Flexural buckling resistance of welded HSS box section members

Thin-Walled Structures 119 (2017) 266–281

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Full length article

Flexural buckling resistance of welded HSS box section members B. Somodi, B. Kövesdi

⁎

MARK

Budapest University of Technology and Economics, Department of Structural Engineering, Műegyetem rkp. 3., 1111 Budapest, Hungary

A R T I C L E I N F O

A B S T R A C T

Keywords: High strength steel Flexural buckling Buckling curve determination Welded box section

The exact consideration of the global buckling behaviour of HSS steel structures is very important in the design because due to the higher yield strength smaller cross sections can be used, which might be more sensitive to stability failure. According to the previous research results, the flexural buckling behaviour of HSS and NSS columns can be significantly different, however, these differences are not considered in the current design methods. The application range of the current EN 1993-1-1 [1] is limited for steel grades up to S460. The EN 1993-1-12 [2] gives design rules for materials up to steel grades of S700, however, for the determination of the flexural buckling resistance of welded box section members there are no improved design guidelines for HSS structures. Therefore, the purpose of the current study is to investigate the column buckling behaviour of HSS welded box section columns based on previous and current experimental investigations and on numerical simulations. This paper focuses on the effect of the different material properties, imperfections and residual stresses on the global buckling behaviour of HSS members and gives design proposal for the applicable column buckling curves for steel grades between S420 and S960.

1. Introduction The application field of high strength steel (HSS – S420 and higher steel grades up to S960) is growing nowadays in the civil engineering praxis due to the numerous advantages of the HSS members compared to the normal strength steel (NSS – S235, S275, S355) structures: economic design, material saving, possibility of creation of lighter and more aesthetic structures. The application range of the current EN 1993-1-1 [1] is limited to ordinary steel materials up to the steel grade of S460. EN 1993-1-12 [2] gives design background for materials up to steel grades of S700, however, for the determination of the column buckling resistance of welded box sections the same rules are given for materials between steel grades of S460–S700 as for S460. However, several previous research results [3,10,15] prove that the flexural buckling behaviour of HSS structures is more favourable than that of NSS structures. The differences in the flexural buckling behaviour come from the (i) different residual stresses, (ii) different material properties and (iii) geometric imperfections. Numerous previous investigations and residual stress measurements prove that the residual stress amplitudes are smaller for HSS structures compared to the yield strength than for NSS members [11,13]. This results in significant benefits in the flexural buckling resistance, and it can be expected that higher column buckling curves can be used for HSS members than for NSS members. The general aim of the current research is to investigate the flexural buckling behaviour of HSS welded box section columns within the steel ⁎

Corresponding author. E-mail address: [email protected] (B. Kövesdi).

http://dx.doi.org/10.1016/j.tws.2017.06.015 Received 8 July 2016; Received in revised form 12 June 2017; Accepted 14 June 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.

grade range of S235–S960. Based on the experimental and numerical investigation our aim is to propose safe and economic design buckling curve for HSS welded box section columns. There are only a limited number of column buckling tests available in the international literature dealing with HSS welded box section columns. The objective of our current research is to collect the available results from the international literature and to perform a new experimental and numerical research program to investigate the flexural buckling behaviour of HSS structures. A total of 49 large-scale tests are carried out in the Structural Laboratory of the Budapest University of Technology and Economics Department of Structural Engineering in 2015. Test specimens are manufactured having steel grades of S235–S960. The aim of the tests with the wide analysed steel grade range is to give reference and comparison background to the test results made on HSS members (S500–S960) and to cover a wide range of steel grades used in the civil engineering praxis. All the previous test programs investigated the flexural buckling behaviour of one specific steel grade. Our aim is to determine the flexural buckling resistance of columns having different steel grades using the same manufacturing process, test equipment and evaluation process. Parallel to the experimental research program a detailed numerical investigation is also carried out to determine the flexural buckling resistance of HSS box section columns and to investigate the differences between NSS and HSS members. The numerical research program concluded that the differences in the flexural buckling behaviour come

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B. Somodi, B. Kövesdi

2.2. Results of the previous investigations on HSS structures

from (i) the different yield strength, (ii) the different residual stress pattern and (iii) the different material properties. Based on the current and the previous experimental results and based on the current numerical investigations the differences in the flexural buckling behaviour of HSS and NSS members are identified, studied and evaluated. Finally applicable design buckling curves are proposed for welded HSS box section members. The objectives of the research program are achieved by the following research strategy:

Nishino and Tall performed tests on columns having rolled or welded box section and H-type cross sections made from high strength steel ASTM514 with a nominal yield strength of 690 MPa [3]. Four tests were executed on welded box sections with pinned support conditions loaded by compression force. The tested columns were positioned in the test such a way that the effect of the end-eccentricity and the initial outof-straightness counteract approximately each other. Therefore the columns failed on a very high load level nearly by the perfect bifurcation point. The test setup, and the elimination of the geometric imperfections was in accordance with the American design philosophy which was based on the resistance of the straight and centrically loaded column and the effect of the imperfections were covered by a relatively large safety factor. Consequently, these tests cannot be used in the current evaluation process and to select a flexural buckling curve of HSS box section members, thus the EN1993-1-1 [1] is based on the resistances of columns having overall geometric imperfections of L/ 1000 and residual stresses coming from the manufacturing process. Fukumoto and Itoh collected a total of 1665 individual column test results stored in an experimental database [4]. The database contained cross sections of rolled and welded H-profiles, welded box section, square and circular tubes, circular solid sections and T-profiles. This huge number of test results are statistically evaluated and compared to the EN 1993-1-1 column buckling curves separately for all the cross section types. Among the test specimens there were 316 box and square tubes manufactured from annealed NSS and HSS materials. The statistical evaluation is conducted for the whole population of the test results without considering the steel grade. Authors observed significant difference between the nominal and the actual yield strength of the investigated materials, therefore the statistical evaluation is executed based on the actual as well as on the nominal yield strength. The authors compared the test results to the column buckling curve b. The mean value of all the tests considering the actual yield strength is equal to 1.4 with a 2.3% lower quantile value of 0.877. Considering the nominal yield strength the mean value is equal to 1.147 with a 2.3% lower quantile of 0.817. In the same time the test results showed that nine of the annealed welded box columns with HSS material would be allocated to the buckling curve a0. Based on the statistical evaluation the authors concluded that the annealed box section columns possess strengths about 20–30% higher than the NSS columns and even 10% higher than HSS columns. But HSS columns showed definitely larger resistances (normalized by the yield strength) than NSS columns. Rasmussen and Hancock performed flexural buckling tests on 13 welded box and I-section specimens investigating the local and global buckling behaviour [5]. The specimens are manufactured from BISALLOY80 steel plates with nominal yield strength of 690 MPa. The aim of the tests was to characterise the buckling behaviour of high strength steel columns and to determine the appropriate slenderness dependent reduction factor and to propose applicable buckling curve from the multiple buckling curves used in different standards. From the 6 specimens two short (700 mm), two medium (1950 mm) and two long (3451 mm) specimens are investigated analysing dominantly different global slenderness ranges. From each specimen group one specimen was loaded centrically, and the other eccentrically. The residual stress pattern of the test specimens were also measured and evaluated. The evaluation of the test results showed that the eccentrically loaded specimen resulted 6% lower resistance, than the centrically loaded columns. The authors concluded that the European buckling curve b is conservative for the investigated columns especially at the intermediate and large slenderness region. The design buckling curve of the American and Australian standards give closer approximation of the test results that is closer to the European buckling curve a. Therefore, it is proposed to use curve a for the box section columns manufactured from S690 steel material. Four buckling tests were executed on centrically loaded rectangular-

1. literature review on buckling tests of HSS welded box section members, 2. flexural buckling tests on welded NSS and HSS members with different global slenderness and steel grades (S235, S355, S420, S460, S500, S700 and S960), 3. evaluation and documentation of the test results, 4. numerical parametric study to investigate the flexural buckling behaviour of HSS structures compared to NSS members, 5. investigation of the differences in the structural behaviour coming from the different (i) material properties, (ii) geometric imperfections and (iii) residual stress patterns and magnitudes, 6. proposal for applicable column buckling curve based on present and previous test results and based on the numerical parametric study. 2. Literature overview 2.1. General observations There are a relative large number of previous investigations available in the international literature dealing with the global (flexural) buckling behaviour of NSS box section members. But there are only a limited number of previous investigations dealing with HSS square box sections. The following publications listed below are found by the authors dealing with HSS box or hollow section columns:

• 1970 – Nishino and Tall [3]; • 1983 – Fukumoto and Itoh [4]; • 1994 – Rasmussen and Hancock [5]; • 2012 – Pavlovčič, Froschmeier, Kuhlmann and Beg [6]; • 2012 – Ban, Shi, Shi and Wang [7]; • 2013 – Ban, Shi, Shi and Bradford [8]; • 2014 – Wang, Li, Chen and Sun [9]; • 2014 – Design guidelines of the SSAB [10]. It can be observed from the list, that there are several early publications from 1970 to 1994 investigating the flexural buckling resistance of HSS columns. All the results showed that based on the yield strength the normalized flexural buckling resistance is larger for HSS members compared to specimens manufactured from NSS. The experiments also showed that the nature of the buckling phenomenon of the HSS columns are different from the NSS structures, which comes from the different residual stress pattern, geometric imperfections (manufacturing quality) and different material properties. Between 2012 and 2014 an intensive research activity was started in this subject especially in China, Australia and also in Europe in frame of the RUOSTE: Rules On High-Strength Steel RFCS Project (RFSR-CT-201200036). The main aim of these investigations are not only the characterisation of the flexural buckling resistance for HSS box sections, but also the standardization and the selection of applicable column buckling curves. The specialties of our current investigation is that all the previous research activities were focusing on a specific new steel grade (S420, S460, S690 or S960) and the flexural buckling behaviour was characterised for that specific steel material. Our current investigation focuses on a wide range of steel grades (S235, S355, S420, S460, S500; S700 and S960) ensuring the comparison possibility using the same loading equipment, support conditions and the same evaluation process. 267

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behaviour in a wide parameter range analysing 8 different cross sections within a large slenderness range. The comparison showed that all the calculation results are above the column buckling curve c that is proposed for the welded box section column with b/t ratio smaller than 30. Ban et al. also observed that all the results are above the column buckling curve a as well. Therefore they proposed the application of the buckling curve a of the EN1993-1-1 [1] for the calculation of flexural buckling resistance of welded box sections made of 960 MPa material. Wang et al. investigated the buckling behaviour of centrically loaded welded box section columns made from S460 material [9]. The purpose of the study was to investigate the overall buckling resistance of HSS welded box section columns. The experimental program included 6 centrically loaded welded box sections with pinned support conditions. Prior the tests the initial out-of-straightness of the specimen and the end-eccentricities during the loading were measured and recorded. The residual stresses of all the test specimens are also measured using sectioning technique. The applied geometric imperfections used in the tests varied between 0.19 · L/1000 – 1.5 · L/1000. The measured flexural buckling resistances were compared to the European buckling curves, and it was stated that the average resistances are underestimated with cca. 20% using the buckling curve c. The main reason of the difference is the lower detrimental effect of residual stresses in HSS columns than those in conventional steel columns. However, the limited test results are not sufficient to find an appropriate design curve for the S460 steel material, therefore additional numerical investigations were executed. The numerical calculations showed that the flexural buckling resistance of the welded box section columns made from S460 steel material may be determined using the column buckling curve b instead of curve c.

shaped sections by Pavlovčič et al. in 2011 [6]. Two welded and two cold-formed section specimens were tested with 4000 mm and 5200 mm lengths. The average yield strength of the specimens was 373.4 MPa, which does not belong to the range of the ultra-high strength steel, but these investigations give good reference points to the results achieved on HSS structures. From the 8 full scale tests 4 specimens were loaded by pure compression and 4 by combination of compression and bending by varying the loading eccentricity. The shorter specimens failed by interaction of local and global buckling, the longer ones by flexural buckling. Based on the experimental and numerical investigations the authors concluded, that the residual stresses have significant impact on the column buckling resistance, which can reduce the column buckling resistance up to 37%. In case of eccentric loading the complete imperfection combination reduced the resistance up to 45%. Based on the verified numerical model a numerical parametric study was completed to investigate the relevant equivalent geometric imperfections which represent the real structural behaviour of the structures studied in the tests. The numerical calculations showed that the measured geometric imperfections and residual stresses may be suitably replaced by equivalent geometric imperfections with amplitudes according to the recommendations of the EN1993-1-1 [1] and EN1993-1-5 [14]. Ban et al. investigated the flexural buckling behaviour of welded box section columns made from S460 material [7]. The complete test program contained I-profile and box section columns as well. The general aim of the investigation was to investigate the buckling resistance of the test specimens made from HSS material and to propose applicable column buckling curve for the S460 steel grade. Additional to the experiments a numerical model has been developed and verified based on the test results. Large number of parametric studies were carried out to study elastic and inelastic buckling behaviour, and all the results are compared to the design buckling curves of the EN1993-1-1 [1]. The experimental research program contained 5 welded box section specimens. Prior to the buckling tests the residual stresses, the geometric imperfections, the loading eccentricities and the material properties of all the specimens are determined. The specimens were supported by hinges which are intended to behave as pinned supports. However, during the tests the hinges provided moment restraints unintentionally and they behaved as rotational springs. The authors determined the spring characteristic, and the buckling lengths are modified based on a modified mechanical model. Parallel to the laboratory tests numerical analyses were also executed. A numerical parametric study was performed investigating 5 different cross sections in a large global slenderness range. Numerical calculations showed that all the results provided larger resistances in the whole analysed slenderness range than the buckling curve c that is proposed for this cross section type in the EN 1993-1-1. Thus, all the numerical calculations resulted in larger resistances in the whole slenderness domain, than the column buckling curve b, Ban et al. proposed the application of it for welded box section columns made of S460 material. Ban et al. investigated the flexural buckling behaviour of welded Isection and box section columns made from S960 HSS material in 2013 [8]. The experimental research program contained 3 welded box section specimens which are tested using pinned support conditions under axial compression. Prior to the column buckling tests the residual stresses, the geometric imperfections and the material properties are measured on the test specimens. The initial out-of-straightness of the specimens and the loading eccentricity during the tests were also measured and reported. The failure mode of all the specimens was overall flexural buckling without any interaction of local buckling. The measured resistances were compared to the European buckling curves. The specimens B2 and B3 resulted in higher resistances than buckling curve a0, but the resistance of the specimen B1 was under the buckling curve d due to its large geometric imperfection. The executed tests were the basis of the numerical model development and verification. A numerical parametric study is executed to investigate the buckling

2.3. Conclusions and research aims The previous experimental and numerical investigations prove that the welded HSS box section columns provide significantly larger resistances than the columns having the same geometry made from NSS. It means, that the buckling behaviour of the HSS columns are more favourable than for NSS columns which can lead to the application of higher buckling curves. The following three reasons from the previous investigations explain the differences: – the ratio between the compression residual stress and the yield stress is smaller for HSS members compared to NSS members, – geometric imperfections are not larger for HSS member than for NSS sections, they are even smaller due the better manufacturing control and improved quality, – different material properties that contain increased yield and ultimate strength, but also different character for the stress – strain relationship. These differences can have large impact on the buckling behaviour of welded box section columns and therefore improved column buckling curves are required for HSS members. These differences are also valid for welded and for cold-formed sections as well, however, the importance of the above mentioned three terms have different weights depending on the manufacturing process. All the previous and the current residual stress measurements proved that HSS members have significantly smaller residual stresses compared to the yield strength than NSS members having the same geometry. An improved residual stress model has been developed for welded box sections by the authors [12], which is implemented in the current global buckling resistance evaluation process. It can be also concluded that there is only a limited number of previous test results available for HSS welded box section members. On the other side all the previous investigations are focusing on one specific steel grade (S460, S690, S960) separately. Due to the large effect of the loading equipment, applied hinges and loading 268

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eccentricities in the tests the comparison of the previous test results can be questionable. Therefore a large experimental and numerical research program is designed and executed consisting of steel grades between S235 and S960. In the framework of the current research the same loading equipment, laboratory stuff and cross sections geometries are tested, which ensures the comparability of the test results and the determination of the effect of the steel grade on the flexural buckling resistance. 3. Experimental research program 3.1. Strategy of the experiments and test setup A total of 49 global buckling tests are carried out at the Budapest University of Technology and Economics, Department of Structural Engineering in 2015. All the test specimens fulfil the requirements of the cross section class 3, therefore no local buckling occurred during the tests. The investigated global slenderness range is between λ = 0.51–1.43. A total of 18 different cross sections made from welded square box sections are investigated by using 7 different steel grades (S235, S355, S420, S460, S500, S700 and S960). For each specimen the following data are measured to be able to evaluate the buckling test results and to determine the appropriate column buckling curve: – residual stresses of all cross section geometries for all analysed steel grades, – global geometric imperfections (out-of-plane straightness) for each specimen, – loading eccentricities calculated for each specimen from strain gauge measurements, – material properties for each analysed steel grades and plate thicknesses measured by coupon tests. – load-displacement diagrams regarding longitudinal and lateral displacements, – stress distribution within the plate parts to check the local buckling phenomenon.

Fig. 1. Flexural buckling test specimen and test layout.

gauges in the middle cross section are used to determine the load level of the first yielding and to check the stress distribution in the middle cross section during the loading process. 3.2. Geometry and material properties of the test specimens Before the buckling tests the width and thickness values of the test specimens are measured and summarized in Table 1. The definition of the different widths and the side numbering of the specimens are shown in Fig. 3. The specimen lengths are presented by the parameter Lcolumn and the distance between the hinges in the test (buckling length of the specimen) is given by the value of Leff. The material properties of all the test specimen are measured and considered in the evaluation procedure. Table 2 summarizes the measured average values for all the test specimen. The Young's modulus (E) is considered equal to 210 GPa in the numerical study.

The specimens are tested between cylindrical testing rigs which provided hinged support condition in one direction and fixed support in the perpendicular direction. The test set-up is shown in Fig. 1. The flexural buckling failure mode always occurred in the hinged direction. During all the tests the following data are measured: – stress distribution at the top and bottom cross-sections using 4 and 4 strain gauges placed at the plate middle points, respectively (20 cm away from the column end), – stress distribution in middle cross-section using 8 strain gauges, – rotation of the top and bottom hinges using 4 displacement transducers, – axial displacements at two sides of the specimen (uz1, uz2), – lateral displacement in the direction of the buckling plane (uy1, uy2), – lateral displacement in the direction perpendicular to buckling plane (ux).

3.3. Geometrical imperfections and loading eccentricities 3.3.1. Geometric imperfections The exact shape of the test specimens are measured prior to the buckling tests. The shape and magnitude of the imperfection is measured on each corner of all test specimens. The measurement is accomplished by using special equipment based on a moving inductive displacement transducer. The imperfection shape of the specimens is measured at least 4 times (2 times forward and 2 times backward) on each specimen. Using this measuring method the regular errors could be filtered out with adequate safety and the global shape of each specimen are derived for both direction perpendicular to the column's axis. The initial deformation of the measuring device has been eliminated from the real column imperfections. The magnitude of the imperfections are determined for all the specimens in both directions and normalized to the total length of the test specimen. The out-of-straightness imperfection magnitudes are summarized in Fig. 4. It can be seen that the measured out-ofstraightness imperfections are significantly smaller than the manufacturing tolerance given by the Eurocode (L/750), and its values vary

The location of the applied strain gauges is shown in Fig. 2a. The arrangement of the displacement transducers is presented in Fig. 2b. The axial displacement (uz) is measured between two points of the specimen. Based on the results of the displacement transducers the average axial deformation (εz) and the rotation of the hinges are recorded and evaluated. The rotation of the hinges is measured to be sure that they cannot provide significant clamping effect during the tests. Inductive transducers placed in the middle of the specimen, these are used to determine the load - deformation diagrams for the in-plane and out-of-plane deformations. The strain gauges located at the end of the specimens are used to measure the applied eccentricity of the loading equipment. The strain 269

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Fig. 2. Position of (a) strain gauges, (b) displacement transducers.

considered as a hinge. After the determination of the end-eccentricity the total imperfections are calculated by the out-of-straightness plus the average of the upper and lower end-eccentricities. The total geometric imperfection magnitudes are presented in Fig. 5 depending on the global slenderness and the steel grade. The results show that there is a large scatter in the applied total geometric imperfections. A total of 23 columns had larger imperfection magnitudes than L/1000, and the average imperfection magnitude is equal to L/945. It means that the total imperfection in average is close to the expected L/1000 value. However, it should be mentioned that the exact placement of the specimens into the loading equipment is quite difficult, and therefore it is almost impossible to guarantee the exact L/1000 imperfection magnitude. The exact imperfection values are measured, and its effect is considered in the evaluation process of the test results.

between 0 and L/1000. No clear tendencies in the imperfection magnitudes can be observed depending on the steel grade and depending on the global slenderness. The measured average out-of-straightness imperfection magnitude is about L/3000. No clear tendencies could be observed in the imperfection magnitude depending on the steel grade, therefore the same out-of-straightness imperfection shape and magnitudes are expected for HSS than for NSS members. This observation suggests that choosing the same geometric imperfection shape and magnitude in the FE calculations for HSS and for NSS members is appropriate. 3.3.2. Loading eccentricity and total imperfection The loading eccentricities are also measured during the tests. The bending moment at the end cross-sections of the specimens are determined from the strain gauge measurements during the loading process. The measured bending moment still can have two origins: (i) loading eccentricity and (ii) rotational restraint of the loading device, if it does not behave as a pure hinge. In the current tests rollers are placed at the column ends, which are expected to behave as a pure hinge without any rotational restraint. However, to separate the effect of the loading eccentricity and the rotational restraint the loading eccentricity is measured in another way as well. The first method to obtain the loading eccentricity is the evaluation of the initial slope of the normal force – end-moment diagrams that were plotted for both ends of all the specimens. The second method is the optical determination of the distance between the centre of the specimen and the centre of the hinge. The centre line of the specimen and the centre line of the hinge are signed by red lines. A photo has been made for both ends of all the specimens and the end-eccentricities are determined from the distance between the red lines by a digital evaluation process. The end eccentricities are determined by both methods and finally they are compared. The comparison showed good agreement between the end-eccentricities derived from the in-situ measurements and based on the evaluation of the initial slope of the M-N curves. It means that the initial slope of the M-N curve characterises the end-eccentricity and the support can be

3.4. Flexural buckling resistances The measured flexural buckling resistances are summarized in Table 3. The relevant column lengths, yield strengths and global slenderness ratios are also given in the table together with the buckling direction. From the measured ultimate resistances the buckling reduction coefficients (χc) are determined using the expression of Eq. (1).

χtest =

Ntest Atest ∙ f y, test

(1)

where χtest is calculated by the actual measured geometries and material properties. The calculated buckling reduction factor (χtest) is plotted in Fig. 6 in the function of the global slenderness and steel grade together with the column buckling curves of the EN1993-1-1 [1]. The previous test results on HSS members [5,8,9] are also shown in the diagrams. The results show that the lowest reduction factor belongs to the specimens made of S355 and S420 steel grades. It can also be observed that there is a good correlation between the new test results and the test results found in the international literature. However, the total 270

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Table 1 Measured width and length of the test specimens, [mm]. Name

Code

bA-C,e

bB-D,e

bA-C,m

bB-D,m

tnom

treal

Lcolumn

Leff

W2–80 × 5–2500A W2–80 × 5–2500B W2–80 × 5–2000A W2–80 × 5–2000B W2–120 × 6–2800A W2–120 × 6–2800B W2–120 × 6–2500A W2–120 × 6–2500B W2–150 × 6–2800A W2–150 × 6–2800B W2–180 × 8–2800A W2–180 × 8–2800B W3–80 × 5–2000A W3–80 × 5–2000B W3–120 × 6–2800A W3–120 × 6–2800B W3–120 × 6–2500A W3–120 × 6–2500B W3–120 × 6–1800A W3–120 × 6–1800B W3–150 × 6–2800A W3–150 × 6–2800B W3–180 × 8–2800A W3–180 × 8–2800B W4–120 × 6–2800A W4–120 × 6–2800B W4–120 × 6–2500A W4–120 × 6–2500B W4–120 × 6–2000A W4–120 × 6–2000B W4–120 × 6–1500A W4–120 × 6–1500B W4–180 × 8–2800A W4–180 × 8–2800B W4–80 × 5–2500A W4–80 × 5–2500B W4–80 × 5–2000A W4–80 × 5–2000B W4–150 × 6–2800A W4–150 × 6–2800B W5-R120 × 6–2800 W5-R120 × 6–2300 W5-R150 × 6–2000 W7-S140 × 6–2800 W7-S140 × 6–1500 W7-R180 × 8–2800 W9-S120 × 6–2800 W9-S120 × 6–1200 W9-R160 × 8–2600

21/1 21/2 34/2 34/3 23/1 23/2 24/1 24/2 25/1 25/2 26/1 26/2 27/1 27/2 28/1 28/2 29/1 29/2 30/1 30/2 31/1 31/2 32/1 32/2 35/1 35/2 36/1 36/2 37/1 37/2 38/1 38/2 40/1 40/2 33/1 33/2 34/1 34/4 39/1 39/2 6 7 8 17 18 9 10 11 12

80.7 80.1 81.0 80.3 120.2 120.4 119.8 119.7 150.1 150.6 179.4 179.8 80.0 80.0 121.1 118.3 119.2 118.4 118.4 118.1 147.8 148.1 178.1 179.4 119.5 119.3 119.5 119.7 119.9 120.3 120.5 120.5 180.0 179.9 80.3 80.0 80.1 80.4 149.3 148.8 119.3 119.5 149.3 140.2 140.3 179.2 116.8 117.0 159.0

79.7 80.2 79.7 80.1 119.7 120.4 119.3 119.1 149.0 150.1 162.7 178.0 79.6 79.6 120.0 120.1 120.5 120.0 120.1 119.8 150.9 151.4 180.2 178.7 120.6 120.2 119.6 119.7 120.6 120.3 120.5 120.8 179.8 180.0 79.5 80.3 79.9 79.9 150.2 150.1 119.8 120.0 149.3 140.2 140.1 180.0 119.7 119.4 160.0

80.7 80.4 80.4 80.3 121.2 120.4 119.5 119.9 150.8 150.1 179.4 179.8 80.2 80.7 117.8 118.9 119.3 118.9 118.8 118.2 148.8 147.9 178.6 179.1 120.1 120.4 120.1 119.7 120.4 120.1 119.8 120.6 180.5 178.2 80.0 80.2 80.7 80.8 148.9 148.9 119.3 119.5 149.3 140.3 140.3 179.3 116.4 117.7 159.0

79.6 80.3 79.5 80.2 120.5 120.4 119.3 119.2 148.8 149.0 177.9 178.2 79.9 79.5 119.0 120.0 120.0 119.6 120.7 119.3 151.0 151.5 179.4 179.5 120.6 120.0 119.5 120.2 121.1 119.8 120.6 120.1 179.8 179.6 79.5 79.4 80.3 80.1 150.5 149.7 120.1 119.7 149.8 140.1 140.2 179.9 120.0 119.2 159.9

5 5 5 5 6 6 6 6 6 6 8 8 5 5 6 6 6 6 6 6 6 6 8 8 6 6 6 6 6 6 6 6 8 8 5 5 5 5 6 6 6 6 6 6 6 8 6 6 8

5.2 5.4 5.3 5.4 6.1 6.3 6.2 6.2 6.2 6.1 8.1 8.1 5.1 5.3 6.3 6.3 6.2 6.1 6.2 6.2 6.1 6.1 8.0 8.1 6.2 6.0 6.1 6.1 6.0 6.0 6.0 6.0 7.8 7.9 5.1 5.1 5.3 5.1 5.9 6.1 6.00 6.00 5.90 6.02 6.04 8.05 6.05 6.05 8.00

2498 2498 1997 1999 2798 2798 2500 2500 2799 2798 2800 2800 1998 1998 2801 2800 2501 2501 1802 1801 2798 2800 2800 2799 2801 2798 2500 2498 2000 2001 1502 1501 2809 2809 2499 2499 2000 2000 2798 2798 2800 2300 2000 2800 1501 2800 2800 1200 2600

2638 2638 2137 2139 2938 2938 2640 2640 2939 2938 2940 2940 2138 2138 2941 2940 2641 2641 1942 1941 2938 2940 2940 2939 2941 2938 2640 2638 2140 2141 1642 1641 2949 2949 2639 2639 2140 2140 2938 2938 2940 2440 2140 2940 1641 2940 2940 1340 2740

column buckling curve b according to EN1993-1-1 [1] using the test values for the geometry and the material properties. In case of the test results regarding S500–S960, all the reduction factors are located over the column buckling curve a. The test results prove that the resistance of the HSS members is higher than the resistance of the specimens made from S235, S355 and S420 steel grades, and significantly larger than the standard resistances developed for NSS structures. 3.5. Statistical evaluation of measured column resistances and buckling curve development The purpose of the statistical evaluation is to obtain the appropriate buckling curve for the welded HSS box section columns based on the test results. Two different safety approaches are used for the determination of the appropriate buckling curve:

Fig. 3. Definition of different widths of the cross-section.

imperfection magnitude strongly influences the flexural buckling resistance. Therefore, in the further evaluation process the specimens with larger imperfection than 2 × L/1000 and with smaller imperfection than 0.1 × L/1000 are omitted from the evaluation process. The rest of the test results are presented in Fig. 7. All the test results (except one S420 specimen) are located over the

1. Lower 2.3% quantile method: The lower 2.3% quantile of the experimental results is determined and proposed as the relevant column buckling curve. This evaluation method is the basis of the

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S235 and S460 steel grades. The comparison of the test results (re) and the standard resistances (rt) are shown in Fig. 8a) for all the HSS specimens (S500–S960) and in Fig. 8b) for all the NSS specimens (S235–S460). It can be observed that the average re/rt ratio is significantly higher for HSS than for the NSS specimens. The results of the statistical evaluations are summarized in Tables 4–5 for the buckling curve c and buckling curve b, respectively. Results show that for all the studied steel grades (S235–S960) the relevant safety using the buckling curve c is clearly met. The application of the buckling curve c is proper for S235–S460 material grades, since the γM1* based on the NSS experiments is equal to 0.99. The lower 2.3% quantile of the test results is 10% higher in case of S235–S460 specimens than the resistance calculated by the buckling curve c. However, the application of the buckling curve c is conservative for S500–S960 material grades, since the γM1* based on the HSS specimens is equal to 0.90 and the lower 2.3% quantile of the experiments is 18% higher than the resistance calculated by the buckling curve c. Based on the test results the buckling curve b sufficiently met the safety criteria for the S500–S960 specimens; while γM1* is 1.00 and the 2.3% lower quantile of the experimental results is 6% higher than the resistances of the buckling curve b. According to the current results the column buckling curve b can be used for welded box section members using S500–S960 steel grades. Moreover the tendency can be observed that higher steel grades could be considered using higher buckling curves. This tendency becomes clearly visible if the resistances of the S960 specimens are evaluated separately, because these resistances are always the highest among the specimens having similar imperfection values.

Table 2 Measured material properties [MPa]. Material

S235

S355

S420 S460

S500 S700 S960

Specimen

21/1 21/2 34/2 34/3 23/1 25/1 26/1 27/1 28/1 31/2 32/1 35/2 40/2 33/1 33/2 34/1 34/4 39/1 6, 7, 8 17, 18 9 10, 11 12

Plate fy

fu

326 326 324 322 326 325 307 415 411 393 397 464 458 514 508 507 499 476 546 670 741 1005 1073

470 466 473 468 451 482 445 574 565 511 545 533 529 575 581 577 571 535 636 735 803 1047 1153

current column buckling curves used in the EN1993-1-1 [1]. This method is applied to deliver comparable values to the current buckling curves for NSS structures. 2. Buckling curve determination using predefined partial safety factor: This method ensures that the design value of the test population is determined with the proposed buckling curve and the predefined partial safety factor. The relevant safety factor for column buckling in EN1993-1-1 is equal to 1.0, thus the proposed buckling curve ensures that the curve gives the design value of the population using γM1* equal to 1.0. The determination of the predefined partial safety factor is done in the same manner as described in [16,17].

4. Numerical modelling of welded HSS box section columns 4.1. Description of the numerical model The aim of the numerical study is to investigate the structural behaviour and to determine the flexural buckling resistance of welded box section columns using numerical modelling. The numerical model is developed using the finite element software Ansys 14.5 [20]. The model is based on a full shell model using four node thin shell elements, as shown in Fig. 9. The ultimate loads are determined by geometrical and material nonlinear analysis using geometric imperfections and residual stresses (GMNIA). The Newton-Raphson approach is used in the nonlinear analysis. In the numerical model quadrilateral shell finite elements are applied. The mesh size is governed by the width of the cross-section, each single plate consists of at least 16 finite elements within one cross section. The number of the finite elements within the individual plates is regulated to ensure the correct (fine enough) application of the residual stress pattern. The width of the cross-section in the numerical

The statistical evaluation according to both methods is executed based on the rules of the EN1990 Annex D [18]. The partial safety factor is calculated by grouping all the data into subgroups based on the global slenderness. The same evaluation process for all the subgroups is executed and the final partial safety factor is determined by averaging of the safety factors determined for the subgroups. Evaluation of all test results together without subdividing them into subgroups results in a conservative result, as shown in [19]. The specimens in the range of S500–S960 are separately evaluated from the specimens made from

Fig. 4. Maximum out-of-plane imperfection magnitudes.

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Fig. 5. Maximum total imperfection magnitudes.

Table 3 Load carrying capacities and reduction factors of the tested columns. Name

#

Lhinge [mm]

fy [MPa]

λgl

Ntest [kN]

χtest

Buckling direction

W2–80 × 5–2500A W2–80 × 5–2500B W2–80 × 5–2000A W2–80 × 5–2000B W2–120 × 6–2800A W2–120 × 6–2800B W2–120 × 6–2500A W2–120 × 6–2500B W2–150 × 6–2800A W2–150 × 6–2800B W2–180 × 8–2800A W2–180 × 8–2800B W3–80 × 5–2000A W3–80 × 5–2000B W3–120 × 6–2800A W3–120 × 6–2800B W3–120 × 6–2500A W3–120 × 6–2500B W3–120 × 6–1800A W3–120 × 6–1800B W3–150 × 6–2800A W3–150 × 6–2800B W3–180 × 8–2800A W3–180 × 8–2800B W4–120 × 6–2800A W4–120 × 6–2800B W4–120 × 6–2500A W4–120 × 6–2500B W4–120 × 6–2000A W4–120 × 6–2000B W4–120 × 6–1500A W4–120 × 6–1500B W4–180 × 8–2800A W4–180 × 8–2800B W4–80 × 5–2500A W4–80 × 5–2500B W4–80 × 5–2000A W4–80 × 5–2000B W4–150 × 6–2800A W4–150 × 6–2800B W5-R120 × 6–2800 W5-R120 × 6–2300 W5-R150 × 6–2000 W7-S140 × 6–2800 W7-S140 × 6–1500 W7-R180 × 8–2800 W9-S120 × 6–2800 W9-S120 × 6–1200 W9-R160 × 8–2600

21/1 21/2 34/2 34/3 23/1 23/2 24/1 24/2 25/1 25/2 26/1 26/2 27/1 27/2 28/1 28/2 29/1 29/2 30/1 30/2 31/1 31/2 32/1 32/2 35/1 35/2 36/1 36/2 37/1 37/2 38/1 38/2 40/1 40/2 33/1 33/2 34/1 34/4 39/1 39/2 6 7 8 17 18 9 10 11 12

2638 2638 2137 2139 2938 2938 2640 2640 2939 2938 2940 2940 2138 2138 2941 2940 2641 2641 1942 1941 2938 2940 2940 2939 2941 2938 2640 2638 2140 2141 1642 1641 2949 2949 2639 2639 2140 2140 2938 2938 2940 2440 2140 2940 1641 2940 2940 1340 2740

326 326 324 322 325 325 325 325 325 325 307 307 415 415 411 411 411 411 411 411 393 393 397 397 464 464 464 464 464 464 464 464 458 458 514 508 507 499 476 476 558.5 524 551.5 670 670 741 1005 1005 1073

1.07 1.08 0.87 0.87 0.79 0.79 0.71 0.71 0.63 0.62 0.51 0.51 0.99 0.99 0.90 0.90 0.80 0.81 0.59 0.60 0.69 0.70 0.58 0.58 0.95 0.94 0.85 0.85 0.69 0.69 0.53 0.52 0.62 0.63 1.36 1.35 1.09 1.08 0.76 0.76 1.04 0.84 0.60 0.96 0.54 0.79 1.43 0.65 1.01

322 417 479 436 761 745 719 824 1019 971 1644 1544 467 423 742 707 847 950 990 995 1114 1069 2014 2193 905 1125 975 1152 1051 1055 1191 1158 1976 2085 390 544 564 549 1364 1281 1168 1418 1849 1942 2347 3386 1598 2615 3730

0.63 0.79 0.93 0.84 0.84 0.80 0.79 0.90 0.88 0.85 0.99 0.91 0.74 0.64 0.63 0.60 0.73 0.84 0.86 0.87 0.81 0.78 0.93 1.00 0.69 0.89 0.76 0.89 0.82 0.83 0.94 0.91 0.80 0.84 0.50 0.70 0.70 0.72 0.84 0.77 0.77 0.99 0.99 0.90 1.08 0.83 0.59 0.96 0.72

A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C A-C B-D B-D B-D B-D B-D B-D B-D B-D B-D

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Fig. 6. Buckling reduction factors (χtest) based on actual values.

Fig. 7. Buckling reduction factors (χtest) based on actual values, imperfection restriction.

Fig. 8. Statistical analysis for welded a) HSS and b) NSS specimens – using curve c.

axis and permitted in the other direction. The load is applied through displacement-control. In the numerical model both the global and the local imperfection shapes are modelled using user-defined imperfection shapes. The shape of the global imperfection is a half-sinus wave shape. The shape of the local imperfection is a continuous sinus wave on each side along the longitudinal axis. The number of the half waves is equal to the L/b ratio

model is considered as the actual outer width of the cross-section reduced by the wall thickness. Pinned supports are applied on the two ends of the specimens, the rotation centre of the hinge is considered 70 mm far from the specimen end. Therefore, the buckling length of a specimen is always 140 mm longer than its geometrical length. This model corresponds to the loading situation used in the laboratory tests. The rotation of the supporting nodes are allowed around one symmetry 274

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Table 4 Statistical evaluation for welded specimens – curve c.

Table 5 Statistical evaluation for welded specimens – curve b.

amplitude has opposite signs on the adjacent sides. The enlarged imperfection shape is shown in Fig. 9. Previous research results showed that a numerical simulation using residual stresses and global imperfection with an amplitude of L/1000 gives appropriate results to the Monte Carlo based column buckling curve development [21,22]. In the parametric study the global imperfection is always considered as the leading imperfection, the local imperfection is considered as the accompanying imperfection, and its magnitude is multiplied by 0.7 according to EN1993-1-5 [14]. The applied imperfection magnitudes in the numerical simulations are L/ 1000 for the global and 0.7×b/1000 for the local imperfections. It should be mentioned that the local imperfection has a minor effect on the calculated flexural buckling resistance. The residual stress model shown in Fig. 10 is applied in the numerical simulations, which model has a test based origin. The development of the residual stress model can be found in [12]. Based on it the compression stress can be calculated by Eqs. (2)–(3) and the tension residual stress can be taken equal to the yield strength for all analysed steel grades. In the equations t and b should be used in mm dimension and the stress is provided in MPa.

Fig. 9. Finite element model: mesh, imperfection shape, support conditions.

rounded up, where L is the actual length of the specimen, b is the crosssection width. The shape of the local imperfection along the crosssection is a half sinus wave on each side of the cross-section, its 275

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Fig. 10. Applied residual stress model. −1

b If t ≤ 5 mm : σrc = 70 − 21t + t 2 − (2900 − 3600 ∙ (t −5)) ∙ ⎛ ⎞ ⎝t ⎠

(2)

−1

b If t ≥ 5 mm : σrc = 70 − 21t + t 2 − (2900 − 290 ∙ (t −5)) ∙ ⎛ ⎞ ⎝t ⎠

(3)

For HSS grades the Ramberg-Osgood material model is applied in the numerical simulations. The character of the Ramberg-Osgood material model is a non-linear elastic - plastic material model using vonMises yield criterion. This material model has an isotropic hardening behaviour in the plastic domain. The parameter n of the RambergOsgood model is validated based on the measured material properties determined by coupon tests. For S500, S700 and S960 steel grades n = 14 gave the best fit to the results of the coupon tests. The material is assumed to behave according to Eq. (4) up to 1.10·fy. Thereafter, in case of the HSS steel grades the material behaves linearly with a reduced modulus of E/1000 up to ε = 10%. The yield strength (fy) is considered equal to the nominal yield strength of the analysed steel grades. The Young's modulus (E) is considered equal to 210 GPa in the numerical study.

Fig. 11. Observed failure mode in the test and in the numerical model.

results the following properties are different between NSS and HSS columns which might have significant effect on the buckling behaviour:

• the yield strength, • the residual stress pattern (value of the compression residual stresses), • the material model law (character of the stress – strain curve).

n

ε=

σ ⎛σ ⎞ + 0.002 ∙ ⎜ ⎟ E f ⎝ y⎠

(4)

The effect of these properties are discussed and evaluated separately in the numerical investigations. In each calculation step only one property is changed, all the other relevant properties are kept as constant. A second parametric study is also executed using the realistic material and residual stress properties, and based on these results appropriate buckling curves for all the investigated steel grades between S235 and S960 are proposed.

4.2. Validation of the numerical model The numerical model is verified based on the previous and the current tests results. Thus the applied numerical model should work with high accuracy for different steel grades, therefore the numerical model is validated for S355, S500 and S960 steel grades separately using different test bases. This model verification strategy proves that the flexural buckling resistance of welded square box section columns can be determined by using the developed numerical model with adequate accuracy. In the model validation the measured geometric imperfections, loading eccentricities, residual stresses and material properties of the analysed specimens are applied. The typical failure modes observed in the test and in the numerical model are presented in Fig. 11. The measured and computed load – axial deformation diagrams are also compared and presented in Fig. 12 for one specific specimen. The comparison of the measured and computed flexural buckling resistances are given in Table 6. The load – deformation curves and the observed resistances show a good agreement, which proves the reliability of the developed numerical model for the analysed steel grades.

4.3.1. Effect of the yield strength on the buckling resistance In the current section only the effect of the yield strength is investigated, all the other material properties and residual stresses are kept as constant in order to get a simple understanding of the effect of the yield strength on the flexural buckling behaviour. Columns with a cross-section size of 120 × 6 with and without residual stresses are investigated using steel grades of S235, S355, S500, S700 and S960 materials. The length of the investigated columns is varied to obtain results in the global slenderness range between 0.2 and 2.0. The obtained column buckling curves are shown in Fig. 13 and Fig. 14. Without using residual stresses the effect of the yield strength on the flexural buckling resistance can be clearly studied. If residual stresses are also applied, such a residual stress model is used where the tensile and compression residual stresses are linear functions of the yield strength, in order to keep its effect steel grade independent. Results prove that increased yield strength results in an increase of the buckling reduction factor. It can be observed that the effect of the yield strength without residual stresses is the highest if the global slenderness is about 1.0. The increase of the buckling resistance from S235 to S960 can

4.3. Strategy and results of the parametric numerical study The aim of the parametric numerical study is to obtain the appropriate buckling curve for welded box section columns made from HSS material. In order to achieve this goal as a first step the effect of the different material properties between NSS and HSS materials are studied and evaluated separately. According to the previous research 276

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Fig. 12. Load – axial deformation diagrams for specimen W9S120×6–1200.

stresses eliminate their influences. It is also observed that the effect of the residual stress is the largest in the global slenderness range between 0.9 and 1.2, which can reach 17% resistance decrease for S235, but only 5% for S960 steel grade. The explanation for this is that the magnitude of the residual stress does not increase linearly by increasing the yield strength. Therefore, in case of higher steel grades the magnitude of residual stress is proportionally lower than in case of lower steel grades. It can also be observed that the decreasing effect of the residual stresses are smaller in case of the Ramberg-Osgood material models compared to the linear elastic hardening plastic material model.

Table 6 Results of the model validation. #

Specimen number

steel grade

Ntest [kN]

Nnum [kN]

diff.

1 2 3 4 6 7 8 10 11 12

Pavlovčič 1. [6] Pavlovčič 2. [6] Pavlovčič 3. [6] Pavlovčič 4. [6] W5-R120 × 6–2800 W5-R120 × 6–2300 W5-R150 × 6–2000 W9-S120 × 6–2800 W9-S120 × 6–1200 W9-R160 × 8–2600

S355

706 564 630 584 1168 1418 1849 1598 2615 3730

649 568 670 594 1105.6 1358.6 1793.8 1587.2 2511.8 3648.7

−8% 0.7% 6.3% 1.8% 5.3% 4.2% 2.9% 0.67% 3.9% 2.2%

S500

S960

4.3.3. Effect of the material model law on the buckling resistance In this study two different material models, the linear elastic hardening plastic (lin.-hardening) and the Ramberg-Osgood (R-O) material models with a parameter n = 14 are compared. For this study the HSS materials are also calculated using the same linear elastic hardening plastic material model which is used for the NSS grades. In these cases, the value of fu is set equal to 1.10 fy. Results showed that in the low slenderness range the Ramberg-Osgood material model results in higher resistances, because in these cases the plastic region is reached and the hardening effect of the Ramberg-Osgood model is significantly higher. However, if the global slenderness is higher, the buckling starts in the elastic region, and the Ramberg-Osgood model has larger plastic elongations than the linear elastic hardening plastic material model, which results in reduction of the buckling resistance. Several typical calculation results comparing the effect of the material models are shown in Fig. 16. Based on the numerical investigations the following statements are made for the difference on the buckling resistance between NSS and HSS steel columns:

reach about 15%. If residual stresses are also applied this increasing tendency can reach 13%, as shown in Fig. 15. 4.3.2. Effect of the residual stresses on the buckling resistance The effect of the residual stresses is investigated using welded box section columns with a cross-section size of 180 × 8. For NSS grades the results using linear elastic hardening plastic material models are also presented and its comparison to the Ramberg-Osgood material model is also executed. For better visibility only the results of the S235 and S960 steel grades are presented in the current paper. The ratio of the numerically calculated buckling curves with and without the residual stresses are presented and compared in Fig. 15. The results proved that the residual stress has a significant decreasing effect on the buckling resistance if the global slenderness ratio is higher than ~ 0.6. In the lower global slenderness range the specimens have dominant plastic failure mode where the tension and compression residual

Fig. 13. Effect of the yield strength without residual stresses.

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Fig. 14. Effect of the yield strength with residual stresses.

• the buckling reduction factor of HSS columns can be higher due to the increased yield strength, • the buckling reduction factor of HSS columns can be higher due to •

seven different cross-sections (100 × 10, 90 × 6, 120 × 6, 160 × 8, 180 × 8, 150 × 6, 180 × 6). The results showed that the calculated buckling curves are outstandingly close to each other, therefore they are independent from the cross-section geometry. In the further evaluation process all of the numerical results are compared to the EN 1993-1-1 column buckling curves a, b and c. The ratios between the numerical buckling reduction coefficient and the corresponding reduction coefficient of the relevant buckling curves are determined. For each buckling curve the minimum of these ratios is calculated by Eq. (5) and presented in Tables 7–9. Table 10 summarizes the main values and the standard deviations regarding all the analysed steel grades and column buckling curves.

the reduced relative magnitude of the compression residual stresses (in case of HSS grades the residual compression stresses does not increase linearly with the yield strength), the buckling reduction factor of HSS columns could be slightly reduced because the Ramberg-Osgood type material model can have a negative effect on the buckling resistance depending on the slenderness range and steel grade.

The results of the numerical simulations show that there are two phenomena that increase and there is one which decreases the buckling reduction factor of HSS columns compared to NSS.

kmin = min λ gl

χFEM χcurve

(5)

Moreover, the safety parameters that are introduced in Section 3.5 are calculated based on the results of the numerical parametric study. The calculated safety levels are summarized in Table 11. The background is marked by green, if the necessary safety level is met; red, if the necessary safety level is not ensured. The results show that based on all of the analysed cross-sections the EN 1993-1-1 column buckling curve b is applicable for S500 and S700 steel grades, and the EN 1993-1-1 column buckling curve a for S960 steel grade. At the same time the results also prove the applicability of the EN 1993-1-1 column buckling curve c for the steel grades of S235–S355 that is currently proposed in the EN1993-1-1 [1].

4.4. Buckling curve development based on parametric numerical study To propose applicable column buckling curve for HSS columns a second parametric study is also executed where the buckling resistance of 7 different welded square box section columns are calculated using realistic properties (material model, residual stresses, imperfection magnitude L/1000). For comparison purposes S235 and S355 material grades are also studied using linear-elastic hardening plastic and Ramberg-Osgood material models as well. The buckling resistances are calculated using different column lengths to obtain results in the global slenderness range between 0.2 and 2.0. The obtained buckling curves for the cross-section 120 × 6 are shown in Fig. 17. The results show that the applicable buckling curve increases, if higher steel grades are used. The obtained buckling curves are compared to the buckling curves a, b and c. The comparison with the buckling curve b is presented in Fig. 18. The numerical buckling curves are determined for

4.4.1. Evaluation of alternative buckling curve - αmod method In the international literature [23] an alternative design method can be found to determine the column buckling resistance which is proposed for high strength steel grades. According to the proposed method Fig. 15. Effect of the residual stress – 180 × 8 using different materials.

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Fig. 16. Buckling curve with the two material models – specimen 120 × 6 S960.

the imperfection parameter α should be replaced in the function of the relevant steel grade by Eq. (6):

which have not. The proposed modification given by Eq. (7) is only applicable for steel grades between S500 and S960.

n

⎛ 235 MPa ⎞ α = α0 ∙ ⎜ ⎟ fy ⎝ ⎠

5. Conclusions and buckling curve proposal (6) 5.1. Results based on the laboratory tests

where α0 is the original α parameter based on EN 1993-1-1 [1] and the parameter n is suggested to be equal to 1.0. The numerically calculated buckling curves are compared to the αmod method using the EN 1993-11 column buckling curve c as the basis of this method (α0 = 0.49). The current results showed that this method can effectively consider the resistance increase depending of the steel grade. However, in the slenderness range between 0.5 and 1.5, where the effect of the residual stress is the largest, the proposed method slightly overestimates the flexural buckling resistance. However, good agreement could be found if the parameter n would be set equal to 0.6 instead of 1.0. In this case all the results would be on the safe side, and the method would be not so conservative for higher steel grades as the original buckling curve c. Based on the numerical simulations it can be concluded that the αmod method with n = 0.6 is applicable for HSS material grades between S500 and S960.

⎛ 235 MPa ⎞ α = 0.49 ∙ ⎜ ⎟ fy ⎝ ⎠

– In case of the welded specimens the average measured out-ofstraightness imperfection is equal to L/3000, which is significantly smaller than the manufacturing tolerance (L/750). There are no clear tendencies observed towards the dependence of steel grade and global slenderness. This observation suggests choosing the same geometric imperfection shape and magnitude in the FE calculations for HSS members as for NSS members, which has positive consequence on the buckling behaviour. – All the test results (except one) for the analysed steel grades (S235–S960) with smaller geometric imperfections than L/500 are located above the column buckling curve b using the measured values for the geometry and the material properties. – In case of the test results regarding steel grades of S500–S960 are considered all the reduction factors are located over the column buckling curve a. – Based on the statistical evaluation of the current test results the column buckling curve b sufficiently met the safety criteria of the EN1990 Annex D for the S500, S700, S960 specimens. The calculated value of γM1* is 1.00 and the 2.3% lower quantile of the experimental results is 6% higher than the resistances of the buckling curve b.

0.6

(7)

However, it should be mentioned that the design method using modification of the alpha parameter depending on the steel grade can be criticized theoretically. It does not consider the differences in the material characteristic change between the NSS and HSS materials. This fact indicates that a different alpha modification factor should be developed for steel grades, which have yield plateau and for steel grades

Fig. 17. Buckling curves for different material grades – 120 × 6.

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Fig. 18. Comparison of simulation results to buckling curve b – 120 × 6.

Table 7 Final comparison between numerical results and column buckling curve a.

Table 10 Statistical evaluation of the different column buckling curves.

kmin - curve a Section

100 × 10 90 × 6 120 × 6 160 × 8 180 × 8 150 × 6 180 × 6 Min

Curve b/t

10 15 20 20 22.5 25 30

Ramberg-Osgood

Lin-hard.

S235

S355

S500

S700

S960

S235

0.85 0.84 0.84 0.83 0.83 0.84 0.84 0.83

0.89 0.88 0.88 0.87 0.87 0.88 0.88 0.87

0.93 0.93 0.92 0.92 0.92 0.92 0.92 0.92

0.97 0.97 0.96 0.96 0.96 0.96 0.94 0.94

1.01 1.02 1.00 1.00 1.00 1.00 1.05 1.00

0.90 0.89 0.90 0.89 0.90 0.91 0.93 0.89

curve a

S355

curve b 0.92

curve c

100 × 10 90 × 6 120 × 6 160 × 8 180 × 8 150 × 6 180 × 6 Min

10 15 20 20 22.5 25 30

Ramberg-Osgood S355

S500

S700

S960

S235

0.90 0.94 0.93 0.93 0.92 0.93 0.93 0.90

0.95 0.98 0.98 0.97 0.97 0.97 0.97 0.95

1.04 1.04 1.03 1.02 1.02 1.02 1.01 1.01

1.09 1.08 1.06 1.05 1.05 1.06 1.03 1.03

1.11 1.12 1.09 1.09 1.07 1.11 1.15 1.07

0.99 0.99 0.99 0.98 0.99 0.99 0.99 0.98

S355

100 × 10 90 × 6 120 × 6 160 × 8 180 × 8 150 × 6 180 × 6 Min

10 15 20 20 22.5 25 30

Ramberg-Osgood

1.00

S355

S500

S700

S960

S235

0.97 1.01 1.00 1.00 0.99 1.00 0.99 0.97

1.01 1.05 1.04 1.04 1.04 1.04 1.02 1.01

1.08 1.09 1.08 1.07 1.06 1.06 1.04 1.04

1.12 1.13 1.11 1.10 1.07 1.09 1.12 1.07

1.15 1.16 1.09 1.09 1.10 1.22 1.23 1.09

1.00 0.99 1.01 1.04 0.99 0.99 0.99 0.99

S700

S960

S235

S355

0.91 0.054 0.97 0.030 1.03 0.027

0.95 0.054 1.01 0.030 1.08 0.031

0.98 0.046 1.05 0.024 1.13 0.033

1.03 0.042 1.10 0.030 1.18 0.048

1.07 0.038 1.15 0.038 1.23 0.062

0.97 0.039 1.04 0.043 1.11 0.067

1.00 0.044 1.07 0.055 1.15 0.081

– The obtained buckling resistances for the S500 and S700 steel grades are always higher than the EN 1993-1-1 column buckling curve b. – The obtained buckling resistances for the S960 steel grade are always higher than the EN 1993-1-1 column buckling curve a. – Results also prove the applicability of the EN 1993-1-1 column buckling curve c for steel grades S235–S355, as currently proposed by the EN1993-1-1 [1]. – The simulation results are also compared to previous design methods found in the international literature. Based on the numerical simulations an enhanced imperfection modification factor

Lin- hard.

S235

S500

The results of the numerical simulations show two phenomena that increase and one phenomenon that decreases the buckling reduction factor of HSS columns compared to NSS material grades. Using the verified numerical model a further parametric study is executed analysing different square box section columns which are the bases of the column buckling curve development. Based on the numerical simulations the following final conclusions are drawn:

kmin - curve c b/t

S355

cause the Ramberg-Osgood type material model can have a negative effect on the buckling resistance depending on the slenderness range and steel grade.

1.00

Table 9 Final comparison between numerical results and column buckling curve c.

Section

S235

• the buckling reduction factor of HSS columns can be higher due to the increased yield strength, • the buckling reduction factor of HSS columns can be higher due to the reduced magnitude of the compression residual stresses, • the buckling reduction factor of HSS columns could be smaller be-

Lin- hard.

S235

Lin-hard.

According to the numerical parametric study the following conclusions are made for the difference on the buckling resistance between NSS and HSS steel welded box section columns:

kmin - curve b b/t

Mean St. dev. Mean St. dev. Mean St. dev.

Ramberg-Osgood

5.2. Results based on the numerical simulations

0.92

Table 8 Final comparison between numerical results and column buckling curve b.

Section

Statistical parameter

S355

1.01

1.01

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[2] EN 1993-1-12. Eurocode 3: design of steel structures – Part 1-12: Additional rules for the extension of EN 1993 up to steel grades S700. CEN, 2007. [3] F. Nishino, L. Tall, Experimental investigation of the strength of T-1 steel columns, Lehigh University, Fritz Engineering Laboratory Report No. 290(9), 1970. [4] Y. Fukumoto, Y. Itoh, Evaluation of multiple column curves using the experimental data-base approach, J. Constr. Steel Res. 3 (3) (1983). [5] K.J.R. Rasmussen, G.J. Hancock, Tests of high strength steel columns, J. Constr. Steel Res. 34 (1995) 27–52. [6] L. Pavlovčič, B. Froschmeier, U. Kuhlmann, D. Beg, Finite element simulation of slender thin-walled box columns by implementing real initial conditions, Adv. Eng. Softw. 44 (2012) 63–74. [7] H. Ban, G. Shi, Y. Shi, Y. Wang, Overall buckling behaviour of 460 MPa high strength steel columns: experimental investigation and design method, J. Constr. Steel Res. 74 (2012) 140–150. [8] H. Ban, G. Shi, Y. Shi, M.A. Bradford, Experimental investigation of the overall buckling behaviour of 960 MPa high strength steel columns, J. Constr. Steel Res. 88 (2013) 256–266. [9] Y.B. Wang, G.Q. Li, S.W. Chen, F.F. Sun, Experimental and numerical study on the behavior of axially compressed high strength steel box-columns, Eng. Struct. 58 (2014) 79–91. [10] M. Heinisuo, Axial resistance of double grade (S355, S420) hollow sections manufactured by SSAB, 2014. [11] M. Clarin, High strength steel local buckling and residual stresses, Licentiate thesis, Lulea University of technology, Dept. of Civil and Environmental Engineering, 2004. [12] B. Somodi, B. Kövesdi, Residual stress measurements on welded HSS square box sections, Thin-Walled Struct. (Submitted for publication). [13] B. Somodi, B. Kövesdi, Residual stress measurements on cold-formed HSS hollow section columns, J. Constr. Steel Res. 128 (2017) 706–720. [14] EN1993-1-5. Eurocode 3: design of steel structures – Part 1.5: Plated structures. CEN, 2006. [15] B. Somodi, B. Kövesdi, Flexural buckling resistance of cold-formed HSS hollow section members, J. Constr. Steel Res. 128 (2017) 179–192. [16] B. Johansson, R. Maquoi, G. Sedlacek, New design rules for plated structures in Eurocode 3, J. Constr. Steel Res. 57 (2001) 279–311. [17] Document CEN/TC250-CEN/TC135 – report on the consistency of the equivalent geometric imperfections used in design and the tolerances for geometric imperfections used in execution, 2010. [18] EN 1990. Eurocode: Basis of structural design. CEN, 2005. [19] L. da Silva, T. Tankova, J. Canha, L. Marques, C. Rebelo, Safety assessment of EC3 stability design rules for flexural buckling of columns. Evolution Group EC3-1-1, CEN TC 250-SC3-EvG-1-1, TC8-Technical Committee 8, 2014. [20] ANSYS® v14. 5, Canonsburg, Pennsylvania, USA. [21] A. Taras, R. Greiner, New design curves for lateral–torsional buckling—proposal based on a consistent derivation, J. Constr. Steel Res. 66 (2010) 648–663. [22] D. Sfintesco, A. Carpena, Experimental bases of the ECCS column curves, 2nd Intern. Coll. on Stability—Introductory report, Tokyo, (Liege, Washington). 1977. pp. 68–75. [23] IABSE, AIPC, IVBH: Use and application of high-performance steels for steel structures. Structural Engineering Documents 8. Chapter 5.3.

Table 11 Calculated safety parameters of the analysed steel grades.

according to Eq. (8) is applicable for steel grades between S500 and S960 for welded box section columns.

⎛ 235 MPa ⎞ α = 0.49 ∙ ⎜ ⎟ fy ⎝ ⎠

0.6

(8)

The results of the current numerical simulations are in harmony with the results of the laboratory tests and the statistical evaluation of the measured column buckling resistances. At the same time all the results are also in harmony with the previous test results and numerical simulations found in the international literature. However, the current investigations enlarge the investigated parameter range (S235–S960), use the same methodology for the evaluation of all the steel grades, make possible to observe the differences in the buckling behaviour between steel grades, and determine its reasons, which are unique in this topic. Acknowledgement The presented research results are part of the RUOSTE: Rules On High-Strength Steel; RFCS Project RFSR-CT-2012-00036 and the STEELBEAM Hungarian R & D project (No. PIAC_13-1-2013-0160). The financial support is gratefully acknowledged. This paper was also supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. References [1] EN 1993-1-1. Eurocode 3: design of steel structures – Part 1.1: General rules and rules for buildings. CEN, 2009.

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