• Structural Engineering and Mechanics
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• v.84 no.1
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• pp.77-83
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• 2022

In the last ten years, many researchers have studied the vibrations of carbon nanotubes using different beam theories. The nano- and micro-scale systems have wavy shape and there is a demand for a powerful tool to mathematically model waviness of those systems. In accordance with the above mentioned lack for the modeling of the waviness of the curved tiny structure, a novel approach is employed by implementing the Timoshenko-beam model. Owing to the small size of the micro beam, these structures are very appropriate for designing small instruments. The vibrations of double walled carbon nanotubes (DWCNTs) are developed using the Timoshenko-beam model in conjunction with the wave propagation approach under support conditions to calculate the fundamental frequencies of DWCNTs. The frequency influence is observed with different parameters. Vibrations of the double walled carbon nanotubes are investigated in order to find their vibrational modes with frequencies. The aspect ratios and half axial wave mode with small length are investigated. It is calculated that these frequencies and ratios are dependent upon the length scale and aspect ratio.

• Journal of the Korean Society of Manufacturing Process Engineers
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• v.19 no.7
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• pp.35-40
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• 2020

This work is concerned with various planar deformations of an isotropic sandwich beam, which generally consists of three layers: two stiff skin layers and one soft core layer. When one layer of the sandwich beam is modeled as a beam, the variational-asymptotic method is rigorously used to construct a zeroth-order beam model, which is similar to a generalized Timoshenko beam model capable of capturing the transverse shear deformations but still carries out the zeroth-order approximation. To analyze the planar sandwich beam, the sum of the energies of the two skin layers and one core layer is then formulated with different material and geometric properties and represented by a universal beam model in terms of the core-layer kinematics through interface displacement and stress continuity conditions. As a preliminary validation, two extreme examples are presented to demonstrate the capability and accuracy of this present approach.

• 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.

• Computers and Concrete
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• v.31 no.1
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• pp.1-7
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• 2023

In this paper, Timoshenko-beam model is developed for the vibration of double carbon nanotubes. The resulting frequencies are gained for axial wave mode and length-to-diameter ratios. The natural frequency becomes more prominent for lower length-to-diameter ratios and diminished for higher ratios. The converse behavior is observed for axial wave mode with clamped-clamped and clamped-free boundary conditions. The frequencies of clamped-free are lower than that of clamped-clamped boundary condition. The eigen solution is obtained to extract the frequencies of double walled carbon nanotubes using Galerkin's method through axial deformation function. Computer softer MATLAB is used for formation of frequency values. The frequency data is compared with available literature and found to be in agreement.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.23 no.11 s.170
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• pp.2058-2066
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• 1999

The present paper proposes a new dynamic analysis method for multi-span Timoshenko beam structures supported by joints with damping subject to moving loads. An exact dynamic element matrix method is adopted to model Timoshenko beam structures. A generalized modal analysis method is applied to derive response formulae for beam structures subject to moving loads. The proposed method offers an exact and closed form solution. Two numerical examples are provided for validating and illustrating the proposed method. In the first numerical example, a single span beam with multiple moving loads is considered. A dynamic analysis on a multi-span beam under a moving load is considered as the second example, in which the flexibility and damping of supporting joints are taken into account. The numerical study proves that the proposed method is useful for the vibration analysis of multi-span beam-hype structures by moving loads.

• Steel and Composite Structures
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• v.26 no.6
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• pp.733-743
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• 2018

This paper presents post-buckling responses of a simply supported laminated composite beam subjected to a non-follower axially compression loads. In the nonlinear kinematic model of the laminated beam, total Lagrangian approach is used in conjunction with the Timoshenko beam theory. In the solution of the nonlinear problem, incremental displacement-based finite element method is used with Newton-Raphson iteration method. There is no restriction on the magnitudes of deflections and rotations in contradistinction to von-Karman strain displacement relations of the beam. The distinctive feature of this study is post-buckling analysis of Timoshenko Laminated beams full geometric non-linearity and by using finite element method. The effects of the fibber orientation angles and the stacking sequence of laminates on the post-buckling deflections, configurations and stresses of the composite laminated beam are illustrated and discussed in the numerical results. Numerical results show that the above-mentioned effects play a very important role on the post-buckling responses of the laminated composite beams.

• Advances in aircraft and spacecraft science
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• v.3 no.4
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• pp.379-397
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• 2016

In this manuscript, free vibrations of a unidirectional composite orthotropic Timoshenko beam based on finite strain have been studied. Using Green-Lagrange strain tensor and comprising all of the nonlinear terms of the tensor and also applying Hamilton's principle, equations of motion and boundary conditions of the beam are obtained. Using separation method in single-harmonic state, time and locative variables are separated from each other and finally, the equations of motion and boundary conditions are gained according to locative variable. To solve the equations, generalized differential quadrature method (GDQM) is applied and then, deflection and cross-section rotation of the beam in linear and nonlinear states are drawn and compared with each other. Also, frequencies of carbon/epoxy and glass/epoxy composite beams for different boundary conditions on the basis of the finite strain are calculated. The calculated frequencies of the nonlinear free vibration of the beam utilizing finite strain assumption for various geometries have been compared to von Karman one.

• Structural Engineering and Mechanics
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• v.43 no.1
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• pp.15-30
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• 2012

A damped Timoshenko beam element is introduced for the DOF-efficient forced vibration analysis of beam-like structures coated with viscoelastic damping layers. The rotary inertia as well as the shear deformation is considered, and the damping effect of viscoelastic layers is modeled as an imaginary loss factor in the complex shear modulus. A complex composite cross-section of structures is replaced with a homogeneous one by means of the transformed section approach in order to construct an equivalent single-layer finite element model capable of employing the standard $C^{0}$-continuity basis functions. The numerical reliability and the DOF-efficiency are explored through the comparative numerical experiments.

• Structural Engineering and Mechanics
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• v.51 no.6
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• pp.955-971
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• 2014

Large post-buckling behavior of Timoshenko beams subjected to non-follower axial compression loads are studied in this paper by using the total Lagrangian Timoshenko beam element approximation. Two types of support conditions for the beams are considered. In the case of beams subjected to compression loads, load rise causes compressible forces end therefore buckling and post-buckling phenomena occurs. It is known that post-buckling problems are geometrically nonlinear problems. The considered highly non-linear problem is solved considering full geometric non-linearity by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. There is no restriction on the magnitudes of deflections and rotations in contradistinction to von-Karman strain displacement relations of the beam. The beams considered in numerical examples are made of lower-Carbon Steel. In the study, the relationships between deflections, rotational angles, critical buckling loads, post-buckling configuration, Cauchy stress of the beams and load rising are illustrated in detail in post-buckling case.

• Structural Engineering and Mechanics
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• v.66 no.1
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• pp.27-36
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• 2018

The objective of this work is to analyze geometrically nonlinear static analysis a simply supported laminated composite beam subjected to a non-follower transversal point load at the midpoint of the beam. In the nonlinear model of the laminated beam, total Lagrangian finite element model of is used in conjunction with the Timoshenko beam theory. The considered non-linear problem is solved considering full geometric non-linearity by using incremental displacement-based finite element method in conjunction with Newton-Raphson iteration method. There is no restriction on the magnitudes of deflections and rotations in contradistinction to von-Karman strain displacement relations of the beam. In the numerical results, the effects of the fiber orientation angles and the stacking sequence of laminates on the nonlinear deflections and stresses of the composite laminated beam are examined and discussed. Convergence study is performed. Also, the difference between the geometrically linear and nonlinear analysis of laminated beam is investigated in detail.