Abstract
The idea of Green quasifunction method is clarified in detail by considering a free vibration problem of simply-supported trapezoidal shallow spherical shell on Winkler foundation. A Green quasifunction is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem. The mode shape differential equation of the free vibration problem of simply-supported trapezoidal shallow spherical shell on Winkler foundation is reduced to two simultaneous Fredholm integral equations of the second kind by Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equation, the irregularity of the kernel of integral equations is avoided. Finally, natural frequency is obtained by the condition that there exists a nontrivial solution in the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the Green quasifunction method.
Similar content being viewed by others
References
Rvachev, V.L., Theory of R-function and Some of its Application. Kiev: Nauk Dumka, 1982, 415–421 (in Russian).
Yuan, H., Green quasifunction method for thin plates on Winkler foundation. Chinese Journal Computational Mechanics, 1999, 16(4): 478–482 (in Chinese).
Wang, H. and Yuan, H., Application of Green quasifunction method in elastic torsion. Journal of South China University of Technology (Nature Science Edition), 2004, 32(11): 86–88 (in Chinese).
Wang, H. and Yuan, H., Application of R-function theory to the problem of elastic torsion with trapezium sections. Journal of Huazhong University of Science and Technology (Nature Science Edition), 2005, 33(11): 99–101 (in Chinese).
Yuan, H., Li, S.Q. and Liu, R., Green quasifunction method for vibration of simply-supported thin polygonic plates on Pasternak foundation. Applied Mathematics and Mechanics (English Edtion), 2007, 28(7): 847–853.
Cheng, X.S., On problems of optimal design of shallow shell with double curvature on elastic foundation. Applied Mathematics and Mechanics, 1986, 7(3): 259–263.
Huang, Y. and He, F.S., Beams, Plates and Shells on Elastic Foundation. Beijing: Science Press, 2005 (in Chinese).
Ortner, V.N., Regularisierte faltung von distributionen. Teil 2: Eine tabelle von fundamentallocunngen. Zeitschrift für angewandte Mathematik und Physik, 1980, 31: 155–173.
Kurpa, L.V., Solution of the problem of deflection and vibration of plates by the R-function method. Soviet Appllied Mechanics, 1984, 20(5): 470–473.
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by Foundation of MOE Key Laboratory of Disaster Forecast and Control in Engineering.
Rights and permissions
About this article
Cite this article
Li, S., Yuan, H. Green Quasifunction Method for Free Vibration of Simply-Supported Trapezoidal Shallow Spherical Shell on Winkler Foundation. Acta Mech. Solida Sin. 23, 370–376 (2010). https://doi.org/10.1016/S0894-9166(10)60038-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1016/S0894-9166(10)60038-9