Composite Timoshenko Beam Research Articles

Timber-timber composite (TTC) beams subjected to hogging moment

This study aims to investigate the structural behaviour of timber-timber composite (TTC) systems subject to hogging moment. A total of thirty-one TTC and six bare timber beam-to-column subassemblies (including fourteen replicates) were fabricated and tested to failure under hogging moment. The TTC beams were fabricated by connecting a cross laminated timber (CLT) slab to the top edge of a pair of laminated veneer lumber (LVL) or glued laminated timber (GLT) beams/joists. The effect of the CLT slab thickness, width, and orientation (i.e., loaded lengthwise or crosswise), column penetration in the CLT slab, bending moment to shear force ratio (span length), degree of shear interaction between the slabs and beams (controlled by the size of the shear connectors), and type of beams (LVL or GLT) on the structural performance of the TTC subassemblies were investigated experimentally. In addition, an analytical model was adopted for composite Timoshenko beams and modified to predict the stiffness and load carrying capacity of the TTC beams under hogging moment. The width of the CLT slab and column penetration in the slab (resulting in the reduction of the slab area of up to 20%) had minor (less than 9%) influence on the stiffness and peak load. However, the CLT slab thickness (in the range of 60–100 mm) had a major (up to 48%) influence on the peak load and stiffness of the TTC beam subject to hogging moment. The size of shear connectors (degree of shear connection provided by 8–12 mm screws) and orientation of the CLT slab had a moderate (max. 18–23%) influence on the load carrying capacity and stiffness of the samples.

Free vibration analysis of rotating composite Timoshenko beams with bending-torsion couplings

The free vibration of rotating bending-torsional composite Timoshenko beams (CTBs) with arbitrary boundary conditions is analyzed. The composite material coupled rigidity, Coriolis effects and the separation of the cross-section’s mass and shear centers are considered and they will cause the bending-torsion coupled vibration of the beam. Based on the Hamilton’s principle, the governing partial differential matrix equation of motion of the beam are formulated with variable coefficients. Those equations containing variable coefficients are expressed in a special matrix form in terms of the flexural translation, rotation of cross section and torsional rotation of the beam. The differential transform matrix method (DTMM), which is an improved approach from the traditional differential transform method (DTM), is proposed to deal with the governing matrix equation with corresponding boundary conditions. The mode shapes and natural frequencies of the beam are obtained. Cantilevers are calculated as examples to verify the present theory and investigate the dynamic characteristics of the rotating CTB. The present results are consistent with those reported in the literature and those calculated from COMSOL. The influence of the composite material rigidity, rotation speed, hub radius and axial load on the natural frequencies and mode shapes of the beam are studied, while the frequency veering and mode shift phenomena are observed. Furthermore, the beam’s critical buckling loads corresponding to the axial load are found.

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.

Comparison and experimental verification of simplified one-dimensional linear elastic models of multilayer sandwich beams

Three analytical one-dimensional linear elastic models of composite laminated beams are considered – composite Bernoulli-Euler beam (BE), composite Timoshenko beam (T) and multilayer sandwich beam model (MS). They are compared with results obtained via finite element method for a two-dimensional model in plane stress state. Overall system stiffness is verified with experimental data obtained for two statical configurations – three-point bending and four-point bending. The first configuration concerned 8 types of three-panel cross-laminated timber (CLT) beams accounting for various materials and thickness of timber panels as well as various materials and thickness of adhesive layer, while the second one concerned 5 types of two-panel aluminium laminated beams accounting for different thickness of adhesive layer. Simplified multilayer sandwich model is found to be in good accordance with FEA results and with experimental data, while simple BE and T models are shown to provide erroneous estimates.

Static deflection of fully coupled composite Timoshenko beams: An exact analytical solution

The purpose of this paper is to present the exact analytical solution for the static deflection analysis of fully coupled composite Timoshenko beams. The system of governing equations and the boundary conditions are derived from variational principles. Using the method of direct integration, the exact analytical solution of the static deflection of a Timoshenko beam is obtained by solving this system of differential equations in terms of transverse displacements and cross-sectional rotations. Static deflection analyses of Timoshenko beams, subject to various boundary conditions and uniformly distributed and tip loads, are performed and the results are compared to those obtained from classical Euler–Bernoulli theory by using different values of length-to-thickness ratio. In addition, it is shown that for the case of a cantilevered composite beam subject to tip loads, the proposed exact analytical solution is equivalent to the exact solution from the intrinsic formulation. The Chebyshev collocation method is also employed to validate the obtained exact analytical solution. In the proposed formulation the stiffness properties of the composite beam are expressed by engineering constants, therefore is not limited by the cross-sectional shape of the beam, type of material and thus can be utilised for engineering applications and design purposes. The exact analytical solution can also be used as a benchmark for validating results obtained from various numerical methods.

Dynamic modeling for rotating composite Timoshenko beam and analysis on its bending-torsion coupled vibration

A formulation is presented for steady-state dynamic responses of rotating bending-torsion coupled composite Timoshenko beams (CTBs) subjected to distributed and/or concentrated harmonic loadings. The separation of cross section's mass center from its shear center and the introduced coupled rigidity of composite material lead to the bending-torsion coupled vibration of the beams. Considering those two coupling factors and based on Hamilton's principle, three partial differential non-homogeneous governing equations of vibration with arbitrary boundary conditions are formulated in terms of the flexural translation, torsional rotation and angle rotation of cross section of the beams. The parameters for the damping, axial load, shear deformation, rotation speed, hub radius and so forth are incorporated into those equations of motion. Subsequently, the Green's function element method (GFEM) is developed to solve these equations in matrix form, and the analytical Green's functions of the beams are given in terms of piecewise functions. Using the superposition principle, the explicit expressions of dynamic responses of the beams under various harmonic loadings are obtained. The present solving procedure for Timoshenko beams can be degenerated to deal with for Rayleigh and Euler beams by specifying the values of shear rigidity and rotational inertia. Cantilevers with bending-torsion coupled vibration are given as examples to verify the present theory and to illustrate the use of the present formulation. The influences of rotation speed, bending-torsion couplings and damping on the natural frequencies and/or shape functions of the beams are performed. The steady-state responses of the beam subjected to external harmonic excitation are given through numerical simulations. Remarkably, the symmetric property of the Green's functions is maintained for rotating bending-torsion coupled CTBs, but there will be a slight deviation in the numerical calculations.

The Deflection of Rotating Composite Tapered Beams with an Elastically Restrained Root in Hygrothermal Environment

Abstract This article aims to study the static deflection of a rotating composite Timoshenko beam subjected to the laterally distributed load and restrained by the elastic root and affected by the various cross-section, installation mode, and hygrothermal environment. The governing equation is established according to the force equilibrium condition and solved by a semianalytical power series solution. To verify the correctness, the results of differential quadrature method are introduced to make a comparison. Then, several parameters that can affect the static deflection of the beam, such as the rotating speed, temperature variation, elastic root, and so on, are investigated. Results indicate that (1) pitch angle, rotating speed, and hub radius can result in the centrifugal stiffening effect; (2) setting angle, fibre orientation angle, taper ratio, and elastic root affect the static deflection by changing the rigidity of the rotating composite tapered beam; and (3) temperature variation and moisture concentration can cause the expansion deformation and the change of material properties.

Nonlinear free vibration analysis of functionally graded rotating composite Timoshenko beams reinforced by carbon nanotubes

In this paper, the linear and nonlinear free vibrations of functionally graded rotating composite Timoshenko beams reinforced by carbon nanotubes are presented. The formulation is based on the assumptions of Timoshenko beam theory in addition to consideration of the nonlinear von Karman strain–displacement relationship. The effective material properties of carbon nanotube reinforced composites are determined employing the Mori–Tanaka micromechanics model and the extended mixture rule. For the carbon nanotube reinforced composite beams, uniform distribution and four types of functionally graded distribution patterns of single-walled carbon nanotube reinforcements are considered. A differential transform method is applied on the nondimensionalized equations of motion to release the flapping modeshapes and the associated natural frequencies. The direct method of multiple scales is implemented to derive the effective nonlinearity and the corresponding nonlinear natural frequency. The accuracy of the present outcomes is validated by the comparison with the results given in the literature. The numerical results are presented in both tabular and graphical forms to investigate the effects of nanotube volume fractions, distribution types of the carbon nanotubes, and rotation speed on linear and nonlinear free vibration characteristics of carbon nanotube reinforced composite beam. The results demonstrate the important role of carbon nanotube distribution profile on linear and nonlinear free vibration features.

A new approach for determination of interlaminar normal/shear stresses in micro and nano laminated composite beams

In this article, a displacement model for micro and nano laminated composite Timoshenko beam is presented. The layer-wise displacement model is refined in order to calculate interlaminar stresses c...

Bending–torsional-coupled vibrations and buckling characteristics of single and double composite Timoshenko beams

The stability and free vibration analyses of single and double composite Timoshenko beams have been investigated. The closed-section beams are subjected to constant axially compressive or tensile forces. The double beams are assumed to be connected by a layer of elastic translational and rotational springs. The coupled governing partial differential equations of motion are discretized, and the resulted eigenvalue problem is solved numerically by applying the Chebyshev spectral collocation method. The effects of the elastic layer parameters, the axial forces, the slenderness ratio, the bending–torsional coupling, and the boundary conditions on the critical buckling loads, mode shapes, and natural transverse frequencies have been studied. A parametric study was performed, and the obtained results revealed different features, which hopefully can be useful for single- and double-beam-like engineering structures.

Free vibration analysis of carbon nanotube reinforced composite Timoshenko beam

The paper deals with the free vibration analysis of nano-composite beams. The nano-composite beams are a combination of single walled carbon nano-tubes (SWCNT) as reinforcement fibre and are dispersed in the epoxy matrix. Material properties of the resulting nano-composite material are extracted using the Timoshenko beam theory. Finite element method (FEM) is employed to solve the governing equations of motion. The effect of various parameters such as boundary conditions, carbon nanotube (CNT) volume fraction, slenderness ratio and the distribution of nano-material on the natural frequency of the composite structure are studied and presented. The obtained results reveal that CNT volume fraction and distribution have substantial effect on the properties of the nano-composite beam.

Free vibration analysis of carbon nanotube reinforced composite Timoshenko beam

The paper deals with the free vibration analysis of nano-composite beams. The nano-composite beams are a combination of single walled carbon nano-tubes (SWCNT) as reinforcement fibre and are dispersed in the epoxy matrix. Material properties of the resulting nano-composite material are extracted using the Timoshenko beam theory. Finite element method (FEM) is employed to solve the governing equations of motion. The effect of various parameters such as boundary conditions, carbon nanotube (CNT) volume fraction, slenderness ratio and the distribution of nano-material on the natural frequency of the composite structure are studied and presented. The obtained results reveal that CNT volume fraction and distribution have substantial effect on the properties of the nano-composite beam.

The effect of finite strain on the nonlinear free vibration of a unidirectional composite Timoshenko beam using GDQM

In this manuscript, free vibrations of a unidirectional composite orthotropic Timoshenko beam based on finite strain have been studied. Using Green-Lagrange strain tensor and comprising all of the nonlinear terms of the tensor and also applying Hamilton's principle, equations of motion and boundary conditions of the beam are obtained. Using separation method in single-harmonic state, time and locative variables are separated from each other and finally, the equations of motion and boundary conditions are gained according to locative variable. To solve the equations, generalized differential quadrature method (GDQM) is applied and then, deflection and cross-section rotation of the beam in linear and nonlinear states are drawn and compared with each other. Also, frequencies of carbon/epoxy and glass/epoxy composite beams for different boundary conditions on the basis of the finite strain are calculated. The calculated frequencies of the nonlinear free vibration of the beam utilizing finite strain assumption for various geometries have been compared to von Karman one.

Effects of axial load and structural damping on wave propagation in periodic Timoshenko beams on elastic foundations under moving loads

The propagation and attenuation properties of waves in ordered and disordered periodic composite Timoshenko beams, which consider the effects of axial static load and structural damping, resting on elastic foundations are studied when the system is subjected to moving loads of constant amplitude with a constant velocity. The transfer matrix methodology is adopted to formulate the model in a reference coordinate system moving with the load. The localization factor is calculated to determine the wave velocity pass bands and stop bands. The interactions between the static axial load and moving load, structural damping and disorder on the bands are analyzed.

Vibration analysis of a rotating closed section composite Timoshenko beam by using differential transform method

This study introduces the Differential Transform Method (DTM) in the analysis of the free vibration response of a rotating closed section composite, Timoshenko beam, which features material coupling between flapwise bending and torsional vibrations due to ply orientation. The governing differential equations of motion are derived using Hamilton’s principle and solved by applying DTM. The natural frequencies are calculated and the effects of the bending-torsion coupling, the slenderness ratio and several other parameters on the natural frequencies are investigated using the computer package, Mathematica. Wherever possible, comparisons are made with the studies in open literature.

Non-Linear Thermal Stability Analysis of SMA Wire-Embedded Hybrid Laminated Composite Timoshenko Beams on Non-Linear Hardening Elastic Foundation

In the present study, non-linear thermal post-buckling analysis of hybrid laminated composite Timoshenko beams embedded with the shape memory alloy (SMA) wires resting on a non-linear hardening elastic foundation were studied. Mechanical properties of composite media are considered temperature-dependent. The theory of Timoshenko beams and von Kármán's strain-displacement relations are applied simultaneously in virtual work principles to derive the system of non-linear equilibrium equations. Various types of boundary conditions such as clamped, simply supported, and rolled edges were studied for edge supports. Generalized Differential Quadrature Method (GDQM) was utilized to discrete the equilibrium equations in space domain. Different types of lay-ups, such as symmetric and asymmetric, were considered. Post-buckling paths are depicted for different values of non-linear elastic foundation parameters, volume fractions, pre-strains of the SMA fibers, and boundary conditions. The one-dimensional thermo-mechanical constitutive law suggested by Brinson [1] is applied to model the SMA wires. Numerical results make it possible to recognize that increases in the volume fraction and pre-strain of SMA lead to a dramatic enhancement in thermal buckling and post-buckling capacity of the beam. Pre-buckling, buckling, and post-buckling behavior of the beam are totally different and this is due to variations among critical buckling, austenite start and finish temperatures. Due to the recovery stress of SMA wires, particular consequences are shown.

Dynamics of two-layer smart composite Timoshenko beams: Frequency domain spectral element analysis

Ultrasonic guided waves in a laminated composite structure can be excited or measured using piezoelectric transducers (PZTs). In this study, we propose a spectral element model for two-layer smart composite Timoshenko beams that consist of a host composite beam and a PZT layer. Contrary to the previous study (2012) where the Bernoulli-Euler beam theory was applied to the host composite beam, the Timoshenko beam theory and the Mindlin-Herrmann rod theory were applied in this study. The proposed spectral element model was evaluated first by comparison with standard finite element analyses, and it was then used to investigate the guided waves in some laminated composite beams.

Free vibration and postbuckling of laminated composite Timoshenko beams

Abstract A numerical solution method was developed to investigate the postbuckling behavior and vibrations around the buckled configurations of symmetrically and unsymmetrically laminated composite Timoshenko beams subject to different boundary conditions. The Hamilton principle was employed to derive the governing equations and corresponding boundary conditions which are then discretized by introducing a set of matrix differential operators. The pseudo-arc-length continuation method was used to solve the postbuckling problem. To study the free vibration that takes place around the buckled configurations, the corresponding eigenvalue problem was solved by means of the postbuckling configuration modes obtained in the previous step. The static bifurcation diagrams for composite beams with different lay-up laminates are given, and it is shown that the lay-up configuration considerably affects the magnitude of critical buckling load and postbuckling behavior. The study of the vibrations of composite beams with different laminations around the buckled configurations indicates that the natural frequency in the prebuckling domain increases as the stiffness of a beam increases, while there is no specific relation between the lay-up lamination and natural frequency in the postbuckling domain which necessitates conducting an accurate analysis in this area.

Exact Solution for Nonlinear Thermal Stability of Geometrically Imperfect Hybrid Laminated Composite Timoshenko Beams Embedded with SMA Fibers

AbstractThe nonlinear analysis of shape memory alloy (SMA) hybrid composite beams under in-plane thermal load is presented in this paper. The properties of the constituents are assumed to be temperature dependent. The initial imperfection of the beam is also taken into account as an initial deflection function prior to heating. Geometrical nonlinearity is formulated in the von Karman sense. It is assumed that the kinematics of the beam obeys the Timoshenko first-order beam theory and the SMA fibers follow the one-dimensional Brinson constitutive law. These basic assumptions are inserted into the static version of the virtual displacements principle to derive the nonlinear equilibrium equations. The resulting system of equations is uncoupled and solved exactly. Closed-form expressions are presented to trace the equilibrium path of the beam as a function of the uniform heating parameter. It is shown that, in special cases, the induced tensile recovery stress of the SMA fibers may stabilize an imperfect beam...

Finite strain analysis of nonlinear vibrations of symmetric laminated composite Timoshenko beams using generalized differential quadrature method

In this study, free vibrations of laminated composite Timoshenko beams undergoing finite strain have been studied. The Green-Lagrange strain tensor according to finite strain assumption has been used to obtain governing equations and boundary conditions of the beam. The generalized differential quadrature method has been employed for solving the governing equations and boundary conditions. Frequencies of the beam have been calculated for carbon/epoxy material subjected to different boundary conditions. The dimensionless frequencies of the beam for different lay-ups based on finite strain and von Karman assumptions have been obtained and compared to each other. Also, the effects of fiber orientation change and lay-ups sequence of angle-ply composite beams have been investigated on the lateral frequency of the beam.