Nonlinear vibration of FGM moderately thick toroidal shell segment within the framework of Reddy’s third order-shear deformation shell theory

  • Pham Minh Vuong
  • Nguyen Dinh DucEmail author


Nonlinear vibration and dynamic response of functionally graded moderately thick toroidal shell segments resting on Pasternak type elastic foundation are investigated in this paper. Functionally graded materials are made from ceramic and metal, and the volume fraction of constituents are assumed to vary through the thickness direction according to a power law function. Reddy’s third order shear deformation, von Karman nonlinearity, Airy stress function method and analytical solutions are used to derive the governing equations. Galerkin method is used to convert the governing equation into nonlinear differential equation, then the explicit expressions of natural frequencies and nonlinear frequency–amplitude relations are obtained. Using Runge–Kutta method, the nonlinear differential equation of motion is solved, and then nonlinear vibration and dynamic response of shells are analyzed. The effects of temperature, material and geometrical properties, and foundation parameters on nonlinear vibration and dynamic characteristics are investigated and discussed in detail.


Nonlinear vibration FGM toroidal shell segment Reddy’s third order shear deformation shell theory 



This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 107.02-2018.04. The authors are grateful for this support.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interests.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Faculty of Civil and IndustrialNational University of Civil EngineeringHai Ba Trung DistrictVietnam
  2. 2.NTT Institute of High TechnologyNguyen Tat Thanh UniversityHo Chi Minh CityVietnam
  3. 3.Advanced Materials and Structures LaboratoryVNU Hanoi - University of Engineering and TechnologyCau Giay DistrictVietnam
  4. 4.National Research Laboratory, Department of Civil and Environmental EngineeringSejong UniversitySeoulKorea

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