Analytical solutions for Euler-Bernoulli beam on visco-elastic foundation subjected to moving load

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.

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Investigation on the Analysis of Bending and Buckling for FGM Euler-Bernoulli Beam Resting on Winkler-Pasternak Elastic Foundation

Journal of Physics Conference Series ◽

10.1088/1742-6596/1773/1/012027 ◽

2021 ◽

Vol 1773 (1) ◽

pp. 012027

Author(s):

Ali Taha Mohammed ◽

Maroa Ali Hareb ◽

Asaad Kadhem Eqal

Keyword(s):

Elastic Foundation ◽

Bernoulli Beam ◽

Pasternak Elastic Foundation ◽

Euler Bernoulli Beam

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DYNAMIC STABILITY OF A BEAM ON AN ELASTIC FOUNDATION INCLUDING STRESS WAVE EFFECTS

International Journal of Applied Mechanics ◽

10.1142/s1758825112500172 ◽

2012 ◽

Vol 04 (02) ◽

pp. 1250017 ◽

Cited By ~ 6

Author(s):

YING LIU ◽

G. LU

Keyword(s):

Dynamic Stability ◽

Elastic Foundation ◽

Stress Wave ◽

Axial Load ◽

Elastic Beam ◽

Beam Theory ◽

Stress Wave Propagation ◽

Bernoulli Beam ◽

Wave Effects ◽

Euler Bernoulli Beam

This paper examines the dynamic stability of an elastic beam on the elastic foundation, in which the stress wave effect is taken into account. Based on Euler–Bernoulli beam theory, the dynamic response of the elastic beam on the elastic foundation to a small transverse perturbation is analyzed. By considering the stress wave propagation in the beam and the constraint of the elastic foundation, the critical bifurcation condition of elastic beam is derived, and the critical axial load of the elastic beam is predicted. Furthermore, the effects of the elastic foundation and the beam length on buckling condition are discussed by using numeric examples. Finally, an approximate solution of critical axial load for elastic beam on the elastic foundation is provided, which may be used to investigate elastic beam buckling problem.

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Modal Analysis of Vibration of Euler-Bernoulli Beam Subjected to Concentrated Moving Load

Iraqi Journal of Science ◽

10.24996/ijs.2020.61.9.19 ◽

2020 ◽

pp. 2324-2334

Author(s):

Usman M. A ◽

Makinde T. A. ◽

Daniel D. O.

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Numerical Methods ◽

Modal Analysis ◽

Series Solution ◽

Moving Load ◽

Bernoulli Beam ◽

Partial Differential ◽

Concentrated Load ◽

Euler Bernoulli Beam

This paper investigates the modal analysis of vibration of Euler-Bernoulli beam subjected to concentrated load. The governing partial differential equation was analysed to determine the behaviour of the system under consideration. The series solution and numerical methods were used to solve the governing partial differential equation. The results revealed that the amplitude increases as the length of the beam increases. It was also found that the response amplitude increases as the foundation increases at fixed length of the beam.

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