Single-span Beam Research Articles

Differential transform method and numerical assembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number of intermediate lumped masses and rotary inertias

Multiple-step beams carrying intermediate lumped masses with/without rotary inertias are widely used in engineering applications, but in the literature for free vibration analysis of such structural systems; Bernoulli-Euler Beam Theory (BEBT) without axial force effect is used. The literature regarding the free vibration analysis of Bernoulli-Euler single-span beams carrying a number of spring-mass systems, Bernoulli-Euler multiple-step and multi-span beams carrying multiple spring-mass systems and multiple point masses are plenty, but that of Timoshenko multiple-step beams carrying intermediate lumped masses and/or rotary inertias with axial force effect is fewer. The purpose of this paper is to utilize Numerical Assembly Technique (NAT) and Differential Transform Method (DTM) to determine the exact natural frequencies and mode shapes of the axial-loaded Timoshenko multiple-step beam carrying a number of intermediate lumped masses and/or rotary inertias. The model allows analyzing the influence of the shear and axial force effects, intermediate lumped masses and rotary inertias on the free vibration analysis of the multiple-step beams by using Timoshenko Beam Theory (TBT). At first, the coefficient matrices for the intermediate lumped mass with rotary inertia, the step change in cross-section, left-end support and right-end support of the multiple-step Timoshenko beam are derived from the analytical solution. After the derivation of the coefficient matrices, NAT is used to establish the overall coefficient matrix for the whole vibrating system. Finally, equating the overall coefficient matrix to zero one determines the natural frequencies of the vibrating system and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the differential equations of the motion. The calculated natural frequencies of Timoshenko multiple-step beam carrying intermediate lumped masses and/or rotary inertias for the different values of axial force are given in tables. The first five mode shapes are presented in graphs. The effects of axial force, intermediate lumped masses and rotary inertias on the free vibration analysis of Timoshenko multiple-step beam are investigated.

Finite element formulation for earthquake analysis of single-span beams involving forced deformation caused by multi-support motions

This study formulates a finite element (FE) equation of motion of a beam involving a forced deformation induced by multi-support motions with the use of the quasi-static decomposition method. A set of relevant FEs for the dynamic analysis is derived. The elements are based on absolute displacements and are thus simple and convenient to use for building an FE model. In this study, a single-span beam with fixed support at one end and hinged support at the other end is given as an example to illustrate how an FE model is built and how its accuracy is verified. Numerical results of the beam subjected to asynchronous seismic motion at supports show good agreement with the analytical results.

Simplified modeling of beam vibrations induced by a moving mass by regression analysis

In this paper, we propose a fast computation of beam-type dynamic response to a moving mass. Dynamics of a single-span beam is first accurately computed with a semi-analytical procedure based on characteristic orthogonal polynomials (COPs), in the case of a force or a mass traveling across. A regression analysis is then adopted to automatically define the effects of the moving mass once the effects of the moving force are known. Best-fitting multivariable correlation splines, with strictly given coefficients, are obtained. The provided correlation splines conveniently interpolate the maximum design parameters of the base beam over a wide range of load inertia and velocity. Results provided are valid for any geometry and elastic stiffness of the beam, thanks to a properly set normalization factor here defined. Hence, practitioners are provided with an effortless and highly simplified direct prediction of design parameters when inertial interactions are taken into consideration.

Mechanical Characters of Six Engineering Timoshenko Beams

Timoshenko beam is an extension of Euler-Bernoulli beam to interpret the transverse shear impact. The more refined Timoshenko beam relaxes the normality assumption of plane section that remains plane and normal to the deformed centerline. The manuscript presents some exact concise analytic solutions on deflection and stress resultants of NET single-span Timoshenko beam with general distributed force and 6 kinds of standard boundary conditions, adopting its counterpart Euler-Bernoulli beam solutions. Engineering example shows that scale impact would not unveil itself for micro structure with micrometer μm-order length, yet will be prominent for nanostructure with nanometer nm-order length. When simply supported CNTs is undergone to a concentrative force at the median and complete bend moment, scale action is observed along the ensemble CNTs, while it unfurls itself the most at the position of the concentrated strength. When a clamped-free CNTs is exposed to a centralized force at the mesial and distributed force, there is no scale impact about the deflection at all positions on the left border of the concentrated strength position, while such operation inspires at once at all positions on the right margin of the concentrated strength position. When a clamped-clamped CNTs is lain under a concentrative strength at the middle, the deflection of NET Euler-Bernoulli CNTs reflects scale effect completely. Notable differences between the deflection of Euler-Bernoulli CNTs and that of Timoshenko CNTs are reflected at large ratio of diameter versus length. The deflection of NET clamped-free and simply supported Timoshenko beam doesn’t introduce surplus scale process in terms of its counterpart, NET Euler-Bernoulli beam. However, the deflection of NET clamped-clamped Timoshenko beam does involve additional scale impact solely including the method when the concentrated strength position is at the midway in the beam-length direction.

New damage localization indicator based on curvature for single-span beams

Most existing damage detection methods based on curvature should investigate the indicator value changes between the intact and damaged state. However, the 'footprint' in the intact state is usually not available for most actual beams. Therefore, a new damage localization indicator called Difference of Nearby Difference Curvature Indicator (DNDCI) was introduced for single-span beams. This indicator does not require prior information of the intact beam and is sensitive to the damage occurs on or nearby the measuring points. Practical and detailed application method of DNDCI has been proposed. Besides the idealized simply supported beams, it was validated by example that DNDCI is also applicable to the actual single-span beams supported by bearings.

A load capacity of cyclic loaded reinforced concrete beams

In this paper, an effect of cyclic and variable loads on the capacity of reinforced concrete beams is investigated. The study was conducted on 28 reinforced concrete beams including 12 two-span beams - statically indeterminate and 16 simply supported single-span beams. Those beams were differentiated by the structure and degree of longitudinal reinforcement (two degrees of reinforcement) and load applied programs (three programs types) including the monotonic load, load with unloadings and cyclic load. The last one was taken as a sinusoidal cyclic load with 100,000 cycles carried out for three ranges of loading forces. On the basis of a comparative analysis it can be concluded that in the case of the two-span beams, the imposition of 252,000 load cycles results in 20% drop of the beam capacity. Such an effect is not observed for load with unloadings and in the case of simply supported beams.

Dynamic effect of high-speed trains on simple bridge structures

In this paper the overall dynamic response of simple railway bridges subjected to high-speed trains is investigated numerically based on the mechanical models of simply supported single-span and continuous two-span Bernoulli-Euler beams. Each axle of the train, which is composed of rail cars and passenger cars, is considered as moving concentrated load. Distance, magnitude, and maximum speed of the moving loads are adjusted to real high-speed trains and to load models according to Eurocode 1. Nondimensional characteristic parameters of the train-bridge interaction system are identified. These parameters permit a spectral representation of the dynamic peak response. Response spectra assist the practicing engineers in evaluating the expected dynamic peak response in the design process of railway bridges without performing time-consuming time history analyses.

Review and analysis of fire resistance tests of timber members in bending, tension and compression with respect to the Reduced Cross-Section Method

The Reduced Cross-Section Method (RCSM) provides a popular method for the design of timber members exposed to fire, using an effective cross-section and mechanical properties at normal temperature. The RCSM was adopted from a single publication and was originally developed for single-span beams exposed to bending load. It has been introduced in Eurocode 5 for a large range of timber members, including columns under compression and members under tension. Recently, the applicability of the method and its extended applications were called into question on the basis of limitations and contradictions found by advanced simulations. This paper analyses a total of 153 fire resistance tests (117 members in bending, six members in compression and 30 members in tension) with respect to the RCSM. The analysis shows that (i) most of the references are of too poor quality, or are incomplete, to validate a design model, (ii) results with adequate information content showed a significant deviation from the RCSM, and (iii) that the RCSM may lead to a non-conservative design. For members in tension, results fit well with the RCSM, while significant deviations were found for members in compression. Members in bending show very large scatter. It is therefore recommended that the existing design approach in Eurocode 5 should be revised in order to include the results of advanced calculations and appropriate tests.

Natural oscillations of single span beam placed on cylindrical supports

Oscillations of a single spanbeam placed on two circular cylindrical supports are investigated. Contact line of the beam and the support surface changes its location therefore the span length also alternates. Nonlinear equa-tion of motion is transformed to linear differential equa-tion, where displacement instead of time is the independent variable and the first integral in a phase plane velocity-displacement is deduced. Dependence of displacement on time is expressed in Jacobian elliptic functions in the second integral. The complete period of the oscillations is almost the same as when supports are hinges but the half-periods and the half-amplitudes in upper and lower portions of the motion are significantly distorted. The level of the distortion is increasing when amplitude of oscillations and radius of the support circle are growing. The beam, placed on cylindrical supports, oscillates about a center shifted up compared with the same hinged beam. DOI: http://dx.doi.org/10.5755/j01.mech.20.3.6365

Static response of thermally loaded sandwich beams with confined horizontal displacements of faces at the supports

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

Prestress Force Identification for Externally Prestressed Concrete Beam Based on Frequency Equation and Measured Frequencies

A prestress force identification method for externally prestressed concrete uniform beam based on the frequency equation and the measured frequencies is developed. For the purpose of the prestress force identification accuracy, we first look for the appropriate method to solve the free vibration equation of externally prestressed concrete beam and then combine the measured frequencies with frequency equation to identify the prestress force. To obtain the exact solution of the free vibration equation of multispan externally prestressed concrete beam, an analytical model of externally prestressed concrete beam is set up based on the Bernoulli-Euler beam theory and the function relation between prestress variation and vibration displacement is built. The multispan externally prestressed concrete beam is taken as the multiple single-span beams which must meet the bending moment and rotation angle boundary conditions, the free vibration equation is solved using sublevel simultaneous method and the semi-analytical solution of the free vibration equation which considered the influence of prestress on section rigidity and beam length is obtained. Taking simply supported concrete beam and two-span concrete beam with external tendons as examples, frequency function curves are obtained with the measured frequencies into it and the prestress force can be identified using the abscissa of the crosspoint of frequency functions. Identification value of the prestress force is in good agreement with the test results. The method can accurately identify prestress force of externally prestressed concrete beam and trace the trend of effective prestress force.

ON THE DYNAMICS OF A BEAM PARTIALLY SUPPORTED BY AN ELASTIC FOUNDATION: AN EXACT SOLUTION-SET

Free vibrations of straight beams which are partially supported by an elastic foundation are analyzed. For the sake of simplicity, only the Euler–Bernoulli beam model coupled with a Winkler-type elastic foundation is considered. This structural system can be used to study, in a rather accurate way, the dynamic response of partially embedded piles (like those used for telecommunications) when dealing with the problem of identifying their mechanical properties during operative conditions. The study makes clear that different kinds of vibration modes may occur in the part of the beam which is supported by the continuous elastic foundation: indeed apart from the classical modes, corresponding to the dynamics of a free beam, it is possible to have vibration modes which are similar to the static deflection of a beam on an elastic support or even corresponding to rigid-body modes. For the same beam it is shown that transition between these vibration modes can appear when switching from the fundamental natural frequency to subsequent ones. This effect is the focus of the presented numerical examples. In particular, the analytic expression of the transcendental functions governing the vibration modes, and of the coefficients of the eigenfunctions for all occurring cases, are given here — to the best of the author's knowledge — for the first time. From the practical point of view, the reported results allow to define a suitable range of the elastic stiffness parameter such that the behavior of a partially supported beam can be conveniently approximated with that of a single-span beam, having one built-in end and the other free.

Analytic Solution of Single Span Beam under Non Uniform Load

During the process of structure design, non uniform load on beam usually is considered as similar uniform load. Some analytic solution formulas of bending moment about beams acted on symmetric load are deduced. Table of equivalent uniform load coefficient according to the principle of bending moment equivalency is listed. The feasibility and beware are also discussed.

Force Characteristic Analysis of Overlying Stratum Based on Elastic Foundation Beam Theory

Regard the mechanics model of key rock stratum in overlying strata as the single-span beam with semi-infinite length elastic foundation support. Based on elastic foundation beam theory, derive the analytic solution about inner force distribution of overlying strata. Then analyze the variation law of maximum value of bending moment and its location of overlying strata followed by beam space and relative rigidity. Through the research and analysis, the relationship between the elasticity of elastic foundation and the stress field of digging area is elaborated.

Free Vibrations of a Reddy-Bickford Multi-Span Beam Carrying Multiple Spring-Mass Systems

The structural elements supporting motors or engines are frequently seen in technological applications. The operation of machine may introduce additional dynamic stresses on the beam. It is important, then, to know the natural frequencies of the coupled beam-mass system, in order to obtain a proper design of the structural elements. The literature regarding the free vibration analysis of Bernoulli-Euler and Timoshenko single-span beams carrying a number of spring-mass system and multi-span beams carrying multiple spring-mass systems are plenty, but the free vibration analysis of Reddy-Bickford multi-span beams carrying multiple spring-mass systems has not been investigated by any of the studies in open literature so far. This paper aims at determining the exact solutions for the natural frequencies and mode shapes of Reddy-Bickford beams. The model allows analyzing the influence of the shear effect and spring-mass systems on the dynamic behavior of the beams by using Reddy-Bickford Beam Theory (RBT). The effects of attached spring-mass systems on the free vibration characteristics of the 1–4 span beams are studied. The natural frequencies of Reddy-Bickford single-span and multi-span beams calculated by using the numerical assembly technique and the secant method are compared with the natural frequencies of single-span and multi-span beams calculated by using Timoshenko Beam Theory (TBT); the mode shapes are presented in graphs.

Free vibrations of a Reddy-Bickford multi-span beam carrying multiple spring-mass systems

The structural elements supporting motors or engines are frequently seen in technological applications. The operation of machine may introduce additional dynamic stresses on the beam. It is important, then, to know the natural frequencies of the coupled beam-mass system, in order to obtain a proper design of the structural elements. The literature regarding the free vibration analysis of Bernoulli-Euler and Timoshenko single-span beams carrying a number of spring-mass system and multi-span beams carrying multiple spring-mass systems are plenty, but the free vibration analysis of Reddy-Bickford multi-span beams carrying multiple spring-mass systems has not been investigated by any of the studies in open literature so far. This paper aims at determining the exact solutions for the natural frequencies and mode shapes of Reddy-Bickford beams. The model allows analyzing the influence of the shear effect and spring-mass systems on the dynamic behavior of the beams by using Reddy-Bickford Beam Theory (RBT). The effects of attached spring-mass systems on the free vibration characteristics of the 1-4 span beams are studied. The natural frequencies of Reddy-Bickford single-span and multi-span beams calculated by using the numerical assembly technique and the secant method are compared with the natural frequencies of single-span and multi-span beams calculated by using Timoshenko Beam Theory (TBT); the mode shapes are presented in graphs.

Finite element method based Monte Carlo filters for structural system identification

The paper proposes a strategy for combining two powerful computational procedures, namely, the finite element method (FEM) for structural analysis and particle filtering for dynamic state estimation, to tackle the problem of structural system parameter identification based on a set of noisy measurements on static and (or) dynamic structural responses. The proposed identification method automatically inherits the wide ranging capabilities of both FEM and particle filtering procedures thereby enabling the treatment of many complicated features of system identification problems, namely, imperfections in mathematical models for structural behavior, noisy and spatially incomplete measurements, nonlinear models for process and measurement equations, possible non-Gaussian nature of system and measurement noises, and dynamic and (or) static behaviors of the structure. Additionally, the authors propose a method that permits assimilation of measurements from multiple static and (or) dynamic tests and from multiple sensors in the system identification procedure in a unified manner. The paper describes how finite element models residing in commercially available softwares can be made to communicate with a database of measurements via a particle filtering algorithm developed on the Matlab platform. This leads to a probabilistic description of system parameters to be identified. Illustrative examples consider measurements from computational models, laboratory and field tests. These illustrations include studies on a rubber sheet with a hole undergoing large deformations, laboratory investigations on a single span beam and field investigations on an existing multi-span masonry arch bridge subjected to diagnostic moving loads.

Wear Prediction of Steam Generator Tubes due to Shell Side Foreign Objects

A method to evaluate tube wear due to foreign objects on a steam generator tube sheet is introduced. In order to produce the database for the evaluation, a simplified vibration model based on the zonal approach concept and dissected tube model is introduced in this paper. The entire tube sheet region of the steam generator is divided into 30 zones. The data of a representative tube is used in a zone that includes hundreds of tubes. Therefore, the number of tubes analyzed to produce the database is dramatically reduced by the zonal approach concept. The dissected tube model simplifies the mode analysis by adopting a single-span beam instead of a multispan beam idealizing the full-length U-tube of a steam generator. It is shown that the natural frequencies obtained from the dissected model are about 4 to 5% lower than those of the dominant modes between the tube sheet and the first tube support structure obtained from the analysis of the full-length U-tubes. If the concerned region is between the tube sheet and the first tube support structure of the steam generator, using the zonal approach concept and the dissected tube model is very effective to save time and efforts to calculate the vibration data. In this paper, a method to predict the wear depth of the tubes due to the fixed foreign objects on the tube sheet in the secondary side of a steam generator with the simplified vibration model is introduced.

Interaction response of maglev masses moving on a suspended beam shaken by horizontal ground motion

As a maglev transport route has to cross a region with occasional earthquakes, the train/guideway interaction is an issue of great concern in dominating safety of the maglev system. This paper intends to present a computational framework of interaction analysis for a maglev train traveling over a suspension bridge shaken by horizontal earthquakes. The suspended guideway girder is modeled as a single-span suspended beam and the maglev train traveling over it as a series of maglev masses. Due to motion- dependent nature of magnetic forces in a maglev suspension system, appropriate adjustments of the magnetic forces between magnets and guide-rail require the air gaps be continuously monitored. Thus an on-board hybrid LQR+PID controller with constraint rule base is designed to control the dynamic response of a running maglev mass. Then the governing equations of motion for the suspended beam associated with all the controlled maglev masses are transformed into a set of generalized equations by Galerkin's method, and solved using an incremental-iterative procedure. Numerical investigations demonstrate that when a controlled maglev train travels over a suspended guideway shaken by horizontal earthquakes, the proposed hybrid controller has the ability to adjust the levitation gaps in a prescribed stable region for safety reasons and to reduce the vehicle's acceleration response for ride quality.

Vehicle/bridge interactions of a rail suspension bridge considering support movements

This paper is intended to investigate interaction response of a train running over a suspension bridge undergoing support settlements. The suspension bridge is modeled as a single-span suspended beam with hinged ends and the train as successive moving oscillators with identical properties. To conduct this dynamic problem with non-homogeneous boundary conditions, this study first divides the total response of the suspended beam into two parts: the static and dynamic responses. Then, the coupled equations of motion for the suspended beam carrying multiple moving oscillators are transformed into a set of nonlinearly coupled generalized equations by Galerkin`s method, and solved using the Newmark method with an incremental-iterative procedure including the three phases: predictor, corrector, and equilibrium-checking. Numerical investigations demonstrate that the present iterative technique is available in dealing with the dynamic interaction problem of vehicle/bridge coupling system and that the differential movements of bridge supports will significantly affect the dynamic response of the running vehicles but insignificant influence on the bridge response.