Free Axisymmetric Flexural Vibrations of Circular Plate with Symmetrically Varying Mechanical Properties Supported on Elastic Foundation

In this article, the geometrically nonlinear free and forced vibrations of a beam made of functional gradient materials are analytically analyzed, in which piezoelectric actuators are bound. Based on the Euler–Bernoulli beam theory and the nonlinear Von-Karman deformation field, the analytical solutions of the nonlinear free vibrations, and forced vibrations of a beam with two clamped ends and resting on an elastic foundation are predicted. Using Hamilton's principle and an approximate multimodal dominant mode method, the nonlinear governing equations are obtained. Due to the lack of forced vibration results of the functionally graded piezoelectric beam subjected to thermal loading and resting on elastic foundations, the results obtained by the current solution were compared with their counterparts. The good agreement between the current solution and equivalent data existing in the open literature proves the efficiency and accuracy of the current resolution analysis method. Finally, the new parametric studies covered in this research include the effects of several parameters such as elastic foundation, thermal loading, and thermal properties of constituent materials on the free and forced nonlinear vibration of the piezoelectric functional gradient beam.

This present paper concerned with the analytic modelling for vibration of the functionally graded (FG) plates resting on non-variable and variable two parameter elastic foundation, based on two-dimensional elasticity using higher shear deformation theory. Our present theory has four unknown, which mean that have less than other higher order and lower theory, and we denote do not require the factor of correction like the first shear deformation theory. The indeterminate integral are introduced in the fields of displacement, it is allowed to reduce the number from five unknown to only four variables. The elastic foundations are assumed a classical model of Winkler-Pasternak with uniform distribution stiffness of the Winkler coefficient (kw), or it is with variables distribution coefficient (kw). The variable\'s stiffness of elastic foundation is supposed linear, parabolic and trigonometry along the length of functionally plate. The properties of the FG plates vary according to the thickness, following a simple distribution of the power law in terms of volume fractions of the constituents of the material. The equations of motions for natural frequency of the functionally graded plates resting on variables elastic foundation are derived using Hamilton principal. The government equations are resolved, with respect boundary condition for simply supported FG plate, employing Navier series solution. The extensive validation with other works found in the literature and our results are present in this work to demonstrate the efficient and accuracy of this analytic model to predict free vibration of FG plates, with and without the effect of variables elastic foundations.

A new version differential quadrature method is proposed to obtain the vibration characteristics of rectangular plates resting on elastic foundations and carrying any number of sprung masses. The accuracy of the technique is demonstrated by comparing the calculated results with the published data. The non-uniform grid spacing is used in this work. The results also demonstrate the efficiency of the method in treating the vibration problem of the rectangular plates carrying any number of sprung masses and resting on the elastic foundations.

This paper carries out free and forced vibration analysis of piezoelectric FGM plates resting on two‐parameter elastic foundations placed in thermal environments. By employing the third‐order shear deformation theory and the finite element method, this work establishes free and forced vibration equations of piezoelectric FGM plates, where the materials are assumed to be varied in the thickness directions, and the mechanical properties depend on the temperature. Then, comparative examples are conducted to verify the proposed theory and mathematical model, and the results of this study and other methods meet a very good agreement. Then, effects of geometrical and material properties such as the feedback coefficient, voltage, volume fraction index, temperature as well as the parameters of elastic foundations on free and forced vibration of the plates are investigated, and the conclusions are given out to provide the effective direction for the design and practical use of these structures.

A train moving on the ground can be modified as moving forces acting on abeam with elastic foundation. This paper is concerned with the vibration responses of an infinite, fluid-loaded, elastic foundation of Bernoulli-Euler beam under the convected forces. The formulation of equation is developed by taking account of the whole structure in a wavenumber domain through the Fourier transform and distinguishes itself from the space-harmonic analysis. The effects of convected loading speed, elastic foundation stiffness, fluid loading and a point or line force excitation by high frequency on a Bernoulli-Euler beam are discussed. Furthermore, the behaviors of a periodically spring-supported beam how to approximate the beam on the elastic foundation are also illustrated.

This work investigates a new type of quasi-3D hyperbolic shear deformation theory is proposed in this study to discuss the statics and free vibration of functionally graded porous plates resting on elastic foundations. Material properties of porous FG plate are defined by rule of the mixture with an additional term of porosity in the through-thickness direction. By including indeterminate integral variables, the number of unknowns and governing equations of the present theory is reduced, and therefore, it is easy to use. The present approach to plate theory takes into account both transverse shear and normal deformations and satisfies the boundary conditions of zero tensile stress on the plate surfaces. The equations of motion are derived from the Hamilton principle. Analytical solutions are obtained for a simply supported plate. Contrary to any other theory, the number of unknown functions involved in the displacement field is only five, as compared to six or more in the case of other shear and normal deformation theories. A comparison with the corresponding results is made to verify the accuracy and efficiency of the present theory. The influences of the porosity parameter, power-law index, aspect ratio, thickness ratio and the foundation parameters on bending and vibration of porous FG plate.

Based on a four-variable shear deformation shell theory, the free vibration analysis of functionally graded graphene platelet-reinforced composite (FGGPRC) doubly-curved shallow shells with different boundary conditions is investigated in this work. The doubly-curved shells are composed of multi nanocomposite layers that are reinforced with graphene platelets. The graphene platelets are uniformly distributed in each individual layer. While, the volume faction of the graphene is graded from layer to other in accordance with a novel distribution law. Based on the suggested distribution law, four types of FGGPRC doubly-curved shells are studied. The present shells are assumed to be rested on elastic foundations. The material properties of each layer are calculated using a micromechanical model. Four equations of motion are deduced utilizing Hamilton's principle and then converted to an eigenvalue problem employing an analytical method. The obtained results are checked by introducing some comparison examples. A detailed parametric investigation is performed to illustrate the influences of the distribution type of volume fraction, shell curvatures, elastic foundation stiffness and boundary conditions on the vibration of FGGPRC doubly-curved shells.

Based on the Winkler assumption for the elastic foundation, a double Euler-Bernoulli beam system was presented in the form of two partial differential equations, and the analytic solutions of the steady vibration were obtained under a moving load with uniform velocity. Then, by an example, the analytic solution would be compared with the solutions of FLAC, and results showed their consistent rules. Dynamic analytical solutions were then degenerated into the static solution. When the upper beam stiffness tended to infinity, static solution was simplified as the form of double elastic foundation, when the lower beam stiffness tends to infinity, static solution was simplified as common single-layer elastic foundation form. Accordingly, this method provided theory basis for coupled vibration analysis of soil - structure caused by the traffic load in elastic half-space.

In this paper, a closed form solution for bending and free vibration analyses of simply supported rectangular laminated composite plates is presented. The static and free vibration behavior of symmetric and antisymmetric laminates is investigated using a refined first-order shear deformation theory. The Winkler–Pasternak two-parameter model is employed to express the interaction between the laminated plates and the elastic foundation. The Hamilton’s principle is used to derive the governing equations of motion. The accuracy and efficiency of the theory are verified by comparing the developed results with those obtained using different laminate theories. The laminate theories including the classical plate theory, the classical first-order shear deformation theory, the higher order shear deformation theory and a three-dimensional layerwise theory are selected in order to perform a comprehensive comparison. The effects of the elastic foundation parameters, orthotropy ratio and width-to-thickness ratio on the bending deflection and fundamental frequency of laminates are investigated.

In this paper, the effect of the homogenization models on buckling and free vibration is presented for functionally graded plates (FGM) resting on elastic foundations. The majority of investigations developed in the last decade, explored the Voigt homogenization model to predict the effective proprieties of functionally graded materials at the macroscopic-scale for FGM mechanical behavior. For this reason, various models have been used to derive the effective proprieties of FGMs and simulate thereby their effects on the buckling and free vibration of FGM plates based on comparative studies that may differ in terms of several parameters. The refined plate theory, as used in this paper, is based on dividing the transverse displacement into both bending and shear components. This leads to a reduction in the number of unknowns and governing equations. Furthermore the present formulation utilizes a sinusoidal variation of displacement field across the thickness, and satisfies the stress-free boundary conditions on the upper and lower surfaces of the plate without requiring any shear correction factor. Equations of motion are derived from Hamilton’s principle. Analytical solutions for the buckling and free vibration analysis are obtained for simply supported plates. The obtained results are compared with those predicted by other plate theories. This study shows the sensitivity of the obtained results to different homogenization models and that the results generated may vary considerably from one theory to another. Comprehensive visualization of results is provided. The analysis is relevant to aerospace, nuclear, civil and other structures.

This study presents a comprehensive nonlinear dynamic approach to investigate the linear and nonlinear vibration of sandwich plates fabricated from functionally graded materials (FGMs) resting on an elastic foundation. Higher-order shear deformation theory and Hamilton's principle are employed to obtain governing equations. The Runge–Kutta method is employed together with the commercially available mathematical software MAPLE 14 to solve the set of nonlinear dynamic governing equations. Method validity is evaluated by comparing the results of this study and those of previous research. Good agreement is achieved. The effects of temperature change on frequencies are investigated considering various temperatures and various volume fraction index values, N. As the temperature increased, the plate frequency decreased, whereas with increasing N, the plate frequency increased. The effects of the side-to-thickness ratio, c/h, on natural frequencies were investigated. With increasing c/h, the frequencies increased nonlinearly. The effects of foundation stiffness on nonlinear vibration of the sandwich plate were also studied. Backbone curves presenting the variation of maximum displacement with respect to plate frequency are presented to provide insight into the nonlinear vibration and dynamic behavior of FGM sandwich plates.

Dynamic analysis of beams by the boundary element method

Nonlinear dynamic analysis of damaged Reddy–Bickford beams supported on an elastic Pasternak foundation

In the present research, the free vibration analysis of functionally graded (FG) nanocomposite deep spherical shells reinforced by graphene platelets (GPLs) on elastic foundation is performed. The elastic foundation is assumed to be Winkler-Past ernak-type. It is also assumed that graphaene platelets are randomly oriented and uniformly dispersed in each layer of the nanocomposite shell. Volume fraction of the graphene platelets as nanofillers may be different in the layers. The modified HalpinTsai model is used to approximate the effective mechanical properties of the multilayer nanocomposite. With the aid of the first order shear deformation shell theory and implementing Hamilton\'s principle, motion equations are derived. Afterwards, the generalized differential quadrature method (GDQM) is utilized to study the free vibration characteristics of FG-GPLRC spherical shell. To assess the validity and accuracy of the presented method, the results are compared with the available researches. Finally, the natural frequencies and corresponding mode shapes are provided for different boundary conditions, GPLs volume fraction, types of functionally graded, elastic foundation coefficients, opening angles of shell, and thickness-to-radius ratio.

In this study, the effect of steel fiber utilization, boundary conditions, different beam cross-section, and length parameter are investigated on the free vibration behavior of fiber reinforced self-compacting concrete beam on elastic foundation. In the analysis of the beam model recommended by Euler-Bernoulli, a method utilizing Stokes transformations and Fourier Sine series were used. For this purpose, in addition to the control beam containing no fiber, three SCC beam elements were prepared by utilization of steel fiber as 0.6% by volume. The time-dependent fresh properties and some mechanical properties of self-compacting concrete mixtures were investigated. In the modelled beam, four different beam specimens produced with 0.6% by volume of steel fiber reinforced and pure (containing no fiber) SCC were analyzed depending on different boundary conditions, different beam cross-sections, and lengths. For this aim, the effect of elasticity of the foundation, cross-sectional dimensions, beam length, boundary conditions, and steel fiber on natural frequency and frequency parameters were investigated. As a result, it was observed that there is a noticeable effect of fiber reinforcement on the dynamic behavior of the modelled beam.