• Journal of Korean Society of Steel Construction
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• v.16 no.1 s.68
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• pp.81-89
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• 2004

The vibration analysis of the circular arch as a member of a structure has been an important subject of mechanics due to its various applications to many industrial fields. In particular, circular arches with variable cross section are widely used to optimize the distribution of weight and strength and to satisfy special architectural and functional requirements. The Generalized Differential Quadrature Method (GDQM) and Differential Transformation Method (DTM) were recently proposed by Shu and Zou, respectively. In this study, GDQM and DTM were applied to the vibration analysis of circular arches with variable cross section. The governing equations of motion for circular arches with variable cross section were derived. The concepts of Differential Transformation and Generalized Differential Quadrature were briefly introduced. The non-dimensionless natural frequencies of circular arches with variable cross section were obtained for various boundary conditions. The results obtained using these methods were compared with those of previous works. GDQM and DTM showed fast convergence, accuracy, efficiency, and validity in solving the vibration problem of circular arches with variable cross section.

• Structural Engineering and Mechanics
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• v.68 no.6
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• pp.713-720
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• 2018

In this paper a new numerical method to determine the elastic curve of the simply supported beams of variable cross-section is demonstrated. In general case it needs to solve linear or small nonlinear second order differential equations with prescribed boundary conditions. For numerical solution the initial values of the slope and the deflection of the end cross-section of the beam is necessary. For obtaining the initial values a lively procedure is developed: it is a special application of the shooting method because boundary value problems can be transformed into initial value problems. As a result of these transformations the initial values of the differential equations are obtained with high accuracy. Procedure is applied for calculating of elastic curve of a simply supported beam of variable cross-section. Results of these numerical procedures, analytical solution of the linearized version and finite element method are compared. It is proved that the suggested procedure yields technically accurate results.

• Structural Engineering and Mechanics
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• v.75 no.6
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• pp.737-746
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• 2020

This paper attempts to investigate the free vibration of functionally graded material beams with nonuniform width based on the nonlocal elasticity theory. The theoretical formulations are established following the Euler-Bernoulli beam theory, and the governing equations of motion of the system are derived from the minimum total potential energy principle using the nonlocal elasticity theory. In addition, the Differential Quadrature Method (DQM) is applied, along with the Chebyshev-Gauss-Lobatto polynomials, in order to determine the weighting coefficient matrices. Furthermore, the effects of the nonlocal parameter, cross-section area of the functionally graded material (FGM) beam and various boundary conditions on the natural frequencies are examined. It is observed that the nonlocal parameter and boundary conditions significantly influence the natural frequencies of the functionally graded material beam cross-section. The results obtained, using the Differential Quadrature Method (DQM) under various boundary conditions, are found in good agreement with analytical and numerical results available in the literature.

• Journal of KSNVE
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• v.9 no.6
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• pp.1193-1199
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• 1999

This paper deals with the free vibrations of circular curved beams with thin-walled cross-section. The differential equation for the coupled flexural-torsional vibrations of such beams with warping is solved numerically to obtain natural frequencies and mode shapes. The Runge-Kutta and determinant search methods, respectively, are used to solve the governing differential equation and to compute the eigenvalues. The lowest three natural frequencies and corresponding mode shapes are calculated for the thin-walled horizontally curved beams with hinged-hinged, hinged-clamped, and clamped-clamped end constraints. A wide range of opening angle of beam, warping parameter, and two different values of slenderness ratios are considered. Numerical results are compared with existing exact and numerical solutions by other methods.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2003.05a
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• pp.858-863
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• 2003

In the practical engineering fields, the horizontally curved beams are frequently erected as the major/minor structural components. The effects of both variable curvature and variable cross-section on structural behavior are very important and therefore these effects should be included in structural analyses. From this viewpoint, this paper deals with the free vibrations of horizontally curved beams with variable curvature and variable cross-section. In this study, the parabola as the curvilinear shape and stepped beam as the variable cross-section are considered. The ordinary differential equation governing free vibrations of such beams are derived. For calculating the natural frequencies, the governing equations are solved by numerical methods. The Runge-Kutta and Determinant search Methods are used for integrating the differential equations and for calculating the natural frequencies, respectively. With regard to numerical results, the relationships between frequency parameters and various beam parameters are presented in the forms of Table and Figures.

• Structural Engineering and Mechanics
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• v.2 no.4
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• pp.345-357
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• 1994

The differential equations governing free, in-plane vibrations of linearly elastic circular arches with variable cross-sections are derived and solve numerically for quadratic arches with three types of rectangular cross sections. Frequencies, mode shapes, cross-sectional load distributions, and the effects of rotatory inertia on frequencies are reported. Experimental measurements of frequencies and their corresponding mode shapes agree closely with those predicted by theory. The numerical methods presented here for computing frequencies and mode shapes are efficient and reliable.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.29 no.1A
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• pp.53-60
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• 2009

This paper deals with the geometrical nonlinear analyses of post-buckled columns with variable cross-section. The objective columns having variable cross-section of the width, depth and square tapers are supported by both hinged ends. By using the Bernoulli-Euler beam theory, differential equations governing the elastica of post-buckled column and their boundary conditions are derived. The solution methods of these differential equations which have two unknown parameters are developed. As the numerical results, equilibrium paths, elasticas and stress resultants of the post-buckled columns are presented. Laboratory scaled experiments were conducted for validating the theories developed in this study.

• Proceedings of the Korean Institute of Electrical and Electronic Material Engineers Conference
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• 2002.05c
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• pp.11-14
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• 2002

Accurate sets of electron collision cross sections for atoms and molecules are necessary for quantitative understanding and modeling of plasma phenomena. So, in this study, we explains the concept of electron collision cross sections for gases, and the principle of determination of the electron collision cross sections for atoms and molecules by using the present electron swarm method.

• Steel and Composite Structures
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• v.11 no.6
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• pp.489-504
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• 2011

This paper presents a theoretical investigation in free vibration of sigmoid functionally graded beams with variable cross-section by using Bernoulli-Euler beam theory. The mechanical properties are assumed to vary continuously through the thickness of the beam, and obey a two power law of the volume fraction of the constituents. Governing equation is reduced to an ordinary differential equation in spatial coordinate for a family of cross-section geometries with exponentially varying width. Analytical solutions of the vibration of the S-FGM beam are obtained for three different types of boundary conditions associated with simply supported, clamped and free ends. Results show that, all other parameters remaining the same, the natural frequencies of S-FGM beams are always proportional to those of homogeneous isotropic beams. Therefore, one can predict the behaviour of S-FGM beams knowing that of similar homogeneous beams.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2002.05a
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• pp.417-422
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• 2002

The main purpose of this paper is to determine the dynamic optimal shapes of tapered column with constant volume. The linear, parabolic and sinusoidal tapers with the regular polygon cross-section are considered, whose material volume and span length are always held constant. The ordinary differential equation including the effect of axial load is applied to calculate the natural frequencies. The Runge-Kutta method and Regula-Falsi methods are used to integrate the differential equation and compute the frequencies, respectively. Then the dynamic optimal shape whose lowest natural frequency is highest is determined by reading the critical value of the frequency versus section ratio curve plotted by the frequency data. In the numerical examples, the tapered columns are analysed and the numerical result of this study are shown in table and figures.