123,041 research outputs found
Size effect in the bending of a Timoshenko nanobeam
No full textThe size effect should be considered due to the large ratio of surface area to volume when the characteristic length of a beam lies in the nanoscale. The size effect in the bending of a Timoshenko nanobeam is investigated in this paper based on a recently developed elastic theory for nanomaterials in which only the bulk surface energy density and the surface relaxation parameter are involved as independent parameters to characterize the surface property of nanomaterials. In contrast to the Euler nanobeams and the classical Timoshenko beam not only the size effect but also the shear deformation effect in Timoshenko nanobeams is included. Closed-form solutions of the deflection and the effective elastic modulus for both a fixed-fixed Timoshenko nanobeam and a cantilevered one are achieved. Comparing to the classical solution of Timoshenko beams the size effect is obviously significant in Timoshenko nanobeams. The shear deformation effect in nanobeams cannot be neglected in contrast to the solution of Euler-Bernoulli nanobeams when the aspect ratio of a nanobeam is relatively small. Furthermore the size effect exhibits different influences on the bending behavior of nanobeams with different boundary conditions. A nanobeam with a fixed-fixed boundary would be stiffened while a cantilevered one is softened by the size effect compared to the classical solution. All the findings are consistent with the existing experimental measurement. The results in this paper should be very useful for the precision design of nanobeam-based devices
Timoshenko beams
No full textUnlike the Bernoulli beam formulation, the Timoshenko beam formulation accounts for transverse shear deformation. It is therefore capable of modeling thin or thick beams. In this chapter we perform the analysis of Timoshenko beams in static bending, free vibrations and buckling. We present the basic formulation and show how a MATLAB code can accurately solve this problem
Perturbation theory and the Rayleigh quotient
No full textThe characteristic frequencies ? of the vibrations of an elastic solid subject to boundary conditions of either zero displacement or zero traction are given by the Rayleigh quotient expressed in terms of the corresponding exact eigenfunctions. In problems that can be analytically expanded in a small parameter ?, it is shown that when an approximate eigenfunction is known with an error O(?N), the Rayleigh quotient gives the frequency with an error O(?2N), a gain of N orders. This result generalizes a well-known theorem for N=1. A non-trivial example is presented for N=4, whereby knowledge of the 3rd-order eigenfunction (error being 4th order) gives the eigenvalue with an error that is 8th order; the 6th-order term thus determined provides an unambiguous derivation of the shear coefficient in Timoshenko beam theor
Surface effect on the resonant frequency of Timoshenko nanobeams
No full textThe dynamic behavior of a Timoshenko nanobeam would be significantly different from a macro-one due to the large ratio of surface area to volume of nanomaterials. Furthermore, the shear deformation effect would be obvious for a Timoshenko nanobeam in contrast to an Eulerian one. In this paper, a recently developed elastic theory is adopted in order to predict the resonant frequency of a Timoshenko nanobeam, in which not only the surface effect but also the shear deformation effect and the rotary inertia one are considered. In contrast to the existing surface effect theories, surface effect of nanomaterials is characterized by the surface energy density in the adopted theory. The resonant frequency of both a fixed-fixed nanobeam and a cantilevered one is analyzed. It is found that the dynamic behavior of nanobeams deviates significantly from the one predicted by both the classical Timoshenko beam theory and the Euler-Bernoulli one due to the surface effect. Furthermore, the shear deformation effect and the rotary inertia effect cannot be neglected in nanobeams with a relative small aspect ratio, which cannot be precisely characterized by the Euler-Bernoulli beam theory. In addition, the influencing mechanism of surface effect on the dynamic behavior of nanobeams would depend on the boundary conditions. The resonant frequency of a fixed-fixed Timoshenko nanobeam would be improved, while that of a cantilevered one would be weakened by the surface effect in contrast to the corresponding classical solutions. The results in this paper should be useful for precise design of nano-devices and helpful for reasonable assessment of test results of nano-instruments. (C) 2017 Elsevier Ltd. All rights reserved
A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams
No full textThis paper presents a nonlocal sinusoidal shear deformation beam theory for the bending, buckling, and vibration of nanobeams. The present model is capable of capturing both small scale effect and transverse shear deformation effects of nanobeams, and does not require shear correction factors. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion as well as the boundary conditions of the beam are derived using Hamilton’s principle. Analytical solutions for the deflection, buckling load, and natural frequency are presented for a simply supported beam, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory. The comparison firmly establishes that the present beam theory can accurately predict the bending, buckling, and vibration responses of short nanobeams where the small scale and transverse shear deformation effects are significant
Analysis of a double Timoshenko beam model
No full textIn this work we consider a double beam system modeled in the theory of Timoshenko. An existence and uniqueness result is achieved by using the standard theory of linear semigroup. The exponential stability is also proved. Then, fully discrete approximations are introduced and a prior error estimates are shown. Finally, some numerical simulations are presented. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC license (http://creativecommons .org /licenses /by -nc /4 .0/)
Timoshenko N. Modelling of electric arc furnace off-gas removal system. / N. Timoshenko, A. Semko, S. Timoshenko. Ironmaking and Steelmaking, vol. 41, № 4, 2014. Р. 257-261.
No full textA mathematical model of an electric arc furnace off-gas removal system, aimed at improving the
energy efficiency of the smelting process, was created and verified by methods of physical
modelling and numerical simulations. For an industrial-scale furnace, equipped with annular
distributing off-gas suction system, the possibility to reduce fugitive emissions by 40%, air inflow
by 24%, total emissions by 13% and the melting dust removal by 33% was shown. The obtained
data will allow, according to estimations, specific energy consumption to reduce by at least 16 –
20kW ht -1 and increase the environmental performance of the furnace
Computational model of vibration of Timoshenko beam under axial loads
No full textThe free vibration characteristics of a cantilever tapered Timoshenko beam is analyzed in this study. First, the strain displacement relationship for the Timoshenko beam is formulated and used to derive the kinetic and strain energies in explicit analytical form. Second, Lagrange’s variational principle is used to derive the governing differential equation of motion and the associated boundary conditions. Third, the Rayleigh-Ritz Method (R-R) is applied to the equation of motion and the boundary conditions to form a set of algebraic equations from which the frequency equation is derived. Next, a numerical algorithm implemented in the software package Mathematica is used to compute the natural frequencies. Also, the variation of the natural frequencies of vibration with respect to variations in the taper ratio and also the slenderness ratio is studied. The results obtained from the Timoshenko theory are compared with results obtained in literature to demonstrate the accuracy and relevance of the their application
Analysis of a double Timoshenko beam model
No full textIn this work we consider a double beam system modeled in the theory of Timoshenko. An existence and uniqueness result is achieved by using the standard theory of linear semigroup. The exponential stability is also proved. Then, fully discrete approximations are introduced and a prior error estimates are shown. Finally, some numerical simulations are presented
The Collected Papers of Stephen P. Timoshenko
No full textDugas R. The Collected Papers of Stephen P. Timoshenko. In: Revue d'histoire des sciences et de leurs applications, tome 8, n°2, 1955. pp. 184-186
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