The factors affecting the bistable behavior of anti-symmetric cylindrical shells are investigated by the NUCM and FE simulation. The results from EUCM and IUCM are also given for a direct comparison. The bistable performances of anti-symmetric cylindrical shells are mainly affected by: (i) layup parameters containing angle of layup α, number of plies
p, (ii) geometrical parameters including the angle of embrace
γ, the longitudinal length
l, the initial nature curvature
h0 and planform shapes. Some factors have been studied in the previous literature [
28], but a few extra considerations are reported with the NUCM. In essence, these factors directly influence the bistability of an anti-symmetric cylindrical shell in numerical model, but yet some of which do not affect straightforwardly in the NUCM. The direct factors for the bistability of anti-symmetric cylindrical shell in NUCM are: (i) the geometrical parameters used to describe the planform shape with the initial nature curvature, (ii) the dimensionless stiffness parameters, (3) the scalar factor
ψij and characteristic radius
R0 related to the dimensional transformation, both of which are affected by the former two factors. The relationships of the direct factors between the numerical simulation and NUCM are shown in the
Figure 3. The geometric parameters relationship between them are:
a =
l/2,
b = sin(
γ/2)/
h0, in which
γ cannot exceed 180 degrees due to the duplication of
b. Three examples are selected, referred to as No.1, No.2 and No.3, respectively. No.1 represents the classical anti-symmetric cylindrical shell as stated before. No.2 and No.3 are the same as No.1 except for a change in layup parameters (replaced with
p = 5) or geometric parameters (replaced with
a = 21.56 mm), respectively.
4.2.1. Influence of Scalar Factor and Characteristic Radius
The scalar factor
ψij is influenced by the layup parameters, material parameters and the geometrical parameters except for the initial curvature. When the scalar factor is determined, the bistability of anti-symmetric cylindrical shells can be investigated with the variation of dimensionless curvatures of the natural stress-free configuration (the initial twist curvature is assumed to be zero), (see bistability curves in
Figure 4). When the characteristic radius
R0 is determined, the principal curvature
kx2 can be determined through dimensional transformation, corresponding to a point, i.e., the black dot shown in the
Figure 4. All the bistability curves show similar trends with the different critical value, which relates to the vanishing of bistability. The scalar factor changed by geometrical parameters shows a slightly deviation for the trend and the initial dimensionless curvature
H, but an obvious difference in the dimensionless curvature
KX2. It is contrary to the scalar factor changed by layup parameters. In addition, the bistability curves of NUCM tend to be lower than that of EUCM, and the critical value is greater, which conforms to the trend shown in the Figure 9 in Ref. [
35]. The bistability of anti-symmetric cylindrical shells exists when the point is on the curve, and disappears when the point deviates from the curve. Note that the corresponding point and the bistability curve vary for different cases. In this sense, the bistability range predicted by NUCM is narrower than that by EUCM. It can be concluded that the scalar factor
ψij and characteristic radius
R0 are the most important direct factors in the NUCM affecting the bistability of anti-symmetric cylindrical shells.
4.2.2. Influence of Layup Parameters
The influence of the angle of layup α is investigated by varying α in the layup of [α/−α/α/−α], (see
Figure 5). The IUCM, EUCM, NUCM and FEA predict that the bistability exists at the layup angle ranging from 30° to 60°, 30° to 60°, 34° to 58°and 33° to 51°, respectively. All of them show a gradually increasing trend of principle curvature
kx2 and demonstrate that the angle of layup has significant influence on the bistability of anti-symmetric cylindrical shells. The results from NUCM agree well with FE results within the overlap range of layup angle. The discrepancy between results of NUCM and FEA reaches the minimum value 1% at α = 45°, and increases as α is drawn apart from 45°, not exceeding 10%.
The results of cases with
p from 4 to 8 are shown in
Figure 6. The influence of the number of plies
p on principal curvature
kx2 is based on the layup with α = 45°. It should be noted that the ply angle of the middle ply is zero when
p is odd. For example, when
p = 4 and 5, the layups are [α/−α/α/−α] and [α/−α/0°/α/−α], respectively. In these two cases, the total thickness is the same, but the thickness of each single layer is different. It is seen in
Figure 6 that the analytical and FE results show a similar trend and the number of plies
p has a little influence on the principal curvature
kx2. The discrepancy between the results of NUCM and FEA is approximately within 1%, as seen in
Table 3, where the error represents the relative error between results from FEA and NUCM.
Additionally, the influence of material parameters has not been discussed because they affect the bistability of anti-symmetric cylindrical shell through the dimensionless stiffness parameters, the same as layup parameters do (see
Figure 3).
4.2.3. Influence of Geometrical Parameters
Figure 7 shows the influence of the angle of embrace
γ and comparison the analytical and FE results. For IUCM, the principal curvature
kx2 are constant irrespective of the variation of
γ and the disappearance of bistability is unable to be predicted. As the extensible deformation assumption applied, the disappearance of bistability can be predicted, with the critical value of
γ = 63°, 88° and 94° for EUCM, NUCM and FEA, respectively, and then the principal curvature
kx2 gradually increases with the increase in
γ (see
Figure 7). The results from NUCM and FEA agree well with the relative error below 5%, as seen in
Table 4 where the error represents error between results from FEA and NUCM.
For IUCM, the principal curvature
kx2 are constant, irrespective of the variation of
l and the disappearance of bistability is unable to be predicted as well. As the extensible deformation assumption is applied, the influence of the longitudinal length
l on the principal curvature can be predicted, as shown in
Figure 8, with the critical values of
l = 30, 40 and 32 mm for EUCM, NUCM and FEA, respectively. In fact, the anti-symmetric cylindrical shell with sufficiently large length tends to roll up in the second equilibrium configuration, and thus
kx2 tends to be a constant. As the increase in
l occurs, the principal curvatures
kx2 of EUCM and FEA show a similar trend and tend to be constant when the length is over 60 and 100 mm, respectively. In contrast, the
kx2 of NUCM gradually increases and converges at a length greater than 160 mm. Nonetheless, difference between the results of NUCM and numerical results (1.0 m
−1 on average) is closer to that between the results of EUCM and numerical results (2.6 m
−1 on average), as shown in
Figure 8.
The influence of the initial natural curvature
h0 is shown in
Figure 9 in which the initial natural curvature changes while keeping the aspect ratio
r unchanged. According to IUCM, the principal curvature
kx2 can be simply calculated as
kx2 = vdh0. In addition, as the initial curvature increases, the principal curvature
kx2 grows, with the critical values of
h0 = 0.140, 0.085 and 0.137 mm
−1 for EUCM, NUCM and FEA, respectively. A sensible increment of the principal curvature is observed in
Figure 9 for numerical results and results obtained by EUCM and NUCM, which show the disappearance of bistability. It can be seen that the numerical results agree well with the results obtained by NUCM for small initial natural curvature, as listed in
Table 5 where the error-1 and error-2 represent the relative error between the results of FEA and EUCM and that of FEA and NUCM, respectively. Whilst for large initial natural curvature the results obtained by EUCM are more reliable in terms of the relative error and critical value. The possible reason for the increasing error is that a large value of
h0 may violate the thin shallow shell assumption [
34] when the total thickness is unchanged.
The influence of the anti-symmetric cylindrical shell with different planform shapes is investigated by EUCM, NUCM and FEA in
Table 6, where the error-1 and error-2 represent the relative error between results from FEA and EUCM and that from FEA and NUCM, respectively. Three different planform shapes are considered, as shown in
Figure 1. The geometric parameters are:
a = 60 mm,
b = 25 mm,
c = 5 mm. Theoretically, for NUCM, the planform shapes influence the bistability of anti-symmetric cylindrical shells through the integral region in Equation (8) and the scalar factor
ψij (
i,
j = 1,2,3,4), but only through a single scalar factor
ψ11 for EUCM. With the change in planform shape, the scalar factor
ψ11 changes slightly, resulting in the slightly deviation of the results of EUCM. The decline and increase in principal curvature
kx2 are found for results of EPS and TPS from EUCM, respectively, compared to the principal curvature of RPS. However, the contrary results are found based on the FEA results as listed in
Table 6, which indicates the lower inaccuracy of EUCM for the prediction of planform effects of anti-symmetric cylindrical shells. Compared to principal curvature of RPS, the increase in principal curvature
kx2 of EPS is correctly predicted using NUCM. In addition, a more accurate result is obtained from NUCM in contrast with the result from EUCM, as shown in error-1 and error-2 in
Table 6. Although the decline in principal curvature
kx2 of TPS is not predicted, the relative error between the results from the analytical model and simulation are reduced by utilizing NUCM, as listed in
Table 6.
The possible reason for the deviation between results of TPS from analytical model may lay on the difference of bending boundary effect for different planform shapes [
44,
45]. The bending boundary effect develops for any configuration other than the base state and is due to the disequilibrium of the bending moment. The change in local curvature occurs as the bending moment is equilibrated by the out-of-plane shear stress, which results in the change in average curvature. The local curvatures discussed later are absolute values. Due to the bending boundary effect, the local curvatures in the corner and edge of RPS decrease and increase, respectively (see
Figure 10a), where A
1, A
2, A
3, A
4 are labeled as the region near the corner; the local curvatures of EPS decrease near the region B
1, B
2, B
3, B
4 and increase in the edge between them (see
Figure 10b); the local curvatures of TPS decrease obviously near the region C
3, C
4 and less obviously near the region C
1, C
2 (see
Figure 10c). The increase in local curvature of EPS is greater than that of RPS, as shown in
Figure 10, which demonstrates the greater average principal curvature
kx2 of EPS. The curvature distribution of TPS is similar to that of RPS, and the decrease in local curvature of TPS is apparent, which demonstrates the smaller average principal curvature
kx2 of TPS, compared to that of RPS. However, in the NUCM, the transverse displacement for TPS is chosen to be the same as that for RPS for simplicity, which may be the reason why the results of TPS from NUCM are large.