# Dynamic Analysis of Multi-Stepped Functionally Graded Carbon Nanotube Reinforced Composite Plate with General Boundary Condition

Authors: Kwanghun Kim; Songhun Kwak; Yonguk Ri; Yongsong Paek; Wonjin Han; Kumchol Ri
Journal: Shock and Vibration
 (2021)

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## Abstract

This study presents the multi-stepped functionally graded carbon nanotube reinforced composite (FG-CNTRC) plate model for the first time, and its free and forced vibration is analyzed by employing the domain decomposition method. The segmentation technique is employed to discretize the structure along the length direction. The artificial spring technique is applied to the structural boundary and piecewise interface for satisfying the boundary conditions and the combined conditions between subplates. Based on this, the boundary conditions of subdomains could be considered as a free boundary constraint, reducing the difficulty in constructing the allowable displacement function. Since all the structures of subdomains are identical, the allowable displacement functions of them can be uniformly constructed using the two-dimensional ultraspherical polynomial expansion. The potential energy function of the plate is derived from the first-order shear deformation theory (FSDT). The allowable displacement function is substituted into the potential energy function, and then the natural frequencies and mode shapes of the multi-stepped FG-CNTRC plate are decided by using the Rayleigh–Ritz method. The accuracy and reliability of the proposed method are confirmed by the results of the previous literature and finite element method (FEM). On this basis, the influences of the geometric and material parameters on free and forced vibration of the multi-stepped FG-CNTRC plate are also studied.


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## Body

## 1. Introduction

As the advanced manufacturing technology is rapidly developed, the FG-CNTRC has appeared as a prospective kind of composites in the past few years. The FG-CNTRC is composed of carbon nanotubes (CNTs) and functionally graded materials (FGMs) and considered as the advanced material with extraordinary mechanical, optical, thermal, and electrical features. Because of its excellent features, a lot of experimental and theoretical research studies have been conducted to study its mechanical and thermomechanical characteristics [1–5]. The basic research of Shen [6] on the bending behavior of CNT reinforced composite plates indicated that the bending moments of the plates can be significantly enhanced by introducing the functionally graded distribution of CNTs in a polymeric matrix. In addition, lots of efforts were made to study the FG-CNTRC beams, plates, and shells of various forms. Liew et al. [7] brilliantly summarized these investigations, in which the mechanical behavior of FG-CNTRC structures was described in detail including static vibration, dynamic vibration, free vibration, buckling and post-buckling, and linear and non-linear analysis. The wide range of investigations on the free vibration analysis has firstly paid attention to the analysis of the vibrational behavior of functionally graded materials [8–15]. Then, the research was enlarged to the analysis of the FG-CNTRC. The following paragraphs illustrate several research studies related to the analysis of the free vibration of FG-CNTRC shell structures. Applying the FSDT, Zhu et al. [16] studied the bending and free vibration behavior of thin-to-moderately thick FG-CNTRC plates by employing the finite element method. Zhang and his co-authors introduced several results on the vibrational behavior of FG-CNTRC rectangular [17], triangular [18], and skew [19] plates. In addition, Zhang et al. analyzed the free vibration characteristics of FG-CNTRC cylindrical panels [20]. In these articles, the FSDT theory and the kp-Ritz method were employed to obtain the governing equations of the cylindrical panels, and the influence of the distribution and volume fraction of CNTs was also studied.Recently, an interesting study concerning the free vibration analysis of arbitrarily shaped FG-CNTRC plates was introduced by Fantuzzi et al. [21]. In this study, the FSDT theory was applied for the approximation of the displacement field of nanoplates, and the mapping of arbitrary shapes with holes and discontinuities of nanoplates reinforced by CNTs was conducted using the Non-Uniform Rational B-Spline (NURBS) curves. Based on the FSDT theory, Malekzadeh et al. [22, 23] analyzed the free vibration characteristics of laminated plates with FG-CNTRC layers using the differential quadrature method (DQM) for the numerical solutions. The comparative results of the natural frequencies, which were obtained by applying the extended rule of mixture, Eshelby–Mori–Tanaka method, and FSDT theory, respectively, were presented by Mehrabadi et al. [24]. The parametric studies were also provided, in this research, to analyze the effects of different kinds of CNTs and geometrical parameters on the free vibration characteristics of FG-CNTRC plates. In addition, Kiani studied the free vibration behavior of different types of FG-CNTRC plates [25] and skew plates [26]. Using the FSDT theory to describe the kinematics of the considered structure, Mirzaei and Kiani [27] employed the Ritz method to obtain the vibrational solutions, and they summarized the research studies mentioned above which were studying the influence of carbon nanotube reinforcements on the improvement of the vibrational behavior of FG-CNTRC structures. Mirzaei and Kiani applied the framework of the FSDT theory to investigate the natural frequencies of FG-CNTRC plates with cutout. In this research, the Ritz method with Chebyshev basis polynomials was used to find the numerical solutions, and it was shown that this solution method was effective for the arbitrary in-plane and out-of-plane boundary conditions of the FG-CNTRC plate. In other studies, Kiani applied the piezoelectric effect to the CNTRC plates [28], and the effect of electrical parameters on the fundamental frequency of FG-CNTRC piezoelectric plates was considered under the two kinds of electrical boundary conditions such as closed circuit and open circuit. Wang et al. [29] proposed a stepped functionally graded piezoelectric material (FGPM) plate model for the first time and studied its free and forced vibration by using the domain energy decomposition method. Based on the widely used FSDT, lots of other research studies on the free and forced vibrations analysis of FG-CNTRC plate structures have also been conducted [30–33]. Selim et al. [34] analyzed the free vibration behavior of FG-CNTRC plates based on Reddy’s higher-order shear deformation theory (HSDT) and element-free kp-Ritz method in the thermal environment. Parametric effects such as CNT distribution, boundary conditions, plate aspect ratio, plate thickness-to-width ratio, and CNT volume fraction on the dimensionless frequencies were also examined. In the same way, Mehar et al. [35] conducted the vibration analysis of carbon nanotube reinforced composite plates. In addition, Wattanasakulpong and Chaikittiratana [36] studied the static and dynamic analysis of FG-CNTRC plates resting on the Pasternak elastic foundations. The governing equations were derived from the HSDT theory, and the accurate solutions were obtained to study the static as well as the vibrational behavior of such behavior. In addition, the vibration analysis of FG-CNTRC structures such as beam [37–41], panel [42–50], and shells [34, 51–58] has been also widely conducted.As can be seen from the previous works, until now, the research on the FG-CNTRC plate has been mainly focused on the non-stepped plates with uniform thickness, and the multi-stepped FG-CNTRC plate has not been studied yet. Moreover, from the consideration of previous research studies, it can be known that it is still important to develop the simple and efficient integrated solution method for the free and forced vibration analysis of the multi-stepped plates. Therefore, the purpose of this study is to provide the integrated solution method for the free and forced vibration analysis of the multi-stepped FG-CNTRC plates. In this research, a unified modeling method is employed to construct the dynamic characteristic analysis model of multi-stepped FG-CNTRC plate. Within the framework of the domain decomposition method, the rectangular plates are segmented along the length direction using the segmenting technology, and then, the thickness of each subplate is taken differently, so that the stepped rectangular plates are constructed successfully. The potential boundary and combined conditions of segmented interfaces are obtained through the application of artificial spring technique. The allowable displacement functions in subdomains are established using the two-dimensional ultraspherical polynomials. In addition, the global potential energy functional of the multi-stepped FG-CNTRC plate is constructed by employing the FSDT. The polynomials’ unknown coefficient is treated using the standard variational operation to study the dynamic characteristics of the FG-CNTRC plate. The convergence and accuracy of the proposed model are validated using numerical examples.


## 2. Formulation

### 2.1. Model of the Multi-Stepped FG-CNTRC Plate

Figure1 shows the calculation model of the multi-stepped FG-CNTRC plate. As can be seen in Figure 1(a), the multi-stepped plate consists of several subplates with different thicknesses in x and y directions, and the lengths of individual subplates are expressed as ai (i = 1, 2, …) and bj (j = 1, 2, …) in x and y directions, respectively.Figure 1
Multi-stepped FG-CNTRC plate model. (a) Geometric relationship. (b) Boundary condition. (c) Combined condition.
(a)(b)(c)Also, the heights of individual subplates are indicated ashi. In this research, the heights of individual subplates are expressed as h1 and h2 for the convenient calculation, and u,v,andw are the displacements in x, y, z directions, respectively. The artificial spring technique is introduced for the generalization of boundary conditions, and the four sides of the plate are modeled to be supported by the artificial springs (Figure 1(b)).Each side has a boundary spring group consisting of three artificial springs (ku,kv,kw) and two rotating artificial elastic springs (kφ, kθ), and the boundary condition can be generalized by adjusting the stiffness of individual springs. The boundary conditions at four sides are expressed by adding subscript 0 at x = 0 and y = 0 boundary and subscript 1 at x = a and y = b boundary. The stepped plates can be seen to be made up of the strong combinations of the individual subplates, and the connective condition can be modeled in a similar way with the boundary condition. That is, by setting the stiffness of connective springs as infinity, the strong connective condition of individual plates can be accomplished. Figure 1(c) shows the connective conditions of individual plates.


### 2.2. Material Properties

The stepped plate considered here is composed of the isotropic matrix reinforced with CNTs. The distribution of CNTs in the matrix can be either uniform or functionally graded according to the thickness of plates. Figure2 shows five types of CNT distributions such as UD-CNTRC, FG-Λ CNTRC, FG-V CNTRC, FG-X CNTRC, and FG-O CNTRC.Figure 2
Distribution patterns of CNTs in FG-CNTRC plates. (a) UD-CNTRC. (b) FG-Λ CNTRC. (c) FG-V CNTRC. (d) FG-O CNTRC. (e) FG-X CNTRC.
(a)(b)(c)(d)(e)The CNT volume fractionsVCNT in the various kinds of FG-CNTRC plates are indicated as follows [57, 58]:(1)VCNTz=VCNT∗,UD−CNT,1+2zhVCNT∗,FGV−CNT,1−2zhVCNT∗,FGΛ−CNT,4zhVCNT∗,FGX−CNT,21−2zhVCNT∗,FGO−CNT,in which(2)VCNT∗=wCNTwCNT+ρCNT/ρm−ρCNT/ρmwCNT,where wCNT is the mass fraction of nanotube and ρCNT and ρm are the mass densities of CNT and matrix constituents, respectively.Figure3 shows the variation characteristics of CNT volume fractions VCNT according to z/h in five patterns. It shows that the value of VCNT is constant regardless of thickness of plates in UD-CNTRC, while it increases or decreases in FG-Λ CNTRC and FG-V CNTRC, respectively. In cases of FG-X CNTRC and FG-O CNTRC, the value of VCNT changes symmetrically about the middle surface.Figure 3
Change of the CNT volume fractionsVCNT on the thickness of plate. (a) VCNT∗=0.1. (b) VCNT∗=0.2.
(a)(b)Based on the improved rule of mixtures, effective Young’s modulus, shear modulus, Poisson’s ratio, and mass density for CNTRC materials can be written as follows [56, 58]:(3)E11z=η1VCNTzE11CNT+VmzEm,η2E22z=VCNTzE22CNT+VmzEm,η3G12z=VCNTzG12CNT+VmzGm,Gm=Em21−μm,μ12z=VCNTzμ12CNT+Vmzμm,μ21z=μ12zE11zE22z,ρz=VCNTzρCNT+Vmzρm,Vmz=1−VCNT,where E11CNT,E22CNT,G12CNT,Em,Gm represent Young’s modulus and shear modulus of CNT and matrix, η1, η2, and η3 are CNT/matrix efficiency parameters, and μ12CNT,μm denote Poisson’s ratios of CNT and matrix.


### 2.3. Energy Function

The displacement components of the individual subplate can be expressed with the displacements of the midsurface and the rotations of the cross section using the FSDT as follows:(4)Ui,jx,y,z,t=u0i,jx,y,t+zφi,jx,y,t,Vi,jx,y,z,t=v0i,jx,y,t+zθi,jx,y,t,Wi,jx,y,z,t=w0i,jx,y,t,where u0,v0,andw0 represent the middle surface displacements of the i, jth subplate in the x, y, and z directions, φ and θ indicate the transverse normal rotations in regard to x and y axes, and t denotes the time variable. The strain elements at a random point of the ith FG-CNTR subplate can be written as follows:(5)εxxi,j=εxx,0i,j+zχxxi,j,εyyi,j=εyy,0i,j+zχyyi,j,γxyi,j=γxy,0i,j+zχxyi,j,γxzi,j=γxz,0i,j,γyzi,j=γyz,0i,j,where εxx,0i,j,εyy,0i,j,γxy,0i,j,γxz,0i,j, and γyz,0i,j indicate the middle surface strains and χxxi,j,χyyi,j, and χxyi,j denote the surface curvatures of i, jth subplate, respectively; these surface strains and curvatures are defined as follows:(6)εxx,0i,j=∂u0i,j∂x,εyy,0i,j=∂v0i,j∂y,γxy,0i,j=∂v0i,j∂x+∂u0i,j∂y,γxz,0i,j=∂w0i,j∂x+φi,j,γyz,0i,j=∂w0i,j∂y+θi,j,χxxi,j=∂φi,j∂x,χyyi,j=∂θi,j∂y,χxyi,j=∂θi,j∂x+∂φi,j∂y.Following the state of generalized Hooke’s law, the constitutive relations of FG-CNTRC plate can be written as follows:(7)σxxi,jσyyi,jτyzi,jτxzi,jτxyi,j=Q11zQ12z000Q12zQ22z00000Q44z00000Q55z00000Q66zεxxi,jεyyi,jγyzi,jγxzi,jγxyi,j,where Qij (i, j = 1, 2, 4, 5, 6) are the reduced material stiffness coefficients compatible with plane-stress conditions and expressed in terms of elastic and shear moduli and Poisson’s ratio as(8)Q11z=E111−μ12μ21,Q22z=E221−μ12μ21,Q12z=μ21E111−μ12,Q44=G23,Q55=G13,Q66=G12,where G13 and G23 indicate the shear moduli of CNTRC materials. The relationship between the shear moduli is supposed to be G13 = G12, G23 = 1.2 G12 [57, 58]. Through the integration of the stresses and moments of the in-plane stresses across the plate thickness, the force and moment resultants can be expressed as follows:(9)Nxxi,jNyyi,jNxyi,jMxxi,jMyyi,jMxyi,j=A11A120B11B120A12A220B12B22000A6600B66B11B120D11D120B12B220D12D22000B6600D66εxx,0i,jεyy,0i,jγxy,0i,jχxxi,jχyyi,jχxyi,j,QxziQyzi=κA5500A44γxz0,iγyz0,i,where Nxxi,j,Nyyi,j, and Nxyi,j indicate the in-plane force resultants, Mxxi,j,Myyi,j, and Mxyi,j represent the bending and twisting moment resultants, and Qxzi and Qyzi denote the transverse shear force resultants. In addition, κ = 5/6 stands for the shear correction factor, and Aij, Bij, and Dij (i, j = 1, 2, 4, 5, 6) represent the stretching, coupling, and bending stiffnesses defined as follows:(10)Aij,Bij,Dij=∫−h/2h/2Qijz1,z,z2dz.The strain energy of stretching and bending of the FG-CNTRC cylindrical shell is expressed as follows. The strain energyUSi,j stored in i, jth subplate can be written as(11)USi,j=12∫0a∫0bNxxi,jεxx,0i,j+Nyyi,jεyy,0i,j+Nxyiγxy,0i,j+Mxxiχxxi,j+Myyi,jχyyi,j+Mxyi,jχxyi,j+Qxzi,jγxzi,j+Qyzi,jγyzi,jdydx.By substituting equations (6) and (9) into equation (11), the strain energy of the FG-CNTRC plate can be represented using the displacements (u0,v0,w0) and rotation components (φ, θ).For the simplification of the equation, equation (11) is represented as USi,j=UTi,j+UTBi,j+UBi,j, where UTi,j,UBi,j and UTBi,j indicate stretching, bending, and bending-stretching coupling energy expressions, respectively.(12)UTi,j=12∫0a∫0bA11∂u0i,j∂x2+A66∂u0i,j∂y2+2A12∂u0i,j∂x∂v0i,j∂y+2A66∂u0i,j∂y∂v0i,j∂x+A22∂v0i,j∂y2+A66∂v0i,j∂x2+κA66∂w0i,j∂x2+κA66∂w0i,j∂y2+2κA66∂w0i,j∂xφi,j+2κA66∂w0i,j∂yθi,j+κA66φi,j2+κA66θi,j2dydx,UTBi,j=12∫0a∫0b2B11∂u0i,j∂x∂φi,j∂x+2B12∂u0i,j∂x∂θi,j∂y+2B12∂v0i,j∂y∂φi,j∂x+2B22∂v0i,j∂y∂θi,j∂y+2B66∂v0i,j∂x∂θi,j∂x+2B66∂v0i,j∂x∂φi,j∂y+2B66∂u0i,j∂y∂θi,j∂x+2B66∂u0i,j∂y∂φi,j∂ydydx,UBi,j=12∫0a∫0bD11∂φi,j∂x2+2D12∂φi,j∂x∂θi,j∂y+2D66∂φi,j∂y∂θi,j∂x+D66∂φi,j∂y2+D66∂θi,j∂x2+D22∂θi,j∂y2dydx.The kinetic energy of a certain segment could be obtained as follows:(13)T=12∬Aρu˙0i,j+zφ˙i,j2+v˙0i,j+zθ˙i,j2+w˙0i,j2dA=12∬A∫−h/2h/2ρu˙0i,j2+2u˙0i,jφ˙i,jz+z2φ˙i,j2+v˙0i,j2+2v˙0i,jθ˙i,jz+z2θ˙i,j2+w˙0i,j2dzdA=12∬AI0u˙0i,j2+v˙0i,j2+w˙0i,j2+2I1u˙0i,jφ˙i,j+v˙0i,jθ˙i,j+I2φ˙i,j2+θ˙i,j2dA.The dots on the symbols indicate the differentiation of displacement components with respect to time.(14)I0,I1,I2=∫−h/2h/2ρz1,z,z2dz.The potential energy stored in the boundary springs is expressed as follows:(15)Ub=12∫0bku,x0u02+kv,x0v02+kw,x0w02+kφ,x0φ02+kθ,x0θ02x=0+ku,x1u02+kv,x1v02+kw,x1w02+kφ,x1φ02+kθ,x1θ02x=Ldy+12∫0aku,y0u02+kv,y0v02+kw,y0w02+kφ,y0φ02+kθ,y0θ02y=0+ku,y1u02+kv,y1v02+kw,y1w02+kφ,y1φ02+kθ,y1θ02y=Ldx,where kt,0 (t = u, v, w, φ, θ) and kt,1 represent the boundary spring stiffness of the both ends of FG-CNTRC plate, respectively.The potential energy stored in the connective springs is represented as follows:(16)Uci,j=12∫0bkuu0i−u0i+12+kvv0i−vi+12+kww0i−wi+12+kφφi−φi+12+kθθi−θi+12dy+12∫0akuuj−uj+12+kvvj−vj+12+kwwj−wj+12+kφφj−φj+12+kθθj−θj+12dx,where ku,kv,kw, kφ, and kθ represent the stiffnesses of the springs between individual subplates and the superscripts i and i + 1 denote the ith and i + 1th subplates.Therefore, total potential energy including boundary conditions and connective conditions can be represented as follows:(17)U=∑i=1Nx∑j=1NyUsi,j+∑i=1Nx−1∑j=1Ny−1Uci,j+Ub.As a result, the arbitrary boundary conditions can be freely modeled in the proposed model by setting the stiffness of the springs as proper values.It is supposed that the external force act on the entire middle surface of the FG-CNTRC plate. The virtual work done on thei, jth subplate by the distributed load components can be expressed as follows [59]:(18)Wi,j=12∭Vfui,ju0i,j+fvi,jv0i,j+fwi,jw0i,j+mφi,jφ0i,j+mθi,jθ0i,jdV.


### 2.4. Solution Procedure

The convergence and accuracy of the analysis results rely on the selection of the displacement. In this research, the vibration characteristics of FG-CNTRC plate are studied using the suitable allowable displacement function. Finally, all the displacement functions including boundary and continuous conditions are chosen as ultraspherical polynomials. The ultraspherical polynomial is a special case of the Jacobi orthogonal polynomial, and main advantage is that it can guarantee the very high accuracy and robustness of computation [60–62]. When the polynomial’s parameter λ = 0, the ultraspherical polynomials Pmλξ are called as first kind of Chebyshev polynomials (Cm (ξ)), while they are represented as Legendre polynomials Lm (ξ) if λ = 0.5. When λ = 1, Pmλξ is indicated as second kind of Chebyshev polynomials. The ultraspherical polynomials Pmλξ are defined in the interval ξ ∈ [−1, 1].(19)ℑξ=1−ξ2λ−1/2,λ>−12.The orthogonality condition is(20)∫−111−ξ2λ−1/2PmλξPnλξdx=πn!Γ2λΓλ+1/2Γn+2λn+λΓλ,m=n,0,m≠n.The ultraspherical polynomialsPmλξ can be also expressed with the recurrence relation [63, 64].(21)m+2λPmλξ=2m+λξPmλξ−mPm−1λξ,where m = 1, 2, 3, ….Therefore, the allowable displacement function of FG-CNTRC plate can be more generalized using the ultraspherical polynomials and written as(22)u=∑mx=0Mx∑ny=0NyUmnPmxλξxPnyλξyeiωt,v=∑mx=0Mx∑ny=0NyVmnPmxλξxPnyλξyeiωt,w=∑mx=0Mx∑ny=0NyWmnPmxλξxPnyλξyeiωt,φ=∑mx=0Mx∑ny=0NyΦmnPmxλξxPnyλξyeiωt,θ=∑mx=0Mx∑ny=0NyΘmnPmxλξxPnyλξyeiωt,where Umn,Vmn,Wmn, Φmn, and Θmn indicate the unknown coefficients of the ultraspherical polynomials and Mx, Ny are maximum m-order and n-order, respectively. Pmλξx,Pnλξy denote the m-order and n-order ultraspherical polynomials in regard to displacement in x and y directions, and ω, t represent the angular frequency and time, respectively. As the ultraspherical polynomials are complete and the orthogonal polynomials are defined at interval of ξ ∈ [−1, 1], the linear transformation statute should be applied for the coordinate conversion from the interval x ∈ [0, L] of the divided beam to the interval ξ (ξ ∈ [−1, 1]) of the ultraspherical polynomials, that is, ξ = 2x/L − 1.The total Lagrangian energy functions of FG-CNTRC plate can be written as follows:(23)L=∑i=1Nx∑i=1NyTi,j−Ui,j+Wi,j.The total Lagrangian energy function can be minimized in regard to the unknown coefficients based on the Rayleigh–Ritz method.(24)∂L∂I=0,I=Um,Vm,Wm,Φm,Θm.Therefore, the vibration governing equation of FG-CNTRC plate can be indicated as follows:(25)K−ω2MA=F.The stiffness matrixK, mass matrix M, and unknown coefficient matrix A are represented by the following equations. The natural frequencies of the FG-CNTRC plate can be calculated when the right term F in equation (25) equals zero. The detailed expression of stiffness matrix K and mass matrix M in equation (25) can be found in the Appendix.


## 2.1. Model of the Multi-Stepped FG-CNTRC Plate

Figure1 shows the calculation model of the multi-stepped FG-CNTRC plate. As can be seen in Figure 1(a), the multi-stepped plate consists of several subplates with different thicknesses in x and y directions, and the lengths of individual subplates are expressed as ai (i = 1, 2, …) and bj (j = 1, 2, …) in x and y directions, respectively.Figure 1
Multi-stepped FG-CNTRC plate model. (a) Geometric relationship. (b) Boundary condition. (c) Combined condition.
(a)(b)(c)Also, the heights of individual subplates are indicated ashi. In this research, the heights of individual subplates are expressed as h1 and h2 for the convenient calculation, and u,v,andw are the displacements in x, y, z directions, respectively. The artificial spring technique is introduced for the generalization of boundary conditions, and the four sides of the plate are modeled to be supported by the artificial springs (Figure 1(b)).Each side has a boundary spring group consisting of three artificial springs (ku,kv,kw) and two rotating artificial elastic springs (kφ, kθ), and the boundary condition can be generalized by adjusting the stiffness of individual springs. The boundary conditions at four sides are expressed by adding subscript 0 at x = 0 and y = 0 boundary and subscript 1 at x = a and y = b boundary. The stepped plates can be seen to be made up of the strong combinations of the individual subplates, and the connective condition can be modeled in a similar way with the boundary condition. That is, by setting the stiffness of connective springs as infinity, the strong connective condition of individual plates can be accomplished. Figure 1(c) shows the connective conditions of individual plates.


## 2.2. Material Properties

The stepped plate considered here is composed of the isotropic matrix reinforced with CNTs. The distribution of CNTs in the matrix can be either uniform or functionally graded according to the thickness of plates. Figure2 shows five types of CNT distributions such as UD-CNTRC, FG-Λ CNTRC, FG-V CNTRC, FG-X CNTRC, and FG-O CNTRC.Figure 2
Distribution patterns of CNTs in FG-CNTRC plates. (a) UD-CNTRC. (b) FG-Λ CNTRC. (c) FG-V CNTRC. (d) FG-O CNTRC. (e) FG-X CNTRC.
(a)(b)(c)(d)(e)The CNT volume fractionsVCNT in the various kinds of FG-CNTRC plates are indicated as follows [57, 58]:(1)VCNTz=VCNT∗,UD−CNT,1+2zhVCNT∗,FGV−CNT,1−2zhVCNT∗,FGΛ−CNT,4zhVCNT∗,FGX−CNT,21−2zhVCNT∗,FGO−CNT,in which(2)VCNT∗=wCNTwCNT+ρCNT/ρm−ρCNT/ρmwCNT,where wCNT is the mass fraction of nanotube and ρCNT and ρm are the mass densities of CNT and matrix constituents, respectively.Figure3 shows the variation characteristics of CNT volume fractions VCNT according to z/h in five patterns. It shows that the value of VCNT is constant regardless of thickness of plates in UD-CNTRC, while it increases or decreases in FG-Λ CNTRC and FG-V CNTRC, respectively. In cases of FG-X CNTRC and FG-O CNTRC, the value of VCNT changes symmetrically about the middle surface.Figure 3
Change of the CNT volume fractionsVCNT on the thickness of plate. (a) VCNT∗=0.1. (b) VCNT∗=0.2.
(a)(b)Based on the improved rule of mixtures, effective Young’s modulus, shear modulus, Poisson’s ratio, and mass density for CNTRC materials can be written as follows [56, 58]:(3)E11z=η1VCNTzE11CNT+VmzEm,η2E22z=VCNTzE22CNT+VmzEm,η3G12z=VCNTzG12CNT+VmzGm,Gm=Em21−μm,μ12z=VCNTzμ12CNT+Vmzμm,μ21z=μ12zE11zE22z,ρz=VCNTzρCNT+Vmzρm,Vmz=1−VCNT,where E11CNT,E22CNT,G12CNT,Em,Gm represent Young’s modulus and shear modulus of CNT and matrix, η1, η2, and η3 are CNT/matrix efficiency parameters, and μ12CNT,μm denote Poisson’s ratios of CNT and matrix.


## 2.3. Energy Function

The displacement components of the individual subplate can be expressed with the displacements of the midsurface and the rotations of the cross section using the FSDT as follows:(4)Ui,jx,y,z,t=u0i,jx,y,t+zφi,jx,y,t,Vi,jx,y,z,t=v0i,jx,y,t+zθi,jx,y,t,Wi,jx,y,z,t=w0i,jx,y,t,where u0,v0,andw0 represent the middle surface displacements of the i, jth subplate in the x, y, and z directions, φ and θ indicate the transverse normal rotations in regard to x and y axes, and t denotes the time variable. The strain elements at a random point of the ith FG-CNTR subplate can be written as follows:(5)εxxi,j=εxx,0i,j+zχxxi,j,εyyi,j=εyy,0i,j+zχyyi,j,γxyi,j=γxy,0i,j+zχxyi,j,γxzi,j=γxz,0i,j,γyzi,j=γyz,0i,j,where εxx,0i,j,εyy,0i,j,γxy,0i,j,γxz,0i,j, and γyz,0i,j indicate the middle surface strains and χxxi,j,χyyi,j, and χxyi,j denote the surface curvatures of i, jth subplate, respectively; these surface strains and curvatures are defined as follows:(6)εxx,0i,j=∂u0i,j∂x,εyy,0i,j=∂v0i,j∂y,γxy,0i,j=∂v0i,j∂x+∂u0i,j∂y,γxz,0i,j=∂w0i,j∂x+φi,j,γyz,0i,j=∂w0i,j∂y+θi,j,χxxi,j=∂φi,j∂x,χyyi,j=∂θi,j∂y,χxyi,j=∂θi,j∂x+∂φi,j∂y.Following the state of generalized Hooke’s law, the constitutive relations of FG-CNTRC plate can be written as follows:(7)σxxi,jσyyi,jτyzi,jτxzi,jτxyi,j=Q11zQ12z000Q12zQ22z00000Q44z00000Q55z00000Q66zεxxi,jεyyi,jγyzi,jγxzi,jγxyi,j,where Qij (i, j = 1, 2, 4, 5, 6) are the reduced material stiffness coefficients compatible with plane-stress conditions and expressed in terms of elastic and shear moduli and Poisson’s ratio as(8)Q11z=E111−μ12μ21,Q22z=E221−μ12μ21,Q12z=μ21E111−μ12,Q44=G23,Q55=G13,Q66=G12,where G13 and G23 indicate the shear moduli of CNTRC materials. The relationship between the shear moduli is supposed to be G13 = G12, G23 = 1.2 G12 [57, 58]. Through the integration of the stresses and moments of the in-plane stresses across the plate thickness, the force and moment resultants can be expressed as follows:(9)Nxxi,jNyyi,jNxyi,jMxxi,jMyyi,jMxyi,j=A11A120B11B120A12A220B12B22000A6600B66B11B120D11D120B12B220D12D22000B6600D66εxx,0i,jεyy,0i,jγxy,0i,jχxxi,jχyyi,jχxyi,j,QxziQyzi=κA5500A44γxz0,iγyz0,i,where Nxxi,j,Nyyi,j, and Nxyi,j indicate the in-plane force resultants, Mxxi,j,Myyi,j, and Mxyi,j represent the bending and twisting moment resultants, and Qxzi and Qyzi denote the transverse shear force resultants. In addition, κ = 5/6 stands for the shear correction factor, and Aij, Bij, and Dij (i, j = 1, 2, 4, 5, 6) represent the stretching, coupling, and bending stiffnesses defined as follows:(10)Aij,Bij,Dij=∫−h/2h/2Qijz1,z,z2dz.The strain energy of stretching and bending of the FG-CNTRC cylindrical shell is expressed as follows. The strain energyUSi,j stored in i, jth subplate can be written as(11)USi,j=12∫0a∫0bNxxi,jεxx,0i,j+Nyyi,jεyy,0i,j+Nxyiγxy,0i,j+Mxxiχxxi,j+Myyi,jχyyi,j+Mxyi,jχxyi,j+Qxzi,jγxzi,j+Qyzi,jγyzi,jdydx.By substituting equations (6) and (9) into equation (11), the strain energy of the FG-CNTRC plate can be represented using the displacements (u0,v0,w0) and rotation components (φ, θ).For the simplification of the equation, equation (11) is represented as USi,j=UTi,j+UTBi,j+UBi,j, where UTi,j,UBi,j and UTBi,j indicate stretching, bending, and bending-stretching coupling energy expressions, respectively.(12)UTi,j=12∫0a∫0bA11∂u0i,j∂x2+A66∂u0i,j∂y2+2A12∂u0i,j∂x∂v0i,j∂y+2A66∂u0i,j∂y∂v0i,j∂x+A22∂v0i,j∂y2+A66∂v0i,j∂x2+κA66∂w0i,j∂x2+κA66∂w0i,j∂y2+2κA66∂w0i,j∂xφi,j+2κA66∂w0i,j∂yθi,j+κA66φi,j2+κA66θi,j2dydx,UTBi,j=12∫0a∫0b2B11∂u0i,j∂x∂φi,j∂x+2B12∂u0i,j∂x∂θi,j∂y+2B12∂v0i,j∂y∂φi,j∂x+2B22∂v0i,j∂y∂θi,j∂y+2B66∂v0i,j∂x∂θi,j∂x+2B66∂v0i,j∂x∂φi,j∂y+2B66∂u0i,j∂y∂θi,j∂x+2B66∂u0i,j∂y∂φi,j∂ydydx,UBi,j=12∫0a∫0bD11∂φi,j∂x2+2D12∂φi,j∂x∂θi,j∂y+2D66∂φi,j∂y∂θi,j∂x+D66∂φi,j∂y2+D66∂θi,j∂x2+D22∂θi,j∂y2dydx.The kinetic energy of a certain segment could be obtained as follows:(13)T=12∬Aρu˙0i,j+zφ˙i,j2+v˙0i,j+zθ˙i,j2+w˙0i,j2dA=12∬A∫−h/2h/2ρu˙0i,j2+2u˙0i,jφ˙i,jz+z2φ˙i,j2+v˙0i,j2+2v˙0i,jθ˙i,jz+z2θ˙i,j2+w˙0i,j2dzdA=12∬AI0u˙0i,j2+v˙0i,j2+w˙0i,j2+2I1u˙0i,jφ˙i,j+v˙0i,jθ˙i,j+I2φ˙i,j2+θ˙i,j2dA.The dots on the symbols indicate the differentiation of displacement components with respect to time.(14)I0,I1,I2=∫−h/2h/2ρz1,z,z2dz.The potential energy stored in the boundary springs is expressed as follows:(15)Ub=12∫0bku,x0u02+kv,x0v02+kw,x0w02+kφ,x0φ02+kθ,x0θ02x=0+ku,x1u02+kv,x1v02+kw,x1w02+kφ,x1φ02+kθ,x1θ02x=Ldy+12∫0aku,y0u02+kv,y0v02+kw,y0w02+kφ,y0φ02+kθ,y0θ02y=0+ku,y1u02+kv,y1v02+kw,y1w02+kφ,y1φ02+kθ,y1θ02y=Ldx,where kt,0 (t = u, v, w, φ, θ) and kt,1 represent the boundary spring stiffness of the both ends of FG-CNTRC plate, respectively.The potential energy stored in the connective springs is represented as follows:(16)Uci,j=12∫0bkuu0i−u0i+12+kvv0i−vi+12+kww0i−wi+12+kφφi−φi+12+kθθi−θi+12dy+12∫0akuuj−uj+12+kvvj−vj+12+kwwj−wj+12+kφφj−φj+12+kθθj−θj+12dx,where ku,kv,kw, kφ, and kθ represent the stiffnesses of the springs between individual subplates and the superscripts i and i + 1 denote the ith and i + 1th subplates.Therefore, total potential energy including boundary conditions and connective conditions can be represented as follows:(17)U=∑i=1Nx∑j=1NyUsi,j+∑i=1Nx−1∑j=1Ny−1Uci,j+Ub.As a result, the arbitrary boundary conditions can be freely modeled in the proposed model by setting the stiffness of the springs as proper values.It is supposed that the external force act on the entire middle surface of the FG-CNTRC plate. The virtual work done on thei, jth subplate by the distributed load components can be expressed as follows [59]:(18)Wi,j=12∭Vfui,ju0i,j+fvi,jv0i,j+fwi,jw0i,j+mφi,jφ0i,j+mθi,jθ0i,jdV.


## 2.4. Solution Procedure

The convergence and accuracy of the analysis results rely on the selection of the displacement. In this research, the vibration characteristics of FG-CNTRC plate are studied using the suitable allowable displacement function. Finally, all the displacement functions including boundary and continuous conditions are chosen as ultraspherical polynomials. The ultraspherical polynomial is a special case of the Jacobi orthogonal polynomial, and main advantage is that it can guarantee the very high accuracy and robustness of computation [60–62]. When the polynomial’s parameter λ = 0, the ultraspherical polynomials Pmλξ are called as first kind of Chebyshev polynomials (Cm (ξ)), while they are represented as Legendre polynomials Lm (ξ) if λ = 0.5. When λ = 1, Pmλξ is indicated as second kind of Chebyshev polynomials. The ultraspherical polynomials Pmλξ are defined in the interval ξ ∈ [−1, 1].(19)ℑξ=1−ξ2λ−1/2,λ>−12.The orthogonality condition is(20)∫−111−ξ2λ−1/2PmλξPnλξdx=πn!Γ2λΓλ+1/2Γn+2λn+λΓλ,m=n,0,m≠n.The ultraspherical polynomialsPmλξ can be also expressed with the recurrence relation [63, 64].(21)m+2λPmλξ=2m+λξPmλξ−mPm−1λξ,where m = 1, 2, 3, ….Therefore, the allowable displacement function of FG-CNTRC plate can be more generalized using the ultraspherical polynomials and written as(22)u=∑mx=0Mx∑ny=0NyUmnPmxλξxPnyλξyeiωt,v=∑mx=0Mx∑ny=0NyVmnPmxλξxPnyλξyeiωt,w=∑mx=0Mx∑ny=0NyWmnPmxλξxPnyλξyeiωt,φ=∑mx=0Mx∑ny=0NyΦmnPmxλξxPnyλξyeiωt,θ=∑mx=0Mx∑ny=0NyΘmnPmxλξxPnyλξyeiωt,where Umn,Vmn,Wmn, Φmn, and Θmn indicate the unknown coefficients of the ultraspherical polynomials and Mx, Ny are maximum m-order and n-order, respectively. Pmλξx,Pnλξy denote the m-order and n-order ultraspherical polynomials in regard to displacement in x and y directions, and ω, t represent the angular frequency and time, respectively. As the ultraspherical polynomials are complete and the orthogonal polynomials are defined at interval of ξ ∈ [−1, 1], the linear transformation statute should be applied for the coordinate conversion from the interval x ∈ [0, L] of the divided beam to the interval ξ (ξ ∈ [−1, 1]) of the ultraspherical polynomials, that is, ξ = 2x/L − 1.The total Lagrangian energy functions of FG-CNTRC plate can be written as follows:(23)L=∑i=1Nx∑i=1NyTi,j−Ui,j+Wi,j.The total Lagrangian energy function can be minimized in regard to the unknown coefficients based on the Rayleigh–Ritz method.(24)∂L∂I=0,I=Um,Vm,Wm,Φm,Θm.Therefore, the vibration governing equation of FG-CNTRC plate can be indicated as follows:(25)K−ω2MA=F.The stiffness matrixK, mass matrix M, and unknown coefficient matrix A are represented by the following equations. The natural frequencies of the FG-CNTRC plate can be calculated when the right term F in equation (25) equals zero. The detailed expression of stiffness matrix K and mass matrix M in equation (25) can be found in the Appendix.


## 3. Convergence and Validation Study

To ensure the validity and accuracy of the suggested method, the calculation examples for the free and forced vibration analysis of FG-CNTRC under the several boundary conditions are presented. The calculation results from the suggested method are compared with those of the previous works or obtained by the finite element analysis software ABAQUS. Based on the validation results, the effects of geometric and material parameters on the free or forced vibration response are studied. MATLAB is applied for the calculation process of the proposed method, which is run on a Intel(R) Core(TM) i7-7500 2.20 GHz PC.


### 3.1. Convergence Study

As can be seen from the theoretical formula, the accuracy of the solution calculated by the proposed method is determined by the degree of the ultraspherical polynomial and polynomial parameters. Therefore, it is necessary to conduct the convergence study to determine these parameters. It is certain that the accuracy of solution becomes higher as the degree of polynomial increases infinitely. However, in this case, as it requires high level of hardware and increased amount of calculation time, it is important to determine the reasonable degree of polynomial. For the convergence study, it is supposed that the material has the characteristics of uniform distribution and the material and geometric properties are set as follows:(26)Em=3GPa,ρm=1000kg/m3,μm=0.3,E11CNT=5TPa,E22CNT=7TPa,G12CNT=2TPa,ρCNT=1500kg/m3,μ12CNT=0.2,VCNT∗=0.1,η1=0.1,η2=1,η3=0.7,a=1m,b=1m,h=0.05m.Also, in all the following processes, the dimensionless frequency is calculated by the formulaΩ=ωa12/h1ρm/Em.Table1 shows the convergence characteristics of non-dimensional frequency of the FG-CNTRC plates, in which four sides are fully clamped, according to the increase of ultraspherical polynomial degree Mx × Ny. From Table 1, it can be known that as the degree of polynomial increases, the dimensionless frequency of FG-CNTRC plates approximate to a certain value, and then it does not change any more after the degree of polynomial is beyond Mx × Ny = 10 × 10. Therefore, in this research, in all the calculation of the numerical examples, the degree of Mx × Ny is set as Mx × Ny = 10 × 10. In addition, the calculation time by the method proposed in Table 1 and the calculation time by FEM (ABAQUS) are shown. As shown in Table 1, when the number of elements is 40000, it is the most similar to the result of the proposed method, and the calculation time required at this time is 30.3 s. In the case of the proposed method, the calculation time is 0.77 s (in the case of Mx × Ny = 10 × 10), and it can be seen that the calculation time is much shorter than that of the FEM. In other words, it can be seen that the proposed method has the advantage of very high calculation accuracy and calculation efficiency.Table 1
Convergence of dimensionless frequencies of stepped FG-CNTRC plate.
ModeTruncated number12345678910Time (s)Present3 × 36.697111.12813.26516.00018.29423.37126.20126.57228.04028.6050.5444 × 45.35257.363611.75512.91815.00918.22018.43420.14320.89222.720.5425 × 55.33826.815610.03511.47612.35814.53316.89818.21818.86819.4870.5486 × 65.33796.81339.901311.47312.35414.34414.43417.90918.21718.7700.5557 × 75.33786.81219.871211.47212.35314.27414.41517.85718.21718.7600.5638 × 85.33786.81219.870711.47212.35314.23814.41517.82918.21718.7600.5689 × 95.33776.81219.870411.47212.35314.23714.41517.82918.21718.7600.59010 × 105.33796.81209.870411.47212.35314.23714.41517.82818.21718.7600.77011 × 115.33786.81209.870411.47212.35314.23714.41517.82818.21718.7600.82912 × 125.33786.81209.870411.47212.35314.23714.41417.82818.21718.7601.00713 × 135.33786.81209.870411.47212.35314.23714.41417.82818.21718.7601.158Number of elementFEM (ABAQUS)16005.34296.82559.927511.49912.37614.39514.45817.94618.23018.8361.325005.34116.82069.906811.48912.36814.33814.44217.90418.22718.8091.9100005.33876.81429.879411.47612.35614.26214.42117.84718.21918.7726.7225005.33816.81309.874511.47412.35414.24814.41717.83618.21718.76514.5400005.33806.81269.872611.47312.35314.24314.41617.83218.21718.76330.3As mentioned above, as the ultraspherical polynomial is characterized by the polynomial’s parameterλ, it is necessary to conduct the study on the determination of polynomial parameters. Figure 4 presents the percentage error (Ωλ − Ωλ = 0)/Ωλ = 0 of the solution of the ultraspherical polynomial parameter λ in the FG-CNTRC plate. Figure 4 indicates that the error of dimensionless frequency in the FG-CNTRC plate does not exceed 1.5 × 10−4 regardless of the change of polynomial parameter λ. Therefore, in this research, the polynomial parameter λ is set as 0.Figure 4
Percentage error of the dimensionless frequencies for the ultraspherical polynomial parametersλ. (a) CCCC. (b) FFFF.
(a)(b)As mentioned in the previous section, the artificial elastic spring technique is introduced to generalize the boundary conditions for the vibration analysis of the stepped FG-CNTRC plate. The boundary condition is changed according to the selection of the stiffness of artificial elastic spring. Therefore, it is necessary to conduct the determination study of the boundary conditions.Figure5 shows the variation characteristics of the non-dimensional frequency in the FG-CNTRC plate according to the change of stiffness of the boundary elastic spring. In order to analyze the variation characteristics of the FG-CNTRC plate according to the change of stiffness of the individual boundary spring, except for the considered boundary spring, the stiffness of all other boundary springs is set as zero, and then the stiffness of the considered spring is changed from 102 to 1016. As can be seen from Figure 5, the frequency shows almost no change when the stiffness of boundary spring is below 104, and then it increases dramatically beyond this value until the stiffness reaches 1011. When the stiffness is beyond 1011, the frequency does not change again. Based on these results, the classic and elastic boundary conditions for the calculation of vibration characteristics in the FG-CNTRC plate can be set as shown in Table 2. In Table 2, F, S, C, and E indicate the free, simple, clamped, and elastic boundary conditions. In addition, in the following processes, the boundary condition CFSE1 represents the clamped boundary condition at x = 0 , free boundary condition at x = a, simple-supported boundary condition at y = 0, and elastic boundary condition at y = b, respectively.Figure 5
Convergence characteristics of dimensionless frequencies on the boundary spring stiffness. (a) Mode1. (b) Mode2. (c) Mode3. (d) Mode4.
(a)(b)(c)(d)Table 2
The spring stiffness values of the elastic boundary conditions.
Boundary conditionsku,0,ku,1kv,0,kv,1kw,0,kw,1kφ,0, kφ,1kθ,0, kθ,1F00000S01014101401014C10141014101410141014E110810610610141014E2101410141014108108E3108108108108108


### 3.2. Validation

#### 3.2.1. Free Vibration

In the previous section, through the convergence study, parameters including the boundary parameter for the vibration analysis of multi-stepped FG-CNTRC plates are determined. Based on these results, the accuracy of the suggested method is validated. The accuracy is validated by comparing the results from the proposed method with those from the previous works or finite element method. Table3 shows the comparison results of the dimensionless frequency of the non-stepped plates with isotropic materials. Here, Poisson’s ratio is 0.3. Table 3 indicates that the results from the proposed method agree well with those of the previous works.Table 3
Comparison of dimensionless frequencies of non-stepped isotropic plate.
Boundary conditionshRefs.Modes1234567SSSS0.1Present19.06545.48245.48264.36864.36869.79385.523Reference [65]19.09845.63645.63664.38464.38470.14985.5Reference [66]19.0945.62245.62264.38364.38370.11285.502Reference [67]19.0945.61945.61964.38364.38370.10485.4880.2Present17.44832.17632.17638.14938.14945.52655.144Reference [65]17.52832.19232.19238.48838.48845.52655.802Reference [66]17.52832.19232.19238.50238.50245.52655.843Reference [67]17.52632.19232.19238.48338.48345.52655.787CCCC0.1Present32.5362.05162.05186.963103.89104.94123.11Reference [65]33.00963.04363.04388.411104.28105.29123.73Reference [66]32.79762.67262.67287.941103.71104.7123.6Reference [67]32.78262.6362.6387.869103.61104.6123.590.2Present26.52246.2946.2961.53161.53163.75671.962Reference [65]27.06547.34647.34662.006263.63572.604Reference [66]26.97447.25347.25361.94461.94463.5772.568Reference [67]26.90647.10347.10361.91761.91763.34872.286FFFF0.1Present12.73418.94523.32831.99431.99455.37755.377Reference [65]12.72818.95623.34631.96531.96555.49355.493Reference [68]12.72618.95523.34731.96531.96555.49355.4930.2Present11.70217.421.19427.5827.5840.19442.81Reference [65]11.7117.43321.25227.64827.64840.19242.775Reference [68]11.7117.43321.25227.64727.64740.19142.776Next, the natural frequency result of stepped plates with isotropic materials is compared with that obtained by the finite element method. The stepped plate consists of three subplates inx and y directions, respectively, and the geometric parameters are a1 = 0.5 m, a2 = 0.1 m, a3 = 0.5 m, b1 = 0.5 m, b2 = 0.1 m, b3 = 0.5 m, hx1 = 0.1 m, hx2 = 0.2 m, hx3 = 0.1 m, hy1 = 0.1 m, hy2 = 0.2 m, and hy3 = 0.1 m. The finite element analysis software ABAQUS is used in the finite element method, and the element type is S4R and the number of elements is 8476.Through the comparison results of Tables3 and 4, it can be known that the suggested method is suitable for the free vibration analysis of non-stepped or stepped plates. The main purpose of this research is to study the vibration analysis method of the multi-stepped FG-CNTRC plate; therefore, the accuracy of the proposed method is validated by comparing the results of natural frequency in the FG-CNTRC plate using the suggested method. Table 5 shows the comparison results of natural frequency in the non-stepped FG-CNTRC plate. The material and geometric parameters are as follows [16]:(27)Em=2.1GPa,ρm=1150kg/m3,μm=0.34,E11CNT=5.6466TPa,E22CNT=7.08TPa,G12CNT=1.9445TPa,VCNT∗=0.11,η1=0.149,η2=0.934,η3=0.934,a=1m,b=1m.Table 4
Comparison of natural frequency of non-stepped isotropic plate.
FrequencyBoundary conditionsCCCCSSSSCCFFPresentFEMError (%)PresentFEMError (%)PresentFEMError (%)1812.7764813.050.0337439.6185439.620.0003467.8344468.130.063221513.9041516.10.14511100.1061101.40.1176488.3953488.530.027631513.9541516.10.14181100.1151101.40.1168909.7355909.70.003941998.5491999.90.06761551.6291552.60.06261190.2181192.40.183452479.5512486.70.28832028.0332032.90.241232.62512340.111562485.87624930.28662043.132048.40.25791263.8751265.40.120772603.52605.80.08832308.1512312.30.17981668.4721670.30.109582603.7872605.80.07732308.1532312.30.17971736.6551738.30.094792783.3032789.10.20832603.52605.30.06912148.3472155.90.3516102783.3382789.10.2072603.7872605.30.05812165.6322172.70.3264Table 5
Comparison of dimensionless frequency of non-stepped FG-CNTRC plate.
Boundary conditionsb/hΩFG-VFG-XFG-OReference [16]Reference [16] (ANSYS)PresentReference [16]Reference [16] (ANSYS)PresentReference [16]Reference [16] (ANSYS)PresentSSSS10112.45212.49512.48614.61614.65914.65511.5511.611.581217.0617.2117.05518.64618.82418.65116.26516.40916.256319.49919.47919.56919.49919.47919.55419.49919.47919.595419.49919.47919.56919.49919.47919.55419.49919.47919.595527.3427.52427.00928.51928.74228.226.51326.68726.17920115.1115.10315.13619.93919.91619.97513.52313.53113.542219.90319.87919.88223.77623.77323.7718.48618.46418.458331.56131.23331.1234.38934.12333.97730.16629.8329.719438.99838.95839.13838.99838.95839.10838.99838.95839.19538.99838.95839.13838.99838.95839.10838.99838.95839.19CCCC10117.21117.24717.17518.08318.12918.04716.70716.74416.677222.81822.92222.6523.60623.72823.43822.25322.35322.089333.0733.0632.87634.33834.33833.66132.37832.37232.188433.55233.53432.90334.46734.46434.30332.85732.83732.228536.52836.80436.35937.44737.437.57835.80936.08335.65620126.30426.24826.19630.42130.3830.32724.48624.44324.375231.49631.40531.26235.03635.00334.83929.79529.69929.55343.58942.99142.63546.4845.98445.59141.89541.27940.93456.24955.89455.7661.9861.69261.55853.55753.20453.058559.22159.0958.7964.56263.32162.556.61756.46456.164As can be seen from Table5, the dimensionless frequency in the non-stepped FG-CNTRC plate with uniform thickness agrees well with the previous works.Next, the natural frequency results of the multi-stepped FG-CNTRC plate are compared with that obtained by the finite element method. The material properties are the same as the case mentioned above, and the geometric properties and the parameters for the finite element analysis are set as shown in Table3. Tables 5 and 6 show that the results from the proposed method agree well with those of the previous works or obtained by the finite element method, and through the convergence and validity study, it can be known that the suggested method is the accurate method for the free vibration analysis for not only the isotropic materials but also the non-stepped and multi-stepped plates of FG-CNTRC. The finite element analysis software ABAQUS is used in the finite element method, and the element type is S4R and the number of elements is 10868.Table 6
Comparison of natural frequency of multi-stepped FG-CNTRC plate.
FrequencyBoundary conditionsCCCCSSSSCCFFPresentFEMError (%)PresentFEMError (%)PresentFEMError (%)1369.1226369.190.0182264.3662264.360.0023304.0064304.060.01762529.1282529.650.0986397.9645398.280.0793311.0388311.120.02613718.3087718.740.06675.8775676.340.0684377.6699377.610.01594794.5131794.790.0349678.0683679.450.2038383.4401383.590.03915802.1168803.970.231714.4617714.720.0362571.7157572.120.07076858.3181858.660.0398858.3178858.530.0247685.7125686.270.08137976.337977.740.1437881.0082881.90.1012686.7083687.170.067281059.8321061.10.1196995.2035998.670.3483733.3032733.490.025591093.591095.80.20211021.291022.60.1282827.5539827.370.0222101118.3681121.20.25321041.6171042.60.0944856.6614857.660.1166


#### 3.2.2. Forced Vibration

In engineering applications, the external loads act on the plate which is the foundation construction, so it is needed to consider the forced response of structures. Considering the forced response, there mainly exist two parts such as the stability response analysis in frequency domain and transient response analysis in time domain. In this section, the accuracy of the proposed method is validated by comparing the forced vibration results obtained by the proposed method with those from the finite element method. For the analysis of the forced vibration, in the following research processes, the external forces are considered as three cases such as point force, line force, and area force, and it is assumed that the uniform load (fw = 1 N) is applied in the Z direction [63, 69].Figure6 represents the comparison results between steady-state responses in the non-stepped and multi-stepped FG-CNTRC plate under the CCCC boundary condition. The geometric and material parameters are the same as those shown in Tables 5 and 6. The ranges of natural frequency are from 300 to 900 in the non-stepped plates and from 300 to 800 in the multi-stepped plates, respectively. The interval is set as 1 Hz. Here, the harmonic point force fw acts in the thickness direction. The point load is fw=f¯wδx−lx1y−lx1, in which f¯w=−1N and δ indicates the Dirac delta function. The applied point of force and response measurement point are all set as A = (0.3 m, 0.3 m) and B = (0.3 m, 0.3 m). The comparison is made with the data obtained from the finite element analysis software ABAQUS, and the calculation is conducted under the same conditions.Figure 6
The comparison of vertical displacement of FG-CNTRC plate. (a) Non-stepped FG-CNTRC plate. (b) Multi-stepped FG-CNTRC plate.
(a)(b)The comparison results in Figure6 indicate that the proposed method is suitable for the steady-state vibration analysis of multi-stepped FG-CNTRC plate.Next, the comparison study for the transient response analysis is conducted. In this case, it is supposed that four kinds of transient loads are applied. Figure7 shows four types of transient loads used in this paper.Figure 7
Load pulse shapes. (a) Rectangular pulse; (b) triangular pulse; (c) half-sine pulse; (d) exponential pulse.
(a)(b)(c)(d)Four kinds of transient loads employed in this research are represented in Figure7, and the load functions are written as follows.Rectangular pulse:(28a)ft=ft,0≤t≤τ,0,t>τ.Triangular pulse:(28b)ft=2tτft,0≤t≤τ2,ft−2τt−τ2ft,τ2≤t≤τ,0,t>τ.Half-sine pulse:(28c)ft=ftsinπtτ,0≤t≤τ,0,t>τ.Exponential pulse:(28d)ft=fte−ξt,0≤t≤τ,0,t>τ,where ft is the load amplitude; τ is the pulse width; and t is the time variable.Figure8 presents the comparison results of transient response data between the proposed method and finite element software ABAQUS. The geometric parameters, material constants, and boundary conditions are set as shown in Figure 6. Here, the transient load f (t) is set as rectangular pulse and the amplitude of the rectangular pulse is set as ft = −1 N. The calculating time step is ∆t = 0.02 ms, and the loading time τ and calculating time are set as 20 ms, respectively. The results indicate that the prediction accuracy of the suggested method agrees well with that of the finite element method; therefore, the accuracy of the proposed method is validated.Figure 8
The comparison of normal displacement response of FG-CNTRC plate. (a) Non-stepped FG-CNTRC plate. (b) Multi-stepped FG-CNTRC plate.
(a)(b)


## 3.1. Convergence Study

As can be seen from the theoretical formula, the accuracy of the solution calculated by the proposed method is determined by the degree of the ultraspherical polynomial and polynomial parameters. Therefore, it is necessary to conduct the convergence study to determine these parameters. It is certain that the accuracy of solution becomes higher as the degree of polynomial increases infinitely. However, in this case, as it requires high level of hardware and increased amount of calculation time, it is important to determine the reasonable degree of polynomial. For the convergence study, it is supposed that the material has the characteristics of uniform distribution and the material and geometric properties are set as follows:(26)Em=3GPa,ρm=1000kg/m3,μm=0.3,E11CNT=5TPa,E22CNT=7TPa,G12CNT=2TPa,ρCNT=1500kg/m3,μ12CNT=0.2,VCNT∗=0.1,η1=0.1,η2=1,η3=0.7,a=1m,b=1m,h=0.05m.Also, in all the following processes, the dimensionless frequency is calculated by the formulaΩ=ωa12/h1ρm/Em.Table1 shows the convergence characteristics of non-dimensional frequency of the FG-CNTRC plates, in which four sides are fully clamped, according to the increase of ultraspherical polynomial degree Mx × Ny. From Table 1, it can be known that as the degree of polynomial increases, the dimensionless frequency of FG-CNTRC plates approximate to a certain value, and then it does not change any more after the degree of polynomial is beyond Mx × Ny = 10 × 10. Therefore, in this research, in all the calculation of the numerical examples, the degree of Mx × Ny is set as Mx × Ny = 10 × 10. In addition, the calculation time by the method proposed in Table 1 and the calculation time by FEM (ABAQUS) are shown. As shown in Table 1, when the number of elements is 40000, it is the most similar to the result of the proposed method, and the calculation time required at this time is 30.3 s. In the case of the proposed method, the calculation time is 0.77 s (in the case of Mx × Ny = 10 × 10), and it can be seen that the calculation time is much shorter than that of the FEM. In other words, it can be seen that the proposed method has the advantage of very high calculation accuracy and calculation efficiency.Table 1
Convergence of dimensionless frequencies of stepped FG-CNTRC plate.
ModeTruncated number12345678910Time (s)Present3 × 36.697111.12813.26516.00018.29423.37126.20126.57228.04028.6050.5444 × 45.35257.363611.75512.91815.00918.22018.43420.14320.89222.720.5425 × 55.33826.815610.03511.47612.35814.53316.89818.21818.86819.4870.5486 × 65.33796.81339.901311.47312.35414.34414.43417.90918.21718.7700.5557 × 75.33786.81219.871211.47212.35314.27414.41517.85718.21718.7600.5638 × 85.33786.81219.870711.47212.35314.23814.41517.82918.21718.7600.5689 × 95.33776.81219.870411.47212.35314.23714.41517.82918.21718.7600.59010 × 105.33796.81209.870411.47212.35314.23714.41517.82818.21718.7600.77011 × 115.33786.81209.870411.47212.35314.23714.41517.82818.21718.7600.82912 × 125.33786.81209.870411.47212.35314.23714.41417.82818.21718.7601.00713 × 135.33786.81209.870411.47212.35314.23714.41417.82818.21718.7601.158Number of elementFEM (ABAQUS)16005.34296.82559.927511.49912.37614.39514.45817.94618.23018.8361.325005.34116.82069.906811.48912.36814.33814.44217.90418.22718.8091.9100005.33876.81429.879411.47612.35614.26214.42117.84718.21918.7726.7225005.33816.81309.874511.47412.35414.24814.41717.83618.21718.76514.5400005.33806.81269.872611.47312.35314.24314.41617.83218.21718.76330.3As mentioned above, as the ultraspherical polynomial is characterized by the polynomial’s parameterλ, it is necessary to conduct the study on the determination of polynomial parameters. Figure 4 presents the percentage error (Ωλ − Ωλ = 0)/Ωλ = 0 of the solution of the ultraspherical polynomial parameter λ in the FG-CNTRC plate. Figure 4 indicates that the error of dimensionless frequency in the FG-CNTRC plate does not exceed 1.5 × 10−4 regardless of the change of polynomial parameter λ. Therefore, in this research, the polynomial parameter λ is set as 0.Figure 4
Percentage error of the dimensionless frequencies for the ultraspherical polynomial parametersλ. (a) CCCC. (b) FFFF.
(a)(b)As mentioned in the previous section, the artificial elastic spring technique is introduced to generalize the boundary conditions for the vibration analysis of the stepped FG-CNTRC plate. The boundary condition is changed according to the selection of the stiffness of artificial elastic spring. Therefore, it is necessary to conduct the determination study of the boundary conditions.Figure5 shows the variation characteristics of the non-dimensional frequency in the FG-CNTRC plate according to the change of stiffness of the boundary elastic spring. In order to analyze the variation characteristics of the FG-CNTRC plate according to the change of stiffness of the individual boundary spring, except for the considered boundary spring, the stiffness of all other boundary springs is set as zero, and then the stiffness of the considered spring is changed from 102 to 1016. As can be seen from Figure 5, the frequency shows almost no change when the stiffness of boundary spring is below 104, and then it increases dramatically beyond this value until the stiffness reaches 1011. When the stiffness is beyond 1011, the frequency does not change again. Based on these results, the classic and elastic boundary conditions for the calculation of vibration characteristics in the FG-CNTRC plate can be set as shown in Table 2. In Table 2, F, S, C, and E indicate the free, simple, clamped, and elastic boundary conditions. In addition, in the following processes, the boundary condition CFSE1 represents the clamped boundary condition at x = 0 , free boundary condition at x = a, simple-supported boundary condition at y = 0, and elastic boundary condition at y = b, respectively.Figure 5
Convergence characteristics of dimensionless frequencies on the boundary spring stiffness. (a) Mode1. (b) Mode2. (c) Mode3. (d) Mode4.
(a)(b)(c)(d)Table 2
The spring stiffness values of the elastic boundary conditions.
Boundary conditionsku,0,ku,1kv,0,kv,1kw,0,kw,1kφ,0, kφ,1kθ,0, kθ,1F00000S01014101401014C10141014101410141014E110810610610141014E2101410141014108108E3108108108108108


## 3.2. Validation

### 3.2.1. Free Vibration

In the previous section, through the convergence study, parameters including the boundary parameter for the vibration analysis of multi-stepped FG-CNTRC plates are determined. Based on these results, the accuracy of the suggested method is validated. The accuracy is validated by comparing the results from the proposed method with those from the previous works or finite element method. Table3 shows the comparison results of the dimensionless frequency of the non-stepped plates with isotropic materials. Here, Poisson’s ratio is 0.3. Table 3 indicates that the results from the proposed method agree well with those of the previous works.Table 3
Comparison of dimensionless frequencies of non-stepped isotropic plate.
Boundary conditionshRefs.Modes1234567SSSS0.1Present19.06545.48245.48264.36864.36869.79385.523Reference [65]19.09845.63645.63664.38464.38470.14985.5Reference [66]19.0945.62245.62264.38364.38370.11285.502Reference [67]19.0945.61945.61964.38364.38370.10485.4880.2Present17.44832.17632.17638.14938.14945.52655.144Reference [65]17.52832.19232.19238.48838.48845.52655.802Reference [66]17.52832.19232.19238.50238.50245.52655.843Reference [67]17.52632.19232.19238.48338.48345.52655.787CCCC0.1Present32.5362.05162.05186.963103.89104.94123.11Reference [65]33.00963.04363.04388.411104.28105.29123.73Reference [66]32.79762.67262.67287.941103.71104.7123.6Reference [67]32.78262.6362.6387.869103.61104.6123.590.2Present26.52246.2946.2961.53161.53163.75671.962Reference [65]27.06547.34647.34662.006263.63572.604Reference [66]26.97447.25347.25361.94461.94463.5772.568Reference [67]26.90647.10347.10361.91761.91763.34872.286FFFF0.1Present12.73418.94523.32831.99431.99455.37755.377Reference [65]12.72818.95623.34631.96531.96555.49355.493Reference [68]12.72618.95523.34731.96531.96555.49355.4930.2Present11.70217.421.19427.5827.5840.19442.81Reference [65]11.7117.43321.25227.64827.64840.19242.775Reference [68]11.7117.43321.25227.64727.64740.19142.776Next, the natural frequency result of stepped plates with isotropic materials is compared with that obtained by the finite element method. The stepped plate consists of three subplates inx and y directions, respectively, and the geometric parameters are a1 = 0.5 m, a2 = 0.1 m, a3 = 0.5 m, b1 = 0.5 m, b2 = 0.1 m, b3 = 0.5 m, hx1 = 0.1 m, hx2 = 0.2 m, hx3 = 0.1 m, hy1 = 0.1 m, hy2 = 0.2 m, and hy3 = 0.1 m. The finite element analysis software ABAQUS is used in the finite element method, and the element type is S4R and the number of elements is 8476.Through the comparison results of Tables3 and 4, it can be known that the suggested method is suitable for the free vibration analysis of non-stepped or stepped plates. The main purpose of this research is to study the vibration analysis method of the multi-stepped FG-CNTRC plate; therefore, the accuracy of the proposed method is validated by comparing the results of natural frequency in the FG-CNTRC plate using the suggested method. Table 5 shows the comparison results of natural frequency in the non-stepped FG-CNTRC plate. The material and geometric parameters are as follows [16]:(27)Em=2.1GPa,ρm=1150kg/m3,μm=0.34,E11CNT=5.6466TPa,E22CNT=7.08TPa,G12CNT=1.9445TPa,VCNT∗=0.11,η1=0.149,η2=0.934,η3=0.934,a=1m,b=1m.Table 4
Comparison of natural frequency of non-stepped isotropic plate.
FrequencyBoundary conditionsCCCCSSSSCCFFPresentFEMError (%)PresentFEMError (%)PresentFEMError (%)1812.7764813.050.0337439.6185439.620.0003467.8344468.130.063221513.9041516.10.14511100.1061101.40.1176488.3953488.530.027631513.9541516.10.14181100.1151101.40.1168909.7355909.70.003941998.5491999.90.06761551.6291552.60.06261190.2181192.40.183452479.5512486.70.28832028.0332032.90.241232.62512340.111562485.87624930.28662043.132048.40.25791263.8751265.40.120772603.52605.80.08832308.1512312.30.17981668.4721670.30.109582603.7872605.80.07732308.1532312.30.17971736.6551738.30.094792783.3032789.10.20832603.52605.30.06912148.3472155.90.3516102783.3382789.10.2072603.7872605.30.05812165.6322172.70.3264Table 5
Comparison of dimensionless frequency of non-stepped FG-CNTRC plate.
Boundary conditionsb/hΩFG-VFG-XFG-OReference [16]Reference [16] (ANSYS)PresentReference [16]Reference [16] (ANSYS)PresentReference [16]Reference [16] (ANSYS)PresentSSSS10112.45212.49512.48614.61614.65914.65511.5511.611.581217.0617.2117.05518.64618.82418.65116.26516.40916.256319.49919.47919.56919.49919.47919.55419.49919.47919.595419.49919.47919.56919.49919.47919.55419.49919.47919.595527.3427.52427.00928.51928.74228.226.51326.68726.17920115.1115.10315.13619.93919.91619.97513.52313.53113.542219.90319.87919.88223.77623.77323.7718.48618.46418.458331.56131.23331.1234.38934.12333.97730.16629.8329.719438.99838.95839.13838.99838.95839.10838.99838.95839.19538.99838.95839.13838.99838.95839.10838.99838.95839.19CCCC10117.21117.24717.17518.08318.12918.04716.70716.74416.677222.81822.92222.6523.60623.72823.43822.25322.35322.089333.0733.0632.87634.33834.33833.66132.37832.37232.188433.55233.53432.90334.46734.46434.30332.85732.83732.228536.52836.80436.35937.44737.437.57835.80936.08335.65620126.30426.24826.19630.42130.3830.32724.48624.44324.375231.49631.40531.26235.03635.00334.83929.79529.69929.55343.58942.99142.63546.4845.98445.59141.89541.27940.93456.24955.89455.7661.9861.69261.55853.55753.20453.058559.22159.0958.7964.56263.32162.556.61756.46456.164As can be seen from Table5, the dimensionless frequency in the non-stepped FG-CNTRC plate with uniform thickness agrees well with the previous works.Next, the natural frequency results of the multi-stepped FG-CNTRC plate are compared with that obtained by the finite element method. The material properties are the same as the case mentioned above, and the geometric properties and the parameters for the finite element analysis are set as shown in Table3. Tables 5 and 6 show that the results from the proposed method agree well with those of the previous works or obtained by the finite element method, and through the convergence and validity study, it can be known that the suggested method is the accurate method for the free vibration analysis for not only the isotropic materials but also the non-stepped and multi-stepped plates of FG-CNTRC. The finite element analysis software ABAQUS is used in the finite element method, and the element type is S4R and the number of elements is 10868.Table 6
Comparison of natural frequency of multi-stepped FG-CNTRC plate.
FrequencyBoundary conditionsCCCCSSSSCCFFPresentFEMError (%)PresentFEMError (%)PresentFEMError (%)1369.1226369.190.0182264.3662264.360.0023304.0064304.060.01762529.1282529.650.0986397.9645398.280.0793311.0388311.120.02613718.3087718.740.06675.8775676.340.0684377.6699377.610.01594794.5131794.790.0349678.0683679.450.2038383.4401383.590.03915802.1168803.970.231714.4617714.720.0362571.7157572.120.07076858.3181858.660.0398858.3178858.530.0247685.7125686.270.08137976.337977.740.1437881.0082881.90.1012686.7083687.170.067281059.8321061.10.1196995.2035998.670.3483733.3032733.490.025591093.591095.80.20211021.291022.60.1282827.5539827.370.0222101118.3681121.20.25321041.6171042.60.0944856.6614857.660.1166


### 3.2.2. Forced Vibration

In engineering applications, the external loads act on the plate which is the foundation construction, so it is needed to consider the forced response of structures. Considering the forced response, there mainly exist two parts such as the stability response analysis in frequency domain and transient response analysis in time domain. In this section, the accuracy of the proposed method is validated by comparing the forced vibration results obtained by the proposed method with those from the finite element method. For the analysis of the forced vibration, in the following research processes, the external forces are considered as three cases such as point force, line force, and area force, and it is assumed that the uniform load (fw = 1 N) is applied in the Z direction [63, 69].Figure6 represents the comparison results between steady-state responses in the non-stepped and multi-stepped FG-CNTRC plate under the CCCC boundary condition. The geometric and material parameters are the same as those shown in Tables 5 and 6. The ranges of natural frequency are from 300 to 900 in the non-stepped plates and from 300 to 800 in the multi-stepped plates, respectively. The interval is set as 1 Hz. Here, the harmonic point force fw acts in the thickness direction. The point load is fw=f¯wδx−lx1y−lx1, in which f¯w=−1N and δ indicates the Dirac delta function. The applied point of force and response measurement point are all set as A = (0.3 m, 0.3 m) and B = (0.3 m, 0.3 m). The comparison is made with the data obtained from the finite element analysis software ABAQUS, and the calculation is conducted under the same conditions.Figure 6
The comparison of vertical displacement of FG-CNTRC plate. (a) Non-stepped FG-CNTRC plate. (b) Multi-stepped FG-CNTRC plate.
(a)(b)The comparison results in Figure6 indicate that the proposed method is suitable for the steady-state vibration analysis of multi-stepped FG-CNTRC plate.Next, the comparison study for the transient response analysis is conducted. In this case, it is supposed that four kinds of transient loads are applied. Figure7 shows four types of transient loads used in this paper.Figure 7
Load pulse shapes. (a) Rectangular pulse; (b) triangular pulse; (c) half-sine pulse; (d) exponential pulse.
(a)(b)(c)(d)Four kinds of transient loads employed in this research are represented in Figure7, and the load functions are written as follows.Rectangular pulse:(28a)ft=ft,0≤t≤τ,0,t>τ.Triangular pulse:(28b)ft=2tτft,0≤t≤τ2,ft−2τt−τ2ft,τ2≤t≤τ,0,t>τ.Half-sine pulse:(28c)ft=ftsinπtτ,0≤t≤τ,0,t>τ.Exponential pulse:(28d)ft=fte−ξt,0≤t≤τ,0,t>τ,where ft is the load amplitude; τ is the pulse width; and t is the time variable.Figure8 presents the comparison results of transient response data between the proposed method and finite element software ABAQUS. The geometric parameters, material constants, and boundary conditions are set as shown in Figure 6. Here, the transient load f (t) is set as rectangular pulse and the amplitude of the rectangular pulse is set as ft = −1 N. The calculating time step is ∆t = 0.02 ms, and the loading time τ and calculating time are set as 20 ms, respectively. The results indicate that the prediction accuracy of the suggested method agrees well with that of the finite element method; therefore, the accuracy of the proposed method is validated.Figure 8
The comparison of normal displacement response of FG-CNTRC plate. (a) Non-stepped FG-CNTRC plate. (b) Multi-stepped FG-CNTRC plate.
(a)(b)


## 3.2.1. Free Vibration

In the previous section, through the convergence study, parameters including the boundary parameter for the vibration analysis of multi-stepped FG-CNTRC plates are determined. Based on these results, the accuracy of the suggested method is validated. The accuracy is validated by comparing the results from the proposed method with those from the previous works or finite element method. Table3 shows the comparison results of the dimensionless frequency of the non-stepped plates with isotropic materials. Here, Poisson’s ratio is 0.3. Table 3 indicates that the results from the proposed method agree well with those of the previous works.Table 3
Comparison of dimensionless frequencies of non-stepped isotropic plate.
Boundary conditionshRefs.Modes1234567SSSS0.1Present19.06545.48245.48264.36864.36869.79385.523Reference [65]19.09845.63645.63664.38464.38470.14985.5Reference [66]19.0945.62245.62264.38364.38370.11285.502Reference [67]19.0945.61945.61964.38364.38370.10485.4880.2Present17.44832.17632.17638.14938.14945.52655.144Reference [65]17.52832.19232.19238.48838.48845.52655.802Reference [66]17.52832.19232.19238.50238.50245.52655.843Reference [67]17.52632.19232.19238.48338.48345.52655.787CCCC0.1Present32.5362.05162.05186.963103.89104.94123.11Reference [65]33.00963.04363.04388.411104.28105.29123.73Reference [66]32.79762.67262.67287.941103.71104.7123.6Reference [67]32.78262.6362.6387.869103.61104.6123.590.2Present26.52246.2946.2961.53161.53163.75671.962Reference [65]27.06547.34647.34662.006263.63572.604Reference [66]26.97447.25347.25361.94461.94463.5772.568Reference [67]26.90647.10347.10361.91761.91763.34872.286FFFF0.1Present12.73418.94523.32831.99431.99455.37755.377Reference [65]12.72818.95623.34631.96531.96555.49355.493Reference [68]12.72618.95523.34731.96531.96555.49355.4930.2Present11.70217.421.19427.5827.5840.19442.81Reference [65]11.7117.43321.25227.64827.64840.19242.775Reference [68]11.7117.43321.25227.64727.64740.19142.776Next, the natural frequency result of stepped plates with isotropic materials is compared with that obtained by the finite element method. The stepped plate consists of three subplates inx and y directions, respectively, and the geometric parameters are a1 = 0.5 m, a2 = 0.1 m, a3 = 0.5 m, b1 = 0.5 m, b2 = 0.1 m, b3 = 0.5 m, hx1 = 0.1 m, hx2 = 0.2 m, hx3 = 0.1 m, hy1 = 0.1 m, hy2 = 0.2 m, and hy3 = 0.1 m. The finite element analysis software ABAQUS is used in the finite element method, and the element type is S4R and the number of elements is 8476.Through the comparison results of Tables3 and 4, it can be known that the suggested method is suitable for the free vibration analysis of non-stepped or stepped plates. The main purpose of this research is to study the vibration analysis method of the multi-stepped FG-CNTRC plate; therefore, the accuracy of the proposed method is validated by comparing the results of natural frequency in the FG-CNTRC plate using the suggested method. Table 5 shows the comparison results of natural frequency in the non-stepped FG-CNTRC plate. The material and geometric parameters are as follows [16]:(27)Em=2.1GPa,ρm=1150kg/m3,μm=0.34,E11CNT=5.6466TPa,E22CNT=7.08TPa,G12CNT=1.9445TPa,VCNT∗=0.11,η1=0.149,η2=0.934,η3=0.934,a=1m,b=1m.Table 4
Comparison of natural frequency of non-stepped isotropic plate.
FrequencyBoundary conditionsCCCCSSSSCCFFPresentFEMError (%)PresentFEMError (%)PresentFEMError (%)1812.7764813.050.0337439.6185439.620.0003467.8344468.130.063221513.9041516.10.14511100.1061101.40.1176488.3953488.530.027631513.9541516.10.14181100.1151101.40.1168909.7355909.70.003941998.5491999.90.06761551.6291552.60.06261190.2181192.40.183452479.5512486.70.28832028.0332032.90.241232.62512340.111562485.87624930.28662043.132048.40.25791263.8751265.40.120772603.52605.80.08832308.1512312.30.17981668.4721670.30.109582603.7872605.80.07732308.1532312.30.17971736.6551738.30.094792783.3032789.10.20832603.52605.30.06912148.3472155.90.3516102783.3382789.10.2072603.7872605.30.05812165.6322172.70.3264Table 5
Comparison of dimensionless frequency of non-stepped FG-CNTRC plate.
Boundary conditionsb/hΩFG-VFG-XFG-OReference [16]Reference [16] (ANSYS)PresentReference [16]Reference [16] (ANSYS)PresentReference [16]Reference [16] (ANSYS)PresentSSSS10112.45212.49512.48614.61614.65914.65511.5511.611.581217.0617.2117.05518.64618.82418.65116.26516.40916.256319.49919.47919.56919.49919.47919.55419.49919.47919.595419.49919.47919.56919.49919.47919.55419.49919.47919.595527.3427.52427.00928.51928.74228.226.51326.68726.17920115.1115.10315.13619.93919.91619.97513.52313.53113.542219.90319.87919.88223.77623.77323.7718.48618.46418.458331.56131.23331.1234.38934.12333.97730.16629.8329.719438.99838.95839.13838.99838.95839.10838.99838.95839.19538.99838.95839.13838.99838.95839.10838.99838.95839.19CCCC10117.21117.24717.17518.08318.12918.04716.70716.74416.677222.81822.92222.6523.60623.72823.43822.25322.35322.089333.0733.0632.87634.33834.33833.66132.37832.37232.188433.55233.53432.90334.46734.46434.30332.85732.83732.228536.52836.80436.35937.44737.437.57835.80936.08335.65620126.30426.24826.19630.42130.3830.32724.48624.44324.375231.49631.40531.26235.03635.00334.83929.79529.69929.55343.58942.99142.63546.4845.98445.59141.89541.27940.93456.24955.89455.7661.9861.69261.55853.55753.20453.058559.22159.0958.7964.56263.32162.556.61756.46456.164As can be seen from Table5, the dimensionless frequency in the non-stepped FG-CNTRC plate with uniform thickness agrees well with the previous works.Next, the natural frequency results of the multi-stepped FG-CNTRC plate are compared with that obtained by the finite element method. The material properties are the same as the case mentioned above, and the geometric properties and the parameters for the finite element analysis are set as shown in Table3. Tables 5 and 6 show that the results from the proposed method agree well with those of the previous works or obtained by the finite element method, and through the convergence and validity study, it can be known that the suggested method is the accurate method for the free vibration analysis for not only the isotropic materials but also the non-stepped and multi-stepped plates of FG-CNTRC. The finite element analysis software ABAQUS is used in the finite element method, and the element type is S4R and the number of elements is 10868.Table 6
Comparison of natural frequency of multi-stepped FG-CNTRC plate.
FrequencyBoundary conditionsCCCCSSSSCCFFPresentFEMError (%)PresentFEMError (%)PresentFEMError (%)1369.1226369.190.0182264.3662264.360.0023304.0064304.060.01762529.1282529.650.0986397.9645398.280.0793311.0388311.120.02613718.3087718.740.06675.8775676.340.0684377.6699377.610.01594794.5131794.790.0349678.0683679.450.2038383.4401383.590.03915802.1168803.970.231714.4617714.720.0362571.7157572.120.07076858.3181858.660.0398858.3178858.530.0247685.7125686.270.08137976.337977.740.1437881.0082881.90.1012686.7083687.170.067281059.8321061.10.1196995.2035998.670.3483733.3032733.490.025591093.591095.80.20211021.291022.60.1282827.5539827.370.0222101118.3681121.20.25321041.6171042.60.0944856.6614857.660.1166


## 3.2.2. Forced Vibration

In engineering applications, the external loads act on the plate which is the foundation construction, so it is needed to consider the forced response of structures. Considering the forced response, there mainly exist two parts such as the stability response analysis in frequency domain and transient response analysis in time domain. In this section, the accuracy of the proposed method is validated by comparing the forced vibration results obtained by the proposed method with those from the finite element method. For the analysis of the forced vibration, in the following research processes, the external forces are considered as three cases such as point force, line force, and area force, and it is assumed that the uniform load (fw = 1 N) is applied in the Z direction [63, 69].Figure6 represents the comparison results between steady-state responses in the non-stepped and multi-stepped FG-CNTRC plate under the CCCC boundary condition. The geometric and material parameters are the same as those shown in Tables 5 and 6. The ranges of natural frequency are from 300 to 900 in the non-stepped plates and from 300 to 800 in the multi-stepped plates, respectively. The interval is set as 1 Hz. Here, the harmonic point force fw acts in the thickness direction. The point load is fw=f¯wδx−lx1y−lx1, in which f¯w=−1N and δ indicates the Dirac delta function. The applied point of force and response measurement point are all set as A = (0.3 m, 0.3 m) and B = (0.3 m, 0.3 m). The comparison is made with the data obtained from the finite element analysis software ABAQUS, and the calculation is conducted under the same conditions.Figure 6
The comparison of vertical displacement of FG-CNTRC plate. (a) Non-stepped FG-CNTRC plate. (b) Multi-stepped FG-CNTRC plate.
(a)(b)The comparison results in Figure6 indicate that the proposed method is suitable for the steady-state vibration analysis of multi-stepped FG-CNTRC plate.Next, the comparison study for the transient response analysis is conducted. In this case, it is supposed that four kinds of transient loads are applied. Figure7 shows four types of transient loads used in this paper.Figure 7
Load pulse shapes. (a) Rectangular pulse; (b) triangular pulse; (c) half-sine pulse; (d) exponential pulse.
(a)(b)(c)(d)Four kinds of transient loads employed in this research are represented in Figure7, and the load functions are written as follows.Rectangular pulse:(28a)ft=ft,0≤t≤τ,0,t>τ.Triangular pulse:(28b)ft=2tτft,0≤t≤τ2,ft−2τt−τ2ft,τ2≤t≤τ,0,t>τ.Half-sine pulse:(28c)ft=ftsinπtτ,0≤t≤τ,0,t>τ.Exponential pulse:(28d)ft=fte−ξt,0≤t≤τ,0,t>τ,where ft is the load amplitude; τ is the pulse width; and t is the time variable.Figure8 presents the comparison results of transient response data between the proposed method and finite element software ABAQUS. The geometric parameters, material constants, and boundary conditions are set as shown in Figure 6. Here, the transient load f (t) is set as rectangular pulse and the amplitude of the rectangular pulse is set as ft = −1 N. The calculating time step is ∆t = 0.02 ms, and the loading time τ and calculating time are set as 20 ms, respectively. The results indicate that the prediction accuracy of the suggested method agrees well with that of the finite element method; therefore, the accuracy of the proposed method is validated.Figure 8
The comparison of normal displacement response of FG-CNTRC plate. (a) Non-stepped FG-CNTRC plate. (b) Multi-stepped FG-CNTRC plate.
(a)(b)


## 4. Numerical Example

Based on the convergence and validity study of the proposed method, in this section, the results of free and forced vibration analysis of the FG-CNTRC plate are suggested. The material properties for the CNTRC plate studied here and the CNT efficient parameters are as follows [56]:(29)Em=2.5GPa,ρm=1150kg/m3,μm=0.34,ρCNT=1400kg/m3,μCNT=0.175,E11CNT=5.6446TPa,E22CNT=7.08TPa,G12CNT=1.9445TPa,VCNT∗=0.12,η1=0.137,η2=1.022,η3=0.715,VCNT∗=0.17,η1=0.142,η2=1.626,η3=1.138,VCNT∗=0.28,η1=0.141,η2=1.585,η3=1.109.


### 4.1. Free Vibration

Here, according to the several classic and elastic boundary conditions and material parameters, new natural frequency results and mode types of FG-CNTRC plate are suggested. The geometric parameters of FC-CNTRC plate area = 1 m, b = 1 m, and h = 0.1 m. Table 7 shows the results of dimensionless frequency in the non-stepped FG-CNTRC plate according to different material distribution characteristics under the four-side fully clamped boundary condition. As can be seen from Table 7, as VCNT∗ increases, the dimensionless frequency of the FG-CNTRC plate is also increased. Also, when the material distributions are FG-Λ and FG-V, the non-dimensional frequencies are same; therefore, FG-Λ is not considered in the following process. In the non-stepped FG-CNTRC, under the same VCNT∗ condition, the dimensionless frequency is the highest when the material distribution is FG-X, while it is the lowest in case of FG-O.Table 7
The dimensionless frequencies of non-stepped FG-CNTRC plate.
VCNT∗Modes123456789100.12UD3.97035.39057.45697.90748.34799.387610.19711.08111.40411.927FG-Λ3.89125.35547.32477.91148.25699.426410.15511.11411.27711.923FG-V3.89125.35547.32477.91148.25699.426410.15511.11411.27711.923FG-X4.06945.52167.61468.08198.52019.479910.39811.29711.56112.009FG-O3.78555.22887.18267.75388.10359.44179.979410.92711.12711.7660.17UD5.08796.93789.563410.20110.72912.12913.13514.30814.66315.407FG-Λ4.98576.9089.403510.24110.63412.22913.11914.40414.5215.375FG-V4.98576.9089.403510.24110.63412.22913.11914.40414.5215.375FG-X5.25937.18859.824910.55411.03712.35213.52114.75714.94715.616FG-O4.82466.68859.20269.947510.39312.26412.81614.04614.30815.1350.28UD5.43217.333310.19710.72411.37912.70813.85715.01215.53816.155FG-Λ5.44727.439710.24510.9511.50213.05314.09115.36515.73316.575FG-V5.44727.439710.24510.9511.50213.05314.09115.36515.73316.575FG-X5.72667.863910.65411.54411.99513.16814.71716.10516.10916.676FG-O5.28777.100110.08610.4111.21413.17313.61914.66615.61616.385Table8 shows the results of dimensionless frequency in the multi-stepped FG-CNTRC plate according to different material distributions under the several classic classical and elastic boundary conditions. The material properties are considered when VCNT∗=0.17, η1 = 0.142, η2 = 1.626, and η3 = 1.138 and the geometric parameters are as follows: a1 = 0.5 m, a2 = 0.1 m, a3 = 0.5 m, b1 = 0.5 m, b2 = 0.1 m, b3 = 0.5 m, hx1 = 0.02 m, hx2 = 0.05 m, hx3 = 0.02 m, hy1 = 0.02 m, hy2 = 0.05 m, and hy3 = 0.05 m. Table 8 shows that the dimensionless frequencies of multi-stepped FG-CNTRC plate are different according to the boundary conditions.Table 8
The dimensionless frequencies of multi-stepped FG-CNTRC plate with different boundary conditions.
ModesBoundary conditionsCCCCSSSSCCFFCCSSCFCFE1E1CCE2E2CCE3E3CCUD111.8786.31018.436910.4071.76310.41611.73510.354212.3557.25588.660610.7733.209511.92312.32611.922319.46513.21411.80416.696.643818.1719.34818.124424.72617.84212.57522.5319.314921.3924.66621.382FG-V110.715.39417.15069.50771.71699.610310.6049.5606211.6266.857.41799.63813.267611.33411.60411.345318.87212.91710.62215.9286.856917.88418.817.857422.01615.5311.85521.3927.836619.83221.97219.823FG-X113.2997.36999.867411.7952.236711.30613.12511.23213.7058.043810.08412.1944.042213.01313.66613.007320.9614.34413.19518.1547.409219.28920.81919.255427.99120.75713.87922.69310.87222.95627.91222.953FG-O19.74024.87336.33918.61411.448.88579.68348.8171210.7756.46736.60118.69442.938110.56510.79510.55317.73812.0849.651814.8586.316516.96917.66216.937419.90613.9810.97919.2736.930218.44819.90218.418In the multi-stepped FG-CNTRC plate, similar to Table7, the dimensionless frequency is the highest in case of FG-X, while it is the lowest in case of FG-O. Figure 9 shows the mode types of the multi-stepped FG-CNTRC plate corresponding to the FG-X under the boundary conditions including CCCC and CCFF of Table 8.Figure 9
Some mode shapes of multi-stepped FG-CNTRC plate. (a) CCCC. (b) CCFF.
(a)(b)To help readers understand the mode shapes of the multi-stepped FG-CNTRC plate, Figures10–12 show visually the different shapes of modes in the multi-stepped FG-CNTRC plate with different kinds of boundary conditions, material properties, material distribution, and number of steps.Figure 10
Some mode shapes of multi-stepped FG-CNTRC plate with CCCC boundary condition (FG-V,VCNT∗=0.28, η1 = 0.141, η2 = 1.585, η3 = 1.109). (a) Mode1. (b) Mode2. (c) Mode3.
(a)(b)(c)Figure 11
Some mode shapes of multi-stepped FG-CNTRC plate with SSSS boundary condition (FG-O,VCNT∗=0.12, η1 = 0.137, η2 = 1.022, η3 = 0.715). (a) Mode1. (b) Mode2. (c) Mode3.
(a)(b)(c)Figure 12
Some mode shapes of multi-stepped FG-CNTRC plate with CCFF boundary condition (FG-X,VCNT∗=0.17, η1 = 0.142, η2 = 1.626, η3 = 1.138). (a) Mode1. (b) Mode2. (c) Mode3.
(a)(b)(c)


### 4.2. Forced Vibration

#### 4.2.1. Steady-State Vibration Analysis

This section is mainly focused on the frequency-displacement characteristics of the multi-stepped FG-CNTRC plate. Figure13 shows the frequency-displacement characteristics of the multi-stepped FG-CNTRC plate with different material distributions under the four-side fully clamped boundary conditions when three types of loads are applied.Figure 13
The frequency-displacement characteristics of multi-stepped FG-CNTRC plate under three types of load.
(a)(b)(c)(d)The material and geometric properties are the same as Table6, and VCNT∗=0.12, η1 = 0.137, η2 = 1.022, and η3 = 0.715, respectively. The applied points of force are at A (x, y) = (0.3 m, 0.3 m), A (x1, y1-x2, y2) = (0.3 m, 0.3 m-0.4 m, 0.3 m), and A (x1, y1-x2, y2) = (0.3 m, 0.3 m-0.4 m, 0.4 m) in cases of point force, line force, and area force, respectively. The external force is assumed that the uniform load (fw = −1 N) is applied in the rectangular direction. It can be clearly seen that the external load cannot alter the natural frequency of the multi-stepped FG-CNTRC plate itself. However, regardless of the material distribution, at the same frequency, the displacement is the largest in the application of point force, while it is the smallest in case of area force. When the line force is applied, the displacement is medium. Figure 14 shows the frequency-displacement characteristics of the multi-stepped FG-CNTRC plate according to the different VCNT∗ when the point force is applied under the several boundary conditions. The material and geometric properties are set the same as Figure 13. The examples of free vibration analysis clearly show that as VCNT∗ increases, the frequency of multi-stepped FG-CNTRC plate also increases.Figure 14
The frequency-displacement characteristics of multi-stepped FG-CNTRC plate under three types ofVCNT∗. (a) CCCC. (b) SSSS. (c) CCFF. (d) E1E1CC.
(a)(b)(c)(d)As the last example of the steady-state vibration analysis, the frequency-displacement characteristics in the non-stepped and stepped plants with same size are considered. Materials have FG-X distribution and the study is conducted whenVCNT∗=0.12, η1 = 0.137, η2 = 1.022, and η3 = 0.715. The geometric parameters are a = 1.1 m, b = 1.1 m, and h = 0.02 m in the non-stepped plates and a1 = 0.5 m, a2 = 0.1 m, a3 = 0.5 m, b1 = 0.5 m, b2 = 0.1 m, b3 = 0.5 m, hx1 = 0.02 m, hx2 = 0.05 m, hx3 = 0.02 m, hy1 = 0.02 m, hy2 = 0.05 m, and hy3 = 0.02 m in stepped plates, respectively.The external load is assumed that the unit point force (fw = −1 N) is applied in the vertical direction. The applied points of force are all set as A (x, y) = (0.55 m, 0.55 m) in two cases, and the displacement measurement points are all taken as B (x, y) = (0.3 m, 0.3 m) in all cases. Figure 15 shows the frequency-displacement characteristics curve in two cases under different boundary conditions.Figure 15
The frequency-displacement characteristics of non-stepped and multi-stepped FG-CNTRC plates. (a) CCCC. (b) SSSS. (c) E1E1CC.
(a)(b)(c)Figure15 shows that the frequency is expressed largely in the stepped plates while the displacement is seen largely in the non-stepped plates, which is mainly due to the resisting force of the steps.


#### 4.2.2. Transient Response

Figure16 shows the displacement of multi-stepped FG-CNTRC plate under CCCC boundary condition when three types of load are applied. The material and geometric parameters are set as shown in Table 6, and the size and applied point of force and response measurement point of displacement are the same as Figure 13. The transient load f (t) is set as rectangular pulse, and the calculating time step, loading time, and calculating time are taken as ∆t = 0.02 ms, τ = 10 ms, and t = 20 ms. As can be seen from Figure 16, in the application of point force, the displacement change according to the time is the largest, while the displacement change is the smallest in case of area force.Figure 16
The displacement response of multi-stepped FG-CNTRC plate under different load types.Figure17 shows the displacement of multi-stepped FG-CNTRC plate when various transient loads are applied. The material has FG-V distribution characteristics; the study is conducted when VCNT∗=0.12, η1 = 0.137, η2 = 1.022, and η3 = 0.715. The type of applied force is point force, and the size, applied point of force, and displacement-response measurement point are set as shown in Figure 16. The calculating time step is taken as ∆t = 0.02 ms, and the loading time and calculation time are τ = 10 ms and t = 20 ms, respectively. As can be seen from Figure 17, the transient response exhibited by the exponential pulse is the largest, while it is the smallest in case of the rectangular pulse. In addition, the change of displacement is very slow when the half-sine pulse and triangular pulse are applied. Overall results indicate that the variation of transient response of the multi-stepped FG-CNTRC plate has a close relationship with the type of applied loads.Figure 17
The displacement response of multi-stepped FG-CNTRC with various boundary conditions. (a) CCCC. (b) SSSS. (c) CCFF. (d) E1E1CC.
(a)(b)(c)(d)Figure18 represents the influence of VCNT∗ on the transient response of multi-stepped FG-CNTRC plate when different kinds of transient loads are applied. The material is assumed to have FG-V distribution characteristics, and the clamped boundary condition is considered. The material and geometric parameters and the parameters of displacement-response characteristics including the applied force and applied time are set as shown in Figure 17. Figure 18 shows that as VCNT∗ increases, the displacement also increases. Also, if the only absolute variation of displacement is considered, it is the largest when VCNT∗=0.12 regardless of the type of transient loads.Figure 18
The displacement response of stepped FGPE plate under different pulse types. (a) Rectangular pulse. (b) Triangular pulse. (c) Half-sine pulse. (d) Exponential pulse.
(a)(b)(c)(d)


## 4.1. Free Vibration

Here, according to the several classic and elastic boundary conditions and material parameters, new natural frequency results and mode types of FG-CNTRC plate are suggested. The geometric parameters of FC-CNTRC plate area = 1 m, b = 1 m, and h = 0.1 m. Table 7 shows the results of dimensionless frequency in the non-stepped FG-CNTRC plate according to different material distribution characteristics under the four-side fully clamped boundary condition. As can be seen from Table 7, as VCNT∗ increases, the dimensionless frequency of the FG-CNTRC plate is also increased. Also, when the material distributions are FG-Λ and FG-V, the non-dimensional frequencies are same; therefore, FG-Λ is not considered in the following process. In the non-stepped FG-CNTRC, under the same VCNT∗ condition, the dimensionless frequency is the highest when the material distribution is FG-X, while it is the lowest in case of FG-O.Table 7
The dimensionless frequencies of non-stepped FG-CNTRC plate.
VCNT∗Modes123456789100.12UD3.97035.39057.45697.90748.34799.387610.19711.08111.40411.927FG-Λ3.89125.35547.32477.91148.25699.426410.15511.11411.27711.923FG-V3.89125.35547.32477.91148.25699.426410.15511.11411.27711.923FG-X4.06945.52167.61468.08198.52019.479910.39811.29711.56112.009FG-O3.78555.22887.18267.75388.10359.44179.979410.92711.12711.7660.17UD5.08796.93789.563410.20110.72912.12913.13514.30814.66315.407FG-Λ4.98576.9089.403510.24110.63412.22913.11914.40414.5215.375FG-V4.98576.9089.403510.24110.63412.22913.11914.40414.5215.375FG-X5.25937.18859.824910.55411.03712.35213.52114.75714.94715.616FG-O4.82466.68859.20269.947510.39312.26412.81614.04614.30815.1350.28UD5.43217.333310.19710.72411.37912.70813.85715.01215.53816.155FG-Λ5.44727.439710.24510.9511.50213.05314.09115.36515.73316.575FG-V5.44727.439710.24510.9511.50213.05314.09115.36515.73316.575FG-X5.72667.863910.65411.54411.99513.16814.71716.10516.10916.676FG-O5.28777.100110.08610.4111.21413.17313.61914.66615.61616.385Table8 shows the results of dimensionless frequency in the multi-stepped FG-CNTRC plate according to different material distributions under the several classic classical and elastic boundary conditions. The material properties are considered when VCNT∗=0.17, η1 = 0.142, η2 = 1.626, and η3 = 1.138 and the geometric parameters are as follows: a1 = 0.5 m, a2 = 0.1 m, a3 = 0.5 m, b1 = 0.5 m, b2 = 0.1 m, b3 = 0.5 m, hx1 = 0.02 m, hx2 = 0.05 m, hx3 = 0.02 m, hy1 = 0.02 m, hy2 = 0.05 m, and hy3 = 0.05 m. Table 8 shows that the dimensionless frequencies of multi-stepped FG-CNTRC plate are different according to the boundary conditions.Table 8
The dimensionless frequencies of multi-stepped FG-CNTRC plate with different boundary conditions.
ModesBoundary conditionsCCCCSSSSCCFFCCSSCFCFE1E1CCE2E2CCE3E3CCUD111.8786.31018.436910.4071.76310.41611.73510.354212.3557.25588.660610.7733.209511.92312.32611.922319.46513.21411.80416.696.643818.1719.34818.124424.72617.84212.57522.5319.314921.3924.66621.382FG-V110.715.39417.15069.50771.71699.610310.6049.5606211.6266.857.41799.63813.267611.33411.60411.345318.87212.91710.62215.9286.856917.88418.817.857422.01615.5311.85521.3927.836619.83221.97219.823FG-X113.2997.36999.867411.7952.236711.30613.12511.23213.7058.043810.08412.1944.042213.01313.66613.007320.9614.34413.19518.1547.409219.28920.81919.255427.99120.75713.87922.69310.87222.95627.91222.953FG-O19.74024.87336.33918.61411.448.88579.68348.8171210.7756.46736.60118.69442.938110.56510.79510.55317.73812.0849.651814.8586.316516.96917.66216.937419.90613.9810.97919.2736.930218.44819.90218.418In the multi-stepped FG-CNTRC plate, similar to Table7, the dimensionless frequency is the highest in case of FG-X, while it is the lowest in case of FG-O. Figure 9 shows the mode types of the multi-stepped FG-CNTRC plate corresponding to the FG-X under the boundary conditions including CCCC and CCFF of Table 8.Figure 9
Some mode shapes of multi-stepped FG-CNTRC plate. (a) CCCC. (b) CCFF.
(a)(b)To help readers understand the mode shapes of the multi-stepped FG-CNTRC plate, Figures10–12 show visually the different shapes of modes in the multi-stepped FG-CNTRC plate with different kinds of boundary conditions, material properties, material distribution, and number of steps.Figure 10
Some mode shapes of multi-stepped FG-CNTRC plate with CCCC boundary condition (FG-V,VCNT∗=0.28, η1 = 0.141, η2 = 1.585, η3 = 1.109). (a) Mode1. (b) Mode2. (c) Mode3.
(a)(b)(c)Figure 11
Some mode shapes of multi-stepped FG-CNTRC plate with SSSS boundary condition (FG-O,VCNT∗=0.12, η1 = 0.137, η2 = 1.022, η3 = 0.715). (a) Mode1. (b) Mode2. (c) Mode3.
(a)(b)(c)Figure 12
Some mode shapes of multi-stepped FG-CNTRC plate with CCFF boundary condition (FG-X,VCNT∗=0.17, η1 = 0.142, η2 = 1.626, η3 = 1.138). (a) Mode1. (b) Mode2. (c) Mode3.
(a)(b)(c)


## 4.2. Forced Vibration

### 4.2.1. Steady-State Vibration Analysis

This section is mainly focused on the frequency-displacement characteristics of the multi-stepped FG-CNTRC plate. Figure13 shows the frequency-displacement characteristics of the multi-stepped FG-CNTRC plate with different material distributions under the four-side fully clamped boundary conditions when three types of loads are applied.Figure 13
The frequency-displacement characteristics of multi-stepped FG-CNTRC plate under three types of load.
(a)(b)(c)(d)The material and geometric properties are the same as Table6, and VCNT∗=0.12, η1 = 0.137, η2 = 1.022, and η3 = 0.715, respectively. The applied points of force are at A (x, y) = (0.3 m, 0.3 m), A (x1, y1-x2, y2) = (0.3 m, 0.3 m-0.4 m, 0.3 m), and A (x1, y1-x2, y2) = (0.3 m, 0.3 m-0.4 m, 0.4 m) in cases of point force, line force, and area force, respectively. The external force is assumed that the uniform load (fw = −1 N) is applied in the rectangular direction. It can be clearly seen that the external load cannot alter the natural frequency of the multi-stepped FG-CNTRC plate itself. However, regardless of the material distribution, at the same frequency, the displacement is the largest in the application of point force, while it is the smallest in case of area force. When the line force is applied, the displacement is medium. Figure 14 shows the frequency-displacement characteristics of the multi-stepped FG-CNTRC plate according to the different VCNT∗ when the point force is applied under the several boundary conditions. The material and geometric properties are set the same as Figure 13. The examples of free vibration analysis clearly show that as VCNT∗ increases, the frequency of multi-stepped FG-CNTRC plate also increases.Figure 14
The frequency-displacement characteristics of multi-stepped FG-CNTRC plate under three types ofVCNT∗. (a) CCCC. (b) SSSS. (c) CCFF. (d) E1E1CC.
(a)(b)(c)(d)As the last example of the steady-state vibration analysis, the frequency-displacement characteristics in the non-stepped and stepped plants with same size are considered. Materials have FG-X distribution and the study is conducted whenVCNT∗=0.12, η1 = 0.137, η2 = 1.022, and η3 = 0.715. The geometric parameters are a = 1.1 m, b = 1.1 m, and h = 0.02 m in the non-stepped plates and a1 = 0.5 m, a2 = 0.1 m, a3 = 0.5 m, b1 = 0.5 m, b2 = 0.1 m, b3 = 0.5 m, hx1 = 0.02 m, hx2 = 0.05 m, hx3 = 0.02 m, hy1 = 0.02 m, hy2 = 0.05 m, and hy3 = 0.02 m in stepped plates, respectively.The external load is assumed that the unit point force (fw = −1 N) is applied in the vertical direction. The applied points of force are all set as A (x, y) = (0.55 m, 0.55 m) in two cases, and the displacement measurement points are all taken as B (x, y) = (0.3 m, 0.3 m) in all cases. Figure 15 shows the frequency-displacement characteristics curve in two cases under different boundary conditions.Figure 15
The frequency-displacement characteristics of non-stepped and multi-stepped FG-CNTRC plates. (a) CCCC. (b) SSSS. (c) E1E1CC.
(a)(b)(c)Figure15 shows that the frequency is expressed largely in the stepped plates while the displacement is seen largely in the non-stepped plates, which is mainly due to the resisting force of the steps.


### 4.2.2. Transient Response

Figure16 shows the displacement of multi-stepped FG-CNTRC plate under CCCC boundary condition when three types of load are applied. The material and geometric parameters are set as shown in Table 6, and the size and applied point of force and response measurement point of displacement are the same as Figure 13. The transient load f (t) is set as rectangular pulse, and the calculating time step, loading time, and calculating time are taken as ∆t = 0.02 ms, τ = 10 ms, and t = 20 ms. As can be seen from Figure 16, in the application of point force, the displacement change according to the time is the largest, while the displacement change is the smallest in case of area force.Figure 16
The displacement response of multi-stepped FG-CNTRC plate under different load types.Figure17 shows the displacement of multi-stepped FG-CNTRC plate when various transient loads are applied. The material has FG-V distribution characteristics; the study is conducted when VCNT∗=0.12, η1 = 0.137, η2 = 1.022, and η3 = 0.715. The type of applied force is point force, and the size, applied point of force, and displacement-response measurement point are set as shown in Figure 16. The calculating time step is taken as ∆t = 0.02 ms, and the loading time and calculation time are τ = 10 ms and t = 20 ms, respectively. As can be seen from Figure 17, the transient response exhibited by the exponential pulse is the largest, while it is the smallest in case of the rectangular pulse. In addition, the change of displacement is very slow when the half-sine pulse and triangular pulse are applied. Overall results indicate that the variation of transient response of the multi-stepped FG-CNTRC plate has a close relationship with the type of applied loads.Figure 17
The displacement response of multi-stepped FG-CNTRC with various boundary conditions. (a) CCCC. (b) SSSS. (c) CCFF. (d) E1E1CC.
(a)(b)(c)(d)Figure18 represents the influence of VCNT∗ on the transient response of multi-stepped FG-CNTRC plate when different kinds of transient loads are applied. The material is assumed to have FG-V distribution characteristics, and the clamped boundary condition is considered. The material and geometric parameters and the parameters of displacement-response characteristics including the applied force and applied time are set as shown in Figure 17. Figure 18 shows that as VCNT∗ increases, the displacement also increases. Also, if the only absolute variation of displacement is considered, it is the largest when VCNT∗=0.12 regardless of the type of transient loads.Figure 18
The displacement response of stepped FGPE plate under different pulse types. (a) Rectangular pulse. (b) Triangular pulse. (c) Half-sine pulse. (d) Exponential pulse.
(a)(b)(c)(d)


## 4.2.1. Steady-State Vibration Analysis

This section is mainly focused on the frequency-displacement characteristics of the multi-stepped FG-CNTRC plate. Figure13 shows the frequency-displacement characteristics of the multi-stepped FG-CNTRC plate with different material distributions under the four-side fully clamped boundary conditions when three types of loads are applied.Figure 13
The frequency-displacement characteristics of multi-stepped FG-CNTRC plate under three types of load.
(a)(b)(c)(d)The material and geometric properties are the same as Table6, and VCNT∗=0.12, η1 = 0.137, η2 = 1.022, and η3 = 0.715, respectively. The applied points of force are at A (x, y) = (0.3 m, 0.3 m), A (x1, y1-x2, y2) = (0.3 m, 0.3 m-0.4 m, 0.3 m), and A (x1, y1-x2, y2) = (0.3 m, 0.3 m-0.4 m, 0.4 m) in cases of point force, line force, and area force, respectively. The external force is assumed that the uniform load (fw = −1 N) is applied in the rectangular direction. It can be clearly seen that the external load cannot alter the natural frequency of the multi-stepped FG-CNTRC plate itself. However, regardless of the material distribution, at the same frequency, the displacement is the largest in the application of point force, while it is the smallest in case of area force. When the line force is applied, the displacement is medium. Figure 14 shows the frequency-displacement characteristics of the multi-stepped FG-CNTRC plate according to the different VCNT∗ when the point force is applied under the several boundary conditions. The material and geometric properties are set the same as Figure 13. The examples of free vibration analysis clearly show that as VCNT∗ increases, the frequency of multi-stepped FG-CNTRC plate also increases.Figure 14
The frequency-displacement characteristics of multi-stepped FG-CNTRC plate under three types ofVCNT∗. (a) CCCC. (b) SSSS. (c) CCFF. (d) E1E1CC.
(a)(b)(c)(d)As the last example of the steady-state vibration analysis, the frequency-displacement characteristics in the non-stepped and stepped plants with same size are considered. Materials have FG-X distribution and the study is conducted whenVCNT∗=0.12, η1 = 0.137, η2 = 1.022, and η3 = 0.715. The geometric parameters are a = 1.1 m, b = 1.1 m, and h = 0.02 m in the non-stepped plates and a1 = 0.5 m, a2 = 0.1 m, a3 = 0.5 m, b1 = 0.5 m, b2 = 0.1 m, b3 = 0.5 m, hx1 = 0.02 m, hx2 = 0.05 m, hx3 = 0.02 m, hy1 = 0.02 m, hy2 = 0.05 m, and hy3 = 0.02 m in stepped plates, respectively.The external load is assumed that the unit point force (fw = −1 N) is applied in the vertical direction. The applied points of force are all set as A (x, y) = (0.55 m, 0.55 m) in two cases, and the displacement measurement points are all taken as B (x, y) = (0.3 m, 0.3 m) in all cases. Figure 15 shows the frequency-displacement characteristics curve in two cases under different boundary conditions.Figure 15
The frequency-displacement characteristics of non-stepped and multi-stepped FG-CNTRC plates. (a) CCCC. (b) SSSS. (c) E1E1CC.
(a)(b)(c)Figure15 shows that the frequency is expressed largely in the stepped plates while the displacement is seen largely in the non-stepped plates, which is mainly due to the resisting force of the steps.


## 4.2.2. Transient Response

Figure16 shows the displacement of multi-stepped FG-CNTRC plate under CCCC boundary condition when three types of load are applied. The material and geometric parameters are set as shown in Table 6, and the size and applied point of force and response measurement point of displacement are the same as Figure 13. The transient load f (t) is set as rectangular pulse, and the calculating time step, loading time, and calculating time are taken as ∆t = 0.02 ms, τ = 10 ms, and t = 20 ms. As can be seen from Figure 16, in the application of point force, the displacement change according to the time is the largest, while the displacement change is the smallest in case of area force.Figure 16
The displacement response of multi-stepped FG-CNTRC plate under different load types.Figure17 shows the displacement of multi-stepped FG-CNTRC plate when various transient loads are applied. The material has FG-V distribution characteristics; the study is conducted when VCNT∗=0.12, η1 = 0.137, η2 = 1.022, and η3 = 0.715. The type of applied force is point force, and the size, applied point of force, and displacement-response measurement point are set as shown in Figure 16. The calculating time step is taken as ∆t = 0.02 ms, and the loading time and calculation time are τ = 10 ms and t = 20 ms, respectively. As can be seen from Figure 17, the transient response exhibited by the exponential pulse is the largest, while it is the smallest in case of the rectangular pulse. In addition, the change of displacement is very slow when the half-sine pulse and triangular pulse are applied. Overall results indicate that the variation of transient response of the multi-stepped FG-CNTRC plate has a close relationship with the type of applied loads.Figure 17
The displacement response of multi-stepped FG-CNTRC with various boundary conditions. (a) CCCC. (b) SSSS. (c) CCFF. (d) E1E1CC.
(a)(b)(c)(d)Figure18 represents the influence of VCNT∗ on the transient response of multi-stepped FG-CNTRC plate when different kinds of transient loads are applied. The material is assumed to have FG-V distribution characteristics, and the clamped boundary condition is considered. The material and geometric parameters and the parameters of displacement-response characteristics including the applied force and applied time are set as shown in Figure 17. Figure 18 shows that as VCNT∗ increases, the displacement also increases. Also, if the only absolute variation of displacement is considered, it is the largest when VCNT∗=0.12 regardless of the type of transient loads.Figure 18
The displacement response of stepped FGPE plate under different pulse types. (a) Rectangular pulse. (b) Triangular pulse. (c) Half-sine pulse. (d) Exponential pulse.
(a)(b)(c)(d)


## 5. Conclusion

In this paper, using the domain decomposition method, dynamic behavior of multi-stepped FG-CNTRC plate with random boundary conditions is analyzed based on the FSDT. Within the framework of the domain decomposition method, the rectangular plates are segmented along the length direction using the segmenting technology and the thickness of each subdomain is taken differently. In this way, the multi-stepped FG-CNTRC plate can be constructed simply. The artificial spring technique is employed to satisfy the boundary conditions and the continuity conditions of the piecewise interface. The displacement admissible function of the multi-stepped FG-CNTRC plate is constructed using ultraspherical polynomials in a unified form. The ultraspherical polynomial expansion coefficient is considered as an unknown independent variable, and the dynamic solution equation of the multi-stepped FG-CNTRC plate can be constructed through the calculation of the extremum of the unknown independent variable. As a result, a complex system of partial differential equations is converted into the standard system of linear equations. The accuracy and convergence of the proposed method are validated using numerical examples. Next, the effect of geometric and material parameters on the free vibration characteristics of multi-stepped FG-CNTRC plate is investigated. Several natural frequency parameters and mode shapes which have not been published yet are also introduced in this paper, and they can be referred as comparative data by future researchers. Based on these results, the forced response of the multi-stepped FG-CNTRC plate is also parameterized. The proposed method will be expanded in a further work to include multi-stepped cylindrical, conical, and double-curved shell with various boundary conditions.



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*Source: 5579439-2021-07-14.xml*