9.2 Literature Review
335
In addition, the investigations are focused in determination of the free frequencies
corresponding to different modes of the beam vibrations [2–8]. It should be emphasized that analysis of free vibrations of beams with non-homogenous transverse cross
sections and with tapered material properties along the beam longitudinal direction is
more complex and difficult to study than those with homogenous cross sections and
homogenous material [9, 10]. This is due to the occurrence of variable coefficients
in the governing equations, which requires specific and sometimes novel approaches
for modelling the system dynamics and for solving the derived differential equations.
Only in rare cases, analytical solutions can be obtained for the case of tapered beams
in their axial direction [11, 12]. Therefore, majority of case studies were carried out
with the help of the numerical methods. Hemmatnezhad et al. [13] studied nonlinear
behaviour of functionally graded beams with an account of Kármán nonlinearity and
with the help of finite element method (FEM). Kien [14] employed FEM to analyze
tapered cantilever beams made of FGM. Alshorbagy et al. [15] investigated free
vibrations of FGM beams using FEM. Shaba et al. [10, 16, 17] analyzed stability
and vibration of FGM tapered Timoshenko and Euler-Bernoulli beams using a few
numerical approaches. Ke et al. [18] investigated free vibrations of geometrical FGM
beams using Galerkin procedure in its first-order approximation.
In order to study a tapered beam made from homogenous/non-homogenous material various mathematical models based on either Euler-Bernoulli [10, 19–24] or Timoshenko [17, 25–28] theories were constructed. Free vibrations as well as dynamic
characteristics of a beam made from FGM and with tapered and/or non-uniform cross
section were analyzed in references [29–31].
On the other hand, in order to investigate FGM mechanical structural systems
with an account of the size-dependent behaviour, the following novel theories were
developed: modified couple stress theory [32–38], nonlocal polar theory of elasticity [39], strain gradient theory [40] and the general theory of curved deformable
interfaces [41].
Arbind and Reddy [42] carried out nonlinear study of the functionally graded
microstructure-dependent beams made from non-homogenous material properties
along the beams thickness and with the von Kármán non-linearity. The modified
couple stress theory was used to follow the size-dependent beam behavior. The
derived governing PDEs are based on the Euler-Bernoulli and Timoshenko theories.
The static beam deflections versus the length size-dependent parameter, the character of change of beam thickness properties, the shear deformations and geometric
nonlinearity were investigated. Arbind et al. [43] employed Hamilton principle and
the modified couple stress-based third-order theory to study nonlinear behaviour of
FGM beams.
More recently the beam-type structures are strengthened by composite materials tailored by the methods of topological optimization. Topological optimization
of composite structures is widely used while fabricating materials with improved
physical characteristics. In order to average the complex dynamical phenomena and
make it computationally feasible, the complex microstructure behaviour of an elastic
medium undergoes the so-called asymptotic homogenization [44–46]. In addition, in
the case of design of microstructure materials other methods are employed including
335
In addition, the investigations are focused in determination of the free frequencies
corresponding to different modes of the beam vibrations [2–8]. It should be emphasized that analysis of free vibrations of beams with non-homogenous transverse cross
sections and with tapered material properties along the beam longitudinal direction is
more complex and difficult to study than those with homogenous cross sections and
homogenous material [9, 10]. This is due to the occurrence of variable coefficients
in the governing equations, which requires specific and sometimes novel approaches
for modelling the system dynamics and for solving the derived differential equations.
Only in rare cases, analytical solutions can be obtained for the case of tapered beams
in their axial direction [11, 12]. Therefore, majority of case studies were carried out
with the help of the numerical methods. Hemmatnezhad et al. [13] studied nonlinear
behaviour of functionally graded beams with an account of Kármán nonlinearity and
with the help of finite element method (FEM). Kien [14] employed FEM to analyze
tapered cantilever beams made of FGM. Alshorbagy et al. [15] investigated free
vibrations of FGM beams using FEM. Shaba et al. [10, 16, 17] analyzed stability
and vibration of FGM tapered Timoshenko and Euler-Bernoulli beams using a few
numerical approaches. Ke et al. [18] investigated free vibrations of geometrical FGM
beams using Galerkin procedure in its first-order approximation.
In order to study a tapered beam made from homogenous/non-homogenous material various mathematical models based on either Euler-Bernoulli [10, 19–24] or Timoshenko [17, 25–28] theories were constructed. Free vibrations as well as dynamic
characteristics of a beam made from FGM and with tapered and/or non-uniform cross
section were analyzed in references [29–31].
On the other hand, in order to investigate FGM mechanical structural systems
with an account of the size-dependent behaviour, the following novel theories were
developed: modified couple stress theory [32–38], nonlocal polar theory of elasticity [39], strain gradient theory [40] and the general theory of curved deformable
interfaces [41].
Arbind and Reddy [42] carried out nonlinear study of the functionally graded
microstructure-dependent beams made from non-homogenous material properties
along the beams thickness and with the von Kármán non-linearity. The modified
couple stress theory was used to follow the size-dependent beam behavior. The
derived governing PDEs are based on the Euler-Bernoulli and Timoshenko theories.
The static beam deflections versus the length size-dependent parameter, the character of change of beam thickness properties, the shear deformations and geometric
nonlinearity were investigated. Arbind et al. [43] employed Hamilton principle and
the modified couple stress-based third-order theory to study nonlinear behaviour of
FGM beams.
More recently the beam-type structures are strengthened by composite materials tailored by the methods of topological optimization. Topological optimization
of composite structures is widely used while fabricating materials with improved
physical characteristics. In order to average the complex dynamical phenomena and
make it computationally feasible, the complex microstructure behaviour of an elastic
medium undergoes the so-called asymptotic homogenization [44–46]. In addition, in
the case of design of microstructure materials other methods are employed including