Table 1:
Frequency associated with the first mode of vibration of the corner point supported thin hyperbolic paraboloid with curved edges. Comparison with Narita and Leissa (1984Narita, Y. and Leissa, A.W., (1984). Vibrations of corner point supported shallow shells of rectangular planform. Earthquake Engineering Structures. 12(5):651-61.), Chakravorty and Bandyopadhyay (1995Chakravorty, D., and Bandyopadhyay, J., (1995). On the free vibration of shallow shells. Journal of Sound and Vibration, 185(4):673-684.), and the degenerated solid approach of Dvorkin and Bathe (1984).
Table 2:
Frequency parameter λ for simply supported thin hyperbolic paraboloid shells and various values of and .
Table 3:
Mesh convergence analysis of the proposed formulation. Parameter for simply supported thin hyperbolic paraboloid shells and various values of and . (ν=0.3, a⁄b=1, and b⁄h=100)
Table 4:
Vibration analysis for distorted meshes, for the simply supported thin hyperbolic paraboloid shells. Parameter λ for the first frequency mode (ν=0.3, a⁄b=1, and b⁄h=100). 50 x 50 element mesh distorted with of 0.1L, 0.2L, and 0.4L. Comparison with the deg. solid approach of Dvorkin and Bathe (1984Dvorkin E. N., and Bathe K.-J., (1984). A continuum mechanics based four-node shell element for general nonlinear analysis. Eng. Comput. 1(1):77-88.).
Table 5:
Frequency parameter for completely clamped thin hyperbolic paraboloid shells and various values of and . (ν=0.3, a⁄b=1, b⁄h=100). Comparison with the degenerated solid approach of Dvorkin and Bathe (1984Dvorkin E. N., and Bathe K.-J., (1984). A continuum mechanics based four-node shell element for general nonlinear analysis. Eng. Comput. 1(1):77-88.).
Table 6:
Parameter for completely clamped thin shells with circular planform (Example 4): is a spherical, is hyperbolic paraboloid, is cylindrical, and is elliptic paraboloid. Comparison with Dvorkin and Bathe (1984Dvorkin E. N., and Bathe K.-J., (1984). A continuum mechanics based four-node shell element for general nonlinear analysis. Eng. Comput. 1(1):77-88.).
Table 7:
Frequency parameter for completely clamped moderately thick hyperbolic paraboloid shells and various values of and (Example 5), (ν=0.3, a⁄b=1, b⁄h=10). Comparison with the degenerated solid approach of Dvorkin and Bathe (1984Dvorkin E. N., and Bathe K.-J., (1984). A continuum mechanics based four-node shell element for general nonlinear analysis. Eng. Comput. 1(1):77-88.).
Table 8:
Frequency parameter λ for simply supported thin cylindrical shells and spherical shells (Example 6) for various values ofand (ν=0.3, a⁄b=1, b⁄h=100). Comparison with Liew and Lim (1996Liew, K., and Lim, C., (1996). Vibration of doubly-curved shallow shells. Acta Mechanica, 114:95-119.) and the degenerated solid approach of Dvorkin and Bathe (1984Dvorkin E. N., and Bathe K.-J., (1984). A continuum mechanics based four-node shell element for general nonlinear analysis. Eng. Comput. 1(1):77-88.).
Table 9:
Frequency parameter λ´ (=) for clamped (at the corners) thin spherical shells and various values of . Comparison with ANSYS (ANSYS, 2011ANSYS, (2011). Academic Research Mechanical, Release 14, Help System. ) and the degenerated solid approach of Dvorkin and Bathe (1984Dvorkin E. N., and Bathe K.-J., (1984). A continuum mechanics based four-node shell element for general nonlinear analysis. Eng. Comput. 1(1):77-88.).
Table 10:
Frequency parameter λ´ (=) for clamped (at the corners) thin cylindrical shells and various values of . Comparison with ANSYS (ANSYS, 2011ANSYS, (2011). Academic Research Mechanical, Release 14, Help System. ) and the degenerated solid approach of Dvorkin and Bathe (1984Dvorkin E. N., and Bathe K.-J., (1984). A continuum mechanics based four-node shell element for general nonlinear analysis. Eng. Comput. 1(1):77-88.).