242
7 Mathematical Models of Functionally Graded Beams in Temperature Field
2
1
2
1
3
w
150
100
50
0
0 q
2
1
5
4
6
w
150
100
50
0
0 q
2
1
4
1
7
w
150
100
50
0
0 q
8
0
5
4
2
w
27
53
79
-1.6
1
-0.4
-0.8
-1.2
n
6
1
3
7
8
a)
b)
c)
d)
Fig. 7.13 Solutions to static problems of Timoshenko beam: a w(0.5; q) for variants 1–3, γ 2 = 0.3;
b w(0.5; q) for variants 4–6, γ 2 = 0; c w(0.5; q) for homogenous beams; d solution for w(n) and
q = 150 [reprinted with permission from the Journal of Computational and Nonlinear Dynamics
publishers]
Let us discuss the meaning of the so far introduced parameters. P = 1 corresponds
to a homogeneous material, P = 2 states for a functionally graded material, i.e. the
material with 2E 0 located on the upper part of the beam. For P = 0.5, the beam
is also made of a functionally graded material, but the situation is reversed (2E 0 is
now on the bottom beam part). Figure 7.13a reports the deflection-load dependence
for variants 1–3; Fig. 7.13b shows the deflection-load relations for the variants 4–6;
Fig. 7.13c presents the deflection-load curves for the variants 1, 4, 7, 8; Fig. 7.13d
shows deflections for q = 50, 100, 150 and for all studied variants.
Figure 7.13a shows that for the same loads (q > 100), the largest deflection values are observed for the functionally graded beam with location of the stiffest layer
on the bottom side of the beam. The difference between the homogenous beam and
the functionally graded beam with stiffer layer located on the bottom (upper) side of
the beam is on amount of 15% (20%) for q = 200. The same observation holds for
the beam without the size-dependent effect (Fig. 7.13b). The difference between the
values of deflection reaches 13 and 18%, respectively. On the other hand, the investigation of homogenous beams with various stiffness and size-dependent coefficients
(Fig. 7.13c) shows that the largest deflection corresponds to the variant with the initial
stiffness and no size-dependent coefficient (γ 2 = 0, P E = 1, E = E 0 ). The differ-
7 Mathematical Models of Functionally Graded Beams in Temperature Field
2
1
2
1
3
w
150
100
50
0
0 q
2
1
5
4
6
w
150
100
50
0
0 q
2
1
4
1
7
w
150
100
50
0
0 q
8
0
5
4
2
w
27
53
79
-1.6
1
-0.4
-0.8
-1.2
n
6
1
3
7
8
a)
b)
c)
d)
Fig. 7.13 Solutions to static problems of Timoshenko beam: a w(0.5; q) for variants 1–3, γ 2 = 0.3;
b w(0.5; q) for variants 4–6, γ 2 = 0; c w(0.5; q) for homogenous beams; d solution for w(n) and
q = 150 [reprinted with permission from the Journal of Computational and Nonlinear Dynamics
publishers]
Let us discuss the meaning of the so far introduced parameters. P = 1 corresponds
to a homogeneous material, P = 2 states for a functionally graded material, i.e. the
material with 2E 0 located on the upper part of the beam. For P = 0.5, the beam
is also made of a functionally graded material, but the situation is reversed (2E 0 is
now on the bottom beam part). Figure 7.13a reports the deflection-load dependence
for variants 1–3; Fig. 7.13b shows the deflection-load relations for the variants 4–6;
Fig. 7.13c presents the deflection-load curves for the variants 1, 4, 7, 8; Fig. 7.13d
shows deflections for q = 50, 100, 150 and for all studied variants.
Figure 7.13a shows that for the same loads (q > 100), the largest deflection values are observed for the functionally graded beam with location of the stiffest layer
on the bottom side of the beam. The difference between the homogenous beam and
the functionally graded beam with stiffer layer located on the bottom (upper) side of
the beam is on amount of 15% (20%) for q = 200. The same observation holds for
the beam without the size-dependent effect (Fig. 7.13b). The difference between the
values of deflection reaches 13 and 18%, respectively. On the other hand, the investigation of homogenous beams with various stiffness and size-dependent coefficients
(Fig. 7.13c) shows that the largest deflection corresponds to the variant with the initial
stiffness and no size-dependent coefficient (γ 2 = 0, P E = 1, E = E 0 ). The differ-