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Artykuły w czasopismach na temat "Non-classical Euler-Bernoulli beam"
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Samayoa, Didier, Andriy Kryvko, Gelasio Velázquez i Helvio Mollinedo. "Fractal Continuum Calculus of Functions on Euler-Bernoulli Beam". Fractal and Fractional 6, nr 10 (29.09.2022): 552. http://dx.doi.org/10.3390/fractalfract6100552.
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A new approach for solving the fractal Euler-Bernoulli beam equation is proposed. The mapping of fractal problems in non-differentiable fractals into the corresponding problems for the fractal continuum applying the fractal continuum calculus (FdH3-CC) is carried out. The fractal Euler-Bernoulli beam equation is derived as a generalization using FdH3-CC under analogous assumptions as in the ordinary calculus and then it is solved analytically. To validate the spatial distribution of self-similar beam response, three different classical beams with several fractal parameters are analysed. Some mechanical implications are discussed.
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Stempin, Paulina, i Wojciech Sumelka. "Dynamics of Space-Fractional Euler–Bernoulli and Timoshenko Beams". Materials 14, nr 8 (7.04.2021): 1817. http://dx.doi.org/10.3390/ma14081817.
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This paper investigates the dynamics of the beam-like structures whose response manifests a strong scale effect. The space-Fractional Euler–Bernoulli beam (s-FEBB) and space-Fractional Timoshenko beam (s-FTB) models, which are suitable for small-scale slender beams and small-scale thick beams, respectively, have been extended to a dynamic case. The study provides appropriate governing equations, numerical approximation, detailed analysis of free vibration, and experimental validation. The parametric study presents the influence of non-locality parameters on the frequencies and shape of modes delivering a depth insight into a dynamic response of small scale beams. The comparison of the s-FEBB and s-FTB models determines the applicability limit of s-FEBB and indicates that the model (also the classical one) without shear effect and rotational inertia can only be applied to beams significantly slender than in a static case. Furthermore, the validation has confirmed that the fractional beam model exhibits very good agreement with the experimental results existing in the literature—for both the static and the dynamic cases. Moreover, it has been proven that for fractional beams it is possible to establish constant parameters of non-locality related to the material and its microstructure, independent of beam geometry, the boundary conditions, and the type of analysis (with or without inertial forces).
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Yin, Shuohui, Zhibing Xiao, Gongye Zhang, Jingang Liu i Shuitao Gu. "Size-Dependent Buckling Analysis of Microbeams by an Analytical Solution and Isogeometric Analysis". Crystals 12, nr 9 (9.09.2022): 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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Zhang, GY, i X.-L. Gao. "A new Bernoulli–Euler beam model based on a reformulated strain gradient elasticity theory". Mathematics and Mechanics of Solids 25, nr 3 (25.11.2019): 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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Ghorbanpourarani, A., M. Mohammadimehr, A. Arefmanesh i 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, nr 3 (19.10.2009): 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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Ishaquddin, Md, i S. Gopalakrishnan. "Differential quadrature-based solution for non-classical Euler-Bernoulli beam theory". European Journal of Mechanics - A/Solids 86 (marzec 2021): 104135. http://dx.doi.org/10.1016/j.euromechsol.2020.104135.
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Mei, C. "Vibrations in a spatial K-shaped metallic frame: an exact analytical study with experimental validation". Journal of Vibration and Control 23, nr 19 (20.01.2016): 3147–61. http://dx.doi.org/10.1177/1077546315627085.
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In a spatial K-shaped metallic frame, there exist in- and out-of-plane bending, axial, and torsional vibrations. A wave-based vibration analysis approach is applied to obtain free and forced vibration responses in a space frame. In order to validate the analytical approach, a steel K-shaped space frame was built by welding four beam elements of rectangular and square cross-section together. Bending vibrations are modeled using both the classical Euler–Bernoulli theory and the advanced Timoshenko theory. This allows the effects of rotary inertia and shear distortion, which were neglected in the classical Euler–Bernoulli theory, to be studied. In addition, the effect of torsional rigidity adjustment for structures of rotationally non-symmetric cross-section is also examined.
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Zhang, G. Y., X. L. Gao, C. Y. Zheng i C. W. Mi. "A non-classical Bernoulli-Euler beam model based on a simplified micromorphic elasticity theory". Mechanics of Materials 161 (październik 2021): 103967. http://dx.doi.org/10.1016/j.mechmat.2021.103967.
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König, Paul, Patrick Salcher, Christoph Adam i Benjamin Hirzinger. "Dynamic analysis of railway bridges exposed to high-speed trains considering the vehicle–track–bridge–soil interaction". Acta Mechanica 232, nr 11 (2.10.2021): 4583–608. http://dx.doi.org/10.1007/s00707-021-03079-1.
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AbstractA new semi-analytical approach to analyze the dynamic response of railway bridges subjected to high-speed trains is presented. The bridge is modeled as an Euler–Bernoulli beam on viscoelastic supports that account for the flexibility and damping of the underlying soil. The track is represented by an Euler–Bernoulli beam on viscoelastic bedding. Complex modal expansion of the bridge and track models is performed considering non-classical damping, and coupling of the two subsystems is achieved by component mode synthesis (CMS). The resulting system of equations is coupled with a moving mass–spring–damper (MSD) system of the passing train using a discrete substructuring technique (DST). To validate the presented modeling approach, its results are compared with those of a finite element model. In an application, the influence of the soil–structure interaction, the track subsystem, and geometric imperfections due to track irregularities on the dynamic response of an example bridge is demonstrated.
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Yin, Shuohui, Yang Deng, Tiantang Yu, Shuitao Gu i Gongye Zhang. "Isogeometric analysis for non-classical Bernoulli-Euler beam model incorporating microstructure and surface energy effects". Applied Mathematical Modelling 89 (styczeń 2021): 470–85. http://dx.doi.org/10.1016/j.apm.2020.07.015.
Pełny tekst źródłaRozprawy doktorskie na temat "Non-classical Euler-Bernoulli beam"
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Migotto, Dionéia. "Autofunções e Frequências de Vibração do Modelo Euler-Bernoulli para Vigas Não-Clássicas". Universidade Federal de Santa Maria, 2011. http://repositorio.ufsm.br/handle/1/9971.
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This paper presents a methodology for determining eigenfunctions and frequencies
of the Euler-Bernoulli model for elastic beams that can include damping and devices
located at intermediate or end points of the beam. The eigenfunctions or vibration
modes of the beam are obtained by using solution basis generated by the dynamic
solution of a fourth-order differential equation, through a block matrix formulation
of the boundary and compatibility conditions. The use of the dynamic basis has been
often used to reduce the calculations in obtaining the modes and frequencies. Forced
responses are obtained with the Galerkin method by modifying the classical modal
analysis with the inclusion of new conditions of orthogonality between modes that
are suitable for problems with viscous damping or non-classical boundary conditions.Este trabalho apresenta uma metodologia para determinar as autofunções e as frequências de um modelo Euler-Bernoulli para vigas elásticas que podem incluir amortecimento e dispositivos localizados num ponto intermediário ou nos extremos da viga. As autofunções ou modos de vibração da viga são obtidos usando uma base de solução gerada pela solução dinâmica de uma equação diferencial de quarta ordem, através de uma formulação matricial em blocos para as condições de contorno e de compatibilidade. O uso da base dinâmica tem sido frequentemente utilizada para reduzir os cálculos na obtenção dos modos e das frequências. Respostas forçadas são obtidas usando o método de Galerkin, modificando a análise modal clássica com a inclusão de novas condições de ortogonalidade entre os modos que são adequadas para problemas com amortecimento viscoso ou com condições de contorno não-clássicas
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Ishaquddin, Mohammed. "Numerical solution of non-classical beam and plate theories using di erential quadrature method". Thesis, 2019. https://etd.iisc.ac.in/handle/2005/4586.
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For effcient design of the nano/micro scale structural systems, detailed analysis and through
understanding of size-dependent mechanical behaviour at nano/micro scale is very critical. Various
approaches have been used to investigate the mechanical behaviour of small scale structures,
for instance, experimental approach, atomistic and molecular dynamics simulations, multi-scale
modelling, etc. However, the application of these methods for practical problems have their
own limitations, some are very cumbersome and expensive, others need high computational resources
and remaining are mathematically involved. The non-classical continuum theories with
micro-structural behaviour have proven to be very efficient alternative, which assures reasonable
accuracy with less complexity and computational efforts as compared to other approaches.
The non-classical theories are governed by higher order differential equations and introduce
additional degrees of freedom (related to curvature and triple derivative of displacements) and
material parameters to account for scale effects. A considerable amount of analytical work on
beams and plates is conducted based on these theories, however, numerical treatment is limited
to only few speci fic applications.
The primary objective of this research is to develop a comprehensive set of novel and effcient
differential quadrature-based elements for non-classical Euler-Bernoulli beam and Kirchhoff
plate theories. Both strong and weak form differential quadrature elements are developed,
which are fundamentally different in their formulation. The strong form elements are formulated
using the governing equation and stress resultant equations, and the weak form elements are
based on the variational principles. Lagrange interpolations are used to formulate the strong
form beam elements, while the weak form beam elements are constructed for both Lagrange
and Hermite interpolations. The plate elements (strong and weak) are developed using two
different combinations of interpolation functions in the orthogonal directions, in the first choice,
Lagrange interpolations are assumed in both orthogonal directions and in the second case
Lagrange interpolation are assumed in one direction and Hermite in the another. The capability
of these elements is demonstrated through non-classical Mindlin's simpli ed fi rst and second
strain Euler-Bernoulli beam / Kirchhoff plate theories, which are governed by sixth
and eighth order differential equations, respectively. The accuracy and applicability of the
beam elements is veri fied for bending, free-vibration, stability, dynamic/transient and wave
propagation analysis, and the plate elements for bending, free-vibration and stability analysis.
The strong form differential quadrature element developed for first strain gradient Euler-
Bernoulli theory demonstrated excellent agreement with the exact solutions with less number of
nodes for static, free vibration and buckling analysis of prismatic and non-prismatic beams for
different combinations of boundary conditions, loading and length scale parameters. Similar
performance was demonstrated by the weak form quadrature element which was formulated
using Hermite interpolation functions. The Lagrange interpolation based weak form quadrature
element exhibited inferior performance as compared to the above two elements, and needed more
number of nodes to obtain the accurate results. Good performance was shown by both strong
and weak form differential quadrature elements for dynamic and wave propagation analysis.
With fewer number of elements and nodes the velocity response and the group speeds were
predicted accurately using these elements. Based on the finding it was concluded that the
beam elements produced accurate results with reasonable number of nodes and can be effciently
applied for different analysis of non-classical Euler-Bernoulli prismatic and non-prismatic beams
for any choice of loading, boundary conditions and length scale parameters. The performance of
strong and weak form beam elements developed for second strain gradient Euler-Bernoulli beam
theory was also validated for static, free vibration, stability, dynamic and wave propagation
analysis. Similar performance was demonstrated by the weak and strong form beam elements
developed for second strain gradient Euler-Bernoulli beam theory.
The strong form elements developed for fi rst strain gradient Kirchhoff plate theory demonstrated
excellent performance for static bending, free vibration and stability analysis. Deflections,
frequencies and buckling loads obtained using the single element with fewer number of
nodes compare well with the exact solutions for different loading, boundary conditions and
length scale values. The results obtained using the weak form quadrature elements also compared
well with available literature results, however, for the plates which include one or more
clamped edges need more number of nodes to obtain converged solutions as compared to the
strong form elements. This aspect of weak form quadrature elements needs further investigation. Similar set of strong and weak form DQ elements developed for second strain gradient
Kirchhoff plate theory also exhibited similar performance for static bending, free vibration and
stability analysis.
Streszczenia konferencji na temat "Non-classical Euler-Bernoulli beam"
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Claeyssen, Julio R., Rosemaira Dalcin Copetti i Teresa Tsukazan. "Matrix Vibration Formulation of Damped Multi-Span Beams". W ASME 2006 International Mechanical Engineering Congress and Exposition. ASMEDC, 2006. http://dx.doi.org/10.1115/imece2006-14933.
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In this work we consider segmented Euler-Bernoulli beams that can have an internal damping of the type Kelvin-Voight and external viscous damping at the discontinuities of the sections. In the literature, the study of this kind of beams has been sufficiently studied with proportional damping only, however the effects of non-proportional damping has been little studied in terms of modal analysis. The obtaining of the modes of segmented beams can be accomplished with a the state space methodology or with the classical Euler construction of responses. Here, we follow a newtonian approach with the use of the impulse response of beams subject both types of damping. The use of the dynamical basis, generated by the fundamental solution of a differential equation of fourth order, allows to formulate the eigenvalue problem and the shapes of the modes in a compact manner. For this, we formulate in a block manner the boundary conditions and intermediate conditions at the beam and values of the fundamental matrix at the ends of the beam and in the points intermediate. We have chosen a basis generated by a fundamental response and it derivatives. The elements of this basis has the same shape with a convenient translation for each segment. This choice reduce computations with the number of constants to be determined to find only the ones that correspond to the first segment. The eigenanalysis will allow to study forced responses of multi-span Euler-Bernoulli beams under classical and non-classical boundary conditions as well as multi-walled carbon nanotubes (MWNT) that are modelled as an assemblage of Euler-Bernoulli beams connected throughout their length by springs subject to van der Waals interaction between any two adjacent nanotubes.
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Taati, E., M. Nikfar i M. T. Ahmadian. "Formulation for Static Behavior of the Viscoelastic Euler-Bernoulli Micro-Beam Based on the Modified Couple Stress Theory". W ASME 2012 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/imece2012-86591.
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In this work an analytical solution is presented for a viscoelastic micro-beam based on the modified couple stress theory which is a non-classical theory in continuum mechanics. The modified couple stress theory has the ability to consider small size effects in micro-structures. It is strongly emphasized that without considering these effects in such structures the solution will be wrong and not suitable for designing systems in micro-scales. In this study correspondence principle is used for deriving constitutive equations for viscoelastic material based on the modified couple stress theory. Governing equilibrium equations are obtained by considering an element of micro-beam. Closed-form solution for the static deflection of simply supported micro-beam is presented. Numerical results show that when the size of system is near the length scale parameter, the classical response will intensely be deviated from the correct solution observed in laboratories contrary to the modified couple stress which reflects the size effects.
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Cuhat, Daniel, i Patricia Davies. "An Experimental Approach to the Design of PVDF Modal Sensors". W ASME 2001 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/detc2001/vib-21537.
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Abstract The principle of modal sensing is based on the use of a shaped PVDF piezoelectric film measuring strains on the surface of a bending beam and acting as a modal filter. So far, the use of this type of sensors has remained confined to studies involving uniform structures with classical boundary conditions. The goal of this paper is to present an experimental methodology for the design of a shaped modal sensor applicable to an non-uniform Euler-Bernoulli beam with arbitrary boundary conditions. This approach is illustrated with test data collected on a cantilever beam structure with a laser Doppler velocimeter.
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Kheiri, Mojtaba, Michael P. Païdoussis i Giorgio Costa del Pozo. "Dynamics of a Pipe Conveying Fluid Flexibly Supported at the Ends". W ASME 2014 Pressure Vessels and Piping Conference. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/pvp2014-28335.
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The subject of this paper is the study of dynamics and stability of a pipe flexibly supported at its ends and conveying fluid. First, the equation of motion of the system is derived via the extended form of Hamilton’s principle for open systems. In the derivation, the effect of flexible supports, modelled as linear translational and rotational springs, is appropriately considered in the equation of motion rather than in the boundary conditions. The resulting equation of motion is then discretized via the Galerkin method in which the eigenfunctions of a free-free Euler-Bernoulli beam are utilized. Thus, a general set of second-order ordinary differential equations emerge, in which, by setting the stiffness of the end-springs suitably, one can readily investigate the dynamics of various systems with either classical or non-classical boundary conditions.
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Saadatnia, Zia, i Ebrahim Esmailzadeh. "Nonlinear Forced Vibration Analysis of a Non-Local Carbon Nanotube Carrying Intermediate Mass". W ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/detc2015-46855.
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The aim of this study is to model and investigate the nonlinear transversal vibration of a carbon nanotube carrying an intermediate mass along the structure considering the nonlocal and non-classical theories. Due to the application of the proposed system in sensors, actuators, mass detection units among others, the analysis of forced vibration of such systems is of an important task being considered here. The governing equation of motion is developed by combining the Euler-Bernoulli beam theory and the Eringen non-local theory. The Galerkin approach is employed to obtain the governing differential equation of the system and the transient beam response for the clamped-hinged boundary condition. A strong perturbation method is utilized to solve the equation obtained and the system responses subjected to a harmonic excitation is examined. The steady-state motion is studied and the frequency response in an analytical form is obtained. Finally, results are evaluated for some numerical parameter values and their effect on the frequency responses are presented and fully discussed.
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Boivin, Nicolas, Christophe Pierre i Steven W. Shaw. "Non-Linear Normal Modes, Invariance, and Modal Dynamics Approximations of Non-Linear Systems". W ASME 1993 Design Technical Conferences. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/detc1993-0028.
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Abstract Non-linear systems are here tackled in a manner directly inherited from linear ones, i.e., by denning proper normal modes of motion. These are defined in terms of invariant manifolds in the system’s phase space, on which the uncoupled system dynamics can be studied. Two different methodologies which were previously developed to derive the non-linear normal modes of continuous systems — one based on a purely continuous approach, and one based on a discretized approach where the theory developed for discrete systems can be applied — are simultaneously applied to the same study case — an Euler-Bernoulli beam constrained by a non-linear spring —, and compared as regards accuracy and reliability, resulting in the abandonment of the continuous approach for lack of reliability. Numerical simulations of purely non-linear modal motions are performed using these approaches, and compared to simulations of equations obtained by a classical projection onto the linear modes. The invariance properties of the nonlinear normal modes are demonstrated, and it is also found that, for a purely non-linear modal motion, the invariant manifold approach achieves the same accuracy as that obtained using several linear normal modes, but with significantly reduced computational cost. This is mainly due to the possibility of obtaining high-order accuracy in the dynamics by solving only one non-linear ordinary differentia] equation.
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João Fernandes da Silva, Lucas Allende Dias do Nascimento i Simone dos Santos Hoefel. "Free vibration analysis of Euler-Bernoulli beams under non-classical boundary conditions". W IX Congresso Nacional de Engenharia Mecânica. Rio de Janeiro, Brazil: ABCM Associação Brasileira de Engenharia e Ciências Mecânicas, 2016. http://dx.doi.org/10.20906/cps/con-2016-1053.
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Chen, Boyang, Simon Jones i Matt Riley. "Stochastic Finite Element Modeling of Laminated Fiber-Reinforced Composite Beams Under Transverse Loading". W ASME 2021 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/imece2021-69851.
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Abstract It is common in analytic fiber-reinforced composite theory to assume uniformly distributed material properties across the fiber direction to minimize computational expense. However, manufacturing processes introduce imperfections during the construction of composite materials, such as localized delamination, non-uniform distribution in matrix and fibers, pre-existing stress, and tolerance issues [1]. These imperfections make it more difficult to predict the behavior of composite materials under loading. As a result, manufacturers and designers must use conservative estimates of material strength. This study aims to quantify the uncertainty in laminated fiber-reinforced composite beams subjected to cantilever loads on a macroscopic scale and to provide an all-inclusive introduction to stochastic composite modeling using the finite element method. This introduction is intended for upper undergraduates or new graduate students how are already familiar with structural mechanics and the finite element method. The goal of the paper is to introduce the key topics related to stochastic composite modeling and have validation material with which they can develop and verify custom finite element code. The system investigated herein is a composite cantilever beam subjected to a transverse tip displacement. Classical Lamination Theory (CLT) is first employed to predict the transverse tip displacement of a beam composed of four lamina at adjustable fiber orientations. A finite element model is then created using a CLT approach to simulate the composite beam’s deformation under tip loading. The Euler-Bernoulli beam elements contain two nodes with two degrees of freedom each: transverse deflection and rotation. These elements are relatively simplistic relative to other composite finite elements, but are sufficient to demonstrate the effect of stochastic material property variation on the overall response of the beam without obfuscating the approach. The finite element results are validated against the analytic predictions for multiple fiber direction layups to ensure the numerical predictions are accurate. The stochastic approach for varying material properties is then added to the validated finite element code. A Karhunen–Loève expansion of a modified exponential kernel is used to produce spatially-varying elastic modulus profiles for each lamina in the composite beam. The predicted tip displacement for the beam with varying properties is computed, and then CLT is used to determine the effective uniform elastic modulus that is required to produce the same tip displacement. This comparison allows the reader to quantify the impact of the spatially varying properties to a single design property: the effective flexural modulus. A Monte Carlo simulation of 1000 composite beams is then used to determine the statistical distribution of the effective flexural modula. Results suggest that the “averaging effect” of bonding multiple laminas with varying material properties together into composite beams produces effective flexural modula for the beams that do not vary as significantly as the laminas’ elastic modula. Standard deviations of the effective flexural modula are found to be an order of magnitude smaller than that of the variation imposed on the laminas’ elastic modulus.
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Nunes, Afonso Willian, Samuel da Silva, Paulo Gonçalves i Jean-Mathieu Mencik. "Stable computation of mode shapes of uniform Euler-Bernoulli beams subject to classical and non-classical boundary conditions via Lie symmetries". W 8th International Symposium on Solid Mechanics. ABCM, 2022. http://dx.doi.org/10.26678/abcm.mecsol2022.msl22-0132.
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