Abstract
This paper presents a discrete physical model to approach the problem of nonlinear vibrations of beams resting on elastic foundations. The model consists of a beam made of several small bars, evenly spaced. The bending stiffness is modeled by spiral springs, and the Winkler soil stiffness is modeled using linear vertical springs. Concentrated masses, presenting the inertia of the beam, are located at the bar ends. Finally, the nonlinear effect is presented by the axial forces in the bars, assumed to behave as longitudinal springs, due to the change in their length induced by the Pythagorean Theorem. This model has the advantage of simplifying parametric studies, because of its discrete nature, allowing any modification in the mass matrix, the stiffness matrix, and the nonlinearity tensor to be made separately. Therefore, once the model is established, various practical applications may be performed without the need of going through all the formulation again. The study of the nonlinear behavior makes the solution of the movement equation rise in complexity. By considering this discrete model and using the linearization method, one can achieve an idealized approach to this nonlinear problem and obtain quite easily approximate solutions.
Highlights
The analysis of the vibration of beams resting on elastic foundations wears a practical and theoretical interest in many fields such as civil, mechanical, and transportation engineering
Malekzadeh and Karami [7] gather the advantages of both previous methods (DQM and Finite Element Method (FEM)) to perform the free vibration and buckling analysis of thick beams on twoparameter elastic foundations
The aim of this work is the development of a flexible general discrete model for linear and nonlinear vibrations of beams resting on elastic foundations, via the adaptation and the extension of the approach presented in [9]
Summary
The analysis of the vibration of beams resting on elastic foundations wears a practical and theoretical interest in many fields such as civil, mechanical, and transportation engineering. Analytical and numerical methods applied to the modeling of such a problem are extensively addressed in the literature. The combination between the emergence of high-performance computers and problems complexity made the discrete methods more appealing. A discrete method such as the Finite Element Method (FEM) is the first to address the problem numerically [1,2,3]. The Differential Quadrature Method (DQM) is employed for the solution to similar engineering problems involving beam vibrations and foundations [4,5,6]. Malekzadeh and Karami [7] gather the advantages of both previous methods (DQM and FEM) to perform the free vibration and buckling analysis of thick beams on twoparameter elastic foundations
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