Timoshenko Beam Model Research Articles

Energy decay for damped Shear beam model and new facts related to the classical Timoshenko system

In this paper, we consider a beam model known as Shear beam model (no rotary inertia). We assure, based on behavior of the wave speeds, that the Shear beam model corresponds to a part of the classical Timoshenko beam model which is governed only by one wave speed. Unlike the classical dissipative Timoshenko type system with viscous damping acting on transverse displacement, we prove that the corresponding dissipative Shear beam model has an energy exponential decay regardless any relationship between coefficients of the system. This happens because such model has only one finite wave speed for all wave numbers.

Modal-Based Nonlinear Model Predictive Control for 3-D Very Flexible Structures

In this article a novel nonlinear model predictive control (NMPC) scheme is derived, which is tailored to the underlying structure of the intrinsic description of geometrically exact nonlinear beams (in which velocities and strains are primary variables). This is an important class of partial differential equation (PDE) models whose behavior is fundamental to the performance of flexible structural systems (e.g., wind turbines, high-altitude long-endurance aircraft). Furthermore, this class contains the much-studied Euler–Bernoulli and Timoshenko beam models, but has significant additional complexity (to capture 3-D effects and arbitrarily large displacements) and requires explicit computation of rotations in the PDE dynamics to account for orientation-dependent forces. A challenge presented by this formulation is that uncontrollable modes necessarily appear in any finite dimensional approximation to the PDE dynamics. We show, however, that an NMPC scheme can be constructed in which the error introduced by the uncontrollable modes can be explicitly controlled. It is demonstrated that the asymptotic error can be made insignificant (from a practical perspective) using our NMPC scheme and excellent performance is obtained even when applied to a highly resolved numerical simulation of the PDEs. We also present a generalization of Kelvin–Voigt damping to the intrinsic description of geometrically exact beams. Finally, special emphasis is placed on constructing a framework suitable for real-time NMPC control, where the particular structure of the underlying PDEs is exploited to obtain both efficient finite-dimensional models and numerical schemes.

Dynamics of Space-Fractional Euler–Bernoulli and Timoshenko Beams

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).

Closed-form solutions for vibrations analysis of cracked Timoshenko beams on elastic medium: An analytically approach

In this paper a novel analytically method for analysing the dynamic behaviour of Timoshenko beam model, resting on Winkler-type elastic soil, under simply-supported boundary condition and with a crack is proposed. The beam is also supposed to be constrained at the ends by elastically flexible springs, with transverse stiffness and rotational stiffness. Applying the Timoshenko beam theory and employing the auxiliary functions, the equation of motion is derived. The natural frequencies are obtained by applying the Euler–Bernoulli method and are derived by the corresponding auxiliary functions of the governing equation of the Euler–Bernoulli beam in free vibration. For different values of soil parameter, taking into account the effects of rotational and shear deformation and considering the presence of crack, typical results are presented, in order to demonstrate the efficiency of the proposed approach. Finally the obtained results are compared with some results available in the literature. It is shown that very good results are obtained. This approach is very effective for the study of vibration problems of Timoshenko beams. The novelty of the proposed approach is that although the auxiliary functions, used to find the solution to the dynamic problem of a Timoshenko beam, are different for the two theories applied, in both cases, the dynamic problem is traced to the study of an Euler - Bernoulli beam subjected to an axial load.

Formulation and experimental validation of space-fractional Timoshenko beam model with functionally graded materials effects

In this study, the static bending behaviour of a size-dependent thick beam is considered including FGM (Functionally Graded Materials) effects. The presented theory is a further development and extension of the space-fractional (non-local) Euler–Bernoulli beam model (s-FEBB) to space-fractional Timoshenko beam (s-FTB) one by proper taking into account shear deformation. Furthermore, a detailed parametric study on the influence of length scale and order of fractional continua for different boundary conditions demonstrates, how the non-locality affects the static bending response of the s-FTB model. The differences in results between s-FTB and s-FEBB models are shown as well to indicate when shear deformations need to be considered. Finally, material parameter identification and validation based on the bending of SU-8 polymer microbeams confirm the effectiveness of the presented model.

On bending, buckling and free vibration analysis of 2D-FG tapered Timoshenko nanobeams based on modified couple stress and surface energy theories

In this study, the static bending, buckling, and free vibration behaviors of two-dimensional functionally graded (2D-FG) tapered micro/nanobeams are modelled and analyzed. For the first time, the nonclassical equations of motion and corresponding boundary conditions of 2D-FG beam are simultaneously derived in the framework of the modified couple stress and Gurtin-Murdoch surface elasticity theories in conjunction with the Timoshenko beam theory. Here, all the material properties of the beam are graded along the thickness and length directions according to the power-law. Both thickness and width of the beam are considered to vary linearly along the length direction. Unlike existing Timoshenko beam models, the current formulation accounts for the axial and bending deformations, Poisson’s effect, and exact position of the physical neutral plane. Differential quadrature method is employed to estimate the variable coefficients of the derived equations. The bending deflection, critical buckling load, and free vibration frequency of simply supported 2D-FG Timoshenko nanobeam are obtained employing the Navier's method. A parametric study is accompanied to investigate the effects of the different material and geometrical parameters on the mechanics of 2D-FG micro/nanobeams. It is observed that these parameters are very important in investigating the static and dynamic responses of 2D-FG micro/nanobeam.

Analysis of a curved Timoshenko nano-beam with flexoelectricity

A Timoshenko beam model is applied for the analysis of the flexoelectric effect for a curved nano-beam. This theory transforms the beam to a 1-d problem along the beam axis. The electric intensity vector for the open-circuit condition can be expressed by mechanical quantities, namely strains and strain gradients. The variational principle is applied to derive the system of ordinary differential equations (ODEs) for the beam deflection, cross-section rotation, and in-plane displacement. A numerical solution based on the Taylor series expansion with Cartesian coordinate along the beam is proposed for the system of ODEs. A convergence analysis with respect to the number of terms in the Taylor series expansion is performed. Numerical results for the beam deflection, rotation, in-plane displacement, and induced electric intensity vector are presented for various flexoelectric coefficients and the beam curvature.

Stability and critical spinning speed of a flexible liquid-filled rotor in thermal environment with nonlinear variable-temperature

This paper deals with the stability and critical spinning speed of a liquid-filled rotor in thermal environment with nonlinear variable-temperature. The nonlinear temperature field model of the rotor is developed by using Laplace transform. The thermal axial force exerted on the rotor as a result of the thermal effect is calculated as functions of temperature rise rate and rotor thickness ratio. Spinning Timoshenko beam model is employed to establish the structural dynamics equations of the rotor system. The governing equation of the motion is derived by using the Hamilton principle. The validity of the developed model is confirmed by comparing with the numerical solutions available in the literature. The numerical results based on the obtained analytical solutions are given for a better understanding of the effects of the shear deformation, rotary inertia, filling parameters and thermal effect on the system stability and critical spinning speed. The results show that the system stability is dependent on the thermal axial force and cavity ratio. Moreover, the results also highlight the role of thermal effect, rotary inertia, cavity ratio and mass ratio on the critical spinning speed.

Viscous inertial mass damper (VIMD) for seismic responses control of the coupled adjacent buildings

Many research studies have focused on a novel kind of damper involved with an inertial/gyro mass component, called the inertial mass damper (IMD). Different realizations of the IMD share common standard features: a sizeable apparent mass and the pseudo-negative dynamic stiffness. The connected control method (CCM) has been proven effective for mitigating vibration between adjacent flexible structures. This paper explores the efficacy of using an IMD in parallel with a viscous damper (termed VIMD) configured in the sky bridge to control the vibration of two adjacent towers under random earthquake excitation. First, numerical models of two buildings are developed using a Timoshenko beam model, which can accommodate both bending and shear deformation. Key damper parameters are explored through a parametric study. Then, stochastic optimization is performed to determine optimum damper parameters and the most effective configuration. The optimization results are analyzed in detail, and guidelines for the damper configuration are recommended. By comparing the optimal control performance between the VIMD and traditional viscous damper (VD), the following conclusions can be drawn: due to its pseudo-negative stiffness characteristics and large damper force, the VIMD can achieve better control performances for a wide range of the axial stiffness of the sky bridge; the VIMD requires a smaller damping coefficient and axial stiffness, which can result in smaller damper sizes and member sections of the sky bridge. Therefore, the VIMD is demonstrated to be a useful alternative for engineering practice in CCM vibration mitigation strategies.

Effect of rotational inertia on building response to earthquakes via a closed-form solution

In this article, a widely used building model, comprised of uniform coupled flexural and shear beams, herein improved by allowing for the effects of rotational inertia. Closed-form solutions in terms of trigonometric and hyperbolic functions are obtained, allowing for the explicit formulation of period ratios, modal participation factors (MPFs), mode shapes, and mode shape derivatives based solely on displacement response, without coupling with chord rotations, which is the case for the Timoshenko beam model. This makes the model proposed in this study more convenient for assessing building behavior to ground motion, by explicitly highlighting the effect of rotational inertia in their response to earthquakes, making the case for studying its beneficial effects in mitigating response of buildings to ground motion. It is observed that rotational inertia induces mild fundamental period lengthening, while notably reducing period ratios between higher modes and the fundamental one. This can lead to an enhanced response to ground motion showcasing narrow-band characteristics. However, the most severe effect is found to be on the MPF, which is significantly diminished. This leads to important reductions on the overall response, as a consequence of attenuation of the first-mode response, along with severe undercutting of the effects of higher modes. The results demonstrate that without considering the rotational inertia in the assessment of in-plane structural response to horizontal ground motion can lead to conservative results. Moreover, the results showcase the advantages of providing supplemental rotational inertia as a way to improve the seismic behavior of buildings.

Lateral dynamic response of pile group embedded in unsaturated soil

A mathematical formulation is proposed for the analysis of a pile group embedded in unsaturated soil subjected to horizontal time-harmonic vibration. The unsaturated soil is governed by the motion theory of three-phase poroelastic medium, and the piles with fixed connection to cap are modeled as Timoshenko beam models. By introducing Helmholtz decomposition to the governing equations, the resistance and attenuation function of displacement of the surrounding soil are theoretically derived. Then, the impedances of pile group in frequency domain are evaluated by utilizing the superposition approach based on the dynamic interaction factors. The internal forces of piles induced by both kinematic effects arising from the adjacent pile motion and inertial effect from pile cap, as well as the degree of contribution of the two effects have been investigated. The accuracy of the proposed method was validated by its comparison with a benchmark solution for the corresponding single phase medium problem. A comprehensive parametric study is performed to depict the effects of saturation, permeability, and pile group arrangement, and indicating that the saturation and permeability exert considerable effects on the nearly saturated condition, by contrast, the unsaturated case is much less affected.

Frequency study of porous FGPM beam on two-parameter elastic foundations via Timoshenko theory

The goal of this study is to fill this apparent gap in the area about investigating free vibration of Functionally Graded Piezoelectric Materials (FGPMs) nanobeams with porosity resting on two-parameter elastic foundations, under voltage load considering Timoshenko beam model and nonlocal theory. The elastic foundation is considered as a Pasternak model with adding a shear layer to the Winkler model. The electromechanical and mechanical properties of the nanobeam (such as elastic, piezoelectric, dielectric coefficients and mass density) are FG in the thickness direction of the beam. Based on Hamilton principle, governing equations of the problem are derived. The Differential Quadrature Method (DQM) for solution of these equations are employed to determine the natural frequencies of the FGPM nanobeams at different Boundary Conditions (B.C.s). The influences of supporting conditions, the porosity coefficient and patterns including even and uneven, nonlocal parameter, Winkler foundation modulus, shear elastic foundation modulus, external voltage and power-law index on the electromechanical vibration characteristics of the FGPM nanobeams are discussed in details. It is found that the FG index and nonlocal parameter will reduce the natural frequencies of the FG nanobeam, while the Winkler and Pasternak moduli of the foundation show an opposite tendency.

Rammed earth as bi-modulus material. Experimental and analytical investigations through Euler-Bernoulli and Timoshenko beam models

Rammed earth as bi-modulus material. Experimental and analytical investigations through Euler-Bernoulli and Timoshenko beam models

The nano scale buckling properties of isolated protein microtubules based on modified strain gradient theory and a new single variable trigonometric beam theory

Microtubules (MTs) are the main part of the cytoskeleton in living eukaryotic cells. In this article, a mechanical model of MT buckling, considering the modified strain gradient theory, is analytically examined. The MT is assumed as a cylindrical beam and a new single variable trigonometric beam theory is developed in conjunction with a modified strain gradient model. The main benefit of the present formulation is shown in its new kinematic where we found only one unknown as the Euler-Bernoulli beam model, which is even less than the Timoshenko beam model. The governing equations are deduced by considering virtual work principle. The effectiveness of the present method is checked by comparing the obtained results with those reported by other higher shear deformation beam theory involving a higher number of unknowns. It is shown that microstructure-dependent response is more important when material length scale parameters are closer to the outer diameter of MTs. Also, it can be confirmed that influences of shear deformation become more considerable for smaller shear modulus and aspect ratios.

2D magneto-mechanical vibration analysis of a micro composite Timoshenko beam resting on orthotropic medium

In the present study, the free vibration analysis of a size-dependent micro composite Timoshenko beam model reinforced by various distributions of carbon nanotubes under temperature changes and two-dimensional magnetic field is investigated based on modified strain gradient theory. Also, the effects of environment are simulated by orthotropic elastic foundation and it is assumed that the material properties are temperature-dependent. Mathematical formulations are obtained using Hamilton's principle and the governing equations of motion are derived based on energy approach and variation method. These equations are solved using semi-analytical and numerical methods such as Navier's type solution, finite element method and generalized differential quadrature method for various boundary conditions. The obtained results of this study are compared with the other previous researches and there is a good agreement between them. The main purpose of this work is the comparison of various solution methods on the problem outputs. Thus, the results are compared together and the effects of solution approach on the dimensionless natural frequencies is developed. Moreover, the effects of length-to-thickness ratio, magnetic field, temperature changes, elastic foundation and carbon nanotubes volume fractions on the dimensionless natural frequencies are studied. The results of this article demonstrate that the micro composite Timoshenko beam reinforced by FG-O and FG-X CNTs have lowest and highest dimensionless natural frequency, respectively. It is investigated that the dimensionless natural frequency enhances by increasing the magnetic field in x and z-directions.

Shear and flexural factors for static analysis of homogenized beam models of planar frames

One-dimensional equivalent beam models, embedded in a two-dimensional space, are developed in this work for static analysis of beam-like structures, such as planar frames (fine models). Extensible shear and Timoshenko beam models (coarse models) are formulated in the framework of the direct one-dimensional approach, under the assumption of rigid cross-section, while their constitutive laws are determined through a homogenization procedure. A linear constitutive equation is obtained, also taking into account for the presence of bracing elements in the frame, with axial and shear forces coupled with bending. The warping of the cross-section, caused both by shear and bending, is taken into account, in an approximate way, by resorting to the concept of shear and flexural factors. These factors are introduced to correct the constitutive coefficients of the equivalent beam. They are determined by equating the elastic potential energy of the fine model and that of the coarse models, once the same macroscopic displacement is given. The limits of applicability of the corrected homogenized beam models are discussed with reference to the linear static response of some planar frames, taken as case-studies. Numerical results obtained by the homogenized models are compared with finite element analyses on planar frames, in order to show the effectiveness of the procedure.

A new stabilization scenario for Timoshenko systems with thermo-diffusion effects in second spectrum perspective

In this work, we analyze a truncated version for the Timoshenko beam model with thermal and mass diffusion effects derived by Aouadi et al. (Z Angew Math Phys 70:117, 2019). In particular, we study some issues related to the second spectrum of frequency according to a procedure due to Elishakoff (in: Advances in mathematical modelling and experimental methods for materials and structures, solid mechanics and its applications, Springer, Berlin, 2010). In Aouadi et al. (2019), the lack of exponential stability for the classical Timoshenko beam with thermodiffusion effects without assuming the nonphysical condition of equal wave speeds has be proved. By using the classical Faedo–Galerkin method combined with the a priori estimates, we prove the existence and uniqueness of a global solution of the truncated version of this problem. Then we prove that this solution is exponentially stable without assuming the condition of equal wave speeds.

A damaged non-homogeneous Timoshenko beam model for a dam subjected to aging effects

A hemi-variational formulation for a damaged non-homogeneous Timoshenko beam is proposed here for the purpose of fast simulation of the properties of a dam. The dam is therefore modeled as a damaged non-homogeneous Timoshenko beam embedded in 2D space. The damage evolution and the mechanics of the beam are governed both by the hemi-variational principle and by the assumption on the form of the deformation energy functional, that comprehends the dissipative part owing to damage-aging effects. In the formulation, the Karush–Kuhn–Tucker (KKT) condition is derived and the damage, which locally decreases the stiffness (i.e., axial, bending, and shear stiffness) of the beam, becomes relevant once a measure of the elastic energy, called the normalized undamaged elastic energy, reaches a certain threshold that is constitutively prescribed ab initio. The aging effect is assumed by reducing such a threshold in the neighborhood of the bottom of the beam, where the concentration of the chemical and physical attacks is higher. As a result, we show a method for the calculation of the mechanical failure condition. In addition, we observe that, even though this threshold is assumed to decrease in a large region far from the bottom of the dam, the damage is confined at the bottom of it. Thus, we have proved that the trapezoidal shape that is used in the dam design is useful not only to decrease the state of stress in the elastic regime, but also to confine the effects of damage evolution owing to the aging effects.

Static and Dynamic Responses of Micro-Structured Beams

In this study, we developed a one-dimensional Timoshenko beam model, embedded in a 3D space for static and dynamic analyses of beam-like structures. These are grid cylinders, that is, micro-structured bodies, made of a periodic and specifically designed three-dimensional assembly of beams. Derivation is performed in the framework of the direct 1D approach, while the constitutive law is determined by a homogenization procedure based on an energy equivalence between a cell of the periodic model and a segment of the solid beam. Warping of the cross-section, caused by shear and torsion, is approximatively taken into account by the concept of a shear factor, namely, a corrective factor for the constitutive coefficients of the equivalent beam. The inertial properties of the Timoshenko model are analytically identified under the hypothesis, and the masses are lumped at the joints. Linear static and dynamic responses of some micro-structured beams, taken as case studies, are analyzed, and a comparison between the results given by the Timoshenko model and those obtained by Finite-Element analyses on 3D frames is made. In this framework, the effectiveness of the equivalent model and its limits of applicability are highlighted.

Buckling of Planar Micro-Structured Beams

In this paper, a Timoshenko beam model is formulated for buckling analysis of periodic micro-structured beams, uniformly compressed. These are planar grid beams, whose micro-structure consists of a square lattice of equal fibers, modeled as Timoshenko micro-beams. The equivalent beam model is derived in the framework of a direct one-dimensional approach and its constitutive law, including the effect of prestress of the longitudinal fibers, is deduced through a homogenization approach. Accordingly, micro–macro constitutive relations are obtained through an energy equivalence between a cell of the periodic model and a segment of the equivalent beam. The model also accounts for warping of the micro-structure, via the introduction of elastic and geometric corrective factors of the constitutive coefficients. A survey of the buckling behavior of sample grid beams is presented to validate the effectiveness and limits of the equivalent model. To this purpose, results supplied by the exact analyses of the equivalent beam are compared with those given by finite element models of bi-dimensional frames.