• Steel and Composite Structures
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• v.19 no.5
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• pp.1259-1277
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• 2015

In this work, a trigonometric refined beam theory for the bending, buckling and free vibration analysis of carbon nanotube-reinforced composite (CNTRC) beams resting on elastic foundation is developed. The significant feature of this model is that, in addition to including the shear deformation effect, it deals with only 3 unknowns as the Timoshenko beam (TBM) without including a shear correction factor. The single-walled carbon nanotubes (SWCNTs) are aligned and distributed in polymeric matrix with different patterns of reinforcement. The material properties of the CNTRC beams are assessed by employing the rule of mixture. To examine accuracy of the present theory, several comparison studies are investigated. Furthermore, the effects of different parameters of the beam on the bending, buckling and free vibration responses of CNTRC beam are discussed.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.3 no.4
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• pp.109-122
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• 1983

An accurate thick beam element (TB4) which includes the effects of the shear deformation and rotatory inertia has been degenerated from the three dimensional continuum by employing the Timoshenko beam assumptions. The proposed TB4 element has four nodes and two degrees of freedom at each node, totally eight degrees of freedom. The transverse deflection W and plane rotation ${\theta}$ with the cubic interpolation functions are selected as nodal variables. The element characteristics are formulated by discretizing the beam equations of motion, using the Galerkin weighted residual method, and are numerically integrated by the reduced shear integration technique, using the three-point Gauss quadrature with the various shear coefficients. Several numerical examples are analyzed to demonstrate the accuracy and the monotonic convergence behavior of the proposed TB4 beam element. The result indicates that the TB4 element shows the more excellent performance and the monotonic convergence behavior than the other existing Timoshenko beam type elements for the whole range of the beam aspect ratios, in both static and free vibration analyses.

• Journal of the Society of Naval Architects of Korea
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• v.41 no.4
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• pp.45-52
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• 2004

Exact solution on the anti-symmetric response of ships having uniform sectional properties in waves is derived. Boundary value problem consisted of Timoshenko beam equation and free-free end condition is solved analytically. The responses are assumed as linear and wave loads are calculated by using strip method. Horizontal bending moment, shear force and torsional moment are calculated. The developed analysis model is used for the benchmark test of the numerical codes in this problem. Also the application on the preliminary design of barge-like ships and VLFS (Very Large Floating Structure) is expected

• Journal of the Society of Naval Architects of Korea
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• v.41 no.2
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• pp.47-54
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• 2004

Exact solution on the vertical responses of ships having uniform sectional properties in waves is derived. Boundary value problem consisted of Timoshenko beam equation and free-free end condition is solved analytically. The responses are assumed as linear and wave loads are calculated by using strip method. Vertical bending moment, shear force and deflection are calculated. The developed analysis model is used for the benchmark test of the numerical codes in this problem. Also the application on the preliminary design of barge-like ships and VLFS (Very Large Floating Structure) is expected.

• 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.

• Advances in nano research
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• v.13 no.4
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• pp.351-368
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• 2022

Recently, promising structural technologies like multi-function, ultra-load bearing capacity and tailored structures have been put up for discussions. Finite Element (FE) modelling is probably the best-known option capable of treating these superior properties and multi-domain behavior structures. However, advanced materials such as Functionally Graded Material (FGM) and nanocomposites suffer from problems resulting from variable material properties, reinforcement aggregation and mesh generation. Motivated by these factors, this research proposes a unified shape function for FGM, nanocomposites, graded nanocomposites, in addition to traditional isotropic and orthotropic structural materials. It depends not only on element length but also on the beam's material properties and geometric characteristics. The systematic mathematical theory and FE formulations are based on the Timoshenko beam theory for beam structure. Furthermore, the introduced element achieves C1 degree of continuity. The model is proved to be convergent and free-off shear locking. Moreover, numerical results for static and free vibration analysis support the model accuracy and capabilities by validation with different references. The proposed technique overcomes the issue of continuous properties modelling of these promising materials without discarding older ones. Therefore, introduced benchmark improvements on the FE old concept could be extended to help the development of new software features to confront the rapid progress of structural materials.

• Nuclear Engineering and Technology
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• v.50 no.6
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• pp.971-980
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• 2018

Using the concept of quasi-static decomposition and using three-noded isoparametric locking-free element, this article presents a formulation of the finite element method for Timoshenko beam subjected to spatially different time-dependent motions at supports. To verify the validity of the formulation, three fixed-hinged beams excited by the real seismic motions are examined; one is a slender beam, another is a stocky one, and the other is an intermediate one. The numerical results of time histories of motions of the three beams are compared with corresponding analytical solutions. The internal loads such as bending moment and shearing force at a specific time are also compared with analytic solutions. These comparisons show good agreements. The comparisons between static components of the internal loads and the corresponding total internal loads show that the static components predominate in the stocky beam, whereas the dynamic components predominate in the slender one. Thus, the total internal loads of the stocky beam, which is governed by static components, can be predicted simply by static analysis. Careful numerical experiments indicate that the fundamental frequency of a beam can be used as a parameter identifying such a stocky beam.

• Proceedings of the KSME Conference
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• 2000.04a
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• pp.596-601
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• 2000

The paper presents free vibration and stability analyses of a non-uniform Timoshenko beam resting on a two-parameter elastic soil. The soil parameters can vary along the spat and is assumed to be two-parameter model including the effects of both transverse shear deformation and elastic foundation Governing equations related to the vibration and the stability of the beam are derived from Hamilton's principle, and the resulting eigen-value problems can be solved to give natural frequencies and critical force by finite element method. Numerical results for both vibration and stability of beams under an axial force are presented and compared with other available solutions. Finally, vibration frequencies, mode shapes and critical forces are investigated for various thickness ratios, shear foundation parameter, Winkler foundation parameter and boundary conditions of tapered Timoshenko beams.

• Smart Structures and Systems
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• v.19 no.3
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• pp.243-257
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• 2017

In this paper, free vibration of functionally graded (FG) size-dependent nanobeams is studied within the framework of nonlocal Timoshenko beam model. It is assumed that material properties of the FG nanobeam, vary continuously through the thickness according to a power-law form. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The non-classical governing differential equations of motion are derived through Hamilton's principle and they are solved utilizing both Navier-based analytical method and an efficient and semi-analytical technique called differential transformation method (DTM). Various types of boundary conditions such as simply-supported, clamped-clamped, clamped-simply and clamped-free are assumed for edge supports. The good agreement between the presented DTM and analytical results of this article and those available in the literature validated the presented approach. It is demonstrated that the DTM has high precision and computational efficiency in the vibration analysis of FG nanobeams. The obtained results show the significance of the material graduation, nonlocal effect, slenderness ratio and boundary conditions on the vibration characteristics of FG nanobeams.

• Steel and Composite Structures
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• v.26 no.5
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• pp.607-620
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• 2018

Vibration analysis of deep curved FG nano-beam has been carried out based on modified couple stress theory. Material properties of curved Timoshenko beam are assumed to be functionally graded in radial direction. Governing equations of motion and related boundary conditions have been obtained via Hamilton's principle. In a parametric study, influence of length scale parameter, aspect ratio, gradient index, opening angle, mode number and interactive influences of these parameters on natural frequency of the beam, have been investigated. It was found that, considering geometrical deepness term leads to an increase in sensitivity of natural frequency about variation of aforementioned parameters.