• KSCE Journal of Civil and Environmental Engineering Research
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• v.13 no.4
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• pp.251-258
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• 1993

The main purpose of this paper is to present both the natural frequencies and the buckling loads of beam-columns on Winkler-type foundations. The ordinary differential equations governing the free vibrations and the buckling loads of beam-columns on Winkler-type foundation are derived as nondimensional forms. The Runge-Kutta method and Determinant Search method are used to perform the integration of the differential equations and to determine the eigenvalues(natural frequencies and buckling loads), respectively. Hinged-hinged and damped-clamped end constraints are applied in numerical examples. The relation between frequency parameter and elastic foundation parameter is presented in figure. The effects of axial loads on the natural frequencies of beam-columns on elastic foundations are investigated and the relation between buckling load parameter and elastic foundation parameter is also analyzed. The relation between foundation rested ratio and frequency parameter, buckling load parameter are investigated. The beam-columns on non-homogeneous elastic foundation are analyzed and typical mode shapes are also presented.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.11 no.3
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• pp.37-46
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• 1991

A method is developed for solving the natural frequencies and mode shapes of linearly variable tapered beams. The governing differential equation for the tapered beam is derived. Three kinds of cross sectional shape are considered in differential equation. The Runge-Kutta method and the determinant search method are used to perform the integration of the differential equation and to determine the natural frequencies, respectively. The hinged-hinged, hinged-clamped, damped-clamped and free-damped end constraints are investigated in numerical examples. The lowest four nondimensional natural frequencies are obtained as functions of $d_b/d_a$. ratio. The effects of end constraints and cross sectional shapes on frequencies are analyzed and typical mode shapes are also presented.

• Structural Engineering and Mechanics
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• v.9 no.5
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• pp.449-467
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• 2000

The natural frequencies and the corresponding mode shapes of a non-uniform beam carrying multiple point masses are determined by using the analytical-and-numerical-combined method. To confirm the reliability of the last approach, all the presented results are compared with those obtained from the existing literature or the conventional finite element method and close agreement is achieved. For a "uniform" beam, the natural frequencies and mode shapes of the "clamped-hinged" beam are exactly equal to those of the "hinged-clamped" beam so that one eigenvalue equation is available for two boundary conditions, but this is not true for a "non-uniform" beam. To improve this drawback, a simple transformation function ${\varphi}({\xi})=(e+{\xi}{\alpha})^2$ is presented. Where ${\xi}=x/L$ is the ratio of the axial coordinate x to the beam length L, ${\alpha}$ is a taper constant for the non-uniform beam, e=1.0 for "positive" taper and e=1.0+$|{\alpha}|$ for "negative" taper (where $|{\alpha}|$ is the absolute value of ${\alpha}$). Based on the last function, the eigenvalue equation for a non-uniform beam with "positive" taper (with increasingly varying stiffness) is also available for that with "negative" taper (with decreasingly varying stiffness) so that half of the effort may be saved. For the purpose of comparison, the eigenvalue equations for a positively-tapered beam with five types of boundary conditions are derived. Besides, a general expression for the "normal" mode shapes of the non-uniform beam is also presented.

• 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 Computational Structural Engineering Institute Conference
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• 2002.10a
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• pp.11-16
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• 2002

The differential equations governing free vibrations of the elastic, horizontally curved beams with unsymmetric axis are derived in Cartesian coordinates rather than in polar coordinates, in which the effect of torsional inertia is included. Frequencies are computed numerically for the sinusoidal curved beams with both clamped ends and both hinged ends. Comparisons of natural frequencies between this study and SAP 2000 are made to validate theories and numerical methods developed herein. The convergent efficiency is highly improved under the newly derived differential equations in Cartesian coordinates. The lowest four natural frequency parameters are reported, with and without torsional inertia, as functions of three non-dimensional system parameters: the horizontal rise to chord length ratio, the span length to chord length ratio, and the slenderness ratio.

• Structural Engineering and Mechanics
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• v.27 no.2
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• pp.243-257
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• 2007

A numerically efficient superelement is proposed as a low degree of freedom model for dynamic analysis of rotating tapered beams. The element uses a combination of polynomials and trigonometric functions as shape functions in what is also called the Fourier-p approach. Only a single element is needed to obtain good modal frequency prediction with the analysis and assembly time being considerably less than for conventional elements. The superelement also allows an easy incorporation of polynomial variations of mass and stiffness properties typically used to model helicopter and wind turbine blades. Comparable results are obtained using one superelement with only 14 degrees of freedom compared to 50 conventional finite elements with cubic shape functions with a total of 100 degrees of freedom for a rotating cantilever beam. Excellent agreement is also shown with results from the published literature for uniform and tapered beams with cantilever and hinged boundary conditions. The element developed in this work can be used to model rotating beam substructures as a part of complete finite element model of helicopters and wind turbines.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2005.05a
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• pp.832-835
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• 2005

The free vibration of tapered piles embedded in soil is investigated. The pile model is based on the Bernoulli-Euler beam theory and the soil is idealized as a Winkler model for mathematical simplicity. The governing differential equations for the free vibrations of such members are solved numerically. The square tapered piles with one free and the other hinged end with rotational spring are applied in numerical examples. The lowest two natural frequencies are obtained over a range of non-dimensional system parameters: the rotational spring parameter, the embedded ratio, the foundation parameter, the width ratio of the contact area and the section ratio.

• Steel and Composite Structures
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• v.49 no.6
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• pp.603-614
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• 2023

In this paper, frequency analysis of curved functionally graded (FG) nanobeam by consideration of deepness effect has been studied. Differential transform method (DTM) has been used to obtain frequency responses. The nonlocal theory of Eringen has been applied to consider nanoscales. Material properties are supposed to vary in radial direction according to power-law distribution. Differential equations and related boundary conditions have been derived using Hamilton's principle. Finally, by consideration of nonlocal theory, the governing equations have been derived. Natural frequencies have been obtained using semi analytical method (DTM) for different boundary conditions. In order to study the effect of deepness, the deepness term is considered in strain field. The effects of the gradient index, radius of curvature, the aspect ratio, the nonlocal parameter and interaction of aforementioned parameters on frequency value for different boundary conditions such as clamped-clamped (C-C), clamped-hinged (C-H), and clamped-free (C-F) have been investigated. In addition, the obtained results are compared with the results in previous literature in order to validate present study, a good agreement was observed in the present results.

• Journal of Korean Society of Steel Construction
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• v.19 no.4
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• pp.367-381
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• 2007

A transmission tower was designed using the structural methodology to assume a simple truss behavior. However, there is a big difference between a simple truss behavior and a real one. A suitable explanation of structural stability is that it is a semi-rigid connection and not the assumed hinged connection. This study proposes an alternative structural-analysis modeling strategy for the transmission tower design. The element models that were considered were the truss element model, the beam element model, and the combined beam-truss element model. This study includes linear static analysis, free-vibration analysis, and elastic buckling analysis with respect to the design load. The results of the analysis indicate that the axial forces, axial stresses, and maximum displacements of the three analytical models are very similar. However, the bending moments and stresses of the beam element model and of the combined beam-truss element model are significantly high. The results of the free-vibration and elastic buckling analyses show that the beam-truss model can be conservatively used for the transmission tower design.

• Journal of the Society of Naval Architects of Korea
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• v.44 no.3 s.153
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• pp.267-278
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• 2007

This experimental study investigated on the eddy making effect on the roll motion of a rectangular barge in a two-dimensional wave tank. The structure was used to simulate a simplified rectangular barge in the beam sea condition. The structure with a draft one half of its height was hinged at the center of gravity and free to roll by waves. The rectangular barge was tested with regular waves with a range of wave periods that are shorter, equal to, and longer than its roll natural period. Particle image velocimetry (PIV) was employed to obtain the velocity field in the vicinity of the structure. The coupled interactions between the incident wave and the barge were demonstrated by examining the vortical flow fields to elucidate the eddy making effect during the roll motion. For incoming wave with a wave period same as the roll natural period, the barge roll motion was reduced by the eddy making damping effect. At the wave period shorter than the roll natural period, the structure roll motion was slightly reduced by the vertical flow around the barge. However, at the wave period longer than the roll natural period, the eddy making effect due to flow separation at structure corners indeed amplifies the roll motion. This indicates that not only can the eddy making effect damp out the roll motion, it can also increase the roll motion.