Chapter Nine - Vibration of FG beams

Abstract

The related studies on vibration of isotropic and different composite beams have been setting rigorous trends among researchers throughout the world. As such, the very first objective here is to outline a few investigation performed by various researchers on the beam vibration problem. The free vibration behavior of a simply supported functionally graded (FG) beam has been studied in Aydogdu and Taskin (2007) by using Euler–Bernoulli beam theory, parabolic shear deformation theory and exponential shear deformation theory. A new beam theory is considered in Sina et al. (2009), different from traditional first-order shear deformation beam theory, to analyze the free vibration of FG beams with an analytical approach. In Şimşek (2010a) the vibration response is examined of a simply supported FG beam to a moving mass by using Euler–Bernoulli, Timoshenko and the third-order shear deformation beam theories. Using different higher-order shear deformation beam theories, in Şimşek (2010b) the fundamental frequencies of FG beams subjected to different boundary conditions are studied. In Mahi et al. (2010), exact solutions to study the free vibration of FG beams based on a unified higher-order shear deformation theory are presented, in which material properties are temperature-dependent. An improved third-order shear deformation theory is introduced in Wattanasakulpong et al. (2011) to study thermal buckling and elastic vibration of FG beams. In Alshorbagy et al. (2011a), the Finite Element Method (FEM) is used to study the free vibration characteristics of an FG beam. In Shahba et al. (2011), free vibration and stability analysis of axially FG Timoshenko tapered beams are investigated using classical and nonclassical boundary conditions through the finite element approach. Using an analytical method, in Thai and Vo (2012) the bending and free vibration of FG beams is developed using various higher-order shear deformation beam theories. Free vibration and stability of axially FG tapered Euler–Bernoulli beams are investigated in Shahba and Rajasekaran (2012) using FEM. Size-dependent linear free flexural vibration behavior of FG nanoplates is investigated in Natarajan et al. (2012) using the isogeometric-based FEM. In Şimşek (2012), the free longitudinal vibration of axially FG tapered nanorods is found with the nonlocal elasticity theory. Free vibration analysis of size-dependent FG nanobeams is studied in Eltaher et al. (2012) using FEM. Static and buckling behaviors of nonlocal FG Timoshenko nanobeams is investigated in Eltaher et al. (2014) based on FEM. In Vo et al. (2014a), static and vibration analysis of FG beams is presented using refined shear deformation theory by using finite element formulation. Static and free vibration of axially loaded rectangular FG beams is developed in Nguyen et al. (2013) based on the first-order shear deformation beam theory. The plane stress problem of an orthotropic FG beam with arbitrary graded material properties along the thickness direction has been investigated recently by the displacement function approach in Nie et al. (2013). In Huang et al. (2013), the vibration behavior of axially FG Timoshenko beams is investigated with nonuniform cross-section by introducing an auxiliary function in the coupled governing equations. Also, the study of dynamic characteristics of FG beams based on different shear deformation theories is discussed in Kahrobaiyan et al. (2012); Lei et al. (2013); Rahmani and Pedram (2014); Kien (2014); Komijani et al. (2014); Sharma et al. (2014) and the articles provided therein.