# New Shape Function for the Bending Analysis of Functionally Graded Plate

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of the Problem

_{c}represents the material property at the top of the plate z = h/2 − ceramic, and P

_{m}represents the material property at the bottom of the plate z = −h/2 − metal. Index p is the exponent of the equation which defines the volume fraction of the constituents in FGM. Practically, by varying the index p, homogenous plates as well as FGM plate with precisely determined gradient structure could be obtained, as it is presented in Figure 4b:

- when p = 0 the plate is homogenous, made of ceramics,
- when 0 < p < ∞ the plate has a gradient structure,
- theoretically, when p = ∞ the plate becomes homogenous again, made of metal, although the plate can be considered homogenous even when p > 20.

## 3. Kinematic Displacement-Strain Relations and Constitutive Equation of Elasticity for FGM

## 4. Bending of FGM Plates and FGM Plates on Elastic Foundation

## 5. Analytical Solution of the Equilibrium Equations

## 6. Numerical Results

_{0}) and Pasternak (k

_{1}) coefficient of the elastic foundation. Apart from the normalization given in (26), it is necessary to apply the normalization of the coefficients k

_{0}and k

_{1}, in the following form:

_{0}and k

_{1}coefficients, as well as for two different ratios length/thickness of the plate (a/h = 10 and a/h = 5). In order to determine the effect of the elastic foundation on the displacements and stresses of the plate, the values of displacements and stresses for k

_{0}= 0 and k

_{1}= 0 are first shown, which practically matches the case of the plate without the elastic foundation. Afterwards, the values of the given coefficients are varied in order to conclude which of the two has greater influence. Based on the results, it is concluded that the introduction of the coefficient k

_{0}has less influence on the change of the displacements and stresses values then when only k

_{1}coefficient is introduced. By introducing k

_{0}and k

_{1}coefficients, bending stiffness of the plate increases, i.e., displacement and stresses values decrease and the influence of the Winkler coefficient is smaller than the influence of the Pasternak coefficient. This phenomenon is especially noticeable in the diagram dependency which is to be shown later.

_{0}on the distribution of the normal stress ${\overline{\sigma}}_{xx}$, shear stress ${\overline{\tau}}_{xy}$ and transversal shear stresses ${\overline{\tau}}_{xz}$ and ${\overline{\tau}}_{yz}$ across the thickness of the plate on the elastic foundation. By analyzing the diagram, it can be seen that the value of the stresses ${\overline{\sigma}}_{xx}$ and ${\overline{\tau}}_{xy}$ equals zero for z/h = 0.15. On the other hand, the maximum values of ${\overline{\tau}}_{xz}$ and ${\overline{\tau}}_{yz}$ stresses are at z/h = 0.2 when the new proposed shape function is applied, while the maximum values of mentioned stresses is respectively at z/h = 0.15 i.e., z/h = 0.25 for Karama’s shape function.

_{0}and k

_{1.}By comparing the two diagrams, it can be seen that the change of the displacement value $\overline{w}$ is higher with the increase of the coefficient k

_{1}value than with the increase of the coefficient k

_{0}. For example, for the FGM plate when p = 5, and the increase of the coefficient from k

_{0}= 0 to k

_{0}= 100, the value of deflection changes twice its value. In the other case, with the change of the coefficient from k

_{1}= 0 to k

_{1}= 100, the value of deflection changes 8 times its value.

## 7. Conclusions

- the values of the vertical displacement $\overline{w}$ (deflection) and the corresponding stresses, which were obtained in this paper by using HSDT theory based on the new shape function, match the results of the same values obtained in the reference papers by using TSDT theory [58], quasi 3D theory of elasticity [59] and HSDT theories based on 13 different shape functions. However, in contrast to that, there are significant deviations of the results obtained for the values of the vertical displacement, especially for stresses ${\overline{\sigma}}_{xx}$, from the results obtained by CPT theory from the reference papers [60].
- the diagram of the distribution of transverse shear stresses ${\overline{\tau}}_{xz}$ and ${\overline{\tau}}_{yz}$ across the thickness of the plate shows the difference in behavior between a homogenous, ceramic or metal, plate and FGM plate. A basic property of FGM can be clearly seen, and that is the asymmetry of the stress distribution in relation to the middle plane of the plate (z = 0). The maximum values of stresses, depending on the volume fraction of certain constituents, are shifted in relation to the plane z=0, which represents a neutral plane in homogenous plates.
- the highest values of the displacement $\overline{w}$ are obtained in a metal plate, the lowest in a ceramic plate and in an FGM plate, the values are somewhere in between and they depend on the volume fraction of the constituents. Based on that, it can be concluded that by varying the volume fraction of metal and ceramic, a desired bending rigidity of the plate can be achieved.
- a comparative analysis of the change of transverse shear stresses ${\overline{\tau}}_{xz}$ and ${\overline{\tau}}_{yz}$ across the thickness of the plate shows that, unlike the stress ${\overline{\tau}}_{xy}$, their values do not match for all the shape functions.
- by introducing FG plate on Winkler–Pasternak model of elastic foundation is shown that the influence of the Winkler coefficient (k
_{0}) is smaller than the influence of the Pasternak coefficient (k_{1}).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Golak, S.; Dolata, A.J. Fabrication of functionally graded composites using a homogenised low-frequency electromagnetic field. J. Compos. Mater.
**2016**, 50, 1751–1760. [Google Scholar] [CrossRef] - Watanabe, Y.; Inaguma, Y.; Sato, H.; Miura-Fujiwara, E. A Novel Fabrication Method for Functionally Graded Materials under Centrifugal Force: The Centrifugal Mixed-Powder Method. Materials
**2009**, 2, 2510–2525. [Google Scholar] [CrossRef] [Green Version] - El-Hadad, S.; Sato, H.; Miura-Fujiwara, E.; Watanabe, Y. Fabrication of Al-Al
_{3}Ti/Ti_{3}Al Functionally Graded Materials under a Centrifugal Force. Materials**2010**, 3, 4639–4656. [Google Scholar] [CrossRef] [PubMed] - Jha, D.K.; Kant, T.; Singh, R.K. A critical review of recent research on functionally graded plates. Compos. Struct.
**2013**, 96, 833–849. [Google Scholar] [CrossRef] - Bharti, I.; Gupta, N.; Gupta, K.M. Novel Applications of Functionally Graded Nano, Optoelectronic and Thermoelectric Materials. IJMMM Int. J. Mater. Mech. Manuf.
**2013**, 1, 221–224. [Google Scholar] [CrossRef] - Saiyathibrahim, A.; Nazirudeen, M.S.S.; Dhanapal, P. Processing Techniques of Functionally Graded Materials—A Review. In Proceedings of the International Conference on Systems, Science, Control, Communication, Engineering and Technology 2015, Coimbatore, India, 10–11 August 2015. [Google Scholar]
- Birman, V.; Byrd, L.W. Modeling and analysis of functionally graded materials and structures. Appl. Mech. Rev.
**2007**, 60, 195–216. [Google Scholar] [CrossRef] - Akbarzadeh, A.H.; Abedini, A.; Chen, Z.T. Effect of micromechanical models on structural responses of functionally graded plates. Compos. Struct.
**2015**, 119, 598–609. [Google Scholar] [CrossRef] - Kirchhoff, G. Uber das gleichgewicht und die bewegung einer elastischen scheibe. J. Die Reine Angew. Math.
**1850**, 1850, 51–88. [Google Scholar] [CrossRef] - Mindlin, R.D. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J. Appl. Mech.
**1951**, 18, 31–38. [Google Scholar] - Reissner, E. On bending of elastic plates. Q. Appl. Math.
**1947**, 5, 55–68. [Google Scholar] [CrossRef] [Green Version] - Reissner, E. The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech.
**1945**, 12, 69–72. [Google Scholar] - Zenkour, A.M. An exact solution for the bending of thin rectangular plates with uniform, linear, and quadratic thickness variations. Int. J. Mech. Sci.
**2003**, 45, 295–315. [Google Scholar] [CrossRef] - Mohammadi, M.; Saidi, A.R.; Jomehzadeh, E. Levy solution for buckling analysis of functionally graded rectangular plates. Appl. Compos. Mater.
**2010**, 17, 81–93. [Google Scholar] [CrossRef] - Bhandari, M.; Purohit, K. Static Response of Functionally Graded Material Plate under Transverse Load for Varying Aspect Ratio. Int. J. Met.
**2014**, 2014, 980563. [Google Scholar] [CrossRef] - Yang, J.; Shen, H.S. Non-linear analysis of FGM plates under transverse and in plane loads. Int. J. Nonlinear Mech.
**2003**, 38, 467–482. [Google Scholar] [CrossRef] - Alinia, M.M.; Ghannadpour, S.A.M. Nonlinear analysis of pressure loaded FGM plates. Compos. Struct.
**2009**, 88, 354–359. [Google Scholar] - Liu, D.Y.; Wang, C.Y.; Chen, W.Q. Free vibration of FGM plates with in-plane material inhomogeneity. Compos. Struct.
**2010**, 92, 1047–1051. [Google Scholar] [CrossRef] - Ruan, M.; Wang, Z.M. Transverse vibrations of moving skew plates made of functionally graded material. J. Vib. Control
**2016**, 22, 3504–3517. [Google Scholar] [CrossRef] - Woo, J.; Meguid, S.A.; Ong, L.S. Nonlinear free vibration behavior of functionally graded plates. J. Sound Vib.
**2006**, 289, 595–611. [Google Scholar] [CrossRef] - Hu, Y.; Zhang, X. Parametric vibrations and stability of a functionally graded plate. Mech. Based Des. Struct.
**2011**, 39, 367–377. [Google Scholar] - Taczała, M.; Buczkowski, R.; Kleiber, M. Nonlinear free vibration of pre- and post-buckled FGM plates on two-parameter foundation in the thermal environment. Compos. Struct.
**2016**, 137, 85–92. [Google Scholar] [CrossRef] - Xing, C.; Wang, Y.; Waisman, H. Fracture analysis of cracked thin-walled structures using a high-order XFEM and Irwin’s integral. Comput. Struct.
**2019**, 212, 1–19. [Google Scholar] [CrossRef] - Kim, K.D.; Lomboy, G.R.; Han, S.C. Geometrically non-linear analysis of functionally graded material (FGM) plates and shells using a four-node quasi-conforming shell element. J. Compos. Mater.
**2008**, 42, 485–511. [Google Scholar] [CrossRef] - Wu, T.L.; Shukla, K.K.; Huang, J.H. Nonlinear static and dynamic analysis of functionally graded plates. IJAME Int. J. Appl. Mech. Eng.
**2006**, 11, 679–698. [Google Scholar] - Reddy, J.N. A simple higher-order theory for laminated composite plates. J. Appl. Mech.
**1984**, 51, 745–752. [Google Scholar] [CrossRef] - Phan, N.D.; Reddy, J.N. Analysis of laminated composite plates using a higher-order shear deformation theory. Int. Numer. Methods Eng.
**1985**, 21, 2201–2219. [Google Scholar] [CrossRef] - Reddy, J.N. Analysis of functionally graded plates. Int. Numer. Methods Eng.
**2000**, 47, 663–684. [Google Scholar] [CrossRef] - Kim, J.; Reddy, J.N. A general third-order theory of functionally graded plates with modified couple stress effect and the von Kármán nonlinearity: Theory and finite element analysis. Acta Mech.
**2015**, 226, 2973–2998. [Google Scholar] [CrossRef] - Dung, D.V.; Nga, N.T. Buckling and postbuckling nonlinear analysis of imperfect FGM plates reinforced by FGM stiffeners with temperature-dependent properties based on TSDT. Acta Mech.
**2016**, 227, 2377–2401. [Google Scholar] [CrossRef] - Bodaghi, M.; Saidi, A.R. Thermoelastic buckling behavior of thick functionally graded rectangular plates. Arch. Appl. Mech.
**2011**, 81, 1555–1572. [Google Scholar] [CrossRef] - Yang, J.; Liew, K.M.; Kitipornchai, S. Dynamic stability of laminated FGM plates based on higher-order shear deformation theory. Comput. Mech.
**2004**, 33, 305–315. [Google Scholar] [CrossRef] - Akbarzadeh, A.H.; Zad, S.H.; Eslami, M.R.; Sadighi, M. Mechanical behaviour of functionally graded plates under static and dynamic loading. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
**2011**, 225, 326–333. [Google Scholar] [CrossRef] - Thai, H.T.; Choi, D.H. Efficient higher-order shear deformation theories for bending and free vibration analyses of functionally graded plates. Arch. Appl. Mech.
**2013**, 83, 1755–1771. [Google Scholar] [CrossRef] - Thai, H.T.; Park, T.; Choi, D.H. An efficient shear deformation theory for vibration of functionally graded plates. Arch. Appl. Mech.
**2013**, 83, 137–149. [Google Scholar] - Lo, K.H.; Christensen, R.M.; Wu, E.M. A high-order theory of plate deformation part 1: Homogeneous plates. J. Appl. Mech.
**1977**, 44, 663–668. [Google Scholar] [CrossRef] - Lo, K.H.; Christensen, R.M.; Wu, E.M. A high-order theory of plate deformation part 2: Laminated plates. J. Appl. Mech.
**1977**, 44, 669–676. [Google Scholar] [CrossRef] - Kant, T.; Swaminathan, K. Analytical solutions for free vibration of laminated composite and sandwich plates based on a higher-order refined theory. Compos. Struct.
**2001**, 53, 73–85. [Google Scholar] [CrossRef] [Green Version] - Kant, T.; Swaminathan, K. Analytical solutions for the static analysis of laminated composite and sandwich plates based on a higher order refined theory. Compos. Struct.
**2002**, 56, 329–344. [Google Scholar] [CrossRef] [Green Version] - Xiang, S.; Kang, G.; Xing, B. A nth-order shear deformation theory for the free vibration analysis on the isotropic plates. Meccanica
**2012**, 47, 1913–1921. [Google Scholar] [CrossRef] - Song, X.; Kang, G. A nth-order shear deformation theory for the bending analysis on the functionally graded plates. Eur. J. Mech. A-Solid
**2013**, 37, 336–343. [Google Scholar] - Song, X.; Kang, G.; Liu, Y. A nth-order shear deformation theory for natural frequency of the functionally graded plates on elastic foundations. Compos. Struct.
**2014**, 11, 224–231. [Google Scholar] - Soldatos, K. A transverse shear deformation theory for homogeneous monoclinic plates. Acta Mech.
**1992**, 94, 195–220. [Google Scholar] [CrossRef] - Aydogdu, M. A new shear deformation theory for laminated composite plates. Compos. Struct.
**2009**, 89, 94–101. [Google Scholar] [CrossRef] - Mantari, J.L.; Oktem, A.S.; Soares, S.G. A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates. Int. J. Solids Struct.
**2012**, 49, 43–53. [Google Scholar] [CrossRef] - Mantari, J.L.; Bonilla, E.M.; Soares, C.G. A new tangential-exponential higher order shear deformation theory for advanced composite plates. Compos. Part B-Eng.
**2014**, 60, 319–328. [Google Scholar] [CrossRef] - El Meichea, N.; Tounsia, A.; Zianea, N.; Mechaba, I.; El Abbes, A.B. A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate. Int. J. Mech. Sci.
**2011**, 53, 237–247. [Google Scholar] [CrossRef] - Mechab, B.; Mechab, I.; Benaissa, S. Analysis of thick orthotropic laminated composite plates based on higher order shear deformation theory by the new function under thermo-mechanical loading. Compos. Part B-Eng.
**2014**, 43, 1453–1458. [Google Scholar] [CrossRef] - Akavci, S.S. Two new hyperbolic shear displacement models for orthotropic laminated composite plates. Mech. Compos. Mater.
**2010**, 46, 215–226. [Google Scholar] [CrossRef] - Viola, E.; Tornabene, F.; Fantuzzi, N. General higher-order shear deformation theories for the free vibration analysis of completely doubly-curved laminated shells and panels. Compos. Struct.
**2013**, 95, 639–666. [Google Scholar] [CrossRef] - Grover, N.; Maiti, D.K.; Singh, B.N. Flexural behavior of general laminated composite and sandwich plates using a secant function based shear deformation theory. Lat. Am. J. Solids Strut.
**2014**, 11, 1275–1297. [Google Scholar] [CrossRef] [Green Version] - Ambartsumyan, A.S. On the Theory of Anisotropic Shells and Plates. In Proceedings of the Non-Homogeneity in Elasticity and Plasticity: Symposium, Warsaw, Poland, 2–9 September 1958; Olszak, W., Ed.; Pergamon Press: London, UK, 1958. [Google Scholar]
- Reissner, E.; Stavsky, Y. Bending and Stretching of Certain Types of Heterogeneous Aeolotropic Elastic Plates. J. Appl. Mech.
**1961**, 28, 402–408. [Google Scholar] [CrossRef] - Stein, M. Nonlinear theory for plates and shells including the effects of transverse shearing. AIAA J.
**1986**, 24, 1537–1544. [Google Scholar] [CrossRef] - Mantari, J.L.; Oktem, A.S.; Soares, G.C. Bending and free vibration analysis of isotropic and multilayered plates and shells by using a new accurate higherorder shear deformation theory. Compos. Part B-Eng.
**2012**, 43, 3348–3360. [Google Scholar] [CrossRef] - Karama, M.; Afaq, K.S.; Mistou, S. Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. Int. J. Solids Struct.
**2003**, 40, 1525–1546. [Google Scholar] [CrossRef] - Reddy, J.N. Mechanics of Laminated Composite Plates and Shells: Theory and Analysis; CRC Press LLC: New York, NY, USA, 2004. [Google Scholar]
- Wu, C.P.; Li, H.Y. An RMVT-based third-order shear deformation theory of multilayered functionally graded material plates. Compos. Struct.
**2010**, 92, 2591–2605. [Google Scholar] - Wu, C.P.; Chiu, K.H.; Wang, Y.M. RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D analysis of multilayered composite and FGM plates. Compos. Struct.
**2011**, 93, 923–943. [Google Scholar] [CrossRef] - Carrera, E.; Brischetto, S.; Robaldo, A. Variable kinematic model for the analysis of functionally graded material plates. AIAA J.
**2008**, 46, 194–203. [Google Scholar] [CrossRef]

**Figure 1.**Schematic of continuously graded microstructure with metal-ceramic constituents: (

**a**) smoothly graded microstructure; (

**b**) enlarged view; (

**c**) ceramic-metal functionally graded materials (FGM).

**Figure 2.**Different types of functionally graded materials based on nature of gradients: (

**а**) fraction gradient type; (

**b**) shape gradient type; (

**c**) orientation gradient type; (

**d**) size gradient type.

**Figure 4.**The comparison of homogenous plates (ceramic or metal) and FGM plates: (

**a**) volume fraction of the material; (

**b**) homogenous and FGM plates.

**Figure 5.**Shape function diagrams: (

**a**) shape function from literature; (

**b**) new proposed shape function.

**Figure 6.**Distribution of the normalized values of the normal stresses ${\overline{\sigma}}_{xx}$ and ${\overline{\sigma}}_{yy}$ across the thickness of the plate for different values of the index p: (

**a**) a/h = 10, a/b = 1; (

**b**) a/h = 10, a/b = 1.

**Figure 7.**Distribution of the normalized values of the shear stress ${\overline{\tau}}_{xy}$ across the thickness of the plate for different values of the index p and different shape function: (

**a**) a/h = 10, a/b = 1; (

**b**) a/h = 10, a/b = 1, p = 5.

**Figure 8.**Distribution of the normalized values of the transverse shear stresses ${\overline{\tau}}_{xz}$ and ${\overline{\tau}}_{yz}$ across the thickness of the plate for different values of the index p and different shape functions: (

**a**) a/h = 10, a/b = 1; (

**b**) a/h = 10, a/b = 1, p = 5; (

**c**) a/h = 10, a/b = 1; (

**d**) a/h = 10, a/b = 1, p = 5.

**Figure 9.**Normalized values of the displacement $\overline{w}$ for different a/h and a/b ratios and the values of the index p: (

**a**) a/h = 10; (

**b**) a/b = 1.

**Figure 10.**Distribution of the normalized values of the normal stresses ${\overline{\sigma}}_{xx}$, the shear stress ${\overline{\tau}}_{xy}$ and the transverse shear stresses ${\overline{\tau}}_{xz}$ and ${\overline{\tau}}_{yz}$ across the thickness of the plate on elastic foundation for different values of the coefficients k

_{0}: (

**a**) a/h = 10, a/b = 1, p = 2, k

_{1}= 10; (

**b**) a/h = 10, a/b = 1, p = 2, k

_{1}= 10; (

**c**) a/h = 10, a/b = 1, p = 2, k

_{1}= 10; (

**d**) a/h = 10, a/b = 1, p = 2, k

_{1}= 10.

**Figure 11.**Distribution of the normalized values of the transverse shear stresses ${\overline{\tau}}_{xz}$ and ${\overline{\tau}}_{yz}$ across the thickness of the plate on elastic foundation for different shape functions: (

**a**) a/h = 10, a/b = 1, p = 2, k

_{0}= 100, k

_{1}= 10; (

**b**) a/h = 10, a/b = 1, p = 2, k

_{0}= 100, k

_{1}= 10.

**Figure 12.**Normalized values of the displacement $\overline{w}$ of plate on elastic foundation for different values index p and coefficients k

_{0}and k

_{1}: (

**a**) a/h = 10, a/b = 0.2, k

_{1}= 10; (

**b**) a/h = 10, a/b = 0.2, k

_{0}= 10.

Number of Shape Function (SF) | Names of Authors | Shape Function f(z) |
---|---|---|

SF 1 | Ambartsumain [52] | $\left(z/2\right)\left({h}^{2}/4-{z}^{2}/3\right)$ |

SF 2 | Kaczkowski, Panc and Reissner [53] | $\left(5z/4\right)\left(1-4{z}^{2}/3{h}^{2}\right)$ |

SF 3 | Levy, Stein, Touratier [54] | $\left(h/\pi \right)\mathrm{sin}\left(\pi z/h\right)$ |

SF 4 | Mantari at al. [55] | $\mathrm{sin}\left(\pi z/h\right){e}^{\mathrm{cos}\left(\pi z/h\right)/2}+\left(\pi z/2h\right)$ |

SF 5–6 | Mantari at al. [45] | $\mathrm{tan}\left(mz\right)-zm{\mathrm{sec}}^{2}\left(mh/2\right),\text{}m=\left\{1/5h,\pi /2h\right\}$ |

SF 7 | Karama at al. [56], Aydogdu [44] | $z\mathrm{exp}\left(-2{\left(z/h\right)}^{2}\right),z\mathrm{exp}\left(-2{\left(z/h\right)}^{2}/\mathrm{ln}\alpha \right),\text{}\forall \alpha 0$ |

SF 8 | Mantari at al. [46] | $z\cdot {2.85}^{-2{\left(z/h\right)}^{2}}+0.028z$ |

SF 9 | El Meiche at al. [47] | $\xi \left[\left(h/\pi \right)\mathrm{sin}\left(\pi z/h\right)-z\right],\xi =\left\{1,1/\mathrm{cosh}(\pi /2)-1\right\}$ |

SF 10 | Soldatos [43] | $h\mathrm{sinh}\left(z/h\right)-z\mathrm{cosh}\left(1/2\right)$ |

SF 11 | Akavci and Tanrikulu [49] | $z\mathrm{sec}h\left({z}^{2}/{h}^{2}\right)-z\mathrm{sec}h\left(\pi /4\right)\left[1-\left(\pi /2\right)\mathrm{tanh}\left(\pi /4\right)\right]$ |

SF 12 | Akavci and Tanrikulu [49] | $\left(3\pi /2\right)h\mathrm{tanh}\left(z/h\right)-\left(3\pi /2\right)z\mathrm{sec}{h}^{2}\left(1/2\right)$ |

SF 13 | Mechab at al. [48] | $\frac{z\mathrm{cos}\left(1/2\right)}{-1+\mathrm{cos}\left(1/2\right)}-\frac{h\mathrm{sin}\left(z/h\right)}{-1+\mathrm{cos}\left(1/2\right)}$ |

Material | Material Properties | |
---|---|---|

Elasticity Modulus, E[GPa] | Poisson’s Ratio, ν | |

Aluminum (Al) | ${E}_{m}=70$ | $\nu =0.3$ |

Alumina (Al_{2}O_{3}) | ${E}_{c}=380$ | $\nu =0.3$ |

**Table 3.**Normalized values of displacement and stresses of square plate for different values of the index p and the ratio a/h (a/b = 1). CPT: classical plate theory; TSDT: third-order shear deformation theory.

p | Theory | $\overline{\mathit{w}}$ | ${\overline{\mathit{\sigma}}}_{\mathit{x}\mathit{x}}(\mathit{h}/3)$ | ${\overline{\mathit{\tau}}}_{\mathit{x}\mathit{y}}(-\mathit{h}/3)$ | ${\overline{\mathit{\tau}}}_{\mathit{x}\mathit{z}}(\mathit{h}/6)$ | ||||
---|---|---|---|---|---|---|---|---|---|

a/b = 1 | |||||||||

a/h = 10 | a/h = 5 | a/h = 10 | a/h = 5 | a/h = 10 | a/h = 5 | a/h = 10 | a/h = 5 | ||

1 | Present study | 0.5889 | 0.6687 | 1.4899 | 0.7345 | 0.6111 | 0.3034 | 0.2604 | 0.2599 |

CPT [60] | 0.5623 | ----- | 2.0150 | ----- | ----- | ----- | ----- | ----- | |

Quasi 3D [59] | 0.5876 | ----- | 1.5061 | ----- | 0.6112 | ----- | 0.2511 | ----- | |

TSDT [58] | 0.5890 | ----- | 1.4898 | ----- | 0.6111 | ----- | 0.2599 | ----- | |

SF 1 | 0.5889 | 0.6687 | 1.4898 | 0.7344 | 0.6111 | 0.3034 | 0.2607 | 0.2602 | |

SF 2 | 0.5889 | 0.6687 | 1.4898 | 0.7344 | 0.6111 | 0.3034 | 0.2607 | 0.2602 | |

SF 3 | 0.5889 | 0.6685 | 1.4894 | 0.7336 | 0.6110 | 0.3033 | 0.2621 | 0.2615 | |

SF 4 | 0.5880 | 0.6648 | 1.4888 | 0.7323 | 0.6109 | 0.3030 | 0.2566 | 0.2554 | |

SF 5 | 0.5889 | 0.6687 | 1.4898 | 0.7344 | 0.6111 | 0.3034 | 0.2607 | 0.2601 | |

SF 6 | 0.5888 | 0.6683 | 1.4908 | 0.7363 | 0.6113 | 0.3038 | 0.2551 | 0.2547 | |

SF 7 | 0.5887 | 0.6679 | 1.4891 | 0.7330 | 0.6109 | 0.3031 | 0.2624 | 0.2616 | |

SF 8 | 0.5887 | 0.6679 | 1.4891 | 0.7330 | 0.6109 | 0.3031 | 0.2623 | 0.2615 | |

SF 9 | 0.5887 | 0.6679 | 1.4891 | 0.7330 | 0.6109 | 0.3031 | 0.2623 | 0.2615 | |

SF 10 | 0.5889 | 0.6687 | 1.4898 | 0.7344 | 0.6111 | 0.3034 | 0.2605 | 0.2600 | |

SF 11 | 0.5887 | 0.6679 | 1.4902 | 0.7352 | 0.6112 | 0.3036 | 0.2569 | 0.2566 | |

SF 12 | 0.5889 | 0.6686 | 1.4895 | 0.7338 | 0.6110 | 0.3033 | 0.2617 | 0.2611 | |

SF 13 | 0.5889 | 0.6687 | 1.4898 | 0.7343 | 0.6111 | 0.3034 | 0.2609 | 0.2603 | |

2 | Present study | 0.7572 | 0.8670 | 1.3961 | 0.6838 | 0.5442 | 0.2696 | 0.2732 | 0.2726 |

CPT [60] | ----- | ----- | ----- | ----- | ----- | ----- | ----- | ----- | |

Quasi 3D [59] | 0.7571 | ----- | 1.4133 | ----- | 0.5436 | ----- | 0.2495 | ----- | |

TSDT [58] | 0.7573 | ----- | 1.3960 | ----- | 0.5442 | ----- | 0.2721 | ----- | |

SF 1 | 0.7573 | 0.8671 | 1.3960 | 0.6836 | 0.5442 | 0.2695 | 0.2736 | 0.2730 | |

SF 2 | 0.7573 | 0.8671 | 1.3960 | 0.6836 | 0.5442 | 0.2695 | 0.2736 | 0.2730 | |

SF 3 | 0.7573 | 0.8671 | 1.3954 | 0.6824 | 0.5440 | 0.2693 | 0.2763 | 0.2755 | |

SF 4 | 0.7563 | 0.8629 | 1.3940 | 0.6797 | 0.5437 | 0.2687 | 0.2741 | 0.2726 | |

SF 5 | 0.7572 | 0.8671 | 1.3961 | 0.6836 | 0.5442 | 0.2695 | 0.2735 | 0.2729 | |

SF 6 | 0.7568 | 0.8656 | 1.3975 | 0.6865 | 0.5444 | 0.2701 | 0.2653 | 0.2649 | |

SF 7 | 0.7572 | 0.8667 | 1.3949 | 0.6813 | 0.5439 | 0.2691 | 0.2777 | 0.2767 | |

SF 8 | 0.7572 | 0.8666 | 1.3948 | 0.6812 | 0.5439 | 0.2691 | 0.2777 | 0.2768 | |

SF 9 | 0.7572 | 0.8666 | 1.3948 | 0.6812 | 0.5439 | 0.2691 | 0.2777 | 0.2768 | |

SF 10 | 0.7572 | 0.8670 | 1.3961 | 0.6837 | 0.5442 | 0.2696 | 0.2733 | 0.2727 | |

SF 11 | 0.7567 | 0.8649 | 1.3969 | 0.6854 | 0.5444 | 0.2699 | 0.2667 | 0.2663 | |

SF 12 | 0.7573 | 0.8672 | 1.3956 | 0.6827 | 0.5441 | 0.2694 | 0.2755 | 0.2748 | |

SF 13 | 0.7573 | 0.8671 | 1.3960 | 0.6835 | 0.5442 | 0.2695 | 0.2739 | 0.2733 | |

4 | Present study | 0.8814 | 1.0406 | 1.1795 | 0.5707 | 0.5669 | 0.2799 | 0.2529 | 0.2523 |

CPT [60] | 0.8281 | ----- | 1.6049 | ----- | ----- | ----- | ----- | ----- | |

Quasi 3D [59] | 0.8823 | ----- | 1.1841 | ----- | 0.5671 | ----- | 0.2362 | ----- | |

TSDT [58] | 0.8815 | ----- | 1.1794 | ----- | 0.5669 | ----- | 0.2519 | ----- | |

SF 1 | 0.8814 | 1.0409 | 1.1794 | 0.5704 | 0.5669 | 0.2798 | 0.2537 | 0.2529 | |

SF 2 | 0.8814 | 1.0409 | 1.1794 | 0.5704 | 0.5669 | 0.2798 | 0.2537 | 0.2529 | |

SF 3 | 0.8818 | 1.0423 | 1.1783 | 0.5684 | 0.5667 | 0.2795 | 0.2580 | 0.2571 | |

SF 4 | 0.8815 | 1.0402 | 1.1756 | 0.5630 | 0.5662 | 0.2784 | 0.2623 | 0.2606 | |

SF 5 | 0.8814 | 1.0408 | 1.1794 | 0.5705 | 0.5669 | 0.2799 | 0.2535 | 0.2528 | |

SF 6 | 0.8802 | 1.0360 | 1.1816 | 0.5749 | 0.5673 | 0.2807 | 0.2421 | 0.2417 | |

SF 7 | 0.8820 | 1.0429 | 1.1774 | 0.5666 | 0.5665 | 0.2791 | 0.2612 | 0.2601 | |

SF 8 | 0.8820 | 1.0429 | 1.1773 | 0.5664 | 0.5665 | 0.2791 | 0.2614 | 0.2603 | |

SF 9 | 0.8820 | 1.0429 | 1.1773 | 0.5664 | 0.5665 | 0.2791 | 0.2614 | 0.2603 | |

SF 10 | 0.8814 | 1.0407 | 1.1795 | 0.5706 | 0.5669 | 0.2799 | 0.2532 | 0.2525 | |

SF 11 | 0.8798 | 1.0346 | 1.1811 | 0.5739 | 0.5672 | 0.2805 | 0.2427 | 0.2423 | |

SF 12 | 0.8817 | 1.0420 | 1.1786 | 0.5690 | 0.5668 | 0.2796 | 0.2568 | 0.2559 | |

SF 13 | 0.8815 | 1.0411 | 1.1793 | 0.5702 | 0.5669 | 0.2798 | 0.2541 | 0.2534 | |

8 | Present study | 0.9745 | 1.1828 | 0.9478 | 0.4544 | 0.5858 | 0.2886 | 0.2082 | 0.2076 |

CPT [60] | ----- | ----- | ----- | ----- | ----- | ----- | ----- | ----- | |

Quasi 3D [59] | 0.9739 | ----- | 0.9622 | ----- | 0.5883 | ----- | 0.2261 | ----- | |

TSDT [58] | 0.9747 | ----- | 0.9747 | ----- | 0.5858 | ----- | 0.2087 | ----- | |

SF 1 | 0.9746 | 1.1832 | 0.9476 | 0.4541 | 0.5858 | 0.2886 | 0.2087 | 0.2081 | |

SF 2 | 0.9746 | 1.1832 | 0.9476 | 0.4541 | 0.5858 | 0.2886 | 0.2087 | 0.2081 | |

SF 3 | 0.9749 | 1.1845 | 0.9465 | 0.4520 | 0.5856 | 0.2881 | 0.2120 | 0.2113 | |

SF 4 | 0.9739 | 1.1794 | 0.9435 | 0.4461 | 0.5850 | 0.2871 | 0.2139 | 0.2125 | |

SF 5 | 0.9745 | 1.1831 | 0.9477 | 0.4542 | 0.5858 | 0.2886 | 0.2086 | 0.2080 | |

SF 6 | 0.9730 | 1.1774 | 0.9500 | 0.4589 | 0.5863 | 0.2895 | 0.1995 | 0.1991 | |

SF 7 | 0.9751 | 1.1848 | 0.9455 | 0.4500 | 0.5854 | 0.2877 | 0.2143 | 0.2134 | |

SF 8 | 0.9751 | 1.1848 | 0.9454 | 0.4498 | 0.5854 | 0.2877 | 0.2145 | 0.2135 | |

SF 9 | 0.9751 | 1.1848 | 0.9454 | 0.4498 | 0.5854 | 0.2877 | 0.2145 | 0.2135 | |

SF 10 | 0.9745 | 1.1830 | 0.9477 | 0.4543 | 0.5858 | 0.2886 | 0.2084 | 0.2078 | |

SF 11 | 0.9727 | 1.1763 | 0.9496 | 0.4581 | 0.5861 | 0.2893 | 0.2006 | 0.2003 | |

SF 12 | 0.9749 | 1.1842 | 0.9469 | 0.4526 | 0.5856 | 0.2883 | 0.2111 | 0.2104 | |

SF 13 | 0.9746 | 1.1833 | 0.9475 | 0.4539 | 0.5858 | 0.2885 | 0.2091 | 0.2084 | |

20 | Present study | 1.1377 | 1.3727 | 0.7710 | 0.3721 | 0.6079 | 0.2993 | 0.2011 | 0.2005 |

CPT [60] | ----- | ----- | ----- | ----- | ----- | ----- | ----- | ----- | |

Quasi 3D [59] | ----- | ----- | ----- | ----- | ----- | ----- | ----- | ----- | |

TSDT [58] | ----- | ----- | ----- | ----- | ----- | ----- | ----- | ----- | |

SF 1 | 1.1377 | 1.3727 | 0.7709 | 0.3720 | 0.6078 | 0.2993 | 0.2013 | 0.2008 | |

SF 2 | 1.1377 | 1.3727 | 0.7709 | 0.3720 | 0.6078 | 0.2993 | 0.2013 | 0.2008 | |

SF 3 | 1.1374 | 1.3712 | 0.7702 | 0.3707 | 0.6076 | 0.2989 | 0.2025 | 0.2019 | |

SF 4 | 1.1338 | 1.3561 | 0.7687 | 0.3677 | 0.6073 | 0.2982 | 0.1979 | 0.1966 | |

SF 5 | 1.1377 | 1.3727 | 0.7709 | 0.3720 | 0.6078 | 0.2993 | 0.2013 | 0.2007 | |

SF 6 | 1.1375 | 1.3723 | 0.7723 | 0.3748 | 0.6083 | 0.3002 | 0.1963 | 0.1960 | |

SF 7 | 1.1368 | 1.3686 | 0.7697 | 0.3696 | 0.6075 | 0.2986 | 0.2028 | 0.2019 | |

SF 8 | 1.1367 | 1.3683 | 0.7696 | 0.3695 | 0.6075 | 0.2986 | 0.2027 | 0.2019 | |

SF 9 | 1.1367 | 1.3683 | 0.7696 | 0.3695 | 0.6075 | 0.2986 | 0.2027 | 0.2019 | |

SF 10 | 1.1377 | 1.3727 | 0.7709 | 0.3721 | 0.6079 | 0.2993 | 0.2012 | 0.2006 | |

SF 11 | 1.1375 | 1.3722 | 0.7720 | 0.3741 | 0.6081 | 0.2998 | 0.1983 | 0.1979 | |

SF 12 | 1.1375 | 1.3718 | 0.7704 | 0.3711 | 0.6077 | 0.2990 | 0.2022 | 0.2016 | |

SF 13 | 1.1377 | 1.3726 | 0.7708 | 0.3718 | 0.6078 | 0.2993 | 0.2015 | 0.2009 |

**Table 4.**Normalized values of displacement and stresses of square plate on elastic foundation for p = 5, different values of the k

_{0}и k

_{1}and the ratio a/h (a/b = 1).

p | k_{0} | k_{1} | Theory | $\overline{\mathit{w}}$ | ${\overline{\mathit{\sigma}}}_{\mathit{x}\mathit{x}}$ | ${\overline{\mathit{\tau}}}_{\mathit{x}\mathit{y}}$ | ${\overline{\mathit{\tau}}}_{\mathit{x}\mathit{z}}$ | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

a/b = 1 | |||||||||||

a/h = 10 | a/h = 5 | a/h = 10 | a/h = 5 | a/h = 10 | a/h = 5 | a/h = 10 | a/h = 5 | ||||

5 | 0 | 0 | Present study | 0.9113 | 1.0882 | 4.2441 | 2.2107 | 0.5757 | 0.2840 | 0.1916 | 0.1911 |

SF 1 | 0.9113 | 1.0885 | 4.2447 | 2.2118 | 0.5756 | 0.2839 | 0.1929 | 0.1924 | |||

SF 2 | 0.9113 | 1.0885 | 4.2447 | 2.2118 | 0.5756 | 0.2839 | 0.1929 | 0.1924 | |||

SF 3 | 0.9118 | 1.0902 | 4.2488 | 2.2199 | 0.5754 | 0.2835 | 0.2016 | 0.2009 | |||

SF 4 | 0.9115 | 1.0885 | 4.2612 | 2.2443 | 0.5748 | 0.2824 | 0.2329 | 0.2313 | |||

SF 5 | 0.9113 | 1.0884 | 4.2445 | 2.2116 | 0.5756 | 0.2839 | 0.1927 | 0.1921 | |||

SF 6 | 0.9098 | 1.0826 | 4.2359 | 2.1945 | 0.5761 | 0.2848 | 0.1759 | 0.1756 | |||

SF 7 | 0.9121 | 1.0911 | 4.2527 | 2.2276 | 0.5752 | 0.2831 | 0.2104 | 0.2095 | |||

SF 8 | 0.9121 | 1.0911 | 4.2531 | 2.2284 | 0.5752 | 0.2831 | 0.2113 | 0.2104 | |||

SF 9 | 0.9121 | 1.0911 | 4.2531 | 2.2284 | 0.5752 | 0.2831 | 0.2113 | 0.2104 | |||

SF 10 | 0.9112 | 1.0883 | 4.2443 | 2.2110 | 0.5756 | 0.2839 | 0.1921 | 0.1916 | |||

SF 11 | 0.9094 | 1.0810 | 4.2359 | 2.1945 | 0.5760 | 0.2846 | 0.1668 | 0.1665 | |||

SF 12 | 0.9117 | 1.0898 | 4.2476 | 2.2175 | 0.5755 | 0.2836 | 0.1991 | 0.1985 | |||

SF 13 | 0.9114 | 1.0887 | 4.2450 | 2.2126 | 0.5756 | 0.2839 | 0.1937 | 0.1932 | |||

100 | 0 | Present study | 0.4967 | 0.5450 | 2.3135 | 1.1073 | 0.3138 | 0.1422 | 0.1045 | 0.0957 | |

SF 1 | 0.4967 | 0.5451 | 2.3137 | 1.1076 | 0.3137 | 0.1422 | 0.1051 | 0.0963 | |||

SF 2 | 0.4967 | 0.5451 | 2.3137 | 1.1076 | 0.3137 | 0.1422 | 0.1051 | 0.0963 | |||

SF 3 | 0.4969 | 0.5455 | 2.3154 | 1.1108 | 0.3136 | 0.1418 | 0.1098 | 0.1005 | |||

SF 4 | 0.4968 | 0.5451 | 2.3225 | 1.1239 | 0.3133 | 0.1414 | 0.1269 | 0.1158 | |||

SF 5 | 0.4967 | 0.5451 | 2.3136 | 1.1076 | 0.3137 | 0.1422 | 0.1050 | 0.0962 | |||

SF 6 | 0.4963 | 0.5436 | 2.3107 | 1.1019 | 0.3142 | 0.1430 | 0.0960 | 0.0882 | |||

SF 7 | 0.4969 | 0.5457 | 2.3172 | 1.1142 | 0.3134 | 0.1416 | 0.1146 | 0.1048 | |||

SF 8 | 0.4969 | 0.5457 | 2.3174 | 1.1146 | 0.3134 | 0.1416 | 0.1151 | 0.1052 | |||

SF 9 | 0.4969 | 0.5457 | 2.3174 | 1.1146 | 0.3134 | 0.1416 | 0.1151 | 0.1052 | |||

SF 10 | 0.4967 | 0.5450 | 2.3135 | 1.1074 | 0.3138 | 0.1422 | 0.1047 | 0.0959 | |||

SF 11 | 0.4961 | 0.5432 | 2.3112 | 1.1028 | 0.3143 | 0.1430 | 0.0910 | 0.0837 | |||

SF 12 | 0.4968 | 0.5454 | 2.3148 | 1.1098 | 0.3136 | 0.1419 | 0.1085 | 0.0993 | |||

SF 13 | 0.4967 | 0.5451 | 2.3138 | 1.6370 | 0.3137 | 0.1421 | 0.1056 | 0.0967 | |||

0 | 10 | Present study | 0.3442 | 0.3668 | 1.6032 | 0.7451 | 0.2175 | 0.0957 | 0.0724 | 0.0644 | |

SF 1 | 0.3442 | 0.3667 | 1.6033 | 0.7453 | 0.2174 | 0.0956 | 0.0728 | 0.0648 | |||

SF 2 | 0.3442 | 0.3667 | 1.6033 | 0.7453 | 0.2174 | 0.0956 | 0.0728 | 0.0648 | |||

SF 3 | 0.3443 | 0.3669 | 1.6043 | 0.7472 | 0.2172 | 0.0954 | 0.0761 | 0.0676 | |||

SF 4 | 0.3442 | 0.3667 | 1.6093 | 0.7562 | 0.2171 | 0.0951 | 0.0879 | 0.0779 | |||

SF 5 | 0.3442 | 0.3667 | 1.6033 | 0.7452 | 0.2174 | 0.0956 | 0.0727 | 0.0647 | |||

SF 6 | 0.3440 | 0.3661 | 1.6017 | 0.7421 | 0.2178 | 0.0963 | 0.0665 | 0.0594 | |||

SF 7 | 0.3443 | 0.3670 | 1.6055 | 0.7494 | 0.2171 | 0.0952 | 0.0794 | 0.0704 | |||

SF 8 | 0.3443 | 0.3671 | 1.6057 | 0.7496 | 0.2171 | 0.0952 | 0.0797 | 0.0707 | |||

SF 9 | 0.3443 | 0.3671 | 1.6057 | 0.7496 | 0.2171 | 0.0952 | 0.0797 | 0.0707 | |||

SF 10 | 0.3442 | 0.3667 | 1.6032 | 0.7451 | 0.2174 | 0.0957 | 0.0725 | 0.0645 | |||

SF 11 | 0.3439 | 0.3659 | 1.6022 | 0.7428 | 0.2178 | 0.0963 | 0.0631 | 0.0563 | |||

SF 12 | 0.3442 | 0.3669 | 1.6040 | 0.7466 | 0.2173 | 0.0955 | 0.0752 | 0.0668 | |||

SF 13 | 0.3442 | 0.3668 | 1.6034 | 1.2283 | 0.2174 | 0.0956 | 0.0731 | 0.0650 | |||

100 | 10 | Present study | 0.2617 | 0.2745 | 1.2190 | 0.5578 | 0.1654 | 0.0716 | 0.0550 | 0.0482 | |

SF 1 | 0.2617 | 0.2745 | 1.2190 | 0.5579 | 0.1653 | 0.0716 | 0.0554 | 0.0485 | |||

SF 2 | 0.2617 | 0.2745 | 1.2190 | 0.5579 | 0.1653 | 0.0716 | 0.0554 | 0.0485 | |||

SF 3 | 0.2617 | 0.2746 | 1.2197 | 0.5592 | 0.1652 | 0.0714 | 0.0578 | 0.0506 | |||

SF 4 | 0.2617 | 0.2745 | 1.2236 | 0.5661 | 0.1650 | 0.0712 | 0.0668 | 0.0583 | |||

SF 5 | 0.2617 | 0.2745 | 1.2190 | 0.5661 | 0.1653 | 0.0716 | 0.0553 | 0.0484 | |||

SF 6 | 0.2616 | 0.2741 | 1.2180 | 0.5578 | 0.1656 | 0.0721 | 0.0506 | 0.0444 | |||

SF 7 | 0.2617 | 0.2747 | 1.2206 | 0.5608 | 0.1651 | 0.0712 | 0.0604 | 0.0527 | |||

SF 8 | 0.2617 | 0.2747 | 1.2207 | 0.5610 | 0.1651 | 0.0712 | 0.0606 | 0.0529 | |||

SF 9 | 0.2617 | 0.2747 | 1.2207 | 0.5610 | 0.1651 | 0.0712 | 0.0606 | 0.0529 | |||

SF 10 | 0.2617 | 0.2745 | 1.2190 | 0.5578 | 0.1653 | 0.0716 | 0.0551 | 0.0483 | |||

SF 11 | 0.2615 | 0.2740 | 1.2184 | 0.5564 | 0.1656 | 0.0721 | 0.0479 | 0.0422 | |||

SF 12 | 0.2617 | 0.2746 | 1.2195 | 0.5588 | 0.1652 | 0.0714 | 0.0571 | 0.0500 | |||

SF 13 | 0.2617 | 0.2745 | 1.2191 | 0.9722 | 0.1653 | 0.0716 | 0.0556 | 0.0487 |

**Table 5.**Normalized values of displacement and stresses of square plate on elastic foundation for p = 10, different values of the k

_{0}и k

_{1}and the ratio a/h (a/b = 1).

p | k_{0} | k_{1} | Theory | $\overline{\mathit{w}}$ | ${\overline{\mathit{\sigma}}}_{\mathit{x}\mathit{x}}$ | ${\overline{\mathit{\tau}}}_{\mathit{x}\mathit{y}}$ | ${\overline{\mathit{\tau}}}_{\mathit{x}\mathit{z}}$ | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

a/b =1 | |||||||||||

a/h = 10 | a/h = 5 | a/h = 10 | a/h = 5 | a/h = 10 | a/h = 5 | a/h = 10 | a/h = 5 | ||||

10 | 0 | 0 | Present study | 1.0086 | 1.2273 | 5.0843 | 2.6423 | 0.5896 | 0.2904 | 0.2101 | 0.2095 |

SF 1 | 1.0087 | 1.2275 | 5.0848 | 2.6434 | 0.5895 | 0.2903 | 0.2113 | 0.2107 | |||

SF 2 | 1.0087 | 1.2275 | 5.0848 | 2.6434 | 0.5895 | 0.2903 | 0.2113 | 0.2107 | |||

SF 3 | 1.0089 | 1.2282 | 5.0890 | 2.6515 | 0.5893 | 0.2899 | 0.2198 | 0.2190 | |||

SF 4 | 1.0071 | 1.2201 | 5.1006 | 2.6742 | 0.5888 | 0.2889 | 0.2488 | 0.2472 | |||

SF 5 | 1.0086 | 1.2275 | 5.0847 | 2.6431 | 0.5895 | 0.2903 | 0.2111 | 0.2104 | |||

SF 6 | 1.0074 | 1.2229 | 5.0758 | 2.6255 | 0.5900 | 0.2913 | 0.1944 | 0.1940 | |||

SF 7 | 1.0088 | 1.2277 | 5.0928 | 2.6590 | 0.5891 | 0.2895 | 0.2281 | 0.2272 | |||

SF 8 | 1.0088 | 1.2275 | 5.0931 | 2.6597 | 0.5891 | 0.2895 | 0.2290 | 0.2280 | |||

SF 9 | 1.0088 | 1.2275 | 5.0931 | 2.6597 | 0.5891 | 0.2895 | 0.2290 | 0.2280 | |||

SF 10 | 1.0086 | 1.2274 | 5.0845 | 2.6426 | 0.5896 | 0.2903 | 0.2105 | 0.2099 | |||

SF 11 | 1.0072 | 1.2222 | 5.0762 | 2.6263 | 0.5899 | 0.2910 | 0.1852 | 0.1849 | |||

SF 12 | 1.0088 | 1.2281 | 5.0877 | 2.6491 | 0.5894 | 0.2900 | 0.2174 | 0.2166 | |||

SF 13 | 1.0087 | 1.2276 | 5.0852 | 2.6442 | 0.5895 | 0.2903 | 0.2121 | 0.2115 | |||

100 | 0 | Present study | 0.5243 | 0.5779 | 2.6430 | 1.2440 | 0.3065 | 0.1367 | 0.1092 | 0.0986 | |

SF 1 | 0.5243 | 0.5779 | 2.6432 | 1.2444 | 0.3064 | 0.1366 | 0.1098 | 0.0992 | |||

SF 2 | 0.5243 | 0.5779 | 2.6432 | 1.2444 | 0.3064 | 0.1366 | 0.1098 | 0.0992 | |||

SF 3 | 0.5244 | 0.5780 | 2.6451 | 1.2479 | 0.3063 | 0.1364 | 0.1142 | 0.1030 | |||

SF 4 | 0.5239 | 0.5762 | 2.6534 | 1.2630 | 0.3063 | 0.1364 | 0.1294 | 0.1167 | |||

SF 5 | 0.5243 | 0.5779 | 2.6432 | 1.2443 | 0.3064 | 0.1367 | 0.1097 | 0.0990 | |||

SF 6 | 0.5240 | 0.5768 | 2.6401 | 1.2385 | 0.3069 | 0.1374 | 0.1011 | 0.0915 | |||

SF 7 | 0.5243 | 0.5779 | 2.6471 | 1.2517 | 0.3062 | 0.1363 | 0.1186 | 0.1069 | |||

SF 8 | 0.5243 | 0.5779 | 2.6474 | 1.2521 | 0.3062 | 0.1363 | 0.1190 | 0.1073 | |||

SF 9 | 0.5243 | 0.5779 | 2.6474 | 1.2521 | 0.3062 | 0.1363 | 0.1190 | 0.1073 | |||

SF 10 | 0.5243 | 0.5778 | 2.6431 | 1.2441 | 0.3064 | 0.1367 | 0.1094 | 0.0988 | |||

SF 11 | 0.5239 | 0.5767 | 2.6405 | 1.2392 | 0.3068 | 0.1373 | 0.0963 | 0.0872 | |||

SF 12 | 0.5243 | 0.5780 | 2.6445 | 1.2468 | 0.3063 | 0.1365 | 0.1130 | 0.1019 | |||

SF 13 | 0.5243 | 0.5779 | 2.6434 | 1.2447 | 0.3064 | 0.1366 | 0.1102 | 0.0995 | |||

0 | 10 | Present study | 0.3573 | 0.3813 | 1.8008 | 0.8209 | 0.2088 | 0.0902 | 0.0744 | 0.0651 | |

SF 1 | 0.3572 | 0.3813 | 1.8010 | 0.8212 | 0.2088 | 0.0902 | 0.0748 | 0.0654 | |||

SF 2 | 0.3572 | 0.3813 | 1.8010 | 0.8212 | 0.2088 | 0.0902 | 0.0748 | 0.0654 | |||

SF 3 | 0.3572 | 0.3814 | 1.8022 | 0.8234 | 0.2087 | 0.0900 | 0.0778 | 0.0680 | |||

SF 4 | 0.3570 | 0.3806 | 1.8084 | 0.8342 | 0.2087 | 0.0901 | 0.0882 | 0.0771 | |||

SF 5 | 0.3572 | 0.3813 | 1.8009 | 0.8211 | 0.2088 | 0.0902 | 0.0747 | 0.0653 | |||

SF 6 | 0.3571 | 0.3809 | 1.7992 | 0.8177 | 0.2091 | 0.0907 | 0.0689 | 0.0604 | |||

SF 7 | 0.3572 | 0.3813 | 1.8036 | 0.8259 | 0.2086 | 0.0899 | 0.0808 | 0.0705 | |||

SF 8 | 0.3572 | 0.3813 | 1.8038 | 0.8262 | 0.2086 | 0.0899 | 0.0811 | 0.0708 | |||

SF 9 | 0.3572 | 0.3813 | 1.8038 | 0.8262 | 0.2086 | 0.0899 | 0.0811 | 0.0708 | |||

SF 10 | 0.3572 | 0.3813 | 1.8009 | 0.8210 | 0.2088 | 0.0902 | 0.0745 | 0.0652 | |||

SF 11 | 0.3570 | 0.3808 | 1.7995 | 0.8183 | 0.2091 | 0.0906 | 0.0656 | 0.0576 | |||

SF 12 | 0.3572 | 0.3814 | 1.8018 | 0.8227 | 0.2087 | 0.0900 | 0.0770 | 0.0672 | |||

SF 13 | 0.3572 | 0.3813 | 1.8011 | 0.9376 | 0.2088 | 0.0901 | 0.0751 | 0.0657 | |||

100 | 10 | Present study | 0.2692 | 0.2826 | 1.3569 | 0.6084 | 0.1574 | 0.0669 | 0.0561 | 0.0482 | |

SF 1 | 0.2691 | 0.2826 | 1.3570 | 0.6086 | 0.1573 | 0.0668 | 0.0564 | 0.0485 | |||

SF 2 | 0.2691 | 0.2826 | 1.3570 | 0.6086 | 0.1573 | 0.0668 | 0.0564 | 0.0485 | |||

SF 3 | 0.2692 | 0.2826 | 1.3579 | 0.6102 | 0.1572 | 0.0667 | 0.0586 | 0.0504 | |||

SF 4 | 0.2690 | 0.2822 | 1.3628 | 0.6186 | 0.1573 | 0.0668 | 0.0664 | 0.0571 | |||

SF 5 | 0.2691 | 0.2826 | 1.3570 | 0.6086 | 0.1573 | 0.0668 | 0.0563 | 0.0484 | |||

SF 6 | 0.2691 | 0.2823 | 1.3558 | 0.6062 | 0.1576 | 0.0672 | 0.0519 | 0.0448 | |||

SF 7 | 0.2692 | 0.2826 | 1.3590 | 0.6121 | 0.1572 | 0.0666 | 0.0608 | 0.0523 | |||

SF 8 | 0.2692 | 0.2826 | 1.3591 | 0.6123 | 0.1572 | 0.0666 | 0.0611 | 0.0525 | |||

SF 9 | 0.2692 | 0.2826 | 1.3591 | 0.6123 | 0.1572 | 0.0666 | 0.0611 | 0.0525 | |||

SF 10 | 0.2691 | 0.2826 | 1.3569 | 0.6085 | 0.1573 | 0.0668 | 0.0562 | 0.0483 | |||

SF 11 | 0.2690 | 0.2823 | 1.3561 | 0.6067 | 0.1575 | 0.0672 | 0.0494 | 0.0427 | |||

SF 12 | 0.2692 | 0.2826 | 1.3576 | 0.6097 | 0.1572 | 0.0667 | 0.0580 | 0.0498 | |||

SF 13 | 0.2692 | 0.2826 | 1.3571 | 0.6087 | 0.1573 | 0.0668 | 0.0566 | 0.0486 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Čukanović, D.; Radaković, A.; Bogdanović, G.; Milanović, M.; Redžović, H.; Dragović, D.
New Shape Function for the Bending Analysis of Functionally Graded Plate. *Materials* **2018**, *11*, 2381.
https://doi.org/10.3390/ma11122381

**AMA Style**

Čukanović D, Radaković A, Bogdanović G, Milanović M, Redžović H, Dragović D.
New Shape Function for the Bending Analysis of Functionally Graded Plate. *Materials*. 2018; 11(12):2381.
https://doi.org/10.3390/ma11122381

**Chicago/Turabian Style**

Čukanović, Dragan, Aleksandar Radaković, Gordana Bogdanović, Milivoje Milanović, Halit Redžović, and Danilo Dragović.
2018. "New Shape Function for the Bending Analysis of Functionally Graded Plate" *Materials* 11, no. 12: 2381.
https://doi.org/10.3390/ma11122381