Nonlinear free vibration of reticulated shallow spherical shells taking into account transverse shear deformation

doi:10.1007/s10483-018-2399-9

Abstract

Abstract: This paper deals with nonlinear free vibration of reticulated shallow spherical shells taking into account the effect of transverse shear deformation. The shell is formed by beam members placed in two orthogonal directions. The nondimensional fundamental governing equations in terms of the deflection, rotational angle, and force function are presented, and the solution for the nonlinear free frequency is derived by using the asymptotic iteration method. The asymptotic solution can be used readily to perform the parameter analysis of such space structures with numerous geometrical and material parameters. Numerical examples are given to illustrate the characteristic amplitudefrequency relation and softening and hardening nonlinear behaviors as well as the effect of transverse shear on the linear and nonlinear frequencies of reticulated shells and plates.

Key words: singular perturbation, reaction diffusion equation, delay, upperand lower solutions, uniformly validity, amplitude-frequency relation, nonlinear free vibration, reticulated shallow spherical shell, transverse shear effect, asymptotical iteration method

2010 MSC Number:

74K25
74H45

Rong WANG, Guohua NIE. Nonlinear free vibration of reticulated shallow spherical shells taking into account transverse shear deformation. Applied Mathematics and Mechanics (English Edition), 2018, 39(12): 1825-1836.

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References

[1] MCDANIEL, T. J. and CHANG, K. J. Dynamics of rotationally periodic large space structures. Journal of Sound and Vibration, 68(3), 351-368(1980)
[2] WILLIAMS, F. W. An algorithm for exact eigenvalue calculations for rotationally periodic structures. International Journal for Numerical Methods in Engineering, 23, 609-622(1986)
[3] NOOR, A. K. Continuum modeling for repetitive structures. Applied Mechanics Reviews, 41, 285-296(1988)
[4] MOREAU, G. and CAILLERIE, D. Continuum modeling of lattice structures in large displacement applications to buckling analysis. Computers and Structures, 68, 181-189(1998)
[5] LIU, R. H., LI, D., NIE, G. H., and CHENG, Z. Q. Non-linear buckling of squarely-latticed shallow spherical shells. International Journal of Non-Linear Mechanics, 26(5), 547-565(1991)
[6] NIE, G. H. and LIU, R. H. Non-linear elastic theory of rectangular reticulated shallow shell structures. Applied Mathematics and Mechanics (English Edition), 15(5), 413-423(1994) https://doi.org/10.1007/BF02451491
[7] NIE, G. H. Non-linear vibration of rectangular reticulated shallow shell structures. Applied Mathematics and Mechanics (English Edition), 15(6), 525-535(1994) https://doi.org/10.1007/BF02450765
[8] NIE, G. H. and CHEUNG, Y. K. A non-linear model for stability analysis of reticulated shallow shells with imperfections. International Journal of Space Structures, 10(4), 215-230(1995)
[9] NIE, G. H. An asymptotic analysis on non-linear free vibration of squarely-reticulated circular plates. Structural Engineering and Mechanics, 8(6), 547-560(1999)
[10] NIE, G. H. Non-linear free vibration of single-layer reticulated shallow spherical shells. International Journal of Space Structures, 15(1), 53-58(2000)
[11] NIE, G. H. and LI, Z. W. Nonlinear analysis of imperfect squarely-reticulated shallow spherical shells. Science in China Series G, 50(1), 109-117(2007)
[12] NIE, G. H. On the buckling of imperfect squarely-reticulated shallow spherical shells supported by elastic media. Thin-Walled Structures, 41(1), 1-13(2003)
[13] FAN, F., YAN, J. C., and CAO, Z. G. Stability of reticulated shells considering member buckling. Journal of Constructional Steel Research, 77, 32-42(2012)
[14] FAN, F., WANG, D. Z., ZHI, X. D., and SHEN, S. Z. Failure modes of reticulated domes subjected to impact and the judgment. Thin-Walled Structures, 48, 143-149(2010)
[15] LÓPEZ, A., PUENTE, I., and SERNA, M. Numerical model and experimental tests on single-layer latticed domes with semi-rigid joints. Computers and Structures, 85, 360-374(2007)
[16] LÓPEZ, A., PUENTE, I., and SERNA, M. Direct evaluation of the buckling loads of semi-rigidly jointed single-layer latticed domes under symmetric loading. Engineering Structures, 29, 101-109(2007)
[17] MA, H. H., FAN, F., CHEN, G. B., CAO, Z. G., and SHEN, S. Z. Numerical analyses of semirigid joints subjected to bending with and without axial force. Journal of Constructional Steel Research, 90, 13-28(2013)
[18] MA, H. H., FAN, F., WEN, P., ZHANG, H., and SHEN, S. Z. Experimental and numerical studies on a single-layer cylindrical reticulated shell with semi-rigid joints. Thin-Walled Structures, 86, 1-9(2015)
[19] LIU, H. B., CHEN, Z. H., HAN, Q. H., CHEN, B. B., and BU, Y. D. Study on the thermal behavior of aluminum reticulated shell structures considering solar radiation. Thin-Walled Structures, 85, 15-24(2014)
[20] SABZIKAR, B. M. and ESLAMI, M. R. Axisymmetric snap-through behavior of piezo-FGM shallow clamped spherical shells under thermo-electro-mechanical loading. International Journal of Pressure Vessels and Piping, 120-121, 19-26(2014)
[21] KATO, S., UEKI, T., and MUKAIYAMA, Y. Study of dynamic collapse of single layer reticular dome subjected to earthquake motion and the estimation of statically equivalent seismic forces. International Journal of Space Structures, 12, 191-204(1997)
[22] KUMAGAI, T. and OGAWA, T. Dynamic buckling behavior of single layer lattice domes subjected to horizontal step wake. Journal of the International Association for Shell and Spatial Structures, 44(3), 167-174(2003)
[23] YU, Z. W., ZHI, X. D., FAN, F., and LU, C. Effect of substructures upon failure behavior of steel reticulated domes subjected to the severe earthquake. Thin-Walled Structures, 49(9), 1160-1170(2011)
[24] LI, Y. G., FAN, F., and HONG, H. P. Effect of support flexibility on seismic responses of a reticulated dome under spatially correlated and coherent excitations. Thin-Walled Structures, 82, 343-351(2014)
[25] KATO, S., KIM, J. M., and CHEONG, M. C. A new proportioning method for member sections of single layer reticulated domes subjected to uniform and non-uniform loads. Engineering Structures, 25(10), 1265-1278(2003)
[26] LI, Y. Q., TAMURA, Y., YOSHIDA, A., KATSUMURA, A., and CHO, K. Wind loading and its effects on single-layer reticulated cylindrical shells. Journal of Wind Engineering and Industrial Aerodynamics, 94(12), 949-973(2006)
[27] HE, Y. J., ZHOU, X. H., and ZHANG, X. T. Finite element analysis of the elastic static properties and stability of pretensioned cylindrical reticulated mega-structures. Thin-Walled Structures, 60, 1-11(2012)
[28] LI, P. C., WU, M., and XING, P. J. Novel cable-stiffened single-layer latticed shells and their stabilities. Journal of Constructional Steel Research, 92, 114-121(2014)
[29] HE, Y. J., ZHOU, X. H., and LIU, D. Research on stability of single-layer inverted catenary cylindrical reticulated shells. Thin-Walled Structures, 82, 233-244(2014)
[30] MALEK, S., WIERZBICKI, T., and ANDOCHSENDORF, J. Buckling of spherical cap gridshells:a numerical and analytical study revisiting the concept of the equivalent continuum. Engineering Structures, 75, 288-298(2014)
[31] WANG, R. and NIE, G. H. Non-linear modeling and analysis of reticulated cylindrical shells taking into account the lateral bending moment. Chinese Quarterly of Mechanics, 38(1), 34-42(2017)
[32] LI, Q. S., LIU, J., and TANG, J. Buckling of shallow spherical shells including the effects of transverse shear deformation. International Journal of Mechanical Sciences, 45(9), 1519-1529(2003)
[33] GROH, R. M. J. and WEAVER, P. M. Buckling analysis of variable angle tow, variable thickness panels with transverse shear effects. Composite Structures, 107, 482-493(2014)