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Vibration and nonlinear dynamic response of eccentrically stiffened functionally graded composite truncated conical shells surrounded by an elastic medium in thermal environments

Abstract

A semi-analytical approach to eccentrically stiffened functionally graded truncated conical shells surrounded by an elastic medium in thermal environments is presented. Based on the classical thin shell theory with geometrical nonlinearity in von Karman Donnell sense, the smeared stiffeners technique and the Galerkin method, this paper deals with vibration and nonlinear dynamic problems. The truncated conical shells are reinforced by ring stiffeners made of full metal or full ceramic depending on the situation of the stiffeners at the metal-rich or ceramic-rich side of the shell, respectively. In addition, the study not only assume that the material properties depend on environment temperature variation, but also consider the thermal stresses in the stiffeners. Numerical results are given to evaluate effects of inhomogeneous, dimensional parameters, outside stiffeners, temperatures and elastic foundations on the vibration and nonlinear dynamic response of the structures.

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Abbreviations

N :

The volume fraction index (nonnegative number)

w :

The deflection of the truncated conical shell

\(K_{\mathrm{w}} \) :

The Winkler foundation modulus

\(K_\mathrm{p}\) :

The shear layer foundation stiffness of the Pasternak model

\(\varepsilon _S^0 , \varepsilon _\theta ^0 \) :

The normal strains

\(\gamma _{S\theta }^0 \) :

The shear strain at the middle surface of the truncated conical shell

\(k_S ,k_\theta ,k_{S\theta } \) :

The changes of curvatures and twist

\(A_1 ,A_2 \) :

The cross-sectional area of eccentrically longitudinal and latitude stiffeners, respectively

\(d_1 , d_2 , h_1 , h_2 \) :

The width and height of eccentrically longitudinal and latitude stiffeners, respectively

\(n_1 , n_2 \) :

The numbers of eccentrically longitudinal and latitude stiffeners, respectively

\(E_0 \) :

Young’s modulus of the stiffeners; \(E_0 =E_\mathrm{c} \) if the stiffeners are reinforced at the surface of the ceramic-rich, \(E_0 =E_\mathrm{m} \) if the stiffeners are reinforced at the surface of the metal-rich side

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Acknowledgements

This work has been supported/partly supported by Vietnam National University, Hanoi (VNU), under Project No. QG.18.37. The authors are grateful for this support.

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Correspondence to Nguyen Dinh Duc.

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Chan, D.Q., Anh, V.T.T. & Duc, N.D. Vibration and nonlinear dynamic response of eccentrically stiffened functionally graded composite truncated conical shells surrounded by an elastic medium in thermal environments. Acta Mech 230, 157–178 (2019). https://doi.org/10.1007/s00707-018-2282-4

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  • DOI: https://doi.org/10.1007/s00707-018-2282-4