**American Journal of Mechanical Engineering**

## A Theoretical Analysis of Static Response in FG Rectangular Thick Plates with a Four-Parameter Power-Law Distribution

Mechanical Engineering department, Islamic Azad University Khomeinishahr Branch, Isfahan, IranAbstract | |

1. | Introduction |

2. | Displacement Field and Constitutive Equation |

3. | Stress Resultants |

4. | Governing equation of Mindlin Plates |

5. | Numerical results |

6. | Conclusions |

Acknowledgments | |

References |

### Abstract

In this paper, we proposed a simple mathematical procedure to solve the differential equations governing the buckling and bending analysis of FG thick rectangular plates resting on two-parametric foundation based on Mindlin assumption. All edges are set on the simply supported conditions. Young modulus of the FG plate was assumed to vary according to a simple four-parameter power law across the thickness direction. For bending analysis, the plate was subjected to two kinds of loading: sinusoidal and uniform. For bucking analysis, two kinds of in-plane loading were applied to the plate: uniaxial and biaxial. Variations of FG material variation profile, thickness ratio, and foundation parameters on buckling critical load and out-plane displacement were examined. The distribution of axial and shear stress across the thickness, when the plate is exposed to uniform transverse loading, was further studied.

**Keywords:** Mindlin rectangular plates, power law FG distribution, two parametric elastic foundations

**Copyright**© 2016 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Fatemeh Farhatnia. A Theoretical Analysis of Static Response in FG Rectangular Thick Plates with a Four-Parameter Power-Law Distribution.
*American Journal of Mechanical Engineering*. Vol. 4, No. 1, 2016, pp 11-20. http://pubs.sciepub.com/ajme/4/1/3

- Farhatnia, Fatemeh. "A Theoretical Analysis of Static Response in FG Rectangular Thick Plates with a Four-Parameter Power-Law Distribution."
*American Journal of Mechanical Engineering*4.1 (2016): 11-20.

- Farhatnia, F. (2016). A Theoretical Analysis of Static Response in FG Rectangular Thick Plates with a Four-Parameter Power-Law Distribution.
*American Journal of Mechanical Engineering*,*4*(1), 11-20.

- Farhatnia, Fatemeh. "A Theoretical Analysis of Static Response in FG Rectangular Thick Plates with a Four-Parameter Power-Law Distribution."
*American Journal of Mechanical Engineering*4, no. 1 (2016): 11-20.

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### At a glance: Figures

### 1. Introduction

Functionally graded materials (FGM) are the new composites that are microscopically heterogeneous. Mechanical properties of this kind of new composite change gradually from metal to ceramics in arbitrary directions. Ceramic can tolerate high temperature, and metal provides good machinery ability, high hardness, and flexibility ^{[1]}. Many structures can be modeled with rectangular FG plates in the aerospace engineering. Investigating the buckling critical loads of these structures is of great importance in achieving better performance. Plates with ratio of length to plate thickness less than10 times are known as the thick plates Since the classical plate theory underestimates the out-plane displacement and overestimates the buckling loads and frequencies of the thick plate due to ignoring the effect of shear deformation, many studies have focused on considering the behavior of thick plates with higher order shear deformation theory when subjected to static and dynamic loadings [2-8]^{[2]}. Since the theoretical analysis is carried out in this paper using First Shear Deformation Theory (FSDT), we introduce some papers related to it here. Levy, Reissner, and Mindlin were the first ones who tried to rectify this deficiency existing in classical plate theory ^{[9]}. In his first attempt in this area, Levy ^{[10]} achieved a solution for governing equations by employing the three-dimensional elasticity. Reissner ^{[11]} proposed the theory of thick plate by considering the influence of shear deformation. He performed his approach based on stress analysis for bending the elastic plates. Mindlin ^{[12]} assumed that the transverse shear stress is constant along the thickness, whereas this assumption makes to take the shear strain constant, too. To remove this deficiency, he introduced the shear factor in his formulation to predict the shear stress resultant. Levinson ^{[13]} refined the theory of Mindlin plate by omitting the shear correction factor from his approach based on the displacement theory. Lanhe ^{[14]} investigated the thermal buckling of FG rectangular moderately thick plates by first shear order theory using simply supported edge condition. Liew and Chen ^{[15]} proposed the buckling analysis of Mindlin rectangular plates subjected to partial in-plane loading, using radial point interpolation method. Shimpi et al. ^{[16]} proposed two new displacement-based, first-order shear deformation theories involving only two unknown functions to analyze static and dynamic problems.

Solving the governing equations based on first and higher-order shear deformation theories brings about computational complexity. In addition, while the neutral plane is not coincident to the middle one, there are extension-bending couplings in FG plates; therefore, to present an efficient and simply solving procedure in order to derive governing equations, are always attractive for researchers ^{[17, 18, 19]}.

Solving the governing equations based on first and higher-order shear deformation theories brings about computational complexity. In addition, while the neutral plane is not coincident to the middle one, there are extension-bending couplings in plates made of functionally graded materials; therefore, to present an efficient and simply solving procedure, the derived governing equations always attract researchers ^{[17, 18, 19]}.

Furthermore, considering plates resting on elastic foundation is of great importance in modern engineering structures, aerospace, biomechanics, petrochemical, construction, electronics, and nuclear and civil engineering ^{[20]}. Winkler simulated an elastic foundation by using a set of linear elastic springs which worked independently without taking into account the effects of shear coupling between them. It was known as the one-parameter model. Pasternak proposed the two-parameter model considering the influence of shear layer between springs ^{[21]}. Gupta et al. ^{[22]} investigated the buckling and vibration of orthotropic plate in Winkler elastic foundation. Rashed ^{[23]} studied the bending analysis of thick plates in two-parametric elastic foundation by exploiting boundary integral transformation. Additionally, in another paper Wen ^{[24]} proposed the Laplace transform for analysing moderately thick plate resting on two-parametric elastic foundation. Civalek ^{[25]} reported the discrete singular convolution method to solve the governing equation of bending, buckling, and vibration in Mindlin plates resting on two-parametric elastic foundation. Akhavan et al. ^{[26]} presented an exact solution for the buckling analysis of rectangular plates on two simply supported opposite edges resting on Pasternak foundation subjected to uniform and non-uniform in-plane loading. Hosseini-Hashemi et al. ^{[27]} studied the Hydro-elastic vibration and buckling rectangular plates resting on two-parametric elastic foundation for various edge conditions. In their research the plate was subjected to linear distributed in-plane loading. Bouderba et al. ^{[28]} presented thermo-mechanical bending analysis of FG rectangular plate resting on two-parametric elastic foundation. They exploited a developed trigonometric shear deformation theory. The benefit of their approach was dealing with four unknowns as against five in case of the other shear deformation theories. Zidi et al. ^{[29]} studied the bending behavior of FG plates, using a four-variable, refined plate theory. In another work, to reduce the number of unknowns in shear deformable theories Hamidi et al. ^{[30]} employed a sinusoidal plate theory to study the thermo-mechanical bending of sandwich plates. They dealt with 5 unknowns in the governing equations of this kind of plate. Bennoun et al. ^{[31]} proposed a simple method for considering the vibration analysis of FG sandwich plates. They divided the displacement into three parts. The unknowns were diminished to five, as opposed to six or more in the other shear deformation theories.

With respect to this important issue that the governing equation of Mindlin plates are included in three displacement components in three axes of rectangular Cartesian system and two rotations in in-plane directions, solving the governing equations simultaneously to obtain the exact solution is a complicated task. Therefore, in this research in the first stage, we managed to reduce three differential equilibrium equations of plates to one equation in terms of lateral displacement. For this purpose, we rewrote the equations based on the neutral plane of FG plates. Consequently, the stress resultants of FG plates were formulated as isotropic, homogenous ones. In the next stage, by exploiting some algebraic operations, these equations were reduced to one in terms of lateral displacement. Following this method, the influences of elastic foundation parameters, thickness ratio, loading factor, and various FG power indices were further investigated on deflection, normal and shear stresses, and critical buckling load.

The rest of the paper is organized as follows: In Section 1, the governing equilibrium equations are derived based on Mindlin theory and neutral plane. In Section 2, the solving procedure is devoted to decreasing the three differential equilibrium equations from three to one. By employing this procedure, the quantities of unknowns are diminished from three (rotation in two directions and lateral displacement) to one. Section 3 presents some numerical examples to illustrate the accuracy and precision of the present approach by comparing it with other related approaches in the literature.

### 2. Displacement Field and Constitutive Equation

Consider the rectangular plate of length a, width b and thickness h, defined in xyz rectangular coordinate system. Its center is located in the middle plane of plate, whereas z is representative for distance of any arbitrary points in the thickness of the plate with respect to the middle plane. Here, we consider a relatively thick plate. In this category of plates, the validation of hypothesis of straight normal is not established. This is due to the existence of shear deformation effects that cannot be ignored. In this paper, we assumed the shear deformation behavior is based on Mindlin model; therefore it can be described by first shear deformation theory (FSDT). The displacement field can be written as follows:

(1) |

Where *u*_{α}* *and *w* are the displacement components in *x*, *y* and *z* directions, (*u*_{0})_{α} and *w*_{0} are mid-plane displacement and are rotation components in in-plane directions. By appropriate differentiation with respect to the coordinate axes, the strain components of Mindlin plate in any arbitrary point in state of the infinitesimal deformation are obtained as follows:

(2) |

In the above relations is middle plane strain. In two-dimensional stress state, the stress-strain relations of Hook’s law are defined as follows:

(3.1) |

(3.2) |

In above equation:. Based on equations (2) and (3), the stress components can be obtained in terms of strains of the middle plans and rotation in two directions x and y, as follows:

(4) |

where is Young modulus of the FG plate that assumed to vary according to a simple power law, across the thickness coordinate z. It can be expressed as:

(5) |

*Vc, Em *and* Ec* represent ceramic volume fraction, Young modulus of metal and ceramics, respectively. In this study, it is assumed ceramic volume fraction of functionally graded material follows two simple four-parameter power-law distributions. Therefore, Young modulus is defined as follows^{[32]}:

model 1:

(6-1) |

model 2:

(6-2) |

In above equation, *p* is functionally graded power index. When p equals to zero and infinity, it corresponds to pure metal and pure ceramic plate, respectively. The parameters *a*, b*, c* corresponds for the material variation profile through the thickness of functionally graded plate.

### 3. Stress Resultants

The stress resultants can be expressed as follows:

(7) |

is the transverse shear correction coefficient, applied to the transverse shear forces based on the parabolic distribution of shear transverse strains across thickness. By inserting relations (4) into (7) and integration across the thickness, the stress resultants are evaluated, as follows:

(8a) |

(8b) |

where,

(9) |

It is clear that the extension-bending coupling existing in the above relations makes the solution procedure get more complicated. For anisotropic plates, the neutral plane doesn’t coincide with the middle one. For an isotropic plate or a composite one with symmetrical mechanical properties with respect to mid-plane of the plate, B gets value of zero and consequently the displacement of the middle plane equals zero, too. Abrate ^{[18]} showed that instead of expressing the stress resultant relations in base of middle plane, writing them with respect to the neutral one, the governing equations are derived as those of ones for homogenous materials. To achieve the aim, we introduced *z*_{0} as the position of the neutral plane, as follows ^{[33]}:

(10) |

It is obvious that for an isotropic or symmetric laminated composite plate, the neutral plane is coincident with the middle surface. After substituting into relations (9) and by employing relation:

(11) |

equals to zero. Consequently the bending-extension coupling is removed.

Therefore, the stress resultants are expressed as:

(12) |

### 4. Governing equation of Mindlin Plates

The governing equation can be derived through principal of virtual work. It can be stated as follows:

(13) |

In above equation *F*_{k }and* q *are elastic foundation reaction force and external loading, respectively. Substituting equation (2) and (4) into equation (13) and employing this principle leads to the following governing equations of Mindlin plate, as follow:

(14a) |

(14b) |

Where *k*_{1} and *k*_{2} are Pasternak elastic foundation parameters. By assumption to infinitesimal deformation and writing the equilibrium equations based on the neutral surface, the in-plane displacement *u*_{0}*, v*_{0} are eliminated. Then by substituting the relations (12) into equilibrium equations (14), the governing equations for a Mindlin plate are expressed as:

(15a) |

(15b) |

Where ψ*=*ψ_{α,α} , *q**^{ }in the equation (15b) is defined as follow:

(16) |

where *q*,* k*_{1},* k*_{2} are the transverse loading, elastic foundation reaction parameters of Pasternak model, respectively. Now By differentiation of equation (15a) with respect to α and rearranging it, then:

(17) |

In state of buckling analysis, *q* is expressed as:

(18) |

In above equation, *N*_{α}β* *denotes to in-plane loading. By eliminating of rotation parameters from governing equation (17) by substituting from equation (15b) into it, bi-harmonic resultant equation is obtained as:

(19) |

As observed, the above equation assembles the same one that employed for thin plate with classical plate theory (CPT). By introducing the non-dimensional parameters and operator *L, *as follows:

Using in equation (20), the governing equation of Mindlin plate resting on Pasternak elastic foundation exposed on transverse loading, can be written in non-dimensional form as follow:

(20) |

where,

In state of buckling analysis, the governing differential equation is derived in non-dimensional form as:

(21) |

where,

(22a) |

(22b) |

(22c) |

where

**4.1. Bending Analysis**

By considering the transverse loading in two form of sinusoidal and uniform distribution, as:

(23) |

By utilizing aforementioned function and equation (22) in governing equation (20), the coefficient of *W*_{mn} is determined, as follows:

-type 1

(24a) |

-type 2

(24b) |

To find the rotation in direction x, firstly ψ* *can be determined based on equation (15b) and then by substituting into equation (15a), ψ_{x} may be derived from the following equation:

(25) |

The non-dimensional form of equation (26) is expressed as follows:

(26) |

where,

By substituting the equation (22a) into aforementioned differential equation, may be expressed in terms of out-plane displacement component as follows:

-type 1

(27a) |

-type 2

(27b) |

By carrying out the similar steps to determineψη , this parameter can be obtained, too.

-type 1

(28a) |

-type 2

(28b) |

The distribution of shear stresses through thickness can be obtained from equilibrium equations as follows:

(29) |

Upon substituting of the in-plane stress equation (4) based on neutral plane, into equation (29):

(30) |

The distribution of shear stress *σ*_{xz}_{ }and *σ*_{yz}_{ }can be obtained as:

(31) |

where , δ_{α}β is Kronecker delta and *µ*^{+}^{ }and *µ*^{-}^{ }are defined as:

**4.2. Buckling Analysis**

In bucking problem, an analytical solution for critical bucking loads of FGM rectangular plate with four-side simply supported boundaries can be obtained by utilizing equation (20a) into governing equation (21), as follows:

- Biaxial in-plane loading (r=1):

(32) |

- Uniaxial in-plane loading (r=0):

(33) |

### 5. Numerical results

In this section, the author represented some numerical results. A computer program has been prepared in MATLAB. These results are categorized in two subsections, one for bending analysis and the other for buckling analysis. In both, the following is assumed:

*E*_{m} and *E*_{c}* *are Aluminum (metal) and Alumina (ceramic) Young modulus, respectively. The simply supported FG plate resting on Pasternak foundation parameters: *k*_{1}=0.1*q*_{0}, *k*_{2}=0.1*k*_{1}. In all numerical results, a shear correction factor of is used. Depending on the FG model in equations (6), three categorized cases can be considered: case1 (*a**=1,*b**=0), case2(*a**=*b**=1), case3(*a**=1,*b**=0.5). In all cases, c=2 is kept as constant. The FG power index *p* is chosen: (0.125 0.25 0.5 1 2 5 10 15 20 50 100).

As observed in Figure 1- Figure 3, the FG material distribution obeys asymmetric profile in cases 1 and 3, whereas in case 2, the material distribution through the thickness is symmetric through the thickness. In cases 1 and 3, the distributions of metal and ceramic as constitution are characterized by the fact that bottom surface and top surface is ceramic rich in model and model 2, respectively. However, there is a mixture of two constituents through the thickness on the one surface when ceramic rich is obtained in one of two surfaces.

As observed in Figure 1a - Figure 1f, the FG material distribution obeys asymmetric profile in cases 1 and 3, whereas in case 2, the material distribution through the thickness is symmetric through the thickness. In cases 1 and 3, the distributions of metal and ceramic as constitution are characterized by the fact that bottom surface and top surface is ceramic rich in model and model 2, respectively. However, there is a mixture of two constituents through the thickness on the one surface when ceramic rich is obtained in one of two surfaces.

**Fig**

**ure**

**1**

**.**Variations of the ceramic volume fraction

*V*

_{C}through the thickness for cases 1,2,3

**5.1. Bending Analysis**

The plate is subjected to two forms of transverse loading: Sinusoidal and uniformly distributed model. The variation of maximum deflection (at the center of the plate) in non-dimensional form for the aforementioned cases is tabulated in Table 1 and Table 2. The following non-dimensional parameters are used in presenting the numerical results in figures and tables:

#### Table 1. variation of non-dimensional displacement with respect to variation of FG law index for uniform loading

As observed, in case of symmetric volume fraction profile, the deflection has less value than that of other two cases. Also, by raising p, the volume fraction of ceramic increased, thereby decreasing the values of deflection. The validity of present results was verified for the non-dimensional deflection (), as shown in Table 3. It can be concluded that the present yields very good results compared to that presented by Civalek ^{[24]}.

**Fig**

**ure**

**2**

**.**Distribution of non-dimensional stress

*σ*

_{xx}across the thickness for case 1,

*k*

_{1}=

*k*

_{2}=0

**Fig**

**ure**

**3**

**.**Distribution of non-dimensional stress

*σ*

_{xx}across the thickness for case 2, model 2

**Fig**

**ure**

**4**

**.**Distribution of nondimensional stress across thickness for case 3, model 1

In Figure 2 – Figure 4 to show the capability of the presented method, distribution of bending stress component across the thickness with and without influence of elastic foundation under simply supported boundary condition, are illustrated for cases (1-3).** **As observed, stress component does not take value of zero in the middle surface for a FG plate. In addition, the amount of bending stress decreases effectively in the presence of elastic foundation.

**Fig**

**ure**

**5**

**.**Distribution of nondimensional shear stressτ

_{xz}across the thickness for case 1,

*k*

_{1}

*=k*

_{2}

*=0*

**Fig**

**ure**

**6**

**.**Distribution of nondimensional shear stress across the thickness for case 2, model 1

**Fi**

**gure**

**7**

**.**Distribution of nondimensional shear stress across the thickness for case 3, model 1

In Figure 5-Figure 7, the distribution of shear stress τ_{xz }across the thickness is depicted. As observed, the maximum of shear stress doesn’t occur at the middle surface for cases 1 and 3, due to be not symmetric distribution of mixture of ceramic and metal throughout the thickness. On the other hand, as shown in Figure 6, in case 2, this distribution seems to be symmetric as profile of ceramic volume fraction has symmetric pattern, too.

**5.2. Buckling Analysis**

The buckling analysis of FG rectangular thick plate is carried out for simply supported boundary condition. The results are given for case 1, n=1 as tabulated (Table 4). As observed, increasing the FG power law decreases the critical buckling load. This is due to the fact that increasing n raises the volumetric percentage of metal in FG plate, and as a result decreasing the bending rigidity of FG plate. Thus, when the plate is exposed to biaxial compression in-plane loads, the decline in the resistance against buckling is more diminished in comparison with the state of uniaxial compression loading. Also, the results are compared with those have been reported to be obtained by Thai and Kim ^{[34]}.

#### Table 5. Comparison the critical buckling load in state of biaxial compression loading with Thai and Kim [34] for case 1, *k*_{1}*=k*_{2 }*=0*

### 6. Conclusions

In this paper, the buckling and bending responses of rectangular thick plates made of functionally graded materials resting on two-parametric foundation were investigated. Since in FSDT theory the governing equations involve five unknown functions including three displacements and two rotations, the author aimed to represent a simple and effective procedure to reduce the complexity of the solution. As a result, three coupled governing equations were diminished to one ordinary differential equation in terms of out-plane displacement. The influences of power law index, thickness ratio, and foundation parameters on the critical buckling load were investigated in two cases of uniaxial and biaxial in-plane loadings of FG plates. The distribution of bending and shear stresses across the thickness direction was studied as well as out-plane displacement. The result showed that in the case of symmetric volume fraction profile, the deflection had less value than those of the other two cases. It is suggested that this approach should be used in vibration and post buckling of Mindlin FG plates.

### Acknowledgments

The research described in this paper was financially supported by Islamic Azad University, Khomeinishahr branch.

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