Non-homogeneous Elastic Foundations Research Articles

Vibration response of FG-CNTRC plate resting on non-homogeneous elastic foundation via a closed-form dynamic stiffness formulation

In this work, a closed-form dynamic stiffness matrix has been formulated to study the influence of various configurations of non-homogeneous elastic foundation on the free vibration characteristics of carbon nanotube-reinforced functionally graded (FG-CNTRC) plate. The plate kinematic variables are defined based on the first order shear deformation theory. Through the application of Hamilton’s principle, frequency dependent dynamic stiffness (DS) matrix is formulated for a Levy type plate element supported on the Winkler-Pasternak foundation. Appropriate assembly of these DS matrices leads to the formation of global dynamic stiffness matrix, for the non-homogeneously supported plate, which is finally solved using the well-established Wittrick–Williams algorithm to compute the mode shapes and natural frequencies to a desired accuracy. Thus, the present work exploits the applicability of the dynamic stiffness method (DSM) for the free vibration study of the FG-CNTRC plate supported by partial and non-homogeneous elastic foundations. A comparative study is also carried out to ascertain the exactness and the computational efficiency of the dynamic stiffness method. Results from the parametric studies show that the non-homogeneous supports greatly influence the frequency as well as the modeshape of the plate. It is observed that the Pasternak parameter has a profound effect on the response of the plates as compared to that of the Winkler parameter. Effect of different CNT distributions on the frequency behavior are also discussed in detail. It can be emphasized that the new set of DSM computed results reported in the paper for diverse plate configurations may serve as reference results for the future research studies.

Free Vibration Analysis of Rectangular Thin Plates Resting on Nonhomogeneous Elastic Foundations by Using Matched Interface and Boundary Algorithms and their Multi-Domain Formulations

Matched interface and boundary (MIB) algorithms and their multi-domain formulations are developed to solve free vibration problems of thin plates resting on unidirectionally and bidirectionally nonhomogeneous elastic foundations. Winkler and Pasternak types of elastic foundations are considered. Three types of MIB algorithms are introduced to handle unidirectional interfaces caused by nonhomogeneous foundations. On the basis of in-depth understanding of MIB method and Kirchhoff plate theory, new multi-domain MIB (MD-MIB) algorithms are proposed to handle bidirectional interfaces caused by nonhomogeneous foundations that cannot be handled by using the existing MIB method. Three types of MD-MIB algorithms are developed by using three domain decomposition strategies. A number of examples are chosen to validate the applicabilities of different MIB and MD-MIB algorithms to the title problems. Numerical results obtained by using MIB and MD-MIB algorithms are compared with the existing solutions or finite element solutions. Numerical studies show that MIB and MD-MIB algorithms are efficient approaches to deal with nonhomogeneous elastic foundations.

A series of multi-domain matched interface and boundary algorithms for dynamic and static responses of annular sectorial plates

A series of multi-domain matched interface and boundary (MDMIB) algorithms are proposed to analyze the free vibration and bending of annular sectorial thin plates. Through an in-depth understanding of the matched interface and boundary (MIB) method, six types of MDMIB algorithms are developed herein by using various domain decomposition techniques for handling bi-directional crossing interfaces due to non-homogeneous elastic foundations and discontinuous loads, which cannot be treated by the conventional MIB method. The rationale of the MDMIB method is to divide the entire computational domain into multiple subdomains that can be connected by using the interconnecting conditions between crossing interfaces and mesh lines. Annular sectorial plates, having non-uniform boundary conditions, partially supported elastic foundations, and patch loads, are all considered. Numerous examples are chosen to examine the computational performance of the MDMIB algorithms. For verification, the present results are checked against existing and finite element solutions. It is shown that the proposed MDMIB algorithms are highly effective to deal with the bi-directional interface problems.

Dynamic Response of Railway Track Resting on Variable Foundation Using Finite Element Method

At high train speed, differential settlement of railway track arises due to the variation in stiffness and yield characteristics of foundation along its length. In the present study, the effect of nonhomogeneous elastic foundation on dynamic response of railway track due to a moving load is studied. The railway track is modeled as an infinitely long beam on nonhomogeneous elastic foundation subjected to a moving load. The foundation stiffness is varying harmonically, about a mean value, along the length of beam. The dynamic response is obtained using a finite element (FE) approach. Adequacy of the proposed FE model is ascertained by comparing the FE results with the results available in the literature. The comprehensive new results are obtained to quantify the effect of amplitude (e) and wavelength (λ) of harmonic foundation on the dynamic response of beam. The peak displacement of beam increases with e, λ and velocity. It is further shown that for harmonically varying foundation, higher-order harmonics are significantly present in the response making it irregular. Also, the critical velocity (Vcr) is considerably affected by e and λ. The variation of peak displacement with velocity is found to be irregular and instead of a distinct peak at Vcr, a plateau is obtained, implying that for a range of velocities peak displacement is quite high. This is also demonstrated through modal mass contribution by a large number of modes. The FE model used in this study can be naturally extended to predict the differential track settlement by including yielding characteristics of foundation.

Energy-Based Solution for Bending Analysis of Thin Plates on Nonhomogeneous Elastic Foundation

In this paper, the application of Ritz method to the bending of thin plates having mixed boundary conditions and resting on nonhomogeneous foundation with varied subgrade reaction is presented. A typical application of the considered problem is in the analysis and design of foundations resting on a varying soil bed. The proposed model can tackle different kinds of changes in the subgrade reaction including zero subgrade reaction at some locations (cavity zones) as well as rigid zones. The variation of the subgrade reaction modulus is represented by an appropriate mathematical expression that accounts for its possible discontinuous variation. A systematic approach for deriving Ritz trial functions that satisfy the geometric boundary conditions is illustrated for different plate supporting conditions. Three numerical examples are solved to verify the accuracy of the proposed method.

Free and forced vibration analysis of moderately thick orthotropic plates in thermal environment and resting on elastic supports

This paper investigates the free and forced vibration of moderately thick orthotropic plates under thermal environment and resting on elastic supports. Three kinds of elastic supports, namely non-homogeneous elastic foundations, point elastic supports and line elastic supports, are considered in the present study. The first-order shear deformation theory is employed to formulate the strain and kinetic energy functions of the structures, and then the stiffness and mass matrices can be obtained by applying the Hamilton’s principle. The modified Fourier method is adopted to solve the dynamic problems of moderately thick orthotropic plates with different combinations of temperature variations, elastic supports and boundary conditions. The accuracy and reliability of the proposed formulation are validated by comparing the obtained results with the finite element method results. Finally, the effects of some key parameters including temperature variation and stiffness values of the elastic supports on the modal and dynamic characteristics of the plates are analyzed in detail. In views of the versatility of the developed method, it offers an efficient tool for the structural analysis of moderately thick orthotropic plates under thermal environment and resting on elastic supports.

Free vibration analysis of stepped rectangular plates resting on non-homogeneous elastic foundations

Abstract A Half Boundary Method (HBM) is proposed for analyzing the free vibration problem of rectangular plates with stepped thickness resting on non-homogeneous elastic foundations. The unknown quantities of the method exist only on half of the boundary. The non-homogeneous elastic foundation discussed here consists of two-segment elastic Winkler foundation. The fundamental differential equations are established for the bending problem of the plate on elastic foundations. The Green function, which is obtained by transforming these differential equations into integral equations and using numerical integration, is used to establish the characteristic equation of the free vibration. The effects of the modulus of the foundation, the stepped thickness and aspect ratio on the frequency parameters are considered. By comparing the present numerical results with those previously published, the efficiency and accuracy of the present method are investigated.

Vibration of a beam on continuous elastic foundation with nonhomogeneous stiffness and damping under a harmonically excited mass

In this paper, a method of analysis of a beam that is continuously supported on a linear nonhomogeneous elastic foundation and subjected to a harmonically excited mass is presented. The solution is obtained by decomposing the nonhomogeneous foundation properties and the beam displacement response into double Fourier summations which are solved in the frequency–wavenumber domain, from which the space–time domain response can be obtained. The method is applied to railway tracks with step variation in foundation properties. The validity of this method is checked, through examples, against existing methods for both homogeneous and nonhomogeneous foundation parameters. The effect of inhomogeneity and the magnitude of the mass are also investigated. It is found that a step variation in foundation properties leads to a reduction in the beam displacement and an increase in the resonance frequency for increasing step change, with the reverse occurring for decreasing step change. Furthermore, a beam on nonhomogeneous foundation may exhibit multiple resonances corresponding to the foundation stiffness of individual sections, as the mass moves through the respective sections along the beam.

Free vibration of thick composite plates on non-homogeneous elastic foundations by dynamic stiffness method

This research presents a new model for vibration analysis of thick laminated plates on a non-homogeneous elastic foundation by the Dynamic Stiffness Method(DSM). The non-homogeneous foundation consists of multi-segment Winkler-type and Pasternak-type elastic foundation. The Dynamic Stiffness Matrices using the First ShearDeformation Theory (FSDT) are constructed for cross-ply thick composite plates. A computer program is written using the present formulation for calculating natural frequencies and harmonic response of composite plates subjected to various types of boundary conditions. Numerical results are validated by comparison with available results in the literature and with Finite Element Method (FEM). Different test cases make evidence the advantages of the present model: higher precision, less data storage, less computing time and studied frequency range extended.

Dynamic stiffness formulation for vibration analysis of thick composite plates resting on non-homogenous foundations

This research presents new continuous elements for thick laminated plates on a Pasternak or a non-homogeneous foundation using the Dynamic Stiffness Method (DSM). The non-homogeneous foundation consists of multi-section Winkler-type and Pasternak-type elastic foundation. The Dynamic Stiffness Matrices using the First Shear Deformation Theory (FSDT) are constructed based on the exact closed form solutions of the governing differential equations of both cross-ply and angle-ply thick composite plates on non-homogeneous elastic foundations and subjected to various types of boundary conditions. A computer program is written using the present formulation for calculating the natural frequencies and harmonic response of composite plates without contact with elastic foundation and composite plates resting on non-homogenous foundations. The validation is done by comparison of continuous element model with available results in the literature and with Finite Element Method (FEM). Different test cases confirm the advantages of the present model which is supposed to be especially efficient in dynamics analysis.

不均-弾性地盤上の正方形板の自由振動の-解析法

不均-弾性地盤上の正方形板の自由振動の-解析法

The stability of uniform-friction piles inhomogeneous and non-homogeneous elastic foundations

The stability of piles that are supported vertically by a frictional force and laterallyalong their entire length by an elastic Winkler foundation is investigated for the case when thecoefficient of subgrade reaction is either constant or varies linearly with depth and the friction isconstant along the embedded length of the pile. A comparison between frictional and end-bearingmodels is made. Finally, a concise summary of the buckling loads for fully embedded friction pilesis produced.

Analysis of general elastically end restrained non-uniform beams using differential transform

This study introduces using differential transform to solve the free and forced vibration problems of a general elastically end restrained non-uniform beam, resting on a non-homogeneous elastic foundation and subjected to axial tensile and transverse forces. First, differential transform is briefly introduced. Second, taking differential transform of non-uniform beam vibration problems, a set of difference equations is derived. Doing some simple algebraic operations on these equations, we can determine any ith natural frequency, the closed form series solution of any ith mode shape and the dynamic response. Finally, two vibration problems are given to illustrate the accuracy and efficiency of the present method.

Stability of end-bearing piles in a non-homogeneous elastic foundation

The stability of end-bearing piles that are supported laterally along their entire length by an elastic Winkler foundation is investigated for the case when the coefficient of horizontal subgrade reaction varies linearly with depth. A pattern of clustering of buckling modes is shown to occur and the approximate modelling of the elastic foundation by averaging the stiffness of the subgrade is discussed. © 1997 by John Wiley & Sons, Ltd.

Free vibration of plates with stepped variations in thickness on non-homogeneous elastic foundations

A finite element model is presented for the analysis of the free vibration of plates with multiple stepped variations in thickness resting on non-homogeneous elastic foundations. Based on Mindlin plate theory, the model includes transverse shear deformation as well as bending-extension coupling in cases of plates with stepped sections eccentrically located with respect to the mid-planes. The section of elastic foundation under a plate element is treated as a separate foundation element. The transverse deformation of these foundation elements is made to be consistent with the deflection of plate elements being supported, resulting in a consistent stiffness matrix for the elastic foundation. Numerical results are in good agreement with the available reported results. The effect of eccentricity of the locations of stepped sections on the natural frequencies is found to be not negligible. The elastic foundations are found to have a significant effect on the fundamental natural frequencies of both uniform and stepped plates. Natural frequencies and mode shapes of rectangular plates and circular plates with multiple eccentrically stepped sections resting on non-homogeneous elastic foundations are presented.

Analysis of beams on non-homogeneous elastic foundation

An approximate method is developed to study the bending of the beam on non-homogeneous elastic foundation. The solutions are obtained by translating the differential equations into the integral equations and applying the numerical integrations. It is shown that we can analyze the beams with arbitrary boundary conditions and load conditions resting on non-homogeneous elastic foundation.

Vibration of beams on non-homogeneous elastic foundations : Pavlovic, M N; Wylie, G B Earthq Engng Struct DynamV11, N6, Nov–Dec 1983, P797–808

Vibration of beams on non-homogeneous elastic foundations : Pavlovic, M N; Wylie, G B Earthq Engng Struct DynamV11, N6, Nov–Dec 1983, P797–808

Flexure of slabs of variable rigidity on a nonhomogeneous elastic foundation

Flexure of slabs of variable rigidity on a nonhomogeneous elastic foundation