A two variable refined plate theory for the bending analysis of functionally graded plates (original) (raw)
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Bending response of functionally graded plates by using a new higher order shear deformation theory
Composite Structures
This paper presents an analytical solution to the static analysis of functionally graded plates, using a recently developed higher order shear deformation theory (HSDT) and provides detailed comparisons with other HSDT's available in the literature. These theories account for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surfaces, thus a shear correction factor is not required. The mechanical properties of the plates are assumed to vary in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded (FG) plate and boundary conditions are derived by employing the principle of virtual work. Navier-type analytical solution is obtained for FG plates subjected to transverse bi-sinusoidal and distributed loads for simply supported boundary conditions. Results are provided for thick to thin FG plates and for different volume fraction distributions. The accuracy of the present code is verified by comparing it with known results in the literature.
Generalized shear deformation theory for bending analysis of functionally graded plates
Applied Mathematical Modelling, 2006
In this study, the static response is presented for a simply supported functionally graded rectangular plate subjected to a transverse uniform load. The generalized shear deformation theory obtained by the author in other recent papers is used. This theory is simplified by enforcing traction-free boundary conditions at the plate faces. No transversal shear correction factors are needed because a correct representation of the transversal shearing strain is given. Material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The equilibrium equations of a functionally graded plate are given based on a generalized shear deformation plate theory. The numerical illustrations concern bending response of functionally graded rectangular plates with two constituent materials. The influences played by transversal shear deformation, plate aspect ratio, side-to-thickness ratio, and volume fraction distributions are studied. The results are verified with the known results in the literature.
Static Bending Behavior of Functionally Graded Plates Subjected to Mechanical Loading
2014
This paper presents analytical formulations and solutions for the bending behavior of simply supported functionally graded plates (FGPs) using Higher Order Shear Deformation Theory (HSDT) without enforcing zero transverse shear stresses on the top and bottom surfaces of the plate. It does not require shear correction factors. Material properties of the plate are assumed to vary in the thickness direction according to a power law distribution interns of the volume fractions of the constituents. The governing equations of motion and boundary conditions are derived using the principle of virtual work. Solutions are obtained for FGPs in closed-form using Navier's technique. The results of deflections and stresses are presented for simply supported boundary conditions. The present numerical results are compared with the available solutions in the literature for deflections and stresses, from which it can be concluded that the proposed theory results are very close agreement to the published ones. After validating the present theory results for FGM plates, the effect of side-to-thickness ratio, aspect ratio, modulus ratio, the volume fraction exponent,and through-the-thickness on the deflections and stresses are studied. The shear deformation effect and inhomogeneities played a greater role in estimating the deflctions and stress distribution in the functionally graded material plates.
Navier’s Approach for Bending Analysis of Functionally Graded Square Plates
Materials, methods & technologies, 2016
In last decades functionally graded materials become very popular for lots of industries such as automotive, naval, railroad, aerospace, etc. There are many papers in literature for bending, stability and vibration analysis of functionally graded plates. In this paper, bending of simply supported functionally graded square plates are studied. For functionally graded plates many shear deformation theory is offered. Various shape functions for defining displacement fields are used in these theories. In this study, several shape functions are discussed for deflection and stress distribution of functionally graded plate with sinusoidal loading. The exponential gradient form is assumed for change of material properties through thickness direction. Refined plate theory with different shape functions is used. Governing equations are derived from the principle of virtual displacements. The solution is obtained by Navier’s double trigonometric series approach. Numerical results of deflection...
A displacement-based, higher order shear and normal deformations theory (HOSNT) is used to analyse the thick functionally graded (FG) plates in which mechanical properties are changing smoothly with the plate's thickness coordinate. A more realistic parabolic distribution of transverse shear strains through thickness of plate is ensured in the presented formulation. The influence of transverse normal strain on bending responses of FG plates is examined in this study. Functionally graded materials (FGMs), although heterogeneous, are idealized as continua with their mechanical properties changing smoothly with respect to spatial coordinates. The material properties of FG plates are assumed to be varying through thickness of plate in a continuous manner. Poisson's ratio of FG plates is assumed to be constant, but their Young's modulii are considered to vary continuously in thickness direction according to the volume fraction of its constituents which is mathematically modelled as an exponential function. The governing equations of equilibrium for static analysis of FG plates are obtained using principle of minimisation of potential energy (PMPE) employing HOSNT. Navier solution method is used to solve the governing differential equations of equilibrium. It is thought that the presented results would be a reference for other researchers to compare their results.
A Theoretical Analysis for Static and Dynamic Behavior of Functionally Graded Plates
2012
Theoretical formulation, Navier’s solutions of rectangular plates based on a new higher order shear deformation model are presented for the static and dynamic analysis of functionally graded plates (FGPs). This theory enforces traction free boundary conditions at plate surfaces. Shear correction factors are not required because a correct representation of transverse shearing strain is given. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The mechanical properties of the plate are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. Numerical illustrations concern flexural behavior of FG plates with Metal–Ceramic composition. Parametric studies are performed for varying ceramic volume fraction, volume fraction profiles, aspect ratios and length to thickness ratios. Results are verified with available ...
Electronic Journal of Vocational Colleges, 2018
In this research, an efficient shear deformation plate theory for a functionally graded plate has been investigated by the use of the new four variable refined plate theory. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory account for higher-order variation of transverse shear strain through the depth of the plate and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. Based on the present higher-order shear deformation plate theory, the equations of the motion are derived from Hamilton's principal. The plate faces are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity, Poisson's ratio of the faces, and thermal expansion coefficients are assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. The validity of the present theory is investigated by comparing some of the present results with those of the classical, the first-order and the other higher-order theories. The influences played by the transverse shear deformation, aspect ratio, side-to-thickness ratio, and volume fraction distribution are studied. Numerical results for deflections and stresses of functionally graded plate are investigated.
An efficient and simple refined theory for bending and vibration of functionally graded plates
Composite Structures, 2009
A two-dimensional theory of functionally graded plates is presented using a mixed variational approach. The theory accounts for a displacements field in which the inplane displacements vary linearly through the plate thickness, while the out-of-plane displacement is a second-degree function of thickness coordinate. The advantages of the present theory are that it contains both the transverse normal strain and stress in complete consistence with the boundary conditions at the top and bottom surfaces of the plates without loss of its simplicity. Therefore, the rationale for the shear correction factor used in such theories is obviated. The bending and free vibration problems of isotropic plates with material properties varying in the thickness direction are solved. Numerical results for frequencies are presented for two-phase graded material with a power-law through the plate thickness variation of the volume fractions of the constituents based on Mori-Tanaka scheme. In addition, numerical results of transverse deflections are obtained for FG simply supported isotropic plates with Young's modulus varying exponentially through the thickness and constant Poisson's ratio. The validity of the present theory is investigated by comparing some of the present results with their counterparts obtained due to three-dimensional approaches by Qian et al. and by Kashtalyan. The influence of the transverse normal strain on the bending and vibration of the FG plates is illustrated.
Analysis of functionally graded plates using higher order shear deformation theory
This work addresses a static analysis of functionally graded material (FGM) plates using higher order shear deformation theory. In the theory the transverse shear stresses are represented as quadratic through the thickness and hence it requires no shear correction factor. The material property gradient is assumed to vary in the thickness direction. Mori and Tanaka theory (1973) [1] is used to represent the material property of FGM plate at any point. The thermal gradient across the plate thickness is represented accurately by utilizing the thermal properties of the constituent materials. Results have been obtained by employing a C° continuous isoparametric Lagrangian finite element with seven degrees of freedom for each node. The convergence and comparison studies are presented and effects of the different material composition and the plate geometry (side-thickness, side–side) on deflection and temperature are investigated. Effect of skew angle on deflection and axial stress of the plate is also studied. Effects of material constant n on deflection and the temperature distribution are also discussed in detail.
Static analysis of orthotropic functionally graded (FG) elastic, rectangular, and simply supported (diaphragm) plates under transverse loads is presented based on a higher order shear and normal deformation theory (HOSNT). Although functionally graded materials (FGMs) are highly heterogeneous in nature, they are generally idealized as continua with mechanical properties changing smoothly with respect to the spatial coordinates. The material properties of FG plates are assumed here to be varying through the thickness of the plate in a continuous manner. The Poisson's ratios of the FG plates are assumed to be constant, but their Young's moduli vary continuously in the thickness direction according to the volume fraction of constituents, which are mathematically modeled as an exponential function. The governing equations of equilibrium for the FG plates are derived on the basis of a HOSNT assuming varying material properties. Numerical solutions are obtained by the use of the Navier solution method. Several examples of isotropic, orthotropic, and FG plates are presented. The accuracy of the numerical solutions has been compared with the solutions obtained by other models and the exact three-dimensional (3D) elasticity solutions.