9.7 Size-Dependent Euler-Bernoulli Beams with Topologically Optimized Microstructure 375
been employed for this purpose [2–8]. It has been observed that the problems of
free vibrations of beams with non-homogeneous transverse cross sections or with
material properties changing in the longitudinal direction exhibit more complexity
comparing to the classical investigations of beams with homogeneous transverse
cross sections and made of homogeneous materials due to the occurrence of timedependent coefficients in the governing differential equations [9, 10]. This is why the
majority of the carried out investigations rely on numerical methods [11, 12]. For
instance, nonlinear behaviour of functionally graded beams with the von Kármán
geometric nonlinearity has been studied in reference [13] by using finite element
method (FEM). Kien [14] has considered large displacement response of the tapered
cantilever beams made of axially functionally graded material. Alshorbagy et al.
[15] have studied free vibration characteristics of a functionally graded beam also
by using FEM. Shahba et al. [10, 16, 17] have carried out investigations of free
vibration and stability of axially functionally graded tapered Euler-Bernoulli and
Timoshenko beams with classical/non-classical boundary conditions by using a few
numerical techniques. An analytical study of nonlinear vibrations of functionally
graded beams has been carried out by Ke et al. [18] with the use of the first-order
Galerkin method.
In order to analyze vibrations of non-uniform and exponentially functionally
graded and tapered beams, numerous mathematical models based on the EulerBernoulli [10, 19–24, 116] and Timoshenko [17, 25–28] hypotheses have been
employed. Authors of the references [29–31] have studied free vibrations of clamped
tapered beams on linear elastic foundations, axially functionally graded tapered
Euler-Bernoulli beams, and axially functionally graded Timoshenko beams having
non-uniform cross sections.
It should be noted that the classical mechanics of rigid bodies do not allow one
to interpret or forecast the size-dependent behaviour of the MEMS/NEMS structural members due to a lack of a size-dependent parameter responsible for quantifying micron and submicron scale processes playing a crucial role in the final global
dynamic behaviour of the mentioned structural elements. So far, the following theories have been used for modelling of the scale effects occurred in a continuum: the
couple stress theory of elasticity [37, 117], the nonlocal theory of elasticity [39], the
strain gradient theory of elasticity [40] and the general theory of curved deformable
interfaces in solids [41].
A theoretical foundation for the couple stress-based strain gradient theory for elasticity has been proposed by Yang et al. [118]. In that paper, the governing equations
contain, in spite of two classical Lame constants, an additional higher order material
constant. This theory has been employed and validated by numerous researchers to
get a reliable interpretation of the size-dependent dynamic behaviour of microstructures [32–34, 119].
Rajabi and Ramezani [36] have derived PDEs for a geometrically nonlinear homogeneous Euler-Bernoulli beam by using the von Kármán relations. The Galerkin
method in the first approximation has been employed and the influence of a sizedependent coefficient of the magnitude equal to the fundamental frequency of nonlinear vibrations has been studied.
been employed for this purpose [2–8]. It has been observed that the problems of
free vibrations of beams with non-homogeneous transverse cross sections or with
material properties changing in the longitudinal direction exhibit more complexity
comparing to the classical investigations of beams with homogeneous transverse
cross sections and made of homogeneous materials due to the occurrence of timedependent coefficients in the governing differential equations [9, 10]. This is why the
majority of the carried out investigations rely on numerical methods [11, 12]. For
instance, nonlinear behaviour of functionally graded beams with the von Kármán
geometric nonlinearity has been studied in reference [13] by using finite element
method (FEM). Kien [14] has considered large displacement response of the tapered
cantilever beams made of axially functionally graded material. Alshorbagy et al.
[15] have studied free vibration characteristics of a functionally graded beam also
by using FEM. Shahba et al. [10, 16, 17] have carried out investigations of free
vibration and stability of axially functionally graded tapered Euler-Bernoulli and
Timoshenko beams with classical/non-classical boundary conditions by using a few
numerical techniques. An analytical study of nonlinear vibrations of functionally
graded beams has been carried out by Ke et al. [18] with the use of the first-order
Galerkin method.
In order to analyze vibrations of non-uniform and exponentially functionally
graded and tapered beams, numerous mathematical models based on the EulerBernoulli [10, 19–24, 116] and Timoshenko [17, 25–28] hypotheses have been
employed. Authors of the references [29–31] have studied free vibrations of clamped
tapered beams on linear elastic foundations, axially functionally graded tapered
Euler-Bernoulli beams, and axially functionally graded Timoshenko beams having
non-uniform cross sections.
It should be noted that the classical mechanics of rigid bodies do not allow one
to interpret or forecast the size-dependent behaviour of the MEMS/NEMS structural members due to a lack of a size-dependent parameter responsible for quantifying micron and submicron scale processes playing a crucial role in the final global
dynamic behaviour of the mentioned structural elements. So far, the following theories have been used for modelling of the scale effects occurred in a continuum: the
couple stress theory of elasticity [37, 117], the nonlocal theory of elasticity [39], the
strain gradient theory of elasticity [40] and the general theory of curved deformable
interfaces in solids [41].
A theoretical foundation for the couple stress-based strain gradient theory for elasticity has been proposed by Yang et al. [118]. In that paper, the governing equations
contain, in spite of two classical Lame constants, an additional higher order material
constant. This theory has been employed and validated by numerous researchers to
get a reliable interpretation of the size-dependent dynamic behaviour of microstructures [32–34, 119].
Rajabi and Ramezani [36] have derived PDEs for a geometrically nonlinear homogeneous Euler-Bernoulli beam by using the von Kármán relations. The Galerkin
method in the first approximation has been employed and the influence of a sizedependent coefficient of the magnitude equal to the fundamental frequency of nonlinear vibrations has been studied.