Residual stress of 460 MPa high strength steel welded box section: Experimental investigation and modeling

Thin-Walled Structures 64 (2013) 73–82

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Residual stress of 460 MPa high strength steel welded box section: Experimental investigation and modeling Huiyong Ban, Gang Shi n, Yongjiu Shi, Yuanqing Wang Department of Civil Engineering, Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Tsinghua University, Beijing 100084, PR China

a r t i c l e i n f o

abstract

Article history: Received 10 November 2011 Received in revised form 5 November 2012 Accepted 19 December 2012 Available online 21 January 2013

An experimental study is presented to investigate and model the residual stress in 460 MPa high strength steel welded box sections using the sectioning method. A total of six square box sections with various width-thickness ratios and thicknesses of steel plates were tested and over 2000 original readings were obtained to quantify the magnitude and distribution of both compressive and tensile residual stresses. The effects of width-thickness ratio and plate thickness on residual stresses, the human error and interaction of the four component plates were clarified. The compressive residual stress magnitude was found to be significantly correlated with the sectional dimensions; while for the tensile stresses near the weld region no clear correlation was found. The human error obtained through the comparison of experimental results measured by two persons for the same section was quite small. No residual stress interaction among four component plates was identified due to the residual stress equilibrium within each plate. Finally, a distribution model and its simplified form were established in this study, and a good agreement was found in the comparison with experimental results. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Residual stress 460 MPa high strength steel Welded box section Sectioning method Experiment Modeling

1. Introduction Sectional residual stress has significant influence on the mechanical performance of steel structural members especially the buckling capacity of compression columns because of the cause of premature yielding and loss of stiffness. An accurate and reliable establishment of residual stress is very important both for numerical simulations and design theories especially for high strength steel (HSS) structures which have been more and more widely applied in the world [1–4]. Although the residual stress models for normal strength steel (NSS) sections were well developed based on previous investigations such as [5–14], and some further adopted in steel structure design specifications like in Eurocode 3 [15,16], ANSI/AISC 360-10 [17,18] and Chinese code GB50017-2003 [19,20] etc., they may not be still reliable and accurate for HSS sections due to the following three major reasons: (1) the material properties and manufacturing processes for HSS sections are different from those of NSS ones; (2) the maximum tensile residual stress near the weld region may be lower than the yield strength (fy) resulting from its higher yield strength while for NSS section it is normally taken as fy; (3) most current models have not taken into account

n Correspondence to: Department of Civil Engineering, Tsinghua University, Beijing 100084, PR China. Tel.: þ86 10 6279 7420; fax: þ86 10 6278 8623. E-mail address: [email protected] (G. Shi).

0263-8231/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2012.12.007

the effects from section dimensions and no detailed calculation formula were established. Consequently, it becomes necessary to investigate and model the residual stress in HSS sections, and welded box section shape is the focus of this study. A few previous tests have been carried out for the residual stress in HSS welded box section members. Four welded box sections made of 690 MPa HSS with the thickness of 5 mm were studied in Refs. [21,22], from which only the compressive residual stress was obtained through strain gauges attached at each component plate near the half-width. Two welded box sections made of A514 (690 MPa) HSS 1/4 in. thick plates were examined in Ref. [23] using the sectioning method, and the magnitude and distribution on the outside surface of the box shape was obtained because direct measurements of strains inside were impossible. It was found that the ratio of compressive residual stress divided by yield strength was much smaller than that of A7 (250 MPa), and the maximum tensile residual stresses near the weld regions for the former were 419 MPa–683 MPa with an average of 542 MPa. Three welded box sections made of HT80 (690 MPa) HSS 6 mm thick plates and another three ones made of SM58 (460 MPa) 4.5 mm HSS plates were measured in Refs. [24,25], respectively. The maximum tensile residual stress near weld region was about 60% and 80% of the yield strength for the former and latter respectively; while the average compressive residual stresses were quite different for sections with various dimensions. Other section shapes and steel grades for HSS were also investigated such as 690 MPa steel welded I and cruciform sections [21,22,26]

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Plate thickness t (mm)

h0/t

RB1-460 RB2-460 RB3-460 RB4-460 RB5-460 RB6-460

100 140 150 240 330 380

100 140 150 240 330 380

10 14 10 12 12 10

8 8 13 18 25.5 36

8

Sectional width B (mm)

t

Sectional height H (mm)

h0

Section label

H

Table 1 Specimen sectional dimensions.

Fig. 1. Schematic diagram for specimen section shape.

and 420 MPa hot-rolled equal angle section [27]. All these residual stress measurements were mainly employed in associated column buckling analyses; however no models were established for HSS welded box sections, let alone the identification of effects from section dimensions. The authors investigated the correlation of the compressive residual stress with sectional geometry based on the experimental results reviewed above [28], but only effects from width-thickness ratios were taken into account. As a result, the existing investigations are not sufficient for a comprehensive understanding of the residual stress in welded box sections made of HSS. The present study describes an experimental program to investigate the residual stress distributions of six welded box sections fabricated from 460 MPa HSS plates which differed in widththickness ratios and thicknesses of component plates. Longitudinal residual stress measurements were obtained by the sectioning method at whole section locations including the weld regions, and the effects of section dimensions, human error and interaction among four component plates were identified. A model and its simplified form to describe the residual stress distribution and magnitudes as well as effects of sectional dimensions are established for 460 MPa HSS welded box sections, and compared with the experimental results with a good agreement.

2. Experimental program 2.1. Section specimen preparation Six welded square box section shapes were measured in this program as summarized in Table 1 where the geometrical parameters are defined in Fig. 1. All component steel plates of Q460C (460 MPa) HSS were flame cut with moderate thicknesses of 10 mm, 12 mm and 14 mm, and welded together using butt welds. During the welding process, the manual metal arc welding (MMAW) was used, and the shielding gasses were Ar80% þCO220%. The voltage of the welding gun was around 25 V–27 V, the amps were 230 A– 235 A and the welding velocity was about 32 cm/min. The filler wire type was JM-60, which were made to the American specification AWS A5.28-2007 ER80S-G. The yield strength of the filler wire material was 545 MPa, and the elongation after fracture was 26%. The number of runs of all butt welds was three, and there was no backing plate. The detail of the weld dimension is shown in Fig. 1. This welding procedure was qualified according to the Chinese standard JGJ81-2002 Technical specification for welding of steel structure of building [29]. The average static yield strength was 505.7 MPa and the elastic modulus was 2.11  105 MPa according to tension coupon tests consisting of nine standard coupons. 2.2. Sectioning process The sectioning method [9] was employed in this experimental program, and the Wire-cut Electron Discharge Machining

(WEDM) was used to slice the section into strips so that thermal effects were minimum. The Whittemore strain gage with a gage length of 10 in. (i.e. 254 mm) and a sensitivity of 1/10,000 in. (i.e. 1/254,000 mm) was used to obtain the strains of each strip during the sectioning process. Fig. 2 represents a typical cutting arrangement for the welded square box sections (taking RB3-460 by way of example), where the length of the adopted central portion was 500 mm (i.e. no less than 3.0 times of the sectional lateral size) and the distance from both ends was 350 mm (i.e. no less than 1.5–2.0 times of the sectional lateral size) so as to ensure a representative initial residual stress distribution over the entire steel profile [9]. The number of measurement points of section RB3-460 was fifteen and thirteen in one flange and one web plate respectively. For other section sizes, the width of the strip specimens was also taken as 10 mm. Two holes were prepared on each strip specimen with a distance of 254 mm (equal to the Whittemore strain gage length) in between ends for installation of the Whittemore stain gage as shown in Fig. 2. The sectioning process includes three steps as shown in Fig. 3: in step1, holes for the Whittemore gage were drilled and initial readings were taken for each couple of holes only on the outside surface because it is impossible to obtain inside readings; in step 2, the four component plates were separated by WEDM, the readings for each couple of gage holes on both surfaces were measured again, and the inside measurements in this step were taken as the initial ones in the calculation process of residual stresses on the inside surface; in step 3, the four component plates were completely sliced into strips by WEDM and the final readings on both surfaces were taken. Fig. 4 shows all the strips after being sliced from section specimen RB3-460. 2.3. Data interpretation For the outside surface of the component plate, the released strain of each strip specimen was obtained by comparing the initial readings taken in step 1 and final ones taken in step 3, while for the inside surface, it was obtained through the initial readings in step 2 and final ones in step 3. The temperature corrections were considered by using a reference bar and the detailed approach was reported in Ref. [9]. No strip was curved visibly in step 3 so that the corresponding correction was not taken into account. The residual stress was calculated by using Hooke’s Law, i.e. the released strain multiplied by the elastic modulus of the steel.

3. Test results and analysis 3.1. Residual stress test results In this experimental program over 2000 readings were taken from all of the strip specimens which are sufficient to construct the corresponding distribution over the entire box section. Fig. 5 summarizes the magnitude and distribution of residual stresses

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10mm×13

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Fig. 2. Arrangement of specimens resulted from a typical box section RB3-460 (the same for the others).

Fig. 3. Sectioning method steps.

the residual stress on the outside surface was taken to represent the residual stress in such sections which was also the basic data for following discussions in this study. Similar to previous studies [23,24], an analogous distribution shape of the residual stress over welded box section was found for 460 MPa HSS compared to NSS, where nearly constant compressive residual stresses were observed over the central portion of each component plate and rather high tensile stresses were obtained near the flange-web junctions. This distribution is due only to the welding at the edges of the plates during fabrication [7]. Such a distribution shape can be characterized in Fig. 6 where src1 to src4 represent the average of nearly constant compressive stresses from four component plates, and srt1 to srt8 denote the maximum tensile stresses near the weld region. Table 2 summarizes all these representative residual stress values based on the experimental results on the outside surface of the box shape, where ‘‘ * ’’ denotes that the values were measured by another different operator from the section RB5-460. Based on Fig. 5 and Table 2, it was found that the residual stresses in 460 MPa HSS welded box sections had different features from those of NSS sections: firstly the compressive residual stress was significantly different among those with various section dimensions (i.e. width-thickness ratio and steel plate thickness) and these effects were not taken into account in exiting models for NSS sections, which will be discussed in Sections 3.2 and 3.3 in detail; secondly most of the maximum tensile residual stresses near the weld region was found to be much lower than the steel yield strength, while the latter was normally taken as the maximum tensile one for NSS box sections. These primary research findings suggest that it may be necessary to establish a more detailed and accurate model for HSS welded box sections. 3.2. Effects of plate width-thickness ratio

Fig. 4. Strip specimen RB3-460 after being sectioned.

for all six sections including measured results at both surfaces of strips and their average values. It should be noted that all of the residual stresses on the inside surface were much lower than those on the outside surface because the former were obtained by using the initial readings taken in step 2 after separating the component plates. During step 2, the constraints from the adjacent plates were removed after the plate separation; as a result, a certain amount of the initial residual stress was released. The regions near the weld shrank and the central location of the component plate expanded due to the initial stress status within the plate, which led to the reduction of both the compressive and tensile residual stress magnitude. The measured results at the inside surface did not consider the stress release in step 2, the compressive and tensile residual stress magnitudes were therefore both smaller than those at the outside surface, as shown in Fig. 5. This demonstrated that the residual stresses at the inside surface obtained in this way were not accurate. As a result, only

From Fig. 5 and Table 2, it was found that the average compressive residual stress (src) over the central portion of component plates significantly correlated with the section dimension. As shown in Fig. 7(a), it decreased with the increase of the plate width-thickness ratio (h0/t). However, no clear correlation between the maximum tensile stress (srt) near the weld regions and width-thickness ratio (h0/t) was identified in Fig. 7(b). 3.3. Effects of plate thickness Experimental results from sections RB1-460 (t ¼10 mm) and RB2-460 (t ¼14 mm) with the same width-thickness ratios (h0/ t¼ 8) for component plates but different thicknesses (10 mm and 14 mm respectively) were categorized to identify the effects of plate thickness on the residual stress, as shown in Fig. 8. An analogous distribution shape between these two sections was indicated in Fig. 8(a). From Fig. 8(b), it was found that the compressive residual stress src (i.e. the average value of src1, src2, src3, and src4 in Fig. 6) of RB2-460 was quite close to that of section RB1-460, with 0.6% less than the latter. This may be due to

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300

300

300

0

0

-300

-300

0 -300

Outside Inside Average

-300

-300

-300

0

0

300

300 300 0 -300

Outside Inside Average

Outside Inside Average

-300 0 300

0 300 300 0 -300

-300 0 300

300 0 -300

-300 0 300

500 0 -500 300 0 -300 Outside Inside Average

Outside Inside Average

-500 -300 0

0 300

500 300 0 -300

-300 0 300

500

0

-500

-500

0

5

500 0 -500

Outside Inside Average

-500 0 500 500

0

-500

-500

0

500

Fig. 5. Residual stress test results for all sections. (a) RB1-460; (b) RB2-460; (c) RB3-460; (d) RB4-460; (e) RB5-460 and (f) RB6-460 compressive residual stress in negative and tensile ones in positive.

the insignificant difference for plate thicknesses between these two sections, i.e. 10 mm and 14 mm. To clarify the effects of plate thickness more intensively, Fig. 8(b) further supplied the measured results from section RB5-460 (t ¼12 mm, h0/t ¼25.5) in this study and RES-29 (t¼ 4.5 mm, h0/t ¼29.1) also made of 460 MPa HSS in Ref. [25].

It was found that the compressive residual stress src of RB5-460 was much lower (i.e. 62.3%) than that of RES-29 even the former had smaller width-thickness ratio. In addition, it is true that increased distribution area of compressive residual stress leads to reduced stress magnitude when the welding heat as the energy input is constant. Consequently, it can be concluded that the

H. Ban et al. / Thin-Walled Structures 64 (2013) 73–82

77

Fig. 9(a) represents nearly coincident distribution shapes for the measured results from two operators. From Fig. 9(b), it was found that the differences between the residual stresses from RB5-460 and RB5*-460 were slight, with relative error values of 18.1%,  19.4%, 4.7% and  2.5% for the compressive residual stresses src1, src2, src3, and maximum tensile stress srt, respectively. For the average of compressive stresses over the three component plates (for RB5*-460 only three were measured) the relative error was 3.4%. Consequently, it was indicated that the human error during the measurement in sectioning method process was small, and the experimental data was reliable and accurate.

thickness of plate has significant effects on the compressive residual stress. However, the maximum tensile residual stress srt (i.e. the maximum value of srt1 to srt8 in Fig. 6) seems to hardly correlate with the plate thickness according to Fig. 8(b); the stress srt of RB2-460 was 7.9% larger than that of RB1-460, while the stress srt of RES-29 was 7.2% less than that of RB5-460. 3.4. Human error analysis Experimental results measured by different operators for section RB5-460 are compared in Fig. 9 to quantify the human error, and the residual stress magnitudes were characterized in Table 2 labeled as ‘‘RB5-460’’ and ‘‘RB5*-460’’ in which only three of the component plates were measured for the latter.

3.5. Interaction of component plates To clarify the interaction of four component plates in the welded box shape, i.e. whether the four plates are self-balanced separately or just true for the entire section, it is necessary to examine the stress equilibrium within both the whole section and each component plate. The self-balancing deviation stress serr was employed to evaluate the satisfaction of stress equilibrium, and can be obtained by using Eq. (1) as follow—a closer value of serr to zero means a better stress equilibrium status, " # n X serr ¼ sri Ai =A ð1Þ i¼1

where, n is the amount of strip specimens, sri is the residual stress magnitude at one strip location which is positive for tensile ones and negative for compressive ones, Ai denotes the cross-sectional area of the strip, and A is the whole sectional area. The calculation results for each component plate and whole section are shown in Fig. 10. From Fig. 10 it was found that most of the deviations were negative and much less than the steel yield strength (fy). For each component plate, the deviation stress was around 2%–5% of the

Fig. 6. Illustration of residual stresses distribution and major characteristics over entire section.

Table 2 Representative residual stresses at typical locations for HSS welded box sections. Compressive stresses (MPa)

srt1

srt2

srt3

srt4

srt5

srt6

srt7

srt8

src1

src2

src3

src4

142.5 132.9 259.4 312.7 289.3 258.0 168.8

207.8 76.4 306.2 202.0 263.8 247.7 246.0

246.5 192.6 294.9 219.6 239.6 243.2 269.2

141.1 54.2 296.5 278.3 246.4 247.8 220.2

212.9 312.8 342.4 341.3 408.3 386.6 455.6

163.8 254.7 366.2 418.3 489.9 477.7 292.5

201.8 252.6 332.2 431.4 473.2 – 353.7

289.8 222.7 348.2 343.8 223.7 – 383.3

 203.5  144.9  144.8  78.5  74.5  88.0  83.0

 187.5  191.0  149.2  67.2  53.6  43.2  67.8

 201.6  251.1  165.7  115.5  99.6  104.3  81.5

 146.9  147.7  197.2  112.7  46.2 –  81.5

Compressive stress σrc (MPa)

RB1-460 RB2-460 RB3-460 RB4-460 RB5-460 RB5*-460 RB6-460

Tensile stresses (MPa)

-300

Tensile stress σrt (MPa)

Specimen label

-250 -200 -150 -100 -50 0 5

10 15 20 25 30 35 Width-thickness ratio h0/t

40

600 500 400 300 200 100 0 5

10 15 20 25 30 35 Width-thickness ratio h0/t

40

Fig. 7. Correlation between residual stress magnitudes and width-thickness ratios of component plates: (a) compressive stresses and (b) tensile stresses.

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H. Ban et al. / Thin-Walled Structures 64 (2013) 73–82

500 Residual stress (MPa)

300 0 -300 RB1-460 RB2-460 -300 0

400 300

RB5-460 RES-29 [25]

200 100

300 0 -300

289.8

312.8

σrc

0 -100

σrt

-68.5

-200

300

489.9 454.4

RB1-460 RB2-460

-184.9 -183.7 -181.8

Typical residual stress label

-300 0 300

Fig. 8. Comparison of experimental results between section RB1-460 and RB2-460 with the same width-thickness ratio but various thicknesses: (a) distribution and (b) magnitude.

500

Residual stress (MPa)

0 -500 Operator1 Operator2 -500

400 300 200 100 0

0

-100 500

500

0

-500

-500

0

489.9 477.7

RB5-460 RB5*-460

500

500

σrc1

σrc2

σrc3

σrc4

-46.2 -53.6 -43.2 -74.5 -88.0 -99.6 -104.3

σrf

Typical residual stress label

Fig. 9. Comparison of experimental results of section RB5-460 from different operators: (a) distribution and (b) average magnitude.

Fig. 11. Existing models of residual stress in NSS welded box sections: (a) ECCS model and (b) Chinese model.

Fig. 10. Residual stress self-balancing deviation calculation results.

yield strength, thus the flanges and webs can be considered as selfbalancing separately. This indication was also adopted in existing models for NSS welded box section in which the distribution shapes in both flanges and webs were symmetric [16,18,20]. 4. Modeling of residual stress for 460 MPa HSS welded box sections 4.1. Existing models for NSS welded box sections In 1960s and 1970s, Dwight et al. at Cambridge University investigated the welding-induced residual stress in plate

structures [30,31]. The Cambridge research developed the concept of a tendon force, which is equilibrated by compressive residual stresses. The tendon force is determined as a function of the heat input rather than weld size. This research significantly contributed in understanding the development of welding-induced residual stresses. For the residual stress distribution within welded box sections, in Europe [16], residual stress distributions for welded square box sections derived experimentally and theoretically were used as shown in Fig. 11(a). The distribution ranges and compressive stress magnitudes were presented of a range of plate widththickness ratios (H/t). Table 3 summarizes the parameter values in this distribution model for five different sections mentioned in [16]. It was found that the compressive stress calculation considered the effects from plate width-thickness ratios but excluded the thickness, and the tensile stress was determined as a fixed

H. Ban et al. / Thin-Walled Structures 64 (2013) 73–82

value, i.e. the steel yield strength (fy). Besides, the determination of the distribution range of tensile stress near the weld region considered the effects of the section dimension. In America, a complete model was not available [18] for welded box sections; the maximum tensile residual stress at the weld was equal to or greater than the steel yield strength. In China [20], the model used in column buckling calculation is shown in Fig. 11(b), in which the maximum tensile residual stress was equal to the yield strength (fy) and the maximum compressive residual stress was a fixed value of 0.53fy; no specified distribution range was given. None of the existing models above considers the effects from both the width-thickness ratio and plate thickness as identified in this investigation. The following section is to establish a new detailed model to represent the residual stress distribution for 460 MPa HSS welded square box section. 4.2. Proposed model for 460 MPa HSS welded box sections On the basis of shapes of residual stress distribution characterized in Fig. 6 and other existing models for NSS sections reviewed in Section 4.1, the proposed residual stress distribution

79

shape for 460 MPa welded square box sections was simplified as the multiple stepped functions (see Fig. 6), where constant tensile and compressive residual stresses extend over a certain length near the weld regions and central portion of plates respectively, and residual stresses in other regions between the constant ones change linearly. This distribution shape is symmetrical about the two axes due to the equilibrium of both axial force and bending moment within the section, and the magnitudes were labeled as follows: constant tensile stress in flanges sfrt ¼ srt1–4, constant tensile one in webs swrt ¼ srt5–8 and constant compressive one in both flanges and webs src ¼ src1–4. The multiple stepped functions of the distribution shape in flanges sfr(x) and webs swr(y) are described in Eqs. (2) and (3), respectively. 8 s , 0 rx r a > < frt src sfrt s þ ð xa Þ, a ox o a þ b ð2Þ sfr ðxÞ ¼ sfr ðBxÞ ¼ frt b > :s , a þb r x rB=2 rc 8 s , > < wrt swr ðyÞ ¼ swr ðh0 yÞ ¼ swrt þ src vswrt ðyuÞ, > :s , rc

0 r yr u u o yo u þ v

ð3Þ

u þ v ry r h0 =2

Table 3 Parameter values for ECCS model. Welding type

srt/fy

src/fy

a

b

0

10 20 20 40 40

– Heavy weld Light weld Heavy weld Light weld

1.0 1.0 1.0 1.0 1.0

 0.60  0.82  0.29  0.29  0.13

0 3t 1.5t 3t 1.5t

– 3t 1.5t 3t 1.5t

1

RB2-460, h0/t = 8.0 RB3-460, h0/t = 13.0

Residual Stress (MPa)

200

RB4-460, h0/t = 18.0 RB5-460, h0/t = 25.5 RB5*-460, h0/t = 25.5 RB6-460, h0/t = 36.0

2 3 4 5 6

RB1-460, h0/t = 8.0

7

RB3-460, h0/t =13.0

8

RB4-460, h0/t =18.0

9

Flange

RB5*-460, h0/t =25.5

RB5-460, h0/t =25.5 RB6-460, h0/t =36.0

10

0

450

-100

Web

RB2-460, h0/t = 8.0

100 Flange midline

Flange

Weld

460

300

Web midline

RB1-460, h0/t = 8.0

345 300

Normalised Position in Half Web

H/t

150 0 Residual stress (MPa)

-150

-300

Fig. 13. Residual stress test results distribution in the web.

-200

Table 5 Experimental results of other 460 MPa welded section specimens [25].

Web

-300 0

1

2 3 4 5 6 7 8 Normalised Position in Half Flange

9

10

Fig. 12. Residual stress test results distribution in the flange.

Specimen label

h0/t

t (mm)

src.t (MPa)

RES-29 RES-44 RES-58

29.1 44.6 58.2

4.49 4.43 4.48

 181.8  125.0  85.2

Table 4 Residual stress ranges in proposed model. Model

a

b

c

u

v

w

Proposed Simplified

t t þ0.05h0

Eqs. (4) and (6)

Eqs. (4) and (6)

0.05h0

Eqs. (5) and (7)

Eqs. (5) and (7)

H. Ban et al. / Thin-Walled Structures 64 (2013) 73–82

-500

-500

-400

-400

-300

-300

σrc (MPa)

σrc (MPa)

80

-200 -100 0 10

-200 -100

8

20 25 Test results h0 /t 30 35 Fitted results

0 10

10

15

14 16

12 m) t (m

8 10

15

20 25 Test results h0 /t 30 Simplified formula results35

12

14 16

m) t (m

Fig. 14. Comparison between test results and fitted nonlinear surface: (a) proposed model and (b) simplified model.

where the residual stress extension lengths (a, b, u and v) are illustrated in Fig. 6; B is the flange width and h0 is the web height as shown in Fig. 1; x is the coordinate along flange width B and y is the coordinate along web height h0 as shown in Fig. 6. The residual stress equilibrium was valid individually for each component plate as discussed in Section 3.5, and the corresponding equations were established in Eqs. (4) and (5) ZZ sfr dA ¼ 0 ð4Þ Af

ZZ

swr dA ¼ 0

ð5Þ

Aw

where Af and Aw are the cross-sectional area of the flange and web, respectively. Eqs. (6) and (7) describe the geometry shown in Fig. 6, where B is the flange width and h0 is the web height as shown in Fig. 1. 2ða þ bÞ þ c ¼ B

ð6Þ

2ðu þ vÞ þw ¼ h0

ð7Þ

Table 6 Fitted results of compressive residual stresses. Specimen label

src.c (MPa)

src.c/src.t

src.cs (MPa)

src.cs/src.t

RB1-460 RB2-460 RB3-460 RB4-460 RB5-460 RB5*-460 RB6-460 RES-29 RES-44 RES-58

 212.4  184.9  170.1  127.4  99.2  99.2  87.7  186.9  155.7  134.5

1.15 1.01 1.04 1.36 1.45 1.26 1.12 1.03 1.25 1.58

 252.5  236.8  180.4  139.2  114.7  114.7  106.7  184.0  167.8  158.5

1.37 1.29 1.10 1.49 1.67 1.46 1.36 1.01 1.34 1.86

Table 7 Residual stress magnitudes in proposed model. Model

sfrt (MPa)

swrt (MPa)

src

Proposed Simplified

345 460

460 460

Eq. (9) Eq. (10)

4.3. Parameter identification for tensile residual stress The parameters in the distribution shape functions Eqs. (2) and (3) and equilibrium equations Eqs. (4)–(7) were determined on the basis of experimental results introduced in Section 3.1, such as the residual stress magnitudes (sfrt, swrt and src) and their extension lengths (a, b, c, u, v and w) as shown in Fig. 6. As demonstrated in Sections 3.2 and 3.3, no clear correlation was found between the tensile stresses (sfrt and swrt) and sectional dimensions, therefore they were determined to be constant for both magnitudes (sfrt and swrt) and corresponding extension ranges (a and u). Fig. 12 summarizes the experimental results from flanges of all section specimens where the position along the half flange was represented by 11 portions. Based on this figure as well as the conclusion of negative equilibrium deviation stress (which may indicate that measured tensile stresses were underestimated) discussed in Section 3.5 and conservative considerations, the tensile stress sfrt was taken as 345 MPa (i.e. 75% of the steel nominal yield strength) to cover 100% of the corresponding experimental results and marked in Fig. 12. Its distribution range (a) was determined as a constant value of the plate thickness (t) as summarized in Table 4, and the other two ranges (b and c) in flanges can be obtained by Eqs. (4) and (6). The experimental results from webs of all sections were summarized in Fig. 13 where the position along the half web was represented by 10 portions. The nominal yield strength 460 MPa was determined for the tensile stress swrt to cover 90%

of corresponding experimental data. The distribution range (u see Fig. 6) was taken as 0.05h0 as summarized in Table 4, and the other two ranges (v and w) in webs can be obtained by Eqs. (5) and (7). 4.4. Parameter identification for compressive residual stress As demonstrated in Sections 3.2 and 3.3, the compressive residual stress significantly correlated with the sectional dimensions such as width-thickness ratio h0/t and plate thickness ratio t. An empirical equation was suggested to represent the correlation of experimental data (src) with the sectional dimensions above, as shown in Eq. (8).   src ¼ s0 þM h0 =t a1 þNðtÞa2 ð8Þ The empirical coefficients (s0, M, N, a1 and a2) in Eq. (8) were determined by fitting the experimental data including those summarized in Table 2 and those obtained from three 460 MPa welded box sections in [25] (reviewed in Table 5), and to ensure the calculated envelop could cover the experimental data. The resulting empirical equation was given in Eq. (9), where the units of stress and thickness are in MPa and mm, respectively.   src ¼ 9001100 h0 =t 0:1 550t0:4 , & Z 460, & r 46 ð9Þ It is noteworthy that the calculated value for src by using Eq. (9) may become extremely large (e.g. larger than the yield

Web Weld

200 0 -200 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Normalized position in flange Normalized position in web Experimental data

Residual stress (MPa)

Weld

Model

Web

Web Weld

Residual stress (MPa)

Web

400

Web

Weld

400 200 0 -200 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Weld

200 0 -200 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Normalized position in flange Normalized position in web

Web

Weld

400 200 0 -200 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Normalized position in flangeNormalized position in web

Web Weld

Weld

400 200 0 -200 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Normalized position in flange Normalized position in web

Residual stress (MPa)

Residual stress (MPa)

Web Weld

Web Weld

400

Normalized position in flange Normalized position in web Web

81

Simplified model

Residual stress (MPa)

Residual stress (MPa)

H. Ban et al. / Thin-Walled Structures 64 (2013) 73–82

Web

Web Weld

Weld

400 200 0 -200 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Normalized position in flange Normalized position in web

Fig. 15. Comparison of proposed models with experimental results: (a) RB1-460; (b) RB2-460; (c) RB3-460; (d) RB4-460; (e) RB5-460 and (f) RB6-460.

strength) if width-thickness ratio or thickness of steel plates is very small. Therefore, the steel nominal yield strength fy (460 MPa) was adopted as the maximum magnitude for the stress src. On the other hand, a minimum magnitude was also advised as 0.1fy (46 MPa) conservatively. A comparison was shown in Fig. 14(a) where the fitted nonlinear surface for the compressive stress src was plotted according to Eq. (9). For further quantitative comparison, Table 6 gives the calculated results of the compressive residual stress (src.c) for each section specimen by using Eq. (9) and their ratios divided by experimental data src.t (the average of sfrc1, sfrc2, sfrc3 and sfrc4 listed in Table 2 or the value of src.t listed in Table 5). It was found that Eq. (9) could describe the experimental data very well. Considering that Eq. (9) seems too complex for engineering application especially for the indices of power functions, it was further simplified by assuming the indices (a1 and a2, see Eq. (8)) as a fixed value of  1, and other empirical coefficients (s0, M, N) were re-determined through the fitting of the experimental data in Tables 2 and 5. The resulting simplified function was shown in Eq. (10), and the corresponding simplified fitted nonlinear surface was plotted in Fig. 14(b) to be compared with the experimental data. Calculated results (src.cs) by using Eq. (10) for all sections were given in Table 6. It was found that the simplified formula could also describe the experimental data well.

4.5. Comparison Based on governing equations and parameters determined above (see Fig. 6, Tables 4 and 7 and Eqs. (2)–(10)), the proposed model and its simplified form can be used to represent the residual stress of 460 MPa HSS welded box sections. The results of the model are compared with experimental measurements in Fig. 15 where a satisfactory agreement was found for each section investigated in this study. Considering that the effects from sectional geometric properties were well described by the proposed models as indicated in Fig. 15, it was concluded that the model could be further extended for 460 MPa HSS welded square box sections with other dimensions than those investigated.

5. Conclusions

ð10Þ

The residual stress of six welded square box sections made of 460 MPa high strength steel were investigated in this experimental program. On the basis of measured residual stress magnitudes and distributions, the effects from sectional geometric properties, human error during measuring process and interaction among component plates were clarified in detail. Furthermore, a complete residual stress model and its simplified form were established and compared with experimental data from all section specimens. Based on this study, the following conclusions were made

Besides, in the simplified model the constant tensile stresses in both flanges (sfrt) and webs (swrt) were taken as the same value of 460 MPa (summarized in Table 7) and the distribution range a for sfrt (see Fig. 6) was determined as tþ 0.05h0 (summarized in Table 4) so that the residual stresses in flanges and webs are symmetric with each other.

(1) The distribution shape of residual stress over the entire welded 460 MPa HSS box sections was found to be similar to that of NSS welded box sections and a multiple stepped function was established to describe it. (2) The tensile residual stress near welds in flanges was much lower than that in webs which may result from the welding

src ¼ 101500

1 1 550 , h0 =t t

& Z460,

& r46

82

(3)

(4)

(5)

(6)

H. Ban et al. / Thin-Walled Structures 64 (2013) 73–82

detail. In the proposed model the former was suggested as 345 MPa (i.e. 75% of the nominal steel yield strength) while the latter was suggested as the yield strength like the NSS. Both of stresses were also suggested to have the same value of 460 MPa (i.e. the nominal steel yield strength) in the simplified model for conservative and convenient considerations. The compressive residual stress in each component plate for 460 MPa high strength steel box sections was found to be significantly correlated with the section geometric including width-thickness ratios and plate thicknesses. The compressive residual stress was reduced with the increase in the width-thickness ratio and steel plate thickness. Human error was investigated by considering measured residual stress obtained from the same section specimen (RB5-460) by two different operators. Very small errors were found which indicated that the sectioning method used in this study was reliable and accurate enough. Only one typical type of welding process (butt weld) was adopted in this experimental program and further studies are needed to clarify the effects of the welding when other types of welding processes are used. A model and its simplified form were established to represent the residual stress over box sections welded from 460 MPa HSS, in which effects of section dimensions including widththickness ratios and thicknesses were taken into account. The models were in good agreement with experimental results. Both models can be further applied to 460 MPa high strength steel welded box sections with other section geometric properties and easily employed in a finite element analysis to consider the effects of residual stress in nonlinear buckling simulations, for example. Consequently, this investigation is valuable for further researches on high strength steel structures.

Acknowledgment The authors gratefully acknowledge the financial supports from the National Natural Science Foundation of China (No. 51078205). Thanks are also extended to the Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry for the test equipment and conditions provided. References [1] International association for bridge and structural engineering. Use and application of high-performance steels for steel structures. Zurich: IABSE; 2005. [2] Pocock G. High strength steel use in Australia, Japan and the US. The Structural Engineer 2006;84(21):27–30. [3] Griffis GL, Axmann G, Patel BV, Waggoner CM, Vinson J. High-strength steel in the long-span retractable roof of Reliant Stadium, 2003 NASCC Proceedings, Baltimore, MD, 2003: 1–9. [4] Shi G, Ban HY, Shi YJ, Wang YQ Recent research advances on high strength and high performance steel structures in Tsinghua University. Proceedings of

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