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Buckling resistance predictions resulting from flexural and flexural–torsional buckling of double tee section members subjected to compression and bending are considered. A novel analytical model is developed for establishing design criteria based on decomposition of the member buckling behaviour into in-plane and out-of-plane resistance. The former is based on second-order bending relationships of load effects of structural members with in-plane equivalent imperfections, while the latter is based on the stability theory of thin-walled open sections. First part of this study presents an analytical formulation of the in-plane buckling resistance of beam–columns. In this regard, further decomposition is postulated for the in-plane first-order bending moment diagram that results in the loading state to be the superposition of two components. The first component is related to symmetrical loading and the second to antisymmetric loading. To consider the second-order effects, prebuckling displacements generated by the abovementioned loading components are amplified with regard to the inversion of the residuum of the buckling force utilisation ratio in order to obtain approximate values of second-order displacements and internal moments. As a result, the in-plane interaction curve, expressed in dimensionless coordinates, that describes the beam–column in-plane flexural buckling resistance without considering lateral–torsional buckling effects, is obtained. The results of nonlinear finite element simulations are used for the verification of the developed analytical formulation. It is concluded that this proposal yields less conservative predictions than those based on the interaction relationships of clause 6.3.3 of Eurocode 3, Part 1–1.

The accuracy of the interaction methods for combined flexural and lateral torsional buckling are investigated using statistical data, which has become available after the introduction of Eurocode EN1993-1-1:2005. The freely available statistical data for geometric and material parameters for standard profiles are quite limited and as background documents are based on IPE 160 profile, our investigations are also based on this profile. A semi-probabilistic first order reliability approach is used and the resistance of the member is treated as a stochastic variable. Latin Hypercube Sampling is used for population sampling. This simulation approach for determination of the buckling interaction surface has not previously been reported. The Eurocode handles buckling interaction through two interaction equations. These equations include a number of interaction factors. The calculation of the interaction factors may be performed by one of two methods, referred to as Method 1 and Method 2. Both interaction methods make use of the buckling curves for determination of reduction factors for both flexural and lateral torsional buckling. The flexural buckling curve is well calibrated; however, this is not the case for lateral torsional buckling. It turns out that the methods may lead to unsafe designs when a lateral torsionally slender column is loaded predominantly in bending. The present paper investigates how the Eurocode emulate the complex behavior also for very slender beam-columns. The 0.1% quantile interaction curves are compared to those resulting from the use of Method 1 and Method 2.

A comprehensive experimental and numerical study on flexural buckling behaviour of welded stainless steel box-section beam-columns is presented in this paper. A total of ten test specimens were prepared and tested subjected to eccentric compression. The cross-section and length of the test specimens were carefully designed by referring to existing design specifications to be insusceptible to local buckling. The full experimental load versus deformation and load versus strain curves were obtained. Prior to member testing, both material tests and geometric imperfection measurements were conducted and reported in detail. Finite element (FE) models were developed and validated against the obtained beam-column test results afterwards, which were utilised to perform parametric studies on the flexural buckling behaviour, generating sufficient numerical data points. Based on the obtained test and FE results, the design methods provided in EN 1993-1-4+A1 and AISC DG 27 were assessed. It was shown that both EN 1993-1-4+A1 and AISC DG 27 provided generally conservative and scattered predictions for the buckling resistance of welded stainless steel box-section beam-columns. Moreover, a newly revised proposed interaction factor was proposed in the framework of EN 1993-1-4+A1, and strength predictions with reduced scatter were therefore achieved.

Researchers have not yet reached a consensus on which buckling curve to use for the calculation of the lateral torsional buckling (LTB) resistance of cellular steel beams in the current version of EC3. In addition, there is a new proposal for the future update of EC3, in which the imperfection factor is a function of the elastic section modulus, as well as material imperfections. The aim of this study is to investigate the LTB resistance of cellular steel beams by modifying the distribution of residual stresses after the manufacturing process. Geometrical and material nonlinear analysis are performed using the software ABAQUS 6.12. The cellular steel beams are simply supported, with fork-supports at the end, and subjected to uniform bending, mid-span concentrated load and uniformly distributed loads. In the latter two cases, neutral and destabilizing effects of loading are considered. The results are compared to analytical procedures, international standards, and also, with the possible EC3 updating. It was concluded that the calculation prescription of the new proposal, considering the imperfection factor that meets the magnitude of the residual stresses after the manufacturing process of cellular steel beams, is effective, accurate and conservative for LTB resistance in inelastic and elastic behavior. However, for the case of stocky cellular steel beams, whose shear stress is preponderant, ultimate limit states, such as Vierendeel mechanism and web post buckling, must be verified first.

The steel-concrete (SC) or steel-concrete-steel (SCS) composite structures are widely used in large-scale structures like immersed tunnels, protection structures, etc., due to their superior performances in capacity, ductility, waterproofness, construction efficiency, etc. The steel plates in these structures are usually stiffened with ribs to increase the out-of-plane rigidity and to improve the local buckling performance. While there have been several applications, the buckling design of the composite stiffened plates, based on the methods for steel structures, is still immature and conservative. To deal with this problem, a selected series method (SSM) based the energy theory is proposed, which could both be utilized in composite stiffened plates and steel stiffened plates. The special series shape functions, which could efficiently and accurately represent the 3-dimentional spatial buckling deformation of the stiffened plates with multiple stiffeners, are selected. The explicit high-order solutions for the buckling stress of stiffened plates are derived. Furthermore, a finite element model (FEM) is established. Based on the FEM, a numerical database including most common design parameters is derived, and extensive numerical analyses are conducted to modify and verify the proposed SSM. It is found that, for steel structures, the SSM has a 10%–20% improvement in accuracy compared with the existing methods, and for composite structures, the SSM fills the blank in this area and has acceptable accuracy for design. Finally, from the above studies, the design suggestions are given and discussed.

A novel analytical model is proposed for establishing design criteria based on the decomposition of the in-plane deformation and out-of-plane stability states. First part of this study considered the in-plane buckling resistance of beam–columns. This study uses the results from Gizejowski et al. [42] to develop an analytical model for the inelastic out-of-plane buckling resistance of beam–columns subjected to a moment gradient. The elastic flexural–torsional buckling solution is combined with the in-plane solution via the generalised Ayrton–Perry model. In order to unify the recommendations for the resistance evaluation of beams, columns, and beam–columns, the model is customised to conform to the standard Eurocode technique of modelling buckling resistance of steel elements in compression or bending about the cross-sectional axis with the greater moment of inertia. As a result, the out-of-plane resistance interaction curves, expressed in dimensionless coordinates, which describe the beam–column flexural–torsional buckling resistance and consider lateral–torsional buckling effects, are obtained. The results of finite element simulations are used for the verification of the developed analytical formulation. Two numerical techniques of imperfection modelling are used: an equivalent geometric imperfection approach with the Maquoi–Rondal generalised initial imperfection and an approach that individually considers geometric and material imperfections. The results obtained from the proposed analytical model and those obtained using the Eurocode design criteria and other recent analytical proposals are compared. Finally, concluding remarks and directions of future studies are also presented.

Buckling-restrained braces (BRBs) are widely used in engineering especially for braced frames as load-bearing as well as excellent energy dissipating members. However, the conventional BRBs are designed in elastic state under frequent earthquakes for avoiding low-cycle fatigue, and cannot contribute to seismic energy dissipation. In order to overcome this defect, conventional BRBs are recommended to combine with metal tube dampers to form a novel type of two-level yielding buckling-restrained braces (TYBRBs). The tube damper in TYBRBs yields under frequent earthquakes to consume seismic energy. The core plate of TYBRBs yields under rare earthquakes to further dissipate energy and reduce seismic effect. In this paper, the configuration and mechanical models of TYBRBs are first presented. Two TYBRB specimens are designed and cyclic loading tests are carried out. It is found that: (1) the low-cycle fatigue resistance of tube dampers can meet the requirements of TYBRBs for frequent earthquake resistance; (2) The functions of TYBRBs can be realized with rational design; (3) The performance of BRBs in TYBRBs does not degrade with combination of tube dampers.

This paper presents an investigation of the optimum locations of multiple longitudinal web stiffeners (up to six) for steel plates under pure bending by using the gradient-based interior point optimization algorithm. Performing eigenvalue buckling analyses, buckling coefficients are calculated accounting for the lowest eigenvalues. The optimization algorithm is then employed to maximize the buckling coefficients until the optimum stiffener locations of the plates are attained. Based on this research, the optimum locations of multiple longitudinal stiffeners are suggested for plates under pure bending. Moreover, equations to determine minimum flexural rigidities of multiple longitudinal stiffeners for steel plates are also proposed. The results obtained from this study are compared with the AASHTO requirement and previous works. It is proved that the minimum flexural rigidities of stiffeners recommended by AASHTO and the previous work are conservative.

This paper presents a new buckling-restrained steel plate shear wall system with assembled multi reinforced concrete (Multi-RC) panels (MBRSPSW). The finite element model of MBRSPSW system with hinged frame, validated by existing experimental results, is developed to investigate the general mechanical behaviors, such as hysteresis property, horizontal shear force, internal force demands of boundary columns and damage of RC panels. Three key parameters, including steel plate slenderness ratio, gap width between RC panels and number of RC panels, are discussed. To investigate the general behavior of MBRSPSW system and conventional buckling-restrained steel plate shear wall (BRSPSW) system, two test specimens with rigid frame corresponding to actual moment-resisting frame are further conducted. The results show that MBRSPSW system provide stable bearing capacity and good energy dissipation capacity by dividing single RC panel into multi-RC panels. The internal force demands of the boundary column of MBRSPSW system, especially for the bending moment, decrease significantly compared to that of BRSPSW system. Damage of RC panels of MBRSPSW system is greatly decreased compared with that of BRSPSW system. Number of RC panels and the gap width of MBRSPSW system have significant effect on the tension damage of RC panels, especially for the number of RC panels. BRSPSW system is found to fail prior to MBRSPSW system with a plastic hinge at the bottom of boundary column in test investigation and damage of RC panels of the latter is much lower than that of the former. The experimental observations agree well with the numerical results.

When it is subjected to a fire, creep strain developed in a steel column can significantly influence the behaviour of the column. As the result, the column deformation will be increased and the strength will be decreased. However, the effect of creep in steel structures has not been fully considered in current fire resistance design codes. Presented in this paper is a study on creep buckling of Q460 steel columns subjected to elevated temperatures via experimental and numerical investigations. Six high-strength Q460 steel columns with welded H shape were tested for the determination of creep buckling strength at elevated temperatures. The effects of the slenderness ratio, load ratio, and elevated temperature on failure time corresponding to creep buckling (also called as creep buckling time) were taken into consideration in the investigations. The evolutions of furnace and column temperatures, column lateral and axial displacements associated with fire duration were measured. The creep buckling time of the columns were obtained based on the magnitudes of axial and lateral displacement of the columns subjected to constant elevated temperature and compressive load. The test results showed that the creep deformation of high strength Q460 steel columns became significant at temperature 800 °C even the applied load level was as low as 10%. The creep buckling time of Q460 steel columns are sensitive to load ratio, slenderness and temperature. Finite element analysis program ANSYS was employed to establish a structural model of Q460 steel column subjected to the constant elevated temperature and compressive load, with mechanical properties at elevated temperature, creep strains and residual stresses induced by welding being taken into account. The finite element model was validated by the test results. Parametric studies were carried out to quantitatively determine the effects of the slenderness ratio, load ratio, temperature and geometric imperfections on the creep buckling time of Q460 steel columns.

The fire-resistant steel as a type of high performance steel has significant application potential in building structures. The overall buckling behavior of fire-resistant steel columns at elevated temperatures is remarkably different from those of mild steel and high strength steel. However, structural behavior at ambient temperature had to be understood first before the investigation at elevated temperatures. This paper presents compressive tests on WGJ high performance fire-resistant steel welded I-section columns. The experimental results are compared with suggested values of Chinese, European and American specifications. Finite element models are established and the parameter analysis are carried out, showing current Chinese specification, Eurocode and American specification cannot be directly applied to determine the buckling resistance of high performance fire-resistant steel welded I-section columns. In addition, the modification on the design methods specified in current Chinese specification is also proposed.

The steel structures of conveyance guiding systems, particularly steel guides used in mine shafts, are commonly made of box-section beams welded from hot-rolled channel sections. Hot-rolled profiles are generally resistant to local buckling; however, long-term exposure to the aggressive environment of a mine shaft leads to high corrosive loss of the material and reduction of the wall thickness. Some industrial regulations on shaft steelwork design and maintenance allow more than 50% corrosion loss for guides; however, they do not require local buckling calculations. This study investigated the buckling resistance of box-section beams welded from hot-rolled channel sections under bending, in terms of uniform corrosion loss. Laboratory bending tests were conducted on five beams for three levels of web thickness reduction achieved by etching and the results were compared with those obtained from finite element (FE) simulations. Linear modal analyses and nonlinear Riks simulations with imperfections were performed. The hot-rolled box-sections, which are generally resistant to local buckling, were found to be susceptible to local buckling at 46 and 60% corrosion losses in the plastic state and at 71% corrosion loss in the elastic state. This study highlights the need to consider local buckling when using hot-rolled closed profile beams at high corrosion loss.

A buckling-restrained steel plate shear wall with inclined slots (simply referred as “slotted SPSW”) consisting of an inner steel plate sandwiched between two precast concrete panels, has been developed as an energy dissipation device of structures. The steel strips between inclined slots behave like a series of buckling-restrained braces to dissipate the energy through inelastic axial deformation when subjected to cyclic loading. This paper experimentally investigated the seismic behavior of slotted SPSWs. Three half-scale specimens were fabricated and tested under quasi-static cyclic loading. Details for proper fabrication of slotted SPSWs were firstly configured. The results illustrated that the slotted SPSW could sustain the target lateral drift ratio (2%) without a reduction of shear force and energy dissipation capacity. Moreover, the tested specimens exhibited stable fatigue hysteresis loops when the cyclic loadings were repeated 30 times at 1.5% peak lateral drift ratio. The maximum drift ratio of tested specimens reached 3.5%, and the cumulative plastic ductility factor was larger than 200. The experimental results indicate that the slotted SPSW is capable of providing high lateral force-resisting and energy dissipation, which can be widely used in engineering structures.

This paper provides an insight into the buckling behavior of high strength concrete encased steel (CES) columns through a comprehensive investigation including experimental, numerical and analytical analyses. Three long CES column specimens made of high strength concrete C100 and S355 H-section were tested under axial compression. The maximum test loads obtained from these tests were compared with the buckling resistance predicted by EN 1994-1-1, AISC 360-10 and ACI 318-08. Nonlinear finite element analyses were performed to predict the buckling resistance and trace the load displacement behaviour of these columns. In the finite element model, the column initial imperfections were carefully chosen to predict the maximum resistance and the load-displacement response, and compared with the equivalent imperfection values stipulated in the modern design codes. In order to examine the validity and limitation of the current design approaches in predicting the buckling resistance of CES columns made of high strength concrete, a statistical study was undertaken based on the established database covering a wider range of material strength and geometric configurations. In the statistical study, the buckling resistance and effective flexural stiffness obtained from the tests were compared with the codes' predictions from EN 1994-1-1, AISC 360-10 and ACI 318-08. Finally, the reliability of the current design methods is assessed by correlating their accuracy with respect to the variation of material strength, column slenderness ratio, load eccentricity and steel contribution ratio.

High-strength structural steel plates are being increasingly used as composite columns in tall buildings, bridges and large infrastructure. The presence of concrete infill in these composite sections enhances their local buckling strength, and thus very slender steel plates can be used in their fabrication. This paper presents the results of an experimental study and numerical investigation of the local buckling slenderness limits for high-strength steel plates. A set of sixteen tests were conducted on both hollow steel and steel-concrete composite sections to explore their local and post-local buckling behaviour under axial compression. A numerical model which accounts for the effects of residual stresses and initial geometric imperfections was developed to predict the local buckling and post-local buckling response of box and I-section columns. This model was verified against the test results. Yield slenderness limits obtained from numerical results were compared with existing codes of practice for both hollow steel and composite sections incorporating high-strength steel plates.

The present paper shows an experimental and numerical modelling investigation of the flexural-torsional buckling behaviour and load-carrying capacities of fixed-ended hot-rolled austenitic stainless steel equal-leg angle section columns. The testing programme was conducted on four austenitic stainless steel angle sections and involved material testing, initial global and torsional imperfection measurements and sixteen fixed-ended column tests. This was followed by a numerical simulation study, where the numerical models were firstly developed to validate against the experimental data and then utilised to perform numerical parametric studies to generate additional results over a broader range of member lengths and cross-section geometric dimensions. The derived test and finite element results were carefully analysed and then utilised to evaluate the accuracy of the established design rules in Europe, America and Australia/New Zealand. Comparisons of the failure loads derived from the experimental and numerical studies with the corresponding codified flexural-torsional buckling strength predictions revealed an unduly high level of conservatism and scatter. Extension of the direct strength method to carbon steel equal-leg angle section columns has been recently made, and the applicability of the approach to fixed-ended hot-rolled austenitic stainless equal-leg steel angle section columns was also assessed. The new approach was generally shown to yield more precise flexural-torsional buckling resistance predictions than the existing design standards, but a large portion of the predictions were unsafe, indicating that further improvements are required.

The local buckling behaviors of welded π-shaped stub columns were investigated experimentally and numerically in this study. Nine welded π-shaped columns made from Q345 steel with different width-to-thickness ratios were tested under axial compression. The material properties, initial geometric imperfections and welding residual stress distribution of specimens were measured. Results from the tests, including the strengths, the load-deformation responses, and the failure modes were presented. The strengths determined by tests were compared with the design strengths predicted by the current steel design specifications. The experimental studies were complemented by the numerical investigations, in which the finite element models were initially verified against the test results and subsequently used for parametric studies covering a wide range of cross-sections. The modifications on the Bleich formulas were derived to calculate the buckling coefficient of π-shaped section. Based on the experimental and parametric analyses, formula for direct strength method (DSM) was proposed to predict the compressive strengths of welded π-shaped stub columns, which offered reasonable predictions.

This paper describes a restraint tube local bulging study of newly developed small-type buckling-restrained braces (SBRBs). An analytical calculation model for the restraint tube stress distribution is established by the shell theory and the Fourier series. It is demonstrated that the stress concentration lies mainly in a region with a length of 2R0, which is approximately equal to the tube middle surface diameter. A simplified design method including a safety factor and recommended thickness-radius ratio is then proposed based on the analytical study. To verify the accuracy of this calculation method, four SBRB specimens with different tube thicknesses were tested under quasi-static cyclic loading. The calculation results are in reasonable agreement with the test results and can predict the local bulging occurring instant with acceptable accuracy.

It is commonly known that compression resistance of longitudinally stiffened orthotropic plates depends on both the column and plate-like behavior. Large number of previous investigations studied the structural behavior of stiffened plates subjected to compression. However, a comprehensive investigation and a parametric study is not available in the international literature about applicable interpolation function for plates having closed section stiffeners. It is known from previous research results that the interpolation function can be different depending on the calculation method of the critical stresses (analytical or numerical). It is also known, that the application of the numerically computed critical stresses in the Eurocode based design method can lead to overestimation of the compression resistance through the increased critical stress resulting in smaller slenderness ratio. However, there is no recommendation for applicable interpolation formula when computed critical stresses are used by the designers. Therefore, the current research program focuses on the investigation of longitudinally stiffened orthotropic plates, studying the interpolation curve between plate and column-like buckling behavior. The applicability of the EN 1993-1-5 based interpolation curve is discussed and evaluated. The reason of the differences by using analytical and numerical critical stresses are explained and applicable interpolation function is proposed for both cases.

Buckling-Restrained Braced steel frames (BRBFs) exhibit well-controlled seismic behavior due to their excellent lateral load resisting capacity and energy dissipation capacity. However, they always suffer from concentration of inter-storey drifts and excessive residual deformations in weak/soft stories under strong earthquakes. A novel system of Energy-Dissipative Rocking Column (EDRC) is proposed in this paper to improve the seismic performance of multi-storey BRBFs. It consists of dual hinge supported steel column branches and replaceable steel strip dampers between them. Nonlinear time historey analyses are conducted on a six-storey prototype BRBFs with EDRCs to investigate the effectiveness of EDRCs on improving the seismic performance of BRBFs under different levels of earthquakes. It is found that the EDRC can not only mitigate the maximum and residual inter-storey drifts, but facilitate more uniform distribution of inter-storey drifts. A rational range of EDRC stiffness ratio less than 0.2 is recommended for the seismic design of BRBFs with EDRCs.

This paper presents a numerical investigation on elastic buckling and hysteretic behavior of a steel angles assembled buckling-restrained brace (SAA-BRB). The SAA-BRB is composed of a cruciform-sectional steel core encased by an external restraining system, which consists of four steel angles connected together by high-strength bolts and spacers. It is found that the SAA-BRB may fail in a global buckling mode or in a local buckling mode. In order to predict the global and local buckling behavior of SAA-BRBs, two design parameters: the restraining ratio and segment restraining ratio are proposed. To determine these two design parameters, a simplified theoretical model of SAA-BRBs and an accurate finite element (FE) model are used to investigate the global and local elastic buckling behaviors of SAA-BRBs. Based on more than 500 FE results, explicit expressions of the restraining ratio and segment restraining ratio are obtained and validated to be sufficiently accurate for practical design applications. In addition, 20 refined FE models considering material and geometrical nonlinearities are used to carry out parametric studies for comprehensively investigating the hysteretic behavior of SAA-BRBs. Finally, the following design recommendations for fix-ended SAA-BRBs are proposed: (1) the restraining ratio should be >2.2 to prevent a SAA-BRB from the global buckling failure, and (2) the segment restraining ratio should be >6.0 to prevent a SAA-BRB from the local buckling failure.

An inner-stiffened double-tube buckling-restrained brace (BRB) is proposed for boiler steel plants in the low-to-medium seismicity regions. The test specimen is of 15.0 m length, consisting of a 15.0 m inner tube with inner stiffeners at both ends and a 12.2 m outer tube for lateral deformation restraint. The full-scale test was conducted to reproduce the cyclic behaviour of the proposed double-tube steel BRB in the plant structures at a practical manufacturing level. The structural performance of the proposed BRB under extremely low cycle fatigue (ELCF), including the strength, deformation capacity, hysteretic response and the failure mode, was investigated. The finite element (FE) model incorporating a material fracture model was established and verified, which could predict the fracture initiation and propagation accurately. The parametric study was carried out to further assess the influence of the stiffener arrangement, material property, gap size and controlling metric of the imperfection on the cyclic behaviour of the proposed BRB. The experimental and numerical investigations demonstrated that the proposed braces could achieve a stable and saturated hysteretic curve. The failure mode of the test specimen was the stress concentration induced fracture which initiated at the stiffener tips of the inner tube. On the basis of the parametric study, the optimized BRB could achieve a high equivalent ratio of critical viscous damping at the loading amplitude of 1/75 L.

Cable-stiffened steel columns can significantly enhance the stability behaviour of ordinary compression columns because of the additional stiffness offered by the pre-tensioned cables and crossarms. Most of the previous studies aimed to investigate the behaviour of stiffened steel columns with four branch crossarm systems; however, the current work focuses on investigating the interactive buckling of cable-stiffened steel columns with pin-connected three branch crossarm system via buckling analysis. The analysis shows that the crossarms remain straight in the antisymmetric buckling mode, which distinguishes them from stiffened columns with rigidly connected crossarms. In addition, the interactive buckling must be considered in nonlinear buckling analyses in some cases to obtain the actual capacities of the stiffened columns. The method to consider the interactive buckling is to introduce asymmetric initial geometric imperfections during buckling analysis; the principle to form the asymmetric initial imperfections is suggested.

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

The accurate and economical consideration of the flexural buckling resistance of high strength steel (HSS) structures is highly important in the design. According to the previous research results the flexural buckling behaviour of HSS and NSS columns can be significantly different, which differences are not considered in the current design rules. The aim of the current study is to obtain a reliable design method for the flexural buckling resistance of HSS welded box section columns. The buckling resistance is previously determined by the authors based on an experimental research program and based on stochastic numerical simulations . The current paper focuses on the investigation of the theoretical background of the flexural buckling phenomena and on the implementation of the specialties regarding HSS materials into the buckling curve formulation. The analytically derived Ayrton-Perry type formulation is studied and modified based on the theoretical and numerical investigations implementing the effect of residual stresses and geometrical imperfections into the design method. Using the revised buckling curve, the flexural buckling resistance of welded box columns can be predicted with larger accuracy for steel grades between S420 – S960.

Design of cold-formed stainless steel tubular columns undergoing local-overall flexural interactive buckling is investigated in this paper. A numerical study has been carried out to study the interactive buckling behavior of square hollow section (SHS) and rectangular hollow section (RHS) columns of various stainless steel grades that are employed in structural applications. A finite element model was developed and verified against available experimental cold-formed stainless steel SHS/RHS column data. Extensive parametric studies, with cross-section aspect ratio, plate slenderness, overall slenderness and stainless steel grade selected as key variable parameters, were carried out thereafter to generate further numerical data featuring the tubular columns undergoing local-flexural interactive buckling. The obtained interactive buckling strengths were used to assess the design provisions in current European, American, Australian/New Zealand and Chinese design standards for stainless steel structures as well as available Direct Strength Method (DSM) design rules in the literature. It is shown that better predictions can be achieved when material nonlinearities of different material grades are accounted for. A modified DSM, which relies on simple hand calculations, is developed in this paper. It is demonstrated that the modified DSM is a suitable and efficient design alternative for cold-formed stainless steel SHS/RHS columns subjected to local-flexural interactive buckling.

This research study presents an investigation of the structural behaviour and local web buckling resistance of single-coped beam connections with slender web. Single-coped beams with slender web (SBSWs) can be used in long-span structures where the top flange of the beam is removed to provide clearance to connect the beam to the main girder. Eleven (11) full-scale tests of single-coped beam connections with slender web were carried out. In general, all the test specimens failed by local web buckling. It was observed that the ultimate connection resistance was decreased with increasing web slenderness ratio, cope length and cope depth. However, the slender web with significant cope length and cope depth could promote evident post-buckling strength for the connection. Finite element (FE) analyses were subsequently conducted to provide further understandings of the test results. The FE analysis results matched the test data reasonably, and the influences of the test parameters on the performance and ultimate resistance of the test connections were carefully studied. The FE results indicated that the ultimate resistances of the models with high web slenderness were insensitive to initial imperfection amplitude, and such trend was more pronounced for cases with large cope length and depth. Based on the test results, the current design methods for predicting the local web buckling resistance of single-coped beam connections were revisited. In general, overly conservative estimates were obtained using the available design guidelines for quantifying the ultimate connection resistance of the test connections with slender web.

This research presents a study about the structural behavior of steel cellular beams, focused on the web-post buckling. The main objective of this study is to propose a new formulation to calculate the shear resistance of cellular beams for this phenomenon, based on laboratory tests and numerical analysis. Series of tests were performed in this work, with full-scale steel cellular beams. In these experiments, vertical and lateral displacements were measured, as well as web-post deformations. The steel mechanical properties of these beams were determined by tensile testing. A numerical model was proposed, developed in ABAQUS software, to perform parametric analysis. From these numerical models, processing 597 cases, a new formulation to determine the shear resistance in cellular beams for the web-post buckling was proposed, based on resistance curves. The proposed formulation was verified in several situations of geometry and material properties, presenting compatible results with those obtained numerically, and showing better accuracy than those available in the literature.

This paper presents the behavior and design concept of an efficient structural steel system based on the applications of buckling-restrained knee braces (BRKBs). The advantages of a buckling-restrained knee-braced frame (BRKBF) include relatively simple connections, reparability after an earthquake, and fewer obstructions than conventional bracing systems. Various BRKBF configurations can be designed and detailed for different levels of strength, stiffness, and ductility. BRKBFs are designed so that all inelastic activities are confined to the BRKB. The key design concepts for ensuring the ductile behaviors of BRKBFs are first summarized. Cyclic tests of large-scale BRKBF sub-assemblage specimens are then carried out. The results from both experimental and analytical studies of the behavior of BRKBFs show that they can be a viable alternative to conventional structural systems.

This paper investigates the global stability of welded singly symmetric I-shaped beams fabricated from Q460GJ steel, a typical high-performance structural steel. Experimental tests and numerical simulations were carried out on a total of eight laterally unrestrained steel beams under a concentrated point load. A special loading system was designed through which the beam could develop lateral deflections at the mid-span. Load-displacement curves and strains of beams were measured during testing. In the finite element model, initial geometric imperfections and residual stresses were considered. The model was verified through comparisons with experimental results, and parametric studies were conducted to investigate the effects of non-dimensional slenderness and height-to-width ratio. Meanwhile, design curves for lateral-torsional buckling of welded singly symmetric beams were calculated from GB50017-2003, GB50017-201X, Eurocode 3 and ANSI/AISC360-10. Comparisons between numerical results and design curves suggested that the design methods in GB50017-2003 and ANSI/AISC360-10 were unsafe for Q460GJ steel beams while GB50017-201X and Eurocode 3 could provide reasonably conservative results for global buckling design of singly symmetric Q460GJ steel beams.

We proposed a novel buckling-restrained multi-stiffened low-yield-point steel plate shear wall (BR-LYP-SPSW). To study the seismic behavior of BR-LYP-SPSW, a 1:3 scale model shaking-table test was conducted. The increase in the peak ground acceleration decreased the natural frequency and acceleration dynamic magnification factor and increased the damping ratio. The advantage of the sandwiched LYP infill plate is the higher order buckling formed at the LYP infill plate, which played an important role in “the first line of defense.” The maximum inter-storey drifts during frequent and rare earthquakes were 1/694 and 1/88, respectively, which met the demands of the seismic design code of buildings.

This paper reports on experiments addressing the buckling and collapse behavior of back-to-back lipped channel built-up cold-formed steel (CFS) columns assembled using 16 different CFS lipped channel sizes. The lipped channel sections are connected at the web using a pair of self-drilling screw fasteners at a specified spacing along the column length of 1.83 m (6 ft). These experiments aim to quantify the effect of two web fastener layouts on composite action for each section size, study member end fixity, observe buckling and collapse behavior, and provide benchmarks for design that includes specific considerations for thin-walled member buckling. A total of 32 monotonic, displacement-controlled, concentric compression tests are completed with up to 17 position transducers monitoring displacements at key locations. All tests are conducted with the built-up member seated in CFS tracks, as would be found in practice. Local–global interaction is shown to be a prevalent failure mode, and the stud-to-track end condition is determined to be semi-rigid, but generally closer to a fixed condition. End rigidities are estimated using a Southwell approach. Rational design approaches extending the application of the Direct Strength Method (DSM) and employing current state-of-the-art numerical modeling techniques are proposed and validated with test data. In addition, the development of definitive design recommendations that help reduce the complexity of fastener designs and incorporates the DSM framework when predicting built-up member strength is underway.

Cold-formed steel hollow flange beams have been commonly used as flexural members in buildings due to their structural efficiency. The rivet fastened Rectangular Hollow Flange Channel Beam (RHFCB) is a new type of cold-formed steel section, characterized by two torsionally rigid hollow flanges connected to a web element. The RHFCB takes advantage of an intermittent fastening method using self-pierce riveting. Buckling analyses have shown that the moment capacity of short-span RHFCB is governed by local buckling while intermediate span RHFCB is governed by the unique lateral distortional buckling. An experimental investigation of RHFCB flexural members was carried out based on a quarter point loading method to investigate their lateral distortional buckling behaviour and moment capacities. It considered the effects of intermittent rivet spacing on the member moment capacity of RHFCBs. The results from the experiments were compared with the member moment capacities predicted using the Australian cold-formed steel structures standard. Following the experimental study, finite element models of tested RHFCBs were developed to simulate the experiments. The developed finite element models were validated by comparison of ultimate moment capacities, applied moment versus deflection curves and failure modes. This paper presents the details of the experimental investigation, the development of finite element models of RHFCBs subject to lateral distortional buckling effects, and the results.

An innovative frame system called rocking buckling-restrained brace frame (RBRBF), which is characterized by allowing the column base uplifting during strong earthquakes, has been proposed to decrease seismic damage and reduce repair cost of building structures after the disaster. According to the working mechanism and functional objectives of this new type of structure system, four buckling-restrained brace steel frames including two rocking ones were designed as half-scale specimens considering various parameters. Pseudo-static tests were conducted on the model specimens to investigate their seismic performance, the yielding mechanism and the damage mode. The experimental results illustrated that the lateral resisting capacity and initial stiffness between the rocking structural models and non-rocking ones were nearly same at the elastic stage due to the reasonable design of rocking column base. The reduced beam sections (RBS) used at the beam-column joints could effectively control the damage location, and the liftable column bases with friction dampers could improve the energy absorption capacity of the rocking structure model. Comparing with the non-rocking models, the plastic damage on rocking specimens was greatly reduced under large lateral deformation. Furthermore, there was no local buckling on the structural elements of the rocking steel frame even under 2.5% drift ratio, which showed that the rocking structures with rational design would perform well under server seismic.

Two specimens were tested to investigate the effects of constructional details, including panel buckling-restrained braces (panel BRBs), connections, strengthening measures, etc., on the hysteretic behavior of two-story chevron-shaped panel buckling-restrained braced frame (PBRBF). Each panel BRB is an unbonded steel plate brace encased in either an assembled steel panel or a steel-concrete composite panel. Tests revealed that obvious yielding of panel BRBs and framing members occurred after an average inter-story drift of approximately 1/375 and 1/75 respectively. Generally, both specimens from the capacity design exhibited stable hysteretic behavior within story drifts of 1/50, while large local bucking, low cyclic fatigue cracks and tension fracture of steel components, which result in eventual failure of the PBRBFs, occurred at the drifts far larger than 1/50. The PBRBF1 with moment-resisting beam-to-column connections has trilinear skeleton curves within story drifts of 1/30. The PBRBF2 with non-moment-resisting beam-to-column connections, along with the panel BRBs, has bilinear skeleton curves prior to failure of panel. Although local failure of a composite panel occurred at story drifts of 1/68 in the first test, the PBRBF2 still exhibited stable hysteretic behavior in the second test and finally failed at story drifts of 1/33. The constructional details are generally acceptable to ensure good performance of the PBRBFs. Furthermore, if 1/200 is considered as a permissible residual drift level, the residual drifts would challenge function of use when the PBRBFs underwent peak inter-story drifts larger than approximately 1/100.

Practical use of high-strength (HS) steel in contemporary construction has become one of the most important design solutions. Loading capacities of columns may benefit enormously from the HS steel, whilst their overall buckling behaves differently compared with conventional mild (CM) steel columns due to varying effects of initial imperfections and inelastic properties of the HS steel materials. Despite a number of investigations regarding the HS steel columns being undertaken, there is lack of research focused on variation of HS steel grades and their effects. To deepen understanding of overall buckling behaviour of the HS steel columns, a comprehensive review of an extensive body of column test data available in the literature is carried out in the present paper, based on which a three-dimensional finite element (FE) model developed herein is validated. Parametric analyses are subsequently undertaken with various HS steel grades, welded cross-sectional geometric parameters, slenderness values and initial imperfections being involved. The FE analysis results are also compared with calculation values in accordance with national standards. It has been demonstrated that with an increase of the grade of HS steel, effects of imperfections decrease whilst that of Y/T ratios are rather limited; reduction effects on the overall buckling strength become less severe, and therefore higher column curves available in current national standards may be selected and imperfection factors in the alternative column curve equations proposed herein descend accordingly. In addition, new theoretical column curves based on Perry-Robertson formula are developed by introducing imperfection parameters independent on the steel strength.

This paper describes an experimental and numerical study on the global buckling behaviour of welded Q460GJ steel box columns. In the experimental programme, seven steel columns with different cross sections and wall thicknesses were tested under axial compression. The load capacity of steel columns was quantified. Comparisons were made between experimental results and design values calculated in accordance with national standards. Furthermore, numerical models were established in which initial geometric imperfections and residual stress distributions were considered. The model was validated against test data with reasonably good accuracy. A parametric study was conducted on the effects of initial geometric imperfections and normalised slenderness on the load capacity of box columns. Experimental and numerical results indicated that Q460GJ steel box columns could develop higher global buckling resistances than the values calculated from GB50017-2003 and Eurocode 3, but ANSI/AISC360-10 might not be safe for welded box columns with small width-thickness ratios. Therefore, the design approaches for conventional steel columns were modified so that the buckling behaviour of box columns fabricated of Q460GJ steel could be accurately evaluated.

The accurate consideration of the flexural buckling resistance of high strength steel (HSS) structures is highly important in the design. Higher yield strength indicates the applicability of smaller cross-sections, which might be more sensitive for stability problems. The purpose of the current study is (1) to investigate the flexural buckling behaviour of HSS welded box section columns and (2) to determine a reliable column buckling curve. The characteristic and design values of the buckling resistances for HSS welded box section column are determined by using Monte Carlo simulation technique for a wide range of relative slenderness and steel grades. Based on the simulation results buckling curves are proposed for all the analysed steel grades. Required value for the partial safety factor is also determined considering the design resistance level of the Eurocode. The proposed buckling curves are applicable for HSS welded box section columns made from steel grades between S420–S960.

This paper presents stability analysis of steel frames composed of linearly tapered columns and partially tapered restraining beams. In the first part of the stability analysis, closed-form equations for the bending stiffness of the symmetrical and unsymmetrical partially tapered restraining beam are derived and presented in form of design charts. Subsequently, slope deflection equations of linearly tapered column subjected to axial compressive load are then assembled along with the bending stiffness equations of partially tapered beams to determine the elastic critical load and the effective buckling length factor for non-sway and sway steel frames. Numerical examples and efficient design charts are presented for several braced and unbraced steel frames. Moreover, the effect of gravity loads acting on leaning columns connected to sway frames is investigated. Based on this investigation it is concluded that correction of effective buckling length factor based on story buckling concept can be extended to cover steel frames with tapered column as well.

This paper investigates the in-plane inelastic buckling of high strength steel (HSS) circular arches with pinned or fixed end supports. A finite element (FE) model is developed to analyse in plane behaviour and buckling loads of HSS circular arches under general loading conditions which considers geometric non-linearity, material inelasticity, initial imperfection and residual stresses of welded HSS I-sections. The FE model is validated by comparison to analytical elastic buckling loads for arches in uniform compression, and experimental results reported in the literature on the out-of-plane elastic-plastic buckling of steel arches. It is found that current design codes for normal strength steel (NSS) structures require modification before application to HSS arches. Based on the results of the FE analysis, design equations are proposed which are in-line with existing provisions for NSS structures for both fixed and pin-ended arches under uniform compression and combined compressive and bending actions. The presented design rules give good lower-bound predictions for in-plane buckling strength of fixed and pinned HSS arches. Additionally, the influence of residual stresses and size of initial geometric imperfections on in-plane buckling strength are explored.

A new partially buckling-restrained brace with dual-plate cores (DPBRB) was proposed in this paper to improve the flexural stiffness of an all-steel BRB with a plate core. The core member was composed of two identical and separated plates in parallel. The restraining member consisted of steel channels and steel plates that were assembled with high-strength bolts. The restraining member was located at the periphery of the core member to restrain the edge part of the outside and the inside plates in the core member. The restraining member can be reused and easily disassembled by removing the bolts, which effectively improves the utilization. The test results and numerical simulation indicated that all specimens presented stable energy dissipation capacities and low-cycle fatigue properties. The yield segments exhibited inwards buckling deformation, and the deformations at the ends were more evident than those in the middle. Insufficient stiffness of the end of the restraining member caused large deformation and fatigue fracture of the end of the yield segment. A new all-steel assembled restraining plate was provided. The calculation formula for the design length of the restraining member was given based on the results of finite element analysis.

This paper focuses on the hysteretic behavior of a proposed all-steel assembled double-cores buckling-restrained braces (DCBRB). The DCBRB employs an I-shaped steel and two channels to resist the global instability of the brace and hence to force the double-cores into high-order buckling mode for the purpose of energy dissipation. Six DCBRB specimens were fabricated, where both Q195 low-yield core and Q235B core were concerned for comparison. The quasi-static cyclic loading was conducted on the DCBRB specimens to investigate the failure modes, as well as the hysteretic behavior from different perspectives, such as hysteretic curves, skeleton curves, compressive strength adjustment factor, energy dissipation capacity and plastic deformation performance. The experimental results demonstrate that the insufficient constraining of the external restraining member could cause the global instability of the buckling-restrained braces. The specimens exhibited great and stable hysteretic responses until the axial strains of cores researched 1.5%. Finally, the cumulative plastic deformation capacities were greater than 300 for all specimens. The finite element models were established to further investigate the influence of constraint ratio and core's width-to-thickness ratio on the hysteretic behavior of the DCBRBs. The numerical results reveal that better global stability is expected to be obtained with a larger constraint ratio ζ, and the limit value of ζ avoiding the global buckling almost increases linearly with the increase of core's width-to-thickness ratio.

Thin-wall structural sections and members are susceptible to three fundamental instabilities: local, distortional and lateral-torsional buckling. For cold-formed purlins or girts commonly used in roof and wall systems, distortional buckling may govern the predominant failure mode when the compression flange is not attached to sheathing or panels. The paper presents an experimental and numerical investigation on the behaviour and strength of cold-rolled aluminium beams subjected to distortional buckling. A total of eighteen specimens were tested in a four-point bending configuration with six various types of cross-sections and two different thicknesses. In the pure bending region between the two loading points, the upper compression flanges were kept free to allow distortional buckling to form. Subsequently, detailed finite element (FE) models of cold-rolled aluminium beams using the software ABAQUS were developed with the incorporation of measured material inelasticity and initial geometric imperfections. Good agreement between the experimental results and the FE models was achieved. The reliable and verified FE models along with the test data in this paper will be used in a separate publication to undertake a detailed parametric study and calibration of new proposed design rules for cold-rolled aluminium beams subjected to distortional buckling.

This paper studied buckling of parallel Z- and C-purlin systems inter-braced by sag-rods at mid-span under pure hogging moment. The purlins are connected to metal sheeting by self-drilled screws and the lateral movement of the top flange is prevented. The paper studied first in detail the buckling of a single purlin, and then buckling of parallel purlin systems. The alternate offset of the sag-rods in adjacent bays and the local deformation of the purlin web have been taken into consideration in parallel purlin systems. For one-purlin system, the buckling moments are given, valid for different combinations of sag-rod and distributed rotational stiffness. The threshold stiffness of the sag-rod, at which the buckling mode changes from symmetrical buckling to anti-symmetrical buckling, is also given. For parallel purlin systems, the buckling moments are calculated by the formulas of one-purlin system, but with a reduced equivalent sag-rod stiffness used in the calculation. Numerical examples for the reduced sag-rod stiffness reveal that an astonishingly large reduction of the sag-rod effective stiffness occurs when the purlin number and the sag-rod offset increase, and the sag-rod stiffness is maximum if the sag-rod is set as close to the bottom flange as possible. And finally some strengthening practices and details are recommended.

An all-steel buckling-controlled brace (BCB) with two different configurations is studied and its behavior is compared with the conventional braces in terms of energy dissipation and ductility capacities. A parametric study was first conducted on an ensemble of all-steel BCBs in a general purpose finite-element (FE) software in order to study the influential parameters of these braces. The select types of BCBs were then experimentally investigated. Finally, seismic performance of buckling-controlled braced frames (BCBFs) was compared with that of special concentrically braced frames (SCBFs) as well as that of buckling-restrained braced frames (BRBFs). The study concludes that (1) the BCB with round-in-square tube section has stable hysteretic behavior either when thickness ratio of the outer tube to inner tube is greater than one or when an enhanced gusset plate is employed. Furthermore, due to much increased compressive strength in square in round BCBs, it is necessary to utilize an enhanced gusset plate in order to achieve ductile behavior; (2) BCBs have a stable and symmetrical hysteretic behavior in tension and compression with little post-yielding strength decrease or increase, avoiding the significant unbalanced force on the brace-intersected beams in SCBFs and BRBFs; (3) BCBFs are capable of sustaining larger story drift ratio response without considerable strength loss in comparison with SCBFs; (4) Inelastic deformation demand distributes throughout the height of BCBF floors, preventing the occurrence of weak story often observed in SCBFs.

Introducing the concept of Tube-in-Tube buckling restrained brace (TiTBRB) members, a comprehensive parametric investigation has been carried out involving the influential parameters affecting the behaviour and modes of failure of the TiTBRBs during cyclic loading through detailed finite element analysis procedures accounting for material and geometric nonlinearities as well as the effects of gaps and contacts. Several BRB related parameters have been considered here for parametric analysis that had not been investigated previously: the strength and stiffness of external restraining tube, the core strength, the core diameter to thickness ratio, the friction coefficient between the core and the restraining tube, the gap size between the core and the external tube, the gap size between rings and the external tube, the magnitude of initial imperfection, the number of intermediate rings along the length of the member and the particulars of the end collars. On the basis of the finite element analyses results, it has been demonstrated that the proposed TiTBRB - if well designed - would be quite competent in accomplishing the intended tasks as a buckling restrained bracing member. Properly designed TiTBRBs can exhibit stable cyclic behaviour and satisfactory cumulative plastic ductility capacity, so that they can serve as effective hysteretic dampers. At the same time, in such all-steel TiTBRBs concreting has been eliminated and hence much lighter members are obtained. This is also associated with ease and speed of fabrication, erection, inspection, replacement and hence a more economical and environmentally friendly design.

Design formulae of buckling-restrained braces, considering the stiffening part of the core and the effect of friction, are proposed, based on theoretical analyses of the buckling mechanism of the steel core in the presence of a gap between the core and the restraining member. The theoretical analyses are validated by finite element analyses. Compared to existing methods, the results of the proposed analyses are generally better and show the closest correlation with the results of the finite element analysis.

Eurocode allows for finite element modelling of plated steel structures, however the information in the code on how to perform the analysis or what assumptions to make is quite sparse. The present paper investigates the deterministic modelling of flexural column buckling using plane shell elements in advanced non-linear finite element analysis (GMNIA) with the goal of being able to reestablish the European buckling curves. A short comprehensive historical review is given on the development of the European buckling curves and the related assumptions made with respect to deterministic modelling of column buckling. The European buckling curves allowing deterministic analytical engineering analysis of members are based on large experimental and parametric measurement programs as well as analytical, numerical and probabilistic investigations. It is of enormous practical value that modern numerical deterministic analysis can be performed based on given magnitudes of characteristic yield stress, material stress–strain relationship, and given characteristic values for imperfections and residual stresses. The magnitude of imperfections and residual stresses are discussed as well as how the use of equivalent imperfections may be very conservative if considered by finite element analysis as described in the current Eurocode code. A suggestion is given for a slightly modified imperfection formula within the Ayrton-Perry formulation leading to adequate inclusion of modern high grade steels within the original four bucking curves. It is also suggested that finite element or frame analysis may be performed with equivalent column bow imperfections extracted directly from the Ayrton-Perry formulation.

A novel assembled steel double-stage yield buckling restrained brace (DYB) was experimentally studied. The DYB core plate comprises one small core plate and one large core plate connected in series. The deformation capacity of the small core plate is restricted by a stopper mechanism. The deformation of the DYB is first concentrated at the small core plate and then shifted to the large core plate once the stopper mechanism is triggered, resulting in double-stage yield behavior of the DYB. Three specimens were fabricated with core plates of different size to study the main influencing factors in the design of DYBs. Quasi-static tests were conducted. Results from a comprehensive experimental study demonstrate that the DYB has a ductile, stable, and repeatable hysteretic behavior. It is a reliable and practical alternative to conventional framing systems in that it can enhance the earthquake resistance of existing and new structures and provide the rigidity needed to satisfy structural drift limits while delivering a stable and substantial energy absorption capacity. The proposed hysteretic model of the DYB was calibrated with the test results, showing good agreement and the capacity to capture characteristic points of hysteresis curves.

Built-up tapered I-beams, which are mostly loaded so as to be under bending about their strong axis for the economical use of the structural material, are commonly used in wide span steel structures. Lateral-torsional buckling, excessive deflection and partial or complete yielding of the material are the important failure modes, which should be considered in design of these members. However, today's “modern design” concept not only requires the satisfaction of certain constraints but also a lightweight structure. This study presents the shape optimization of flange and/or web tapered doubly-symmetric I-beams and discusses the contribution of flange and/or web tapering and inflection point location to the economical design. The optimization procedure is constructed by slightly modifying the Big Bang - Big Crunch algorithm, which is a simple and effective population-based metaheuristic, and installing the Deb's constraint handling method. Four beam configurations with various loading and restraint conditions are optimized and final designs are validated by finite element analysis software. It is shown that tapering, and therefore the inflection point location may not have a significant effect on the material economy in certain conditions.