Shear Center Research Articles

Lateral-torsional buckling behavior of hot-rolled steel beams with flange upstands

The lateral-torsional buckling behavior of hot-rolled steel beams has been studied extensively. Much of the research carried out has been limited to doubly symmetric beams and a few other common geometries. The study of monosymmetric and non-symmetric geometries has not been as extensive. This paper presents a study of the lateral-torsional buckling behavior of monosymmetric beams having the monosymmetry introduced by flange end upstands. The study is carried out using finite element modelling. The behavior is investigated for various spans and with various heights of upstands. It is observed that the upstands increase the critical buckling moment of the section but that this increase diminishes with increase in upstand height. It is further observed that this unique behavior could be influenced by a movement in the relative position of the shear center to the geometric centroid as the flange upstands are adjusted. In addition there exists for these geometries unique points where the monosymmetric sections have properties typically associated with doubly symmetric sections. It is shown that the relative position of the shear center and centroid have some influence on the strength gain pattern observed.

Coupled bending and torsional vibrations of non-uniform thin-walled beams by the transfer differential transform method and experiments

This paper describes a new methodology capable of analyzing coupled bending and torsional vibrations in non-uniform thin-walled beams. The transfer matrix method (TMM) and the differential transform method (DTM) are combined to create the proposed transfer differential transform method (TDTM) for solving equations coupling bending and torsional vibrations in non-uniform beams. Compared with the finite element method (FEM), the TDTM utilizes a type of changeable mode shape functions so that the number of meshed elements can be reduced greatly when the beam's geometric size is uniform. The equations of motion of the non-uniform thin-walled beams are established using Hamilton's principle. The method considers both the warping and rotary effects of the beam section. The natural frequencies and mode shapes of the bending and torsional components are obtained using the TDTM. The accuracy of solutions can be controlled by the mesh density and the series expansion order of the mode shape whose ranges are also discussed. For illustrative purposes, the natural frequencies and the frequency response curves of a uniform beam and a non-uniform beam are studied respectively and are validated by experiments. The effects of warping and the distance between the centroid of a given section and the corresponding shear center on the vibration properties of the beam are also investigated.

Torsion sensing based on patterned piezoelectric beams

In this study, we investigated the sensing characteristics of piezoelectric beams under torsional loads. We used partially patterned piezoelectric beams to sense torsion. In particular, the piezoelectric patches are located symmetrically with respect to the line of the shear center of the beam. The patterned piezoelectric beam is modeled as a slender beam, and its electrical responses are obtained by piezoelectric electromechanical equations. To validate the modeling framework, experiments are performed using a setup that forces pure torsional deformation. Three different geometric configurations of the patterned piezoelectric layer are used for the experiments. The frequency and amplitude of the forced torsional load are systematically varied in order to study the behavior of the piezoelectric sensor. Experimental results demonstrate that two voltage outputs of the piezoelectric beam are approximately out of phase with identical amplitude. Moreover, the length of the piezoelectric layers has a significant influence on the sensing properties. Our theoretical predictions using the model support the experimental findings.

Interactive geometric interpretation and static analysis of thin-walled bi-directional functionally graded beams

The paper aims to present an analysis of static behaviors and interactive geometric interpretation for modeling of thin-walled bi-directional functionally graded beams regarding monosymmetric I- and channel-sections. Several material distributions based on a typical power law are provided. The model of thin-walled bi-directional FG beams was originally developed from Vlasov’s theory in which warping has been taken into account. Since the variation of constituent phases such as metal or ceramic in both longitudinal and through-the-thickness directions, properties of the thin-walled beam have considerably changed. It is shown that in each cross section, center of gravity and shear center are highly sensitive to geometries, materials and thus need to be reconsidered. For that reason the external load cannot be randomly applied. Effects of gradient parameters on the static behavior of a thin-walled bi-directional FG beam under various loadings and boundary conditions have also been parametrically studied.

GBTul 2.0 − A second-generation code for the GBT-based buckling and vibration analysis of thin-walled members

This paper presents the fundamentals and illustrates the application and potential of the recent 2.0 release of the software GBTul – a computer program developed by the authors and made available as freeware on the website of the Department of Civil Engineering of the University of Lisbon. The program is based on Generalised Beam Theory (GBT), a bar theory accounting for cross-section in-plane and out-of-plane (warping) deformation, and performs linear buckling and undamped free vibration analyses of prismatic thin-walled members. Its domain of application is much wider than that of the previous release (1.0β), making it possible to analyse single or multi-span members (i) with various support conditions, namely those due to discrete bracing systems, (ii) exhibiting arbitrary (open, closed or “mixed”) flat-walled cross-sections and (iii) acted by fairly general loadings, including concentrated and/or distributed transverse forces applied away from the member shear centre axis. After providing a brief overview on the GBT fundamentals, the program capabilities and innovative aspects are addressed, and its application is illustrated by means of a few relevant numerical examples. Moreover, the program Graphical User Interface is described and the procedures and/or options associated with its main commands are mentioned.

Static analysis of thin-walled space frame structures with arbitrary closed cross-sections using transfer matrix method

This paper develops a new transfer matrix method for the static analysis of thin-walled space frame structures with arbitrary closed single-cell sections, that is, thin-walled beam transfer matrix method (TBTMM), which expands the advantages of the general transfer matrix method (TMM). In this method, the bimoment B and the rate of twist of the beam φ(x) are taken into account based on the Vlasov's beam theory and Kollbrunner-Hajdin assumption, and the bending and twisting can be calculated separately combined with the characteristic of Maillart's shear center. Using this method, the transfer matrix of single beam is obtained through Laplace transform and inverse Laplace transform at first, and then gets the coupling equations of the whole structure with respect to coordinate transformation matrices and joint coupling matrices. Some numerical examples are given and compared with the general TMM and the finite element method (B-FEM and S-FEM) results to validate the developed method.

GBT buckling analysis of generally loaded thin-walled members with arbitrary flat-walled cross-sections

This paper deals with extending the domain of applicability of a recently developed Generalised Beam Theory (GBT) formulation intended to perform elastic linear buckling analyses of thin-walled members (i) exhibiting arbitrary flat-walled cross-sections (including those combining closed cells and open branches), and (ii) acted by general loadings. These loadings, which include transverse forces acting away from the member shear centre axis, are termed “general” in the sense that they may involve the presence of pre-buckling stress distributions associated with any possible combination of all the stress tensor membrane components (σxx, σss and τxs, for a plane stress state), including cell shear flows – therefore, all the relevant geometrically non-linear effects need to be taken into consideration. After briefly presenting the main concepts and procedures involved in the development and implementation of the above GBT formulation, this same formulation is employed to analyse the buckling behaviour of beams with different types of cross-section geometry (containing closed cells) and exhibiting different loading and support conditions. In particular, they consist of (i) a RHS cantilever acted by two tip point loads, (ii) a closed-flange I-section simply supported beam subjected to a uniformly distributed load and (iii) a two-cell RHS section cantilever acted by tip transverse forces and couples. In all cases, the loads are applied both along the shear centre axis and also along axes parallel to it and located at the beam top and bottom surfaces. The results presented and discussed, which consist of pre-buckling stress fields, buckling curves and buckling mode shapes, are obtained by means of the newly released code GBTul 2.0 and validated by means of the comparison with shell finite element values obtained with the code Ansys.

Numerical modeling of ice transport in channels with river restoration structures

Cross vane structures aim to reduce near-bank shear and increase center channel flow intensity to retain flood flow and maintain sediment transport capacity in stream channels. These in-stream structures have been widely applied for river management, but how they interact with the local ice regime is poorly understood. This paper presents a numerical model study of the interaction of cross vane structures with sediment and ice using a two-dimensional coupled river ice-sediment dynamic model. The model was used to simulate a variety of ice and flow conditions to show how cross vanes affect sediment and ice processes. The study showed the design of cross vane structures in cold region rivers should consider the ice effects, especially the increasing potential of ice jam formation and related bed change.

Lateral buckling of tapered members

• Centroidal and shear centre axes vary in tapered mono-symmetric members. • Using a web centre line axis simplifies loading and restraint data. • Using lateral deflections and twist rotations for this axis simplifies buckling energy equation. • Members are modelled as a series of tapered lines which form fans between plate edges. • Finite element program is verified for a wide range of numerical and test results. • Program can be used to explore optimisation.

P05.02: Different cross‐section in lateral‐torsional buckling

ABSTRACTThe goal of the research presented in this paper is the experimental and theoretical analysis of cross‐section which can be asymmetric. Particularly, there are experimental tests of several beams of selected cross‐section shapes described and analysed. The main idea of those experiments was to observe the differences in behaviour between monosymmetric cross‐section and the asymmetric one. Currently used design methods for a solution of lateral‐torsional buckling are based on critical moment approach. This solution is also implemented in design codes but it is limited for use of double and monosymmetric cross‐sections.Another part of the paper is focused on the numerical evaluation of transversally loaded beam of one monosymmetric and four asymmetric cross‐section shapes susceptible to lateral‐torsional buckling. Several analytic methods used for evaluation of the beam bending resistance in design software were evaluated.This paper presents the comparison of numerical and experimental results of transversally loaded beams with monosymmetric and asymmetric cross‐sections. Stresses in selected parts of the cross‐section are evaluated and compared together. This comparison shows that the normal stress on both (monosymmetric and asymmetric) cross sections is the subject of the same rules – a combination of stresses inflicted by bending and torsion caused by an eccentricity between the point of load application and the centre of shear.

Effect of moment gradient and load height with respect to centroid on the reliability of wide flange steel beams subject to elastic lateral torsional buckling

The reliability of doubly-symmetric wide flange steel beams designed to the AISC Specification for Structural Steel Buildings subjected to elastic lateral torsional buckling was evaluated when considering variation in moment gradient and load height. The analysis considers continuous loads on spans subjected to various end moments with supports that are torsionally fixed and laterally supported, without additional intermediate restraints. Dead load, occupancy live load, and beam resistance random variables were considered. Beam lateral torsional buckling resistance was evaluated from numerical solution of a fundamental differential equation that accounts for the effect of moment gradient and load height. In some cases, it was found that use of the AISC design procedure results in significant inaccuracies for estimation of elastic lateral torsional buckling resistance, where underestimations occur in regions of reverse curvature bending and when loads are placed below the beam shear center, while large overestimations can occur when loads are placed above the beam shear center. These discrepancies result in significant variation in beam reliability. However, the use of accurate equivalent uniform moment factors can restore uniformity in notional reliability level.

Staff Radiation Exposure Reduction

This two-part contribution presents a novel theory of bending of thin-walled beams with influence of shear (TBTS). The theory is valid for general open thin-walled cross-sections. The loads are reduced to the cross-section principal pole, i.e. the cross-section shear center. In part I, the TBTS is established, in this part II, the theory is used to analyze various beam cross-section shapes, loads and boundary conditions. Beams with the extreme low ratios of beam length to beam cross-section contour dimensions are analyzed. The stress predictions of the TBTS, stresses as functions of the longitudinal coordinate as well as the cross-section curvilinear coordinate, are compared to those obtained by finite elements computations (FEM) as well as to exact solutions of the theory of elasticity. The results of the TBTS are given in closed analytic forms, i.e. parametric forms, suitable for general studies of thin-walled beam behavior under transvers bending loads, as well in the early design stage of thin-walled structures.

Modeling and analysis for coupled flexural-torsional spinning beams with unsymmetrical cross sections

The structural modeling and dynamic properties of a spinning beam with an unsymmetrical cross section are studied. Due to the eccentricity and spinning, transverse deflections along the two principal directions and the torsional motion about the longitudinal axis are coupled. The structural model of the beam is established based on the Hamilton principle and by incorporating the torsional inertia. Moreover, because of its significant influence on characteristics for the non-circular cross-sectional beam, the warping effect is considered in the formulation. The proposed model is effectively validated in two cases: the spinning beam with a symmetric cross section and the cantilevered beam with an unsymmetrical cross section. Then the effects of the spinning speed on natural frequencies and mode shapes are investigated. Numerical results reveal that the critical speed is altered with respect to noncoincidence of the centroid and the shear center. For the beams with strong warping rigidities, the warping effect cannot be neglected due to significant influence on natural frequencies.

Inelastic buckling capacity of beams considering load height effect, stepped flanges, and compact or noncompact sections

This study aims to investigate the effects of load height, stepped beam configuration and section compactness on the inelastic lateral buckling strength of doubly and singly stepped I-beams. To study the effects of load height, the loads are situated at three points: at the center of the top flange, at the shear center and at the center of the bottom flange. To check for the effect of stepped beam configuration, beams were stepped either at both top and bottom flanges or at top flange only. Meanwhile, to investigate the effects of section compactness, two sections are analyzed: one section with compact flanges and web and another section with non-compact flanges and a compact web. The loadings are limited to those having an inflection point of zero. To also check the effect of steps, stepped parameters α, β and γ are varied. The buckling strengths of the beams investigated are obtained by conducting nonlinear analysis using the finite element program, ABAQUS. In conducting the nonlinear analysis, the residual stresses and geometric imperfections are taken into consideration to clearly simulate the inelastic behavior of the beams. The results of the analysis would then determine if the location of the loads, configuration of stepped flanges and compactness of flanges have significant effect on the inelastic buckling strength of the stepped beams.

Bending of thin-walled beams of open section with influence of shear—Part II: Application

This two-part contribution presents a novel theory of bending of thin-walled beams with influence of shear (TBTS). The theory is valid for general open thin-walled cross-sections. The loads are reduced to the cross-section principal pole, i.e. the cross-section shear center. In part I, the TBTS is established, in this part II, the theory is used to analyze various beam cross-section shapes, loads and boundary conditions. Beams with the extreme low ratios of beam length to beam cross-section contour dimensions are analyzed. The stress predictions of the TBTS, stresses as functions of the longitudinal coordinate as well as the cross-section curvilinear coordinate, are compared to those obtained by finite elements computations (FEM) as well as to exact solutions of the theory of elasticity. The results of the TBTS are given in closed analytic forms, i.e. parametric forms, suitable for general studies of thin-walled beam behavior under transvers bending loads, as well in the early design stage of thin-walled structures.

Flexural-torsional analysis of functionally graded sandwich I-beams considering shear effects

Abstract This paper presents a flexural-torsional analysis of shear flexible thin-walled sandwich I-beams with functionally graded materials (FGMs). The transverse shear and warping shear deformations are considered in this study. Based on the power law distribution of volume fraction of ceramic or metal, the mechanical properties of beam such as Young's and shear moduli are assumed to continuously vary in the thickness direction. The locations of center of gravity and shear center for FG beams are derived. Governing equations are also derived from the principle of minimum total potential energy. To solve the static problems of FG beams, the isoparametric beam element using the reduced Gauss numerical integration scheme is employed with the scope to discretize the governing equations. In order to demonstrate the validity of this study, the numerical solutions for the flexural-torsional problem of beams with bisymmetric and mono-symmetric I-sections are presented and compared with the previously published results. The effects of shear deformation, gradient index, thickness ratio of ceramic, material ratio, boundary condition and span-to-height ratio on the flexural-torsional response of FG I-beams are parametrically investigated.

A geometrically exact Kirchhoff beam model including torsion warping

This paper presents a geometrically exact beam model that includes the Kirchhoff constraint (null shear deformation), torsion-related warping and Wagner effects, and aims at capturing the flexural-torsional behaviour of thin-walled beams undergoing large displacements. The parametrization of the cross-section rotation tensor is based on the composition of a torsional rotation and the so-called smallest rotation to the tangent vector of the beam axis, as proposed in Boyer and Primault (2004). The novel aspects of the proposed model are the following: (i) closed-form expressions for the variations of the strain measures are presented and the equilibrium equations and their linearization are completely written in terms of the independent kinematic parameters, (ii) torsion-warping is allowed, as well as Wagner effects, and (iii) arbitrary cross-sections are considered, namely cross-sections where the shear centre and centroid do not coincide. Moreover, several fundamental aspects related to the cross-section rotation parametrization are discussed. The accuracy and the computational efficiency of the finite element implementation of the proposed model is demonstrated in several numerical examples.

Analytical Solutions of Lateral–Torsional Buckling of Castellated Beams

The majority of the existing literature on the lateral stability of castellated beams deals with experimental and/or numerical studies. This paper presents a comprehensive analytical study of the lateral–torsional buckling of simply supported castellated beams subject to pure bending and/or a uniformly distributed load. Using the principle of total potential energy, analytical expressions for the critical buckling moments and loads are derived and applied for various beam lengths. The three different locations of the applied load are used: At the top flange, shear center and bottom flange. The results show that the influence of web openings on the critical buckling moments and loads are mainly due to the reduction of the torsional constant caused by the web openings. Web shear effects and web shear buckling become important only when the beam is short and the flange is wide. The critical moments and loads will be overestimated or underestimated if the full or reduced section properties are used. The accurate critical moment or load should be calculated based on the average torsional constant of the full and reduced sections rather than simply taking the average of the critical moments or loads calculated from the full and reduced section properties. The present analytical solutions are verified using 3D finite element analysis results.

The experimental and analytical investigations of torsion phenomenon of thin-walled cold formed channel beams subjected to four-point bending

Thin-walled beams have many practical applications. This stimulates practical and scientific researches on them. The recent developments in metal cold forming makes it possible to produce more complicated shapes to reduce the mass of structural members. However, vast variety of cross-section of such beams make their analysis difficult. In the paper the authors have investigated thin-walled cold formed steel beams with open section loaded subjected to pure bending. Experimental results were compared with analytical ones. This gave the answer why the actual tested beams rotated around the shear centre though they were subjected to pure bending. The authors described the factors which are responsible for the torsion moment in the investigated thin-walled steel beams with open sections subjected to pure bending. Two different cross-sections of thin-walled beams were investigated and the more resistant to torsion phenomena was indicated.

Structural characterization of rotor blades through photogrammetry

This paper deals with the use of photogrammetry for the experimental identification of structural and inertial properties of helicopter rotor blades. The identification procedure is based upon theoretical/numerical algorithms for the evaluation of mass and flexural stiffness distributions which are an extension of those proposed in the past by Larsen, whereas the torsional properties (stiffness and shear center position) are determined through the Euler–Bernoulli beam theory. The identification algorithms require the knowledge of the blade displacement field produced by known steady loads. These data are experimentally obtained through photogrammetric detection technique, which allows the identification of 3D coordinates of labeled points (markers) on the structure through the correlation of 2D digital photos. Indeed, the displacement field is simply evaluated by comparing the markers positions on the loaded configuration with those on the reference one. The proposed identification procedure, numerically and experimentally validated in the past by the authors, has been here applied to the structural characterization of two main rotor blades, designed for ultra-light helicopters. Strain gauges measurements have been used to assess the accuracy of the identified properties through natural frequencies comparison as well as to evaluate the blades damping characteristics.