Classical Timoshenko Research Articles

A rigid‐flexible coupled rocking structure with nonlinear stress distribution along its base: Analytical modeling, validation and parametric investigation

AbstractStructures allowing their bases to uplift and rock during an earthquake event usually experience less damage; thus, they are more resilient to seismic hazards. Accurate simulation is critical to design and seismic performance evaluation of these flexible rocking structures. Motivated from this, this article presents a new rigid‐flexible coupled rocking model for numerical evaluation of flexible rocking structures under both cyclic and dynamic loadings. The key idea in the model formulation is that the total motion of flexible rocking structure can be decomposed into a rigid‐body motion and an associated flexible deformation. The flexible deformation of the rocking body is described using displacement field of classical Timoshenko beam. Using this approach and appropriate degrees‐of‐freedom (DOFs), the governing equations‐of‐motion for flexible rocking structures are formulated using the variational principle. Multiple springs with appropriate constitutive model are distributed along rocking base to represent the inelastic behavior of rocking body. Potential post‐tensioned tendons are also considered in this model. In addition, nonlinear stress distribution along the rocking base, which cannot be considered in existing flexible rocking model, is also incorporated in the developed model. The proposed model is subsequently validated against published experimental data through quasi‐static and shake table tests, showing good accuracy when the rocking structure is relatively stockier. Finally, a parametric investigation on the effect of flexibility, nonlinear stress distribution and yield stress of rocking body on the rocking response is performed. This preliminary investigation shows that influence of flexibility and yield stress is significant while impact of nonlinear stress distribution is minor.

An efficient numerical method for the quasi-static behaviour of micropolar viscoelastic Timoshenko beams for couple stress problems

A viscoelastic Timoshenko micro-beam is investigated using micropolar viscoelasticity theory. The governing equations are derived by applying the principle of virtual displacements, and can be used in a variety of problems. The equations are then reduced to the special case of a homogeneous uniform beam and solved for quasi-static elastic and viscoelastic cases. The viscoelastic material is modeled via standard linear solid model. For the elastic case, a closed-form solution is reached, but for the viscoelastic case, the inverse Laplace transform is calculated numerically by utilizing De Hoog's algorithm. It is shown that the proposed model can capture the size effects, and converges to the classical Timoshenko beam if the beam is thick enough. The model also converges to the micropolar Euler-Bernoulli model if the thickness of the beam is small. The role of distributed couple stress is also investigated in this paper. The results indicate that in a cantilever beam, the distributed couple stress has a high influence on the deflection of the beam, micro-rotation, stress components and couple stress. However, in a beam resting on three supports, it only affects the micro-rotation and the shear stress component in transverse direction (σxz). Further investigation shows an extremum in some of the results over time, illustrating the importance of taking viscoelastic behaviour into account.

Exact vibration solution for three versions of Timoshenko beam theory: A unified dynamic stiffness matrix method

This paper introduces a unified and exact method for the vibration solution of three versions of Timoshenko beam theory, namely the classical Timoshenko beam theory (TBT), truncated Timoshenko beam theory (T-TBT), and slope inertia Timoshenko beam theory (S-TBT). The proposed unified method enables the free vibration analysis of these three beam theories performed by only one equation, which avoids separate modeling and solution with different beam theories. Firstly, the comparison of three beam theories is conducted with a special focus on deriving the governing differential equation of T-TBT based on Hamilton’s principle. Then, two parameters are introduced to unify the three governing differential equations. The dynamic stiffness matrix is derived in a unified form by reducing the order of inverse matrix operation from 2n to n, allowing the dynamic properties of three beam theories to be characterized with only a single model. Additionally, this paper also derives the unified frequency-dependent mass and stiffness matrices based on the unified dynamic stiffness matrix, which eliminates the limitation of DSM in studying the independent influence of the structure's mass and stiffness characteristics. Finally, the discrepancies of the three beam theories are discussed under different boundary conditions and aspect ratios.

Comparative study on free vibration analysis of rotating bi-directional functionally graded beams using multiple beam theories with uncertainty considerations

The present study investigates the free vibration behavior of rotating beams made of functionally graded materials (FGMs) with a tapered geometry. The material properties of the beams are characterized by an exponential distribution model. The stiffness and mass matrices of the beams are derived using the principle of virtual energy. These matrices are then evaluated using three different beam theories: Bernoulli–Euler (BE) or Classical Beam Theory (CBT), Timoshenko (T) or First-order Shear Deformation Theory (FSDT), and Reddy (R) or Third-order Shear Deformation Theory (TSDT). Additionally, the study incorporates uncertainties in the model parameters, including rotational velocity, beam material properties, and material distribution. The mean-centered second-order perturbation method is employed to account for the randomness of these properties. To ensure the robustness and accuracy of the probabilistic framework, numerical examples are presented, and the results are compared with those obtained through the Monte Carlo simulation technique. The investigation explores the impact of critical parameters, including material distribution, taper ratios, aspect ratio, hub radius, and rotational speed, on the natural frequencies of the beams is explored within the scope of this investigation. The outcomes are compared not only with previously published research findings but also with the results of 3-Dimensional Finite Element (3D-FE) simulations conducted using ANSYS to validate the model’s effectiveness. The comparisons demonstrate a strong agreement across all evaluations. Specifically, it is observed that for thick beams, the results obtained from FSDT and TSDT exhibit a greater agreement with the 3D-FE simulations compared to CBT. It is shown that the coefficient of variation (C.O.V.) of first mode eigenvalue of TSDT, FSDT and CBT are approximately identical for random rotational velocity and discernible deviations are noted in CBT compared to FSDT and TSDT in the case of random material properties. The findings suggest that TSDT outperforms FSDT by eliminating the need for a shear correction coefficient, thereby establishing its superiority in accurately predicting the natural frequencies of rotating, tapered beams composed of FGMs.

Exponential Stability for Damped Shear Beam Model and New Facts Related to the Classical Timoshenko System With a Distributed Delay Term

We consider in this manuscript  a Timoshenko type beam model with a distributed delay term.  If the distributed delay term is small enough, we prove the global existence of solutions by using the Faedo-Galerkin approximations together with some energy estimates. Under suitable assumptions, we prove exponential stability of the solution. This result is obtained by introducing a suitable Lyapounov function.

General Theory of Nano Trilayer Film Bending

Multilayer sheets or films have many potential applications in micro-nanoelectromechanical systems. When surface and scale effects are not considered, the bending of multilayer film systems can theoretically be discussed by the classical Stoney formula or Timoshenko formula. When the system has anisotropic surface stress or mismatch strain, the four-parameter bending model proposed by Narsu et al. can be used. However, if the thickness of the film is several nanometers and the bending radius of curvature is less than 1micron, the existing theoretical model is no longer applicable. For this reason, a bending formulation for the nanomulti-layer film system is derived and the structure of the multilayer film is optimized in this paper.

Surface Effects on the Frequency Dispersion of Flexural Waves in Timoshenko Nanobeams

The surface effects play an important role in nanobeams. Based on a recently developed theory of surface elastodynamics, a model of the flexural wave propagation in Timoshenko nanobeams is established, in which the surface effects characterized by surface energy and surface inertia are introduced. It is found that when the size of a beam is comparable with nanometers, the surface energy effect would enhance the wave speed, while the surface inertial one would reduce it. An interesting phenomenon discovered is that with an increasing wave frequency, the dominant role transits from the surface energy effect to the surface inertial effect. The two kinds of surface effects exhibit a frequency-dependent competitive mechanism. In contrast to the macroscopic beams, due to the surface effects, the frequency dispersion of flexural waves in nanobeams becomes size-dependent. Furthermore, a comparison of the Timoshenko nanobeam and the Euler one indicates that the shear deformation effect and rotary inertial one cannot be neglected for a large wave number, which would prominently decrease the wave speed. Besides, when the size of the beam is large enough, the surface effects can be neglected and the present results can degenerate to the classical Timoshenko ones. The present results should be helpful not only for deep understanding of the dispersive mechanism of flexural waves in nanobeams, but also for optimal design of nanobeam-based acoustic wave devices.

Forced vibration of a novel beam model considering the shear deformation induced rotary inertia

Since the Timoshenko beam was introduced to engineering fields, many efforts have been devoted and many problems are yet to handle. Based on the classical Timoshenko beam theory, the rotary inertia caused by the shear deformation is further considered and the dynamic response under forced vibration is derived in this paper. Numerical analysis on beams with different lengths and different boundary conditions are conducted by four different beam theories including the Euler beam theory, shear beam theory, Timoshenko beam theory, and novel modified Timoshenko beam theory. It is found that the effect of the shear deformation induced rotary inertia is small for thin and slender beams; however, it is evident for short and deep beams with high stiffness. Meanwhile, the modification of the shear rotary inertia is obvious when the external load frequency is close to the first natural frequency of the beam. This study shows that the proposed modified Timoshenko beam theory is a more reasonable and accurate model to calculate the responses of the beam under forced vibration.

Spectral Chebyshev method coupled with a high order continuation for nonlinear bending and buckling analysis of functionally graded sandwich beams

AbstractThe main objective of the present work is to couple the spectral Chebyshev differential quadrature method (SCDQM) to the high order continuation method (HOCM) which was proposed in previous works with several discretization techniques. This new approach (SCDQM‐HOCM) is proposed to analyze the nonlinear bending and buckling analysis of functionally graded sandwich beams. The originality of this work consists also to use a beam model which taken into account the nonlinear term neglected in several works of the literature. This term makes it possible to have the buckling and bending problems by using the classical Timoshenko model and to handle the boundary conditions that several works could not to take into account. A strong form of nonlinear equations is established based on the first order shear deformation theory of beams with the von‐Kármán kinematic hypothesis. Regarding functionally graded materials (FGM), two typical types are investigated: sandwich beam with FGM faces and ceramic core (Type‐A), and sandwich beam with FGM core and uniform faces (Type‐B). The accuracy and efficiency of the SCDQM‐HOCM compared with finite element method coupled with high order continuation method (FEM‐HOCM) are illustrated on numerical examples of the FGM beams and then the FG sandwich beams. Furthermore, a parametric study is led to carry out the influence of different skin‐core‐skin thickness ratios, span‐to‐height ratio, and volume fraction on the bending and buckling behavior of FG sandwich beams subjected to different loadings and various boundary conditions.

A New Theoretical Framework for Couple Stress Analysis of Reddy–Levinson Micro-Beams

In this paper a new scheme, other than the modified couple stress theory (M-CST) and consistent couple stress theory (C-CST), is proposed for the analysis of micro-beams and based on the main kinematical assumptions in the beam theories, the couple-stress tensor is constrained to be skew-symmetric. This new consideration also makes it possible to revise the governing equations and boundary conditions, derived from Hamilton’s principle into a more convenient form. Then according to the third-order Reddy–Levinson beam theory, a revised couple stress model similar to the classical Timoshenko beam theory is presented in the current study. The validity of the model is demonstrated by static bending and free vibration investigations of isotropic and FG porous micro-beams subjected to a concentrated load at the mid-span.

A new nonlinear 5-parameter beam model accounting for the Poisson effect

In this paper, a new kinematic beam model based on a five-parameter displacement field is proposed in a geometric nonlinear framework. The proposed displacement field enriches the classical Timoshenko beam displacement field with two additional parameters, accounting for the Poisson effect. The equilibrium equations have been derived through a variational approach, and the linearized equations are solved analytically. The adoption of the linear solution as approximation functions for the nonlinear case allows prediction of nonlinear response of problems involving complex geometries with a relatively small computational effort. Several numerical examples of benchmark problems are analyzed, highlighting the characteristic features of the proposed five-parameter model and comparing the results with those obtained using the classical Bernoulli beam model and 3D finite element model. Numerical results show that, although for thin beams the differences in generalized displacement are generally negligible, the proposed model predicts a comparable stress field only when the Poisson ratio is equal to zero. Conversely, the stress field is meaningfully enriched by the new parameters, with significant differences where the Poisson effect is more pronounced.

Design and Characterization of Asymmetric Cell Structure of Auxetic Material for Predictable Directional Mechanical Response.

A three-dimensional auxetic structure based on a known planar configuration including a design parameter producing asymmetry is proposed in this study. The auxetic cell is designed by topology analysis using classical Timoshenko beam theory in order to obtain the required orthotropic elastic properties. Samples of the structure are fabricated using the ABSplus fused filament technique and subsequently tested under quasi-static compression to statistically determine the Poisson’s ratio and Young’s modulus. The experimental results show good agreement with the topological analysis and reveal that the proposed structure can adequately provide different elastic properties in its three orthogonal directions. In addition, three point bending tests were carried out to determine the mechanical behavior of this cellular structure. The results show that this auxetic cell influences the macrostructure to exhibit different stiffness behavior in three working directions.

Geometrical Nonlinearity for a Timoshenko Beam with Flexoelectricity.

The Timoshenko beam model is applied to the analysis of the flexoelectric effect for a cantilever beam under large deformations. The geometric nonlinearity with von Kármán strains is considered. The nonlinear system of ordinary differential equations (ODE) for beam deflection and rotation are derived. Moreover, this nonlinear system is linearized for each load increment, where it is solved iteratively. For the vanishing flexoelectric coefficient, the governing equations lead to the classical Timoshenko beam model. Furthermore, the influence of the flexoelectricity coefficient and the microstructural length-scale parameter on the beam deflection and the induced electric intensity is investigated.

Design of an auxetic cellular structure with different elastic properties in its three orthogonal directions

Auxetic structures have become one of the most studied types of cellular structure in recent years, thanks to their highly specific mechanical properties. Most microstructures are designed with homogeneous properties in their main axes; however, in this research, we propose an innovative asymmetric 3D auxetic structure based on a 2D configuration of cells drawn from the literature. The new auxetic cell is designed by topology analysis using classical Timoshenko beam theory. To validate the design, samples were constructed by fused filament fabrication with ABSplus, and tested under quasi-static compression to determine Young's modulus in the three orthogonal directions. The experimental results show good agreement with the topological analysis and reveal that the proposed cell can transmit a different mechanical behavior to the macrostructure in its three orthogonal directions.

Analytical solution for a 5-parameter beam displacement model

In this paper, a new beam model based on a 5-parameter displacement field, accounting for an enhanced kinematics and able to reproduce the Poisson effect, is proposed. The displacement field enriches the classical three-parameter Timoshenko beam with two new parameters capable to simulate the shortening effect over the thickness. The related differential equations, derived from a variational formulation, are analytically solved and implemented in a deformation method approach, capable of solving structural problems of beams with general geometry, boundary and load conditions. Several numerical applications are developed, highlighting the characteristic of the proposed 5-parameters model and comparing the results with those obtained using the classical Bernoulli, Timoshenko and Reddy beam models. Numerical results show that, although for homogeneous beam the differences in terms of generalized stress and displacement are generally very small, the proposed model returns local stress enriched by the new parameters, with more significant differences where the Poisson effect is more pronounced.

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.

New refined model for curved linear anisotropic rods with circular cross section

An asymptotic reduction method is introduced to construct a curved rod theory for a general anisotropic linearized elastic material. For the sake of simplicity, the cross section is assumed to be circular. The starting point is Taylor expansions about the mean-line in curvilinear coordinates, and the goal is to eliminate the two spatial variables in the cross section in a pointwise manner in order to obtain a closed system for the displacement coefficients. We achieve this by using a Fourier series for the lateral traction condition together with the use of cylindrical coordinates in the cross section and by considering exact tridimensional equilibrium equation. We get a closed differential system of ten vector unknowns, and after a reduction process we obtain a differential system of the vector of the mean line displacement and twist angle. Six boundary conditions at each edge are obtained from the edge term in the tridimensional virtual work principle, and a unidimensional virtual work principle is also deduced from the weak forms of the rod equations. Through one example, we show that our theory gives more accurate results than the ones of both classical Euler-Bernoulli rod theory and Timoshenko rod theory. The displacement field is computed for two types of material symmetry : isotropy and transverse isotropy.

Thermo-responsive and shape-morphing CF/GF composite skin: Full-field experimental measurement, theoretical prediction, and finite element analysis

Shape morphing is an attractive functionality for fibre reinforced composites. Shape morphing composites can adopt various shapes and undergo different shape morphologies in response to a set of external stimuli. One of the approaches to attain shape morphing materials is through fabrication of multi-layered and asymmetric composites where morphing stems from structural anisotropy. In this work, asymmetric hybrid carbon fibre/glass fibre/epoxy composites are manufactured in which a mismatch in the coefficient of thermal expansion between carbon and glass fibre layers and different fibre directions at each layer resulted in a thermo-responsive morphing behaviour. The full-field displacement of laminate surfaces at the temperature range of −30 °C to 60 °C are monitored using digital image correlation technique. Classical laminate theory and Timoshenko bimetallic strip formula are coupled with experimental observations to predict the radius of curvature for laminates at different temperatures. Furthermore, finite element analyses are performed to uncover the stress state in the laminates and identify the contributing mechanisms. This study contributes to the state of the art by elaborating on the relations between morphing performance with stiffness and thermal expansion of anisotropic fibre reinforced laminates and their connections to the microstructure.

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.

Higher order adhesive effects in composite beams

Composite adhesive bonded joints are widely used in various industrial and technological applications, including aerospace, electronics, biomedical, automotive, ship building and construction. In this paper, the attention is focused on layered structures consisting of two adherent beams bonded together by an adhesive layer. For such structures, a modeling approach based on the classical Timoshenko beam theory in conjunction with an adhesive model of imperfect interface is introduced. This imperfect interface approach, recently proposed by the authors in the contest of linear elastic adhesive and adherents materials, small strains and small displacements theory, models the asymptotic behavior of a thin interphase at higher orders for both the cases of hard and soft interface materials in a unified approach (Rizzoni et al., 2014). Accounting for higher order terms of the asymptotic expansions in the adhesive, the proposed approach generalizes simpler models based on the classical spring-type interface law or on the case of perfect contact between the adherent layers.The proposed methodology is used to evaluate stresses in two adhesive bonded joint configurations subjected to bending moment and transverse shear loading. Numerical simulations are produced and the results show good agreements with those obtained through finite element analysis.