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.

Mechanical Characters of Six Engineering Timoshenko Beams

2014 ◽  
Vol 668-669 ◽  
pp. 201-204
Author(s):  
Hong Liang Tian
Keyword(s):  
Timoshenko Beam ◽  
Analytic Solutions ◽  
Timoshenko Beams ◽  
Bernoulli Beam ◽  
Simply Supported ◽  
Normality Assumption ◽  
Length Direction ◽  
The Right ◽  
Euler Bernoulli Beam ◽  
Mechanical Characters

Timoshenko beam is an extension of Euler-Bernoulli beam to interpret the transverse shear impact. The more refined Timoshenko beam relaxes the normality assumption of plane section that remains plane and normal to the deformed centerline. The manuscript presents some exact concise analytic solutions on deflection and stress resultants of NET single-span Timoshenko beam with general distributed force and 6 kinds of standard boundary conditions, adopting its counterpart Euler-Bernoulli beam solutions. Engineering example shows that scale impact would not unveil itself for micro structure with micrometer μm-order length, yet will be prominent for nanostructure with nanometer nm-order length. When simply supported CNTs is undergone to a concentrative force at the median and complete bend moment, scale action is observed along the ensemble CNTs, while it unfurls itself the most at the position of the concentrated strength. When a clamped-free CNTs is exposed to a centralized force at the mesial and distributed force, there is no scale impact about the deflection at all positions on the left border of the concentrated strength position, while such operation inspires at once at all positions on the right margin of the concentrated strength position. When a clamped-clamped CNTs is lain under a concentrative strength at the middle, the deflection of NET Euler-Bernoulli CNTs reflects scale effect completely. Notable differences between the deflection of Euler-Bernoulli CNTs and that of Timoshenko CNTs are reflected at large ratio of diameter versus length. The deflection of NET clamped-free and simply supported Timoshenko beam doesn’t introduce surplus scale process in terms of its counterpart, NET Euler-Bernoulli beam. However, the deflection of NET clamped-clamped Timoshenko beam does involve additional scale impact solely including the method when the concentrated strength position is at the midway in the beam-length direction.

scholarly journals Vibrational Energy Flow Analysis of Corrected Flexural Waves in Timoshenko Beam – Part I: Theory of an Energetic Model

2006 ◽  
Vol 13 (3) ◽  
pp. 137-165 ◽  
Author(s):  
Young-Ho Park ◽  
Suk-Yoon Hong
Keyword(s):  
Timoshenko Beam ◽  
Energy Flow ◽  
Flow Model ◽  
Vibrational Energy ◽  
Classical Solutions ◽  
Timoshenko Beams ◽  
Governing Equations ◽  
Bernoulli Beam ◽  
Vibrational Energy Flow ◽  
Euler Bernoulli Beam

In this paper, an energy flow model is developed to analyze transverse vibration including the effects of rotatory inertia as well as shear distortion, which are very important in the Timoshenko beam transversely vibrating in the medium-to-high frequency ranges. The energy governing equations for this energy flow model are newly derived by using classical displacement solutions of the flexural motion for the Timoshenko beam, in detail. The derived energy governing equations are in the general form incorporating not only the Euler-Bernoulli beam theory used for the conventional energy flow model but also the Rayleigh, shear, and Timoshenko beam theories. Finally, to verify the validity and accuracy of the derived model, numerical analyses for simple finite Timoshenko beams were performed. The results obtained by the derived energy flow model for simple finite Timoshenko beams are compared with those of the classical solutions for the Timoshenko beam, the energy flow solution, and the classical solution for the Euler-Bernoulli beam with various excitation frequencies and damping loss factors of the beam. In addition, the vibrational energy flow analyses of coupled Timoshenko beams are described in the other companion paper.

Medium Frequency Vibration Analysis of Beam Structures Modeled by the Timoshenko Beam Theory

2020 ◽  
Author(s):  
Yichi Zhang ◽  
Bingen Yang
Keyword(s):  
Shear Deformation ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Bernoulli Beam ◽  
Beam Structure ◽  
Medium Frequency ◽  
Beam Structures ◽  
Euler Bernoulli Beam

Abstract Vibration analysis of complex structures at medium frequencies plays an important role in automotive engineering. Flexible beam structures modeled by the classical Euler-Bernoulli beam theory have been widely used in many engineering problems. A kinematic hypothesis in the Euler-Bernoulli beam theory is that plane sections of a beam normal to its neutral axis remain normal when the beam experiences bending deformation, which neglects the shear deformation of the beam. However, as observed by researchers, the shear deformation of a beam component becomes noticeable in high-frequency vibrations. In this sense, the Timoshenko beam theory, which describes both bending deformation and shear deformation, may be more suitable for medium-frequency vibration analysis of beam structures. This paper presents an analytical method for medium-frequency vibration analysis of beam structures, with components modeled by the Timoshenko beam theory. The proposed method is developed based on the augmented Distributed Transfer Function Method (DTFM), which has been shown to be useful in various vibration problems. The proposed method models a Timoshenko beam structure by a spatial state-space formulation in the s-domain, without any discretization. With the state-space formulation, the frequency response of a beam structure, in any frequency region (from low to very high frequencies), can be obtained in an exact and analytical form. One advantage of the proposed method is that the local information of a beam structure, such as displacements, bending moment and shear force at any location, can be directly obtained from the space-state formulation, which otherwise would be very difficult with energy-based methods. The medium-frequency analysis by the augmented DTFM is validated with the FEA in numerical examples, where the efficiency and accuracy of the proposed method is present. Also, the effects of shear deformation on the dynamic behaviors of a beam structure at medium frequencies are illustrated through comparison of the Timoshenko beam theory and the Euler-Bernoulli beam theory.

scholarly journals Analysis of static forces generated in-track on a railway sleeper resting on an elastic foundation due to structural imperfections using the PONTOS system

2019 ◽  
Vol 262 ◽  
pp. 11001
Author(s):  
Włodzimierz Andrzej Bednarek
Keyword(s):  
Theoretical Calculations ◽  
Railway Track ◽  
Bernoulli Beam ◽  
Main Concept ◽  
Field Investigations ◽  
Railway Sleeper ◽  
Foundation Parameters ◽  
Euler Bernoulli Beam ◽  
Structural Imperfections ◽  
Two Parameter

In the paper a considered railway sleeper was analysed as an Euler-Bernoulli beam and a Timoshenko beam of finite length resting on a oneand two-parameter foundation. The foundation parameters were determined based on a modified and analogue Vlasov soil model and field investigations. The main concept for the executed investigations was to induce an intentional imperfection in an actual railway track, propose a way of appropriate measurement (e.g. the PONTOS system by GOM mbh), and utilize author’s field investigations results to calibrate necessary parameters for theoretical calculations. An experimental formula describing the value of the force transferred from the rail to the railway sleeper on the grounds of the survey site caused by a locomotive was given. Furthermore, the deflection of the chosen railway sleeper due to the generated imperfection was analysed. Finally the objective of the present analysis was to resolve the calculations into the beam element such that the results can be utilised in computational railway practice. In the presented paper also the computational examples, diagrams and tables reflecting influence of analyzed parameters on obtained a CWR track’s displacements are enclosed.

Influence of Axial Force and Initial Curvature on the Flexural Deflection of Euler-Bernoulli Beam Structures Solved by DQEM Using EDQ

2006 ◽  
Author(s):  
Chang-New Chen
Keyword(s):  
Axial Force ◽  
Offshore Structures ◽  
Differential Quadrature ◽  
Analysis Model ◽  
Bernoulli Beam ◽  
Beam Structures ◽  
Initial Curvature ◽  
Differential Quadrature Element Method ◽  
Governing Differential Equation ◽  
Euler Bernoulli Beam

The influence of axial force and initial curvature on the flexural deflection of Euler-Bernoulli beam structures is analyzed by differential quadrature element method (DQEM) using extended differential quadrature (EDQ). The DQEM uses the differential quadrature to discretize the governing differential equation defined on each element, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions of the beam. Numerical results solved by the developed numerical algorithm are presented. The convergence of the developed DQEM analysis model is efficient. The theory presented in the paper can be used to effectively obtain accurate results in analyzing the offshore structures.

Influence of Axial Force on the Vibration of Euler-Bernoulli Beam Structures Solved by DQEM Using EDQ

2006 ◽  
Author(s):  
Chang-New Chen
Keyword(s):  
Boundary Conditions ◽  
Axial Force ◽  
Differential Quadrature ◽  
Analysis Model ◽  
Bernoulli Beam ◽  
Beam Structures ◽  
Differential Quadrature Element Method ◽  
Quadrature Element Method ◽  
Euler Bernoulli Beam ◽  
Element Boundary

The influence of axial force on the vibration of Euler-Bernoulli beam structures is analyzed by differential quadrature element method (DQEM) using extended differential quadrature (EDQ). The DQEM uses the differential quadrature to discretize the governing differential eigenvalue equation defined on each element, the transition conditions defined on the inter-element boundary of two adjacent elements and the boundary conditions of the beam. Numerical results solved by the developed numerical algorithm are presented. The convergence of the developed DQEM analysis model is efficient.

scholarly journals Macaulay's Method for a Timoshenko Beam

2007 ◽  
Vol 35 (4) ◽  
pp. 285-292 ◽  
Author(s):  
N. G. Stephen
Keyword(s):  
Timoshenko Beam ◽  
Bending Moment ◽  
Shearing Force ◽  
Bernoulli Beam ◽  
Deflection Curve ◽  
Axial Coordinate ◽  
Deflection Analysis ◽  
Euler Bernoulli Beam

The Macaulay bracket notation is familiar to many engineers for the deflection analysis of a Euler–Bernoulli beam subject to multiple or discontinuous loads. An expression for the internal bending moment, and hence curvature, is valid at all locations along the beam, and the deflection curve can be calculated by integrating twice with respect to the axial coordinate. The notation obviates the need for matching of multiple constants of integration for the various sections of the beam. Here, the method is extended to a Timoshenko beam, which includes the additional deflection due to shear. This requires an additional expression for the shearing force, also valid at all locations along the beam.

Effects of Different Models on Natural Frequencies of Short Span Bridges Used in High Speed Rail

2015 ◽  
Author(s):  
Said I. Nour ◽  
Mohsen A. Issa
Keyword(s):  
Timoshenko Beam ◽  
High Speed ◽  
Natural Frequencies ◽  
Bernoulli Beam ◽  
High Speed Rail ◽  
Vertical Stiffness ◽  
Short Span ◽  
Elastically Supported ◽  
Euler Bernoulli Beam ◽  
Track Modulus

The natural frequencies of vibration of short span bridges used in high-speed rail were investigated. Three different models of increasing complexity were evaluated and their effects on the vibration frequency were compared to the first basic model of simply supported Euler-Bernoulli beam. In the second and third cases, the bridge was modeled as an Euler-Bernoulli and Timoshenko beam supported at its two ends by identical spring elements with an equivalent vertical stiffness to simulate elastomeric bearings and soil foundation. The boundary value problem was solved numerically to extract the bridge eigenfrequencies. In the case of Euler-Bernoulli beam, curve fitting techniques were used to deduce accurate simple empirical formulae to calculate the first six natural frequencies of an elastically supported bridge. In the case of a Timoshenko beam, graphical solutions were proposed to compute the fundamental frequency. Results confirmed that the use of Timoshenko beam theory reduces the natural frequency and the consideration of flexible supports further decreases the natural frequency. In the fourth model, the interaction of the track and the bridge was included. The bridge was modeled as an elastically supported beam and the track was modeled as a spring-damper element with an equivalent vertical stiffness resulting from track components like rail pads, cross-ties and ballast. A parametric study was performed to analyze the effects of the track stiffness on the natural frequencies of the bridge. Graphical solutions were presented to quantify the change of the normalized natural frequencies of the system with the increase in the track modulus. Results indicated that the changes in the track modulus have no significant effects in models with rigid supports. A decrease in the fundamental frequency was noticeable with softer track modulus as the support flexibility increased.

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
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