Lateral bracing of I-girder with corrugated webs under uniform bending

Lateral-torsional buckling may occur in an unrestrained beam where its compression flange is free to displace laterally and rotate. This paper presents the results of the theoretical and finite element analyses of the lateral-torsional buckling of I-girders with corrugated webs and lateral bracing, under uniform bending. It is well known that an elastic lateral brace restricts partially the lateral buckling of slender beams and increases the elastic buckling moment. However, a full study of the effect of lateral braces on lateral-torsional buckling has not been made especially for I-girder with corrugated webs. This paper develops a three-dimensional finite element model using ANSYS [User’s manual, version 10.0] for the lateral-torsional buckling analysis of I-girder with corrugated webs and uses it to investigate the effects of elastic lateral bracing stiffness on the critical moment of simply supported I-girders with corrugated webs under pure bending. It was found that for plastic and inelastic I-girder with corrugated webs, the effect of bracing initially is increased to some extent as the lateral unbraced length increases and then decreased until the beam behaves as an elastic beam. In other words, the effect of bracing depends not only on the stiffness of the restraint but also on the modified slenderness of the I-girder. Also, the results show that Winter’s simplified method to determine full brace requirements cannot be applied to I-girders with corrugated webs. Therefore, a general equation is proposed to determine the value of optimum stiffness (K_opt) in terms of the I-girder’s slenderness.     

Interaction of buckling modes in castellated steel beams

This paper investigates the behaviour of normal and high strength castellated steel beams under combined lateral torsional and distortional buckling modes. An efficient nonlinear 3D finite element model has been developed for the analysis of the beams. The initial geometric imperfection and material nonlinearities were carefully considered in the analysis. The nonlinear finite element model was verified against tests on castellated beams having different lengths and different cross-sections. Failure loads and interaction of buckling modes as well as load-lateral deflection curves of castellated steel beams were investigated in this study. An extensive parametric study was carried out using the finite element model to study the effects of the change in cross-section geometries, beam length and steel strength on the strength and buckling behaviour of castellated steel beams. The parametric study has shown that the presence of web distortional buckling causes a considerable decrease in the failure load of slender castellated steel beams. It is also shown that the use of high strength steel offers a considerable increase in the failure loads of less slender castellated steel beams. The failure loads predicted from the finite element model were compared with that predicted from Australian Standards for steel beams under lateral torsional buckling. It is shown that the Specification predictions are generally conservative for normal strength castellated steel beams failing by lateral torsional buckling, unconservative for castellated steel beams failing by web distortional buckling and quite conservative for high strength castellated steel beams failing by lateral torsional buckling.     

A new prediction model for the load capacity of castellated steel beams

In this study, a robust variant of genetic programming, namely gene expression programming (GEP), is utilized to build a prediction model for the load capacity of castellated steel beams. The proposed model relates the load capacity to the geometrical and mechanical properties of the castellated beams. The model is developed based on a reliable database obtained from the literature. To verify the applicability of the derived model, it is employed to estimate the load capacity of parts of the test results that were not included in the modeling process. The external validation of the model was further verified using several statistical criteria recommended by researchers. A multiple least squares regression analysis is performed to benchmark the GEP-based model. A sensitivity analysis is further carried out to determine the contributions of the parameters affecting the load capacity. The results indicate that the proposed model is effectively capable of evaluating the failure load of the castellated beams. The GEP-based design equation is remarkably straightforward and useful for pre-design applications.     

Semi-empirical design of complex metal beams by buckling analysis

Structural elements with complex geometries, boundary conditions and load patterns cannot be designed against buckling using empirical formulae because of uncertain elastic buckling moments or unknown buckling effective lengths, which are basic parameters for these equations. This article proposes a shell finite element procedure for buckling design of metal beams of complex configurations with codified initial imperfections assumed in the Perry–Robertson formula. The advantage of the proposed method lies in the use of elastic buckling moment with empirical design formulae for determination of design moment capacity of a beam; thereby eliminating the uncertainty of modelling initial imperfections. More importantly, the moment modification factor and assumption of effective length can be avoided because all second-order and yield effects have been considered in the computer model. Numerical examples demonstrate that the simplified method has a high level of accuracy, versatility and flexibility for the design of complex beams.     

Buckling of laterally braced beams

This paper reports on thteral buckling tests of H-shaped beams unbraced or braced laterally by purlins or a sub-beam. These tested beams were subjected to uniform moment or moment gradient. The restraining effect of adjacent members, bracing effect of purlins or a sub-beam and effect of moment gradient on load carrying capacity and the post-lateral-buckling behaviour were investigated. Elastic-plastic analysis by means of the finite element method was performed to simulate the experimental behaviour. Analytical results were found to match well with experimental behaviour for the most part. Effective length factors that incorporate the effects of moment gradient, restraining, and bracing into design of H-shaped beams are discussed and proposed, based mainly on the analytical results.     

Ultimate strength of submarine pressure hulls with failure governed by inelastic buckling

The analysis of submarine pressure hull assumes great importance among structural engineers due to the complexity involved in the collapse mechanism of stiffened shell structures. In most of the cases, the failure of stiffened shell structures occurs due to elastic buckling. But for some combinations of shell-stiffener geometry and material characteristics, the structure can fail by inelastic buckling, for which the methods of analysis are meagre. In this paper, the analysis of submarine pressure hull structure in which the failure gets governed by inelastic buckling is demonstrated. Three different approaches have been employed to investigate the ultimate strength of the ring stiffened submarine pressure hull structure with inelastic buckling modes of failure. The methods used are ‘Johnson–Ostenfeld inelastic correction’, ‘imperfection method’ and ‘finite element approach’. A typical submarine shell structure has been analysed for the inelastic buckling failure using these three approaches and the results are discussed.     

竖向荷载作用下厂房纵向支撑的受力分析

从保证厂房柱列纵向稳定性的角度,对柱顶撑杆的设计要求进行了研究。采用蒙特卡罗方法,考虑实际工程中柱子和柱顶撑杆初始缺陷的随机遇合,应用有限元程序ANSYS对承受竖向荷载作用的柱列纵向支撑体系进行了大量的仿真分析,得到了柱列纵向支撑体系的三种失稳模式;通过概率统计得到柱顶撑杆所受内力的三峰正态概率密度函数,发现柱子和撑杆初始缺陷的随机遇合作用导致柱顶撑杆受压、拉或零受力的随机性,降低了支撑力,据此确定了可用于实际工程设计的支撑力的大小,为传统的柱顶撑杆由纵向受力决定的设计方法补充了竖向受力分析的验算方法。     

Assessment of various kinematic models for instability analysis of sandwich beams

The aim of this paper is to present and compare various one-dimensional (1D) finite element (FE) models which are based on different kinematic models. The kinematic models presented take into account thickness variations, but differ in the inclusion of the shear deformation effects in their kinematics formulations. These 1D FE models are used to study global and local instability phenomenon in sandwich beams. Simulations of the buckling and three-point bending tests are performed to verify the capability of each model to reproduce the linear and nonlinear mechanical response of sandwich beams. The predicted results using 1D FE models are compared to a two-dimensional (2D) FE analysis, which is considered as a reference solution.     

Lateral buckling resistance of inelastic I-beams under off-shear center loading

Lateral-torsional buckling (LTB) strength of steel I-beams subjected to moment gradient loading is scaled by the moment gradient factor, C_b. The C_b factor depends on the non-uniformity of moment diagram, the height of the applied transverse loads within the unbraced length and end conditions. Generally, the C_b factors given by codes have been derived from elastic LTB analysis theory. However, the same C_b factors are used for beams that buckle inelastically. This paper develops a three dimensional finite element model using ANSYS for the inelastic nonlinear flexural-torsional analysis of I-beams and uses it to investigate the effects of unbraced length and central off-shear center loading (located at center, top flange and bottom flange) on the moment gradient factor in inelastic behavioral zone. It is found that the C_b factors given by AISC-LRFD in Specification for structural steel buildings (AISC 360-05) and Structural Stability Research Council Guidelines are not accurate for the point load cases applied at center and bottom flange in which I-beam buckles inelastically. It is seen also that the AISC-LRFD flexural resistance equations overestimate the actual moment capacity of inelastic I-beams under moment gradient. Therefore, a simple equation is proposed to be used instead of the code equation in inelastic zone for the investigated load cases in this paper.     

Simulation of cold-formed steel beams in local and distortional buckling with applications to the direct strength method

A nonlinear finite element (FE) model is developed to simulate two series of flexural tests, previously conducted by the authors, on industry standard cold-formed steel C- and Z-section beams. The previous tests focused on laterally braced beams with compression flange details that lead predominately to local buckling failures, in the first test series, and distortional buckling failures, in the second test series. The objectives of this paper are to (i) validate the FE model developed for simulation of the testing, (ii) perform parametric studies outside the bounds of the original tests with a particular focus on variation in yield stress and influence of moment gradient on failures, and (iii) apply the study results to examine and extend the Direct Strength Method of design. The developed FE model shows good agreement with the test data in terms of ultimate bending strength. Extension of the tested sections to cover yield stresses from 228 to 506 MPa indicates that the Direct Strength Method is applicable over this full range of yield stresses. The FE model is also applied to analyze the effect of moment gradient on distortional buckling. It is found that the distortional buckling strength of beams is increased due to the presence of moment gradient. Further, it is proposed and verified that the moment gradient effect on distortional buckling failures can be conservatively accounted for in the Direct Strength Method by using an elastic buckling moment that accounts for the moment gradient. An empirical equation, appropriate for use in design, to predict the increase in the elastic distortional buckling moment due to moment gradient, is developed.     

Nonlinear free vibration analysis of asymmetric thin-walled circularly curved beams with open cross section

A finite element formulation is present for the nonlinear free vibration of thin-walled curved beams with non-symmetric open across section. The kinetic and potential energies are derived by the virtual principle. The energy functional includes the effect of flexural–torsional coupling, the torsion warping and the shear center location. For finite element analysis, cubic polynomials are utilized as the shape functions of the two nodal thin-walled curved elements. Each node possesses seven degrees freedom including the warping degree of freedom. The nonlinear eigenvalue problem has been solved by the direct iteration technique. The results are compared with those for straight beams as available in the literature. The results for nonlinear free vibration analysis of curved beams for various radii and subtended angle are presented.     

Fire analysis of timber composite beams with interlayer slip

The purpose of this paper is to model the behaviour of timber composite beams with interlayer slip, when simultaneously exposed to static loading and fire. A transient moisture-thermal state of a timber beam is analysed by the Luikov equations, and mechanical behaviour of timber composite beam is modelled by Reissner's kinematic equations. The model can handle layers of different materials. Material properties are functions of temperature. The thermal model is validated against the experimental data presented in the literature. Generally, the model provides excellent agreement with the experimental data. It is shown that the material properties of timber play an important role in the fire resistance analysis of timber structures when exposed to fire.     

Mechanically-fastened hybrid composites for flexural strengthening of steel beams

Recent experimental studies by Sweedan et al. [17] and Alhadid et al. [2] on the behavior of mechanically fastened (MF) steel-FRP lap connections and steel beams strengthened with MF-FRP, respectively, revealed a promising efficiency of the fastening system in retrofitting deteriorated steel beams. The study demonstrated that the dominant failure mode, in the tested connections and beams, was due to excessive bearing in the FRP laminate at the locations of the fasteners as long as sufficient number of fasteners is used. The current study describes a three-dimensional nonlinear finite element (FE) model that accounts for the interfacial slip between the FRP laminates and the steel beam. The FE model is validated against the experimental results reported by Alhadid et al. [2], and excellent agreement is found. The FE model is then used to shed more light on the mechanical behavior of the tested composite steel-FRP beams including force distribution in steel fasteners especially during spread of yielding in the steel section, and the stress distribution in the FRP laminates. The study concludes that as the length of the FRP increases, the degree of composite action in the elastic range increases indicating higher efficiency of the FRP laminate. The FRP laminate contributes significantly in carrying the mid-span loads after yielding of the steel section.     

Quadruply coupled linear free vibrations of thin-walled beams with a generic open section

The coupled vibration of thin-walled beams with a generic open section induced by the boundary conditions is investigated using the finite element method. If the axial displacement of the pin end is restrained at another point rather than the centroid of the asymmetric cross section, the axial vibration, two bending vibrations, and torsional vibration may be all coupled. The element developed here has two nodes with seven degrees of freedom per node. The shear center axis is chosen to be the reference axis and the element nodes are chosen to be located at the shear centers of the end cross sections of the beam element. Different sets of element nodal degrees of freedom corresponding to different pin ends are considered here. The relation between element matrices referred to different sets of element nodal degrees of freedom is derived.

Numerical examples are presented to demonstrate the accuracy of the proposed method and to investigate the effects of different pin ends on the coupled vibrations of the thin-wall beam.     

Finite element formulation for inflatable beams

The discretized nonlinear equations for bending and buckling of inflatable beams are written by use of the virtual work principle with Timoshenko's kinematics, finite rotations and small strains. The linearized equations around a pre-stressed reference configuration are then deduced, giving rise to a new inflatable beam finite element. The stiffness matrix contains the shear coefficient and the internal pressure. Use is made of the particular 3-node beam element to investigate the bending and the buckling of a cantilever beam, the deflection of a pinched torus and the buckling of a torus submitted to a radial compressive force. The numerical results obtained with the beam element are shown to be close to analytical and three-dimensional (3D) membrane finite element results. The validity of the numerical results is discussed, in connection with the concepts of the crushing force or the wrinkling pressure of the inflated beam.     

空间梁单元切线刚度矩阵的精确分析方法

为有效进行空间刚架结构后屈曲分析,提出一种新的空间梁单元切线刚度矩阵的精确分析方法。首先用直接法建立梁单元杆端力与杆端位移的增量关系式,然后根据矩阵微分理论求出单元杆端力关于杆端位移的导数,在求导结果表达式中令杆端位移增量为0,即可得到梁单元切线刚度矩阵。对六层和二十层空间刚架结构进行了后屈曲分析。结果表明:所得的空间梁单元切线刚度矩阵具有足够精度,可有效用于大型空间刚架结构的后屈曲分析。     

Improvement in accuracy of the finite element method in analysis of stability of Euler–Bernoulli and Timoshenko beams

The problem of buckling of the Euler–Bernoulli and Timoshenko beams is analyzed by the finite element method. Significant improvement in accuracy of the method is obtained by replacing the discontinuous function of the bending moment related to the approximation of the eigenfunction obtained by FEM by a “smoothed” function in the Rayleigh quotient. The smoothed function is obtained by fitting the discontinuous one using the least square technique.     

Non-linear finite element analysis of axially restrained steel beams at elevated temperatures in a fire

A method is presented for the analysis of the non-linear structural behavior of axially restrained steel beams at elevated temperatures, which employs the axis arc-length and section rotation of the deformed beam as basic variables. The novelty of the proposed formulation is an inclusion of a balance function that measures the error of the equilibrium between the internal- and external-forces in a cross-section of the beam. This strategy can easily deal with the geometric non-linearity and elasto-plasticity of steel at elevated temperatures. Each node for representing the section of the beam has two degrees of freedom in the proposed method. It is more computationally economical than the traditional beam element, which has three degrees of freedom. An example beam is studied to verify the proposed method. Parameters including the load ratio, axial restraint stiffness ratio, transversal and longitudinal temperature gradient, are studied. The middle-span’s deflection, axial force and moment, along with the strain and stress distributions across the section, are calculated at elevated temperatures. The comparison with results from the finite element method employing shell elements shows that the method presented here is precise.     

Behaviour of headed stud shear connectors for composite steel-concrete beams at elevated temperatures

The behaviour of composite steel-concrete beams at elevated temperatures is an important problem. A three-dimensional push test model is developed herein with a two-dimensional temperature distribution field based on the finite element method (FEM) and which may be applied to steel-concrete composite beams. The motivation for this paper is to increase the awareness of the structural engineering community to the concepts behind composite steel-concrete structural design for fire exposure. The behaviour of reinforced concrete slabs under fire conditions strongly depends on the interaction of the slabs with the surrounding elements which include the structural steel beam, steel reinforcing and shear connectors. This study was carried out to consider the effects of elevated temperatures on the behaviour of composite steel-concrete beams for both solid and profiled steel sheeting slabs. This investigation considers the load-slip relationship and ultimate load behaviour for push tests with a three-dimensional non-linear finite element program ABAQUS. As a result of elevated temperatures, the material properties change with temperature. The studies were compared with experimental tests under both ambient and elevated temperatures. Furthermore, for the elevated temperature study, the models were loaded progressively up to the ultimate load to illustrate the capability of the structure to withstand load during a fire. It is concluded that finite element analysis showed that the shear connector strength under fire exposure was very sensitive. It is also shown that profiled steel sheeting slabs exhibit greater fire resistance when compared with that of a solid slab as a function of their ambient temperature strength.     

Two-dimensional FE model for evaluation of composite beams, I: Formulation and validation

This paper develops a two-dimensional finite element model for composite beams, based on the use of the commercial package ANSYS. Different degrees of continuity can be taken into account, permitting the investigation of systems ranging from simply-supported to fully-continuous. Beams with either full or partial shear connection can be considered, as well as different slab typologies. The model represents the behaviour of all components of the composite arrangement, based on the best available current understanding, including the reinforcing bars (considering the tension-stiffening effect), the load-slip characteristic of the shear connectors, and the key components of the beam to column connection. In addition, material nonlinearity for all components is taken into account. The model has been fully verified by extensive comparisons against experimental and numerical results and it has been demonstrated that it is suitable for comprehensive parametric studies of the behaviour of composite beams. A sensitivity study focusing on both the required and the available rotation capacities and their importance for moment redistribution in semi-continuous composite beams is presented in the companion paper.