Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and multiquadrics discretizations

In this paper, the static and free vibration analysis of composite plates are performed, using a layerwise deformation theory and multiquadrics discretization. This meshless discretization method considers radial basis functions as the approximation method for both the equations of motion and the boundary conditions. The combination of this layerwise theory and the multiquadrics discretization method allows a very accurate prediction of the natural frequencies.     

Free vibration response of composite sandwich cylindrical shell with flexible core

This study deals with the free vibration analysis of composite sandwich cylindrical shell with a flexible core using a higher order sandwich panel theory. The formulation uses the classical shell theory for the face sheets and an elasticity theory for the core and includes derivation of the governing equations along with the appropriate boundary conditions. The model consists of a systematic approach for the analysis of sandwich shells with a flexible core, having high-order effects caused by the nonlinearity of the in-plane and the vertical displacements of the core. The behavior is presented in terms of internal resultants and displacements in the faces, peeling and shear stresses in the face–core interface and stress and displacement field in the core. The accuracy of the solution is examined by comparing the results obtained with the analytical and numerical results published in the literatures. The parametric study is also included to investigate the effect of geometrical properties such as radius of curvature, length and sector angle of the shell.     

Numerical and exact models for free vibration analysis of cylindrical and spherical shell panels

The present paper shows a comparison between classical two-dimensional (2D) and three-dimensional (3D) finite elements (FEs), classical and refined 2D generalized differential quadrature (GDQ) methods and an exact three-dimensional solution. A free vibration analysis of one-layered and multilayered isotropic, composite and sandwich cylindrical and spherical shell panels is made. Low and high order frequencies are analyzed for thick and thin simply supported structures. Vibration modes are investigated to make a comparison between results obtained via the FE and GDQ methods (numerical solutions) and those obtained by means of the exact three-dimensional solution. The 3D exact solution is based on the differential equations of equilibrium written in general orthogonal curvilinear coordinates. This exact method is based on a layer-wise approach, the continuity of displacements and transverse shear/normal stresses is imposed at the interfaces between the layers of the structure. The geometry for shells is considered without any simplifications. The 3D and 2D finite element results are obtained by means of a well-known commercial FE code. Classical and refined 2D GDQ models are based on a generalized unified approach which considers both equivalent single layer and layer-wise theories. The differences between 2D and 3D FE solutions, classical and refined 2D GDQ models and 3D exact solutions depend on several parameters. These include the considered mode, the order of frequency, the thickness ratio of the structure, the geometry, the embedded material and the lamination sequence.     

Free vibration analysis of nano-plate using three-dimensional theory of elasticity

Vibration analysis of a nano-plate, based on three-dimensional theory of elasticity, is studied employing non-local continuum mechanics. By using state-space method in the thickness direction and Fourier series in the in-plane directions, a closed form solution for the natural frequencies of a rectangular simply supported nano-plate is obtained. To verify the accuracy of the present approach, numerical results are compared with the results available in the literature. The effect of the non-local parameter, aspect ratio, thickness-to-length ratio and half-wave numbers in the frequency behavior is examined.     

Static,free vibration and thermal analysis of composite plates and shells using a flat triangular shell element

Finite element static, free vibration and thermal analysis of thin laminated plates and shells using a three noded triangular flat shell element is presented. The flat shell element is a combination of the Discrete Kirchhoff Theory (DKT) plate bending element and a membrane element derived from the Linear Strain Triangular (LST) element with a total of 18 degrees of freedom (3 translations and 3 rotations per node). Explicit formulations are used for the membrane, bending and membrane-bending coupling stiffness matrices and the thermal load vector. Due to a strong analogy between the induced strain caused by the thermal field and the strain induced in a structure due to an electric field the present formulation is readily applicable for the analysis of structures excited by surface bonded or embedded piezoelectric actuators. The results are presented for (i) static analysis of (a) simply supported square plates under doubly sinusoidal load and uniformly distributed load (b) simply supported spherical shells under a uniformly distributed load, (ii) free vibration analysis of (a) square cantilever plates, (b) skew cantilever plates and (c) simply supported spherical shells; (iii) Thermal deformation analysis of (a) simply supported square plates, (b) simply supported-clamped square plate and (c) simply supported spherical shells. A numerical example is also presented demonstrating the application of the present formulation to analyse a symmetrically laminated graphite/epoxy laminate excited by a layer of piezoelectric polyvinylidene flouride (PVDF). The results presented are in good agreement with those available in the literature.The work was partly sponsored by a grant (DAAHO4-95-1-0175) from the army research office with Dr. Gary Anderson as the grant monitor.     

Free vibration analysis of composite sandwich plates based on Reddy's higher-order theory

Two new C⁰ assumed strain finite element formulations of Reddy's higher-order theory are used to determine the natural frequencies of isotropic, orthotropic, and layered anisotropic composite and sandwich plates. The material properties typical of glass fibre polyester resins for the skin and HEREX C70 PVC (polyvinyl chloride) foam materials for the core are used to show the parametric effects of plate aspect ratio, length-to-thickness ratio, degree of orthotropy, number of layers and lamination scheme on the natural frequencies. A consistent mass matrix is adopted in the present formulation. The results presented in this investigation could be useful for a better understanding of the behaviour of sandwich laminates under free vibration conditions and potentially beneficial for designers of sandwich structures.     

Static and dynamic analysis of micro beams based on strain gradient elasticity theory

The static and dynamic problems of Bernoulli-Euler beams are solved analytically on the basis of strain gradient elasticity theory due to Lam et al. The governing equations of equilibrium and all boundary conditions for static and dynamic analysis are obtained by a combination of the basic equations and a variational statement. Two boundary value problems for cantilever beams are solved and the size effects on the beam bending response and its natural frequencies are assessed for both cases. Two numerical examples of cantilever beams are presented respectively for static and dynamic analysis. It is found that beam deflections decrease and natural frequencies increase remarkably when the thickness of the beam becomes comparable to the material length scale parameter. The size effects are almost diminishing as the thickness of the beam is far greater than the material length scale parameter.     

Static and free vibration analyses of continuously graded fiber-reinforced cylindrical shells using generalized power-law distribution

In this study, based on the three-dimensional theory of elasticity, static and free vibration characteristics of continuously graded fiber-reinforced (CGFR) cylindrical shells are considered by making use of a generalized power-law distribution. In the present formulation, the cylindrical shell is assumed to be made of an orthotropic material. The CGFR cylindrical shells have a smooth variation of matrix volume fraction in the radial direction. Symmetric and asymmetric volume fraction profiles are presented in this paper. Suitable displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which can be solved by a generalized differential quadrature method. The fast rate of convergence of the method is demonstrated, and comparison studies are carried out to establish its very high accuracy and versatility. The main contribution of this work is to illustrate useful results for a cylindrical shell continuously graded fiber reinforced in the radial direction. Finally, these results are compared with a similar discrete laminated composite cylindrical shell.     

Static analysis of delaminated composite shell panels using layerwise theory

In this paper, a layerwise theory and associated finite element model are presented to study the bending deformations of delaminated composite shell panels. The layerwise theory is based on the assumption of first-order shear deformation theory in each layer, and it satisfies the displacement continuity at layer interfaces. The presence of multiple discrete delaminations is modeled through the use of Heaviside step functions. Delamination movements (relative slip and opening) are included in the theory as additional degrees of freedom. The finite element model developed is validated by comparing the present results with those available in the literature. This model based on the layerwise theory is subsequently used for the first time in the literature to study the static response and delamination movements such as interfacial slipping and opening of delaminated composite shell panels.     

Static,free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique

In this paper the authors derive a higher-order shear deformation theory for modeling functionally graded plates accounting for extensibility in the thickness direction.The explicit governing equations and boundary conditions are obtained using the principle of virtual displacements under Carrera’s Unified Formulation. The static and eigenproblems are solved by collocation with radial basis functions.The efficiency of the present approach is assessed with numerical results including deflection, stresses, free vibration, and buckling of functionally graded isotropic plates and functionally graded sandwich plates.     

Static and free vibration analyses of composite and sandwich plates by an edge-based smoothed discrete shear gap method (ES-DSG3) using triangular elements based on layerwise theory

An edge-based smoothed stabilized discrete shear gap method (ES-DSG3) based on the first-order shear deformation theory (FSDT) was recently proposed for static and dynamic analyses of Mindlin plates. In this paper, the ES-DSG3 is extended and incorporated with a layerwise theory for static and free vibration analyses of composite and sandwich plates. In the layerwise theory, the behavior of each layer follows the first-order shear deformation theory and the condition of displacement continuity is imposed at the interfaces of layers. This hence does not require shear correction factors and improves significantly the accuracy of transverse shear stresses. The stiffness formulation of the ES-DSG3 is performed by using the strain smoothing technique over the smoothing domains associated with edges of elements for each layer. The accuracy and reliability of the proposed method are confirmed in several numerical examples.     

Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and RBF-PS discretizations with optimal shape parameter

A study of static deformations and free vibration of shear flexible isotropic and laminated composite plates is presented. A layerwise theory for laminated or sandwich plates is used. The analysis is based on a new numerical scheme, where collocation by radial basis functions is viewed as a pseudospectral method to produce highly accurate results. A cross-validation technique is used to optimize the shape parameter for the basis functions. Numerical results for symmetric laminated composite and sandwich plates are presented and discussed.     

Anisotropic cylindrical shell element based on discrete Kirchhoff theory

A triangular cylindrical shell element based on discrete Kirchhoff theory is developed. It is a three-node, 27-degrees-of-freedom element using cubic polynomials for the tangential and normal displacement interpolations. The normal rotations are independently interpolated by quadratic polynomials. The formulation is capable of modelling general anisotropy representative of multi-layered, multi-directionally oriented composite construction. The numerical results indicate that the solution for displacements and stresses of cylindrical shells converge monotonically and rapidly to those based on deep shell theory.     

Free vibration analysis of carbon nanotubes by using three-dimensional theory of elasticity

This paper studies vibration behavior of single-walled carbon nanotubes based on three-dimensional theory of elasticity. To accounting for the size effect of carbon nanotubes, nonlocal theory is adopted to the shell model. The nonlocal parameter is incorporated into all constitutive equations in three dimensions. Governing differential equations of motion are reduced to the ordinary differential equations in thickness direction by using Fourier series expansion in axial and circumferential direction. The state equations obtained from constitutive relations and governing equations are solved analytically by making use of the state space method. A detailed parametric study is carried out to show the influences of the nonlocal parameter, thickness-to-radius ratio and length-to-radius ratio. Results reveal that excluding small-scale effects caused decreasing accuracy of natural frequencies. Furthermore, the obtained closed form solution can be used to assess the accuracy of conventional two-dimensional theories.     

Static analysis of functionally graded cylindrical shell with piezoelectric layers using differential quadrature method

Three-dimensional solution for static analysis of functionally graded (FG) cylindrical shell with bonded piezoelectric layers is presented using differential quadrature method (DQM) and state-space approach. Applying the DQM to the governing differential equations and to the edges boundary conditions, new state equations about state variables at discrete points are derived. The stress, displacement, and electric potential distributions are obtained by solving these state equations. The convergence and accuracy of the present method is validated by comparing numerical results for the hybrid FG cylindrical shell with simply-supported edges with the analytical solution that has been published in the literature. Both the direct and the inverse piezoelectric effects are investigated and the influence of piezoelectric layers and gradient index on the mechanical behavior of shell is studied.     

Consistent higher-order free vibration analysis of composite sandwich plates

The consistent higher-order dynamic formulation for foam-type (soft) core sandwich beams was extended to the case of composite sandwich plates. Eight dynamic governing equations and the corresponding boundary conditions were derived through the application of Hamilton’s principle. The extended formulation was applied to the free vibration analysis of soft-core and honeycomb-core sandwich plates with anti-symmetric and symmetric lay-ups. The vibration results for the thin and thick composite sandwich plates obtained using the extended formulation were consistent with the predictions of the higher order mixed layerwise theory for laminated and sandwich plates. To simplify the formulation for the case of symmetric sandwich plates, the general dynamic formulation was decoupled into two systems of equations representing symmetric and anti-symmetric vibrations. The numerical study demonstrates the importance of the present formulation for the prediction of higher mode vibration response of composite sandwich plates.     

Free vibration analysis of composite and sandwich plates using an improved discrete Kirchhoff quadrilateral element based on third-order zigzag theory

The new improved discrete Kirchhoff quadrilateral element based on the third-order zigzag theory developed earlier by the present authors for the static analysis of composite and sandwich plates is extended for dynamics and assessed for its performance for the free vibration response. The element is free from the shear locking. The finite element formulation is validated by comparing the results for simply supported plates with the analytical Navier solution of the zigzag theory. Comparison of the present results for the natural frequencies with those of a recently developed triangular element based on the zigzag theory, for composite and sandwich plates, establishes the superiority of the present element in respect of simplicity, accuracy and computational efficiency. The accuracy of the zigzag theory is assessed for composite and sandwich plates with various boundary conditions and aspect ratio by comparing the finite element results with the 3D elasticity analytical and finite element solutions.     

On the forced vibration analysis of Timoshenko nanobeams based on the surface stress elasticity theory

This paper aims to present nonlinear forced vibration characteristics of nanobeams including surface stress effect. By considering the local geometrical nonlinearity based on von Karman relation, a new formulation of the Timoshenko beam model is developed through the Gurtin–Murdoch elasticity theory in which the effect of surface stress is incorporated. By using a variational approach on the basis of Hamilton’s principle, the size-dependent equations of motion and associated boundary conditions are obtained. The generalized differential quadrature (GDQ) method is employed to discretize the non-classical governing differential equations over the spatial domain by using the shifted Chebyshev–Gauss–Lobatto grid points. Subsequently, a Galerkin-based numerical approach is put to use in order to reduce the set of nonlinear equations into a time-varying set of ordinary differential equations of Duffing-type. In the next step, the time domain is discretized via spectral differentiation matrix operators which are defined based on the derivatives of a periodic base function. Finally, the pseudo arc-length method is employed to solve the resulting nonlinear parameterized algebraic equations. The frequency–response curves for forced vibration behavior of nanobeams including the effect of surface stress are predicted corresponding to various values of beam thickness, length to thickness ratio and surface elastic constants. It is revealed that by incorporating the surface stress effect, the maximum amplitude occurs at lower excitation frequencies and the wide of region of the response tends to decrease.     

An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nanobeams using a nonlocal elasticity model

This paper is concerned with the free transverse vibration of cracked nanobeams modeled after Eringen's nonlocal elasticity theory and Timoshenko beam theory. The cracked beam is modeled as two segments connected by a rotational spring located at the cracked section. This model promotes discontinuities in rotational displacement due to bending which is proportional to bending moment transmitted by the cracked section. The governing equations of cracked nanobeams with two symmetric and asymmetric boundary conditions are derived; then these equations are solved analytically based on concerning basic standard trigonometric and hyperbolic functions. Besides, the frequency parameters and the vibration modes of cracked nanobeams for variant crack positions, crack ratio, and small scale effect parameters are calculated. The vibration solutions obtained provide a better representation of the vibration behavior of short, stubby, micro/nanobeams where the effects of small scale, transverse shear deformation and rotary inertia are significant.     

An accurate higher-order theory and C finite element for free vibration analysis of laminated composite and sandwich plates

At present, it is difficult to accurately predict natural frequencies of sandwich plates with soft core by using the C⁰ plate bending elements. Thus, the C¹ plate bending elements have to be employed to predict accurately dynamic response of such structures. This paper proposes an accurate higher-order C⁰ theory which is very different from other published higher-order theory satisfying the interlaminar stress continuity, as the first derivative of transverse displacement has been taken out from the in-plane displacement fields of the present theory. Therefore, the C⁰ interpolation functions is only required during its finite element implementation. Based on the Hamilton’s principle and Navier’s technique, analytical solutions to the natural frequency analysis of simply-supported laminated plates have been presented. To further extend the ranges of application of the proposed theory, an eight-node C⁰ continuous isoparametric element is used to model the proposed theory. Numerical results show the present C⁰ finite element can accurately predict the natural frequencies of sandwich plate with soft core, whereas other global higher-order theories are unsuitable for free vibration analysis of such soft-core structures.