• Bulletin of the Society of Naval Architects of Korea
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• v.19 no.2
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• pp.27-31
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• 1982

The practical utilization of the frequency and normal mode equations of uniform Timoshenko beams such as those presented by Huang is not simple due to their highly trancendental nature. In this note largely simplified equations obtained for the fixed-fixed and the free-free boundary conditions, the modes of which are separable into symmetric modes and antisymmetric ones, are given. Numerical results obtains for six common-type boundary conditions show that the quantitative measure of the effect of rotary inertia and shear deformation on the natural frequency is greatly dependent upon the boundary conditions as well as the order number.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2000.06a
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• pp.712-717
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• 2000

The main purpose of this paper is to present both the fundamental and some higher natural frequencies of axially loaded Timoshenko beams resting on the elastic foundation. The non-dimensional differential equation governing the free vibrations of such beam is derived in which the effects of rotatory inertia and shear deformation are included. The Improved Euler method and 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 and clamped-clamped end constraints are applied in numerical examples. The relations between frequency parameters and both the foundation parameter and slenderness ratio are presented in figures. The effect of cross-sectional shapes is also investigated.

• Journal of the Korean Society for Precision Engineering
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• v.26 no.9
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• pp.112-120
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• 2009

In this paper the purpose is to investigate the stability and variation of natural frequency of a cracked Timoshenko cantilever beams subjected to subtangential follower force. In addition, an analysis of the stability of a cantilever beam as the crack effect and slenderness ratio is investigated. The governing differential equations of a Timoshenko beam subjected to an end tangential follower force are derived via Hamilton's principle. The two coupled governing differential equations are reduced to one fourth order ordinary differential equation in terms of the flexural displacement. By using the results of this paper, we can obtain the judgment base that the choice of beam models for the effect of slenderness ratio and crack.

• Proceedings of the KSR Conference
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• 2008.11b
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• pp.1403-1406
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• 2008

Periodic lattice structures such as the large space lattice structures and carbon nanotubes may take the extension-transverse shear-bending coupled vibrations, which can be well represented by the extended Timoshenko beam theory. In this paper, the spectrally formulated finite element model (simply, spectral element model) has been developed for extended Timoshenko beams and applied to some typical periodic lattice structures such as the armchair carbon nanotube, the periodic plane truss, and the periodic space lattice beam.

• Journal of KSNVE
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• v.9 no.5
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• pp.1019-1028
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• 1999

Free vibration characteristics of cantilevered laminated composite beams with a transverse non0propagating open carck are investigated. In the present analysis a special ply-angle distribution referred to as asymmetric stiffness configuration inducing the elastic coupling between chord-wise bending and extension is considered. The open crack is modelled as an equivalent rotational spring whose spring constant is calculated on the basis of fracture mechanics of composite material structures. Governing equations of a composite beam with a open crack are derived via Hamilton's Principle and Timoshenko beam theory encompassing transverse shear and rotary inertia effect. the effects of various parameters such as the ply angle, fiber volume fraction, crack depth, crack position and transverse shear on the free vibration characteristics of the beam with a crack is highlighted. The numerical results show that the natural frequencies obtained from Timoshenko beam theory are always lower than those from Euler beam theory. The presence of intrinsic cracks in anisotropic composite beams modifies the flexibility and in turn free vibration characteristics of the structures. It is revealed that non-destructive crack detection is possible by analyzing the free vibration responses of a cracked beam.

• Nuclear Engineering and Technology
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• v.52 no.5
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• pp.1066-1078
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• 2020

The present paper describes an analytic solution procedure for flexural vibration of a rotationally restrained hinged-hinged Timoshenko beam at the supports during an earthquake. Focusing on maximal magnitudes of internal loads such as bending moment and shearing force under wide variations of two parameters, kL/EI and kGAL²/EI, various beams under synchronous and asynchronous support motions are simulated. The simulations under asynchronous support motions show the following facts. The variations of the maximal magnitudes of internal loads of stocky beams due to the variation of kL/EI from zero to infinity show much wider variations than those of slender beams as kGAL²/EI decreases. The maximal magnitudes of internal loads of a beam tend to be governed by their static components as kL/EI increases and kGAL²/EI decreases. When the internal loads are governed by their static components, maximal magnitudes of internal loads of the stocky tend to increase monotonically as the value of kL/EI increases. However, the simulations under synchronous support motions show the static components of the internal loads vanish and the internal loads are governed by dynamic components irrespective of the two parameters.

• Structural Engineering and Mechanics
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• v.54 no.3
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• pp.433-451
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• 2015

This paper focuses on large deflection static behavior of edge cracked simple supported beams subjected to a non-follower transversal point load at the midpoint of the beam by using the total Lagrangian Timoshenko beam element approximation. The cross section of the beam is circular. The cracked beam is modeled as an assembly of two sub-beams connected through a massless elastic rotational spring. It is known that large deflection problems are geometrically nonlinear problems. The considered highly nonlinear 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 Aluminum. In the study, the effects of the location of crack and the depth of the crack on the non-linear static response of the beam are investigated in detail. The relationships between deflections, end rotational angles, end constraint forces, deflection configuration, Cauchy stresses of the edge-cracked beams and load rising are illustrated in detail in nonlinear case. Also, the difference between the geometrically linear and nonlinear analysis of edge-cracked beam is investigated in detail.

• Structural Engineering and Mechanics
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• v.62 no.2
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• pp.247-258
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• 2017

Beam-like structures such as bridge, high building and tower, pipes, flexible connecting rods and some robotic manipulators are often excited by support motions. These structures are important in machines and structures. So, this study proposes an analytic method to accurately predict the dynamic behaviors of the structures during support motions or an earthquake. Using Timoshenko beam theory which is valid even for non-slender beams and for high-frequency responses, the analytic responses of fixed-fixed beams subjected to a real seismic motions at supports are illustrated to show the principled approach to the proposed method. The responses of a slender beam obtained by using Timoshenko beam theory are compared with the solutions based on Euler-Bernoulli beam theory to validate the correctness of the proposed method. The dynamic analysis for the fixed-fixed beam subjected to support motions gives useful information to develop an understanding of the structural behavior of the beam. The bending moment and the shear force of a slender beam are governed by dynamic components while those of a stocky beam are governed by static components. Especially, the maximal magnitudes of the bending moment and the shear force of the thick beam are proportional to the difference of support displacements and they are influenced by the seismic wave velocity.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 2002.10a
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• pp.788-791
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• 2002

The Timoshenko beam model has been acknowledged as the most accurate model for representing beam structures. However, the Timoshenko beam model may give rise to significant error when it is applied to multi-stepped beam structures. This paper is intended to demonstrate and improve the modeling error of Timoshenko beam theory for multi-stepped team structures. A tentative bending spring is introduced to represent the stiffness change around a step in beams. This paper proposes a finite element modeling method in the light with the bending spring. The proposed method is rigorously compared with commercial finite element codes. The validity of the proposed method is also demonstrated through an experiment..

• Journal of Korean Society of Steel Construction
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• v.18 no.3
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• pp.349-359
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• 2006

This paper presents a new block stiffness matrix for the analysis an orthogonal Cartesian coordinate system. The displacement fields are defined using the first order shear deformable beam theory. The longitudinal displacement can be expressed as the sum of the projected plane deformation of the cross-section due to Timoshenko's beam theory and axial warping deformation due to modified Vlasov's thin-waled beam theory. The derived element takes into account flexural shear deformation and torsional warping deformation. Three different types of beam elements, namely, the two-noded, three-noded, and four-noded beam elements, are developed. The quadratic and cubic elements are found to be very efficient for the flexural analysis of laminated composite beams. The versatility and accuracy of the new element are demonstrated by comparing the numerical results available in the literature.