116
6 Nonlinear Analysis of Piezoceramic Laminated Structures
LRT56 model meshed by 5 × 1 or 10 × 1 SH851URI elements is almost identical.
This implies that the model with 5 elements is converged. In order to compare the
results of Yi et al. [17], the model with 5 × 1 SH851FI elements are considered
in the simulation, whose results are in good agreement with those in [17]. In the
reference, Yi et al. [17] used fully geometrically nonlinear model meshed by 5 twentynode solid elements for the host beam and 1 solid element for piezoelectric patch
without avoiding the locking effects, which is equivalent to use LRT56 model with
5 × 1 SH851FI elements. Because of the locking effects in the computation, the
nonlinear model meshed by 5 SH851FI elements is not well converged. By increasing
the number of elements, the solution approaches to the one of LRT56 model with
SH851URI elements.
6.3.2 Fully Clamped Smart Plate
The second example of smart structure is a piezoelectric patch bonded plate with
four edges fixed, as shown in Fig. 6.20, first studied by Yi et al. [17]. The host plate
is made up of steel and piezoelectric is made of PZT, similar to the smart beam. The
material properties can be found in Table 6.3. The boundary conditions and geometry
are symmetric about the center point. For simplicity, only a quarter of the plate is
considered, and the structures is meshed by 5 × 5 SH851URI shell elements.
1 0 0 ×
1 0 0 ×
1
m
m
2 0 × 2 0 × 1 m m
Θ
3
Θ
2
Θ
1
Fig. 6.20 Fully clamped plate with one piezoelectric patch centrally bonded
Table 6.3 Material properties of the cantilevered smart beam
Steel
PZT
E = 197 GPa
E = 67 GPa
ν = 0.33
ν = 0.33
ρ = 7900 kg/m 3
ρ = 7800 kg/m 3
d 31 = d 32 = −1.7119 × 10 −10 C/N
33 = 2.03 × 10 −8 F/m
6 Nonlinear Analysis of Piezoceramic Laminated Structures
LRT56 model meshed by 5 × 1 or 10 × 1 SH851URI elements is almost identical.
This implies that the model with 5 elements is converged. In order to compare the
results of Yi et al. [17], the model with 5 × 1 SH851FI elements are considered
in the simulation, whose results are in good agreement with those in [17]. In the
reference, Yi et al. [17] used fully geometrically nonlinear model meshed by 5 twentynode solid elements for the host beam and 1 solid element for piezoelectric patch
without avoiding the locking effects, which is equivalent to use LRT56 model with
5 × 1 SH851FI elements. Because of the locking effects in the computation, the
nonlinear model meshed by 5 SH851FI elements is not well converged. By increasing
the number of elements, the solution approaches to the one of LRT56 model with
SH851URI elements.
6.3.2 Fully Clamped Smart Plate
The second example of smart structure is a piezoelectric patch bonded plate with
four edges fixed, as shown in Fig. 6.20, first studied by Yi et al. [17]. The host plate
is made up of steel and piezoelectric is made of PZT, similar to the smart beam. The
material properties can be found in Table 6.3. The boundary conditions and geometry
are symmetric about the center point. For simplicity, only a quarter of the plate is
considered, and the structures is meshed by 5 × 5 SH851URI shell elements.
1 0 0 ×
1 0 0 ×
1
m
m
2 0 × 2 0 × 1 m m
Θ
3
Θ
2
Θ
1
Fig. 6.20 Fully clamped plate with one piezoelectric patch centrally bonded
Table 6.3 Material properties of the cantilevered smart beam
Steel
PZT
E = 197 GPa
E = 67 GPa
ν = 0.33
ν = 0.33
ρ = 7900 kg/m 3
ρ = 7800 kg/m 3
d 31 = d 32 = −1.7119 × 10 −10 C/N
33 = 2.03 × 10 −8 F/m