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Dynamic analysis of stepped composite cylindrical shells surrounded by Pasternak elastic foundations based on the continuous element method

Le Thi Bich Nam, Nguyen Manh Cuong, Thinh Ich Tran, Le Quang Vinh


This research presents a continuous element model for solving vibration problems of stepped composite cylindrical shells surrounded by Pasternak foundations with various boundary conditions. Based on the First Order Shear Deformation Theory (FSDT), the equations of motion of the circular cylindrical shell are introduced and the dynamic stiffness matrix is obtained for each segment of the uniform shell. The interesting assembly procedure of continuous element method (CEM) is adopted to analyze the dynamic behavior of the stepped composite cylindrical shell surrounded by an elastic foundation. Free vibrations and harmonic responses of different configurations of stepped composite cylindrical shells on elastic foundations are examined. Effects of structural parameters, stepped thickness and elastic foundations on the free vibration responses of stepped composite cylindrical shells are also presented. Comparisons with previously published results and finite element (FE) analyses show that the proposed technique saves data storage volume and calculating time, and is accurate and efficient for widening the studied frequency range.


stepped shell; vibration of cylindrical shell; composite shell; elastic foundation; Pasternak foundation; continuous element method; dynamic stiffness method

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M. K. Ahmed. Natural frequencies and mode shapes of variable thickness elastic cylindrical shells resting on a Pasternak foundation. Journal of Vibration and Control, 22, (1), (2016), pp. 37–50. doi:10.1177/1077546314528229.

Y. Qu, Y. Chen, X. Long, H. Hua, and G. Meng. Free and forced vibration analysis of uniform and stepped circular cylindrical shells using a domain decomposition method. Applied Acoustics, 74, (3), (2013), pp. 425–439. doi:10.1016/j.apacoust.2012.09.002.

S.Wu, Y. Qu, and H. Hua. Free vibration of laminated orthotropic conical shell on Pasternak foundation by a domain decomposition method. Journal of Composite Materials, 49, (1), (2015), pp. 35–52. doi:10.1177/0021998313514259.

A. H. Sofiyev and N. Kuruoglu. Natural frequency of laminated orthotropic shells with different boundary conditions and resting on the Pasternak type elastic foundation. Composites Part B: Engineering, 42, (6), (2011), pp. 1562–1570. doi:10.1016/j.compositesb.2011.04.015.

A. H. Sofiyev, H. Halilov, and N. Kuruoglu. Analytical solution of the dynamic behavior of non-homogenous orthotropic cylindrical shells on elastic foundations under moving loads. Journal of Engineering Mathematics, 69, (4), (2011), pp. 359–371. doi:10.1007/s10665-010-9392-x.

E. Bagherizadeh, Y. Kiani, and M. R. Eslami. Mechanical buckling of functionally graded material cylindrical shells surrounded by Pasternak elastic foundation. Composite Structures, 93, (11), (2011), pp. 3063–3071. doi:10.1016/j.compstruct.2011.04.022.

Y. W. Kim. Free vibration analysis of FGM cylindrical shell partially resting on Pasternak elastic foundation with an oblique edge. Composites Part B: Engineering, 70, (2015), pp. 263–276. doi:10.1016/j.compositesb.2014.11.024.

D. V. Dung and N. T. Nga. Nonlinear buckling and post-buckling of eccentrically stiffened functionally graded cylindrical shells surrounded by an elastic medium based on the first order shear deformation theory. Vietnam Journal of Mechanics, 35, (4), (2013), pp. 285–298. doi:10.15625/0866-7136/35/4/3116.

J. R. Banerjee and A. J. Sobey. Dynamic stiffness formulation and free vibration analysis of a three-layered sandwich beam. International Journal of Solids and Structures, 42, (8), (2005), pp. 2181–2197. doi:10.1016/j.ijsolstr.2004.09.013.

J. R. Banerjee and F. W. Williams. Coupled bending-torsional dynamic stiffness matrix for Timoshenko beam elements. Computers & Structures, 42, (3), (1992), pp. 301–310. doi:10.1016/0045-7949(92)90026-v.

J. B. Casimir, M. C. Nguyen, and I. Tawfiq. Thick shells of revolution: Derivation of the dynamic stiffness matrix of continuous elements and application to a tested cylinder. Computers & Structures, 85, (23), (2007), pp. 1845–1857. doi:10.1016/j.compstruc.2007.03.002.

T. I. Thinh and M. C. Nguyen. Dynamic stiffness matrix of continuous element for vibration of thick cross-ply laminated composite cylindrical shells. Composite Structures, 98, (2013), pp. 93–102. doi:10.1016/j.compstruct.2012.11.014.

T. I. Thinh, M. C. Nguyen, and D. G. Ninh. Dynamic stiffness formulation for vibration analysis of thick composite plates resting on non-homogenous foundations. Composite Structures, 108, (2014), pp. 684–695. doi:10.1016/j.compstruct.2013.10.022.

L. T. B. Nam, N. M. Cuong, and T. I. Thinh. Continuous element formulation for vibration of thick composite annular plates and rings. In Proceeding of International Conference on Engineering Mechanics (ICEMA 3), Vol. 2, Hanoi, Vietnam, (2014). pp. 319–324.

L. T. B. Nam, N. M. Cuong, and T. I. Thinh. Continuous element formulation for thick composite annular plates and rings surrounded by elastic foundation. In Proceeding of International Conference on Engineering Mechanics (ICEMA 3), Vol. 2, Hanoi, Vietnam, (2014). pp. 387–394.

N. M. Cuong, L. T. B. Nam, and T. I. Thinh. A new continuous element for vibration analysis of stepped composite annular plates and rings. In Proceedings of National Conference on Composite Material and Structure, Nha Trang, Vietnam, (2016). pp. 103–110.

N. M. Cuong, L. Q. Vinh, T. I. Thinh, and N. T. T. Hoan. Continuous element formulation for composite combined conical-cylindrical shells on elastic foundations. In Proceeding of the 12th National Conference on Mechanics, Da Nang, Vietnam, (2015). pp. 281–288.





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