Tae-Eun Lee

Free vibrations of horizontally curved beams with unsymmetric axes in Cartesian coordinates

The differential equations governing the free vibrations of elastic, horizontally curved beams with unsymmetric axes were derived from Cartesian coordinates rather than polar coordinates, in which the effect of torsional inertia is included. Frequencies and mode shapes were computed numerically for parabolic curved beams with both clamped ends and both hinged ends. Comparisons of natural frequencies between this study and SAP 2000 were made to validate theories and numerical methods developed herein. The convergent efficiency significantly improved under the newly derived differential equations in Cartesian coordinates. The lowest four natural frequency parameters were 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. Typical mode shapes of vertical displacement were also presented.

Free vibrations of tapered beams laterally restrained by elastic springs

This paper discusses the development of numerical methods for calculating the natural frequencies of tapered beams that are laterally restrained by elastic springs. In formulating the governing equation of the beam, each elastic spring is modeled as a discrete Winkler foundation of the finite longitudinal length, and the effect of axial load is included. By using this model, the differential equation governing the free vibration of the beam is derived, which is solved numerically. The Runge-Kutta method is used to integrate the differential equation, and the determinant search method combined with the Regula-Falsi method is used to determine the eingenvalues, namely, the natural frequencies. In the numerical examples, clamped-clamped, clamped-hinged, hinged-clamped and hinged-hinged end constraints are considered. The numerical results, including the frequency parameters and mode shapes of free vibrations are presented in non-dimensional forms.

In-plane Free Vibration Analysis of Parabolic Arches with Hollow Section

The differential equations governing free vibrations of the elastic arches with hollow section are derived in polar coordinates, in which the effect of rotatory inertia is included. Natural frequencies is computed numerically for parabolic arches with both clamped ends and both hinged ends. Comparisons of natural frequencies between this study and reference are made to validate theories and numerical methods developed herein. The lowest four natural frequency parameters are reported, with the rotatory inertia, as functions of three non-dimensional system parameters: the breadth ratio, the thickness ratio and the rise to span length ratio.

Geometrical Non-linear Analyses of Tapered Variable-Arc-Length Beam subjected to Combined Load

This paper deals with geometrical non-linear analyses of the tapered variable-arc-length beam, subjected to the combined load with an end moment and a point load. The beam is supported by a hinged end and a frictionless sliding support so that the axial length of the deformed beam can be increased by its load. Cross sections of the beam whose flexural rigidities are functionally varied with the axial coordinate. The simultaneous differential equations governing the elastica of such beam are derived on the basis of the Bernoulli-Euler beam theory. These differential equations are numerically solved by the iteration technique for obtaining the elastica of the deformed beam. For validating theories developed herein, laboratory scaled experiments are conducted.

Free Vibration Analysis of Parabolic Hollowed Beam-columns with Constant Volume

This paper deals with free vibrations of the parabolic hollowed beam-columns with constant volume. The cross sections of beam-column taper are the hollowed regular polygons whose depths are varied with the parabolic functional fashion. Volumes of the objective beam-columns are always held constant regardless given geometrical conditions. Ordinary differential equation governing free vibrations of such beam-columns are derived and solved numerically for determining the natural frequencies. In the numerical examples, hinged-hinged, hinged-clamped and clamped-clamped end constraints are considered. As the numerical results, the relationships between non-dimensional frequency parameters and various beam-column parameters such as end constraints, side number, section ratio, thickness ratio and axial load are reported in tables and figures.

Geometrical Non-linear Analyses of Tapered Cantilever Column Subjected to Sub-tangential Follower Force

This paper deals with geometrical non-linear analyses of the tapered cantilever column subjected to the sub-tangential follower force at the free end. Cross-sections of the column whose flexural rigidities are functionally varied with the axial coordinate. The differential equations governing the elastica of such column are derived on the basis of the large deformation theory. These differential equations have three unknown parameters of the vertical and horizontal deflections and rotation at the free end. These differential equations are numerically solved by the iteration technique for obtaining three unknowns and elastica of the deformed column. For validating theories developed herein, laboratory scaled experiments are conducted.

Free Vibrations of Tapered Timoshenko Beam by using 4th Order Ordinary Differential Equation

This paper deals with free vibrations of the tapered Timoshenko beam in which both the rotatory inertia and shear deformation are included. The cross section of the tapered beam is chosen as the rectangular cross section whose depth is constant but breadth is varied with the parabolic function. The fourth order ordinary differential equation with respect the vertical deflection governing free vibrations of such beam is derived based on the Timoshenko beam theory. This governing equation is solved for determining the natural frequencies corresponding with their mode shapes. In the numerical examples, three end constraints of the hinged-hinged, hinged-clamped and clamped-clamped ends are considered. The effects of various beam parameters on natural frequencies are extensively discussed. The mode shapes of both the deflections and stress resultants are presented, in which the composing rates due to bending rotation and shear deformation are determined.

Free Vibrations of Circular Curved Beams with Constant Volume

This paper deals with free vibrations of the circular curved beams with constant volume, whose cross sectional shapes are the circular solid cross-sections. Volumes of the objective beam are always held in constant regardless shape functions of the cross-sectional radius. The shape functions are chosen as the linear, parabolic and sinusoidal ones. Ordinary differential equations governing free vibrations of such beam are derived and solved numerically for determining the natural frequencies. In numerical examples, the hinged-hinged, hinged-clamped and clamped-clamped end constraints are considered. As the numerical results, relationships between frequency parameters and various beam parameters such as rise ratio, section ratio, elasticity ratio, volume ratio, slenderness ratio and taper type are reported in tables and figures.

Stability Analysis of Stiffened Plates on Elastic Foundations

This research analyzes the dynamic stability of stiffened plates on elastic foundations using the finite element method. For analyzing the stiffened plates, both the Mindlin plate theory and Timoshenko beam-column theory were applied. In application of the finite element method, 8-nodes serendipity element system and 3-nodes finite element system were used for plate and beam elements, respectively Elastic foundations were modeled as the Pasternak foundations in which the continuity effect of foundation is considered. In order to verify the theory of this study, solutions obtained by this analysis were compared with the classical solutions in open literature and experimental solutions. The dynamic stability legions of stiffened plates on Pasternak foundations were determined according to changes of in-plane stresses, foundation parameters and dimensions of stiffener.