7.5 Stability of the Size-Dependent Graded Curvilinear Timoshenko Beams
235
the symmetric part of the stress tensor are obtained (Poisson’s effect is neglected)
σ xx = E (x, ˜
z)
u , x +
1
2
w , x
2 + zψ , x + k x w
, σ xz = k s μ (x, ˜
z)
w , x + ψ
,
where E(x, ˜
z) and μ(x, ˜
z) denote Young’s and shear moduli, respectively, k s is the
shear corrector factor, which has been introduced due to the grading shear deformation with respect to the transverse cross section of the curvilinear beam (it depends
on the form of the cross section of the curvilinear beam) [57, 135].
In what follows, we assume that the beam properties are changed only along the
beam thickness, and the components of the symmetric part of the stress tensor curve
are expressed in terms of the kinematic parameters in the following way
σ xx = E (˜ z)
u , x +
1
2
w , x
2 + zψ , x + k x w
, σ xz = k s μ (˜ z)
w , x + ψ
.
(7.64)
Substituting (7.63) into (7.58), the components of the derivative part of the higher
order moment can be expressed by the kinematic parameter χ xy :
m xy = m yx =
1
4
β (˜ z)
ψ , x − w , xx
.
(7.65)
For further analysis, it is convenient to introduce a reference line for the zcoordinate, using the following condition
B 11 =
A
E (˜ z) zd A =
A
E (˜ z) (˜ z − ˜
z) d A = 0.
(7.66)
From Eq. (7.66), we obtain the coordinate ˜
z c of the reference line
˜
z =
A
E (˜ z) ˜
zd A
A
E (˜ z) d A
.
(7.67)
The obtained formula (7.67) allows one to exclude the terms with coefficients B 11
from the system of equations, which significantly simplifies the system of equations
obtained in Refs. [57, 101].
7.5.3 Derivation of Equations of Motion
In order to define the equation governing the dynamics of the FG curvilinear Timoshenko beam based on the modified couple stress theory, we substitute (7.61)–(7.65)
235
the symmetric part of the stress tensor are obtained (Poisson’s effect is neglected)
σ xx = E (x, ˜
z)
u , x +
1
2
w , x
2 + zψ , x + k x w
, σ xz = k s μ (x, ˜
z)
w , x + ψ
,
where E(x, ˜
z) and μ(x, ˜
z) denote Young’s and shear moduli, respectively, k s is the
shear corrector factor, which has been introduced due to the grading shear deformation with respect to the transverse cross section of the curvilinear beam (it depends
on the form of the cross section of the curvilinear beam) [57, 135].
In what follows, we assume that the beam properties are changed only along the
beam thickness, and the components of the symmetric part of the stress tensor curve
are expressed in terms of the kinematic parameters in the following way
σ xx = E (˜ z)
u , x +
1
2
w , x
2 + zψ , x + k x w
, σ xz = k s μ (˜ z)
w , x + ψ
.
(7.64)
Substituting (7.63) into (7.58), the components of the derivative part of the higher
order moment can be expressed by the kinematic parameter χ xy :
m xy = m yx =
1
4
β (˜ z)
ψ , x − w , xx
.
(7.65)
For further analysis, it is convenient to introduce a reference line for the zcoordinate, using the following condition
B 11 =
A
E (˜ z) zd A =
A
E (˜ z) (˜ z − ˜
z) d A = 0.
(7.66)
From Eq. (7.66), we obtain the coordinate ˜
z c of the reference line
˜
z =
A
E (˜ z) ˜
zd A
A
E (˜ z) d A
.
(7.67)
The obtained formula (7.67) allows one to exclude the terms with coefficients B 11
from the system of equations, which significantly simplifies the system of equations
obtained in Refs. [57, 101].
7.5.3 Derivation of Equations of Motion
In order to define the equation governing the dynamics of the FG curvilinear Timoshenko beam based on the modified couple stress theory, we substitute (7.61)–(7.65)