• Proceedings of the Computational Structural Engineering Institute Conference
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• 2008.04a
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• pp.45-48
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• 2008

Periodic lattice structures such as the large space lattice structures and carbon nanotubes may take the extension-transverse shear-bending coupled vibrations, which can be well represented by the extended Timoshenko beam theory. In this paper, the spectrally formulated finite element model (simply, spectral element model) has been developed for extended Timoshenko beams and applied to some typical periodic lattice structures such as the armchair carbon nanotube, the periodic plane truss, and the periodic space lattice beam.

• Journal of the Korean Society for Precision Engineering
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• v.22 no.1
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• pp.143-151
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• 2005

This paper study the effect of open cracks on the dynamic behavior of simply supported Timoshenko beam with a moving mass. The influences of the depth and the position of the crack in the beam have been studied on the dynamic behavior of the simply supported beam system by numerical method. Using Lagrange's equation derives the equation of motion. The crack section is represented by a local flexibility matrix connecting two undamaged beam segments i.e. the crack is modeled as a rotational spring. This flexibility matrix defines the relationship between the displacements and forces on the crack section and is derived by the applying fundamental fracture mechanics theory. As the depth of the crack is increased the mid-span deflection of the Timoshenko beam with the moving mass is increased. And the effects of depth and position of crack on dynamic behavior of simply supported beam with moving mass are discussed.

• Steel and Composite Structures
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• v.11 no.1
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• pp.59-76
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• 2011

Dynamic analysis of an embedded single-walled carbon nanotube (SWCNT) traversed by a moving nanoparticle, which is modeled as a moving load, is investigated in this study based on the nonlocal Timoshenko beam theory, including transverse shear deformation and rotary inertia. The governing equations and boundary conditions are derived by using the principle of virtual displacement. The Galerkin method and the direct integration method of Newmark are employed to find the dynamic response of the SWCNT. A detailed parametric study is conducted to study the influences of the nonlocal parameter, aspect ratio of the SWCNT, elastic medium constant and the moving load velocity on the dynamic responses of SWCNT. For comparison purpose, free vibration frequencies of the SWCNT are obtained and compared with a previously published study. Good agreement is observed. The results show that the above mentioned effects play an important role on the dynamic behaviour of the SWCNT.

• Structural Engineering and Mechanics
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• v.53 no.3
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• pp.537-573
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• 2015

Multiple-step beams carrying intermediate lumped masses with/without rotary inertias are widely used in engineering applications, but in the literature for free vibration analysis of such structural systems; Bernoulli-Euler Beam Theory (BEBT) without axial force effect is used. The literature regarding the free vibration analysis of Bernoulli-Euler single-span beams carrying a number of spring-mass systems, Bernoulli-Euler multiple-step and multi-span beams carrying multiple spring-mass systems and multiple point masses are plenty, but that of Timoshenko multiple-step beams carrying intermediate lumped masses and/or rotary inertias with axial force effect is fewer. The purpose of this paper is to utilize Numerical Assembly Technique (NAT) and Differential Transform Method (DTM) to determine the exact natural frequencies and mode shapes of the axial-loaded Timoshenko multiple-step beam carrying a number of intermediate lumped masses and/or rotary inertias. The model allows analyzing the influence of the shear and axial force effects, intermediate lumped masses and rotary inertias on the free vibration analysis of the multiple-step beams by using Timoshenko Beam Theory (TBT). At first, the coefficient matrices for the intermediate lumped mass with rotary inertia, the step change in cross-section, left-end support and right-end support of the multiple-step Timoshenko beam are derived from the analytical solution. After the derivation of the coefficient matrices, NAT is used to establish the overall coefficient matrix for the whole vibrating system. Finally, equating the overall coefficient matrix to zero one determines the natural frequencies of the vibrating system and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the differential equations of the motion. The calculated natural frequencies of Timoshenko multiple-step beam carrying intermediate lumped masses and/or rotary inertias for the different values of axial force are given in tables. The first five mode shapes are presented in graphs. The effects of axial force, intermediate lumped masses and rotary inertias on the free vibration analysis of Timoshenko multiple-step beam are investigated.

• Structural Engineering and Mechanics
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• v.62 no.2
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• pp.247-258
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• 2017

Beam-like structures such as bridge, high building and tower, pipes, flexible connecting rods and some robotic manipulators are often excited by support motions. These structures are important in machines and structures. So, this study proposes an analytic method to accurately predict the dynamic behaviors of the structures during support motions or an earthquake. Using Timoshenko beam theory which is valid even for non-slender beams and for high-frequency responses, the analytic responses of fixed-fixed beams subjected to a real seismic motions at supports are illustrated to show the principled approach to the proposed method. The responses of a slender beam obtained by using Timoshenko beam theory are compared with the solutions based on Euler-Bernoulli beam theory to validate the correctness of the proposed method. The dynamic analysis for the fixed-fixed beam subjected to support motions gives useful information to develop an understanding of the structural behavior of the beam. The bending moment and the shear force of a slender beam are governed by dynamic components while those of a stocky beam are governed by static components. Especially, the maximal magnitudes of the bending moment and the shear force of the thick beam are proportional to the difference of support displacements and they are influenced by the seismic wave velocity.

• Steel and Composite Structures
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• v.29 no.4
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• pp.483-491
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• 2018

Timoshenko beam model is widely exploited in the literature to examine the mechanical behavior of stubby beam-like components. Timoshenko beam theory is well-known to require the shear correction factor in order to recognize the nonuniform shear distribution at a section. While a variety of shear correction factors are appeared in the literature so far, there is still no consensus on the most appropriate form of the shear correction factor. The Saint-Venant's flexure problem is first revisited in the frame work of the classical theory of elasticity and a highly accurate approximate closed-form solution is presented employing the extended Kantorovich method. The resulted approximate solution for the elasticity field is then employed to introduce two shear correction factors consistent with the Cowper's and energy approaches. The mathematical form of the proposed shear correction factors are then simplified and compared with the results available in the literature over an extended range of Poisson's and aspect ratios. The proposed shear correction factors do not exhibit implausible issue of negative values and do not result in numerical instabilities too. Based on the comprehensive discussion on the shear correction factors, a piecewise definition of shear correction factor is introduced for rectangular cross-sections having excellent agreement with the numerical results in the literature for both shallow and deep cross-sections.

• Advances in nano research
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• v.10 no.5
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• pp.493-507
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• 2021

This article focused on studying the buckling behavior of two-dimensional functionally graded (2D-FG) nanosize tubes, including porosity based on first shear deformation and higher-order theory of tube. The nano-scale tube is simulated based on the nonlocal gradient strain theory, and the general equations and boundary conditions are derived using Hamilton's principle for the Zhang-Fu's tube model (as higher-order theory) and Timoshenko beam theory. Finally, the derived equations are solved using a numerical method for both simply-supported and clamped boundary conditions. The parametric study is performed to study the effects of different parameters such as axial and radial FG power indexes, porosity parameter, nonlocal gradient strain parameters on the buckling behavior of di-dimensional functionally graded porous tube.

• Advances in nano research
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• v.8 no.4
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• pp.293-305
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• 2020

In the current research, the free vibrational behavior of the FG nano-beams integrated in the hygro-thermal environment and reposed on the elastic foundation is investigated using a novel integral Timoshenko beam theory (ITBT). The current model has only three variables unknown and requires the introduction of the shear correction factor because her uniformed variation of the shear stress through the thickness. The effective properties of the nano-beam vary according to power-law and symmetric sigmoid distributions. Three models of the hygro-thermal loading are employed. The effect of the small scale effect is considered by using the nonlocal theory of Eringen. The equations of motion of the present model are determined and resolved via Hamilton principle and Navier method, respectively. Several numerical results are presented thereafter to illustrate the accuracy and efficiency of the actual integral Timoshenko beam theory. The effects of the various parameters influencing the vibrational responses of the P-FG and SS-FG nano-beam are also examined and discussed in detail.

• Proceedings of the KSR Conference
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• 2008.11b
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• pp.1403-1406
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• 2008

Periodic lattice structures such as the large space lattice structures and carbon nanotubes may take the extension-transverse shear-bending coupled vibrations, which can be well represented by the extended Timoshenko beam theory. In this paper, the spectrally formulated finite element model (simply, spectral element model) has been developed for extended Timoshenko beams and applied to some typical periodic lattice structures such as the armchair carbon nanotube, the periodic plane truss, and the periodic space lattice beam.

• Advances in aircraft and spacecraft science
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• v.5 no.1
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• pp.141-164
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• 2018

In this paper, the thermal effect on buckling and free vibration characteristics of functionally graded (FG) size-dependent Timoshenko nanobeams subjected to an in-plane thermal loading are investigated by presenting a Navier type solution for the first time. Material properties of FG nanobeam are supposed to vary continuously along the thickness according to the power-law form and the material properties are assumed to be temperature-dependent. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived based on Timoshenko beam theory through Hamilton's principle and they are solved applying analytical solution. According to the numerical results, it is revealed that the proposed modeling can provide accurate frequency results of the FG nanobeams as compared to some cases in the literature. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as thermal effect, material distribution profile, small scale effects, aspect ratio and mode number on the critical buckling temperature and normalized natural frequencies of the temperature-dependent FG nanobeams in detail. It is explicitly shown that the thermal buckling and vibration behaviour of a FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.