Plastic strength of thin-walled plated members—Alternative solutions review

The paper presents results of the comparative theoretical study into plastic strength of thin-walled plated structures (plates and columns) using two different approaches: energy method and equilibrium strip method. Two types of members, a thin plate and a channel section column subjected to compression, were under investigation. Numerical results obtained using both methods, together with FE simulation are presented in load-deformation diagrams. Experimental verification of theoretical analysis (using two methods under investigation together with standards strength predictions) focused on stub columns (lipped channel sections) is presented.     

Web crippling of thin-walled cold formed steel cassettes

The resistance of cold-formed thin-walled steel cassettes against local transverse forces, i.e. the web crippling capacity, was investigated both experimentally and numerically using finite element analysis. Both unreinforced (flat) cassette webs and webs with longitudinal stiffeners situated on only one side of the web mid-line were studied. The calculation of the web crippling capacity of this type of stiffened webs is not included in current design codes. However, if cassettes are designed as continuous over two or more spans, the resistance against local transverse forces has to be verified. The study included a total of 52 structural tests. Finite element models were developed and validated on the basis of the test results and very good agreement was achieved. This was followed by an extensive parametric study of the influence of different cross-sectional parameters on the resistance against local transverse forces of longitudinally stiffened webs. Recommendations concerning the design of the cassette web section are given based on the results.     

Distortional buckling of thin-walled inclined lipped channel beams bending about the minor axis

Based on the Generalised Beam Theory (GBT), two computing models are presented to analyse the distortional critical stress of cold-formed thin-walled inclined lipped channel beams bending about the minor axis. The computing model of rigid-body motion ignores the transverse bending deformation of the flange. However, the bending deformation of the flange is accounted for in the transverse bending computing model. Based on the transverse bending computing model, this paper puts up a simple method to take into account the in-plane bending of the flange. The results given by the rigid-body motion computing model does not correlate as well as those given by the transverse bending computing model with the results available in the literature. The accuracy of the transverse bending computing model is verified through comparison of its results with the known results. The comparisons demonstrate the importance of the bending deformation of the flange on the distortional buckling of cold-formed thin-walled channel beams bending about the minor axis.     

Distortion of thin-walled beams

First order Generalized Beam Theory (GBT) describes the behaviour of prismatic thin-walled structural members by using system of ordinary differential equations. Solution can lead to separation of the load components and then to subsequent combination of the stresses and deformations. Application of GBT to a steel, cold-formed, thin-walled cantilever beam with complex non-symmetrical open cross section is presented. Theoretical values are compared with experimental ones.     

Equivalent thickness of cold-formed thin-walled channel sections with perforated webs under compression

Cold-formed thin-walled channel sections with perforated webs (thermal studs) are widely used in external wall panels in cold regions to reduce the cold bridging effect. However, no design method appears to be available for this type of structure. A possible method is to convert the perforated web of a thermal stud into a solid one with a reduced thickness (which is referred to as the equivalent thickness) and then adopt an existing design method for solid sections (e.g. EN 1993-1-3). This paper presents the development of a method to calculate the equivalent web thickness. The equivalent thickness calculation equation is based on regression analysis of a large number of finite element simulation results of elastic local buckling strength of perforated plates under compression, considering the effects of a number of different design variables such as plate depth, thickness, perforation patterns and dimensions of the plate. The FE simulations were carried out using a general FE software. This study suggests that the equivalent thickness is mainly related to the plate width to thickness ratio, the total width of perforation at the critical section and the width of the perforation zone (total plate width between the first and last perforation). A regression equation has been proposed to relate the equivalent thickness to these parameters. To demonstrate the validity of the proposed equivalent thickness method, the compression strengths of a large range of perforated columns have been simulated by using either the original perforated sections or the equivalent solid section; and a comparison of the simulation results shows good agreement between the two sets of results.     

Web crippling behaviour of laterally restrained cold-formed steel re-entrant profiled deckings

Web crippling failure is often found to be critical in cold-formed steel profiled deckings during construction of composite slabs. Therefore, accurate prediction to the web crippling resistances of profiled deckings over internal supports under hogging moment is highly desirable. This paper presents an extensive experimental investigation into the structural behaviour of laterally restrained re-entrant profiled deckings under concentrated loads. A total of 104 web crippling tests on fully supported re-entrant profiled deckings with nominal yield strengths at 235 and 550 N/mm² are carried out to provide data for direct comparison with design resistances obtained from codified design rules. It should be noted that in the past, little attention has been paid to establish the lateral restraint condition of profiled deckings under concentrated loads in tests. Hence, local section distortion or ‘section spreading’ is often observed in tests but seldom dealt with rigorously during design development. In the present study, effective lateral restraints are provided to the test specimens in order to fully mobilize the web crippling resistances of the profiled deckings. It is found that the measured web crippling resistances are typically 20%–40% higher than those obtained from the codified design rules given in BS5950: Part 6, Eurocode 3: Part 1.3 and the North America Specification, depending on the steel grades and thicknesses, the load bearing lengths as well as the loading conditions.

In general, both BS5950 and Eurocode 3 give conservative web crippling resistances for re-entrant profiled deckings with both low and high strength steel under internal and end loading conditions. Moreover, the corresponding resistance factors determined according to a codified reliability analysis are considerably larger than the required values. Thus, the design rules are reliable and conservative, although they tend to be very conservative for profiled deckings under end loading condition. However, an examination on the design rules given in the NAS shows that only some of them are applicable to predict the web crippling resistances of low and high strength steel re-entrant profiled steel deckings. The design rule for IOF condition always gives both conservative and reliable resistances for both low and high strength steel profiled deckings, compared to the measured values. The design rule for EOF condition is also found to be both conservative and reliable for low strength steel profiled deckings only, but not for high strength steel profiled deckings. The design rules for ITF and ETF conditions are not applicable for both low and high strength steel profiled deckings according to the reliability analyses.

It is demonstrated that a set of new design rules specifically for re-entrant profiled deckings is needed for both improved efficiency and reliability. Moreover, the comprehensive set of test data is readily adopted to calibrate both finite element models and design expressions in subsequent studies.     

Distortional theory of thin-walled beams

The classic thin-walled beam theory for open and closed cross-sections is generalized to include one distortional mode of deformation. Distortional cross-section parameters are introduced and the new orthogonality conditions for uncoupling of the axial displacement modes are given. A normalization technique for the distortional modes leads to unique distortional cross-section properties. The theoretical formulations for torsion and distortion are nearly similar and result in nearly identical equilibrium equations. However, for closed single- or multi-cell cross-sections the torsional and distortional shear flows may couple. A study of the order of magnitude of the governing torsional and distortional parameters shows the difference between open and closed cross-sections and the related solution types. The difference in the order of magnitude of the governing cross-section parameters also leads to approximate solution techniques. In the examples, section three cross-sections are used to illustrate variations of the theoretical parameters.     

Application of surfaces of ultimate strength for thin-walled beams

This paper deals with the problem of ultimate load-carrying capacity of thin-walled sections subject to combined load. That has direct implementation in sizing and design of thin-walled structures. It is solved using the ultimate strength method based on the theory of plastic analysis of structures. It is assumed that the elastic strains are negligible in comparison to the plastic strains and that justifies the application of a fully plastic model. The following problems have to be analyzed before the sizing and design is completed:

• Load vectors and load components

• Locations of the plastic neutral axes

• The surface of ultimate strength

The most important achievement presented in this paper is an improvement for the location of the plastic neutral axis. Until now, the position of the plastic neutral axis has been localized by iterations, starting with the position from the elastic model. That led, in some cases, to a statically inadmissible model and lack of equilibrium in case of asymmetric sections or asymmetric loads.A successful solution to the problem consists in covering the whole section with a mesh of plastic neutral axes and a cluster of corresponding points on the surface of ultimate strength.One of the points on the surface has load components in proportion with the load vector. The corresponding location of the plastic neutral axis is precisely the one we are seeking.The load vector may be extended to pierce one of the triangles that the surface is made of. The coordinates of the point where the load vector is piercing the triangle is a weighted-average of the coordinates of three vertices of the triangle. The same weight is used to localize the plastic neutral axis corresponding to the piercing point of the surface.     

Hysteretic behaviour of tubular joints under cyclic loading

This paper examines the cyclic performance of CHS joints used in steel tubular structures. Quasi-static experimental study into the response of eight T-joint specimens is described. Four of them are subjected to cyclic axial load, and the other four are subjected to cyclic in-plane bending. The general test arrangement, specimen details, and most relevant results (failure modes and load-relative deformation hysteretical curves) are presented. Some indexes to assess the seismic performance of tubular joints, including strength, ductility and energy dissipation, are synthetically analyzed and compared. Test results show that failure modes of axially loaded joints mainly contain weld cracking in tension and chord plastification in compression. But for joints under cyclic in-plane bending, both punching shear and chord plastification become regular failure modes accompanied by ductile fracture of the welds. Hysteretic curves take on a plump form in general. Ultimate strengths of joints are also compared with equation values for monotonic loading from various design codes. Results indicate the strength at a certain deformation limit can be regarded as the ultimate strength of a T-joint under cyclic loading and existing codes can be used to check it. It is also found that there is a significant distinction in the energy dissipation mechanism for tubular joints under different loading conditions. Finite element analyses are performed by taking into account weld geometry to facilitate the interpretation of the test results. It is identified that high tensile stress triaxiality can be one primary cause of weld cracking which happened under low cyclic load level.     

Load-capacity estimation and collapse analysis of thin-walled beams and columns—recent advances

The present paper is devoted to the recent results of research in the area of load-capacity and post-failure behaviour of thin-walled beams and columns (among them thin-walled cold-formed profiles). It deals with ultimate load and collapse of box-section girders (tubes) of different cross-sections under bending, as well as of lipped and plain channel-section beam-columns. The paper contains the presentation of theoretical analysis and experimental investigation of plastic mechanisms of failure and collapse behaviour of these thin-walled sections. The short review of results obtained in recent years in general precedes those obtained by the Department of Strength of Materials and Structures, TUL. The problem of post-failure behaviour is solved using the rigid-plastic theory adopted and modified for the purposes of the solution taking into consideration strain-hardening of the member’s material. On the basis of experimental investigations theoretical models of plastic mechanisms of failure are produced for different sections. Theoretical analysis is based on the principle of virtual velocities. The problem is solved in an analytical–numerical way. The particular attention has been paid to the influence of the strain-hardening of the material after yielding upon the collapse structural behaviour and also to the influence of cross-section shape and dimensions on the character of collapse. The upper-bound estimation of the load-carrying capacity of analysed thin-walled sections by combining results of non-linear, elastic post-buckling analysis with the results of plastic mechanism analysis is carried out. Results are presented in diagrams showing post-failure curves as well as curves representing structural behaviour in the whole range of loading up to and beyond the ultimate load. Some results are compared with experimental results and those obtained from FE analysis. A comparison of lower- and upper-bound estimation of the load-carrying capacity is discussed and illustrated in diagrams. Conclusions dealing with the influence of strain-hardening phenomenon displayed by the material upon the load-carrying capacity and collapse behaviour of examined sections are derived. Also conclusions concerning different upper- and lower-bound estimations of the load-carrying capacity of analysed sections are presented.     

Web crippling of cold-formed unlipped channels with flanges restrained

The paper describes a series of web crippling tests on cold-formed unlipped channels with flanges restrained (fastened) as well as channels with flanges unrestrained (unfastened). The tests were performed under end and interior two-flange loading conditions specified in the North American Specification and Australian/New Zealand Standard for cold-formed steel structures, namely end-two-flange and interior-two-flange loading conditions. The concentrated load was applied by a bearing plate at the top flange of the channels, and the reaction force applied by an identical bearing plate at the bottom flange of the channels. The bearing plates acted across the full flange widths of the channels. The flanges of the channels were either bolted to one or two bearing plates for the specimens with flanges restrained. The web crippling test strengths are compared with the design strengths obtained using the North American Specification, Australian/New Zealand Standard and American Iron and Steel Institute Specification for cold-formed steel structures. It is shown that the design strengths predicted by the North American Specification using the unfastened design rules are generally conservative, but unconservatively predicted using the fastened design rules, even when the flanges of the specimens were restrained. The design strengths predicted by the Australian/New Zealand Standard and American Iron and Steel Institute Specification are unconservative.     

Strength of cold-formed lipped channel beams under interaction of local,distortional and lateral torsional buckling

Cold-formed thin-walled lipped channel steel beams may undergo buckling modes such as short half-wavelength local buckling, intermediate half-wavelength distortional buckling and long half-wavelength lateral-torsional buckling or a combination of these before failure. ABAQUS software based on finite element analysis is used to analyse the interaction behaviour of these buckling modes in this study. The finite element model, after calibration with experimental results available in the literature, is used to perform parametric studies, to evaluate the behaviour and strength of such beams under different types of interactions due to variation of material and member properties. The large volume of synthetic data thus generated over a range of failure modes along with the available test results are used to evaluate different equations for calculating the strength of such cold-formed lipped channel beams. Based on the comparison, a method for the design of lipped channel beams failing under the interaction of local, distortional and overall lateral torsional buckling is recommended.     

Effective shaping of cold-formed thin-walled channel beams with double-box flanges in pure bending

The paper is devoted to cold-formed thin-walled channel beams with double-box flanges. Geometric properties of the C-section are described in terms of dimensionless parameters. The warping function and the warping inertia moment are analytically determined. The optimization criterion and the dimensionless objective functions are defined as a quality measure. The space of feasible solutions is constrained by the strength, global, local buckling, and geometric conditions. Analytical solutions of the problems of global and local buckling for thin-walled beams are presented. Results of the numerical calculations of the optimal shaping problem are presented in tables and figures.     

Load-carrying capacity and energy absorption of thin-walled profiles with edge stiffeners

The paper presents results of the study into plastic mechanisms (collapse behaviour) of short stub thin-walled profiles with edge stiffeners under compression. The aim of the study was to investigate an applicability of such profiles as energy absorbers. Solutions of other researchers and the author are presented. The new plastic mechanism solution for hat-section column is derived.     

Coupled stability analysis of thin-walled composite beams with closed cross-section

For the coupled stability analysis of thin-walled composite beam with closed cross-section subjected to various forces such as eccentric constant axial force, end moments, and linearly varying axial force, the efficient numerical method to evaluate the element stiffness matrix is newly presented based on the homogeneous form of simultaneous ordinary differential equations. The general bifurcation type of buckling theory for thin-walled composite box beam is developed based on the energy functional corresponding to semitangential rotations and semitangential moments. The coupled stability equations including variable coefficients and the force–displacement relationships are derived from the energy principle and explicit expressions for displacement functions are presented based on power series expansions of displacement components. The element stiffness matrix is evaluated by applying member force–displacement relationships to these displacement functions. In addition, the finite element model based on the cubic Hermitian interpolation polynomial is presented. In order to verify the accuracy and validity of this study, numerical solutions are presented and compared with the finite element solutions using the Hermitian beam elements and the available results from other researchers. Particularly, the influence of the eccentricity and the force ratio of axial forces, the fiber orientation, and the boundary conditions on the buckling behavior of composite box beam are parametrically investigated. Also the emphasis is given in showing the phenomenon of buckling mode change.     

Plastic bending behaviour and section moment capacities of mono-symmetric LiteSteel beams with web openings

Recently developed cold-formed LiteSteel beam (LSB) sections have found increasing popularity in residential, industrial and commercial buildings due to their light weight and cost-effectiveness. Currently, there is significant interest in the use of LSB sections as flexural members in floor joist systems, although they can be used as flexural and compression members in a range of building systems. The plastic bending behaviour and section moment capacity of LSB sections with web holes can be assumed to differ from those without, but have yet to be investigated. Hence, no appropriate design rules for determining the section moment capacity of LSB sections with web holes are yet available. This paper presents the results of an investigation of the plastic bending behaviour and section moment capacity of LSB sections with circular web holes. LSB sections with varying circular hole diameters and degrees of spacing were considered. The paper also describes the simplified finite element (FE) modelling technique employed in this study, which incorporates all of the significant behavioural effects that influence the plastic bending behaviour and section moment capacity of these sections. The numerical and experimental test results and associated findings are also presented.     

Coupled instabilities with distortional buckling in cold-formed steel lipped channel columns

The paper addresses the elastic post-buckling behaviours of cold-formed steel lipped channel simply supported columns affected by mode interaction phenomena involving distortional buckling, namely local/distortional, distortional/global (flexural-torsional) and local/distortional/global mode interaction. The results presented were obtained by means of Abaqus shell finite element analyses adopting column discretisations into fine 4-node element meshes. In order to enable a thorough assessment of all possible mode interaction effects, the column lengths and cross-section dimensions were carefully selected to ensure similar local, distortional and/or global buckling loads. One analyses otherwise identical (elastic) columns having initial geometrical imperfections (i) with various configurations (combinations of the competing critical buckling mode shapes) and (ii) sharing the same overall amplitude.     

Nonlinear inelastic analysis of building frames with thin-walled cores

A second-order inelastic analysis by combining the theories of stability and plasticity is proposed for studying frames with thin-walled cores. In the proposed approach, steel frameworks surrounding the cores are modelled by using the plastic hinge beam-column approach, and core walls modelled by using the thin-walled beam-column approach. Transformation procedures are proposed to consider the kinematic relationship between beams, columns, core walls and floor diaphragm. Nonlinear solution procedures are incorporated for the incremental analysis. The proposed inelastic analysis is used to investigate the inelastic behaviour and ultimate strength of core-braced frames.     

Distortional warping functions and shear distributions in thin-walled beams

In the analysis of thin-walled beams it is often necessary to consider the effects of distortion of the cross-section. The distortion in the plane of the cross-section generates axial warping displacements. On the basis of a known in-plane distortional displacement mode it is possible to derive a unique warping function and the related shear stress distributions. Local axial equilibrium is used to derive the main differential equation for determination of the distortional warping function and shear distributions. In closed single- or multi-celled cross-sections it is necessary to introduce circulation shear force flows around the cells to achieve compatibility of the axial displacement. Methods for analysis of open and closed cross-sections are generalized to include distortional displacement modes. It is shown that axial extension, flexure and torsional warping are included as special cases of distortion. A generalization of the conventional orthogonalization procedure and a normalization technique for distortional modes are also presented. A triple cell cross-section is used to illustrate the generalized calculation procedure and computed results are presented.