Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations

The Generalized Differential Quadrature (GDQ) method is applied to study the dynamic behavior of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature from the beginning of the theory formulation a generalization of the kinematical model is adopted for the Reissner–Mindlin and Toorani–Lakis theory. By so doing a generalization of the theory of anisotropic doubly-curved shells and panels of revolution is proposed. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. The Differential Quadrature (DQ) rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the general formulation and the Classical Reissner–Mindlin and Classical Toorani–Lakis theory are presented. New results are presented in order to investigate the effects of the Winkler modulus, the Pasternak modulus and the inertia of the elastic foundation on the free vibrations of anisotropic shells of revolution with a free-form meridian.     

General higher-order layer-wise theory for free vibrations of doubly-curved laminated composite shells and panels

ABSTRACT

The present article illustrates a general formulation for a higher-order layer-wise theory related to the analysis of the free vibrations of thick doubly-curved laminated composite shells and panels. The theoretical framework relates to the dynamic analysis of shell structures by using a general displacement field based on the Carrera Unified Formulation (CUF), including the stretching effect for each layer. The order of the expansion along the thickness direction is taken as a free parameter. The starting point of the present general higher-order layer-wise formulation is to propose a kinematic assumption, with an arbitrary number of degrees of freedom. The main aim of this work is to determine the explicit fundamental operators that can be used for the layer-wise (LW) approach. These fundamental operators are obtained for the first time by the author and are related to motion equations of doubly-curved shells described in an orthogonal curvilinear co-ordinate system. The free vibration shell and panel problems are computationally solved using the generalized differential quadrature (GDQ) and generalized integral quadrature (GIQ) techniques. The numerical results are compared with recent papers in the literature and commercial finite element codes.     

Nonlinear vibrations of laminated circular cylindrical shells: Comparison of different shell theories

The geometrically nonlinear forced vibrations of laminated circular cylindrical shells are studied by using the Amabili–Reddy higher-order shear deformation theory. An energy approach based on Lagrange equations, retaining modal damping, is used in order to obtain the equations of motion. An harmonic point excitation is applied in radial direction and simply supported boundary conditions are assumed. The equations of motion are studied by using the pseudo-arclength continuation method and bifurcation analysis. A one-to-one internal resonance is always present for a complete circular cylindrical shell, giving rise to pitchfork bifurcations of the nonlinear response with appearance of a second branch with travelling wave response and quasi-periodic vibrations. The numerical results obtained by using the Amabili–Reddy shell theory are compared to those obtained by using an higher-order shear deformation theory retaining only nonlinear term of von Kármán type and the Novozhilov classical shell theory.     

Four-parameter functionally graded cracked plates of arbitrary shape: A GDQFEM solution for free vibrations

The dynamic behavior of moderately thick FGM plates with geometric discontinuities and arbitrarily curved boundaries is investigated. The Generalized Differential Quadrature Finite Element Method (GDQFEM) is proposed as a numerical approach. The irregular physical domain in Cartesian coordinates is transformed into a regular domain in natural coordinates. Several types of cracked FGM plates are investigated. It appears that GDQFEM is analogous to the well-known Finite Element Method (FEM). With reference to the proposed technique the governing FSDT equations are solved in their strong form and the connections between the elements are imposed with the inter-element compatibility conditions. The results show excellent agreement with other numerical solutions obtained by FEM.     

Exact characteristic equations for free vibrations of thin orthotropic circular cylindrical shells

This paper presents an analytical procedure and closed-form vibration solutions with analytically determined coefficients for orthotropic circular cylindrical shells having classical boundary conditions. This analysis is based upon the Donnell-Mushtari shell theory. This is the simplest thin shell theory and its results for the lowest frequencies of a closed cylinder may not be as accurate. It is known that the exact procedure is complicated for orthotropic shells and this complexity has apparently prevented most researchers from getting results. Using the separation of variables method, the closed-form natural frequencies are successfully obtained in this work. They are found in a compact form. Moreover, the characteristics of the eigenvalues are examined. The exact solutions are validated through numerical comparisons with available solutions in literatures and the semi-analytical differential quadrature finite element method (S-DQFEM) solutions calculated by the authors.     

A semi-analytical finite element model for the analysis of laminated 3D axisymmetric shells: Bending, free vibration and buckling

A general semi-analytical finite element model is developed for bending, free vibration and buckling analysis of shells of revolution made of laminated orthotropic elastic material. The 3D elasticity theory is used and the equations of motion are obtained by expanding the displacement field and load in the Fourier series in terms of the circumferential coordinate, θ. The coefficients of the expansion are functions of (r, z), and they are approximated using the finite element method. This leads to a semi-analytical finite element in the (r, z) plane. The element is validated by comparing the present results with the analytical and numerical solutions available in the literature.     

Transient wave propagation and early short time transient responses of laminated composite cylindrical shells

The generalized ray method (GRM) and the method of reverberation ray matrix (MRRM) have been successfully used to study the transient elastic wave transmission in the beams, planar trusses, space frames and infinite layered media. In this paper, the GRM and MRRM are extended to investigate the early short time transient responses of laminated composite cylindrical shells with finite size under impact load. Using the Laplace transformation, the ray groups transmitting in the laminated cylindrical shells under the shock load are yielded by means of the boundary conditions. The reverberation-ray matrix representing the multi-reflected and scattered waves in the laminated cylindrical shells with finite structural size is deduced. Using FFT algorithm, the transient response corresponding to each ray group can be derived. Through the numerical simulations, it is seen that the early short time transient accelerations of the laminated composite cylindrical shells under impact loads are very large, but the early short time transient shear strain and displacement are very small. Furthermore, the effects of the stacking sequence, thickness of the composite cylindrical shells and different shock loads on the early short time transient responses of the laminated composite cylindrical shells with finite size are also analyzed.     

Three‐dimensional theory of elasticity for free vibration analysis of composite laminates via layerwise differential quadrature modelling

In this paper, the free vibration analysis of simply‐supported and clamped composite laminates, especially thick laminates, is carried out. The three‐dimensional theory of elasticity is integrated into a layerwise model via differential quadrature discretization. All physical governing equations are satisfied, including the additional constraints of the characteristics of continuity and discontinuity of interfacial transverse and in‐plane strains and stresses along the interfaces of composite laminates. Effects of plate aspect and thickness ratios on the free vibration of these laminates are examined in detail. This study demonstrates the applicability, accuracy, and stability of the present methodology, for vibration analyses of composite structures of thick laminated constitution. Copyright © 2003 John Wiley & Sons, Ltd.