Direct Strength Method Research Articles

Design Methods of Aluminium Pin-Ended Columns with Topology-Optimised Cross-Sections

This paper presents a numerical study of topology-optimised pin-end aluminium alloy columns using finite element analysis (FEA). The FEA models integrate geometric imperfections and material nonlinearity, and are validated against experimental findings from the existing literature. ABAQUS v.6.15 (release 2020) is used in preparing the FEA models and obtaining the analysis results. Furthermore, modern design methodologies including Eurocode 9, the direct strength method (DSM), and the continuous strength method (CSM) are employed to assess the maximum load capacity of such columns. Parametric investigations encompass diverse parameters such as varied cross-sections, column lengths, and global and local imperfections. By analysing a total of 288 FE models, incorporating 16 column cross-sections across two lengths with nine distinct imperfections, this study compares results with those derived from modern design methodologies. Thus, this research elucidates the behaviour of novel cross-sections and the application of contemporary design techniques in their analysis.

Plastic Mechanism Models for use in DSM Localised Loading Design of Hat Sections

The Direct Strength Method (DSM) of design has been recently developed for the design of cold-formed steel members under localised loading. The method requires a yield load (Py) and an elastic buckling load (Pcr) as input variables to the DSM design equations. This paper summarises test results used to develop plastic mechanism models for calculating Py for hat sections subject to practical loading cases. All four loading cases, including Interior Two Flange (ITF), End Two Flange (ETF), Interior One Flange (IOF) and End One Flange (EOF) loading cases were investigated. The yield/plastic mechanism behaviour of multiple web sections is not yet fully understood, while several publications in the literature have discussed the mechanical behaviour of hat sections under localised loading. Hence, this paper explains and proposes a plastic mechanism model based on the experimental data for calculating the yield load (Py) to be used in the DSM for localised loading design of hat sections in conjunction with the elastic buckling load Pcr.

Stub column behaviour of high strength steel square hollow sections with semi-circular stiffeners

This paper proposes a novel high strength steel square hollow section with semi-circular stiffeners, aiming to delay or eliminate premature local buckling failure of the steel tube and lead to improved compressive performance. A total of 9 stiffened and unstiffened stub columns were tested to investigate the failure modes and compressive responses. The test results showed that the stiffened sections achieved significantly higher compressive strength with delayed local buckling failure compared to the unstiffened counterparts. Tensile coupons were extracted from flat, corner and curve portions of the steel tubes in order to determine the corresponding material properties. Finite element (FE) models were established to numerically simulate the tested stub columns, in which both failure modes and load-end shortening curves could well replicate the test results. Parametric studies were then implemented using the validated FE model to extend the dataset. The obtained test and numerical results were used to assess the applicability of existing design standards and methods. It was found that EC3, ANSI/AISC 360-22, AS 4100 and Direct Strength Method (DSM) could be safely adopted in cross-section classifications and cross-section capacity predictions.

Post fire flexural behavior of mild steel based cold-formed built-up beams exposed to elevated temperature

The use of back-to-back built-up channel beams in cold-formed steel (CFS) structures is steadily rising. The growing demand for CFS sections as a cost-effective design solution has driven the development of these CFS built-up sections. Despite this, there has been limited research on the performance of mild steel (MS) based CFS at high temperatures, particularly regarding its flexural behavior. This study thoroughly explores the behavior of MS-based CFS beams with different spans under high temperatures, followed by cooling with air or water. It assesses the impact of thermal loading and evaluates the effectiveness of these cooling methods. Experimental findings are validated and analyzed in conjunction with Finite Element Modeling (FEM) using ABAQUS and the Direct Strength Method (DSM). The study also conducts a parametric analysis to determine how the varying span that affects flexural capacity of beam. Among beams heated to the same temperature, those cooled with water exhibit slightly lower load capacities than those cooled with air. The maximum load observed is 91.21 kN for the reference specimen, while the minimum load is 39.82 kN for the specimen heated for 90 min and cooled with water, resulting in a 78.45% difference between these values. Additionally, as heating duration increases, ductility of beam also increases. Various failure modes are observed based on different heating and cooling conditions across different beam spans. This study offers valuable insights into the performance of MS-based CFS beams under thermal stress and different cooling conditions, providing important data for structural design and safety in construction.

Axial buckling resistance of cold-formed steel built-up I-section wall panels with single- and double-layer sheathing configurations: experimental and analytical studies

ABSTRACT This paper presents experimental investigation of the axial behaviour of built-up (BU) (I-section) cold-formed steel doubly-symmetric channel with single-/double-layer sheathing on both sides configuration. Fifteen (15) monotonic concentric axial compression tests were performed on 7-BU sections with single-layer sheathing, 7-BU sections with double-layer sheathing, 1-BU section stud without sheathing. The experiments aim to enumerate ultimate axial strength, buckling interactions, failure pattern and strength enhancement due to sheathing for BU section stud members. The novelty of the study is that, for the first time an attempt is made to investigate the axial behaviour of single-/double-layer sheathed BU section panels using seven different sheathing boards. Results indicate large range of deformation behaviour, with local-global and distortional-global interaction buckling phenomena. Failure is also observed due to failing of screws in specimens with sheathing of thickness ≥ 9 mm and modulus of elasticity > 6400MPa. Also, for the first time, efficacies of four semi-analytical methods based on rational extension of direct strength method (DSM) are presented. Results suggest the insufficiency of current AISI design specifications, whereas predicted results obtained using the new modified DSM methodology are within the adequate variation range as the adopted methodology considers desired buckling interaction. The efficacy of the recommended design methodology is verified through reliability analysis.

Explainable machine learning models for predicting the ultimate bending capacity of slotted perforated cold-formed steel beams under distortional buckling

This study develops explainable machine learning (ML) models to predict the ultimate bending capacity of cold-formed steel (CFS) beams with staggered slotted perforations, focusing on distortional buckling behavior. Utilizing a dataset from 432 non-linear finite element analysis simulations of CFS Lipped channels, ten ML algorithms, including four basic and six ensemble models, were evaluated. Ensemble models, specifically CatBoost and XGBoost, demonstrated superior accuracy, with test-set performances reaching a coefficient of determination (R2) of 99.9%, outperforming traditional analytical methods such as the Direct Strength Method (DSM). SHapley Additive Explanations (SHAP) were applied to highlight how features like plate thickness and root radius critically influence predictions. The findings underscore the enhanced predictive capabilities of ML models for structural performance, suggesting a significant potential to refine traditional design methodologies and optimize CFS beam designs.

Axial capacity of cold-formed steel channel sections with slits

Cold-formed steel (CFS) studs often include slits in their webs to improve thermal performance. However, the inclusion of such slits reduces their axial capacity. The literature contains limited experimental studies on the axial capacity of these studs with slits, and no non-linear elasto-plastic finite element studies have been reported. This research aims to address this gap. Non-linear elasto-plastic finite element models were developed and validated against experimental test results. A parametric study consisting of 960 finite element analysis (FEA) models was then conducted, covering the effects of different channel dimensions, slit sizes, section thicknesses, and stud lengths. Ultimate axial capacities obtained from these analyses were compared to the current codified direct strength method (DSM) predictions as per the Australian/New Zealand Standard AS/NZS 4600. Finally, design recommendations in the form of strength reduction factors are proposed. Based on these recommendations, modified DSM design equations with a reliability index above 2.5 are presented.

New Buckling Curve for a Compressed Member with Cold-Formed Channel Cross-Section

The verification of a column made from a lipped cold-formed channel section, subjected to pure axial compression relative to the gross cross-section, often results in a combined verification of bending and compression due to the appearance of a shift of the centroid of its effective cross-section. Following Eurocode 3 rules, this requires the determination of two distinct effective cross-sections and various interaction factors. This paper, based on an analytic approach, offers a modification to the actual buckling curve, based on Ayrton–Perry formulation, to include the second-order effects raised by the eventual shift of the effective centroid due to local buckling of the compressed web plate. This eliminates the need to use an interaction formula. The modified buckling curve is verified based on a GMNIA analysis performed on a numerical parametric model, which was previously validated by laboratory tests. In addition, the results are compared with strength results provided by appropriate Eurocode 3 formulas and AISI Direct Strength Method for global-local interaction and with classic experimental results.

Bending response of cover-plated CFS built-up sections: Testing, numerical parametric analysis and design

This study investigates the bending response of cover-plated cold-formed steel (CPCFS) built-up sections comprising of plain channel members under pure bending achieved through four-point loading, both experimentally and numerically. Channels were appropriately spaced back-to-back and screw-fastened at the flanges using cover-plates on the tension and compression sides. This novel design for the CPCFS built-up section combines the benefits of box-profile (closed sections) and I-profile (open sections), including high strength-to-weight ratio, simplified fabrication, and enhanced torsional rigidity. Eight CPCFS built-up sections with different cross-sectional aspect ratios were tested for bending about the major axis under simply supported end conditions. Flexural strengths, failure modes, and moment-curvature plots were discussed. Subsequently, a finite element model was developed in ABAQUS and validated against the test response obtained. Afterwards, an extensive parametric analysis was conducted to assess the variational effect of aspect ratios, cross-sectional slenderness, and plate slenderness on the ultimate moment capacity of the CPCFS built-up beams. Finally, the adequacy of the codified Continuous Strength Method (CSM) and Direct Strength Method (DSM) for CPCFS built-up beams was evaluated. Generally, it was observed that the flexural strengths predicted by the codified CSM and DSM for CPCFS built-up beams were unconservative and conservative, respectively. Suitable modifications were proposed to both DSM and CSM to accurately predict the flexural strength of CPCFS built-up beams, which were eventually verified through a reliability analysis.

Buckling behavior of thin-walled cold-formed steel latticed column with lacings under axial compression

Extensive research has been undertaken on built-up cold-formed steel (CFS) column with I-shaped or box-shaped section. However, there has been little attention given to lattice CFS columns, particularly regarding their buckling performance and the design methods for determining buckling capacity. This paper proposes a latticed CFS column, formed by combining two G-shaped (complex edge stiffener) channels using lacings (LCCFS). Eighteen LCCFS columns were tested under axial compression, divided into two groups with web widths of 75 mm and 200 mm. The experimental results demonstrated that no distortional buckling failure occurred in the G-shaped channels of all specimens, mainly due to the constraint provided by lacings on the flanges. A general consistency in the ultimate bearing capacities and failure modes was observed among duplicate specimens. Specimens with a small slenderness ratio predominantly experienced local buckling failure, leading to a more abrupt failure. In contrast, specimens characterized by a large slenderness ratio primarily demonstrated failure due to global buckling. The specimens with larger web width-to-thickness ratios exhibited earlier and more pronounced local buckling, resulting in a reduction in axial compressive stiffness. Furthermore, FE models have been developed to analyze the effects of different parameters on the axial compressive buckling performance of LCCFS columns. The existing direct strength method exhibited notable deviations in predicting axial compressive buckling capacity of LCCFS column, which are deemed unsatisfactory. Based on direct strength method, a method was proposed for determining the axial compressive buckling capacity of LCCFS column.

Lateral-Torsional Failure and DSM Design of CFS Lipped Channel Beams at Room and Elevated Temperatures

About a decade ago, in-depth investigations on the behaviour of cold-formed steel (CFS) single-span simply supported lipped channel beams failing in lateral-torsional modes at both room and elevated temperatures were reported at QUT (Queensland Technical University). They revealed a significant underestimation of the lateral-torsional (LT) failure moments by the Direct Strength Method (DSM) global design curve at room temperature, then codified in the Australian/New Zealand and North American specifications. In order to remedy this situation, these authors proposed two novel DSM-based beam strength curve sets, which were found to substantially improve the LT failure moment prediction quality, both at room and elevated temperatures. This work aims at extending the scope of the above investigations, by analysing a visibly larger set of CFS lipped channel beams at elevated temperatures (up to 800 °C), exhibiting (i) various cross-section dimensions and yield stresses, selected to cover wider LT slenderness ranges, (ii) two end support conditions, differing only in the end cross-section wall displacement/rotation and warping restraints (either fully free or fully prevented), and (iii) temperature-dependent steel material properties according with the model prescribed in Part 1-2 of Eurocode 3. The results presented and discussed consist of beam LT post-buckling equilibrium paths and failure moments, obtained through Abaqus shell finite element GMNIA that include critical-mode (LT) initial geometrical imperfections. Lastly, the numerical failure moment data obtained in this work and reported in the literature are used to develop and propose a new unified set of DSM-based strength curves capable of adequately handling beam LT failures at room or elevated temperatures. Since it is shown that the proposed strength curve set provides a quite good LT failure moment prediction quality, it is fair to argue that it constitutes a good starting point to search for an efficient general DSM-based design approach for CFS beams failing in LT modes at room and elevated temperatures.

Moment capacity of cold-formed steel channel beams with edge-stiffened and unstiffened elongated web holes

Cold-formed steel channel beams (CFSCB) often incorporate web holes to accommodate building services, but this reduces their moment capacity due to decreased web area. Recent studies have shown that a new type of edge-stiffened web holes can enhance moment capacity, especially for circular ones, and can now be extended to elongated web holes. However, there hasn't been research on the moment capacity of such CFSCB with elongated web holes. This paper uses finite element analysis (FEA) to examine the moment capacity and flexural behaviour of such beams with both elongated edge-stiffened web holes (EEH) and elongated unstiffened web holes (EUH). After validating the FEA models against experimental results available in the literature, a comprehensive parametric study involving 2160 FEA models was conducted. Results showed that CFSCB with EEH exhibited, on average, a 9% increase in moment capacity compared to those with EUH. Additionally, the parametric results were compared with the design capacities calculated from existing standards for CFSCB with unstiffened web holes. It was found that the current design equations for CFSCB with EUH were overly conservative by 9% and 66%, on average, for distortional or local buckling failure and lateral-torsional buckling failure, respectively. Consequently, modified Direct Strength Method (DSM) design equations were proposed to calculate the moment capacity of CFSCB with both EEH and EUH. Finally, a reliability analysis was conducted to assess the accuracy of the proposed design equations.

Validation of AISI Design of Cold-Formed Steel Beams Using Non-Linear Finite Element Analysis

In the local building industry of Pakistan, pre-engineered steel building manufacturers mainly employ their own self-developed software and Excel sheets. These systems are based on empirical formulas mentioned in the AISI manual. Under this scenario, a need was found to validate AISI flow charts using commercial software like CUFSM 5.04 and ABAQUS R2019x. This study presents a validation of the CUFSM software and the American Iron and Steel Institute (AISI) Direct Strength Method (DSM) results of channel section flexural members using the non-linear finite element method employing ABAQUS. In this study, eight standard cold-formed channel-section (C-section) steel members were modeled and analyzed using ABAQUS to simulate realistic behavior under four-point loading conditions. The non-linear finite element models incorporated material and geometric non-linearities to capture the actual response of the steel elements. The results obtained from ABAQUS were compared with those predicted by the CUFSM and DSM, focusing on critical parameters such as nominal strength, buckling modes, and deformation patterns. During this study, it was observed that out of the selected sections, the AISI charts predict conservative and even unsafe flexural capacities in some of the cases concerning other methods, with a maximum difference of 14.03%. The differences obtained using DSM and ABAQUS when compared with the results of the AISI charts varies on both the plus and minus sides. This study will not only affect the industry in terms of innovative designs for efficient structures but also the community in regards to low-budget construction.

Experimental and numerical investigations on cold-formed titanium-clad bimetallic steel angle and channel section stub columns

Titanium-clad bimetallic steel (TCBS) exhibits outstanding corrosion resistance, and can be advantageously applied in corrosive environments. This study investigated an innovative application of TCBS in cold-formed stub columns. During the cold-forming process, TCBS demonstrated no discernible separation at the bonding interface of the Q235 substrate layer and TA1 cladding layer, indicating the maintenance of synergistic deformation between the cladding and substrate layers. The compressive behavior and ultimate resistance of cold-formed TCBS angle and channel section stub columns were investigated using both experimental and numerical methods. These tests included material tensile flat and corner coupon tests, initial local geometric imperfection measurements, and stub column tests. The stub column test comprised four cold-formed TCBS angle section stub columns and three cold-formed TCBS channel section stub columns covering the non-slender and slender cross-sections. The experimental results were utilized to validate the finite-element (FE) model, which was then used in a parametric study to obtain further numerical data at different cross-sections. A comparative analysis of the ultimate resistance from the tests and numerical analyses with the predicted results from design approaches in EN 1993-1-1, AISI S100, and the direct strength method (DSM) was revealed. The evaluation findings generally showed that the EN 1993-1-1 design method forecasts for cold-formed TCBS channel section stub columns were dangerous. The non-slender cross-section classification coefficient, which was inapplicable to cold-formed TCBS channel steel stub columns, was the primary cause of the above unsafe prediction. As for the design approaches in AISI S100 and DSM for cold-formed TCBS angle and channel section stub columns, as well as those in EN 1993-1-1 for cold-formed TCBS angle section stub columns, they were judged safe but conservative. The above prediction error was mainly caused by ignoring the enhancement in strength properties of the TCBS after the cold-forming process. The considerations mentioned above led to the proposal and demonstration of modifying recommendations that offered safe, accurate, and consistent design approaches to ultimate resistance for the cold-formed TCBS angle and channel section stub columns.

Flexural behaviour and design of hybrid cold-formed steel built-up I-section beams

The current study investigates the flexural performance and design of hybrid cold-formed steel built-up I-section beams, which are increasingly popular in the construction industry. These beams combine the benefits of both an "I" section and a closed-box section in their cross-section profile. They are constructed from two identical plain channels placed back-to-back with space between them and top and bottom cover plates, fastened together by screws. A nonlinear finite element (FE) model was developed to consider initial geometric imperfections, material and geometric non-linearities. The model was validated against the test results of cold-formed steel (CFS) built-up beams from the companion paper and literature. Once validation was successful, a comprehensive parametric study using finite element modelling was conducted to generate data on hybrid cold-formed steel built-up I-section beams across a broader range of cross-sectional slenderness, hybrid ratio, aspect ratio of the cross section and thickness of the cover plates. The parametric results were used to analyse the influence of key parameters on the moment carrying capacity and buckling modes. The applicability of the current Direct Strength Method in the AISI S100–16 specifications was evaluated based on these results, revealing that it is unconservative. Additionally, modified design procedures and equations were suggested for hybrid CFS built-up I-section beams vulnerable to local buckling in major axis bending, which were then evaluated through reliability analysis.

Local buckling behaviour of web perforated cold-formed steel lipped channel columns

To connect beams and bracings with storage rack uprights, closely spaced perforations are provided along the web, flanges, and rear flanges of uprights. These perforations can significantly lower the ultimate capacity of such compression members with the possible influence of its natural buckling modes. This capacity reduction can depend on various parameters such as (a) geometrical shape, proportioning of cross-section, and stiffeners; (b) perforation shape, size, spacing, and location; (b) slenderness of member, cross-section, and elements of cross-section; and (d) material properties. A thorough understanding of the influence of the above-mentioned factors is necessary for the accurate strength prediction of perforated Cold-formed steel (CFS) compression members. Even though the current design standards are updated for the accurate strength prediction of unperforated CFS compression members, they do not collectively account for the influence of all the aforementioned factors on the load-carrying capacity of the perforated CFS members, particularly for the local buckling capacity. Though the Direct Strength Method (DSM) of design is the most accepted method for accurate strength prediction of CFS members even for complex cross-sectional shapes, recent research on the strength evaluation of perforated CFS members using DSM has emphasized the need for refinement in DSM. The Modified Direct Strength Method (MDSM), which accounts for the simultaneous buckling of flanges and web, includes the cross-section aspect ratio and cross-section slenderness to predict more accurately the local buckling design strength. However, it was developed only for unperforated specimens. Hence, a systematic experimental and numerical investigation was done to understand the influence of the perforation in the local buckling behavior of the lipped channel section. In total, 14 specimens, including 2 unperforated and 12 web perforated CFS lipped channel stub columns were physically tested with fixed support conditions. The Finite Element Analysis using ABAQUS software was used to conduct an extensive parametric numerical study. The results were used to compare the strength curves of DSM and MDSM and the modification in the design curves has been proposed by considering the erosion in strength due to the presence of perforation.

Regression-classification ensemble machine learning model for loading capacity and bucking mode prediction of cold-formed steel built-up I-section columns

This paper presents a regression-classification ensemble machine learning (ML) model for loading capacity and, in particular, buckling mode prediction with respect to cold-formed steel (CFS) I-section columns. The ML model comprises two sub-models, including the regression sub-model for load capacity prediction and the classification sub-model for buckling mode prediction. A total of 541 experimental and numerical data for CFS built-up I-section columns with varies geometric dimensions, material properties and buckling modes were collected from the literature to construct a dataset. To improve the accuracy and the explainability of the ML model on relative small-scale dataset, physical information-enhanced features based on the knowledge of the direct strength method were adopted as the input features of the ML model, including the yield strength, the elastic local buckling load, the elastic distortional buckling load and the elastic global buckling load, instead of directly using the parameters of geometric dimensions and material properties as the input features. The appropriateness of seven classical machine learning algorithms was compared, namely Decision Tree (DT), Random Forest (RF), Support Vector Machine (SVM), K-Nearest Neighbour (KNN), Adaptive Boosting (ADA), Extreme Gradient Boosting (XG), and Categorical Boosting (CB). The statistical analysis showed that the XG algorithm had the highest level of accuracy for both loading capacity and buckling mode prediction. In addition, the explainability analysis indicated that the developed ML model successfully learned the mechanical characteristic of CFS built-up I-section columns. Finally, the developed ML model was further compared with the existing codified method in the literature. The comparison results indicated that the developed ML model can significantly improve the level of accuracy for loading capacity and buckling mode prediction, which provides a promising alternative choice for the design of CFS built-up I-section columns

Flexural behaviour of Q1100 ultra high strength steel welded I-sections

An experimental and numerical program investigating the in-plane flexural behaviour of Q1100 ultra high strength steel (UHSS) welded I-sections is presented in the current paper. The testing program comprised of material coupon tests, measurements of residual stresses, and in-plane 3-point bending tests. Subsequently, a numerical program was executed, involving the development of finite element (FE) models. The FE models validated against the experiments were subsequently employed for parametric studies, aiming at assessing the flexural behaviour of Q1100 welded I-sections across a broader spectrum of section geometries. The data obtained from the experiments and numerical simulations were used to assess the applicability of prevailing design provisions outlined in the European, Chinese and Australian codes for high-strength steel structures. These design provisions encompass classification limits for cross-sections and predicted methods for cross-sectional resistance in bending. Via reliability analyses, adjustments were made to the cross-section classification limits. Finally, a modified direct strength method was proposed to predict the in-plane bending resistance for Q1100 welded I-sections.

Simplified Direct Strength Method for the design of cold-formed steel channels with circular web openings under two-flange loadings

Previous studies have put forward reduction factor equations to determine the web crippling strength of cold-formed steel (CFS) channels with circular web openings. However, these equations overlook vital parameters such as material yield strength, flange width, and fillet radius of CFS channels. Consequently, this paper introduces simplified Direct Strength Method (DSM) equations, which account for plastic loads (Py) and elastic buckling loads (Pcr), integrating the aforementioned crucial parameters. Eight unique yield line models were devised, considering two-flange loadings: interior-two-flange (ITF) and end-two-flange (ETF), as well as the positioning of circular web openings, to compute the yield line length for Py. Similarly, new Pcr equations were also proposed. The reliability of the proposed equations was evaluated using a dataset of 236 test results sourced from the literature. This assessment aims to demonstrate the suitability of the equations for incorporation into future revisions of international design codes.

Numerical Investigation of the Axial Load Capacity of Cold-Formed Steel Channel Sections: Effects of Eccentricity, Section Thickness, and Column Length

Cold-formed steel channel (CFSC) sections have gained widespread adoption in building construction due to their advantageous properties, including superior energy efficiency, expedited construction timelines, environmental sustainability, material efficiency, and ease of transportation. This study presents a numerical investigation into the axial compressive behavior of CFSC section columns. A rigorously developed finite element model for CFSC sections was validated against existing experimental data from the literature. Upon validation, the model was employed for an extensive parametric analysis encompassing a dataset of 208 CFSC members. Furthermore, the efficacy of the design methodologies outlined in the AISI Specification and AS/NZS Standard were evaluated by comparing the axial load capacities obtained from the numerically generated data with the results of four previously conducted experimental tests. The findings reveal that the codified design equations, based on nominal compressive resistances determined using the current direct strength method, exhibit a conservative bias. On average, these equations underestimate the actual load capacities of CFSC section columns by approximately 11.5%. Additionally, this investigation explores the influence of eccentricity, cross-sectional dimensions, and the point-of-load application on the axial load capacity of CFSC columns. The results demonstrate that a decrease in section thickness, an increase in column length, and a higher degree of eccentricity significantly reduce the axial capacity of CFSC columns.