Timoshenko Beam Theory Research Articles

An analytical approach for the analysis of stress wave transmission and reflection in waveguide systems based on Timoshenko beam theory

The paper presents an analytical continuous solution for the analysis of wave refraction and transmission/reflection caused by arbitrarily complex waveguide subsystems. It allows analyzing the way stress wave components are altered as they go through a substructure. The method is new in that it investigates the wave phenomena which govern the dynamic behavior of a structural system explicitly and studies the effectiveness of different serrated and fractal subsystem designs functioning as wave content filters. Timoshenko beam theory is used to model stress wave propagation in waveguides because of its accuracy and consistency over a wide range of vibration frequencies. The versatility of the method is demonstrated using a number of numerical examples characterizing stress wave refraction in single input/single output and single input/multiple output subsystems to study periodic, fractal, and serrated forms. The low computation cost and stable accuracy of the method over a very wide range of frequencies, make it an effective tool for metaheuristic optimization approaches which require massive calculations over many generations. The method is demonstrated to perform seamlessly in coupling with a genetic algorithm optimization framework to determine the optimal serrated configuration of a wave filter.

Nonlinear modeling and analysis of spatial pipeline with arbitrary shapes supported by clamps considering displacement-dependent characteristic

Considering the displacement-dependent characteristics of the clamp, a general nonlinear semi-analytical modeling method for the spatial pipeline with arbitrary shapes supported by multiple clamps is proposed to study the vibration characteristics of the pipeline-clamp system. The coupling of the complex spatial pipeline is realized by introducing the connecting springs at the connection region of the straight pipe and curved pipe. The equation of motion for spatial pipeline structure is established based on the Lagrange energy equation and Timoshenko beam theory. High-order polynomials are used to characterize the stiffness and damping characteristics of the clamp, and the simple straight pipe is used as the basic model to reverse the clamp parameters. The influence of clamps on pipeline vibration is skillfully introduced into the nonlinear dynamic model of the spatial pipeline. According to the Newton-Raphson method, the nonlinear equation of motion is solved. Taking the spatial pipeline supported by double clamps as an example, the related experiment tests and ANSYS simulation are carried out, and the effectiveness of the established model is verified by comparing the results. Finally, the nonlinear vibration mechanism of the pipeline-clamp system is analyzed. The modeling method proposed in this paper can provide positive guidance for modeling and vibration analysis of the aero-engine pipeline with arbitrary shapes.

Electro-mechanical transfer matrix modeling of piezoelectric actuators and application for elliptical flexure amplifiers

Mechanically amplified piezoelectric actuators (APAs) can be found in many scenarios of precision engineering and instrumental devices. However, the electro-mechanical design of curvilinear APAs is difficult due to the existence of curved flexure beams and multi-domain electro-elasto dynamics. In this paper, the electro-mechanical behavior of elliptical APAs is studied by exploiting a novel electro-mechanical transfer matrix method. The motivation is to facilitate both the static and dynamic analyses of such a complex APA with curvilinear flexure beams by considering the multi-domain dynamics of piezoelectric stacks and compliant mechanisms from an electro-mechanical viewpoint. To this end, an analytical electro-elasto transfer matrix of piezoelectric stacks operating at the d33 mode is derived in the form of Taylor's series based on Timoshenko beam theory. The dynamic response spectrum of displacement and impedance for elliptical APAs are insightfully captured by such an electro-mechanical model. Different topologies of elliptical APAs are also compared and the optimal configuration is suggested. At last, a proof-of-concept prototype is fabricated and tested with a special focus on experimental evaluation of the electro-mechanical coupling model.

Exact closed-form solutions for the free vibration analysis of multiple cracked FGM nanobeams

Exact closed-form solutions for the free vibration analysis of multiple cracked FGM nanobeams with arbitrary boundary conditions based on the Nonlocal Elasticity Theory (NET) and the Timoshenko beam theory are presented. The NET considering the size effect of nanostructures is applied. FGM properties vary nonlinearly along the height of the beam. A crack model using two springs with stiffness depending on the crack depth is applied. The proposed solutions are provided explicitly as functions of integration constants determined from the standard boundary conditions. Frequency equations, that established in the form of the determinant of the matrix 3 × 3 order for arbitrary boundary conditions, are applied to analyse free vibration. Expressions for the mode shapes are given explicitly. The effects of geometry, material, nonlocal and crack parameters on the free vibration of the multiple cracked nanobeam are then analysed in detail.

Particle detachment in reservoir flows by breakage due to induced stresses and drag

Suspension-colloidal-nano transport in porous media encompasses the detachment of detrital fines against electrostatic attraction and authigenic fines by breakage, from the rock surface. While much is currently known about the underlying mechanisms governing detachment of detrital particles, including detachment criteria at the pore scale and its upscaling for the core scale, a critical gap exists due to absence of this knowledge for authigenic fines. For the first time, we integrate the 3D version of Timoshenko's beam theory of elastic cylinder deformation with a CFD-based model for viscous flow around the attached particle and with strength failure criteria for particle-rock bond. This results in a novel explicit criterium for fines detachment by breakage at the pore scale. The criterium includes analytical expressions for tensile and shear stress maxima along with two geometric diagrams which allow determining the breaking stress. This leads to an explicit formula for the flow velocity that provides the particle-rock bond breakage. Its upscaling yields a novel mathematical model for fines detachment by breakage, expressed in the form of the maximum retained concentration of attached fines versus flow velocity – maximum retention function (MRF) for breakage. We performed corefloods with piecewise constant increasing flow rates, measuring breakthrough concentration and pressure drop across the core. It was found out that the behaviour of the measured data is consistent with two-population colloidal transport, attributed to detrital and authigenic fines migration. Indeed, the laboratory data show high match with the analytical model for two-population colloidal transport, which validates the proposed mathematical model for fines detachment by breakage.

Insight into numerical characteristics of embedded finite elements for pile‐type structures employing an enhanced formulation

AbstractA number of problems encountered in the field of computational geotechnics requires the modelling of numerous pile‐type structures embedded in a three‐dimensional soil continuum. Traditional modelling approaches to this problem result in computational costs that must be regarded as unbearable for most practical purposes. As an attractive alternative, we propose an enhanced embedded FE model with implicit interaction surface (EB‐I) where coupling between the contacting domains is realized by an implicit surface‐to‐volume (2D‐to‐3D) coupling scheme. As a novelty, the latter implements a non‐linear interface constitutive model that allows for explicit consideration of endpoint interaction, but does not represent a constraint for the solid mesh generation. As the slender structure is discretized employing the Timoshenko beam theory, shear deformability is explicitly considered, as opposed to earlier EB‐I‐type models reassessed in this paper. The credibility of the proposed EB‐I is numerically validated on the basis of comparative studies. It is found that the 2D‐to‐3D coupling scheme generally improves the well‐posedness of the resultant global stiffness matrix, making the proposed EB‐I computationally competitive to geometrically simplified line‐to‐volume coupling schemes. Future lines of research are carefully addressed throughout this work and include the normal stress recovery technique as well as applications to large‐scale simulations.

Dynamic Analysis of a Timoshenko–Ehrenfest Single-Walled Carbon Nanotube in the Presence of Surface Effects: The Truncated Theory

The main objective of this paper is to study the free vibration of a Timoshenko–Ehrenfest single-walled carbon nanotube based on the nonlocal theory and taking surface effects into account. To model these effects on frequency response of nanotubes, we use Eringen’s nonlocal elastic theory and surface elastic theory proposed by Gurtin and Murdoch to modify the governing equation. A modified version of Timoshenko nonlocal elasticity theory—known as the nonlocal truncated Timoshenko beam theory—is put forth to investigate the free vibration behavior of single-walled carbon nanotubes (SWCNTs). Using Hamilton’s principle, the governing equations and the corresponding boundary conditions are derived. Finally, to check the accuracy and validity of the proposed method, some numerical examples are carried out. The impacts of the nonlocal coefficient, surface effects, and nanotube length on the free vibration of single-walled carbon nanotubes (SWCNTs) are evaluated, and the results are compared with those found in the literature. The findings indicate that the length of the nanotube, the nonlocal parameter, and the surface effect all play important roles and should not be disregarded in the vibrational analysis of nanotubes. Finally, the results show how effective and successful the current formulation is at explaining the behavior of nanobeams.

Static and buckling behaviors analysis of FG beams using a three unknowns finite element based on enhanced Timoshenko theory

A two-node beam element based on enhanced Timoshenko beam theory with three unknowns is developed for static and buckling behaviors of functionally graded beams. The model considers a quadratic variation of shear strain through the depth, and satisfies zero-shear strain at the top and bottom surfaces of the beam. The concept of a physical neutral axis is introduced to avoid stretching-bending phenomenon. The elementary matrices are derived utilizing the minimum potential energy. The accuracy and performance of the present model are verified through a series of examples. Furthermore, the effect of different parameters on behaviors of beams is discussed.

An improved analytical model of a thick defective phononic crystal for bending wave excitation

This research aims to advance the existing analytical model, which was based on the Euler–Bernoulli beam theory, for predicting the velocity amplitude of bending waves excited while amplified by defective phononic crystals (PnCs) with piezoelectric elements. The previous analytical model falls short for thick defective PnCs and is limited to low frequencies. Hence, the Timoshenko beam theory is newly incorporated to address these issues, considering shear deformations and rotational inertia. Its performance is validated against a finite element model. It confirms that the proposed analytical model outperforms its predecessor, which was suitable for thin defective PnCs with a slenderness ratio above 10 and frequencies around 2 kHz. Remarkably, the improved analytical model demonstrates excellent predictive capabilities up to 40 kHz for thick defective PnCs with a slenderness ratio of around 2.5. With a relative error of less than 1% compared to the finite element model, this advanced approach also demonstrates time-efficient and stable computational performance. This work introduces three contributions. First, it pioneers the proposal of an electromechanically coupled analytical model for predicting the performance of defect-mode-enabled bending wave excitation. Second, this enhanced analytical model surpasses existing counterparts in terms of both accuracy and computational stability, particularly for thick defective PnCs, even at high frequencies. Last, the analytical model dramatically reduces the computation time.

Thermal vibration analysis of cracked nanobeams submerged in elastic foundations by nonlocal continuum mechanics

Understanding the mechanical behavior of nanoscale structures is crucial in the development of advanced nanotechnologies. In this study, a novel approach to investigate the thermal lateral vibration of cracked nanobeams immersed in an elastic matrix is investigated. For this purpose, Reddy’s third-order shear deformation theory (TSDT) is considered. In contrast to Timoshenko beam theory (First-order Shear Deformation Theory, FSDT), TSDT does not depend on a shear correction coefficient. The nano-scale effect is modeled using Eringen’s nonlocal continuum mechanics theory. The nonlocal form of the governing equation is obtained through the application of Hamilton’s principle. The weak form of the finite element global mass and stiffness matrices are obtained using Lagrange linear and Hermitian cubic interpolation. To model the crack in bending vibration, two rotational springs are used for TSDT, unlike the use of a single rotational spring in traditional Bernoulli–Euler (Classical Beam Theory, CBT) and FSDT. The stiffness of the springs is adjusted based on the severity of the crack. The influences of the nonlocal parameter, beam slenderness, position of crack, crack severity, Pasternak and Winkler foundation parameters, thermal effects and boundary conditions on the natural frequencies are investigated. The model’s outcomes are compared with findings from prior publications, demonstrating a strong level of agreement. This study contributes to the growing research on nanostructures by presenting a novel approach to understanding the dynamics of cracked nanobeams using Reddy beam analysis-based solutions.

Coupled bandgap properties and wave attenuation in the piezoelectric metamaterial beam on periodic elastic foundation

In recent years, the innovative design of metamaterials and the study of bandgap characteristics have become the focus of research. The flexible setting of small added mass in locally resonant metamaterials can effectively regulate bandgap characteristics and wave transmission. In this paper, a novel tunable piezoelectric metamaterial beam on periodic elastic foundation is proposed for realizing the wider bandgap region and the stronger attenuation capability. Based on Timoshenko beam theory, wave equation of the metamaterial beam considering the piezoelectric and the inertial amplification effects is obtained from the transfer matrix method, and the calculational results are compared to that of numerical simulation. The results show that periodic elastic foundation can improve the overall stiffness of the structure, and the bending wave attenuation frequency domain will broaden with the increase of the elastic foundation stiffness. The metamaterial beam exhibits the strong wave attenuation in the bandgap induced by external negative capacitance shunt circuit, and the wave transmission loss is improved about 20 dB than the short circuit case. The coupling effects between the local resonance and the inertial amplification mechanism can cause the coupled bandgaps, which expand the original bandgap range by at least 25%. In addition, the coupled bandgaps boundary can be extended to lower frequencies by adjusting the angle of rigid rod and the additional mass separately, resulting in broadband wave attenuation. This work will provide valuable ideas for the design of advanced smart materials and structures in the field of vibration and noise reduction.

Explicit analytical solutions for arbitrarily laminated composite beams with coupled stretching-bending and transverse shear deformation

Consider the coupling of stretching-bending deformation induced by the arbitrarily laminated composites, and the transverse shear deformation caused by the thick beams. A mathematical model based upon Timoshenko beam theory is established for the general laminated composite beams, and a proper replacement of stiffness is suggested for narrow or wide beam. By using the state-space approach, a general solution is expressed in terms of matrix exponential, which is applicable for any possible loading and boundary conditions. Expand the matrix exponential through Taylor series, an explicit analytical solution is obtained for the arbitrarily laminated composite beams under general loading conditions. With the obtained general solutions, several analytical solutions are derived explicitly such as the Green's functions for an infinite laminated composite thick beam, and the solutions for some typical beam problems. For example, a cantilever beam, a simply supported beam, or a fixed-end beam, subjected to uniformly distributed load, concentrated force/moment, or sinusoidal load, etc. To cover more complicated beam structures, we develop the boundary element method by using the obtained Green's functions. Three numerical examples are implemented to investigate the influences of stacking sequences and beam geometry, and to validate the applicability of boundary element method.

Buckling analysis of multi-span non-uniform beams with functionally graded graphene-reinforced foams

This paper conducts the buckling analysis of multi-span functionally graded (FG) beams reinforced by three-dimensional graphene foams (3D-GFs) restrained by elastic supports. The graphene foams are considered as graded variation in the thickness direction according to four different porosity distributions. The width and height of the non-uniform beams vary in the longitudinal direction in accordance with the power-law principle, and the multi-span beams consist of two types of optional beams. The governing equations of buckling behavior are derived based on the Timoshenko beam theory and the principle of virtual displacements, and solved by the discrete singular convolution (DSC) method. The purpose of this paper is to obtain the critical buckling load of FG 3D-GFs reinforced beams under the consideration of variable cross-section and multi-span beams with 2–4 spans. The effect of porosity coefficients, slenderness ratio, and spring constants on the buckling characteristic also are studied. The results suggest that the taper ratio of beams and the configuration of multi-span beams lead to remarkable changes in the critical buckling load of beams.

Vibration response of viscoelastic nanobeams including cutouts under moving load

This manuscript presents a mathematical model and numerical solution to predict vibration responses of nonlocal strain gradient (NLS) perforated viscoelastic nanobeam under dynamic moving loads. The microstructure and size length scales are considered in the frame of NLS theory. The viscoelastic damping of the structure is considered by linear Kelvin Voigt viscoelastic model. Timoshenko and Euler Bernoulli beam theories are developed to consider both thin and thick structure response. The moving load is portrayed by point load and harmonic type. The dynamic equations of motion incorporating viscoelasticity and size dependency effects is derived by using Hamilton principle. A differential quadrature method (DQM) with Chebyshev–Gauss–Lobatto formula is used to discretize the space domain and solve the model numerically. The obtained results are validated and compared with available literature. Numerical studies are conducted to show the influence of the material viscosity parameter, perforation parameters, shear deformation effect, nonlocal strain gradient, moving load velocity, and beams slenderness ratio on the dynamic behavior of viscoelastic perforated nanobeams under moving load. The material viscosity parameter can effectively control the magnitude and stability of the magnification factor profiles. Higher filling ratios result in larger values of the dynamic magnification factor, whereas larger numbers of hole rows lead to smaller values. The proposed procedure is supportive in the analysis and design of perforated viscoelastic NEMS structures under moving load.

Physics-informed discrete element modeling for the bandgap engineering of cylinder chains

We propose an efficient method to build a simple discrete element model (DEM) that accurately simulates the oscillation of a continuum beam. The DEM is based on the Timoshenko beam theory of slender cylindrical members and their corresponding wave dynamics in assembly. This physics-informed DEM accounts for multiple vibration modes of the constituting beam elements in wide frequency ranges. We construct various DEMs mimicking cylinder chains and compare their wave dynamics with those measured in experiments to validate the proposed method. Furthermore, we construct a graded woodpile chain of slender cylinders. We experimentally and numerically investigate the frequency bandgaps of the system and demonstrate the possibility of constructing a wide bandgap by consecutively superposing multiple stop bands generated from cylinders of various lengths. This system is highly efficient in blocking propagating waves by leveraging the vibration isolation effect stemming from the local resonance of the cylinders. The proposed DEM method can be useful for investigating and designing complex vibration systems in an efficient and accurate manner. Moreover, the design approach of manipulating the frequency bandgap can be exploited for developing vibration filters and impact mitigation devices.

Dynamic Analysis of Non-Uniform Functionally Graded Beams on Inhomogeneous Foundations Subjected to Moving Distributed Loads

Inhomogeneous materials, variable foundations, non-uniform cross-sections, and non-uniformly distributed loads are common in engineering structures and typically complicate their mechanical analysis considerably. This paper presents an accurate and efficient numerical method for the dynamic analysis of non-uniform functionally graded beams resting on inhomogeneous viscoelastic foundations subjected to non-uniformly distributed moving load and investigates the effects of non-uniformities and inhomogeneities on material, foundation, and load. Based on the Timoshenko beam theory and a Chebyshev spectral method, a consistent discrete dynamic model is derived, which can deal with all axially varying properties. A series of numerical experiments are carried out to validate the convergence and accuracy of the proposed method. The results are compared with those obtained through finite element analysis or in the literature, and excellent agreement is observed. Then, the dynamic response of an axially functionally graded beam resting on an inhomogeneous viscoelastic foundation and subjected to a non-uniformly distributed moving load is investigated. The results show that the material gradient and the inhomogeneous foundation can alter the vibration amplitudes and critical speeds of the beam significantly. Compared with more realistic non-uniformly distributed moving load models, idealized concentrated and uniformly distributed moving load models produce apparent computation errors in vibration amplitudes.

Mechanical predictive modeling of stereolithographic additive manufactured alumina microlattices

Alumina microlattices with stretch- and bending-dominated structures were manufactured to establish the intercorrelation between the quasi-static microlattice mechanical properties and the strut aspect ratio using the stereolithographic additive manufacturing method. In the present work, two kinds of novel mechanical prediction models were calibrated and validated through the experiment and simulation data, aiming to realize the topological optimization and effective design of the weight, strut size, and spatial geometrical structures of the microlattice according to the given load bearing and stiffness demands. First, the mathematical prediction models of the compression strength and Young’s modulus for the microlattices with different aspect ratios were developed based on the modified Gibson and Ashby’s law and Timoshenko beam theory. Then, the prediction models exhibited an excellent prediction achievement of the experiment and simulation results with the assistance of data fitting. Subsequently, the tensile and compressive damage models perfectly reappeared and tracked the actual fracture behaviors of the microlattice, demonstrating different collapse trends for the Simple cubic (SC) and Body-centered cubic (BCC) microlattices. Finally, the quasi-static mechanical properties of SC were far stronger than that of BCC under the same aspect ratio and from the experiment data, simulation results, and theory formulas. This work laid a solid foundation for the microlattice structure design in lightweight engineering.

Nonlinear vibration analysis of Z-shaped pipes with CLD considering amplitude-dependent characteristics of clamps

Under external excitation, the clamp-pipe structures may exhibit nonlinear vibration behavior due to stiffness softening and damping energy dissipation of the clamp. To implement vibration suppression on pipes, a semi-analytical modeling method is proposed for the Z-shaped pipe structure with partial constrained damping layer (CLD) treatment under the constraint of the clamp. The nonlinear vibration characteristics of the Z-shaped pipe with CLD in both in-plane and out-of-plane directions are further investigated by considering the nonlinear mechanical properties of the clamp. Specifically, the energy expressions of the laminated straight and curved pipe segments are deduced based on the Timoshenko beam theory. The coupling of pipe segments is introduced by the improving artificial spring technology. A novel equivalent mechanical model is developed to simulate the nonlinear mechanical properties of the clamp, which are expressed by nonlinear stiffness and damping with the amplitude-dependent characteristic. Next, an effective algorithm is presented for solving nonlinear dynamic equations in the frequency domain with the amplitude-dependent characteristic. The proposed linear and nonlinear models are verified by the FE analysis and experimental test results. The comparison demonstrates that the proposed method can accurately replicate the nonlinear vibration of the Z-shaped pipe with CLD treatment. Finally, the suppression mechanism of the CLD on the vibration of Z-shaped pipes and the nature of the amplitude-dependent characteristic are further investigated, which can guide the design and vibration reduction of pipes.

Numerical study on damped nonlinear dynamics of cracked FG-GNPRC dielectric beam with active tuning

Functionally graded (FG) graphene nanoplatelets (GNPs) reinforced composite (FG-GNPRC) is widely used in various engineering fields due to its high-performance and multifunctional features. Cracks generated in complex environments have a significant impact on the structural behavior of FG-GNPRC structures. This paper studies the damped nonlinear dynamics of cracked FG-GNPRC dielectric beam subjected to mechanical excitation and electrical field. The effective material properties of the composites are determined by the effective medium theory (EMT). Based on Timoshenko beam theory and nonlinear von Kármán strain–displacement relationship, the governing equations which incorporate damping and dielectric properties are derived by using energy method. Stress intensity factor (SIF) of cracked FG-GNPRC beam at the crack tip is calculated via finite element method. Differential quadrature (DQ) and incremental harmonic balance (IHB) methods combined with arc-length algorithm are utilized to discretize and solve the nonlinear system. After validation of the model and solution, the effects of crack depth and location, damping, FG distribution and the attributes of GNP and the electric field on the dynamic response of the system are comprehensively investigated. The study indicates that when an electric field is applied, the amplitude ratio of cracked FG-GNPRC beam with profile V is the largest when the crack is close to the edge, whereas profile A exhibits the largest value when the crack is near the mid-span. In addition, the generated bending moment enables profiles A and V to transit into profile U at a certain voltage or GNP aspect ratio due to the existence of electrostatic stress.

Large Deflection Effects on the Energy Release Rate and Mode Partitioning of the Single Cantilever Beam Sandwich Debond Configuration

Abstract This paper investigates the effects of large deflections on the energy release rate and mode partitioning of face/core debonds for the single cantilever beam sandwich composite testing configuration, which is loaded with an applied shear force and/or a bending moment. Studies in this topic have been done by employing geometrically linear theories (either Euler–Bernoulli or Timoshenko beam theory). This assumes that the deflection at the tip of the loaded debonded part is small, which is not always the case. To address this effect, we employ the elastica theory, which is a nonlinear theory, for the debonded part. An elastic foundation analysis and the linear Euler Bernoulli theory are employed for the “joined” section where a series of springs exists between the face and the substrate (core and bottom face). A closed form expression for the energy release rate is derived by the use of J-integral. Another closed-form expression for the energy release rate is derived from the energy released by a differential spring as the debond propagates. Furthermore, a mode partitioning angle is defined based on the displacement field solution. Results for a range of core materials are in very good agreement with the corresponding ones from a finite element analysis. The results show that large deflection effects reduce the energy release rate but do not have a noteworthy effect on the mode partitioning. A small deflection assumption can significantly overestimate the energy release rate for relatively large applied loads and/or relatively long debonds.