Design rules for stainless steel welded I-columns based on experimental and numerical studies

Engineering Structures 172 (2018) 850–868

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Design rules for stainless steel welded I-columns based on experimental and numerical studies Shameem Ahmeda, Safat Al-Deena, Mahmud Ashrafb, a b

T

⁎

School of Engineering and Information Technology, The University of New South Wales, Canberra, ACT 2600, Australia Deakin University Geelong, School of Engineering, VIC 3216, Australia

A R T I C LE I N FO

A B S T R A C T

Keywords: Buckling stress Continuous strength method Perry curve Stainless steel Welded I-section

Stainless steel is characterised by its nonlinear stress-strain behaviour with significant strain hardening, although current design codes treat it as an elastic, perfectly plastic material like carbon steel. The continuous strength method (CSM) is a newly developed strain based design approach which was proposed for nonlinear metallic materials. With recent developments, CSM can be used to predict the cross-section resistance for stocky and slender sections, and CSM design rules have recently been proposed for predicting the buckling resistance of cold-formed RHS and SHS columns. Welded sections, however, could behave differently from cold-formed sections due to the presence of residual stresses. Despite offering more economic options in many design cases, research on stainless steel welded sections is very limited to date. In this study, the behaviour of stainless steel welded I-sections was investigated through a test program, and the investigation was complemented by finite element (FE) modelling. The test program covered tensile coupon tests, residual stress and initial geometric imperfection measurements, stub column tests and flexural buckling tests of pin-ended long columns. FE models were developed for both major and minor axis buckling based on test results, and the verified FE modelling technique was used to investigate the effects of cross-section slenderness λp, section height-to-width ratio H/ Band the ratio of flange thickness-to-web thickness tf/tw on column curves of welded I-sections. Buckling formulas for welded I-columns were eventually proposed following the same philosophy recently adopted by the authors for cold-formed hollow section columns. The imperfection parameter was recalibrated appropriately to incorporate special features of welded I-sections. Two sets of equations were proposed to tackle the observed variation in buckling behaviour against major and minor axis buckling. Buckling resistance predictions obtained from the proposed method were deemed reliable showing good accuracy and consistency with test and FE results.

1. Introduction Welded sections are often used to meet the high load bearing capacity required for buildings and bridges as this type of sections can be fabricated to meet the exact design requirements, and may yield more economic design solution for a structure. During the last decade, research on structural stainless steel was mainly focused on cold-formed sections due largely to their easy availability. Limited number of studies were conducted on welded sections, and very few design codes [1,2] have design guidelines for welded sections. In recent years, a number of research projects were reported on stainless steel welded sections. Kuwamura [3], Saliba and Gardner [4] and Yuan et al. [5] studied the local buckling behaviour of stainless steel welded I-sections. Real et al. [6], Saliba and Gardner [7], Hassanein [8] and Fortan et al. [9] studied the shear response of stainless steel plate girders. Wang et al. [10] and

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Yang et al. [11] investigated the lateral torsional buckling of stainless steel welded I-section beams. Yuan et al. [12] measured the residual stresses of welded box sections and I-sections, and observed that the magnitudes and the distribution of longitudinal residual stresses of stainless steel welded sections were different from those observed in carbon steel welded sections. They also proposed a model for residual stress distribution in stainless steel welded sections. Investigation on the compression resistance of stainless steel welded sections is scarce. Recently, Yuan et al. [13] studied the local-overall interactive buckling of welded box sections by testing eight specimens. Yuan et al. [14] also tested welded I-section columns produced from austenitic and duplex grades of stainless steel to study a similar behaviour. They also performed numerical analysis, and observed that residual stresses significantly affect the buckling resistance of welded I-section columns. Yang et al. [15] tested stainless steel welded I-section columns for

Corresponding author. E-mail addresses: [email protected] (S. Ahmed), [email protected] (S. Al-Deen), [email protected] (M. Ashraf).

https://doi.org/10.1016/j.engstruct.2018.06.080 Received 22 September 2017; Received in revised form 20 June 2018; Accepted 20 June 2018 0141-0296/ © 2018 Elsevier Ltd. All rights reserved.

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tests and flexural buckling tests on austenitic grade stainless steel welded I-sections. Based on the results obtained from this test program, nonlinear FE models were developed and verified, and a comprehensive parametric study was carried out to identify the influential key parameters onthe flexural buckling of welded I-columns. Design formulas were developed using test and FE results for welded I section columns, and finally, the performance of the proposed CSM flexural buckling formulas was verified and compared with other standards.

flexural buckling, and showed that EN 1993-1-4 [1] and AS/NZS 4673 [16] predictions were conservative for predicting the buckling resistance of stainless steel welded I-columns, and ASCE 8-02 [17] predictions were very scattered. It should, however, be noted that AS/NZS 4673 and ASCE 8-02 design rules are proposed for cold-formed stainless steel structures. Recently, Gardner et al. [18] investigated the behaviour of laser welded stainless steel columns for local buckling and flexural buckling, and observed that the carrying capacity of laser welded sections were higher than the conventionally welded sections due to lower residual stress magnitudes. It is evident that there is significant lack of test data for appropriate understanding of the behaviour of stainless steel welded sections, and current design standards produce conservative or erroneous predictions for the buckling resistance of welded sections. This paper aims to fill up the knowledge gaps through an experimental program, and proposes design formulations for stainless steel welded I-section columns based on comprehensive numerical analysis. Current design codes [1,16,17] treat stainless steel like carbon steel ignoring its nonlinear behaviour and, hence, its strain hardening benefits are not fully exploited. For nonlinear metallic materials like stainless steel, a new design technique named the Continuous Strength Method (CSM) [19,20] was proposed. CSM is a strain based design method that incorporates material nonlinearity, exploits strain hardening and incorporates element interactions in predicting resistances at the cross-section level. With the recent development of CSM [21,22], cross-section capacity for both stocky and slender cross-sections can be predicted through simple formulas using a bilinear material model and without calculating effective cross-sectional properties. Therefore, there is a clear scope for using CSM philosophies for predicting the buckling resistance of columns. The buckling resistance of stainless steel columns are normally calculated following two different approaches: tangential stiffness method and Perry formulas. SEI/ASCE8-02 [17] and AS/NZS 4673 [16] codes use the tangential stiffness method, which recognises material nonlinearity through an iterative process to calculate the instantaneous tangent modulus but does not consider geometric imperfections of the member. This method is not applicable for welded sections as there is no provision of considering residual stresses for welded sections. On the other hand, Eurocode [1] follows Perry curves, which is a direct method specifying separate curves for different types of cross-sections based on an imperfection parameter but the technique does not incorporate material nonlinearity. Through numerical analysis, Rasmussen and Rondal [23] showed that different column curves are necessary to predict the buckling resistance of different grades of stainless steel as their nonlinearity varies significantly between grades. Hradil et al. [24] tried to include the material nonlinearity in Perry curves by defining transformed slenderness but their suggested procedure uses tangent modulus, which is iterative. Shu et al. [25] proposed two base curves and some complicated transfer formulas for hollow sections which could be used to develop multiple curves from two base curves to cover different grades of stainless steel. All of the aforementioned methods use effective areas for slender cross-sections. Huang and Young [26] proposed a method using full cross section area with material properties taken from stub column tests to predict the column capacity. Recently Ahmed and Ashraf [27] proposed new buckling formulas for predicting the buckling capacity of cold-formed stainless steel RHS and SHS columns following CSM. This proposal successfully incorporated all the characteristics of stainless steel through simple equations. However, the behaviour of welded sections is different from cold-formed sections due to the presence of residual stresses [13–15]. Further investigations are required to investigate the suitability of the proposed CSM based technique for stainless steel welded sections. In this study, the structural behaviour of stainless steel welded Icolumns are investigated through a comprehensive test program as well as FE analysis. The test program included material test, initial geometric imperfection and residual stress measurements, stub column

2. Test program A test program was conducted to investigate the structural behaviour of stainless steel welded I-sections produced from 316L austenitic grade stainless steel. Flanges and webs were connected by Tungsten Inert Gas (TIG) welding. Most of the recent studies used shielded metal arc welding (SMAW) for fabricating welded sections [5,13–15]. But compared to SMAW, TIG welding offers better quality and precision. TIG welding is aesthetically good with smaller seam size, and the thermal distortion is also significantly low for TIG welded members. Material properties of the considered stainless steel were determined through tensile coupon tests. Three stub column tests were performed to examine the local buckling behaviour of stainless steel welded Isections. To examine the buckling behaviour of the welded members, 16 long columns were tested. Prior to the test, local and global geometric imperfections were measured. In addition, residual stresses of two representative members were also determined by sectioning method. Details of the test program are described in the following sections. Designation system adopted for the considered specimens are as follows: “I D × B × tf × tw−L”, where I stands for I-section, D is the nominal depth of the section, B is the nominal width of the section, tf is the nominal flange thickness, tw is the nominal web thickness and L is the nominal length of column.

3. Tensile coupon test Tensile coupon tests were performed to evaluate accurate material properties for the plate materials used to fabricate the considered Isections. Plates of five different thicknesses were used for different cross-sections, and plate thicknesses varied from 2 to 6 mm. Five coupons, each representing a specific thickness, were cut from 200 × 200 mm plates taken from the same batch as the welded I-columns. Tensile coupons were produced according to EN ISO 6892-1 [28], and all tensile coupons were necked in the middle. Submersible wire cutting technology was used to prepare the test coupons to minimise heat effects during the cutting process. All tensile coupon tests were performed using a Shimadzu Z100 kN electromechanical universal testing machine (UTM), and video extensometer was used to measure the longitudinal strain over a specified gauge length. A linear electrical resistance strain gauge was also attached to the face of each tensile coupon to record more accurate measurements for the initial elastic part of stress-strain curves. In the tests, strain rate was maintained at 0.001/s throughout the test. The plastic strain at fractures was also measured over a gauge length of 5.65 A c , where Ac is the cross-sectional area of the coupon. Key material parameters such as Young’s modulus E, 0.2% proof stress σ0.2 and ultimate tensile stress σu, strain corresponding to the ultimate tensile stress εu, and plastic strain at fracture εpl,f were extracted from the recorded stress-strain curves. The best-fit Young's modulus E was calculated based on the strain gauge measurements. Compound Ramberg–Osgood nonlinearity parameters n and m were also calculated from the strain gauge data. Results obtained from all tested coupons are summarised in Table 1. Stress-strain curves are shown in Fig. 1.

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Table 1 Material properties of austenitic stainless steel plates used to fabricate welded Icolumn specimens. Plate thickness

E MPa

σ0.2 MPa

σu MPa

n

m

εpl,f %

6 mm 5 mm 4 mm 3 mm 2 mm

198,340 197,480 199,445 206,090 201,200

310 301 290 311 309

620.5 611.01 597.8 628.5 648

9.0 8.0 6.5 16.0 25.0

2.1 2.2 2.2 2.1 2.1

54 58 60 65 68

4. Residual stresses observed in stainless steel welded I-sections Residual stresses are unavoidable in welded sections. In this study, residual stresses were measured by sectioning method, which has been widely used for many years and is reported to produce accurate and reliable results. In 1888, Kalakoutsky [29] first proposed this method for measuring longitudinal stresses in steel bars. He slit the bars into longitudinal strips; measured the change in length of each strip before and after slitting, and residual stresses were calculated by applying Hooke’s law. In this study, residual stresses were measured for the following two welded I-sections – I 80 × 80 × 5 × 4–500 and I 60 × 120 × 6 × 4–450. Consistent and uniform welding throughout the member was ensured to obtain reliable results. The strips were cut at least at a distance ‘d’ from both ends to minimise the end effects, where d is the maximum dimension of the cross-section [30]. Each element of the I-section (flange and wed) was divided into nine strips, which were marked by a pair of indentations on each strip over a gauge length of 200 mm using setting out bar to ensure better contact between punch holes and demac gauge. The initial gauge length Li of each strip was measured using a standard 200 mm demac gauge as shown in Fig. 2. All specimens were kept in an environmentally controlled room for 24 h before taking any measurement; the room temperature was fixed at 23 ± 0.5 °C for minimising the influence of temperature change. After recording the initial gauge length Li, flanges and webs were separated from each other, and were subsequently cut into pre-designed strips. All cutting operations were conducted using a submergible wire-cutting machine to avoid additional heat input during the sectioning process, which could otherwise affect residual stresses. All strips cut from within I 80 × 80 × 5 × 4 section are presented in Fig. 3. After sectioning, the final gauge readings Lf were recorded. Using the initial and the final gauge lengths, the relieved longitudinal residual strains were determined using Eq. (1). A negative relived strain indicated a tensile residual stress, whereas a positive relived strain indicated a

Fig. 2. Taking measurements with demac gauge.

Fig. 3. All strips cut from I 80 × 80 × 5 × 4 section to measure residual stresses.

compressive residual stress.

ε=

L f −L i Li

(1)

Due to the presence of through thickness residual stress gradient, strips exhibited longitudinal curvature which was more prominent in strips nearer to the weld. For this curvature, the gauge lengths were chord lengths rather than arc lengths. Chord lengths were corrected to arc length using the measured offset value δ. The corrected relieved

Fig. 1. Stress-strain curves of different stainless steel plates used to fabricate I-sections: (a) complete stress-strain curves, (b) Stress-strain curves enlarged up to 1% strain. 852

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Fig. 4. Residual stress distributions observed in welded I-sections: (a) I 80 × 80 × 5 × 4–500 section, (b) I 60 × 120 × 6 × 4-450 section.

modulus with the measured relieved residual strains. The computed residual stresses and their distribution patterns observed in two stainless steel welded I-sections are shown in Fig. 4. The observed pattern was in-line with the typical distribution reported in welded I-sections; tensile residual stresses developed near the weld and compressive residual stresses developed away from the weld. In I 80 × 80 × 5 × 4–500, the maximum tensile residual stress of 257 MPa was recorded in the web, which is 85.4% of the material yield stress (0.2% proof stress σ0.2). The maximum compressive residual stress of 194 MPa (64.5% of σ0.2) was recorded in the top flange. In the case of I 60 × 120 × 6 × 4–450 section, the maximum tensile residual stress was 94% of σ0.2 and the maximum compressive residual stress was 53% of σ0.2. In both sections, maximum tensile residual stresses in flanges were significantly lower than those observed in the web. 5. Initial geometric imperfection measurements Fig. 5. Measurement of the initial local geometric imperfections of I 120 × 60 × 4 × 2–500 column.

A laser scanner (scanCONTROL 2710-100(500)) was used to measure the initial local geometric imperfections in all stainless steel columns. Column specimens were mounted on the table of a milling machine, which provided a flat reference surface for accurate measurement as shown in Fig. 5. The laser scanner was set on top of the milling machine, and 3D data were collected automatically using a built-in software. The table of the milling machine was moved at a constant rate of 5 mm/s and the resolution of the measurements was 10 Hz; this allowed obtaining cross profiles approximately at 0.5 mm interval. It is worth noting that the milling table was moved manually, which inevitably produced some variability on the speed of the table.

strain εc can be approximated as Eq. (2) where δ/L is the ratio of the offset δ to the initial gauge length Li. The curvature correction may be neglected until this ratio exceeds 0.001 [30].

εc = ε +

(δ/Li)2 6(δ/Li) 4 + 1

(2)

Residual stresses were calculated by multiplying the Young’s 853

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6. Testing of stub columns 6.1. Test procedure Three stub columns were tested in axial compression, whose lengths were taken equal to three to four times the larger nominal dimension of the cross section to ensure representative distribution of geometric imperfection and residual stress but to avoid global buckling [32]. Two 200 × 200 mm plates of 16 mm thickness were welded at the top and the bottom end of stub columns. The geometric dimensions of stub column specimens were measured and are reported in Table 2, where D is the depth of the section, B is the width of the section, tf is the flange thickness and tw is the web thickness. The initial local geometric imperfections W0 were measured prior to testing as outlined in Section 5. Tests were conducted in a Shimadzu 1000 kN hydraulic UTM. The displacement of the bottom plate was measured by four LVDTs, whilst additional four LVDTs were used to measure the displacement of the top base plate. Ten linear electric resistance strain gauges were attached at the mid-length. The schematic diagram of the instrumentation is shown in Fig. 8, and positions of the electric strain gauges are shown in Fig. 9. A hemispherical swivel head was used at the top of test specimens to ensure appropriate contact and concentric axial loading. After applying 5 kN load, the rotations of swivel head were locked using metal wedges; the test set-up is shown in Fig. 10. Compression tests were performed under displacement controlled condition where a constant displacement of 0.5 mm/min was applied throughout the test. All relevant data such as time, load, displacements and strains were recorded at 0.1 s interval.

Fig. 6. Distribution of the initial local geometric imperfection of I 80 × 60 × 4 × 2–320 column at mid-height. Table 2 Geometric dimensions and the maximum local imperfections of the stub column specimens considered for experimental study. Section ID

D mm

B mm

tf mm

tw mm

L mm

W0 mm

I 80 × 60 × 4 × 2–320 I 80 × 60 × 6 × 4–320 I 120 × 60 × 5 × 3–360

79.3 79.6 118.9

59.4 59.8 59.6

4.01 6.08 4.93

2.00 3.95 2.99

316 315 355

1.05 0.56 0.98

However this was not significant, and hence the distance between each of the obtained cross profiles was also assumed to be constant. Distance between two horizontal measurement points in a profile typically varied from 0.2 to 0.25 mm as it depends on the distance between the surface and the scanner. The raw data were post-processed by Matlab using ‘meshgrid’ option; collected raw data were converted to 1 mm by 1 mm grid data, where value of each grid point was determined as a weighted average of the values of surrounding points. The cross-section profile at mid length of I 80 × 60 × 4 × 2–320 is shown in Fig. 6. The maximum deviation of elements from the centre of that element was recorded as the maximum amplitude of the initial local geometric imperfection W0. Obtained results are presented in Table 2 for stub column specimens and in Table 4 for long column specimens. The maximum local imperfection value varied from 0.31 mm to 1.05 mm, which are much higher than the EN 1993-1-5 [31] recommended value of b/200, where b is the unsupported length of the element. Initial global geometric imperfections of long column specimens were measured by a digital dial gauge. The specimens were set on a milling table and a dial gauge was mounted in the place of the drill bit. Measurements were taken along the centre line of flanges and web. Due to the presence of end plates, the first measurement was recorded at 50 mm away from the end and the other measurements were taken at 50 mm intervals including the middle points (for I 80 × 60 × 4 × 2–1500 column interval was 100 mm). Global imperfections about both major and minor axes were measured. Deviation of the column from the central axis was recorded as the global imperfection. The maximum imperfections recorded at the mid-height of members are reported as the global imperfection in Table 4. The global imperfection values about major axis varied from L/490 to L/6250 and those about minor axis varied from L/794 to L/4286. For most of the columns, the maximum global imperfection amplitudes were less than L/1000, which is the recommended global imperfection value of a member according to EN 1993-1-5 [31]. The distribution of global imperfections of I 80 × 60 × 4 × 2–1500 column about major and minor axis are presented in Fig. 7; these are typical distributions observed in the considered welded I-columns.

6.2. Discussion on results obtained from stub column tests Key results such as the ultimate load Nu and the corresponding end shortening obtained from stub column tests are presented in Table 3. The end shortening of each specimen was calculated as the difference between the average displacement of the bottom plate and the top plate. All tested stub column specimens failed by local buckling of the comprising elements, as expected. Failure modes of the stub columns are shown in Fig. 11, and complete load vs. end shortening curves are shown in Fig. 12. From electric strain gauge reading, the strain distribution at mid-height of stub columns at the ultimate load Nu and at 50% of Nu are shown in Fig. 13. In the case of I 80 × 60 × 6 × 4–320, electric strain gauges lost contact before reaching ultimate load, and hence strain distribution for this specimen is shown at 90% of Nu. From Fig. 13, it is clear that strains at the ultimate load are well above the yield strain for all the stub columns. 7. Test of long columns for flexural buckling 7.1. Test procedure To investigate the flexural buckling behaviour of stainless steel welded I-sections, 16 long columns were tested for minor axis buckling in pin-ended support condition. Considered columns represented six different cross-sections covering a wide range of tf/tw ratio, height-towidth ratio, cross-section slenderness λp and member slenderness λ. Cross-section slenderness varied from 0.26 to 0.83, whilst geometric lengths of the columns varied from 500 mm to 1500 mm providing a wide spectrum of λ, as calculated in accordance with EN 1993-1-4 [1], ranging from 0.45 to 1.39. The geometric dimensions of the column specimens were measured prior to testing. The measured dimensions are provided in Table 4. Initial local and global geometric imperfections for all column specimens were measured prior to testing and are reported in Table 4. The specimens were tested using the Shimadzu 1000 kN UTM, which was used for stub columns. Hardened steel knife-edge support systems were used at both ends to provide pin end support conditions 854

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Fig. 7. Distribution of the initial global geometric imperfections of I 80 × 60 × 4 × 2–1500 column: (a) about major axis, (b) about minor axis.

Fig. 8. Schematic diagram of the instrumentation of stub column test. Fig. 9. Positions of electric strain gauges at the mid-length of stub column specimens of stainless steel welded I-section.

about the axis of buckling but fixed end support conditions about the orthogonal axis. The schematic diagram of the test set-up is shown in Fig. 14. For appropriate transfer of the applied loading, two plates were welded at the ends of each column. Two fastener plates were used to attach the column specimens to a knife-edge support at each end. Fastener plates can be adjusted to position the column at the centre of the knife-edges. A laser guided levelling machine was used to align the centre line of the column with the centre of the knife-edges as well as that of the UTM. The instrumentation consisted of ten LVDTs as shown in Fig. 14. Four LVDTs were installed to measure the displacements of the bottom plate and additional four LVDTs were used to measure the displacements of the top plate. The other two LVDTs were used to measure the lateral deflection at mid-height as shown in Fig. 15. Similar to the stub columns (Fig. 8), ten linear electric resistance strain gauges were installed at mid-height cross-section. The test set-up is shown in Fig. 16. Tests were performed in displacement control condition with a constant rate of 0.5 mm/min, and all relevant data were recorded at 0.1 s intervals. A typical buckling shape for column I 80 × 60 × 4 × 2–750 is shown in Fig. 17.

7.2. Discussion of results obtained from testing of long columns Key results obtained from the conducted buckling tests such as the ultimate load Nu, end shortening at the ultimate load, lateral defection at the ultimate load and failure modes of column specimens, are presented in Table 5, where ‘FB’ and ‘LB’ refers to flexural buckling and local buckling respectively. The load vs. end shortening curves and the load vs. lateral deflection curves are shown in Figs. 18–23. Strain distributions at the mid-height level at different loading stages for columns with I 80 × 60 × 6 × 4 and I 80 × 80 × 5 × 4 sections are presented in Figs. 24 and 25 respectively. To obtain the strain distribution of a cross-section, readings recorded from strain gauges SG2, SG3, SG5 and SG6 were averaged, and were considered as strain at the top fibre. Similarly, the average of strain gauge readings of SG4 and SG9 was considered as strain at the middle, and the average of strain gauge readings of SG1, SG7, SG8 and SG10 was considered as the strain at the bottom fibre. Figs. 24 and 25 clearly show that for all column specimens, strain distributions at 50% of Nu were uniform confirming the concentric loading condition for the tested columns. As the load was increased, cross-sections rotated due to flexural buckling. In the case of 855

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Fig. 12. Complete axial load vs. end shortening curves of the stub columns.

8. Summary of the testing scheme In the conducted test program, the behaviour of stainless steel welded I-sections was studied through test of tensile coupons, measurement of residual stresses and initial geometric imperfections, and testing of stub and long columns. It was observed that the measured maximum tensile residual stresses were higher than the values recommended by Yuan et al. [12]. The amplitudes of local geometric imperfection were also higher than the code [31] recommended values; however, for most of the columns the amplitudes of global imperfection were lower than the codified values. All three stub columns failed in local buckling and their stress distributions showed that the average strains at failure were higher than the yield strain. Overall buckling behaviour was observed through flexural buckling tests about minor axis. The buckling test results along with material test data and measured imperfection amplitudes are used in the following section to develop FE models.

Fig. 10. Test set-up for stub column test of I 80 × 60 × 6 × 4–320 column. Table 3 Stub column test results. Section ID

Ultimate load (Nu) kN

End shortening mm

I 80 × 60 × 4 × 2–320 I 80 × 60 × 6 × 4–320 I 120 × 60 × 5 × 3–360

220.4 375.4 274.1

3.22 2.79 9.08

9. Current design methods for buckling resistance Tangential stiffness method and Perry-Robertson formulas are two widely used methods to determine the buckling resistance of steel columns. Like carbon steel, the Perry type equations were adopted in EN 1993-1-4 [1] for stainless steel columns. Buckling equations currently used in the European code are presented in Eqs. (3)–(7), where Ag is the gross cross-sectional area, fy is the 0.2% proof stress (σ0.2), χ is the buckling reduction factor, Aeff is the effective cross-sectional area, λ is the non-dimensional slenderness of the column and Ncr is the elastic buckling load of the column based on gross area. Effective cross-section properties are used to deal with the local buckling of slender class 4 cross-sections. Four column curves were proposed for different crosssection and loading types, and they differ from each other by varying a linear function of imperfection parameter η. The suggested imperfection parameter η is expressed using a linear relationship, η = α(λ−λ0) where α and λ0 factors depend on cross-section types. The effect of residual stress on welded sections is also included in η. Stainless steel is a highly nonlinear material and its nonlinearity significantly varies from grade to grade. Rasmussen and Rondal [23] and Ahmed and Ashraf [27] showed that material nonlinearity has a significant effect on column resistance. However, in the current EN 1993-1-4 [1] the effect of material nonlinearity is not recognised. Tangential stiffness method is based on Euler formulas. The Australian and the American codes [16,17] follow the tangential stiffness approach to determine column resistances. This method involves a simple equation as given in Eq. (8), where Fn is the buckling stress. Fn can be calculated by using Eq. (9) but the process requires iteration as Fn and the tangent modulus Et are interdependent. The effect of material nonlinearity is incorporated through Et but member imperfections

Fig. 11. Failure modes of the stub columns of stainless steel welded I-sections.

I 80 × 80 × 5 × 4–450, due to the initiation of local buckling, crosssection was not plane after deformation. It was observed that the average strain of a cross-section decreased with increase in column length. As shown in Fig. 24, for I 80 × 60 × 6 × 4–750 column, strains at the lower half of the cross-section were greater than the material yield strain at Nu. Whereas, in the case of I 80 × 60 × 6 × 4–1000 column, strains at the bottom was slightly over the material yield strain, and in the case of I 80 × 60 × 6 × 4–1200 column, strains of the full cross-section were below the material yield strain. Strain distributions observed in columns, as shown in Fig. 25, also show a similar trend. 856

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Fig. 13. Strain distributions of the stub columns at mid-height at different stages of loading.

with buckling stress. Gardner and Nethercot [19] first proposed this method. Recently Afshan and Gardner [21] proposed a new base curve as shown in Eq. (10), where normalized deformation capacity εcsm/εy was expressed as a function of cross-section slenderness λp for stocky sections. They set the limiting cross-section slenderness value as 0.68 to differentiate between stocky and slender sections. They also proposed to use the elastic buckling capacity of a full cross-section to determine λpto incorporate element interactions. Instead of using a compound Ramberg-Osgood model, they used a simple bi-linear material model, which is able to explore the strain hardening benefit for stocky crosssections and, hence, simplify the calculation process. Once the normalised deformation capacity of across-section is determined from the design base curve, the proposed material model can be used to obtain the buckling stress fcsm using Eq. (11), which a cross-section can achieve prior to local buckling.

are not considered in this approach. All design codes limit the maximum compression capacity of a section to its squash load ignoring the strain hardening strength of stainless steel.

Nu = χA g fy for Class 1, 2 and 3 cross-sections

(3)

Nu = χA eff fy for Class 4 cross-sections

(4)

λ=

λ=

χ=

Agfy

for Class 1, 2 and 3 cross-sections

Ncr A eff f y Ncr

for Class 4 cross-sections

1 ϕ+

(5)

ϕ2−λ2

(6)

⩽ 1.0 where, ϕ = 0.5[1 + η + λ2] and η = α(λ−λ 0) (7) (8)

Nu = A eff Fn Fn =

π2Et ⩽ fy (KL/r)2

(9)

ε csm 0.25 ε 0.1ε u = 3.6 but csm ⩽ 15, for λ p ⩽ 0.68 εy εy εy λp

(10)

ε f csm = f y + Esh ε y ⎛⎜ csm −1⎟⎞ for λ p ⩽ 0.68 ⎝ εy ⎠

(11)

Ahmed and Ashraf [22] introduced a new parameter Equivalent elastic deformation capacity εe,ev to extended CSM for slender crosssections. They considered the same base curve (Eq. (10)) for the full range of cross-section slenderness, and developed a relationship between εe,ev and εcsm through Eq. (12) where C is a constant that depend on the type of cross-section (for I-section a = 3.05 and b = 3.0). For

10. The Continuous Strength Method (CSM) The Continuous Strength Method (CSM) is a strain based design approach where a base curve relates to the deformation capacity of a given section, and a material model relates to the deformation capacity

Table 4 Geometric dimensions and initial geometric imperfections of long column specimens of stainless steel welded I-sections. Section ID

I I I I I I I I I I I I I I I I

80 × 60 × 4 × 2–750 80 × 60 × 4 × 2–1000 80 × 60 × 4 × 2–1500 80 × 60 × 6 × 4–750 80 × 60 × 6 × 4–1000 80 × 60 × 6 × 4–1200 80 × 80 × 5 × 4–500 80 × 80 × 5 × 4–900 80 × 80 × 5 × 4–1200 100 × 60 × 6 × 4–450 100 × 60 × 6 × 4–900 100 × 60 × 6 × 4–1200 1200 × 60 × 5 × 3–720 1200 × 60 × 5 × 3–1200 1200 × 60 × 4 × 2–500 1200 × 60 × 4 × 2–1000

D

B

tf

tw

L

Le

W0

Global imperfection

mm

mm

mm

mm

mm

mm

mm

Major axis mm

Minor axis mm

79.39 79.5 79.3 79.7 79.6 79.9 79.3 79.3 79.6 100.2 100.0 100.4 119.0 119.5 119.9 119.2

59.44 59.4 59.5 59.7 59.8 59.8 79.6 79.8 80.2 60.0 59.8 60.0 59.7 59.6 59.6 59.6

3.93 4.0 3.9 6.0 6.0 6.0 4.9 4.9 4.9 6.1 6.0 6.0 4.8 4.9 4.0 3.9

2.00 2.0 2.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 3.0 3.0 2.0 2.0

748 996.0 1496.0 743.0 997.0 1194.0 496.0 895.0 1194.0 446.0 897.0 1197.0 718.0 1196.0 496.0 996.0

965 1213.0 1713.0 960.0 1214.0 1411.0 713.0 1112.0 1411.0 663.0 1114.0 1414.0 935.0 1413.0 713.0 1213.0

0.67 0.38 – 0.60 0.31 0.75 0.50 0.48 0.52 0.65 0.65 0.78 0.48 0.44 0.37 0.88

0.26 0.31 2.67 0.12 0.44 2.45 0.2 0.34 2.28 0.22 0.32 0.92 0.39 0.5 0.18 0.37

0.47 1.26 1.68 0.4 0.35 0.28 0.34 0.42 0.46 0.5 0.42 0.44 0.34 0.38 0.36 0.81

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Fig. 16. Test set-up for flexural buckling test. Fig. 14. Schematic diagram of the test set-up for flexural buckling test of welded I-columns.

Fig. 15. Position of two lateral LVDTs at mid-height of the column.

λp > 0.68, the buckling stress fcsm can be calculated by multiplying εe,ev with the Young’s modulus E as shown in Eq. (13).

ε e,ev = Cε csm for λ p > 0.68,

where C= aλ pb

f csm = ε e,ev E= Cε csm E for λ p > 0.68

(12) (13)

Recently Ahmed and Ashraf [27] proposed new buckling formulas for stainless steel cold-formed RHS and SHS columns using fcsm instead of fy in the Perry formulas adopted in EN 1993-1-4 [1]. In their proposal, buckling resistance of a column may be determined directly by Eq. (14). Non-dimensional column slenderness λcsm may be obtained using the modified definition as shown in Eq. (15), where Nu is the buckling resistance of a member, χ is the reduction factor for buckling, Ag is the gross cross-sectional area, and Ncr is the elastic critical buckling capacity of the member based on Ag. They observed that non-dimensional proof stress (e = fy/E), strain hardening exponent n and cross-section slenderness λp have significant effects on column curves. They modified λcsm to reduce the effect of e and n, and introduced a

Fig. 17. Flexural buckling of I 80 × 60 × 4 × 2–750 column showing a typical failure mode.

modified non-dimensional slenderness λm. Modifications of λcsm are shown in Eq. (16), where Ce is the modification factor for e and Cn is the modification factor for n. The reduction factor of the Perry curves similar to the current EN 1993-1-4 [1] was adopted in their study replacing the non-dimensional slenderness λ by λm as given in Eq. (17). They proposed a sigmoidal function of λm for the imperfection factor η as shown in Eq. (18) where A, B and W are the coefficients of the sigmoidal function. Values of A and B were expressed as functions of λp 858

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Table 5 Key results obtained from the flexural buckling of pin-ended welded I section columns. Section ID

Ultimate load kN

End shortening mm

Lateral deflection mm

Failure mode

I I I I I I I I I I I I I I I I

112.9 92.9 56.3 189.8 149.4 123.8 288.0 216.0 178.3 260.1 171.9 127.0 167.0 104.6 150.9 93.7

1.21 1.24 1.15 1.67 1.45 1.22 2.11 2.12 1.39 1.54 1.61 1.25 1.46 1.10 1.89 1.43

2.73 5.05 1.58 2.39 2.40 1.87 1.06 2.18 4.92 4.28 5.88 7.12 3.06 4.00 0.98 2.55

FB FB FB FB FB FB LB + FB FB FB LB + FB FB FB FB FB LB + FB FB

80 × 60 × 4 × 2–750 80 × 60 × 4 × 2–1000 80 × 60 × 4 × 2–1500 80 × 60 × 6 × 4–750 80 × 60 × 6 × 4–1000 80 × 60 × 6 × 4–1200 80 × 80 × 5 × 4–500 80 × 80 × 5 × 4–900 80 × 80 × 5 × 4–1200 100 × 60 × 6 × 4–450 100 × 60 × 6 × 4–900 100 × 60 × 6 × 4–1200 1200 × 60 × 5 × 3–720 1200 × 60 × 5 × 3–1200 1200 × 60 × 4 × 2–500 1200 × 60 × 4 × 2–1000

giving separate column curves for different λp values.

Nu = χA g f csm λ csm =

Ncr

1 ϕ+

η=

(14)

A g f csm

λ m = λ csm + Ce + χ=

FF models were developed by following the guidelines of Ashraf et al. [33]. In FE models, four-node doubly curved shell elements with reduced integration were used with mesh sizes not greater than 5 mm along the transverse direction and 15 mm along the longitudinal direction of I-sections. The two-stage Ramberg–Osgood (R–O) [34] material model proposed by Rasmussen [35], with recent modifications proposed by Arrayago et al. [36], was used in the developed FE models. In this study, both local and global geometric imperfections were considered. Eigenvalue analysis was performed and the corresponding elastic buckling modes were used to simulate the distribution of local and global imperfections. In their reported parametric study, Yuan et al. [14] showed that the amplitude of global geometric imperfection has moderate effect on column resistance but the amplitude of local imperfection has no significant effect on column resistance. In this study, four different amplitudes for global imperfection such as measured imperfection, L/1000, L/1500 and L/2000, where L is the length of the column, were considered. In the case of local geometric imperfection, an amplitude of b/200 was used, where b is the unsupported width of the flange. L/1000 and b/200 are the code recommended values for the global geometric imperfection and the local geometric imperfection respectively [31]. Membrane residual stresses develop in I-sections due to welding, which induce tensile stress in the vicinity of the welds with compressive stress away from those regions. Residual stresses were incorporated in the developed FE models following to the proposal of Yuan et al. [12], where the maximum tensile stress was taken as 0.8σ0.2. Columns were simulated as pin supported at both ends. All nodes at the bottom end and at the top end were coupled with two reference

ϕ2−λ 2m

(15)

5−n Cn ⩾ 0 5

(16)

⩽ 1.0 where, ϕ = 0.5[1 + η + λ 2m]

A −0.25 1 + exp−(λm−B)/W

(17)

(18)

11. Numerical model for long columns Commercial finite element analysis package ABAQUS was used for numerical simulation of the behaviour observed in stub column and long column testing. Initially, the developed FE models were validated against test results carried out as part of the current study, and, once verified, the developed FE models were used to perform a comprehensive parametric study to identify the influence of different parameters on the flexural buckling resistance of stainless steel welded Isection columns. Finally, results obtained from FE analysis were used to propose new column curves for welded I columns produced from stainless steel.

Fig. 18. Load-deflection curves of columns with I 80 × 60 × 4 × 2 cross-section: (a) load vs. end shortening, (b) load vs. lateral deflection. 859

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Fig. 19. Load-deflection curves of columns with I 80 × 60 × 6 × 4 cross-section: (a) load vs. end shortening, (b) load vs. lateral deflection.

over prediction was 16% for the considered columns. Fig. 28 shows the deflected shape for the FE model of I 80 × 60 × 4 × 2–750 column. In Fig. 29, complete load-deflection responses of FE models for four columns are compared with test responses showing good agreement. Overall comparisons showed that the developed FE models were capable of predicting the behaviour of welded I-section stainless steel columns, and could be used to generate additional results to develop reliable column curves. Although the amplitude of global imperfection value L/2000 with residual stress showed marginally better agreement with test results, an amplitude of L/1500 with residual stresses was conservatively used in the parametric study.

points located at the centroid of the corresponding sections. The pin support conditions were applied to those reference points allowing for longitudinal translation at the top end. Columns subjected to buckle around major axis were laterally supported at the quarter-lengths and at the mid-length to prevent minor axis buckling. Typical support conditions adopted in FE models for major axis buckling and minor axis buckling are shown in Figs. 26 and 27 respectively. Displacement was applied at the top reference point to simulate the column test. 12. Verification of FE models for long columns The accuracy of the FE modelling technique was verified using the test results obtained from the welded I-section columns considered in this study. The comparison of ultimate load (Nu) of FE models with those obtained from tests is shown in Table 6. FE models that included measured local and global imperfections with residual stress, and the FE models that included global imperfection of L/2000 and local imperfection of b/200 with residual stress showed good agreement with test results. In both cases, the average ratio of the ultimate loads obtained from FE models and test results was 1.00 with a coefficient of variation (COV) of 0.04 and 0.02. Finite element models using a global imperfection amplitude of L/1500 also showed good agreement with test results with an average of 0.99. However, FE models with L/1000 global imperfection amplitude slightly under predicted the column capacity as the average of the ratio of FE results and test results is 0.96. On the other hand, if the residual stresses were not considered in the FE models, they always over predicted the column capacity; the average

13. Parametric study for welded I columns The verified FE modelling technique was used to identify the parameters that could significantly affect the buckling resistance of stainless steel columns through a parametric study. Ahmed and Ashraf [27] recently reported that cross section slenderness λp has a significant effect on column curves for cold-formed RHS and SHS columns. Therefore, the effect of λp on column curves of welded I-section buckling about the major and the minor axis was thoroughly investigated herein. Additionally, effects of other cross-sectional properties such as the ratio of height-to-width (H/B) of the section and the ratio of flange thickness-to-web thickness (tf/tw) were also examined. Three values of H/B varying between 1.0 and 2.0 (1.0, 1.5 and 2.0) and three values of tf/tw (1.0, 1.5 and 2.0) were considered to cover a practical range of I-sections. Effects of H/B and tf/tw were observed on

Fig. 20. Load-deflection curves of columns with I 80 × 80 × 5 × 4 cross-section: (a) load vs. end shortening, (b) load vs. lateral deflection. 860

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Fig. 21. Load-deflection curves of columns with I 100 × 60 × 6 × 4 cross-section: (a) load vs. end shortening, (b) load vs. lateral deflection.

Fig. 22. Load vs end shortening of columns with I 120 × 60 × 5 × 3 cross-section: (a) load vs. end shortening, (b) load vs. lateral deflection.

Fig. 23. Load-deflection curves of columns with I 120 × 60 × 4 × 2 cross-section: (a) load vs. end shortening, (b) load vs. lateral deflection.

both stocky and slender cross-sections where λp of the considered stocky section was 0.48 and that for slender section was 0.88. Seven values of λp ranging between 0.38 and 0.98 were considered to evaluate the effect of λp on column curves. λp was calculated according to the proposal of Afshan and Gardner [21] where the elastic buckling capacity of the full cross-section (σcr,cs) was determined using CUFSM [37]. A total of 23 different I-sections were analysed for the major and the minor axis buckling, and their cross-sectional properties are shown

in Table 7. Basic material properties such as Young’s modulus (E), proof stress (σ0.2) and strain hardening exponent (n) were taken as 200 GPa, 400 MPa and 7 respectively. Effect of residual stress on column resistance was also observed on the stocky and the slender cross-section for major and minor axis buckling. For each I-section, columns with 15 different slenderness λcsm (0.2, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8 and 2.0) were analyzed to get a full range of column curves. A total of 720 models were analysed to obtain a 861

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Fig. 24. Strain distributions at mid-height of columns with I 80 × 60 × 6 × 4 section at different loading stages.

Fig. 25. Strain distributions at mid-height of columns with I 80 × 80 × 5 × 4 section at different loading stages.

Fig. 27. Support conditions of a I-column subjected to minor axis buckling.

Fig. 26. Support conditions of a I-column subjected to major axis buckling.

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Table 6 Comparison of FE results with tests performed on pin-ended stainless steel welded I-columns. Section ID

I 80 × 60 × 4 × 2−750 I 80 × 60 × 4 × 2−1000 I 80 × 60 × 4 × 2−1500 I 80 × 60 × 6 × 4−750 I 80 × 60 × 6 × 4−1000 I 80 × 60 × 6 × 4−1200 I 80 × 80 × 5 × 4−500 I 80 × 80 × 5 × 4−900 I 80 × 80 × 5 × 4−1200 I 100 × 60 × 6 × 4−450 I 100 × 60 × 6 × 4−900 I 100 × 60 × 6 × 4−1200 I 1200 × 60 × 5 × 3−720 I 1200 × 60 × 5 × 3−1200 I 1200 × 60 × 4 × 2−500 I 1200 × 60 × 4 × 2−1000 Average COV

Test result

FE result with residual stress

FE results without residual stress

Measured imperfection

L/1000

L/1500

L/2000

Measured imperfection

Nu,test kN

Nu,FE kN

Nu,FE/Nu,test

Nu,FE kN

Nu,FE/Nu,test

Nu,FE kN

Nu,FE/Nu,test

Nu,FE kN

Nu,FE/Nu,test

Nu,FE kN

Nu,FE/Nu,test

112.9 92.9 56.3 189.8 149.4 123.8 288.0 216.0 178.3 260.1 171.9 127.0 167.0 104.6 150.9 93.7

112.9 85.6 54.1 186.4 150.5 132.9 281.6 220.2 183.0 254.2 170.1 134.6 169.2 109.4 153.3 92.1

1.00 0.92 0.96 0.98 1.01 1.07 0.98 1.02 1.03 0.98 0.99 1.06 1.01 1.05 1.02 0.98 1.00 0.04

109.4 85.9 54.8 178.1 139.5 119.0 275.9 210.7 172.6 251.6 160.4 123.1 160.3 98.3 150.2 90.5

0.97 0.92 0.97 0.94 0.93 0.96 0.96 0.98 0.97 0.97 0.93 0.97 0.96 0.94 1.00 0.97 0.96 0.02

112.2 88.4 56.9 182.4 143.8 123.5 279.4 215.2 177.0 255.9 164.9 127.9 164.5 102.5 152.5 93.2

0.99 0.95 1.01 0.96 0.96 1.00 0.97 1.00 0.99 0.98 0.96 1.01 0.99 0.98 1.01 0.99 0.99 0.02

113.9 89.9 58.2 184.9 146.4 126.3 281.4 217.9 179.7 258.4 167.8 130.8 166.9 105.0 153.8 94.8

1.01 0.97 1.03 0.97 0.98 1.02 0.98 1.01 1.01 0.99 0.98 1.03 1.00 1.00 1.02 1.01 1.00 0.02

124.4 98.8 66.5 213.7 185.5 165.8 287.5 238.7 212.2 266.9 204.7 167.9 192.7 134.3 162.0 107.5

1.10 1.06 1.18 1.13 1.24 1.34 1.00 1.10 1.19 1.03 1.19 1.32 1.15 1.28 1.07 1.15 1.16 0.10

thorough understanding of the buckling behaviour of stainless steel welded I-section columns. 14. Analysis of FE results To study the effect of different parameters, column curves are carefully plotted in Figs. 30–33 to observe variation in non-dimensional N column strength or reduction factor χ, which was calculated as u,FE . In Agf csm

Fig. 30, the influence of λp on column curves is shown for major axis buckling and minor axis buckling respectively. It is clearly observed that cross-section slenderness λp has a significant effect on the column curves for welded I-section, and it is similar to the effect recently reported for cold-formed RHS and SHS by the authors [27]. In both cases, column curves move upward with increasing values of λp i.e. the reduction factor χ is higher for relatively slender cross-sections with higher values of λp. Shapes of column curves also depend on λp. For slender cross-sections (with higher value of λp), the column resistance approaches to its cross-section capacity at a faster rate (i.e. χ approaches 1.00 at a relatively higher value of λm) when compared against stocky cross-sections (with smaller λp). The effect of λp is more prominent at the intermediate portion of column curves and diminishes with the increase of λm. Figs. 31 and 32 show the effect of H/B and tf/tw on column curves for both stocky and slender cross-sections. It was observed that H/B and tf/tw have no significant effect on column curves for stocky sections (λp = 0.48). In the case of slender sections (λp = 0.88), however, H/B showed some minor effects on column curves at low λm values. From Figs. 31 and 32, it is also clear that column resistance for major axis buckling is higher than that of minor axis buckling predominantly in the intermediate portion of column curves. From this analysis, it is clear that separate column curves are required for major axis buckling and minor axis buckling, and the effect of λp should be included in the formulas. The effect of residual stress on column resistance is shown in Fig. 33. Both the figures showed that column capacity was considerably reduced due to the presence of residual stresses in welded I-sections. This reduction is more significant for columns buckling about minor axis. In the case of minor axis buckling, maximum compressive stress developed at the tip of flanges, which are already in compression due to the presence of residual stress, which may lead to early failure. On the other hand, in case of major axis buckling, full flange goes to

Fig. 28. Deflected shape of I 80 × 60 × 4 × 2–750 specimen.

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Fig. 29. Comparison of experimental load-deflection curves with FE results for welded I-section columns: (a) load vs. end shortening, (b) load vs. lateral deflection.

imperfection parameter η was expressed as a sigmoidal function of λm, where the coefficients of that function depend on λp. Similar approach has been adopted herein to predict the buckling resistance of stainless steel welded I-section columns. Eqs. (14)–(18) were adopted to predict the resistance of stainless steel I-columns, and the imperfection parameter η was calibrated for I-sections. Generated FE results were used to determine appropriate values for imperfection parameter η for the considered stainless steel I-section columns, and were plotted against λm for different values of λp as shown in Fig. 34 for major axis and minor axis buckling. From these figures, it is obvious that λp has a significant influence on η, and the variation of η with λm can be well represented by a sigmoidal function as described in Eq. (18). The coefficients A, B and W of the sigmoidal function were determined for stainless steel welded I-section columns for both major and minor axis buckling. For major axis buckling, the values of A and B can be calculated by using Eqs. (19) and (20). For minor axis buckling, the values of A and B can be calculated following Eqs. (21) and (22). A constant value of W = 0.4 is proposed for both major axis buckling and minor axis buckling.

Table 7 Cross-sectional properties of I-sections. Sl. no.

H (mm)

B (mm)

tf/tw (mm/mm)

1–6

150

150

7–17

225

150

18–23

300

150

8.05/8.05, 4.54/4.54, 9.0/6.0, 5.1/3.4, 9.45/4.73, 5.47/2.74 9.6/9.6, 5.38/5.38, 14.15/9.43, 11.5/7.67, 9.65/ 6.43, 8.35/5.57, 7.35/4.9, 6.56/4.37, 5.93/3.95, 13.83/6.92, 8.03/4.02 12.17/12.17, 6.83/6.83, 15.48/10.32, 8.82/5.88, 19.4/9.7, 11.22/5.61

compression, which is in self-equilibrium under the effect of residual stress. Hence, the effect of residual stress is less severe for major axis buckling. From Fig. 33, it was also observed that if residual stresses were not considered, column curves for major axis buckling and minor axis buckling of a cross-section overlapped each other. Unlike for the case of cold-formed stainless steel members, the residual stress should be incorporated in the FE models of stainless steel welded members for predicting the flexural buckling resistance of columns. 15. Imperfection factor η for welded I-sections Ahmed and Ashraf [27] successfully used fcsm in Perry formulas to predict the buckling resistance of stainless steel cold-formed hollow compression members. In that proposal, CSM buckling stress fcsm was used instead of the material yield stress in basic Perry curves, and the

A= −0.32λ p + 1.06 ⩾ 0.74

(19)

B= 0.3λ p + 0.4 ⩽ 0.65

(20)

A= −0.5λ p + 1.55 ⩾ 1.05

(21)

B= 0.1λ p + 0.72 ⩽ 0.80

(22)

Fig. 30. Column curves of stainless steel welded I-section for different λp values: (a) for major axis buckling, (b) for minor axis buckling. 864

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Fig. 31. Column curves of stocky stainless steel welded I-sections for different H/B ratios: (a) for stocky sections (λp = 0.48), (b) for slender sections (λp = 0.88).

Fig. 32. Column curves of stainless steel welded I-sections for different tf/tw ratios: (a) for stocky sections (λp = 0.48), (b) for slender sections (λp = 0.88).

Fig. 33. Comparison of column curves of welded I-sections buckling with residual stress and without residual stress: (a) for stocky section (λp = 0.48), (b) for slender section (λp = 0.88).

16. Performance of the proposed method

19 columns were tested for major axis buckling and 43 columns were tested for minor axis buckling. The performance of the proposed method was also compared with those obtained using EN 1993-14 + A1 [39], SEI/ASCE 8-02 [17] and AS/NZS 4673 [16]. For AS/NZS 4673, Rasmussen and Rondal’s [23] proposal was considered where the values of the parameters α, β, λ1 and λ0 were calculated using their suggested equations. The key features of the comparison are shown in

The accuracy of the proposed method was verified using available test results of stainless steel welded I-columns and the FE results generated in the current parametric study. A total of 46 test results, collected from different studies in the literature [14,15,38], and 16 test results performed in this study were considered. Among the test results, 865

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Fig. 34. η calculated from FE results and proposed curves of η for different λp values: (a) for major axis buckling, (b) for minor axis buckling. Table 8 Comparison of the performance of the proposed CSM method and other standards. Loading type

Major axis buckling Minor axis buckling

FE data Test data FE data Test data

CSM

EN 1993-1-4+A1

ASCE

AS/NZS

Ncsm/NFE/test

NEC3/NFE/test

NASCE/NFE/test

NAS/NZS/NFE/test

Average

COV

Average

COV

Average

COV

Average

COV

0.99 1.01 0.98 0.98

0.03 0.06 0.04 0.10

0.95 0.98 0.92 0.86

0.05 0.07 0.06 0.08

1.09 1.10 1.13 1.14

0.09 0.08 0.13 0.06

1.01 1.03 1.05 1.03

0.05 0.07 0.08 0.06

Fig. 36. Comparison of CSM and SEI/ASCE 8-02 [17] predictions for major axis buckling of stainless steel welded I-columns.

Fig. 35. Comparison of CSM and EN1993-1-4+A1 [39] predictions for major axis buckling of stainless steel welded I-columns.

stainless steel and did not consider the effect of residual stress of welded sections, and hence ASCE [17] over-predicted the buckling resistance of stainless steel welded I-columns. The effect of residual stress is more prominent for columns subjected to minor axis buckling, and hence the average ratio of ASCE [17] prediction to test or FE results is much higher for minor axis buckling than major axis buckling. For similar reason, AS/NZS [16] also over-predicted the buckling resistance of the welded I-columns. Figs. 35–40 illustrates the comparison of CSM predictions with the predictions of other codes for test and FE results.

Table 8, which clearly shows that the predictions obtained using the proposed method are more accurate and less scattered than the predictions of other standards. For major axis buckling, the average of the ratio of CSM predictions and FE results is 0.99 and the average of the ratio of CSM predictions and test results is 1.0. For minor axis buckling, the average is 0.98 for both FE results and test results. The coefficient of variation (COV) for CSM predictions is always less than those of other standards predictions except for one case. Comparing test results of minor axis buckling, the COV of the CSM prediction is lightly higher than the COV other standards predictions. The prediction of the EC3 + A1 [39] standard shows good agreement with test and FE results for major axis buckling but for minor axis buckling this code is more conservative. ASCE [17] guidelines were proposed for cold form

17. Reliability analysis Reliability analysis of the buckling formulas proposed for stainless steel welded I-section columns was performed following the guidelines 866

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Fig. 40. Comparison of CSM and AS/NZS 4673 [16] predictions for minor axis buckling of stainless steel welded I-columns.

Fig. 37. Comparison of CSM and AS/NZS 4673 [16] predictions for major axis buckling of stainless steel welded I-columns.

Table 9 Results of the reliability analysis. Loading types

Material

n

Kd,n

b

Vδ

Vr

γM1

Major axis buckling

Austenitic Duplex Austenitic Duplex

386 383 407 386

3.115 3.115 3.114 3.114

1.025 1.024 1.049 1.059

0.031 0.031 0.043 0.043

0.084 0.066 0.089 0.072

0.97 1.09 0.97 1.07

Minor axis buckling

the buckling resistance of stainless steel welded I-columns. Therefore, FE results generated for austenitic grades can also conservatively represent the behaviour of weld I-columns of duplex grades stainless steel. Hence, FE results were considered in both austenitic and duplex grades stainless steel for the reliability analysis. The material over strength factors for austenitic and duplex grades were considered as 1.30 and 1.10 respectively according to the proposal of Afshan et al. [41]. They also proposed the coefficient of variation of material strength for austenitic and duplex grades as 0.06 and 0.03, respectively, and the coefficient of variation in geometric properties was considered as 0.05. The results of the reliability analysis are presented in Table 9, where n is the total number of tests and FE results, kd,n is the design (ultimate limit state) fractile factor for n number of data, b is the average ratio of experimental (or FE) results to model resistance based on a least squares fit for each set of data, Vδ is the coefficient of variation of tests and FE simulations relative to the resistance model, Vr is the combined coefficient of variation incorporating both model and basic variable uncertainties, and γM1 is the partial safety factor for buckling resistance. It is observed from Table 9 that the partial safety factor for all cases are below 1.10, which is agreement with EN 1993-14 [1] recommended value. So, the proposed formulas can be used with confidence to predict the buckling resistance of stainless steel welded Icolumns produced from austenitic and duplex grades.

Fig. 38. Comparison of CSM and EN1993-1-4+A1 [39] predictions for minor axis buckling of stainless steel welded I-columns.

18. Conclusions Fig. 39. Comparison of CSM and SEI/ASCE 8-02 [17] predictions for minor axis buckling of stainless steel welded I-columns.

Welded sections play a significant role in structural engineering when readily available sections fail to meet the required high load carrying capacities. The structural response of welded members differs from that of cold-formed members due to the presence of longitudinal residual stresses. In case of stainless steel, very limited number of test evidences are available on welded sections as most of the studies are focused on commonly used cold-formed hollow sections. In this paper, the behaviour of stainless steel welded I-columns has been studied through experimental investigation and FE analysis as well as buckling formulas were proposed for welded I-columns. The experimental program involved measurements of material

of EN1990-Annex D [40]. Although the same material model can be adopted for both austenitic and duplex stainless steel, the magnitudes of residual stresses are different for welded sections produced from different grades of stainless steel. In FE models, the maximum tensile residual stress was taken as 0.8σ0.2 considering austenitic grades, however, the maximum tensile residual stress for duplex grades of stainless steel is recommended as 0.6σ0.2 [12]. In the analysis presented in Section 14, it was found that the residual stress has a negative effect on 867

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properties of 316L stainless steel, residual stress and initial geometric imperfections. A number of stub columns and long columns were tested as part of the current study. From longitudinal residual stress measurement, it was observed that the maximum tension residual stress may be higher than the currently suggested value of 0.8σ0.2 [12]. Initial global geometric imperfections were found less than the code recommended values; however, local geometric imperfections were higher than the code recommended values. Local and global buckling characteristics were examined through testing of stub columns and long columns. The behaviour of stainless steel welded I-columns was further investigated using nonlinear FE models. A parametric study was carried out to observe the effects of λp, H/B, tf/tw and residual stress on the buckling resistance of welded columns. It was observed that H/B and tf/ tw had no significant effect on column curves. Presence of longitudinal residual stresses had a negative impact on the buckling resistance of welded columns, and this effect was more significant for columns buckling around minor axis. The effect of λp on the buckling resistance of welded I-columns was similar to the effect observed on RHS and SHS columns. Finally, two sets of equations were proposed for major and minor axis buckling of stainless steel welded I -columns. The performance of the proposed method was assessed using available test results and FE results generated in this study. Compared to EN 1993-1-4+A1 [39], SEI/ASCE 8-02 [17] and AS/NZS 4673 [16] standards, this method produced more accurate and more consistent predictions for buckling resistance of stainless steel welded I-columns. Overall, the experimental work presented in this study added useful experimental evidences for welded I-section columns, and the scope of CSM based design technique was further extended in predicting flexural buckling resistance for stainless steel welded columns.

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