Free vibration characteristic of laminated conical shells based on higher-order shear deformation theory
Introduction
The increasingly usage of composite laminates for the high-tech applications in aerospace, naval, architecture and chemical industries is owing to their ability to tailor the mechanical properties. Moreover, the composites offer high stiffness to weight ratio, strength to weight ratios, better temperature resistant and shock absorbent characteristics than the homogeneous ones. In fact, the vibration behaviour of any structure is definitely affected by its dynamic characteristics. Thus, during numerical analysis, there should be relation with the theory to help in modelling the structure [31], [11]. Therefore, to model the composite structures there are various theories such as the Classical Shell Theory (CST) based on Love-Kirchhoff assumptions, which ignore transverse shear strains [28]. However, classical theories are only applicable to the thin elastic shells. The importance of the transverse shear and normal stresses are highlighted by Reissner [36]. As, Reissner [37] and Mindlin [26] included the shear deformation in the shell theory, which later is named as the First-order Shear Deformation Theory (FSDT). The First-order Shear Deformation Theory assumes constant transverse shear stress and require a shear correction factor. The restrictions of the Classical Theory (CST) and the First-order Shear Deformation Theory (FSDT) motivated researchers to develop Higher-order Shear Deformation Theories (HSDT), which can accurately evaluate the transverse shear stresses, effectively exist in thick plates and shells. Several Higher-order Shear Deformation Theories are developed to accurately evaluate the transverse shear stresses. In these theories the displacements are expanded up to any desired degree in terms of thickness coordinates [43], [30] (Vinson, 2001; Noor et al., 1996). Such as, the Second-order Shear Deformation formulation by Whitney and Sun [47] and the Third-order Shear Deformation Theory (TSDT) in which, the displacements are expanded up to the cubic term in thickness coordinates to have quadratic variation of transverse shear strains and transverse shear stresses through the shell thickness. There are number of Third-order Shear Deformation Theories depending on different number of unknowns. Among them the Third-order Shear Deformation Theory by Lo et al., [24] with 11 unknowns, Kant [19] with six unknowns, Bhimaraddi and Stevens [1] with five unknowns, Hanna and Leissa [13] with four unknowns, and the Third-order Shear Deformation Theory of Reddy [34], [35] with five unknowns is widely used because it can represent transverse shear stresses in simple and efficient way without the need for shear correction coefficient [33].
The efficiency of aforementioned higher-order shear deformation theories depends on analysis of composite structures. The detail explanation of composite structures that require these kinematic models can be seen in work of Reddy [35], Tornabene et al. [42], Carrera et al. [2] and Carrera [3], [4], [5] contributions to the development of these HSDTs for the analysis of composite structures is considered to be most significant.
Usually, the solution of vibrational problem must require a numerical method. Therefore, the literature narrates that the most popular available approach is the finite element method (FEM), which is the weak formulation of the governing equations. Finite element formulation was used by Kumar et al. [21] to investigate the vibration of composite skew hypar shells. Moreover, Kumar et al. [23] studied free vibration of skew cylindrical shells using finite element formulation. However, different alternative methods based on strong form of the governing equations such as generalized differential quadrature method (GDQ) prove to be precise and reliable tools for analysis of composite structure [10], [42], [7]. Furthermore, Viola et al. [45] studied static analysis of functionally graded conical shells using generalized differential quadrature method. Moreover, Viola et al. [44] did a comparison study of available literature on free vibration of laminated shell for first order shear deformation theory and higher order shear deformation theory. Tonabene et al. [41] analyse the stress and strain recovery shells resting on elastic foundation using generalized differential quadrature technique (GDQ) based on HSDT. Apart from FEM and GDQ methods some other methods are used to analyse shell structures. Among them Galerkin method was used by Duc [8] to analyse the nonlinear dynamic response of higher order shear deformable cylindrical shells resting on elastic foundations. The discrete singular convolution method and differential quadrature method are used to examine free vibration of curved structural components with different material properties such as isotropic, laminated and functionally graded material (FGM) based on HSDT [9]. Moreover, Hwu et al. [14] studied the free vibration of composite plates and cylindrical shells based on HSDT using Navier’s solution, Levy’s solution and Ritz method. It should be noted that most of the studies are based on higher-order models analyse shell panels not complete shells. Among them, Zghal et al. [49] used finite element method to examine the free vibration of functionally graded cylindrical shells panels. Mehar and Panda [25] analysed functionally graded doubly curved shell panels. Moreover, Nasihatgozar et al. [29] studied free vibration of doubly curved panels using Galerkin Method and Zine et al. [48] used Navier-type, closed form solutions to analyse the bending and free vibration of plates and shell panels. Also, differential quadrature method was used for the free vibration analysis of functionally graded composite spherical panels by Setoodeh et al. [38]. Hajlaoui et al. [12], Punera and Kant [32], Mohammadimehr et al. [27], Sharma et al. [39] and Thakur et al. [40] are also among those to analyse shell elements. Apart from vibration Correia et al. [6], Kumar et al. [22] and Tornabene et al. [41] examined the static analyses of conical shells using HSDT. Furthermore, Jedari Salami [18] and Khare et al. [20] analyse beams and shell elements using HSDT.
Apart from all the above mentioned methods present study used spline method to find the solutions of linear differential equations elegantly and accurately. It yields fast convergence and better accuracy for lower-order approximation as compared to a global higher-order approximation.
From the comprehensive review of the literature it is clear that the isotropic and composite structures using different methods incorporating with HSDT and different types of end conditions have been extensively studied for their static and vibration responses. Most of studies considered shell panels not complete shell. However, no studies investigated the free vibration of the conical shells using spline approximation by implementing HSDT been reported till date. Further, numerical studies on the free vibration behavior of laminated composite conical shells of different number of layers with each layer consisting of different material is yet to be reported. This study aims at filling this knowledge gap by investigating the free vibration of composite conical shells by approximating the displacement fields using cubic and quintic splines based on HSDT.
Therefore, present study is unique in the sense to analyse free vibration of conical shells using spline method. The spline method was not used by any of the above mentioned researchers however used by Javed et al. [15], [16], [17] to solve the free vibrational problems of plates based on First-order Shear Deformation Theory and Higher-order Shear Deformation Theory. The aim of present study is to extend the free vibrational research of plates to shells using HSDT. Therefore, purpose of this research is to examine the free vibration of cross-ply laminated conical shells based on HSDT using spline approximation. The kinematics of shell is based on TSDT. Displacement and rotational functions are approximated by cubic and quintic splines. Collocation with these splines yields a set of field equations which along with the equations of boundary conditions, reduce to system of homogeneous simultaneous algebraic equations on the assumed spline coefficients. Then the problem is solved using eigensolution technique to obtain the frequency parameter. The eigenvector are the spline coefficients from which the mode shapes are constructed. Frequency of conical shells was studied by varying cone angle, circumferential node number, length ratio, stacking sequence, and different lamination materials for simply supported boundary condition. Graphs and tables narrates the obtained results.
Section snippets
Formulation
The displacement field considered according to third order shear deformation theory [33]were and are the displacement components in the and directions respectively, and and are the in-plane displacements of the middle plane and and are the shear rotations of any point on the middle surface.
Results and discussion
The higher order shear deformation theory is used to investigate the free vibration of conical shells for simply supported boundary condition. All numerical computations, unless otherwise stated, two materials are considered: Kevlar-49/epoxy (KE) and Graphite/Epoxy (AS4/3501-6) (GE). Conical shells of two, three, four, five and six layered having cross-ply orientations are considered for analysis.
Conclusion
This study is about free vibrational analysis of cross-ply conical shells under higher-order shear deformation theory. The vibration characteristic of the conical shells is examined for circumferential node number, length ratio, cone angle, different number of layers and stacking sequences are analyzed for simply supported boundary conditions. It is concluded that variation of the geometric parameters and materials effect the frequency of conical shells. The applicability and accuracy of
Acknowledgments
This work was completed by Dr. Saira Javed supported by King Faisal University, Kingdom of Saudi Arabia.
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