Exact solutions for the vibration of circumferentially stepped orthotropic circular cylindrical shells

Abstract

The combination of Flügge's shell theory, the transfer matrix approach and the Romberg integration method are used to investigate the free vibration behaviour of stepped orthotropic cylindrical shells. The hoop step on the shell surface is described by a reduced thickness over part of its circumference. Modal displacements of the shell can be described by trigonometric functions and Fourier's approach is used to separate the variables. The vibration equations of the shell are reduced to eight first-order differential equations in the circumferential coordinate, and by using the transfer matrix of the shell, these equations can be written in a matrix differential equation. The transfer matrix is derived from the non-linear differential equations of the cylindrical shells by introducing the trigonometric functions in the longitudinal direction and applying the numerical integration in the circumferential direction. The proposed model is used to get the vibration frequencies and the corresponding mode shapes for symmetrical and antisymmetrical type-modes. Computed results indicate the sensitivity of the frequency parameters and the bending deformations to the geometry of stepped shell, and also to the axial and circumferential rigidities of the shell.

Thermal gradient effect on the free vibration of a cardioid cross-section cylindrical shell

“…In references (Khalifa, 2011a(Khalifa, , b, 2014(Khalifa, , 2015, number of comparisons of numerical results is presented to verify the validity of the used framework and approach method.…”

“…Many pioneering studies have been carried out on the vibration analysis of the elastic cylindrical shells and structural systems, and some of them can be found, for instance, in (Smith, 1968;Chung, 1981;Joseph et al, 1987;Zhang et al, 2001;Toorani and Lakis, 2004;Gokhan and Burak, 2007;Xuebin, 2008;Liu et al, 2012;Xin et al, 2011;Khalifa, 2011a;El-Kaavazi and David, 2012;Lu et al, 2013;Yuehua et al, 2013;Tomabene et al, 2014a). Unlike the circular cylindrical shells, the study of vibration behavior for the noncircular cross-section shells has been received little attention in the past decade, but only a few important studies are concerned with the vibration analysis of the shells with three-lobed, four-lobed, oval, panel and elliptical cross-section (see for example (Culberson and Boyd, 1971;Sewel and Pusey, 1971;Chen and Kempner, 1978;Koumousis and Armenakas, 1983;Yamada et al, 1984Yamada et al, , 1985Sodatos, 1999;Ganapathi et al, 2003Ganapathi et al, , 2004Tomabene and Fantuzzi, 2013;Tomabene et al 2014b;Khalifa, 2010Khalifa, , 2011bKhalifa, ,c, 2014Khalifa, , 2015 ).…”

Buckling behavior of a radially loaded corrugated orthotropic thin-elliptic cylindrical shell on an elastic foundation

“…As is evident by Khalifa [34][35][36][37][38], the author is presented a number of comparisons of numerical results to verify the validity of the used framework and approach method and accuracy of the results. The comparison studies are calculated by different approaches and have confirmed the correctness of the proposed shell theory and solution method for studying the mechanical behavior for such thin-shell structures.…”

Winkler and Pasternak foundations effect on the free vibration of an orthotropic oval cylindrical shell with variable thickness

“…In references, Khalifa [27][28][29], number of comparisons of numerical results is presented to verify the validity of the used framework and approach method. The comparison studies of natural frequencies calculated by different approaches have confirmed the correctness of the proposed shell theory and solution method for studying the vibration behavior for such shells resting on a Pasternak-type foundation.…”