Behaviour of stainless steel press-braked channel sections under compression

Journal of Constructional Steel Research 139 (2017) 236–253

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Journal of Constructional Steel Research

Behaviour of stainless steel press-braked channel sections under compression Jelena Dobrić ⁎, Dragan Buđevac, Zlatko Marković, Nina Gluhović University of Belgrade, Faculty of Civil Engineering, B. k. Aleksandra 73, 11000 Belgrade, Serbia

a r t i c l e

i n f o

Article history: Received 21 May 2017 Received in revised form 27 August 2017 Accepted 5 September 2017 Available online xxxx Keywords: Stainless steel Damage plasticity Press-braking Channel section Stub-column test Numerical modelling Cross-section resistance

a b s t r a c t This paper describes an experimental and numerical investigation of stainless steel material response and behaviour of press-braked channel sections under pure axial compression. A material test programme that covers austenitic stainless steel EN 1.4301 was carried out to study the nonlinear stress–strain relationship and changes of basic mechanical properties due to the press-braking processes. The key experimental results were used to estimate the appropriateness of existing analytical material models and to determinate strain-hardening exponents. The validation of recently proposed models for predicting the strength enhancements in cold-formed sections was also performed. Additionally, corresponding Finite Element (FE) models were built for flat and corner coupons to match the tensile test results and to establish the parameters of a ductile damage model in Abaqus. The susceptibility to local buckling of the channel section was determined by stub column tests. The FE model, calibrated and validated against the experiments, was used to perform a parametric study over a wide range of section slenderness. This allowed the quantitative assessments of design procedures stated in Eurocode 3 and American Specifications, and the Continuous Strength Method (CSM). The comparisons between generated data and predicted strengths reveal the conservatism of the Eurocode 3 design method for both non-slender and slender channels. In contrast, the CSM reflects significantly better the nonlinear buckling behaviour of non-slender channels. Although this method gives more accurate results comparing to effective with method employed in Eurocode 3, the slight unsafe predictions were found for slender channels in the intermediate cross-section slenderness. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Stainless steel belongs to the group of contemporary, sustainable and renewable structural materials. It is characterized by high corrosion resistance, superior appearance, nonlinear behaviour, pronounced ductility and material strengthening effects due to cold-working, retention of mechanical properties at high temperatures, good toughness at low temperatures, harmlessness to the natural environment and good recycling potential. Its application in civil engineering has become synonymous with luxurious and architecturally attractive structures, while its utilization is still limited in conventional structures. The reason for this is a very high cost of stainless steel and, sometimes, the lack of recognition of its long-term benefits by design engineers. Numerous comparative studies on the effects of basic material choice on a structure's life-cycle, including initial and maintenance costs, reflected in corrosion protection, fire protection and restoration activities, demonstrate that stainless steel has an economic advantages in a wide variety of ⁎ Corresponding author. E-mail addresses: [email protected] (J. Dobrić), [email protected] (D. Buđevac), [email protected] (Z. Marković), [email protected] (N. Gluhović).

https://doi.org/10.1016/j.jcsr.2017.09.005 0143-974X/© 2017 Elsevier Ltd. All rights reserved.

structural applications. Responding to market demand and the permanent improvement of the manufacturing process, the metalworking industry initiated the production of new, low alloys of stainless steel: the ferritic and lean duplex grades, which contain b1.5% of nickel and can simultaneously ensure primary properties of stainless steel with an economically competitive price [1,2,3,4]. Changing the views within civil engineering and following a global transition to sustainable development, reductions in environmental impact as well as the availability of a wide range of stainless steel products, together with extension of current design codes, represent crucial elements for increasing the use of structural stainless steel. Apart from the numerous similarities in the design of stainless steel and carbon steel structural elements and connections, the differences in mechanical and thermal properties of these two materials require a modification of the carbon steel design rules for their implementation in stainless steel structural design. The part of Eurocode 3 for design of stainless steel structural elements EN 1993-1-4 [5] is, to a great extent, harmonised with the basic Eurocode 3 for design of carbon steel structures EN 1993-1-1 [6], but the disparity in scope and content of these two standards is very considerable. The lack of experimental data in different design fields of

J. Dobrić et al. / Journal of Constructional Steel Research 139 (2017) 236–253

stainless steel structures results in the fact that the current design provisions in EN 1993-1-4 [5] rely solely on assumed analogies with equivalent carbon steel structures. However, such an approach does not fully identify the specific performances of stainless steel that are strongly associated with the overall behaviour of a structural element. In order to complete but also revise the existing regulations in the domain of stainless steel structural design, several research programmes were initiated in recent years. Providing new and reliable data for a better understanding of stainless steel structural behaviour, some of them resulted in innovative design methods and recommendations [7]. Unlike carbon steel, stainless steel exhibits a nonlinear response with gradual yielding. A precise mathematical description of the stainless steel stress–strain relationship is essential for use in research that does not include an experimental part. Various analytical material models exist in literature [8,9,10,11,12] and all of them are based on the Ramberg–Osgood [13] and modified Hill models [14]. Material properties of stainless steel considerably change through the coldworking process. As a response to plastic deformation, the material exhibits a hardening effect that is manifested through an increase in the yield and tensile strength, but also in a decrease in ductility and in the formation of residual stresses. Although stainless steel is widely used in the construction industry as a cold-formed product, the influence of the cold-working process on the improvement of mechanical properties is not analytically covered in EN 1993-1-4 [5]. It is clear that for a material with a high initial cost, full exploitation of its properties in structural design is essential. Several predictive models for determining the strength enhancements of stainless steel were developed in the previous period [15,16,17,18,19]. In case of compressed stainless steel elements, the majority of experimental programmes included coldformed hollow sections, while experimental data on different types of open cross-sections are still limited. Table 1 provides a summary of the gathered database for stainless steel stub column tests under axial compression. The collected database covers a wide range of structural section types, structural materials and numbers of tests.

237

This paper presents the first part of an extensive investigation addressing the load carrying capacity of stainless steel built-up columns with closely spaced chords that was conducted at the Faculty of Civil Engineering, University of Belgrade [37]. The paper aims to provide reliable experimental and numerical data associated with the material behaviour and cross-section resistance. The experimental programme consisted of tensile and compressive tests on coupons extracted from the flat sheet and final press-braked channel section. The generated results enabled the validation of different Ramberg–Osgood material models from literature. In addition, the results from the corner coupon tests were used to evaluate the quality of existing prediction equations for determining the strength enhancements observed in cold-formed sections and to make comparisons between them. The Finite Element (FE) models of the tensile flat and corner coupon tests were built in order to establish parameters of a ductile damage model and predict the full stress–strain relationship. The ultimate resistance and deformation capacity of press-braked channel sections were determined by stub column tests. Numerical modelling was used to simulate stub column tests after which a parametric study was performed in order to generate data over a wide range of section slenderness and identified dominant impact parameters on the failure mode. The results enabled the assessment of the class 3 slenderness limit and the validation of design rules according to EN 1993-1-4 [5], SE/ASCE 8-02 [38] and the Continuous Strength Method (CSM) [39,40]. The overall aim of this study is to acquire further knowledge about material and cross-section structural behaviour as the first important step towards the development of a suitable design procedure for stainless steel built-up columns [37]. 2. Materials This investigation was concentrated on the most commonly used austenitic stainless steel grade EN 1.4301 (X5CrNi18-10). All test specimens were formed from cold-rolled wide strips with nominal

Table 1 Summary of database for stub column tests. Reference

Material

Section type

No. of tests

Johnson and Winter [20] Johnson and Winter [21] Rasmussen and Hancock [22]

1.4301 1.4301 1.4301 1.4301 1.4512 1.4301

Cold-formed hat members Two press-braked back-to-back channels SHS RHS Welded I section SHS RHS CHS

10 16 2 2 2 1 2 3

CHS Press-braked angle Press-braked lipped channel section Press-braked channel section Built-up welded I section SHS CHS SHS RHS SHS RHS CHS SHS RHS SHS RHS OHS SHS RHS Welded I section Welded I section Welded RHS Welded SHS Cold-formed lipped C section

4 12 12 11 16 12 10 4 8 17 16 4 6 3 4 4 6 2 4 4 28

Bredenkamp and van den Berg [23] Talja and Salmi [24] Burgan et al. [25] Young and Hartono [26] Kuwamura [27]

1.4435 1.4541 1.4301 1.4301 1.4318

Young and Liu [28]

1.4301

Gardner and Nethercot [29]

1.4301

Young and Lui [30]

Duplex stainless steel

Gardner et al. [31]

1.4318

Theofanous et al. [32] Huang and Young [33]

1.4401 1.4162

Saliba and Gardner [34] Yuan et al. [35]

1.4162 1.4301 1.4462

Fan et al. [36]

S30408

10

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(a) Tensile coupon specimens

(b) Typical failure mode

Fig. 1. Tensile coupon test.

dimensions 3002 × 1501 × 4 mm. According to the mill certificate, which is agreeable with EN 10088-2 [41], the type of process route and surface finish of strips were denoted with symbol 2B. Experimental tests of the material properties were performed in the Metals laboratory at the Faculty of Technology and Metallurgy, University of Belgrade. In order to study the asymmetry and anisotropy of stainless steel and the influence of cold working on the material response of the final pressbraked sections, two test series were carried out, with a total of 24 coupons. The first series included flat coupons taken from the flat steel sheets. The second series included coupons used from the faces and corner regions of the press-braked channel. All tests were performed in the servo-hydraulic testing machine Instron – 1332, with a capacity of ±100 kN. The elongations of the coupons, having a gauge length L0 of 50 mm, were monitored using a digital extensometer with a measuring range of 100%. All data were recorded using a data acquisition system. 2.1. Properties of the flat sheet material The material tests were performed at room temperature under both tensile and compressive loading. Tensile tests were performed on nine flat coupons: six in the longitudinal direction (labelled as LT) and three in the transverse direction relative to the cold-rolling direction of the steel strip (labelled as TT). The coupons were designed and tested in accordance with EN 10002-1 [42]. The nominal dimensions of the tensile coupons were 280 × 20 mm. A uniform strain rate of 0.001 s−1 was adopted in tensile tests until coupon fracture. Fig.1 shows the

tensile coupons prior to and during testing. The compressive coupon tests were performed on six flat coupons: three in the longitudinal direction (labelled as LC) and three in the transverse direction (labelled as TC). A supporting jig [22], as shown in Fig.2, was specially built for the compression coupon tests. The coupons were rectangular, with nominal dimensions of 20 × 63 mm. Electronic strain gauges having a ±10 mm range were mounted on two laminated faces of the coupons, for the purpose of ensuring pure axial application of the compression load. The coupons were coated with machinery oil, mounted onto the jig and then fastened by manually turning the bolts. In this way, the buckling of the coupon around its minor axis was prevented, and transverse strains caused by the Poisson effect were unconstrained. The height of the clamping device was around 3 mm lower than the coupon height, in order to provide the measured strain range of around 2% during the test. The strain rate in the compressive tests was 0.0005 s−1. The tests provided nominal stress–strain curves and determined the most important material properties: different proof stresses (σ0.01, σ0.05, σ1.0), 0.2% proof stress f0.2, ultimate tensile strength fu, strain corresponding to the ultimate tensile strength εu, total strain at fracture εf and modulus of elasticity E. For the determination of the modulus of elasticity, the method proposed by Afshan et al. [43], which includes the application of linear regression in the initial range of the representative data set, was used. Average values of key mechanical properties of the flat sheet material, including the strain hardening parameters n and m are compared with equivalent mechanical properties provided by the mill certificate document and reported in Table 2. In

(a) Compressive coupon specimens Fig. 2. Compressive coupon test.

(b) Supporting jig

J. Dobrić et al. / Journal of Constructional Steel Research 139 (2017) 236–253

239

Table 2 Material properties from the flat coupons tests and according to mill certificates. Coupon

Thickness (mm)

f0.2 (N/mm2)

σ0.01 (N/mm2)

σ 0.05 (N/mm2)

σ1.0 (N/mm2)

fu (N/mm2)

E (N/mm2)

εu(%)

εf (%)

Strain hardening parameters n

m

LT TT LC TC Mill certificate values (TT)

4.00 4.01 4.01 4.00 4.00

307 311 279 306 329

181 221 165 198 –

243 271 221 256 –

350 347 334 353 364

634 623 – – 634

192,202 195,813 197,667 202,150 –

53 52

62 60 – – 59

6.3 9.8 5.8 7.5 –

2.2 2.2 – – –

mathematical interpretations of the nonlinear behaviour of different stainless steel families, the strain hardening parameters n and m depict the degree of nonlinearity of the stress–strain curve through two different stages. Further information about the above stated strain hardening exponents are given in Section 2.4.

–

material properties at the corner regions. Conversely, press-braking as the cold-forming method does not have any noticeable impact on the improvement of material properties in the flat regions of the crosssection. 2.3. Finite element modelling of tensile tests

2.2. Properties of the press-braked corners In order to analyse the effects of cold-working on the increase of material strength, nine coupons were cut from the cross-section and tested in tension: three coupons were taken from the web (labelled as LTFw), three from the flange (labelled as LTFf) and three from the corner regions (labelled as LTC) of the press-braked channel sections. The nominal dimensions of the cross-section were 100 × 40 × 4 mm and the nominal internal corner radius was 8 mm. The average measured internal corner radius-to-thickness ratio (ri/t) of the coupons is 2.03. A pair of flat surface clamps was used to grip the corner coupons at each end in the tensile test machine jaws. Therefore, the coupon ends were straightened by avoiding shift of the centroid of the necking zone relative to the centre of gravity of the coupon ends. All data were recorded using the data acquisition system. Dimensions, labels and location of the coupons in the cross-section as well as the test setup are depicted in Fig.3. The averaged values of the obtained results are given in Table 3, including the modulus of elasticity, various proof stresses, ultimate tensile strength and strain corresponding to the ultimate tensile strength, strain at fracture and strain hardening parameters. The results of the corner coupon tests confirm that the coldforming process leads to a significant strength enhancement of the

Although the local buckling failure of cross-sections, as the goal of investigation of this paper, is not followed by damage, the definition of tension damage parameters may enable advanced analyses of other structural applications of stainless steel in which destructive failure modes should be considered. Finite Element Analysis (FEA) was performed to attempt replication of standard tensile flat and corner coupon tests. The aims of the numerical investigation were to determine the FE material model's ability to simulate the ductile fracturing of the austenitic stainless steel and to find material parameters critical for fracture related to the mesh type and mesh size of the FE models, using guidelines given by Pavlović et al. [44]. The main feature of ductile fracture is the occurrence of extensive plastic deformations in the local necking region before fracture. In this case, there is a slow propagation of voids with a relatively large dissipation of released strain energy via plastic deformations. The Abaqus/Explict solver [45] with variable, non-uniform mass scaling was employed to obtain the quasi-static solution. Appropriate smoothing was adopted as time-dependent amplitude functions for the displacement controlled failure loading in order to avoid large inertia forces in the quasi-static analysis. The flat tensile coupon was represented by two FE models with different FE types: 3-node triangular shell elements S3 and solid tetrahedral elements C3D4, while the corner

(a) Symbols and locations of coupons in the cross-section

(b) Corner coupons after tensile failure Fig. 3. Tensile corner coupon test.

(c) Test setup

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Table 3 Average measured material properties for the tensile flat and corner coupons in press-braked section. Coupon

t (mm)

ri (mm)

ri/t

f0.2 (N/mm2)

σ0.01 (N/mm2)

σ0.05 (N/mm2)

σ1.0 (N/mm2)

fu (N/mm2)

E (N/mm2)

εu (%)

εf (%)

LTFf LTFw LTC

4.0 4.01 3.98

– – 8.1

– – 2.03

310 313 458

224 209 246

274 266 345

350 354 511

629 642 680

191,033 201,100 193,340

47 49 37

54 56 45

tensile coupon was built only with S3 shell elements. Isotropic plasticity was used with a modulus of elasticity E = 200,000 N/mm2, and Poisson's ratio ν = 0.3. Parameters of damage mechanics for ductile fracture prediction were derived by implementing basic principles of progressive damage and the failure model presented in Abaqus [45]. By using experimental data gained in the tensile tests, three distinct components of material definition were developed: undamaged constitutive behaviour (parts p-n-r′-f′), damage initiation criterion (point n) and damage evolution law (parts n-r-f). These components are highlighted in the curves for the corner coupon shown in Fig. 4a. The characteristic points marked in the curves are: p – onset of plasticity; n – onset of necking; r – rupture point; f – fracture point. The development of damage represents a gradual loss of material strength in local material points. It starts with the damage initiation criterion and does not affect the previous plastic behaviour. This allows for the calibration of the perfectly plastic undamaged response and the damaged-plastic response of the material separately. In the context of damage mechanics, the damaged curve can be viewed as a degraded response of the curve representing perfect plasticity, beyond the onset of necking that is followed by fictive material behaviour in the absence of damage. Undamaged and damaged material responses were evolved using the incremental procedure described in detail in the investigation by Pavlović et al. [44]. This procedure is based on an engineering approach that considers the localization of plasticity in the necking zone through a detailed assessment of several parameters for ductile damage calculation. The procedure implies the processing of raw experimental data in the i-th stage of the loading sequence. Hence, the set of identified damage parameters, summarized in Table 4, was evaluated through the calibration and validation of numerical results against corresponding experimental data for both flat and corner coupons. Firstly, the 0.01% proof strength σ0.01 was used for the onset of plasticity. The damage initiation for the ductile criterion is used to predict the onset of damage caused by formation, growth and coalescence of voids inside the material. The growth of voids is followed by the reduction of the coupon's effective cross-section, since only the material between the voids has resistance under an external tensile load. The longitudinal plastic strains resulting from the increased material elongation are concentrated in the walls between the voids. This effect causes the necking region of the coupon material. After the onset of necking, further deformations are localized inside this region, while the outside material recovers elasticity. The growth of voids and fracture of the material is governed by the increasing plastic strains and the stress triaxiality ratio. The stress triaxiality ratio θ is defined as the ratio of hydrostatic stress to the von Mises equivalent stress. In the localized necking zone, the equivalent stresses have approximately the same values, but its central region is under higher hydrostatic stress and, consequently, a higher value of stress triaxiality than the outer parts of the region. Thus, the ductile fracture due to growth and coalescence of voids is initiated in the central region of the necking zone. The ductile criterion provided in Abaqus [45] assumes that the equivalent plastic strain at the onset of damage εpl 0 is a function of the stress triaxiality ratio and strain rate. Based on outcomes of Trattnig et al. [46] and Rice and Tracey [47], Pavlović et al. [44] proposed an exponential dependency of the equivalent plastic strain at the onset of damage εpl 0 on the uniaxial

Strain hardening parameters n

m

9.5 8.3 4.9

2.2 2.2 2.5

true plastic strain at the onset of damage εpl n and triaxiality, as given by Eq. (1) for the case of uniaxial tension. pl εpl 0 ðθÞ ¼ ε n exp½−1:5ðθ−1=3Þ

ð1Þ

Once the strain εpl n was defined from experimental results, the damage initiation criterion was developed according to Eq. (1) as shown in Fig. 4b. The hardening part of the plasticity curve for the undamaged material response up to point n, was defined by converting the experimental stress–strain curve to the true stress–strain curve:
 εi ¼ ln 1 þ ε nom i
nom

σi ¼ σi

 nom

1 þ εi

ð2Þ ð3Þ

are given by the elementary Eq. (4). where nominal strains εnom i ¼ Δli =li ; i b n ε nom i

ð4Þ

The second part of the undamaged curve, beyond point n, is developed assuming a perfectly plastic material response, as established in Eqs. (5) and (6):  
 nom − ln 1 þ εnom ε pl ; i≥n p i ≅ ln 1 þ ε i

ð5Þ


 σ i ¼ σ nom 1 þ εnom ; i≥n n i

ð6Þ

Eqs. (2) and (3) are also used to represent the damaged material response in the stress domain after the onset of necking, where input nominal strains are obtained using Eq. (7): ¼ εnom ε nom i i‐1 þ ðΔli −Δli−1 Þ=li ; i ≥ n

ð7Þ

In the aforementioned Eqs. (4) and (7), li is the variable gauge length at integration point “i” defined by Eq. (8) as a function of elongation Δli: ( li ¼

0

l ; ib n  0 loc 0 α l þ l −l ½ðΔli −Δln Þ=ðΔlr −Δln Þ L ; i ≥ n

ð8Þ

where l0 is the initial gauge length, lloc is the average length of the necking zone and αLis the localization rate factor given in Table 4. The damage evolution law based on the Lemaitre damage model [48] describes the rate of degradation of the material stiffness in the region of strain localization after the damage initiation criterion is reached and represents the softening part of the damaged stress–strain material response. The scalar damage variable D is adopted to describe the relationship between the undamaged and damaged material response by assuming an equal distribution of microvoids in all directions and strain equivalence: σ ¼ σ=ð1−DÞ

ð9Þ

J. Dobrić et al. / Journal of Constructional Steel Research 139 (2017) 236–253

(Dn = 0) n 900 Dr

r

800

True stress (MPa)

r

(Dr = Dcr)

700

1.00 corner coupon shell flat coupon shell flat coupon solid

0.80

uniaxial tension

Equivalent plastc strain at the onset of damage (-)

f' r'

1000

241

0.60 0.40 0.20

pl n

0.00 -1.00 -0.33 0.33 1.00 1.67 2.33 3.00 3.67 4.33

600

Stress triaxiality (-) 500

(b) damage initiation criterion Damage variable (-)

400 300

E

p 200 100 pl n

0 0

0.05

0.1

pl f

f (Df = 1)

0.15 0.2 0.25 0.3 0.35 0.4 Localized true plastic strain (-)

0.45

0.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

n

Dr

r

corner coupon shell flat coupon shell flat coupon solid f 0.0

0.1

(a) plasticity curve

0.2 0.3 0.4 0.5 Plastic displacement (mm)

0.6

0.7

(c) damage evolution law

Fig. 4. Plasticity and ductile damage parameters for flat and corner stainless steel coupons.

where σ is the undamaged, effective stress that would exist in the material in the absence of damage, and σ is the true stress. At the point of initiation of material damage that corresponds to the onset of necking of the test coupon, the damage variable is D = 0. As the element is deformed and the strain value increases, the increase of D is followed by a reduction of the initial material modulus E to the value of (1 − D) ∙ E. When the damage reaches a critical value Dcr, a macro crack is initiated. Critical damage is in the range of 0.2 to 0.8, depending on the material type [48]. At the fracture point with a total degradation of stiffness, the damage variable is D = 1. Based on Eq. (9), the damage variable Di is mathematically interpreted as the difference between the undamaged and damaged material model:  Di ¼

1−σ i =σ i ; n ≤ i b r 1; i ¼ f

ð10Þ

and was input in the FE material model in tabular form. Values of upl i were obtained using Eq. (11).     pl pl pl pl pl upl i ¼ u f ε i −ε n = ε f −ε n ; i ≥ n

ð11Þ

Damage evolution laws are presented in Fig. 4c. The damage evolution is defined as mesh independent because it does not directly use any parameters that are dependent on the element size in the model. Hence, the total equivalent plastic displacement at fracture upl is defined f as a product of the plastic strain accumulated during damage, the characteristic element length Lchar and the factor of FE size λs:   upl ¼ λS Lchar εpl −ε pl n f f

ð12Þ

The characteristic element length depends on the size and type of

Note that the damage eccentricity factor αD, introduced in the expression for the damage variable Di by Pavlović et al. [44], is not considered in this framework. The damage evolution law was specified as a dependency of the damage variable Di on the equivalent plastic dis-

FE:

placement upl i at the integration point “i” after the onset of damage,

where LE is the element size and λE is the element type factor.

Lchar ¼ λE LE

ð13Þ

Table 4 Identified parameters for ductile damage calculation. Material

Damage

Element type

Austenitic stainless steel 1.4301

Initiation strain

Type

εpl n Flat coupon Flat coupon Corner coupon

0.351 0.351 0.273

Shell S3 Solid C3D4 Shell S3

Element size

Localization

Factor

Size (mm)

Factor

Length (mm)

Factor

λE

LE

λs

lloc

αL

1 1 1

2.1 2.1 2.1

2.1 1.4 3.0

5 5 5

0.35 0.35 0.35

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700

600

Nominal stress (MPa)

500 Corner coupon test Corner coupon FE shell S3 Flat coupon test Flat coupon FE shell S3 Flat coupon FE solid C3D4

400

300

l0 l loc

200

100

0

0

5

10

15

20

25 30 35 40 45 Nominal strain (%)

50

55

60

65

(a) Experimental and nominal FEA curves

(b) experimental and FEA

(c) experimental and FEA

fracture of flat coupon

fracture of corner coupon

Fig. 5. Experimental and numerical tensile test results.

In Fig. 5a, the obtained numerical stress–strain material curves are compared with corresponding experimental data. It can be noted that experimental curves are accurately reproduced by FE curves over the whole strain range. Fractured shapes and damage variable values are shown in Fig. 5b and Fig. 5c for flat and corner coupons, respectively. Besides, Table 5 provides comparisons between key material properties obtained experimentally and using FEA. From Table 5, it can be seen that close agreement was achieved between all individual material parameters.

which is defined in a new coordinate system with the onset point at 0.2% proof stress, ensures the continuity of the curve slope through a different strain hardening exponent m. The initial modulus is the tangent modulus of the stress–strain curve at the 0.2% proof stress E0.2, and ε0.2 is the total strain at the 0.2% proof stress.

2.4. Evaluation of analytical two-stage material models

ε¼

The experimental data were used in order to validate two various two-stage Ramberg–Osgood predictive models: the material model provided in Annex C of EN 1993-1-4 [5] that is based on the model proposed by Rasmussen [9] and the model based on recent research by Arrayago et al. [12]. The accuracy of these models is assessed through a comparison of experimental and predicted values of key material parameters. Both analytical models are interpreted using the same principal expressions given by Eqs. (14) and (15). Eq. (14), which is presented by the well-known Ramberg–Osgood's expression, depicts the initial stress–strain relationship in the range up to f0.2. The first strain hardening exponent n is valid in this stress range. The second part of the stress– strain curve between the 0.2% proof stress and the ultimate tensile strength is described by Eq. (15). This part of the analytical curve,

ε¼

  σ σ n for σ b f 0:2 þ 0:002 E f 0:2  m σ −f 0:2 σ −f 0:2 þ εu þ ε0:2 E0:2 f u − f 0:2

ð14Þ

for σ N f 0:2

ð15Þ

The main discrepancy between these two analytical models is in the different predictive expressions for determining the strain hardening exponents n and m. According to EN 1993-1-4 [5], the following equations may be used for calculating these exponents:

n¼

ln ð20Þ   f 0:2 ln σ 0:01

m ¼ 1 þ 3:5

ð16Þ

f 0:2 fu

ð17Þ

Table 5 Comparison between experimental and numerical material properties. Coupon

f0.2 (N/mm2)

σ0.05 (N/mm2)

fu (N/mm2)

E (N/mm2)

εu (%)

εf (%)

Strain hardening parameters n

m

Exp. flat coupon FE flat coupon (shell S3) Exp.-to-FE ratio FE flat coupon (solid C3D4) Exp.-to-FE ratio Exp. corner coupon FE corner coupon (shell S3) Exp.-to-FE ratio

296 303 0.98 304 0.97 461 465 0.99

244 234 1.04 239 1.02 341 355 0.96

641 657 0.98 651 0.98 679 684 0.99

201,969 190,040 1.06 189,050 1.07 193,830 188,010 1.03

54 59 0.92 56 0.96 38 42 0.90

63 62 1.02 64 0.98 44 45 0.98

7.9 5.6 1.41 5.8 1.36 5.0 5.1 0.98

2.5 2.3 1.09 2.3 1.09 2.4 3.0 0.80

J. Dobrić et al. / Journal of Constructional Steel Research 139 (2017) 236–253

243

Table 6 Assessment of ultimate strength fu, ultimate strain εu and strain hardening exponents n and m. EN 1993-1-4, Annex C [5]; Arrayago, Real and Gardner's model [12]

LT

Mean CoV (%) Mean CoV (%) Mean CoV (%) Mean CoV (%) Mean CoV (%)

TT LTFf LTFw LTC

εu,exp./εu,pred

nexp/npred

mexp/mpred

nexp/npred

mexp/mpred

1.02 2.1 0.99 2.9 1.02 1.3 1.00 2.6 0.95 5.4

0.97 4.0 0.97 4.7 1.08 1.9 1.05 2.2 0.88 3.2

1.09 10.1 1.11 10.7 1.02 6.5 1.11 6.1 1.03 3.4

0.82 9.7 0.78 3.4 0.81 8.0 0.81 3.0 0.75 4.2

1.04 7.9 0.97 5.4 0.83 2.4 0.96 1.7 1.01 7.9

0.94 9.6 0.90 3.4 0.93 8.0 0.93 3.0 0.87 4.5

ln ð4Þ   f 0:2 ln σ 0:05

m ¼ 1 þ 2:8

ð18Þ

f 0:2 fu

ð19Þ

On the other hand, in case of austenitic stainless steel, the predictive expressions for determining the ultimate strength fu and the ultimate strain εu are the same in both models, and are specified by Eqs. (20) and (21), respectively. f 0:2 f ¼ 0:20 þ 185 0:2 fu E

ð20Þ

f 0:2 fu

ð21Þ

ε u ¼ 1−

In order to obtain the experimental strain hardening exponents n and m, the computer programme based on the least squares regression method was developed. Values of n and m were determinate so that every experimental stress–strain curve closely matches the predictive curve using Eqs. (14) and (15), by minimizing the error between them. The pronounced nonlinearity of the curve caused the focus on the initial strain part of the curve up to 10% in order to achieve negligible deviations in this structural domain. For the calculation of the first strain

hardening exponent n all experimental curves were analysed. However, when considering the second strain hardening exponent m, only the tensile curves were used in the analysis. The comparisons of the two analytical material models provided in Annex C of EN 1993–1-4 [5] and proposed by Arrayago et al. [12], in terms of experimental-topredicted ratios of the strain hardening exponents nexp/npred and mexp/ mpred, including the mean and coefficient of variation (CoV), are presented in Table 6, respectively. The assessment of predictive expressions (20) and (21) for ultimate strength and the ultimate strain is also presented in Table 6. It can be seen that the new predictive eqs. (18) and (19) more accurately estimate the strain hardening parameters than Eqs. (16) and (17) given in existing EN 1993-1-4 [5]. The results show a less accurate prediction of key material parameters when the properties of corner coupons are considered. The larger deviations between experimental and predicted values of the ultimate strain εu and the second hardening exponent m are observed for both analytical models. The mean value of the ratio εu,exp./εu,pred is 0.88 and CoV is 3.2%. The analytical model given in EN 1993-1-4 [5] results in mexp/mpred ratio with mean value of 0.75 and CoV equalling 4.2%, while Arrayago's model [12] gives mean value of 0.87 and CoV of 4.5%. The analytical stress–strain curves were developed using corresponding predictive expressions according to aforementioned models [5,12], respectively and measured parameters for each individual coupon: modulus of elasticity E, 0.01% proof stress σ0.01, 0.05% proof stress σ0.05 and 0.2% proof stress f0.2. The comparison between experimental curves and analytical, predictive curves in the strain range up to 10% is shown in Fig.6 where “EN 1993-1-4” is a label for the model provided in Annex C of EN 1993–1-4 [5] and “A-R-G” is a label for the Arrayago, Real and Gardner analytical material model [12].

500

500

400

400

Stress (MPa)

Stress (MPa)

Arrayago, Real and Gardner's model [12]

fu,exp./fu,pred

Based on the analysis of stainless steel material behaviour through the comprehensive experimental database, Arrayago et al. [12] recommend the following modification of Eqs. (16) and (17): n¼

EN 1993-1-4, Annex C [5]

300 R-O material model

200

A-R-G material model EN 1993-1-4 material model

100

300 R-O material model A-R-G material model EN1993-1-4 material model TT2

200 100

LT2

0

0 0.0

2.0

4.0 6.0 Strain (%)

8.0

(a) Longitudinal tensile coupon LT2

10.0

0.0

2.0

4.0 6.0 Strain (%)

8.0

10.0

(b) Transverse tensile coupon TT2

Fig. 6. Comparison between the experimental and predictive stress–strain curves in the strain range up to 10%. (a) Longitudinal tensile coupon LT2; (b) Transverse tensile coupon TT2.

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Table 7 Comparison of the predictive models and test data for the 0.2% proof strength of corner regions.

where A is the gross cross-sectional area of the press-braked section and Ac,pb is the cross-sectional area of the corner region.

Cruise and Gardner [17]

Rossi, Afshan and Gardner [19]

LTC

f0.2,c,exp./ f0.2,c,pred

f0.2,c,exp./ f0.2,c,pred

3. Cross-section resistance

Mean CoV (%)

0.97 0.57

1.08 0.57

3.1. Stub column tests

2.5. Evaluation of models for predicting material strength increases The average value of the 0.2% proof strength f0.2,c, obtained by testing the corner coupons, was compared with the results of two predictive models: Cruise and Gardner [17] and the more recent model developed by Rossi et al. [19]. Cruise and Gardner [17] performed a comprehensive investigation of the behaviour of cold-formed stainless steel sections from a variety of fabrication processes and proposed predictive expressions to determine strength enhancements in the flat faces and corner regions of the cold-formed cross-sections. The authors recalibrated the Ashraf et al. [16] expression based on additional stainless steel experimental data and proposed a simple power model to predict the 0.2% proof stress of the corners in press-braked sections. Besides, it was indicated that the corner strength enhancement is confined only to the corner region of press-braked sections. Rossi et al. [19] gathered the generated and all available experimental data, which cover a wide range of stainless steel grades, and expanded the scope of research to carbon steel cold-formed structural sections. Through a thorough examination of the material response during the entire cold-working process of structural elements, the authors defined an innovative and unique analytical approach for predicting the increased 0.2% proof stress around stainless steel cold-formed sections. Starting from the assumption of a linear strain distribution over the material thickness, the authors developed predictive analytical expressions that employ the through-thickness averaged plastic strain for the flat faces and corner regions, respectively. The proposed model is based on the inverted compound Ramberg–Osgood material model [10] which allows for the prediction of stresses that correspond to associated plastic strains which are the result of the fabrication process. In order to assess the applicability of these two above stated predictive models, the average value of the 0.2% proof strength for longitudinal tensile flat coupon tests for flat sheet material f0.2,v = 307.3 N/mm2 was used for calculation, considering that this value is lower than the value provided by the mill certificate f0.2,mill. The predicted strength-to-test strength ratios of the corner region f0.2,c,pred/ f0.2,c,exp. are presented in Table 7. Analysis of the results shows that for the corner regions of pressbraked sections, both models offer accurate predictions of the tensile 0.2% proof strength. The power material model proposed by Cruise and Gardner [17] shows a higher degree of agreement in comparison with the measured strength, but with an over-prediction. On the other hand, the linear hardening material model developed by Rossi et al. [19] gives safer predictions and more reliability then the power material model. It should be noted that Rossi's model [19] introduces a factor of 0.85 that includes stainless steel asymmetry effects and required reliability level associated with the prediction of material strength. Assuming the distribution of cross-section material strengthening according to the patterns defined by Cruise and Gardner [17], Rossi et al. [19] developed an expression that provides the average enhanced 0.2% proof stress for press-braked sections. In the case of observed press-braked cross-section (see Fig.3) the enhanced average yield strength can be calculated as:
f 0:2;section ¼



 f 0:2;c Ac;pb þ f 0:2;v A−Ac;pb ¼ 336:3 N=mm2 A

ð22Þ

A total of four press-braked channel section specimens were tested in pure axial compression to assess their structural behaviour and confirm the class 3 limit for the compressed cross-sections. The tests were carried out in the Materials and Structures laboratory at the Faculty of Civil Engineering, University of Belgrade. The steel sheets were cut into strips and press-braked into channel sections having nominal dimensions 100 × 40 × 4 mm and a nominal internal corner radius of 8 mm. The nominal length of specimens was 300 mm. This dimension satisfied the following two conditions: (1) according to Annex A, EN 1993-1-3 [49], the specimen should have a length of at least three times the width of the widest cross-section dimension; (2) according to EN 1993-1-4 [5], the non-dimensional slenderness λ of the specimen should be lower than the limit slenderness λ0 ¼ 0:4in order to prevent flexural buckling failure. The longitudinal axis of the specimens was oriented parallel to the direction of the rolling of the flat strip. The ends of specimens were milled flat, so as to achieve full contact with the testing machine. The cross-section dimensions of stub column specimens were measured using a Vernier calliper at four randomly selected cross-sections along the section length, while the internal corner radius of crosssections was measured using radius gauges. Based on the measured dimensions and density of stainless steel (ρ = 7900 kg/m3), a theoretical mass of each specimen was determined. The theoretical values were compared with the actual masses of each individual specimen in order to assess of accuracy of measured geometric dimensions, primarily of internal corner radius. The obtained deviations were not N 1% which verified the measuring procedure's accuracy. The nominal cross-section geometry is defined in Fig.7a. Table 8 presents the average measured dimensions of the specimens. The designation letters of the specimens denote the type of the test (SC – Stub Column) while the numerical designations refer to the ordinal specimen number. The measuring of initial geometrical imperfections and residual stresses was not included in the experiment. The compressive load was applied using a strain-controlled Amsler hydraulic testing machine, with a capacity of 3500 kN. The strain accretion rate was 0.001 s− 1. The end plates of the testing machine were fixed flat and parallel. End shortening of the specimens was continuously monitored during the test by LVDT sensors mounted on four points, and placed on the upper plate of the testing machine. Six linear electrical strain gauges were affixed to each specimen at its midheight. The strain gauges were used for the purpose of alignment and comparison with the end shortening data from the LVDTs (see Fig.7b). All experimental results—load, displacement and strain values—were recorded in one-second intervals on the data acquisition device MGC+. The test setup of stub column specimens is shown in Fig.7c. The key experimental results are summarized in The compressive load was applied using a strain-controlled Amsler hydraulic testing machine, with a capacity of 3500 kN. The strain accretion rate was 0.001 s−1. The end plates of the testing machine were fixed flat and parallel. End shortening of the specimens was continuously monitored during the test by LVDT sensors mounted on four points, and placed on the upper plate of the testing machine. Six linear electrical strain gauges were affixed to each specimen at its mid-height. The strain gauges were used for the purpose of alignment and comparison with the end shortening data from the LVDTs (see Fig.7b). All experimental results—load, displacement and strain values—were recorded in one-

J. Dobrić et al. / Journal of Constructional Steel Research 139 (2017) 236–253

245

b1=b2=40 mm h=100 mm t=4 mm ri=8 mm

(a) Nominal and measured cross-section

(b) Instrumentation configurations

(c) Test set-up

dimensions of the specimens Fig. 7. Test setup for stub column specimens.

second intervals on the data acquisition device MGC+. The test setup of stub column specimens is shown in Fig.7c. Table 9, where Nc,u is the ultimate load and δu is the end shortening at ultimate load. Based on experimental results, the stress–strain curves for each individual specimen were developed. For each test rate, the longitudinal compressive stresses were determined as a ratio of the compression load to the measured cross-sectional area, while the strains were defined by the ratio of longitudinal displacements δ to the specimen length L. Fig.8 compares the average stress–strain curve obtained by longitudinal compression coupon tests of the flat sheet material with the average stress–strain curve obtained by stub column tests, while the particular curves for each stub column specimen are shown in the background. The following parameters are highlighted in Fig.8: σlb is the average value of the local buckling stress obtained in stub column tests, εlb is the average strain value which corresponds to the local buckling stress σlb and f0.2 is the average value of the 0.2% proof stress obtained in the compressive coupon tests of flat material. The ultimate crosssection failure of specimens was caused by local buckling. Buckling was localized in the middle part of the specimen height and characterized by the concave shape of web buckling and the convex shape of flange buckling. The scatter of experimental data is negligible; CoV for ultimate loads is only 2.4%. The stress–strain relationship gained in the stub column test is pronouncedly nonlinear and indicates a considerable degree of material strengthening caused by increasing plastic strains (see Fig.8). The ratio of the local buckling stress σlb-to-the 0.2% proof stress f0.2 is 1.36, which clearly demonstrates that the cross-section

buckling occurred in the nonlinear stress domain, after reaching the conventional yield strength. 3.2. Finite element modelling of stub column test The numerical simulation was performed by using the Abaqus FE software package [45] with the aim to replicate the full experimental load–deformation curve and to investigate the influences of various input parameters such as the material stress–strain relationship, enhanced corner properties, initial geometric imperfections and residual stresses on the structural cross-section response. A geometrically and materially nonlinear analysis was performed using the modified Riks algorithm based on a static incremental solver. The FE model was built according to nominal geometrical dimensions of stub column specimens and meshed with S4R shell elements with a size of 5 mm. In the FE model, the material properties obtained from respective flat and corner coupon tests were assigned to the flat and corner parts of a crosssection. Experimental stress–strain curves were transformed to stresses and plastic logarithmic strains for input in the Abaqus material model. Boundary conditions for the support reference points were set to allow the model to behave as a fully restrained member: the ends of the FE model were fixed against all degrees of freedom except for the vertical displacement at the loaded end where the failure loading was applied as controlled displacement. The FE model was analysed with and without initial geometrical imperfections. Three imperfection amplitudes were considered: (1) 1/100 of the cross-section thickness; (2) according to modified Dawson and Walker predictive model [50,51], as given by Eq. (23) where σcr,min is the minimum elastic

Table 8 Measured dimensions and masses of stub column specimens. Specimen

Specimen length L (mm)

Web depth h (mm)

Width of upper flange b1 (mm)

Width of bottom flange b2 (mm)

Thickness t (mm)

Internal radius r (mm)

Cross-sectional area A (mm2)

Mass of specimen M (g)

SC1 SC2 SC3 SC4 Mean CoV (%)

300.1 299.8 300.0 299.9 300.0 b0.1

100.2 100.0 100.0 100.1 100.1 0.1

40.0 40.1 40.0 40.0 40.0 0.1

40.1 39.9 40.2 39.9 40.0 0.4

4.0 4.0 4.1 4.1 4.1 1.4

8.0 8.1 8.0 7.9 8.0 1.0

654.9 653.3 669.8 669.0 661.8 1.3

1555.3 1551.7 1559.5 1555.9 1555.6 0.2

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the final product. In addition, the research outcomes [52,53,54] indicated that membrane residual stresses induced by press-braking have a low impact on the buckling response of the cross-section. Thus, the residual stress distribution was not explicitly taken into account in the FE model.

Table 9 Summary of stub column test results. Specimen

Ultimate load Nc,u (kN)

End shortening at ultimate load δu (mm)

SC1 SC2 SC3 SC4 Mean CoV (%)

244.4 241.6 255.0 249.2 247.6 2.4

2.02 2.19 2.43 2.26 2.23 7.5

3.3. Validation of the FE model The created FE model was validated by comparing the full load– deformation curves obtained from stub column tests with corresponding FE curve (see Fig.9a). Excellent matching was observed between the experimental results and the FE predictions of the cross-section response over the whole stress domain. Besides, the local buckling failure mode from the FEA was in good agreement with that from the experiment, as displayed in Fig.9b. Additionally, the ultimate loads and end shortenings at ultimate loads from the FEA, associated with considered imperfection amplitudes, were normalised by the corresponding experimental results from the stub column test and summarized in Table 10. It can be observed that the FE ultimate loads Nc,u, FEA are less sensitive to the changes of the imperfection amplitude than the corresponding deformation

buckling stress of all cross-section parts and f0.2 is the 0.2% proof stress; and (3) a selected value in order to achieve the closest agreement between experimental and FE results.
 ϖ D&W ¼ 0:023t f 0:2 =σ cr; min

ð23Þ

The distribution of normalised imperfections for the first critical buckling mode obtained in the eigenvalue elastic buckling analysis was multiplied with considered amplitudes and introduced in the subsequent nonlinear analysis. Bending residual stresses were reintroduced in coupon tests and consequently implicitly included in the experimental stress–strain curve of the material extracted from

lb=Nc,u/A=378.7

400

MPa

350

(MPa)

250

Stress

300

150

f0.2=279.2 MPa SC1 SC2 SC3 SC4 Average Stub Column test Average Longitudinal Compression test

200

100 E

50

lb= u/L=0.74%

0 0.00

0.50

1.00 Strain (%)

1.50

2.00

Fig. 8. Average stress-strain curves for the stub column test and longitudinal compression of flat coupons.

250

Axial load (kN)

200 150 100

SC1 SC2 SC3 SC4 FEA

50 0 0.0

2.0

4.0 6.0 End shortening (mm)

(a) Experimental and FE load–end shortening curves

8.0

(b) Experimental and numerical failure mode

Fig. 9. Comparison of experimental and FE results for stub column specimen.

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247

Table 10 Comparison of the stub column test results with FE results for varying imperfection amplitudes. ϖ = t/100 = 0.04 mm

Specimen

Without imperfection

Modified Dawson and Walker model ϖ = 0.038 mm

Specimen

Nc,u,FE/Nc,u,exp

δu,FE/δu,exp

Nc,u,FE/Nc,u,exp

δu,FE/δu,exp

Nc,u,FE/Nc,u,exp

δu,FE/δu,exp

Nc,u,FE/Nc,u,exp

δu,FE/δu,exp

SC1 SC2 SC3 SC4 Mean

1.03 1.04 0.99 1.01 1.02

1.44 1.33 1.20 1.29 1.32

1.02 1.03 0.98 1.00 1.01

1.27 1.17 1.06 1.14 1.16

1.02 1.03 0.97 1.00 1.00

1.18 1.09 0.98 1.05 1.08

1.01 1.03 0.97 0.99 1.00

1.17 1.08 0.97 1.05 1.07

parameters δu,FE. The best agreement between the experimental and FE results was achieved for the selected amplitude value of 0.06 mm. This value is about 58% higher than the imperfection amplitude predicted by the modified Dawson and Walker expression which is usually used in FEA of cold-formed stainless steel structural elements. 3.4. FE parametric study The FE parametric study was conducted with reference to a wideranging set of channel section slenderness to simulate different levels of the sections' structural behaviour and assess the applicability of existing design methods for cross-section resistance given in EN 1993–1-4 [5], SE/ASCE 8–02 [38], as well as the CSM [39,40]. The study encompasses FE models calibrated and validated against stub-column tests, considering variability in the depth and width of channel section: the nominal thickness of 4 mm and nominal internal corner radius of 8 mm were not changed. The cross-section sizes were chosen to cover all four cross-section classes in accordance with EN 1993-1-4 [5], focusing on the range of the cross-section width-todepth ratio: 0.3 ≤ b / h ≤ 0.55. Thus the results of local web-to-flange interaction were considered. A total of 103 parametric FE simulations were made; Fig.10 shows the analysed range of the width-tothickness ratios of internal elements and outstand flanges of the channel section. The overall length of all FE models was set equal to three times the largest cross-section dimension. Note that the initial local imperfection amplitude of ω = 0.036 t(f0.2 / σcr,min) was applied to all FE models. The full cross-section slenderness λp was obtained using the software CUFSM [55]. 3.4.1. Results and discussions In Fig.11, the numerical ultimate loads obtained in the FE parametric study are normalised with yield loads taken as the product of cross-

Selected value ϖ ¼ 0:06 mm ≅0:036tðf 0:2 =σ cr; min Þ

sectional area and enhanced average yield strength (see Eq. (22)). There is a continuous decrease in the transition from full to reduced compressive capacities of channel sections, with a transition point of approximately λp ¼ 0:68.An abrupt loss of full section capacity can be seen in the range of intermediate cross-section slenderness λp = 0.2–1.2, subsequently followed by a gradual strength reduction of the slender channel sections. The significant drop of full section strength by varying section slenderness in the intermediate domain is simultaneously caused by impacts of initial geometric imperfections, material nonlinearity and the level of element interaction that depends upon the stiffness of adjacent parts of the channel section. 3.5. Evaluation of design predictive models for cross-section resistance The accuracy of the codified design approaches for cross-section resistance given in the American Specification SEI/ASCE 8-02 [38] and Eurocode EN 1993-1-4 [5] was estimated by comparing the generated experimental and numerical ultimate buckling loads Nc,u with corresponding unfactored predicted strengths. The CSM [39,40] was also assessed herein both for stocky and slender channel sections. The predicted design values were calculated using the nominal cross-section dimensions of specimens and FE models and, in accordance with the corresponding design method, the measured material properties obtained in the longitudinal compression coupon test (for ASCE resistance calculation) and the enhanced average yield strength (for Eurocode 3 and CSM resistance calculation), to account for higher strengths in the corner regions of the cross-sections. The CSM predicted strengths were based on the full cross-section slenderness λp . The predicted values of the critical buckling stresses in the nonlinear stress domain fcr,inel according to SEI/ASCE 8-02 [38] were evaluated for stocky cross-

1.4 FE data

70

1.2

Nc,u / Nc,u,pred

Flange slenderness b/(t )

60 50 40 30 20

1.0 0.8 0.6 0.4

10 0.2

0

0.0

0 10 20 30 40 50 60 70 80 90 100 110 120 Web slenderness (h-2t)/(t ) Fig. 10. Range of plate element slenderness in parametric study.

0.4 0.8 1.2 1.6 Cross-section slenderness

2.0

2.4

Fig. 11. Transition of normalised ultimate buckling loads over the cross-section slenderness range.

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sections through the comparison with experimental and FE local buckling stresses σlb. Besides, the FE and experimental data enabled the accuracy assessment of the Class 3 width-to-thickness limits for internal and outstand elements in compression and the corresponding effective width formulas for Class 4, as given in EN 1993-1-4 [5]. 3.5.1. Cross-section resistance according to SEI/ASCE 8-02 The elementary formula for determining the cross-section resistance stated in clause 3.4.2 of SEI/ASCE 8-02 [38] takes into account the effect of nonlinear behaviour of stainless steel as a function of cross-sectional area A and permissible compressive stress fb. According to clause 3.3.1, the permissible compressive stress fb directly depends on the critical buckling stress fcr,inel whose predictive formula reflects the outcomes of research [56,57]. In order to consider inelastic buckling, the plastic reduction coefficient η is introduced to modify the basic design formula for elastic buckling: f cr;inel ¼ k

w2 π2 E
 η ¼ f cr;el η 12 1−ν 2 t

ð24Þ

where w is the width of the flat part of the cross-section, up to where the rounded part of the cold-formed section starts (in Eurocode 3 denoted as b), t is the cross-section thickness, ν (= 0.3) is the Poisson's ratio, E is the modulus of elasticity and k is the buckling coefficient (in Eurocode 3 denoted as kσ). The plastic reduction coefficient is η = (Et/ E)0.5 for stiffened compressed cross-section elements and η = Es/E for unstiffened compressed cross-section elements. The secant modulus Es, defined as the ratio of stress-to-strain for the stress value which corresponds to the critical buckling stress, can be determined using the analytical expression B-1, Appendix B of SEI/ASCE 8-02 [38]. The tangent modulus Et, defined as the tangent slope on the stress–strain curve at the stress value which corresponds to the critical buckling stress, can be determined using the analytical expression B-2, Appendix B of SEI/ ASCE 8-02 [38]. Because of the nonlinear nature of the stress–strain curve, the critical buckling stress should be determined through an iterative process. However, it should be noted that the ASCE design method for cross-section resistance sets the 0.2% proof stress f0.2 (denoted as specified yield strength fy) as an upper limit of the permissible compressive stress fb, without considering the pronounced strain hardening exhibited by austenitic stainless steel. Thus by following guidelines in clause 3.4, the design formula for cross-section resistance, expressed as the product of effective cross-sectional area Ae and the specified yield strength fy, becomes identical to that in Eurocode 3 [5]. The concept of effective width given in clause 2.2 is employed to determinate the cross-section effectiveness. The reduction factor ρ, which reflects loss of stiffness and a redistribution of compressive stresses along plate elements, is given as the following: ρ¼

1−0:22=λ λ

ð25Þ

where λ is slenderness of the plate element (in Eurocode 3 denoted as λp ): 1:052 w λ ¼ pffiffiffi k t

sffiffiffiffiffi fy E

ð26Þ

In Eq. (26), the flat width of the internal element is w = d − 2 t − 2 ∙ ri and the flat width of the outstand element is w = b − t − ri. It should be also emphasized that the buckling coefficients k = 4.0 for an internal compressed element and to k = 0.5 for an outstand compressed element are specified in SEI/ASCE 8-02 [38]. 3.5.2. Cross-section resistance according to EN 1993-1-4 For the purpose of harmonisation with the basic Eurocode for structural carbon steel design [6], in the current Eurocode for stainless steel

[5], local buckling is accounted for by using the concept of crosssection classification and effective width which is based on an elastic, perfectly-plastic material model such as ordinary carbon steel. The cross-section is classified through individual comparison of each plate element's width-to-thickness ratio with a specified slenderness limit ratio. A cross-section that is capable of reaching their cross-section yield Afy is fully effective in compression and classified as Class 1, 2 or 3, while cross-sections which fail due to local buckling prior to achieving cross-section yield Afy, are classified as Class 4. The yield strength fy is taken as the minimum specified value for the 0.2% proof strength. The effective width method focuses on the isolated plate elements that comprise a cross-section and accounts for the reduction of compressive cross-section capacity due to local buckling through a reduction of plate element width. Taking into account the outcomes of comprehensive research conducted by Gardner and Theofanous [58], the slenderness limits for different section classes and analytical expressions for the local buckling reduction factor ρ were recently corrected in EN 1993–1-4 [5]. The relationships between the reduction factor ρ and non-dimensional plate element slenderness λp are defined as. ρ¼

ρ¼

1 λp

−

0:188 λp

2

≤1:0 for outstand elements with λp ≥ 0:748

0:772 0:079 − ≤1:0 for internal elements with λp ≥ 0:651 2 λp λp

ð27Þ

ð28Þ

where: λp ¼

b=t pffiffiffiffiffiffi kσ

28:4ε

ð29Þ

In Eq. (29), b is the flat width of the plate element, t is the plate thickness, ε = [(235/fy)·(E/210000)]0.5 is the material parameter and kσ is the plate buckling, taken as 0.43 for outstand elements and 4.0 for internal elements in uniform compression. Fig.12a and Fig.12b compare the EN 1993-1-4 [5] Class 3 limits for outstand and internal elements in compression, respectively with the obtained FE and experimental data. In Fig.12, the ultimate load Nc,u was normalised by its cross-section yield load and plotted versus the slenderness parameters c / (t ∙ ε), where c is the nominal width of the flat cross-section part. The cross-section yield load was taken as the product of the cross-sectional area and the enhanced average yield strength, allowing for the higher strength material in the corner regions of the press-braked sections. The material parameter ε was also based on the enhanced average yield strength. It can be seen from Fig.12a that the Class 3 slenderness limit for outstand compressed elements of c / (t ∙ ε) = 14 is closely related with the intersection point of lower limit data with horizontal axis passing through unity on the ordinate, and therefore may be safely applied to flanges of channel sections made from austenitic stainless steel. Similarly, Fig.12b shows that the Class 3 slenderness limit for internal compressed elements corresponds to the mean data trend passing through unity on the vertical axis at approximately of c / (t ∙ ε) = 37 value. Thus, the value of c / (t ∙ ε) = 37 appears to be adequate for application to austenitic stainless steel internal elements. It should be pointed out that experimental data for the outstand flanges of specimens lie between the Class 2 slenderness limit of c / (t ∙ ε) = 10 and the Class 3 slenderness limit for outstand compressed elements (see Fig.12a), while the experimental data for the internal elements of specimens are on left side regarding to Class 1 slenderness limit of c / (t ∙ ε) = 33 for internal compressed elements (see Fig.12b). Thus, the cross-section of the specimens is class 3. Based on the same generated data, the effective width equations for slenderness Class 4 outstand and internal elements in compression, given by Eqs. (28) and (27) were also examined.

J. Dobrić et al. / Journal of Constructional Steel Research 139 (2017) 236–253

1.3 1.2

FE data Experiment data

Nc,u/(Af0.2)

1.0 0.9

EN1993-1-4 Class 3 limit (14)

Nc,u/(Af0.2)

1.1

0.8 0.7 0.6 0.5 0.4

0.3 0

10

20

30 c/(t )

40

50

1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3

Experiment data

0

60

(a) Class 3 limit for outstand elements in compression

FE data

EN1993-1-4 Class 3 limit (37)

1.4

249

10 20 30 40 50 60 70 80 90 100 110 120 c/(t )

(b) Class 3 limit for internal elements in compression

Fig. 12. Assessment of Class 3 slenderness limits for cross-section's elements in compression according to [5].

3.5.3. Cross-section resistance according to the CSM The CSM [39,40] represents a new approach in the design of compact, semi-compact and slender cross-sections under compression, bending and combined loading that takes into account the benefits of strain hardening and element interaction for determining crosssection resistances. The CSM is based on the continuous relationship between the cross-section's deformation capacity expressed in terms of the strain ratio εcsm/εy and the full cross-section slenderness λp as given by Eqs. (30) and (31) for non-slender and slender crosssections, respectively:   εcsm 0:25 εcsm 0:1εu ¼ 3:6 but ≤ min 15; for λp ≤0:68 and austenitic stainless steel εy εy εy λp

ð30Þ εcsm ¼ εy

1−

0:222 λp

1:050

!

1 λp

1:050

for λp N0:68

ð31Þ

where εy = f0.2 / E is the yield strain and εu = 1 − f0.2 / fu is the predicted strain corresponding to the material ultimate strength fu. A significant difference between the effective width method and the CSM is the replacement of elastic plate buckling stress with cross-section elastic

1.4

1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

FE data

1.3

FE data

Experiment data

1.2

Experiment data

1.1

Nc,u/(Af0.2)

Nc,u/(Af0.2)

In Fig.13a and Fig.13b, the ultimate loads reached in the experiment an FE study were normalised in the same manner as previously explained and plotted against the plate element slenderness λp for outstand and internal elements, respectively. From Fig.13, it is evident that the codified effective width equations fit the generated data for both outstand and internal elements in compression, with the majority of the database above empirical curves for the reduction factor ρ. The analytical expression for outstand elements, given by Eq. (27), offers unsafe prediction of the reduction factor ρ in the case of channel sections whose flange width-to-section depth ratio is approximately 0.3 ≤ b / h ≤ 0.35 in the flange slenderness range from λp ≈ 0:748 to λp ≈ 2:0. Conversely, it was found that the normalised database lies under the analytical curve given by Eq. (28) for internal elements, in the case of channel sections whose flange width-to-section depth ratio is approximately b / h ≥ 0.45 in the web slenderness range from λp ≈ 0:6 to λp ≈ 1:4. These effects are primarily the result of the local web-to-flange interaction. The effective width method uses an element plate approach and therefore the web slenderness does not affect the flange behaviour; by increasing the web slenderness, the effective width of flange is slightly changed. Hence, a statistical reliability analysis is required to determine the partial safety factor for cross-section resistance in conjunction with the effective width equations given in EN 1993-1-4 [5].

1.0 0.9 0.8 0.7

0.6 0.5 0.4 0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

(a) Outstand elements in compression

3.2

0.3

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

(b) Internal elements in compression

Fig. 13. Assessment of effective width equations for cross-section's elements in compression according to [5].

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local bucking stress, including consideration of interaction among the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cross-section elements. The slenderness λp is taken as f 0:2 =σ cr in which the full cross-section elastic buckling stress σcr may be determined using a numerical method such as the one used in the CUFSM software [55], or conservatively calculated as the elastic buckling stress of the most slender constituent plate element of the cross-section in accordance with EN 1993-1-5 [59]. The second key feature of the CSM is the utilization of an elastic, linear hardening material model which better reflects the strain hardening effects and actual material behaviour of stainless steels relative to the elastic, perfectly-plastic material model used in the concept of cross-section classification. For austenitic stainless steel, the strain hardening slope Esh is determined as Esh ¼

f u− f y 0:16ε u −ε y

ð32Þ

For non-slender sections with λp ≤0:68 and εcsm / εy ≥ 1.0, the CSM cross-section compression resistance Ncsm,Rd. is determined by using the following equation: Ncsm;Rd ¼

Af csm γM0

ð33Þ

where A is the gross cross-sectional area, fcsm is the CSM design stress given by Eq. (34) and γM0 is the material partial safety factor as recommended in EN 1993-1-4 [5]. f csm ¼ f y þ Esh εy

  εcsm −1 εy

ð34Þ

For slender sections that fail by local buckling before yielding, with slenderness λp N0:68and εcsm / εy b 1.0, the CSM cross-section compression resistance Ncsm,Rd. is determined by considering a linear load– deformation structural response: Ncsm;Rd ¼

εcsm Af y εy γM0

ð35Þ

3.5.4. Summary In order to assess the accuracy of the existing design methods, a comparative analysis was performed in which predicted cross-sections

2.0

(1) The analytical expression given by Eq. (5) in SEI/ASCE 8-02 [38] provides a good prediction of the inelastic local buckling stress of stainless steel cross-sections: the mean experiment (or FE)to-predicted inelastic buckling stress ratio σlb / fcr,inel is 1.05 and the Coefficient of Variation (CoV) is 7.1%. However, it should be pointed out that iterative process may be cumbersome for hand calculating the inelastic buckling stress. The ASCE design method [38] very highly overestimates the compressive capacity of nonslender cross-sections: the mean experiment (or FE)-to-predicted buckling load ratio Nc,u / Nc,u,pred is 1.27 and the CoV is 11.5%. However, the ASCE resistance calculation given herein was based on material properties in compression without using the beneficial strength enhancement due to cold-work hardening. Considering slender cross-sections, SEI/ASCE 8-02 [38] demonstrates slightly unsafe errors (see Fig.14a): the mean resistance ratio is 0.97 and the CoV is 5.9%. Note that the ASCE design guide [38] and EN 1993-1-4 [5] utilise different reduction factors for outstand plate elements in compression and a different definition of the flat width of a plate element. (2) EN 1993-1-4 [5] offers conservative resistance predictions over the full range of cross-section slenderness (see Fig.14b). Even though the enhanced average yield strength was introduced in calculation, there is unduly overestimating of the compressive strengths of compact and semi-compact cross-sections. This outcome is associated with neglecting gradual yielding and strain

2.0 FE data

1.8

1.8

Experiment data

1.6

1.6

1.4

2.0

FE data Experiment data

1.0

0.8

1.4

1.2

1.0 0.8

1.2 1.0 0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0.0

0.0 0.2

0.6 1.0 1.4 1.8 2.2 Cross-section slenderness

(a) Comparison with SEI/ASCE 8-02 [38] strength predictions

Experiment data

1.6

Nc,u / Nc,u,pred

1.2

FE data

1.8

1.4

Nc,u / Nc,u,pred

Nc,u / Nc,u,pred

resistances, in accordance with SEI/ASCE 8-02 [38], EN 1993-1-4 [5] and the CSM [39,40], were compared with generated experimental and numerical section strengths. The comparisons are presented in terms of the ratio of experimental (or FE)-to-predicted cross-section resistance (Nc,u / Nc,u,pred). The distinctions between codified design methods [38,5] and the CSM [39,40] are graphically emphasized in Fig.14 where the resistance ratios both for non-slender and slender sections are plotted against the full cross-section slenderness λp. In order to provide an indication of the variation in aforementioned design procedures, the summary of the obtained results is reported in Table 11. The comparisons between the predicted values of the inelastic critical buckling stresses according to SEI/ASCE 8-02 [38] and experimental and FE local buckling stresses σlb = Nc,u / A are also summarized in Table 11. According to the results of the comparative analysis, the following remarks can be stated:

0.0 0.2 0.6 1.0 1.4 1.8 Cross-section slenderness

2.2

(b) Comparison with EN 1993-1-4 [5] strength predictions

0.2

0.6 1.0 1.4 1.8 2.2 Cross-section slenderness

(c) Comparison with CSM [39], [40] strength predictions

Fig. 14. Comparison between design resistance predictions and experimental and FE results.

J. Dobrić et al. / Journal of Constructional Steel Research 139 (2017) 236–253

251

Table 11 Comparisons of experimental with design cross-section resistances. No. of experiment data: 4 No. of FE data: 103 Non-slender cross-sections Slender cross-sections

SEI/ASCE 8-02 [38]

Mean Cov Mean Cov

EN 1993-1-4 [5]

CSM [39,40]

σlb / fcr,inel

Nc,u / Nc,u,pred

Nc,u / Nc,u,pred

Nc,u / Nc,u,pred

1.05 0.071

1.27 0.115 0.97 0.059

1.16 0.064 1.13 0.037

1.05 0.028 0.95 0.042

hardening in the nonlinear stress domain beyond the yield strength in the design procedure. The mean value of Nc,u / Nc,u,pred ratio is 1.16 and the CoV is 6.4%. Additionally, by ignoring element interaction, the effective width method used in EN 19931-4 [5] offers conservative resistance prediction of slender cross-sections: the mean value of Nc,u / Nc,u,pred ratio is 1.13 and the CoV is 3.7%. (3) The CSM [39,40] provides a considerably precise strength prediction of the non-slender cross-sections and a low degree of scatter in comparison with others design methods: the mean value of Nc,u / Nc,u,pred ratio is 1.05 and the CoV is 2.8%. Furthermore, although the CSM [39,40] indicates more accurate results in comparison with EN 1993-1-4 [5], the unconservative predictions were obtained in the slenderness range approximately from 0.8 to 1.8 (see Fig.14c) for slender sections: the mean value of Nc,u / Nc,u,pred ratio is 0.95 and the CoV is 4.2%. It should be noted that the CSM is practical and includes simpler resistance calculations than that of the effective width method which is a sequence of more complex steps for obtaining the design resistance by using hand calculations.

4. Reliability analysis In order to identify the values of the partial factors for cross-section resistance γM0 both for the EN 1993-1-4 [5] design procedure and the CSM [39,40], a statistical analysis based on provisions stated in Annex D, of EN 1990 [60] was performed. Table 12 lists the key statistical parameters for comparisons between predicted design resistances and experimental and numerical data, respectively: the design (ultimate limit state) fractile factor kd,n, the correction factor representing the average experiment or FE resistance-to-design model resistance ratio based on a least squares best fit to the slope of all data b, the CoV of the experimental and FE data relative to the design model resistance Vδ, the combined CoV incorporating both model and basic variable uncertainties Vr and the partial factor for cross-section resistance γM0. For yield strength, an over-strength value of 1.3 and a CoV of 0.06 for austenitic stainless steel are used, as recommended by Afshan et al. [61]. It can be seen from Table 12 that the obtained partial safety factors γM0 both for the EN 1993-1-4 [5] design procedure and the CSM [39, 40] exceed the currently adopted value of 1.1 in EN 1993-1-4 [5]. It should be noted that the reliability assessment of the EN 1993-1-4 [5] design provision for cross-section resistance, performed by Afshan et al. [61], provided value γM0 = 1.15 for limited number of experimental data for channel sections. Hence, this value together with value γM0 = 1.125 identified herein, indicates a decreasing trend in the value of coefficient γM0 that is used in EN 1993-1-4 [5]. Besides, the reliability analyses of the CSM for slender stainless steel cross-sections performed by Zhao et al. [39] resulted with to resistance factor γM0 = 1.05 for

cross-sections from austenitic stainless steel. This statistical analysis was based on comprehensive experimental and numerical data including tubular SHS and RHS, but also open section profiles, such as I-, T-, channel and angle sections. Considering relatively small number of experiments on channel sections, it appears that the result of γM0 = 1.135, based herein primarily on FE data, would change if further testing was carried out. 5. Conclusions In this paper, experimental data gained from the material coupon tests and stub column tests were generated and presented. Experimental results of the tensile coupon tests were used to validate the predicted analytical models for the material response and strength enhancements in press-braked structural sections. The FE parametric study based on the FE simulation of sub column tests was performed to extend the gathered experimental and numerical outcomes to a wider range of geometric variations affecting the compressive capacity of channel sections. The FE and experimental results were used to evaluate the current design guidelines for the prediction of cross-section resistance. The reliability assessment of the EN 1993-1-4 [5] design provisions and the CSM [39,40] for cross-section resistance under compression was carried out in order to identified partial safety factors γM0. The following conclusions may be drawn: 1) Austenitic stainless steel EN 1.4301 exhibits a rounded nonlinear stress–strain relationship with gradual yielding and ductility almost twice as high as that of the most used ordinary carbon steels as S275 or S355. The re-evaluated material model according to Arrayago et al. [12] represents the measured stress–strain curves with higher accuracy than the model given in Annex C, EN 1993–1-4 [5]. It was indicated that both material models offer a lower level of accuracy for the analytical description of stress–strain curves obtained from corner coupon tests, the mean value of the ratio εu,exp./εu,pred is 0.88; 2) Significant improvement of material strength occurred in the corner regions of the channel section due to press-braking. The experimental value of the 0.2% proof stress in the corner regions was 49% higher than the 0.2% proof strength of the flat sheet materials. As expected, no improvement of the material strength was recorded in the flat regions of the sections. Two proposed predictive models developed by Cruise and Gardner [17] and Rossi et al. [19] were assessed. Both models are able to precisely predict the corner strength enhancement in a press-braked section. However, in contrast to Cruise's and Gardner model [17], the Rossi's model [19] gives a safe and reliable prediction; 3) Continuous damage FE models for stainless steel were successfully built based on tensile coupon tests, by using the engineering approach for the determination of the damage evolution law according to Pavlović et al. [44];

Table 12 Reliability analysis results calculated according to EN 1990 [60]. Dataset

Material

No. of experiment data: 4 No. of FE data: 103

Austenitic stainless steel

EN 1993-1-4 [5] CSM [39,40]

kd,n

b

Vδ

Vr

γM0

3.072 3.072

1.127 0.961

0.041 0.056

0.084 0.093

1.125 1.135

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4) In the stub column testes, the failure mode of the compressed channel section was governed by inelastic local buckling, which occurred at a stress value that is 36% higher than the 0.2% proof stress. 5) The numerical results obtained in FE parametric study together with experimental results enabled the assessment of the applicability of design methods for cross-section resistance according to SEI/ASCE 8-02 [38], EN 1993-1-4 [5] and the CSM [39,40]. The ASCE [38] and Eurocode 3 [5] design methods significantly underestimate the actual strengths of compact and semi-compact channel sections. The mean experiment (or FE)-to-predicted buckling load ratio is 1.27 for ASCE [38] and 1.16 for EN 1993-1-4 [5]. This outcome is closely associated with the limited ability of codified design methods to interpret the realistic nonlinear behaviour and effects of strength enhancement of stainless steel structural elements, which undergo the cold-working process. On the other hand, the CSM provides a higher predictive resistance accuracy of non-slender channels, owing to the fact that this deformation-based design approach formulates a relationship between the strength and deformation capacity of cross-section and exploits the beneficial strain hardening effects through the use of an elastic, linear hardening material model; the mean experiment (or FE)-to-predicted buckling load ratio is 1.05. However, the slight unsafe predictions were obtained for slender channel sections: the mean experiment (or FE)-to-predicted buckling load ratio is 0.95. 6) The reliability analysis of the proposed design method performed on 4 experimental and 103 numerical results indicates a higher value of the partial safety factors both for the EN 1993-1-4 [5] design procedure and the CSM [39,40] in comparison with the codified value of 1.1 in EN 1993-1-4 [5] and suggests that an increase in the number of reliable data for more precise statistical analysis is necessary.

Acknowledgements This investigation is supported by the Serbian Ministry of Education, Science and Technological Development through the TR-36048 project. The authors are grateful to “Exing-Inox” Ltd. in Novi Sad, the Institute for Testing of Materials in Belgrade and the Institute for Materials and Structures, University of Belgrade for their technical support. References [1] N.R. Baddoo, Stainless steel in construction: a review of research, applications, challenges and opportunities, J. Constr. Steel Res. 64 (11) (2008) 1199–1206. [2] G. Graham, Structural uses of stainless steel – building and civil engineering, J. Constr. Steel Res. 64 (11) (2008) 1194–1198. [3] L. Gardner, The use of stainless steel in structures, Prog. Struct. Eng. Mater. 7 (2) (2005) 45–55. [4] B. Rossi, Discussion on the use of stainless steel in constructions in view of sustainability, Thin-Walled Struct. 83 (2014) 182–189. [5] Eurocode 3: Design of steel structures – part 1–4: General rules – supplementary rules for stainless steels, including amendment A1 (2015), EN 1993-1-4:2006+ A1:2015, Brussels, Belgium, CEN 2015. [6] Eurocode 3: Design of steel structures – Part 1–1: General rules and rules for buildings EN 1993–1-1, Brussels, Belgium, CEN 2005. [7] J. Dobrić, Z. Marković, D. Buđevac, Z. Flajs, Specific features of stainless steel compression elements, Građevinar 67 (2) (2015) 143–150. [8] E. Mirambell, E. Real, On the calculation of deflections in structural stainless steel beams: an experimental and numerical investigation, J. Constr. Steel Res. 54 (2000) 109–133. [9] K.J.R. Rasmussen, Full–range stress–strain curves for stainless steel alloys, J. Constr. Steel Res. 59 (1) (2003) 47–61. [10] K. Abdella, Inversion of a full-range stress–strain relation for stainless steel alloys, Int. J. Non-Linear Mech. 41 (2006) 456–463. [11] P. Hradil, A. Talja, E. Real, E. Mirambell, B. Rossi, Generalized multistage mechanical model for nonlinear metallic materials, Thin-Walled Struct. 63 (2013) 63–69. [12] I. Arrayago, E. Real, L. Gardner, Description of stress–strain curves for stainless steel alloys, Mater. Des. 87 (2015) 540–552. [13] W. Ramberg, W.R. Osgood, Description of stress–strain curves by three parameters, Technical Note No. 902, 1943 (Washington, USA). [14] H.N. Hill, Determination of stress–strain relations from offset yield strength values, Technical Note No. 927, 1944 (Washington, USA).

[15] G.J. van den Berg, P. van der Merwe, Prediction of corner mechanical properties for stainless steels due to cold forming, Eleventh International Speciality Conference on Cold-Formed Steel Structures 1992, pp. 571–586 St. Louis, Missouri, USA. [16] M. Ashraf, L. Gardner, D. Nethercot, Strength enhancement of the corner regions of stainless steel cross-sections, J. Constr. Steel Res. 61 (1) (2005) 37–52. [17] R.B. Cruise, L. Gardner, Strength enhancements induced during cold forming of stainless steel sections, J. Constr. Steel Res. 64 (11) (2008) 1310–1316. [18] W. Quach, J. Teng, K. Chung, Residual stresses in press-braked stainless steel sections, II: press-braking operations, J. Constr. Steel Res. 65 (8) (2009) 1816–1826. [19] B. Rossi, S. Afshan, L. Gardner, Strength enhancements in cold-formed structural sections - part II: predictive models, J. Constr. Steel Res. 83 (2013) 189–196. [20] A.L. Johnson, G. Winter, Behaviour of stainless steel columns and beams, J. Struct. Div. ASCE 92 (ST5) (1966) 97–118. [21] A.L. Johnson, G. Winter, The Structural Performance of Austenitic Stainless Steel Members (Report No. 327) (1966) Department of Structural Engineering, School of Civil Engineering, Cornell University. [22] K.J.R. Rasmussen, G.J. Hancock, Stainless Steel Tubular Columns - Tests and Design, Tenth International Specialty Conference on Cold-Formed Steel Structures, 1990 (St. Louis, Missouri, USA). [23] P.J. Bredenkamp, G.J. van den Berg, The strength of stainless steel built up I section, J. Constr. Steel Res. 34 (1995) 131–144. [24] A. Talja, P. Salmi, Design of Stainless Steel RHS Beams, Columns and Beam-Columns (Research Note 1619), VTT Building Technology, Finland, 1995. [25] B.A. Burgan, N.R. Baddoo, K.A. Gilsenan, Structural design of stainless steel members-comparison between Eurocode 3, Part 1.4 and test results, J. Constr. Steel Res. 54 (1) (2000) 51–73. [26] B. Young, W. Hartono, Compression tests of stainless steel tubular members, J. Struct. Eng. ASCE 128 (6) (2002) 754–761. [27] H. Kuwamura, Local buckling of thin-walled stainless steel members, Steel Struct. 3 (2003) 191–201. [28] B. Young, Y. Liu, Experimental investigation of cold-formed stainless steel columns, J. Struct. Eng. ASCE 129 (2) (2003) 169–176. [29] L. Gardner, D.A. Nethercot, Experiments on stainless steel hollow sections – part 1: material and cross-sectional behaviour, J. Constr. Steel Res. 60 (2004) 1293–1318. [30] B. Young, W.M. Lui, Tests on cold formed high strength stainless steel compression members, Thin-Walled Struct. 44 (2006) 224–234. [31] L. Gardner, A. Talja, N. Baddoo, Structural design of high-strength austenitic stainless steel, Thin-Walled Struct. 44 (5) (2006) 517–528. [32] M. Theofanous, T.M. Chan, L. Gardner, Structural response of stainless steel oval hollow section compression members, Eng. Struct. 31 (2009) 922–934. [33] Y. Huang, B. Young, Material properties of cold-formed lean duplex stainless steel sections, Thin-Walled Struct. 54 (2012) 72–81. [34] N. Saliba, L. Gardner, Cross-section stability of lean duplex stainless steel welded Isections, J. Constr. Steel Res. 80 (2013) 1–14. [35] H.X. Yuan, Y.Q. Wang, Y.J. Shi, L. Gardner, Stub column tests on stainless steel builtup sections, Thin-Walled Struct. 83 (2014) 103–114. [36] S. Fan, F. Liu, B. Zheng, G. Shu, Y. Tao, Experimental study on bearing capacity of stainless steel lipped C section stub columns, Thin-Walled Struct. 83 (2014) 70–84. [37] J. Dobrić, Behaviour of Built-up Stainless Steel Members Subjected to Axial CompressionPhD thesis University of Belgrade, Faculty of Civil Engineering, 2014. [38] Specification for the Design of Cold-Formed Stainless Steel Structural Members [SEI/ ASCE 8-02]. , American Society of Civil Engineers, 2002. [39] L. Gardner, S. Afshan, The continuous strength method for structural stainless steel design, Thin-Walled Struct. 68 (2013) 42–49. [40] O. Zhao, S. Afshan, L. Gardner, Structural response and continuous strength method design of slender stainless steel cross-sections, Eng. Struct. 140 (2017) 14–25. [41] EN 10088: Stainless steels; EN 10088-1:2005: List of stainless steels; EN 10088-2: 2005: Technical delivery conditions for sheet/plate and strip of corrosion resisting steels for general purposes; EN 10088-3:2005: Stainless steels, Technical delivery conditions for semifinished products, bars, rods and sections for general purposes, (2005) European Standard CEN. [42] EN 10002–1: Metallic materials – tensile testing part 1, method of test at ambient temperature (2001) CEN. [43] S. Afshan, B. Rossi, L. Gardner, Strength enhancements in cold-formed structural sections - part I: material testing, J. Constr. Steel Res. 83 (2013) 177–188. [44] M. Pavlović, Z. Marković, M. Veljković, D. Buđevac, Bolted shear connectors vs. headed studs behaviour in push-out tests, J. Constr. Steel Res. 88 (2013) 134–149. [45] ABAQUS, User Manual, Version 6.12, DS SIMULIA Corp, Providence RI, USA, 2012. [46] G. Trattnig, T. Antretter, R. Pippan, Fracture of austenitic steel subject to a wide range of stress triaxiality ratios and crack deformation modes, Eng. Fract. Mech. 75 (2) (2008) 223–235. [47] J.R. Rice, D.M. Tracey, On the ductile enlargement of voids in triaxial stress fields, J. Mechanics and Phys. Solids 17 (1969) 201–217. [48] J.A. Lemaitre, Continuous damage mechanics model for ductile fracture, J. Eng. Mater. Technol. 107 (1) (1985) 83–90. [49] Eurocode 3, Design of Steel Structures – Part 1-3: General Rules - Supplementary Rules for Cold-Formed Members and Sheeting EN 1993-1-3, Brussels, Belgium, 2006. [50] L. Gardner, D.A. Nethercot, Numerical modelling of stainless steel structural components – a consistent approach, J. Struct. Eng. ASCE 130 (10) (2004) 1586–1601. [51] R.G. Dawson, A.C. Walker, Post-buckling of geometrically imperfect plates, J. Struct. Div. ASCE 98 (1) (1972) 75–94. [52] M. Jandera, L. Gardner, J. Machacek, Residual stresses in cold-rolled stainless steel hollow sections, J. Constr. Steel Res. 64 (11) (2008) 1255–1263. [53] S. Niu, K.J.R. Rasmussen, F. Fan, Distortional–global interaction buckling of stainless steel C-beams: part II - numerical study and design, J. Constr. Steel Res. 96 (2014) 40–53.

J. Dobrić et al. / Journal of Constructional Steel Research 139 (2017) 236–253 [54] O. Zhao, B. Rossi, L. Gardner, B. Young, Behaviour of structural stainless steel crosssections under combined loading – part II: numerical modelling and design approach, Eng. Struct. 89 (2015) 247–259. [55] B.W. Schafer, S. Ádány, Buckling Analysis of Cold-Formed Steel Members Using CUFSM: Conventional and Constrained Finite Strip Methods, in: Proceedings of the Eighteenth International Specialty Conference on Cold-Formed Steel Structures, 2006 (Orlando, USA). [56] Y. Bruitendag, G.J. van den Berg, The strength of partially stiffened stainless steel compression members, In Proceedings of the 12th International Specialty Conference on Cold-Formed Steel Structures, University of Missouri-Rolla, 1994.

253

[57] W. Reyneke, G.J. van den Berg, The strength of partially stiffened stainless steel compression members, In Proceedings of the 13th International Specialty Conference on Cold-Formed Steel Structures, University of Missouri-Rolla, 1996. [58] L. Gardner, M. Theofanous, Discrete and continuous treatment of local buckling in stainless steel elements, J. Constr. Steel Res. 64 (11) (2008) 1207–1216. [59] Eurocode 3, Design of steel structures – part 1–5, Plated structural elements EN 1993-1-5, Brussels, Belgium, CEN, 2006. [60] Eurocode – Basis of Structural Design EN 1990. , CEN, Brussels, Belgium, 2012. [61] S. Afshan, P. Francis, N.R. Baddoo, L. Gardner, Reliability analysis of structural stainless steel design provisions, J. Constr. Steel Res. 114 (2015) 293–304.