Citation Thin-walled Structures, 2013, v. 73, p

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1 Title Experimental and Nmerical Investigation of Cold-formed Lean Dplex Stainless Steel Flexral embers Athor(s) Hang, Y; Yong, B Citation Thin-walled Strctres, 2013, v. 73, p Issed Date 2013 URL Rights NOTICE: this is the athor s version of a work that was accepted for pblication in Thin-walled Strctres. Changes reslting from the pblishing process, sch as peer review, editing, corrections, strctral formatting, and other qality control mechanisms may not be reflected in this docment. Changes may have been made to this work since it was sbmitted for pblication. A definitive version was sbseqently pblished in Thin-walled Strctres, 2013, v. 73, p DOI: /j.tws

2 EXPERIENTAL AND NUERICAL INVESTIGATION OF COLD-FORED LEAN DUPLEX STAINLESS STEEL FLEXURAL EBERS Yner Hang and Ben Yong* Department of Civil Engineering, The University of Hong Kong, Pokflam Road, Hong Kong, China Abstract: Experimental and nmerical investigation of cold-formed lean dplex stainless steel flexral members is presented in this paper. The test specimens were cold-rolled from flat plates of lean dplex stainless steel with the nominal 0.2% proof stress of 450 Pa. Specimens of sqare and rectanglar hollow sections sbjected to both major and minor axes bending were tested. A finite element model has been created and verified against the test reslts sing the material properties obtained from copon tests. It is shown that the model can accrately predict the behavior of lean dplex stainless steel flexral members. An extensive parametric stdy was carried ot sing the verified finite element model. The test and nmerical reslts as well as the available data on lean dplex beams are compared with design strengths predicted by varios existing design rles, sch as the American Specification, Astralian/New Zealand Standard, Eropean Code and direct strength method for cold-formed stainless steel. Reliability analysis was performed to evalate the reliability of the design rles. It is shown that these crrent design rles provide conservative predictions to the design strengths of lean dplex stainless steel flexral members. In this stdy, modified design rles on the American Specification, Astralian/New Zealand Standard, Eropean Code and direct strength method are proposed, which are shown to improve the accracy of these design rles in a reliable manner. Keywords: Beam; Cold-formed steel; Design strength; Finite element modeling; Flexral members; For-point bending tests; Hollow sections; Lean dplex; Stainless steel. * Corresponding athor. Tel.: ; fax: address: yong@hk.hk (B. Yong). 1

3 1. Introdction Cold-formed stainless steel is gaining increasing applications as a constrction material serving both architectral and strctral needs. It provides aesthetic and modern shining appearance, sperior corrosion resistance, longer service life with easy maintenance, and convenience in constrction. Therefore, extensive research has been carried ot on the strctral performance of stainless steel strctres. Design specifications for stainless steel strctres were developed for varios types of stainless steel, inclding ferritic, astenitic and dplex stainless steel. Nevertheless, the high cost of stainless steel material constrains its wider application. In recent years, a relatively new type of stainless steel, called lean dplex stainless steel of grade EN (LDX 2101), with strctral and economical advantages was developed. It is becoming an attractive choice as a constrction material de to its low cost compared to dplex stainless steel, and the strength of the material is comparable with dplex stainless steel. However, the lean dplex stainless steel is crrently not covered in any design specification, and the investigation on sch new material is also limited. Theofanos and Gardner [1] carried ot three-point bending tests on 8 specimens and finite element analysis on 36 specimens of lean dplex stainless steel rectanglar hollow section (RHS) and sqare hollow section (SHS). It was fond that the Eropean Code is overly conservative, while the Astralian/New Zealand Standard and the American Specification provided more accrate prediction to the strengths of flexral members. The modified classification limits that proposed by Gardner and Theofanos [2] and the continos strength method (CS) provided better prediction to the flexral members. Hang and Yong [3] investigated the material properties of lean dplex stainless steel by condcting copon tests, stb colmn tests and measrement of residal stresses. Colmn tests were condcted on cold-formed lean dplex stainless steel members by Hang and Yong [4]. It was fond that the crrent design specifications are generally conservative for colmns, and a new design approach of sing stb colmn property & fll cross-sectional area in calclation compression capacity has been recommended. Frthermore, finite element analysis on lean dplex stainless steel colmns was also performed by Hang and Yong [5]. A total nmber of 259 colmn strengths were compared with design vales predicted by varios design rles. It is shown that the existing design rles are generally conservative. odifications are proposed for the AS/NZS Standard, EC3 Code and direct strength method in order to obtain a more accrate prediction for the cold-formed lean dplex stainless steel colmns. Saliba and Gardner [6] performed experimental and nmerical investigation on the strctral behavior of lean dplex stainless steel welded I-sections. The investigation inclded copon tests, stb 2

4 colmn tests and bending tests as well as parametric stdy on welded I-sections sing finite element analysis. The experimental and nmerical data were compared with design predictions by Eropean Code for stainless steel and continos strength method (CS). It is shown that the crrent Class limits in the Eropean Code can be relaxed. In addition, the continos strength method is shown to provide better prediction than the crrent Eropean Code prediction. The objective of this stdy is mainly to investigate the strctral performance of cold-formed lean dplex stainless steel flexral members. A series of bending tests and a wide range of parametric stdy on lean dplex stainless steel flexral members were carried ot. The 180 nmerical and experimental data obtained from this stdy and previos research [1] were compared with design predictions by the American Specification (ASCE) [7], Astralian/New Zealand Standard (AS/NZS) [8], Eropean Code (EC3) [9], the design rle proposed by Gardner and Theofanos [2] and the direct strength method (DS) described in the North American Specification (AISI) [10]. Reliability analysis was condcted for each of the crrent design rles, and design recommendations are proposed in this stdy. It shold be noted that the lean dplex stainless steel is not covered in the ASCE, AS/NZS nor EC3. 2. Experimental Investigation 2.1 Test specimens Cold-formed lean dplex stainless steel flexral members were tested sbjected to pre bending. The nominal 0.2% proof stress of the lean dplex stainless steel is 450 Pa. There are six different sections inclding two sqare hollow sections (SHS) and for rectanglar hollow sections (RHS). The test specimens sed in this stdy are the same batch of specimens as those investigated by Hang and Yong [3]. The material properties as shown in Table 1 are also reported by Hang and Yong [3]. The specimens of RHS were tested nder bending abot both the major and minor axes. The specimens were labelled sch that the depth of the web (D), width of the flange (B), thickness (t) of the cross-section as well as the specimen length (L) can be recognized. The arrangement of the cross-sectional dimensions also refers to the bending axis. For example, the label L900 defines the following specimen. The nmbers before the letter L is refer to the cross-sectional dimension. The dimensions of the web (D), flange (B) and thickness (t) of the cross-section are eqal to 30, 50 and 2.5 mm, respectively. The nmbers after the letter L indicates the specimen length of 900 mm. 3

5 The dimension of the web (D) is smaller than the flange (B), ths the beam is sbjected to minor axis bending. On the other hand, the specimen L900 is of the same cross-sectional dimension and length, bt sbjected to major axis bending. 2.2 Test setp and procedre The for-point bending tests were condcted to obtain the moment capacity of each test specimen. The relationship between the bending moment and crvatre of the specimens can also be obtained. A total of ten for-point bending tests was condcted. The test setp is shown in Fig. 1. The pin-ended bondary conditions were simlated by a half-ronded spport and a roller spport located at 70 mm from the two ends of the specimen. ajor and minor axes bending tests were carried ot on the specimens of RHS. The specimens of SHS were placed so that the srface with the weld is located at the web of the sections. The moment span between the two loading points, and the shear span between the end spports and the loading points were careflly designed, so that the section moment capacity cold be obtained withot the occrrence of shear failre. Vertical loading was applied throgh a lockable ball bearing connecting to a spreader beam. The fnction of the lockable ball bearing is to eliminate any possible gaps between the spreader beam and the two loading points. The bearing was locked by for bolts and restrained from rotation prior to testing, as shown in Fig. 1. Web stiffening plates were clamped at the two loading points and the ends of each specimen. In addition, wooden blocks were inserted at these locations to prevent any possible local bearing failre dring testing. Three displacement transdcers (LVDTs) were placed along the centerline of the tension face of each specimen at the two loading points and at the mid-span of the specimen. The vertical deflections of the specimen at these three locations (two loading points and mid-span) were recorded, and the crvatre of the specimen was calclated from the recorded deflections. Concentrated compressive force was applied by a hydralic testing machine sing displacement control with a constant loading rate of 1.0 mm/min for all test specimens. The static load was recorded by pasing the applied staining for two mintes at the ltimate load. A data acqisition system was sed to record the applied load and the readings of the LVDTs at reglar intervals dring the tests. 4

6 2.3 Test reslts The experimental ltimate moments ( Exp ) and the corresponding crvatres (k Exp, ) of the test specimens are smmarized in Table 3. The static moment-crvatre crve for each specimen is plotted in Fig. 2(a). The static moment () and the crvatre (k) of each specimen are normalized with the plastic moment ( pl ) and the crvatre corresponding to plastic moment (k pl ), respectively, as shown in Fig. 2(b). The moments were obtained sing half of the static applied load from the actator mltiplied by the shear span of the specimens. Ot-of-plane bending was not observed in the tests. In addition to flexral behavior (F), local bckling (L) is also observed at ltimate load of specimens L900, L1100, L1500 and L1500. The failre modes observed at ltimate load of the specimens L900 and L900 involved flexral behavior (F) and combination of flexral behavior and local bckling (L+F) are shown in Figs 3 and 4, respectively. The crvatres (k Exp, ) of test specimens were calclated from displacements measred from the three LVDTs. A constant crvatre between the transdcer locations was assmed, and the crvatre was calclated sing the radis (r) of the crved beam specimen between the LVDTs located at the two loading points, sch that k Exp, = 1/r. The experimental ltimate moments ( Exp ) are compared with the theoretical elastic ( el ) and plastic ( pl ) bending moments, as shown in Table 3. The elastic and plastic bending moments were calclated sing the measred 0.2% proof stress (σ 0.2 ) obtained from the flat copon tests, as shown in Table 1, mltiplied by the elastic and plastic section modli of the fll sections, respectively. Generally, conservative predictions to moment capacity of the test specimens were fond, especially for those sbjected to major axis bending. The mean vale of Exp / el and Exp / pl ratios is eqal to 1.33 and 1.10 with the corresponding coefficients of variation (COV) of and 0.178, respectively. The moment capacity of specimen L1500 was over predicted by elastic and plastic bending moments with the Exp / el and Exp / pl ratios of 0.76 and 0.68, respectively. This is de to the early occrrence of local bckling in the compression flange before the specimen reached yielding. 3. Finite element model Finite element model was developed sing the program ABAQUS version 6.11 [11] to simlate the cold-formed lean dplex stainless steel flexral members. The initial local geometric imperfections and materials properties obtained from tensile copon tests of flat 5

7 portions and corners measred by Hang and Yong [3] were incorporated in the finite element model. A for-noded dobly crved shell element with redced integration (S4R) with a mesh size of 10 mm 10 mm (length by width) in the flat portions of the cross-sections and a finer mesh at the corners were sed. In the experiments, the concentrated compressive load was applied vertically throgh the lockable bearing and spreader beam, and then the load transferred to the roller and half-ronded bar onto the load transferring plates to the specimen, as shown in Fig. 1. Therefore, the two loading points were modelled by two reference points located at the middle of the contact srfaces between the load transferring plates and the specimen. The reference points were copled to the contact srfaces between the transferring plates and the specimen, and restrained against all degrees of freedom except for displacement in the vertical and longitdinal directions along the flexral member as well as the rotation abot the bending axis. Similarly, the two spports were modelled by copling the contact srfaces with two reference points located at the bottom flange of the specimen in the middle of the corresponding contact srfaces. The pin spport (half-ronded) was modeled by restraining against all degrees of freedom except for the rotation abot the bending axis, while the roller spport was modeled by allowing an extra degree of freedom for longitdinal displacement along the specimen. The loading was applied by displacement control method, which is identical to the tests of flexral members, by specifying an axial displacement at the two reference loading points. In the finite element model, the loading was applied by a static RIKS step available in the ABAQUS library. The nonlinear geometric parameter (*NLGEO) was inclded to deal with the large displacement analysis. Hang and Yong [3] condcted the tensile copon tests to obtain the material properties of the cold-formed lean dplex stainless steel specimens. These specimens are of the same batch as the beam specimens in this stdy. The copons were extracted from the flat portions and corners of each section, and the measred stress-strain crves were sed in the finite element model. The material properties inclding the 0.2% proof strength (yield strength) (σ 0.2 ), ltimate strength (σ ), strain at fractre (ε f ), initial Yong s modls (E o ), and Ramberg-Osgood parameter (n) of flat and corner copon tests for each section are smmarized in Table 1. A mlti-linear stress-strain crve containing the elastic part p to the proportional limit stress with the measred Yong s modls, and the plastic part with the tre stress and logarithmic tre plastic strain crve, which is converted from a static stress-strain crve, was sed. The tre plastic stress-strain crves converted from flat copon test reslts were sed as the material properties in modelling the flat portions of the 6

8 specimens, while those converted from the corner copon test reslts were sed in modelling the corner regions of the specimens. The local geometric imperfections for each section that measred by Hang and Yong [3] were inclded in the finite element model. The local bckling mode, which was obtained by carrying ot Eigenvale analysis with a large D/t ratio and sing a BUCKLE procedre, was sperposed on the flexral member model. The first bckling mode predicted by the ABAQUS Eigenvale analysis are normalized to 1.0, ths the bckling mode was factored by the measred magnitdes of the initial local geometric imperfections for each member. In addition, the membrane residal stresses of section , which was measred by Hang and Yong [3], was inclded in the model of specimens L1500 and L1500 sing the ABAQUS (*INITIAL CONDITIONS, TYPE = STRESS) parameter to assess the inflence of the residal stresses on the beam capacities of cold-formed lean dplex stainless steel flexral members. The section was partitioned into strips of the same width as those measred by Hang and Yong [3]. It shold be noted that the bending residal stresses effect has been taken into accont by the material properties obtained from flat and corner copon tests. The ltimate moments predicted by finite element analysis ( FEA ) inclding residal stresses are compared with the test reslts ( Exp ). The nmerical reslts obtained by the model exclding residal stresses are also compared with the test reslts. The moment ratio Exp / FEA of specimen L1500 eqals to 0.96 for the finite element model incldes residal stresses, while the Exp / FEA ratio eqals to 0.94 for the model withot considering residal stresses. It is shown that the finite element model inclding residal stresses is 2% more accrate than that withot considering residal stresses. For the specimen L1500, the Exp / FEA ratios are both eqal to 1.00 for the finite element models with and withot residal stresses. The finite element model with residal stresses is 0.3% more accrate than that withot residal stresses. The moment-crvatre crves of the test and finite element analysis for specimen L1500 are shown in Fig. 5. It is shown that the effect of residal stresses on the beam are qite small, ths the residal stresses are not inclded in the parametric stdy. 7

9 4. Verification of Finite Element odel The moment capacities ( FEA ) predicted by the finite element analysis are compared with the test reslts ( Exp ), as shown in Table 3. The mean vale of Exp / FEA ratio is 1.00 with the coefficient of variation (COV) of A maximm difference of 7% is fond between the experimental and nmerical moment capacities for specimen L900. The failre modes predicted by the finite element analysis are identical to those obtained from the tests at ltimate load. The failre modes of flexral bckling (F) and interaction of local and flexral bckling (L+F) obtained from the FEA compare well with the experimental failre modes for specimens L900 and L900 as shown in Figs 3 and 4, respectively. The comparison of crvatres of the test specimens (k Exp, ) with those of the finite element reslts are smmarized in Table 3. The mean vale of k Exp, /k FEA, eqals to 1.04 with the COV of Fig. 5 shows the comparison of the experimental and nmerical reslts for specimen L1500. It is observed that good agreement has been achieved in terms of the ltimate flexral strength and corresponding crvatre. 5. Parametric Stdy An extensive parametric stdy was performed sing the verified finite element model with a total of 126 flexral members of lean dplex stainless steel. The material properties adopted in the parametric stdy were based on the stress-strain crves obtained from the flat and corner copon tests of section reported in Hang and Yong [3]. The averaged measred local geometric imperfection for the tested specimens reported in Hang and Yong [3, 4] was t/11, where t is the thickness of each section. Ths, a slightly conservative ronded nmber of t/10 was sed as the local imperfection in the parametric stdy. The residal stresses of the flexral members are not inclded in the finite element model, considering its negligible effect on the moment capacity. The 126 specimens in the parametric stdy were SHS and RHS, with 6 different overall profiles (overall depth overall width) of SHS ranged from to , and 7 different overall profiles of RHS ranged from to The thicknesses of each profile varied to cover a wide range of slenderness ratio from stocky to slender sections. The aspect ratio (D/B) for the specimens was ranged from 0.25 to 4. The moment span between the two loading points was 500 mm for all specimens, and the shear span between the loading points to the spports were careflly designed so that the section flexral capacity can be 8

10 reached withot shear failre. The RHS specimens were sbjected to both major and minor axes bending. The specimens in the parametric stdy sed the same labelling system as that of the test specimens, as shown in Table 4. The ltimate moment capacities and the corresponding crvatres predicted by the finite element analysis are smmarized in Table Reliability Analysis The sitability of the crrent design rles, inclding ASCE [7], AS/NZS [8], EC3 [9], modified EC3 by Gardner and Theofanos [2], direct strength method (DS) in the AISI [10] and continos strength method (CS) [6] for the cold-formed lean dplex stainless steel flexral members is evalated sing reliability analysis, which is detailed in the Commentary of the ASCE Specifications [7]. A target reliability index (β 0 ) of 2.5 for stainless steel strctral members is sed as a lower limit. The design rles are considered to be reliable if the reliability index is greater than or eqal to 2.5. The resistance factors (φ 0 ) of 0.90 for members with stiffened compression flanges sbjected to bending is recommended by ASCE [7], AS/NZS [8], and AISI Standard [10] for direct strength method (DS), while the resistance factors of 0.91 is sed by the EC3 [9], the modified EC3 by Gardner and Theofanos [2] as well as the continos strength method (CS) [6]. The load combinations of 1.2DL+1.6LL, 1.25DL+1.5LL and 1.35DL+1.5LL were sed for design rles in ASCE, AS/NZS and EC3 in the reliability analysis, respectively, where DL is the dead load and LL is the live load. The load combination of 1.35DL+1.5LL was sed for reliability analysis of modified EC3 by Gardner and Theofanos [2] and continos strength method (CS), while the load combination of 1.2DL+1.6LL was sed for the direct strength method (DS). The Eq in the ASCE Specification [7] was sed in calclating the reliability index. The statistical parameters m = 1.10, F m = 1.00, V m = 0.10 and V F = 0.05, which are the mean vales and coefficients of variation for material properties and fabrication factors for flexral members in Clase of the commentary of the ASCE Specification were adopted. The mean vale (P m ) and coefficient of variation (V P ) of tested-to-predicted load ratio or nmerical reslts to design predictions ratio are shown in Table 5. In calclating the reliability index, Eq. F1.1-3 in the North American cold-formed steel Specification AISI S100 [10] was sed to calclate the correction factor, in order to take into accont for the inflence by the nmber of data. For the prpose of direct comparison, a constant resistant factor (φ 1 ) of 0.90 and a load combination of 1.2DL+1.6LL were sed to calclate the 9

11 reliability index (β 1 ) for the design rles, and the vales of the reliability index are also shown in Table Crrent Design Rles and Comparison of oment Capacities 7.1 General The experimental and nmerical moment capacities ( ) are compared with the nfactored design flexral strengths (nominal strength) predicted by ASCE [7], AS/NZS [8], EC3 [9], modified EC3 by Gardner and Theofanos [2], DS [10] and CS [6]. The comparison of the experimental and nmerical moment capacities with the design moment capacities is shown in Table 5 and 6. The flat copon test reslt of section , reported in Hang and Yong [3], is sed in calclating the design flexral strengths for specimens in the parametric stdy. The design rles in ASCE [7], AS/NZS [8], EC3 [9] and modified EC3 by Gardner and Theofanos [2] sed the effective width method for the sections when local bckling occrs. Therefore, the calclation procedre sing these design rles involved iterative process, as the location of the netral axis shifts with the effective width when the sections sbjected to bending. However, sch tedios iterative process is not reqired in the DS [10] and CS [6], as the flexral strength is calclated by the fll section instead of effective section. It shold be noted that the ASCE, AS/NZS and EC3 do not cover the material of lean dplex stainless steel. 7.2 American Specification and Astralian/New Zealand Standard The ASCE [7] and AS/NZS [8] se the same design rles to calclate the moment capacity and the effective width of the section. According to Clase in the ASCE Specification and Clase 3.3 of the AS/NZS Standard, the two design specifications allow the calclation base on initiation of yielding and inelastic reserve capacity. Therefore, both approaches are assessed in this stdy. For the approach by initiation of yielding, the moment capacities ( yielding ) were calclated by the effective section modls (S e ) mltiplied by the yield strength (f y ). The effective width was calclated in accordance with Clase 2.2 in ASCE and AS/NZS, where a yield strength at the extreme fibre of compressive flange and the stress distribtion vary linearly in the 10

12 section were assmed. It is shown in Table 5 that the initiation of yielding approach in ASCE [7] and AS/NZS [8] provide qite conservative and scattered predictions for moment capacities of the specimens, especially to those stocky sections with low slenderness ratio d/(tε), where d is the flat portion of the web and ε is the material factor, as shown in Table 5 and Fig. 6. The mean vale of the / yielding ratio is 1.50 and the coefficient of variation (COV) eqals to This approach is considered to be reliable for ASCE [7] and AS/NZS [8] with the reliability indices (β 0 ) of 2.85 and 2.71, respectively. For the approach by inelastic reserve capacity, the moment capacities ( inelastic ) were calclated by the eqivalent force mltiplying the lever arm within the section, considering eqilibrim of stresses in the effective section and assming an ideally elastic-plastic stress distribtion in the section. The compression strain factor (C y ) was calclated to determine the stress distribtion in the section. However, the ASCE [7] and AS/NZS [8] did not state clearly the calclation of effective widths that involve elastic-plastic stress distribtion in the section. According to Y and Labobe [12] for elastic-plastic stress distribtion in the section, the effective widths were calclated sing the eqations for elements with stress gradient in Clase of the ASCE [7] and AS/NZS [8], as shown in Fig. 7(a). It is reqired in both the ASCE [7] and AS/NZS [8] that the ratio of the depth of compressed portion of the web to its thickness does not exceed the slenderness ratio. In this stdy, there are 47 ot of 180 specimens that exceeded this limitation. Ths, 133 specimens within the limit were compared sing the inelastic reserve capacity design approach, as shown in Table 5. It is shown that the mean vale of / inelastic ratio is 1.34 and the COV eqals to 0.200, which is less conservative and less scattered than the approach by initiation of yielding. The reliability indices (β 0 ) for the ASCE [7] and AS/NZS [8] are 2.92 and 2.76, respectively, which are larger than the target vale of The comparison of experimental and nmerical reslts with design strengths by ASCE [7] and AS/NZS [8] is shown in Fig. 6, where the flat width b is obtained by the flat portion of the flange. It shold be noted that both approaches provide qite conservative predictions to specimens with the compression strain factor (C y ) eqal to 3, which is the elastic-plastic stress distribtion occrred in the section. The comparison of the test and nmerical reslts with the design predictions to the 54 specimens with compression strain factor eqal to 3 are shown in Table 6. It is shown that the approach by inelastic reserve capacity is very conservative for these specimens with the mean vale of / inelastic eqal to 1.54 and COV of for both ASCE [7] and AS/NZS [8]. Therefore, design rles for stainless steel flexral members need to be modified. 11

13 Frthermore, the limitation of the approach by inelastic reserve capacity, that the ratio of the depth of the compressed portion in the web (d w ) to its thickness does not exceed the slenderness ratio (λ l ), was assessed by comparing the design flexral strengths ( * inelastic) of the 47 specimens which exceeded the limitation (d w /t > λ l ) with the test and nmerical reslts, as shown in Table 5, where λ l is defined in Clase 3.3 of ASCE [7] and AS/NZS [8]. The reslt shows that the approach provides good prediction on these specimens even they are beyond the limitation. The mean vale of / * inelastic is 1.17 with the COV of 0.153, and the reliability indices (β 0 ) of 2.72 and 2.55 for ASCE [7] and AS/NZS [8], respectively. 7.3 Eropean Code According to Clase 5.1 of the EC3 Part 1.4 [9], the provisions given in Section 5 and 6 of EC3 Part 1.1 [13] shold be applied for stainless steel, except where modified or sperseded by the special provisions given in EC3 Part 1.4. Therefore, the moment capacity ( EC3 ) was calclated by the eqations 6.13 to 6.15 of Clase in EC3 Part 1.1 [13]. Classification was reqired according to Table 5.2 in EC3 Part 1.4, where the flat portions of the web and compressive flange were classified since these elements sbjected to bending and compression. For specimens classified as Class 4 section, the effective widths were calclated by Table 4.1 of EC3 Part 1.5 [14] together with the redction factor calclated by Clase in EC3 Part 1.4. The classification is governed by the larger Class of element in the section. In this stdy, 80 specimens had a larger class in the flange compared to the class in the web, while 24 specimens had a larger class in the web instead of the flange. The remaining 76 sections had the same class in either the flange or web. It shold be noted that there are not many investigation covered lean dplex stainless steel flexral members for section having a larger class in the web. The 24 specimens from the parametric stdy are sed to assess the classification of elements sbject to bending in the EC3 Code. The comparison of the experimental and nmerical data with the design strengths is shown in Fig. 8 and Table 5. It is shown that the EC3 predictions are qite conservative with the mean vale of / EC3 eqal to 1.25 and COV of The reliability index (β 0 ) is 2.83, which is greater than the target reliability index of 2.5. Gardner and Theofanos [2] analysed the classification limits of the Eropean Code [9] for stainless steel. A set of classification limits for compression elements was proposed, and the effective width eqations [2] were modified as shown in Eqs. (1-2): 12

14 ρ = 2 1 λ for otstand elements (1) p λ p ρ = 2 1 λ for internal elements (2) p λ p where ρ is the redction factor for local bckling and λ p is the element slenderness. In this stdy, the design moment capacities ( G&T ) predicted by Gardner and Theofanos [2] in modifying the Eropean Code were compared with the test and nmerical reslts obtained from this stdy and available data. It is shown that the modified EC3 by Gardner and Theofanos [2] provides a more accrate and less scattered prediction, with the mean vale of / G&T eqal to 1.19 with the COV of as well as the reliability index (β 0 ) of In Eropean Code, a section is classified as Class 1 when it reaches plastic moment capacity for fll section and be able to maintain sfficient deformation capacity, which is calclated by Eq. (3), ^ k pl R = 1 (3) k pl where k pl is the crvatre corresponding to the plastic moment ( pl ) on the ascending branch of moment-crvatre crve, while k^pl is the crvatre on the descending branch at the plastic moment after the ltimate moment. The locations of k, k^pl and k pl on the moment-crvatre crve of specimen L1500 is shown in Fig. 8. For specimens L900, L900, L900 and L1100, the crvatre (k^pl) was not recorded, becase the deformations of the specimens were very large and tests had to stop before reaching k^pl. Therefore, the maximm recorded crvatres for these for specimens were sed as the vale of k^pl in Eq. (3) to calclate the rotation capacities (R). Since there is no reqired deformation capacity for Class 1 in the EC3 Part 1.4 [9], the reqirement of R = 3 for carbon steel are adopted in this stdy to assess the Class 1 limit [15, 16], and Gardner and Theofanos [2] also adopted the same reqirement for stainless steel. The rotation capacity R of the specimens with moment capacity exceeds the plastic moment capacity are plotted against the slenderness of the flange (b/(tε)) and web (d/(tε)) in Figs 10 and 11, respectively. The specimens governed by the flange are those with a larger class of the flange than the web, as plotted in Fig. 10, while the specimens governed by the web are those with a larger class of 13

15 the web than the flange, as plotted in Fig. 11. It is shown in Figs 10 and 11 that the crrent Class 1 limits for carbon steel in EC3 Part 1.1 [13] is more appropriate than those for stainless steel in EC3 Part 1.4 [9], as the limit of moment capacity exceeds the plastic moment while maintaining rotation capacity R larger than 3, in designing for lean dplex stainless steel internal elements sbjected to compression and bending. A section is classified as Class 2 and Class 3 when it reaches plastic and elastic moment capacities, respectively. Therefore, the moment capacities obtained from the test and nmerical reslts are normalized with plastic moment capacity for Class 2 and elastic moment capacity for Class 3 that plotted against the flange and web slenderness, in order to assess the Class 2 and Class 3 limits in EC3 Code, as shown in Figs 12 to 15. It is observed that the crrent Class 2 and Class 3 limits for stainless steel are generally conservative for lean dplex stainless steel beams. Hence, the crrent Class 2 and Class 3 limits for stainless steel can be relaxed for lean dplex material. 7.4 Direct Strength ethod The direct strength method sed in this stdy was based on the clase of Appendix 1 in the North American Specification for the Design of Cold-Formed Steel Strctral embers [10]. The nominal flexral strength ( DS ) shall be determined by the minimm of the nominal flexral strength for lateral-torsional bckling ( ne ), local bckling ( nl ) and distortional bckling ( nd ). The lateral-torsional bckling and distortional bckling did not occr for SHS and RHS. According to the Clase in the Appendix 1 of Commentary on the North American Cold-formed Steel Specification [17], for flly braced beams which are restrained against lateral-torsional bckling, the maximm ne vale is taken as the yield moment ( y ). According to Fig. C in the Appendix 1 of Commentary on the North American Cold-formed Steel Specification [17], the flexral strength for local bckling ( nl ) is calclated by Eqs , and in the North American Cold-formed Steel Specification [10], for which the nominal flexral strength for lateral-torsional bckling ( ne ) is replaced by the yield moment ( y ). Therefore, the nominal flexral strength ( DS ) eqation [10] is shown in Eq. (4): y for λ l DS = crl crl 0.15 y for λ y l > (4) y 14

16 where the yield moment ( y ) is eqal to S f f y, and S f = gross section modls, f y = yield strength, and λ l = ( y / crl ) 0.5. The critical elastic local bckling moment ( crl ) of the cross-section was obtained from a rational elastic finite strip bckling analysis [18] with a 5 mm half-wave length interval. It is shown that the DS also provided conservative and scattered predictions to the flexral members of SHS and RHS considered in this stdy, as shown in Table 5. The mean vale of / DS ratio is 1.35 with the corresponding COV and the reliability index (β 0 ) of and 3.12, respectively. The comparison of test and nmerical reslts with design strengths by DS is also shown in Fig. 16. It is fond that the DS provides a very conservative prediction for the 81 specimens with λ l smaller than 0.776, as shown in Table 6. The mean vale of / DS ratio is 1.47 with COV of On the other hand, the DS provides good prediction to the 99 specimens with λ l greater than or eqal to 0.776, with the mean vale of / DS ratio eqals to 1.16 and COV of Therefore, the DS for specimens with λ l small than shold be modified, in order to provide a more accrate prediction for cold-formed lean dplex stainless steel flexral members. 7.5 Continos strength method The continos strength method (CS) is a deformation-based design method. Gardner and Theofanos [2] and Saliba and Gardner [6] have shown that the CS is capable of providing accrate predictions for stainless steel flexral members. Similar to the DS, the cross-section classification and effective width calclation are not reqired in CS. The continos strength method sed in this stdy was based on the CS eqations for bending resistances presented in Saliba and Gardner [6]. In addition, the CS allows for strain hardening in determining the cross-section resistance. It is stated in Saliba and Gardner [6] that the CS does not apply to cross-sections where the slenderness ( λ ) larger than 0.748, becase there is no significant benefit to be derived from strain hardening beyond this limit [6]. Therefore, the flexral strengths of 96 specimens that meet the reqirement of the CS are compared with the design vales ( CS ) by the continos strength method, as shown in Fig. 5. It is fond that the CS provides the best prediction among the existing design rles. The mean vale of / CS ratio is 1.13 with COV of The reliability index (β 0 ) is eqal to 2.75, which is greater than the target vale of 2.5. p 15

17 It is noted that the continos strength method is capable of providing good prediction for the cold-formed lean dplex stainless steel flexral members, while the other existing design rles investigated in this stdy provide qite conservative predictions for the moment capacities of lean dplex stainless steel flexral members. Therefore, modifications to the ASCE and AS/NZS as well as EC3 and DS are proposed in the following section. 8. odified Design Rles & Comparison of Beam Strengths 8.1 General The ASCE Specification [7], AS/NZS Standard [8], EC3 Code [9] and DS [10] do not cover the design of lean dplex stainless steel. It was fond that these design rles provide qite conservative predictions to the moment capacity of lean dplex stainless steel. Therefore, modifications to these design rles are proposed for lean dplex stainless steel flexral members. A total nmber of 180 test and nmerical reslts as well as the available data was sed in the modifications of the design rles. The design strengths calclated by the modified inelastic reserve capacity approach in the ASCE Specification and AS/NZS Standard as well as the modified EC3 Code and DS are represented by # inelastic, # EC3, and # DS, respectively. 8.2 ASCE Specification and AS/NZS Standard The approach by initiation of yielding is considered to be too conservative in designing flexral members of lean dplex stainless steel, de to the assmption of linear elastic stress distribtion in the section for all specimens inclding those reached elastic-plastic stress distribtion. In this stdy, modifications are made to the approach by inelastic reserve capacity, especially to those stocky sections with compression strain factor (C y ) eqals to 3.0. In the approach by inelastic reserve capacity, the stress distribtion in the section is governed by the compression strain factor (C y ). If the factor is eqal to 1.0, the stress distribtion is linearly elastic p to the yield strength (f y ) at the extreme fibre of the compressive flange. If the factor is larger than 1.0, the location of the threshold of linear elastic stress distribtion having the strain eqal to yield strain (e y ), beyond which a stress block of yield strength is formed in the section. Therefore, the stress distribtion in the section is elastic-plastic. However, the calclation procedre of effective width is not clearly stated in the ASCE 16

18 Specification [7] and AS/NZS Standard [8] for sections having elastic-plastic stress distribtion (C y > 1). Therefore, Y and Labobe [12] sggested that the effective width eqations for sections with linear elastic stress distribtion can be sed for those having elastic-plastic stress distribtion. However, sch design calclation leads to a qite conservative prediction for stocky members, especially those with C y eqals to 3.0. In this stdy, the modified design rles for moment capacities ( # inelastic) consist of three parts, namely for (i) effective width calclation; (ii) pper bond limit of moment capacity; and (iii) limitation of web slenderness (d w /t ratio). Firstly, the effective width calclation for specimens with compression strain factor (C y ) larger than 1.0, it is recommended that the height of the stress gradient in the compression portion of the web (d g ), as shown in Fig. 7(b), is assmed to be flly effective, while that of the effective width for the plastic compressive stress block is calclated by the eqations for webs with niform compression, as indicated in the Clase of the ASCE Specification [7] and AS/NZS Standard [8]. The effective width and the stress distribtion in the sections calclated by Y and Labobe [12] method and the proposed method are shown in Figs 7(a) and 7(b), respectively. Secondly, the pper bond limit of moment capacity from the approach by inelastic reserve capacity shall not exceed 1.25 times that obtained from the approach by initiation of yielding ( inelastic 1.25 yielding ). Hence, this provides conservative predictions for specimens with low d/(tε) vale, de to the overly conservative predictions by initiation of yielding approach. Therefore it is recommended that the reqirement of inelastic 1.25 yielding is not reqired in designing lean dplex stainless steel flexral members. Thirdly, the limitation of web slenderness of d w /t < λ l in calclating the moment capacity can be removed. It is shown in Table 5 that the approach by inelastic reserve capacity provided good predictions to the 47 specimens exceeded the web slenderness limit (d w /t λ l ). The comparison of the test and nmerical reslts with the predications from the modified design rles sing the inelastic reserve capacity approach is smmarized in Table 5 and Fig. 6. The mean vale of the / # inelastic ratio eqals to 1.18 with COV of 0.116, and is considered to be reliable with the reliability index (β 0 ) of 2.98 and 2.79 for ASCE Specification and AS/NZS Standard, respectively. Fig. 6 shows the improvement of moment capacity predictions by the modified design rles, especially for those specimens with compression strain factor larger than 1.0. It is also observed from Table 6 that the accracy of the prediction to the specimens with compression strain factor (C y ) eqals to 3.0 improves considerably compared to the crrent design rle. The mean vale of the / # inelastic ratio eqals to 1.20 with the COV of 0.119, and the reliability index (β 0 ) of 3.01 and 2.82 for 17

19 ASCE Specification and AS/NZS Standard, respectively. The modified design rles for moment capacities ( # inelastic) are validated against the nmerical and experimental data of specimens with B/t ratio ranged from 6.25 to 140, and D/t ratio ranged from 6.25 to EC3 Code The rotational capacities (R) of the specimens, where the ltimate moments ( ) are larger than the plastic moment ( pl ), are plotted against the web slenderness (d/tε) and flange slenderness (b/tε) in Figs 10 and 11. As discssed in Section 7.3 of this paper, the Class 1 limits for carbon steel in EC3 Part 1.1 [13] are more appropriate than those in EC3 Part 1.4 [9] for stainless steel. In addition, the moment capacities of the flange governed and web governed specimens that normalized with the elastic and plastic moment capacities are shown in Figs It is shown that the Class 2 and Class 3 limits for carbon steel in EC3 Part 1.1 [13] are more appropriate for the cold-formed lean dplex stainless steel flexral members. Therefore, it is recommended that the Class 1, Class 2 and the Class 3 limits for carbon steel in EC3 Part 1.1 [13] are adopted for lean dplex stainless steel flexral members. Therefore, it is sggested that the class limits (b/tε) of 33, 38 and 42 are sed for Classes 1, 2 and 3 for element sbject to compression, while the class limits (d/tε) of 72, 83 and 124 are sed for Classes 1, 2 and 3 for element sbject to bending, respectively. The effective width formla for internal element is modified and shown in Eq. (5): ρ = 1 (5) 2 λ p λ p The design strengths calclated by the proposed class limits and effective width formla are represented by # EC3, and the comparison of the test and nmerical reslts with the design strengths predicted by the modified design rles is smmarized in Table 5 and Fig. 8. The mean vale of the / # EC3 ratio eqals to 1.15 and the COV eqals to It is shown that the modified design rles are less conservative than the crrent EC3 Code predictions. The EC3 modified design rles are considered to be reliable with the reliability index (β 0 ) eqal to Table 6 shows that the experimental and nmerical reslts-to-prediction moment ratio of Classes 1, 2 and 3 sections redced from 1.35 to 1.23, while that of the Class 4 sections redced from 1.17 to 1.09 when the crrent design predictions compared with the modified design predictions. The EC3 modified design rles are considered to be reliable for all for Classes 1, 2, 3 and 4 sections, with the reliability index larger than or eqal to the target vale 18

20 of The modified design rles are validated against the nmerical and experimental data of specimens with D/t ratio ranged from 6.25 to 140, and B/t ratio ranged from 6.25 to Direct Strength ethod The nominal flexral strength ( DS ) in the direct strength method [10] as calclated by Eq. (4) shows that the nominal flexral strength is eqal to a constant vale of yield moment ( DS = y ), when λ l is less than or eqal to (λ l 0.776). In this stdy, based on a total nmber of 81 data with λ l less than or eqal to 0.776, it is shown that the flexral strengths generally decrease linearly as λ l increases. Frthermore, the flexral strength predictions sing the crrent direct strength method are generally conservative. Therefore, it is recommended to modify the crrent direct strength eqation [10] to Eq. (6) as shown below: (( l ) 1) y 1.1 λ + for λ l # DS = crl crl y 1.1 for λ y 1.1 l > (6) y The comparison of the experimental and nmerical data with the design vales calclated by the modified DS in Eq. (6) is shown in Tables 5-6 and Fig. 16. The design strengths calclated by the modified DS are represented by # DS. The mean vale of the / # DS is 1.07 with COV of and the reliability index (β 0 ) of 2.55, as shown in Table 5. It is shown that the modified DS provides more accrate and less scattered predictions compared to the crrent DS for the cold-formed lean dplex stainless steel flexral members. The comparison of the experimental and nmerical data with the design vales for the 81 specimens with λ l less than or eqal to 0.776, and the 99 specimens with λ l greater than 0.776, is smmarized in Table 6. The mean vales of / # DS for specimens with λ l and λ l > are eqal to 1.04 and 1.07 with COV of and 0.142, respectively. The modified DS is considered to be reliable for these two grops of specimens with reliability index (β 0 ) of 2.70 and 2.50, respectively. It is shown that the modified DS provides the most accrate predictions among the design rles discssed earlier. Frthermore, the tedios iterative process is not reqired. The modified direct strength eqation in Eq. (6) is validated against the experimental and nmerical data of specimens with λ l ranged from 0.13 to

21 9. Conclsions Experimental and nmerical investigation on the strctral performance of cold-formed lean dplex stainless steel flexral members has been presented in this paper. A series of for-point bending tests was condcted on sqare and rectanglar hollow sections. A finite element model of flexral members was developed and verified with the experimental reslts. A wide range of parametric stdy was performed sing the verified finite element model. The experimental and nmerical reslts obtained from this stdy and the available data were compared with the design strengths predicted by the American Specification [7], Astralian/New Zealand Standard [8], Eropean Code [9], modified Eropean Code by Gardner and Theofanos [2], direct strength method [10] and continos strength method [6]. It is shown that the continos strength method is capable of providing good prediction for cold-formed lean dplex stainless steel sqare and rectanglar hollow sections flexral members, while the other design rles are qite conservative. odifications on the design rles in the AS/NZS, ASCE, EC3 and DS are proposed and compared with the experimental and nmerical reslts. It is shown that the modified design rles provide more accrate and less scatter predictions than the existing design rles. The modified direct strength method provides the most accrate predictions compared to the other design rles, and this method is relatively simple in calclating the flexral strengths. Therefore, it is recommended that the modified DS is sed in designing cold-formed lean dplex stainless steel flexral members of sqare and rectanglar hollow sections. The modified DS design eqations are capable of prodcing reliable limit state designs when calibrated with resistance factor of 0.9. Acknowledgements The writers are gratefl to STALA Tbe Finland for spplying the test specimens. The research work described in this paper was spported by a grant from the Research Grants Concil of the Hong Kong Special Administrative Region, China (Project No. HKU718612E). 20