Reliability analysis of shear design provisions for cold formed steel sections

This study focuses on the structural reliability analysis of cold-formed steel (CFS) sections under the ultimate limit state of shear. It considers two design models: the EN 1993-1-3 standard and its recent modification proposed by a previous study. The bias and uncertainty in these models were calibrated by comparing the design models’ prediction to 67 experimental results. Reliability analyses for the CFS beams, designed according to both models, were conducted using Monte Carlo Simulation (MCS) and the First Order Reliability Method (FORM). This analysis incorporated the model uncertainty and other parameters describing these models into the stochastic description. The 10% fractile of the reliability values for beams designed using the existing EN 1993-1-3 provisions showed values that were more conservative than those of the modified EN 1993-1-3 provisions, compared to the reliability target of 3.8. A FORM sensitivity analysis identified the yield strength of steel 𝑓 𝑦𝑏 and the resistance model uncertainty 𝛤 𝑅 as the main positive drivers of uncertainty in the computed reliability indices of the design models. Additionally, a multiplicative modification factor was proposed for both the existing and modified versions of EN 1993-1-3, ensuring that these models optimally meet the specified reliability targets of 3.8 and 4.3 for Eurocode reliability classes 2 and 3, respectively. The proposed modifications maintain the partial factor 𝛾 𝑀 0 at 1.0, as stipulated by the EN 1993-1-3 provisions for the resistance of the cross-section.


Introduction
The design of cold-formed steel (CFS) sections is a crucial aspect of structural engineering, due to their extensive use as load-bearing elements in various applications.Renowned for their high strengthto-weight ratio, these sections serve as integral elements in modern modular construction [1].The complex shear behavior of these slender sections, under a variety of loading conditions, has been the subject of thorough research (e.g.[2][3][4][5][6]), informing design guidelines encompassed within various international codes such as the European EN1993-1-3 [7], the North American AISI S100 [8], and the Australasian AS/NZS 4600 [9].It is worth noting that a study by Pham and Hancock [10] laid the ground work for the development of the direct strength method (DSM) shear design rules for in both the AISI S100 and the AS/NZS 4600 design standards.Despite the comprehensive nature of these standards, the progression of theoretical knowledge and practical applications necessitates ongoing refinement of these shear provisions.
A succession of investigative work has progressively shed light on the understanding of the shear behavior of CFS section.Initial studies by researchers like Schafer and Peköz [11]  Partial factor related modification factor  Importance vector and Meimand and Schafer [14] have augmented the literature by integrating the variability of material properties into the safety assessments of cold formed elements.The pioneering work in probabilistic design by Ellingwood and Galambos [15] has also been vital in promoting a shift towards reliability-based design approaches.The concept of structural reliability provides a standardized metric for evaluating the safety of various structures by comparing their resistance capabilities.This approach establishes the most objective criterion for determining the safety of structures [16].Modern structural design guidelines commonly incorporate this methodology, calibrating partial factors to ensure they meet fundamental and optimal safety requirements [17].
This work integrates the deterministic foundations laid by prior research [12] with a probabilistic approach to the shear strength assessment of shear provisions for CFS sections.By collating and analyzing experimental results from a spectrum of scholarly sources, model uncertainty in the existing EN 1993-1-3 model and the proposed modified version is calibrated.Thereafter, the structural reliability of CFS beams designed to the two considered models is computed by incorporating model uncertainty, geometric, and material uncertainty into the stochastic description of the problem.This study seeks to reinforce the deterministic rigor of existing provisions conducted by Gatheeshgar et al. [12] with the robustness required to withstand the inherent aleatory and epistemic uncertainties present in practical applications, thus proposing modifications that uphold structural integrity within a probabilistic framework [18].

Analysis of CFS sections under shear failure
This section outlines the analysis of shear behavior in CFS sections, incorporating both deterministic and probabilistic approaches.It begins with a deterministic analysis of shear strength as per EN1993-1-3 [7], followed by an examination of the modifications proposed by Gatheeshgar et al. [12].The section concludes with an introduction to the performance function, a key component in reliability analysis, bridging deterministic findings with probabilistic assessments.

Existing eurocode shear design provisions
The design guidelines outlined in EN1993-1-3 contain a methodology for calculating the shear resistance in CFS elements.The design values of the shear resistance  , is given by the equation: In Eq. ( 1), ℎ  denotes the distance between the midpoints of the flange edges, and  denotes the web's angle in relation to the flanges (see Fig. 1).The variable  stands for the web thickness, while   signifies the shear strength, accounting for the potential of buckling effects.The term  0 is the partial factor for the ultimate limit state of resistance of the cross-section.
In order to compute the shear buckling strength   , the EN1993-1-3 recommends taking into account the web's relative slenderness   and the basic yield strength of steel   .For webs without of longitudinal stiffeners (focus of this study),   is computed via Eq.( 2): Herein,  signifies the material's modulus of elasticity, and   measures the web's height from the corner midpoints.EN1993-1-3 also notes that the shear buckling strength   is a function of the presence of web stiffening at the support points.In the experiments considered for the calibration of model uncertainty in Section 3.1, stiffening was typically achieved by affixing cleat plates or side plates to the web, which served to mitigate web distortion and effectively resist support reactions.For scenarios with web stiffening at the supports, EN1993-1-3 proposes the following formulae to approximate the shear buckling strength   :

Modified eurocode shear design provisions
The current shear design guidelines in EN1993-1-3 have limitations, particularly in accounting for the web-flange juncture restraint level and the aspect ratio [12].Gatheeshgar et al. [12] proposed an enhanced approach to integrate these factors into the Eurocode's shear design rules.This approach introduces a new formula for calculating the modified web slenderness, denoted as  ,new , as shown in Eq. ( 4).This new formula incorporates a novel shear buckling coefficient,   , allowing the incorporation of the level of restraint at web-flange juncture and aspect ratio into the formulation.
Following the approach of Lee et al. [19], Keerthan and Mahendran [4]   is computed as shown in Eq. (5), where   and   represent shear buckling coefficients under simple-simple and simplefixed conditions, respectively.The factor   quantifies the restraint level at the web-flange juncture, with values assigned as 23% for lipped channel sections (  = 0.23) and 87% for hollow flange sections (  = 0.87).Eqs. ( 6) and (7) detail the computation of   and   , which are influenced by the aspect ratio, the ratio of shear span () to clear web height ( 1 ). ) This updated methodology calculates shear buckling strength   for sections with web stiffening at the support as illustrated in Eq. (8).Notably, the analysis of experimental data reveals an inelastic reserve for small web slenderness values.Hence, for  ,new less than or equal to 0.83, a linear trend reflecting this reserve is considered.
The revised approach is noteworthy for its inclusion of aspect ratio and web-flange juncture constraints, factors critical to accurately determining shear response [12], particularly in CFS sections without longitudinal stiffeners and with web stiffening at the support.

Performance function
The demarcation of the failure region is a prerequisite for performing reliability analysis, enabling the distinction between safety and failure conditions.In line with the Eurocode 0 (EN 1990 [17]) standards, the performance function for the ultimate limit state of shear buckling of CFS channels is formulated in Eq. ( 9), inclusive of parameters representing model error (also known as model uncertainty) on both the resistance and action variables.
In this expression,   and   encapsulate the stochastic variables that characterize the model uncertainty associated with shear strength of CFS sections and applied shear actions, respectively. , is the shear buckling resistance computed at the mean values of geometric and material parameters, while   and   symbolize the permanent and imposed variable action effects of the applied shear action, taken also at their mean values.The Subsequent section will present the uncertainty quantification of the stochastic variables involved in this performance function.

Uncertainty quantification
This section delves into the uncertainty quantification of stochastic variables and models relevant to the study's reliability analysis.Given the inherent lack of complete and precise data for all input variables in a typical engineering problem, it becomes imperative to use probabilistic models to encapsulate the uncertainties in these variables.The analysis specifically considers uncertainties in sectional dimensions, material properties, and the predictive models used for calculating ultimate shear strength of CFS sections.

Model uncertainty for shear resistance
In addressing the unpredictability inherent in engineering problems, it is essential to acknowledge that complete and precise data for all input variables is often unattainable [20].Consequently, probabilistic models become indispensable tools for encapsulating uncertainties in these variables, particularly in the context of reliability analysis [20].This analysis focuses on the uncertain aspects of sectional dimensions, material properties, and predictive models used for ultimate strength estimation.
Model uncertainties arise primarily from the design equations used to forecast ultimate strength, which are based on a series of hypotheses and approximations.As a result, the estimation of potential model errors becomes a crucial aspect of ensuring the accuracy and reliability of these predictive models.In many cases of reliability analysis, one of the fundamental steps involves identifying and quantifying these modeling uncertainties [21].In this study, the model uncertainty for the resistance variable is denoted as   and is conceptualized as the discrepancy between experimental results,  expt , and model predictions,  pred , computed using Equations described in Sections 2.1 and 2.2, as provided in Eq. (10), (10) To statistically characterize   , statistical moments such as mean  and standard deviation  are computed over a sample or an experimental design of   .To gather statistical data on   , information from 67 CFS beams was compiled from multiple sources [22][23][24][25][26][27].Among these, 23 were hollow flange sections (Litesteel beam) and remaining 34 were lipped channel sections.Fig. 2 in the study illustrates the variation of  EN1993-1-3 and  Modified EN1993-1-3 as a function of the relative web slenderness   and revised relative web slenderness  ,new , respectively.Further details about this aggregated data are available in Table 1.Additionally, the sectional configuration of the experimental CFS beams beam is depicted in Fig. 3.
To ascertain the distribution functions representing model errors in the two analytical models being considered, this investigation employed the Kolmogorov-Smirnov (K-S) statistic [28].The K-S test is proficient at assessing the conformity of empirical data to a specified probability distribution, with a pronounced efficiency for normal distributions [29,30].It encompasses a non-parametric test that deliberately avoids assumptions about the underlying distribution, allowing for a versatile evaluation of sample populations against continuous probability distribution functions [31].
Typically, the K-S statistic   is determined by the equation: where sup  is the supremum of the set of distances,   () is the empirical cumulative distribution function of the sample, and  () is the theoretical cumulative distribution function (CDF) of the reference distribution.The critical values for the K-S statistic, which indicate the significance of the results, are well-established in the statistical domain [32].For this analysis, the significance level was set at 5% [33], an effective threshold for a wide array of data distributions, including those with tail-end deviations [34].
Complementary to the K-S test, the -value was computed to yield an intuitive gauge for the statistical hypotheses at play.This index, predicated on the null hypothesis, streamlines the decision-making process concerning its rejection [35].A substantial -value indicates a robust alignment between the posited and the actual distributions [36].The inquiry pinpointed that all the three distributions (normal, lognormal and gamma) with specified parameters ( Tables 2 and 3) most suitably captures the uncertainties for shear failure of CFS sections.The corresponding graphical representations for these probability distributions in terms of joint probability density function (PDF), and CDF are illustrated in Figs. 4 and 5.The findings from the K-S tests and the associated p-values, and the statistical moments of the model uncertainty for the EN 1993-1-3 and modified EN 1993-1-3 models are documented in Tables 2 and 3, respectively.The study adopts the lognormal distribution, along with its associated statistical moments, to represent the uncertainty of model uncertainty in accordance with the recommendations of the Joint Committee on Structural Safety (JCSS) probabilistic model code [37].This distribution, alongside its statistical moments is then used in the subsequent reliability analyses for incorporating model uncertainty.

Statistical characteristics of material parameters
Cold-formed steel's material properties, including yield strength   and elastic modulus , exhibit variations due to several factors.These factors include inherent material variability stemming from the use of various steel alloys in production.Manufacturing processes L. Simwanda et al.   - is spatial position factor ( = 1.05 for webs of hot rolled sections and  = 1 otherwise).
- is found to be in the range of −1.5 to −2.0.
- is taken at a recommended value of 20 MPa.
- is the coefficient of variation taken as 7%.
involved in the cold-forming can also introduce deviations from design assumptions.Inconsistent quality control measures during production and environmental factors during storage, transport, and handling can further influence mechanical properties.Variability in testing and certification procedures can contribute to differences between assumed and actual properties.The JCSS model code [37] addresses uncertainty by incorporating the lognormal PDF.Table 4 contains data extracted from JCSS, specifying the mean and standard deviation of the PDF for the two random variables.

Statistical characteristics of dimensions
The discrepancy between nominal and as-built dimension values was employed as the error variable for statistical investigation.In accordance with the JCSS probabilistic model guidelines [37], the deviation of a dimension, denoted as , is defined as the difference between its statistical deviation characteristic, represented as  , and the nominal value   .As per the JCSS guidelines [37], preliminary findings for steel profiles suggest that the mean and standard deviation of  for fundamental dimensions (such as height, width, and thickness) are less than 1 mm.Since cold-formed sections are somewhat 10 times thinner than hot-rolled sections, a tolerance deviation of 0.1 mm between mean and characteristic value of dimensions is reasonable.Table 5 displays the statistical models for geometric parameter variations.These statistical models are utilized for subsequent reliability analysis.It is worth noting, however, that while JCSS does not provide probabilistic models for cold-formed steel, this study reasonably adopts probabilistic models for that the JCSS recommends for hot-rolled steel by adjusting these tolerance deviations to account for the differences in thickness of dimensions between hot-rolled and cold-formed steel sections.As shown in Fig. 6, the probability density functions (PDFs) illustrate the distribution of different dimensions with their corresponding mean () and standard deviation () values.The mean  values are extracted as minimum values of the dataset in Table 7, and the standard deviation  values are extracted from Table 5.   7 and  values extracted from Table 5.

Table 5
Stochastic data for geometric dimensions.

Action uncertainties
When designing a CFS section using the limit state design philosophy outlined in EN 1990 [17], two categories of actions affecting the CFS beam are considered: permanent  and variable  actions.The permanent action is largely due to the self-weight of the structure, with a high degree of predictability, often modeled as a normally distributed variable [38,39].The variable action, which takes into account the presence and movement of individuals and objects within the structure, displays variability based on the size of the area they influence; an increase in area typically leads to a reduction in action intensity.Specific statistical models for these actions are adopted from authoritative literature [40] and the information is listed in Table 6.
Adhering to the guidelines of Turkstra's rule [38], the analysis typically holds the permanent action as a baseline and factors in the peak of the variable action for action combinations.This approach is grounded in the rationale that the simultaneous peaking of multiple actions has a probability of occurrence approximately zero.Eqs. ( 12)-( 14) establish a framework for computing the design actions.The idea is to combine the adjusted permanent and variable actions, which are essential for the structural design process, ensuring that the structural component can withstand both expected and unexpected forces throughout its design life.
=     +     (12) Herein,   and   are the characteristic values of permanent and variable actions, respectively, with value are calculated using the following equations, where   is the design value for shear capacity considering the partial factor  0 , and  = is the ratio of the characteristic variable action to the characteristic permanent action.Moreover,   and   were taken 1.35 and 1.5, respectively [41].

Reliability analysis
Reliability analysis, a branch of probability theory, addresses uncertainties at the level of both epistemic and aleatory, in engineering using mathematical methods, facilitating real-world safety evaluations and aiding in risk informed design decisions [39].Reliability problems are typically solved by representing uncertainty in parameters via a random vector  following some joint PDF   () and then defining the so called performance function .A positive value of this function indicates a safe state, while a negative value represents failure.The performance function, as expressed in Eq. ( 9) in detail, is generally written as: where () represents the resistance variable, and () denotes the action variable, both of which are functions of random variables.The failure probability is calculated via the multi-dimensional integral: where   () is the joint PDF of random variables , and 1   () is the indicator function indicating failure, In the context of structural reliability for civil engineering applications, it is common to assess the probability of failure   over the expected service life of a structure, typically set at 50 years [18,37].These failure probabilities are generally of a magnitude within the range of 10 −3 to 10 −4 .For a more nuanced analysis, these probabilities are often translated into their standard normal counterparts, referred to as the reliability index [16,42].The reliability index, denoted as , is defined through the inverse standard normal CDF,  −1 , such that  is equal to the negative inverse of the CDF evaluated at the failure probability   , i.e.,  = − −1 (  ), where  symbolizes the standard normal CDF.

Reliability methods
Direct evaluation of the multiple integral in Eq. ( 16) is challenging, so approximate and simulation methods are often used.The analytical evaluation of the performance function  via closed-form expressions, that is Eqs.( 1)-(3) and Eqs. ( 4)-( 8), facilitate the execution of Monte Carlo simulations (MCS), which with a sufficient number of iterations, can accurately estimate the failure probability   .Moreover, this study also examines the application of the First Order Reliability Method (FORM).Extensive discussions on both methods are available in the body of structural reliability [43,44].The forthcoming section provides only the critical details foundational to the reliability analyses conducted and discussed in Section 5.

Monte Carlo simulation (MCS)
When MCS is executed with an adequately large number of trials , a robust estimation of the failure probability   can be attained.This method reinterprets Eq. ( 16) to calculate the expected value of the indicator function 1   ().An ample sample size, for instance  = 10 7 , is utilized, consisting of independent and identically distributed sample vectors  = { (1) ,  (2) , … ,  () } from the joint PDF   of the random variables.The expected value of the indicator function with respect to   is then approximated by the sample mean of 1   (), thereby providing an estimate for   as: where  fail is the number of samples that satisfy  ≤ 0. Consequently, the associated reliability index   can be deduced as: 4.1.2.First order reliability method (FORM) FORM condenses the multivariate probability problem into a univariate format by assuming that the performance equation is linear when projected onto the standard normal space.It calculates the nearest point on this linear approximation to the origin of the standard normal space [16,42].The Euclidean distance from this origin to the design point, denoted as  * , quantifies the reliability index   .The design point  * is defined as the point in the transformed standard normal space that minimizes the distance to the origin while simultaneously satisfying the failure condition (()) ≤ 0.
The variable  represents the transformed variables in the standard normal space, derived from the physical space variables  through an isoprobabilistic transformation.This transformation typically involves normalizing the original variables  to have zero mean and unit variance.The failure probability   is then deduced from the reliability index as   = (−  ), where  is the cumulative distribution function of a standard normal distribution.

Design building set
To conduct a comprehensive reliability analysis, it is essential to gather a diverse collection of design examples that cover a wide range of potential conditions encountered in practical applications.In this study, a dataset comprising 67 experimental shear strength results for CFS sections is compiled.This dataset served as the basis for calibrating mode uncertainty in Section 3.1.
The 67 experimental shear strength results are set in this study as a set of design cases.For each CFS beam in this dataset, the characteristic imposed action required to make each CFS beam to critical under shear is adjusted .In other words, the characteristic load required to reach the limit state design for each CFS beam is determined.Subsequently, the corresponding mean imposed action for each case is computed, taking into account the coefficient of variation provided in Table 6.It is worth noting that since the beams in the database were specifically tested to fail in shear, cases where flexure or other failure modes dominate to significantly impact the computed reliability indices are not expected.In Table 7, summary statistics of the database is given, including the range considered for various parameters such as aspect ratio, web height, thickness, and yield strength.

Reliability results
Figs. 7 and 8 illustrate how the distribution of the reliability index  changes with increasing values of  for both the current and modified EN 1993-1-3 models, according to calculations using the FORM.Additionally, the  values calculated by FORM closely match those determined by crude MCS, as evidenced in Table 8 for  = 1.0, with a mean absolute percentage error (MAPE) averaging 0.87%.This finding suggests that FORM's linear approximation of the performance function is appropriate for the present reliability problem.
In Figs. 7 and 8, one can observe a positive correlation between the reliability index  and the relative web slenderness parameter   .The correlation appears to be more pronounced for the original EN 1993-1-3 model than for the modified version.This difference may stem from the fact that the original EN 1993-1-3 does not consider the level of restraint at the web-flange juncture and the aspect ratio when computing the web slenderness   , resulting in a less robust reliability assessment compared to the modified version.
According to EN 1990 [17], a reliability target   of 3.8 is recommended for structures under reliability class (RC) 2, into which most structural elements are categorized.In reliability studies, it is standard practice to measure against this RC2 benchmark.These reliability targets are established through optimization to strike a balance between safety and economic efficiency, facilitating cost-effective structural design.As evidenced in Figs.7 and 8, the modified EN 1993-1-3 model tends towards more economically optimal designs than the existing EN 1993-1-3, which yields overly conservative designs with  values significantly exceeding 3.8.

Sensitivity analysis based on FORM
One of the key merits of FORM analysis lies in its proficiency to pinpoint the variables that predominantly influence the computed  values.This is achieved via an importance vector of sensitivity coefficients, denoted as , which quantifies the impact of each variable.This is done by approximating the performance function () near the design point with a linear function ĝ() = ‖∇‖( −  T ), so that the variance of this linear approximation is given by: where ).This signifies that the magnitude of ‖  ‖ reflects the variable's importance to the performance function.Variables with a larger ‖  ‖ are considered more critical to the considered model when used in reliability analyses.
Furthermore, the sign of each   is indicative of its nature.In the Eurocode [17], negative   corresponds to action variables, and positive   is related to resistance variables.The components of  are determined by the sensitivity of the reliability index  with respect to the standardized random variables , expressed as: Empirical values of  provided in the EN 1990 [17] give a guidance on the recommendations for variables associated with the present shear failure of CFS sections, for the EN 1993-1-3 and modified EN 1993-1-3 models as shown in Figs. 9 and 10, respectively.
Based on both the EN 1993-1-3 and the modified EN 1993-1-3 models (refer to Figs. 9 and 10), the resistance random variable   is the most influential on the computed reliability indices, followed by the yield strength of steel   .The remaining resistance variables display desirable because it can lead to cost-ineffective structures, contrary to the principles of reliability optimization embedded in structural design codes [45].These observations align with findings from related studies that recommend cost-optimal design approaches amidst uncertainty [46][47][48].
Reducing the  values closer to   would imply a decrease in  0 below the typical baseline of 1.0, which is generally not favored due to the potential for underestimating the effects of uncertainties [49].Therefore, we maintain  0 at 1.0 and propose the calibration of a modification factor   in Eq. ( 22), aiming to achieve the specified reliability targets while also considering the effect of the load ratio .This approach is in line with the methodologies applied in recent reliability studies that employ modification factors for resistance models [50,51].
The modification factor   modifies Eq. (1) as follows, To ensure that the revised expressions satisfy the designated reliability targets, the   parameter is carefully adjusted.This adjustment is implemented to both the existing EN 1993-1-3 and its modified version.Typically, for structural components displaying ductile characteristics [52] and classified under Reliability Class 2 (RC2), a reliability target of   = 3.8 is advised, as per EN 1990 [17].
For the specific case of CFS channels under shear stress, it may be advisable to aim for a higher target reliability of   = 4.3, aligning with Reliability Class 3 (RC3).This is due to the potential instability risks, such as web buckling, associated with the thin walls of CFS channels.In the following, distinct values for the   factor are calibrated and suggested, tailored for both the existing EN 1993-1-3 and its modified version.These adjustments are intended to ensure that these models meet the specified reliability targets of 3.8 and 4.3, corresponding to different ratios of variable to permanent actions denoted by .
Fig. 11 illustrates the relationship between the reliability index () at the 10% fractile and the proposed modification factor (  ), for various  values ranging from 1.0 to 2.0 in increments of 0.5.The study particularly emphasizes the critical intersection where the line representing  = 2.5 meets the prescribed reliability targets   = 3.8 for Reliability Class 2 (RC2) and   = 4.3 for Reliability Class 3 (RC3).At this point, based on the trend line analysis, the estimated   values are identified to be approximately   = 1.5 for RC2 and   = 1.3 for RC3.
The potential threshold effect observed at   = 2.0 for the EN 1993-1-3 model may indicate a physical limit in the material behavior or an inflection point within the resistance model that necessitates further investigation.This observation could be compared with findings from other studies exploring the behavior of CFS channels under varying load conditions and resistance modifications [53][54][55].
In the context of the modified EN 1993-1-3 (refer to Fig. 12), the estimated   values are approximately   = 1.06 for Reliability Class 2 (RC2) and   = 0.925 for Reliability Class 3 (RC3).For the modified EN 1993-1-3 model, when considering the RC3 target , the existing format in Eq. (1) remains applicable, but with a modification to the  0 value, changing it from 1.0 to 1.08.
Furthermore, in the case of the modified EN 1993-1-3, the change in the rate of  values with respect to the action ratios  is not observed for the considered values of   in Fig. 12.However, extrapolating the trend indicates that this significant value is likely to be greater than 1.25.
In conclusion, the calibration of   based on the reliability targets of   = 3.8 and   = 4.3 provides a pathway to designs of CFS channels that meet both safety requirements and economic considerations [47].

Concluding synopsis
The study advances the understanding of reliability in shear design for CFS sections, evaluating both the existing and modified EN 1993-1-3 [12] models.Incorporating the FORM and MCS, key insights and recommendations are presented as follows: • A substantial correlation is identified between the reliability index  and the relative web slenderness   .This correlation is more pronounced in the original EN 1993-1-3 model, indicating enhanced reliability assessment in the modified version due to its inclusion of restraint levels in   calculations.• It is recommended to maintain  0 at 1.0 and adjust the modification factor   according to the load ratio , aiming to strike a balance between safety and economic efficiency.
In conclusion, the findings contribute to the enhancement of reliability in shear design of CFS sections, identify improvements for design standards, and propose solutions for balanced, economically viable designs and leveraging approaches endorsed by the JCSS.By adapting probabilistic models from hot-rolled steel, we address the lack of specific models for CFS sections, emphasizing the importance of considering material properties such as thickness in structural reliability analysis.Furthermore, our findings highlight the need for further research on the execution tolerances of thin-walled CFS sections to enhance structural performance and inform accurate design practices.These insights are anticipated to guide future revisions of design standards, fostering the development of resilient and cost-effective structural systems.

Fig. 1 .
Fig. 1.General sectional configuration of a CFS section with a sloping web.

Fig. 2 .
Fig. 2. Comparison between model error   and perfect prediction as a function of   and  ,new , for the (a) EN 1993-1-3 and (b) modified EN 1993-1-3 shear provisions, respectively, for CFS channels with different aspect ratios in the collected database.

Fig. 3 .
Fig. 3. Sectional configuration of the CFS sections and the notations used for the different geometric parameters.Source: Adapted from Gatheeshgar et al. [12].

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Fig. 6 .
Fig. 6.Probability density functions (PDFs) for different dimensions with Mean () and Standard Deviation () Values.Each subplot represents a normal distribution curve fitted to the respective dimension, with mean  values extracted as minimum values of the dataset in Table7and  values extracted from Table5.

Fig. 11 .
Fig. 11.10% fractile of the reliability indices  as a function of   for the EN 1993-1-3 model.

Fig. 12 .
Fig. 12. 10% fractile of the reliability indices  as a function of   for the modified EN 1993-1-3 model.

Table 1
Statistical data of model uncertainty (  ) for CFS channels subjected to shear.

Table 2
Results of the considered distributions of the model error for shear buckling of the EN1993-1-3 model.

Table 3
Results of the considered distributions of the model error for shear buckling of the modified EN1993-1-3 model.

Table 4
Stochastic data for material properties.
-Subscript () is used in Eurocode to denote characteristic value for variable under consideration.

Table 6
Statistical data of loads.

Table 7
Summary statistics of parameters in building design set.Variations of distribution of the reliability index with relative web slenderness   when the existing EN 1993-1-3 model is used for design.
• The modified EN 1993-1-3 model aligns more effectively with the reliability targets set forth by EN 1990, particularly for structures classified under RC2.The existing model tends to produce designs that are more conservative than necessary.•Sensitivity analysis underscores the importance of precise parameter selection for reliability assessments, advocating for future design codes to incorporate considerations of restraint levels in   calculations.