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

Abstract

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.

1. 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 have mainly been dedicated to hollow sections [18,19,20,21,22,23,24] to study the bearing strength of such sections under local buckling effect (distortional and global flexural-torsional buckling is not relevant for such members) and crippling and openings in web. Relatively limited research was seen on stainless steel open-section beams. Welded I-sections were studied experimentally [25,26] and numerically [27], most of which were stocky and demonstrated inelastic strength reserve. Zhou et al. [28] studied the strength of stainless steel compact I-sections failed by shear buckling, and proposed methods to amend the current design guidance. Stainless steel plate girders (IPGs) featuring stocky flanges and slender web elements with or without transverse stiffeners were studied, the failure mode of which was dominated by lateral-distortional buckling [29] and shear buckling [30] respectively. Cold-formed lipped channels were tested in bending by [31,32], which demonstrated predominant distortional or distortional-global interaction buckling mode. Karthik et al. [33] studied close-section beams built-up with face-to-face channels, for which no global buckling was incurred, and the sectional buckling involved both local and distortional buckling modes of the constitutive channels. A systematic experimental and numerical study was made in [34,35] on stainless steel I-section beams (made up of cold-formed back-to-back channels) which demonstrated purely local-global interaction buckling. An evaluation was also made by [35] on the existing design formula, including the existing DSM formula in design codes and literature. It was found that the EWM design formula failed to reflect the local-global interaction buckling behavior and predicted generally optimistic strengths for sections of high slenderness. The existing DSM formulae were not satisfactory either, rendering obviously conservative predictions for medium- and short-length members.

Based on the above review, this paper intends to propose revised DSM formula for local-global interaction buckling of stainless steel I-section beams. The paper is organized as follows. An interpretation of the concept and format of DSM is given in Section 2, which is important for understanding the newly proposed formula in the following section. A databank from [35,36] is collected and new DSM design formula is proposed in Section 3 for improved design predictions. A traditional format DSM formula is first proposed with a prescribed reliability level, which makes a better predictor than the currently existing DSM formula. Furthermore, novel two-phase DSM formulae are proposed which offer an excellent predictor for local-global interaction buckling of stainless steel I-section beams. Conclusion remarks are made in Section 4.

2. Concept of DSM Design Method

2.1. 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.

$$M_u=\left(\frac{a}{{\lambda }_s^c}-\frac{b}{{\lambda }_s^{2c}}\right)M_{ref }\le { M}_{ref}$$

(1)

$${ \lambda }_s=\sqrt{\frac{M_{ ref}}{M_{ crs}}}$$

(2)

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.

2.2. 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].

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, (G₀, E₀) 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.

$$M_{ref}=\mathrm{m}\mathrm{i}\mathrm{n}\left(M_{y }{, M}_{o }\right)$$

(3)

$$M_y{=f}_{0.2}W$$

(4)

$$M_o{=C}_b\left(\frac{E_t}{E_0}\right)\sqrt{\frac{{\pi }^2{E}_0{I}_y}{L_y^2}+\left(G_{0}J+\frac{{\pi }^2{E}_0{I}_w}{L_z^2}\right)}$$

(5)

$$f_o=\frac{M_o}{W}$$

(6)

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.

$${\lambda }_{ l}=\sqrt{ \frac{M_y}{M_{crl}}}=\sqrt{ \frac{f_{0.2}}{f_{crl}}}$$

(7)

$${ \lambda }_{o l}=\sqrt{ \frac{M_o}{M_{crl}}}=\sqrt{ \frac{f_o}{f_{crl}}}$$

(8)

3. DSM Formula for Stainless Steel I-Beams

3.1. 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 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 λ_l = 1.1. The sections used for the parametric study are listed in

Table 1. I-sections for FE parametric study.

Section	H (mm)	B (mm)	T (mm)	R (mm)	fcrl (MPa)	E0 (GPa)	f0.2 (MPa)	n
304-I-L11	180	130	2.30	3.5	216	210	246	5.1
304-I-L17	180	130	1.48	3.5	89
304-I-L20	180	130	1.25	3.5	63
304-I-L23	180	97	0.85	3.5	50
443-I-L11	180	100	1.90	3.0	227	197	286	8.2
443-I-L17	180	100	1.22	3.0	93
443-I-L20	180	100	1.05	3.0	69
443-I-L24	180	100	0.90	3.0	51
2101-I-L11	176.5	130	3.00	2.6	350	200	436	5.0
2101-I-L17	176.5	130	1.98	2.6	152
2101-I-L20	176.5	130	1.65	2.6	105
2101-I-L24	176.5	130	1.40	2.6	76

, using the dimensional nomenclature illustrated in Figure 2. The E₀, 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.

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. Parametric study results for cold-formed stainless steel beams [36].

Section	Parametric Study Results
443-I-L11 (λl = 1.1)	L = 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5 m M-FE05 = 10.81, 9.06, 8.4, 6.33, 5.54, 4.21, 3.62, 2.96 kNm M-FE10 = 10.29, 8.66, 7.74, 6.0, 5.17, 4.04, 3.48, 2.87 kNm
443-I-L17 (λl = 1.7)	L= 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0 m M-FE05 = 5.19, 4.08, 3.28, 2.65, 2.22, 1.88, 1.59 kNm M-FE10 = 4.94, 3.96, 3.17, 2.54, 2.11, 1.80, 1.52 kNm
443-I-L20 (λl = 2.0)	L = 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0 m M-FE05 = 5.19, 4.08, 3.28, 2.65, 2.22, 1.88, 1.59 kNm M-FE10 = 4.94, 3.96, 3.17, 2.54, 2.11, 1.80, 1.52 kNm
443-I-L24 (λl = 2.4)	L = 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0 m M-FE05 = 3.03, 2.40, 1.89, 1.51, 1.27, 1.06, 0.86 kNm M-FE10 = 2.94, 2.34, 1.85, 1.48, 1.25, 1.05, 0.85 kNm
304-I-L11 (λl = 1.1)	L = 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5 m M-FE05 = 14.78, 13.12, 11.42, 10.14, 9.75, 8.79, 7.26, 6.38 kNm M-FE10 = 14.47, 12.74, 11.03, 9.75, 9.26, 8.22, 7.02, 6.18 kNm
304-I-L17 (λl = 1.7)	L= 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.5, 5.5, 6.5 m M-FE05 = 7.34, 6.57, 5.60, 4.76, 4.24, 3.79, 3.17, 2.34, 1.94 kNm M-FE10 = 7.18, 6.42, 5.48, 4.65, 4.11, 3.62, 2.93, 2.25, 1.87 kNm
304-I-L20 (λl = 2.0)	L= 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.5, 5.0 m M-FE05 = 5.5, 4.92, 4.23, 3.61, 3.17, 2.76, 2.15, \ kNm M-FE10 = \, \, \, \, 3.10, 2.70, 2.11, 1.85 kNm
304-I-L23 (λl = 2.3)	L = 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5 m M-FE05 = 2.32, 1.92, 1.53, 1.26, 1.06, 0.9, 0.78, 0.68 kNm M-FE10 = 2.26, 1.87, 1.49, 1.23, 1.03, 0.87, 0.76, 0.67 kNm
2101-I-L11 (λl = 1.1)	L = 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5 m M-FE05 = 32.55, 26.95, 23.23, 19.51, 16.58, 15.2, 12.26, 10.47, 9.14, 8.39 kNm M-FE10 = 31.72, 26.03, 22.19, 18.83, 15.94, 14.42, 11.90, 10.22, \, 8.10 kNm
2101-I-L17 (λl = 1.7)	L = 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5 m M-FE05 = 16.97, 13.35, 10.9, 8.86, 7.43, 7.14, 5.83, 5.06, 4.45, 4.33 kNm M-FE10 = 17.43, 13.08, 10.65, 8.66, 7.23, 6.68, 5.61, 4.89, 4.31, 4.07 kNm
2101-I-L20 (λl = 2.0)	L = 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0 m M-FE05 = 12.27, 9.85, 8.04, 6.51, 5.41, 4.82, 4.03, 3.51, 3.11 kNm M-FE10 = 11.88, 9.65, 7.88, 6.39, 5.33, 4.63, 3.98, 3.45, 3.03 kNm
2101-I-L24 (λl = 2.4)	L = 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5 m M-FE05 = 9.21, 7.39, 6.07, 4.92, 4.13, 3.6, 3.11, 2.68, 2.34, 2.23 kNm M-FE10 = 9.04, 7.25, 5.96, 4.84, 4.07, 3.56, 3.08, 2.65, 2.31, 2.09 kNm

Note: a couple of simulations failed to converge, where a “\” was marked.

, 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.

3.2. 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 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.

$$M_u=\left\{\begin{array}{c}\begin{array}{cc}{M}_{o }\hfill & for {\lambda }_{ol}\le 0.776\end{array}\\ \begin{array}{cc}\left(\frac{1}{{\lambda }_{ol}^{0.8}}-\frac{0.15}{{\lambda }_{ol}^{1.6}}\right)M_{o }\hfill & for {\lambda }_{ol}>0.776\end{array}\end{array}\right.$$

(9)

$$M_u=\left\{\begin{array}{c}\begin{array}{cc}{M}_{o }\hfill & for {\lambda }_{ol}\le 0.463\end{array}\\ \begin{array}{cc}\left(\frac{0.9}{{\lambda }_{ol}^{0.9}}-\frac{0.2}{{\lambda }_{ol}^{1.8}}\right)M_{o }\hfill & for {\lambda }_{ol}>0.463\end{array}\end{array}\right.$$

(10)

3.3. 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).

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

Curve	a	b	c	Pm	VP	β	φ
DSM-uniform	0.60	0.0	0.70	1.15	0.14	2.5	0.90

. 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.

$$\begin{array}{c}{M}_u=\left\{\begin{array}{cc}{M}_{o }\hfill & for {\lambda }_{ol}\le 0.48\hfill \\ \frac{0.6}{{\lambda }_{ol}^{0.7}}{M}_{o }\hfill & for {\lambda }_{ol}>0.48\hfill \end{array}\right.\hfill \end{array}$$

(11)

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 DSM-uniform 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.

3.4. 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.

$$\frac{M_u}{M_{crl}}=\frac{M_{ref}}{M_{crl}}\left(\frac{a}{{\lambda }_{ol}^c}-\frac{b}{{\lambda }_{ol}^{2c}}\right){=\lambda }_{ol}^2\left(\frac{a}{{\lambda }_{ol}^c}-\frac{b}{{\lambda }_{ol}^{2c}}\right)\stackrel{ c=1 }{\to } a{\lambda }_{ol}-b$$

(12)

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_ol²). 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.

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

Section	λl	λTrans	λTrans/λl
443-I-L11	1.10	0.85	0.77
443-I-L17	1.70	1.35	0.79
443-I-L20	2.00	1.56	0.78
443-I-L24	2.40	1.85	0.77
304-I-L11	1.10	0.83	0.75
304-I-L17	1.70	1.23	0.72
304-I-L20	2.00	1.40	0.70
304-I-L23	2.30	1.53	0.67
2101-I-L11	1.10	0.86	0.78
2101-I-L17	1.70	1.31	0.77
2101-I-L20	2.00	1.50	0.75
2101-I-L24	2.40	1.85	0.77
Average	/	/	0.75

. 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 Figure 6 and Figure 7. In these figures, the data points with the cross-section are interconnected with thin lines.

It follows from Figure 6 and Figure 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. Parameters for two-phase DSM of I-section beams.

Curve	a	b	c	Pm	VP	β	φ
Long members	0.70	0.11	0.93	1.08	0.08	2.5	0.90
Short members	1.74	0.47 + 0.47λl	0.43	1.08	0.05	2.5	0.90

. 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.

$$M_{long}=\left\{\begin{array}{c}\begin{array}{cc}{ M}_o\hfill & for {\lambda }_{ol}\le 0.44\hfill \end{array}\\ \begin{array}{cc}\left(\frac{0.7}{{\lambda }_{ol}^{0.93}}-\frac{0.11}{{\lambda }_{ol}^{1.86}}\right)M_o\hfill & for {\lambda }_{ol}>0.44\hfill \end{array}\end{array}\right.$$

(13)

$$M_{short}=\left(\frac{1.74}{{\lambda }_{ol}^{0.43}}-\frac{(0.47+0.47{\lambda }_l)}{{\lambda }_{ol}^{0.86}}\right)M_o\le M_o$$

(14)

$$M_u{=\mathrm{m}\mathrm{a}\mathrm{x}(M}_{long}{ , M}_{short})$$

(15)

where $M_{o }\le M_{y }$, ${\lambda }_{ol}=\sqrt{M_{o }/M_{crl }}$, ${\lambda }_l=\sqrt{M_{y }/M_{crl }}$

The 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 two-phase DSM gives a single curve for long members and multiple curves for 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.

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.

3.5. 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 two-phase 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).

$$M_u=\left\{\begin{array}{c}\begin{array}{cc}{ M}_o\hfill & for {\lambda }_{ol}\le 0.44\end{array}\hfill \\ \begin{array}{cc}\left(\frac{0.7}{{\lambda }_{ol}^{0.93}}-\frac{0.11}{{\lambda }_{ol}^{1.86}}\right)M_o\hfill & for 0.44<{\lambda }_{ol}\le 0.75{\lambda }_l\end{array}\\ \begin{array}{cc}\left(\frac{0.7}{{\left(0.75{\lambda }_l\right)}^{0.93}}-\frac{0.11}{{\left(0.75{\lambda }_l\right)}^{1.86}}\right)M_o& for 0.75{\lambda }_l<{\lambda }_{ol}\end{array}\end{array}\right.$$

(16)

where $M_{o }\le M_{y }$, ${\lambda }_{ol}=\sqrt{M_{o }/M_{crl }}$, ${\lambda }_l=\sqrt{M_{y }/M_{crl }}$.

4. 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:

• 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.
• 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.
• 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.
• 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.
• 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.