A Steadily Moving Load on an Elastic Beam Resting on a Tensionless Winkler Foundation

1979 ◽  
Vol 46 (1) ◽  
pp. 175-180 ◽  
Author(s):  
J. Choros ◽  
G. G. Adams
Keyword(s):  
Moving Load ◽  
Elastic Beam ◽  
Energy Criterion ◽  
Winkler Foundation ◽  
Coordinate Systems ◽  
Bernoulli Beam ◽  
Concentrated Force ◽  
Closed Form Solutions ◽  
Euler Bernoulli Beam ◽  
Steady State Solutions

An infinitely long Euler-Bernoulli beam resting on a tensionless Winkler foundation is considered. Steady-state solutions are obtained for a downward directed concentrated force moving with constant speed. First, the critical load necessary to initiate separation of the beam from the foundation is determined for a range of speed. For loads greater than critical, one or more regions of noncontact can be expected to occur. Closed-form solutions of the differential equations are obtained in terms of local coordinate systems which significantly reduces the coupling among the various regions. The extent and location of the noncontact regions, as well as the corresponding beam deflections, are then determined for a range of force and speed. The results show that many solutions are possible and the final determination is based on an energy criterion.

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.

Spectral finite element for vibration analysis of cracked viscoelastic Euler–Bernoulli beam subjected to moving load

2015 ◽  
Vol 226 (12) ◽  
pp. 4259-4280 ◽  
Author(s):  
Vahid Sarvestan ◽  
Hamid Reza Mirdamadi ◽  
Mostafa Ghayour ◽  
Ali Mokhtari
Keyword(s):  
Finite Element ◽  
Vibration Analysis ◽  
Moving Load ◽  
Bernoulli Beam ◽  
Spectral Finite Element ◽  
Euler Bernoulli Beam

The elastochrone: the descent time of a sphere on a flexible beam

2009 ◽  
Vol 465 (2107) ◽  
pp. 2293-2311 ◽  
Author(s):  
Jeffrey M. Aristoff ◽  
Christophe Clanet ◽  
John W.M. Bush
Keyword(s):  
Gravitational Field ◽  
Theoretical Model ◽  
Elastic Beam ◽  
Beam Theory ◽  
Flexible Beam ◽  
Bernoulli Beam ◽  
Horizontal Beam ◽  
Theoretical Predictions ◽  
Static Regime ◽  
Euler Bernoulli Beam

We present the results of a combined experimental and theoretical investigation of the motion of a sphere on an inclined flexible beam. A theoretical model based on Euler–Bernoulli beam theory is developed to describe the dynamics, and in the limit where the beam reacts instantaneously to the loading, we obtain exact solutions for the load trajectory and descent time. For the case of an initially horizontal beam, we calculate the period of the resulting oscillations. Theoretical predictions compare favourably with our experimental observations in this quasi-static regime. The time taken for descent along an elastic beam, the elastochrone, is shown to exceed the classical brachistochrone, the shortest time between two points in a gravitational field.

scholarly journals A system of two infinite beams separated by an elastic core under moving load

2018 ◽  
Vol 19 (12) ◽  
pp. 285-288
Author(s):  
Magdalena Ataman ◽  
Wacław Szcześniak
Keyword(s):  
Differential Equations ◽  
Moving Load ◽  
Fourier Integral ◽  
Winkler Foundation ◽  
Eighth Order ◽  
Concentrated Force ◽  
Moving Force ◽  
Elastic Core ◽  
Lower Beam ◽  
Extensive List

The paper discussed the analytical solution of a dynamic problem of a system of two infinite beams separated by an elastic core. The beams’ system rests on the Winkler foundation and is loaded with a moving concentrated force. Because the problem is stationary for an observer moving with the load, partial differential equations, describing the vibrations of the system, were transformed into ordinary differential equations in the coordinate system related to the moving force. The system of equations was transformed to one differential equation of an eighth order. The equation defines deflection of the lower beam. The solution of the problem was resulted to the simple infinite Fourier integral. An extensive list of publications on the related literature is presented in the paper [1-45].

Case Study of the Effect of Combined Axial and Lateral Loadings on the Critical Buckling Capacity of Piping Supports

2020 ◽  
Author(s):  
Phillip Wiseman ◽  
Alex Mayes ◽  
Shreeya Karnik
Keyword(s):  
Large Deformation ◽  
Axial Loading ◽  
Beam Theory ◽  
Second Order ◽  
Lateral Acceleration ◽  
Bernoulli Beam ◽  
Deformation Method ◽  
Closed Form Solutions ◽  
Bernoulli Beam Theory ◽  
Euler Bernoulli Beam

Abstract Snubbers are used in industry to restrain piping in dynamic events which can see significant axial loading as well as lateral acceleration. Snubbers are often employed with an extension when required to bridge gaps between the piping and building structure. As a result, they are susceptible to buckling instability issues. The pipe support and restraint design by analysis buckling criteria for supports given within the American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Code, Section III, Division 1, Subsection NF is investigated to determine the behavior of snubber assemblies under combined axial and lateral loadings. Four types of analyses are performed on the assemblies under the action of axial loading to demonstrate finite element and closed form solutions. These include the following: linear Eigen buckling, nonlinear second order large deformation method, energy method and Euler Bernoulli beam theory. In addition, a variety of snubber assembly sizes are subjected to combined axial and lateral loading in the form of multiple magnitudes of lateral acceleration. The behavior was analyzed by the Euler Bernoulli beam theory and nonlinear second order large deformation method. The techniques of each method are compared providing explanations of the assumptions taken, relevant limitations and recommended applications.

scholarly journals On the Equivalence between Static and Dynamic Railway Track Response and on the Euler-Bernoulli and Timoshenko Beams Analogy

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Wlodzimierz Czyczula ◽  
Piotr Koziol ◽  
Dorota Blaszkiewicz
Keyword(s):  
Axial Force ◽  
Timoshenko Beam ◽  
Linear Models ◽  
Moving Load ◽  
Railway Track ◽  
Variable Load ◽  
Static Case ◽  
Timoshenko Beams ◽  
Bernoulli Beam ◽  
Euler Bernoulli Beam

The paper tries to clarify the problem of solution and interpretation of railway track dynamics equations for linear models. Set of theorems is introduced in the paper describing two types of equivalence: between static and dynamic track response under moving load and between the dynamic response of track described by both the Euler-Bernoulli and Timoshenko beams. The equivalence is clarified in terms of mathematical method of solution. It is shown that inertia element of rail equation for the Euler-Bernoulli beam and constant distributed load can be considered as a substitute axial force multiplied by second derivative of displacement. Damping properties can be treated as additional substitute load in the static case taking into account this substitute axial force. When one considers the Timoshenko beam, the substitute axial force depends additionally on shear properties of rail section, rail bending stiffness, and subgrade stiffness. It is also proved that Timoshenko beam, described by a single equation, from the point of view of solution, is an analogy of the Euler-Bernoulli beam for both constant and variable load. Certain numerical examples are presented and practical interpretation of proved theorems is shown.

On the Dynamics of Fractional Visco-Elastic Beams

2012 ◽  
Author(s):  
Salvatore Di Lorenzo ◽  
Francesco P. Pinnola ◽  
Antonina Pirrotta
Keyword(s):  
Elastic Beam ◽  
Constitutive Law ◽  
Dynamic Loads ◽  
Elastic Behavior ◽  
Advanced Manufacturing ◽  
Suitable Model ◽  
Bernoulli Beam ◽  
Static Behavior ◽  
Creep Tests ◽  
Euler Bernoulli Beam

With increasing advanced manufacturing process, visco-elastic materials are very attractive for mitigation of vibrations, provided that you may have advanced studies for capturing the realistic behavior of such materials. Experimental verification of the visco-elastic behavior is limited to some well-known low order models as the Maxwell or Kelvin models. However, both models are not sufficient to model the visco-elastic behavior of real materials, since only the Maxwell type can capture the relaxation tests and the Kelvin the creep tests, respectively. Very recently, it has been stressed that the most suitable model for capturing the visco-elastic behavior is the spring-pot, characterized by a fractional constitutive law. Based on this assumption, the quasi-static behavior has been investigated very recently, however for noise control there is a need of exploiting the dynamic behavior of such a fractional visco-elastic beam. The present paper introduces the dynamic response of fractional visco-elastic Euler-Bernoulli beam under dynamic loads.

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