Winkler-type Foundation Research Articles

On the refined boundary condition at the edge of a thin elastic strip supported by a Winkler-type foundation under antiplane shear deformation

The derivation of the boundary conditions is the most challenging part of the asymptotic techniques underlying low-dimensional models for thin elastic structures. At the moment, these techniques do not take into consideration the effect of the environment, e.g., a Winkler foundation, when tackling boundary conditions, and have to be amended. In this paper as an example we consider an antiplane problem for a thin elastic strip contacting with a relatively compliant Winkler foundation. Refined boundary conditions at an edge loaded by prescribed stresses are established using a properly adjusted Saint-Venant’s principle. They appear to be useful for advanced structure modelling including analysis of the static equilibrium under self-equilibrated loading.

Dynamic analysis of anisotropic doubly-curved shells with general boundary conditions, variable thickness and arbitrary shape

In the present contribution a general formulation is proposed to account for general boundary conditions within the dynamic analysis of anisotropic laminated doubly-curved shell having arbitrary shape and variable thickness. Different analytical expressions are considered for the shell thickness variation along the geometrical principal directions, and the distortion of the physical domain is described by a mapping procedure based on Non-Uniform Rational Basis Spline (NURBS) curves. Mode frequencies and shapes are determined employing higher-order theories within an Equivalent Single Layer (ESL) framework. The related fundamental relations are tackled numerically by means of the Generalized Differential Quadrature (GDQ) method. The dynamic problem is derived from the Hamiltonian Principle, leading to a strong formulation of the governing equations. General external constraints are enforced along the edges of the shell employing a distribution of linear springs distributed on the faces of the three-dimensional solid and accounting for a spatial coordinate-dependent stiffness along both in-plane and out-of-plane directions. Moreover, a Winkler-type foundation with general distribution of linear springs is modelled on the top and bottom surfaces of the shell. A systematic set of numerical examples is carried out for the validation of the proposed theory by comparing mode frequencies with predictions from refined three-dimensional finite element analyses. Finally, we perform a sensitivity analysis of the dynamic response of mapped curved structures for different spring stiffnesses and general external constraints, according to various kinematic assumptions.

A homotopy-based wavelet method for extreme large bending analysis of heterogeneous anisotropic plate with variable thickness on orthotropic foundation

A novel homotopy-based decoupled wavelet numerical strategy is developed to investigate the large-deflection bending of variable-thickness anisotropic thin plate constituted by heterogeneous material resting on orthotropic foundation. The anisotropic plates are described undergoing sinusoidally initial imperfections with inhomogeneous thickness and exponentially varying elastic modules, while the subgrade reaction involves the nonlinear support effects by Winkler-type foundation along with Pasternak-type orthotropy. Geometric nonlinearity by uneven thickness and large deformation, material nonlinearity by heterogeneity and contact nonlinearity of orthotropic foundation are simultaneously considered with the highly coupled and strongly nonlinear variable-coefficient Von Kármán equations derived. The accuracy and convergence of the present formulation are validated in excellent agreement with published results, while the new results of extreme bending solutions are provided, which are few given by other methods and further verify the effectiveness and superiority of proposed wavelet procedures. Parametric investigations of influences of material orthotropy and inhomogeneity, unevenness of thickness, amplitude of initial imperfection along with orthotropy of nonlinear Winkler–Pasternak foundation have been studied, which are of great significance on the extremely large bending properties.

Localized folding of thick layers

We describe the localized folding of thick layers embedded in a viscoelastic framework. Higher-order partial differential equations such as the Swift-Hohenberg equation are standard for modelling the folding process. Using a high-order shear theory, we modify the Swift-Hohenberg equation to describe the buckling of thick layers and consider the folded layer's viscoelastic behavior. The use of thick layers enables us to consider shear strains parallel to the layers during folding. Our model naturally captures the softening-stiffening behavior by including a non-linear viscoelastic description using a Winkler-type foundation. Next, we study the linear stability behavior of the system and derive the dispersion relations. Finally, we simulate this new model using a robust custom-built isogeometric analysis solver, which allows us to describe thick folded layers with localized folding. The numerical results show that the folding of an elastic layer produces periodic patterns while a viscoelastic layer deflects locally. When the horizontal forces are unequal, the periodic folds initiate in the direction of the smaller force and the localized deformations occur parallel to the larger force. Later, the nonlocal deformation occurs in the direction of a smaller force. Domes and basins or more linear ridges and valleys are formed according to the relative magnitudes of the applied forces. Domes and basins are the results of equal horizontal applied forces, and non-equal forces result in ridges and valleys.

FORCE DRIVEN VIBRATIONS OF NONLINEAR PLATES ON A VISCOELASTIC WINKLER FOUNDATION UNDER THE HARMONIC MOVING LOAD

In the present paper, the forced driven nonlinear vibrations of an elastic plate in a viscoelastic medium and resting on a viscoelastic Winkler-type foundation are studied. The damping features of the surrounding medium and foundation are described by the Kelvin-Voigt model and standard linear solid model with fractional derivatives, respectively. The dynamic response of the plate is described by the set of nonlinear differential equations with due account for the fact that the plate is being under the conditions of the internal resonance accompanied by the external resonance. The expressions for the stress function and nonlinear coefficients for different types of boundary conditions are presented.

Ultimate bearing capacity of plate on Winkler foundation subjected to a circular uniform load

The present study investigated the problem of a large slab on a Winkler-type foundation subjected to a circular load. The slab is modeled as a ductile Kirchhoff plate and obeys Rankine’s yield criterion with associative flow rule. A circular uniform load is applied onto the top of the plate. As the load increases, the radial cracks occur and spread out at the bottom of the slab. The slab’s load-carrying capacity is defined by the onset of a negative yield line, namely a circumferential crack at the upper side of the slab. Depending on whether the circumferential bending curvature of the plate reaches its elastic limit, the slab was divided into two regions: the inner rigid-plastic region and the outer elastic region. A closed-form solution of the governing equations for each region is found. The influence of the material and geometrical parameters, size of the loaded area, and temperature gradient on the plate’s load-carrying capacity is then investigated. The load-carrying capacities found in previous investigations are compared with the predictions of the present analysis. Based on the analytical results, a regression formula for calculating the plate’s ultimate bearing capacity resting on a Winkler foundation is proposed.

Effect of temperature on vibration of cracked single-walled carbon nanotubes embedded in an elastic medium under different boundary conditions

This study investigates thermal effect on free vibration of cracked single-walled carbon nanotubes (SWCNTs) embedded in an elastic medium under different boundary conditions. The nonlocal Euler–Bernoulli beam equations are derived using the Hamiltonian principle. The interaction of the elastic medium and the SWCNT is modeled using Winkler-type foundation. The effect of low and high temperatures, crack severity, crack position, scale coefficient, and elastic medium are investigated. The results show that the temperature at some cases have the significant effects on the frequencies. The elastic medium has a great influence on the frequencies of cracked and non-cracked beams at any temperature.

Hollow Core Slabs on Winkler Foundation

This research dealt with the linear elastic behavior of hollow core slabs resting on a linear Winkler type foundation. A finite difference method was used to model the slabs as wide beams and the foundation as elastic springs. The finite element method was also used to model the problem using ABAQUS 6.10 software program. A comparison between the method proposed in this paper and methods from previous studies was made to check the accuracy of the solutions. Several important parameters were incorporated in the analysis, viz. the hollow core size and shape, subgrade reaction and slab depth, to trace their effects on deflections, bending moments and shear forces. A computer program coded in Fortran was developed for the analysis of hollow core slabs on an elastic foundation. It was found that the maximum difference in deflection between the present study and the exact solution was 3% for the finite difference and 7% for the finite element method.

Measurement of Vibrations of a Plate on Elastic Foundation

Vibrations of a plate on elastic foundation of Winkler type are investigated. A two-dimensional element having six nodal degrees of freedom (three displacements of the lower surface and three displacements of the upper surface) is used. Elastic foundation of Winkler type on the lower surface is used in the numerical model. Eigenmodes are calculated. Experimental investigations of vibrations were performed using a special experimental setup and typical experimental results are presented. Using the presented results places for recommended locations of Braille elements are determined. It is recommended to locate them at the places were the amplitude of vibrations is smallest. The stiffness parameters of the elastic foundation must be chosen in such a way that enables to shift the eigenfrequencies of the elastic structure from the excitation frequencies of the investigated mechanical device. DOI: http://dx.doi.org/10.5755/j01.mech.24.4.20742

Dynamics of a beam on a bilinear elastic foundation under harmonic moving load

The present paper is concerned with the numerical modelization of the transient dynamic response of a simply supported Euler–Bernoulli elastic beam resting on a Winkler-type foundation, under the action of a transverse concentrated load, moving at a constant velocity along the beam, displaying an harmonic-varying magnitude in time. The elastic foundation, assumed as homogeneous in space, behaves according to a bilinear constitutive law, characterized by two different stiffness coefficients in compression and in tension. A finite element method approach coupled with a direct integration algorithm is developed for efficiently tracing the nonlinear dynamic response of the beam-foundation system. An original automated procedure is set, as being apt to resolve all required space/time discretization issues. Extensive parametric numerical analyses are performed to investigate how the frequency of the harmonic moving load amplitude and the ratio between the foundation’s moduli in compression and in tension affect the so-called critical velocities of the moving load, leading to high transverse beam deflections. Analytical interpolating expressions are proposed and fitted for the achieved two-branch critical velocity trends. The present outcomes shall reveal potential practical implications in scenarios of contemporary railway engineering, especially in terms of lowering down the admissible high-speed train velocities, as for structural requirement or preventing potential passenger discomfort.

Vibrations of Circular Plates Resting on Elastic Foundation with Elastically Restrained Edge Against Translation

The present paper deals with exact solutions for the free vibration characteristics of thin circular plates resting on Winkler-type elastic foundation based on the classical plate theory elastically restrained against translation. Parametric investigations are carried out for estimating the influence of edge restraint against translation and stiffness of the elastic foundation on the natural frequencies of circular plates. The elastic edge restraint against translation and the presence of elastic foundation has been found to have a profound influence on vibration characteristics of the circular plate undergoing free transverse vibrations. Computations are carried out for natural frequencies of vibrations for varying values of translational stiffness ratio and stiffness parameter of Winkler-type foundation. Results are presented for twelve modes of vibration both in tabular and graphical form for use in the design. Extensive data is tabulated so that pertinent conclusions can be arrived at on the influence of translational edge restraint and the foundation stiffness ratio of the Winkler foundation on the natural frequencies of uniform isotropic circular plates.

Stability and optimization of a clamped beam elastically restrained against translation on one end resting on Winkler foundation

In this study, stability and bimodal optimization of clamped beam elastically restrained against translation on one end subjected to a constant axially load are analyzed. The beam is positioned on elastic Winkler type foundation. The Euler method of adjacent equilibrium configuration is used in deriving the nonlinear governing equations. The critical load parameters, axial force and stiffness of foundation, are obtained for beam with the unit cross-sectional area.The shape of the beam stable against buckling that has minimal volume is determined by using Pontryagin’s maximum principle. The optimality conditions for the case of bimodal optimization are derived. The cross-sectional area for optimally designed beam is found from the solution of a nonlinear boundary value problem. New numerical results are obtained. A first integral (Hamiltonian) is used to monitor accuracy of integration. It is shown that there is the saving in material for the same buckling force.

Vibrations of Circular Plates with Elastically Restrained Edge against Translation and Resting on Elastic Foundation

The present paper deals with exact solutions for the free vibration characteristics of thin circular plates elastically restrained against translation and resting on Winkler-type elastic foundation based on the classical plate theory. Parametric investigations are carried out for estimating the influence of edge restraint against translation and stiffness of the elastic foundation on the natural frequencies of circular plates. The elastic edge restraint against translation and the presence of elastic foundation has been found to have a profound influence on vibration characteristics of the circular plate undergoing free transverse vibrations. Computations are carried out for natural frequencies of vibrations for varying values of translational stiffness ratio and stiffness parameter of Winkler-type foundation. Results are presented for twelve modes of vibration both in tabular and graphical form for use in design. Extensive data is tabulated so that pertinent conclusions can be arrived at on the influence of translational edge restraint and the foundation stiffness ratio of the Winkler foundation on the natural frequencies of uniform isotropic circular plates.

Parametric stability of symmetrically laminated composite super-elliptical plates

Static and parametric stability of thin symmetrically laminated composite super-elliptical plates resting on Winkler-type foundation and subjected to uniform in-plane harmonic loads, under clamped, simply supported and free boundary conditions, are investigated based on the classical laminated plate theory. The governing equations are obtained from a variational approach and then the classical Ritz method is used to reduce the problem into a set of coupled Mathieu–Hill equations. Hsu’s technique is utilized to determine the dynamic instability regions of principal and combination resonance frequencies. Extensive numerical data are provided to examine the effects of plate aspect ratio, super-ellipticity power, foundation stiffness parameter, stacking sequence, and fiber orientation on the vibrational, static, and parametric stability characteristics of symmetrically laminated super-elliptical composite plates. Furthermore, three-dimensional buckling mode shapes are illustrated. The accuracy of formulation is checked by performing convergence studies and the validity of results is established by comparison with the existing results in the literature, exact results obtained from the analytical approach, and as well as from FEM results.

Comparative dynamic analysis of axially loaded beams on modified Vlasov foundation

Vibration analysis of the beams on elastic foundation has gained the great interest of many researchers. In the literature, there are many studies that focus on the free vibration analysis of the beams on one or two parameter elastic foundations. On the other hand, there are no sufficient studies especially focus on the comparison of dynamic response including the bending moment and shear force of the beams resting on Winkler and two parameter foundations. In this study, dynamic response of the axially loaded Timoshenko beams resting on modified Vlasov type elastic soil was investigated by using the separation of variables method. Governing equations were obtained by assuming that the material had linear elastic behaviour and mass of the beam was distributed along its length. Numerical analysis were provided and presented in figures to find out the differences between the modified Vlasov model and conventional Winkler type foundation. Furthermore, the effect of shear deformation of elastic soil on the dynamic response of the beam was investigated.

Effect of foundation rigidity on stratified roadway roof stability in underground coal mines

The stability of roadway roof has been a long-standing issue in underground coal mines and thus has received a great deal of attention by both researchers and practitioners over the past few decades. This paper presents a simple analytical model for stratified roadway roof stability analysis. The model treats the immediate roof stratum of the roadway as an elastic beam resting on Winkler-type foundation and subjected to uniform roof loading that is determined from the lithology and depth of overlying strata. The developed model calculates bending stresses and deflections of the roof beam. Two case studies from an underground coal mine are used to demonstrate the model. A model parametric study reveals that the coal foundation elasticity plays a key factor in roof deformation and potential failure. Field monitoring data is used to validate analytical model results, and the comparison shows a good agreement.

Nano- and viscoelastic Beck’s column on elastic foundation

Beck's type column on Winkler type foundation is the subject of the present analysis. Instead of the Bernoulli-Euler model describing the rod, two generalized models will be adopted: Eringen non-local model corresponding to nano-rods and viscoelastic model of fractional Kelvin-Voigt type. The analysis shows that for nano-rod, the Herrmann-Smith paradox holds while for viscoelastic rod it does not.

Buckling and vibration of laminated composite circular plate on winkler-type foundation

Buckling and vibration characteristics of circular laminated plates under in-plane edge loads and resting on Winkler-type foundation are solved by the Ritz method. Inclusive numerical data are presented for the first three eigen-frequencies as a function of in-plane load for different classical edge conditions. Moreover, the effects of fiber orientation on the natural frequencies and critical buckling loads of laminated angle-ply plates with stacking sequence of ((β / -β / β / -β))s, are studied. Also, selected deformation mode shapes are illustrated. The correctness of results is established using finite element software as well as by comparison with the existing results in the literature.

Instability of heated circular FGM plates on a partial Winkler-type foundation

Thermal buckling analysis of a transversely graded circular plate attached to a centric partial elastic foundation is studied, analytically. Thermomechanical properties of the circular plate are distributed across the thickness based on a power law function. The governing equations of the plate are obtained by means of the classical plate theory. A conventional Winkler-type foundation is assumed to be in contact with the plate which acts in compression as well as in tension. Proper boundary conditions are chosen after pre-buckling analysis of the plate, and stability equations are established via the adjacent equilibrium criterion. To analyze the thermal stability problem, the plate is divided into two sections, a foundation-less domain and an in-contact region. An exact procedure is presented to accurately predict the critical buckling temperature as well as the buckled configuration of the plate. Analysis of various involved parameters including the Winkler parameter, foundation radius, power law index, and loading type is presented. It is concluded that while the loading is symmetric, in many cases, the buckled configuration of the plate is asymmetric.

Sound wave propagation in zigzag double-walled carbon nanotubes embedded in an elastic medium using nonlocal elasticity theory

In the present work, nonlocal Euler–Bernoulli beam theory is used to investigate the wave propagation in zigzag double-walled carbon nanotube (DWCNT) embedded in an elastic medium. Winkler-type foundation model is employed to simulate the interaction of the DWCNT with the surrounding elastic medium. The DWCNTs are considered as two nanotube shells coupled through the van der Waals interaction between them. It is noticed in the presented study that the equivalent Young’s modulus for zigzag DWCNT is derived using an energy-equivalent model. Influences of nonlocal effects, the chirality of zigzag DWCNT, Winkler modulus parameter, and aspect ratio on the frequency of DWCNT are analyzed and discussed. The new features of the vibration behavior of zigzag DWCNTs embedded in an elastic medium and some meaningful results in this paper are helpful for the application and the design of nanostructures in which zigzag DWCNTs act as basic elements.