Proposed Design Equations Research Articles

Notch and Fracture Mechanics‐Based Assessment of Multiaxial Fatigue Thresholds of Defects and Sharp Notches in Metallic Materials

ABSTRACTThis investigation focuses on the constant amplitude (CA) multiaxial fatigue limit of components made of metallic materials weakened by defects, cracks, and sharp U‐ and/or V‐notches. To estimate the multiaxial fatigue thresholds of plain, sharply notched, and cracked materials and defects, a novel theoretical framework based on the well‐known averaged strain energy density (SED) criterion is proposed, which extends the Atzori–Lazzarin–Meneghetti (ALM) diagram to multiaxial loading. The proposed design equations are validated against 128 experimental multiaxial fatigue limits, taken from the literature.

Local-distortional-global buckling interaction os lipped channel steel cold-formed columns

It is well-established that axially loaded thin-walled steel cold-formed members are susceptible to elastic buckling. This has been a focal point of numerous research studies, leading to insights into various buckling interaction cases, including local-global (LG), local-distortional (LD), distortional-global (DG), and local-distortional-global (LDG) interactions. This investigation specifically focuses on the triple buckling interaction LDG, building upon the authors' previously proposed design solution for LD buckling interaction. The study was conducted using experimental and finite element method (FEM) column results, creating a comprehensive database to validate the proposed solutions. The first research step involved verifying the accuracy of the shell FEM model by calibrating with experimental results of lipped channel (LC) columns developing the LDG buckling. Once the FEM model's accuracy was confirmed, a parametric study was conducted to support the development of the proposed LDG design solution for LC columns. The proposed design equations adhere to the core principle of the direct strength method, capable of incorporating all single buckling modes (L, D, and G) and their combinations (LG, LD, and LDG). Comparisons were made using LRFD-based reliability analysis, which confirmed that the proposed solution is reliable, easy to apply, and aligns with the usual design principles and parameters found in current codes and standards for steel cold-formed structures. Additionally, DG and LDG design approaches from other authors are mentioned and discussed in the context of the findings of the present investigation.

Simulation and cross-section design of steel beams under moment gradients

A systematic study into the effect of moment gradients on the cross-section resistance of hot-rolled steel square hollow section (SHS), rectangular hollow section (RHS) and I-section beams has been conducted and is presented in this paper. Finite element (FE) models were first developed and validated against test results from the literature; parametric studies covering different steel grades, cross-section geometries and moment distributions were then carried out. It was found that cross-section bending resistances increase with increasing moment gradient in the low to intermediate range, despite the necessary rise in the level of co-existent shear. Under high moment gradients, the negative impact of the high co-existent shear outweighs the positive influence of the moment gradient, and cross-section bending resistances fall. It was found that the benefits from moment gradients vary with the cross-section slenderness and the aspect ratio of the cross-section, and increase when intermediate web stiffeners are present. A new design method that captures the observed behaviour is devised, featuring a new parameter to describe the local moment gradient in beams under different loading conditions. The proposed design equations are shown to be more accurate than existing provisions in EC3 and to meet the reliability requirements set out in EN 1990. The new method is therefore deemed to be suitable for implementation within the EC3 framework.

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.

Displacement prediction equations for seismic design of single friction pendulum base‐isolated structures

AbstractWe propose a set of equations for preliminary risk‐targeted seismic design of structures base‐isolated using single friction pendulum (SFP) bearings. The equations offer statistical estimates of the maximum displacement of the superstructure relative to the base and the maximum displacement of the isolated base response quantities of interest (RQIs), given ground motion intensity measures and the essential dynamic properties of the SFP bearing base‐isolated structure. The set of proposed equations enables preliminary seismic design of SFP base‐isolated structures using the mean annual frequency of exceeding displacement‐defined limit states related to the two RQIs. To develop the design equations, we define and use a two‐degree‐of‐freedom (2DOF) surrogate model that features a few essential dynamic parameters of the SFP bearing base‐isolated structure. We first verify that the 2DOF surrogate model is accurate enough to represent the base and superstructure displacements compared to the complete multiple‐degree‐of‐freedom model of the prototype SFP bearing base‐isolated structure. Next, we perform more than 5 million non‐linear dynamical analyses of 4900 distinct 2DOF surrogates subjected to 210 different recorded ground motions scaled five times. We fit the response data of each 2DOF surrogate to a linear model and use two different modeling techniques to derive the design equations. One model is based on Polynomial Chaos Expansion (PCE), and the other model is based on Linear Regression (LR). The PCE model is more accurate than the LR model, with the trade‐off regarding its simplicity. Nevertheless, both the PCE and the LR models are accurate enough to estimate the design quantities of interest of the SFP bearing base‐isolated structure for preliminary design purposes. Lastly, we exemplify the use of the proposed design equations and we show that the estimate of base displacement is conservative in the cases when the superstructure of the SPF bearing base‐isolated structure yields.

Estimation of stress concentration factor at stop-hole repair strengthened by bonded patches

Fatigue cracks detected in existing steel structures during periodic inspections are often tentatively repaired by drilling holes, called stop-holes, at crack tips. However, in some cases, fatigue cracks may reoccur at the edge of stop-holes, depending on the crack length and applied stress range. Although numerous studies have demonstrated the efficacy of patch plates in strengthening cracked steel plates and extending fatigue life of steel structures, there is a limited literature on the effectiveness of stop-hole repair strengthened by bonded patches. Particularly, the impact of the bonded patch configuration and position on the fatigue strength improvement remains unexplored. This study investigated the influence of the bonded patch configuration and position on the stress concentration factor of a stop-hole by finite element analysis. In this study, the material of bonded patch plates is assumed to be steel. Furthermore, this study presents a fatigue design equation for stop-hole repair strengthened by bonded patches, considering bonded patch configuration and position based on the fracture mechanics. The proposed design equation enables the design of patch configuration and the evaluation of remaining fatigue life to repair fatigue cracks in existing structures.

Eccentric compression behavior of circular concrete‐filled steel tubes reinforced with internal latticed steel angles

AbstractThe eccentric compression behavior of the CFSTs reinforced with inner latticed steel angles was experimentally investigated, and test results, such as the load–displacement curves, load‐strain curves, distributions of lateral deflection, and ultimate load, were obtained and analyzed. Test results showed the influence of the latticed steel angles on the ultimate load was not obvious when the steel consumption was relatively small, and the lateral deformation was not obvious when the compression load was smaller than 80% of the ultimate load. The corresponding finite element (FE) model was established and validated by the test results, and parametric analysis was also conducted subsequent to validation. The analysis results showed the contribution of latticed steel angles to the ultimate load would be diminished as the eccentric distance increases. Moreover, the design equations for the ultimate load were proposed based on the FE results, and the calculation results indicated the proposed design equations can predict the ultimate load conservatively and safely.

Numerical Modelling and Proposed Design Rules of 7075-T6 and AA-6086 High-Strength Aluminium Alloy Channels under Concentrated Loading

This study presents a detailed numerical investigation into the web buckling behaviour exhibited by high-strength aluminium alloy channels, namely 7075-T6 and AA-6086, when subjected to concentrated loading. A nonlinear finite element (FE) model was established and verified using the experimental data reported by other researchers, and the material properties of 7075-T6 and AA-6086 high-strength aluminium alloy were obtained through the literature. A parametric study comprising 1024 models was performed using the validated FE models. Variables examined in this work included web slenderness ratio, internal corner radii, bearing lengths, and aluminium alloy grades. The numerical results generated by the parametric investigation were used to evaluate the applicability and reliability of the most recent design specifications given in the Australian and New Zealand Standards (AS/NZ S4600) (2018) and Australian Standards (AS/NZS 1664.1) (1997). The comparison indicated that the calculated design strength using AS/NZ S4600 was over-conservative by 41% and 43% for 7075-T6 and AA-6086 aluminium alloy, correspondingly, while the design strength computed using AS/NZS 1664.1 was marginally unconservative, compared to numerical results. Finally, using bivariate linear regression analysis, new design formulas with new coefficients for determining the web buckling behaviour of 7075-T6 and AA-6086 high-strength aluminium alloy channels were proposed. A reliability analysis was then undertaken, indicating that the proposed design equations possess the capability of accurately predicting the web buckling behaviour of these members.

Retrofitting of post-heated R.C. columns using steel fiber reinforced self-compacting concrete jackets

This work aims to examine the effectiveness of using steel fiber-reinforced self-compacting concrete (SFRSCC) jackets in retrofitting RC columns subjected to high temperatures. In order to do that, eleven RC columns were cast and tested under axial loading. One column served as an unheated control column, and ten columns were heated to 400 °C and 600 °C for two hours. Subsequently, eight columns were retrofitted using SFRSCC jackets. The volume fraction of steel fibers and the thickness of the SFRSCC jackets were the main parameters considered for the retrofitting technique. According to the experimental findings, SFRSCC jackets can significantly enhance the structural performance of post-heated columns as evaluated by failure load, stiffness, and toughness. The percentage increase in the failure load, stiffness, and toughness of retrofitted post-heated columns is greatly impacted by the thickness of the SFRSCC jackets. The retrofitted columns with 25 mm and 50 mm SFRSCC jackets increased their failure load by 63.9% and 211.8%, respectively, relative to that of the heated control column at 600 °C. It is obvious that the failure load of the retrofitted columns was not significantly impacted by increasing the volume fraction of steel fiber in the SFRSCC jackets. Finally, a proposed design equation was suggested to predict the failure load of retrofitted post-heated columns with SFRSCC jackets.

Experimental investigation and design consideration on pull-through capacity of a C-shaped purlin section

A small-scale experimental study was carried out to investigate the pull-through failure of the C-shaped Cold-Formed Steel (CFS) roof purlin sections subjected to wind uplift force. The parameters investigated to comprehend the pull-through behavior of roof purlin are thickness, web depth, flange width of the CFS purlin section, and size of the screw head diameter. The test specimens experienced two different modes of pull-through failure (rupture and bearing) along with the prying effect. It was observed that the pull-through failure mode transitions from rupture of the purlin section around the screw fastener in 1 mm thickness to bearing type in 2 mm and 3 mm thicknesses. The test results were compared with existing design limits and previous studies from the literature which concluded that the existing elastic design approach for the service limit state is unconservative. Hence, a suitable design limit to predict the critical pull-through capacity of C-shaped roof purlins under high wind load is proposed. Additionally, reliability studies were carried out to determine the resistance and safety factors for the proposed design equations. The suggested design specifications are applicable and conservative for the material strength in the range of 350 MPa.

Discussion: Design for Local Member Shear at Brace and Diagonal-Member Connections: Full-Height and Chevron Gusset

The paper “Design for Local Member Shear at Brace and Diagonal-Member Connections: Full-Height and Chevron Gusset” (Sabelli and Saxey, 2021) develops equations for checking local member shear demands using a Concentrated Stress Method (CSM) and presents a design example. This discussion presents results from a finite element (FE) model, based on the design example in the paper, to quantify the accuracy of the proposed design equations in predicting beam yielding. The FE model confirmed that the beam yielding state would not have occurred for the example frame under the design forces but would have occurred for forces about 4% higher. The stress and strain distributions observed in the FE model were consistent with those assumed in the CSM, although two points for potential refinement were noted. If beam yielding in a concentrically braced frame is forced, through oversized braces, the plastic mechanism would be like an eccentrically braced frame (EBF) but with higher yield force, and higher beam inelastic rotations and strains for the same inelastic story drift.

Axial-flexural strength of steel-concrete composite shear walls

A predictive equation is proposed for the axial-flexural strength of rectangular and flanged Composite Plate Shear Walls-Concrete Filled (C-PSW/CF), with and without endplates and boundary elements. The proposed equation is based on plastic stress distribution, adjusted by coefficients to account for influential design parameters. The adjustment coefficients are established by a regression analysis performed on the data developed using an extensive numerical parametric study. The continuum-based finite element model used for the simulations is first validated using the available test data. The effects of various design variables, including the aspect ratio, slenderness ratio, reinforcement ratio, axial load ratio, infill concrete compressive strength, and yielding strength of the steel faceplates on axial-moment are then numerically examined. Four cross-sections, including double skin plate, double skin plate with endplates, double skin plate with boundary elements, and double skin plate with flanges, are considered. The results are used to develop a new predictive equation to obtain the flexural strength of C-PSW/CFs in presence of axial loading. The proposed equation is finally validated using the experimental test data and its accuracy is evaluated by comparing it to available predictive equations and the method proposed by the 2016 AISC Seismic Provisions. The results show that the proposed design equation can well predict the flexural moment capacity of C-PSW/CFs with various cross-sections in presence of axial loading.

Statistical modeling and safety analysis for developing design resistance of deconstructable HSFGB shear connectors in composite beam

In recent years, the high-strength friction-grip bolt (HSFGB) shear connectors in steel-concrete composite beams as deconstructable connectors are of great research interest since they are aligned with sustainable construction. However, the design strength equation for this type of connector has not yet been provided by the design codes such as AISC and Eurocode 4. In this paper, with regard to the reliability aspect, the design strength equation of HSFGB shear connectors is investigated which is projected into a partial safety factor according to Eurocode 4. Accordingly, the three-dimension finite element (FE) push-out model of the composite connection is developed and verified via experimental results to predict the load-slip behavior and failure modes in this type of bolted shear connector. Then One-Factor-at-a-Time (OFAT) method as a sensitivity analysis and Plackett-Burman (PB) design method as a screening analysis are utilized to identify the effective factors of the ultimate strength of HSFGB connectors. Finally, based on a complete database of all existing experimental findings, the results obtained from 55 three-dimension finite element models conducted in this paper, and appropriate statistical analysis, a high-precision equation for predicting ultimate strength is provided. Then the value of the partial safety factor is determined through a reliability study in accordance with EN1990, which provides an acceptable level of safety for the proposed design equation based on Eurocodes. Reliability analysis found that a partial factor of 1.52 was justified for the design strength of HSFGB connectors, which differed from the partial factor of 1.25 stated in the EC for welded stud shear connectors.

Proposed Stiffener Spacing Requirements for the Seismic Design of Short Links in Eccentrically Braced Steel Frames

Links in eccentrically braced frames (EBFs) are designed with transverse web stiffeners to achieve the desired seismic performance targets established in current design standards. Prior studies have suggested that stiffener spacing requirements in the standards may be conservative. This paper proposes new stiffener spacing design requirements for short EBF links. The proposed limits were established on the basis of a methodology that combines a numerical solution of the classical problem of inelastic plate buckling of a thick plate under shear thrusts along with comparisons with available experimental data. The resultant design equation was verified through complementary continuum finite-element simulations of 18 wide-flange short links under symmetric cyclic loading. The finite-element simulations suggested that the proposed design equation reduce the stiffener requirements by about 35%, on average, relative to the current stiffener spacing requirements for short EBF links. Limitations and suggestions of future work are discussed.

Experimental and Numerical Investigation of Integrated Steel Shear Walls with Varying Web Width-to-Thickness Ratios

The cold-formed steel-framed shear wall sheathed with steel sheet is widely used as a lateral force-resisting system for light-framed construction. However, the boundary conditions of the screw joint are not clear when steel is screwed to the frame. To address this issue, experimental methods have been used to calculate shear strength, but these methods have many limiting factors. In this study, we investigate the structural performance of integrated steel shear walls with varying web width-to-thickness ratios as an alternative to steel-sheathed shear walls used in light-gauge steel frames. The proposed design equation for cold-formed steel integrated shear walls is validated by three specimens and 72 finite element models. This equation allows designers to determine the strength of integrated steel shear walls without conducting full-scale shear-wall tests.

Static performance and design of cold-formed high strength steel rectangular hollow section X-joints

Cold-formed high strength steel (CFHSS) X-joints made of square and rectangular hollow sections (SHS and RHS) brace and chord members were investigated in this study. The steel grades of tubular members were S900 and S960 with the nominal 0.2% proof stresses of 900 and 960 MPa, respectively. The authors carried out tests on cold-formed S900 and S960 steel grades RHS X-joints. The test results were used to develop accurate finite element (FE) models in this study. Using the validated FE models, a comprehensive FE parametric study was then performed. The validity ranges of critical geometric parameters were extended beyond the current limits mentioned in international codes and guides. The nominal resistances predicted from existing design rules given in European code and Comité International pour le Développement et ĺEtude de la Construction Tubulaire (CIDECT) were compared with a total of 726 test and FE joint resistances, including 684 numerical data generated in this study. Chord face failure, chord side wall failure and a combination of these two failure modes were reported. It is shown that design rules given in European code and CIDECT are not suitable for the range of cold-formed S900 and S960 steel grades RHS X-joints investigated in this study. Therefore, user-friendly, accurate and reliable design equations are proposed in this study. Moreover, reliability analysis was also performed for the existing and proposed design equations.

A unified design equation for square and rectangular concrete-filled steel tubular short columns

This paper develops a new unified design equation to estimate the axial compressive strength of square and rectangular concrete-filled steel tubular (CFST) short columns. The following four steps are involved: (i) compiling an experimental database consisting of 348 square and 113 rectangular CFST short columns, (ii) conducting 245 parametric analyses using benchmarked finite element (FE) models to address the gaps in the database, (iii) evaluating existing design equations and identifying their insufficiencies, and (iv) proposing and validating the new design equation. As compared with the existing design equations, the proposed design equation has the following advantages: (i) allowing a fast and convenient estimation of the axial compressive strength of square and rectangular CFST short columns, (ii) getting rid of the calculation of the section slenderness, and (iii) extending the codified material strength limits of the concrete infill and steel tube, i.e., it is suitable to square and rectangular CFST short columns with the compressive strength of the concrete infill up to 150 MPa and yield stress of steel tube up to 1000 MPa.

Three-dimensional undrained face stability of circular tunnels in non-homogeneous and anisotropic clays

The advanced 3D finite element limit analysis is employed to examine three-dimensional (3D) face stability of circular tunnels in undrained clays with the impact of strength non-homogeneity and anisotropy. The soil behaviour for all analyses is governed by the anisotropic undrained shear (AUS) strength model, which requires three anisotropic undrained shear strengths that are simply evaluated from conventional laboratory tests. The upper and lower bound solutions of 3D face stability of circular tunnels are accurately analyzed by considering four dimensionless parameters, including the cover-to-depth ratio, the overburden stress factor, the strength gradient ratio, and the anisotropic strength ratio. The significance of four dimensionless parameters on the stability load factor of the 3D face stability of circular tunnels, and the predicted failure mechanisms due to the influence of cover depth ratios, are also illustrated. A nonlinear regression analysis is employed to the computed data, which enables a new design equation of the factor of safety incorporating the conventional stability number. Thus, the new tunnel stability factors are proposed for the first time in the literature. A comprehensive application of the proposed safety factor equation is also presented and compared. The 3D effects of the tunnel face configuration between 3D geometry and 2D plane strain heading on the tunnel stability factors are also discussed. The new proposed design equation of the factor of safety provides an accurate and practical tool in assessing face stability of circular tunnels in non-homogeneous and anisotropic clays.

More accurate design equations for cold-formed steel members subjected to combined axial compressive load and bending

Cold-formed steel (CFS) load-bearing members in multi-storey frame systems are subjected to combined actions sourced from gravity and lateral loads. However, limited information is available on the complex interaction behaviour of such elements affected by different buckling modes. This study aims to provide a better understanding of the behaviour and design of CFS sections under various combinations of compression and bending about both major- and minor-axes. Experimentally validated finite element (FE) models of CFS elements were developed in ABAQUS software, accounting for material nonlinearity and geometric imperfections. The validated models were then used to conduct a parametric study to assess the structural performance and failure modes of over 500 CFS elements with various lengths, thicknesses and cross-sectional dimensions under 19 different load eccentricities. It was demonstrated that the element and web slenderness ratios and the magnitude and direction of eccentricity are the key factors affecting the behaviour of the CFS beam–column elements. The accuracy of current design specifications, including American Iron and Steel Institute (AISI-S100), Australian/New Zealand Standard (AS/NZS-4600) and European standard (Eurocode-3) was then investigated. Subsequently, the results were used to propose a new design interaction equation as a function of element and web slenderness ratios. It was shown that the proposed equation could considerably improve the accuracy of the code strength predictions, especially in the case of medium to high slenderness elements. Finally, a reliability analysis was conducted within the framework of the AISI-S100 to ensure that the proposed design equation provides the required level of safety.

End-One-Flange Web Crippling Behavior of Cold-Formed High-Strength Steel Channels with Web Holes at Elevated Temperatures

This paper investigates the web crippling strength of cold-formed high-strength steel (CHS) channels with centered web holes subjected to end-one-flange (EOF) loading at elevated temperatures, considering both flanges fastened and unfastened to load plates conditions. The stress-strain curve and material properties for CHS (S690QL steel grade) channels were adopted from the literature, where the temperatures ranged from 20 to 800 °C. The material characteristics were incorporated into finite element (FE) models using ABAQUS. The developed FE model was then validated against the published test results to evaluate the effects of various parameters including web hole diameter, bearing length, cross-section sizes, and flange fastening conditions of such channels at elevated temperatures, and a comprehensive parametric investigation including a total of 1710 validated finite element models was performed. From the parametric study results, it was found that the web crippling strength reduction factor is sensitive to the changes of the hole size and the bearing length, with the parameters of hole size having the largest effect on the web crippling reduction factor; however, the web crippling strength reduction factor remains stable when the temperature is changed from 20 to 800 °C. According to the FEA results, new reliable web crippling strength reduction factor equations for such CHS channels were proposed. In the comparison of proposed design strengths to the numerical failure load, the proposed design equations are suitable to predict the web crippling strength for CHS channels subject to EOF loading at ambient and elevated temperatures.