Classical Beam Theory Research Articles

Response of functionally graded beams subjected to a moving load and different types of attachments

This paper develops an analytical solution for the dynamic response of a Euler-Bernoulli functionally graded (FG) beam subjected to a load moving with constant velocity and different types of attachments. The classical beam theory (CBT) is used to derive the equation of motion. The material properties of the beam are considered to vary continuously in the transverse direction according to an exponential and a power law models. The stress distribution is evaluated and the location of the neutral axis of the FG beam is obtained. Different types of attachments connected to the beam at specific locations are analysed. The effect of using FG materials to enhance the properties of the beam, resulting in a decreased response of the beam, is made clear. Also, the effect of the restoring force of attachments in reducing the dynamic response of the beam is discussed.

Flax/Epoxy Composite Beams - Simulation of Flexural Stresses

Bio-based fibre reinforcements (hemp, jute, flax) derived from agricultural waste and forestry are increasing its potential towards advanced composites, promoting environmental benefits and thermal recycling (including a reduction in CO2 emissions and the fossil fuel depletion). Currently, there is a growing interest of natural fibres due to its lower density, and higher modulus-to-density ratioscompared to glass and carbon fibre reinforcements. The present work was intended to understand how the flax fibre layups and orientation affect the behaviour of laminated composites in bending. Unidirectional [00]2S, cross-ply [0/90]S, and angle-ply [+45/-45]S laminates made up of flax fibre reinforced epoxy composites are considered to study flexural stresses and mid-span deflections. Basic principles of the classical beam theory (CBT) are applied for obtaining analytical solutions, which were also compared with the finite element simulation results.

Stress and damping of wide cantilever beams under free vibration

Research has shown that the damping of a vibrating structure is highly dependent on its stress function. In this study, the bending stress and damping of wide cantilever beams under free vibration were analyzed using the classical plate and beam theory. The damping stress equation for cantilever beams under free vibration was derived based on the empirical function of unit dissipating energy, whereas the plate bending equation was derived using the double finite integral transform method. The bending stress and damping ratio results from the beam and the plate theory were compared with simulation results from finite element analysis (FEA) for different length-to-width ratios. Results show that the plate theory displayed a good agreement with FEA results in terms of estimated value and trending curve shape when a significantly large number of terms were used. Using a small number of terms resulted in large errors at high length-to-width ratios, but provided sufficient estimates when the length-to-width ratio dropped below four. It was found that the beam theory was only valid for beams with very high length-to-width ratios or square plates. Beyond this ratio, the beam theory recorded a higher error estimate than the plate theory. Overall, the most accurate stress and damping estimations come from the use of plate theory with a very high number of terms.

An analytical solution for static problems of curved composite beams

Abstract An analytical solution is presented for the determination of deformation of curved composite beams. Each cross-section is assumed to be symmetrical and the applied loads are acted in the plane of symmetry of curved beam. In-plane deformations are considered of composite curved beams. Assumed form of the displacement field assures the fulfillment of the classical Bernoulli-Euler beam theory. The curvature of beam is constant and the internal forces in a cross-section is replaced by an equivalent forcecouple system at the origin of the cylindrical coordinate system used. The internal forces are expressed in terms of two kinematical variables, which are the radial displacement and the rotation of the cross-sections. The determination of the analytical solutions of the considered static problems are based on the fundamental solutions. Linear combination of the fundamental solutions which are filling to the given loading and boundary conditions, gives the total solution. Closed form formulae are derived for the radial displacement, cross-sectional rotation, nomral and shear forces and bending moments. The circumferential and radial normal stresses and shear stresses are obtained by the integration of equilibrium equations. Examples illustrate the developed method.

An approximate solution for the inverted four-point bending test in symmetric specimens

An approximate solution is derived for the interfacial energy release rate of the inverted four-point bending test. The analysis builds on a previous model developed by one of the authors for an orthotropic edge-cracked layer subject to arbitrary generalized end forces. Contact forces exerted from the upper on the lower layer of the edge-cracked portion behind the delamination tip are introduced. Their value is chosen such that the two layers undergo the same deflection. The effects of both shear deformations along the layers and friction at the point of contact are taken into account within beam theories and approximate Coulomb model with a prescribed friction coefficient. The delamination energy release rate is derived for homogeneous and symmetric specimens. A parametric analysis is performed on varying the mechanical parameters of the model to analize the influence of shear deformations and friction. The results show that both shear deformations and friction affect the value of the energy release rate for short/intermediate interfacial cracks. For very long interfacial cracks the delamination energy release rate tends to a constant limit value which corresponds to that obtained within classical Euler-Bernoulli beam theory in absence of friction.

Deformation of rotating two-layer curved composite beams

Abstract An analytical solution is presented for the determination of the deformation of rotating two-layer composite beams. The direction of axis of rotation is vertical and the speed of rotation is constant. The axis of rotation is in the plane of symmetry of curved beam. The source of the in-plane deformation is the stationary rotation of the curved beam. The plane of the curvature is the symmetry plane of the curved beam for its material, geometrical and supporting properties. Assumed form of the displacement field meets the prescriptions of the classical Euler-Bernoulli beam theory. Examples illustrate the applications of the presented analytical solution.

Dynamic Amplification Factor of Shear Force on Bridge Columns under Impact Load

Shear failure is a common mode for bridge column collapse during a vehicle‐column collision. In current design codes, an equivalent static load value is usually employed to specify the shear capacity of bridge columns subject to vehicle collisions. But how to consider the dynamic effect on bridge columns induced by impact load needs further research. The dynamic amplification factor (DAF) is generally used in the analysis and design to include the dynamic effect, which is usually determined using the equivalent single degree of freedom (SDOF) method. However, SDOF method neglects the effect of the higher‐order modes, leading to big difference between the calculated results and the real induced forces. Therefore, a novel method to obtain dynamic response under concentrated impact load including the effect of higher‐order modes is proposed in the paper, which is based on the modified Timoshenko beam theory (MTB) and the classical Timoshenko beam theory (CTB). Finite element models are conducted to validate the proposed method. The result comparisons show that the results from the proposed method have more accuracy compared with the results from the CTB theory. Additionally, the proposed method is employed to calculate the maximum DAF of shear forces for bridge columns under impact load. Parametric studies are conducted to investigate the effect on the DAF of shear forces including slenderness ratio, boundary condition, and shape and position of impact load. Finally, a simplified formula for calculating the maximum DAF of shear force is proposed for bridge column design.

Analysis and Optimization of a Piezoelectric Energy Harvester

The paper aims to assess and improve the performances of a multilayer piezoelectric MEMS device for vibrations harnessing. Two operating modes are possible: at resonance and outside resonance. In some applications it is not possible to operate at resonance, functioning being mostly at low frequencies in a quasi-static regime. An Euler-Bernoulli classic beam theory mathematical model was studied for estimating the behaviour of multilayer piezoelectric generators, in terms of deflection and voltage, at functioning under resonance frequency. The analytical results were compared with the finite element method simulation in COMSOL Multiphysics. The main goal of this study is to obtain an accurate model for engineering design purposes, with simple analytical equations and ease of use, but with predictable errors. The study proved the usefulness of the derived model but also its limitations. It also proves the need to improve the model using plate theory, for sensors with high width/height ratio.

Optimized design of RC deep beams based on performance metrics applied to strut and tie model and in-plane stress conditions

Abstract Deep beams do not behave according to classical beam theory. The nonlinearity of strain distribution within these elements requires application of strut and tie models (STM) or other alternatives to evaluate the complex stress field. Although the design of these elements is a common task for structural engineers, limited research is found on assessing effectiveness of the results. The purpose of this work is to compare, in a systematic approach, different design solutions for a deep beam using selected performance metrics which are strain energy, reinforcement ratio, maximum load, structural efficiency, safety factor and cracking behavior. A deep beam (2.85 m of height, 4.20 m of length and 0.2 m of thickness) with a square opening (0.7 m x 0.7 m) close to one of the supports was subjected to a uniform loading at the top surface while resting on supports at both ends. A simplified finite element model (FEM) of this beam was developed simulating concrete with elastic linear stress-strain behavior and disregarding steel reinforcements. This model allowed determination of elastic stress fields necessary to subsequent analyses. Four STM were then developed, supposing the total load respectively represented by one (STM-1), two (STM-2), four (STM-4) or eight (STM-8) concentrated loads equally spaced along the top of the beam. Additionally, an in-plane stress field method (SFM) was applied to the design of the same beam subjected to uniform loading on the top surface. After design and detailing of the reinforcement for each situation, nonlinear FEMs were used to predict the ultimate conditions. The strain energy reduced significantly comparing results from STM-1 to STM-2 and subsequently to STM-4 and remained at a low level in STM-8 and SFM. The reinforcement ratio reduced systematically from STM-1 to STM-8, was minimum with the SFM and the same behavior was followed by maximum load. The structural efficiency (maximum load/reinforcement ratio) increased from STM-1 to STM-8, with maximum efficiency at STM-8 and was slightly below with SFM. The safety factor reduced systematically from STM-1 to STM-8 and was slightly lower with SFM, but in all cases was above acceptable limits found in design codes.

Magnetic field effects on thermally affected propagation of acoustical waves in rotary double-nanobeam systems

ABSTRACT Herein, the thermo-magneto-elastic wave dispersion answers of rotary functionally graded (FG) double-nanobeam systems (DNBSs) are surveyed implementing a nonlocal strain gradient theory (NSGT). The kinematic relations are derived employing the classical beam theory. Also, scale influences are covered precisely in the framework of NSGT. Moreover, Mori-Tanaka homogenization model is introduced in order to obtain the effective material properties of FG nanobeams. Meanwhile, effects of external forces such as thermal and Lorentz forces are included in this research. Also, based upon the Hamilton's principle, the Euler–Lagrange equations are developed; afterwards, these equations are incorporated with those of NSGT to reach the nonlocal governing equations of FG-DNBSs. Furthermore, according to an analytical approach, the governing equations are solved to obtain the wave frequencies and phase velocities of FG-DNBSs. At the end, some illustrations are rendered to clarify the influences of a wide range of involved parameters.

Dynamic response of axially functionally graded beam with longitudinal–transverse coupling effect

This study investigates the dynamic response of an axially functionally graded (AFG) beam with longitudinal–transverse coupling effect. The beam is acted by a moving transverse harmonic load and a moving longitudinal harmonic load simultaneously. Both the classical beam theory (CBT) and the Timoshenko beam theory (TBT) are employed and written in a unified form. The system of equations of motion is obtained using Lagrange's equations. The nonlinear formulations of the longitudinal–transverse coupling AFG beams are presented. These formulations are solved using the Newmark method in conjunction with a direct iteration procedure; subsequently, the free and forced dynamic behavior characteristics of the AFG beams are investigated. A comparison of the results obtained using the TBT and CBT is performed. The nonlinear effects caused by different excitation frequencies and amplitudes both in the transverse and longitudinal directions of the moving loads on the dynamic responses of the AFG beam are discussed. Further, different nonlinear dynamic behaviors of the AFG beam caused by different longitudinal–transverse coupling coefficients are revealed. The numerical results indicate that the longitudinal–transverse coupling effect plays an important role in the dynamic behavior of the AFG beam.

The vibration of nanobeam resting on elastic foundation using modified couple stress theory

In this paper, the vibration of nanobeams resting on the Winkler foundation is proposed using the modified couple stress theory. Hamilton’s principle is utilized to construct the governing equations. The size effect of the nanobeam cannot be captured by using classical Euler-Bernoulli beam theory, but the modified couple stress theory model can capture it because it includes material length scale parameter that a newly developed model has. Once the material length scale parameter is assumed to be zero, the classical Euler-Bernoulli beam theory equation is obtained. Multiple scale method is employed to obtain the result. Simply supported boundary condition is used to study natural frequencies. The influence of material length scale parameter and the Winkler elastic foundation parameter on the fundamental frequencies of the nanobeam is investigated and tabulated. Also, in the present study, Poisson’s ratio is taken as constant. Nanobeam resting on the Winkler foundation which is simply supported is analyzed to illustrate the size effects on the free vibration. Numerical results for the simply supported nanobeam indicate that the first fundamental frequency calculated by the presented model is higher than the classical one. Moreover, it is obtained that the size influence is more substantial for higher vibration modes. The results indicate that the significant importance of the size influences the analysis of nanobeams. The vibration of nanobeam exhibits a hardening spring behavior, and the newly developed models are the beams stiffer than according to the classical beam theory. Modified couple stress theory tends to be more helpful in describing the size-dependent mechanical properties of nanoelectromechanical systems (NEMS).

Analytical Study of Elastic-Plastic Fracture in the Crack-Lap Shear Multilayered Beam Configuration

Abstract This paper reports an analytical study of delamination fracture in the Crack-Lap Shear (CLS) multilayered beam configuration with taking into account the material non-linearity. A delamination crack was located arbitrary along the beam height. It was assumed that the CLS mechanical response can be described by using a power-law stress-strain relation. It should be mentioned that each layer may have different material constants in the stress-strain relation. Besides, the thickness of each layer may be different. The classical beam theory was applied in the present study. The non-linear fracture behaviour was analyzed by the J-integral. Analytical solutions of the J-integral were obtained for homogeneous as well as for multilayered CLS beams. In order to verify the solutions obtained, analyses of the strain energy release rate were developed with considering material non-linearity. Material properties and crack location effects on the non-linear fracture behaviour were investigated. The analysis revealed that the J-integral value increases when the material non-linearity is taken into account. It was found also that the J-integral value decreases with increasing the lower crack arm thickness. The approach developed here is very convenient for parametric fracture analyses. The solutions derived can be used for optimization of the CLS multilayered beams with respect to their fracture performance.

On the fundamental natural frequency of bridge decks: review and applications

The fundamental frequency of vibration of a bridge deck is a key parameter that represents the dynamic characteristics and the condition of the superstructure. This paper reviews the different empirical and/or theoretical formulae that have been proposed in the past. The classical vibration theory of beams and plates was simplified in order to clarify and review the general philosophy behind the previously proposed formulae. On the basis of the results obtained from limited dynamic tests performed by the authors on a number of reinforced concrete bridges and the results of limited tests conducted by other researchers, a formula for the first bending natural frequency of a bridge deck was derived. Moreover, as new applications, new practical formulae are introduced that can be used for estimating the initial dimensions of a deck cross-section for simple span bridges, which are particularly suited to bridge-widening projects. The first bending natural frequency/period of the bridge deck and its corresponding cross-sectional properties are considered here as functions of the bridge main span length.

Optimal locations of magnetorheological fluid pockets embedded in an elastically supported honeycomb sandwich beams for supersonic flutter suppression

In this paper, the optimal locations of finite number of embedded magneto-rheological fluid (MRF) pockets are calculated by using Single Objective Genetic Algorithm (SOGA) to suppress the supersonic flutter of an elastically supported sandwich beam with regular honeycomb interlayer. The structural formulation is based on the classical beam theory along with the Winkler-Pasternak foundation model, the MRF is modeled as a first order Kelvin-Voigt material, and the quasi-steady first order supersonic piston theory is employed to define the aerodynamic loading. The aeroelastic time response of honeycomb sandwich beams with different number of embedded MRF pockets are measured with/without foundation and flow stream under impact load. The simulation shows that by using MRF pockets in optimized locations in honeycomb sandwich structure, not only both lightness of honeycombs and variability of structural damping of MRF are considered, but also the supersonic flutter can be postponed to higher dynamic pressure compared to entirely honeycomb or MRF sandwich beams. Finally, in order to achieve the optimum compromise between mass and the flutter suppression, the best number of MR pockets along with the optimum magnetic field values in honeycomb sandwich structure is presented.

Size-dependent electro-thermo-mechanical analysis of multilayer cantilever microactuators by Joule heating using the modified couple stress theory

The size-dependent electro-thermo-mechanical responses of multilayer cantilever microactuators by Joule heating is investigated. An analytical expression of the deflection of three-layered cantilever microactuators subjected to non-uniform temperature variations and mechanical loading is derived by using the modified couple stress theory and the two-variable method. The predicted result is in good agreement with the experimental data. Compared with the results from the classical beam theory, the influence of the size-dependency on the deformation, equivalent bending rigidity and average thin film stress are discussed. The results reveal that the equivalent bending rigidity of multilayer cantilever microactuators predicted by the present model is larger than that from the classical beam model, which decreases the size-dependent bending response of the microactuator. In addition, the size-dependent average stresses of thin film is more likely to produce mechanical failure. The current study is helpful for the design of micro/nano-actuators.

A Finite Element Analysis of the Stability of Composite Beams With Arbitrary Curvature

A finite element approximation of a theory recently proposed for the geometrically nonlinear analysis of laminated curved beams is developed. The application of the given finite element model to the computation of stability points and post-buckling behavior of beams with arbitrary curvature is also carried out, on taking into account the influences of shear deformation and warping effects on the in-plane and out-plane responses of the beam. The stability analysis is performed through a path-following procedure and a bordering algorithm. Several numerical results are given and comparisons with classical beam theories and other theories available in the relevant literature are established. The given results highlight that the proposed finite element model is well suited to study the stability of structures that incorporate laminated composite beams, such as, e.g., light-weight roof structures and arch bridges.

Flexural buckling and post-buckling of columns made of aluminium alloy

The paper concerns flexural buckling and initial post-buckling of axially compressed columns made of aluminium alloy described by the Ramberg-Osgood relationship. The non-linear differential equation of the problem is derived using the stationary total energy principle and the assumptions of classical beam theory within a finite range. The approximate analytical solution of the equation leading to the buckling loads and initial post-buckling equilibrium path is determined by means of the perturbation approach. Numerical examples dealing with simply supported and clamped I-columns are presented, the effect of the material non-linearity on the critical loads and initial post-buckling behaviour in comparison to linear one is discussed too. The analytical results are compared with the FEM solutions to present a good agreement.

Multiscale topology optimization using neural network surrogate models

We are concerned with optimization of macroscale elastic structures that are designed utilizing spatially varying microscale metamaterials. The macroscale optimization is accomplished using gradient-based nonlinear topological optimization. But instead of using density as the optimization decision variable, the decision variables are the multiple parameters that define the local microscale metamaterial. This is accomplished using single layer feedforward Gaussian basis function networks as a surrogate models of the elastic response of the microscale metamaterial. The surrogate models are trained using highly resolved continuum finite element simulations of the microscale metamaterials and hence are significantly more accurate than analytical models e.g. classical beam theory. Because the derivative of the surrogate model is important for sensitivity analysis of the macroscale topology optimization, a neural network training procedure based on the Sobolev norm is described. Since the SIMP method is not appropriate for spatially varying lattices, an alternative method is developed to enable creation of void regions. The efficacy of this approach is demonstrated via several examples in which the optimal graded metamaterial outperforms a traditional solid structure.

Mechanical properties and buckling analysis of woven fabric

The buckling of a woven fabric influences its performance throughout processing and during actual use. In addition, this phenomenon reflects on the recovery of the textile–polymer composites. The mechanics of the buckling behavior of woven, knitted or non-woven fabrics have a large number of different approaches, and the obvious inconsistency of applying the classical beam theory to the textile problem remains. The buckling of textile materials is a deciding factor of their drape and conformability. Fabric buckling also defines the ability of a fabric to fit to a different shape object without wrinkling. An empirical approach foundation to quantitatively compare the buckling of existing and new flexible textile materials, no matter what their composition, was developed. The buckling coefficient was found to be highly correlated to the fabric modulus of rigidity, fabric Young's modulus and fabric weave design. For the twill structure, the value of twill line angle to the direction of buckling force has a significant effect. In this paper, the buckling of woven fabric with or without holes was analyzed. It has been proven that major modes of deformation in the buckling of fabrics with holes are fabric bending rigidity, fabric design and the ratio of the volume of the hole to sample volume.