Multi-span Beam Research Articles

A system-based approach for the design of laterally unbraced multi-span steel columns and beams

This paper discusses aspects related to the determination of the buckling resistance of multi-span steel columns and beams in accordance with Eurocode 3 (EN 1993-1-1:2005, 2005). In particular, the paper reviews and extends the method previously proposed by the authors (Gonçalves and Camotim, 2005) for uniformly compressed members (columns) integrated in structural systems, which means that it is (i) system-based, i.e. relies on system parameters, and (ii) consistent with the Eurocode 3 buckling curves. It is shown that, besides making it possible to avoid individual member checks, the proposed approach provides relevant insight into the structural efficiency of the complete system. In order to illustrate the application of the proposed approach, several examples are presented and discussed throughout the paper. For comparison purposes, “exact” buckling (collapse) loads are provided, which are obtained from non-linear elastoplastic finite element analyses including member imperfections.

A Comparative Study of First-Order Reliability Method-Based Steepest Descent Search Directions for Reliability Analysis of Steel Structures

Three algorithms of first-order reliability method (FORM) using steepest descent search direction are compared to evaluate the reliability index of structural steel problems which are designed by the Iranian National Building code. The FORM formula is modified based on a dynamic step size which is computed based on the merit functions named modified Hasofer-Lind and Rackwitz-Fiessler (MHL-RF) method. The efficiency of the gradient, HL-RF, and MHL-RF method was compared for a bar structure under tensile capacity, a multispan beam under bending capacity, a connection under tension load, and a column under axial force. The results illustrated that the MHL-RF method is more efficient than the HL-RF and gradient method. The designed steel components by the Iranian National Building code showed good confidence levels with the reliability index in the range from 2.5 to 3.0.

Basic Study on Vibration Reduction Effect due to Multi-span Beam Type Absorber

Basic Study on Vibration Reduction Effect due to Multi-span Beam Type Absorber

On the dynamics of two multi-span continuous beam bridges model coupled by their close environment

In this paper, we sought to develop a valid theoretical model of two bridges coupled via their dynamic environment modelled as a linear viscoelastic Winkler foundation. Analytical, numerical and experimental study of the dynamic response of the two bridges are explored in the cases where they are submitted to sinusoidal excitation and periodic impulsive force. The effects of the close environment and the distance between the two bridges on the amplitude of vibration of each beam bridge is pointed out.

Inverse method to identify crack parameters in multi-span beam using genetic algorithm

The increased flexibility due to the presence of an open crack in the multi-span beam is made bases for identification of crack location and size. The genetic algorithm is coupled with the inverse dynamics of the cracked beam to get effective solution. The modal parameters, obtained through non-destructive vibration testing, are the input to the code and the location and size of crack are the output. The crack is modeled as rotational spring, and support conditions are generalized in the formulation for cracked multi-span beam.

Random seismic response and sensitivity analysis of uncertain multi-span continuous beams subjected to spatially varying ground motions

An analytical method is formulated for the seismic analysis of multi-span continuous beams with random structural parameters subjected to spatially varying ground motions. An earthquake-induced ground motion is modelled as a stationary random process defined by power spectral density function, and the spatial variation is considered. The physical parameters of the multi-span beams are random and modelled as continuous random Gaussian variables. The stationary random responses are determined as approximate explicit functions of the structural parameters. Direct differentiation of these functions with respect to the structural parameters provides analytical expressions of the sensitivities of the stationary responses. On the basis of Taylor expansion, the statistic moments of the random responses are obtained. Taking the four-span beam as an illustrative example, the mean value and standard deviation of the random responses are computed and compared with those from Monte Carlo simulation to demonstrate the accuracy of the proposed method. Results are illustrated for the influence of different structural parameters on the statistic moments of the random responses. It is found that randomness in Young's modulus and the mass per unit length has approximate equivalent and significant influence on the random responses, while that of damping is negligible.

The Structure Analytical Research of Porous Permeable Wire Material

Abstract The details of making technology of porous permeable material with use of wire are allowed to carry out the analytical research of structure and structural characteristics of wire winding body. Its permit for prognostication the final properties of material, that is produced by the following deformation treatment (diameter reduction). Due to the regular organized arrangement of wire, the coil of winding body is considered as a multispan continuous beam, but a contact of coils - as interaction of two cylinders. Possibility of exactly calculation of the contacts between coils is allowed to go over the single fragment displacements into deformation of whole winding body. During research of deformation processes in regards of winding body geometry and used wire mechanical properties, the structural characteristics of porous permeable wire material are expected. The optimal number of winding layers, eliminating the distortion of organized final structure, is established. The material pressure-compactness relation is obtained in order to control the technological conditions of winding and drafting for guarantee the product required properties.

Exact free vibration analysis of Timoshenko stepped shaft carrying elastically supported eccentric masses with application on SWRO mechanical system

The aim of the study is to investigate the exact natural frequencies and their associated modal shapes for an axially loaded Timoshenko stepped multi-span shaft or beam system. Each span is carrying an elastically supported eccentric mass and resting on Pasternak elastic foundation. The end and intermediate conditions governing the lateral deflections and slope due to bending of the system have been formulated. Application models have been chosen to verify the reliability of the generalized system frequency equation. Some results of previous works are validated using the present analysis. Real application example of a combined shaft assembly for high pressure boosting device for sea water reverse osmosis (SWRO) desalination plant is investigated. Experimental tests for extended real shaft are carried out. The comparison between the analytical results and the experimental ones, for the first two modes of vibration, shows good agreement with minimal errors. These errors do not exceed 5%. In addition, the results of the proposed shaft model can be easily obtained for any number of spans and attachments. The present model and analysis is important for industrial applications, such as mechanical, civil, naval and for wider range of sea water desalination machinery.

Application of Volterra Integral Equations in Dynamics of Multispan Uniform Continuous Beams Subjected to a Moving Load

The dynamic behavior of multispan uniform continuous beam arbitrarily supported on its edges subjected to various types of moving noninertial loads is studied. Problem is solved by replacing a multispan structure with a single-span beam loaded with a given moving load and redundant forces situated in the positions of the intermediate supports. Redundant forces are obtained by solving Volterra integral equations of the first or the second order (depending on the stiffness of the intermediate supports) which are consistent deformation equations corresponding to each redundant. Solutions for the beam arbitrarily supported on its edges (pinned or fixed) due to a moving concentrated force and moving distributed load are given. The difficulty of solving Volterra integral equations analytically is bypassed by proposing a simple numerical procedure. Numerical examples of two- and three-span beam have been included in order to show the efficiency of the presented method.

Dynamic behavior of a multispan continuous beam traversed by a moving oscillator

In this article, dynamic behavior of a multispan continuous beam influenced by a moving oscillator is studied. It is assumed that each span of the continuous beam obeys uniform Euler–Bernoulli beam theory. By exploiting a series of beam natural mode shapes, the governing differential equation of beam vibration is transformed into the time domain for which a time integration method is applied. The dynamic amplification factor (DAF) of the base beam is extensively analyzed for different moving oscillator’s weight, stiffness and velocity. At the asymptotic states, the presented results indicate a close agreement with previously published benchmark solutions for moving force and moving mass models. In addition, it is shown that the structural response can be considerably altered by the variation of the moving oscillator stiffness within a notable range of influential parameters.

Lyapunov-based boundary control of a multi-span beam subjected to moving masses

This paper considers a vibration control problem for a multi-span beam under moving masses by using boundary control method. The vibration of the multi-span beam under active control is governed by both a partial differential equation (PDE) and several ordinary differential equations (ODEs) which are derived from Hamilton’s principle. In order to suppress its vibration, boundary control strategy is proposed based on Lyapunov’s direct method. The closed-loop system stability of the multi-span beam with the proposed boundary control is proved. It can avoid spill-over effects which may occur in those popular methods that discretise the system model through modal expansion, as it directly acts on the PDE-ODEs of the system model in control design. Moreover, the sensors and actuators in the proposed boundary control strategy can be easily placed, as they are installed at the boundary of the system. The external excitations applied in simulation are a rectangular impulse, a moving mass and two moving masses, respectively. Numerical results demonstrate the effectiveness of the proposed method and good control performance in suppressing vibration of moving mass problems. This investigation has wide applications in engineering, notably, train-bridge dynamic interaction in high-speed railways.

A strain measurement model using a limited number of sensors for steel beam structures subjected to uncertain loadings

The maximum stress of a structural member has been extensively adopted as a safety assessment indicator in structural health monitoring. Due to construction errors in the field and changes in the loading conditions during or after construction, it is impractical to accurately predict the location and magnitude of the maximum strain of a member a priori. To avoid the dependency of strain sensing methods on information of the structural and loading conditions, this paper proposes a strain distribution measurement model for steel beam structures subjected to uncertain loadings with uncertainties in magnitudes and shapes. With strains measured from a limited number of sensors, a general form equation of the strain distribution is determined for the estimation of the strain distribution. The performance of the strain distribution measurement model is verified by comparing estimated strain values from the proposed method and measured strains directly from fiber Bragg grating sensors or electrical strain gauges during static loading tests on single- and multi-span beam structures.

Vibration-based non-destructive technique to detect crack in multi-span beam

This article presents the study on identification of a single open crack in a straight multi-span beam using natural frequency-based non-destructive technique. The crack is assumed to be transverse and one-dimensional partial cut of a beam, and is modelled as equivalent elastic rotational spring. The effects of crack location and depth on the natural frequency for multi-span uniform beam are demonstrated. The reduction in natural frequency due to presence of crack is utilised to detect crack location and its severity. The formulation is validated experimentally.

Flexural–torsional buckling assessment of steel beam–columns through a stiffness reduction method

In this paper, a stiffness reduction method for the flexural–torsional buckling assessment of steel beam–columns subjected to major axis bending and axial compression is presented. The proposed method is applied by reducing the Young’s E and shear G moduli through the developed stiffness reduction functions and performing Linear Buckling Analysis. To account for second-order forces induced prior to buckling, the in-plane (in the plane of bending) and out-of-plane analyses of a member are separated and stiffness reduction for the out-of-plane instability assessment is applied on the basis of member forces determined from the in-plane analysis. Since the developed stiffness reduction functions fully take into account the detrimental influence of imperfections and spread of plasticity, the proposed method does not require the use of member design equations, thus leading to practical design. For the purpose of verifying this approach, the strength predictions determined through the proposed stiffness reduction method are compared against those obtained from nonlinear finite element modelling for a large number of regular, irregular, single and multi-span beam–columns.

Validation of numerical models of concrete box bridges based on load test results

A Results of experimental load tests are used for the validation of numerical models applied in analyses of the concrete box bridges by means of the Finite Element Method. Six different models were used in the analyses: two created of 1-dimensional finite elements in a 2- and 3-dimensional space and four hybrid models composed of 1- and 2-dimensional elements in a 3-dimensional space. Validation of the theoretical models was based on the case of a cable-stayed bridge structure (max. span length 256m) and a continuous multi-span beam structure (max. span length 60m). Details of the numerical models, results of static and dynamic analyses as well as the procedure and effects of static and dynamic load tests are presented, compared and discussed.

Strength of multi-span composite beams subjected to combined flexure and torsion

An improved understanding of the performance of composite beams subjected to combined flexure and torsion is essential for practical design as such loading states are very common in bridges for the main girder and in buildings for edge beams. Multi-span steel–concrete composite beams consisting of two steel joists and one concrete slab are concerned herein, on which an experimental study and finite element (FE) modelling analyses are presented. Eight reduced scale beams under combined flexure and torsion to varying degrees are tested, with both full and partial shear connection being incorporated. Either flexure-dominated or torsion-dominated failure modes are observed in each test depending on the ratio between the bending and torsional moments applied externally. Load-deformation responses, ultimate strengths as well as flexure–torsion interaction of all specimens are also reported and discussed. Based on the experimental outcomes, a three-dimensional FE model is developed and further validated against the test results, and an extensive range of parametric analyses are undertaken by using the FE model. It is indicated that in addition to well-known parameters relating to the concrete slab and stirrups, the torsional strength of multi-span composite beams is also sensitive to the degree of shear connection, spacing of the joists and span-to-depth ratio. Contributions of the steel joist and its constraint on the slab towards the torsional strength are increased. With respect to their flexure–torsion interaction relation, no increase in the torsional strength is observed within the presence of combined flexure, and vice versa. The interaction is insensitive generally to various parameters considered herein and the test results exhibit consistent phenomenon. Design guidance and new equations for predicting the torsional strength and the flexure–torsion interaction relation are proposed based on the test and FE analysis results as well as a comprehensive review of existing approaches in the literature.

Modal identification of damaged frames

The paper investigates the possibility of identifying localised damages for multi-span and multi-floor linear elastic frames using only natural frequencies measured in the undamaged and damaged configurations. Namely, frames of increasing complexity are studied by exploring one by one their significant substructures (i.e. multi-span beams, floor by floor); the error function is defined and minimised on a database of finite element damaged models that only includes the natural frequencies of the local modes of the substructure, that is, the only modes significantly affected by the localised damage considered here. The performances and limits of the procedure are here discussed by means of numerical simulations on steel frames of increasing complexity; a particular attention is also devoted to the role of noise on the identification procedure. Copyright © 2015 John Wiley & Sons, Ltd.

Maximum stress estimation model for multi-span waler beams with deflections at the supports using average strains.

The safety of a multi-span waler beam subjected simultaneously to a distributed load and deflections at its supports can be secured by limiting the maximum stress of the beam to a specific value to prevent the beam from reaching a limit state for failure or collapse. Despite the fact that the vast majority of accidents on construction sites occur at waler beams in retaining wall systems, no safety monitoring model that can consider deflections at the supports of the beam is available. In this paper, a maximum stress estimation model for a waler beam based on average strains measured from vibrating wire strain gauges (VWSGs), the most frequently used sensors in construction field, is presented. The model is derived by defining the relationship between the maximum stress and the average strains measured from VWSGs. In addition to the maximum stress, support reactions, deflections at supports, and the magnitudes of distributed loads for the beam structure can be identified by the estimation model using the average strains. Using simulation tests on two multi-span beams, the performance of the model is evaluated by estimating maximum stress, deflections at supports, support reactions, and the magnitudes of distributed loads.

Lateral–torsional buckling assessment of steel beams through a stiffness reduction method

This paper presents a stiffness reduction approach utilising Linear Buckling Analysis (LBA) with developed stiffness reduction functions for the lateral–torsional buckling (LTB) assessment of steel beams. A stiffness reduction expression is developed for the LTB assessment of beams subjected to uniform bending and modified for the consideration of moment gradient effects on the development of plasticity. The proposed stiffness reduction method considers the influence of imperfections and plasticity on the response through the reduction of the Young's modulus E and shear modulus G and obviates the need of using LTB buckling curves in design. The accuracy and practicality of the method are illustrated for regular, irregular, single and multi-span beams. In all of the considered cases, the proposed method is verified against the results obtained through nonlinear finite element modelling.

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.