scholarly journals The edge waves on a Kirchhoff plate bilaterally supported by a two-parameter elastic foundation

2015 ◽  
Vol 23 (12) ◽  
pp. 2014-2022 ◽  
Author(s):  
J Kaplunov ◽  
A Nobili
Keyword(s):  
Phase Velocity ◽  
Elastic Foundation ◽  
Moving Load ◽  
Kirchhoff Plate ◽  
Winkler Foundation ◽  
Edge Waves ◽  
Bending Waves ◽  
Pasternak Elastic Foundation ◽  
Two Parameter ◽  
Limiting Value

In this paper, the bending waves propagating along the edge of a semi-infinite Kirchhoff plate resting on a two-parameter Pasternak elastic foundation are studied. Two geometries of the foundation are considered: either it is infinite or it is semi-infinite with the edges of the plate and of the foundation coinciding. Dispersion relations along with phase and group velocity expressions are obtained. It is shown that the semi-infinite foundation setup exhibits a cut-off frequency which is the same as for a Winkler foundation. The phase velocity possesses a minimum which corresponds to the critical velocity of a moving load. The infinite foundation exhibits a cut-off frequency which depends on its relative stiffness and occurs at a nonzero wavenumber, which is in fact hardly observed in elastodynamics. As a result, the associated phase velocity minimum is admissible only up to a limiting value of the stiffness. In the case of a foundation with small stiffness, asymptotic expansions are derived and beam-like one-dimensional equivalent models are deduced accordingly. It is demonstrated that for the infinite foundation the related nonclassical beam-like model comprises a pseudo-differential operator.

scholarly journals Transfer matrix formulation for dynamic response of Timoshenko beams resting on two-parameter elastic foundation subjected to moving load

2021 ◽  
Vol 4 (2) ◽  
pp. 99-110
Author(s):  
Baran Bozyigit
Keyword(s):  
Dynamic Response ◽  
Shear Layer ◽  
Elastic Foundation ◽  
Natural Frequencies ◽  
Moving Load ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Winkler Foundation ◽  
Foundation Model ◽  
Two Parameter

In this study, the dynamic response of beams resting on two-parameter elastic foundation subjected to moving load is investigated by using the transfer matrix method (TMM). The Timoshenko beam theory (TBT) which considers shear deformation and rotational inertia is used to model the beam. The two-parameter elastic foundation model is selected as Pasternak foundation that takes into account a shear layer at the end of linear springs of Winkler foundation. The TMM which uses the relation between analytically obtained state vectors of each end of the beam is applied to solve the free vibration problem. After performing the free vibration analysis, the mathematical model is simplified into an equivalent single degree of freedom (SDOF) system by using the exact mode shapes to obtain dynamic responses. The generalized displacement is calculated for each mode by using the Runge-Kutta algorithm. A numerical case study is presented for a simply-supported Timoshenko beam on the Pasternak foundation subjected to a concentrated load. The natural frequencies obtained from finite element method (FEM) results of SAP2000 are presented with the results of TMM for comparison purposes using the Winkler foundation. The effects of shear layer on the natural frequencies of the model are revealed. The mode shapes are plotted. The proposed approach for calculating dynamic responses is validated by using the results of FEM for Winkler foundation model. Then, the effects of Winkler springs and shear layer of the foundation model on the dynamic responses are presented in figures. The effects of modal damping are discussed. Finally, the critical velocities for the model are calculated for various elastic foundation scenarios and the effects of elastic foundation parameters on the dynamic response of beam model subjected to moving load with high velocity are observed.

Accurate solution for functionally graded beams with arbitrarily varying thicknesses resting on a two-parameter elastic foundation

2020 ◽  
Vol 55 (7-8) ◽  
pp. 222-236 ◽  
Author(s):  
Zhiyuan Li ◽  
Yepeng Xu ◽  
Dan Huang
Keyword(s):  
Elastic Foundation ◽  
Accurate Solution ◽  
Functionally Graded ◽  
Fourier Series Expansion ◽  
Two Dimensional ◽  
Stress Distributions ◽  
Varying Thickness ◽  
Pasternak Elastic Foundation ◽  
First Time ◽  
Two Parameter

This work presents analytical solutions for bending deformation and stress distributions in functionally graded beams with arbitrarily and continuously variable thicknesses and resting on a two-parameter Pasternak elastic foundation. Based on two-dimensional elasticity theory directly, the general solutions of displacements and stresses which completely satisfy the differential equations governing the equilibrium for arbitrarily varying thickness functionally graded beams are derived for the first time. The undetermined coefficients in the general solution are obtained using Fourier series expansion along the upper and lower surfaces. The accuracy and efficiency of the proposed method are verified through several typical examples. The effects of mechanical and geometry parameters on the stress and displacement distributions of varying thickness functionally graded beams resting on a two-parameter Pasternak elastic foundation are discussed further.

Steady-State Vibrations of Beam on Elastic Foundation for Moving Load

1954 ◽  
Vol 21 (4) ◽  
pp. 359-364
Author(s):  
J. T. Kenney
Keyword(s):  
Steady State ◽  
Elastic Foundation ◽  
Critical Velocity ◽  
Constant Velocity ◽  
Analytic Solution ◽  
Moving Load ◽  
Viscous Damping ◽  
Beam On Elastic Foundation ◽  
Bending Waves ◽  
Free Bending

Abstract This paper presents an analytic solution and resonance diagrams for a constant-velocity moving load on a beam on an elastic foundation including the effect of viscous damping. The limiting cases of no damping and critical damping are investigated. The possible velocities for the propagation of free bending waves are found and their relation to the critical velocity of the beam is studied.

Asymptotic derivation of a refined equation for an elastic beam resting on a Winkler foundation

2021 ◽  
pp. 108128652110238
Author(s):  
Barış Erbaş ◽  
Julius Kaplunov ◽  
Isaac Elishakoff
Keyword(s):  
Ad Hoc ◽  
Moving Load ◽  
Low Frequency ◽  
Elastic Beam ◽  
Winkler Foundation ◽  
One Dimensional ◽  
Beam Equations ◽  
Dimensional Equation ◽  
Phase And Group Velocities ◽  
Group Velocities

A two-dimensional mixed problem for a thin elastic strip resting on a Winkler foundation is considered within the framework of plane stress setup. The relative stiffness of the foundation is supposed to be small to ensure low-frequency vibrations. Asymptotic analysis at a higher order results in a one-dimensional equation of bending motion refining numerous ad hoc developments starting from Timoshenko-type beam equations. Two-term expansions through the foundation stiffness are presented for phase and group velocities, as well as for the critical velocity of a moving load. In addition, the formula for the longitudinal displacements of the beam due to its transverse compression is derived.

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