Winkler-type Foundation Research Articles

Effect of an elastic foundation on axisymmetric vibrations of polar orthotropic annular plates of variable thickness

Free axisymmetric vibrations of a polar orthotropic annular plate of linearly varying thickness resting on an elastic foundation of Winkler type are studied on the basis of classical theory of plates. The fourth order linear differential equation with variable coefficients governing the motion is solved by using the quintic spline interpolation technique for three different combinations of boundary conditions. The effect of the elastic foundation together with the orthotropy on the natural frequencies of vibration is illustrated for different values of the radii ratio and the thickness variation parameter for the first three modes of vibration. Transverse displacements and moments are presented for a specified plate. The validity of the spline technique is demonstrated by presenting a comparison of present results with those available in the literature.

Response spectra for torsion, rocking and rigid foundations

AbstractSimple procedures are proposed for computing response spectra for torsional and rocking input ground motions assuming horizontally travelling waves of constant shape. It is shown that harmonic relationships exist between the rotational spectra and the corresponding translational spectra, and that SV rather than PSV is the correct basis for deriving the angular displacement, velocity and acceleration response spectra. An approximation enabling the use of the standard tripartite logarithmic response spectra is discussed. Simple expressions for ‘accidental’ eccentricity and rocking input effects are presented. Also proposed are multipliers to spectral ordinates to account for the filtering effects of rigid base mats resting on Winkler type foundations. For wave transit times shorter than half the natural period of the structure, these multipliers can be approximated by the frequency dependent averaging coefficients given in the literature, which are dependent, however, on the response, rather than the input, frequency.

Flexure of beams resting on hyperbolic elastic foundations

This article examines the static problem of the flexure of a Bernoulli-Euler beam resting on a nonlinear Winkler-type foundation. A perturbation technique is used to solve the nonlinear differential equation associated with the problem. Using this technique, the initially nonlinear problem is reduced to the solution of a set of linearised equations. For the successive solution of these equations, some analytical methods are outlined. These methods are applicable to either finite or infinite beams. As an example of the applications of the proposed analysis, the problem of the flexure of a finite beam subjected to a concentrated line load, applied at an arbitrary point of the beam, is solved. In a second example, the solution of the problem of the flexure of a finite beam having free edges and subjected to an initial displacement at its middle point is presented.

Buckling and vibrations of circular plates of variable thickness

Axisymmetric vibrations of a circular plate of linearly varying thickness under the action of an hydrostatic in-plane force and resting on elastic foundation of Winkler type are discussed on the basis of the classical theory of plates. The fourth order linear differential equation governing the transverse motion is solved by Frobenius' method. Frequency parameters of clamped as well as simply supported plates for the first three modes of vibration are computed for various values of the taper parameter, the foundation modulus and the in-plane force. Transverse displacements and moments have also been computed.

Elastic-Plastic Slabs on Elastic Foundation

A method of analysis and design of rectangular and circular elastoplastic, reinforced concrete, and slabs resting on a Winkler-type foundation is proposed in this paper. The method is an application of the yield-line theory of structures and gives upper bounds of the true limit loads. It is very simple and flexible so that realistic slab and loading configurations can be easily accounted for. The results of the method for a circular, centrally loaded slab agree with the ones reported by other researchers for a Winkler foundation, while of much interest is the comparison with the results for an elastic half space.

On the effect of different edge flexibility coefficients on transverse vibrations of thin, rectangular plates

To solve problems of transverse vibration of thin, rectangular plates with different edge flexibility coefficients polynomial co-ordinate functions are used which identically satisfy the boundary conditions. It is shown that by a proper combination of the polynomials several modes of vibrations can be analyzed with a minimum amount of labour. A variational formulation is used to generate the frequency equation. Eigenvalues calculated by using a two-term approximation seem to possess extremely good accuracy, at least from an engineering viewpoint. It is also shown that the effect of in-plane forces or the case where the plate is supported on a Winkler-type foundation can be studied without any formal difficulties.

Boundary Value Problems in the Improved Theory of Elastic Plates. I: Existence of Eigenvibrations for Plates of Arbitrary Shape

Boundary value problems governing the vibrations of elastic plates are formulated within the improved (Timoshenko) plate theory. The free vibrations of finite plates of arbitrary shape are considered, and the existence of an infinite, discrete spectrum of eigenfrequencies and of a set of corresponding eigenmodes of vibrations is established for plates with clamped, simply-supported, and free edges. The investigation is done through the formulation of a variational problem and the use of functional-analytic techniques. The existence of eigenfrequencies and eigenmodes is first established for clamped plates supported on Winkler-type foundations. The difficulty in extending this result to plates with simply-supported or free edges is then studied and finally resolved by attaching torsional springs to a surface of the plate.

The buckling of transversely isotropic beams on elastic foundations

An analysis of buckling for beams made of transversely isotropic material and resting on Winkler-type foundation is presented. The analysis, which includes the effect of shear deformation, shows that for a high E G and a low slenderness ratio, the effect of shear can be significant for beams with ends simply supported or fixed. In both cases including the shear effect lowers the critical load value.

Finite difference solution for circular plates on elastic foundations

AbstractIn the present work the authors have developed a finite difference method of analysis for any circular plate with any kind of loading on semi‐infinite elastic foundations. No assumption regarding the contact pressure distribution has been made. The equations have been developed in non‐dimensional form and also the results have been obtained in non‐dimensional form. These results have been compared with the available experimental results and the agreement between them is found to be much better than that of the previous works. The same method with slight modification can be applied for Winkler type foundations and problems of circular plates with varying thickness.

A triangular plate bending element for contact problems

A refined triangular plate bending element is developed for the problem of a thin elastic plate resting on an elastic half-space. The behaviour of the elastic half-space is described in terms of displacements at ten points in each triangular region into which the surface in contact with the plate is divided. The plate finite element similarly has ten nodes. The requirements of continuity of displacement and slope between adjacent plate elements are met by using a seventh order polynomial with 33 degrees of freedom. The behaviour of this element is tested on various plate bending problems and found to be satisfactory. The element is then applied to the problems of square, triangular and circular plates resting on an elastic half-space and on a foundation of Winkler type, and some comments are made on the problem of finding an equivalent Winkler foundation for a given elastic half-space.

Beam Columns on Elastic Foundations

A buckling analysis, bearing out the prominent differences in buckling loads of a pin-ended rod embedded in a Winkler and Wieghardt-type foundation, is given by T. E. Smith. Herein it is shown that existing results in the case of beams resting on a Winkler-type foundation and subjected to simultaneous action of axial loads, compressive or tensile and transverse loading, can be utilized after suitable modification to represent the same structure resting on a Wieghardt-type foundation. Modified results in one case are shown.

Steady-state response of a finite beam on a Pasternak-type foundation

A uniform Bernoulli-Euler beam of finite length is supported by a Pasternak-type foundation and subjected to a harmonic force F = F0ejwt, concentrated at the midpoint. The influence of the “shear layer” is exhibited by comparison with the response of a beam supported by a Winkler-type foundation. At the first resonant frequency, unbounded values occur for the bending moment, which are not expected according to Winkler's hypothesis. Observations are made on the influence of damping and inertia of the foundation. Analytical expressions as well as frequency-response curves are presented for the beam deflections and the bending moments.