7.5 Stability of the Size-Dependent Graded Curvilinear Timoshenko Beams
243
ence with respect to the remaining values is as follows: (γ 2 = 0.3, P E = 1)—15%,
(γ 2 = 0, P E = 1, E = 2E 0 )—35%, (γ 2 = 0.3, P E = 1, E = 2E 0 )—55%.
Finally, plots of the beam forms are reported in Fig. 7.13d for the same P values.
The smaller and larger values of the deflection are observed for the variant 8 (γ 2 =
0.3, P E = 1, E = 2E 0 ) and 5 (γ 2 = 0, P E = 0.5), respectively.
We have shown that the size-dependent behaviour decreases the deflection value
for the same value of the functional grading parameter. Furthermore, if the most rigid
layer is located in the upper part of the beam, the decrease in the deflection value is
implied, taking into account the same size-dependent coefficient.
7.5.5.2 Stability Loss
In what follows, we consider the problems of a static stability loss of the curvilinear
beam subjected to a constant load. In literature, there exists a large number of stability
loss criterions. We will consider only the simple Lyapunov criterion, i.e. the beam is
stable/unstable if the LLE estimated for the governing equation is negative/positive.
Due to the fact that there is no G (x, t) and (x, t), the buckling occurs when the
beam is exposed to transverse forces q (x) just because of the initial curvature. This
means that the transverse buckling is not taken into account (pitchfork bifurcation),
and our study is restricted to follow the snap-through bifurcation.
In what follows, we study the dependence of the beam centre on the external load.
We have employed all of the mentioned stability criterions, which coincide with
each other well. The computational examples for k x = 24 and for all eight variants
of Table 7.12 are shown in Fig. 7.14.
The carried out numerical analysis of the curvilinear Timoshenko beams for k x =
24 yields the following observations.
1. Beams with the size-dependent behaviour (Fig. 7.13a). The smallest value of
the critical load, at which the stability loss occurs, takes place for the variant
2 of the functionally graded beam, where the most rigid layer (2E 0 ) is located
on the bottom side of the beam. If the most rigid layer is located on the upper
beam part, the stability loss occurs for a larger value of the load (500 instead of
190). It means that a change in the layers position essentially changes the beam
stress-strain state.
2. Beams without the size-dependent behaviour (Fig. 7.13b). The qualitative picture of the change in the critical load follows the previously described case. However, a difference in the critical load values between the two locations of the rigid
layer is decreased.
3. In the case of the homogenous beams, the largest load, at which stability loss
occurs, is exhibited by the homogenous beam with a doubled stiffness (2E 0 ) and
with the size-dependent effect (see variant 8). The difference between the largest
and smallest values of the critical load estimations, at which stability loss occurs,
for the variants 8 and 4, achieves the value of 200%.
243
ence with respect to the remaining values is as follows: (γ 2 = 0.3, P E = 1)—15%,
(γ 2 = 0, P E = 1, E = 2E 0 )—35%, (γ 2 = 0.3, P E = 1, E = 2E 0 )—55%.
Finally, plots of the beam forms are reported in Fig. 7.13d for the same P values.
The smaller and larger values of the deflection are observed for the variant 8 (γ 2 =
0.3, P E = 1, E = 2E 0 ) and 5 (γ 2 = 0, P E = 0.5), respectively.
We have shown that the size-dependent behaviour decreases the deflection value
for the same value of the functional grading parameter. Furthermore, if the most rigid
layer is located in the upper part of the beam, the decrease in the deflection value is
implied, taking into account the same size-dependent coefficient.
7.5.5.2 Stability Loss
In what follows, we consider the problems of a static stability loss of the curvilinear
beam subjected to a constant load. In literature, there exists a large number of stability
loss criterions. We will consider only the simple Lyapunov criterion, i.e. the beam is
stable/unstable if the LLE estimated for the governing equation is negative/positive.
Due to the fact that there is no G (x, t) and (x, t), the buckling occurs when the
beam is exposed to transverse forces q (x) just because of the initial curvature. This
means that the transverse buckling is not taken into account (pitchfork bifurcation),
and our study is restricted to follow the snap-through bifurcation.
In what follows, we study the dependence of the beam centre on the external load.
We have employed all of the mentioned stability criterions, which coincide with
each other well. The computational examples for k x = 24 and for all eight variants
of Table 7.12 are shown in Fig. 7.14.
The carried out numerical analysis of the curvilinear Timoshenko beams for k x =
24 yields the following observations.
1. Beams with the size-dependent behaviour (Fig. 7.13a). The smallest value of
the critical load, at which the stability loss occurs, takes place for the variant
2 of the functionally graded beam, where the most rigid layer (2E 0 ) is located
on the bottom side of the beam. If the most rigid layer is located on the upper
beam part, the stability loss occurs for a larger value of the load (500 instead of
190). It means that a change in the layers position essentially changes the beam
stress-strain state.
2. Beams without the size-dependent behaviour (Fig. 7.13b). The qualitative picture of the change in the critical load follows the previously described case. However, a difference in the critical load values between the two locations of the rigid
layer is decreased.
3. In the case of the homogenous beams, the largest load, at which stability loss
occurs, is exhibited by the homogenous beam with a doubled stiffness (2E 0 ) and
with the size-dependent effect (see variant 8). The difference between the largest
and smallest values of the critical load estimations, at which stability loss occurs,
for the variants 8 and 4, achieves the value of 200%.