Resistance factor calibration for perforated cold-formed steel compression members Calibración del factor de resistencia a la compresión en perfiles de acero conformado en frío perforados

Cold-formed Steel profiles are structural profiles widely used in civil construction. They are often manufactured with perforations. The designing can be performed using the direct resistance method. Formulations were adapted by Moen and Schafer (2008) to consider the presence of perforations in these profiles. The objective of this study is to investigate the structural safety of columns with web perforations. The calculation of the resistance capacity was performed using the formulations proposed by the authors. The reliability indexes were determined using the First Order Reliability Method (FORM), First Order Second Moment (FOSM) and Monte Carlo Method (MCM), which are reliability methods for the Load and Resistance Factor Design (LRFD) and Limit States Design (LSD) philosophies. Following the same criteria performed by AISI S100, the resistance factors were obtained from the FOSM method. Based on the results, it was found that the desired security level for the LSD philosophy was not achieved. The calculated resistance factors are predominantly lower than the target. However, for the LRFD philosophy, the safety level was achieved, and the resistance factors were higher than the target. de confiabilidad para los Factores de Carga y de Resistencia (LRFD) y Estados Límite (LSD). Siguiendo los mismos criterios realizados por el AISI S100, los factores de resistencia se obtuvieron del método FOSM. Con base en los resultados, se encontró que no se logró el nivel de seguridad deseado para la filosofía LSD. Los factores de resistencia calculados son predominantemente más bajos que el objetivo. Sin embargo, para la filosofía LRFD, se logró el nivel de seguridad y los factores de resistencia fueron más altos que el objetivo. Palabras clave: Fiabilidad, acero conformado en frio, perforación, método de resistencia directa, compresión


Cold-formed Steel (CFS) profiles have less weight and greater width / thickness ratio plates. For this reason, section instabilities and instabilities along the length of the profiles can occur. Among the existing procedures for dimensioning CFS profiles, the Direct Strength Method (DSM) stands out. This is a method originally developed by
, whose resistance calculation is based on analyses of elastic buckling and stands out for its ease and functionality. CFS profiles have commonly been manufactured with perforations along their length, on flanges and webs. The perforations permit the accommodation and passage of pipes in the walls and ceilings of the buildings, in addition to the connection between construction profiles. Several sections can be manufactured from CFS members, providing advantages for their use. This study gathered profiles with perforated lipped C-section type, but studies have been developed with other perforated sections, such as rack profiles and closed profiles, due to the importance and use of these components in other construction systems. The existing demand for profiles with perforations has led to studies by (Moen and Schafer, 2008), who presented proposals for changes to the original DSM buckling curves.
Revista Ingeniería de Construcción Vol 37 Nº1 Abril de 2022 www.ricuc.cl The United States, Mexico and Canada use the North American standard (North American Specification -NAS) for the design of steel structures consisting of CFS sections. The North American standard includes design provisions for LRFD (Load and Resistance Factor Design), used by the United States and Mexico, and LSD (Limit States Design), used in Canada (AISI, 2016).
A limit state is represented as a condition for which a structural member or structural system fails to perform the function for which it was designed (Hsiao, 1989). For the limit state of resistance, the usual format of the LRFD method is represented by (Equation 1). LSD and LRFD are based on the same philosophy: the design load effect does not exceed the design resistance.
where # is the nominal resistance, is the resistance factor, $ is the load factor and $ is the load effect. # is obtained based on an appropriate analytical model, using the properties of the nominal section and the specified minimum material properties. The resistance factor involves the uncertainties and variability inherent in the nominal resistance. The load factor involves the uncertainties and variability of the loads and the effects of the load (Ellingwood et al., 1980).
In the LRFD and LSD formats, structural reliability is characterized in terms of a reliability index, , determined by a statistical analysis of loads and resistances. Load and resistance factors are obtained so that the reliability of a structure is at the desired level, using the proposed normative provisions. The reliability index is related directly to the load and resistance factors used in the project, and consequently, to the structural reliability of the project. The technical committee responsible for developing the design standards must calibrate the resistance factors, so that the reliability index reaches a required target value % .
A reliability method aims to assess a reliability index or a probability of failure. When the method is used in standard calibration, resistance factors are proposed so that the calculated reliability indexes approximate an established target reliability index % . Procedures for calibrating the first standards in limit states (Ravindra and Galambos, 1978); (Ellingwood et al., 1980); (Hsiao, 1989), are still used in the structural verification of propositions for new design equations, or adaptations that may result in updating standards.
∅ = calibration coefficient (1.52 for LRFD; 1.42 for LSD); . = mean value of material factor M; . = mean value of fabrication factor F; . = mean value of professional factor P; / = coefficient of variation of material factor M; 0 = coefficient of variation of fabrication factor F; 1 = coefficient of variation of professional factor P; 2 = coefficient of variation of load effect; 1 = correction factor (for a large number of tests 1 close to 1).
The basis of the project for the LRFD and LSD formats is the same. However, the values of the target reliability index, as well as the load ratio are different for each design philosophy. As the calibration coefficient depends mainly on the # # ⁄ ratio and the load combination, different values are obtained for LRFD and LSD, so that ∅ is 1.52 for LRFD and 1.42 for LSD.
The objective of this article is to evaluate the reliability of members in cold-formed, perforated web profiles submitted to axial compression force. The resistance calculation followed criteria proposed by (Moen and Schafer, 2008), based on a database with 183 columns. The values for the professional factor calculated in this study were obtained from the ratio between the experimental results of the database and the calculated theoretical results. To obtain the reliability indexes, the following reliability methods were employed: FOSM -First Order Second Moment, FORM -First Order Reliability Method and the Monte Carlo Method (MCM). The same calibration data as the American standard was used. The results obtained were compared with the target indexes of the LRFD and LSD design philosophies. The resistance factors were calibrated using the FOSM reliability method.

The Direct Strength Method
The Direct Strength Method can be used to obtain axial compressive forces in cold-formed profiles. This is a method that uses properties of elastic buckling to calculate resistance (Toledo, 2021). To identify the buckling modes and their respective critical loads, software based on the finite strip method can be used, such as the CUFSM (Constrained and Unconstrained Finite Strip Method) adopted in this study. Formulations for global ( #3 ), local ( #ℓ ) and distortional ( #5 ) buckling, without perforations, are shown below.

Global
Where: = L M . @ is the slenderness factor of global buckling for column; A is yield load; @B3 is the global buckling. Where: F is the slenderness factor of local buckling for column; @BF is the local buckling force. Where:

= L M
. 5 is the slenderness factor of distortional buckling for column; @B5 is the distortional buckling force.
The extension of DSM to columns with perforations was performed based on adaptations of the formulations of the original method presented. (Moen and Schafer, 2008) developed 6 modifications, but only four formulation options were evaluated in this study: DSM 2, DSM 3, DSM 4, DSM 5. In all options developed by the authors, the influence of the perforations should be considered in determining the global ( @B3 ) local ( @Bℓ ) and distortional ( @B5 ) elastic buckling force in compression.
DSM 1 -The original MRD equations for obtaining #3 , #ℓ and #5 (AISI S100) are normally applied, but the influence of the perforation is considered in the buckling analysis. Where: 5#3I is the reduced slenderness factor of the perforated section associated with the distortional buckling; O is the gross area of the cross-section of the column; #3I is the net area of the cross-section of the column; A is the yield stress of the material; @B5 is the critical axial bending force of the perforated profile; #5 is the characteristic compressive strength of the perforated profile, associated with distortional buckling.
DSM 5 -The same transition defined for DSM 4 is considered in the distortional buckling. In addition, a transition was introduced in the formulation of local buckling, from #ℓ to A#3I . Where: @BF is the critical axial force of the local buckling for of perforated profiles DSM 6 -The same transition defined for DSM 4 is considered in the distortional buckling. A transition in the formulation of the local buckling is also considered, but with a modified formulation in relation to DSM 5.

The FOSM, FORM and SMC Methods
Structural reliability is assessed by the relationship between the measures of failure probability, P , and the reliability index (Ditlevsen and Madsen, 2007) and can be resolved using approximating analytical methods (FORM and FOSM) and simulation methods, like MCM. FOSM is based on the first-order Taylor series approach and the required statistical parameters are the mean and standard deviations. The AISI S100 standard uses the FOSM to calibrate the resistance factors in force. FORM was initially proposed by (Hasofer and Lind, 1974). It is applied in a standardized normal space whose random variables are uncorrelated. For nonlinear functions, the design point determination is a nonlinear minimization constraint problem. Optimization techniques that aim to determine the design point include the Rackwitz and Fiessler method 1978. Formulations to transform the distributions of random variables into normal distributions are also presented by the authors.
The analysis carried out by the Monte Carlo Method (MCM) generates random numbers, based on their respective probability distributions. The evaluation of the structural response is given from the probability of failure, calculated by the ratio between the number of trials n for which the limit-state function is less than zero and the total number of simulations (Melchers and Beck, 2018).

Performance function
The safety condition for each ultimate limit state is expressed by the inequality that relates the nominal values of resistance ( # ) and load effect ( # ), such that: # is calculated by design formulation, is the resistance factor, whose numerical value depends on the limit state under analysis and type of load effect that the member is requested, S and T are the load factors of the permanent and variable actions taken in the (AISI S100, 2016) and # and # are, respectively, the nominal values of dead and live loads.
The (AISI, 2016) standard covers two design philosophies in limit states, the LRFD and the LSD. FOSM was the reliability method used for the calibration of the (AISI, 2016), but the calibration data, the resistance factors, the combinations of actions, the # # ⁄ ratio, and the target reliability indexes, % , were specific to each design philosophy. (Table 2) shows the data used for the calibration of the (AISI, 2016) standard.

Professional Factor
The professional factor, P, is a random variable, whose analysis includes the uncertainties inherent in the model. It is the ratio between results obtained experimentally and theoretical results. The P factor provides information about the real performance of the model studied, revealing how conservative or insecure it is. In this case, the experimental values correspond to values of resistance of column tests, obtained from studies of several authors. The theoretical values were obtained from the calculation of resistances by the formulations of (Moen and Schafer, 2008) presented above.
A total of 183 tests performed by different authors were used in this study. Cold-formed lipped channel member with perforations in the webs were subjected to centered compression. The database provides a wide range of lengths and dimensions for sections and perforations. The perforations have rectangular, circular, square, or oblong shapes. Information about the database is shown in (Table 3). The P values were calculated, the mean parameters ( . ), standard deviation ( 1 ) and coefficient of variation ( U ) were determined. The P calculations were grouped based on the failure modes obtained and the (Moen and Schafer, 2008) formulations: DSM 2, DSM 3, DSM 4 and DSM 5. The results are shown in (Table 4).
Revista Ingeniería de Construcción Vol 37 Nº1 Abril de 2022 www.ricuc.cl The best-fitting probability density functions (pdf) were determined using the Minitab software and Anderson-Darling fit tests. Analyses of the statistical parameters of the variable P, carried out by (Moen and Schafer, 2011), demonstrated that the methodology used in DSM 4 resulted in the best performance.
When analyzing the entire data set, it is observed that the DSM 2 option presented more conservative results, with high averages and high reliability indexes. Better values for the statistical parameters, i.e., averages close to the unit value and better coefficients of variation, were observed in the DSM 4 and DSM 5 formulations. Their reliability indexes showed very close values, due to the great similarity between the two formulations. For MCM, 10,000 simulations were performed with the aid of Excel software. (Figure 1) and (Figure 2) show that the reliability indices calculated by the FORM method indicated a good approximation with MCM. The indices calculated by FOSM did not show a better approximation with the MCM and were, in general, superior to the others. The exceptions are the groups of distortional buckling, which showed a different pattern for the Gumbel probability distributions.
It can be seen from (Figure 1)  When calibration is performed from LSD data (Figure 2), with a target index of 3.0, most of the results obtained are not satisfactory, with predominantly lower than the target. This behavior was observed in works developed by (Ganesan, 2010); (Freitas et al., 2013) and (Capanema, 2018). The values calculated by the FOSM method were those that came closest to the target.

Calibration of the resistance factor
(AISI S100 2016) used the FOSM method as the basis for calibrating its standard. Resistance factors, , of 0.85 and 0.8 were presented for the LRFD and LSD philosophies and profiles without perforations. In this study, the resistance factors were also obtained by the FOSM method. (Figure 3) shows the results of the calculated resistance factors. It appears that for the LRFD philosophy, the resistance factors vary between 0.86 and 0.98, with less than for the distortional buckling modes. LSD shows resistance factors varying between 0.72 and 0.84.
It appears that for the LRFD philosophy, all values were higher than those recommended by AISI for profiles without perforations. Regarding the LSD philosophy, the observed behavior is the opposite. Most of the calculated values are lower than the value presented by the standard, for profiles without perforations. These are expected behaviors when evaluating the calculated reliability indices. The distortional buckling modes also showed lower values for .

Conclusions
The objective of this study was to evaluate the reliability of columns in cold-formed steel profiles with perforated webs, following the criteria proposed by (Moen and Schafer, 2008). The results obtained from this analysis allowed the following conclusions: • The formulations DSM 4 and DSM 5 showed better results for the statistics of the professional factor.
• The DSM 2 formulation presents conservative results, with high values for the means and the reliability indices. • For the LRFD philosophy, all data groups, regardless of the reliability method used, presented reliability indexes higher than the target % of 2.5. When was set at 2.5, the results calculated for were greater than 0.85, varying between 0.86 and 0.98. • The LSD philosophy, the reliability indices did not reach the target of 3.0 in all analyzed groups. When setting at 3.0, the results calculated for were, in general, lower than the value adopted by the standard, with equal to 0.8.