DSM Formula for Local-Global Interaction Buckling of Cold-Formed Stainless Steel I-Beams

: The paper presents a set of Direct Strength Method (DSM) formulae for cold-formed stainless steel I-section beams. On basis of a carefully calibrated FE model for stainless steel open-section beams featuring local-global interaction buckling, an extensive database was ﬁrst established. The existing DSM formula of the best prediction performance was commented on for its drawbacks. For improved strength prediction, a traditional-form DSM formula was proposed ﬁrst, and novel two-phase DSM formulae were derived on basis of a deep insight into the varying trend of the member strengths. The two-phase DSM formulae were found to be very powerful predictors yielding more accurate strength predictions over the whole range of slenderness and avoiding undue conservatism for intermediate and short members. A considerable boost in design strength by up to 39% can be achieved with the two-phase DSM formula. The DSM curves were developed with a due reliability level to ﬁt into the current design codes


Introduction
Direct Strength Method (DSM), which originated from the research of Hancock et al. [1] and was fully developed and formally coined with the current name by Schafer et al. [2], is a new generation method for design of cold-formed steel members after the traditional Effective Width Methods (EWM).Compared with EWM, DSM avoids the iterative and cumbersome calculation process for effective section properties, and makes a highly efficient and convenient design process by (i) directly predicting the strength of a member from proper design curves in combination with the material yielding stress and section buckling stresses, (ii) making a more rational and accurate buckling analysis for the whole section instead of on a discrete element basis and (iii) incorporating calculation power of the computer by adopting numerical tools such as ThinWall [3], CUFSM [4] and GBTUL [5] in buckling analysis of sections.The simplicity and efficiency of DSM make it widely suitable (e.g., for complex cross-sections which would be too cumbersome or impossible to design using EWM) and adaptable to various conditions.Until now, researchers have kept extending the scope of DSM, from the very first local, distortional, and global buckling DSM formulae for cold-formed steel members [2,6,7], to those covering shear buckling [8,9], for perforated members [10], and those covering stainless [11][12][13][14] and aluminum members [15].This list goes on.
While the DSM method gained rapid acceptance in cold-formed steel design codes, e.g., in North America standard AISI S100 [16] and Australia/New Zealand standard AS/NZS 4600 [17], the development of the DSM formula for stainless steel members is still underway.Among other structural members, stainless steel columns have received extensive study, and the corresponding DSM formulae for local, distortional, and sectional-global interaction buckling [11][12][13] were proposed.As for stainless steel beams, research efforts

General Format of a DSM Formula
The underlying concept of DSM is that the strength of a section can be predicted directly with some properly chosen design formulae f u = F(f crl , f crd , f cro , f y ), where f u is the ultimate strength of a member, f crl , f crd , f cro are critical buckling stresses of the local, distortional and overall buckling modes as relevant for the member considered, and f y is the material yield stress.The universally accepted column curves and beam curves in design codes can be deemed as an original version of the DSM formula, by which the inelastic overall buckling strength of a column/beam can be determined from a slenderness value defined in terms of the material-yielding stress f y and the member elastic overall buckling stress f cro .
A general DSM formula further accounts for the sectional (local or distortional) buckling effect, as exemplified in Equation (1) for beams, where M u is the ultimate capacity of a beam, a, b and c are properly determined constant parameters, M ref is the reference capacity of a member without influence from sectional buckling, λ s is normalized slenderness defined with the ratio between reference capacity M ref and the sectional (local or distortional) buckling moment M crs as shown in Equation (2).The normalized slenderness λ s is essentially an index of the level of sectional buckling experienced by the member before attaining the reference capacity M ref .It can be appreciated that the general format DSM in Equations ( 1) and (2) express the erosion effects of sectional (local or distortional) buckling on the reference member capacity M ref .Therefore, the ultimate strength M u calculated from the formula is not allowed to exceed M ref .
A practical DSM formula, e.g., those codified in AISI S00 [16] and AS/NZS 4600 [17], is typically derived by fitting a set of parameters a, b, and c to a strength database, which is composed of experimental and/or numerical results reflecting the relevant buckling modes.The geometric and material limitations of sections included in the database are used to define 'prequalified sections' for the proposed DSM formula.In practice, if sections outside the scope of 'prequalified sections' are to be designed with the DSM formula, a more conservative resistance factor is generally required to be used [16,17], i.e., a higher safety margin is used in comparison with the prequalified counterparts.

Accounting for Interaction Buckling Effect with DSM
For a beam section without sectional buckling, the upper bound of its bearing capacity is the lesser of the section yielding moment M y and the inelastic overall buckling moment M o , as shown in Figure 1.Note that the first yielding moment M y is generally adopted as an upper bound of cold-formed steel sections, although some extent of plastic reserve beyond M y can be obtained in cold-formed sections of low section slenderness [18].member before attaining the reference capacity Mref.It can be appreciated that the general format DSM in Equations ( 1) and ( 2) express the erosion effects of sectional (local or distortional) buckling on the reference member capacity Mref.Therefore, the ultimate strength Mu calculated from the formula is not allowed to exceed Mref.
A practical DSM formula, e.g., those codified in AISI S00 [16] and AS/NZS 4600 [17], is typically derived by fitting a set of parameters a, b, and c to a strength database, which is composed of experimental and/or numerical results reflecting the relevant buckling modes.The geometric and material limitations of sections included in the database are used to define 'prequalified sections' for the proposed DSM formula.In practice, if sections outside the scope of 'prequalified sections' are to be designed with the DSM formula, a more conservative resistance factor is generally required to be used [16,17], i.e., a higher safety margin is used in comparison with the prequalified counterparts.

Accounting for Interaction Buckling Effect with DSM
For a beam section without sectional buckling, the upper bound of its bearing capacity is the lesser of the section yielding moment My and the inelastic overall buckling moment Mo, as shown in Figure 1.Note that the first yielding moment My is generally adopted as an upper bound of cold-formed steel sections, although some extent of plastic reserve beyond My can be obtained in cold-formed sections of low section slenderness [18].It is evident the upper bound strengths, My or Mo, shall be used for the reference member capacity Mref in Equation (1).For members of short span or closely spaced bracings, the material yielding effect is dominant and My should be taken for Mref.In this case, the resultant strength prediction Mu from Equation (1) reflects the section strength of a member.As the span of a member increases, the overall buckling effect starts dominating the member failure, and a span-dependent inelastic buckling moment Mo should be used for Mref.In this case, the DSM prediction Mu from Equation (1) reflects the overall buckling strength of a member under the influence of sectional buckling, i.e., the member strength under the sectional-overall interaction buckling effect.
Throughout this paper, the Mref term is taken as the lesser of My and Mo, as mathematically expressed in Equation (3).In contrast to carbon steels, stainless steel demon-Moment member (inelastic) overall buckling Mo Member Length section yielding My It is evident the upper bound strengths, M y or M o , shall be used for the reference member capacity M ref in Equation (1).For members of short span or closely spaced bracings, the material yielding effect is dominant and M y should be taken for M ref .In this case, the resultant strength prediction M u from Equation (1) reflects the section strength of a member.As the span of a member increases, the overall buckling effect starts dominating the member failure, and a span-dependent inelastic buckling moment M o should be used for M ref .In this case, the DSM prediction M u from Equation (1) reflects the overall buckling strength of a member under the influence of sectional buckling, i.e., the member strength under the sectional-overall interaction buckling effect.
Throughout this paper, the M ref term is taken as the lesser of M y and M o , as mathematically expressed in Equation (3).In contrast to carbon steels, stainless steel demonstrates rounded stress-strain curves without a distinct yielding point, from which yielding stress f y can be identified.Therefore, the yielding moment M y of stainless steel sections is calculated from Equation (4) where f 0.2 is the proof stress corresponding to 0.2% plastic strain, and W is the section modulus of the gross cross-section.The inelastic overall (flexural-torsional) buckling moment M o should also be calculated with due consideration of the gradual yielding feature of stainless steel.The tangent modulus expression in Australian standard AS/NZS 4673 [37], see Equation (5), is adopted for M o in this paper, in which C b is a moment modification factor, (L y , L z ) are effective lengths for lateral bending and twisting, (I y , I w , J) are the inertia moment about the minor axis, warping constant, and torsional constant of the gross cross-section, (G 0 , E 0 ) are the initial shear modulus and Young's modulus of the material, and E t is the tangent modulus corresponding to a stress level of f o , as calculated from Equation (6).Iteration is necessary to determine the M o and f o values because E t is to be calculated at M o , which is the object of the calculation.
From Equation (2), two types of slenderness can be defined depending on which value of (M y , M o ) is used for M ref , see Equations ( 7) and (8).Note that the subscript 's' in Equation ( 2) is replaced with 'l' representing local buckling, e.g., M crl and f crl = M crl /W represent the elastic local buckling moment of a cross-section and the corresponding local buckling stress.In the context of this paper, the slenderness λ l defined in Equation ( 7) is termed section slenderness reflecting the erosion effect of local buckling on the section strength, and the slenderness λ ol defined in Equation ( 8) is termed relative slenderness reflecting the erosion effect of local buckling on the member overall buckling load.The two slenderness can be defined for the same member, reflecting different aspects of the member's behavior.

A Database for Local-Global Interaction Buckling of Stainless Steel Beams
Detailed FE models based on the shell element of ABAQUS were developed in [35] to replicate the local-global interaction buckling of stainless steel I-beams.The FE models were calibrated accurately against a full series of test results in the companion paper [34], showing excellent agreement in terms of ultimate capacities and post-buckling deformations.Parametric studies were therefore carried out to expand the available structural performance data over a wider range of sectional and overall slenderness values.The resultant databank of member failure strengths is adopted here as a basis for DSM formula derivation.Regarding the experimental results in [34], it's noted that a couple of inevitable but important affecting factors, for example, the warping constraints introduced by localized clamping of the web at loading points, were involved in the experimental process, the effect of which could not be accurately accounted for in the existing design provisions [35].Therefore, the parametric study results obtained with more clearly defined boundary conditions, serve as a better basis for assessment and derivation of the design formula.
Sections adopted for the parametric study [35] were named using the convention "Alloy-I-L××", where the Alloy was austenitic S30401, ferritic S44330, or lean duplex S32101 as per the ASTM unified numbering system [38] as briefly referred here as 304, 443, and 2101 re-spectively, and the L×× denoted the nominal local buckling slenderness.For example, 304-I-L11 represents an I-section made of 304 alloys with a nominal slenderness λ l = 1.1.The sections used for the parametric study are listed in Table 1, using the dimensional nomenclature illustrated in Figure 2. The E 0 , f 0.2, and n in Table 1 are respectively Young's modulus, nominal yielding stress, and Ramberg-Osgood index of the alloy used, which are also employed in the later sections in strength prediction formulae.2101 respectively, and the L×× denoted the nominal local buckling slenderness.For example, 304-I-L11 represents an I-section made of 304 alloys with a nominal slenderness λl = 1.1.The sections used for the parametric study are listed in Table 1, using the dimensional nomenclature illustrated in Figure 2. The E0, f0.2, and n in Table 1 are respectively Young's modulus, nominal yielding stress, and Ramberg-Osgood index of the alloy used, which are also employed in the later sections in strength prediction formulae.For each of the sections, a series of member lengths were chosen to cover a target overall slenderness range of 0.5~2.0(a lower bound of 1.0 m was used for the member length to avoid failure by shear buckling).Flexural and torsional geometric imperfections were introduced in the parametric study.To introduce a reasonable variation in the effect of geometric imperfections, an initial twist rotation of 0.5° and 1.0° were determined on basis of the measured imperfection statistics [35] and were used for each specimen.For each of the sections, a series of member lengths were chosen to cover a target overall slenderness range of 0.5~2.0(a lower bound of 1.0 m was used for the member length to avoid failure by shear buckling).Flexural and torsional geometric imperfections were introduced in the parametric study.To introduce a reasonable variation in the effect of geometric imperfections, an initial twist rotation of 0.5 • and 1.0 • were determined on basis of the measured imperfection statistics [35] and were used for each specimen.
The ultimate strengths obtained from the parametric study are collectively reproduced in Table 2, where L is the member length, and M-FE05 and M-FE10 are ultimate strengths corresponding to the 0.5 • and 1.0 • initial rotation imperfections.

Current DSM Formula of the Best Performance
The ultimate strengths obtained from the parametric study were compared in [35] with the strength predictions from the EWM design formula in Australian/American and European codes for stainless steel structures.It was found that the methods were incapable of capturing the interaction buckling effect properly, affording overly optimistic strength predictions for beams of high section slenderness.
Existing DSM formulae were also evaluated in [35] about their applicability to localglobal interaction buckling of stainless steel I-section beams, including the codified formula in AS/NZS 4600 [17] for carbon steel beams, see Equation (9), and the formula proposed in the literature [12] for stainless steel columns.The DSM formula proposed in [12], which was converted into beam format as Equation (10), was found to be the best predictor among others.However, it was still not working well in that (i) the formula makes overly conservative predictions for short members but unsafe predictions for long members, it's especially so in cases of high sectional slenderness, and (ii) the overall reliability level offered by the formula was not adequate.A reliability analysis [35] showed that for a target reliability index β = 2.5, the required strength reduction factor ϕ was 0.78, which was considerably lower than the typically prescribed value of 0.9 by current design codes.This implied that the design formulae (10) were improperly optimistic due to their inability to properly capture the local-global interaction buckling effect in the stainless steel I-section beams.

Proposed DSM for Stainless Steel I-Section Beams
The data of I-section member capacity in Table 2 were normalized against the reference capacity M ref , which should be the lesser of the yield moment M y (see Equation ( 4)) and the inelastic overall buckling capacity M o (see Equation ( 5)).Note that for the I-section specimens involved in the parametric study, the inelastic overall buckling moments M o were all below the yield moment M y , and therefore the reference moment M ref actually took the value of M o .Figure 3 plots the normalized strengths M/M o against the relative slenderness λ ol (see Equation ( 8)).This graph demonstrates how the member capacity is reduced from the inelastic overall buckling strength (M o ) with increasing relative slenderness λ ol .A globally descending trend in the data band is observed which reflects a further reduction in strength as a result of increased relative slenderness (λ ol ).There is no obvious clustering per alloy type, and the collection of data points demonstrates quite a high scatter (variance).
Existing DSM formulae were also evaluated in [35] about their applicability to local-global interaction buckling of stainless steel I-section beams, including the codified formula in AS/NZS 4600 [17] for carbon steel beams, see Equation (9), and the formula proposed in the literature [12] for stainless steel columns.The DSM formula proposed in [12], which was converted into beam format as Equation (10), was found to be the best predictor among others.However, it was still not working well in that (i) the formula makes overly conservative predictions for short members but unsafe predictions for long members, it's especially so in cases of high sectional slenderness, and (ii) the overall reliability level offered by the formula was not adequate.A reliability analysis [35] showed that for a target reliability index β = 2.5, the required strength reduction factor φ was 0.78, which was considerably lower than the typically prescribed value of 0.9 by current design codes.This implied that the design formulae (10) were improperly optimistic due to their inability to properly capture the local-global interaction buckling effect in the stainless steel I-section beams.

Proposed DSM for Stainless Steel I-Section Beams
The data of I-section member capacity in Table 2 were normalized against the reference capacity Mref, which should be the lesser of the yield moment My (see Equation ( 4)) and the inelastic overall buckling capacity Mo (see Equation ( 5)).Note that for the I-section specimens involved in the parametric study, the inelastic overall buckling moments Mo were all below the yield moment My, and therefore the reference moment Mref actually took the value of Mo. Figure 3 plots the normalized strengths M/Mo against the relative slenderness λol (see Equation ( 8)).This graph demonstrates how the member capacity is reduced from the inelastic overall buckling strength (Mo) with increasing relative slenderness λol.A globally descending trend in the data band is observed which reflects a further reduction in strength as a result of increased relative slenderness (λol).There is no obvious clustering per alloy type, and the collection of data points demonstrates quite a high scatter (variance).A new DSM design formula was derived by fitting Equation (1) to the whole database, during which a reliability analysis process was carried out conforming to the principles established by Lin et al. [39].Note that a reliability index β was taken as a measure of the reliability level, the calculation of which involved a series of random variables.It's worth noting that a random variable directly related to the design formula was the professional factor P, defined as the ratio of actual member strength to the predicted strength from the design formula.The mean value and coefficient of variation of the professional factor (P m , V P ) were obtained by statistical calculation on basis of the whole strength database, and together with a prescribed strength reduction factor ϕ, the reliability index β could be worked out.When deriving a new design formula, target values of (ϕ = 0.9, β = 2.5) were adopted which fit into the existing LRFD framework.A trial and error process was used to search for a suitable design formula until the target (ϕ, β) values were achieved.A minimum coefficient of variation V p was also aimed for in the search process.The resulting design formula is shown in Equation (11), with its key parameters listed in Table 3.In contrast with the novel "two-phase DSM" formula in the later sections, Equation ( 11) is also termed a "Uniform DSM" for local-global interaction buckling of stainless steel beams.
where For comparison, the existing DSM formulae Equations ( 9) and ( 10), and the proposed Equation (11) in this section are collectively plotted in Figure 3, which are labeled as "DSM-Equation ( 9)", "DSM-Equation ( 10)" and "DSM-uniform" respectively.As seen from the figure, the Equation ( 9) curve, codified in AS/NZS4600 for carbon steel beams, nearly follows the upper boundary of the data band and therefore is much too optimistic for stainless steel beams design.The DSM-Equation ( 10) curve, as transformed from the stainless steel column formula in [12], lies approximately in the middle of the data band with slightly more data points below it and fewer above it.The curve, therefore, represents the general strength varying trend reasonably well but is on the unsafe side.The DSMuniform curve derived here for stainless steel beams lies just above the lower bound of the data band with the majority of data points above it.It serves as a proper design curve with an adequate safety margin.

A Novel Two-Phase DSM Formula
The data processing method in Section 3.3 was commonly used by researchers to derive DSM formulae.However, as observed in Figure 3 and Table 3, the resultant data points normally scatter widely and result in high variance (V P ) when fitting a single design curve to the whole database.A different data processing method is used in this section to further explore the strength database.As per the definition of M u in Equation ( 1), the expressions in Equation ( 12) can be derived.It is seen that when parameter c takes a value close to unity (which is a very common practice), the DSM prediction M u normalized with the critical local buckling moment M crl follows a linearly varying trend against λ ol .
Beams of the same cross-section and different member lengths were collected.Recalling the slenderness definitions in Equations ( 7) and (8), the members should have the same section slenderness λ l but different relative slenderness λ ol .Their strengths were normalized with M crl and plotted against λ ol in Figure 4 (two typical sections are given as an example, and only the member strengths for 0.5 • initial twist rotation were plotted for simplicity).For comparison, the inelastic overall buckling moment M o is also plotted in the figure, which takes the shape of a parabolic curve (appreciate this by recalling that M o /M crl = λ ol 2 ).The deviation of data points from the M o curve implies the deteriorating effect of section local buckling.The DSM curve of Equation ( 10) is also plotted in Figure 4, which demonstrates a nearly linear trend as can be expected from the above analysis.Note should be taken about Figure 4 in that the member strengths normalized with elastic local buckling moment (M/M crl ) can exceed unity due to the post-local buckling strength reserve.This is different from Figure 3 where the strengths were normalized with the inelastic overall buckling strength, which was a theoretical upper bound of strength, and therefore the values (M/M o ) were always lower than unity.Note also that the lower values of λ ol correspond to lower values of M o which implies longer members.simplicity).For comparison, the inelastic overall buckling moment Mo is also plotted in the figure, which takes the shape of a parabolic curve (appreciate this by recalling that Mo/Mcrl = λol 2 ).The deviation of data points from the Mo curve implies the deteriorating effect of section local buckling.The DSM curve of Equation ( 10) is also plotted in Figure 4, which demonstrates a nearly linear trend as can be expected from the above analysis.Note should be taken about Figure 4 in that the member strengths normalized with elastic local buckling moment (M/Mcrl) can exceed unity due to the post-local buckling strength reserve.This is different from Figure 3 where the strengths were normalized with the inelastic overall buckling strength, which was a theoretical upper bound of strength, and therefore the values (M/Mo) were always lower than unity.Note also that the lower values of λol correspond to lower values of Mo which implies longer members.As seen from Figure 4, the normalized strengths Mu/Mcrl of members with the same cross-section varies essentially bilinearly.It is obvious that the bi-linear variation trend cannot be well represented with a linearly varying design curve such as the Equation (10) curve displayed in Figure 4.This explains the high scatter when fitting a traditional DSM formula to the database of FE strengths in Section 3.3.It was therefore conceived that the data points from each linearly varying region should be represented with a separate DSM curve.The transition slenderness λTrans at which the slope of trend line changes was identified from each section (see Figure 4 for example), as listed in Table 4.It was found that the ratio of λTrans/λl value for all the sections was quite close to each other, with an average of 0.75.It was therefore decided that members should be distinguished at the point λol ≈ 0.75λl (this corresponds to Mo ≈ 0.56 My, see Equations ( 7) and ( 8)) and be treated separately in design.Accordingly, members of the same cross-section can be divided into 'short' and 'long' members as (i) short members λol > 0.75λl or Mo > 0.56 My, (ii) long members λol < 0.75λl or Mo < 0.56 My.As seen from Figure 4, the normalized strengths M u /M crl of members with the same cross-section varies essentially bilinearly.It is obvious that the bi-linear variation trend cannot be represented with a linearly varying design curve such as the Equation (10) curve displayed in Figure 4.This explains the high scatter when fitting a traditional DSM formula to the database of FE strengths in Section 3.3.It was therefore conceived that the data points from each linearly varying region should be represented with a separate DSM curve.The transition slenderness λ Trans at which the slope of trend line changes was identified from each section (see Figure 4 for example), as listed in Table 4.It was found that the ratio of λ Trans /λ l value for all the sections was quite close to each other, with an average of 0.75.It was therefore decided that members should be distinguished at the point λ ol ≈ 0.75λ l (this corresponds to M o ≈ 0.56 M y , see Equations ( 7) and ( 8)) and be treated separately in design.Accordingly, members of the same cross-section can be divided into 'short' and 'long' members as (i) short members λ ol > 0.75λ l or M o > 0.56 M y , (ii) long members λ ol < 0.75λ l or M o < 0.56 M y .All member strength results were collected as plotted in Figure 5.The database was then divided into two groups of 'short' and 'long' members as per the above-mentioned criterion, as plotted separately in Figures 6 and 7.In these figures, the data points with the cross-section are interconnected with thin lines.

304-I-L23
2 All member strength results were collected as plotted in Figure 5.The database was then divided into two groups of 'short' and 'long' members as per the above-mentioned criterion, as plotted separately in Figures 6 and 7.In these figures, the data points with the cross-section are interconnected with thin lines.304-I-L20 All member strength results were collected as plotted in Figure 5.The database was then divided into two groups of 'short' and 'long' members as per the above-mentioned criterion, as plotted separately in Figures 6 and 7.In these figures, the data points with the cross-section are interconnected with thin lines.It follows from Figures 6 and 7 that (i) all the long member data points cluster closely near a straight line, and no obvious variance was caused by different alloy types nor by different section slenderness values, and (ii) for the short members, the data points cluster as per the section slenderness (λl).The data were lying on different lines, with similar It follows from Figures 6 and 7 that (i) all the long member data points cluster closely near a straight line, and no obvious variance was caused by different alloy types nor by different section slenderness values, and (ii) for the short members, the data points cluster as per the section slenderness (λ l ).The data were lying on different lines, with similar slopes but offset from each other.Based on this observation, a novel two-phase DSM was proposed.For 'long' members, the traditional form of the DSM in Equation ( 1) was adopted, and for 'short' members, the section slenderness λ l was incorporated into the parameter b of the DSM expression in Equation (1).The geometric interpretation of the b term is the y-intercept of the line in Equation (12).The 'long members' and 'short members' databases were used separately for the regression of a corresponding design formula.Similar to the work in Section 3.3, target values of (ϕ = 0.9, β = 2.5) were adopted when searching for a suitable design formula.A minimum coefficient of variation V p was also aimed for in the search process.The finally proposed two-phase DSM is shown in Equations ( 13)-( 15), with key parameters listed in Table 5. Considerably smaller coefficients of variation V P = 0.08 (for long members) and 0.05 (for short members) were obtained in comparison with that V P = 0.14 for the uniform DSM as shown in Table 3.This reflects a generally lower deviation between the predicted strength and the strength database, as a result of the two-phase DSM accurately fitting the varying trend of actual member strengths.Note that the M long and M short expressions were separately regressed from the corresponding database, and the two design curves might not accurately intersect at the point of λ ol = 0.75λ l .Instead of directly specifying the scope of application for the M long and M short expressions, Equations ( 13)-( 15) calculates both M long and M short values first and picks a proper one by comparison.A smooth transition between the M long and M short design curves can be achieved this way.
M short = 1.74The parametric study strengths and the strength curves predicted by Equations ( 13)-( 15) are normalized with M o and plotted against λ ol in Figure 8.As seen from the figure, the twophase DSM gives a single curve for long members and multiple curves short members as a result of including the variable λ l in the DSM expression.Note also that the two-phase design curves in Figure 8 demonstrate a "descending + ascending" varying trend with increasing slenderness.This is different from the traditional DSM curves, which feature a monotonically descending trend with increasing slenderness (see Figure 3).The slightly ascending trend in the short-member design curves is probably a result of the development of plastic deformation and corresponding plastic strength reserve in short members.
and is close to the 'DSM-uniform' curve.The multiple curves for sh from the lower bound and pass through the body of the data band upper bound.The shape of the two-phase DSM curves is found to foll very closely.Figure 9 displays the member capacity versus member length gr of the same cross-section (2101-I-L20 for example).The Mlong and M Equations ( 13) and ( 14) are plotted as the 'Mu-DSM-2Phase(long)' and (short)' curves respectively, and the higher part of the two curves are a For comparison, the DSM formula in Equations ( 9)- (11) are also plott 'Mu-Equation (9)', 'Mu-Equation (10)', and 'Mu-DSM -Uniform' curve first pointed out in Figure 9 that the predicted member strength from th descends monotonically with increasing member length, althoug strength curve in Figure 8 ascends slightly for the short member.Th predictions are found to closely follow the varying trend of the strengt a reasonable safety margin.The uniform DSM yields pretty good The traditional form DSM formulae in Equations ( 9)-( 11) are also plotted in Figure 8, labeled as 'DSM-Equation ( 9)', 'DSM-Equation ( 10)', and 'DSM-Uniform' respectively.The two-phase DSM curves are found to lie between these traditional DSM curves.Specifically, the single curve for long members lies around the lower bound of the data band, and is close to the 'DSM-uniform' curve.The multiple curves for short members start from the lower bound and pass through the body of the data band until reaching the upper bound.The shape of the two-phase DSM curves is found to follow the data points very closely.
Figure 9 displays the member capacity versus member length graph for specimens of the same cross-section (2101-I-L20 for example).The M long and M short expressions in Equations ( 13) and ( 14) are plotted as the 'Mu-DSM-2Phase(long)' and 'Mu-DSM-2Phase (short)' curves respectively, and the higher part of the two curves are adopted for design.For comparison, the DSM formula in Equations ( 9)- (11) are also plotted in the figure as 'Mu-Equation ( 9)', 'Mu-Equation (10)', and 'Mu-DSM -Uniform' curves respectively.It is first pointed out in Figure 9 that the predicted member strength from the two-phase DSM descends monotonically with increasing member length, although the normalized strength curve in Figure 8 ascends slightly for the short member.The two-phase DSM predictions are found to closely follow the varying trend of the strength data points with a reasonable safety margin.The uniform DSM yields pretty good predictions in the long-member region, but becomes very conservative in the short-member region.Equation (10), which is obviously better than the other existing DSM Equation ( 9), predicts slightly unsafe results for long members and overly conservative results for short members.Recall that Equation (10) was not of adequate reliability level when fitting into the framework of current design codes (Section 3.2), while the newly proposed uniform and two-phase DSMs in this paper all have an adequate reliability level.For this example, an increase of design strength by up to 39% can be achieved with the two-phase DSM as compared with the uniform DSM.
bers.Recall that Equation (10) was not of adequate reliability level when fittin framework of current design codes (Section 3.2), while the newly proposed un two-phase DSMs in this paper all have an adequate reliability level.For this ex increase of design strength by up to 39% can be achieved with the two-phas compared with the uniform DSM.

Revised Two-Phase DSM Formula
It was appreciated by the authors that a slightly ascending design curve shown in Figure 8 for short members, might be too newfangled to be accepted codes.In this section, a compromise is sought between the beneficial strengt and the easiness of common acceptance.Specifically, a leveling-off trend is ad the short member design curves instead of the slightly ascending trend in F 'short member' design curve is assumed to start at the point λol = 0.75λl, and 'long member' DSM curve exactly at this slenderness value.Therefore, the lon DSM formula can be used directly to work out the intersection point at λol = 0 revised two-phase DSM expression is shown in Equation ( 16), and the corr design curves are plotted in Figure 10.A comparison with Equations ( 13)-( 15) a 8 reflects that the revised DSM formula is more concise and the design cur "descending + leveling off" trend, which might make it easier to be adopted practices.However, it's noted that the strength of intermediate and short me sacrificed to some extent in Equation ( 14), and therefore the reliability level o be higher than those prescribed for Equations ( 11) and ( 13)- (15).

Revised Two-Phase DSM Formula
It was appreciated by the authors that a slightly ascending design curve, as those shown in Figure 8 for short members, might be too newfangled to be accepted in design codes.In this section, a compromise is sought between the beneficial strength increase and the easiness of common acceptance.Specifically, a leveling-off trend is adopted for the short member design curves instead of the slightly ascending trend in Figure 8.A 'short member' design curve is assumed to start at the point λ ol = 0.75λ l , and to join the 'long member' DSM curve exactly at this slenderness value.Therefore, the long-member DSM formula can be used directly to work out the intersection point at λ ol = 0.75λ l .The revised twophase DSM expression is shown in Equation ( 16), and the corresponding design curves are plotted in Figure 10.A comparison with Equations ( 13)- (15) and Figure 8 reflects that the revised DSM formula is more concise and the design curves take a "descending + leveling off" trend, which might make it easier to be adopted in design practices.However, it's noted that the strength of intermediate and short members are sacrificed to some extent in Equation ( 14), and therefore the reliability level of it should be higher than those prescribed for Equations ( 11) and ( 13)- (15).

Conclusion Remarks
This paper aims at developing design formulae in the form of the Direct Strength Method (DSM) for cold-formed stainless steel beams undergoing local-global interaction buckling.A database was first established with extensive parametric studies of stainless steel I-section beams, on basis of the calibrated FE models in [35].The following conclusions could be drawn from the research: 1. Existing DSM formulae in literature were found incapable of accurately predicting the strength of cold-formed stainless steel beams and lacking a due level of reliability.A traditional-form DSM equation was regressed from the collected strength database, offering improved accuracy and adequate reliability level.2. Cold-formed stainless steel I-beams could be distinguished as "long" or "short" members with criteria (i) short members λol > 0.75λl and (ii) long members λol < 0.75λl, where λl and λol are "section slenderness" and "relative slenderness" defined in Equation (7) and Equation (8) of the paper.3. Varying trend of the member strengths differs considerably between the 'long' and 'short' members.A fundamental drawback of the traditional-form DSM formula was observed, viz. the mathematical expression hindered it from accurately following the strength varying trend of stainless steel I-beams and therefore resulting in generally high deviation in predictions and undue conservatism for intermediate and short members.4. Novel two-phase DSM was therefore proposed with a prescribed level of reliability.
The novel two-phase DSM formula was found to predict member strengths more accurately the whole slenderness range, resulting in strength predictions with much lower deviation from actual strengths.Enhanced design strengths by up to 39% could be obtained from the two-phase DSM formula for members with intermediate and short spans.5.A revised version of the two-phase DSM was further proposed, which simplified the formula expression at some cost of the member strength of short members.

Conclusions Remarks
This paper aims at developing design formulae in the form of the Direct Strength Method (DSM) for cold-formed stainless steel beams undergoing local-global interaction buckling.A database was first established with extensive parametric studies of stainless steel I-section beams, on basis of the calibrated FE models in [35].The following conclusions could be drawn from the research: 1.
Existing DSM formulae in literature were found incapable of accurately predicting the strength of cold-formed stainless steel beams and lacking a due level of reliability.A traditional-form DSM equation was regressed from the collected strength database, offering improved accuracy and adequate reliability level.

2.
Cold-formed stainless steel I-beams could be distinguished as "long" or "short" members with criteria (i) short members λ ol > 0.75λ l and (ii) long members λ ol < 0.75λ l , where λ l and λ ol are "section slenderness" and "relative slenderness" defined in Equation (7) and Equation (8) of the paper.

3.
Varying trend of the member strengths differs considerably between the 'long' and 'short' members.A fundamental drawback of the traditional-form DSM formula was observed, viz. the mathematical expression hindered it from accurately following the strength varying trend of stainless steel I-beams and therefore resulting in generally high deviation in predictions and undue conservatism for intermediate and short members.4.
Novel two-phase DSM was therefore proposed with a prescribed level of reliability.The novel two-phase DSM formula was found to predict member strengths more accurately over the whole slenderness range, resulting in strength predictions with much lower deviation from actual strengths.Enhanced design strengths by up to 39% could be obtained from the two-phase DSM formula for members with intermediate and short spans.

5.
A revised version of the two-phase DSM was further proposed, which simplified the formula expression at some cost of the member strength of short members.

Figure 1 .
Figure 1.Upper bound strength of beams without sectional buckling.

Figure 1 .
Figure 1.Upper bound strength of beams without sectional buckling.

Figure 5 .
Figure 5.All data points for I-sections.

Figure 5 .
Figure 5.All data points for I-sections.

Figure 10 .
Figure 10.FE strengths and revised 2-phase DSM of I-sections.

Author Contributions:
Conceptualization, S.N.; methodology, S.N. and Y.Y.; validation, S.N. and X.Z.; investigation, S.N. and Y.Y.; writing-original draft preparation, Y.Y.; writing-review and editing, S.N.; supervision, X.Z.; project administration, S.N.; funding acquisition, S.N.All authors have read and agreed to the published version of the manuscript.Funding: This research was supported by the National Natural Science Foundation of China (Grant No. 51508134) and the Fundamental Research Funds for Central Universities (Grant No. HIT.NSRIF.201676).

Figure 10 .
Figure 10.FE strengths and revised 2-phase DSM of I-sections.

Table 1 .
I-sections for FE parametric study.

Table 1 .
I-sections for FE parametric study.

Table 3 .
Parameters for a uniform DSM of I-section beams.

Table 4 .
Transition point for a bilinear varying trend of member strengths.

Table 5 .
Parameters for two-phase DSM of I-section beams.