48
2 Size-Dependent Theories of Beams, Plates and Shells
ence of the size effects of the FG cantilevers on their deflection and rotations as well
as on the eigenfrequency of simply supported FG. However, geometric nonlinearity
was not taken into account. Ke and Wang [305] employed the TBT model to study
free vibrations, static bending and dynamic stability of the FG microbeams under
different boundary conditions and using the DQ method. Stability and free vibrations of the FG microbeams in the problem of thermoelasticity were investigated
by Nateghi and Salamat-talab [306] with the help of modified couple stress theory
and the TBT model. The DQ method was implemented for estimation of the critical
load and eigenfrequencies of the FG microbeams with various boundary conditions.
Numerical results showed that the influence of the temperature is of more importance
for higher values of the ratio of the beam thickness to the scale length parameter.
Simsek et al. [307] chose the TBT model for investigation of the size effect on deflection of the simply supported FG microbeams subjected to discrete and/or continuous
loads.
Chen et al. [308], Chen and Li [309], Roque et al. [310] and Mohammad-Abadi
and Daneshmehr [311] proposed suitable models for static bending [308, 310], free
vibrations [309] and buckling [311] of the laminated composite microbeams. Thai
et al. [312] considered application of the TBT model for static bending, stability and
free vibrations of the sandwich FG microbeams. Most recently, Krysko et al. [313]
developed the TBT model for static bending and free vibrations of the three-layer
microbeams based on the Grigoluk-Chulkov theory.
One of the earlier models of the FSDT modified theory was developed by Ma et
al. [314] and Ke et al. [315] for isotropic microplates. It should be mentioned that the
FSDT model [314] takes into account both longitudinal and bending deformations,
whereas Ke et al. [315] considered only the bending deformation in their model.
Besides, Ma et al. [314] obtained solutions in the closed form for the problem of
bending and vibrations of the simply supported plates, while Ke et al. [315] obtained
the numerical solutions for the eigenfrequencies of plates with clamped sides with
the help of the Ritz method. Roque et al. [316] presented the numerical solutions of
the FSDT model of modified couple stress theory for analysis of the static bending
of the isotropic microplates using the meshless method of collocation with radial
basic functions. Zhou and Gao [317] worked out the modified couple stress theory
of the FSDT model to carry out the axially symmetric analysis of bending of the
isotropic circular microplates. Alinaghizadeh et al. [318] developed the modified
couple stress theory of the FSDT model for the static analysis of bending of the FG
annular microplates sector. In order to solve the bending problem of the microplates,
the DQ method was used for the different boundary conditions. He et al. [319]
extended the FSDT model to analyse static bending of the laminated composites
microplates, whereas Simsek and Aydin [320] extended the FSDT model to solve
the static bending and vibrations of the externally loaded FG microplates under action
of moving load.
Reddy and Berry [321] generalized the axially symmetric model of FSDT in
order to explain the influence of the geometric nonlinearity, temperature and nonhomogeneous behaviour of the FG materials and the axially symmetric analysis of
nonlinear bending of the circle microplates. Ke et al. [322, 323] also developed
2 Size-Dependent Theories of Beams, Plates and Shells
ence of the size effects of the FG cantilevers on their deflection and rotations as well
as on the eigenfrequency of simply supported FG. However, geometric nonlinearity
was not taken into account. Ke and Wang [305] employed the TBT model to study
free vibrations, static bending and dynamic stability of the FG microbeams under
different boundary conditions and using the DQ method. Stability and free vibrations of the FG microbeams in the problem of thermoelasticity were investigated
by Nateghi and Salamat-talab [306] with the help of modified couple stress theory
and the TBT model. The DQ method was implemented for estimation of the critical
load and eigenfrequencies of the FG microbeams with various boundary conditions.
Numerical results showed that the influence of the temperature is of more importance
for higher values of the ratio of the beam thickness to the scale length parameter.
Simsek et al. [307] chose the TBT model for investigation of the size effect on deflection of the simply supported FG microbeams subjected to discrete and/or continuous
loads.
Chen et al. [308], Chen and Li [309], Roque et al. [310] and Mohammad-Abadi
and Daneshmehr [311] proposed suitable models for static bending [308, 310], free
vibrations [309] and buckling [311] of the laminated composite microbeams. Thai
et al. [312] considered application of the TBT model for static bending, stability and
free vibrations of the sandwich FG microbeams. Most recently, Krysko et al. [313]
developed the TBT model for static bending and free vibrations of the three-layer
microbeams based on the Grigoluk-Chulkov theory.
One of the earlier models of the FSDT modified theory was developed by Ma et
al. [314] and Ke et al. [315] for isotropic microplates. It should be mentioned that the
FSDT model [314] takes into account both longitudinal and bending deformations,
whereas Ke et al. [315] considered only the bending deformation in their model.
Besides, Ma et al. [314] obtained solutions in the closed form for the problem of
bending and vibrations of the simply supported plates, while Ke et al. [315] obtained
the numerical solutions for the eigenfrequencies of plates with clamped sides with
the help of the Ritz method. Roque et al. [316] presented the numerical solutions of
the FSDT model of modified couple stress theory for analysis of the static bending
of the isotropic microplates using the meshless method of collocation with radial
basic functions. Zhou and Gao [317] worked out the modified couple stress theory
of the FSDT model to carry out the axially symmetric analysis of bending of the
isotropic circular microplates. Alinaghizadeh et al. [318] developed the modified
couple stress theory of the FSDT model for the static analysis of bending of the FG
annular microplates sector. In order to solve the bending problem of the microplates,
the DQ method was used for the different boundary conditions. He et al. [319]
extended the FSDT model to analyse static bending of the laminated composites
microplates, whereas Simsek and Aydin [320] extended the FSDT model to solve
the static bending and vibrations of the externally loaded FG microplates under action
of moving load.
Reddy and Berry [321] generalized the axially symmetric model of FSDT in
order to explain the influence of the geometric nonlinearity, temperature and nonhomogeneous behaviour of the FG materials and the axially symmetric analysis of
nonlinear bending of the circle microplates. Ke et al. [322, 323] also developed