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Timoshenko versus Euler beam theory: Pitfalls of a deterministic approach

BECK, Andre Teofilo; SILVA JR., Claudio R. A. da
Fonte: ELSEVIER SCIENCE BV Publicador: ELSEVIER SCIENCE BV
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
76.77%
The selection criteria for Euler-Bernoulli or Timoshenko beam theories are generally given by means of some deterministic rule involving beam dimensions. The Euler-Bernoulli beam theory is used to model the behavior of flexure-dominated (or ""long"") beams. The Timoshenko theory applies for shear-dominated (or ""short"") beams. In the mid-length range, both theories should be equivalent, and some agreement between them would be expected. Indeed, it is shown in the paper that, for some mid-length beams, the deterministic displacement responses for the two theories agrees very well. However, the article points out that the behavior of the two beam models is radically different in terms of uncertainty propagation. In the paper, some beam parameters are modeled as parameterized stochastic processes. The two formulations are implemented and solved via a Monte Carlo-Galerkin scheme. It is shown that, for uncertain elasticity modulus, propagation of uncertainty to the displacement response is much larger for Timoshenko beams than for Euler-Bernoulli beams. On the other hand, propagation of the uncertainty for random beam height is much larger for Euler beam displacements. Hence, any reliability or risk analysis becomes completely dependent on the beam theory employed. The authors believe this is not widely acknowledged by the structural safety or stochastic mechanics communities. (C) 2010 Elsevier Ltd. All rights reserved.; Sao Paulo State Foundation for Research - FAPESP[2008/10366-4]; National Council for Research and Development - CNPq[305120/2006-9]

Galerkin Solution of Stochastic Beam Bending on Winkler Foundations

SILVA, C. R. A.; HEUSI, H. P. Azikri de; MANTOVANI, G. E.; BECK, A. T.
Fonte: TECH SCIENCE PRESS Publicador: TECH SCIENCE PRESS
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
66.44%
In this paper, the Askey-Wiener scheme and the Galerkin method are used to obtain approximate solutions to stochastic beam bending on Winkler foundation. The study addresses Euler-Bernoulli beams with uncertainty in the bending stiffness modulus and in the stiffness of the foundation. Uncertainties are represented by parameterized stochastic processes. The random behavior of beam response is modeled using the Askey-Wiener scheme. One contribution of the paper is a sketch of proof of existence and uniqueness of the solution to problems involving fourth order operators applied to random fields. From the approximate Galerkin solution, expected value and variance of beam displacement responses are derived, and compared with corresponding estimates obtained via Monte Carlo simulation. Results show very fast convergence and excellent accuracies in comparison to Monte Carlo simulation. The Askey-Wiener Galerkin scheme presented herein is shown to be a theoretically solid and numerically efficient method for the solution of stochastic problems in engineering.; Sao Paulo State Foundation for Research - FAPESP[2008/10366-4]; National Council for Research and Development - CNPq[305120/2006-9]

Um método de identificação de fontes de vibração em vigas.; A method of identification of sources of vibrations in beams.

Nunes, Luis Flávio Soares
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
Publicado em 22/11/2012 PT
Relevância na Pesquisa
66.64%
Neste trabalho, procuramos resolver o problema direto da equação da viga de Euler- Bernoulli bi-engastada com condições iniciais nulas. Estudamos o problema inverso da viga, que consiste em identificar a fonte de vibração, modelada como um elemento em L2, usando como dado a velocidade de um ponto arbitrário da viga, durante um intervalo de tempo arbitrariamente pequeno. A relevância deste trabalho na Engenharia encontra-se, por exemplo, na identificação de danos estruturais em vigas.; In this work, we try to solve the direct problem of the clamped-clamped Euler- Bernoulli beam equation, with zero initial conditions. We study the inverse problem of the beam, consisting in the identification of the source of vibration, shaped as an element in L2, using as data the speed from an arbitrary point of the beam, during a time interval arbitrarily small. The relevance of this work in Engineering, for example, is in the identification of structural damage in beams.

A Transmission Problem for Euler-Bernoulli beam with Kelvin-Voigt Damping

Raposo, C. A.; Bastos, W. D.; Avila, J. A. J.
Fonte: Natural Sciences Publishing Corporation Publicador: Natural Sciences Publishing Corporation
Tipo: Artigo de Revista Científica Formato: 17-28
ENG
Relevância na Pesquisa
96.56%
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); In this work we consider a transmission problem for the longitudinal displacement of a Euler-Bernoulli beam, where one small part of the beam is made of a viscoelastic material with Kelvin-Voigt constitutive relation. We use semigroup theory to prove existence and uniqueness of solutions. We apply a general results due to L. Gearhart [5] and J. Pruss [10] in the study of asymptotic behavior of solutions and prove that the semigroup associated to the system is exponentially stable. A numerical scheme is presented,

Non-linear vibration of Euler-Bernoulli beams

Barari,A.; Kaliji,H.D.; Ghadimi,M.; Domairry,G.
Fonte: Associação Brasileira de Ciências Mecânicas Publicador: Associação Brasileira de Ciências Mecânicas
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/06/2011 EN
Relevância na Pesquisa
76.38%
In this paper, variational iteration (VIM) and parametrized perturbation (PPM) methods have been used to investigate non-linear vibration of Euler-Bernoulli beams subjected to the axial loads. The proposed methods do not require small parameter in the equation which is difficult to be found for nonlinear problems. Comparison of VIM and PPM with Runge-Kutta 4th leads to highly accurate solutions.

Static analysis of tapered nanowires based on nonlocal Euler-Bernoulli beam theory via differential quadrature method

Janghorban,Maziar
Fonte: Associação Brasileira de Ciências Mecânicas Publicador: Associação Brasileira de Ciências Mecânicas
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/04/2012 EN
Relevância na Pesquisa
96.53%
As a first endeavor, bending analysis of tapered nano wires with circular cross section is investigated. In this research, nonlocal elasticity theory based on Euler-Bernoulli beam theory is used to formulate the equations. Differential quadrature method (DQM) is employed to solve the governing equations. Different parameters such as nonlocal parameter, length and radius of tapered nano wires are also considered. The results of present work can be used as bench marks for future works.

Harmonic differential quadrature method for static analysis of functionally graded single walled carbon nanotubes based on Euler-Bernoulli beam theory

Janghorban,Maziar; Zare,Amin
Fonte: Associação Brasileira de Ciências Mecânicas Publicador: Associação Brasileira de Ciências Mecânicas
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/12/2012 EN
Relevância na Pesquisa
96.53%
Bending analysis of functionally graded single walled carbon nano tubes is presented in this paper. Carbon nano tubes are modeled as Euler-Bernoulli beam theory in this study. Harmonic differential quadrature (HDQ) method is used to discretize the governing equations. In order to show the accuracy of present work, the results are compared with those of other existing results. Then the effects of different parameters such as power law index, inner and outer radius of nano tubes and length nano tubes of are studied, too.

Application of iteration perturbation method and Hamiltonian approach for nonlinear vibration of Euler-Bernoulli beams

Ahmadi,Morteza; Hashemi,Gholamreza; Asghari,Ali
Fonte: Associação Brasileira de Ciências Mecânicas Publicador: Associação Brasileira de Ciências Mecânicas
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/11/2014 EN
Relevância na Pesquisa
76.38%
This paper is devoted to the new classes of analytical techniques called the Iteration Perturbation Method (IPM)and Hamiltonian Approach(HA) for solving the equation of motion governing the nonlinear vibration of Euler-Bernoulli beams subjected to the axial loads. It has been found that theIPMand HAare very prolific, rapid, functional and do not demand small perturbation and are also sufficiently accurate to both linear and nonlinear problems in engineering. Comparison of the results of these methods with together and with the results of numerical solution reveals that the IPM and HA are very effective and convenient, and can be easily extended to other nonlinear systems so that can be found widely applicable in engineering and other sciences.

Buckling configurations and dynamic response of buckled Euler-Bernoulli beams with non-classical supports

Sinira,B. G.; Özhanb,B. B.; Reddyc,J. N.
Fonte: Associação Brasileira de Ciências Mecânicas Publicador: Associação Brasileira de Ciências Mecânicas
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/01/2014 EN
Relevância na Pesquisa
76.38%
Exact solutions of buckling configurations and vibration response of post-buckled configurations of beams with non-classical boundary conditions (e.g., elastically supported) are presented using the Euler-Bernoulli theory. The geometric nonlinearity arising from mid-plane stretching (i.e., the von Kármán nonlinear strain) is considered in the formulation. The nonlinear equations are reduced to a single linear equation in terms of the transverse deflection by eliminating the axial displacement and incorporating the nonlinearity and the applied load into a constant. The resulting critical buckling loads and their associated mode shapes are obtained by solving the linearized buckling problem analytically. The buckling configurations are determined in terms of the applied axial load and the transverse deflection. The first buckled shape is the only stable equilibrium position for all boundary conditions considered. Then the pseudo-dynamic response of buckled beams is also determined analytically. Natural frequency versus buckling load and natural frequency versus amplitudes of buckling configurations are plotted for various non-classical boundary conditions.

Free Vibration Analysis of DWCNTs Using CDM and Rayleigh-Schmidt Based on Nonlocal Euler-Bernoulli Beam Theory

De Rosa, Maria Anna; Lippiello, Maria
Fonte: Hindawi Publishing Corporation Publicador: Hindawi Publishing Corporation
Tipo: Artigo de Revista Científica
Publicado em 25/02/2014 EN
Relevância na Pesquisa
66.34%
The free vibration response of double-walled carbon nanotubes (DWCNTs) is investigated. The DWCNTs are modelled as two beams, interacting between them through the van der Waals forces, and the nonlocal Euler-Bernoulli beam theory is used. The governing equations of motion are derived using a variational approach and the free frequencies of vibrations are obtained employing two different approaches. In the first method, the two double-walled carbon nanotubes are discretized by means of the so-called “cell discretization method” (CDM) in which each nanotube is reduced to a set of rigid bars linked together by elastic cells. The resulting discrete system takes into account nonlocal effects, constraint elasticities, and the van der Waals forces. The second proposed approach, belonging to the semianalytical methods, is an optimized version of the classical Rayleigh quotient, as proposed originally by Schmidt. The resulting conditions are solved numerically. Numerical examples end the paper, in which the two approaches give lower-upper bounds to the true values, and some comparisons with existing results are offered. Comparisons of the present numerical results with those from the open literature show an excellent agreement.

A NON-UNIFORM, AXIALLY LOADED EULER-BERNOULLI BEAM HAVING COMPLEX ENDS

CHANG, DAQING; POPPLEWELL, NEIL
Fonte: Oxford University Press Publicador: Oxford University Press
Tipo: Artigo de Revista Científica Formato: text/html
EN
Relevância na Pesquisa
66.39%
An operator-based formulation is used to show the completeness of the eigenfunctions of a non-uniform, axially-loaded, transversely-vibrating Euler-Bernoulli beam having eccentric masses and supported by offset linear springs. This result generalizes the classical expansion theorem for a beam having conventional end conditions. Furthermore, the effect of truncating a series approximation of the initial deflection is investigated for the first time. New asymptotic forms of the eigenvalues and eigenfunctions are determined which are themselves often sufficiently accurate for high-frequency calculations.

Análise dinâmica de vigas Euler-Bernoulli pelo método dos elementos de contorno utilzando soluções dependentes do tempo

Scuciato, Raphael Fernando
Fonte: Universidade Federal do Paraná Publicador: Universidade Federal do Paraná
Tipo: Tese de Doutorado Formato: 133 f. : il. (algumas color.) ; 31 cm. +.; application/pdf
PORTUGUêS
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Orientador : Prof. José Antonio Marques Carrer; C-orientador : Prof. Luiz Alkimin de Lacerda; Anexo Cd-Rom; Tese (doutorado) - Universidade Federal do Paraná, Setor de Tecnologia, Programa de Pós-Graduação em Métodos Numéricos em Engenharia. Defesa: Curitiba, 07/11/2014; Inclui bibliografia; Área de concentração: Mecânica computacional; Resumo: O presente trabalho trata da solução da equação de Euler-Bernoulli para flexão dinâmica de vigas através do Método dos Elementos de Contorno utilizando soluções fundamentais dependentes do tempo. Inicialmente, são apresentadas uma breve revisão da teoria de vigas de Euler-Bernoulli e as soluções analíticas utilizadas como referência. A seguir, é introduzida a solução fundamental dependente do tempo e são discutidas algumas de suas propriedades. Na sequência, a formulação integral do problema é deduzida a partir da técnica de resíduos ponderados e são propostas três implementações numéricas diferentes. Finalmente, os resultados numéricos obtidos através dos códigos computacionais desenvolvidos com base nas implementações numéricas propostas são comparados de maneira gráfica às soluções analíticas adotadas. Palavras-chave: Análise dinâmica. Vigas de Euler-Bernoulli. Método dos Elementos de Contorno. Soluções fundamentais dependentes do tempo.; Abstract: This work is concerned with the solution of the Euler-Bernoulli equation for dynamic bending of beams through the Boundary Element Method with time-dependent fundamental solutions. Initially...

An efficient coupled polynomial interpolation scheme to eliminate material-locking in the Euler-Bernoulli piezoelectric beam finite element

Sulbhewar,Litesh N.; Raveendranath,P.
Fonte: Associação Brasileira de Ciências Mecânicas Publicador: Associação Brasileira de Ciências Mecânicas
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/01/2015 EN
Relevância na Pesquisa
56.69%
The convergence characteristic of the conventional two-noded Euler-Bernoulli piezoelectric beam finite element depends on the configuration of the beam cross-section. The element shows slower convergence for the asymmetric material distribution in the beam cross-section due to 'material-locking' caused by extension-bending coupling. Hence, the use of conventional Euler-Bernoulli beam finite element to analyze piezoelectric beams which are generally made of the host layer with asymmetrically surface bonded piezoelectric layers/patches, leads to increased computational effort to yield converged results. Here, an efficient coupled polynomial interpolation scheme is proposed to improve the convergence of the Euler-Bernoulli piezoelectric beam finite elements, by eliminating ill-effects of material-locking. The equilibrium equations, derived using a variational formulation, are used to establish relationships between field variables. These relations are used to find a coupled quadratic polynomial for axial displacement, having contributions from an assumed cubic polynomial for transverse displacement and assumed linear polynomials for layerwise electric potentials. A set of coupled shape functions derived using these polynomials efficiently handles extension-bending and electromechanical couplings at the field interpolation level itself in a variationally consistent manner...

Study of nonlinear vibration of Euler-Bernoulli beams by using analytical approximate techniques

Bagheri,S.; Nikkar,A.; Ghaffarzadeh,H.
Fonte: Associação Brasileira de Ciências Mecânicas Publicador: Associação Brasileira de Ciências Mecânicas
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/01/2014 EN
Relevância na Pesquisa
66.34%
In this paper, nonlinear responses of a clamped-clamped buckled beam are investigated. Two efficient and easy mathematical techniques called He's Variational Approach and Laplace Iteration Method are used to solve the governing differential equation of motion. To assess the accuracy of solutions, we compare the results with the Runge-Kutta 4th order. The results show that both methods can be easily extended to other nonlinear oscillations and it can be predicted that both methods can be found widely applicable in engineering and physics.

A 2D Hopfield Neural Network approach to mechanical beam damage detection

Almeida, Juliana; Alonso, Hugo; Ribeiro, Pedro; Rocha, Paula
Fonte: Springer Publicador: Springer
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
66.48%
The aim of this paper is to present a method based on a 2D Hopfield Neural Network for online damage detection in beams subjected to external forces. The underlying idea of the method is that a significant change in the beam model parameters can be taken as a sign of damage occurrence in the structural system. In this way, damage detection can be associated to an identification problem. More concretely, a 2D Hopfield Neural Network uses information about the way the beam vibrates and the external forces that are applied to it to obtain time-evolving estimates of the beam parameters at the different beam points. The neural network organizes its input information based on the Euler-Bernoulli model for beam vibrations. Its performance is tested with vibration data generated by means of a different model, namely Timonshenko's, in order to produce more realistic simulation conditions.

Flatness-based Deformation Control of an Euler-Bernoulli Beam with In-domain Actuation

Badkoubeh, Amir; Zheng, Jun; Zhu, Guchuan
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
66.39%
This paper addresses the problem of deformation control of an Euler-Bernoulli beam with in-domain actuation. The proposed control scheme consists in first relating the system model described by an inhomogeneous partial differential equation to a target system under a standard boundary control form. Then, a combination of closed-loop feedback control and flatness-based motion planning is used for stabilizing the closed-loop system around reference trajectories. The validity of the proposed method is assessed through well-posedness and stability analysis of the considered systems. The performance of the developed control scheme is demonstrated through numerical simulations of a representative micro-beam.; Comment: Preprint of an original research work

Stability of an Euler-Bernoulli beam with a nonlinear dynamic feedback system

Miletić, Maja; Stürzer, Dominik; Arnold, Anton; Kugi, Andreas
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
66.34%
This paper is concerned with the stability analysis of a lossless Euler-Bernoulli beam that carries a tip payload which is coupled to a nonlinear dynamic feedback system. This setup comprises nonlinear dynamic boundary controllers satisfying the nonlinear KYP lemma as well as the interaction with a nonlinear passive environment. Global-in-time wellposedness and asymptotic stability is rigorously proven for the resulting closed-loop PDE-ODE system. The analysis is based on semigroup theory for the corresponding first order evolution problem. For the large-time analysis, precompactness of the trajectories is shown by deriving uniform-in-time bounds on the solution and its time derivatives.

Logarithmic stability in determining two coefficients in a dissipative wave equation. Extensions to clamped Euler-Bernoulli beam and heat equations

Ammari, Kaïs; Choulli, Mourad
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
66.34%
We are concerned with the inverse problem of determining both the potential and the damping coefficient in a dissipative wave equation from boundary measurements. We establish stability estimates of logarithmic type when the measurements are given by the operator who maps the initial condition to Neumann boundary trace of the solution of the corresponding initial-boundary value problem. We build a method combining an observability inequality together with a spectral decomposition. We also apply this method to a clamped Euler-Bernoulli beam equation. Finally, we indicate how the present approach can be adapted to a heat equation.

On the exponential decay of the Euler-Bernoulli beam with boundary energy dissipation

Lazzari, Barbara; Nibbi, Roberta
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 15/04/2011
Relevância na Pesquisa
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We study the asymptotic behavior of the Euler-Bernoulli beam which is clamped at one end and free at the other end. We apply a boundary control with memory at the free end of the beam and prove that the "exponential decay" of the memory kernel is a necessary and sufficient condition for the exponential decay of the energy.; Comment: 13 pages

A piezoelectric Euler-Bernoulli beam with dynamic boundary control: stability and dissipative FEM

Miletic, Maja; Arnold, Anton
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 09/07/2015
Relevância na Pesquisa
66.34%
We present a mathematical and numerical analysis on a control model for the time evolution of a multi-layered piezoelectric cantilever with tip mass and moment of inertia, as developed by Kugi and Thull [31]. This closed-loop control system consists of the inhomogeneous Euler-Bernoulli beam equation coupled to an ODE system that is designed to track both the position and angle of the tip mass for a given reference trajectory. This dynamic controller only employs first order spatial derivatives, in order to make the system technically realizable with piezoelectric sensors. From the literature it is known that it is asymptotically stable [31]. But in a refined analysis we first prove that this system is not exponentially stable. In the second part of this paper, we construct a dissipative finite element method, based on piecewise cubic Hermitian shape functions and a Crank-Nicolson time discretization. For both the spatial semi-discretization and the full x - t-discretization we prove that the numerical method is structure preserving, i.e. it dissipates energy, analogous to the continuous case. Finally, we derive error bounds for both cases and illustrate the predicted convergence rates in a simulation example.