Journal articles: 'Bernoulli-Euler beam theory'

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LIM, TEIK-CHENG. "ANALYSIS OF AUXETIC BEAMS AS RESONANT FREQUENCY BIOSENSORS." Journal of Mechanics in Medicine and Biology 12, no. 05 (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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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 (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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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 (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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Gul, U., and M. Aydogdu. "Wave Propagation Analysis in Beams Using Shear Deformable Beam Theories Considering Second Spectrum." Journal of Mechanics 34, no. 3 (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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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 (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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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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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 (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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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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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 (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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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 (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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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 (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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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 (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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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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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 (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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Kopmaz, Osman, and Ömer Gündoğdu. "On the Curvature of an Euler–Bernoulli Beam." International Journal of Mechanical Engineering Education 31, no. 2 (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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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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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 (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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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 (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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Vyasarayani, C. P., Sukhpreet Singh Sandhu, and John McPhee. "Nonsmooth Modeling of Vibro-Impacting Euler-Bernoulli Beam." Advances in Acoustics and Vibration 2012 (September 12, 2012): 1–9. http://dx.doi.org/10.1155/2012/268595.

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A new technique to simulate nonsmooth motions occurring in vibro-impacting continuous systems is proposed. Sticking motions that are encountered during vibro-impact simulation are imposed exactly using a Lagrange multiplier, which represents the normal reaction force between the continuous system and the obstacle. The expression for the Lagrange multiplier is developed in closed form. The developed theory is demonstrated by numerically simulating the forced response of a pinned-pinned beam impacting a point-like rigid obstacle.

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

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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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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 (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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Irschik, H. "Analogy between refined beam theories and the Bernoulli-Euler theory." International Journal of Solids and Structures 28, no. 9 (1991): 1105–12. http://dx.doi.org/10.1016/0020-7683(91)90105-o.

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Civalek, Ömer, and Çiğdem Demir. "Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory." Applied Mathematical Modelling 35, no. 5 (2011): 2053–67. http://dx.doi.org/10.1016/j.apm.2010.11.004.

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Mazilu, Traian, Ionuţ Radu Răcănel, Cristian Lucian Ghindea, Radu Iuliu Cruciat, and Mihai-Cornel Leu. "Rail Joint Model Based on the Euler-Bernoulli Beam Theory." Romanian Journal of Transport Infrastructure 8, no. 2 (2019): 16–29. http://dx.doi.org/10.2478/rjti-2019-0008.

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Abstract In this paper, a rail joint model consisting of three Euler-Bernoulli beams connected via a Winkler foundation is proposed in order to point out the influence of the joint gap length upon the stiffness of the rail joint. Starting from the experimental results aiming the stiffness of the rail joint, the Winkler foundation stiffness of the model has been calculated. Using the proposed model, it is shown that the stiffness of the rail joint of the 49 rail can decreases up to 10 % when the joint gap length increases from 0 to 20 mm.

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Ghorbanpourarani, A., M. Mohammadimehr, A. Arefmanesh, and A. Ghasemi. "Transverse vibration of short carbon nanotubes using cylindrical shell and beam models." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 224, no. 3 (2009): 745–56. http://dx.doi.org/10.1243/09544062jmes1659.

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The transverse vibrations of single- and double-walled carbon nanotubes are investigated under axial load by applying the Euler—Bernoulli and Timoshenko beam models and the Donnell shell model. It is concluded that the Euler—Bernoulli beam model and the Donnell shell model predictions have the lowest and highest accuracies, respectively. In order to predict the vibration behaviour of the carbon nanotube more accurately, the current classical models are modified using the non-local theory. The natural frequencies, amplitude coefficient, critical axial load, and strain are obtained for the simply supported boundary conditions.

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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 (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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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 (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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Chang, Tai Ping. "Large Amplitude Free Vibration of Nanobeams Subjected to Magnetic Field Based on Nonlocal Elasticity Theory." Applied Mechanics and Materials 764-765 (May 2015): 1199–203. http://dx.doi.org/10.4028/www.scientific.net/amm.764-765.1199.

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In the present study, nonlinear free vibration behavior of nanobeam subjected to magnetic field is investigated based on Eringen's nonlocal elasticity and Euler–Bernoulli beam theory. The Hamilton's principle is adopted to derive the governing equations together with Euler–Bernoulli beam theory and the von-Kármán's nonlinear strain–displacement relationships. An approximate analytical solution is obtained for the nonlinear frequency of the nanobeam under magnetic field by using the Galerkin method and He's variational method. In the numerical results, the ratio of nonlinear frequency to linear frequency is presented. The effect of nonlocal parameter on the nonlinear frequency ratio is studied; furthermore, the effect of magnetic field on the nonlinear free vibration behavior of nanobeam is investigated.

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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 (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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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 (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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Zheng, Changjie, Lubao Luan, Hongyu Qin, and Hang Zhou. "Horizontal Dynamic Response of a Combined Loaded Large-Diameter Pipe Pile Simulated by the Timoshenko Beam Theory." International Journal of Structural Stability and Dynamics 20, no. 02 (2019): 2071003. http://dx.doi.org/10.1142/s0219455420710030.

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This paper presents an analytical framework for the horizontal dynamic analysis of a large-diameter pipe pile subjected to combined loadings, in which the pipe pile is simulated by the Timoshenko beam theory. The derived solution allows us to evaluate the effects of both the shear deformations and vertical loads on the horizontal dynamic performance of the pipe pile. The proposed solution provides appropriate estimates of complex impedances of large-diameter pipe piles, unlike the earlier solutions based on the Euler–Bernoulli beam theory for describing the pile behavior, which ignores the shear deformation of the pile. The results indicate that the Euler–Bernoulli theory overestimates the pipe pile’s horizontal impedance, while overestimating the effect of vertical loads on its horizontal performance.

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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 (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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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 (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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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 (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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Zhang, Yichi, and Bingen Yang. "Medium-Frequency Vibration Analysis of Timoshenko Beam Structures." International Journal of Structural Stability and Dynamics 20, no. 13 (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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Krommer, Michael. "On the correction of the Bernoulli-Euler beam theory for smart piezoelectric beams." Smart Materials and Structures 10, no. 4 (2001): 668–80. http://dx.doi.org/10.1088/0964-1726/10/4/310.

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Hedayati, Reza, Naeim Ghavidelnia, Mojtaba Sadighi, and Mahdi Bodaghi. "Improving the Accuracy of Analytical Relationships for Mechanical Properties of Permeable Metamaterials." Applied Sciences 11, no. 3 (2021): 1332. http://dx.doi.org/10.3390/app11031332.

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Permeable porous implants must satisfy several physical and biological requirements in order to be promising materials for orthopaedic application: they should have the proper levels of stiffness, permeability, and fatigue resistance approximately matching the corresponding levels in bone tissues. This can be achieved using designer materials, which exhibit exotic properties, commonly known as metamaterials. In recent years, several experimental, numerical, and analytical studies have been carried out on the influence of unit cell micro-architecture on the mechanical and physical properties of metamaterials. Even though experimental and numerical approaches can study and predict the behaviour of different micro-structures effectively, they lack the ease and quickness provided by analytical relationships in predicting the answer. Although it is well known that Timoshenko beam theory is much more accurate in predicting the deformation of a beam (and as a result lattice structures), many of the already-existing relationships in the literature have been derived based on Euler–Bernoulli beam theory. The question that arises here is whether or not there exists a convenient way to convert the already-existing analytical relationships based on Euler–Bernoulli theory to relationships based on Timoshenko beam theory without the need to rewrite all the derivations from the start point. In this paper, this question is addressed and answered, and a handy and easy-to-use approach is presented. This technique is applied to six unit cell types (body-centred cubic (BCC), hexagonal packing, rhombicuboctahedron, diamond, truncated cube, and truncated octahedron) for which Euler–Bernoulli analytical relationships already exist in the literature while Timoshenko theory-based relationships could not be found. The results of this study demonstrated that converting analytical relationships based on Euler–Bernoulli to equivalent Timoshenko ones can decrease the difference between the analytical and numerical values for one order of magnitude, which is a significant improvement in accuracy of the analytical formulas. The methodology presented in this study is not only beneficial for improving the already-existing analytical relationships, but it also facilitates derivation of accurate analytical relationships for other, yet unexplored, unit cell types.

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Farhat, Ali, and Nbila Gwila. "Euler-Bernoulli Beam Theory in the Presence of Fiber Bending Stiffness." IOSR Journal of Mathematics 13, no. 03 (2017): 10–17. http://dx.doi.org/10.9790/5728-1303051017.

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Park, S. K., and X.-L. Gao. "Bernoulli–Euler beam model based on a modified couple stress theory." Journal of Micromechanics and Microengineering 16, no. 11 (2006): 2355–59. http://dx.doi.org/10.1088/0960-1317/16/11/015.

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Amiot, F. "An Euler–Bernoulli second strain gradient beam theory for cantilever sensors." Philosophical Magazine Letters 93, no. 4 (2013): 204–12. http://dx.doi.org/10.1080/09500839.2012.759294.

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Ishaquddin, Md, and S. Gopalakrishnan. "Differential quadrature-based solution for non-classical Euler-Bernoulli beam theory." European Journal of Mechanics - A/Solids 86 (March 2021): 104135. http://dx.doi.org/10.1016/j.euromechsol.2020.104135.

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Elishakoff, Isaac, Menahem Baruch, Liping Zhu, and Raoul Caimi. "Random Vibration of Space Shuttle Weather Protection Systems." Shock and Vibration 2, no. 2 (1995): 111–18. http://dx.doi.org/10.1155/1995/562346.

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The article deals with random vibrations of the space shuttle weather protection systems. The excitation model represents a fit to the measured experimental data. The cross-spectral density is given as a convex combination of three exponential functions. It is shown that for the type of loading considered, the Bernoulli-Euler theory cannot be used as a simplified approach, and the structure will be more properly modeled as a Timoshenko beam. Use of the simple Bernoulli-Euler theory may result in an error of about 50% in determining the mean-square value of the bending moment in the weather protection system.

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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 (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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Sapountzakis, Evangelos, and Amalia Argyridi. "Influence of in-Plane Deformation in Higher Order Beam Theories." Strojnícky casopis – Journal of Mechanical Engineering 68, no. 3 (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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Balaji, G. N., and J. H. Griffin. "Resonant Response of a Tapered Beam and Its Implications to Blade Vibration." Journal of Engineering for Gas Turbines and Power 119, no. 1 (1997): 147–52. http://dx.doi.org/10.1115/1.2815539.

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The resonant response characteristics of a tapered beam are studied using Euler–Bernoulli beam theory. The sensitivity of the beam’s maximum stress to variations in its geometry is studied for three types of harmonic pressure loading. The implications to the response of airfoil chordwise bending modes are discussed.

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Gruber, P. G., K. Nachbagauer, Y. Vetyukov, and J. Gerstmayr. "A novel director-based Bernoulli–Euler beam finite element in absolute nodal coordinate formulation free of geometric singularities." Mechanical Sciences 4, no. 2 (2013): 279–89. http://dx.doi.org/10.5194/ms-4-279-2013.

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Abstract. A three-dimensional nonlinear finite element for thin beams is proposed within the absolute nodal coordinate formulation (ANCF). The deformation of the element is described by means of displacement vector, axial slope and axial rotation parameter per node. The element is based on the Bernoulli–Euler theory and can undergo coupled axial extension, bending and torsion in the large deformation case. Singularities – which are typically caused by such parameterizations – are overcome by a director per element node. Once the directors are properly defined, a cross sectional frame is defined at any point of the beam axis. Since the director is updated during computation, no singularities occur. The proposed element is a three-dimensional ANCF Bernoulli–Euler beam element free of singularities and without transverse slope vectors. Detailed convergence analysis by means of various numerical static and dynamic examples and comparison to analytical solutions shows the performance and accuracy of the element.

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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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Journal articles on the topic 'Bernoulli-Euler beam theory'

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