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Nonlinear dynamic analysis of beams with layered cross sections under moving masses

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Abstract

This paper considers the nonlinear dynamic analysis of moving masses applied on beams with layered cross sections. This mechanical system is often used in passenger transportation systems, such as high-velocity trains and magnetic levitation transport vehicles. The structural systems are modeled numerically with the aid of a formulation based on the Finite Elements Method. The main objective of this paper was to present an original numerical approach for a mass-beam system using the concept of positional geometric nonlinearity, which uses nodal positions rather than nodal displacements to describe the finite elements kinematics. By using this methodology, analyses can be performed for moving masses traveling with constant or variable velocity in dimensionless or physical problems. For the analyzed problems, sets of results are presented for the mechanical behavior analyses, such as the moving mass trajectories along the mass-beam coupled system. Moreover, a comparative study is performed on beams with rectangular and rail cross sections. After the passage of the moving masses, the elastoplastic rectangular cross-sectional beam presents a higher stiffness than the elastoplastic rail cross-sectional beam, as a result of the double plasticity that occurs at the extreme surfaces. Another original numerical example presents a beam made of two different materials under the action of a moving mass.

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Abbreviations

A :

Cross-sectional area

ds :

Longitudinal fiber length related to height positions

E :

Young’s modulus

E E :

External energy of the moving mass

\(\varvec{g}\left( {X_{0} } \right)\) :

Vector of residues

K :

Kinetic energy contribution of the mass-beam in the energy functional

m :

Moving mass value

M :

Mass matrix

P :

Prescribed force due to the mass

\(\varvec{Q}_{t}\) :

Vector of the variables from the previous time step

\(\varvec{R}_{t}\) :

Vector of the variables from the previous time step

t :

Previous time step

t + ∆t :

Current time step

\(U_{\text{T}}\) :

Total strain energy

X :

Vector of nodal positions

V :

Volume

z :

Height position

γ, β :

Newmark time integration parameters

\(\eta , \xi\) :

Non-dimensional parameters used to integrate finite elements

\(\rho\) :

Mass density

\(\delta \left( {\varvec{X} - \varvec{X}_{m} } \right)\) :

Dirac delta operator for moving mass

∆t :

Time integration interval

\(\nabla \varvec{g}\left( {X_{0} } \right)\) :

Hessian matrix

ε :

Engineering strain measure

\(\varepsilon_{\text{P}}\) :

Plastic strain

0:

Initial or estimated time step

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Acknowledgments

The authors would like to acknowledge CNPq (National Council of Scientific and Technological Development) and FAPEMIG (Minas Gerais State Research Foundation) for their financial support, under Grant Numbers 301487/2010-3, 304275/2013-1 and TEC-PPM-00026-13. The authors would also like to acknowledge the grammar revision service provided by the Pró-Reitoria de Pesquisa da Universidade Federal de Minas Gerais (UFMG).

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Authors and Affiliations

  1. Graduate Program in Structural Engineering, Department of Structural Engineering, School of Engineering, Universidade Federal de Minas Gerais, Av. Presidente Antônio Carlos, 6627/Escola de Engenharia/Bloco 1/Sala 4127, Belo Horizonte, MG, 31270-901, Brazil

    Flávio Marcílio de Oliveira & Marcelo Greco

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  1. Flávio Marcílio de Oliveira
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  2. Marcelo Greco
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Correspondence to Marcelo Greco.

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Technical Editor: Marcelo A. Savi.

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de Oliveira, F.M., Greco, M. Nonlinear dynamic analysis of beams with layered cross sections under moving masses. J Braz. Soc. Mech. Sci. Eng. 37, 451–462 (2015). https://doi.org/10.1007/s40430-014-0184-5

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  • DOI: https://doi.org/10.1007/s40430-014-0184-5

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