Nonlinear vibration analysis of tapered Timoshenko beams

The nonlinear flexural vibration analysis of tapered Timoshenko beams is conducted. The equations of motion for tapered Timoshenko beams are established in which the effects of nonlinear transverse deformation, nonlinear curvature as well as nonlinear axial deformation are taken into account. The nonlinear fundamental frequencies of tapered Timoshenko beams with two simply supported or clamped ends are presented.     

A mixed method for free and forced vibration of rectangular plates

This paper presents a very first combined application of Ritz method and differential quadrature (DQ) method to vibration problem of rectangular plates. In this study, the spatial partial derivatives with respect to a coordinate direction are first discretized using the Ritz method. The resulting system of partial differential equations and the related boundary conditions are then discretized in strong form using the DQ method. The mixed method combines the simplicity of the Ritz method and high accuracy and efficiency of the DQ method. The results are obtained for various types of boundary conditions. Comparisons are made with existing analytical and numerical solutions in the literature. Numerical results prove that the present method is very suitable for the problem considered due to its simplicity, efficiency, and high accuracy.     

On the exact solutions of vibration problems of rectangular orthotropic plates

Zusammenfassung Die Frequenzen von transversal schwingenden isotropen und orthotropen rechteckigen Platten sind durch bestimmte Gleichungen verbunden. Deshalb ist eine Auswertung der Frequenzengleichungen der orthotropen Platten nicht notwendig. Wenn die Frequenz einer isotropen Platte bestimmt ist, kann die Frequenz der entsprechenden orthotropen Platte mit Hilfe dieser Formeln berechnet werden.     

An efficient mixed methodology for free vibration and buckling analysis of orthotropic rectangular plates

In this paper, a simple and efficient mixed Ritz-differential quadrature (DQ) method is presented for free vibration and buckling analysis of orthotropic rectangular plates. The mixed scheme combines the simplicity of the Ritz method and high accuracy and efficiency of the DQ method. The accuracy of the proposed method is demonstrated by comparing the calculated results with those available in the literature. It is shown that highly accurate results can be obtained using a small number of Ritz terms and DQ sample points. The proposed method is suitable for the problem considered due to its simplicity and potential for further development.     

Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory

The main objective of this research work is to present analytical solutions for free vibration analysis of moderately thick rectangular plates, which are composed of functionally graded materials (FGMs) and supported by either Winkler or Pasternak elastic foundations. The proposed rectangular plates have two opposite edges simply-supported, while all possible combinations of free, simply-supported and clamped boundary conditions are applied to the other two edges. In order to capture fundamental frequencies of the functionally graded (FG) rectangular plates resting on elastic foundation, the analysis procedure is based on the first-order shear deformation plate theory (FSDT) to derive and solve exactly the equations of motion. The mechanical properties of the FG plates are assumed to vary continuously through the thickness of the plate and obey a power law distribution of the volume fraction of the constituents, whereas Poisson’s ratio is set to be constant. First, a new formula for the shear correction factors, used in the Mindlin plate theory, is obtained for FG plates. Then the excellent accuracy of the present analytical solutions is confirmed by making some comparisons of the results with those available in literature. The effect of foundation stiffness parameters on the free vibration of the FG plates, constrained by different combinations of classical boundary conditions, is also presented for various values of aspect ratios, gradient indices, and thickness to length ratios.     

Nonlinear vibration of moderately thick laminated beams using finite element method

A finite element model is developed to study the large-amplitude free vibrations of generally-layered laminated composite beams. The Poisson effect, which is often neglected, is included in the laminated beam constitutive equation. The large deformation is accounted for by using von Karman strains and the transverse shear deformation is incorporated using a higher order theory. The beam element has eight degrees of freedom with the inplane displacement, transverse displacement, bending slope and bending rotation as the variables at each node. The direct iteration method is used to solve the nonlinear equations which are evaluated at the point of reversal of motion. The influence of boundary conditions, beam geometries, Poisson effect, and ply orientations on the nonlinear frequencies and mode shapes are demonstrated.     

On the free vibration response of rectangular plates,partially supported on elastic foundation

Rectangular plates on distributed elastic foundations are widely employed in footings and raft foundations of variety of structures. In particular, mounted columns and single footings may partially occupy the rectangular plate of any kind.     

Nonlinear vibration of hybrid laminated plates resting on elastic foundations in thermal environments

This paper deals with large amplitude vibration of hybrid laminated plates containing piezoelectric layers resting on an elastic foundation in thermal environments. The motion equation of the plate that includes plate-foundation interaction is based on a higher order shear deformation plate theory and solved by a two-step perturbation technique. The thermo-piezoelectric effects are also included and the material properties of both orthotropic layers and piezoelectric layers are assumed to be temperature-dependent. The numerical illustrations concern nonlinear vibration characteristics of unsymmetric cross-ply and antisymmetric angle-ply laminated plates with fully covered or embedded piezoelectric actuators under different sets of thermal and electrical loading conditions. The results show that the foundation stiffness and stacking sequence have a significant effect on the nonlinear vibration characteristics of the hybrid laminated plate. The results also reveal that the temperature rise reduces the natural frequency, but it only has a small effect on the nonlinear to linear frequency ratios of the hybrid laminated plate. The results confirm that the effect of the applied voltage on the natural frequency and the nonlinear to linear frequency ratios of the hybrid laminated plate is marginal except the plate is sufficiently thin.     

Modified mixed Ritz-DQ formulation for free vibration of thick rectangular and skew plates with general boundary conditions

Recently, the present authors proposed a simple mixed Ritz-differential quadrature (DQ) methodology for free and forced vibration, and buckling analysis of rectangular plates. In this technique, the Ritz method is first used to discretize the spatial partial derivatives with respect to a coordinate direction of the plate. The DQ method is then employed to analogize the resulting system of ordinary or partial differential equations. The mixed method was shown to work well for vibration and buckling problems of rectangular plates with simple boundary conditions. But, due to the use of conventional Ritz method in one coordinate direction of the plate, the geometric boundary conditions of the problem can only be satisfied in that direction. Therefore, the conventional mixed Ritz-DQ methodology may encounter difficulties when dealing with rectangular plates involving adjacent free edges and skew plates. To overcome this difficulty, this paper presents a modified mixed Ritz-DQ formulation in which all the natural boundary conditions are exactly implemented. The versatility, accuracy and efficiency of the proposed method for free vibration analysis of thick rectangular and skew plates are tested against other solution procedures. It is revealed that the proposed method can produce highly accurate solutions for the natural frequencies of thick rectangular plates involving adjacent free edges and skew plates using a small number Ritz terms and DQ sampling points.     

Nonlinear analysis of thin rectangular plates on Winkler–Pasternak elastic foundations by DSC–HDQ methods

This article introduces a coupled methodology for the numerical solution of geometrically nonlinear static and dynamic problem of thin rectangular plates resting on elastic foundation. Winkler–Pasternak two-parameter foundation model is considered. Dynamic analogues Von Karman equations are used. The governing nonlinear partial differential equations of the plate are discretized in space and time domains using the discrete singular convolution (DSC) and harmonic differential quadrature (HDQ) methods, respectively. Two different realizations of singular kernels such as the regularized Shannon’s kernel (RSK) and Lagrange delta (LD) kernel are selected as singular convolution to illustrate the present DSC algorithm. The analysis provides for both clamped and simply supported plates with immovable inplane boundary conditions at the edges. Various types of dynamic loading, namely a step function, a sinusoidal pulse, an N-wave pulse, and a triangular load are investigated and the results are presented graphically. The effects of Winkler and Pasternak foundation parameters, influence of mass of foundation on the response have been investigated. In addition, the influence of damping on the dynamic analysis has been studied. The accuracy of the proposed DSC–HDQ coupled methodology is demonstrated by the numerical examples.     

The obstacle problem for beams and plates

The numerical solution of the obstacle problem for beams and plates by means of variational inequalities and finite elements is examined. Algorithms for the solution of the discrete problem are discussed and particular attention is paid to different methods of approximating the constraint. The results of some numerical experiments for beams and plates are included.     

Buckling and free vibration analysis of thick rectangular plates resting on elastic foundation using mixed finite element and differential quadrature method

In this article, a combination of the finite element (FE) and differential quadrature (DQ) methods is used to solve the eigenvalue (buckling and free vibration) equations of rectangular thick plates resting on elastic foundations. The elastic foundation is described by the Pasternak (two-parameter) model. The three dimensional, linear and small strain theory of elasticity and energy principle are employed to derive the governing equations. The in-plane domain is discretized using two dimensional finite elements. The spatial derivatives of equations in the thickness direction are discretized in strong-form using DQM. Buckling and free vibration of rectangular thick plates of various thicknesses to width and aspect ratios with Pasternak elastic foundation are investigated using the proposed FE-DQ method. The results obtained by the mixed method have been verified by the few analytical solutions in the literature. It is concluded that the mixed FE-DQ method has good convergancy behavior; and acceptable accuracy can be obtained by the method with a reasonable degrees of freedom.     

Exact solutions for free flexural vibration of Lévy-type rectangular thick plates via third-order shear deformation plate theory

In this paper, exact closed-form solutions in explicit forms are presented for transverse vibration analysis of rectangular thick plates having two opposite edges hard simply supported (i.e., Lévy-type rectangular plates) based on the Reddy’s third-order shear deformation plate theory. Two other edges may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. Hamilton’s principle is used to derive the equations of motion and natural boundary conditions of the plate. Several comparison studies with analytical and numerical techniques reported in literature are carried out to demonstrate accuracy of the present new formulation. Comprehensive benchmark results for natural frequencies of rectangular plates with different combinations of boundary conditions are tabulated in dimensionless form for various values of aspect ratios and thickness to length ratios. A set of three-dimensional (3-D) vibration mode shapes along with their corresponding contour plots are plotted by using exact transverse displacements of Lévy-type rectangular Reddy plates. Due to the inherent features of the present exact closed-form solution, the present findings will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future.     

Nonlinear free vibration analysis of simply supported piezo-laminated plates with random actuation electric potential difference and material properties

Studies are made on nonlinear free vibrations of simply supported piezo-laminated rectangular plates with immovable edges utilizing Kirchoff’s hypothesis and von Kármán strain–displacement relations. The effect of random material properties of the base structure and actuation electric potential difference on the nonlinear free vibration of the plate is examined. The study is confined to linear-induced strain in the piezoelectric layer applicable to low electric fields. The von Kármán’s large deflection equations for generally laminated elastic plates are derived in terms of stress function and transverse deflection function. A deflection function satisfying the simply supported boundary conditions is assumed and a stress function is then obtained after solving the compatibility equation. Applying the modified Galerkin’s method to the governing nonlinear partial differential equations, a modal equation of Duffing’s type is obtained. It is solved by exact integration. Monte Carlo simulation has been carried out to examine the response statistics considering the material properties and actuation electric potential difference of the piezoelectric layer as random variables. The extremal values of response are also evaluated utilizing the Convex model as well as the Multivariate method. Results obtained through the different statistical approaches are found to be in good agreement with each other.     

Vibration analysis of rectangular plates coupled with fluid

The approach developed in this paper applies to vibration analysis of rectangular plates coupled with fluid. This case is representative of certain key components of complex structures used in industries such as aerospace, nuclear and naval. The plates can be totally submerged in fluid or floating on its free surface. The mathematical model for the structure is developed using a combination of the finite element method and Sanders’ shell theory. The in-plane and out-of-plane displacement components are modelled using bilinear polynomials and exponential functions, respectively. The mass and stiffness matrices are then determined by exact analytical integration. The velocity potential and Bernoulli’s equation are adopted to express the fluid pressure acting on the structure. The product of the pressure expression and the developed structural shape function is integrated over the structure-fluid interface to assess the virtual added mass due to the fluid. Variation of fluid level is considered in the calculation of the natural frequencies. The results are in close agreement with both experimental results and theoretical results using other analytical approaches.     

Stability of thin rectangular plates under local heating

We study the temperature fields, the precritical thermally stressed state, and the loss of stability of thin rectangular plates under local heating. We study the influence of the thermophysical characteristics on the magnitude of the critical parameters. Four figures. Bibliography: 5 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 1–4.     

Nonlinear analysis of buckling,free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment

Employing Euler–Bernoulli beam theory and the physical neutral surface concept, the nonlinear governing equation for the functionally graded material beam with two clamped ends and surface-bonded piezoelectric actuators is derived by the Hamilton’s principle. The thermo-piezoelectric buckling, nonlinear free vibration and dynamic stability for the piezoelectric functionally graded beams, subjected to one-dimensional steady heat conduction in the thickness direction, are studied. The critical buckling loads for the beam are obtained by the existing methods in the analysis of thermo-piezoelectric buckling. The Galerkin’s procedure and elliptic function are adopted to obtain the analytical solution of the nonlinear free vibration, and the incremental harmonic balance method is applied to obtain the principle unstable regions of the piezoelectric functionally graded beam. In the numerical examples, the good agreements between the present results and existing solutions verify the validity and accuracy of the present analysis and solving method. Simultaneously, validation of the results achieved by rule of mixture against those obtained via the Mori–Tanaka scheme is carried out, and excellent agreements are reported. The effects of the thermal load, electric load, and thermal properties of the constituent materials on the thermo-piezoelectric buckling, nonlinear free vibration, and dynamic stability of the piezoelectric functionally graded beam are discussed, and some meaningful conclusions have been drawn.     

A vibration analysis of composite laminated plates

A finite element formulation of the equations governing laminated anisotropic plates using Reddy's higher-order theory is presented. This simple higher-order shear deformable theory takes into account the parabolic distribution of the transverse shear deformation through the thickness of the plate and contains the same unknowns as in the first-order shear deformation theory. Finite element solutions are presented for rectangular plates of different layups, such as cross-ply, antisymmetric angle-ply, and sandwich plates with various material properties, boundaries, and plate aspect ratios. The numerical results are compared with the available closed-form results, the 3-D linear elasticity theory results, and the other available numerical results. A comparison is also made with test data from a laminated cantilever plate.