Fig 1.
Geometric parameters and coordinate system of the cross-section of a hyperbolic shell element.
Fig 2.
The coordinate and geometric description of unified model of composite laminated stiffened cylindrical and conical shells.
Table 1.
Parameters related to the conversion between cylindrical or conical shells.
Fig 3.
Boundary spring and coupling spring settings for composite laminated stiffened shell.
(a) boundary spring settings. (b) coupling spring setting of shell. (c) coupling spring setting of curved beam. (d) coupling spring of shell and beam.
Table 2.
Convergence analysis of the frequency parameter Ω of laminated stiffened cylindrical shells under fixed boundary conditions.
Table 3.
Convergence analysis of the frequency parameter Ω of laminated stiffened conical shells.
Fig 4.
The variation curve of frequency parameter Ω for stiffened conical shells under different truncation values.
(a) Open stiffened conical shell. (b) Closed stiffened conical shell.
Fig 5.
Mode shapes of the composite laminated stiffened cylindrical shell.
Open stiffened cylindrical shell: Present method - 1st order, 2nd order, 3rd order, 4th order. FEM - 1st order, 2nd order, 3rd order, 4th order; Closed stiffened cylindrical shell: Present method - 1st order, 2nd order, 3rd order, 4th order. FEM - 1st order, 2nd order, 3rd order, 4th order.
Fig 6.
Variation curves of the first four frequency parameter Ω of the open composite cylindrical shell under different boundary spring stiffness values.
(a) Stiffness value of linear springs. (b) Stiffness value of torsional springs.
Table 4.
The stiffness value of the boundary spring under different boundary conditions.
Fig 7.
Variation curves of the frequency parameter Ω of the closed composite cylindrical shell under different shell internal coupling spring stiffness values.
(a) Boundary condition CC. (b) Boundary condition SS.
Fig 8.
Variation curves of the frequency parameter Ω of the open composite stiffened conical shell under different coupling springs stiffness values between the shell and beams.
(a) Boundary condition CCCC. (b) Boundary condition SSSS.
Table 5.
Frequency parameters Ω of the open composite stiffened cylindrical shell under different shell boundary conditions and parameter condition.
Fig 9.
Variation curves of frequency parameter Ω of the open composite stiffened cylindrical shell with the thickness ratio hs/Rs or rotation angle ϑ under different boundary conditions.
(a) ϑ = 90°, 1st order. (b) ϑ = 90°, 3rd order. (c) hs/Rs = 0.01, 1st order. (d) hs/Rs = 0.01, 3rd order.
Fig 10.
variation curves of the first few mode frequency parameters Ω of the closed composite laminated stiffened conical shell with the length ratio Ls/R1 under different layer angles and apex angle φ.
(a) [0°/90°],2nd order. (b) [0°/90°], 4th order. (c) [-90°/0°/90°],2nd order. (d) [-90°/0°/90°],4th order.
Table 6.
Frequency parameter Ω of closed composite laminated stiffened conical shells under different parameters.
Table 7.
Frequency parameter Ω of composite laminated stiffened shells under different number of stiffeners.
Table 8.
Frequency parameter Ω of composite laminated stiffened conical shell under different number of stiffeners.
Fig 11.
Modal diagrams of composite closed stiffened cylindrical shells under different number of stiffeners.
1st order: n = 0, n = 1, n = 2, n = 3; 3rd order: n = 0, n = 1, n = 2, n = 3.
Fig 12.
variation curves of the frequency parameters Ω of the composite laminated rotationally stiffened shell with the width of the laminated curved beam under different thickness-to-width ratio.
(a) Open stiffened cylindrical shell, bn/hn = 0.5. (b) Open stiffened cylindrical shell, bn/hn = 1. (c) Open stiffened conical shell,bn/hn = 0.5. (d) Open stiffened conical shell,bn/hn = 1.
Table 9.
Frequency parameter Ω of isotropic rotationally stiffened shells under different boundary conditions.
Fig 13.
Layout of the closed stiffened cylindrical shell and the experimental setup.
Fig 14.
Natural frequencies and mode shapes of the stiffened cylindrical shell obtained by the test and the present method.
Test result: f3 = 373.42, f5 = 645.31, f9 = 690.87; Present method: f3 = 376.77, f5 = 630.60, f9 = 679.06.