6.2 Literature Review
133
cies is investigated. In the case of static problem, the linear fourth-order equation
with regard to the longitudinal displacement has been analysed. Investigation of the
natural frequencies behaviour for small deflection has been carried out based on
the sixth-order linear equation with regard to the deflection. The Bubnov–Galerkin
method in the first approximation has been used which allowed to transfer the studied PDE to ODE. Rajabi and Ramezani [29] derived the governing equations for the
geometrically nonlinear Euler–Bernoulli beam based on the von Kármán relations. In
order to obtain a numerical solution, the Bubnov–Galerkin method in the first-order
approximation has been employed. The influence of the length scale parameter on
the natural frequency of nonlinear vibrations has been analysed.
Mathematical model of the Timoshenko beam and, on contrary to the mathematical model of the Bernoulli–Euler beam, a rotation of a normal to the beam axis
and deformation of the transversal shear effects are taken into account. It stands for
the next step of approximation to the real behaviour of a 2D and 3D body. Though
the order of the system of differential equations remains unchanged, there is added a
function responsible for the normal rotation. Investigation of the size-dependent models of Timoshenko beam is realized via various approaches including the modified
couple stress theory of elasticity [30], while Asghari [31] derived the geometrically
nonlinear governing PDEs and boundary conditions based on the strain gradient
theory.
A numerical study has been carried by considering simple cases without taking
into account the longitudinal inertia and with the help of first-order approximation
of the Bubnov–Galerkin method. Moreover, a series of simplifications are used to
reduce the problem to that of solving a linear ODE to get the vibration frequencies.
In addition, nonlinear problems of static deformations have been also considered.
Ansan et al. [32] employed strain gradient theory to analyse free vibration of the
Timoshenko beam. The derived governing PDEs have been reduced to a system
composed of two ODEs through the Bubnov–Galerkin method. Numerical results
have been compared with the earlier reported results based on the models of linear
deformation gradient, nonlinear and linear modified couple stress theory of elasticity,
as well as linear and nonlinear classical models. Deflections obtained based on the
mentioned two models differ essentially when the beam is relatively short. This effect
is due to the neglection of the shear deformation in the Bernoulli–Euler model.
Ma et al. [26] employed the SP linear model to solve the static problems and to
detect the natural frequencies. The obtained results have been compared with the
results based on other models with an account (no account) of the size-dependent
behaviour.
The so far carried out analysis of the state of the art of literature devoted to
investigation of the size-dependent beams based on the models of Bernoulli–Euler,
Timoshenko and Sheremetev–Pelekh is mainly reduced to an ordinary differential
equation of the Duffing type using the Bubnov–Galerkin method in its first approximation to the input governing PDEs. Linear problems focused on the frequencies
estimation and static problems aimed at investigation of the influence of parameter
responsible for the size dependence have been considered. However, nonlinear prob-
133
cies is investigated. In the case of static problem, the linear fourth-order equation
with regard to the longitudinal displacement has been analysed. Investigation of the
natural frequencies behaviour for small deflection has been carried out based on
the sixth-order linear equation with regard to the deflection. The Bubnov–Galerkin
method in the first approximation has been used which allowed to transfer the studied PDE to ODE. Rajabi and Ramezani [29] derived the governing equations for the
geometrically nonlinear Euler–Bernoulli beam based on the von Kármán relations. In
order to obtain a numerical solution, the Bubnov–Galerkin method in the first-order
approximation has been employed. The influence of the length scale parameter on
the natural frequency of nonlinear vibrations has been analysed.
Mathematical model of the Timoshenko beam and, on contrary to the mathematical model of the Bernoulli–Euler beam, a rotation of a normal to the beam axis
and deformation of the transversal shear effects are taken into account. It stands for
the next step of approximation to the real behaviour of a 2D and 3D body. Though
the order of the system of differential equations remains unchanged, there is added a
function responsible for the normal rotation. Investigation of the size-dependent models of Timoshenko beam is realized via various approaches including the modified
couple stress theory of elasticity [30], while Asghari [31] derived the geometrically
nonlinear governing PDEs and boundary conditions based on the strain gradient
theory.
A numerical study has been carried by considering simple cases without taking
into account the longitudinal inertia and with the help of first-order approximation
of the Bubnov–Galerkin method. Moreover, a series of simplifications are used to
reduce the problem to that of solving a linear ODE to get the vibration frequencies.
In addition, nonlinear problems of static deformations have been also considered.
Ansan et al. [32] employed strain gradient theory to analyse free vibration of the
Timoshenko beam. The derived governing PDEs have been reduced to a system
composed of two ODEs through the Bubnov–Galerkin method. Numerical results
have been compared with the earlier reported results based on the models of linear
deformation gradient, nonlinear and linear modified couple stress theory of elasticity,
as well as linear and nonlinear classical models. Deflections obtained based on the
mentioned two models differ essentially when the beam is relatively short. This effect
is due to the neglection of the shear deformation in the Bernoulli–Euler model.
Ma et al. [26] employed the SP linear model to solve the static problems and to
detect the natural frequencies. The obtained results have been compared with the
results based on other models with an account (no account) of the size-dependent
behaviour.
The so far carried out analysis of the state of the art of literature devoted to
investigation of the size-dependent beams based on the models of Bernoulli–Euler,
Timoshenko and Sheremetev–Pelekh is mainly reduced to an ordinary differential
equation of the Duffing type using the Bubnov–Galerkin method in its first approximation to the input governing PDEs. Linear problems focused on the frequencies
estimation and static problems aimed at investigation of the influence of parameter
responsible for the size dependence have been considered. However, nonlinear prob-