• Proceedings of the KSME Conference
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• 2005.11a
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• pp.126.2-126.2
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• 2005

• Magazine of the Korean Society of Agricultural Engineers
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• v.42 no.2
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• pp.108-114
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• 2000

This paper explores the non-linear behavior of tapered beam subjected to a floating concentration load. For applying the Bernoulli-Euler beam theory to this beam, the bending moment at any point of elastical is obtained from the final equilibrium stage. By using the bending moment equation and the Bernoulli-Euler beam theory, the differential equations governing the elastica of simple beam are derived , and solved numberically . Three kinds of tapered beam types are considered . The numerical results of the non-linear behavior obtained in this study are agreed quite well to the results obtained from the laboratory-scale experiments.

• Magazine of the Korean Society of Agricultural Engineers
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• v.41 no.1
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• pp.106-114
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• 1999

This paper explores the geometrical non-linear behavior of the simply supported tapered beams subject to the trapezoidal distributed load and end moments. In order to apply the Bernoulli -Euler beam theory to this tapered beam, the bending moment equation on any point of the elastical is obtained by the redistribution of trapezoidal distributed load. On the basis of the bending moment equation and the BErnoulli-Euler beam theory, the differential equations governging the elastical of such beams are derived and solved numerically by using the Runge-Jutta method and the trial and error method. The three kinds of tapered beams (i.e. width, depth and square tapers) are analyzed in this study. The numerical results of non-linear behavior obtained in this study from the simply supported tapered beams are appeared to be quite well according to the results from the reference . As the numerical results, the elastica, the stress resultants and the load-displacement curves are given in the figures.

• Advances in nano research
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• v.16 no.3
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• pp.265-272
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• 2024

In this paper, the problem of static bending of axially functionally graded (AFG) nanobeam is formulated with the local stress(Lσ)- and strain-driven(εD) two-phase local/nonlocal integral models (TPNIMs). The novelty of the present study aims to compare the size-effects of nonlocal integral models on bending deflections of AFG Euler-Bernoulli nano-beams. The integral relation between strain and nonlocal stress components based on two types nonlocal integral models is transformed unitedly and equivalently into differential form with constitutive boundary conditions. Purely LσD- and εD-NIMs would lead to ill-posed mathematical formulation, and Purely εD- and LσD-nonlocal differential models (NDM) may result in inconsistent size-dependent bending responses. The general differential quadrature method is applied to obtain the numerical results for bending deflection and moment of AFG nanobeam subjected to different boundary and loading conditions. The influence of AFG index, nonlocal models, and nonlocal parameters on the bending deflections of AFG Euler-Bernoulli nanobeams is investigated numerically. A consistent softening effects can be obtained for both LσD- and εD-TPNIMs. The results from current work may provide useful guidelines for designing and optimizing AFG Euler-Bernoulli beam based nano instruments.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2000.10a
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• pp.201-208
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• 2000

This paper explores the non-linear behavior of tapered beam subjected to a floating concentrated load. For applying the Bernoulli-Euler beam theory to this beam, the bending moment at any point of elastica is obtained from the final equilibrium state. By using the bending moment equation and the Bernoulli-Euler beam theory, the differential equations governing the elastica of clamped-roller beam are derived, and solved numerically. Three kinds of tapered beam types are considered. The numerical results of the non-linear behavior obtained in this study are agreed quite well to the results obtained from the laboratory-scale experiments.

• Computational Structural Engineering
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• v.10 no.4
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• pp.177-184
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• 1997

In this paper, numerical methods are developed for solving the elastica of simple beams with variable-arc-length subjected to a point loading. The beam model is based on Bernoulli-Euler beam theory. The Runge-Kutta and Regula-Falsi methods, respectively, are used to solve the governing differential equations and to compute the beam's rotation at the left end of the beams. Extensive numerical results of the elastica responses, including deflected shapes, rotations of cross-section and bending moments, are presented in non-dimensional forms. The possible maximum values of the end rotation, deflection and bending moment are determined by analyzing the numerical data obtained in this study.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.30 no.3 s.246
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• pp.318-327
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• 2006

Due to the diversified use of recent Piezoelectric Zirconate Titanate Composite Actuate. (PZTCA), various PZTCAs with the different ply orientation of the fiber layer have been applied. For this reason, the applicable bending moment equation is necessary even though the fiber layer ply orientation and the laminate configuration are changed. The aim of this research is to evaluate the relationship between the total effective moment $(M^E)$ and Bernoulli-Euler bending moment (M) when the ply orientations of UD CFRP are changed. In conclusions, firstly, as the performance test results by the CFRP ply orientation, the performance of [0] and [90] were stable. However, while the performance of [+45] was suddenly decreased after 5 hours. Secondly, the change of $(M^E)$ by the CFRP ply orientation was evaluated. As the CFRP ply orientation was increased from [0] to [+60], the $(M^E)$ were gradually decreased. However, they became a little bit increased from [+60] to [90]. Finally, after the change of M by the CFRP ply orientation was evaluated, it was found that $M^E=2.2M$ was valid for just [0] and that there was a relationship between $M^E$ and M according to the ply orientation.

• Steel and Composite Structures
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• v.39 no.5
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• pp.543-564
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• 2021

In this study free vibration analysis of a cracked Goland composite wing is investigated. The wing is modelled as a cantilevered beam based on Euler- Bernoulli equations. Also, composite material is modelled based on lamina fiber-reinforced. Edge crack is modelled by additional boundary conditions and local flexibility matrix in crack location, Castigliano's theorem and energy release rate formulation. Governing differential equations are extracted by Hamilton's principle. Using the separation of variables method, general solution in the normalized form for bending and torsion deflection is achieved then expressions for the cross-sectional rotation, the bending moment, the shear force and the torsional moment for the cantilevered beam are obtained. The cracked beam is modelled by separation of beam into two interconnected intact beams. Free vibration analysis of the beam is performed by applying boundary conditions at the fixed end, the free end, continuity conditions in the crack location of the beam and dynamic stiffness matrix determinant. Also, the effects of various parameters such as length and location of crack and fiber angle on natural frequencies and mode shapes are studied. Modal analysis results illustrate that natural frequencies and mode shapes are affected by depth and location of edge crack and coupling parameter.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.30 no.2 s.245
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• pp.120-127
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• 2006

The fatigue behavior of LIPCA was so sensitive to the manufacturing condition, the environmental factors and the change of the test apparatus. Therefore, we could be considering not only the relationship between the stroke range $({\Delta}h)$ and actuating frequency but also the relationship between the stroke range $({\Delta}h)$ and the total effective moment $(M^E)$. Thus, this study proposed the calculation method of the applying $M^E$ when the $({\Delta}h)$ of LIPCA was increased from 1.mm to 20mm. To estimate the relationship between the total effective moment $(M^E)$ and the Bernoulli-Euler bending moment (M) was reviewed. And the residual stress distribution of LIPCA and THUNDER using the CLT was evaluated. In conclusions, converting the $({\Delta}h)$ of LIPCA to the radius of curvature (p) and calculating the $(M^E)$, it was found that the p by the $M^E$ changed similarly as the $({\Delta}h)$. It was found that the $M^E$ was 2.2 times as the M. While CFRP and PZT of LIPCA, which had the superior compressive characteristic, had the compressive residual stress, GFRP was subject to the tensile residual stress. Since this reversed configuration between the compressive residuals stress and the tensile one was made, the requirement of the stroke range $({\Delta}h)$ increase was satisfied.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 2005.10a
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• pp.641-644
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• 2005

The aim of this research is to evaluate the relationship between the total effective moment $(M^E)$ and Bemoulli-Euler bending moment (M) when the ply orientations of UD CFRP in Piezoelectric Zirconate Titanate Composite Actuator (PZTCA) are changed. The obtained results as follows. Firstly, as the performance test results by the CFRP ply orientation, the performance of [0] and [90] were stable. However, while the performance of [+45] was suddenly decreased after 5 hours. Secondly, the change of $M^E$ by the CFRP ply orientation was evaluated. As the CFRP ply orientation was increased from [0] to [+60], the $M^E$ were gradually decreased. However, they became a little bit increased from [+60] to [90]. Finally, after the change of M by the CFRP ply orientation was evaluated, it was found that $M^E=2.2M$ was valid for just [0] and that there was a relationship between $M^E$ and M according to the ply orientation.