• Structural Engineering and Mechanics
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• 제67권4호
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• pp.417-425
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• 2018

The main aim of this paper is to investigate the bending of Euler-Bernouilli nano-beams made of bi-directional functionally graded materials (BDFGMs) using Eringen's non-local elasticity theory in the integral form with compare the differential form. To the best of the researchers' knowledge, in the literature, there is no study carried out into integral form of Eringen's non-local elasticity theory for bending analysis of BDFGM Euler-Bernoulli nano-beams with arbitrary functions. Material properties of nano-beam are assumed to change along the thickness and length directions according to arbitrary function. The approximate analytical solutions to the bending analysis of the BDFG nano-beam are derived by using the Rayleigh-Ritz method. The differential form of Eringen's non-local elasticity theory reveals with increasing size effect parameter, the flexibility of the nano-beam decreases, that this is unreasonable. This problem has been resolved in the integral form of the Eringen's model. For all boundary conditions, it is clearly seen that the integral form of Eringen's model predicts the softening effect of the non-local parameter as expected. Finally, the effects of changes of some important parameters such as material length scale, BDFG index on the values of deflection of nano-beam are studied.

• Advances in nano research
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• 제12권2호
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• pp.139-150
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• 2022

This paper presents sets of explicit analytical equations that compute the static displacements of nanobeams by adopting the nonlocal elasticity theory of Eringen within the framework of Euler Bernoulli and Timoshenko beam theories. Castigliano's theorem is applied to an equivalent Virtual Local Beam (VLB) made up of linear elastic material to compute the displacements. The first derivative of the complementary energy of the VLB with respect to a virtual point load provides displacements. The displacements of the VLB are assumed equal to those of the nonlocal beam if nonlocal effects are superposed as additional stress resultants on the VLB. The illustrative equations of displacements are relevant to a few types of loadings combined with a few common boundary conditions. Several equations of displacements, thus derived, matched precisely in similar cases with the equations obtained by other analytical methods found in the literature. Furthermore, magnitudes of maximum displacements are also in excellent agreement with those computed by other numerical methods. These validated the superposition of nonlocal effects on the VLB and the accuracy of the derived equations.

• Smart Structures and Systems
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• 제7권5호
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• pp.417-432
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• 2011

The present paper is devoted to vibration canceling and shape control of piezoelastic slender beams. Taking into account the presence of electric networks, an extended electromechanically coupled Bernoulli-Euler beam theory for passive piezoelectric composite structures is shortly introduced in the first part of our contribution. The second part of the paper deals with the concept of passive shape control of beams using shaped piezoelectric layers and tuned inductive networks. It is shown that an impedance matching and a shaping condition must be fulfilled in order to perfectly cancel vibrations due to an arbitrary harmonic load for a specific frequency. As a main result of the present paper, the correctness of the theory of passive shape control is demonstrated for a harmonically excited piezoelelastic cantilever by a finite element calculation based on one-dimensional Bernoulli-Euler beam elements, as well as by the commercial finite element code of ANSYS using three-dimensional solid elements. Finally, an outlook for the practical importance of the passive shape control concept is given: It is shown that harmonic vibrations of a beam with properly shaped layers according to the presented passive shape control theory, which are attached to an resistor-inductive circuit (RL-circuit), can be significantly reduced over a large frequency range compared to a beam with uniformly distributed piezoelectric layers.

• Structural Engineering and Mechanics
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• 제35권5호
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• pp.645-658
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• 2010

In this study, the Differential Transform Method (DTM) is employed in order to solve the governing differential equation of a moving Bernoulli-Euler beam with axial force effect and investigate its free flexural vibration characteristics. The free vibration analysis of a moving Bernoulli-Euler beam using DTM has not been investigated by any of the studies in open literature so far. At first, the terms are found directly from the analytical solution of the differential equation that describes the deformations of the cross-section according to Bernoulli-Euler beam theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the differential equation of the motion. The calculated natural frequencies of the moving beams with various combinations of boundary conditions using DTM are tabulated in several tables and are compared with the results of the analytical solution where a very good agreement is observed.

• 소음진동
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• 제8권3호
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• pp.524-532
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• 1998

The purpose of this paper is to investigate the free vibrations of horizontally curved beams resting on Winkler-type foundations. Based on the classical Bernoulli-Euler beam theory, the governing differential equations for circular curved beams are derived and solved numerically. Hinged-hinged, hinged-clamped and clamped-clamped end constraints are considered in numerical examples. The free vibration frequencies calculated using the present analysis have been compared with the finite element's results computed by the software ADINA. Numerical results are presented to show the effects on the natural frequencies of curved beams of the horizontal rise to span length ratio, the foundation parameter, and the width ratio of contact area between the beam and foundation.

• 전산구조공학
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• 제10권4호
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• pp.177-184
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• 1997

이 논문은 한개의 집중하중을 받는 단순지지 변화곡선길이 보에 관한 연구이다. Bernoulli-Euler 보 이론에 의하여 정확탄성곡선을 지배하는 미분방정식을 유도하고 이를 수치해석하여 정확탄성곡선의 거동값들을 예측하였다. 미분방정식을 적분하기 위하여 Runge-Kutta method를 이용하고, 단부의 회전각을 산출하기 위하여 Regula-Falsi method를 이용하였다. 본 연구에서의 수치해석 결과들은 문헌값들과 매우 잘 일치하여 본 연구방법의 타당성을 입증하였다. 수치해석의 결과로 정확탄성곡선의 거동값과 하중사이의 관계 및 한계거동값과 하중위치변수 사이의 관계를 각각 그림에 나타내었다. 수치해석의 결과를 분석하여 변화곡선길이 보에서 발생가능한 최대 단부회전각, 최대 처짐 및 최대 휨모멘트를 산정하였다.

• 대한토목학회논문집
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• 제29권1A호
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• pp.53-60
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• 2009

이 논문은 양단회전 후좌굴 변단면 기둥의 기하 비선형 해석에 관한 연구이다. 기둥의 변단면은 변화폭, 변화깊이, 정방형 변단면으로 채택하였다. Bernoulli-Euler 보 이론을 이용하여 후좌굴 기둥의 정확탄성곡선을 지배하는 미분방정식을 유도하였다. 이 미분방정식은 두 개의 미지수를 가지며 이러한 미분방정식을 풀 수 있는 수치해석 방법을 개발하였다. 후좌굴 기둥의 수치해석 결과로 평형경로, 정확탄성곡선 및 합응력을 산정하였다. 실험을 통하여 후좌굴 거동의 이론을 검증하였다.

• 대한토목학회논문집
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• 제29권6A호
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• pp.617-624
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• 2009

이 논문은 선형 변단면 정확탄성곡선형 아치의 자유진동에 관한 연구이다. 정확탄성곡선형 아치의 선형은 Bernoulli-Euler 보 이론을 이용하여 산정하였다. 이러한 선형을 갖는 아치의 자유진동을 지배하는 미분방정식을 유도하고 이를 수치해석하여 무차원 고유진동수를 산출하였다. 수치해석 예에서는 세 종류의 선형 변단면과 두 종류의 지점조건을 채택하였다. 이 연구의 결과를 검증하기 위하여 이 연구와 SAP 2000의 고유진동수를 비교하였다. 수치해석의 결과로 지점조건, 변단면 형상, 세장비 및 단면비가 최저차 4개의 무차원 고유진동수에 미치는 영향을 분석하였다.

• 한국농공학회지
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• 제41권1호
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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.

• Steel and Composite Structures
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• 제21권5호
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• pp.999-1016
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• 2016

This study presents free vibration of beam made of porous material. The mechanical properties of the beam is variable in the thickness direction and the beam is investigated in three situations: poro/nonlinear nonsymmetric distribution, poro/nonlinear symmetric distribution, and poro/monotonous distribution. First, the governing equations of porous beam are derived using principle of virtual work based on Euler-Bernoulli theory. Then, the effect of pores compressibility on natural frequencies of the beam is studied by considering clamped-clamped, clamped-free and hinged-hinged boundary conditions. Moreover, the results are compared with homogeneous beam with the same boundary conditions. Finally, the effects of poroelastic parameters such as pores compressibility, coefficients of porosity and mass on natural frequencies has been considered separately and simultaneously.