Continuous Beam Model Research Articles

On the effectiveness of principal component analysis for decoupling structural damage and environmental effects in bridge structures

An accurate and reliable identification of structural damage is of prime importance to evaluate the structural integrity of civil infrastructure systems. However, the adverse effect of normal fluctuations in the environment on the effectiveness of damage detection techniques remains a continuing challenge in structural health monitoring applications. In this paper, we present the application of principal component analysis (PCA) to temporal damage detection in continuous beam bridge structures subjected to changing environmental effects. For this purpose, we show that sudden discontinuities in the principal components occur at the onset of damage, and that these discontinuities are observed in the projections of the vibration data on the principal components space. The magnitude of the discontinuity is used to define a damage index for damage quantification. A comprehensive numerical study is used to validate the approach on a continuous beam model of highway bridge structures. In particular a sensitivity analysis is conducted to study the effect of both temperature-dependent boundary conditions and material properties on the principal components for multiple damage scenarios. The numerical results show that the approach is robust to mild nonlinearities caused by the effect of temperature on material properties of composite steel-concrete sections and boundary conditions. Furthermore, the approach is experimentally validated using data of the Z24 bridge in Switzerland measured during a period of one year. It is shown that the approach has the capability of tracking the temporal evolution of various damage states induced on the bridge during the testing program.

Assessment of Floor Micro-Vibrations Induced by Moving Vehicles in High Technology Factories Using a Fragility-Based Method

This study presents assessment of floor micro-vibrations induced by moving vehicles in high technology factories by using a fragility-based method. A series of time history analysis of a simplified three-span continuous beam model under moving loads simulated by a modified Kanai-Tajimi power spectral density (PSD) function considering various vehicle weights and moving speeds is performed. The best-fitting surface in terms of the maximum root mean square (RMS) velocity responses and their corresponding pair of the vehicle weight and moving speed is obtained by using the least square regression method. With the regression surface denoting the mean values and the standard deviation obtained, the fragility surfaces for each vibration criterion or limit are further constructed. The assessment of the floor micro-vibration in terms of probability of exceeding a given vibration limit for various pairs of vehicle weight and moving speed can be read accordingly from the fragility surfaces.

Equivalent Timoshenko linear beam model for the static and dynamic analysis of tower buildings

• Development of a general framework for evaluation of the equivalent beam stiffness . • Proposal of an innovative, constitutive linear law for the equivalent model. • Identification of a measure for the flexural-to-shear ratio in building deformation. • Derivation of motion equations generalizing the classic 3D Timoshenko model. • Easy-to-use procedure to regain deformation and stress state on the single element. A continuous Timoshenko linear beam model immersed in a three-dimensional space is introduced to study the static and dynamic behavior of tower buildings. A schematization of the building as a periodic system with rigid floors connected by deformable elements (columns and shear walls) is considered. The rigid floors are endowed with six degrees of freedom (three displacements and three rotations). The constitutive equations of the equivalent beam (coarse model) are identified from a discrete model of the three-dimensional frame (fine model) via a homogenization procedure. A complete linear constitutive law is obtained, with axial force coupled with bending and shear force coupled with torsion. The first aim is to investigate the relative importance of the macro-shear and macro-bending contributions to the deformation of the building. Then, the ability of the coarse model to reproduce the local stress distribution of the fine model is checked. Finally, the representativeness of the coarse model for the detection of the natural frequencies of the fine model is analyzed.

Vibration-based structural health monitoring under changing environmental conditions using Kalman filtering

A Kalman filtering based framework for structural damage assessment under changing environmental conditions is presented. The approach is based on the well-known property that the filtering residual is a realization of a white stochastic process when the filter is operating under optimal conditions. To decouple structural damage and environmental effects two additional properties of the filtering residual are employed: i) under global changes in the structure caused by environmental variations the residual remains a white process, and thus its spectral density is approximately constant; ii) local changes caused by structural damage induce peaks in the residual spectral density at the affected vibration frequencies, and thus the residual is a colored process. A Bayesian whiteness test is employed to discriminate between the two situations under finite length data conditions (damage detection), while a normalized damage measure based on the spectral moments of the residual spectral density is proposed as a quantitative damage-sensitive feature (damage quantification). The proposed approach is numerically verified in a continuous beam model of a bridge under different operating conditions, including a robustness assessment for non-uniform temperature fields. It is shown that the approach has the capability to decouple physical changes caused by structural damage and varying environmental conditions, providing robust damage measures for structural health monitoring applications.

Modeling of flutter stability margins in an aerofoil cantilever wing with multiple engines mounts under inherent structural nonlinearities

This paper presents an aeroelastic analysis of aircraft wing having structural nonlinearities with two power plants subjected to time-dependent thrust loads in the event of failure of one of the engines. The equations of motion are derived based on continuous beam model of the wing system with two engines subjected to time dependent thrust by using Lagrange’s formulation. The placement of engine at various positions along the span within variable distances between the engines enables us to locate the optimum configuration of engine placement. Firstly, the dynamic responses of the clean wing-with and without cubic nonlinearity- are studied, before illustrating the dynamic analysis of wing with engines subjected to thrust. The parametric analysis is carried out to find the most influencing parameter in the prediction flutter of the system. The multiple regression analysis as well as artificial neural network training is carried out to identify the flutter velocity for a given configuration of the system. Then, the system is subjected to various types of time-dependent engine thrusts, before studying its effects on dynamic responses in the event of a sudden failure of an engine. The flutter velocity boundaries are validated with the published data that generate some significant results.

Bending of an elastoplastic Hencky bar-chain: from discrete to nonlocal continuous beam models

The static behavior of an elastoplastic one-dimensional lattice system in bending, also called a microstructured elastoplastic beam or elastoplastic Hencky bar-chain (HBC) system, is investigated. The lattice beam is loaded by concentrated or distributed transverse monotonic forces up to the complete collapse. The phenomenon of softening localization is also included. The lattice system is composed of piecewise linear hardening–softening elastoplastic hinges connected via rigid elements. This physical system can be viewed as the generalization of the elastic HBC model to the nonlinear elastoplasticity range. This lattice problem is demonstrated to be equivalent to the finite difference formulation of a continuous elastoplastic beam in bending. Solutions to the lattice problem may be obtained from the resolution of piecewise linear difference equations. A continuous nonlocal elastoplastic theory is then built from the lattice difference equations using a continualization process. The new nonlocal elastoplastic theory associated with both a distributed nonlocal elastoplastic law coupled to a cohesive elastoplastic model depends on length scales calibrated from the spacing of the lattice model. Differential equations of the nonlocal engineering model are solved for the structural configurations investigated in the lattice problem. It is shown that the new micromechanics-based nonlocal elastoplastic beam model efficiently captures the scale effects of the elastoplastic lattice model, used as the reference. The hardening–softening localization process of the nonlocal continuous model strongly depends on the lattice spacing which controls the size of the nonlocal length scales.

Novel threshold pressure sensors based on nonlinear dynamics of MEMS resonators

Triggering an alarm in a car for low air-pressure in the tire or tripping an HVAC compressor if the refrigerant pressure is lower than a threshold value are examples for applications where measuring the amount of pressure is not as important as determining if the pressure has exceeded a threshold value for an action to occur. Unfortunately, current technology still relies on analog pressure sensors to perform this functionality by adding a complex interface (extra circuitry, controllers, and/or decision units). In this paper, we demonstrate two new smart tunable-threshold pressure switch concepts that can reduce the complexity of a threshold pressure sensor. The first concept is based on the nonlinear subharmonic resonance of a straight double cantilever microbeam with a proof mass and the other concept is based on the snap-through bi-stability of a clamped-clamped MEMS shallow arch. In both designs, the sensor operation concept is simple. Any actuation performed at a certain pressure lower than a threshold value will activate a nonlinear dynamic behavior (subharmonic resonance or snap-through bi-stability) yielding a large output that would be interpreted as a logic value of ONE, or ON. Once the pressure exceeds the threshold value, the nonlinear response ceases to exist, yielding a small output that would be interpreted as a logic value of ZERO, or OFF. A lumped, single degree of freedom model for the double cantilever beam, that is validated using experimental data, and a continuous beam model for the arch beam, are used to simulate the operation range of the proposed sensors by identifying the relationship between the excitation signal and the critical cut-off pressure.

Pull-in-free design of electrically actuated carbon nanotube-based NEMS actuator assuming non-parallel electrodes arrangement

In this numerical investigation, the nonlinear structural response of an electrostatic carbon nanotube (CNT) based nano-actuator assuming a non-parallel (out-of-plane) plates actuation configuration is examined. The actuating force is initiated by the asymmetry of the resultant electric fringing-fields caused mainly due to the non-parallel electrodes scheme. The nano-actuator is designed based on a carbon nanotube flexible moving electrode and two symmetrically located actuating stationary rectangular-shaped out-of-plane electrodes. First, the electrical problem of the nano-actuator is numerically solved to acquire the resultant actuating force. The force is then mathematically approximated from the outcomes of a 2D numerical solution of the electric problem via a finite-element-based numerical analysis. Then, the structural behavior of the CNT-based nano-actuator under the effect the resultant non-parallel electric fields is investigated numerically via a modal expansion process. The electro-mechanical model was constructed based on an Euler–Bernoulli continuous nonlinear beam model, where both the mid-plane stretching (geometric nonlinearity) and the electric non-parallel fringing-fields effects have been taken into consideration. The reduced-order model (ROM) was derived using the Galerkin decomposition via modal decomposition process. The resultant nonlinear equations are solved numerically to get the static response of the considered CNT-based nano-actuator for various DC voltages. The eigenvalue problem is afterward derived and studied to get the variation of the fundamental as well as higher-order natural frequencies when the system is deflected by a non-zero DC amplitude. A detailed parametric study indicates an opportunity of having larger stroke as well as higher fundamental natural frequency for such CNT-based nano-actuator function of the assumed DC voltage amplitude, enabling it to be used as a tunable NEMS-based device without any pull-in scenario.

Frequency and Mode Shapes of Au Nanowires Using the Continuous Beam Models

Free vibration analysis of Au nanowires has been investigated. Au nanowire is modeled via thin beam using the linear theory. Three-different cross-sections such as circular, rectangular and triangular are taken into consideration for ultra thin nanowires. Frequency values have been obtained for different geometric parameters and simply supported of boundary condition. This study is helpful for design of the nanowires based instruments in modern NEMS technology.

Stochastic dynamics for inextensible fibers in a spatially semi-discrete setting

We investigate a spatially discrete surrogate model for the dynamics of a slender, elastic, inextensible fiber in turbulent flows. Deduced from a continuous space-time beam model for which no solution theory is available, it consists of a high-dimensional second order stochastic differential equation in time with a nonlinear algebraic constraint and an associated Lagrange multiplier term. We establish a suitable framework for the rigorous formulation and analysis of the semi-discrete model and prove existence and uniqueness of a global strong solution. The proof is based on an explicit representation of the Lagrange multiplier and on the observation that the obtained explicit drift term in the equation satisfies a one-sided linear growth condition on the constraint manifold. The theoretical analysis is complemented by numerical studies concerning the time discretization of our model. The performance of implicit Euler-type methods can be improved when using the explicit representation of the Lagrange multiplier to compute refined initial estimates for the Newton method applied in each time step.

Dynamic Simulation of Rigid Guide Structure Based on ANSYS

In order to reflect the varying law of the deflection of the rigid guide when the relative motion occur between the rigid guide and the cage roller, transient dynamic simulation is carried out for the commonly used calculation model of rigid guide and bunton by ANSYS. Simulation of the horizontal force through a section of the guide evenly, and the deflection curves of each model are obtained. It is found that the deflection of the simply supported beam model is the largest, and the three-span continuous beam model have similar peak spans in each span with the spatial grid model, but the spatial grid model has obvious fluctuation with the horizontal force.

Effect of pre-stress on natural vibration frequency of the continuous steel beam based on Hilbert-Huang transform

To study the relationship between the natural vibration frequency of continuous steel beam and the pre-stress, based on two methods, namely experiments and numerical simulation, the continuous beam model under different pre-stress values was excited through using the hammering method, and the acquired vibration signal was divided and analyzed dependent on Fast Fourier Transformation (FFT) and Hilbert-Huang Transform (HHT), respectively. The applicability of the two analytical methods was compared, and meanwhile, the relationship between the vibration frequency of the beam and the stress was obtained. As known from the analytical result, the relationship between the vibration frequency of the beam and the pre-stress cannot be explained better based on the conventional mechanic’s theory, and instantaneous frequency of the beam cannot be captured through the currently used FFT method, thus difficult to obtain the relationship between them. Nevertheless, the relationship between the vibration frequency of the beam and the pre-stress can be obtained better through the instantaneous frequency of the beam and other indicators that was calculated dependent on HHT method. The natural vibration frequency of the beam was increased along with the rising pre-stress, and meanwhile, the vibration amplitude of the instantaneous frequency of the beam was also enlarged correspondingly. Finally, the numerical simulation model was also built to calculate the corresponding results and compared with those of HHT method. The relative small error was presenting in the experimental result and numerical simulation result, thus indicating the reliability and accuracy of the HHT method.

CONSIDERATION ON TRUSS AND SPLICE JOINT DESIGN MODELS IN EXTREMAL LOADING

In this study the behavior of elements and joints in hard loaded Warren trusses is analyzed theoretically with the purpose of disclosing the sections and elements which possibly can become extremely stressed under the design load proposed being real before the ultimate limit state. The significance of truss topology selected has been brought forth, since unfavorable loading of a tensioned chord connection is discussed in this paper. Three types of tensioned splice joints have been examined considering expected elongations of bolts in order to determine additional stresses induced due to the redistribution of tensile forces. An assumption of a continuous beam model on elastically deformed discrete supports has been adopted as a design model for the examination of stresses possibly caused in the sections of bolts. It has been proved that a joint with extended end-plates over both flanges of chord I profile may be accepted as the one which is safe and robust enough for use in tensile chords of bearing structures. The joint type with end-plates extended over a more tensioned flange only and the one with connecting bolts all hidden between flanges has been subjected to sharp criticism due to the extremely nonuniform behavior and the overloading of bolts. The results of this numerical case study promote a deeper understanding and help assessing the endplate joint behavior since they lack the uniqueness of the solution recommended by the building codes, which is particularly significant when hard-loaded structures have been designed for covering the spans of public building areas.

Hybrid Model for Railroad Bridge Dynamics

AbstractRailroads carry approximately 40% of the ton-miles of the freight in the United States. On the average, a bridge occurs every 2.25 km (1.4 mi) of track, making them critical elements. The primary load on the railroad bridges is the train, resulting in numerous models being developed to understand the dynamic response of bridges under train loads. However, because the problem is time-dependent and coupled, developing adequate models is challenging. Most of the proposed models fail to provide a simple yet flexible representation of the train, bridge, and track. This paper proposes a new hybrid model that is effective for solving the track–bridge interaction problem under moving trains. The main approach is to couple the finite-element model of the bridge with a continuous beam model of the track using the assumed modes method. Both single-track and multitrack bridges are considered. The hybrid model is validated against field measurements for a double-track bridge. This model is then used to predict...

Dynamic Analysis of the Structure Exposed to the Moving Periodic Force and Viscoelastic Models of the Human Body

Submitted paper is focused on dynamic analysis of the footbridge structure, which is represented by the simply supported beam. The continuous beam model was discretized by the FEM to MDOF system and loaded by pedestrians, walking across the structure. Three states of loading are considered in this paper. Firstly, it is the mathematical description of pedestrian's ordinary traffic, where each pedestrian is randomly generated by the Monte Carlo method. Secondly, the movement of single pedestrian, simplified by viscoelastic models, is introduced. Thirdly, the DLF model of stochastic vandals is investigated as well.

Moment redistribution in cold-formed steel continuous beams

The external envelope of steel framed industrial buildings normally involves the use of purlins and rails spanning between the main hot-rolled frames to support the roofing/cladding. These purlins are typically light-gauge cold-formed steel members of complex shape for which the thin-walled nature of the material means that local instabilities will significantly influence their structural behaviour. Economic design should be based on failure of the system, recognising the opportunity for redistribution of moments. This paper presents the findings from a numerical investigation of the degree of moment redistribution in continuous cold-formed steel beams subjected to a downward (gravity) uniformly distributed load (UDL). Three types of nonlinear finite element analysis were validated against previously reported physical tests: (i) continuous two-span beams subjected to a UDL, (ii) single span beams subjected to a central point load producing a moment gradient and (ii) single span beams subjected to two point loads producing a central region under pure bending. The interior support moments from the continuous beam models were compared against reference moment capacities from the three-point bending models. Based on various different section sizes, covering a range of cross-sectional slenderness, full moment redistribution with no drop-off in moment at the interior support was found to be possible only for stocky sections but not for slender sections. In the case of slender sections, local and distortional buckling caused a reduction in interior support moment prior to failure of the system. Hence a design formula is proposed to estimate the post-peak reduction of interior support moment from its initial peak, and this reduced moment capacity is then used in conjunction with the full span moment to determine the load-carrying capacity of the system. Comparisons show the proposed approach to offer accurate prediction of observed system failure loads.

Effect of Slam Force Duration on the Vibratory Response of a Lightweight High-Speed Wave-Piercing Catamaran

When the surface of a ship meets the water surface at an acute angle with a high relative velocity, significant short-duration forces can act on the hull plating. Such an event is referred to as a slam. Slam loads imparted on ships are generally considered to be of an impulsive nature. As such, slam loads induce vibration in the global hull structure that has implications for both hull girder bending strength and fatigue life of a vessel. A modal method is often used for structural analysis whereby higher order modes are neglected to reduce computational effort. The effect of the slam load temporal distribution on the whipping response and vertical bending moment are investigated here by using a continuous beam model with application to a 112 m INCAT wave-piercing catamaran and correlation to full-scale and model-scale experimental data. Experimental studies have indicated that the vertical bending moment is dominated by the fundamental longitudinal bending mode of the structure. However, it is shown here that although the fundamental mode is dominant in the global structural response, the higher order modes play a significant role in the early stages of the response and may not be readily identifiable if measurements are not taken sufficiently close to the slam location. A relationship between the slam duration and the relative modal response magnitudes is found, which is useful in determining the appropriate truncation of a modal solution.

Free Vibration Analysis for Cracked FGM Beams by Means of a Continuous Beam Model

Based on Euler-Bernoulli beam theory and a continuous stiffness beam model, the free vibration of rectangular-section beams made of functionally graded materials (FGMs) containing open edge cracks is studied. Assuming the material gradients follow exponential distribution along beam thickness direction, the conversion relation between the vibration governing equations of a FGM beam and that of an isotropic homogenous beam is deduced. A continuous function is used to characterize the bending stiffness of an edge cracked FGM beam. Thus, the cracked FGM beam is treated as an intact beam with continuously varying bending stiffness along its longitudinal direction. The characteristic equations of beams with different boundary conditions are obtained by transfer matrix method. To verify the validity of the proposed method, natural frequencies for intact and cracked FGM beams are calculated and compared with those obtained by three-dimensional finite element method (3D FEM) and available data in the literature. After that, further discussions are carried out to analyze the influences of crack depth, crack location, material property, and slenderness ratio on the natural frequencies of the cracked FGM beams.

Treatment of elastically restrained ends for beam buckling in finite difference, microstructured and nonlocal beam models

This paper presents the treatment of elastically restrained ends for the axially, loaded beam-buckling problem for the central finite difference beam model, the microstructured beam model, and Eringen’s nonlocal continuous beam model. The equivalence between the central finite difference beam model and the microstructured beam model is established herein, and these equivalent systems are regarded as belonging to one class of discrete systems since they become indistinguishable. Also, the continualized form of the discrete system is obtained by adopting the continualization method that is based on an exponential displacement function. Three approaches are then proposed for matching the discrete system with Eringen’s nonlocal continuous system for the beam-buckling problem. The approaches depend on the assumption made on the constancy/or varying Eringen’s small length scale coefficient e0 as well as which one of the discrete or continuum system is taken as the reference system.

The nonlinear vibration and stability of a non-uniform continuous spindle system with nonlinear and nonsmooth boundaries

In this paper, the vibrations of a non-uniform continuous Timoshenko beam model for spindle system with nonlinear and nonsmooth boundaries are analyzed. The Galerkin method is used to discretize the governing nonlinear partial differential equations. Then, the averaging technique is applied to obtain the modulation equations. The frequency response curves are presented and variations of the curves with respect to the bearing interference fit value are discussed. As the interference fit value increases, the nonlinearity of system decreases. All curves are found converge asymptotically to the same dimensionless frequency value as the vibration amplitudes increase. Fixed points are also observed near the natural frequencies in the response curves. These above results provide theoretical foundations for further nonlinear dynamic analysis and design of the spindle bearing systems.