Journal articles: 'Euler Beam Theory'

1

LIM, TEIK-CHENG. "ANALYSIS OF AUXETIC BEAMS AS RESONANT FREQUENCY BIOSENSORS." Journal of Mechanics in Medicine and Biology 12, no. 05 (December 2012): 1240027. http://dx.doi.org/10.1142/s0219519412400271.

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The mechanics of beam vibration is of fundamental importance in understanding the shift of resonant frequency of microcantilever and nanocantilever sensors. Unlike the simpler Euler–Bernoulli beam theory, the Timoshenko beam theory takes into consideration rotational inertia and shear deformation. For the case of microcantilevers and nanocantilevers, the minute size, and hence low mass, means that the topmost deviation from the Euler–Bernoulli beam theory to be expected is shear deformation. This paper considers the extent of shear deformation for varying Poisson's ratio of the beam material, with special emphasis on solids with negative Poisson's ratio, which are also known as auxetic materials. Here, it is shown that the Timoshenko beam theory approaches the Euler–Bernoulli beam theory if the beams are of solid cross-sections and the beam material possess high auxeticity. However, the Timoshenko beam theory is significantly different from the Euler–Bernoulli beam theory for beams in the form of thin-walled tubes regardless of the beam material's Poisson's ratio. It is herein proposed that calculations on beam vibration can be greatly simplified for highly auxetic beams with solid cross-sections due to the small shear correction term in the Timoshenko beam deflection equation.

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Gul, U., and M. Aydogdu. "Wave Propagation Analysis in Beams Using Shear Deformable Beam Theories Considering Second Spectrum." Journal of Mechanics 34, no. 3 (May 15, 2017): 279–89. http://dx.doi.org/10.1017/jmech.2017.27.

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AbstractIn this study, wave propagation in beams is studied using different beam theories like Euler-Bernoulli, Timoshenko and Reddy beam theories. Dispersion curves obtained for these beam theories are compared with the exact plane elasticity solutions. It is obtained that, there are two branches for Reddy beam theory similar to the Timoshenko beam theory. However, one branch is obtained for Euler-Bernoulli beam theory. The effects of in-plane load on Timoshenko and Reddy beam theories are examined and dispersion curves of the Timoshenko and Reddy beams are compared with exact plane elasticity solution. In Timoshenko beam theory, qualitative difference between the two spectrums has been lost with in-plane loads for some wave numbers.

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Siva Sankara Rao, Yemineni, Kutchibotla Mallikarjuna Rao, and V. V. Subba Rao. "Estimation of damping in riveted short cantilever beams." Journal of Vibration and Control 26, no. 23-24 (March 20, 2020): 2163–73. http://dx.doi.org/10.1177/1077546320915313.

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In layered and riveted structures, vibration damping happens because of a micro slip that occurs because of a relative motion at the common interfaces of the respective jointed layers. Other parameters that influence the damping mechanism in layered and riveted beams are the amplitude of initial excitation, overall length of the beam, rivet diameter, overall beam thickness, and many layers. In this investigation, using the analytical models such as the Euler–Bernoulli beam theory and Timoshenko beam theory and half-power bandwidth method, the free transverse vibration analysis of layered and riveted short cantilever beams is carried out for observing the damping mechanism by estimating the damping ratio, and the obtained results from the Euler–Bernoulli beam theory and Timoshenko beam theory analytical models are validated by the half-power bandwidth method. Although the Euler–Bernoulli beam model overestimates the damping ratio value by a very less fraction, both the models can be used to evaluate damping for short riveted cantilever beams along with the half-power bandwidth method.

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Ključanin, Dino, and Abaz Manđuka. "The cantilever beams analysis by the means of the first-order shear deformation and the Euler-Bernoulli theory." Tehnički glasnik 13, no. 1 (March 23, 2019): 63–67. http://dx.doi.org/10.31803/tg-20180802210608.

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The effect of the Timoshenko theory and the Euler-Bernoulli theory are investigated in this paper through numerical and analytical analyses. The investigation was required to obtain the optimized position of the pipes support. The Timoshenko beam theory or the first order shear deformation theory was used regarding thick beams where the shearing effect of the beam is considered. The study of the thin beams was performed with the Euler-Bernoulli theory. The analysis was done for stainless steel AISI-440C beams with the rectangular cross-section. The steel beams were a cantilever and stressed under varying point-centred load.

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Li, Anqing, Qing Wang, Ming Song, Jun Chen, Weiguang Su, Shasha Zhou, and Li Wang. "On Strain Gradient Theory and Its Application in Bending of Beam." Coatings 12, no. 9 (September 5, 2022): 1304. http://dx.doi.org/10.3390/coatings12091304.

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The general strain gradient theory of Mindlin is re-visited on the basis of a new set of higher-order metrics, which includes dilatation gradient, deviatoric stretch gradient, symmetric rotation gradient and curvature. A strain gradient bending theory for plane-strain beams is proposed based on the present strain gradient theory. The stress resultants are re-defined and the corresponding equilibrium equations and boundary conditions are derived for beams. The semi-inverse solution for a pure bending beam is obtained and the influence of the Poisson’s effect and strain gradient components on bending rigidity is investigated. As a contrast, the solution of the Bernoulli–Euler beam is also presented. The results demonstrate that when Poisson’s effect is ignored, the result of the plane-strain beam is consistent with that of the Bernoulli–Euler beam in the couple stress theory. While for the strain gradient theory, the bending rigidity of a plane-strain beam ignoring the Poisson’s effect is smaller than that of the Bernoulli–Euler beam due to the influence of the dilatation gradient and the deviatoric stretch gradient along the thickness direction of the beam. In addition, the influence of a strain gradient along the length direction on a bending rigidity is negligible.

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Golushko, Sergey, Gleb Gorynin, and Arseniy Gorynin. "Analytic solutions for free vibration analysis of laminated beams in three-dimensional statement." EPJ Web of Conferences 221 (2019): 01012. http://dx.doi.org/10.1051/epjconf/201922101012.

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In this research we consider free vibrations of laminated beams in terms of three-dimensional linear theory of elasticity. Analytic solutions for natural frequencies of laminated beams are obtained by using an asymptotic splitting method. The results were compared with classical Euler“Bernoulli beam theory and Timoshenko beam theory.

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Wang, Gang. "Analysis of bimorph piezoelectric beam energy harvesters using Timoshenko and Euler–Bernoulli beam theory." Journal of Intelligent Material Systems and Structures 24, no. 2 (September 27, 2012): 226–39. http://dx.doi.org/10.1177/1045389x12461080.

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Single-degree-of-freedom lumped parameter model, conventional finite element method, and distributed parameter model have been developed to design, analyze, and predict the performance of piezoelectric energy harvesters with reasonable accuracy. In this article, a spectral finite element method for bimorph piezoelectric beam energy harvesters is developed based on the Timoshenko beam theory and the Euler–Bernoulli beam theory. Linear piezoelectric constitutive and linear elastic stress/strain models are assumed. Both beam theories are considered in order to examine the validation and applicability of each beam theory for a range of harvester sizes. Using spectral finite element method, a minimum number of elements is required because accurate shape functions are derived using the coupled electromechanical governing equations. Numerical simulations are conducted and validated using existing experimental data from the literature. In addition, parametric studies are carried out to predict the performance of a range of harvester sizes using each beam theory. It is concluded that the Euler–Bernoulli beam theory is sufficient enough to predict the performance of slender piezoelectric beams (slenderness ratio > 20, that is, length over thickness ratio > 20). In contrast, the Timoshenko beam theory, including the effects of shear deformation and rotary inertia, must be used for short piezoelectric beams (slenderness ratio < 5).

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Naghinejad, Maysam, and Hamid Reza Ovesy. "Free vibration characteristics of nanoscaled beams based on nonlocal integral elasticity theory." Journal of Vibration and Control 24, no. 17 (June 28, 2017): 3974–88. http://dx.doi.org/10.1177/1077546317717867.

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In the present article, the total potential energy principle and the nonlocal integral elasticity theory have been used to develop a novel finite element method for studying the free vibration behavior of nano-scaled beams. The formulations are based on Euler-Bernoulli beam theory and this method is able to properly analyze the free vibration of beams with various boundary conditions. By implementing the variational statements, the eigenvalue problem of the free vibration is obtained. The validation investigation is pursued by comparing the results of the current study with those available in the literature. The effects of nonlocal parameter, geometry parameters and boundary conditions on the free vibration of the Euler-Bernoulli beam are then studied.

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Shimpi, Rameshchandra P., Rajesh A. Shetty, and Anirban Guha. "A simple single variable shear deformation theory for a rectangular beam." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 231, no. 24 (September 29, 2016): 4576–91. http://dx.doi.org/10.1177/0954406216670682.

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This paper proposes a simple single variable shear deformation theory for an isotropic beam of rectangular cross-section. The theory involves only one fourth-order governing differential equation. For beam bending problems, the governing equation and the expressions for the bending moment and shear force of the theory are strikingly similar to those of Euler–Bernoulli beam theory. For vibration and buckling problems, the Euler–Bernoulli beam theory governing equation comes out as a special case when terms pertaining to the effects of shear deformation are ignored from the governing equation of present theory. The chosen displacement functions of the theory give rise to a realistic parabolic distribution of transverse shear stress across the beam cross-section. The theory does not require a shear correction factor. Efficacy of the proposed theory is demonstrated through illustrative examples for bending, free vibrations and buckling of isotropic beams of rectangular cross-section. The numerical results obtained are compared with those of exact theory (two-dimensional theory of elasticity) and other first-order and higher-order shear deformation beam theory results. The results obtained are found to be accurate.

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Karaoglu, P., and M. Aydogdu. "On the forced vibration of carbon nanotubes via a non-local Euler—Bernoulli beam model." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 224, no. 2 (February 1, 2010): 497–503. http://dx.doi.org/10.1243/09544062jmes1707.

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This article studies the forced vibration of the carbon nanotubes (CNTs) using the local and the non-local Euler—Bernoulli beam theory. Amplitude ratios for the local and the non-local Euler—Bernoulli beam models are given for single- and double-walled CNTs. It is found that the non-local models give higher amplitudes when compared with the local Euler—Bernoulli beam models. The non-local Euler—Bernoulli beam model predicts lower resonance frequencies.

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11

Pai, P. Frank. "Geometrically exact beam theory without Euler angles." International Journal of Solids and Structures 48, no. 21 (October 2011): 3075–90. http://dx.doi.org/10.1016/j.ijsolstr.2011.07.003.

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ZOHOOR, Hassan, S. Mahdi KHORSANDIJOU, and Mohammad H. ABEDINNASAB. "Modified Nonlinear 3D Euler Bernoulli Beam Theory." Journal of System Design and Dynamics 2, no. 5 (2008): 1170–82. http://dx.doi.org/10.1299/jsdd.2.1170.

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13

Nayfeh, Ali H., S. A. Emam, Sergio Preidikman, and D. T. Mook. "An Exact Solution for the Natural Frequencies of Flexible Beams Undergoing Overall Motions." Journal of Vibration and Control 9, no. 11 (November 2003): 1221–29. http://dx.doi.org/10.1177/1077546304030692.

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We investigate the free vibrations of a flexible beam undergoing an overall two-dimensional motion. The beam is modeled using the Euler-Bernoulli beam theory. An exact solution for the natural frequencies and corresponding mode shapes of the beam is obtained. The model can be extended to beams undergoing three-dimensional motions.

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Yin, Shuohui, Zhibing Xiao, Gongye Zhang, Jingang Liu, and Shuitao Gu. "Size-Dependent Buckling Analysis of Microbeams by an Analytical Solution and Isogeometric Analysis." Crystals 12, no. 9 (September 9, 2022): 1282. http://dx.doi.org/10.3390/cryst12091282.

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This paper proposes an analytical solution and isogeometric analysis numerical approach for buckling analysis of size-dependent beams based on a reformulated strain gradient elasticity theory (RSGET). The superiority of this method is that it has only one material parameter for couple stress and another material parameter for strain gradient effects. Using the RSGET and the principle of minimum potential energy, both non-classical Euler–Bernoulli and Timoshenko beam buckling models are developed. Moreover, the obtained governing equations are solved by an exact solution and isogeometric analysis approach, which conforms to the requirements of higher continuity in gradient elasticity theory. Numerical results are compared with exact solutions to reveal the accuracy of the current isogeometric analysis approach. The influences of length–scale parameter, length-to-thickness ratio, beam thickness and boundary conditions are investigated. Moreover, the difference between the buckling responses obtained by the Timoshenko and Euler–Bernoulli theories shows that the Euler–Bernoulli theory is suitable for slender beams.

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Park, Young-Ho, and Suk-Yoon Hong. "Vibrational Energy Flow Analysis of Corrected Flexural Waves in Timoshenko Beam – Part I: Theory of an Energetic Model." Shock and Vibration 13, no. 3 (2006): 137–65. http://dx.doi.org/10.1155/2006/308715.

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In this paper, an energy flow model is developed to analyze transverse vibration including the effects of rotatory inertia as well as shear distortion, which are very important in the Timoshenko beam transversely vibrating in the medium-to-high frequency ranges. The energy governing equations for this energy flow model are newly derived by using classical displacement solutions of the flexural motion for the Timoshenko beam, in detail. The derived energy governing equations are in the general form incorporating not only the Euler-Bernoulli beam theory used for the conventional energy flow model but also the Rayleigh, shear, and Timoshenko beam theories. Finally, to verify the validity and accuracy of the derived model, numerical analyses for simple finite Timoshenko beams were performed. The results obtained by the derived energy flow model for simple finite Timoshenko beams are compared with those of the classical solutions for the Timoshenko beam, the energy flow solution, and the classical solution for the Euler-Bernoulli beam with various excitation frequencies and damping loss factors of the beam. In addition, the vibrational energy flow analyses of coupled Timoshenko beams are described in the other companion paper.

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16

Schmidt, Bernd. "A Griffith–Euler–Bernoulli theory for thin brittle beams derived from nonlinear models in variational fracture mechanics." Mathematical Models and Methods in Applied Sciences 27, no. 09 (May 23, 2017): 1685–726. http://dx.doi.org/10.1142/s0218202517500294.

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We study a planar thin brittle beam subject to elastic deformations and cracks described in terms of a nonlinear Griffith energy functional acting on [Formula: see text] deformations of the beam. In particular, we consider the case in which elastic bulk contributions due to finite bending of the beam are comparable to the surface energy which is necessary to completely break the beam into several large pieces. In the limit of vanishing aspect ratio we rigorously derive an effective Griffith–Euler–Bernoulli functional which acts on piecewise [Formula: see text] regular curves representing the midline of the beam. The elastic part of this functional is the classical Euler–Bernoulli functional for thin beams in the bending dominated regime in terms of the curve’s curvature. In addition there also emerges a fracture term proportional to the number of discontinuities of the curve and its first derivative.

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ÖZÜTOK, ATİLLA, and EMRAH MADENCİ. "FREE VIBRATION ANALYSIS OF CROSS-PLY LAMINATED COMPOSITE BEAMS BY MIXED FINITE ELEMENT FORMULATION." International Journal of Structural Stability and Dynamics 13, no. 02 (March 2013): 1250056. http://dx.doi.org/10.1142/s0219455412500563.

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In this study, a mixed-finite element method for free vibration analysis of cross-ply laminated composite beams is presented based on the "Euler–Bernoulli Beam Theory" and "Timoshenko Beam Theory". The Gâteaux differential approach is employed to construct the functionals of laminated beams using the variational method. By using these functionals in the mixed-type finite element method, two beam elements CLBT4 and FSDT8 are derived for the Euler–Bernoulli and Timoshenko beam theories, respectively. The CLBT4 element has four degrees of freedom (DOFs), containing the vertical displacement and bending moment as unknowns at the nodes, whereas the FSDT8 element has eight DOFs, containing the vertical displacement, bending moment, shear force and rotation as unknowns. A computer program is developed to execute the analyses for the present study. The numerical results of free vibration analyses obtained for different boundary conditions are presented and compared with results available in the literature, which indicates the reliability of the present approach.

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ZENKOUR, A. M., M. N. M. ALLAM, and MOHAMMED SOBHY. "EFFECT OF TRANSVERSE NORMAL AND SHEAR DEFORMATION ON A FIBER-REINFORCED VISCOELASTIC BEAM RESTING ON TWO-PARAMETER ELASTIC FOUNDATIONS." International Journal of Applied Mechanics 02, no. 01 (March 2010): 87–115. http://dx.doi.org/10.1142/s1758825110000482.

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This article investigates the effect of transverse normal and shear deformations on a fiber-reinforced viscoelastic beams resting on two-parameter (Pasternak's) elastic foundations. The results are obtained by the refined sinusoidal shear deformation beam theory and compared with those obtained by the simple sinusoidal shear deformation beam theory, Timoshenko first-order shear deformation beam theory as well as Euler-Bernoulli classical beam theory. The effects of foundation stiffness on bending of viscoelastic composite beam are presented. The effective moduli methods are used to derive the governing equations of viscoelastic beams. The influences of several parameters, such as length-to-depth ratio, foundation stiffness, time parameter and other parameters on mechanical behavior of composite beams resting on Pasternak's foundations are investigated. Numerical results are presented and conclusions are formulated.

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Dixit, Akash. "Single-beam analysis of damaged beams: Comparison using Euler–Bernoulli and Timoshenko beam theory." Journal of Sound and Vibration 333, no. 18 (September 2014): 4341–53. http://dx.doi.org/10.1016/j.jsv.2014.04.034.

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Lisitano, Domenico, Janko Slavič, Elvio Bonisoli, and Miha Boltežar. "Strain proportional damping in Bernoulli-Euler beam theory." Mechanical Systems and Signal Processing 145 (November 2020): 106907. http://dx.doi.org/10.1016/j.ymssp.2020.106907.

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Ghannadpour, Seyyed Amir Mahdi, and Bijan Mohammadi. "Vibration of Nonlocal Euler Beams Using Chebyshev Polynomials." Key Engineering Materials 471-472 (February 2011): 1016–21. http://dx.doi.org/10.4028/www.scientific.net/kem.471-472.1016.

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This paper is concerned with the free vibration problem for micro/nano beams modelled after Eringen’s nonlocal elasticity theory and Euler beam theory. The small scale effect is taken into consideration in the former theory. The natural frequencies are obtained using the Hamilton’s principle and Chebyshev polynomial functions. The present method, which uses Rayleigh–Ritz technique in this paper, provides an efficient and extremely accurate vibration solution of micro/nano beams where the effects of small scale are significant. Numerical results for a variety of some micro/nano beams with various boundary conditions are given and compared with the available results wherever possible. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is promoted.

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Li, Qing Lu, and Shi Rong Li. "Pre-Buckling Transverse Free Vibration of an Asymmetrically Supported Euler Beam Subjected to Distributed Follower Forces." Applied Mechanics and Materials 217-219 (November 2012): 2640–43. http://dx.doi.org/10.4028/www.scientific.net/amm.217-219.2640.

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Based on the accurate geometrical theory for the extensible elastic beams, an exact mathematical model of post-buckling transverse free vibration of Euler beams subjected to a distributed tangential follower force along the central axis are established. By using shooting method, pre-buckling free vibrations of both simply supported and fixed Euler beam are solved and the responses of small amplitude vibration are obtained. The numerical results show that all the frequencies of unbuckled beams decrease continuously with the increment of the load parameters.

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Ozutok, Atilla, Emrah Madenci, and Fethi Kadioglu. "Free vibration analysis of angle-ply laminate composite beams by mixed finite element formulation using the Gâteaux differential." Science and Engineering of Composite Materials 21, no. 2 (March 1, 2014): 257–66. http://dx.doi.org/10.1515/secm-2013-0043.

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AbstractFree vibration analyses of angle-ply laminated composite beams were investigated by the Gâteaux differential method in the present paper. With the use of the Gâteaux differential method, the functionals were obtained and the natural frequencies of the composite beams were computed using the mixed finite element formulation on the basis of the Euler-Bernoulli beam theory and Timoshenko beam theory. By using these functionals in the mixed-type finite element method, two beam elements, CLBT4 and FSDT8, were derived for the Euler-Bernoulli and Timoshenko beam theories, respectively. The CLBT4 element has 4 degrees of freedom (DOFs) containing the vertical displacement and bending moment as the unknowns at the nodes, whereas the FSDT8 element has 8 DOFs containing the vertical displacement, bending moment, shear force and rotation as unknowns. A computer program was developed to execute the analyses for the present study. The numerical results of free vibration analyses obtained for different boundary conditions were presented and compared with the results available in the literature, which indicates the reliability of the present approach.

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Zhang, GY, and X.-L. Gao. "A new Bernoulli–Euler beam model based on a reformulated strain gradient elasticity theory." Mathematics and Mechanics of Solids 25, no. 3 (November 25, 2019): 630–43. http://dx.doi.org/10.1177/1081286519886003.

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A new non-classical Bernoulli–Euler beam model is developed using a reformulated strain gradient elasticity theory that incorporates both couple stress and strain gradient effects. This reformulated theory is first derived from Form I of Mindlin’s general strain gradient elasticity theory. It is then applied to develop the model for Bernoulli–Euler beams through a variational formulation based on Hamilton’s principle, which leads to the simultaneous determination of the equation of motion and the complete boundary conditions and provides a unified treatment of the strain gradient, couple stress and velocity gradient effects. The new beam model contains one material constant to account for the strain gradient effect, one material length scale parameter to describe the couple stress effect and one coefficient to represent the velocity gradient effect. The current non-classical beam model reduces to its classical elasticity-based counterpart when the strain gradient, couple stress and velocity gradient effects are all suppressed. In addition, the newly developed beam model includes the models considering the strain gradient effect only or the couple stress effect alone as special cases. To illustrate the new model, the static bending and free vibration problems of a simply supported beam are analytically solved by directly applying the general formulas derived. The numerical results reveal that the beam deﬂection predicted by the current model is always smaller than that by the classical model, with the difference being large for very thin beams but diminishing with the increase of the beam thickness. Also, the natural frequency based on the new beam model is found to be always higher than that based on the classical model.

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Eskandari, Amir H., Mostafa Baghani, and Saeed Sohrabpour. "A Time-Dependent Finite Element Formulation for Thick Shape Memory Polymer Beams Considering Shear Effects." International Journal of Applied Mechanics 10, no. 04 (May 2018): 1850043. http://dx.doi.org/10.1142/s1758825118500436.

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In this paper, employing a thermomechanical small strain constitutive model for shape memory polymers (SMP), a beam element made of SMPs is presented based on the kinematic assumptions of Timoshenko beam theory. Considering the low stiffness of SMPs, the necessity for developing a Timoshenko beam element becomes more prominent. This is due to the fact that relatively thicker beams are required in the design procedure of smart structures. Furthermore, in the design and optimization process of these structures which involves a large number of simulations, we cannot rely only on the time consuming 3D finite element analyses. In order to properly validate the developed formulations, the numeric results of the present work are compared with those of 3D finite element results of the authors, previously available in the literature. The parametric study on the material parameters, e.g., hard segment volume fracture, viscosity coefficient of different phases, and the external force applied on the structure (during the recovery stage) are conducted on the thermomechanical response of a short I-shape SMP beam. For instance, the maximum beam deflection error in one of the studied examples for the Euler–Bernoulli beam theory is 7.3%, while for the Timoshenko beam theory, is 1.5% with respect to the 3D FE solution. It is noted that for thicker or shorter beams, the error of the Euler–Bernoulli beam theory even more increases. The proposed beam element in this work could be a fast and reliable alternative tool for modeling 3D computationally expensive simulations.

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Lei, Tuo, Yifei Zheng, Renjun Yu, Yukang Yan, and Ben Xu. "Dynamic Response of Slope Inertia-Based Timoshenko Beam under a Moving Load." Applied Sciences 12, no. 6 (March 16, 2022): 3045. http://dx.doi.org/10.3390/app12063045.

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In this paper, the dynamic response of a simply supported beam subjected to a moving load is reinvestigated. Based on a new beam theory, slope inertia-based Timoshenko (SIBT), the governing equations of motion of the beam are derived. An analytical solution is presented by using a coupled Fourier and Laplace–Carson integral transformation method. The finite element solution is also developed and compared with the analytical solution. Then, a comparative study of three beam models based on the SIBT, Euler–Bernoulli and Timoshenko, subjected to a moving load, is presented. The results show that for slender beams, the dynamic responses calculated by the three theories have marginal differences. However, as the ratio of the cross-sectional size to beam length increases, the dynamic magnification factors for the mid-span displacement obtained by the SIBT and Timoshenko beams become larger than those obtained by the Euler–Bernoulli beams. Furthermore, until the ratio is greater than 1/3, the difference between the calculated results of the SIBT and Timoshenko beams becomes apparent.

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Sørensen, Kasper S., Horia D. Cornean, and Sergey Sorokin. "Optimal profile design for acoustic black holes using Timoshenko beam theory." Journal of the Acoustical Society of America 153, no. 3 (March 2023): 1554–63. http://dx.doi.org/10.1121/10.0017322.

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We revisit the problem of constructing one-dimensional acoustic black holes. Instead of considering the Euler–Bernoulli beam theory, we use Timoshenko's approach, which is known to be more realistic at higher frequencies. Our goal is to minimize the reflection coefficient under a constraint imposed on the normalized wavenumber variation. We use the calculus of variations to derive the corresponding Euler–Lagrange equation analytically and then use numerical methods to solve this equation to find the “optimal” height profile for different frequencies. We then compare these profiles to the corresponding ones previously found using the Euler–Bernoulli beam theory and see that in the lower range of the dimensionless frequency Ω (defined using the largest height of the plate), the optimal profiles almost coincide, as expected.

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Šalinić, Slaviša, Marko Todorović, and Aleksandar Obradović. "An analytical approach for free vibration analysis of Euler-Bernoulli stepped beams with axial-bending coupling effect." Engineering Today 1, no. 4 (2022): 7–17. http://dx.doi.org/10.5937/engtoday2204007s.

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Free vibration of eccentrically stepped beams with one step change in cross-section is considered. It is assumed that the longitudinal symmetry axes of the beam segments are translationally shifted along the vertical direction with respect to each other. The effect of that arrangement of the segments on the coupling of axial and bending vibrations of the stepped beam is analyzed. The beam segments are modeled in the frame of the Euler-Bernoulli theory of elastic beams. Two numerical examples are presented.

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Zhang, Jin-lun, Liao-jun Zhang, Ren-yu Ge, Li Yang, and Jun-wu Xia. "Study on Natural Frequencies of Transverse Free Vibration of Functionally Graded Axis Beams by the Differential Quadrature Method." Acta Acustica united with Acustica 105, no. 6 (November 1, 2019): 1095–104. http://dx.doi.org/10.3813/aaa.919388.

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Functionally gradient materials with special mechanical characteristics are more and more widely used in engineering. The functionally graded beam is one of the commonly used components to bear forces in the structure. Accurate analysis of the dynamic characteristics of the axially functionally graded (AFG) beam plays a vital role in the design and safe operation of the whole structure. Based on the Euler-Bernoulli beam theory (EBT), the characteristic equation of transverse free vibration for the AFG Euler-Bernoulli beam with variable cross-section is obtained in the present work, and the governing equations of the beam are transformed into ordinary differential equations with variable coefficients. Using differential quadrature method (DQM), the solution formulas of characteristic equations under different boundary conditions are derived, and the natural frequencies of the AFG beam are calculated, while the node partition of a non-uniform geometric progression is discussed.

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Maleki-Bigdeli, Mohammad-Ali, Majid Baniassadi, Kui Wang, and Mostafa Baghani. "Developing a beam formulation for semi-crystalline two-way shape memory polymers." Journal of Intelligent Material Systems and Structures 31, no. 12 (May 30, 2020): 1465–76. http://dx.doi.org/10.1177/1045389x20924837.

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In this research, the bending of a two-way shape memory polymer beam is examined implementing a one-dimensional phenomenological macroscopic constitutive model into Euler–Bernoulli and von-Karman beam theories. Since bending loading is a fundamental problem in engineering applications, a combination of bending problem and two-way shape memory effect capable of switching between two temporary shapes can be used in different applications, for example, thermally activated sensors and actuators. Shape memory polymers as a branch of soft materials can undergo large deformation. Hence, Euler–Bernoulli beam theory does not apply to the bending of a shape memory polymer beam where moderate rotations may occur. To overcome this limitation, von-Karman beam theory accounting for the mid-plane stretching as well as moderate rotations can be employed. To investigate the difference between the two beam theories, the deflection and rotating angles of a shape memory polymer cantilever beam are analyzed under small and moderate deflections and rotations. A semi-analytical approach is used to inspect Euler–Bernoulli beam theory, while finite-element method is employed to study von-Karman beam theory. In the following, a smart structure is analyzed using a prepared user-defined subroutine, VUMAT, in finite-element package, ABAQUS/EXPLICIT. Utilizing generated user-defined subroutine, smart structures composed of shape memory polymer material can be analyzed under complex loading circumstances through the two-way shape memory effect.

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31

Babaei, Alireza, and Masoud Arabghahestani. "Free Vibration Analysis of Rotating Beams Based on the Modified Couple Stress Theory and Coupled Displacement Field." Applied Mechanics 2, no. 2 (April 16, 2021): 226–38. http://dx.doi.org/10.3390/applmech2020014.

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In this paper, transverse vibration analysis of rotating micro-beam is investigated based on the modified couple stress theory. The simply-supported micro-beam is modeled utilizing Euler-Bernoulli and Timoshenko beam theories. The system is rotating around a fixed axis perpendicular to the axial direction of the beam. For the first time, displacement filed is introduced as a coupled field to the translational field. In other words, the mentioned rotational displacement field is expressed as a proportional function of translational displacement field using first (axial), second (lateral), and third (angular or rotational) velocity factors. Utilizing Hamilton’s approach as a variational method, dynamic-vibration equations of motion of the proposed model are derived. Galerkin’s method is adopted to solve the equation corresponding to the Euler–Bernoulli and Timoshenko beams. For the case considering shear deformation effects, Navier method is chosen. For evaluation of current results and models, they are compared with those available at the benchmark. In this paper; effects of slenderness ratio, axial, lateral, and angular velocity factors, and rotations of the beam on the frequency are reported. Based on the results presented, mentioned factors should be counted in the analysis and design of such rotating micro-systems.

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32

Ansari, R., M. A. Ashrafi, and S. Hosseinzadeh. "Vibration Characteristics of Piezoelectric Microbeams Based on the Modified Couple Stress Theory." Shock and Vibration 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/598292.

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The vibration behavior of piezoelectric microbeams is studied on the basis of the modified couple stress theory. The governing equations of motion and boundary conditions for the Euler-Bernoulli and Timoshenko beam models are derived using Hamilton’s principle. By the exact solution of the governing equations, an expression for natural frequencies of microbeams with simply supported boundary conditions is obtained. Numerical results for both beam models are presented and the effects of piezoelectricity and length scale parameter are illustrated. It is found that the influences of piezoelectricity and size effects are more prominent when the length of microbeams decreases. A comparison between two beam models also reveals that the Euler-Bernoulli beam model tends to overestimate the natural frequencies of microbeams as compared to its Timoshenko counterpart.

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33

Le Vey, Georges. "Optimal control theory and Newton–Euler formalism for Cosserat beam theory." Comptes Rendus Mécanique 334, no. 3 (March 2006): 170–75. http://dx.doi.org/10.1016/j.crme.2006.01.006.

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Cavacece, M., and L. Vita. "Optimal Cantilever Dynamic Vibration Absorbers by Timoshenko Beam Theory." Shock and Vibration 11, no. 3-4 (2004): 199–207. http://dx.doi.org/10.1155/2004/710924.

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A double-ended cantilever beam as a distributed parameter dynamic vibration absorber has been applied to a single-degree-of-freedom system subjected to harmonic forces.In this investigation, the beam has been analyzed under the well known model of Timoshenko and the computation of best parameters is based on the Chebyshev’s optimality criterion.This is somewhat novel in the field since:The design of cantilever beams as dynamic vibration absorbers is usually made under the hypotheses of the Euler-Bernoulli theory;It is the first time that the Chebyshev’s criterion is applied to the design of a double-ended cantilever beam used as a dynamic vibration absorber.For a ready use of the results herein presented, design charts allow a quick choice of optimal parameters such as tuning ratio and mass ratio.

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35

Bank, L. C., T. D. Gerhardt, and J. H. Gordis. "Dynamic Mechanical Properties of Spirally Wound Paper Tubes." Journal of Vibration and Acoustics 111, no. 4 (October 1, 1989): 489–90. http://dx.doi.org/10.1115/1.3269888.

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The use of experimental modal analysis to obtain the dynamic mechanical properties of spirally wound paper tubes is investigated. Based on experimentally measured natural frequencies in the free-free mode of transverse vibration, tube flexural stiffness properties are predicted using three beam theories: Euler-Bernoulli beam theory, Timoshenko beam theory for isotropic materials, and Timoshenko beam theory for anisotropic materials.

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36

Dao, Cong-Binh, and Viet-Chinh Mai. "Applications of common energy variability methods for establishing the equilibrium equation in mechanics." IOP Conference Series: Materials Science and Engineering 1289, no. 1 (August 1, 2023): 012067. http://dx.doi.org/10.1088/1757-899x/1289/1/012067.

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Abstract The principles of the energy variation method are commonly utilized in mechanics. Energy is a scalar variable, so these are more convenient and simple to establish the equilibrium equations compared to vector-based approaches (i.e. using forces and displacements). The present article applied the theorem of the energy variation method in order to set the equilibrium equations for various complicated problems. Four examples of applying the energy variation method include the differential equation of the Euler – Bernoulli beam based on the energy method, the system of Equilibrium Equations of the Euler–Bernoulli beam with the theorem of Least Work, the principle of maximum work to establish the equation of motions for the Euler–Bernoulli beam and the equation of motion for the Euler–Bernoulli beam by the virtual work theorem, have been implemented. The results obtained from this study open up further research directions on the application of the energy variation method in mechanics as well as in the analysis theory of beam bridges.

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Gladwell, G. M. L. "On the Scattering of Waves in a Non-Uniform Euler-Bernoulli Beam." Proceedings of the Institution of Mechanical Engineers, Part C: Mechanical Engineering Science 205, no. 1 (January 1991): 31–34. http://dx.doi.org/10.1243/pime_proc_1991_205_088_02.

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A scattering theory for waves in non-uniform Euler-Bernoulli beams is developed and shown to be analogous to the corresponding theory for longitudinal waves in rods. Both continuous and abrupt scattering are considered, and the relevant differential equations and matrices are derived. It is shown in particular that if the beam is such that the product of its area of cross-section and its second moment of area is constant, then it is, in a certain sense, equivalent to a rod. In that case known procedures for reconstructing a rod from two spectra can be applied, with the necessary changes, to the beam.

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Sapountzakis, Evangelos, and Amalia Argyridi. "Influence of in-Plane Deformation in Higher Order Beam Theories." Strojnícky casopis – Journal of Mechanical Engineering 68, no. 3 (November 1, 2018): 77–94. http://dx.doi.org/10.2478/scjme-2018-0028.

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AbstractComparing Euler-Bernoulli or Tismoshenko beam theory to higher order beam theories, an essential difference can be depicted: the additional degrees of freedom accounting for out-of plane (warping) and in-plane (distortional) phenomena leading to the appearance of respective higher order geometric constants. In this paper, after briefly overviewing literature of the major beam theories taking account warping and distortional deformation, the influence of distortion in the response of beams evaluated by higher order beam theories is examined via a numerical example of buckling drawn from the literature.

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Banks, H. T., Y. Wang, and D. J. Inman. "Bending and Shear Damping in Beams: Frequency Domain Estimation Techniques." Journal of Vibration and Acoustics 116, no. 2 (April 1, 1994): 188–97. http://dx.doi.org/10.1115/1.2930411.

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In this paper we consider damping mechanisms in the context of dynamic beam models. We summarize previous efforts on various damping models (strain rate or Kelvin-Voigt, time hysteresis (Boltzmann), spatial hysteresis, bending rate/square root) for the Euler-Bernoulli beam theory. The Euler-Bernoulli theory is known to be inadequate for experiments in which high frequency modes have been excited. In such cases the Timoshenko theory may be more appropriate; we consider a number of damping hypotheses for this theory. Corresponding models are proposed and compared to experimental data in the context of parameter estimation or identification problems formulated in the frequency domain. Theoretical results related to the convergence of approximations to these infinite dimensional distributed parameter system estimation problems are presented. Associated computational findings for specific beam experiments are discussed.

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Khater, Mahmoud E. "A Generalized Model for Curved Nanobeams Incorporating Surface Energy." Micromachines 14, no. 3 (March 16, 2023): 663. http://dx.doi.org/10.3390/mi14030663.

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This work presents a comprehensive model for nanobeams, incorporating beam curvature and surface energy. Gurtin–Murdoch surface stress theory is used, in conjunction with Euler–Bernoulli beam theory, to model the beams and take surface energy effects into consideration. The model was validated by contrasting its outcomes with experimental data published in the literature on the static bending of fixed–fixed and fixed–free nanobeams. The outcomes demonstrated that surface stress alters the stiffness of both fixed–fixed and fixed–free nanobeams with different behaviors in each case.

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Arani, A. Ghorbanpour, R. Kolahchi, and M. Hashemian. "Nonlocal surface piezoelasticity theory for dynamic stability of double-walled boron nitride nanotube conveying viscose fluid based on different theories." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 228, no. 17 (March 25, 2014): 3258–80. http://dx.doi.org/10.1177/0954406214527270.

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Based on nonlocal piezoelasticity theory, dynamic stability of double-walled boron nitride nanotubes (DWBNNTs) conveying viscose fluid is studied by incorporating Euler–Bernoulli beam theory, Timoshenko beam theory, and cylindrical shell theory. The surface stress effects are considered based on Gurtin–Murdoch continuum theory. The DWBNNT is embedded in visco-Pasternak medium and the nonlinear van der Waals forces between the inner and outer surface of the DWBNNT is taken into account. Using von Kármán geometric nonlinearity, the governing equations are derived based on Hamilton’s principle. In order to obtain the dynamic instability region of DWBNNT, incremental harmonic balance method is applied. The detailed parametric study is conducted, focusing on the combined effects of the nonlocality, surface stress, fluid velocity, and surrounding medium on the dynamic instability region of DWBNNT. Furthermore, dynamic instability region of Euler–Bernoulli beam theory, Timoshenko beam theory, and cylindrical shell theory are compared to each other. Numerical results indicate that neglecting the surface stress effects, the difference between dynamic instability region of three theories becomes remarkable.

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Berczyński, Stefan, and Tomasz Wróblewski. "Vibration of Steel–Concrete Composite Beams Using the Timoshenko Beam Model." Journal of Vibration and Control 11, no. 6 (June 2005): 829–48. http://dx.doi.org/10.1177/1077546305054678.

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In this paper we present a solution of the problem of free vibrations of steel–concrete composite beams. Three analytical models describing the dynamic behavior of this type of constructions have been formulated: two of these are based on Euler beam theory, and one on Timoshenko beam theory. All three models have been used to analyze the steel–concrete composite beam researched by others. We also give a comparison of the results obtained from the models with the results determined experimentally. The model based on Timoshenko beam theory describes in the best way the dynamic behavior of this type of construction. The results obtained on the basis of the Timoshenko beam theory model achieve the highest conformity with the experimental results, both for higher and lower modes of flexural vibrations of the beam. Because the frequencies of higher modes of flexural vibrations prove to be highly sensitive to damage occurring in the constructions, this model may be used to detect any damage taking place in such constructions.

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43

Li, Xingjia, and Ying Luo. "Flexoelectric Effect on Vibration of Piezoelectric Microbeams Based on a Modified Couple Stress Theory." Shock and Vibration 2017 (2017): 1–7. http://dx.doi.org/10.1155/2017/4157085.

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A novel electric Gibbs function was proposed for the piezoelectric microbeams (PMBs) by employing a modified couple stress theory. Based on the new Gibbs function and the Euler-Bernoulli beam theory, the governing equations which incorporate the effects of couple stress, flexoelectricity, and piezoelectricity were derived for the mechanics of PMBs. The analysis of the effective bending rigidity shows the effects of size and flexoelectricity can greaten the stiffness of PMBs so that the natural frequency increases significantly compared with the Euler-Bernoulli beam, and then the mechanical and electrical properties of PMBs are enhanced compared to the classical beam. This study can guide the design of microscale piezoelectric/flexoelectric structures which may find potential applications in the microelectromechanical systems (MEMS).

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44

Naghinejad, Maysam, and Hamid Reza Ovesy. "Viscoelastic free vibration behavior of nano-scaled beams via finite element nonlocal integral elasticity approach." Journal of Vibration and Control 25, no. 2 (June 29, 2018): 445–59. http://dx.doi.org/10.1177/1077546318783556.

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In this paper, the free-vibration behavior of viscoelastic nano-scaled beams is studied via the finite element (FE) method by implementing the principle of total potential energy and nonlocal integral theory. The formulations are derived based on the Kelvin–Voigt viscoelastic model and Euler–Bernoulli beam theory considering the nonlocal integral theory. The eigenvalue problem of the free vibration is extracted by employing the variational relations. To the best of the authors knowledge it is the first time that the viscoelastic characteristics are implemented in the nonlocal integral FE method to study mechanical behavior of nano-scaled beams. Various boundary conditions can be properly modeled by the current method. Numerical results are compared with literature in order to validate the proposed approach. Then, the effects of nonlocal parameter, viscoelastic parameter, geometrical parameters and different boundary conditions on the complex natural frequencies of the nano-scaled Euler– Bernoulli beams are studied.

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45

Kopmaz, Osman, and Ömer Gündoğdu. "On the Curvature of an Euler–Bernoulli Beam." International Journal of Mechanical Engineering Education 31, no. 2 (April 2003): 132–42. http://dx.doi.org/10.7227/ijmee.31.2.5.

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This paper deals with different approaches to describing the relationship between the bending moment and curvature of a Euler—Bernoulli beam undergoing a large deformation, from a tutorial point of view. First, the concepts of the mathematical and physical curvature are presented in detail. Then, in the case of a cantilevered beam subjected to a single moment at its free end, the difference between the linear theory and the nonlinear theory based on both the mathematical curvature and the physical curvature is shown. It is emphasized that a careless use of the nonlinear mathematical curvature and moment relationship given in most standard textbooks may lead to erroneous results. Furthermore, a numerical example is given for the reader to make a quantitative assessment.

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46

Zhang, Yichi, and Bingen Yang. "Medium-Frequency Vibration Analysis of Timoshenko Beam Structures." International Journal of Structural Stability and Dynamics 20, no. 13 (September 22, 2020): 2041009. http://dx.doi.org/10.1142/s0219455420410096.

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Medium-frequency (mid-frequency) vibration analysis of complex structures plays an important role in automotive, aerospace, mechanical, and civil engineering. Flexible beam structures modeled by the classical Euler–Bernoulli beam theory have been widely used in various engineering problems. A kinematic hypothesis made in the Euler–Bernoulli beam theory is that the plane sections of a beam normal to its neutral axis remain planes after the beam experiences bending deformation, which neglects shear deformation. However, previous investigations found out that the shear deformation of a beam (even with a large slenderness ratio) becomes noticeable in high-frequency vibrations. The Timoshenko beam theory, which describes both bending deformation and shear deformation, would naturally be more suitable for medium-frequency vibration analysis. Nevertheless, vibrations of Timoshenko beam structures in a medium frequency region have not been well studied in the literature. This paper presents a new method for mid-frequency vibration analysis of two-dimensional Timoshenko beam structures. The proposed method, which is called the augmented Distributed Transfer Function Method (DTFM), models a Timoshenko beam structure by a spatial state-space formulation in the [Formula: see text]-domain. The augmented DTFM determines the frequency response of a beam structure in an exact and analytical form, in any frequency region covering low, middle, or high frequencies. Meanwhile, the proposed method provides the local information of a beam structure, such as displacement, shear deformation, bending moment and shear force at any location, which otherwise would be very difficult with energy-based methods. The medium-frequency analysis by the augmented DTFM is validated in numerical examples, where the efficiency and accuracy of the proposed method is demonstrated. Also, the effects of shear deformation on the dynamic behaviors of a beam structure at medium frequencies are examined through comparison of the Timoshenko beam and Euler–Bernoulli beam theories.

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Ma, Xiao, Shuai Wang, Bo Zhou, and Shifeng Xue. "Study on Electromechanical Behavior of Functionally Graded Piezoelectric Composite Beams." Journal of Mechanics 36, no. 6 (August 6, 2020): 841–48. http://dx.doi.org/10.1017/jmech.2020.44.

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ABSTRACTThis paper investigates the electromechanical behavior of functionally graded piezoelectric composite beams containing axially functionally graded (AFG) beam and piezoelectric actuators subjected to electrical load. The mechanical properties of the AFG beam are assumed to be graded along the axial direction. Employing the electromechanical coupling theory and load simulation method, the expression for the simulation load of the piezoelectric actuators is obtained. Based on Euler-Bernoulli beam theory and the obtained simulation load, the differential governing equation of the piezoelectric composite beams subjected to electrical load is derived. The integration-by-parts approach is utilized to solve the differential governing equation, and the expression for the deflection of the piezoelectric composite beams is obtained. The accuracy of the proposed method is validated by the finite element method. The bending response of the functionally graded piezoelectric composite beams is investigated through the proposed method. In the numerical examples, the effects of electrical load, actuator thickness, AFG beam thickness and AFG beam length on the electromechanical behavior of the functionally graded piezoelectric composite beams are studied.

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Duva, J. M., and J. G. Simmonds. "The Usefulness of Elementary Theory for the Linear Vibrations of Layered, Orthotropic Elastic Beams and Corrections Due to Two-Dimensional End Effects." Journal of Applied Mechanics 58, no. 1 (March 1, 1991): 175–80. http://dx.doi.org/10.1115/1.2897145.

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With the aid of formal asymptotic expansions, we conclude not only that elementary (Euler-Bernoulli) beam theory can be applied successfully to layered, orthotropic beams, possibly weak in shear, but also that, in computing the lower natural frequencies of a cantilevered beam, the most important correction to the elementary theory—of the relative order of magnitude of the ratio of depth to length—comes from effects in a neighborhood of the built-in end. We compute this correction using the fundamental work on semi-infinite elastic strips of Gregory and Gladwell (1982) and Gregory and Wan (1984). We also show that, except in unusual cases (e.g., a zero Poisson’s ratio in a homogeneous, elastically isotropic beam), Timoshenko beam theory produces an erroneous correction to the frequencies of elementary theory of the relative order of magnitude of the square of the ratio of depth to length.

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Bonopera, Marco, Kuo-Chun Chang, Chun-Chung Chen, Yu-Chi Sung, and Nerio Tullini. "Feasibility Study of Prestress Force Prediction for Concrete Beams Using Second-Order Deflections." International Journal of Structural Stability and Dynamics 18, no. 10 (October 2018): 1850124. http://dx.doi.org/10.1142/s0219455418501249.

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The safety and sustainability of prestressed concrete bridges can be improved with accurate prestress loss prediction. Considerable loss of the prestress force may imply damages hidden in the bridge. In this study, a prestress force identification method was implemented for concrete beams. Based on the Euler–Bernoulli beam theory, the procedure estimates the prestress force by using one or a set of static displacements measured along the member axis. The implementation of this procedure requires information regarding the flexural rigidity of the beam. The deflected shape of a post-tensioned concrete beam, subjected to an additional vertical load, was measured in a short term in several laboratory experiments. The accuracy of the deflection measurements provided favorable prestress force estimates. In particular, the “compression-softening” theory was validated for uncracked post-tensioned concrete beams.

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Krommer, Michael. "On the correction of the Bernoulli-Euler beam theory for smart piezoelectric beams." Smart Materials and Structures 10, no. 4 (July 18, 2001): 668–80. http://dx.doi.org/10.1088/0964-1726/10/4/310.

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Journal articles on the topic 'Euler Beam Theory'

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