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Journal articles on the topic 'Timoshenko beam theory'

Author: Grafiati
Published: 4 June 2021
Last updated: 6 September 2023

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1

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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2

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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3

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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4

Wan, Chunfeng, Huachen Jiang, Liyu Xie, Caiqian Yang, Youliang Ding, Hesheng Tang, and Songtao Xue. "Natural Frequency Characteristics of the Beam with Different Cross Sections Considering the Shear Deformation Induced Rotary Inertia." Applied Sciences 10, no. 15 (July 29, 2020): 5245. http://dx.doi.org/10.3390/app10155245.

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Based on the classical Timoshenko beam theory, the rotary inertia caused by shear deformation is further considered and then the equation of motion of the Timoshenko beam theory is modified. The dynamic characteristics of this new model, named the modified Timoshenko beam, have been discussed, and the distortion of natural frequencies of Timoshenko beam is improved, especially at high-frequency bands. The effects of different cross-sectional types on natural frequencies of the modified Timoshenko beam are studied, and corresponding simulations have been conducted. The results demonstrate that the modified Timoshenko beam can successfully be applied to all beams of three given cross sections, i.e., rectangular, rectangular hollow, and circular cross sections, subjected to different boundary conditions. The consequence verifies the validity and necessity of the modification.
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5

Hilton, Harry H. "Viscoelastic Timoshenko beam theory." Mechanics of Time-Dependent Materials 13, no. 1 (December 2, 2008): 1–10. http://dx.doi.org/10.1007/s11043-008-9075-4.

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6

Shi, Guangyu, and Qiaorong Guo. "On the Appropriate Rotary Inertia in Timoshenko Beam Theory." International Journal of Applied Mechanics 13, no. 04 (May 2021): 2150055. http://dx.doi.org/10.1142/s1758825121500551.

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The rotary inertia defined by Timoshenko to account for the angular velocity effect in flexural vibration of beams has been questioned by some researchers in recent years, and it caused some confusions. This paper discusses the appropriate rotary inertia in Timoshenko beam theory (TBT) and evaluates the influence of the two forms of the rotary inertia on the prediction of the higher-mode frequencies of transversely vibrating beams. Based on the theory of elasticity and variational principle, this work shows that the rotary inertia in the original TBT, defined in terms of the rotation of beam cross-section induced by bending deformation, is variational consistent and is capable of yielding good results of the phase velocities of transversely vibrating beams even in the case where the wavelength of vibrating beams approaches the beam height. On the other hand, the so-called corrected TBT, in which the rotary inertia is defined in terms of the slope of beam deflection, is neither variational consistent nor accurate when the wavelength of vibrating beams approaches the beam height. Therefore, the rotary inertia in TBT defined by Timoshenko is correct and should be used in the dynamic analysis of beams.
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7

Cao, MS, W. Xu, Z. Su, W. Ostachowicz, and N. Xia. "Local coordinate systems-based method to analyze high-order modes of n-step Timoshenko beam." Journal of Vibration and Control 23, no. 1 (August 9, 2016): 89–102. http://dx.doi.org/10.1177/1077546315573919.

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High-frequency transverse vibration of stepped beams has attracted increasing attention in various industrial areas. For an n-step Timoshenko beam, the governing differential equations of transverse vibration have been well established in the literature on the basis of assembling classic Timoshenko beam equations for uniform beam segments. However, solving the governing differential equation has not been resolved well to date, manifested by a computational bottleneck: only the first k modes ( k ≤ 12) are solvable for i-step ( i ≥ 0) Timoshenko beams. This bottleneck diminishes the completeness of stepped Timoshenko beam theory. To address this problem, this study first reveals the root cause of the bottleneck in solving the governing differential equations for high-order modes, and then creates a sophisticated method, based on local coordinate systems, that can overcome the bottleneck to accomplish high-order mode shapes of an n-step Timoshenko beam. The proposed method uses a set of local coordinate systems in place of the conventional global coordinate system to characterize the transverse vibration of an n-step Timoshenko beam. With the method, the local coordinate systems can simplify the frequency equation for the vibration of an n-step Timoshenko beam, making it possible to obtain high-order modes of the beam. The accuracy, capacity, and efficiency of the method based on local coordinate systems in acquiring high-order modes are corroborated using the well-known exact dynamic stiffness method underpinned by the Wittrick-Williams algorithm as a reference. Removal of the bottlenecks in solving the governing differential equations for high-order modes contributes usefully to the completeness of stepped Timoshenko beam theory.
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8

Hutchinson, J. R. "Shear Coefficients for Timoshenko Beam Theory." Journal of Applied Mechanics 68, no. 1 (August 15, 2000): 87–92. http://dx.doi.org/10.1115/1.1349417.

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The Timoshenko beam theory includes the effects of shear deformation and rotary inertia on the vibrations of slender beams. The theory contains a shear coefficient which has been the subject of much previous research. In this paper a new formula for the shear coefficient is derived. For a circular cross section, the resulting shear coefficient that is derived is in full agreement with the value most authors have considered “best.” Shear coefficients for a number of different cross sections are found.
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9

Hutchinson, J. R., and S. D. Zillmer. "On the Transverse Vibration of Beams of Rectangular Cross-Section." Journal of Applied Mechanics 53, no. 1 (March 1, 1986): 39–44. http://dx.doi.org/10.1115/1.3171735.

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An exact solution for the natural frequencies of transverse vibration of free beams with rectangular cross-section is used as a basis of comparison for the Timoshenko beam theory and a plane stress approximation which is developed herein. The comparisons clearly show the range of applicability of the approximate solutions as well as their accuracy. The choice of a best shear coefficient for use in the Timoshenko beam theory is considered by evaluation of the shear coefficient that would make the Timoshenko beam theory match the exact solution and the plane stress solution. The plane stress solution is shown to provide excellent accuracy within its range of applicability.
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10

Ghayesh, Mergen H., Ali Farajpour, and Hamed Farokhi. "Asymmetric Oscillations of AFG Microscale Nonuniform Deformable Timoshenko Beams." Vibration 2, no. 2 (June 14, 2019): 201–21. http://dx.doi.org/10.3390/vibration2020013.

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A nonlinear vibration analysis is conducted on the mechanical behavior of axially functionally graded (AFG) microscale Timoshenko nonuniform beams. Asymmetry is due to both the nonuniform material mixture and geometric nonuniformity. Using the Timoshenko beam theory, the continuous models for translation/rotation are developed via an energy balance. Size-dependence is incorporated via the modified couple stress theory and the rotation via the Timoshenko beam theory. Galerkin’s method of discretization is applied and numerical simulations are conducted for a size-dependent vibration of the AFG microscale beam. Effects of material gradient index and axial change in the cross-sectional area on the force and frequency diagrams are investigated.
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11

Elishakoff, Isaac. "Who developed the so-called Timoshenko beam theory?" Mathematics and Mechanics of Solids 25, no. 1 (August 12, 2019): 97–116. http://dx.doi.org/10.1177/1081286519856931.

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The use of the Google Scholar produces about 78,000 hits on the term “Timoshenko beam.” The question of priority is of great importance for this celebrated theory. For the first time in the world literature, this study is devoted to the question of priority. It is that Stephen Prokofievich Timoshenko had a co-author, Paul Ehrenfest. It so happened that the scientific work of Timoshenko dealing with the effect of rotary inertia and shear deformation does not carry the name of Ehrenfest as the co-author. In his 2002 book, Grigolyuk concluded that the theory belonged to both Timoshenko and Ehrenfest. This work confirms Grigolyuk’s discovery, in his little known biographic book about Timoshenko, and provides details, including the newly discovered letter of Timoshenko to Ehrenfest, which is published here for the first time over a century after it was sent. This paper establishes that the beam theory that incorporates both the rotary inertia and shear deformation as is known presently, with shear correction factor included, should be referred to as the Timoshenko-Ehrenfest beam theory.
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12

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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13

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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14

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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15

Ghandi, Elham, and Ahmed Ali Akbari Rasa. "Analytical Investigation of the Dynamic Response of a Timoshenko Thin-Walled Beam with Asymmetric Cross Section Under Deterministic Loads." Open Civil Engineering Journal 11, no. 1 (October 18, 2017): 802–21. http://dx.doi.org/10.2174/1874149501711010802.

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Inroduction: The objective of the present paper is to analyze dynamic response of the Timoshenko thin-walled beam with coupled bending and torsional vibrations under deterministic loads. The governing differential equations were obtained by using Hamilton’s principle. The Timoshenko beam theory was employed and the effects of shear deformations, Rotary inertia and warping stiffness were included in the present formulations. Dynamic features of underlined beam are obtained using free vibration analysis. Methods: For this purpose, the dynamic stiffness matrix method is used. Application of exact dynamic stiffness matrix method on the movement differential equations led to the issue of nonlinear eigenvalue problem that was solved by using Wittrick–Williams algorithm . Differential equations for the displacement response of asymmetric thin-walled Timoshenko beams subjected to deterministic loads are used for extracting orthogonality property of vibrational modes. Results: Finally the numerical results for dynamic response in a sample of mentioned beams is presented. The presented theory is relatively general and can be used for various kinds of deterministic loading in Timoshenko thin-walled beams.
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16

Zhao, Fei, Ziyu Zhou, Bo Zhou, and Shifeng Xue. "Modeling the thermomechanical behavior of a shape memory polymer Timoshenko beam." Journal of Mechanics 37 (2021): 302–10. http://dx.doi.org/10.1093/jom/ufaa017.

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Abstract In this paper, a new constitutive model that can model the thermomechanical behavior of a shape memory polymer (SMP) Timoshenko beam is established. Based on previous work, the SMP constitutive model is introduced and extended. According to Timoshenko beam theory and the SMP constitutive model, the constitutive model of an SMP beam was established using the Hamiltonian principle and energy principle. The simply supported beam was taken as an example to be solved and simulated. The results give the deflection and rotation properties, viscosity properties and shape memory properties of the SMP beam, and some conclusions are drawn. This work describes the thermomechanical behavior of an SMP Timoshenko beam, and it provides a theoretical basis for the practical application of SMP beams.
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17

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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18

Zhang, Yan, Zhi-Qiang Ni, Lin-Hua Jiang, Lin Han, and Xue-Wei Kang. "Study of the bending vibration characteristic of phononic crystals beam-foundation structures by Timoshenko beam theory." International Journal of Modern Physics B 29, no. 20 (August 5, 2015): 1550136. http://dx.doi.org/10.1142/s0217979215501362.

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Vibration problems wildly exist in beam-foundation structures. In this paper, finite periodic composites inspired by the concept of ideal phononic crystals (PCs), as well as Timoshenko beam theory (TBT), are proposed to the beam anchored on Winkler foundation. The bending vibration band structure of the PCs Timoshenko beam-foundation structure is derived from the modified transfer matrix method (MTMM) and Bloch's theorem. Then, the frequency response of the finite periodic composite Timoshenko beam-foundation structure by the finite element method (FEM) is performed to verify the above theoretical deduction. Study shows that the Timoshenko beam-foundation structure with periodic composites has wider attenuation zones compared with homogeneous ones. It is concluded that TBT is more available than Euler beam theory (EBT) in the study of the bending vibration characteristic of PCs beam-foundation structures with different length-to-height ratios.
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19

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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20

LIU, Y. P., and J. N. REDDY. "A NONLOCAL CURVED BEAM MODEL BASED ON A MODIFIED COUPLE STRESS THEORY." International Journal of Structural Stability and Dynamics 11, no. 03 (June 2011): 495–512. http://dx.doi.org/10.1142/s0219455411004233.

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A nonlocal Timoshenko curved beam model is developed using a modified couple stress theory and Hamilton's principle. The model contains a material length scale parameter that can capture the size effect, unlike the classical Timoshenko beam theory. Both bending and axial deformations are considered, and the Poisson effect is incorporated in the model. The newly developed nonlocal model recovers the classical model when the material length scale parameter and Poisson's ratio are both taken to be zero and the straight beam model when the radius of curvature is set to infinity. In addition, the nonlocal Bernoulli–Euler curved beam model can be realized when the normal cross-section assumption is restated. To illustrate the new model, the static bending and free vibration problems of a simply supported curved beam are solved by directly applying the formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko curved beam model. Also, the differences in both the deflection and rotation predicted by the current and classical Timoshenko model are very large when the beam thickness is small, but they diminish with the increase of the beam height. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the nonlocal model is higher than that by the classical model, and the difference between them is significantly large only for very thin beams. These predicted trends of the size effect at the micron scale agree with those observed experimentally.
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21

Kondo, Kyohei. "Analysis of Potential Energy Release Rate of Composite Laminate Based on Timoshenko Beam Theory." Key Engineering Materials 334-335 (March 2007): 513–16. http://dx.doi.org/10.4028/www.scientific.net/kem.334-335.513.

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The Timoshenko beam theory is used to model each part of cracked beam and to calculate the potential energy release rate. Calculations are given for the double cantilever beam specimen, which is simulated as two separate beams connected elastically along the uncracked interface.
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22

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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23

Senthilnathan, N. R., and K. H. Lee. "Some Remarks on Timoshenko Beam Theory." Journal of Vibration and Acoustics 114, no. 4 (October 1, 1992): 495–97. http://dx.doi.org/10.1115/1.2930290.

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24

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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25

Jaroszewicz, Jerzy, and Krzysztof Łukaszewicz. "Analysis of natural frequency of flexural vibrations of a single-span beam with the consideration of Timoshenko effect." Technical Sciences 3, no. 21 (October 8, 2018): 215–32. http://dx.doi.org/10.31648/ts.2890.

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This paper presents general solution of boundary value problem for constant cross-section Timoshenko beams with four typical boundary conditions. The authors have taken into consideration rotational inertia and shear strain by using the theory of influence by Cauchy function and characteristic series. The boundary value problem of transverse vibration has been formulated and solved. The characteristic equations considering the exact bending theory have been obtained for four cases: the clamped boundary conditions; a simply supported beam and clamped on the other side; a simply supported beam; a cantilever beam. The obtained estimators of fundamental natural frequency take into account mass and elastic characteristics of beams and Timoshenko effect. The results of calculations prove high convergence of the estimators to the exact values which were calculated by Timoshenko who used Bessel functions. Characteristic series having an alternating sign power series show good convergence. As it is shown in the paper, the error lower than 5% was obtained after taking into account only two first significant terms of the series. It was proved that neglecting the Timoshenko effect in case of short beams of rectangular section with the ratio of their length to their height equal 6 leads to the errors of calculated natural frequency: 5%÷12%.
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26

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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27

Iurlaro, Luigi, Marco Gherlone, Massimiliano Mattone, and Marco Di Sciuva. "Experimental assessment of the Refined Zigzag Theory for the static bending analysis of sandwich beams." Journal of Sandwich Structures & Materials 20, no. 1 (June 12, 2016): 86–105. http://dx.doi.org/10.1177/1099636216650614.

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In the present work, for the first time, the accuracy of the Refined Zigzag Theory in reproducing the static bending response of sandwich beams is experimentally assessed. The theory is briefly reviewed and an analytical solution of the equilibrium equations is presented for the boundary and loading conditions under investigation (four-point bending). The experimental campaign is described, including the material characterization and the bending tests. The experimentally measured deflections and axial strains are compared with those provided by Refined Zigzag Theory and by the Timoshenko Beam Theory with an ad hoc shear correction factor. The Refined Zigzag Theory is shown to be more accurate than the Timoshenko Beam Theory, in particular for beams with higher face-to-core thickness and stiffness ratios and with a reduced slenderness.
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28

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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29

Bank, L. C., and C. H. Kao. "Dynamic Response of Thin-Walled Composite Material Timoshenko Beams." Journal of Energy Resources Technology 112, no. 2 (June 1, 1990): 149–54. http://dx.doi.org/10.1115/1.2905723.

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Thin-walled structural members are used extensively in the offshore industry in applications ranging from marine risers to platforms and frames. Advanced fiber composite structural members may offer advantages over their conventional steel counterparts in certain situations. Use of composite members will require modifications to existing structural analysis codes. This paper presents a beam theory for thin-walled composite beams that can be incorporated into existing codes. Timoshenko beam theory is utilized to account for shear deformation effects, which cannot be neglected in composite beams, and for the variability in material properties in different walls of the beam cross section. The theory is applied to the analysis of the free vibration problem and shows the dependence of the natural frequencies and mode shapes on the in-plane properties of the laminates that form the walls of the beam. Forced periodic and forced arbitrary problems are also discussed and the deflected shapes and maximum deflections are shown as functions of wall layups.
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30

Wang, C. M., S. Kitipornchai, C. W. Lim, and M. Eisenberger. "Beam Bending Solutions Based on Nonlocal Timoshenko Beam Theory." Journal of Engineering Mechanics 134, no. 6 (June 2008): 475–81. http://dx.doi.org/10.1061/(asce)0733-9399(2008)134:6(475).

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31

Widera, G. E. O., and W. C. Zheng. "New Higher Order Engineering Beam Theory." Journal of Pressure Vessel Technology 115, no. 3 (August 1, 1993): 325–27. http://dx.doi.org/10.1115/1.2929535.

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A refined engineering theory for beams is presented. It contains higher order effects not present in such refined theories as the one by Timoshenko. A comparison with the latter theory is carried out.
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32

Khiem, Nguyen Tien, An Ninh Thi Vu, and Hai Thanh Tran. "MODAL ANALYSIS OF MULTISTEP TIMOSHENKO BEAM WITH A NUMBER OF CRACKS." Vietnam Journal of Science and Technology 56, no. 6 (December 17, 2018): 772. http://dx.doi.org/10.15625/2525-2518/56/6/12488.

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Modal analysis of cracked multistep Timoshenko beam is accomplished by the Transfer Matrix Method (TMM) based on a closed-form solution for Timoshenko uniform beam element. Using the solution allows significantly simplifying application of the conventional TMM for multistep beam with multiple cracks. Such simplified transfer matrix method is employed for investigating effect of beam slenderness and stepped change in cross section on sensitivity of natural frequencies to cracks. It is demonstrated that the transfer matrix method based on the Timoshenko beam theory is usefully applicable for beam of arbitrary slenderness while the Euler-Bernoulli beam theory is appropriate only for slender one. Moreover, stepwise change in cross-section leads to a jump in natural frequency variation due to crack at the steps. Both the theoretical development and numerical computation accomplished for the cracked multistep beam have been validated by an experimental study
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33

Demirkan, Erol, and Reha Artan. "Buckling Analysis of Nanobeams Based on Nonlocal Timoshenko Beam Model by the Method of Initial Values." International Journal of Structural Stability and Dynamics 19, no. 04 (April 2019): 1950036. http://dx.doi.org/10.1142/s0219455419500366.

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Investigated herein is the buckling of nanobeams based on a nonlocal Timoshenko beam model by the method of initial values within the framework of nonlocal elasticity. Since the nonlocal Timoshenko beam theory is of higher order than the nonlocal Euler–Bernoulli beam theory, it is known to be superior in predicting the small-scale effect. The buckling determinants and critical loads for bars with various kinds of supports are presented. The Carry-Over matrix (Transverse Matrix) is presented and the priorities of the method of initial values are depicted. To the best of the researchers’ knowledge, this is the first work that investigates the buckling of nonlocal Timoshenko beam with the method of initial values.
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34

Hassanati, Bahman, and Marcus Wheel. "Size effects on free vibration of heterogeneous beams." MATEC Web of Conferences 148 (2018): 07003. http://dx.doi.org/10.1051/matecconf/201814807003.

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In this paper the influence of microstructure on the free vibration of geometrically similar heterogeneous beams with free-free boundary conditions was numerically investigated by detailed finite element analysis (FEA) to identify and quantify any effect of beam size on transverse modal frequencies when the microstructural scale is comparable to the overall size. ANSYS Mechanical APDL was used to generate specific unit cells at the microstructural scale comprised of two isotropic materials with different material properties. Unit cell variants containing voids and inclusions were considered. At the macroscopic scale, four beam sizes consisting of one, two, three or four layers of defined unit cells were represented by repeatedly regenerating the unit cell as necessary. In all four beam sizes the aspect ratio was kept constant. Changes to the volume fractions of each material were introduced while keeping the homogenized properties of the beam fixed. The influence of the beam surface morphology on the results was also investigated. The ANSYS results were compared with the analytical results from solution to Timoshenko beam and nonlocal Timoshenko beam as well as numerical results for a Micropolar beam. In nonlocal Timoshenko beams the Eringen’s small length scale coefficients were estimated for some of the studied models. Numerical analyses based on Micropolar theory were carried out to study the modal frequencies and a method was suggested to estimate characteristic length in bending and coupling number via transverse vibration which verifies the use of Micropolar elasticity theory in dynamic analysis.
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35

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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36

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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37

Samayoa, Didier, Alexandro Alcántara, Helvio Mollinedo, Francisco Javier Barrera-Lao, and Christopher René Torres-SanMiguel. "Fractal Continuum Mapping Applied to Timoshenko Beams." Mathematics 11, no. 16 (August 13, 2023): 3492. http://dx.doi.org/10.3390/math11163492.

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In this work, a generalization of the Timoshenko beam theory is introduced, which is based on fractal continuum calculus. The mapping of the bending problem onto a non-differentiable self-similar beam into a corresponding problem for a fractal continuum is derived using local fractional differential operators. Consequently, the functions defined in the fractal continua beam are differentiable in the ordinary calculus sense. Therefore, the non-conventional local derivatives defined in the fractal continua beam can be expressed in terms of the ordinary derivatives, which are solved theoretically and numerically. Lastly, examples of classical beams with different boundary conditions are shown in order to check some details of the physical phenomenon under study.
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38

Molaaghaie-Roozbahani, Masoud, Navid Heydarzadeh, Mostafa Baghani, Amir Hossein Eskandari, and Majid Baniassadi. "An Investigation on Thermomechanical Flexural Response of Shape-Memory-Polymer Beams." International Journal of Applied Mechanics 08, no. 05 (July 2016): 1650063. http://dx.doi.org/10.1142/s1758825116500630.

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In this paper, the predictions of different beam theories for the behavior of a shape memory polymer (SMP) beam in different steps of a thermomechanical cycle are compared. Employing the equilibrium equations, the governing equations of the deflection of a SMP beam in the different steps of a thermomechanical cycle, for higher order beam theories (Timoshenko Beam Theory and von-Kármán Beam Theory), are developed. For the Timoshenko Beam Theory, a closed form analytical solution for various steps of the thermomechanical cycle is presented. The nonlinear governing equations in von-Kármán Beam theory are numerically solved. Results reveal that in the various beam length to beam thickness ratios, one of the beam theories provides the most accurate results. In other words, employing the Euler–Bernoulli Beam Theory for developing the governing equations, especially in the large and small beam length to beam thickness ratios, leads to erroneous results.
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39

Ghadyani, Ghasem, Mojtaba Akbarzade, and Andreas Öchsner. "On the Finite Element Modelling and Simulation of Carbon Nanotubes." Key Engineering Materials 607 (April 2014): 55–61. http://dx.doi.org/10.4028/www.scientific.net/kem.607.55.

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In this paper, two different beam elements (i.e. according to the Bernoulli beam and Timoshenko beam theory) for the modeling of the behavior of carbon nanotubes are applied. Finite element models are developed for this study with variation of chirality for both zig-zag and armchair configurations of CNTs. The deformations from the finite element simulations are subsequently used to predict the elastic stiffness and the critical buckling load in terms of material and geometric parameters. Furthermore, the dependence of mechanical properties on the kind of beam element and the mesh density is also compared. Based on the obtained results, Youngs modulus and critical buckling load of structures using Timoshenko beams are clearly lower than the Bernoulli beam approach for all chiralities.
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40

Gordaninejad, F., and A. Ghazavi. "Effect of Shear Deformation on Bending of Laminated Composite Beams." Journal of Pressure Vessel Technology 111, no. 2 (May 1, 1989): 159–64. http://dx.doi.org/10.1115/1.3265652.

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A higher-order shear deformation beam theory is utilized to analyze the bending of thick laminated composite beams. This theory accounts for parabolic distribution of shear strain through the thickness of the beam. The predicted displacements show improvement over the Bresse-Timoshenko beam theory. Mixed finite element results are obtained for those cases where closed-form solutions are not available. The finite element and exact solutions are in close agreement. Numerical results are presented for single, two and three-layer beams under uniform and sinusoidal distributed transverse loadings.
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41

Hodges, Dewey H. "Asymptotic Derivation of Shear Beam Theory from Timoshenko Theory." Journal of Engineering Mechanics 133, no. 8 (August 2007): 957–61. http://dx.doi.org/10.1061/(asce)0733-9399(2007)133:8(957).

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42

Santos, J. V. Araújo dos, and J. N. Reddy. "Vibration of Timoshenko Beams Using Non-classical Elasticity Theories." Shock and Vibration 19, no. 3 (2012): 251–56. http://dx.doi.org/10.1155/2012/307806.

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This paper presents a comparison among classical elasticity, nonlocal elasticity, and modified couple stress theories for free vibration analysis of Timoshenko beams. A study of the influence of rotary inertia and nonlocal parameters on fundamental and higher natural frequencies is carried out. The nonlocal natural frequencies are found to be lower than the classical ones, while the natural frequencies estimated by the modified couple stress theory are higher. The modified couple stress theory results depend on the beam cross-sectional size while those of the nonlocal theory do not. Convergence of both non-classical theories to the classical theory is observed as the beam global dimension increases.
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43

Chen, W. R., and L. M. Keer. "Transverse Vibrations of a Rotating Twisted Timoshenko Beam Under Axial Loading." Journal of Vibration and Acoustics 115, no. 3 (July 1, 1993): 285–94. http://dx.doi.org/10.1115/1.2930347.

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Transverse bending vibrations of a rotating twisted beam subjected to an axial load and spinning about its axial axis are established by using the Timoshenko beam theory and applying Hamilton’s Principle. The equations of motion of the twisted beam are derived in the twist nonorthogonal coordinate system. The finite element method is employed to discretize the equations of motion into time-dependent ordinary differential equations that have gyroscopic terms. A symmetric general eigenvalue problem is formulated and used to study the influence of the twist angle, rotational speed, and axial force on the natural frequencies of Timoshenko beams. The present model is useful for the parametric studies to understand better the various dynamic aspects of the beam structure affecting its vibration behavior.
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44

Yin, Shuohui, Yang Deng, Gongye Zhang, Tiantang Yu, and Shuitao Gu. "A new isogeometric Timoshenko beam model incorporating microstructures and surface energy effects." Mathematics and Mechanics of Solids 25, no. 10 (June 3, 2020): 2005–22. http://dx.doi.org/10.1177/1081286520917998.

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A new isogeometric Timoshenko beam model is developed using a modified couple stress theory (MCST) and a surface elasticity theory. The MCST is wildly used to capture microstructure effects that includes only one material length scale parameter, while the Gurtin–Murdoch surface elasticity theory containing three surface elasticity constants is employed to approximate the nature of surface energy effects. A new effective computational approach is presented for the current nonclassical Timoshenko beam model based on isogeometric analysis with high-order continuity basis functions of non-uniform rational B-splines, which effectively fulfills the higher continuity requirements in MCST. To validate the new approach and quantitatively illustrate both the microstructure and surface energy effects, the numerical results obtained from the developed approach for static deflection and natural frequencies of beams are compared with the analytical results available in the literature. Numerical results reveal that both the microstructure effect and surface energy effect should be considered in very thin beams, which also explains the size-dependent behavior.
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45

Falach, Lior, Roberto Paroni, and Paolo Podio-Guidugli. "A justification of the Timoshenko beam model through Γ-convergence." Analysis and Applications 15, no. 02 (January 25, 2017): 261–77. http://dx.doi.org/10.1142/s0219530515500207.

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We validate the Timoshenko beam model as an approximation of the linear-elasticity model of a three-dimensional beam-like body. Our validation is achieved within the framework of [Formula: see text]-convergence theory, in two steps: firstly, we construct a suitable sequence of energy functionals; secondly, we show that this sequence [Formula: see text]-converges to a functional representing the energy of a Timoshenko beam.
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46

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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47

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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48

Norouzzadeh, Amir, Mohammad Faraji Oskouie, Reza Ansari, and Hessam Rouhi. "Integral and differential nonlocal micromorphic theory." Engineering Computations 37, no. 2 (August 19, 2019): 566–90. http://dx.doi.org/10.1108/ec-01-2019-0008.

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Purpose This paper aims to combine Eringen’s micromorphic and nonlocal theories and thus develop a comprehensive size-dependent beam model capable of capturing the effects of micro-rotational/stretch/shear degrees of freedom of material particles and nonlocality simultaneously. Design/methodology/approach To consider nonlocal influences, both integral (original) and differential versions of Eringen’s nonlocal theory are used. Accordingly, integral nonlocal-micromorphic and differential nonlocal-micromorphic beam models are formulated using matrix-vector relations, which are suitable for implementing in numerical approaches. A finite element (FE) formulation is also provided to solve the obtained equilibrium equations in the variational form. Timoshenko micro-/nano-beams with different boundary conditions are selected as the problem under study whose static bending is addressed. Findings It was shown that the paradox related to the clamped-free beam is resolved by the present integral nonlocal-micromorphic model. It was also indicated that the nonlocal effect captured by the integral model is more pronounced than that by its differential counterpart. Moreover, it was revealed that by the present approach, the softening and hardening effects, respectively, originated from the nonlocal and micromorphic theories can be considered simultaneously. Originality/value Developing a hybrid size-dependent Timoshenko beam model including micromorphic and nonlocal effects. Considering the nonlocal effect based on both Eringen’s integral and differential models proposing an FE approach to solve the bending problem, and resolving the paradox related to nanocantilever.
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Kumar, Saurabh. "Natural Frequencies of Beams with Axial Material Gradation Resting on Two Parameter Elastic Foundation." Trends in Sciences 19, no. 6 (March 3, 2022): 3048. http://dx.doi.org/10.48048/tis.2022.3048.

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Free vibration analysis is carried out on axially inhomogeneous beams resting on Winkler-Pasternak elastic foundation. The material properties of the beam like Young’s modulus, modulus of rigidity and material density are considered to be varying along the length direction following constant, linear and exponential material models. The beam is subjected to different combinations of clamped and simply supported boundary conditions. The formulation is based on Timoshenko beam theory and energy method along with Hamilton’s principle is used to derive the governing equations. The effect of material gradation and the 2 parameters of elastic foundation on the natural frequencies are studied in detail. The present results are validated by comparing them with established ones and satisfactory matching is observed. HIGHLIGHTS Free vibration behavior of axially functionally graded beams in investigated Three different material models and three boundary conditions are considered Effect of two parameter elastic foundation is considered Timoshenko beam theory and Rayleigh Ritz method are employed It is found that the stiffness of elastic foundation significantly affects the natural frequency of the beam GRAPHICAL ABSTRACT
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50

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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