Static and Dynamic Analysis of Axially Functionally Graded Beams

1.3 LITERATURE REVIEW

1.3.1 Static and Dynamic Analysis of Axially Functionally Graded Beams

Elishakoff and Candan [103] presented the first closed-form free vibration solution for axially graded Euler-Bernoulli beam using inverse solution techniques and the solution is based on beam theory. By employing Fredholm integral equations, Huang and Li [104] developed a free vibration solution for axially graded non-uniform Euler-Bernoulli beams subjected to arbitrary support condi- tions (simply-supported, clamped, and free ends). Giunta et al. [105] presented linear static analysis of simply-supported beams having axial or bi-directional material gradation based on refined the- ories. Wang and Wang [106] presented exact natural frequency analysis for an Euler-Bernoulli non-uniform beam subjected to arbitrary boundary conditions and having varying flexural rigidity and density along the axis of beam. Sarkar and Ganguli [107] presented an analytical model for free vibration analysis of non-uniform axially graded Euler-Bernoulli free-free beam and further extended their solution to axially graded Timoshenko beams [108]. Li et al. [109] presented exact

Table 1.1: Some pioneered researcher and their contribution to laminated and FGM structural research

Research Group Contribution

Noor and Burden Displacement-based theories for multilayered plates [17–19]

Saravanos and Heyliger Static and free vibration analysis of laminated elastic and piezo- electric plates [20–26]

J. N. Reddy Develop higher order theories for linear and nonlinear analysis of multilayered composite and FGM plates and shells [27–32]

Rakesh Kapania Mechanics multilayered composite plates and shells using a layer- wise theory and nonlinear finite element [33–36]

R . C. Batra Elasticity based analytical solution for static and dynamic analysis through-thickness FGM rectangular plates [37–40]

Tarun Kant Analytical solutions for analysis of composite and FGM, plates and shells based on a higher-order refined theory [41–45].

Dumir et al. Exact piezoelectric solution for piezoelectric composite beams, plates and shells [46–50]

Chih-Ping Wu Three dimensional static and dynamic analysis of multilayered and functionally graded piezoelectric plates and shells [51–54]

Erasmo Carrera Carrera unified formulation (CUF) for static, dynamic and buck- ling analysis of composite plates and shells [55–58]

Weiqiu Chen Exact and Semi-analytical elasticity solution for composite and FG structures (Beam, plate and, shells) [59–67]

A. J. M. Ferreira Static, free vibration and buckling analysis of composite and FGM plates using finite element and Meshless methods [68–71]

Ranjan Ganguli Closed-form one-dimensional solutions for axially functionally graded beams and aeroelastic analysis of rotating blades [72–77].

Sridhar Idapalapati Failure analysis of composite lap and scarf joints [78–81].

Yogesh Desai Mixed theory-based analytical and FE solution for analysis of lam- inated beams and plates [82–85].

R. Lal Vibrations and buckling analysis of non-homogenous circular and rectangular plates using classical plate theory [86–90]

B. P. Patel Nonlinear analysis of composite/sandwich laminates using higher- order theories [91–94]

Kapuria et al. 3D analytical solutions for accurate prediction of interlaminar stresses in the composite and piezolaminated plate [95–98]

Amirtham Rajagopal Nonlocal nonlinear analysis of functionally graded beams and plates using higher-order theories [99–102]

LITERATURE REVIEW

natural frequency analysis of simply-supported Euler-Bernoulli beam by considering exponential variation of density and stiffness along the axis of the beam. Tang and Wu [110] developed exact free vibration solutions for axially graded Timoshenko beams subjected to arbitrary boundary con- ditions. The density and stiffness of the beam are assumed to vary exponentially along the axis of the beam. Nguyen et al. [111] presented analytical solution for Euler-Bernoulli functionally graded beams (axially as well as through-the thickness) having tapered cross-section under static loading considering power-law variation of elastic modulus. Yuan et al. [67] presented exact solutions for the free vibration of arbitrarily supported Timoshenko beams having variable cross-section and material gradation along the axis of the beam. Rezaiee-Pajand and Hozhabrossadati [112] obtained closed- form free vibration solution for double-axially functionally graded Euler-Bernoulli beams subjected to arbitrary boundary conditions. Wang et al. [113] proposed an analytical solution for natural frequency analysis of two-directional functionally graded Euler-Bernoulli beams under clamped-free and hinged-hinged end supports. By using power series method, Huang and Rong [114] developed a free vibration solution for nonuniform axially graded Euler-Bernoulli beam by considering arbitrary gradation of material properties along the axis of beam. Using asymptotic development method (ADM) in conjunction with Euler-Bernoulli beam theory, Cao et al. [115] obtained an analytical solution for free vibration analysis of uniform axially functionally graded (AFG) beams subjected to arbitrary boundary conditions. Very recently, Cao et al. [116] extended this solution to free vibration analysis of non-uniform AFG beams under arbitrary boundary condition.

Static and dynamic analysis of AFG beams is much more challenging than the composite ma- terial beams because the material properties of the beam are function of a spatial coordinate (x) along length of the beam. Therefore, many researchers have developed the numerical solution for dynamic analysis of axially functionally graded beams, based on Euler-Bernoulli beam theory [117–

129] and Timoshenko beam theory [130–138] and other one-dimensional beam theories [139–144].

Very recently, Chen et al. [145] presented an iso-geometric three-dimensional numerical solution for vibration analysis of AFG beams with variable thickness using the non-uniform rational B- splines(NURBS) basis functions. More extensive literature related to this category can be found in recent review articles presented by Sayyad and Ghugal [146, 147], and Zhang et al. [148].

The present review clearly indicates that most of the analytical and numerical solutions are based on one-dimensional beam theories. But, a two-dimensional elasticity solution is always needed for accurate prediction of static/dynamic behavior and edge effects of composite or functionally graded beams.

1.3.2 Analysis of piezoelectric beams and AFG beams integrated with piezoelectric