# Analytical Solution for Bending Analysis of Axially Functionally Graded Angle-Ply Flat Panels

Authors: Agyapal Singh; Poonam Kumari; Rupam Hazarika
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

In this paper, the analytical solution is presented for axially functionally graded (AFG) angle-ply flat panels subjected to arbitrary boundary condition. Material properties of AFG panels are assumed to vary linearly alongx-direction. Reissner-type variation principle is used to derive the governing equations in mixed form. By employing extended Kantorovich method (EKM), a set of nonhomogeneous ordinary differential equations (ODEs) are obtained along the in-plane (x) and thickness (z) direction. The system of ODEs along the z-direction has constant coefficients, solved analytically. However, the system of ODEs along x-direction has variable coefficients, solved using modified power series method. The influence of property variation on the deflection and stresses is studied and discussed comprehensively for different sets of boundary conditions. Numerical results are validated through comparison with 3D FE. The presented analytical solution can serve as a benchmark for assessing the accuracy of the two-dimensional solution or 3D numerical solutions.


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## Body

## 1. Introduction

The variation of material property in the structural components may occur due to environmental effects [1–3] or it may be induced intentionally (FGMs) to achieve optimized characteristics of structures and overcome the limitation of conventional material [4]. The variation in the material property can be unidirectional or multidirectional. Most of the functionally graded components are designed and developed by taking continuous gradation in material properties for a smooth transition of mechanical properties along a particular direction to overcome the problem of delamination and cracks. The fabrication and application of functionally graded materials (FGMs) are discussed thoroughly in the recently published review articles [5, 6]. Theoretical analysis of functionally graded beams, panels, and plates is now an active area of research. Researchers used various types of methods to develop three-dimensional (3D) solution for through-thickness FGMs plate/panel such as extended Pagano’s approach [7, 8], asymptotic approach [9, 10], Plevako method [11–14], Peano-Baker series method [15, 16], power series method [17, 18], and state space/Fourier series method [19–22]. A comprehensive review of research, specifically on functionally graded plates, is presented by Birman and Byrd [23], Jha et al. [24], Thai at al. [25], Swaminathan et al. [26], and Wu and Liu [27]. However, most of the studies reported in literature are related to the category in which material property varies along the thickness direction by following the various law (power law, exponential law, etc.). The property variation along the in-plane direction is significantly important to control the characteristics of structural component effectively [28, 29]. However, a few articles are available in literature considering an in-plane variation of material properties.Lü et al. [30] developed semianalytical elasticity solutions for static analysis of bidirectional FG beams using a hybrid state space-based method in conjunction with differential quadrature method. Further, they extend the same approach to develop 3D elasticity solutions for bidirectional functionally graded Levy-type rectangular plates [31]. The elastic property of plate is assumed to vary exponentially along x- and z-directions. Wang et al. [32] presented the free vibrations solution for two-directional FG beams based on Euler-Bernoulli theory. Most of analytical studies related to in-plane FGM are based on two-dimensional approach, i.e., classical plate theory (CPT) [33–35], first order shear deformation theory (FSDT) [36], and higher-order shear deformation theory (HSDT) [37, 38]. Recently, Amirpour et al. [39, 40] reported analytical solution for the elastic deformation of FG plates based on CPT and HSDT. The stiffness of the plate is assumed to vary along x-direction by following the power law. The three-dimensional analytical solution, the most accurate method of analysis, is very useful for assessing the accuracy and validity of two-dimensional theories or numerical methods [26]. Very recently, Kumari et al. [41] presented a three-dimensional analytical solution for Levy-type functionally graded plate using the extended Kantorovich method. The material compliances of the plate are assumed to vary linearly along the in-plane (x) direction. However, the analytical solution for FGM laminated angle-ply flat panel under cylindrical bending by considering an in-plane variation of material properties has not been reported yet in the literature.In the present work, the extended Kantorovich method is employed to obtain analytical elasticity solution for laminated AFG angle-ply flat panel under cylindrical bending. The linear variation of the material properties along thex-direction is assumed. Reissner-type variation principle is used to obtain the governing equation in mixed form. Further, EKM is implemented along the in-plane direction (x) and the thickness (z), to reduce the weak form of governing equation into a system of ordinary differential equations (ODEs). The system of 6n first order ODEs and 2n algebraic equation with constant coefficient is obtained along the thickness direction of each layer. A similar set of ODEs is obtained along x-direction but with variable coefficients. These systems of ODEs along x- and z-direction are solved analytically by following the approach given in recent article [41]. Numerical results are presented for single layer and multilayered angle-ply AFG panel in conjunction with the various types of boundary condition. The present method has been validated against the 3D FE. The effect of in-plane property variation on deformation and stress distribution is examined intensively by considering various cases.


## 2. Theoretical Formulation

### 2.1. in-Plane Material Property Variation

The compliances of the angle-ply layers are assumed to vary linearly alongx-direction as(1)s-1jm=s-1j1+δ1ξforj=1,2,3,6;s-45m=s-451+δ2ξs-55m=s-551+δ2ξ;s-66m=s-661+δ2ξwhere δ1 and δ2 are parameters that control the material property variation and ξ=x/a are nondimensional quantities. To represent the graded or degraded property, the variation indexes (δ1, δ2) can be positive or negative. The mechanical properties variation along the in-plane direction may be induced deliberately to enhance the structure strength or it may be due to environmental factors, such as high temperature, moisture, corrosion, and diffusion of chemicals like hydrogen.


### 2.2. Basic Governing Equations for Flat Panel

A flat angle-ply laminated panel of dimensionsa×h along x- and z-directions, as depicted in Figure 1, is considered for the analysis. Panel has infinite/very long length along y-direction and two edges corresponding to x=0 and a can have any suitable combination of boundary conditions (i.e., clamped, free, simply supported). Since it is a case of generalized plane strain problem of cylindrical bending, all variables are independent of y coordinate. Panel has L numbers of perfectly bonded layers. Each layer may have fiber orientation angle θ with respect to x-direction. Layers are numbered from bottom to top, and for a typical kth layer, thickness is denoted by t(k) and its bottom surface is denoted as zk-1.Figure 1
Geometry of flat panel.The 3D constitutive equations for the angle-ply laminated panel are as follows:(2)εxεyεzγyzγzxγxy=s¯11ms¯12ms¯13m00s¯16ms¯12ms¯22s¯2300s¯26s¯13ms¯23s¯3300s¯36000s¯44s¯45m0000s¯45ms¯55m0s¯16ms¯26s¯3600s¯66mσxσyσzτyzτzxτxywhere σi and εi denote the normal stress and strains components, respectively, and τij and γij denote the shear stress and strains components, respectively. The transformed elastic compliances are denoted by s¯ij. In this generalized plain strain problems all strains (ϵx,ϵy,ϵz,γyz,γzx,γxy) and stresses (σx,σy,σz,τyz,τzx,τxy) are independent of y. Therefore, the strain-displacement relations reduce to the following form:(3)εx=u,x,εy=0,εz=w,z,γyz=v,z,γzx=w,x+u,z,γxy=v,xwhere a subscript comma denotes differentiation. Using (3)2 in (2)2, σy is obtained as(4)σy=-s¯12s¯22σx-s^12s¯22σx-s¯26s¯22τxy-s¯23s¯22σzand the same is eliminated from (2)1, (2)3, and (2)6 to yield(5)εx=p11+p^11σx+p16+p^16τxy+p13+p^13σz-δ1ξp¯12-δ12ξ2p¯12εz=p31+p^31σx+p36τxy+p33σzγyz=s¯44τyz+s¯45+s^45τzxγzx=s¯45+s^45τyz+s¯55+s^55τzxγxy=p16+p^16σx+p66+s^66τxy+p63σz where(6)pij=s¯ij-s¯i2s¯2js¯22;p^ij=δ1ξs¯ij-s¯i2s¯2js¯22;s^ij=δ2ξs¯ij;p¯12=s¯12s¯12s¯22.The Reissner-type mixed variational principle for an angle-ply laminated panel without body force, for the cylindrical bending case, can be written as(7)∫Vδuσx,x+τxz,z+δvτxy,x+τyz,z+δwτzx,x+σz,z+δσxεx-u,x+δσzεz-w,z+δτyzγyz-v,z+δτzxγzx-u,z-w,x+δτxyγxy-v,xdV=0,∀δu,δv,δw,δσi,δτijwhere V denotes per unit width volume of the angle-ply panel in the y-direction. Substituting the expressions of strain components εx,εz,γxy,γyz and γzx from (5) into (7) yields(8)∫Vδwσz,z+τzx,x+δvτyz,z+τxy,x+δuτxz,z+σx,x+δσxp11+p^11σx-δ1ξp¯12-δ12ξ2p¯12+p16+p^16τxy+p13+p^13σz-u,x+δσzp31+p^31σx+p36τxy+p33σz-w,z+δτyzs¯44τyz+s¯45+s^45τzx-v,z+δτzxs¯45+s^45τyz+s¯55+s^55τzx-u,z-w,x+δτxyp61+p^61σx+p63σz+p66+s^66τxy-v,xdV=0,∀δu,δv,δw,δσi,δτijwhere ξ is dimensionless in-plane coordinate along x-direction and ζ(k) is nondimensional local thickness parameter for the kth layer (ζ(k)=(z-zk-1)/t(k)) which takes value 0 to 1 for each layer. Panel is subjected to uniform distributed pressure (σz = -p1,-p2) at bottom and top surface and there is no shear stress (τzx and τyz) at bottom and top surface of panel. For perfect interlaminar bounding case, displacements (u, v, w) and transverse stresses (σz, τzx, τyz) need to satisfy the following condition at the kth interface:(9)u,v,w,σz,τyz,τzxζ=1k=u,v,w,σz,τyz,τzxζ=0k+1.Along x-axis AFG panel can have any type of support such as simply supported (σx=τxy=w=0), clamped (u=v=w=0), and free (σx=τxy=τxz=0).


## 2.1. in-Plane Material Property Variation

The compliances of the angle-ply layers are assumed to vary linearly alongx-direction as(1)s-1jm=s-1j1+δ1ξforj=1,2,3,6;s-45m=s-451+δ2ξs-55m=s-551+δ2ξ;s-66m=s-661+δ2ξwhere δ1 and δ2 are parameters that control the material property variation and ξ=x/a are nondimensional quantities. To represent the graded or degraded property, the variation indexes (δ1, δ2) can be positive or negative. The mechanical properties variation along the in-plane direction may be induced deliberately to enhance the structure strength or it may be due to environmental factors, such as high temperature, moisture, corrosion, and diffusion of chemicals like hydrogen.


## 2.2. Basic Governing Equations for Flat Panel

A flat angle-ply laminated panel of dimensionsa×h along x- and z-directions, as depicted in Figure 1, is considered for the analysis. Panel has infinite/very long length along y-direction and two edges corresponding to x=0 and a can have any suitable combination of boundary conditions (i.e., clamped, free, simply supported). Since it is a case of generalized plane strain problem of cylindrical bending, all variables are independent of y coordinate. Panel has L numbers of perfectly bonded layers. Each layer may have fiber orientation angle θ with respect to x-direction. Layers are numbered from bottom to top, and for a typical kth layer, thickness is denoted by t(k) and its bottom surface is denoted as zk-1.Figure 1
Geometry of flat panel.The 3D constitutive equations for the angle-ply laminated panel are as follows:(2)εxεyεzγyzγzxγxy=s¯11ms¯12ms¯13m00s¯16ms¯12ms¯22s¯2300s¯26s¯13ms¯23s¯3300s¯36000s¯44s¯45m0000s¯45ms¯55m0s¯16ms¯26s¯3600s¯66mσxσyσzτyzτzxτxywhere σi and εi denote the normal stress and strains components, respectively, and τij and γij denote the shear stress and strains components, respectively. The transformed elastic compliances are denoted by s¯ij. In this generalized plain strain problems all strains (ϵx,ϵy,ϵz,γyz,γzx,γxy) and stresses (σx,σy,σz,τyz,τzx,τxy) are independent of y. Therefore, the strain-displacement relations reduce to the following form:(3)εx=u,x,εy=0,εz=w,z,γyz=v,z,γzx=w,x+u,z,γxy=v,xwhere a subscript comma denotes differentiation. Using (3)2 in (2)2, σy is obtained as(4)σy=-s¯12s¯22σx-s^12s¯22σx-s¯26s¯22τxy-s¯23s¯22σzand the same is eliminated from (2)1, (2)3, and (2)6 to yield(5)εx=p11+p^11σx+p16+p^16τxy+p13+p^13σz-δ1ξp¯12-δ12ξ2p¯12εz=p31+p^31σx+p36τxy+p33σzγyz=s¯44τyz+s¯45+s^45τzxγzx=s¯45+s^45τyz+s¯55+s^55τzxγxy=p16+p^16σx+p66+s^66τxy+p63σz where(6)pij=s¯ij-s¯i2s¯2js¯22;p^ij=δ1ξs¯ij-s¯i2s¯2js¯22;s^ij=δ2ξs¯ij;p¯12=s¯12s¯12s¯22.The Reissner-type mixed variational principle for an angle-ply laminated panel without body force, for the cylindrical bending case, can be written as(7)∫Vδuσx,x+τxz,z+δvτxy,x+τyz,z+δwτzx,x+σz,z+δσxεx-u,x+δσzεz-w,z+δτyzγyz-v,z+δτzxγzx-u,z-w,x+δτxyγxy-v,xdV=0,∀δu,δv,δw,δσi,δτijwhere V denotes per unit width volume of the angle-ply panel in the y-direction. Substituting the expressions of strain components εx,εz,γxy,γyz and γzx from (5) into (7) yields(8)∫Vδwσz,z+τzx,x+δvτyz,z+τxy,x+δuτxz,z+σx,x+δσxp11+p^11σx-δ1ξp¯12-δ12ξ2p¯12+p16+p^16τxy+p13+p^13σz-u,x+δσzp31+p^31σx+p36τxy+p33σz-w,z+δτyzs¯44τyz+s¯45+s^45τzx-v,z+δτzxs¯45+s^45τyz+s¯55+s^55τzx-u,z-w,x+δτxyp61+p^61σx+p63σz+p66+s^66τxy-v,xdV=0,∀δu,δv,δw,δσi,δτijwhere ξ is dimensionless in-plane coordinate along x-direction and ζ(k) is nondimensional local thickness parameter for the kth layer (ζ(k)=(z-zk-1)/t(k)) which takes value 0 to 1 for each layer. Panel is subjected to uniform distributed pressure (σz = -p1,-p2) at bottom and top surface and there is no shear stress (τzx and τyz) at bottom and top surface of panel. For perfect interlaminar bounding case, displacements (u, v, w) and transverse stresses (σz, τzx, τyz) need to satisfy the following condition at the kth interface:(9)u,v,w,σz,τyz,τzxζ=1k=u,v,w,σz,τyz,τzxζ=0k+1.Along x-axis AFG panel can have any type of support such as simply supported (σx=τxy=w=0), clamped (u=v=w=0), and free (σx=τxy=τxz=0).


## 3. The Generalized Multiterm EKM

There are8 primary field variables (u,v,w,σx,σz,τxy,τyz,τzx) which are to be solved. Using the multiterm EKM [42], the field variables for the kth layer are expressed as(10)Xlξ,ζ=∑i=1nfliξgliζ+δl5pa+zpdforl=1,2,…8 where f and g are unknown functions of ξ and ζ, respectively, pa=-(p1+p2)/2, and pd=-(p2-p1)/h. The functions gli(ζ) are dependent on kth layer, while functions fli(ξ) are valid for all layers.


### 3.1. First Iteration Step

The initial guess functionsfli(ξ) along ξ direction for the first iteration are assumed in trigonometric form (cos⁡iπξ or sin⁡iπξ) which identically satisfy the simply supported boundary conditions. However, in multiterm EKM solution, the starting guess functions need not be assumed as per the boundary conditions. The functions gli(ξ) are to be solved in this step. For this purpose, first variation δXl is obtained as(11)δXl=∑i=1nfliξδgli,l=1,2,…8.Functions gli are divided into two column vectors G¯ and G^. G¯ contains all the six variables that appear in the boundary and interface conditions along z-direction, and G^ contains the remaining two variables:(12)G¯=g11…g1ng21…g2ng31…g3ng51…g5ng71…g7ng81…g8nTG^=g41…g4ng61…g6nT. Equations (10) and (11) are substituted in (8) and perform integration along x-axis. Since variation is arbitrary, the coefficient of δgi must vanish, yielding the following set of governing equations:(13)MG¯,ζ=A¯mG¯+A^mG^+Q¯pm(14)KmG^=A~mG¯+Q~pmwhere M,A¯m,A^m,Km, and A~m are 6n×6n, 6n×6n, 6n×2n, 2n×2n, and 2n×6n matrices and A¯m=A¯+A¯v; A^m=A^+A^v; Km=K+Kv; A~m=A~+A~v; Q~pm=Q~p+Q~pv; Q¯pm=Q¯p+Q¯pv. Appendix A contains all the nonzero terms of these matrices.Solving the system of algebraic equations (14) for G^ and substituting back the solution into (13) yield(15)G¯,ζ=AG¯+Qpwith A=M-1[A¯m+A^mKm-1A~m] and Qp=M-1[Q¯pm+A^mKm-1Q~pm]. Equation (15) is system of simultaneous nonhomogeneous first order differential equations of size 6n having constant coefficients. This set of ODEs is solved using the technique given in [43]. This completely determines G¯(ζ). Now, G^(ζ) can be obtained by solving the algebraic equation (14).


### 3.2. Second Iteration Step

Nowgli(ζ) is known from the first step and arbitrary variation is considered along the x-direction; therefore variation for this case is written as(16)δXl=∑i=1ngliζδfli,l=1,2,…,8.Similarly, like first step, the function fli(ξ) is partition into F¯ and F^. F¯ contains all the six variables that appear in the boundary and interface conditions along x-direction, and F^ contains the remaining two variables:(17)F¯=f11…f1nf21…f2nf31…f3nf41…f4nf61…f6nf81…f8nTF^=f51…f5nf71…f7nT. Substitute (16) in (8), and perform integration along z-axis. Since variation is arbitrary, the coefficient of δfli must vanish, yielding the following set of governing equations:(18)NF¯,ξ=B¯fF¯+B^fF^+P¯mf(19)LF^=B~fF¯+P~mwhereB ¯ f = B ¯ + ξ B ¯ v + ξ 2 B ¯ v v ;   B ^ f = B ^ + ξ B ^ v ;   P ¯ m f = P ¯ m + ξ P ¯ m v ;   B ~ f = B ~ + ξ B ~ v; N,B¯f,B^f,L and B~f are 6n×6n, 6n×6n, 6n×2n, 2n×2n, and 2n×6n matrices, respectively; and P¯mf and P~mf are 6n×1 and 2n×1 load column vectors. The nonzero terms of these matrices are given in Appendix B.Rewrite the governing equation (18) and (19) as(20)NF¯,ξ=B¯+ξB¯v+ξ2B¯vvF¯+B^+ξB^vF^+P¯m+ξP¯mv(21)LF^=B~+ξB~vF¯+P~m.Substituting F^ from (21) into (20) yields the following set of first order ODEs with varying coefficients for F¯:(22)F¯,ξ=B0+ξB1+ξ2B2F¯+P0+ξP1where B0=N-1(B¯+B^L-1B~);B1=N-1(B¯v+B^L-1B~v+B^vL-1B~);B2=N-1(B¯vv+B^vL-1B~v); P0=N-1(P¯m+B^L-1P~m);P1=N-1(P¯mv+B^vL-1P~m).The approximate solution of the first order differential equation (22) with variable coefficients (functions of ξ) is obtained by using the modified power series method suggested by Kumari et al. [41]. Thus, final general solution for current system of (22) is expanded in terms of a power series in the dimensionless axial coordinate ξ(0≤ξ≤1) as(23)F¯jξ=∑i=0NpZijξi+∑i=0NpHijξiC0where constants Zi, Hi are obtained from recursive relations and unknown constant C0 is calculated by applying x-direction boundary conditions. Np is the number of terms in power series which is chosen large enough so that the contribution of further terms is negligible and less than a stipulated small number η(=10-10). For detailed solution procedure, one can refer to article [41]. Now F¯ is known functions which is substituted into (19) to solve the functions F^. Now the second step is completed, and further these two steps of thickness and in-plane directions are repeated to achieve the required level of accuracy.


## 3.1. First Iteration Step

The initial guess functionsfli(ξ) along ξ direction for the first iteration are assumed in trigonometric form (cos⁡iπξ or sin⁡iπξ) which identically satisfy the simply supported boundary conditions. However, in multiterm EKM solution, the starting guess functions need not be assumed as per the boundary conditions. The functions gli(ξ) are to be solved in this step. For this purpose, first variation δXl is obtained as(11)δXl=∑i=1nfliξδgli,l=1,2,…8.Functions gli are divided into two column vectors G¯ and G^. G¯ contains all the six variables that appear in the boundary and interface conditions along z-direction, and G^ contains the remaining two variables:(12)G¯=g11…g1ng21…g2ng31…g3ng51…g5ng71…g7ng81…g8nTG^=g41…g4ng61…g6nT. Equations (10) and (11) are substituted in (8) and perform integration along x-axis. Since variation is arbitrary, the coefficient of δgi must vanish, yielding the following set of governing equations:(13)MG¯,ζ=A¯mG¯+A^mG^+Q¯pm(14)KmG^=A~mG¯+Q~pmwhere M,A¯m,A^m,Km, and A~m are 6n×6n, 6n×6n, 6n×2n, 2n×2n, and 2n×6n matrices and A¯m=A¯+A¯v; A^m=A^+A^v; Km=K+Kv; A~m=A~+A~v; Q~pm=Q~p+Q~pv; Q¯pm=Q¯p+Q¯pv. Appendix A contains all the nonzero terms of these matrices.Solving the system of algebraic equations (14) for G^ and substituting back the solution into (13) yield(15)G¯,ζ=AG¯+Qpwith A=M-1[A¯m+A^mKm-1A~m] and Qp=M-1[Q¯pm+A^mKm-1Q~pm]. Equation (15) is system of simultaneous nonhomogeneous first order differential equations of size 6n having constant coefficients. This set of ODEs is solved using the technique given in [43]. This completely determines G¯(ζ). Now, G^(ζ) can be obtained by solving the algebraic equation (14).


## 3.2. Second Iteration Step

Nowgli(ζ) is known from the first step and arbitrary variation is considered along the x-direction; therefore variation for this case is written as(16)δXl=∑i=1ngliζδfli,l=1,2,…,8.Similarly, like first step, the function fli(ξ) is partition into F¯ and F^. F¯ contains all the six variables that appear in the boundary and interface conditions along x-direction, and F^ contains the remaining two variables:(17)F¯=f11…f1nf21…f2nf31…f3nf41…f4nf61…f6nf81…f8nTF^=f51…f5nf71…f7nT. Substitute (16) in (8), and perform integration along z-axis. Since variation is arbitrary, the coefficient of δfli must vanish, yielding the following set of governing equations:(18)NF¯,ξ=B¯fF¯+B^fF^+P¯mf(19)LF^=B~fF¯+P~mwhereB ¯ f = B ¯ + ξ B ¯ v + ξ 2 B ¯ v v ;   B ^ f = B ^ + ξ B ^ v ;   P ¯ m f = P ¯ m + ξ P ¯ m v ;   B ~ f = B ~ + ξ B ~ v; N,B¯f,B^f,L and B~f are 6n×6n, 6n×6n, 6n×2n, 2n×2n, and 2n×6n matrices, respectively; and P¯mf and P~mf are 6n×1 and 2n×1 load column vectors. The nonzero terms of these matrices are given in Appendix B.Rewrite the governing equation (18) and (19) as(20)NF¯,ξ=B¯+ξB¯v+ξ2B¯vvF¯+B^+ξB^vF^+P¯m+ξP¯mv(21)LF^=B~+ξB~vF¯+P~m.Substituting F^ from (21) into (20) yields the following set of first order ODEs with varying coefficients for F¯:(22)F¯,ξ=B0+ξB1+ξ2B2F¯+P0+ξP1where B0=N-1(B¯+B^L-1B~);B1=N-1(B¯v+B^L-1B~v+B^vL-1B~);B2=N-1(B¯vv+B^vL-1B~v); P0=N-1(P¯m+B^L-1P~m);P1=N-1(P¯mv+B^vL-1P~m).The approximate solution of the first order differential equation (22) with variable coefficients (functions of ξ) is obtained by using the modified power series method suggested by Kumari et al. [41]. Thus, final general solution for current system of (22) is expanded in terms of a power series in the dimensionless axial coordinate ξ(0≤ξ≤1) as(23)F¯jξ=∑i=0NpZijξi+∑i=0NpHijξiC0where constants Zi, Hi are obtained from recursive relations and unknown constant C0 is calculated by applying x-direction boundary conditions. Np is the number of terms in power series which is chosen large enough so that the contribution of further terms is negligible and less than a stipulated small number η(=10-10). For detailed solution procedure, one can refer to article [41]. Now F¯ is known functions which is substituted into (19) to solve the functions F^. Now the second step is completed, and further these two steps of thickness and in-plane directions are repeated to achieve the required level of accuracy.


## 4. Numerical Results and Discussions

Four flat panels (a), (b), (c), and (d) are considered for numerical study as shown in Figure2. All the panels are made of Graphite-Epoxy material [42] and the engineering material constants are given as [(Y1, Y2, Y3, G23, G13, G12), ν12, ν13, ν23] = [(181.0, 10.3, 10.3, 2.87, 7.17, 7.17) GPa, 0.28, 0.28, 0.33]. To study the effect of varying material properties on the behavior of flat panel, the following variation indexes have been considered for panels.Figure 2
Configuration of AFG angle-ply flat panels.Panel (a): case (ia) δ1=δ2=0.5; case (iia) δ1=δ2=1.0; case (iiia) δ1=1, δ2=2.Panel (b): case (ib) bottom layer δ1=δ2=0.0, top layer δ1=δ2=0.5; case (iib) bottom layer δ1=δ2=0.5, top layer δ1=δ2=1.0; case (iiib) bottom layer δ1=δ2=1.0, top layer δ1=δ2=1.0.Panel (c) and panel (d): variation indexes of each ply taken as case (icd) δ1=δ2=0.5; case (iicd) δ1=δ2=1.0.Uniformly distributed pressure load (p2) is applied at top surface of panel. The presented results are nondimensionalized with S=a/h, Y0=10.3 GPa, and p0=1 as (u¯,w¯)=100(u,w/S)Y0/(p0hS3);(σ¯x,τ¯xy,τ¯zx)=(σx,10τxy,Sτzx)/(p0S2).The accuracy and efficiency of the present method are verified thoroughly by comparison with existing literature [43] for homogeneous case (constant material properties) and with 3D FE for varying stiffness case. Since it is an angle-ply panel, plane strain element of ABAQUS cannot be used. Therefore, the rectangular plate whose length along y-direction is 20 times the width along x-direction is modeled in ABAQUS using the element type C3D20R with mesh size of 40 (length) ×50 (width)×16 (thickness) [43]. The spatially graded property distribution (at different Gauss points) is implemented by employing user material subroutine (UMAT). For present case also, it is verified that these FE results do not change with a higher value of b/a. Converged results of EKM are presented in this paper.


### 4.1. Single-Layered Panel (a)

Longitudinal variations of deflection (w¯) and stresses (σ¯x, τ¯xy and τ¯zx) for panel (a) have been presented in Figures 3, 4, and 5 for simply supported (S-S), clamped-simply supported (C-S), and clamped-free (C-F) boundary conditions, respectively. Results are plotted for different variation indexes (δ1=δ2=0.5, δ1=δ2=1.0 and δ1=1.0, δ2=2.0) along with homogeneous case to study the effect of in-plane varying material properties. 3D FE results for case (δ1=δ2=0.5) is also plotted in these figures. It is observed that present result are in excellent agreement with 3D FE results except for stress at very clamped edges. This mismatch at very clamped edge is because the FE solution does not ensure conditions of applied normal stress (σ¯z) and shear stress (τ¯zx) [43] at top and bottom surfaces of panel and it can be verified from Figure 7. For simply supported case, single term (n=1, iter.1) gives accurate prediction for all the entities whereas two-term solution (n=2, iter.2) is required for the other boundary conditions. It is evident that for all the boundary conditions (S-S, C-S, C-F) deflection (w¯) is effected significantly with increase in variation index whereas τ¯zx is least effected. As variation indices increase, the point of maximum deflection (w¯) for S-S and C-S boundary condition is shifted gradually toward ξ=1.0. For S-S and C-S case, stresses σ¯x and τ¯xy decrease as the variation index increases. The decrement in τ¯xy is significantly high under S-S boundary conditions. It is revealed that influence of in-plane graded material properties depends significantly on boundary condition of flat panel.Figure 3
Effect of variation in properties on longitudinal variations of deflection and stresses for panel (a) (α=45°, S=5) subjected to S-S boundary condition.Figure 4
Effect of variation in properties on longitudinal variations of deflection and stresses for panel (a) (α=45°, S=5) subjected to C-S boundary condition.Figure 5
Effect of variation in properties on longitudinal variations of deflection and stresses for panel (a) (α=45°, S=5) subjected to C-F boundary condition.Longitudinal variations of in-plane displacement (u¯) are plotted in Figure 6 for four types of support conditions (i.e., S-S, C-F, C-S, and C-C). For all the support conditions, in-plane displacement (u¯) is influenced greatly by an increase in variation index. It is observed that effect of varying material property is more pronounced near to simply supported and free edges than the center of the flat panel. For S-S, C-S, and C-C conditions, the effect is minimum, around ξ=0.6, with respect to the constant case. All the lines cut the homogeneous case line at one point near to ξ=0.6. For C-C case, asymmetry is the highest for case (iiia) and a similar trend is observed for S-S case also. For all the boundary conditions, the in-plane displacement (u¯) increases abruptly for case (iiia) as compared to case (ia) and case (iia). The longitudinal variations of in-plane displacement for S-S and C-C boundary condition become more asymmetrical as the variation index increases.Figure 6
Effect of variation in properties on longitudinal variations of in-plane deflection (u¯) for panel (a) (α=45°, S=5) subjected to different boundary condition (SS, CF, CS, CC).Figure 7
Comparison of in-plane deflection and stresses distributions at differentξ-locations with 3D FE solution for panel (a) (S=5) under C-F boundary condition.Through-thickness variations at differentξ locations have been presented in Figure 7 for case (ia) under C-F support conditions. 3D FE results for all ξ locations are also plotted in the figure. It is revealed that the distribution of in-plane displacement u¯ and stress σ¯x matches well the 3D FE solution near to clamped support, but 3D FE fails in predicting the stresses σ¯x, σ¯z, and τ¯zx at the clamped edge. 3D FE predicted stress distribution does not satisfy shear traction (zero) and condition of applied normal stress at the top and bottom surfaces of panel whereas present EKM solution satisfies these conditions exactly. The FE results for transverse stresses match well the present solution at some distance away from the clamped edge. It is observed that in-plane displacement (u¯) follows linear pattern near to free edges which is quadratic or cubic near the clamped support. Similar trends are observed for σ¯x and σ¯z. Through-thickness variation of τ¯zx is parabolic and its magnitude decreases as we move from clamped edge to free edge.Effect of ply-angle on longitudinal variation of deflection (w¯) and stresses (σ¯x, τ¯xy and τ¯zx) has been shown in Figure 8. It is observed that, with an increase in ply-angle from 0° to 30°, there is a gradual increase in all the entities whereas for 45° ply-angle magnitude of all the entities increases abruptly.Figure 8
Comparison of deflection and stresses distributions for different value ofα under C-F boundary condition.Percentage increment or decrement, with respect to the homogeneous case, of various entities at the location where it is maximum, is presented in Table1. It is found that in-plane displacement (u¯) increases with increase in variation index and it is maximum for simply supported boundary condition which is 175.65% for case (iiia). Percentage increase in deflection (w¯) is also maximum for S-S support condition. The increment in the in-plane stress is maximum for C-S boundary condition, which is −30.55% for case (iiia). Maximum decrement of 47.7% in τ¯xy is observed for case (iiia) under C-S boundary conditions. From Table 1, it is concluded that the effect of material property variation is relatively less under clamped support conditions (C-S, C-F) than that under simply supported condition (S-S).Table 1
Percentage (%) change in deflections (u-, w-) and stresses (σ-x, τ-xy, τ-zx) with respect to homogeneous case, due to variation in material properties for panel (a) with α=45°.
Boundary conditions Entity (location) (x, z) δ1=δ2=0.5 δ 1 = δ 2 = 1.0 δ 1 = 1.0, δ2 = 2.0 S-S u - (1.0, -0.5) 50.47 93.38 175.65 w - (0.5, 0.0) 36.62 72.19 148.84 σ - x (0.5, -0.5) -2.79 -4.15 -4.07 τ - x y (0.5, -0.5) -11.37 -16.58 -39.59 τ - z x (0.1, 0.0) 0.49 0.94 0.96  C-S u - (1.0, -0.5) 36.10 64.79 113.45 w - (0.5, 0.0) 21.87 41.10 82.28 σ - x (0.5, -0.5) -13.36 -21.72 -30.55 τ - x y (0.5, -0.5) -19.23 -29.84 -47.71 τ - z x (0.1, 0.0) 3.34 5.72 8.71  C-F u - (1.0, -0.5) 28.36 51.64 106.17 w - (1.0, 0.0) 17.97 34.85 74.83 σ - x (0.5, -0.5) 2.30 8.00 13.68 τ - x y (0.5, -0.5) -9.28 -15.64 -33.96 τ - z x (0.1, 0.0) 2.87 3.27 4.18


### 4.2. Two-Layered Panel (b)

Figures9 and 10 depict the effect of variation index on longitudinal variation of deflection (w¯) and stresses (σ¯x, τ¯xy and τ¯zx) for C-S and C-F support conditions, respectively. Panel (b) is a two-layered flat panel in which effect of variation of material property is investigated by taking different and same variation index for the top and bottom layer. To study the effect of varying material property on the longitudinal variation of deflection and stresses, the results for the layer-wise homogeneous case are also plotted in the figures. It is observed that deflection and stresses are significantly affected even by considering the variation of material property in one layer. 3D FE results for case (iib) are also presented in Figures 9 and 10. The present results are in good agreement with 3D FE except for stress at very clamped edges, because at the support specifically clamped type, finite element solutions do not satisfy the boundary conditions and interface continuity conditions which can be verified from Figures 7 and 14. For both C-S and C-F support conditions, deflection w¯ is affected significantly by gradation of properties along x-direction whereas stresses σ¯x, τ¯xy, and τ¯zx are least affected. The deflection at the center of the panel for C-S support increased by 9.5% for case (ib), 29.7% for case (iib), and 43.2% for case (iiib) with respect to the homogeneous case. Similarly, for C-F panel, the percentage change in deflection at the free edge of the panel is 9.1% for case (ib), 29% for case (iib), and 35% for case (iiib). For both C-S and C-F case, the percentage change in τ¯zx is less than 5%.Figure 9
Effect of variation in properties on longitudinal variations of deflection and stresses for panel (b) (α=45°, S=5) under C-S boundary condition.Figure 10
Effect of variation in properties on longitudinal variations of deflection and stresses for panel (b) (α=45°, S=5) under C-F boundary condition.


### 4.3. Four-Layered Panels (c) and (d)

The results for panel (c) with C-C, C-S, and C-F boundary condition are plotted in Figures11 and 12 for different cases of property variation. To verify the accuracy and stability of present solution for the multilayered panel, 3D FE results for case (icd) are also plotted. It is observed that present results are in very good agreement with 3D FE. Longitudinal variations of deflection (w¯) and stresses (σ¯x, τ¯xy, and τ¯zx) for C-C support conditions are plotted in Figure 11. It is observed that deflection (w¯) has significant effect as compared to stresses. Similarly, in Figure 12 longitudinal variation of w¯ and σ¯x is plotted for C-S and C-F boundary conditions. For both boundary conditions, the influence of varying material properties on deflection (w¯) is significantly more pronounced as compared to σ¯x.Figure 11
Effect of variation in properties on longitudinal variations of deflection and stresses for panel (c) (α=45°, S=5) under C-C boundary condition.Figure 12
Effect of variation in properties on longitudinal variations of deflection and stresses for panel (c) under C-F and C-S boundary condition.
(a)
C-F boundary condition (b)
C-S boundary conditionSimilarly in Figure13 longitudinal variations of deflections (u¯, w¯) and stress (σ¯x) have been presented for panel (d) under C-C and C-F boundary conditions. Here, 3D FE result for case (icd) is also plotted which is in good agreement with present results. To study the effect of gradation in material properties on deflection and stresses, the results for the layer-wise homogeneous case are also plotted in the same figures. It is observed that, the effect of material properties is more significant for deflections (u¯, w¯) as compared to stress (σ¯x) for both C-C and C-F support. For panel subjected to C-C boundary conditions, the longitudinal variations of deflections become more asymmetrical as the variation index increases.Figure 13
Effect of variation in properties on longitudinal variations of deflection and stresses for panel (d) under C-C and C-F boundary condition.Figure 14
Comparison of stresses distributions at differentξ-locations with 3D FE solution for panel (d) (S=5) under C-F boundary condition (Case (i)cd).Through-thickness distribution of transverse stressesσ¯z and τ¯zx has been presented in Figure 14 for panel (d) in which material properties vary according to case (icd) and panel subjected to C-F boundary conditions. It is observed that 3D FE solution fails to predict the stress distribution at the clamped support. At very clamped support (ξ=0), stress distribution predicted by FE neither satisfies interface continuity condition nor satisfies shear traction (zero) and condition of applied normal stress at top and bottom surface of the panel. However, present EKM solution satisfies all these conditions at the clamped support. It is observed that away from clamped support, FE results match closely the present EKM results.


## 4.1. Single-Layered Panel (a)

Longitudinal variations of deflection (w¯) and stresses (σ¯x, τ¯xy and τ¯zx) for panel (a) have been presented in Figures 3, 4, and 5 for simply supported (S-S), clamped-simply supported (C-S), and clamped-free (C-F) boundary conditions, respectively. Results are plotted for different variation indexes (δ1=δ2=0.5, δ1=δ2=1.0 and δ1=1.0, δ2=2.0) along with homogeneous case to study the effect of in-plane varying material properties. 3D FE results for case (δ1=δ2=0.5) is also plotted in these figures. It is observed that present result are in excellent agreement with 3D FE results except for stress at very clamped edges. This mismatch at very clamped edge is because the FE solution does not ensure conditions of applied normal stress (σ¯z) and shear stress (τ¯zx) [43] at top and bottom surfaces of panel and it can be verified from Figure 7. For simply supported case, single term (n=1, iter.1) gives accurate prediction for all the entities whereas two-term solution (n=2, iter.2) is required for the other boundary conditions. It is evident that for all the boundary conditions (S-S, C-S, C-F) deflection (w¯) is effected significantly with increase in variation index whereas τ¯zx is least effected. As variation indices increase, the point of maximum deflection (w¯) for S-S and C-S boundary condition is shifted gradually toward ξ=1.0. For S-S and C-S case, stresses σ¯x and τ¯xy decrease as the variation index increases. The decrement in τ¯xy is significantly high under S-S boundary conditions. It is revealed that influence of in-plane graded material properties depends significantly on boundary condition of flat panel.Figure 3
Effect of variation in properties on longitudinal variations of deflection and stresses for panel (a) (α=45°, S=5) subjected to S-S boundary condition.Figure 4
Effect of variation in properties on longitudinal variations of deflection and stresses for panel (a) (α=45°, S=5) subjected to C-S boundary condition.Figure 5
Effect of variation in properties on longitudinal variations of deflection and stresses for panel (a) (α=45°, S=5) subjected to C-F boundary condition.Longitudinal variations of in-plane displacement (u¯) are plotted in Figure 6 for four types of support conditions (i.e., S-S, C-F, C-S, and C-C). For all the support conditions, in-plane displacement (u¯) is influenced greatly by an increase in variation index. It is observed that effect of varying material property is more pronounced near to simply supported and free edges than the center of the flat panel. For S-S, C-S, and C-C conditions, the effect is minimum, around ξ=0.6, with respect to the constant case. All the lines cut the homogeneous case line at one point near to ξ=0.6. For C-C case, asymmetry is the highest for case (iiia) and a similar trend is observed for S-S case also. For all the boundary conditions, the in-plane displacement (u¯) increases abruptly for case (iiia) as compared to case (ia) and case (iia). The longitudinal variations of in-plane displacement for S-S and C-C boundary condition become more asymmetrical as the variation index increases.Figure 6
Effect of variation in properties on longitudinal variations of in-plane deflection (u¯) for panel (a) (α=45°, S=5) subjected to different boundary condition (SS, CF, CS, CC).Figure 7
Comparison of in-plane deflection and stresses distributions at differentξ-locations with 3D FE solution for panel (a) (S=5) under C-F boundary condition.Through-thickness variations at differentξ locations have been presented in Figure 7 for case (ia) under C-F support conditions. 3D FE results for all ξ locations are also plotted in the figure. It is revealed that the distribution of in-plane displacement u¯ and stress σ¯x matches well the 3D FE solution near to clamped support, but 3D FE fails in predicting the stresses σ¯x, σ¯z, and τ¯zx at the clamped edge. 3D FE predicted stress distribution does not satisfy shear traction (zero) and condition of applied normal stress at the top and bottom surfaces of panel whereas present EKM solution satisfies these conditions exactly. The FE results for transverse stresses match well the present solution at some distance away from the clamped edge. It is observed that in-plane displacement (u¯) follows linear pattern near to free edges which is quadratic or cubic near the clamped support. Similar trends are observed for σ¯x and σ¯z. Through-thickness variation of τ¯zx is parabolic and its magnitude decreases as we move from clamped edge to free edge.Effect of ply-angle on longitudinal variation of deflection (w¯) and stresses (σ¯x, τ¯xy and τ¯zx) has been shown in Figure 8. It is observed that, with an increase in ply-angle from 0° to 30°, there is a gradual increase in all the entities whereas for 45° ply-angle magnitude of all the entities increases abruptly.Figure 8
Comparison of deflection and stresses distributions for different value ofα under C-F boundary condition.Percentage increment or decrement, with respect to the homogeneous case, of various entities at the location where it is maximum, is presented in Table1. It is found that in-plane displacement (u¯) increases with increase in variation index and it is maximum for simply supported boundary condition which is 175.65% for case (iiia). Percentage increase in deflection (w¯) is also maximum for S-S support condition. The increment in the in-plane stress is maximum for C-S boundary condition, which is −30.55% for case (iiia). Maximum decrement of 47.7% in τ¯xy is observed for case (iiia) under C-S boundary conditions. From Table 1, it is concluded that the effect of material property variation is relatively less under clamped support conditions (C-S, C-F) than that under simply supported condition (S-S).Table 1
Percentage (%) change in deflections (u-, w-) and stresses (σ-x, τ-xy, τ-zx) with respect to homogeneous case, due to variation in material properties for panel (a) with α=45°.
Boundary conditions Entity (location) (x, z) δ1=δ2=0.5 δ 1 = δ 2 = 1.0 δ 1 = 1.0, δ2 = 2.0 S-S u - (1.0, -0.5) 50.47 93.38 175.65 w - (0.5, 0.0) 36.62 72.19 148.84 σ - x (0.5, -0.5) -2.79 -4.15 -4.07 τ - x y (0.5, -0.5) -11.37 -16.58 -39.59 τ - z x (0.1, 0.0) 0.49 0.94 0.96  C-S u - (1.0, -0.5) 36.10 64.79 113.45 w - (0.5, 0.0) 21.87 41.10 82.28 σ - x (0.5, -0.5) -13.36 -21.72 -30.55 τ - x y (0.5, -0.5) -19.23 -29.84 -47.71 τ - z x (0.1, 0.0) 3.34 5.72 8.71  C-F u - (1.0, -0.5) 28.36 51.64 106.17 w - (1.0, 0.0) 17.97 34.85 74.83 σ - x (0.5, -0.5) 2.30 8.00 13.68 τ - x y (0.5, -0.5) -9.28 -15.64 -33.96 τ - z x (0.1, 0.0) 2.87 3.27 4.18


## 4.2. Two-Layered Panel (b)

Figures9 and 10 depict the effect of variation index on longitudinal variation of deflection (w¯) and stresses (σ¯x, τ¯xy and τ¯zx) for C-S and C-F support conditions, respectively. Panel (b) is a two-layered flat panel in which effect of variation of material property is investigated by taking different and same variation index for the top and bottom layer. To study the effect of varying material property on the longitudinal variation of deflection and stresses, the results for the layer-wise homogeneous case are also plotted in the figures. It is observed that deflection and stresses are significantly affected even by considering the variation of material property in one layer. 3D FE results for case (iib) are also presented in Figures 9 and 10. The present results are in good agreement with 3D FE except for stress at very clamped edges, because at the support specifically clamped type, finite element solutions do not satisfy the boundary conditions and interface continuity conditions which can be verified from Figures 7 and 14. For both C-S and C-F support conditions, deflection w¯ is affected significantly by gradation of properties along x-direction whereas stresses σ¯x, τ¯xy, and τ¯zx are least affected. The deflection at the center of the panel for C-S support increased by 9.5% for case (ib), 29.7% for case (iib), and 43.2% for case (iiib) with respect to the homogeneous case. Similarly, for C-F panel, the percentage change in deflection at the free edge of the panel is 9.1% for case (ib), 29% for case (iib), and 35% for case (iiib). For both C-S and C-F case, the percentage change in τ¯zx is less than 5%.Figure 9
Effect of variation in properties on longitudinal variations of deflection and stresses for panel (b) (α=45°, S=5) under C-S boundary condition.Figure 10
Effect of variation in properties on longitudinal variations of deflection and stresses for panel (b) (α=45°, S=5) under C-F boundary condition.


## 4.3. Four-Layered Panels (c) and (d)

The results for panel (c) with C-C, C-S, and C-F boundary condition are plotted in Figures11 and 12 for different cases of property variation. To verify the accuracy and stability of present solution for the multilayered panel, 3D FE results for case (icd) are also plotted. It is observed that present results are in very good agreement with 3D FE. Longitudinal variations of deflection (w¯) and stresses (σ¯x, τ¯xy, and τ¯zx) for C-C support conditions are plotted in Figure 11. It is observed that deflection (w¯) has significant effect as compared to stresses. Similarly, in Figure 12 longitudinal variation of w¯ and σ¯x is plotted for C-S and C-F boundary conditions. For both boundary conditions, the influence of varying material properties on deflection (w¯) is significantly more pronounced as compared to σ¯x.Figure 11
Effect of variation in properties on longitudinal variations of deflection and stresses for panel (c) (α=45°, S=5) under C-C boundary condition.Figure 12
Effect of variation in properties on longitudinal variations of deflection and stresses for panel (c) under C-F and C-S boundary condition.
(a)
C-F boundary condition (b)
C-S boundary conditionSimilarly in Figure13 longitudinal variations of deflections (u¯, w¯) and stress (σ¯x) have been presented for panel (d) under C-C and C-F boundary conditions. Here, 3D FE result for case (icd) is also plotted which is in good agreement with present results. To study the effect of gradation in material properties on deflection and stresses, the results for the layer-wise homogeneous case are also plotted in the same figures. It is observed that, the effect of material properties is more significant for deflections (u¯, w¯) as compared to stress (σ¯x) for both C-C and C-F support. For panel subjected to C-C boundary conditions, the longitudinal variations of deflections become more asymmetrical as the variation index increases.Figure 13
Effect of variation in properties on longitudinal variations of deflection and stresses for panel (d) under C-C and C-F boundary condition.Figure 14
Comparison of stresses distributions at differentξ-locations with 3D FE solution for panel (d) (S=5) under C-F boundary condition (Case (i)cd).Through-thickness distribution of transverse stressesσ¯z and τ¯zx has been presented in Figure 14 for panel (d) in which material properties vary according to case (icd) and panel subjected to C-F boundary conditions. It is observed that 3D FE solution fails to predict the stress distribution at the clamped support. At very clamped support (ξ=0), stress distribution predicted by FE neither satisfies interface continuity condition nor satisfies shear traction (zero) and condition of applied normal stress at top and bottom surface of the panel. However, present EKM solution satisfies all these conditions at the clamped support. It is observed that away from clamped support, FE results match closely the present EKM results.


## 5. Conclusions

An accurate analytical elasticity solution is presented for axially functionally graded angle-ply flat panel under cylindrical bending and subjected to arbitrary boundary condition. The influence of stiffness variation on static response of angle-ply flat panel is investigated comprehensively. Based on the present study, it is established that the variation of material properties along the in-plane direction affects the panel deflections and stresses to a great extent. It is also observed that effects of in-plane stiffness variation, on the behavior of the panel, depend significantly on boundary conditions. The effect of in-plane material property variation on the deflections and stresses is relatively less under clamped support conditions (C-S, C-C) as compared to simply supported case (S-S). This development has shown that, by controlling the in-plane stiffness variation parameters, the desired distribution of the deflection and stress components can be achieved for specific applications. The present method provided benchmark results to assess the validity and accuracy of different plate theories and computational models for analysis of axially functionally graded angle-ply flat panel under cylindrical bending. The current research will also be beneficial to modeled real life panel structures in which material properties of panel deteriorate due to some environmental effect.



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*Source: 2597484-2018-06-07.xml*