• 대한수학회보
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• 제52권4호
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• pp.1225-1240
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• 2015

We provide a complete proof that there are no eigenvalues of the integral operator ${\mathcal{K}}_l$ outside the interval (0, 1/k). ${\mathcal{K}}_l$ arises naturally from the deflection problem of a beam with length 2l resting horizontally on an elastic foundation with spring constant k, while some vertical load is applied to the beam.

• Computers and Concrete
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• 제19권1호
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• pp.1-6
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• 2017

Prestressed concrete I-girders are subject to different load types at their construction stages. At the time of strand release, i.e., detensioning, prestressed concrete girders are under the effect of dead and prestressing loads. At this stage, the camber, total net upward deflection, of prestressed girder is summation of the upward deflection due to the prestressing force and the downward deflection due to dead loads. For the calculation of the upward deflection, it is generally considered that prestressed concrete I-girder behaves linear-elastic. However, the field measurements on total net upward deflection of prestressed I-girder after detensioning show contradictory results. In this paper, camber calculations with the linear-elastic beam and elastic-stability theories are presented. One of a typical precast I-girder with 120 cm height and 31.5 m effective span length is selected as a case study. 3D finite element model (FEM) of the girder is developed by SAP2000 software, and the deflections of girder are obtained from linear and nonlinear-static analyses. Only geometric nonlinearity is taken into account. The material test and field measurement of this study are performed at prestressing girder plant. The results of the linear-elastic beam and elastic-stability theories are compared with FEM results and field measurements. It is seen that the camber predicted by elastic-stability theory gives acceptable results than the linear-elastic beam theory while strand releasing.

• 한국정밀공학회지
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• 제21권3호
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• pp.109-116
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• 2004

This paper investigates the characteristics of deflection for circular plate that has same supporting boundary condition along the width direction of plate according to the length change of supporting end. For two boundary conditions such as simple supporting and clamping on both ends, this study derives maximum deflection formula of circular plate using differential equation of elastic curve, assuming that a circular plate is a beam with different widths along the longitudinal direction. The deflection formula of circular plate is verified by carrying out finite element analysis with regard to the ratio of length of supporting end to radius of circular plate.

• 한국정밀공학회:학술대회논문집
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• 한국정밀공학회 2005년도 춘계학술대회 논문집
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• pp.1695-1698
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• 2005

In this paper we investigate characteristics of deflection for circular plate with the non-symmetric boundary condition that is the boundary condition partly supported along the width direction of plate according to the length change of supporting end. For two boundary conditions such as simple supported and completely clamped boundary conditions, this study derives the maximum deflection formula of the circular plate using differential equation of elastic curve, assuming that a circular plate is a beam with the change of width and thickness along the longitudinal direction. The deflection formula of circular plate is verified by carrying out finite element analysis with regard to the ratio of length of supporting end to radius of circular plate.

• Journal of applied mathematics & informatics
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• 제30권1_2호
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• pp.27-47
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• 2012

It has become apparent from the recent work by Choi et al. [3] on the nonlinear beam deflection problem, that analysis of the integral operator $\mathcal{K}$ arising from the beam deflection equation on linear elastic foundation is important. Motivated by this observation, we perform investigations on the eigenstructure of the linear integral operator $\mathcal{K}_l$ which is a restriction of $\mathcal{K}$ on the finite interval [$-l,l$]. We derive a linear fourth-order boundary value problem which is a necessary and sufficient condition for being an eigenfunction of $\mathcal{K}_l$. Using this equivalent condition, we show that all the nontrivial eigenvalues of $\mathcal{K}l$ are in the interval (0, 1/$k$), where $k$ is the spring constant of the given elastic foundation. This implies that, as a linear operator from $L^2[-l,l]$ to $L^2[-l,l]$, $\mathcal{K}_l$ is positive and contractive in dimension-free context.

• Journal of Mechanical Science and Technology
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• 제20권8호
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• pp.1149-1158
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• 2006

In this paper, sliding mode control (SMC) is designed and applied to an elastic structure to suppress some of its vibration modes. The system is an elastic beam clamped on one end and the designed controller uses only the deflection measurement of the free end. The infinite dimensional mathematical model of the beam is reduced to an ordinary differential equation set to represent the behavior of required modes. Since the states of the finite dimensional model are not physically measurable quantities, an observer is designed to estimate these states by measuring the tip deflection of the beam. The performance of the observer is important because the observed states are used in the SMC design. In this study, by using the output information, an observer is designed and tested to estimate the states of the finite dimensional model of the beam. Then the designed SMC is applied to the experimental beam system which gives satisfactory suppressed vibrations.

• Structural Engineering and Mechanics
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• 제48권4호
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• pp.519-545
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• 2013

An outline of the Timoshenko beam theory is presented. Two differential equations of motion in terms of deflection and rotation are comprised into single equation with deflection and analytical solutions of natural vibrations for different boundary conditions are given. Double frequency phenomenon for simply supported beam is investigated. The Timoshenko beam theory is modified by decomposition of total deflection into pure bending deflection and shear deflection, and total rotation into bending rotation and axial shear angle. The governing equations are condensed into two independent equations of motion, one for flexural and another for axial shear vibrations. Flexural vibrations of a simply supported, clamped and free beam are analysed by both theories and the same natural frequencies are obtained. That fact is proved in an analytical way. Axial shear vibrations are analogous to stretching vibrations on an axial elastic support, resulting in an additional response spectrum, as a novelty. Relationship between parameters in beam response functions of all type of vibrations is analysed.

• 대한수학회보
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• 제58권1호
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• pp.71-111
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• 2021

We consider the boundary value problem for the deflection of a finite beam on an elastic foundation subject to vertical loading. We construct a one-to-one correspondence �� from the set of equivalent well-posed two-point boundary conditions to gl(4, ℂ). Using ��, we derive eigenconditions for the integral operator ��M for each well-posed two-point boundary condition represented by M ∈ gl(4, 8, ℂ). Special features of our eigenconditions include; (1) they isolate the effect of the boundary condition M on Spec ��M, (2) they connect Spec ��M to Spec ����,α,k whose structure has been well understood. Using our eigenconditions, we show that, for each nonzero real λ ∉ Spec ����,α,k, there exists a real well-posed boundary condition M such that λ ∈ Spec ��M. This in particular shows that the integral operators ��M, arising from well-posed boundary conditions, may not be positive nor contractive in general, as opposed to ����,α,k.

• Structural Engineering and Mechanics
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• 제13권2호
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• pp.211-230
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• 2002

The objective of this study is to investigate the dynamic behavior of elastic beams subjected to moving loads. Although analytical methods are available, they have limitations with respect to complicated structures. The use of computer technology in recent years is an effective way to solve the problem; thus using the latest technology this study establishes a finite-element solution procedure to investigate dynamic behaviors of a typical elastic beam having a set of constant geometric properties and various span lengths. Both the dead load of the beam and traffic load are applied in which the traffic load is considered a concentrated moving force with various traveling passage speeds on the beam. Dynamic behaviors including deflection, shear, and bending moment due to moving loads are obtained by both analytical and finite element methods; for simple structures, they have an excellent agreement. The numerical results show that based on analytical methods the fundamental mode is good enough to estimate the dynamic deflection along the beam, but is not sufficient to simulate the total response of the shear force or the bending moment. The linear dynamic behavior of the elastic beams subjected to multiple exciting loads can easily be found by linear superposition, and the geometric nonlinear results caused by large deformation and axial force of the beam are always underestimated with only a few exceptions which are indicated. In order to make the results useful, they have been nondimensionalized and presented in graphical form.

• 대한기계학회논문집
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• 제19권8호
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• pp.1889-1900
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• 1995

In this study, a large deflection analysis of a plane frame composed of a thin-walled tube in investigated. When bent, a thin-walled tube is usually controlled by local buckling and subsequent bending collapse of the section. So load resistance reaches the yield level in a thin-walled rectangular tube. This relationship can be divided into three regimes : elastic, post-buckling and crippling. In this paper, this relationship is theoretically presented to be capable of describing nonlinearities and a stiffness matrix is derived by introducing a compound beam-spring element. A numerical analysis uses a constant incremental energy method and the solution is obtained by modifying stiffness matrix at elastic/inelastic stage. This analytical results, load-deflection paths show a good agreement with the test results.