2.3 Non-classical (Size-Dependent) Models of Beams, Plates and Shells
35
2.3.5 Size-Dependent Theory of Beams, Plates and Shells
2.3.5.1 Nonlocal Models of Euler-Bernoulli Beams (EBT), the
Kirchhoff Plates (CPT) and the Kirchhoff-Love Shells (KLT)
The first models of nonlocal beams based on EBT have been developed by Peddieson
et al. [100] and Sudak [101]. Peddieson et al. [100] employed their own model to
estimate the influence of size dependence on the bending behaviour of isotropic
nanoshells, while Sudak [101] employed his model to investigate stability loss of
MWCNT. Then numerous articles appear devoted to modelling of nanobeams and
CNT with the help of nonlocal EBT model. For instance, Zhang et al. [102] investigated free vibrations of the DWCNT. In order to study the size effect of the characteristics of the DWCNT vibrations, the analytical solutions have been derived for the
eigen frequencies of simply supported DWCNT. Wang et al. [103] obtained a general
solution in a closed form for stability loss of CNT with various boundary conditions.
Aydogdu [104] investigated the influence of size dependence on a basis of vibration of nanorods for various boundary conditions and obtained explicit formulas for
eigenfrequencies. Murmu and Pradhan [105] introduced thermal effects into analysis
of free vibrations under boundary conditions of SWCNTs with the help of method
of differential quadratures (DQ). The similar approach was also used by Civalek and
Demir [106] for getting bending moments and deflections of nanobeams for different boundary conditions. Mustapha and Zhong [107] investigated free vibrations of
axially loaded non-prismatic SWCNTs employing the Bubnov-Galerkin method. Li
et al. [108] obtained closed-form solutions for eigenfrequencies of axially loaded
simply supported nanobeams. Ghannadpour et al. [109] used the Ritz method to
solve the governing equations of nonlocal EBT, i.e. model of bending under critical load and eigenfrequencies for different boundary conditions. Ansari et al. [110]
worked out the nonlocal EBT taking into account the geometric von Kármán nonlinearity based on the MWCNT model with an account of temperature, whereas Fang
et al. [111] proposed the nonlinear nonlocal EBT model for nonlinear vibrations
with DWCNT. Nonlinear free vibrations and excited vibrations of nanobeams with
different boundary conditions were analysed by Simsek [112] and Bagdatli [113].
Nonlocal model of EBT has been also proposed for the nanobeams made from
functionally graded (FG) materials. Simsek [114] investigated nonlocal effect under
axial vibrations of FG nanorods with changeable transversal cross section. It has been
assumed that the elasticity modulus and mass density of the nanorods are changed in
the axial direction. Nguyen et al. [115] proposed analytical solutions of the nonlocal
EBT model for the static analysis of bending of FG beams with different boundary
conditions. Elasticity modulus of FG nanobeams can be changed in both longitudinal
and transverse directions. Galerkin, Niknam and Akhdam [116] obtained solutions
to the nonlocal EBT model for the eigenfrequencies and critical loads of the FG
nanobeams lying on an elastic foundation. Ebrahimi and Salari [117, 118] studied
the influence of temperature on the free vibrations of FG nanobeams with different
boundary conditions with the help of semi-analytical approach. Nejad and Hadi [119]
35
2.3.5 Size-Dependent Theory of Beams, Plates and Shells
2.3.5.1 Nonlocal Models of Euler-Bernoulli Beams (EBT), the
Kirchhoff Plates (CPT) and the Kirchhoff-Love Shells (KLT)
The first models of nonlocal beams based on EBT have been developed by Peddieson
et al. [100] and Sudak [101]. Peddieson et al. [100] employed their own model to
estimate the influence of size dependence on the bending behaviour of isotropic
nanoshells, while Sudak [101] employed his model to investigate stability loss of
MWCNT. Then numerous articles appear devoted to modelling of nanobeams and
CNT with the help of nonlocal EBT model. For instance, Zhang et al. [102] investigated free vibrations of the DWCNT. In order to study the size effect of the characteristics of the DWCNT vibrations, the analytical solutions have been derived for the
eigen frequencies of simply supported DWCNT. Wang et al. [103] obtained a general
solution in a closed form for stability loss of CNT with various boundary conditions.
Aydogdu [104] investigated the influence of size dependence on a basis of vibration of nanorods for various boundary conditions and obtained explicit formulas for
eigenfrequencies. Murmu and Pradhan [105] introduced thermal effects into analysis
of free vibrations under boundary conditions of SWCNTs with the help of method
of differential quadratures (DQ). The similar approach was also used by Civalek and
Demir [106] for getting bending moments and deflections of nanobeams for different boundary conditions. Mustapha and Zhong [107] investigated free vibrations of
axially loaded non-prismatic SWCNTs employing the Bubnov-Galerkin method. Li
et al. [108] obtained closed-form solutions for eigenfrequencies of axially loaded
simply supported nanobeams. Ghannadpour et al. [109] used the Ritz method to
solve the governing equations of nonlocal EBT, i.e. model of bending under critical load and eigenfrequencies for different boundary conditions. Ansari et al. [110]
worked out the nonlocal EBT taking into account the geometric von Kármán nonlinearity based on the MWCNT model with an account of temperature, whereas Fang
et al. [111] proposed the nonlinear nonlocal EBT model for nonlinear vibrations
with DWCNT. Nonlinear free vibrations and excited vibrations of nanobeams with
different boundary conditions were analysed by Simsek [112] and Bagdatli [113].
Nonlocal model of EBT has been also proposed for the nanobeams made from
functionally graded (FG) materials. Simsek [114] investigated nonlocal effect under
axial vibrations of FG nanorods with changeable transversal cross section. It has been
assumed that the elasticity modulus and mass density of the nanorods are changed in
the axial direction. Nguyen et al. [115] proposed analytical solutions of the nonlocal
EBT model for the static analysis of bending of FG beams with different boundary
conditions. Elasticity modulus of FG nanobeams can be changed in both longitudinal
and transverse directions. Galerkin, Niknam and Akhdam [116] obtained solutions
to the nonlocal EBT model for the eigenfrequencies and critical loads of the FG
nanobeams lying on an elastic foundation. Ebrahimi and Salari [117, 118] studied
the influence of temperature on the free vibrations of FG nanobeams with different
boundary conditions with the help of semi-analytical approach. Nejad and Hadi [119]