Showing papers on "Timoshenko beam theory published in 2005"
TL;DR: In this article, the effect of microstructure on microcantilever bending stiffness was examined using an isotropic Hooke's law constitutive relationship, compared to a model based upon a micropolar elasticity constitutive model.
Abstract: This work examines the effect of microstructure upon microcantilever bending stiffness. An existing beam theory model, based upon an isotropic Hooke's law constitutive relationship, is compared to a model based upon a micropolar elasticity constitutive model. The micropolar approach introduces a bending stiffness relation which is a function of any two independent elastic constants of the Hooke's law model (e.g., the elastic modulus and the Poisson's ratio), and an additional material constant (called γ). A consequence of the additional material constant is the prediction of an increased bending stiffness as the cantilever thickness decreases—a stiffening due to the material microstructure which becomes measurable at micron-order thicknesses. Polypropylene microcantilevers, which have a non-homogeneous microstructure due to their semi-crystalline nature, were fabricated via injection molding. A nanoindenter was used to measure their stiffness. The nanoindenter-determined stiffness values, which include the effect of the additional micropolar material constant, are compared to stiffness values obtained from beam theory. The nanoindenter stiffness values are seen to be at least four times larger than the beam theory stiffness predictions. This stiffening effect has relevance in future MEMS applications which employ materials with non-homogeneous microstructures instead of the conventional MEMS materials (e.g., silicon, silicon nitride), which have a very uniform microstructure.
741 citations
TL;DR: In this paper, the small-scale effect of wave propagation dispersion relation on carbon nanotubes (CNTs) wave numbers and diameters was investigated with two nonlocal continuum mechanics models: elastic Euler-Bernoulli and Timoshenko beam models.
Abstract: Wave propagation in carbon nanotubes (CNTs) is studied with two nonlocal continuum mechanics models: elastic Euler-Bernoulli and Timoshenko beam models [Philos. Mag. 41, 744 (1921)]. The small-scale effect on CNTs wave propagation dispersion relation is explicitly revealed for different CNTs wave numbers and diameters by theoretical analyses and numerical simulations. The asymptotic phase velocities and frequency are also derived from nonlocal continuum mechanics. The scale coefficient in nonlocal continuum mechanics is roughly estimated for CNTs from the obtained asymptotic frequency. In addition, the applicability and comparison of the two nonlocal elastic beam models to CNTs wave propagation are explored through numerical simulations. The research findings are proved effective in predicting small-scale effect on CNTs wave propagation with a qualitative validation study based on the published experimental reports in this field.
609 citations
TL;DR: In this article, the authors present a study on the flexural wave propagation in a single-walled carbon nanotube through the use of the continuum mechanics and the molecular dynamics simulation based on the Terroff-Brenner potential.
Abstract: The paper presents the study on the flexural wave propagation in a single-walled carbon nanotube through the use of the continuum mechanics and the molecular dynamics simulation based on the Terroff-Brenner potential. The study focuses on the wave dispersion caused not only by the rotary inertia and the shear deformation in the model of a traditional Timoshenko beam, but also by the nonlocal elasticity characterizing the microstructure of carbon nanotube in a wide frequency range up to THz. For this purpose, the paper starts with the dynamic equation of a generalized Timoshenko beam made of the nonlocal elastic material, and then gives the dispersion relations of the flexural wave in the nonlocal elastic Timoshenko beam, the traditional Timoshenko beam and the Euler beam, respectively. Afterwards, it presents the molecular dynamics simulations for the flexural wave propagation in an armchair (5,5) and an armchair (10,10) single-walled carbon nanotubes for a wide range of wave numbers. The simulation results show that the Euler beam holds for describing the dispersion of flexural waves in the two single-walled carbon nanotubes only when the wave number is small. The Timoshenko beam provides a better prediction for the dispersion of flexural waves in the two single-walled carbon nanotubes when the wave number becomes a little bit large. Only the nonlocal elastic Timoshenko beam is able to predict the decrease of phase velocity when the wave number is so large that the microstructure of carbon nanotubes has a significant influence on the flexural wave dispersion.
479 citations
Book•
17 Nov 2005
TL;DR: In this article, the Equations of Motion of Open/Closed Cross-Section Beams and Linear Beam Theory have been studied in the context of thin-walled beams.
Abstract: Kinematics of Thin Walled Beams.- The Equations of Motion of Open/Closed Cross-Section Beams.- Additional Equations of the Linear Beam Theory.- Several Theorems in Linear Thin-Walled Beam Theory.- Free Vibration.- Dynamic Response to Time-Dependent External Excitation.- Thin-Walled Beams Carrying Stores.- Rotating Thin-Walled Anisotropic Beams.- Spinning Thin-Walled Anisotropic Beams.- Thermally Induced Vibration and Control of Spacecraft Booms.- Aeroelasticity of Thin-Walled Aircraft Wings.- Open-Section Beams.
204 citations
TL;DR: The exponential decay of the solution is shown by using a method developed by Z. Liu and S. Zheng and their collaborators, which is very different from some others in the literature, such as the traditional energy method.
179 citations
TL;DR: In this paper, wave reflection, transmission, and propagation in Timoshenko beams together with wave analysis of vibrations in timoshenko beam structures are derived, and the transmission and reflection matrices for various discontinuities on a Timoshenko beam are derived.
Abstract: This paper concerns wave reflection, transmission, and propagation in Timoshenko beams together with wave analysis of vibrations in Timoshenko beam structures. The transmission and reflection matrices for various discontinuities on a Timoshenko beam are derived. Such discontinuities include general point supports, boundaries, and changes in section. The matrix relations between the injected waves and externally applied forces and moments are also derived. These matrices can be combined to provide a concise and systematic approach to vibration analysis of Timoshenko beams or complex structures consisting of Timoshenko beam components. The approach is illustrated with several numerical examples.
128 citations
TL;DR: In this article, the vibration analysis of cross-ply laminated beams subjected to different sets of boundary conditions is studied, and the analysis is based on a three-degree-of-freedom shear deformable beam theory.
116 citations
TL;DR: In this article, a generalized Vlasov theory for composite beams with arbitrary geometric and material sectional properties is developed based on the variational asymptotic beam sectional analysis.
Abstract: A generalized Vlasov theory for composite beams with arbitrary geometric and material sectional properties is developed based on the variational asymptotic beam sectional analysis. Instead of invoking ad hoc kinematic assumptions, the variational-asymptotic method is used to rigorously split the geometrically-nonlinear, three-dimensional elasticity problem into a linear, two-dimensional, cross-sectional analysis and a nonlinear, one-dimensional, beam analysis. The developed theory is implemented into VABS, a general-purpose, finite-element based beam cross-sectional analysis code. Several problems are studied to compare the present theory with published results and a commercial three-dimensional finite element code. The present work focuses on the issues concerning the use of the Vlasov correction in the context of the accuracy of the resulting beam theory. The systematic comparison with three-dimensional finite element analysis results helps to quantitatively demonstrate both the advantages and limitations of the Vlasov theory.
104 citations
TL;DR: In this article, non-linear and linearized equations are derived for the in-plane stretching and bending of thin-walled cylindrical beams made of a membrane and inflated by an internal pressure.
Abstract: The non-linear and linearized equations are derived for the in-plane stretching and bending of thin-walled cylindrical beams made of a membrane and inflated by an internal pressure. The Timoshenko beam model combined with the finite rotation kinematics enables one to correctly account for the shear effect and all the non-linear terms in the governing equations. The linearization is carried out around a pre-stressed reference configuration which has to be defined as opposed to the so-called natural state. Two examples are then investigated: the bending and the buckling of a cantilever beam. Their analytical solutions show that the inflation has the effect of increasing the material properties in the beam solution. This solution is compared with the three-dimensional finite element analysis, as well as the so-called wrinkling pressure for the bent beam and the crushing force for the buckled beam. New theoretical and numerical results on the buckling of inflatable beams are displayed.
99 citations
TL;DR: In this article, the Timoshenoko-beam model was used for the vibration analysis of double-wall carbon nanotubes with small aspect ratio (between 10 and 20) and it was shown that rotary inertia and shear deformation are significant for higher-order modes of shorter elastic beams.
Abstract: Short carbon nanotubes of smaller aspect ratio (say, between 10 and 50) are finding significant application in nanotechnology. This paper studies vibration of such short carbon nanotubes whose higher-order resonant frequencies fall within terahertz range. Because rotary inertia and shear deformation are significant for higher-order modes of shorter elastic beams, the carbon nanotubes studied here are modeled as Timoshenko beams instead of classical Euler beams. Detailed results are demonstrated for double-wall carbon nanotubes of aspect ratio 10, 20, or 50 based on the Timoshenko-beam model and the Euler-beam model, respectively. Comparisons between different single-beam or double-beam models indicate that rotary inertia and shear deformation, accounted for by the Timoshenko-beam model, have a substantial effect on higher-order resonant frequencies and modes of double-wall carbon nanotubes of small aspect ratio (between 10 and 20). In particular, Timoshenoko-beam effects are significant for both large-diameter and small-diameter double-wall carbon nanotubes, while double-beam effects characterized by noncoaxial deflections of the inner and outer tubes are more significant for small-diameter than large-diameter double-wall carbon nanotubes. This suggests that the Timoshenko-beam model, rather than the Euler-beam model, is relevant for terahertz vibration of short carbon nanotubes.
98 citations
Abstract: This paper presents new finite element methods for the analysis of localized failures in plastic beams and frames in the form of plastic hinges. The hinges are modeled as discontinuities of the generalized displacements of the underlying Timoshenko beam/rod theory. Hinges accounting for a discontinuity in the transversal and longitudinal displacements and the rotation field are developed in this context. A multi-scale framework is considered in the incorporation of the dissipative effects of these discontinuities in the large-scale problem of a beam and a general frame. A localized softening cohesive law relating these generalized displacements with the stress resultants acting at the level of the cross section is effectively introduced in the frame response. The resulting models, referred to as localized models, are then able to capture the localized dissipation observed in the localized failures of these structural members, avoiding altogether the inconsistencies observed for classical models in the stress resultants with strain softening. The constructive approach followed in the development of these models leads naturally to the formulation of enhanced strain finite elements for their numerical approximation. In this context, we develop new finite elements incorporating the singular strains associated to the plastic hinges at the element level. A careful analysis is presented so the resulting finite elements avoid the phenomenon of stress locking, that is, an overstiff response in the softening of the hinge, not allowing for the full release of the stress. The accurate approximation of the kinematics of the hinges requires a strain enhancement linking the jumps in the deflection and the rotation fields, given the coupled definition of the transverse shear strain in these two fields. Different enhanced strain elements, involving different base finite elements and different enhancement strategies, are considered and analyzed in detail. Their performance are then compared in several representative numerical simulations. These analyses identify optimally enhanced finite elements for the accurate modeling the localized failures observed in common framed structures.
TL;DR: In this article, the experimental modal characteristics obtained on regular concrete structures are described successfully by suited Timoshenko beam modelling, taking into account the experimental data, the corresponding beam model, and choosing the maximum tensile strain of concrete as damage criterion for key structural elements, a maximum level of the ground acceleration is determined.
Abstract: The interest of in situ measurements (presented in Part I paper) for a seismic assessment of existing buildings is analysed in this paper. It is shown that the experimental modal characteristics obtained on regular concrete structures are described successfully by suited Timoshenko beam modelling. For a given structure, taking into account the experimental data, the corresponding beam model, and choosing the maximum tensile strain of concrete as damage criterion for key structural elements, a maximum level of the ground acceleration can be determined. This so-called seismic integrity threshold is directly related to the onset of structural damages. This new approach is illustrated on one of the studied buildings. The advantages of using ambient vibrations survey for the vulnerability assessment of existing buildings are discussed.
TL;DR: In this paper, the free vibration analysis of a multi-span beam with an arbitrary number of flexible constraints is studied, where each span of the continuous beam is assumed to obey Timoshenko beam theory.
TL;DR: In this article, a general analytical model applicable to thin-walled I-section composite beams subjected to vertical load is developed, based on the shear-deformable beam theory, and accounts for the flexural response of the thinwalled composites for arbitrary laminate stacking sequence configuration.
TL;DR: In this paper, the dynamic response of a rotating radial cantilever Timoshenko beam clamped at the hub in a centrifugal force field was derived for coupled axial and lateral motions which included the transverse shear and rotary inertia effects, as well as the hyperbolic wave equation governing the axial motion is coupled with the flexural wave equations governing the lateral motion of the beam through the velocity-dependent skew-symmetric Coriolis force terms.
Abstract: Consideration is given to the dynamic response of a Timoshenko beam under repeated pulse loading. Starting with the basic dynamical equations for a rotating radial cantilever Timoshenko beam clamped at the hub in a centrifugal force field, a system of equations are derived for coupled axial and lateral motions which includes the transverse shear and rotary inertia effects, as well. The hyperbolic wave equation governing the axial motion is coupled with the flexural wave equation governing the lateral motion of the beam through the velocity-dependent skew-symmetric Coriolis force terms. In the analytical formulation, Rayleigh–Ritz method with a set of sinusoidal displacement shape functions is used to determine stiffness, mass and gyroscopic matrices of the system. The tip of the rotating beam is subjected to a periodic pulse load due to local rubbing against the outer case introducing Coulomb friction in the system. Transient response of the beam with the tip deforming due to rub is discussed in terms of the frequency shift and non-linear dynamic response of the rotating beam. Numerical results are presented for this vibro-impact problem of hard rub with varying coefficients of friction and the contact-load time. The effects of beam tip rub forces transmitted through the system are considered to analyze the conditions for dynamic stability of a rotating blade with intermittent rub.
TL;DR: In this article, a dynamic stiffness theory of a three-layered sandwich beam is developed and subsequently used to investigate its free vibration characteristics, based on an imposed displacement field so that the top and bottom layers behave like Rayleigh beams, whilst the central layer behaves like a Timoshenko beam.
TL;DR: In this article, a general theory for the shear deformable thin-walled beam with non-symmetric open/closed cross-sections and its exact dynamic and static element stiffness matrices are evaluated.
Abstract: A general theory is proposed for the shear deformable thin-walled beam with non-symmetric open/closed cross-sections and its exact dynamic and static element stiffness matrices are evaluated. For this purpose, an improved shear deformable beam theory is developed by introducing Vlasov's assumption and applying Hellinger–Reissner principle. This includes the shear deformations due to the shear forces and the restrained warping torsion and due to the coupled effects between them, rotary inertia effects and the flexural–torsional coupling effects due to the non-symmetric cross-sections. Governing equations and force–deformation relations are derived from the energy principle and a system of linear eigenproblem with non-symmetric matrices is constructed based on 14 displacement parameters. And then explicit expressions for displacement parameters are derived and the exact dynamic and the static stiffness matrices are determined using force–deformation relationships. In order to verify the validity and the accuracy of this study, the numerical solutions are presented and compared with other numerical solutions available in the literature and results using the thin-walled beam element and the shell element. Particularly the influences of the coupled shear deformation on the vibrational and the elastic behavior of non-symmetric beams with various boundary conditions are investigated.
TL;DR: In this paper, a transfer function method was used to measure the bending and shear stiffness of complex structures and the damping of structural vibration using porous and granular materials, and a proper equation of motion for the structural vibration should be used to obtain reliable data.
TL;DR: In this article, the natural frequencies of composite tubular shafts have been analyzed using equivalent modulus beam theory (EMBT) with shear deformation, rotary inertia and gyroscopic effects.
TL;DR: In this article, a mathematical model based on Timoshenko beam theory applied on curved beams, representing wavy tows in a non-crimp fabric (NCF) composite layer is employed.
Abstract: In this paper, a study set on development and validation of constitutive models to account for out-of-plane fibre waviness in Non-crimp fabric (NCF) composites is presented. For this purpose, a mathematical model based on Timoshenko beam theory applied on curved beams, representing wavy tows in a NCF composite layer is employed. Stiffness knock-down factors operating at the ply level are established and introduced in laminate theory. The developed models are validated on laminates by comparison between predictions and experimental data as well as by comparison with numerical results for a cross-ply laminate. Application of the models on NCF composite laminates (cross-ply and quasi-isotropic) reveals that the models successfully predict laminate elastic properties.
TL;DR: In this paper, a set of analytical formulae are presented to predict the natural frequency of the first five vibration modes, which are torsion, in-plane and out-of-plane sliding modes and in-planar and out of plane rocking modes.
Abstract: A torsional scan mirror suspended with two flexure beams can be used in various display, imaging and other scanning applications. Using various mirror shapes and flexure dimensions as parameters, a set of analytical formulae is presented to predict the natural frequency of the first five vibration modes, which are torsion, in-plane and out-of-plane sliding modes and in-plane and out-of-plane rocking modes. Mode frequencies are compared with the finite element model (FEM) predictions using ANSYS™ for a wide range of flexure beam dimensions. The formulae include the effective inertia of the flexure beams and orthotropic material anisotropy effects. The analytical formulae are verified for both isotropic (e.g. steel) and orthotropic (e.g. silicon) materials. These formulae work very well when the Euler–Bernoulli beam theory assumptions and the rigid mirror assumption are satisfied. The accuracy of analytical predictions is improved by introducing an empirical correction factor to the analytical predictions using non-dimensional flexure beam ratios. The correction factor reduces the error between analytical formulae and FEM predictions to within a few per cent for all five modes for a large range of flexure dimensions. FEM predictions and analytical formulae are partly verified by experimental results.
TL;DR: In this paper, three joint deformation models (i.e., the rigid, semi-rigid, and flexible joint models) describing the different degrees of crack tip deformation are obtained based on three corresponding bi-layer beam theories (e.g., the conventional composite beam, shear deformable bi-layered beam, and interface deformable beam).
TL;DR: In this paper, a continuous damage model and different simplified numerical strategies are proposed to simulate the behaviour of reinforced concrete (R/C) walls subjected to earthquake ground motions, for 2D modelling of R/C walls controlled primarily by bending, an Euler multilayered beam element is adopted.
Abstract: A continuous damage model and different simplified numerical strategies are proposed to simulate the behaviour of reinforced concrete (R/C) walls subjected to earthquake ground motions. For 2D modelling of R/C walls controlled primarily by bending, an Euler multilayered beam element is adopted. For 3D problems, a multifibre Timoshenko beam element having higher order interpolation functions has been developed. Finally, for walls with a small slenderness ratio we use the Equivalent Reinforced Concrete model. For each case, comparison with experimental results of R/C walls tested on shaking table or reaction wall shows the advantages but also the limitations of the approach.
TL;DR: In this article, an efficient and accurate locking-free corotational beam finite element for the analysis of large displacements and small-strain problems is developed, which incorporates the kinematics of all three theories.
Abstract: An efficient and accurate locking-free corotational beam finite element for the analysis of large displacements and small-strain problems is developed in this paper. Three different finite element models based on three different beam theories, namely, the Euler–Bernoulli, Timoshenko, and simplified Reddy theories are presented. In order to develop a single corotational finite element that incorporates the kinematics of all three theories, the unified linear finite element model of beams developed by Reddy (Comm. Numer. Meth. Eng. 1997; 13:495–510) is included in the formulation. An incremental iterative technique based on the Newton–Raphson method is employed for the solution of the non-linear equilibrium equations. Numerical examples that demonstrate the efficiency and large rotation capability of the corotational formulation are presented. The element is validated by comparisons with exact and/or approximate solutions available in the literature. Very good agreement is found in all cases. Copyright © 2005 John Wiley & Sons, Ltd.
TL;DR: In this paper, a method is presented to determine the geometry of tipless microcantilevers by measuring the resonance frequencies of at least one of their bending, lateral and torsional resonance modes, and having knowledge of the beam's elastic modulus, Poisson's ratio and density.
Abstract: A method is presented to determine the geometry of tipless microcantilevers by measuring the resonance frequencies of at least one of their bending, lateral and torsional resonance modes, and having knowledge of the beam’s elastic modulus, Poisson’s ratio and density. Once the geometry is known, the beam’s stiffness and mass can be calculated. Measurement of multiple modes allows for multiple estimates of cantilever geometry. Multiple data points from the experimental results show that this approach yields dimensional values accurate to roughly 2.5% as compared to SEM-determined length, width and thickness. Stiffness values determined with this new technique are roughly 4.7% and 6.5% less than two existing characterization methods (i.e., Sader’s method and Euler–Bernoulli beam theory predictions), and roughly 16% greater than Hutter and Bechhoefer’s stiffness determination method.
TL;DR: In this paper, a parametric study of ride comfort of high-speed trains passing over railway bridges is carried out using a time domain model and the effects of some design parameters are investigated such as damping and stiffness of the suspension system and also ballast stiffness.
Abstract: The ride comfort of high-speed trains passing over railway bridges is studied in this paper. A parametric study is carried out using a time domain model. The effects of some design parameters are investigated such as damping and stiffness of the suspension system and also ballast stiffness. The influence of the track irregularity and train speed on two comfort indicators, namely Sperling's comfort index and the maximum acceleration level are also studied. Two types of railway bridges, a simple girder and an elastically supported bridge are considered. Timoshenko beam theory is used for modelling the rail and bridge and two layers of parallel damped springs in conjunction with a layer of mass are used to model the rail-pads, sleepers and ballast. A randomly irregular vertical track profile is modelled, characterized by its power spectral density (PSD). The 'roughness' is generated for three classes of tracks. Nonlinear Hertz theory is used for modelling the wheel-rail contact. The influences of some nonlinear parameters in a carriage-track-bridge system, such as the load-stiffening characteristics of the rail-pad and the ballast and that of rubber elements in the primary and secondary suspension systems, on the comfort indicators are also studied. Based on Galerkin's method of solution, a new analytical approach is developed for the combination between the rigid and flexural mode shapes, which could be used not only for elastically supported bridges but also other beam-type structures. (A)
TL;DR: In this paper, the boundary feedback stabilization of a Timoshenko beam with both ends free was considered, and boundary feedback control law was proposed to make the closed loop system dissipative.
Abstract: In the present paper we consider the boundary feedback stabilization of a Timoshenko beam with both ends free. We propose boundary feedback control law that makes the closed loop system dissipative. Using asymptotic analysis techniques, we give explicit asymptotic formula of eigenvalues of the closed loop system, and prove the Riesz basis property of eigenvectors and generalized eigenvectors. By a detailed analysis of spectrum of the closed loop system, we show that the closed system is exponentially stable.
TL;DR: In this article, the authors investigated the interlaminar fracture and fiber-bridging in double-cantilever beam specimens from the theoretical and experimental points of view, and developed a novel beam model, which accounts for the effect of fiberbridging.
TL;DR: In this paper, 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.
Abstract: 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 ...
TL;DR: In this paper, the buckling load of the isolators, including the effect of the flexibility of the steel reinforcing plates, is analyzed by a beam theory in which shear deformation and warping of the cross-section are considered.