Z. Celep

Spurious reflection of elastic waves in nonuniform meshes of constant and linear strain unite elements

Abstract A change in finite element size causes spurious reflection of elastic waves passing through a finite element grid when the wavelength is less than about 10-times the largest element in the grid. Extending the previously published study in which the phenomenon was analyzed for the case of constant strain finite elements, the higher-order elements with linear strain distribution are studied herein. Similarly to the previous study, it is found that the consistent mass matrix gives less spurious wave reflection than the lumped mass matrix; however the advantage is smaller for the higher-order elements. For the lumped mass matrix, there is little difference in spurious wave reflection between the constant strain and linear strain elements. The phenomenon of spurious wave reflection is less pronounced when the higher-order elements are used in conjunction with the consistent mass matrix. These results are obtained from exact analytical solutions in complex variables for a planar wave with a planar wave front propagating along grid lines through an infinite grid which is uniform in each half plane.

Forced vibrations of a beam on a tensionless foundation

Abstract This paper is concerned with the response of an elastic beam resting on a tensionless Winkler foundation, subjected to a concentrated load, a uniformly distributed load and a concentrated load. The external load are assumed to be time dependent, and the non-linear governing differential equation of forced vibrations is derived by employing the Galerkin method. The free vibration mode functions of the completely free beam are adopted as the co-ordinate functions of the displacement function of the beam. The results, which represent the static and dynamic responses of the beam, are presented in the figures. It is concluded that the presence of the lift-off affects the response of the beam markedly by changing the shape of the displacement curve as well as the foundation reactions.

Axisymmetric Vibrations of Circular Plates on Tensionless Elastic Foundations

At first the static solution is obtained and contact radius is determined. Later, this is used as an initial configuration of the forced vibrations. The forced vibrations are assumed to be due to the time dependency of the load. The solution is carried out by Galerkins method and by using modal functions of the completely free plate. Numerical results are illustrated in figures for stepwise change of the loading

Free vibration of some circular plates of arbitrary thickness

Abstract In this paper, in order to be self-contained, a brief account of the construction technique of the method of initial functions which has been developed for circular plates by the author is given. Then the new method is applied to investigate the free vibration of two circular plates: i.e., simply supported and completely free plates. Numerical results are obtained and compared with those of the classical, Reissner and Mindlin theories.

On the axially symmetric vibration of thick circular plates

SummaryThe method of initial functions is developed in cylindrical coordinates for axially symmetric elastodynamic deformation of the circular plate. The governing equations are derived from the three dimensional elastodynamic equations. Using a suitable transformation the difficulty in the application of the method of initial functions is removed. The method is used to obtain the free vibration frequencies of thick circular plates. Numerical results are given for the circular plate with clamped edges and compared with those of the known solutions.ÜbersichtDie Methode der Anfangsfunktionen wird in Zylinderkoordinaten für axial symmetrische und elastodynamische Deformationen der Kreisplatte entwickelt. Die Gleichungen der Aufgabe werden aus den Gleichungen der dreidimensionalen dynamischen Elastizitätstheorie hergeleitet. Die Anwendung der Methode der Anfangsfunktionen folgt durch Gebrauch einer geeigneten Transformation. Die Methode wird zur Ermittlung der Eigenfrequenzen der dicken Kreisplatte angewendet. Die numerischen Ergebnisse für die eingespannte Kreisplatte werden mit den bekannten Näherungslösungen verglichen.

In-plane vibrations of circular rings on a tensionless foundation

Abstract The problem of a thin elastic circular ring on a tensionless Winkler foundation is considered. It is assumed that the ring is subjected to time dependent in-plane loads. The analysis is based on the harmonic approximation for the deflection function of the ring. The governing equations of the problem are derived by employing Lagranges equations. The static and dynamic responses of the ring are obtained numerically and presented in figures comparatively for conventional and for tensionless Winkler foundations.

Contact between a circular plate and a tensionless edge support

Abstract In this paper the extent of contact between a circular plate and its tensionless edge support is studied. The solution is obtained by employing the Ritz method. The displacement function is approximated in terms of the functions which satisfy exactly one of the boundary conditions. The other boundary condition is satisfied approximately by taking the harmonic average. The numerical results showing the effects of the configuration of loading and the stiffness of the support on the extent of the contact and on the edge displacement of the plate are presented in various diagrams. The results which correspond to a very low support stiffness are compared with those of a rigid plate, and very good agreement is found.

Transient wave propagation in layered media conducting heat

Abstract Transient wave propagation in thermoelastic layered composites consisting of alternating isotropic, homogeneous and linearly elastic high-strength reinforcing and low-strength matrix layers is investigated. The layers of the composite medium can be plane, cylindrical or spherical. The inner surfaces of the composite bodies are subjected to uniform time dependent dynamic inputs. A common formulation is employed for the three types of layered media. The generalized theory of thermoelasticity is used, with thermal relaxation predicting finite wave speeds for thermal disturbances. The method of characteristics is employed to obtain the solutions. Curves are plotted denoting the variations of normal stresses with time at different locations and variations of stresses along the thicknesses of the bodies at different times. The curves reveal the thermal and geometric dispersions in the wave profiles and the effects of reflections and refractions at the boundaries and the interfaces of the layers.

Transient wave propagation in constant and linear strain finite elements

This paper deals with transient wave propagation in a semi-infinite rod which is represented by constant or linear strain finite elements having two different element sizes. After assembling the system matrices, a matrix differential equation is obtained and integrated numerically. Two different types of end condition, prescribed axial force and displacement, are considered; the corresponding results are displayed in figures and the oscillations in the numerical solution are discussed. The results reveal spurious reflections due to the change in the element size.

Dynamic response of a column with foundation uplift

Abstract The effects of transient foundation uplift on the dynamic response of a column with attached masses supported on a Winkler foundation are investigated. The analysis developed here is based on the assumption that the column, having continuously distributed mass and rigidity, is supported on a tensionless elastic foundation through a rigid circular base plate. The method employed includes use of series expansion of the lateral displacement of the column, application of the Galerkin technique and step-wise solution of the governing equations of the problem. Besides the free vibrations of the column, the forced vibrations of the system are investigated for an assumed harmonic ground motion excitation. Additionally, the dynamic response of the system to the El Centro 1940 ground motion is presented.