Major Axis Bending Research Articles

Bracing FRP I-section beams to avoid lateral-torsional buckling

This article explains as to how lateral-torsional buckling can be avoided in fiber-reinforced plastic (FRP) beams having an I-shaped cross section and subjected to major-axis bending. To enable a FRP beam to be able to develop its maximum bending moment capacity corresponding to material cracking of the extreme fibers, a number of lateral braces need to be provided. Herein, a procedure is outlined for determining the required number of lateral braces for a FRP I-section beam. It is shown that a FRP beam loaded about its major cross-sectional axis can experience a drastic reduction in its bending moment capacity if such lateral bracing is not provided, whereas it can develop its maximum load-carrying capacity up to the cracking of the extreme fibers if a sufficient number of lateral braces are provided.

Experimental study of steel single unequal-leg angles under eccentric compression

A total of 26 steel unequal-leg angle specimens were tested under eccentric compression with respect to either major or minor principal axis of the cross-section to study the steel single angle beam–column behavior. Results suggested that for angles subjected to major principal axis bending causing the angle short leg in compression, the presence of moment, in some cases, resulted in the angle ultimate load higher than its concentric compressive capacity. In the case of the major axis bending causing the long leg in compression and minor axis bending, an increase in eccentricity resulted in a reduction in the angle ultimate load but the extent of the reduction was more pronounced for specimens under minor axis bending. Overall, the effect of eccentricity on the ultimate load decreased as the slenderness increased. The design of angle beam–columns as proposed in AISC Specification 2005 and the Direct Strength Method was evaluated using the test results.

Practical Design Considerations for Lightweight Channels under Combined Compression, Major and Minor Axis Bending

The paper examines the behavior of lightweight channels under combinations of axial compression, major and minor axis bending. Design guidelines are proposed that can be used by practicing engineers to maximize the channel section performance. Results are presented showing the design space of several North American standard cold-formed steel sections subjected to this load combination. A three-dimensional multistory building model is used to demonstrate cost-effective design strategies that cope with challenges encountered in practice. The influence of the flange, web size, and section thickness on the ultimate strength and serviceability of the channel section are also highlighted. The study leads to significant guidelines that can be used by practicing engineers and steel fabricators in order to optimize the section sizes under this load combination.

Behaviour of steel single angles subjected to eccentric axial loads

The response of steel single angles subjected to axial eccentric loading is investigated by means of numerical modeling based on finite element techniques. Results show that in the case of eccentric compression causing major axis bending, a critical eccentricity for each slenderness ratio investigated exists and below this eccentricity, any reduction in the ultimate capacity due to eccentricity is marginal. The critical eccentricity increases with the slenderness ratio of the angle. In contrast, for the case of eccentric compression causing minor axis bending, the reduction in ultimate capacity, as affected by increasing eccentricity, is more pronounced and no similar critical eccentricity is identified. When compared with values determined from the design equations suggested in the AISC specification 2005, results indicate that the AISC equations, in general, give a satisfactory estimate of the ultimate capacity for angles subjected to axial compression and minor axis bending. For the case of eccentric compression causing major axis bending, AISC equations are conservative over a large range of parameters while they overestimate capacities in some other angles.

Buckling analysis of imperfect I-section beam-columns with stochastic shell finite elements

Buckling loads of thin-walled I-section beam-columns exhibit a wide stochastic scattering due to the uncertainty of imperfections. The present paper proposes a finite element based methodology for the stochastic buckling simulation of I-sections, which uses random fields to accurately describe the fluctuating size and spatial correlation of imperfections. The stochastic buckling behaviour is evaluated by crude Monte-Carlo simulation, based on a large number of I-section samples, which are generated by spectral representation and subsequently analyzed by non-linear shell finite elements. The application to an example I-section beam-column demonstrates that the simulated buckling response is in good agreement with experiments and follows key concepts of imperfection triggered buckling. The derivation of the buckling load variability and the stochastic interaction curve for combined compression and major axis bending as well as stochastic sensitivity studies for thickness and geometric imperfections illustrate potential benefits of the proposed methodology in buckling related research and applications.

Finite element study of steel single angle beam–columns

A finite element study was conducted to investigate the response of steel single angles subjected to eccentric compression and the numerical results were used to examine the effectiveness of the beam–column interaction approach suggested in the current AISC design standard. Numerical results demonstrated inconsistencies and overall conservatism inherent in the design equations and the possible sources of this conservatism were also identified and discussed. Results also indicated the existence of a critical eccentricity associated with each slenderness ratio in the case of eccentric compression causing major axis bending. Within this eccentricity, any reduction in the ultimate capacity due to eccentricity was found to be marginal. A new interaction equation for combined compression and principal axis bending was proposed. The proposed equation achieved the target reliability index with an improved test-to-prediction capacity ratio over a wide range of angle geometric parameters when compared with AISC design equations.

Local/distortional mode interaction in cold-formed steel lipped channel beams

This paper reports a numerical investigation concerning the post-buckling behaviour of cold-formed steel lipped channel beams, subjected to uniform major axis bending, affected by local/distortional mode interaction – the results presented and discussed were obtained through shell finite element analyses performed using the code Abaqus. One analyses simply supported beams (locally/globally pinned and free to warp end sections) (i) with cross-section dimensions ensuring coincident local and distortional critical buckling moments and (ii) having critical-mode initial geometrical imperfections with different shapes and the same overall amplitude – local buckling is triggered either by the compressed flange (most common case) or by the web. The results reported consist of (i) elastic and elastic–plastic post-buckling equilibrium paths, (ii) curves and figures showing how the beam deformed configuration evolves along those paths and, for the elastic–plastic beams, (iii) figures making it possible to visualise (iii 1) the location and growth of plastic strains and (iii 2) the nature of the failure mechanisms detected.

An assessment of modeling strategies for composite curved steel I-girder bridges

This paper investigates the qualities and limitations of a number of modeling strategies for design analysis of composite curved steel I-girder bridges. The strategies range from modified line-girder analysis to 3D finite element approaches. Several key modeling considerations, including girder web distortion, the idealization of the cross-frames, support and load height, and displacement compatibility between the girders and slab, are addressed. Approximate approaches are investigated for the capture of composite I-girder web-distortion effects when modeling the girders using open-section thin-walled beam theory. A representative full-scale composite curved I-girder bridge, tested at the Federal Highway Administration (FHWA) Turner-Fairbank Highway Research Center (TFHRC), is utilized for assessment of the different approaches. The predictions of key lateral and vertical displacements, reactions, cross-frame forces, major-axis bending stresses and flange lateral bending stresses are evaluated using the test bridge. The paper highlights a number of subtle but critical blunders that can occur in the context of the different methods. Engineers need to be aware of these pitfalls in order to avoid potential problems due to inaccurate analysis predictions. The paper closes with a discussion of the efficacy of the various approaches.

GBT FORMULATION TO ANALYZE THE BUCKLING BEHAVIOR OF THIN-WALLED MEMBERS SUBJECTED TO NON-UNIFORM BENDING

In this paper, one investigates the local-plate, distortional and global buckling behavior of thin-walled steel beams subjected to non-uniform bending moment diagrams, i.e. under the presence of longitudinal stress gradients. One begins by deriving a novel formulation based on Generalized Beam Theory (GBT), which (i) can handle beams with arbitrary open cross-sections and (ii) incorporates all the effects stemming from the presence of longitudinally varying stress distributions. This formulation is numerically implemented by means of the finite element method: one (i) develops a GBT-based beam finite element, which accounts for the stiffness reduction associated to applied longitudinal stresses with linear, quadratic and cubic variation, as well as to the ensuing shear stresses, and (ii) addresses the derivation of the equilibrium equation system that needs to be solved in the context of a GBT buckling analysis. Then, in order to illustrate the application and capabilities of the proposed GBT-based formulation and finite element implementation, one presents and discusses numerical results concerning (i) rectangular plates under longitudinally varying stresses and pure shear, (ii) I-section cantilevers subjected to uniform major axis bending, tip point loads and uniformly distributed loads, and (iii) simply supported lipped channel beams subjected to uniform major axis bending, mid-span point loads and uniformly distributed loads — by taking full advantage of the GBT modal nature, one is able to acquire an in-depth understanding on the influence of the longitudinal stress gradients and shear stresses on the beam local and global buckling behavior. For validation purposes, the GBT results are compared with values either (i) yielded by shell finite element analyses, performed in the code ANSYS, or (ii) reported in the literature. Finally, the computational efficiency of the proposed GBT-based beam finite element is briefly assessed.

Elastic buckling of I-beam flanges subjected to a linearly varying stress distribution

Lateral forces or torsion combined with major axis bending result in a stress gradient across the compression flanges of I-shaped flexural members. Sources that cause significant stress gradient include situations where unequal reactions come to opposite sides of a girder or beams that frame into one side of a girder, skewed bridge girder supports, wind loading on facia girders, curvature induced warping, etc. In this study, three sets of support conditions are considered for evaluating the elastic buckling capacity of flanges of I-shaped beams subjected to a stress gradient. This includes a Galerkin series solution that considers the full width of the flange plate with a variable rotational stiffness along the center of the plate. For a stress gradient leading to tension over part of the flange, equations for predicting the buckling capacity are developed using finite element analyses. Simplified equations are developed for I-shaped cross sections, and numerical examples are used to demonstrate the accuracy of the solution. A definition of the noncompact limit for flanges of I-shaped beams and girders subjected to stress gradient is presented.

Strength and Behavior of Cold-Formed Steel Z -Sections Subjected to Major Axis Bending

This paper presents a study on the strength and behavior of cold-formed steel Z -sections with simple and complex stiffeners subjected to major axis bending. The study involved both experimental and numerical investigations, in which a series of specimens were examined. All specimens were failed by the combined mode of shear and moment. The results from the investigations were analyzed and compared with the combined moment and shear strengths predicted by international design rules. The suitability of the design rules in predicting the combined capacity of Z -section with complex stiffeners was assessed through a reliability analysis.

Nonlinear Equation for Flexural Stiffness of Slender Composite Columns in Major Axis Bending

The ACI standard 318-02 permits the use of a moment magnifier approach for the design of slender composite steel-concrete columns. This approach is strongly influenced by the effective flexural stiffness (EI) , which varies due to the nonlinearity of the concrete stress-strain curve and the cracking along the column length among other factors. The EI equations given in the ACI code are approximate when compared to the EI values computed from the axial load-bending moment-curvature relationships. This study was undertaken to determine the influence of a full range of variables on EI used for the design of slender, tied, composite columns in which steel shapes are encased in concrete, and also to examine the existing ACI EI equations. Approximately 12,000 isolated square composite columns, each with a different combination of specified properties of variables, were simulated and used to generate the stiffness data. The columns studied were subjected to short-term ultimate loads and equal and opposite end mo...

Tests of cold-formed stainless steel tubular flexural members

A series of tests on cold-formed stainless steel square and rectangular hollow sections subjected to major axis bending is presented in this paper. The tests were performed on sections fabricated by cold-rolling from normal strength material of austenitic stainless steel type 304, and high strength material of duplex and high strength austenitic steel sheets. Tensile coupon tests were conducted to obtain the material properties of the test specimens. The test strengths were compared with the design strengths obtained using the American Specification and Australian/New Zealand Standard for stainless steel structures. The North American Specification for cold-formed carbon steel structural members was also used to predict and compare the bending strengths. In addition, the test strengths were compared with the theoretical elastic and plastic bending moments. It is shown that the design strengths predicted by the three specifications and the theoretical bending moments are generally conservative for the tested specimens. The reliability analysis was performed to evaluate the reliability of the design rules based on the existing resistance factors and load combination specified in the aforementioned specifications. It is shown that the American Specification and North American Specification are reliable for both normal and high strength specimens. The Australian/New Zealand Standard is reliable for normal strength specimens, but slightly unreliable for high strength specimens.

Buckling and Torsion of Steel Unequal Angle Beams

Although steel single angle sections are commonly used as beams to support distributed loads which cause biaxial bending and torsion, their behavior may be extremely complicated, and the accurate prediction of their strengths very difficult. Further, many design codes do not have any design rules for torsion, while some recommendations are unnecessarily conservative, or are of limited application, or fail to consider some effects which are thought to be important. This paper is one of a series on the behavior and design of single angle section steel beams. Two previous papers have studied the biaxial bending behavior of restrained beams, a third has studied the lateral buckling of unrestrained beams, and a fourth the biaxial bending of unrestrained beams. In each paper, simple design methods have been developed. In a companion paper an approximate method of predicting the second-order deflections and twist rotations of steel angle section beams under major axis bending and torsion is developed. The method is then used to determine the approximate maximum biaxial bending moments in such beams, which are then used with the section moment capacity proposals of the first paper of the series and the lateral buckling proposals of the third paper to approximate the member capacities. This method is extended to steel unequal angle beams in the present paper.

Buckling and Torsion of Steel Equal Angle Beams

Although steel single angle sections are commonly used as beams to support distributed loads which cause biaxial bending and torsion, their behavior may be extremely complicated, and the accurate prediction of their strengths very difficult. Further, many design codes do not have any design rules for torsion, while some recommendations are unnecessarily conservative, or are of limited application, or fail to consider some effects which are thought to be important. This paper is one of a series on the behavior and design of single angle section steel beams. Two previous papers have studied the biaxial bending behavior of restrained beams, a third has studied the lateral buckling of unrestrained beams, and a fourth the biaxial bending of unrestrained beams. In each paper, simple design methods have been developed. In this present paper, an approximate method of predicting the second-order deflections and twist rotations of steel equal angle beams under major axis bending and torsion is developed. This method is then used to determine the approximate maximum biaxial bending moments in such beams, which are then used with the section moment capacity proposals of the first paper of the series and the lateral buckling proposals of the third paper to approximate the member capacities. This method is extended to steel unequal angle beams in a companion paper.

Unified Resistance Equations for Design of Curved and Tangent Steel Bridge I-Girders

The provisions of the 2004 AASHTO load and resistance factor design specifications for steel I- and box-girder bridge design have been updated relative to previous specifications to simplify their logic, organization, and application and to improve their accuracy and generality. These provisions provide a unified approach for the flexural design of both tangent and curved I- and box-girder bridges. Updated resistance equations are a key component of this unified approach. An overview is provided of the updated resistance equations for I-section members. The primary focus is on handling coupled major-axis bending, minor-axis bending, and torsion from any source in both straight and horizontally curved I-section members.

Design of Single Angles Bent About the Major Principal Axis

Hot rolled steel angles are frequently used in applications wherein their flexural strength must be quantified. While flexural applications of angle members frequently involves bending within the plane of one of the angle legs, other bending scenarios can arise. One other similar scenario involves bending a single angle member about its major principal centroidal axis as shown in Figure 1. A practical example of when this type of bending may occur involves the case of sign and billboard structures in which an equal leg single angle member may be specified for the girts that support the face of the sign. The effects of gravity loading from the face of the sign result in major axis bending of the girts. In such instances it is obviously important to understand the major axis strength characteristics of the single angle members. \Nonlinear finite element modeling techniques have been employed herein for the study of equal-leg, hot-rolled steel single angle members bent about the major principal centroidal axis. Results from the modeling indicate that a lower bound design approach can be encapsulated in the form of two, linear, nominal moment predictive equations that depend on the angle leg slenderness and the angle beam overall slenderness.

Measurement techniques in the testing of thin-walled structural members

The paper describes methods for measuring displacements and end moments in the testing of thin-walled columns. The complete spatial deformation of a column can be captured by using local and overall deformation measurement frames. The local frame involves spring-loaded levers and roller bearings to mount the frame on to the specimen. The overall frame slides along high-precision shafts using linear ball bearings. Fixed-ended bearings are used to measure minor and major axis bending moments as well as the applied load. The bearings allow the line of action of the applied force to be determined.

Interaction curves for locally buckled I-section beam-columns

The paper presents a study of the strength of thin-walled I-sections in combined compression and major axis bending. The beam-columns are subjected to equal and opposite end moments causing bending in single curvature. Failure occurs by in-plane bending, local buckling and yielding at short lengths, and by flexural–torsional buckling, local buckling and yielding at intermediate and long lengths. The focus is on how the interaction curve is affected by local buckling. A geometric and material nonlinear finite element analysis is used to produce interaction curves for four cross-sections covering the range from slender to compact. The finite element model has been developed by the authors in a separate paper and shown to agree well with tests. For each cross-section, interaction curves are produced for four overall slenderness values including short and long beam-columns. The interaction curves are compared with design strengths obtained using the Australian Steel Structures Standard AS4100, the American Institute of Steel Construction Load and Resistance Factor Design Specification, and Eurocode 3: Part 1.1. The design strengths are shown to be generally conservative, particularly for slender cross-sections. Improved design interaction curves are proposed.

Nonlinear Analysis of Locally Buckled I-Section Steel Beam-Columns

The paper describes the development of an advanced finite element analysis of slender high strength steel I-sections in combined compression and major axis bending. The analysis accounts for geometric and material nonlinearity, and incorporates measured values of geometric imperfections and residual stresses. The failure modes involve local buckling of component plates and out-of-plane flexural-torsional member buckling. The finite element analysis is shown to be in good agreement with tests. A study is included to devise a simple method for measuring and modeling geometric imperfections in advanced analysis. It is shown that the agreement with tests obtained by determining the amplitude of the local geometric imperfection from a small number of measurements taken at midlength is comparable to that obtained by modelling the complete measured geometric imperfection consisting of measurements along the full length of the member at several points in the cross-section. A further study is included to investigate the sensitivity of the finite element analysis to changes in Young’s modulus, yield stress, residual stress and geometric imperfection.