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This communication is to correct the sensitivity analysis formula of frequencies for composite structures in the paper [1]. It is further pointed out that even though the sensitivity analysis formula of frequencies has been corrected, the first derivatives of frequencies with respect to design variables cannot be achieved by direct differentiating the equations of eigenvalues and eigenvectors because the first derivatives of vibration modes with respect to design variables are still unknown. The readers are referred to the analytical solution of the sensitivity analysis of frequencies and modes for composite structures in the literature [2], which is derived using linear space theory.

In this paper, a general vibration analysis of functionally graded porous (FGP) structure elements of revolution with general elastic restraints is performed for the first time. Under the basic framework of the first order shear theory, the moderately thick and thin FGP structure elements of revolution are considered. The specific FGP structure elements include the cylindrical shell, conical shell, spherical shell, cylindrical panel, conical panel and spherical panel. In order to simulate general elastic restraints, the artificial spring boundary technique is adopted this paper. And the modified series solution composed of a standard cosine Fourier series and two auxiliary terms is introduced in the Rayleigh-Ritz method to express the admissible function. A series of numerical examples are conducted, which show that the current model has superior convergence characteristics, computational accuracy and stability. On this basis, a series of innovative results are also given in the paper, which may provide basic data for other algorithm research in the future.

In this paper, exact solutions for the static bending response of laminated composite beams have been derived considering the effect of strain gradient elasticity. Separate solutions for the laminate comprising of isotropic and orthotropic laminae have been presented here. The sixth-order tensor for the higher order elastic coefficients has been evaluated using the generalized first strain gradient elasticity model with three independent material length constants. The significance of the gradient elasticity in low-dimensional structures has been established employing numerical examples of micro and nano-beams, and comparing the current results with the classical elasticity results. A parametric study over the variation of the axial, transverse Cauchy and physical stresses across the thickness of the beam has been conducted. Also, the discontinuity of the inter-laminar transverse Cauchy stresses and higher order stresses across the thickness has been studied in the current framework. The exact solutions developed in this paper may be used as benchmark results for validating further research involving the strain gradient elastic response of low-dimensional laminated composite beams.

This paper contributes to the understanding of a stress – strain diagram (SSD) in the region beyond the elastic limit (Viscoelastic and Plastic region). According to our results, the SSD in this region can be approximated by a second degree parabola. The advantage of such an approach is the possibility to simply calculate the value of stress if the strain is known and vice versa. Another advantage is in the possibility to simply calculate the tangent modulus. In the experimental part we determined the tangent and chord moduli for beech ( Fagus sylvatica L.) and aspen ( Populus tremula L.) “ WS ”. We also investigated the influence of the following factors on their value: material thickness “ MT ” (4, 6, 10, and 18 mm), degree of densification “ DOD ” (10% and 20% of the original thickness), and the number of loading cycles “ NC ” (0 versus 10,000). The paper describes the behavior of the tangent modulus in the whole force – deflection diagram.

In this paper, free vibration behavior of coupled functionally graded (FG) doubly-curved revolution shell structures with general boundary conditions is studied by the using unified Jacobi-Ritz method. The first-order shear deformation theory in conjunction with a multilevel partition technique is adopted to establish the theoretical model. The substructure of the theoretical model mainly includes the FG elliptical, hyperbolical, paraboloidal and cylindrical shells and three kinds of coupled FG shell structures containing paraboloidal-cylindrical shells, elliptical-cylindrical shells and hyperbolical-cylindrical shells are also considered in actual calculation. To obtain the continuous conditions at the interface and satisfy the arbitrary boundary conditions, the boundary and coupling spring techniques are adopted in this paper. In despite of the shell components and the boundary conditions, a mix function which is with the Jacobi polynomials along the meridional direction and the standard Fourier series along the circumferential direction is used as the admissible displacements of each shell segment. The convergence and comparison studies for the FG doubly-curved revolution shell structures with different boundary conditions, coupling parameters and Jacobi parameters are carried out to verify the reliability and accuracy of the present solutions. To enhance the understanding of the titled problem, some mode shapes of coupled FG shell structure are provided. Through comparative analysis, including the experimental and numerical comparison, it is obvious that the current method has good stability and rapid convergence properties and the present results agree closely with those reference results no matter what the frequency parameters or mode shapes are. The influence of the geometric dimensions and material constants on the vibration behavior of coupled FG shell structure is also reported.

This paper investigates the vibro-acoustic characteristics of the moderately thick laminated rectangular plate-cavity coupling system by using a modified Fourier series method. In the established model, two boundary conditions of moderately thick laminated rectangular plate are set as uniform elastic supports and non-uniform elastic supports including multi-points elastic supports or partial elastic supports. Besides, the three-dimensional rectangular cavity with rigid walls or impedance walls is discussed in the studied model. The admissible function of the displacements and sound pressure of the coupling system is set as the superposition function combined with Fourier cosine series and Fourier sin series. The Lagrangian energy principle is used to express the coupling system. The vibro-acoustic coupling characteristics can be obtained based on Rayleigh-Ritz technique. It shows good convergence and agreement of the present method by being compared with the finite element method (FEM) and the related literatures. This paper focuses on the study of free and forced vibro-acoustic analysis for the moderately thick laminated rectangular plate-cavity coupling system. Moreover, some results are achieved by discussing the effects of various plate thicknesses, different types of elastic supports, lamination schemes and various impedance boundary conditions on the natural characteristics and responses of the coupling system.

In this paper, the state space method is used to analyze the static and dynamic behaviors of laminated magneto-electro-elastic rectangular plates with simply supported boundary conditions . Multilayers are considered as well as viscoelastic interfaces. The Kelvin-Voigt model is used to take into accounted the viscoelastic interface effects. The mathematical formulation is elaborated in a general form and an arbitrary number of layers as well as the orthotropic behavior can be considered. The coupling time and thickness variables formalism is presented in an elegant matrix way allowing a versatility and capability of the formalism to handle more general multilayered plates. The solution procedure is based on the transfer relationships coupled with the Lagrange polynomials in discretised time intervals. The static behavior of multilayered magneto-electro-elastic rectangular plates is analyzed and the main results are the viscoelastic interface effect on the dynamic behavior of multifunctional structures. New results have been obtained by this coupling state space-Lagrange polynomial methodology. The mathematical procedure, elaborated in this paper, is original and allows the numerical study of the effects of the viscoelastic interfaces. The effectiveness of the proposed methods has been demonstrated by performing variant numerical tests.

In this paper the numerical procedure to solve the non-linear eigen-value problem describing the free vibrations of the laminate plates with viscoelastic (VE) layers is presented. The derivation of that main equation based on the refined zig-zag theory and using fractional Zener material model to describe VE material is presented in the companion paper Lewandowski R, Litewka P. Wielentejczyk P. Free vibrations of laminate plates with viscoelastic layers using the refined zig-zag theory – Part 1. Theoretical background. The solution obtained using the continuation method provides the required dynamic characteristics of the plates – the frequency and the damping ratio. In the iterative part of each continuation step either the full Newton or the quasi-Newton iterative procedure are used. Several numerical examples are solved to verify accuracy, efficiency and versatility of the proposed formulation. Numerous parametric studies are carried out to determine the influence of various solution variants and combinations of material parameters onto the results describing the damping properties of laminate plates.

In the present study, a single general formulation has been presented for the analysis of various shell-shaped structures. The proposed model is comprehensive and a variety of theories can be used based on it. The cross-section of the shell structure can be arbitrarily analyzed with the presented equations. In other words, various types of shell structures, including cylindrical, conical, spherical, elliptical, hyperbolic, parabolic, and any non-geometric structure with functional cross-section, can be modeled mechanically with only one partial differential equation system. The obtained equations have been solved by applying SAPM semi-analytical solution method. In order to present a comprehensive research, dynamic nonlinear analysis is considered. The variation of material properties through the thickness has been assumed as functionally graded and its effect on the strength of the shell structure with the functional cross-section has been investigated. The numerical results have been compared with available papers and also with FEM results for some structures that there is no paper available for validation. Different types of shell structures have been studied in terms of cross-sectional shape and properties. Finally, the effects of some important factors on the results such as boundary conditions, nonlinear analysis, dynamic analysis, and rotation of the structure around its central axis have been conducted thoroughly. This study and its original governing equations can be considered as a comprehensive reference for mechanical analysis of various shell structures with functional cross-sectional shape.

By combining the first-order shear deformation theory (FSDT) with the spectral-Tchebychev (ST) technique, including the arbitrary boundary conditions, this paper analyzes the free vibration characteristics of composite laminated stepped and stiffened cylindrical shells . Based on artificial spring technique, the boundary conditions at both ends of the structure and the coupling continuity conditions between the shell segments are simulated. Then the Hamilton’s principle is applied to obtain the governing differential equations of each shell segments, the governing equations of the boundary conditions at both ends of the structure and the elastic coupling governing equations between adjacent shell segments. The displacement components of the composite laminated stepped and stiffened cylindrical shells are spectrally expanded into the finite terms of Tchebyshev polynomials, and then discretized according to Gauss-Lobatto points to obtain the matrix form characteristic equations of the structure. The frequency of free vibration of the structure and the corresponding mode shape can be obtained by solving the characteristic equation. In the numerical example, the convergence analysis of the model established by the technique in this paper is carried out, and then the frequency results of composite laminated stepped and stiffened cylindrical shells under arbitrary boundary conditions calculated in this paper are compared with the calculation results of related literature and finite element software. It is fully verified that the proposed spectral-Tchebychev technique not only has the characteristics of rapid convergence, but also can accurately analyze the free vibration behaviors of composite laminated stepped and stiffened cylindrical shells under arbitrary boundary conditions. After proving its effectiveness and accuracy, the parameter studies are finally carried out to analyze the influence of material parameters and rib geometric parameters on the free vibration behaviors of composite laminated stiffened cylindrical shells.

This paper explores the stress concentration effect and the static responses of FGM cylindrical shells with various boundary conditions using the quasi-3D type higher-order shear deformation theory and the analytical approach. The displacement field is expressed by polynomials of the coordinate along the thickness direction. Compared to the polynomial used to analyze the transverse displacement, the order of the in-plane displacement polynomial is increased by one. The equations of equilibrium and their corresponding boundary conditions are derived based on the principle of virtual work. Using simple trigonometric series and the Laplace transform, the solutions of boundary problems with different conditions are derived. The results from the present theoretical models are compared with previously published data using several other models, including the 3D exact model. The paper also exhibits the effects of the boundary condition, the relative thickness, the relative length and the power-law index on the displacements and the stresses of shells. The stress concentration phenomenon is studied, and then the effects of several structural and material parameters on the concentrated stress are shown and assessed.

In this paper, a new kind of chiral three-dimensional honeycomb material was designed by orthogonal assembling based on chiral two-dimensional honeycomb with four ligaments. The analytical formulae of equivalent Young’s modulus and Poisson’s ratio are deduced using the beam theory. The calculations of the analytical formulae can be well consistent with those of finite element method. The theoretical and numerical results show that the honeycomb material proposed in this paper is isotropic at the macroscopic scale and its Poisson’s ratio is close to −1, which means the material have larger ratio of shear modulus to Young’s modulus. Furthermore, the influence of geometries on the equivalent elastic parameters are also discussed.

The object of this paper is to present a novel semi-analytical method and its associated applications for linear vibration analyses of functionally graded carbon nanotube reinforced composite (FG-CNTRC) doubly-curved panels and shells of revolution on with arbitrary boundary conditions. Distribution of the carbon nanotubes through the thickness of the structures may be uniform or functionally graded and four types of the CNTs distribution are considered in this paper. Properties of the composite media are determined by a refined rule of mixtures approach which contains the efficiency parameters. The translation and rotation displacements of the doubly-curved structures are uniformly expressed as the superposition of a standard cosine Fourier series and several auxiliary functions introduced to eliminate all potential discontinuities of the original displacement function and its derivatives at the edges. Based on that, the energy expression of the FG-CNTRC doubly-curved panels and shells of revolution is examined where the first-order shear deformation elasticity theory is considered. Lastly, to solve the natural frequencies as well as the associated mode shapes by means of the Ritz-variational energy method. Unlike other existing methods, the proposed method is capable of handling various combinations of boundary constraints in a unified fashion, including free, simply-supported, clamped and elastic-supported boundary conditions. Comprehensive studies on the convergence, accuracy, stability and efficiency of the method are derived via the comparison with existing results reported in publications. The parametric studies concerning the influence of the geometrical parameters, CNTs distributions, volume fraction of CNTs as well as boundary restraint parameters on free vibration of FG-CNTRC doubly-curved panels and shells of revolution is also investigated in detail.

This paper presents analytical solutions for the effective viscoelastic properties of composite materials based on a homogenization approach. The cases of spherical inclusions and cracks were recently developed. The objective of this paper is to use the same technique to deal with the case of periodic media containing cuboidal inhomogeneities. The viscoelastic behavior of both the matrix and the homogeneous equivalent medium is modeled by the Zener rheological model while inclusions are assumed to be linearly elastic. The viscoelastic Hill tensor required for the calculation of the effective viscoelastic tensors is obtained explicitly in the Laplace-Carson space in terms of Fourier series. The final expressions show that overall behavior depends on the viscoelastic properties of the matrix, the 3D dimensions of the inclusions and the thickness of the matrix layer between two nearby inclusions. Applications to masonry structures are presented to illustrate the theoretical results.

In engineering, the amount of complex geometrical problems, which needs to be solved by using plates and shells theories, is remarkable. This is the reason why there are so many theories, which attempt to simplify three-dimensional problems into simpler ones. Additionally, the current increasing use of laminated and sandwich structures demands a minimum of accuracy from two-dimensional formulations. In the literature, one can find a variety of bi-dimensional theories and solution methods to solve these problems. Laminated and sandwich structure formulations are mainly classified according to the treatment of the variables in the normal direction of the plate/shell surface: equivalent single layer, Zig-Zag and layer-wise theories. The contribution of this paper is to set the stage for new theories and solution methods for laminated and sandwich structures by reviewing over 100 papers. To show the importance of the coupling between plate/shell theories and the respective solution method, a detailed review on theories and the respective solution methods is firstly given to update the current state of art. After that, solution methods based on the Finite Element Method are explained to exhibit how particular and/or complex an approach can get. In fact, this review gives a clearer picture on plate/shell theories.

In this paper, the method of reverberation ray matrix (MRRM) is extended to develop an exact and unified solution for the transient response analysis of composite laminated curved beams with arbitrary lamination schemes and general boundary restraints. The effects of shear deformation, rotary inertia and deepness term as well as the Poisson’s effect are considered in the theoretical formulations. Through the Laplace transformation, the governing differential equations are derived in frequency domain via the Hamilton’s principle, and the exact closed form solutions are used to represent the elastic waves propagating in the curved beam. As the main focus of this paper, the artificial spring boundary technique is adopted to simulate general boundary restraints and the scattering matrix is redefined to make the MRRM suitable for different boundary cases. Then, the transient responses of the composite laminated curved beam under general boundary restraints are obtained directly by using the Neumann series expansion and the Fast-Fourier transform (FFT) algorithm. Several numerical cases are presented to test and verify the accuracy, reliability and efficiency of the present solution. Meanwhile, a systematic parameter investigation is performed regarding the influences of elastic restraint parameters, lamination schemes, material properties, geometry parameters and loading types.

The dynamical analyses of functionally graded graphene nanoplatelet reinforced composite (FG-GPLRC) trapezoidally and sinusoidally corrugated plates on the nonlinear elastic foundations are investigated in this paper. The FG-GPLRCs are assumed to be distributed uniformly and functionally graded through the thickness. The material properties are estimated through the Halpin-Tsai micromechanical model, while the Poisson’s ratio and density mass are implemented by the rule of mixtures. The mathematical model rests upon the classical theory and Von Karman-Donnell geometrical nonlinearity assumption. The dynamical responses of the simply-supported FG-GPLRC corrugated plates are obtained by employing a homogenization-based analytical model and the Galerkin's method. The effects of the environment, GPLs weight fraction, GPLs distribution patterns, geometric parameters are carried out in detail in the paper. The amplitude of the critical force of periodic and chaotic status can be found by applying the bifurcation diagrams 3D and 2D. With the selected critical values, time history, phase plane graphs, Poincare maps, maximum Lyapunov exponent, and Fourier spectrum are presented to observe the periodic and chaotic status of corrugated plates. The obtained results are also compared and validated with those of other studies.

This paper proposes an efficient generalized beam theory (GBT) formulation, which accounts for cross-sectional deformations in slender prismatic structures. It was shown by the authors in a recent publication [1] that in-plane distortional deformations and accompanied out of plane warping deformations of the cross-section influence the accuracy of results in beam dynamics especially if thin-walled cross-sections are applied. The GBT formulation proposed in [1] overcomes the inaccuracies of classical beam mechanics, however, requires a two-dimensional plane discretization of the cross-section. The computational complexity can be reduced vastly, if thin-walled cross-sections can be discretized with one-dimensional elements. Consequently, this paper discusses an approach with a line mesh discretizing the cross-section and having six degrees of freedom at each node. The membrane part consists of massless micro-polar rotations (drilling rotations) and can be derived independently from the bending part, where a shear elastic formulation is selected.

In this paper, a novel higher order cubic-quintic nonlinear model is proposed for the nonlinear free vibration of damped and undamped bi-directional functionally graded (2D FG) beams. To the best of the researchers’ knowledge, no study has focused on damping characteristics of 2D FG beams. Thus, the present paper extends the previous studies in this field. It is assumed that the material properties of the beam change in the axial and lateral directions simultaneously according to exponential and power-law functions, respectively. A new neutral surface is defined to remove the stretching and bending couplings effect. The variational iteration method (VIM) and the Hamiltonian approach (HA) are applied to obtain closed-form analytical solutions for the nonlinear vibration of damped and undamped 2D FG beams, respectively. Based on the results, an increase in the material grading indices can similarly decrease the damping coefficient for an underdamped 2D FG beam, and more time is required for dying out the oscillations. Further, compared to the homogeneous beams, a significant difference is observed between the cubic and cubic-quintic models indicating the importance of applying higher order nonlinear models for the nonlinear analysis of 2D FG beams.

The main issue of this paper is an attempt to predict the elastic properties of PVC/ABS/nano-CaCO 3 polymer nanocomposites by coupling data of microstructure analysis and mechanical tests with a micro-mechanics analysis. The novel numerical implementation of asymptotic homogenization (NIAH) method is compared with the representative volume element (RVE) method under Dirichlet and Neumann boundary conditions. Numerical simulation results show that the NIAH method is more reasonable and accurate for simulating the micro-mechanical properties of periodic composite materials. In order to account for two kinds of particles spatial distribution in nanocomposites, statistical asymptotic homogenization (AH) method is performed by assigning randomness to the spatial distribution for the two kinds of particles in the unit cell. With increase of the number of the unit cell particles, the results of NIAH method converge to a stable value. We also show that in the case of fixed-size of the unit cell, the average mechanical properties of the obtained results converge to the stable value with the increase of the number of random configurations realization. In this paper, we use the unit cell containing the minimum number of particles for different weight ratio of two kinds of particles to obtain the effective mechanical properties by averaging simulation results of many samples of different configurations in the unit cell. Based on the results obtained, it can be concluded that spatial distribution for the two kinds of particles in the unit cell has a significant impact on the material macroscopic elastic properties. Furthermore, the method is more efficient to predict the mean values of the elastic properties than the experimental dispersions.

In this work an approximate analytical solution based on the Mori–Tanaka method for functionally graded thick-walled tube subjected to internal pressure is given. We assume that the tube consists of two isotropic linear elastic constituents and the volume fraction for one phase is a power function varied in the radial direction. Based on the Mori–Tanaka method, this paper obtains the differential equation of the radial displacement, and then an approximate analytical solution is obtained that agrees well with the numerical results of the Mori–Tanaka method. Furthermore, this paper discusses the same problem with two other methods, the Voigt method that we have already presented in Xin et al. (2014) and the Reuss method. All these three different methods can avoid the assumption of the distribution regularities of unknown overall material parameters appeared in existing papers. Further, they are valid for the materials with different Poisson’s ratios rather than constant Poisson’s ratios usually used in the existing references to obtain the effective Young’s modulus.

A semi-analytical methodology is developed for the transient response of FGM annular sector plate with arbitrary circular boundary conditions. The state space method, differential quadrature method and the numerical inversion method of Laplace transform are integrated into this methodology. The FGM annular sector plate is simply supported at the radial edges. At the circular edges, four kinds of boundary conditions are considered in this paper: Clamped–Clamped, Clamped–Simply supported, Clamped–Free and Simply supported–Simply supported. A comparison of the results generated by the present and finite element method is carried out. It shows that the results of the present and the finite element method always agree with each other excellently, regardless of which boundary condition, load and geometry are employed. The present method is further validated by comparing the fundamental frequency parameters with those in the literature. Moreover, convergence studies for different numbers of sampling points along the radial direction and layer numbers along the thickness direction are performed. It is found that this methodology has fast convergence rate. The effects of the boundary conditions, the functionally graded index, and the thickness on the plate response are also investigated. The results obtained in this paper can serve as benchmark data in further research.

This paper develops a two-scale asymptotic expansion solution method for periodic composite Euler beams for the first time. In this method, a two-scale solution in the asymptotic expansion form is achieved for the fourth-order uniformly elliptic differential equation with periodic oscillating coefficients, and the first-order perturbed displacement is zero. The analytical solutions of unit cell problems are found with the help of four normalization conditions proposed in this work, and it follows that the homogenized elastic modulus of periodic composite beam is exactly the harmonic mean of material moduli. Another contribution of this work is to mathematically verify the convergence of the two-scale asymptotic expansion solution through the two-scale convergence method. In addition, the paper reveals the effects of higher-order expansion terms on the two-scale displacements and concludes that the second order or even higher-order perturbed displacements are necessary for obtaining accurate curvatures relevant to micro stresses. Finally, numerical experiments validate that the present method is rigorous in mathematics and physically acceptable.

A novel two-level strategy is employed to solve the multilayer composite cantilever tubes in elasticity theory. On the first level of our strategy, a complete stress field is developed for the inclined winding cantilever tubes. Thus the tube can be considered as a bending-like tube including the proportional moments together with a shear-like tube including the constant shear forces. The pure bending formula is used to obtain the solution for the bending-like tube. However, the shear-like tube includes some additional terms from the bending-like tube. Therefore, on the second level of our strategy, the shear-like tube is broken into two independent systems according to whether the additional terms are included. Thus their different circumferential functions can be satisfactorily separated from the radial functions. In addition, an approach of undetermined coefficients is employed to find the particular solution for the second independent system. In the examples the method in this paper is used to exactly solve the composite cantilever tubes [90°], [45°] and [75°] as well as [45°/75°], [75°/−75°] and [(75°/−75°)40]. The good agreement of our exact results with the numerical simulation with fine finite elements shows the accuracy and effectiveness of the method in this paper.

In this paper, a theoretical stiffness design approach for a body-centered cubic (BCC) lattice structure is proposed. Although numerous studies have focused on lattice structures, particularly BCC lattice structure, there are still some issues that must be further investigated. Homogenization method is always followed that the mechanical properties of lattice structure can be achieved by directly solving the unit cell, since the structure is a periodically repeated sequence of a unit cell. However, the equivalent modulus varies a lot with different array configurations from our study, which indicates that the spatial array configuration also has an important influence on the equivalent mechanical properties. In this paper, “large unit cell” assumption is used to characterize the equivalent modulus of the lattice structure considering array configuration effect, and then validated using the finite element method (FEM) and experiment. The results demonstrate that the mechanical properties vary significantly for different array configurations, even those with the same unit cell topology. Furthermore, the proposed method is extended to an arbitrary BCC unit cell to broaden its application for stiffness design of engineering parts. This research provides a guidance for the stiffness design of lightweight lattice structures.

This paper analyzed the free vibration of functionally graded (FG) doubly-curved shells with un-uniform thickness distribution based on Ritz method. The energy method and first-order shear deformation theory are adopted to derive the formulas. In this paper, the stepped FG doubly-curved shells are divided into their segments in axial direction according to the steps of the structures, and the displacement functions of shell segments are consisted with Jacobi polynomials along axial direction and standard Fourier series along circumferential direction. In addition, the boundary conditions at ends of the stepped FG doubly-curved shells and the continuity conditions at two adjacent segments were enforced by penalty method. Then the final solutions can be obtained based on Ritz method. Finally, to confirm the validity of proposed method, the results with the same conditions are compared by Finite Element Method (FEM) and experiment. The results show that the proposed method has the advantage of fast convergence, high accuracy and simple boundary simulation etc.

The main content of this paper is to establish an analysis model for dynamic analysis of composite laminated doubly-curved revolution shell based on the Higher order Shear Deformation Theory (HSDT). Firstly, the energy functional of shell is established based on higher order shear theory. Then, the multi-segment partitioning technique is introduced to segment the shell along the generatrix direction, which is mainly to relax the boundary conditions of shell, and then reduce requirements for the displacement function. At the boundary position, the boundary spring parameters are used to obtain the corresponding boundary conditions. Similarly, the connecting spring parameters are introduced to simulate the continuity conditions between the segmented shells. The parameters of boundary spring and connecting spring can be regarded as the weight parameter. Thirdly, all displacement components of the composite laminated doubly-curved revolution shell are expressed by Jacobi orthogonal polynomials. Finally, the whole dynamic characteristics are obtained by Rayleigh-Ritz method with respect to unknown Jacobian expansion coefficient. The convergence, validity and dynamic characteristics of the analytical model established in this paper are given by a series of numerical examples.

This paper presents correct results for a paper by Liu and Xing (2011) [1] that had errors, greatly simplifies its formulations, overcomes its numerical stability problem, and provides some new results. In their early paper, Liu and Xing obtained exact closed-form solutions for free vibrations of orthotropic rectangular Mindlin plates by using the separation of variables method. The equations presented therein are correct, but the numerical values of the frequencies listed have some errors because the shear rigidities of the three-ply laminates were not correctly computed. In addition, the exact characteristic equations of the early paper can also be greatly simplified and the compact equations are presented herein. The characteristic equations in the early paper may have computational stability problem when plates become very thin. This paper overcome such computational difficulty. Some new exact frequency parameters are also included in this work.

This paper presents an analytical closed form solution for free vibration analysis of multilayered plates by using a layer-wise displacement assumption based on Carrera’s Unified Formulation. A wide range of boundary conditions are analysed by using a Levy type solution. By using such a solution, the number of the differential equations of motion depends on the number of plies. This means that for plates with different number of plies the equations of motion need to be re-derived and re-solved. In this paper, a procedure has been devised to overcame this problem and the equation of motions can be automatically obtained and solved to obtain natural frequencies and mode shapes for plates with any number of layers. This procedure has been called “L-matrix method”. The novel solutions have been validated by analytical 3D and 2D Navier type solutions for plates simply supported at all edges. A comparison of the results with 2D and 3D NASTRAN finite element models is also carried out showing the superiority of the analytical solution. Furthermore, benchmark solutions have been obtained for plates with different boundary conditions. The presented Levy type solution can be easily extended to static, buckling and response analysis.

A set of linear differential equations of motion for coupled free vibrations of composite beams with asymmetric cross-sections is obtained in this paper based on Euler–Bernoulli beam theory. More coupling terms are included in the governing equations; and hence, both even and odd order spatial derivatives are included. All these features lead to much difficulty in solving the resultant equations. The axially unloaded condition of F1 = 0, instead of e = 0, is adopted to simplify the equations of motion for the case of inextensional beams. An algorithm is developed for solving the resultant equations to obtain natural frequencies and modes of the beams. Numerical examples for validation show that the equations of motion obtained in this paper are correct and the corresponding algorithm is effective and easy to use.