Finite element formulation and analysis of a functionally graded Timoshenko beam subjected to an accelerating mass including inertial effects of the mass

İsmail Esen,Mehmet Akif Koç,Yusuf Çay

doi:10.1590/1679-78255102

Abstract

Abstract This study describes a new finite element method that can be used to analyse transverse and axial vibrations of a Functionally Graded Material (FGM) beam under an accelerating / decelerating mass. The differential equations of the FGM beam are obtained using First-order Shear Deformation Theory (FSDT). In these equations, the interaction terms of mass inertia are derived from the second-order exact differentiation of displacement functions with respect to mass contact point. The FGM beam is made of two different materials (Steel and Alumina Al2O3), which vary in thickness with a power law. Including the effects of neutral axis shift and mass inertia, the proposed method can be used when the dynamic behaviour of the FGM Timoshenko beams is to be analysed in transverse and axial directions, depending on the interaction with the acceleration of the moving loads. After validating this work with literature studies, new investigations and findings are presented for both moving load and mass assumptions. In addition, the obtained results of Timoshenko Beam (TBT) and Euler Bernoulli beam theory (EBT) are compared for FGM beams with various speeds and accelerations of moving mass.

Highlights

In recent years, there has been a new class of composite materials known as Functionally Graded Materials (FGMs), where different components of materials are graded continuously under a force law
Analysis of systems subjected to moving loads, an ongoing long-standing problem, and some studies have investigated the transverse responses of the limited number of FGM beams to moving forces, in the transverse and axial directions, neglecting the effects of inertia (Kadivar and Mohebpour, 1998; Nguyen et al, 2013; Rajabi et al, 2013; Şimşek and Al-shujairi, 2017; Şimşek and Kocatürk, 2009)
A more realistic modelling and analysis method of such systems is provided without neglecting the Coriolis, inertia and centrifugal effects of the moving load accelerating along the deflected shape of the FGM beams

Summary

INTRODUCTION

There has been a new class of composite materials known as Functionally Graded Materials (FGMs), where different components of materials are graded continuously under a force law. Analysis of systems subjected to moving loads, an ongoing long-standing problem, and some studies have investigated the transverse responses of the limited number of FGM beams to moving forces, in the transverse and axial directions, neglecting the effects of inertia (Kadivar and Mohebpour, 1998; Nguyen et al, 2013; Rajabi et al, 2013; Şimşek and Al-shujairi, 2017; Şimşek and Kocatürk, 2009). A more realistic modelling and analysis method of such systems is provided without neglecting the Coriolis, inertia and centrifugal effects of the moving load accelerating along the deflected shape of the FGM beams. This work presents a detailed comparison of moving load and moving mass assumptions for FGM beams, including differences between Timoshenko (TBT) and Euler-Bernoulli beam theories (EBT)

MATHEMATICAL MODELLING

Finite element formulation

RESULTS AND DISCUSSION

Effect of the Sudden Acceleration and Deceleration

CONCLUSIONS