Design of cold-rolled stainless steel rectangular hollow section columns

Journal of Constructional Steel Research 170 (2020) 106072

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Journal of Constructional Steel Research

Design of cold-rolled stainless steel rectangular hollow section columns Baofeng Zheng a,b,⁎, Ganping Shu a,b, Fuzhe Xie c, Qinglin Jiang d a

School of Civil Engineering, Southeast University, Nanjing 210096, China Key Laboratory of Concrete and Prestressed Concrete Structures of Ministry of Education, Southeast University, Nanjing 210096, China c Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, China d Jiangsu Dongge Stainless Steel Ware Co. Ltd., Yancheng 224212, China b

a r t i c l e

i n f o

Article history: Received 6 October 2019 Received in revised form 13 March 2020 Accepted 2 April 2020 Available online xxxx Keywords: Stainless steel Cold-forming effect Strain hardening Rectangular hollow sections Design method for columns Reliability analysis

a b s t r a c t Considering the effects of cold-forming and strain hardening in the design of cold-formed stainless steel columns is an effective way to reduce construction costs. This approach has become a common trend in the revision of relevant design codes. A few studies were conducted in the past on two key aspects: (i) predicting the enhanced material properties in cold-formed stainless steel cross-sections and (ii) modifying the design methods to adapt to stainless steel material characteristics. However, in the existing literature, they had been separately studied, and integration of these two aspects to form a complete design method was rarely reported. In this study, a series of column tests were conducted on 19 stub and 32 long columns in rectangular hollow sections. The measured imperfections, material properties, load-displacement curves, and failure modes were reported. A new design method for columns was developed, which could clearly and accurately determine the column capacity over the space encompassed by member slenderness and cross-sectional slenderness values. The proposed design method and the method laid out in the Design Manual for Structural Stainless Steel were evaluated against the test data. It was demonstrated that the two design methods were both very conservative in predicting the column capacity when material properties in the annealed condition were used. However, their performances improved when the predicted enhanced material properties were used. Generally, the predictions of the proposed design method matched the test data better. Furthermore, a reliability analysis was conducted, and resistance factors were recommended for the corresponding design methods. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction Corrosion in steel structures usually leads to reduced durability and even serious accidents. Stainless steel has the advantage of superior corrosion resistance compared to low carbon steel. Thus, it is a preferred material for structures built in corrosive environments. The corrosion resistance of stainless steel is attributed to the presence of precious metallic elements (e.g. Ni, Cr etc.). This, however, could lead to a high material price turnoff stainless steel. Experimental studies have shown through stress–strain curves, the gradual yielding and considerable strain hardening of stainless steel, especially in austenitic stainless steel. These material characteristics distinguish the design of stainless steel columns from those made of low carbon steel. However, owing to these characteristics, the yield strength of stainless steel becomes sensitive to plastic deformation from coldforming and could be significantly enhanced through fabrication. In addition, the cold-forming process for different cross-sections involves ⁎ Corresponding author at: School of Civil Engineering, Southeast University, Nanjing 210096, China, E-mail address: [email protected] (B. Zheng).

https://doi.org/10.1016/j.jcsr.2020.106072 0143-974X/© 2020 Elsevier Ltd. All rights reserved.

different amounts of plastic deformations and causes a variety of yield strengths (material properties), i.e., more complicated material characteristics. Utilizing the enhanced material yield strength and material characteristics in the design of columns has become a generally accepted direction for revising relevant design codes and specifications to compensate for the high material cost of stainless steel. To achieve the same, in the prediction of column capacity, two key components are necessary for the design method; the first one is to predict the enhanced material properties in cold-formed cross-sections; and the second is to modify the design methods based on the material characteristics for possible failure modes (e.g., local buckling, global buckling, and local–global interactive buckling). The progress in these two areas is reviewed in the following sections. 1.1. Literature review on predicting the enhanced material properties For the prediction of material property enhancement, this study focuses on cold-rolling process, which is one of the popular production methods for producing stainless steel rectangular hollow sections (RHSs). During cold-rolling, stainless steel sheets are gradually bent transversely into an arc and welded closed to give circular hollow

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B. Zheng et al. / Journal of Constructional Steel Research 170 (2020) 106072

sections (CHSs) using tungsten inert gas welding. Next, the CHSs are reshaped into RHSs. The layout for a typical production method for one of the test cross-sections, namely RHS 100 mm × 50 mm × 3 mm (the corresponding CHS is CHS 96 mm × 3 mm), is shown in Fig. 1. Material enhancement in cold-rolled stainless steel RHSs was first reported in the 1990s [1,2]. Subsequently, a few more studies were conducted on cold-rolled stainless steel members and a series of empirical equations were proposed to predict their enhanced yield strength [3–6]. In recent years, several efforts have been made to predict the yield strength from a theoretical aspect. Based on the inversion of the full-range stress–strain expression [7] and the maximum plastic strain experienced during the cold-rolling process, Rossi et al. [8] proposed a theoretical equation for calculating the yield strength. Afshan, Rossi and Gardner [9,10] predicted the yield strengths of flat faces and corner regions in cold-rolled RHSs using equations based on a power law material model and through thickness averaged plastic strain. These equations were adopted in the Design Manual for Structural Stainless Steel (4th edition) [11]. In an accompanying work [12], a further study based on a similar concept as that proposed in [9,10] was reported to predict not only the yield strength, but also all the other key material parameters (including the ultimate yield strength, strain hardening exponent, etc.) and their distributions in cold-rolled cross-sections. 1.2. Literature review on modifying design methods for columns For modifying the design methods for columns incorporating the material characteristics for possible failure modes, three types of failure modes are discussed here, namely local buckling, global buckling, and local–global interactive buckling. For slender cross-sections subjected to local buckling, design formulae for the prediction of cross-sectional capacity in all the current design codes and specifications [13–16] are based on the effective width method, which was first proposed by Winter [17], and used in the design of low carbon steel structures. The previous studies showed that the relation between the reduction factor of the cross-section and the cross-sectional slenderness was affected by the values of the material properties (i.e., E0, σ0.2, n etc.). [18,19]. Thus, to accommodate a variety of material properties owing to cold-rolling and a large number of types of stainless steel, the parameters in the Winter formula were modified for each specific material property [20–22]. Rasmussen et al. [18] modified the Winter formula with its key parameters expressed using the material properties (i.e., E0, σ0.2, and n). Thus, for each type of material, there are a pair of parameters in the Winter formula. Furthermore, the effect of variation in the material properties on the parameters in the Winter formula and the apparent strain hardening after yielding of stainless steel increase the capacity of a stocky cross-section far beyond its squash load of cross-section (Ny = σ0.2 × A) [3,23,24]. To consider the benefit of strain hardening after yielding, two methods are available in the literature. The first method is based on the effective width, which allows for the case when the effective width ratio (also called reduction

factor) exceeds 1.0. It usually extends the effective width ratio from 1.0 to an assumed ultimate value (where the cross-sectional slenderness is zero) using a linear or nonlinear curve [22,25,26]. The second method is the continuous strength method (CSM), which was proposed and modified by Gardner et al. [20,27–29]. The latest version of CSM uses a simplified linear hardening material model and a base curve that defines the level of strain that a cross-section can carry in a normalized form. The CSM method was adopted in the Design Manual for Structural Stainless Steel (4th edition) [11]. For global buckling, Perry formula was adopted in the Eurocode EN1993-1-4 [13], Australian/New Zealand standard AS/NZS 4673 [15], and Chinese specification CECS 410–2015 [16]. In this study, only the global flexural buckling is considered for an RHS member. Owing to the nonlinearity of the stress-strain curve of stainless steel, the reduction factor curve of a column subjected to global buckling is relevant to material properties (i.e. E0, σ0.2, n, etc.) [30]. As the material properties vary significantly because of the cold-rolling fabrication, different values of imperfection parameters (λm0 and α) in Perry formula were recommended in each individual study [11,31–33]. Rasmussen et al. [30] proposed a set of design equations in the form of a modified Perry formula with the imperfection parameters expressed in terms of the basic material properties in Ramberg-Osgood model (i.e. E0, σ0.2, and n). In their study, the reduction factor for global buckling directly depends on the material properties. Shu et al. [34] proposed a unified Perry formula using a member slenderness conversion equation to consider a variety of material properties. To reflect the effect of stainless steel material characteristics on the global stability of columns, Ahmed et al. [35] proposed modifications for both member slenderness and the imperfection parameters in Perry formula. In addition, in a modification of the parameters in Perry formula, SEI/ASCE 8–02 [14] adopted the tangent elastic modulus method. Hradil [36] developed a design method that combined the formulae used in EN1993-1-4 and SEI/ ASCE 8–02. Iterations are needed in these two calculation methods for columns. It should be mentioned that all the above design methods, except for the Perry formula in design codes and specifications involve extensive calculations. For the local–global interactive buckling in the current design codes and specifications for stainless steel structures (such as EN1993-1-4, SEI/ASCE 8–02, and CECS 410–2015), the effect of local–global interactive buckling is considered by combining two reduction factors; one reduction factor accounts for local buckling and the other for global buckling. Both factors are assumed to be less than or equal to 1.0. However, stub columns with stocky cross-sections usually have a higher strength than the squash load of the cross-section (Ny = σ0.2 × A) owing to considerable strain hardening after yielding. Thus, for stainless steel columns, both the local buckling and strain hardening after yielding are possibly coupled with global buckling and should be included in the design of columns. To utilize the benefit of strain hardening after yielding in the design of columns, Ashraf et al. [27] replaced the yield strength σ0.2 in the expressions for the member slenderness and

Fig. 1. Layout of cold-rolling production method for RHS 100 mm × 50 mm × 3 mm.

B. Zheng et al. / Journal of Constructional Steel Research 170 (2020) 106072

member capacity by a new parameter, namely Effective Buckling Stress σeff = (øcσ0,2σLB)0.5, which includes the strain hardening effect through the local buckling stress σLB, calculated using the CSM method. Ahmed et al. [35] directly used the CSM local buckling stress fcsm to replace the yield strength σ0.2 in the expressions for member slenderness and member capacity, and modified the definition of the imperfection factor in Perry formula to include the effect of cross-sectional slenderness. Arrayago et al. [37,38]carried out a thorough study and developed a full slender range direct strength method (DSM) for stainless steel hollow section columns, in which a detailed design method was proposed to use strain hardening for stocky columns with stocky cross-sections, and to consider the reduction of local buckling for columns with slender cross-sections. In their study, the proposed design method was based on the global buckling curves in EN1993-1-4 and the DSM local buckling cures from low carbon steel. This method was found to be more accurate for both columns and beam-columns than the provisions of the current specifications for stainless steel structural members. 1.3. Objective of this study Although significant progress has been made on the two distinct components of the design method for cold-formed columns, i.e., (i) predicting the enhanced material properties in cold-formed stainless steel cross-sections and (ii) modifying the design methods to adapt to stainless steel material characteristics, the design applications are required to consider these two components together. There is still a lack of clarity on how to integrate these two components to form a complete design method for columns. In addition, although reliability analysis has also been conducted for these two components individually, the overall reliability, when considering these two components together has not been demonstrated. The objective of this study is to develop and evaluate a design method for columns considering both the enhanced material properties owing to cold-forming and the stainless steel material characteristics. For stainless steel material characteristics, the effect of variation of material properties on the reduction factor curves for global and the local buckling was ignored in developing a convenient and practical design method, and only the benefit of considerable strain hardening was focused on in this study. Firstly, a series of tests was carried out on cold-rolled RHS columns subjected to local, global, and local–global interactive buckling. Thereafter, a new design method for columns was developed, which could provide clear and accurate capacity predictions for columns over the complete range of member slenderness and crosssectional slenderness. Comparisons between predictions by the proposed design method and test data collected in this study, as well as form the literature were conducted. Finally, the reliability of the design methods was analyzed and recommendations on the resistance factor were provided. 2. Experimental study This experimental study is a part of a research program aimed at supporting the revision process of the Chinese Technical Specification for Stainless Steel Structures [16] for cold-formed members. Two batches of specimens were tested in this study. The test specimens in Batch-1 were tracked during the whole fabrication process from coils to coldrolled RHSs. Virgin stainless steel sheets (from coils), cold-rolled CHSs, and RHSs were taken from the fabrication procedure as test specimens. The residual stresses, material properties, members, and joints capacities were tested. In related studies, the residual stresses [39] and material enhancements [12] in the cross-sections of the same batch specimens were reported. In this study, the behavior of cold-rolled RHS columns was studied and reported. Owing to the difficulty in obtaining specimens with complete information on their fabrication history, the test specimens in Batch-2 were directly purchased from the market without the details of their virgin material properties and

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fabrication process. It should be noted that while the tests on the material properties and stub columns for Batch-2 specimens were reported in [40], the key test results are summarized here. In total, six cold-rolled RHSs of austenitic stainless steel were studied, including four RHSs in Batch-1 and two in Batch-2. The types and nominal dimensions of these specimens were SHS 40 mm × 2 mm, SHS 70 mm × 3 mm, RHS 100 mm × 50 mm × 3 mm, SHS 100 mm × 3 mm, SHS 80 mm × 3 mm, and RHS 75 mm × 45 mm × 1.5 mm, respectively. The measured geometric dimensions of the cross-sections are shown in Table 1 with the definition of nomenclature shown in Fig. 2. 2.1. Material properties For the specimens in Batch-1, the material properties of the virgin sheets, the corresponding CHSs (shown in the first column in Table 2), and the final RHSs were tested, and the results are summarized in Table 2. Details of the material tensile coupon tests could be referred to from [12]. In this table, E0 is the Young's modulus; σ0.2 is the nominal yield strength; σ1.0 is the 1% proof strength; n and n0.2,1.0 are the strain hardening exponents for the strain ranges of (0 to σ0.2) and (σ0.2 to σ1.0), respectively; σu is the ultimate tensile strength; and δ is the elongation after fracture. It should be noted that the averaged material properties for the flat region of each RHS shown in Table 2 are the averaged results for the coupons extracted from each center of the flat face, and those for the CHSs are the averaged values for the coupons except in the case of the coupon from the weld. For specimens in Batch-2, only the material properties of the final cross-sections were obtained, and the results are shown in Table 3. In this table, the elongation after the fracture was missing for all the coupons owing to an oversight in the test arrangement. As a result, the 1% proof strength σ1.0 for corner coupons of SHS 80 mm × 3 mm and RHS 75 mm × 45 mm × 1.5 mm were not obtained because of a premature failure of strain gauges. 2.2. Stub column tests In total, 19 specimens comprising six cross-sections were tested. For most of the cross-sections, three specimens were tested, while four specimens were tested for SHS 80 mm × 3 mm to check the stability of the test rig. The length of the specimen was taken as approximately 4–5 times the cross-sectional height to avoid global buckling. Table 4 shows the measured length L of the specimens. The specimens were labeled appropriately to recognize their salient features directly. For example, in the specimen designation “S100 × 3–1”, the letter “S” means that it is a stub column, the numbers “100” and “3” are the nominal width and thickness of the cross-section, respectively; the number “1” implies that the specimen was the first stub column to be tested for this cross-section. The local geometric imperfections were measured using a milling machine prior to the stub column tests. For more details on measuring the local imperfections [32,40] could be referred to. For each specimen, the imperfections at three cross-sections were recorded, namely at both the ends and the mid-length. The measured amplitudes ω0 of the local geometric imperfections are shown in Table 4. The stub column test was conducted on a 200 T hydraulic test machine. Prior to the test, both the ends of the stub column were milled flat to ensure a full contact with the loading plates of the test machine. Four strain gauges were pasted onto the outer surface at mid-length of the specimen to measure its average compressive strain. One LVDT was installed at each corner of the loading plate (i.e., a total of four LVDTs) to record the end-shortening of the specimen. A data logger instrument TDS303 was used to collect the data from the strain gauges and LVDTs, while the applied load was read directly through the dial of the test machine. More details on the test rig for the stub column test could be obtained from [32,40]. All the stub columns failed in local buckling, and the typical failure mode is shown in Fig. 3. The ultimate load Nu and the end-shortening

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B. Zheng et al. / Journal of Constructional Steel Research 170 (2020) 106072

Table 1 Measured averaged cross-section dimensions of the test cross-sections. Cross-section

Batch-1

Batch-2

SHS 40 mm × 2 mm SHS 70 mm × 3 mm RHS 100 mm × 50 mm × 3 mm SHS 100 mm × 3 mm SHS 80 mm × 3 mm SHS 75 mm × 45 mm × 1.5 mm

h/mm

b/mm

t/mm

40.14 71.61 100.24 101.90 80.40 74.94

40.14 71.61 52.36 101.90 80.40 45.56

1.97 3.10 3.14 3.09 2.68 1.48

Fig. 2. Cross-section symbols and locations of coupons in the cross-section.

Δ at the ultimate load for the specimens are displayed in Table 4. Their load-end shortening curves are presented in Fig. 4. Owing to an error in the data logging system, the displacements of the LVDTs for loads above 490 kN were lost for specimen S70 × 3–1. 2.3. Long columns tests A total of 32 specimens comprising four cross-sections were tested. The nominal cross-sections of the test specimens were SHS 70 mm ×

Outer corner radius/mm Avg.

ro1

ro2

ro3

ro4

2.81 6.25 6.50 5.88 4.63 2.13

2.25 5.5 7.5 8.0 4.9 2.3

2.5 7.0 6.0 4.5 4.9 2.3

4.0 7.0 6.0 4.0 4.9 2.3

2.5 5.5 6.5 7.0 3.9 1.7

3 mm and SHS 100 mm × 3 mm from Batch-1, and SHS 80 mm × 3 mm and RHS 75 mm × 45 mm × 1.5 mm from Batch-2. The length of the specimens varied from 950 to 3500 mm. For the cross-section RHS 75 mm × 45 mm × 1.5 mm, the buckling of the specimens about both the major and the minor axes was tested. The measured lengths L and the overall geometric imperfection magnitudes e0 are shown in Tables 5 and 6 for Batch-1 and Batch-2, respectively. In these tables, the labels of the specimens depict key information related to the specimens. The first three letters of each label show the type of cross-section; for example, “RHS” and “SHS” indicate rectangular and square hollow sections, respectively. After the letters, two or three numbers are connected using a multiplication symbol; these indicate the nominal dimensions of the cross-section. The number after the hyphen is the nominal length of the specimen. Three specimens have letter “a” after the nominal length, which indicates that it is a repeat test. For specimens with a cross-section of RHS 75 mm × 45 mm × 1.5 mm, “Ma” and “Mi” at the end of the label imply that the major and the minor axes of the cross-section were assigned in the pinned direction in the test rig, respectively. The long column specimens were tested in the pinned boundary condition using a 500 T hydraulic test machine modified with a screw jack to obtain the post-buckling load–displacement curves. To obtain the pinned boundary, one-way knife hinges were installed at the top and the bottom of each specimen, respectively. Owing to the existence of the one-way knife hinges, the effective length of each specimen was

Table 2 Measured averaged material properties of the cross-sections in Batch-1. Cross-section

Position

E0/MPa

σ0.2/MPa

σ1.0/MPa

σu/MPa

δ

n

n0.2,1.0

SHS 40 mm × 2 mm CHS 50.8 mm × 2 mm

Virgin sheet CHS Flat face Corner region Virgin sheet CHS Flat face Corner region Virgin sheet CHS Flat face Corner region Virgin sheet CHS Flat face Corner region

201,458 202,147 194,879 187,019 193,409 204,178 202,933 195,303 200,858 198,760 190,181 198,611 194,383 203,015 197,944 194,591

262.81 385.42 462.99 711.16 254.12 368.63 456.72 620.24 247.40 316.67 396.70 608.96 239.78 311.97 358.13 560.63

318.22 439.75 528.13 887.35 303.60 422.49 537.76 781.52 293.63 363.57 464.23 771.32 288.45 357.53 408.46 679.45

646.29 681.93 722.17 936.88 636.15 678.77 748.58 895.98 642.19 659.91 691.05 878.75 631.51 648.44 669.86 822.84

0.60 0.58 0.50 0.18 0.60 0.59 0.47 0.25 0.66 0.62 0.54 0.26 0.63 0.62 0.60 0.31

8.60 6.74 4.63 4.73 6.36 5.45 4.47 4.30 6.55 6.57 4.87 3.82 6.04 6.38 5.19 4.49

2.25 2.60 4.04 4.53 2.29 3.08 3.88 4.09 2.19 2.50 3.73 4.12 2.28 2.53 3.18 3.73

SHS 70 mm × 3 mm CHS 89 mm × 3 mm

RHS 100 mm × 50 mm × 3 mm CHS 96 mm × 3 mm

SHS 100 mm × 3 mm CHS 127 mm × 3 mm

Table 3 Measured averaged material properties of the cross-sections in Batch-2. Cross-section

Position

E0/MPa

σ0.2/MPa

σ1.0/MPa

σu/MPa

δ

n

n0.2,1.0

SHS 80 mm × 3 mm

Flat face Corner region Flat face Corner region

193,127 190,352 203,740 209,355

482.17 644.46 352.14 692.53

545.61 – 406.79 –

698.81 918.50 767.84 977.11

– – – –

4.98 3.55 4.00 4.15

3.40 – 3.60 –

RHS 75 mm × 45 mm × 1.5 mm

B. Zheng et al. / Journal of Constructional Steel Research 170 (2020) 106072 Table 4 Measured length, imperfection amplitude, and test result for stub columns. Specimen

Batch-1

Batch-2

S100 × 3–1 S100 × 3–2 S100 × 3–3 Avg. S100 × 50 × 3–1 S100 × 50 × 3–2 S100 × 50 × 3–3 Avg. S70 × 3–1 S70 × 3–2 S70 × 3–3 Avg. S40 × 2–1 S40 × 2–2 S40 × 2–3 Avg. S80 × 3–1 S80 × 3–2 S80 × 3–3 S80 × 3–4 Avg. S75 × 45 × 1.5–1 S75 × 45 × 1.5–2 S75 × 45 × 1.5–3 Avg.

L

ω0

Nu

Δ

mm

mm

kN

mm

400.0 400.0 400.0 400.0 300.0 300.0 300.0 300.0 280.0 280.5 281.0 280.5 160.0 160.0 160.0 160.0 400.1 400.2 400 400 400 350.3 350.1 350.4 350.3

0.128 0.09 0.106 0.108 0.080 0.094 0.099 0.091 0.081 0.07 0.107 0.086 0.048 0.037 0.053 0.046 0.035 0.034 0.051 0.069 0.047 0.031 0.029 0.061 0.040

538 490 509 512 478 475 480 478 538 527 530 532 203 201 202 202 414 421 452 453 442 116 116 112 114

3.04 3.42 2.74 3.07 3.15 2.92 2.90 2.99 – 4.52 4.62 4.57 3.22 3.26 2.89 3.12 2.58 3.12 3.03 2.85 2.90 1.43 1.47 1.47 1.46

extended at each end. The final effective length was equal to the length of the specimen as given in Tables 5 and 6 plus two times of the distance from the specimen end to the corresponding edge of the knife hinge, i.e. L0 = L + 2 × 45 mm. wAloadcellwasmountedatthetopofthespecimentomeasuretheapplied load. Torecord therotational and axial deformation atboth theends, four LVDTs were fixed on the loading plates at both ends of the specimen. Two LVDTs were used to monitor the displacements at the mid-length of thespecimen.Oneofthemrecordedthedisplacementinthepinneddirection, while the other recorded the displacement along a direction perpendicular to the pinned direction. The layouts of the LVDTs at both the ends and the mid-height are shown in Fig. 5. In addition, four strain gauges were pasted on the surface of specimen at mid-length. TDS303 was used to acquire and store the data of the load cell, LVDTs, and strain gauges.

5

More details of the test rig for the long column test could be obtained from [32]. The test ultimate loads are shown in Tables 5 and 6 for Batch-1 and Batch-2 specimens, respectively. Here Nu, Δm, and Δs are the test ultimate load, displacement at mid-length, and axial deformation of the specimens in the ultimate state, respectively. Figs. 6 and 7 show the load-end shortening curves and the load-lateral displacement curves of the specimens, respectively. Most of the specimens failed in global flexural buckling. Specimens RHS75 × 45 × 1.5–950-Ma, RHS75 × 45 × 1.5–1450-Ma, RHS75 × 45 × 1.5-1450a-Ma, RHS75 × 45 × 1.5–950-Mi, and RHS75 × 45 × 1.5– 1450-Mi failed in local–global interactive buckling, with visible local buckling waves developed along the length of specimens before reaching the ultimate load. Specimen SHS100 × 3–1000 showed local buckling on the compression side flange at the mid-height just after the ultimate load. Failure mode for specimen RHS75 × 45 × 1.5–1450Ma is shown in Fig. 8. 3. Design method for cold-rolled columns This section comprises two parts. In the first part, a new proposed design method for columns considering both the cold-forming effect and strain hardening is presented, which would be used in the revised Chinese Technical Specification for Stainless Steel Structures (abbreviated as Chinese Specification) [16]. The second part presents a design method for columns introduced in the current Design Manual for Structural Stainless Steel (4th edition)(abbreviated as Design Manual) [11], which incorporates the most recent research results, and enables the use of the enhanced material properties and strain hardening. It should be mentioned that the Cold-formed stainless steel structures AS/NZS 4673:2001 [15] allows us to utilize the material enhancement in the design of columns as well. However, the equations given in that design method are applicable only to ferritic stainless steel, which is used to a limited extent in tests and applications. Thus, it is not discussed here. 3.1. Proposed design method The development of the proposed design method for columns was mainly based on the relevant provisions in the Chinese specification, which is currently under revision. To this end, design formulae for predicting the enhanced material properties in cold-formed stainless

Fig. 3. Typical failure mode of stub columns. (a) SHS 100 mm × 3 mm (b) RHS 100 mm× 50 mm × 3 mm

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B. Zheng et al. / Journal of Constructional Steel Research 170 (2020) 106072

500 S40×2-1 S40×2-2 S40×2-3 S70×3-1 S70×3-2 S70×3-3

Load / kN

400 300 200 100 0

0

1

2

3

4

5

6

7

8

End shortening / mm

(a) SHS 40 mm × 2 mm and SHS 70 mm × 3 mm 500

Load / kN

400 300

S100×50×3-1 S100×50×3-2 S100×50×3-3 S100×3-1 S100×3-2 S100×3-3

200 100 0

0

1

2

3

4

5

6

7

8

End shortening / mm

(b) RHS 100 mm × 50 mm × 3 mm and SHS 100 mm × 3 mm 500

Load / kN

400 300 S80×3-1 S80×3-2 S80×3-3 S80×3-4 S75×45×1.5-1 S75×45×1.5-2 S75×45×1.5-3

200 100 0

0

1

2

3

4

5

End shortening / mm

(c) SHS 80 mm× 3 mm and RHS 75 mm × 45 mm × 1.5 mm Fig. 4. Load-end shortening curves for various stub columns.

steel cross-sections have been supplemented, and design formulae for local buckling, global buckling, and local–global interactive buckling in the Chinese specification have been modified. The capacity of column over the complete range of cross-sections and member slenderness values is clarified.

3.1.1. Enhanced material properties For the material enhancement, the predictive model suggested in Ref. [12] was adopted, which was based on the material plastic deformation in the cross-section experienced during cold-rolling. Furthermore, it was similar in concept to the one proposed in Ref. [10]. The material properties of a point at the center of flat face were conservatively used to represent the averaged material properties of the whole cross-

section. The nominal yield strength σ0.2,cr and the ultimate tensile strength σu,cr were calculated using Eqs.(1) and (2), respectively. 8
 0:5 > > > σ 0:2;cr ¼ σ 0:2;v 1 þ α  ε0:2;cp > > >   > > > qv > > σ 1:5 −σ > u;v 0:2;v > > qv þ 1 > > α¼ pffiffiffiffiffiffiffiffi > > σ 0:2;v  εu;v > > > < 1 t ð1Þ þ 0:03 ε 0:2;cp ¼ > > 2 r RHS > > > > > > h þ b−2t þ ðπ−4Þðr i þ t=2Þ > > > > r RHS ¼ > π > > >   > > ln εu;v =ε1:0;v > > >   : qv ¼ ln σ u;v =σ 1:0;v 8 σ t;u > > σ u;cr ¼ > > 1 þ εu;cr > > > > > εt;u −εu;cp > ε ¼e −1 > > > u;cr <   3 t ln 1 þ ¼ ε > u;cp > 4 r RHS > > > >   > > > σ t;u ¼ σ u;v 1 þ εu;v > > > >   : ε t;u ¼ ln 1 þ ε u;v

ð2Þ

where σ0.2,cr, σu,cr, and εu,cr are the enhanced yield strength, ultimate strength, and its corresponding strain at the center of flat face, respectively; σ0.2,v, σ1.0,v, and ε1.0,v are the yield strength, 1% proof strength, and its corresponding strain, respectively; σu,v and εu,v the ultimate strength and the corresponding strain of the virgin material in the annealed condition, respectively; σt,u and εt,u are the true ultimate stress and the corresponding true strain of the virgin material in the annealed condition, respectively; qv and α are material parameters of the virgin material in the annealed condition; ε0.2,cp and εu,cp are the equivalent plastic strains for the yield strength and the ultimate strength, respectively, which are used to represent the complex distribution of plastic strain in the thickness direction at the calculation point (i.e. the center of flat face); h, b, t, and ri are the height, width, thickness, and the corner inner radius of the cross-section, respectively; and rRHS is the minimum radius experienced by the calculation point during cold-rolling. A comparison between the predictions from Eqs. (1) and (2), and the test results presented earlier in Table 2 is shown in Table 7. It can be seen that the predictions agree with the test data in general, although with a comparatively large scatter owing to the small number of specimens. The details of the validation of Eqs. (1) and (2) with a larger database are reported in [12]. 3.1.2. Local and global instabilities For local buckling, the effective width method is used in the Chinese specification, wherein the maximum load that a stub column could sustain is the squash load of the cross-section (Ny = σ0.2 × A), irrespective of how stocky the cross-section is. The effective width ratio curve is expressed in terms of three stages, with the maximum value being less than 1.0. According to the current Chinese specification, the capacity of a stub column subjected to local buckling is calculated using Eqs. (3)–(5). X Nus ¼ Aeff σ 0:2 ¼ ρi bi t i σ 0:2 ð3Þ 8 qffiffiffiffiffiffi > > 2:41−1:63 λp ≤1:0 λp ≤1:0 > > > > > > > < λp 1:0≤λp ≤1:5 ρ ¼ 3:15λ −1:86 p > > > > > > 0:65 > > > 1:5 ≤λp ≤5:0 : 0:09 þ λp

ð4Þ

B. Zheng et al. / Journal of Constructional Steel Research 170 (2020) 106072

7

Table 5 Measured geometric dimensions, imperfection amplitudes, and ultimate loads for long column specimens in Batch-1. Specimen

SHS70 × 3–1000 SHS70 × 3–1500 SHS70 × 3–2000 SHS70 × 3–2500 SHS70 × 3–3000 SHS70 × 3–3500 SHS100 × 3–1000 SHS100 × 3–1500 SHS100 × 3–2000 SHS100 × 3–2500 SHS100 × 3–3000 SHS100 × 3–3500

Length

Imperfection

Ultimate state

L /mm

e0 /mm

Nu /kN

Δm /mm

Δs /mm

1000 1500 2000 2500 3000 3501 1000 1501 2001 2500 3000 3500

0.10 1.00 0.60 3.00 2.50 4.50 0.70 1.60 0.80 1.60 1.80 2.70

326.20 230.42 179.16 129.13 85.83 59.82 442.78 419.68 340.27 258.57 206.09 164.17

9.26 10.38 10.05 15.08 37.72 83.06 4.42 4.92 13.05 23.21 28.12 44.11

5.95 4.61 4.17 3.91 4.22 8.26 5.99 6.95 7.03 6.45 6.35 6.07

L/10000 L/1500 L/3333 L/833 L/1200 L/778 L/1428 L/938 L/2501 L/1563 L/1667 L/1296

Failure mode

Global Global Global Global Global Global Global Global Global Global Global Global

Table 6 Measured geometric dimensions, imperfection amplitude, and ultimate load for long column specimens in Batch-2. Specimens

Length L /mm

e0 /mm

SHS80 × 3–950 SHS80 × 3–1450 SHS80 × 3–1950 SHS80 × 3-1950a SHS80 × 3–2450 SHS80 × 3–2950 SHS80 × 3–3450 RHS75 × 45 × 1.5–950-Ma RHS75 × 45 × 1.5–1450-Ma RHS75 × 45 × 1.5-1450a-Ma RHS75 × 45 × 1.5–1950-Ma RHS75 × 45 × 1.5–2450-Ma RHS75 × 45 × 1.5–2950-Ma RHS75 × 45 × 1.5-2950a-Ma RHS75 × 45 × 1.5–3450-Ma RHS75 × 45 × 1.5–950-Mi RHS75 × 45 × 1.5–1450-Mi RHS75 × 45 × 1.5–1950-Mi RHS75 × 45 × 1.5–2450-Mi RHS75 × 45 × 1.5–2950-Mi

950 1450 1951 1948 2450 2950 3440 950 1451 1451 1949 2450 2952 2951 3452 954 1451 1950 2451 2951

0.11 0.13 0.21 0.55 0.31 0.87 0.89 0.23 0.11 0.29 0.27 0.27 1.01 1.56 1.15 0.31 0.31 0.61 0.39 1.29

λp ¼

sffiffiffiffiffiffiffiffiffiffi σ 0:2 σ cr;L

Imperfection

Ultimate state

L/8636 L/11154 L/9290 L/3542 L/7903 L/3391 L/3865 L/4130 L/13191 L/5003 L/7219 L/9074 L/2923 L/1892 L/3002 L/3077 L/4681 L/3197 L/6285 L/2288

ð5Þ

where Nus is the capacity of the cross-section; Aeff is the effective area of the cross-section, which is the equal to the gross area minus the sum of the ineffective areas of the slender elements in the cross-section; bi and ti are the width and thickness of the each element, respectively; ρi is the reduction factor of the each element; λp is the slenderness of the element in the cross-section; σ0.2 is the yield strength; and σcr,L is the elastic critical stress for local buckling. In the proposed design method, the three-stage equation (Eq. (4)) was retained; however, the reduction factor was allowed to exceed 1.0 (when λp b 0.75) in the first stage to account for strain hardening after the yielding for the stocky cross-section. The maximum value of the reduction factor would assume the value of the ratio of the ultimate strength to the yield strength, i.e., σu/σ0.2. The modified first stage equation of Eq. (4) is shown in Eq. (6). To consider the enhanced material properties owing to cold-forming, σ0.2 and σu in Eqs. (3), (5), and (6) could be replaced by σ0.2,cr and σu,cr from Eqs. (1) and (2), respectively. qffiffiffiffiffiffi σu λp ≤ 1:0 ð6Þ ρ ¼ 2:41‐1:63 λp ≤ σ 0:2 When global buckling is required to be evaluated, Perry formula is used in the Chinese specification, whereby the effect of the variation

Failure mode

Nu /kN

Δm /mm

Δs /mm

368.11 293.77 220.18 217.49 180.70 132.83 104.57 105.16 94.10 98.58 69.42 54.01 43.39 37.56 31.57 93.35 58.95 46.38 30.08 22.60

3.71 7.61 14.64 16.87 10.74 21.28 17.94 1.43 3.85 5.00 16.49 16.32 16.39 24.95 29.58 1.70 12.42 6.30 18.71 15.56

4.23 5.02 4.56 4.74 3.94 3.82 3.22 3.40 3.56 4.12 3.88 3.22 2.75 3.18 4.46 1.65 2.52 1.57 1.81 1.51

Global Global Global Global Global Global Global Local–Global Local–Global Local–Global Global Global Global Global Global Local–Global Local–Global Global Global Global

in the material properties on the column curve is ignored for simplicity. The formulae for the reduction factor after accounting for the global buckling are as shown in Eqs. (7) and (8). 8 1 > > χ¼ h i ≤ 1:0 > > 2 0:5 < 2 ϕ þ ϕ −λm > >
 > > : ϕ ¼ 0:5 1 þ α λ −λ  þ λ 2 m m m0

λm ¼

sffiffiffiffiffiffiffiffiffiffiffi σ 0:2 σ cr;G

ð7Þ

ð8Þ

where χ is the reduction factor of the column considering the global buckling mode; λm is the member slenderness; σ0.2 is the yield strength, which takes σ0.2,cr from Eq. (1) when enhanced material properties are considered; α is the imperfection factor, which assumes a value of 0.60 for the RHSs; λm0 is the limiting slenderness, which assumes a value of 0.56 for the RHSs; and σcr,G is the elastic critical stress for global buckling based on the gross cross-sectional properties. In the proposed design method, Eqs. (7) and (8) were employed; however the parameters in Eq.(7) were refitted based on the collected test data for failure in global buckling. The imperfection factor α and

8

B. Zheng et al. / Journal of Constructional Steel Research 170 (2020) 106072

Fig. 5. Test rig for long columns.

the limiting slendernessλm0 were taken to be 0.55 and 0.51, respectively in the proposed design method. In the above discussions, the formulae for columns subjected to pure local buckling and pure global buckling were dealt with. In actual design applications, a column may fail in local–global interactive buckling in addition to pure local and the pure global buckling. The reduction factor accounting for local buckling is directly related to the cross-sectional slenderness, as shown in Eq. (4) and (6), while the reduction factor accounting for global buckling is directly related to the member slenderness as shown in Eq. (7). Thus, the cross-sectional slenderness and the member slenderness together could determine the failure mode and the capacity of a specific column. In the proposed design method for columns, the failure mode and the capacity of the column with each possible pair of cross-sectional slenderness and member slenderness would be given. The whole space covered by the cross-sectional slenderness and member slenderness was divided into four regions based on the possible failure modes, and they are discussed below.

3.1.2.1. Region one. For a short column (χ = 1 and λm ≤ 0.51, which is the limiting slenderness for the global bucking (Eq. (7)) with a stocky crosssection (Aeff/A ≥ 1.0), the capacity would be higher than the squash load of the cross-section. As the member slenderness becomes smaller, the column capacity becomes higher. The maximum value of the column capacity is assumed to be its cross-sectional capacity, which is only dependent on the cross-sectional slenderness. A linear expression is used to connect the squash load (λm =0.51) and the cross-sectional capacity ( λm =0.10), as shown in Eq.(9). It should be noted that the minimum member slenderness assumes a value of 0.10, which approximately corresponds to the slenderness of the stub column (fixed boundary condition) with a length 20 times the least radius of gyration

of its gross cross-section [13], i.e., 0.1 ≈ (σ0.2/σcr,G)0.5 = (σ0.2/(π2Ei2/ (0.5 × 20i)2))0.5 = 10(σ0.2/E)0.5/π. A similar linear expression was used by Rossi et al. [25] to extend the column capacity beyond the squash load of the cross-section.  Nus −Ny  λm −0:10 Nu ¼ Nus − λm0 −0:10

ð9Þ

where Nu is the column capacity; Nus is the cross-sectional capacity; Ny is the squash load of the cross-section; and λm0 is the limiting slenderness. 3.1.2.2. Region two. A long column (χ b 1 and λm N 0.51) with a stocky cross-section (Aeff/A ≥ 1.0), fails in global buckling. Arrayago et al. [37] carried out a deep study on the criterion to utilize strain hardening after yielding for columns that failed in global buckling, and proposed a criterion as shown in Eq.(10).  

  σu χσ 0:2 1 þ 1−1:29λp −1 ≥σ 0:2 σ 0:2

ð10Þ

This criterion enables some columns with moderate member slenderness and small cross-sectional slenderness to possess a global bucking strength higher than the squash load of the cross-section, i.e., utilizing strain hardening after yielding for global buckling. This criterion is reasonable; however it involves significant amount of calculations. In the proposed design method, the strain hardening after yielding was ignored for columns that failed in global buckling. It is worth noting that in this study, the cross-sectional slenderness λp and the effective area Aeff should be calculated based on the global buckling

B. Zheng et al. / Journal of Constructional Steel Research 170 (2020) 106072

350

SHS70mm×3mm

300

Load / kN

250 200

1500

150

2000 2500 3000

100 50 0

Load / kN

1000

3500

0

2

4

6

8

10

12

450 400 SHS100mm×3mm 350 300 250 200 150 100 50 0 0

2

400

1950

150

2450

100

2

4

3500

6

8

10

12

1450

1950

60

2450 2950

40 2950a

0 0

3000

1450a

3450

20

2950

50 0

Load / kN

Load / kN

1950a

200

2500

RHS75mm×45mm×1.5mm-Ma

80

1450

250

2000

4

950

100

950

300

1500

(b) SHS 100 mm × 3 mm

SHS80mm×3mm

350

1000

End shortening / mm

End shortening / mm

(a) SHS 70 mm × 3 mm

9

6

8

End shortening / mm

10

(c) SHS 80 mm × 3 mm

0

2

4

6

8

End shortening / mm

10

(d) RHS 75 mm × 45 mm × 1.5 mm-Ma

100

RHS75mm×45mm×1.5mm-Mi

950

Load / kN

80 1450

60

1950

40

2450

20 0

2950

0

1

2

3

4

5

End shortening / mm

6

7

(e) RHS 75 mm × 45 mm × 1.5 mm-Mi Fig. 6. Load–end shortening curves for long columns.

stress χσ0.2, where χ is obtained considering the gross cross-sectional properties. 3.1.2.3. Region three. A short column (χ = 1 and λm ≤ 0.51) with a slender cross-section (Aeff/A b 1.0) fails in local buckling. The capacity of the column in this region should be determined according to Eqs. (3)–(5).

3.1.2.4. Region four. A long column (χ b 1 and λm N 0.51) with a slender cross-section (Aeff/A b 1.0) fails in local–global interactive buckling. Firstly, the reduction factor χ for global buckling is obtained considering the gross cross-sectional properties. Next, the reduction factor ρi for local buckling is calculated based on the global buckling stress χσ0.2 for each plate element, and the effective area Aeff is the sum of the

10

B. Zheng et al. / Journal of Constructional Steel Research 170 (2020) 106072

350 300

Load / kN

200

1500 2000

150 100

2500

3000

50

3500

0

20

40

60

80

Lateral displacement / mm

1000

2000 2500 3000 3500

0

20

Load / kN

Load / kN

1950

1950a

200

2450

150

2950 3450

100

RHS75mm×45mm×1.5mm-Ma 1450a 1950

60

2450 2950

40

2950a

3450

20

50 0

0 0

10

20

80

1450

80

1450

250

950

100

SHS80mm×3mm

300

60

(b) SHS 100 mm × 3 mm

400 950

40

Lateral displacement / mm

(a) SHS 70 mm × 3 mm 350

SHS100mm×3mm

1500

Load / kN

1000

250

0

450 400 350 300 250 200 150 100 50 0

SHS70mm×3mm

30

40

0

50

10

(c) SHS 80 mm × 3 mm 100

20

30

40

Lateral displacement / mm

Lateral displacement / mm

50

(d) RHS 75 mm × 45 mm × 1.5 mm-Ma

950

RHS75mm×45mm×1.5mm-Mi

Load / kN

80 60

1450 1950

40 2450

20 0

2950

0

10

20

30

40

Lateral displacement / mm

50

(e) RHS 75 mm × 45 mm × 1.5 mm-Mi Fig. 7. Load–lateral displacement curves for long columns.

effective areas in each plate element. The column capacity is calculated by using χAeffσ0.2. It should be noted that the process for calculating the capacity of a member that failed in local-global buckling is identical to that in the current Chinese specification.

Based on the above descriptions, the flow chart of the proposed design method for RHS columns is shown in Fig. 9. According to the flow chart, the contour plot of the nominalized column capacity Nu/Ny in λm  λp space is shown in Fig. 10(a). In this figure, the parameter σu/σ0.2 assumes a value of 1.50 as an example.

B. Zheng et al. / Journal of Constructional Steel Research 170 (2020) 106072

11

A comparison of the test results from this study and the predictions of the proposed method is shown in Fig. 11. The four curves shown in this figure depict the relation between the normalized capacity and the member slenderness for the four cross-sections in the test. For RHS 75 mm × 45 mm × 1.5 mm, the cross-sectional slenderness was calculated based on the elastic local buckling stress of the whole crosssection. It can be seen that, in general, the curves matched the test data well. For the stub columns, the points of test data were above (i.e., higher than) the corresponding curves, while for long columns with member slenderness ranging from 0.8 to 1.2, the curves lay marginally above the test data points. 3.2. Design method for columns in the design manual The Design Manual includes not only the traditional design method for columns, but also the latest methods considering both cold-forming and the strain hardening effects. The Design Manual presents a method, included in Annex B, to predict the enhanced yield strength for coldformed sections. This method can be used in the determination of the cross-section and member capacities. In addition, the Design Manual also proposes a method (CSM) to consider the benefits of strain hardening in the determination of cross-section capacities. This method is included in Annex C. 3.2.1. Enhanced material properties According to the Design Manual, the material yield strength in annealed condition used in common design applications could be replaced by the enhanced yield strength in the cold-rolled RHSs. Owing to the unevenly distributed yield strength in the cross-section after cold-rolling, the averaged enhanced yield strength fya is used to represent the whole cross-section. This yield strength is calculated by considering the yield strength fyf in the flat face and the yield strength fyc in the extended corner region. Thus, the enhanced yield strength is predicted using Eqs. ((11) and (12). 8 f yc Ac þ f yf ðA−Ac Þ > > > f ya ¼ > > A <  n ð11Þ f yc ¼ 0:85K εc þ εp0:2 p > > > > >   : n f yf ¼ 0:85K ε f þ εp0:2 p

Fig. 8. Failure mode of RHS 75 mm × 45 mm × 1.5 mm-1450-Ma.

Several slices of the contour plot are shown in Fig. 10(b) and (c), which depicts the relation of the member slenderness and the cross-sectional slenderness with the normalized capacity of the columns, respectively.

Table 7 Comparison between the predicted material properties and test results. Cross-section

SHS 40 mm × 2 mm SHS 70 mm × 3 mm RHS 100 mm × 50 mm × 3 mm SHS 100 mm × 3 mm Avg. Cov.

Predicted

Predicted/Test

σ0.2,cr/MPa σu,cr/MPa εu,cr

σ0.2,cr/σ0.2 σu,cr/σu εu,cr/δ

366.31 383.37 394.53

654.84 674.41 670.12

0.58 1.02 0.58 0.97 0.52 0.86

0.98 0.98 0.90

0.96 1.07 1.11

408.05

684.35

0.52 0.88 0.93 0.080

0.95 0.95 0.041

1.03 1.04 0.063

8 t > > > εc ¼ > > 2ð2r i þ t Þ > > > > > > t πt > > þ εf ¼ > > 900 2 ð b þ h−2t Þ > > < fy > > K ¼ np > > ε > > p0:2 > > >
 > > > > ln f y =f u > > > > : np ¼ ln ε =ε  p0:2

ð12Þ

u

where fya is the average yield strength; fyc and fyf are the enhanced yield strengths in the corner and the flat region, respectively of the RHS; εc and εf are the strains induced in the corner and the flat region, respectively during cold-rolling; K and np are material parameters calculated using Eq. (12); b, h, t, and ti are the width, height, thickness, and inner radius of the corner region, respectively of the RHS; fy and εp0.2 are the yield strength and its corresponding strain, respectively; and fu and εu are the ultimate strength and its corresponding strain of the virgin material in annealed condition, respectively. It should be noted that the constant multiplier 0.85 in Eq. (11) is the product of two factors, i.e., 0.95 and 0.90 [10]. The factor 0.95 is designed to allow for asymmetry in the stress–strain response. The factor 0.90 is used to maintain the same level of

12

B. Zheng et al. / Journal of Constructional Steel Research 170 (2020) 106072

Fig. 9. Proposed design method for cold-formed RHS columns.

reliability as the current codified guidelines [13], and should be dropped out if an additional safety factor is used later.

f csm ¼ f y þ Esh εy

3.2.2. Local and global instabilities For calculating the capacity of a cross-section subjected to local buckling, two methods are provided in the Design Manual, namely the effective width method and the continuous strength method (CSM). The equations for the effective width method are shown in Eqs. (13) and (14). 8 Af y > > > for cross‐ sections of Classes 1; 2 or 3 > < γ M0 ð13Þ Nc;Rd ¼ > > Aeff f y > > for cross‐ sections of Class 4 : γ M0

Esh ¼

ρ¼

0:772 0:079 − 2 ≤1:0 λp λp

ð14Þ

where Nc,Rd is the capacity of cross-section, which is less than the squash load of the cross-section; γM0 is the partial factor for cross-section, which assumes a value of 1.1 for RHSs. The other parameters have the same meanings as those in the earlier sections. The continuous strength method (CSM), which was initially developed for stocky cross-sections to utilize the considerable strain hardening characteristics of stainless steel, has been extended to calculate the capacity of slender cross-sections [28,29]. The formulae for predicting the capacity of cross-sections are shown in Eqs. (15) to (19). In these formulae, Eq. (19) is designed to build a relation between the crosssectional slenderness λp and the average compression strain εcsm that the specimens could sustain at the ultimate state. After obtaining the average compression strain εcsm, a simplified linear hardening material model (shown in Eqs. ((16), (17), and (18)) is used to predict the average compression stress fcsm at the ultimate state. Next, the compression capacity of the cross-section Nc,Rd can be calculated using Eq.(15), which could be higher than the squash load of cross-section. 8 Af csm > > > for λp ≤0:68 > γ < M0 ð15Þ Nc;Rd ¼ > > εcsm Af y > > for λp N0:68 : εy γ M0

  εcsm −1 εy

ð16Þ

f u −f y C 2 εu −εy

ð17Þ


 εu ¼ C 3 1−f y =f u

εcsm εy

ð18Þ

  8 0:25 C 1 εu > > for λp ≤0:68 > 3:6 ≤ min 15; > εy > > < λp 1 ¼ 0 > > >@ 0:222A 1 > > for λp N0:68 > 1:050 : 1− 1:050 λp λp

ð19Þ

where εcsm and fcsm are averaged stress and strain, respectively that the specimens could sustain at the ultimate state; εy is the yield strain, i.e., fy/E; Esh is the strain hardening modulus, calculated using Eq.(17); C1, C2, and C3 are parameters that depend on the type of material and are equal to 0.10, 0.16, and 1.00, respectively for austenitic stainless steel. The other parameters have the same meanings as they had in the earlier equations. It should be mentioned that λp is the crosssectional slenderness and can be conservatively replaced using the slenderness of the most slender plate in the cross-section. For the capacity of columns subjected to global buckling, Perry formula is adopted in the Design Manual, as shown in Eqs.(20), (21) and (22). ( Nb;Rd ¼ 8 > > >χ ¼ > <

χAf y =γM1

for cross‐ sections of Classes1; 2 and 3

χAeff f y =γM1 for cross‐ sections of Class 4 1

i 2 0:5

≤1

> >
> > : ϕ ¼ 0:5 1 þ α λm −λ



h

2

ϕ þ ϕ −λm

m0

þ λm

2



8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > Af y > > for cross‐ sections of Classes1; 2; and 3 > > < N cr λm ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > Aeff f y > > > for cross‐ sections of Class 4 : Ncr

ð20Þ

ð21Þ

ð22Þ

B. Zheng et al. / Journal of Constructional Steel Research 170 (2020) 106072

13

Fig. 10. Nominalized capacity Nu/Ny for RHS columns in λm  λp space.

where Nb,Rd is the capacity of the columns subjected to global buckling; γM1 is the partial factor for columns and is equal to 1.1 for RHSs; fy is the yield strength and adopts the averaged enhanced yield strength when material enhancement due to cold-forming is considered; α is the imperfection factor and is equal to 0.49 for RHSs; λm0 is the limiting slenderness and is equal to 0.30 for RHSs; and the remaining parameters have the same meanings as they had in the earlier sections. It should be mentioned that the calculation of the member slenderness, λm , in the proposed method and in the Design Manual are different. In the Design Manual, for a slender cross-section of Class 4, the member slenderness is calculated based on the effective area of the cross-section, while in the proposed design method, it is calculated based on the full and unreduced area. Eqs. (20), (21), and (22) are also applicable for column failures in local–global interactive buckling. The effect of local–global interactive buckling on the member capacity was obtained by first reducing the cross-sectional resistance owing to local buckling, and later calculating

the reduction of the member resistance owing to global buckling based on the reduced cross-section. It should be noted that for the local–global interactive buckling, the effective area of cross-section Aeff was calculated using the effective width method and not the CSM method. This implies that the integration of the cross-sectional capacity determined using the CSM method and the member capacity subjected to pure global buckling has not been used in the Design Manual, even though several research studies have been conducted [27,41].

4. Comparison of results and reliability analysis In this section, a comparison of the test data obtained in this study and data collected from the literature with the predictions of the proposed design method, as well as the method laid out in the Design Manual is presented. Next, a reliability analysis was conducted to obtain the resistance factor for each design method.

14

B. Zheng et al. / Journal of Constructional Steel Research 170 (2020) 106072

Fig. 11. Comparison of the predictions of the proposed method with the test data in this study.

Table 8 Database of cold-formed stainless steel RHS columns. Year

Ref.

Grades

Cross-section

Long column

Stub column

1992 1995

[1] [2]

2002

[3,43]

1.4307 (304 L) 1.4301 (304) 1.4301 (304) 1.4301 (304)

2003 2003 2005 2006

[44] [45] [47,48]

1.4301 (304) 1.4301 (304) Duplex

SHS SHS RHS SHS RHS SHS RHS SHS RHS SHS RHS SHS RHS SHS RHS SHS RHS SHS RHS SHS SHS RHS SHS RHS SHS SHS RHS RHS SHS SHS SHS RHS SHS RHS

6 3 6 8 14 8 16 12 6 – – 12 – 6 6 10 22 3 8 4 – – – – – – – – – – 2 3 19 13 187

2 1 2 17 16 4 8 4 2 2 1 4 4 6 2 2 4 2 4 2 10 9 2 6 1 1 1 1 1 2 – – 9 3 135

HAS 2006

[49]

1.4318 (301LN)

2009

[50]

1.4162 (S32101)

2013

[42]

1.4162 (S32101)

2013

[31]

1.4003 (3Cr12)

2013

[40]

1.4509 (441) 1.4301 (304)

2015

[51]

2015

[52]

2016 2016

[53] [54]

1.4003 (3Cr12) 1.4003 (3Cr12) 1.4301 (304) 1.4571 (316Ti) 1.4307 (304 L) 1.4404 (316 L) 1.4162 (S32101) 1.4509 (441) 1.4003 (3Cr12)

2019

This paper

1.4301 (304)

Total

4.1. Comparisons Test data on cold-formed stainless steel RHS columns were collected from the literature and summarized in Table 8. A total of 187 long columns and 135 stub columns were obtained. The material includes 10 grades in three commonly used classes, i.e. austenitic, ferritic, and duplex stainless steel. Most of the test specimens were made of austenitic stainless steel.

The predicted capacity of each column was calculated using the proposed design method and the one in the Design Manual. To demonstrate the advantage of the proposed design method, three different types of material properties were used for each specimen, i.e., the material properties of the test specimens from the corresponding test reports (named tested material properties), the material properties of the virgin sheet in the annealed condition given in the Design Manual (named virgin material properties), and the predicted material properties considering the effect of cold-forming (named predicted material properties). The material properties in the annealed condition given in the Design Manual for the grades in the test database are shown in Table 9. These material properties were used as the virgin material properties not only to calculate the column capacity, but also to predict the enhanced material properties owing to cold-forming. In the Design Manual, E0, σ0.2, σu, and n are directly given, while the parameters εu and m are not provided. Thus, these two parameters were approximated using Eqs. (23) and (24) given in Annex C in the Design Manual, respectively. For austenitic stainless steel except for “HAS” in Refs [47,48]., the value of σ1.0 in Table 9 was taken from EN 10088–4 [46]. However, for other grades, the value of σ1.0 is not given in EN 10088–4 and hence, it was calculated based on the material model given in Annex C in the Design Manual. As the exact grades were unknown for “HAS” and “duplex stainless steel” in Refs [47,48]., a material property value of 1.4662 was used based on the similarity in the values of the material strength. 8 σ 0:2 > for austenitic and duplex stainless steel < ε u ¼ 1− σ u  σ 0:2 > : ε u ¼ 0:6 1− for ferritc stainless steel σu m ¼ 1 þ 2:8

σu σ 0:2

ð23Þ

ð24Þ

During the calculations of the column capacity, the following assumptions were made: (1) Partial factors, i.e., γM0 for cross-sections, and γM1 for members in the Design Manual were ignored. (2) If the material properties both in tension and compression were available in the corresponding test reports, the ones from the tensile coupon test were used. (3) For Refs [44,45,48]., the effective length for the column with fixed boundary conditions was equal to half of the specimen length. In addition, all the stub column tests were considered to be conducted with the fixed boundary condition. (4) In calculating the local–global buckling capacity according to the Design Manual, the effective area in Eqs.(20) and (22) for the global and the local–global buckling was predicted using the effective width method (Eqs.(13) and (14)) based on the global buckling stress, considering the gross cross-sectional properties. (5) The CSM method in the Design Manual was used in determining the capacity of the cross-section. When using the CSM method, the cross-sectional slenderness λp (in Eq.(19)) was calculated using the elastic local buckling stress of the full cross-section, which was predicted using Eq.(25) [55].
 8 F SS > σ cr;L ¼ σ SS > cr;w þ ζ σ cr;w −σ cr;w > > > αw > > ζ ¼ 0:53− > > < ϕ   h > α w ¼ 0:63−0:1 > > b > >  2 > > > h−t > :ϕ ¼ h≥b b−t

ð25Þ

where σcr,L is the elastic local buckling stress of the full cross-section; F σSS cr,w and σcr,w are the elastic local buckling stress of web in compression

B. Zheng et al. / Journal of Constructional Steel Research 170 (2020) 106072

15

Table 9 Virgin material properties from the Design Manual for structural stainless steel (4th edition). Types

Grades

E0/MPa

σ0.2/MPa

σ1.0/MPa

σu/MPa

εu

n

m

Austenitic

1.4307 (304 L) 1.4301 (304) 1.4318 (301LN) 1.4571 (316Ti) 1.4404 (316 L) HAS Duplex 1.4162 (S32101) 1.4003 (3Cr12) 1.4509 (441)

200,000 200,000 200,000 200,000 200,000 200,000 200,000 200,000 200,000 200,000

220 230 350 240 240 550 550 530 280 230

250 260 380 270 270 608 608 583 319 265

520 540 650 540 530 750 750 700 450 430

0.58 0.57 0.46 0.56 0.55 0.27 0.27 0.24 0.23 0.28

7.00 7.00 7.00 7.00 7.00 8.00 8.00 8.00 14.00 14.00

2.18 2.19 2.51 2.24 2.27 3.05 3.05 3.12 2.74 2.50

Duplex Lean duplex Ferritic

under simply-supported and fixed boundary conditions, respectively; ζ is the interaction coefficient; h, b, and t are the height, width, and thickness of the cross-section, respectively; and αw and ϕ are intermediate parameters. (6) When using the effective width method (for both the proposed method and the one from the Design Manual), the corner region was assumed to be fully effective. The width of the flat face was assumed to be b–2ro, where b is the width and ro is the outer radius of the corner region. The results from the calculations are shown in Table 10. Here, “CSM”, “DM”, and “CECS” imply the ratio of the test values to the predicted values obtained using the CSM method, design method from the Design Manual, and the proposed design method, respectively; “Tested Material Pro.”, “Virgin Material Pro.”, and “Predicted Material Pro.” indicate that the calculations were based on the tested material properties, virgin material properties, and predicted enhanced material properties, respectively; Pm and Vp are the mean values of the tested-to-predicted ratio and the coefficient of variation, respectively; N is the number of the test; β0 is the target reliability; and ϕ is the corresponding resistance factor. It should be mentioned that the test specimens were separated into four regions according to the proposed method, and then the column capacities were calculated according to the DM and CECS methods.

Table 10 Comparison of test results with the predictions of the design methods based on different material properties. Regions

Tested material pro. CSM DM

CECS CSM DM

CECS CSM DM

CECS

Region 1 N Strain hardening Pm Vp β0 ϕ Region 2 N Global Pm Vp β0 ϕ Region 3 N Local Pm Vp β0 ϕ Region 4 N Local-Global Pm Vp β0 ϕ ALL N Pm Vp β0 ϕ

87 1.20 0.08 2.50 1.08 – – – – – 48 1.08 0.07 2.50 0.98 – – – – – – – – – –

99 1.08 0.06 2.50 0.98 127 1.02 0.10 2.50 0.90 64 1.00 0.06 2.50 0.91 32 0.99 0.08 2.50 0.89 322 1.03 0.09 2.50 0.92

139 1.57 0.19 2.50 1.24 126 1.22 0.13 2.50 1.04 42 1.21 0.17 2.50 0.98 15 1.12 0.14 2.50 0.93 322 1.36 0.21 2.50 1.03

109 1.19 0.13 2.50 1.02 125 1.08 0.10 2.50 0.95 56 1.06 0.11 2.50 0.92 32 1.01 0.10 2.50 0.89 322 1.11 0.13 2.50 0.95

99 1.28 0.12 2.50 1.10 127 1.08 0.10 2.50 0.95 64 1.13 0.06 2.50 1.03 32 1.10 0.07 2.50 0.99 322 1.15 0.13 2.50 0.99

Virgin material pro.

106 1.88 0.25 2.50 1.34 – – – – – 29 1.36 0.17 2.50 1.08 – – – – – – – – – –

139 2.03 0.28 2.50 1.36 126 1.32 0.15 2.50 1.09 42 1.42 0.19 2.50 1.10 15 1.33 0.17 2.50 1.05 322 1.64 0.32 2.50 1.01

Predicted material pro.

93 1.49 0.15 2.50 1.23 – – – – – 42 1.26 0.12 2.50 1.09 – – – – – – – – – –

109 1.61 0.19 2.50 1.26 125 1.21 0.13 2.50 1.03 56 1.33 0.12 2.50 1.14 32 1.23 0.11 2.50 1.08 322 1.37 0.20 2.50 1.05

For the CSM method, only the test data of the stub column were considered. From Table 10, the following can be inferred. 4.1.1. For region 1 (1) When using the virgin material properties, the predictions from all the three methods were very conservative. The average ratio of the test result to the prediction reaches up to 2.03, with a large scatter of 0.28. When using the test material properties, the accuracy of all the three methods was improved significantly. For example, the averaged test-to-prediction ratios of the proposed method were 1.57 and 1.08, when the calculations were based on the virgin and tested material properties, respectively. When the predicted enhanced material properties were used, the accuracies of all the three methods were better than that obtained using the virgin material properties, but worse than that obtained using the tested material properties. From a comparison of the results with the predicted material properties and the results with the tested material properties, it can be seen that the predictions using the tested material properties agreed better with the test data than the other, with the average ratio being slightly higher than 1.0, and with a small scatter. This was because the process of predicting the enhanced material properties introduced some variations in the whole design method. (2) The difference between the results based on the predicted material properties and the virgin material properties revealed the benefit of including the material enhancement owing to coldforming. The accuracies were improved by 26% (which was calculated as 1.88/1.49–1), 26%, and 32% for the CSM method, the effective design method from the Design Manual, and the proposed method, respectively. Thus, the material enhancement from the cold-forming effect should be considered in the design to reduce the waste of material. (3) The effect of considering the strain hardening after yielding could be shown by the difference between the results from the CSM method and the effective method in the Design Manual. The accuracies were improved by 7% (which was calculated by 1.28/ 1.20–1), 8%, and 8% when using the tested, virgin, and predicted material properties, respectively. The benefit of considering the material enhancement was larger than that of including the strain hardening after yielding. 4.1.2. For regions 2, 3, and 4 (1) For these three regions, similar conclusions could be drawn for the results based on different material properties, i.e., the order of accuracy of predictions from high to low was the one based on the tested material properties, the one based on predicted enhanced material properties, and the one based on the virgin material properties. (2) Generally, for each combination of material properties and design method, the accuracies for Regions 2 to 4 were better than

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B. Zheng et al. / Journal of Constructional Steel Research 170 (2020) 106072

global interactive buckling, and for columns with comparatively stocky cross-section in the future. (2) In the four regions, the proposed design method showed similar performance for member failures under global, local, and local– global buckling, while it yielded more conservative results for member failures in the strain hardening region. 4.2. Reliability analysis

Fig. 12. Comparison of test results and predictions of the proposed design method based on predicted enhanced material properties.

those for Region 1. For example, the average predicted ratios of the method in the Design Manual based the tested material properties were 1.08, 1.13, and 1.10 for Regions 2 to 4, respectively, which were better than that for Region 1, which was 1.28. (3) For the results based on the same type of material (for example, the predicted material properties), the calculated results from the proposed method had similar averaged ratio in predicting the member capacity subjected to global buckling (1.08) and local buckling (1.06), and a little lower for the local–global buckling (1.01), while the calculation results of the method from the Design Manual had similar averaged ratio in predicting the member capacity subjected to global buckling (1.21) and local–global buckling (1.23), and a little higher for the local buckling (1.33). (4) A comparison of the results with the predicted enhanced material properties and those with the virgin material properties showed that the accuracy for predicting the global buckling capacity were improved by 9% (which was calculated as 1.32/ 1.21–1), and 13% for the design method in the Design Manual and the proposed design method, respectively. The corresponding numbers were 7% and 14% and (8% and 11%) for local buckling and local–global buckling. Thus, the enhancement in the material properties owing to cold-forming had more influence on the capacity of the member in strain hardening in Region 1 than the in the other three regions. 4.1.3. For all regions Considering all the test data, the predictions from the proposed design method had better accuracy owing to the inclusion of the benefit of both cold-forming and strain hardening. The CSM method could provide good predictions for the capacity of cross-sections. Fig. 12 shows the distribution of test data and the test-to-prediction ratio from the proposed design method based on the predicted enhanced material properties. From the results shown in Fig. 12 and Table 10, it can be inferred that: (1) Most of specimens in the database failed in global buckling (Region 2) and material strain hardening (Region 1). Only 32 specimens were in the region of local–global buckling failure mode (Region 4). Most of specimens had cross-sectional slenderness larger than 0.20. Thus, although 322 tests were conducted, more tests should be carried out for column failures in local–

In this section, the reliability of the proposed design method and the one in the Design Manual are presented through statistical analysis according to the provisions in SEI/ASCE-8-02 [14]. A minimum reliability index of 2.5 was used for structural members [56]. In SEI/ASCE-8-02 [14], the reliability index was calculated based on a load combination of 1.2DL + 1.6LL, where DL and LL were the dead load and live load, respectively. The load ratio of the dead load to the live load was 1:5. The mean values of the material factor and the coefficient of variation of the nominal yield strength assume values of 1.1 and 0.10, respectively. For the fabrication factor, the mean value and the coefficient of variation were 1.00 and 0.05, respectively. The resistance factors ϕ at the target reliability index of 2.5 were calculated, and results are shown in Table 10. It can be seen that: (1) For the proposed design method, when using the virgin material properties, the resistance factors ϕ for the four regions were 1.24, 1.04, 0.98, and 0.93, respectively. When using the predicted enhanced material properties to consider the cold-forming effect, the resistance factors ϕ decreased for all the four regions. A maximum value of 1.24 was obtained in Region 1 strain hardening, while a minimum value of 0.89 was obtained for Region 4 local–global buckling. (2) For the design method in the Design Manual, when the design of columns was based on the virgin material properties or the predicted enhanced material properties, the resistance factors ϕ were all higher than 1.0. The maximum value of the resistance factors in Region 1 strain hardening reached 1.36 when virgin material properties were used. As the partial factor γM0 for the cross-section and γM1 for members were approximately equal to 1/ϕ [44,45]; the values of the partial factor γM0 (1.1 for RHSs) and γM1(1.1 for RHSs) in the Design Manual were too big, which would make the design of columns more conservative. (3) When the design of the columns was based on the predicted enhanced material properties, the minimum resistance factors ϕ in the four regions, i.e., 0.89 and 1.03 (or the partial factors γM of 1.12 and 0.97) were recommended for the proposed design method and the one in the Design Manual, respectively. 5. Conclusions This study reported a series of tests on cold-formed stainless steel RHS columns. A design method for columns considering both the cold-forming effect and the strain hardening was proposed and evaluated with test data. The following conclusions could be drawn: (1) A total of 19 stub column and 32 long column specimens comprising six cross-sections were tested. The failure modes of the test specimens included the local buckling, global buckling, local–global buckling, and material strain hardening after yielding. Combined with the test results in the companion studies [12,39], a full set of test data were built, including the information of the cold-rolling fabrication method, material properties of the virgin sheet and final cross-section, residual stresses in cross-section, and behavior of the column. (2) A new design method for columns considering both the coldforming effect and strain hardening was developed, which could provide clear and accurate capacity predictions of columns

B. Zheng et al. / Journal of Constructional Steel Research 170 (2020) 106072

in the member and the cross-sectional slenderness space. The effective width method was used to account for local buckling, wherein the maximum capacity of the cross-section was extended to Aσu. Perry formula was used for column failures in global buckling. For the members with member slenderness in the range of [0.1, λm0 ], a linear formula was used to link the squash load of the cross-section (when λm ¼ λm0) and the capacity of cross-section (when λm ¼ 0:1). (3) Predictions according to the proposed design method and the one in the Design Manual were compared with the test data in this study and data collected from the literature (a total of 187 long columns and 135 stub columns). Predictions based on the virgin material properties were very conservative and the maximum averaged test-to-prediction ratios reached 2.03 for members in Region 1. As the predicted material properties owing to cold-forming were included, the performance of the design methods was significantly improved. On an average, the predictions of the design method in the Design Manual were more conservative than those of the proposed design method. Although 322 columns were tested, the number of test specimens that failed in the local–global interactive buckling was only 32, and more tests should be conducted. (4) A reliability analysis was conducted according to SEI/ASCE 8–02. When the tested material properties, virgin material properties, and predicted material properties were used in the calculations, the resistance factors for the design method in the Design Manual were all higher than 1.0. For the proposed design method, the resistance factors were less than 1.0 when the design was based on the tested material properties and the predicted material properties. When the design of columns was based on the predicted material properties considering the cold-forming effect, resistance factors ϕ of 0.89 and 1.03 were recommended for the proposed design method and the one in the Design Manual, respectively.

Declaration of Competing Interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgement The research work described in this paper is supported by National Science Foundation of China through the projects No. 51578134, No. 51808110 and No. 51608234, Jiangsu Science Foundation through the projects No. BK20180399 and BK. 20160543, National Key Technologies R&D Program through the projects No. 2018YFC0705502-4. The financial supports are highly appreciated. Special thanks to the Jiangsu Dongge Stainless Steel Ware Co., Ltd. and Xinghua Zhaohui Stainless Steel Tube Company for providing the test specimens. References [1] K.J.R. Rasmussen, G.J. Hancock, Design of cold-formed stainless steel tubular members. I: columns, J. Struct. Eng. ASCE 119 (8) (1993) 2349–2367. [2] A. Talja, P. Salmi, Design of stainless steel RHS beams, columns and beam-columns, Research Note 1619, VTT Building Technology, Finland, 1995. [3] L. Gardner, D.A. Nethercot, Experiments on stainless steel hollow sections-part 1: material and cross-sectional behaviour, J. Constr. Steel Res. 60 (2004) 1291–1318. [4] M. Ashraf, L. Gardner, D.A. Nethercot, Strength enhancement of the corner region of stainless steel cross-sections, J. Constr. Steel Res. 61 (2005) 37–52. [5] R.B. Cruise, L. Gardner, Strength enhancements induced during cold forming of stainless steel sections, J. Constr. Steel Res. 64 (2008) 1310–1316.

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