234
7 Mathematical Models of Functionally Graded Beams in Temperature Field
The introduced line, further referred to as the bending line, is the line around
which a pure rotation occurs in a cross section. The coordinate z in Fig. 7.11 denotes
the distance between a cross section point and the bending line. In Fig. 7.11 G(x, t)
denotes the force per unit length acting on the axial cross section along the body
axis O X. The q(x, t) stands for the resultant of the transverse forces located on the
upper beam surface per beam unit length. The parameter C(x, t) stands for the y
component of the resultant volume moment per the unit length of the curvilinear
beam being adjusted to the beam section. A distance between an arbitrary point and
the beam surface is denoted by ˜
z. The distance between the bending line and the
lower surface is denoted by ˜
z c .
Here, we consider the geometrically nonlinear shallow beam, i.e. the beam with
small deformation and rotations but with possibly large deflection w, which imply
nonlinear effects. Employment of (7.3), (7.6) yields. The non-zero physical components of the strain tensor ε i j of the curved beam in the curvilinear coordinate system
of the following form [50]
ε xx = (1 + k x z)
−1
u , x +
1
2
w , x
2 + zψ , x + k x w
,
ε xz =
1
2
(1 + k x z)
−1
w , x + ψ − k x u
,
where k x = 1/R x denotes curvature and R x is a radius of the beam curvature.
Assuming that the thickness of the curvilinear beam h is small in comparison to
curvature radius R x , i.e. h/R x << 1 and |z/R x | << 1, and assuming (1 + k x z) ≈
1, as well considering the low-pitched beams k x u ≈ 0 only, the deformations depend
on the displacements in the following way [135]
ε xx =
u , x +
1
2
w , x
2 + zψ , x + k x w
, ε xz =
1
2
w , x + ψ
.
(7.61)
The relation θ i = (rot (u)) i /2 yields one non-zero rotation vector of components
of the form
θ y =
1
2
ψ +
1
2
(1 + k x z)
−1
k x u − w , x + k x zψ
.
(7.62)
Substituting (7.62) into (7.59), employing the above-given assumption (1 + k x z) ≈
1 and k x u ≈ 0, the components of the symmetric part of the curvature tensor are
found:
χ xy = χ yx =
1
4
ψ , x − w , xx
, χ yz = χ zy =
1
4
k x
ψ + w , x
.
(7.63)
Again, owing to the low-pitched beam, we have χ yz ≈ 0.
In a general case, when the material properties are changed along the beam length
and thickness, after substitution of (7.61) into (7.56), the following components of
7 Mathematical Models of Functionally Graded Beams in Temperature Field
The introduced line, further referred to as the bending line, is the line around
which a pure rotation occurs in a cross section. The coordinate z in Fig. 7.11 denotes
the distance between a cross section point and the bending line. In Fig. 7.11 G(x, t)
denotes the force per unit length acting on the axial cross section along the body
axis O X. The q(x, t) stands for the resultant of the transverse forces located on the
upper beam surface per beam unit length. The parameter C(x, t) stands for the y
component of the resultant volume moment per the unit length of the curvilinear
beam being adjusted to the beam section. A distance between an arbitrary point and
the beam surface is denoted by ˜
z. The distance between the bending line and the
lower surface is denoted by ˜
z c .
Here, we consider the geometrically nonlinear shallow beam, i.e. the beam with
small deformation and rotations but with possibly large deflection w, which imply
nonlinear effects. Employment of (7.3), (7.6) yields. The non-zero physical components of the strain tensor ε i j of the curved beam in the curvilinear coordinate system
of the following form [50]
ε xx = (1 + k x z)
−1
u , x +
1
2
w , x
2 + zψ , x + k x w
,
ε xz =
1
2
(1 + k x z)
−1
w , x + ψ − k x u
,
where k x = 1/R x denotes curvature and R x is a radius of the beam curvature.
Assuming that the thickness of the curvilinear beam h is small in comparison to
curvature radius R x , i.e. h/R x << 1 and |z/R x | << 1, and assuming (1 + k x z) ≈
1, as well considering the low-pitched beams k x u ≈ 0 only, the deformations depend
on the displacements in the following way [135]
ε xx =
u , x +
1
2
w , x
2 + zψ , x + k x w
, ε xz =
1
2
w , x + ψ
.
(7.61)
The relation θ i = (rot (u)) i /2 yields one non-zero rotation vector of components
of the form
θ y =
1
2
ψ +
1
2
(1 + k x z)
−1
k x u − w , x + k x zψ
.
(7.62)
Substituting (7.62) into (7.59), employing the above-given assumption (1 + k x z) ≈
1 and k x u ≈ 0, the components of the symmetric part of the curvature tensor are
found:
χ xy = χ yx =
1
4
ψ , x − w , xx
, χ yz = χ zy =
1
4
k x
ψ + w , x
.
(7.63)
Again, owing to the low-pitched beam, we have χ yz ≈ 0.
In a general case, when the material properties are changed along the beam length
and thickness, after substitution of (7.61) into (7.56), the following components of