38
2 Size-Dependent Theories of Beams, Plates and Shells
of a cantilever epoxidal beam with a discrete load on its free end. It was demonstrated
that an account of the length parameter of material implies an increase of stiffness
under bending of the cantilever beam. The latter effect played an important role when
the beam thickness was small but it decreased with increase of the beam thickness.
These observations were validated by experimental investigations. The modified
couple stress theory of EBT was extended by Kong et al. [156, 157] for the problem
of free vibrations [156] and buckling of isotropic microbeams [157].
The model of nonlinear modified couple stress theory of EBT was first worked
out by Xia et al. [158] for the case of nonlinear bending behaviour, nonlinear analysis of stability and free vibrations of isotropic microbeams on a basis of the von
Kármán nonlinearity. The obtained results showed the importance of account of nonlinearity and size effects during the study of microscale devices and systems such
as biosensors, atomic-force microscopes and MEMS [158]. Simsek [159] also proposed nonlinear EBT model for fitting nonlinear deflections and free vibrations of
isotropic microbeams lying an elastic foundation. Nonlinear EBT model has been
widely employed for analysing the influence of the size effects on nonlinear deflections [160], nonlinear vibrations [161–165] and stability [164, 165] of isotropic
microbeams. It should be noted that Farokhi et al. [162] considered the initial geometric imperfections while studying nonlinear beam vibrations, whereas Togun and
Bagdatli [163] took into account axial extension of the microbeam vibrations. Wang
et al. [160, 164] considered the thermal effects within nonlinear microbeam deflections [160] as well as stability problems and vibrations of the beams [164]. Ansari
et al. [165] obtained analytical solutions while investigating both vibrations and
stability of microbeams under various boundary conditions.
The developed EBT model was also applied to FG microbeams. For example, it
was employed by Asghari et al. [166] to study bending and free vibrations of the FG
microbeams. Free vibrations of FG cantilevers with changeable properties of material
in the longitudinal direction were considered by Akgoz and Civalek [167] and Shafiei
et al. [168]. It should be mentioned that Akgoz and Civalek [167] analysed only cantilever, whereas Shafiei et al. [168] considered beams with account of the geometric
nonlinearity with various boundary conditions. Simsek [169] also investigated nonlinear free vibrations of microbeam functionally graded in the longitudinal direction.
He employed the Galerkin methods and his variational method to get the approximate solutions for beams with simple support and clamping. Dehrouyeh-Semnani et
al. [170] include the initial geometric imperfections while studying vibrations of the
FG microbeams.
The modified couple stress theory under CPT model was first proposed by Tsiatas
[171] for analysis of bending of isotropic microplates of arbitrary form. Yin et al.
[172] and Jomehzadeh et al. [173] expended that theory to include free vibrations of
simply supported microplates [172] and the Levi-type microplates [173]. Akgoz and
Civalek [174] proposed the CPT model within the modified couple stress theory in
order to investigate the influence of the size dependence on free vibrations of simply
supported SLGS included in an elastic matrix. It was shown that the size effect was
observed for higher vibration modes. Akgoz and Civalek [175] also included elastic
medium into the CPT model for the static bending, the stability problem and analysis
2 Size-Dependent Theories of Beams, Plates and Shells
of a cantilever epoxidal beam with a discrete load on its free end. It was demonstrated
that an account of the length parameter of material implies an increase of stiffness
under bending of the cantilever beam. The latter effect played an important role when
the beam thickness was small but it decreased with increase of the beam thickness.
These observations were validated by experimental investigations. The modified
couple stress theory of EBT was extended by Kong et al. [156, 157] for the problem
of free vibrations [156] and buckling of isotropic microbeams [157].
The model of nonlinear modified couple stress theory of EBT was first worked
out by Xia et al. [158] for the case of nonlinear bending behaviour, nonlinear analysis of stability and free vibrations of isotropic microbeams on a basis of the von
Kármán nonlinearity. The obtained results showed the importance of account of nonlinearity and size effects during the study of microscale devices and systems such
as biosensors, atomic-force microscopes and MEMS [158]. Simsek [159] also proposed nonlinear EBT model for fitting nonlinear deflections and free vibrations of
isotropic microbeams lying an elastic foundation. Nonlinear EBT model has been
widely employed for analysing the influence of the size effects on nonlinear deflections [160], nonlinear vibrations [161–165] and stability [164, 165] of isotropic
microbeams. It should be noted that Farokhi et al. [162] considered the initial geometric imperfections while studying nonlinear beam vibrations, whereas Togun and
Bagdatli [163] took into account axial extension of the microbeam vibrations. Wang
et al. [160, 164] considered the thermal effects within nonlinear microbeam deflections [160] as well as stability problems and vibrations of the beams [164]. Ansari
et al. [165] obtained analytical solutions while investigating both vibrations and
stability of microbeams under various boundary conditions.
The developed EBT model was also applied to FG microbeams. For example, it
was employed by Asghari et al. [166] to study bending and free vibrations of the FG
microbeams. Free vibrations of FG cantilevers with changeable properties of material
in the longitudinal direction were considered by Akgoz and Civalek [167] and Shafiei
et al. [168]. It should be mentioned that Akgoz and Civalek [167] analysed only cantilever, whereas Shafiei et al. [168] considered beams with account of the geometric
nonlinearity with various boundary conditions. Simsek [169] also investigated nonlinear free vibrations of microbeam functionally graded in the longitudinal direction.
He employed the Galerkin methods and his variational method to get the approximate solutions for beams with simple support and clamping. Dehrouyeh-Semnani et
al. [170] include the initial geometric imperfections while studying vibrations of the
FG microbeams.
The modified couple stress theory under CPT model was first proposed by Tsiatas
[171] for analysis of bending of isotropic microplates of arbitrary form. Yin et al.
[172] and Jomehzadeh et al. [173] expended that theory to include free vibrations of
simply supported microplates [172] and the Levi-type microplates [173]. Akgoz and
Civalek [174] proposed the CPT model within the modified couple stress theory in
order to investigate the influence of the size dependence on free vibrations of simply
supported SLGS included in an elastic matrix. It was shown that the size effect was
observed for higher vibration modes. Akgoz and Civalek [175] also included elastic
medium into the CPT model for the static bending, the stability problem and analysis