Minor Axis Bending Research Articles

Inelastic behaviour of composite members under combined bending and axial loading

This paper presents an account of the results of a series of tests on partially-encased composite steel/concrete beam–columns. The experimental inelastic behaviour of members is examined under idealised conditions representing extreme lateral loading in combination with co-existing axial gravity loads. Members with three different cross-sectional sizes, utilising Grade S460 steel, were employed in the tests. The specimens were tested under major- or minor-axis bending in conjunction with a constant level of axial loading. The experimental arrangement and test rig constructed for this purpose are described together with the material properties and details of the specimens. The main experimental results from tests carried out on ten composite models are presented and discussed. In addition to providing essential data for validating future analytical and design studies, the experimental results and observations enable a direct assessment of several important factors. Particular emphasis is placed on key parameters related to yield and ultimate capacity as well as on ductility and residual strength considerations.

Flexural Strength for General Lateral–Torsional Buckling

The lateral–torsional buckling behavior of general prismatic and tapered steel members with doubly and singly symmetric sections are investigated based on the general forms of differential equilibrium equations for minor-axis bending and torsion. Fundamental lateral–torsional buckling moments are derived for general tapered steel members that are in forms similar to those for prismatic members. The factor accounting for the effects of level loading application is derived and incorporated into the moment diagram modification factor that is generalized for the prismatic and tapered steel members. American Institute of Steel Construction-type equations for calculating the inelastic lateral–torsional capacity are proposed by using straight-line transition, and limiting unbraced member lengths are explicitly derived. The analytical procedure presented can give reasonable results with consistent accuracy for general design consideration and follows the same philosophy as for prismatic doubly symmetric wide-flan...

Experimental Investigation of Aluminum Alloy Thin-Walled Tubular Members in Combined Compression and Bending

This paper presents a test program on aluminum alloy thin-walled square and rectangular hollow sections in combined axial compression and minor axis bending. A total of six series of tests was conducted that included 27 beam-column and 4 pure bending tests. The test specimens were manufactured by extrusion using 6061-T6 heat-treated aluminum alloy in three different cross sections having the clear width-to-thickness ratio ranged from 31.3 to 81.3. The beam-column specimens were compressed between pinned ends at different eccentricities in order to obtain an interaction curve for each series of tests. Pure bending tests were conducted for each test series to determine the bending capacities of the specimens. Local and overall initial geometric imperfections of the specimens were measured. Material properties were obtained from tensile coupon tests. The observed failure modes of the beam-column tests included local buckling, flexural buckling, interaction of local and flexural buckling, and material yielding in the heat-affected zone. The test strengths were compared with the design strengths calculated using the American, Australian/New Zealand, and European specifications for aluminum structures. Beam-column interaction curves were plotted for each test series. It is shown that the beam-column design strengths and bending capacities predicted by the three specifications are generally conservative.

Inelastic behaviour and design of slender I-sections in minor axis bending

Slender steel sections in bending are generally designed by taking the maximum moment as the yield moment. The assumption for the ultimate condition is thus the point at which first yield is reached in the section. Certain types of slender sections, however, have shown significant post-elastic behaviour in attainment of the ultimate moment. Experiments on I-sections in minor axis bending have shown this to be the case, where significant plastic stress distributions are attained in the tension flanges after the compression flanges have locally buckled. Current international steel specifications are unduly conservative when estimating the bending strength of these sections as the yield moment. This paper quantifies this conservatism and presents inelastic design methods whereby the post-elastic strength may be captured. Design equations are proposed for Australian, American and European hot-rolled and cold-formed steel specifications.

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 Provisions for Sections Containing Unstiffened Elements with Stress Gradient

This paper presents a general design procedure for calculating the moment capacity of sections containing unstiffened elements with stress gradient. The method uses design equations for effective widths of unstiffened elements with stress gradients based on plate test results. Current international design provisions allow the capacity of sections that contain stiffened elements with stress gradients to be calculated on the basis of inelastic reserve capacity, whereas sections that contain unstiffened elements must be designed on the basis of initiation of yielding in the section. This paper presents methods for the calculation of the capacity where inelastic reserve capacity may be taken into account, for sections containing fully effective unstiffened elements with stress gradient. The methods are shown to be in good agreement with experimental data of I-sections and plain channels in minor axis bending. Particular attention is given to the effect of both the elastic buckling coefficient used in the effective width method, and the use of inelastic considerations, on the bending capacity. Specific design proposals are presented in the form of amendments to the current American and Australian standards for cold-formed steel structures, for elements (effective width equations) and sections in bending (inelastic reserve capacity equations).

On the application of beam-column interaction formulae to steel members with arbitrary loading and support conditions

An investigation on dealing with the application of the so-called Level 1 and Level 2 beam-column interaction formulae to isolated steel members with arbitrary loading and end support conditions is reported. The issues addressed concern mostly the importance, both in terms of safety and accuracy, of adopting appropriate “equivalent moment factor” (Cm) values. Only the in-plane behavior of members subjected to axial compression and major or minor axis bending is investigated, but several load combinations and support conditions are dealt with. The strength estimates provided by the Level 1 and Level 2 interaction formulae are presented, discussed and compared with “exact” values, obtained from second-order plastic zone analyses performed by means of the FEM code, ABAQUS. On the basis of such comparative studies, several conclusions are drawn concerning the safety and accuracy of the strength estimates yielded by the two interaction formulae and, in particular, it is shown that significant improvements can be achieved by adopting “exact” Cm values. Moreover, some attention is paid to issues related to the proper choice or determination of these “exact” Cm values.

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.

T-Bracing for Stability of Compression Webs in Wood Trusses

Compression-web members in metal-plate-connected wood trusses may buckle elastically as long slender columns. One common construction technique is to nail a side member on the compression-web member (main member) so that the cross section is “T” shaped, thus increasing the cross-sectional dimensions and the critical buckling load. The bracing efficiency of the T-brace is not known. The objective of this study was to assess the influence of the main-member length, side-member length, and connection method on T-brace efficiency by measuring the first buckling-mode critical load. Finite-element models were developed with simply supported ends and side members that varied from full length to one-quarter length. The models had side members attached with either 16d common nails or phenol-resorcinol adhesive. The results of a finite-element investigation were used to determine which T-brace geometries to test. The testing program included full-length, half-length, and quarter-length side members nailed or glued to 1,830 and 3,050-mm main members. The testing results paralleled the finite-element results. The T-brace assemblies exhibited a linear minor-axis bending deflection from the onset of loading until an instability load was reached; the deflection then increased without bound. Some slipping between the main and side members was observed in the nailed assemblies, and twisting was evident in the deformed shape of the adhesive-attached assemblies. T-bracing was most efficient for very long members; its effectiveness increased with increasing side-member length. Adhesive-attached side members were more effective than nail attached. The effects of end geometry and out-of-plane metal plate bending were not investigated but could have a substantial effect.

Design of steel I-beams curved in plan to BS 5950

This paper is concerned with the strength design of class 1 and class 2 steel I-beams that are curved in plan under vertical (or transverse) loading. It proposes and exemplifies straightforward lower bound interaction equations for designing steel I-beams curved in plan against combined bending and torsion actions, which are both easy to use and consistent with the corresponding methods of BS 5950 for designing steel members subjected to bending. The interaction equations incorporate the significant effects of the included angle and the secondary minor axis bending and torsion actions, and allow for different configurations of lateral bracing. The use of the proposed interaction equations will facilitate the strength design of steel I-beams curved in plan, and lead to safe and more economical structures.

Design of steel I-beams curved in plan to BS 5950

This paper is concerned with the strength design of class 1 and class 2 steel I-beams that are curved in plan under vertical (or transverse) loading. It proposes and exemplifies straightforward lower bound interaction equations for designing steel I-beams curved in plan against combined bending and torsion actions, which are both easy to use and consistent with the corresponding methods of BS 5950 for designing steel members subjected to bending. The interaction equations incorporate the significant effects of the included angle and the secondary minor axis bending and torsion actions, and allow for different configurations of lateral bracing. The use of the proposed interaction equations will facilitate the strength design of steel I-beams curved in plan, and lead to safe and more economical structures.

Strength Design of Steel I-Section Beams Curved in Plan

When a steel I-section beam that is curved in plan is subjected to vertical (or transverse) loading, its primary actions are bending about its major axis and torsion. Secondary torsion and minor axis bending actions are usually significant in steel I-section beams curved in plan under combined bending and torsion. However, the common circular interaction equation used in practice for rectangular solid or hollow section curved beams does not account for these significant effects and so may overestimate the interaction strength of steel I-section beams curved in plan under bending and torsion. This paper investigates the nonlinear elastic-plastic behavior of steel I-section beams curved in plan under vertical loading and proposes straightforward formulas for their design against combined bending and torsion actions. Comparisons with rational finite-element results and existing experimental results show that the proposed interaction equations provide a good lower bound for the design of steel I-section beams curved in plan against combined bending and torsion. The proposed interaction equations are simple and consistent with existing codified design rules.

Thin-Walled Beam-Columns. II: Proportional Loading Tests

This paper presents tests on thin-walled I-sections in combined compression and minor axis bending. The tests were conducted in a rig consisting of two interlinked actuators applying axial force and end moment. The tests involved pure compression, combined compression and bending, and pure bending. The test results are compared with the design strengths of the Australian Standard for Steel Structures AS4100, the AISC-LRFD Specification, the Eurocode3, Part 1.1. It is shown that the experimental interaction curve is close to being linear in the intermediate and long slenderness ranges. The design interaction curves of AS4100, the AISC-LRFD Specification, and Eurocode3 are shown to be conservative in all cases, particularly in the high moment regions. The conservatism is attributed to the definition of the design bending section capacity in the three specifications. Proposals that increase the bending section capacity are presented based on the behavior observed during the tests. The resulting moment–axial force interaction curves are conservative and in reasonable agreement with the tests.

Thin-Walled Beam-Columns. I: Sequential Loading and Moment Gradient Tests

This paper presents tests on thin-walled I-sections in combined compression and minor-axis bending. The tests were conducted in a rig consisting of two interlinked actuators applying axial force and end moment. Tests were performed using sequential application of axial force and moment, as well as proportional loading with a moment gradient. The results are compared with the results of proportionally loaded members described in a companion paper. It is shown that sequential loading produces beam-column strengths that are greater than or equal to those obtained using proportional loading. It is also shown that the beam-column strength of members loaded with a moment gradient exceeds that of beam-columns in uniform bending. The latter result is attributed to the effect of greater amplitude modulation of the local buckling deformations in beam-columns with moment gradient.

Lateral buckling strengths of cold-formed Z-section beams

This paper is concerned with the inelastic lateral buckling strengths of cold-formed Z-section (CFZ) beams. The point symmetry of the cross-section of a CFZ beam introduces characteristics that are not encountered in a doubly symmetric I-beam. Firstly, the effective section rotates after yielding, so that a CFZ beam under in-plane bending about the geometrical major principal axis is subjected to bending moments about the effective minor axis and bimoments. Secondly, the minor axis bending and warping strain distributions and therefore the lateral inelastic buckling behaviour and strengths of CFZ beams are related to the twist rotation and minor axis displacement directions. The stress–strain curves, residual stresses, initial imperfections, and lipped flanges of CFZ beams are all different to those of hot-rolled I-beams. This paper develops a realistic finite element model for the analysis of CFZ beams and uses it to investigate the elastic lateral-distortional buckling, inelastic behaviour, and strengths of CFZ beams with residual stresses and initial imperfections. The results of the study are used to develop improved design rules which are suitable for CFZ beams. The effects of moment distribution and load height on the lateral buckling strengths are also studied.

Lateral Buckling Strengths of Cold-Formed Channel Section Beams

This paper is concerned with the inelastic lateral buckling strengths of cold-formed channel section (CFC) beams. Code rules for designing steel beams against lateral buckling that are based on data for hot-rolled I-section beams may be unsuitable for CFC beams, because they have different stress-strain curves, residual stresses, and initial imperfections. Distortion may reduce the resistance of a CFC beam to lateral buckling. In addition, because of the monosymmetry of the cross section, the effective shear center moves toward the web and the effective section becomes asymmetrical after yielding, so that a CFC beam under major axis bending is subjected to minor axis bending and twisting. Also, the minor axis bending and warping strain distributions, and therefore the strengths of CFC beams, are related to the twist rotation and minor axis deflection directions. This paper deals with these problems by developing an advanced finite element model to investigate the elastic lateral-distortional buckling, inelastic behavior, and strengths of CFC beams with residual stresses and initial imperfections. The results are used to develop improved design rules for CFC beams. The effects of moment distribution and load height on the strengths are also studied.

Inelastic Analysis and Design: Frames with Members in Minor-Axis Bending

To extend previous research on obtaining designs that rely on a second-order inelastic hinge analysis to demonstrate that a structure is adequate in resisting the effects of factored loads, this paper presents a study of a series of two-story planar steel frames where all columns are subjected to combinations of axial force and minor-axis bending. Inelastic designs, which are prepared to satisfy both strength and serviceability limit states requirements, are compared to designs obtained using a conventional second-order elastic analysis in conjunction with the AISC LRFD Specification. The degree of accuracy and reliability of using a concentrated plasticity analysis to model system behavior is also established. The study indicates that for a majority of the frames investigated, a second-order inelastic hinge analysis can be used effectively to design steel structures. It is also shown that in some cases a distributed plasticity analysis may be required.

DESIGN CONSIDERATIONS OF PARTIALLY ENCASED COMPOSITE COLUMNS.

The load-carrying capacity of partially encased composite columns subjected to eccentric load has been studied. Nineteen full-scale columns of partially encased sections were tested in two parallel investigations at the University of Jordan. IPE 200 x 100 x 22 steel sections (German standards) with a buckling length of 2.4 m were encased partially in concrete and tested to failure under eccentric axial load. For comparison, a twentieth column was tested as a bare steel column. All specimens were tested under minor axis bending. The variables studied in this research program were: the eccentricity of the applied load; the eccentricity ratio at the column ends; and the effect of concrete strength. The maximum strength of the tested columns was obtained by deflection methods, using Newmark's technique of numerical integration. Design values of the load-carrying capacity of the tested specimens were calculated by the Bridge Code: BS 5400. This paper presents theoretical and experimental results of the tested columns. Comparisons with design calculations are also presented. Experimental values of failure loads were in good agreement with the theoretical predictions, while the design calculations showed better agreement in the case of equal end eccentricities and for columns of higher strength concrete. Expressions, based on full plastic stress distribution, for calculating the ultimate moment of resistance of the column section are outlined. Moreover, simplified formulation of the moment capacity is suggested. The analytical results verify these simplifications.

Experimental evaluation of the effect of bond on the maximum capacity of composite columns

This paper presents results from an experimental study of the effect of bond between structural steel and concrete on the maximum carrying capacity of composite columns. Fourteen slender battened composite columns were tested under concentric and eccentric loads in major and minor axis bending. The inside face of the steel channels of seven of these columns has been greased to weaken the natural bond between the concrete core and the steel channels. Test results showed that the bond between steel and concrete has no effect on the load carrying capacity of the tested columns. Furthermore, data collected during the tests indicate a stable behaviour of the columns in the absence of bond.

The experimental assessment of the force components within structural steel ‘h’ sections

A technique is described for manipulating strain gauge data obtained from tests on structural steel ‘H’ sections such that individual force components may be identified. The technique recognises the different structural actions present when open sections deform in a full three-dimensional fashion (compression, major and minor axis bending, and warping) covering both elastic and elastic–plastic behaviour. The results obtained from experimental tests on beam columns exhibited excellent agreement with the corresponding predictions of a sophisticated non-linear finite element analysis.