Response Of Beam Research Articles

Nonlinear vibrations of multiscale composite beams on a nonlinear softening foundation

This paper describes a new approach to evaluate the instability of beams on a softening elastic foundation. The basis of this approach is solving a nonlinear frequency problem. The beam is made of multiscale composite materials based on carbon nanotubes (CNTs) and long fibers. Homogenization methods are utilized in a sequential process to obtain the effective material properties of these composites. Furthermore, the thermal environment's effect on the beam's response is examined. Reddy's third-order shear deformation theory approximates the induced displacements in the composite beam. Also, the kinetic and strain energies of the beam are calculated based on the von Kármán nonlinear strains and linear thermoelasticity theories. A recently reported Lagrangian nodal weak formulation (LNWF) method is implemented to obtain the linear and nonlinear stiffness and mass matrices governing the beam's vibrations. The modified coefficients for nonlinear stiffness matrices are obtained using the residual Galerkin method across time. Finally, the nonlinear natural frequencies are calculated by employing the iterative displacement control strategy. In the results section, in addition to examining various parameters on nonlinear free vibrations, the effect of these parameters on the displacement required for the loss of stability of the beam placed on the softening foundation is also examined.

An analytical solution for the vibrational response of stepped smart cross-ply laminated composite beams with experimental validation

This paper investigates the vibrational behaviour of smart orthotropic cross-ply laminated stepped beams. Specifically, an analytical closed-form model based on the first-order shear deformation theory (FSDT) is developed to consider a step change in laminate configuration along the beam axis due to integration of a piezoelectric patch for actuation applications. The 6th order governing equation within each element, along with appropriate continuity, equilibrium, and boundary equations, were formulated. State-space approach was utilised to find the beam's response to piezoelectric sinusoidal actuation. Results of the natural frequency analysis were compared with results in the literature and finite element modelling. An experimental investigation examined the accuracy of the analytical method to find the natural frequencies and forced vibrational response of a smart carbon-fibre/polyetheretherketone (PEEK) cantilever beam. Comparisons demonstrate that the provided closed-form method is capable of efficiently predicting the free and forced vibrational behaviour of smart stepped laminated composite beams with high accuracy.

Integral and differential approaches to Eringen's nonlocal elasticity models accounting for boundary effects with applications to beams in bending

AbstractThe Eringen's fully nonlocal elasticity model is known to lead to ill‐posed boundary‐value problems and to suffer some boundary effects arising from particle interactions impeded by the body's boundary surface. An enhanced model is derived from the original fully nonlocal one by the addition of a regularizing non‐homogeneous local phase which accounts for boundary effects and which leads to a Fredholm integral equation of the second kind, hence to well‐posed boundary‐value problems, without paradoxes, nor other drawbacks. The enhanced integral model applied to a beam in bending proves to be equivalent to a sixth order differential equation with variable coefficients, with extra nonlocality boundary conditions here also derived. Both the integral approach and the differential one lead to a same unique solution of the small‐scale beam problem. An efficient numerical algorithm is presented in which the sixth order differential equation with variable coefficients is reduced to one of the second order, which is addressed by a finite difference method. The proposed theory is applied to a set of engineering beam problems, for each of which the inherent size effects are reported and graphically illustrated. The influence of the length scale parameter upon the beam's response is highlighted by means of a function representing the normalized maximum deflection of the beam as a function of the length scale parameter. It is shown that the enhanced model always predictssoftening size effectsno matter the boundary and loading conditions, and that the related response function generally exhibits awaved pattern with positive slopes first, then negative, as the length scale parameter increases, with a limit asymptotic behavior like an atomic lattice model. A comparison with other theories is also presented together with possible future developments.

An energy harvesting and damage sensing solution based on postbuckling response of nonuniform cross‐section beams

Postbuckling response of elastic beams has been widely used in many systems to develop efficient energy harvesting and damage sensing mechanisms under quasistatic excitations. In particular, the snap-through behavior of bilaterally constrained beams can be used to transform low-frequency and low-rate excitations into high-rate motions. Using a piezoelectric energy harvester, these motions are converted into electric power. However, the efficiency of buckling-based energy harvesters highly depends on the postbuckling behavior of the buckled elements. Inadequate control over the beam's response critically impedes the application of the mechanism. This study aims to control the location of the snap-throughs and the spacing between the transitions in order to increase the levels of the harvested energy and tune the sensitivity of the sensor. As uniform prismatic beams do not allow for such control, nonprismatic cross-section beams are herein investigated. An energy-based theoretical model is developed in this study to investigate the effect of different shapes and geometries on the postbuckling response of nonuniform beams. The total potential energy of the system is minimized under constraints that represent the physical confinement between the lateral boundaries. Experimental results prove that the theoretical model is accurate and can be used to optimize the buckling elements. Results show that the spacing between the transitions can be tuned. Furthermore, an optimal buckling element can improve the energy conversion efficiency by more than 290%.

Nonlinear vibration of viscoelastic beams described using fractional order derivatives

The problem of non-linear, steady state vibration of beams, harmonically excited by harmonic forces is investigated in the paper. The viscoelastic material of the beams is described using the Zener rheological model with fractional derivatives. The constitutive equation, which contains derivatives of both stress and strain, significantly complicates the solution to the problem. The von Karman theory is applied to take into account geometric nonlinearities. Amplitude equations are obtained using the finite element method together with the harmonic balance method, and solved using the continuation method. The tangent matrix of the amplitude equations is determined in an explicit form. The stability of the steady-state solution is also examined. A parametric study is carried out to determine the influence of viscoelastic properties of the material on the beam's responses.

On the dynamic behaviour of masonry beam–columns: An analytical approach

The paper presents an analytical approach to the study of the transverse vibrations of masonry beam–columns. Starting with the constitutive equation for beams made of a masonry–like material and the averaged Lagrangian of the system, some explicit approximate solutions are found to the problem of free damped periodic oscillations and forced oscillations in the case of primary resonance on the beam's first mode. In particular, a set of equations is obtained that gives the modulation over time of the system's energy and of the fundamental frequency of the beam's response. The analytical results are compared to those obtained via the finite element code NOSA–ITACA, developed at ISTI–CNR.

A new FEM approach for analysis of beams with relative movements of masses

The dynamics of beam–moving mass systems is considered. A ‘composite’ beam element is introduced, which explicitly identifies the Coriolis and centripetal effects dependent on the given current relative velocity of the particular mass. These effects are then mimicked in the numerical procedure by applying certain fictitious transversal and axial forces only to the elements being currently traversed by the masses. It is shown that the approach proposed is capable of recreating accurately and efficiently several analytical solutions reported in the literature and obtained mostly for a beam and masses moving with constant velocities, the cases pertaining to simulating a bridge–traveling vehicles interaction. The importance of the Coriolis and centripetal effects on the beam's response is quantified in terms of the parameters representing the dimensionless mass and its dimensionless velocity. Finally the interaction of a disturbed beam with a mass moving in a cyclic pattern is analyzed resulting in attenuation and amplification phases of the beam's vibrations, as expected. This case, for which an accurate recreation of the Coriolis effects is crucial, demonstrates that the approach can also be used for control purposes to explore numerically the possibilities of eliminating the system's vibrations by properly synchronized relative motion of its components.

Spectral density of the bridge beam response with uncertain parameters under a random train of moving forces

The paper presents the spectral analysis of the beam's vibration with uncertain parameters under a random train of moving forces which forms a filtered Poisson process. It is assumed that natural frequencies of the bridge beam are uncertain and are modelled by fuzzy numbers, random variables or fuzzy random variables. In order to obtain general solutions for the spectral density function of the beam's response the normal mode dynamic influence function has been introduced. As an example the spectral density functions of a bridge modelled as a simple supported beam are determined.

Dynamic pulsebuckling and postbuckling of composite laminated beam using higher order shear deformation theory

Abstract The dynamic pulsebuckling and postbuckling response of fiber-reinforced plastic (FRP) laminated beams, having initial geometric imperfections, subjected to an axial impact was simulated by the finite difference method. The Timoshenko beam theory was used to construct the analytical model in which all inertia components of the beam that affect the dynamic response were considered. The von-Karman non-linear strain–displacement relationship was used to describe the beam's response and the higher order shear deformation theory was used to model the displacement field of the beam. The influence of boundary conditions and amplitude of imperfection on the pulsebuckling and postbuckling response of FRP beams was investigated. The results obtained from the proposed formulation agree well with those of the 3D-finite element analysis. The numerical results show that the rotary restraints do not significantly affect the critical pulsebuckling momentum.

Dynamic response of elastic-plastic beams with axial constraints

Dynamic response of axially restrained beams under transverse impulsive loading is discussed. The material of the beam is elastic-plastic with linear strain-hardening. The ends of the beam are clamped against end rotations and axial displacements. The equations of motion are integrated by the method of finite differences. Computations for different material and beam parameters are carried out. These results enable us to investigate the character of the beam's response. After reaching the peak deflection the beam responds with nonlinear vibrations. In order to analyse this phase of motion an approximate solution is proposed. According to this the motion subsequent to the peak deflection is regarded as elastic. More details of the beam's single-degree-of-freedom model are discussed. In spite of its artificiality, this model describes anomalous behavior of the beam, which was found by Symonds et al. for a Shanley-type beam.

Parameters Affecting FE Analysis of Concrete Structures

The effects of the shape of the descending branch of the tensile stress‐strain curve, fracture energy, grid refinement, and load‐step size on the response of finite element models of a notched concrete beam are studied. The width of the process zone and constraints imposed on crack angles are also investigated. Cracking of the concrete in tension is the only nonlinear behavior modeled. Load‐deflection curves and general cracking patterns provide the primary means for evaluating response. Comparisons with discrete crack models are made. The fracture energy, tensile strength, and shape of the descending branch interact to determine the stiffness and general behavior of the specimen. The width of the process zone has a negligible influence on the beam's response. The importance of proper crack orientation is demonstrated. The model is objective with respect to grid refinement and load‐step size. The implications of the results are discussed with respect to modeling reinforced concrete structures.

Design and Non-Linear Analysis of a Parabolic Leaf Spring

Tapered cantilever beams, traditionally termed as leaf springs, undergo much larger deflections in comparison to a beam of constant cross-section that takes their study in the domain of geometric nonlinearity. This paper studies response of a leaf spring of parabolic shape, assumed to be made of highly elastic steel. Numerical simulation was carried out using both the small and large deflection theories to calculate the stress and the deflection of the same beam. Non-linear analysis is found to have significant effect on the beam's response under a tip load. It is seen that the actual bending stress at the fixed end, calculated by nonlinear theory, is 2.30-3.39% less in comparison to a traditional leaf spring having the same volume of material. Interestingly, the maximum stress occurs at a region far away from the fixed end of the designed parabolic leaf spring. Keywords: Parabolic leaf spring, End-shortening, Geometric nonlinearity, Equilibrium Configuration Path, Varying Cross-section.doi:10.3329/jme.v37i0.819Journal of Mechanical Engineering Vol.37 June 2007, pp.47-51