FGP plate is computed byin which

And , and are given by

The kinetic energy is defined aswhere the kinetic energy of the FGP plate is computed byin which , and is the mass matrix which is defined as follows:with .

The kinetic energy of the mass of the foundation is defined as

The work done by applied forces is given by

Substituting equations (14) and (17) into equation (11), the weak-form formulation for the dynamic analysis of FGP plates resting on the EF is finally obtained as

#### 4. Formulation of the ES-MITC3 Method for FGP Plates

##### 4.1. Formulation of the Finite Element Using the MITC3 Element

The middle surface of plate is discretized into finite three-node triangular elements with nodes such that and . The generalized displacements at any point for elements of the FGP plates are approximated aswhere is the number of nodes of the plate element and and are the shape function and the nodal degrees of freedom (DOF) of associated with the node of the element, respectively.

Similar to the standard triangular element, the linear membrane and the bending strains of a MITC3 element can be expressed in matrix forms as follows:where

To avoid the shear locking problem when the thickness of the FGP plates becomes very thin, the formulation of the transverse shear strains of the triangular element based on the FSDT [38] in this paper is given byin whichwithwherein which , and are pointed out in Figure 3, and is the area of the three-node triangular element.

Substituting the discrete displacement fields into equation (22), we obtained the discrete system equations for the dynamic response analysis of the FGP plates resting on the EF, respectively, aswhere is the stiffness matrix of the FGP plates resting on the EF and is the mass matrix, and they are assembled bywhere is the element stiffness matrix of the plates and is computed byin whichand is the element stiffness matrix of the EF and is computed bywith

The mass matrix of the plate is given bywhere is the element mass matrix of the EF and is computed by

In this work, we consider the moving loads depending on the time . This load moves via the direction perpendicular of the FGP plates corresponding to the variable velocity as shown in Figure 1. The equivalent distributed force is a function of the moving load at the position and the Dirac delta function [11]:

The nodal force vector of the element is computed from the distributed force applying on the element as follows:

In the case of taking into account the structural damping, we have the force vibration equation of the plate given byin which and are Rayleigh drag coefficients [11].

Equation (41) includes the linear differential equation with coefficients depending on time. In order to solve the system of equations, we use the Newmark-beta method [11] with the flowchart given in [54].

##### 4.2. Formulation of an ES-MITC3 Method for the FGP Plates

In the ES-MITC3 formulation, the computation of the local stiffness matrix is no longer based on the element domain but on the smoothing domain, in which the smoothing domain is constructed based on the edges of the triangular elements such that and An edge-based smoothing domain associated with the inner edge is formed by connecting two end-nodes of the edge to the centroids of adjacent triangular elements, as shown in Figure 4.

Then, by using the edge-based smoothed technique of the ES-FEM [46], the smoothed membrane, bending strains , and the smoothed shear strain over the smoothing domain are computed bywhere , and are the compatible membrane, bending strain, and the transverse shear strain, respectively, of the MITC3 triangular elements given in equations (5) and (8) and is a smoothing function that satisfies at least the unity property .

In the present work, we use the constant smoothing function:in which is the area of the smoothing domain and is defined bywhere is the number of the adjacent triangular elements in the smoothing domain and is the area of the triangular element attached to the edge .

By substituting equations (42), (43), and (44) into equations (24), (25), and (27), the approximation of the smoothed strains on the smoothing domain can be expressed aswhere is the total number of nodes of the MITC3 triangular elements attached to edge for boundary edges and for inner edges, as shown in Figure 4); is the nodal dof associated with the smoothing domain ; and are the smoothed membrane, bending, and shear strain gradient matrices, respectively, at the node of the elements attached to the edge computed by

The stiffness matrix of the FGP plates using the ES-MITC3 method is assembled bywhere is the ES-MITC3 stiffness matrix of the smoothing domain expressed byin which

The ES-MITC3 element has some advantages such as (1) it is easy and flexible for discretizing the problem domain into three-node triangular elements, even for arbitrary complicated geometry domains; (2) it has the number of degrees of freedom similar to that of the standard triangular element or of the original MITC3 element; (3) it can avoid the shear locking phenomenon even with the ratio of the thickness to the length of the structures reaching 10^{−8}; and (4) it has better accuracy than some available triangular elements in the literature such as MITC3 [38], DSG3 [51], CS-DSG3 [52] and is a good competitor with the quadrilateral element MITC4 [53].

#### 5. Accuracy Study

To evaluate the accuracy and reliability of the proposed method in this paper, we investigate the free vibration of the plate resting on the EF without including the featured-index of the mass of the foundation . The numerical results of the present work are compared with those of published works in the literature. For convenience, the stiffness foundation and dimensionless frequencies of plates are given by

The convergence of the first two dimensionless frequencies in the case of the fully simple support (SSSS) plate and the fully clamped (CCCC) plate with is shown in Figure 5. It can be seen that the natural frequencies of plates under different boundary conditions using mesh size are all converging. Table 1 shows the first three dimensionless frequencies of the plate resting on the EF in the present work, together with the available results in the literature. It can be observed that the obtained results of ES-MITC3 match well with those of the analytical solution in the published works [5, 6] and FEM [4]. Based on these results, the present method will use the mesh size to analyze the dynamic response of the plates in all numerical examples of the upcoming sections.

**(a)**

**(b)**

Let us now consider an SSSS rectangular plate with length *a* = 10 m, width , thickness , modulus of elasticity , Poisson’s ratio , and mass density under concentrated load along the edge of plate as presented in [14]. In this case, the FGP plate is considered as the isotropic plate with , and . From Figure 6, it is observed that the deflections of the central point of the plate in the present work agree excellently with those of Song et al. [14] in both shape and magnitude.

#### 6. Numerical Results and Discussion

In this section, we use the material properties of the FGP plate as presented in [21]. The moduli and mass densities of two components are presented in Table 2.

We now analyze an SSSS FGP plate for case 1 of the porosity distribution. The geometric parameters are given by , , and the material properties of the FGP plate are given as follows: the maximum porosity distribution , power-index , Winkler foundation stiffness , shear layer stiffness of the Pasternak foundation , the density ratio of the foundation , and featured-index . The first six frequencies of the FGP plate are shown in Table 3, and the first six mode shapes are illustrated in Figure 7.

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

##### 6.1. Influence of the Cases of Porosity Distributions

In this section, we examine the effects of porosity distributions on the dynamic response of the SSSS rectangular FGP plate with three different cases of porosity distributions: case 1, case 2, and case 3. A moving load with magnitude moves along via the edge of the plate with velocity . The responses of deflection, velocity, and stress at the center of the plate versus time are shown in Figures 8(a)–8(c). Figure 8(d) displays the stress at the center of the plate through thickness when the load moves to the midpoint of the plate. Maximum values of deflection, velocity, and stress at the central point of the plate are illustrated in Table 4. From these figures and Table 4, it is seen that, with the same geometrical parameter and material properties, case 1 of the porosity distribution gives the minimum value of the deflection response of the FGP plate, while case 3 of the porosity distribution gives the maximum value. In addition, it is also observed that case 1 of the porosity distribution helps reduce the oscillates better than the remaining two cases of the porosity distribution.

**(a)**

**(b)**

**(c)**

**(d)**

##### 6.2. Influences of Velocity of Moving Loads

In this example, we consider the effects of velocity on the response of the SSSS rectangular FGP plate in case 1 of the porosity distribution. The moving load with magnitude travels along the edge of the plate with the variation of velocity . The responses of central deflection, velocity, and stress of the plate versus time are shown in Figures 9(a)–9(c). Figure 9(d) plots the stress at the center of the plate through thickness when the load moves to the midpoint of the plate. The maximum deflections, velocities, and stresses at the central point of the plate are illustrated in Table 5. From Figure 9 and Table 5, it can be seen that the deflection along the *z*-axis of the plate decreases slightly when the velocity of the moving load is less than but increases rapidly when the velocity of the moving load varies from 10 to 25 ms. These results demonstrate that the speed of the moving load affects significantly the dynamic response of the FGP plate.

**(a)**

**(b)**

**(c)**

**(d)**

##### 6.3. Influence of the Parameters of EFTIM

In this example, we study the effect of Winkler foundation stiffness on the dynamic responses of the FGP plate in case 1 of the porosity distribution. We examine varying from 0 to 100, and the shear layer stiffness of the Pasternak foundation is fixed with . The moving load with magnitude moves along the edge of the plate with velocity . The responses of deflection, velocity, and stress at the center of the plate versus time are shown in Figures 10(a)–10(c). Figure 10(d) presents the central stress of the plate through thickness when the load moves to the midpoint of the plate. The maximum deflections, velocities, and stresses at the central point of the plate are illustrated in Table 6. From these numerical results, it can be seen that the increase of the Winkler foundation stiffness leads to the increase of the FGP plate stiffness and hence makes decrease the deflection of the plate.

**(a)**

**(b)**

**(c)**

**(d)**

Next, we consider the effects of shear layer stiffness of Pasternak foundation on the dynamic response of the FGP plate in case 1 of the porosity distribution. The moving load with magnitude moves along the edge of the plate with velocity . The variation of the shear layer stiffness of the Pasternak foundation is given by , and the Winkler foundation stiffness is fixed at . The responses of deflection, velocity, and stress () at the center of the plate versus time are shown in Figures 11(a)–11(c). Figure 11(d) shows the central stress of the plate through thickness when the load moves to the midpoint of the plate. The maximum deflections, velocities, and stresses at the central point of the plate are illustrated in Table 7. From these numerical results, it can be found that the increase of the foundation stiffness leads to the decrease of the deflection of the plate. This implies that the increase of the Winkler foundation stiffness leads to the increase of stiffness of FGP plates. However, when the foundation stiffness factor is less than 50, the response of the deflection of the plate can be negligible, and the effect of factor on the stiffness of the plate is less than the effect of factor .

**(a)**

**(b)**

**(c)**

**(d)**

To investigate the effects of the featured-index of the mass of the foundation *β* on the dynamic response of the FGP plate in case 1 of the porosity distribution, we consider the plate under the moving load traveling along the edge of the plate with the velocity and other parameters given as the Winkler foundation stiffness the shear layer stiffness of Pasternak foundation , and the variation of the mass of the foundation , and 1. The responses of deflection, velocity, and stress () at the center of the plate versus time are shown in Figures 12(a)–12(c). Figure 12(d) plots the central stress of the plate through thickness when the load moves to the midpoint of the plate. The maximum deflections, velocities, and stresses at the central point of the plate are illustrated in Table 8. From these numerical results, it can be seen that the increase of the mass of the foundation varying from 0 to 1 leads to the increase of the deflection of the plate, but this change is not much. This demonstrates that the mass of the EF involved in the plate’s vibration makes decrease the stiffness of FGP plates.

**(a)**

**(b)**

**(c)**

**(d)**

##### 6.4. Influences of the Parameters-FGP to the Dynamic Response of the FGP Plate on the EFTIM

In this example, we study the effects of the material properties on the dynamic response of the FGP plate in case 1 of the porosity distribution. The power-law index is chosen as , and the porosity distribution Ω varies from 0 to 1. The SSSS rectangular FGP plate rests on the EFTIM with parameter , , and . The responses of deflections, velocities, and stresses at the central point of the FGP plate are shown in Figures 13 and 14, and these maximum values are presented in Tables 9 and 10. From Figure 13 and Table 9, it can be seen that the increase of the power-law index leads to the increase of the response of the deflections and velocities of the plate because the increase of the power-law index makes the plate become metal-rich and hence makes decrease the hardness of the plate. However, it is noted that when the porosity distribution increases, the response of deflections, velocities, and stresses of the plate also increases. This is because the increase of the distribution of porosity makes decrease the stiffness of FGP plates. These results are illustrated in Figure 14 and Table 10.

**(a)**

**(b)**

**(c)**

**(d)**

**(a)**

**(b)**

**(c)**

**(d)**

#### 7. Conclusions

This work presents an extension of ES-MITC3 using three-node triangular elements for dynamic analysis of the functionally graded porous (FGP) plates subjected to moving loads resting on the elastic foundation taking into mass (EFTIM). The obtained numerical results of the proposed approach in different contexts are compared with other available solutions in the literature. The effect of the geometrical parameters and material properties on the dynamic response of the FGP plate resting on the EFTIM is also examined in detail. From the present formulation and the numerical results, we can withdraw some following points:(i)The numerical results by the ES-MITC3 element agree excellently with those of some other compared elements and have better accuracy than that of the original MITC3 element(ii)The EFTIM model can be used to accurately describe the EF model(iii)The increase in foundation parameters *k*_{1} and *k*_{2} of the EF leads to the increase in the stiffness of the FGP plates(iv)The increase in the mass of the EF leads to decrease in the stiffness of the FGP plates(v)The material parameters *k* and and the porosity distribution of the FGP plates significantly affect the dynamic response of the FGP plates(vi)The increase in the velocity of the moving load leads to the increase in the vibration of the FGP plates

It is noted that, besides the advantages of using the first-order shear deformation theory (FSDT) regarding the simplicity in formulations and computational efficiency, the proposed method also has a limit regarding using the FSDT to approximate displacement fields of the FGP plates. This limit somehow leads to the limited accuracy and hence should be a topic for improvement in upcoming studies.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This research was funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant no. 107.02-2019.330.