Improved brace fracture model for seismic evaluation of concentrically braced frames

Engineering Structures 206 (2020) 110184

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Improved brace fracture model for seismic evaluation of concentrically braced frames

T

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Mahmoud Faytarounia, Jay Shena, , Onur Sekerb, Bulent Akbasb a b

Department of Civil, Construction and Environmental Engineering, Iowa State University, USA Department of Civil Engineering, Gebze Technical University, Turkey

A R T I C LE I N FO

A B S T R A C T

Keywords: Fracturing process Fracture initiation Square hollow section brace Database of tested specimens Damage measure Brace ductility

Reliably predicting brace fracture is crucial in seismic evaluation of post-fracture performance of concentrically braced frames. This paper evaluates existing fracture models, proposes a new fracture initiation model based on a large amount of experimental data, and compares all models with the test results. The existing ductility-related empirical models and strain-related fiber models are evaluated by experimental tests. A comprehensive database of previously tested square hollow structural sections from different research programs is established to conduct a regression analysis leading to development of a fracture initiation model. The study concludes that the existing fracture models are unable to predict fracture events observed during physical tests. Furthermore, the existing strain-related fracture models, accessible to all users in OpenSees, fail to simulate the cyclic response of braces beyond initial inelastic deformation. The fracture initiation model proposed by this study depends on the reliable measure of ductility, and it has consistently predicted fracture initiations observed during the tests. The fracturing process concept, along with the proposed fracture initiation model, is able to simulate the cyclic response of the braces, whether they are tested as single brace members or in actual braced frames, throughout inelastic deformation, fracture initiation, fracture propagation, and terminal fracture stages, providing a much-improved tool for numerical simulation of the post-fracture response of concentrically braced frames.

1. Introduction The fracture of braces was explicitly included in seismic response analysis of the concentrically braced frame (CBF) in the 1980 s when some of the primary design provisions for the CBF were developed in the modern seismic design specifications. In fact, an empirical fracture criteria of the steel hollow structural section (HSS) proposed in the late 1980 s by Tang and Goel [1,2], and then refined by Lee and Goel [3], and Hassan and Goel [4], played a critical role in the seismic response evaluation of the CBF during the developments that established the basic seismic design requirements for the special CBF (SCBF) in the current seismic design specification, AISC 341–16 [1]. This empirical fracture model was primarily based on a group of six square HSS brace specimens with width-thickness ratios ranging from 14 to 30 [1–4]. The model was used in simulating a scaled six-story CBF that was tested using the pseudo-dynamic method. It demonstrated a good match between analytical and experimental results in terms of displacement time-history responses [2]. Consequently, the seismic performance of a set of six-story CBFs subjected to three earthquake ground motions of moderately low intensity was evaluated using the empirical fracture

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model. This model was based on the seismic design provisions for the SCBF that were recommended in the early 1990 s [4,6,7] and have significantly influenced seismic design practice ever since. In particular, two major conclusions from these developments [1–4,6–7] that directly resulted from the empirical brace fracture model are: (1) Square HSS braces with a small width-thickness ratio of no more than 94 Fy or 0.55 E Fy were defined as ductile due to their ability to avoid fracture during cyclic tests. (2) The CBF with ductile braces exhibited very satisfactory behavior that was characterized by experiencing about a 3% story drift ratio because brace fractures were avoided, as compared to a more than 6% story drift ratio where a much more slender section was used. In addition, the developments [1–4,6–7] demonstrated unequivocally that: (1) The seismic response of CBFs was extremely sensitive to modeling post-elastic behavior of braces, including modeling brace fracture; and

Corresponding author. E-mail address: [email protected] (J. Shen).

https://doi.org/10.1016/j.engstruct.2020.110184 Received 26 March 2019; Received in revised form 30 December 2019; Accepted 6 January 2020 0141-0296/ © 2020 Elsevier Ltd. All rights reserved.

Engineering Structures 206 (2020) 110184

M. Faytarouni, et al.

Table 1 Summary of the cyclically tested SHS braces. Study

Specimenc,d

Shapea,b

b t

KL r

E Fy

Experiment

Nf

μf

Δf

β = Pf Py

∑ (0.1Δ1 + Δ2)

∑ (Δ1 + Δ2)

Δc Δy

Δt Δy

Δc Δcr

7.52 7.52 7.52 7.52

94.0 50.0 115 80.0

525.5 525.5 525.5 525.5

113.3 112.4 109.9 113.3

44.3 68.9 36.4 50.6

113.3 112.4 109.9 113.3

9.5 7.0 9.7 8.5

6.5 9.6 5.5 8.2

19.4 11.7 27.7 15.9

— — — —

14.2 5.60 5.60

80.0 80.0 80.0

491.5 353.7 353.7

55.1 49.5 31.0

23.5 19.3 13.4

55.1 48.1 30.5

6.0 7.7 4.5

7.5 2.4 1.0

10.5 16.6 9.6

— — —

25.0 25.0 30.0 14.0 14.0 14.0

58.0 35.0 43.0 77.0 45.0 45.0

485.4 480.8 517.2 405.4 405.4 405.4

13.9 17.4 35.7 72.4 25.1 23.0

5.20 7.40 3.50 10.5 9.80 15.0

13.9 17.4 13.7 45.8 22.5 20.6

3.0 3.2 7.9 10.5 6.0 5.0

1.0 1.0 1.0 1.0 1.0 1.0

4.0 3.6 9.2 19.5 7.4 6.2

0.75 0.60 0.30 0.50 0.30 0.60

16.8 13.8 16.0 14.0 17.8 13.8 11.4 17.0 13.0

52.3 53.9 53.3 52.4 64.8 65.8 61.6 63.5 59.7

424.9 453.4 457.1 443.5 424.9 453.4 437.9 457.1 443.5

33.3 66.5 32.5 43.6 35.9 65.0 91.0 41.9 49.9

9.40 26.8 17.1 19.6 12.0 26.2 50.1 18.5 25.0

33.3 68.3 33.4 43.6 27.6 64.6 91.1 42.9 41.2

4.8 8.8 5.8 7.3 7.3 6.8 9.6 7.1 8.2

4.1 4.7 3.8 3.6 2.4 3.6 3.5 2.8 2.7

6.2 11.5 7.5 9.5 11.1 10.2 13.8 10.3 11.6

0.40 0.40 0.80 0.45 0.38 0.32 0.30 0.50 0.30

6 × 6 × 3/8b

14.2

51.3

483.3

67.7

25.8

54.1

6.4

6.8

8.1

0.45

S1-CyIS1 S1-CyIS4 S2-CyLS1

40 × 40 × 2.5a 20 × 20 × 2.0a 40 × 40 × 2.5a

13.1 6.70 12.9

33.3 87.8 97.8

745.5 965.4 608.7

195 296 159

83.3 88.3 44.5

164.7 295.9 158.3

16.3 21.1 7.4

12.6 19.4 7.4

17.3 29.6 14.4

— — —

Johnson AM. (2005) [22]

HSS-02e HSS-03e HSS-04e HSS-05e

5 5 5 5

× × × ×

5 5 5 5

× × × ×

11.3 11.3 11.3 11.3

77.6 77.2 77.7 81.0

414.0 414.0 396.2 396.2

— — — —

— — — —

— — — —

6.7 7.3 6.7 6.9

5.1 4.4 5.0 5.9

12.4 13.4 12.7 14.0

0.67 0.67 0.70 0.62

Han et al. (2007) [23]

S77-28c S85-14A S85-14B S85-14C S90-8A S90-8B S90-8C

100 100 100 100 100 100 100

× × × × × × ×

100 100 100 100 100 100 100

28.3 13.7 13.7 13.7 8.11 8.11 8.11

76.9 84.9 84.9 84.9 90.4 90.4 90.4

489.4 493.5 463.1 463.0 463.1 499.6 488.5

15.1 92.6 75.0 112.9 19.8 26.3 22.6

7.60 28.8 30.3 43.4 9.50 14.2 12.8

15.1 76.7 76.0 114.7 19.8 27.2 22.6

2.0 10.0 7.9 8.0 3.0 2.7 2.5

2.0 8.5 8.5 9.0 2.5 2.2 2.0

3.3 18.6 15.3 15.5 6.3 5.4 5.1

0.90 0.35 0.90 0.40 0.90 0.90 0.90

Herman DJ. (2007) [24]

HSS-06e HSS-07e HSS-09e HSS-10e

5 5 5 5

5 5 5 5

× × × ×

3/8b 3/8b 3/8b 3/8b

11.3 11.3 11.3 11.3

81.2 66.1 77.1 77.8

448.2 448.2 448.2 440.7

— — — —

— — — —

— — — —

9.6 11.1 7.6 7.7

4.8 4.8 4.3 5.6

17.9 16.7 13.3 13.8

0.85 0.70 0.70 0.73

Kotulka BA. (2007) [25]

HSS-12e HSS-14e HSS-17e

5 × 5 × 3/8b 5 × 5 × 3/8b 5 × 5 × 3/8b

11.3 11.3 11.3

70.1 81.0 81.7

440.7 440.7 440.7

— — —

— — —

— — —

7.2 5.9 8.2

4.3 5.9 6.2

11.5 11.1 15.6

0.73 — 0.53

Uriz and Mahin (2008) [26]

LSB LNB

6 × 6 × 3/8b 6 × 6 × 3/8b

14.2 14.2

49.0 49.0

478.5 478.5

48.3 38.0

29.0 26.0

48.3 35.2

11.0 11.0

6.5 2.6

13.0 13.0

0.45 0.65

Tremblay et al. (2008) [27]

RHS2 RHS4c RHS19

254 × 254 × 15a 254 × 254 × 13a 254 × 254 × 15a

13.0 17.0 13.0

40.0 40.0 60.0

579.7 579.7 579.7

57.9 33.1 85.9

31.9 15.8 49.3

57.9 33.1 85.9

6.0 4.5 8.5

4.5 3.0 6.0

6.0 3.6 11.3

0.60 0.40 0.40

Richard J. (2009) [28]

RHS10c RHS12c RHS13c

254 × 254 × 9.5a 254 × 254 × 9.5a 254 × 254 × 8.0a

24.7 24.7 30.4

56.0 40.0 40.0

476.0 465.5 584.7

30.1 22.9 36.4

13.5 9.30 12.1

31.0 23.5 30.6

3.0 4.2 5.3

2.0 2.0 2.5

3.9 4.9 6.0

0.50 0.45 0.40

Fell et al. (2009) [29]

HSS1-1 HSS1-3 HSS2-1 HSS2-2

101.6 101.6 101.6 101.6

× × × ×

101.6 101.6 101.6 101.6

14.2 14.2 8.50 8.50

77.0 77.0 80.0 80.0

439.5 439.5 340.9 340.9

48.7 45.2 126.8 120.2

16.9 18.1 27.8 32.9

48.7 45.2 99.1 108.1

6.0 6.6 8.7 7.3

6.0 4.7 8.5 6.9

10.6 11.6 19.3 16.1

0.70 0.46 0.30 0.25

Nip et al. (2010) [30]

2050-CS-CFc 2050-CS-CF 2050-CS-CF 1250-CS-CF

60 40 40 40

60 40 40 40

× × × ×

3a 4a 3a 3a

17.0 7.00 10.5 10.5

40.0 62.0 61.5 34.2

574.1 491.5 471.4 471.4

40.0 48.9 44.3 46.9

20.8 22.4 19.2 27.5

40.1 48.9 44.3 46.9

5.2 4.3 5.5 5.6

4.8 4.2 5.5 5.3

5.8 6.0 7.8 6.3

0.85 0.75 0.70 0.80

Powell JA. (2010) [31]

HSS-18e HSS-20e

5 × 5 × 3/8b 5 × 5 × 3/8b

11.3 11.3

81.3 72.4

580.0 490.0

— —

— —

— —

10.1 8.5

6.6 6.4

16.4 13.3

0.75 0.50

× × × ×

Jain et al. (1978) [17]

6 9 12A 15

25 25 25 25

Black et al. (1980) [18]

17 18 22

4 × 4 × 1/4b 4 × 4 × 1/2b 4 × 4 × 1/2b

Lee and Goel (1987) [3]

Spec.1c,d Spec.2c,d Spec.4c,d Spec.5d Spec.6d Spec.7d

5 5 4 4 4 4

× × × × × ×

5 5 4 4 4 4

× × × × × ×

Shaback B. (2003) [19]

1Ac 1Bd 2Ac,d 2Bd 3Ac,d 3Bd 3Cd 4Ac,d 4Bd

127 127 152 152 127 127 127 152 152

× × × × × × × × ×

127 127 152 152 127 127 127 152 152

Yang et al. (2005) [20]

#5

Goggins et al. (2005) [21]

× × × ×

× × × ×

25 25 25 25

× × × ×

a

2.7 2.7a 2.8a 2.9a

0.188b 0.188b 0.125b 0.25b 0.25b 0.25b × × × × × × × × ×

6.4a 8.0a 8.0a 9.5a 6.4a 8.0a 9.5a 8.0a 9.5a

3/8b 3/8b 3/8b 3/8b × × × × × × ×

3.2a 6a 6a 6a 9a 9a 9a

× × × ×

6.4a 6.4a 9.5a 9.5a

(continued on next page) 2

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Table 1 (continued) Study

Specimenc,d

Shapea,b

b t

KL r

E Fy

Experiment

Nf

HSS-21e HSS-22e HSS-24e

5 × 5 × 3/8b 5 × 5 × 3/8b 5 × 5 × 3/8b

Haddad et al. (2011) [32]

1d 2d 4d 5d 6d 7d

152 152 127 127 127 127

× × × × × ×

152 152 127 127 127 127

Lai and Mahin (2013) [33]

TFB1(1F,W) TFB1(1F,E) TFB1(2F,W) TFB1(2F,E) TFB4(1F,W) TFB4(1F,E) TFB4(2F,W) TFB4(2F,E)

6 6 5 5 6 6 5 5

6 6 5 5 6 6 5 5

× × × × × × × ×

× × × × × × × ×

× × × × × ×

8a 8a 8a 8a 8a 8a

3/8b 3/8b 5/16b 5/16b 3/8b 3/8b 5/16b 5/16b

μf

Δf

β = Pf Py

∑ (0.1Δ1 + Δ2)

∑ (Δ1 + Δ2)

Δc Δy

Δt Δy

Δc Δcr

11.3 11.3 11.3

81.1 84.3 68.8

482.5 479.1 448.2

— — —

— — —

— — —

8.4 7.7 8.9

4.9 4.6 6.4

15.0 14.5 13.9

0.35 0.40 0.20

13.8 13.8 10.7 10.7 10.7 10.7

64.7 64.7 66.2 54.4 49.9 49.9

428.3 428.3 416.4 416.4 416.4 416.4

20.5 24.6 36.8 40.1 11.6 41.8

9.20 12.8 17.9 14.5 7.20 18.6

21.1 26.2 38.5 31.6 11.6 41.8

5.1 4.7 6.0 6.0 3.5 7.5

1.6 2.2 3.4 3.4 2.0 4.5

7.7 7.1 9.4 8.1 4.5 9.7

0.50 0.40 0.35 0.40 0.60 0.40

14.2 14.2 14.2 14.2 14.2 14.2 14.2 14.2

46.0 46.0 51.0 51.0 46.0 46.0 51.0 51.0

626.3 626.3 517.9 517.9 494.0 494.0 468.0 468.0

46.8 123 58.1 66.8 25.9 36.7 26.2 23.3

30.5 50.0 32.5 34.5 11.0 16.0 11.0 14.2

46.8 98.6 58.1 56.7 20.4 36.7 23.3 26.2

11.9 11.2 6.6 5.3 4.3 6.2 5.8 5.8

9.0 11.8 5.3 5.8 7.6 4.3 4.8 5.8

14.0 12.7 8.3 6.7 5.0 7.2 7.5 7.5

0.50 0.38 0.67 0.67 0.50 0.44 0.75 0.72

Note: a Dimensions in millimeters. b Dimensions in inches. c Specimens with b t larger than λhd given in AISC 341–16. d Specimens with reported Δf values. e Specimens without Δf values.

(2) Satisfactory seismic behavior of a CBF was judged by story drift ratio rather than post-elastic behavior, including brace fracture. In other words, the seismic performance of CBFs is considered satisfactory as long as story drift ratio is under 3%, regardless of brace fractures.

(2) What performance criteria shall be used for a CBF to exhibit very satisfactory seismic behavior, story drift, brace fracture, and/or anything else? (3) Are the CBFs with square HSS that meet the provisions in the current seismic design specification considered very satisfactory?

These developments and the resulting conclusions have made a significant contribution to improving the understanding of seismic behavior and design of CBFs, given the limitations on experimental data and computational capability in the late 1980 s. Ever since these pioneering works recorded in [1–4,6–7], extensive experimental data and many empirical fracture models on HSS braces have become available. In retrospect, with more and more experimental studies on brace members with HSSs, the fracture models may have become more reliable. However, it is intriguing to note that the studies spanning 30 years have not been able to update and modify seismic design practice in any meaningful way. In fact, the current edition of seismic design specification, AISC 341–16 [5], still refers to the works of three decades ago. For instance, the most important design parameter, the limit on the width-thickness ratio of square HSS, was kept as 0.64 E Fy (or its numerically equivalent form) between 1990 and 2010. AISC 341–10 [8] began to require braces to be highly ductile with the width-thickness ratio of square HSS limited to 0.55 E Fy (or its numerically equivalent form of 94 Fy ), a ratio that was proposed 20 years earlier in [1]. Furthermore, advanced computational capability has enabled extensive seismic analysis, deterministically as well as probabilistically, for almost all scales of CBFs subjected to a large number of earthquake ground motions. As a result, the conclusions based on the seismic responses of one six-story CBF subjected to only three ground motions using the fracture model derived from very few tested specimens should have been reexamined and amended if needed. Consequently, the following concerns emerged from the preceding discussions:

This paper seeks to address the first concern, which is considered an essential first step toward addressing the next two concerns. A comprehensive literature review found a considerable number of experimental studies on seismic braces, which formed the basis for 12 fracture models that have been proposed in the literature. The goal of this paper is to evaluate these models, and identify or develop a reliable way of introducing brace fracture for seismic performance evaluation of CBFs. The paper is organized as follows: (1) The results of experimental studies on seismic braces with square hollow sections are summarized, focusing on fracture life under cyclic loading; (2) The 12 existing fracture models are evaluated to identify a reliable way of introducing brace fracture for seismic performance study of CBFs; and (3) A fracture initiation model is developed and applied to a three-stage fracturing process model. Its reliable prediction of fracture life is demonstrated by comparison with experimental data.

2. Experimental studies on seismic brace in the literature Numerous experimental studies on steel braces have been conducted since the 1970 s. The experimental programs available in the literature (technical reports, refereed journals, or conference proceedings) have been carefully reviewed to identify the most effective experimental efforts. Relevant experimental data on seismic braces with square hollow sections (SHS) under cyclic loading will be collected from these programs. The experimental programs with potential were identified using the following criteria:

(1) How reliable are the existing fracture models in terms of predicting brace fracture under cyclic load and evaluating seismic performance of modern CBFs? If not, develop one.

3

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M. Faytarouni, et al.

3. Existing ductility-based fracture models of seismic braces

(1) Sufficient information was provided on the tested specimens; (2) The tested specimens were made following the governing design provisions, the cyclic testing program followed a well-accepted procedure [9–13]; and (3) Experimental results were provided in tables, figures, or both to reveal the cyclic axial force versus deformation relationship up to a terminal stage as a perceived fracture.

A number of empirical fracture models of seismic braces with rectangular hollow structural sections have been developed since 1980 s by various researchers, based on experimental studies available at the time when these models were proposed. The experimental studies provided essential information in identifying potential influential factors as well as accumulating test data on brace fracture under cyclic loadings, as summarized in Table 1. On the capacity side, the width-tothickness (b t ) and slenderness (KL r ) ratios have been identified by most experimental studies as primary and secondary parameters influencing fracture resistance. In addition, specified minimum steel yield stress (Fy ) has also been considered to have some impact on fracture life. On the demand side, however, there is no consensus on the response index of a seismic brace leading to its fracture. Many assumptions on the fracture-inducing response index have been made in order to develop fracture models, ranging from accumulative deformation (the sum of deformations of all cycles) to maximum deformation (the maximum deformation at brace fracture). Although the formulations of existing empirical models are substantially different from each other, they were all formulated in the same pattern, as follows:

The review has found that the well-documented experimental programs in the US, Canada, and Europe over 35 years that followed the respective governing design specifications [8,14–21] have tested 79 SHS specimens. The test results of the specimens from these programs [3,22–38] are summarized in Table 1 and are divided into two categories, A and B. Category A includes the testing program and specimen properties, and Category B includes fracture-causing response indices measured from the cyclic test results, either directly reported or evaluated from the hysteresis loops presented in the reference. Items in these two categories are described as follows: (A) The first six columns in Table 1 include: (1) the reference of the experimental study, (2) the specimen’s identity in the reference, (3) SHS shape sizes, (4) width-to-thickness ratio, b t , (5) slenderness ratio, KL r , and (6) elastic modulus (E )-to-minimum yield stress (Fy ) ratio, E Fy . (B) Columns 7 through 12 show the test results in terms of deformation response at the time of fracture. The formats of the test results are organized so that the existing fracture models can be evaluated against the experimental results. Some experimental studies directly reported their measured data in this category together with other testing results. In case the data were not reported in any of these studies, the hysteresis loops of axial force and deformation relation were used to obtain the data such as Nf , Δ1, Δ2 , Δc , Δt , Δcr , Δy , and β . Nf is the equivalent number of standard cycles at fracture, Δ1 and Δ2 are the peak deformations in compression and tension at each cycle, respectively, Δc is the maximum deformation under compression, and Δt is the maximum deformation under tension prior to fracture. Δcr and Δy are the deformations corresponding to buckling and yielding loads, respectively, and β is the remaining tensile strength ratio at fracture. Descriptive illustrations will be presented later in the paper to better reflect on the definitions of Nf , Δ1, Δ2 , and β .

(1) The fracture demand index was based on either the accumulative number of cyclic deformations or maximum cyclic deformations; and (2) A set of design properties such as b t , KL r , and/orE Fy were included in an empirical function as dependent variables. The empirical function introduces various numerical parameters around these design parameters to fit a given group of experimental data on brace fracture life. Based on these observations, the existing empirical fracture models published in the literature were summarized and discussed in three groups below, according to their assumptions on the fracture-inducing response index.

3.1. Group I: Accumulative ductility-based fracture models In the late 1980 s, Tang and Goel [1,2] proposed one of the earliest empirical fracture models to predict the fracture life of square HSS braces. The significant development this fracture model helped to achieve was in verifying that by reducing the limit on b t ratio from 190 Fy = 28, as then prescribed in the 1980 AISC Plastic Design Specification [Refs] for square HSS braces to 95 Fy = 14 [Refs], premature fracture is prevented from occurring in CBFs under severe earthquakes. Tang and Goel [1,2] proposed an accumulative tensile deformation-based fracture model with a procedure converting a cyclic normalized deformation history into a number of standard cycles with tensile deformation. The major assumption of this model is that only the amplitude of the tensile deformation contributes to low-cycle-fatigue fracture. The procedure includes reducing the hysteretic cycles into three types of standard cycles: (i) simple (tension-only) cycles (for Δt > Δy ), (ii) incremental (tension and compression) cycles, and (iii) small cycles (for peak cycle tension deformation Δt < yielding deformation, Δy ). An exemplified illustration of this procedure is presented in Fig. 1. These cyclic responses with hysteretic cycle types (i) and (ii) are considered by Eqs. (1a) and (1b), respectively, that define an equivalent ductility demand, Ns , in terms of the accumulative tensile deformation excursion between peak tension deformation Δt , and peak compression deformation Δc , in each cycle, divided by yielding deformation Δy . Note that the cycles with Δt < Δy i.e., cycle type (c), were ignored in predicting fracture. Based on these assumptions, a set of two empirical fracture life equations, Eqs. (1c) and (1d), representing two effective slenderness ranges, were developed from the experimental results of three rectangular hollow sections (RHSs) by Liu and Goel [41]

The geometric and material parameters collected in Table 1 are considered influential in assessing low-cycle fatigue-induced fracture in seismic braces with HSS, and appear in various existing fracture models that have been used in seismic evaluation of SCBFs. The specimens in Table 1 include a realistic range of design parameters and loading patterns, including section dimensions, material properties, lengths, end conditions, and applied cyclic loading histories. The section dimensions range from 20 × 20 × 2.0 mm to 254 × 254 × 15 mm with the width-to-thickness ratio (b t ) from 5 to 30, the slenderness ratio (KL r ) from 33 to 115, and the E Fy from 340 to 960, respectively. Note that the limiting b t ratio = 13.8 and E Fy = 630 for an SHS with ASTM A500 Gr. B. in AISC 341–16 [5]. The specimens with a b t ratio larger than 13.8 are recognized under the specimen’s identity with a superscript c in Table 1. It is essential to reiterate that the data presented in Table 1 was collected from the experimental studies that square HSS braces were tested under quasi-static cyclic loading. Further, the survey included only the experimental programs that provided sufficient information on the tested specimens, such as measured mechanical and material properties, along with the cyclic brace behavior. Although the literature includes tested brace specimens in braced frames subjected to ground shaking [39,40], the number of these shake table tests with braces made of square HSS is considerably small in comparison to the quasi-static test. Thus, in this paper, the emphasis is placed on the large number of specimens tested under quasi-static loading. 4

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Δt Δ y Hysteretic Cycles a

Number of Cycles

b

Δc Δ y

d

c

a-b

1

2

b-c

4

3

5

6

7

c-d

8

9

10 11

12

13

Δt Δ y 1

2

3

4

5

6

7

8

9

10

11

12

13

Number of Cycles

Standard Cycles

Fig. 1. Example of the conversion of hysteretic cycles to standard cycles used in [1,2].

compressive deformation and 100% of the tensile deformation are considered. Although compressive deformation is still considered to have much less influence on fracture than tensile deformation, the fracture demand expressed in Eq. (2a) recognizes the cyclic impact of both tensile and compressive deformation on low-cycle fatigue-induced fracture. The corresponding fracture-resistance index, Δf , was estimated in Eq. (2b) that was modified from the fracture model proposed by Tang and Goel [1,2]. The fracture occurs when Δd ⩾ Δf .

and seven SHSs tested by Lee and Goel [3]. The brace is fractured when the ductility demand Ns exceeds the proposed ductility capacity Nf , i.e., Ns ⩾ Nf .

Ns =

Δt Δy

Ns =

Σ(Δt − Δc ) Δy

Nf = Cs

(simple cycles)

(1a)

(incremental cycles)

(b d )(KL r ) ((b − 2t ) t )2.0

(b d )(60) Nf = Cs ((b − 2t ) t )2.0

for

KL > 60 r

KL for ⩽ 60 r

(1b)

Δd =

∑ (0.1Δ1 + Δ2)

(1c)

Δf = Cs (1d)

(2a)

(46 Fy )1.2

⎛ 4b d + 1 ⎞ ((b − 2t ) t )1.6 ⎝ 5 ⎠

(2b)

where Cs is 1335, and Fy is the minimum yielding stress of steel. Likewise, Hassan and Goel [4] calibrated the model of Lee and Goel [3] by dividing each cycle into two portions, peak compressive deformation Δ1, and tensile deformation Δ2 , at point B’, as shown in Fig. 3, instead of the point of P Py equals to 1/3, as in Fig. 2. For each cycle, point B’ is defined as the intersection point of a straight line starting from D’ with a slope equals to 1/9, and the reloading line (curve) A’C’ in Fig. 3. Similar to Lee and Goel [3], Hassan and Goel [4] used Eqs. (2a) and (2b) for computing the fracture deformation demand and capacity. However, Cs in Eq. (2b) was recalculated using different experimental results, and a value of 1560 was proposed instead of 1335. In fact, this fracture model proposed by Hassan and Goel [4] was included in the seismic response analysis of CBFs that helped in establishing the basic design requirements for SCBFs in the current seismic design. Archambault et al. [42] used the same deformation demand index, Δd , as expressed in Eq. (2a) by Lee and Goel [3], but with different expressions for the fracture-resistance index, Δf ,as shown in Eqs. (3a) and (3b). The coefficient Cs is equal to 0.0184 based on the regression analysis of their own cyclic tests on nine SHS and fifteen RHS braces in [42], and equal to 0.0257 when the regression analysis included both their own tests in [42] and the tests used in Lee’s model [3]. Note that the fracture-resistance index model proposed by Archambault et al. [42] is independent of the brace slenderness ratio, KL r up to 70, but significantly increased with proportion to (KL r )2 for KL r that is larger than 70. For instance, the fracture life would be doubled if the

where Cs = 262 based on the test results in [3,41]; b d is the crosssection aspect ratio, with b as cross-sectional width, d as cross-sectional depth, and t as wall thickness; KL r is the slenderness ratio, with K as the effective length factor, L the length, and r as the radius of gyration. Note that the ductility capacity was inversely proportional to the widththickness ratio, but linearly proportional to the slenderness ratio of the brace. Lee and Goel [3] refined the fracturing criteria proposed by Tang and Goel [1,2], above discussed, by considering additional factors that have an impact on the fracture life of braces such as material properties and the compressive deformations underwent by the bracing members. By assuming that the fracture demand is directly related to total accumulative deformation (both in tension and compression) of all cycles prior to fracture, Lee and Goel [3] introduced a procedure to define peak compressive deformation Δ1 and peak tensile deformation Δ2 at each cycle. A descriptive illustration is presented in Fig. 2 to highlight how the fatigue life of square HSS braces is predicted following the model of Lee and Goel [3]. The hysteresis of a brace with a square hollow structural section, HSS1-1, tested by Fell et al. [34] are used here in Fig. 2. In each cycle, the peak compressive deformation Δ1 and tensile deformation Δ2 were determined by dividing each cycle into two portions with respect to a point where P Py is equal to 1/3, as shown in Fig. 2(b), (c), and (d). The fracture demand in terms of the accumulative deformation Δd is then expressed in Eq. (2a), in which 10% of the 5

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(a) 1st cycle

2nd cycle C

3rd + n cycles C

C

B

B

B A

A Δ1

Δ1

Δ2

(b)

A

Δ2

Δ2

Δ1

(c)

(d)

Fig. 2. Illustration of Δ1 and Δ2 definitions used in Lee and Goel [3] based on a tested brace [34]: (a) normalized hysteresis cycles, (b) Δ1 and Δ2 in the 1st cycle, (c) Δ1 and Δ2 in the 2nd cycle, and (d) Δ1 and Δ2 in the 3rd cycle.

Δf = Cs

C’ B’

Δf = Cs

Δd =

D’

A’ Δ1

0.55 (50 Fy )−3.5 ⎛ 4b d − 0.5 ⎞ (70)2 ((b − 2t ) t )1.2 ⎝ 5 ⎠

Δ2

0.8 ⎛ 4b d + 1 ⎞ (70)2 ((b − 2t ) t )0.5 ⎝ 5 ⎠

Δf = Cs

(46 Fy )1.2 ((b − 2t )

t )0.5

for

0.8 2 ⎛ 4b d + 1 ⎞ ⎛ KL ⎞ 5 ⎝ ⎠ ⎝ r ⎠

KL ⩽ 70 r

for

KL > 70 r

for

KL ⩾ 70 r

(4b) (4c)

(50 Fy )1.01 ((b − 2t )

(4a)

t )2.5

for

0.55 2 ⎛ 4b d − 0.5 ⎞ ⎛ KL ⎞ 5 ⎝ ⎠ ⎝ r ⎠

KL < 70 r for

KL ⩾ 70 r

(4d)

(4e)

Haddad et al. [37] also considered the accumulative total deformation, as in Eq. (4c), to be fracture-leading demand, and developed a fracture-resistant capacity formula, as shown in Eq. (5), using the regression fitting of the experimental results of ten SHS specimens tested by themselves.

slenderness ratio is changed from 70 to 100, based on Eqs. (3a) and (3b).

(46 Fy )1.2

KL 2 ⎛ ⎞ ⎝ r ⎠

0.55 (50 Fy )1.01 ⎛ 4b d − 0.5 ⎞ (70)2 ((b − 2t ) t )2.5 ⎝ 5 ⎠

Δf = Cs

Δf = Cs

(50 Fy ⎛ 4b d − 0.5 ⎞ ((b − 2t ) t )1.2 ⎝ 5 ⎠

0.55

KL < 70 r

∑ (Δ1 + Δ2)

Δf = Cs

Fig. 3. Per brace cycle determination of Δ1 and Δ2 using the model of Hassan and Goel [4].

)−3.5

for

Δf = 378(λ0.19)(b t )−0.94 (3a)

where λ =

KL r

Fy π 2E

(5)

.

3.2. Group II: Equivalent total ductility-based fracture models (3b) Tremblay [43] assumed that the fracture-leading demand is governed by the sum of maximum tensile and compressive deformation normalized by yielding deformation, an equivalent total ductility demand, μd , in Eq. (6a). The fracture-resistant capacity, μf , is assumed to be related only to the slenderness ratio parameter of a brace, λ , in Eq. (6b) that was formulated with a linear regression analysis of cyclic response of 19 SHS and 22 RHS braces tested by a number of research programs published in the literature. The brace fracture occurs if μd ⩾ μf .

where Cs= 0.0184 in [42] or 0.0257 when combined tests of [42] and [3]. Using the same accumulative deformation demand index of Eq. (2a) proposed by Lee and Goel [3], Shaback [24] introduced the expressions for a fracture-resistance index, Δf , Eqs. (4a) and (4b), based on nine newly tested SHS braces together with the existing experimental data, were used to develop Eqs. (3a) and (3b). Coefficient Cs of 0.065 was recommended for Eqs. (4a) and (4b). In addition, a new set of the accumulative deformation demand index, Δd , and the associated fractureresistance index, Δf , were also proposed, defined by Eqs. (4c), (4d), and (4e) with Cs equal to 29.5.

μd = 6

Δt + Δc Δy

(6a)

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μf = 2.4 + 8.3λ where λ =

KL r

the maximum tensile deformation (Δt ) over yielding deformation (Δy ) ratio, Eq. (9a); and the fracture-resistant capacity of a brace is dependent either on the brace slenderness ratio (KL r ), Eq. (9b); or on the width-to-thickness ratio (b t ) of the brace, Eq. (9c). In both equations, the fracture-resistant capacity is simply linearly related to either KL r , or b t , based on the experimental data of four SHS and two RHS braces. The brace fracture occurs when μd ⩾ μf .

(6b) Fy π 2E

.

Δt and Δc are the peak deformation in tension and compression, respectively, prior to fracture; Δy is the yielding deformation; and E is the steel Young’s Modulus. This fracture model was the first of its kind to relate the brace fracture only to the slenderness ratio, neglecting the effect of the width-thickness ratio. At about the same time, another work by Tremblay et al. [44] proposed a different fracture-resistance capacity model as shown in Eqs. (7a) and (7b), in place of Eq. (6b) to predict brace fracture with the same fracture-leading demand, μd , in Eq. (6a). It is worth noting that the same test data used to formulate Eq. (6b) were also utilized for the development of Eqs. (7a) and (7b), but with a few additional specimens (i.e., 3 SHS and 2 RHS), making a total of 22 SHS and 24 RHS. However, the former considers only the slenderness ratio, and the latter includes both width-to-thickness ratios and slenderness ratios of braces. In other words, the very significant conceptual difference between Eqs. (6b) and (7a) and (7b)) lies in their fundamental assumptions on the critical parameter(s) reconsidering the root cause leading to low-cycle, fatigue-related fracture. Yet both equations were proven to fit the same test data well enough to be recommended to predict brace fracture in seismic evaluation of braced frames. −0.1

b d⎞ θf = 0.091 ⎛ ⎝t t ⎠ μf = 1.0 + θf2

KL 0.3 ⎛ ⎞ ⎝ r ⎠

E 2Fy

μd =

2d d+b

Δrange − predicted =

(9c)

KL rπ E Fy

.

where ε =

235 Fy and Fy is in

(10)

kN/mm2.

(7a) 3.4. Brace fracture predicted by existing ductility-based models and observed experiments The empirical fracture models presented earlier were all developed based on a limited amount of test data, and were often shown to match well with these test data. However, the confidence level for the applicability of these models in predicting brace fractures, in general, has not been clearly demonstrated. The large collection of test results given in Table 1 presents an opportunity to evaluate the applicability of these empirical fracture models. 3.4.1. Group i models: Predicted fracture versus experimentally observed fracture Group I fracture models considered the number of standard cycles, Ns (Tang and Goel [1,2]), and the accumulative deformation demand, Δd (Lee and Goel [3], Hassan and Goel [4], Archambault et al. [42], Shaback [24], and Haddad et al. [37]), responsible for brace fracture. Based on the hysteresis loops recorded during cyclic testing of available specimens, the equivalent number of standard cycles at fracture, Nf , was experimentally calculated, and these Nf (experiment) values were reported in the 7th column of Table 1. In a similar manner, the experimentally observed fracture deformation, Δf (experiment) was also calculated and reported in the 8th column of Table 1 for those = ∑ (0.1Δ1 + Δ2) , and in the 9th column for those = ∑ (1.0Δ1 + Δ2) . The predicted equivalent number of standard cycles at fracture, Nf (prediction), based on the model of Tang and Goel [1,2], and the predicted fracture deformation, Δf (prediction), of Lee and Goel [3], Hassan and Goel [4], Archambault et al. [42], Shaback [24], and Haddad et al. [37] models are compared with the testing results, Nf (experiment) and Δf (experiment), in Fig. 4. Each brace specimen (represented by a square or diamond dot) has two fracture values, Nf (prediction), and Nf (experiment), as in Fig. 4(a), and Δf (prediction) and Δf (experiment), as shown in Fig. 4(b-h). If the two values are equal or close to each other, the dot is on or near the diagonal line. The fracture model would overestimate or underestimate the fracture resistance if the dot is above or below the diagonal line. The braces with a b t ratio larger than λhd = 0.65 E Ry Fy [5] are indicated by a red diamond dot. From the visual comparison of the results shown in Fig. 4, one might notice that the predictions of Group I empirical equations tend to scatter data, which means either over- or underestimation of the fracture actual life.

(8a) (8b)

2 L ⎡ 0.9(d + b) (b t )−0.53⎤ 2⎣ 2d ⎦

μf = 29.1 − 1.07(b t )

μf = 6.45 + 2.28λ − 0.11 b tε − 0.06(λ ) b tε

2Δrange − reported

εf = 0.9(b t )−0.53

(9b)

Nip et al. [35] proposed a model to predict tensile ductility capacity μf , given in Eq. (10), using test results from four SHS braces. With the ductility demand μd in Eq. (9a), Eq. (10) considers the impact of the combined width-to-thickness ratio, slenderness ratio, and steel yielding stress on fracture-resistant capacity. The coefficients in Eq. (10) were derived from curve fitting on the test data of the four tested SHS braces.

(7b)

L

(9a)

μf = 26.2λ − 0.7

where λ =

Fell et al. [45] agreed with Tremblay [43] hat fracture-leading demand is governed by the sum of maximum tensile and compressive deformation, or axial deformation range, Δt + Δc , but considered the width-thickness ratio (b t ) as the sole parameter affecting fracture-resistant capacity of a brace. Fell et al. [45] introduced a semi-analytical, semi-empirical approach to determine the fracture-resistant capacity. The observed fracture strain, εf , at the plastic hinge location of each specimen of the surveyed 63 HSS brace specimens (22 RHS and 41 SHS) was calculated based on axial deformation range Δrange − reported (the summation of peak tensile and compressive deformations during the experiments) and the brace dimensions (i.e., sectional width b , depth d , and length L ), as given in Eq. (8a). A relationship between εf and b t in Eq. (8b), was established with least square regression fitting of the fracture strain εf estimated by Eq. (8a). This relation was back substituted to Eq. (8a) to predict a deformation range Δrange − predicted in Eq. (8c). Equivalent total ductility capacity μf = Δrange − predicted Δy , and the equivalent total ductility demand μd = (Δt + Δc ) Δy , where Δt and Δc are the maximum deformation in tensile and compression, respectively, are part of the situation when brace fracture occurs (i.e., μd ⩾ μf ).

εf =

Δt Δy

(8c)

The model relates the strain at the plastic hinge of a brace to its cross-section dimensions in Eq. (8a). 3.3. Group III: Tension ductility-based fracture model Group III fracture models, discussed in what follows, regard tensile deformations as the only demand parameter leading to fracture and neglect the compressive deformation demands that were considered by Group II models. For examples, Goggins et al. [26] assumed that the fracture was caused by excessive tensile ductility demand μd , defined as 7

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Fig. 4. Group I fracture models’ predictions on fracture, Δf (prediction), vs. experimentally observed fracture, Δf (experiment).

equal to the total number of considered samples – 1. Therefore, χ 2 critical at 62 degrees of freedom, and at a significant level of 5% (90% confidence level), is 79.08. The χ 2 critical value represents the threshold to accept or reject the computed statistical test χ 2 of the considered data. In other words, if χ 2 falls below χ 2 critical , then the prediction can be considered acceptable, otherwise, rejected. In Fig. 4(a) for example, χ 2 value of Tang and Goel’s model [1,2] for 62 degrees of freedom, is equal to 5050 ( χ 2 (62) = 5050 ), where the χ 2 critical , for 62 degrees of fredom, is equal to 79.08 ( χ 2 critical (62) = 79.08) at 5% significance level ( p < 0.05), which indicates that the dispersion is extreme. The mean bias error for

In addition to the visual inspection, the performance of these predictive models was quantitatively evaluated through error metrics, such as dispersion and bias. To evaluate the dispersion between the predictions and the experimentally obtained fracture capacities, the statistical test of Pearson’s chi-squared ( χ 2 ) [46–49] was calculated for each model of Group I using Eqs. (11) and (12) along with the mean error bias [49]. The dispersion and bias results are presented in Fig. 4, under each model’s plot. The total number of specimens used to evaluate Group I fracture models, was 63, as summarized in Table 1. To find the critical value of χ 2 , the degrees of freedom have to be calculated, which are 8

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Pearson’s chi-squared test ( χ 2 ) χ (78) = 355 > χ 2

2

critical

(78) = 98.8, p < 0.05

Reject

Pearson’s chi-squared test ( χ 2 ) χ (78) = 180 > χ 2

2

critical

(78) = 98.8, p < 0.05

Reject

Pearson’s chi-squared test ( χ 2 ) χ (78) = 135 > χ 2critical (78) = 98.8, p < 0.05 2

Reject

Mean error (bias)

Mean error (bias)

Mean error (bias)

ME = 3.8

ME = 3.6

ME = 3.8

(a)

(b)

(c)

Fig. 5. Group II fracture models’ predictions on fracture, μf (prediction), vs. experimentally observed fracture, μf (experiment).

within a narrow range between 2.5 and 7.5 while the observed values are between 2.0 and 20, respectively, as shown in Fig. 6(c). By a statistical point of view, although the dispersion measure of χ 2 in all models of Group III (Fig. 6(a), (b) and (c)) is greater than χ 2 critical , the predictive model of Nip et al. [35] in Fig. 6(c) appear to have less dispersion in comparison to the extreme dispersion Goggin’s et al. [26], models have in Fig. 6(a) and (b). A similar observation can be carried out by comparing the mean error bias that the models of Goggin’s et al. [26] have a high bias compared to that of Nip et al. [35].

Tang and Goel’s model [1,2] was 78, which is considered high. Further, by evaluating other predictive models shown in Fig. 4(b–h), a major conclusion can be obtained that the dispersion of data varied between moderate to extreme with high bias predictions.

∑ 1 n

(e − p)2 p

(11)

e−p p

(12)

∑

where e and p in Eqs. (11) and (12) stand for experimental and predicted values, and n is the total number of sample points.

3.5. Observations and conclusions on existing ductility-based fracture models

3.4.2. Group II models: Predicted fracture versus experimentally observed fracture Group II fracture models were based on the assumption that the equivalent total ductility demand, μd , the sum of maximum tensile and compressive deformation divided by yielding deformation, μd = (Δt + Δc ) Δy , is responsible for brace fracture. Fig. 5 compares the predictions by three fracture models with experimental results. Two models that Tremblay proposed [43,44], Fig. 5(a) and (b), were established based on the cyclic test results of a total of 41 (first model) and 46 (second model) specimens, and the model proposed by Fell et al. [45], Fig. 5(c), was based on 63 tested brace specimens. Fig. 3 shows that they all fail to predict brace fracture within a reasonable margin of error. In addition, by evaluating the predictive models statistically, it is apparent from Fig. 5(a), (b) and (c) that χ 2 test is greater than χ 2 critical , which equal to 98.8 for 78 degrees of freedom, indicating a high dispersion, and when evaluating the mean error bais, these models shown to be low bias. Note that a total number of 79 sample data were used in the evaluation of group II models, as summarized in Table 1.

From the discussions on the existing ductility-based fracture models above, the following observations emerge: (1) The assumptions on what causes low-cycle fatigue-induced brace fracture vary significantly among the existing ductility-based fracture models, ranging from accumulative deformation to maximum deformation, and from tensile deformation only to tensile and compressive deformation. However, one common notion among all models is that fracture occurs under tensile deformation. (2) The major design parameters that affect fracture-resistant capacity in the existing fracture models include width-to-thickness ratio (b t ) as the primary parameters, slenderness ratio (KL r ) and material properties (E Fy ) as the secondary parameters. Some models use the primary parameter, and others use both primary and secondary parameters. In some fracture models, the same group of data was used to develop multiple formulas with different design parameters involved, depending on the nature of experimental data. (3) All ductility-based fracture models were developed with experimental data as the primary source to construct a fracture-resistant capacity formula. Some models were based on a few brace fracture tests, and others were based on as many as between 40 and 60 brace fracture tests. Although the models with the most test data have not shown any reliable fracture prediction ability, they often demonstrate an improved distribution pattern: more of them equally overestimate and underestimate the brace fracture capacity (i.e., the dots are more likely to be on both sides of diagonal lines in Figs. 4, 5, and 6). (4) Despite various assumptions on fracture-causing demand indices and fracture-resisting capacity parameters, all ductility-based existing models failed to predict reliably fracture of a group of 79 braces collected in Table 1, which indicates the highly uncertain nature of low-cycle fatigue-induced fracture.

3.4.3. Group III models: Predicted fracture versus experimentally observed fracture Group III fracture models were based on the assumption that excessive tensile ductility demand causes brace fracture. Fig. 6 compares the predictions by three fracture models with experimental results. Both models proposed by Goggins et al. [26], one considering slenderness ratio only, and the other considering width-to-thickness ratio only, significantly overestimated the fracture-resistance capacity of braces, as indicated by the fact that most square and diamond dots are well above the diagonal line in Fig. 6(a) and (b). Clearly, the models given in Fig. 6 heavily overestimate the capacity. The fracture model by Nip et al. [35] considered slenderness ratio and width-to-thickness ratio, and was based on four cyclic tests. The predicted fracture ductility values are 9

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Pearson’s chi-squared test ( χ 2 ) χ (78) = 1273 > χ 2

2

critical

(78) = 98.8, p < 0.05

Pearson’s chi-squared test ( χ 2 ) χ (78) = 544 > χ 2

2

critical

Reject

(78) = 98.8, p < 0.05

Reject

Pearson’s chi-squared test ( χ 2 ) χ (78) = 107 > χ 2 critical (78) = 98.8, p < 0.05 2

Reject

Mean error (bias)

Mean error (bias)

Mean error (bias)

ME = 19.5

ME = 9.9

ME =1.2

(a)

(b)

(c)

Fig. 6. Group III fracture models’ predictions on fracture, μf (prediction), vs. experimentally observed fracture, μf (experiment).

with much-improved dispersion and bias; and (e) Easily implemented in most popular computer programs used in earthquake engineering and structural dynamics such as OpenSees [50].

These observations lead to the following conclusions: (1) All fracture-causing assumptions may be valid in some cases, and invalid in other cases, largely depending on the actual cyclic loading history during an earthquake ground motion. With the uncertain nature of low-cycle fatigue-induced fracture under earthquake ground motions, an ideal fracture model should be able to reliably predict as many brace fractures as possible from available experiments, and be flexible enough to adapt to more test data as they become available. (2) The notion that fracture occurs under tension (as implied in all existing ductility-based fracture models) may lead to overestimating or underestimating the fracture-resistance capacity of seismic braces. It has been observed in almost all tested braces with hollow sections that the fracture is often initiated at the plastic hinge zone under compression, and grows in the subsequent tension and compression cycles. Even though the final stage of the fracture (the separation of the brace into two pieces) appears to occur under tension, the fracture is often initiated at the compression cycle prior to the final tension cycle leading to the perceived fracture status.

4.1. Definition of fracturing process of initiation, propagation, and termination The existing fracture models all assumed that a “fracture” state occurred when partial separation of a brace was visible under tension. However, experimental studies have shown that cracks at a bulging area under compression become visible on the outer surface of the HSS wall and gradually propagate to become a partial or complete separation. Although fracture first initiates under compressive deformations, it is essential to underline that the tensile cycles are the ones that cause propagation of the initial cracks, and enable them to become visible on the outer HSS wall once the brace is entirely under tensile deformations, leading to partial or complete separation. Many researchers correlated the onset of fracture to tensile deformation and developed their fracture models on the assumption that the brace is considered fractured only under tension, which can be seen from the previously mentioned ductility-related predictions. Using a given tensile deformation to define an isolated fracture status is questionable because the tensile strength of a brace at any given tensile deformation varies dramatically from 20% to 100% of original tensile strength, as seen in the tests included in Table 1. Recent cyclic tests on large-size braces have shown clearly that a fracture is typically initiated in the midlength of braces around the plastic hinge region upon reloading in tension under compressive deformations [24,31–33]. It is fundamentally critical to model the “fracturing process,” a new concept to describe and model a brittle failure phenomenon, rather than focus on a “fracture” that is difficult to be quantified for simulation. A typical fracturing process that was observed from the cyclic tests conducted by Fell et al. [45] is shown in Fig. 7. The first sign of low-cycle fatigue fracture is often observed under compression, in Fig. 7(d); following the global buckling, Fig. 7(a); and with local buckling, Fig. 7(b). The subsequent tensile deformation after fracture initiation starts the reduction of tensile strength at a different rate, depending on the rate of fracture propagation in Fig. 7(c). The fracture would propagate under tension until a perceived terminal stage has been reached. Traditionally, any partial separation, from a modest separation in Fig. 7(d), to 20 to 30% partial separation as shown in Fig. 7(c), to 100% compete separation, has been described as a “fracture” status. Under the proposed fracture process concept, it is logical to define a terminal stage in the fracture process in terms of the remaining strength ratio during the fracturing

4. Proposed fracture model The main reasons that all ductility-based existing fracture models fail to predict brace fracture with any level of confidence may include the following: (1) Low-cycle fatigue-induced fracture of HSS braces is influenced by a large number of demand and resistance parameters, and due to the nature of these uncertainties; many of these parameters are not well understood; (2) All ductility-based models were established with overly simplified assumptions that select a few parameters and ignore others; and (3) Ductility-based models were based on a limited amount of data from low-cycle fatigue tests, and lack the ability to adapt as an everincreasing amount of test data becomes available in the future. A new fracture initiation model of the braces with HSS is proposed with the following characteristics: (a) A new fracture indicator is introduced to redefine fracture; (b) Includes all test data available to date, and is able to adapt to any new test data available in the future; (c) Considers all identified parameters in both demand and resistance; (d) Ability to simulate the fracture response observed in experiments 10

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(a) Global buckling

(b) Local buckling

Fracture propagates through cross-section

Typical fracture initiation at filleted corners

(c) Fracture propagation

(d) Fracture initiation

(2) The effect of the slenderness ratio KL r on the fracture initiation ductility ratio, μf , seems to depend on the b t ratio, as shown in Fig. 8(b). The braces with a larger KL r appear to have higher fracture ductility if their b t ratios are between 8 and 14. This trend is less clear for the braces with a b t less than 8 or larger than 14. (3) The effect of yield strength on fracture ductility is examined in Fig. 8(c). There is a weak trend suggesting that braces with a lower yield strength may have better fracture resistance. In general, the influence of E Fy on fracture ductility is not as strong as that of the b t ratio. Note that the values of b t and KL r ratios presented in Fig. 8 were taken directly from the experimental studies summarized in Table 1, and as such, these ratios could vary depending on how the dimensions b , t , L , and K were estimated in the experiments.

4.3. Proposed fracture initiation model As observed from Fig. 8, although μf has appeared somewhat inversely proportional to b t , and weakly proportional to KL r and E Fy , these relationships are much more complex due to the robust interaction of these parameters and many other uncertainties. In view of the complexity and uncertain nature of the problem, a series of multiple nonlinear regression analyses were conducted to propose an improved equation that predicts the fracture with confidence. The b t , KL r , and E Fy are explicitly included as physically defined parameters, and regression coefficients are introduced to take into account any uncertainties. The regression analyses were conducted on the previously collected data that includes the experimental test results of 66 highly ductile square HSS specimens (i.e., b t ⩽ λhd , where λhd is the limiting width-to-thickness ratio for highly ductile members specified in AISC 341 [5]). After several iterations with different combinations that included several regression approaches, Eq. (13) was found to satisfactorily predict the experimentally obtained ductilities ( μf ) summarized in Table 1. As illustrated in Fig. 9(a), the overall match between the predicted and experimental results of 66 square HSSs with highly ductile sections appears satisfactory (most data are along or close to the diagonal line), considering the levels of complexity and uncertainty involved. In addition, the statistical tests, Pearson’s chisquared ( χ 2 ) and mean error bias, were performed for the predictions obtained from Eq. (13), and are shown in Fig. 9(a). The χ 2 test for 65 degrees of freedom is equal to 66, which falls below 79.08 – the χ 2 critical for 65 degrees of freedom at a 5% significance level, indicating a low dispersion. Furthermore, a low mean bias error of the model has also prevailed, as indicated in Fig. 9(a). Results from the error metrics indicate that the predictions from Eq. (13) are considered acceptable. In comparison with the models shown for example in Fig. 9(b–c), and Figs. 4–6, it is obvious that Eq. (13) has dramatically improved the reliability of the prediction, as shown in Fig. 9(a), which can be attributable to the increased number of specimens included. The adjusted R2 of Eq. (13) was 0.45. Note that the typical statistical measure of the predictive model, fit adjusted R2 that is close to 1.0, specifies that a large proportion of the variability in the response have been predicted by the regression [51]. It should also be noted that Eq. (13) is only applicable to highly ductile square HSS sections, as it was developed based on regression analysis of experimental HSS sections with the following geometric and material parameters: 5.6 ⩽ b t ⩽ 14.2, 33 ⩽ KL r ⩽ 115, and 340 ⩽ E Fy ⩽ 960 .

Fig. 7. Typical progressive deformation sequence in a brace from global buckling to partial/complete separation observed from the cyclic tests conducted by Fell et al. [45]: (a) global buckling, (b) local buckling, (c) fracture initiation, and (d) fracture propagation. Table 2 Mean and standard deviation of β . β = Pf Py Mean ( μ ) Standard Deviation (σ ) μ ± 1σ μ ± 2σ

0.6 0.2 0.6 ± 0.2 0.6 ± 0.4

propagation process, i.e., the ratio β = Pf Py , where Pf is the remaining tensile strength of the brace after fracture is initiated, and Py is the design tensile strength of the brace. The last column in Table 1 recorded β values of most tested braces, showing large variations with a mean value of 0.6 and standard deviation of 0.2, as summarized in Table 2. Due to the high uncertainty of the post-fracture initiation strength, it is reasonable to use the mean value to define the terminal point in the fracturing process. 4.2. Analysis of cyclic test data For the 79 cyclically tested SHS braces included in Table 1, the compression deformations Δc were computed and reported in Table 1. Note that Δc represents the compressive deformation when fracture initiated in the cycle upon reloading in tension during which complete or partial separation occurred in the experimental test. The compression deformation Δc in Table 1 was normalized by buckling deformation Δcr . The ratio Δc Δcr in Table 1 is considered as an indicator of the onset of the fracturing process for SHS braces. The relationships between this ratio, μf = Δc Δcr , and width-to-thickness (b t ) ratio, slenderness ratio (KL r ), and E Fy , respectively, (as collected in Table 1) are plotted in Fig. 8. The following was observed: (1) An overall trend exists in the relationship between fracturing initiation ductility, μf , and b t ratio, as shown in Fig. 8(a), where braces with smaller b t ratios appear to be less vulnerable to fracture initiation (higher μf ). However, this trend is less clear among the braces with a b t ratio less than 14, where fracture initiation ductility can be as small as 5.0 and as large as 20–30 for braces with the same b t ratio. Fig. 8(a) also shows that the difference between upper and lower bounds of fracture ductility is more significant in the braces with a smaller b t ratio.

μf =

11

[(b t )−548 + (KL r ) 4.467 + (E Fy )3.065]0.449 595

(13)

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≤

Upper bound

Lower bound

(a)

(b)

(c)

Fig. 8. Observed relationship between fracture ductility, μf , and (a) width-thickness ratio, b t ; (b) slenderness ratio, KL r ; and (c) E Fy .

5. Numerical implementation of the proposed fracture initiation model for seismic response analysis

developed based on small groups of test results, and might not always be reliable in predicting brace fracture. More recent studies that evaluated seismic performance of CBFs [31,55,56] have turned to softwarespecific numerical models to simulate brace fracture; the most common of which are the fiber-based models in OpenSees [50], a numerical simulation platform for the earthquake engineering research community. These studies [31,55,56] used in their models the axial strain response of a brace as its damage measure, a natural fit with the fiber-based formulation in OpenSees [50]. However, operating at a fiber level with the assumption that plane sections remain plane to monitor localized high compressive strains induced by the formation of local buckling at the plastic hinge locations is not valid due to the limitations of the software (OpenSees [50]). Further, recent work on axial strain in HSS braces has indicated that post-buckling axial strain in braces around plastic zones can be extremely high, and the axial strain values obtained by detailed nonlinear finite element analysis vary substantially with a slight change in a routine modeling decision, such as the size of elements. Moreover, fracture prediction itself is highly uncertain.

5.1. Implementation of the proposed fracture initiation model for postfracture seismic evaluation of braced frames The ultimate goal of developing a reliable way of simulating brace fracture response is for the structural/earthquake engineering community to improve the post-fracture seismic evaluation of CBFs. Seismic demands on braces in CBFs have been shown to be consistently higher than the expected fracture-resisting capacity [52–54] and post-fracture performance needs to be evaluated with a reliable modeling process. The literature has shown a striking contrast between the far-reaching impact of the seismic evaluation results and analytical models used to produce the results [6,7,31,44,55,56]. The early studies on performance of CBFs [6,7] employed the empirical models, as discussed earlier in this paper, in the post-fracture seismic evaluations of CBFs. It has been shown that all empirical fracture models evaluated were

Equation (13)

Pearson’s chi-squared test ( χ 2 ) χ (65) = 66 < χ 2 critical (65) = 79.08, p < 0.05 2

Accept

Mean error (bias) ME = 1.7

(a)

(b)

(c) Fig. 9. Comparison of predicted experimental fracture by various models. 12

(d)

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As presented in Fig. 10, and summarized in Algorithm 1, once any of Stages B, C, and D are predicted, the area of the square HSS section gradually reduces by removing the fibers at the filleted corners and the flat HSS wall. The rate of fiber deletion at each of these stages, particularly at Stages B, and C was assumed half of the fiber layers through the wall thickness. For example, half of the fibers that are under compression are removed in the through-thickness direction at the filleted corners when Stage B (initiation) is predicted, as shown in Fig. 10. Then, once Stage C (propagation) is predicted, half of the remaining fibers at the corner are removed, and consistent with deletion at the corners, the flat HSS wall thickness is also reduced by half. Lastly, when Stage D (termination) is reached, the remaining fibers at the corners are completely removed, as illustrated in Fig. 10.

The proposed fracture initiation model, Eq. (13), used within the fracturing process framework, has been shown to reliably simulate brace fracture response. The model is inclusive for existing and future experimental database work on cyclic behavior of steel braces, and comprehensive to match the significance of the numerical simulation of the post-fracture performance of CBFs, based on the fracturing process concept. The general strategy in applying the fracturing process concept to modeling brace fracture under cyclic loading is described in three distinctive steps, initiation, propagation, and termination. The definition of each stage is as follows: (1) Initiation: An initiation point is specified in terms of deformation status that is experimentally proven and numerically identifiable during a simulation. The fracturing initiation ductility expressed in Eq. (13) is the point that the fracturing process of square HSS braces begins; (2) Propagation: As the fracture propagates gradually through the crosssection from where it was initiated, the cross-sectional area is decreasing, resulting in a reduction in tensile strength; and (3) Termination: The cross-section is considered to progress to the terminal stage of the fracturing process when β = Pf Py reaches a certain value less than 1.0, indicating that the brace is no longer a structural member.

Algorithm 1. Fracturing process algorithm of square HSS braces.

1: 2:

Define a square HSS fiber section including filleted corners, named as Stage A Monitor the mid-span brace element responses, such as μcompression = Δc Δcr and

3:

Ptensile Py if μcompression = μf , where μf is the proposed limit of fracture initiation ductility,

4: 5: 6:

Using the general strategy of the fracturing process concept, Algorithm 1 is developed and incorporated into OpenSees [50]. Fig. 10 reflects the progressive low cycle fatigue fracturing process for square HSS braces proposed in Algorithm 1. Stage A defines a state that the middle section remains intact until fracture initiation is predicted. Then, in Stage B, the fibers at the filleted corners are removed to imitate initial tearing. An increasing number of fibers continue to be removed upon reloading in tension in Stage C, and once β attains 0.6, the state of the terminal stage is reached where separation of the adjacent fibers at the corner occurs, as demonstrated in Stage D shown in Fig. 10.

7: 8: 9:

predicted in Eq. (11) The area at the corners of the round edges is gradually reduced, named as Stage B As the fracture propagates, remove an increasing number of fibers, named as Stage C if Ptensile Py = β = 0.6 , mean value of tensile strength reduction observed from tested specimens in Table 1 Terminate process, named as Stage D end end

Note: when applying Algorithm 1, it is recommended to divide the braces into a minimum of 8-elements, as presented in Fig. 10. Number of elements on the brace response shown to affect the brace strain history but not the displacement axial history. Readers are referred to [31] for more information.

Progressive Response of Brace Middle Section

Stage A

Stage B

Prior to fracturing

Fracture initiation

Stage C

Stage D

Fracture propagation

Terminal stage

Rigid link

Zero-length element (pin)

Brace Element Modeling

Fiber section

Fig. 10. Progressive response of brace middle section following the fracturing process described in Algorithm 1. 13

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Fracture initiation HSS2 -1

Actuator Reaction Wall

Sliding Beam

Cyclic Loading

Tested Specimen

Fracture initiation HSS1 -1

Actuator

(a)

(b) Progressive Response of Brace Middle Section

Stage B

Stage A Prior to fracturing

Fracture initiation

Stage C

Stage D

Fracture propagation

Terminal stage

Brace Element Modeling for HSS1 -1 and HSS2 -1

(c) Terminal stage Stage D

Terminal stage Stage D

Fracture propagation Stage C

Fracture propagation Stage C

Fracture initiation Stage B

Fracture initiation Stage B

(d)

(e)

Fig. 11. Simulations of two braces tested by Fell et al. [45]: (a) test setup and specimens HSS 1–1 and HSS 2–1 [45], (b) loading cycles [45], (c) brace model and progressive response for both specimens, (d) HSS 1–1 response from the test [45] and predicted by the proposed method, and (e) HSS 2–1 response from the test [45] and predicted by the proposed method.

with the cross-section area intact, represents all responses prior to any sign of fracture initiation. Fracture initiation, illustrated as Stage B of Fig. 11(c), was captured at the peak compression deformation, by Eq. (13), matching the test result, as shown in Fig. 11(d) and (e) for HSS1-1 and HSS2-1, respectively. After fracture initiation at Stage B, the crosssection area at the two corners gradually decreases as the fracture propagates, as illustrated by Stage C in Fig. 11(c), as well as Fig. 11(d) and (e). The terminal stage, Stage D in Fig. 11(c), of the fracturing process is reached at tensile strength ratio of 0.6 (i.e., β = Pf Py ), indicated in Fig. 11(d) and (e), in which the simulation-predicted and tested terminal stages match reasonably well for both specimens. The tensile strength ratio is almost identical for the predicted and tested HSS 1–1, and noticeably different in HSS2-1. The predicted and tested terminal tensile strength ratios in HSS 2–1 are 0.6 and 0.3, respectively, with the difference of the mean value minus 1.5 times standard deviation, according to Table 2 that lists the mean value and standard deviation as 0.6 and 0.2, respectively. Note that HSS 1–1 and HSS 2–1 were modeled in OpenSees [50] with the following modeling parameters: 8 force-based elements; 4 integration points defined per

5.2. Verification of the proposed fracture model in OpenSees The proposed fracture modeling process implemented in OpenSees [50] was first evaluated by comparing the model-simulated and experimental cyclic responses of two SHS braces tested by Fell et al. [45]. As shown in Fig. 11(a), two SHS braces, namely HSS1-1 with HSS4 × 4 × 1/4 and HSS2-1 with HSS4 × 4 × 3/8, were tested under cyclic loading as shown in Fig. 11(b) [45]. The fracture ignition (onset) observed in the tests was recorded in Fig. 11(b). The braces modeled in OpenSees [50], as shown in the lower portion of Fig. 11(c), were subjected to the cyclic loading shown in Fig. 11(b). The prediction simulation of the braces applies the methodology of the three-stage fracturing process, i.e., initiation, propagation, and termination. The cyclic responses of each brace under test and prediction are compared in Fig. 11(d) and (e) for HSS1-1 and HSS2-1, respectively. The simulated progressive response of the braces was revealed by tracking the brace middle section, where the plastic hinging location was observed in test and OpenSees [50] simulation as shown in Fig. 11(c). The four representative stages, A, B, C, and D, are illustrated in Fig. 11(c). Stage A, 14

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Cyclic Loading

9’ 0’’

LNB

W 10x45

LSB

9’ 0’’

W24x117

W 10x45

W24x117

Fracture initiation LSB

Fracture initiation LNB

1’ 0’’

W 10x45

W 10x45

Loading Beam

Reaction Beam Reaction Block South

North 20’ 0’’

(a) W 24x117

(b) Progressive Response of Brace Middle Section

Rigid Elements

Stage B

W 10x45

Stage A Pin-Ended Braces

Prior to fracturing

Fracture initiation

W 24x117

W 10x45

Stage C

Stage D

Fracture propagation

Terminal stage

Pin-Ended Braces

LSB

LNB Brace Element Modeling for LSB and LNB

(c)

(d) Terminal stage Stage D

Terminal stage Stage D Fracture propagation Stage C

Fracture propagation Stage C

Fracture initiation Stage B

Fracture initiation Stage B

(e)

(f)

Fig. 12. Simulations of brace fracture in a two-story frame (UCB) tested by Uriz and Mahin [31]: (a) test setup [31], (b) loading cycles [31], (c) OpenSees model of the tested frame, (d) brace model and progressive response for both lower-north brace (LNB) and lower-south brace (LSN), (e) LSB response from the test [31] and predicted by the proposed method, and (f) LNB response from the test [31] and predicted by the proposed method.

element, fiber sections with 12 × 4 mesh applied to each wall of HSS sections and 4 × 4 mesh for the round filleted corners, initial camber of 0.06% of the brace length, strain hardening ratio of 0.002, and material model parameters of RO = 18, cR1 = 0.93, and cR2 = 0.5. The proposed fracture modeling process implemented in OpenSees [50] was also tested by simulating experimental cyclic responses of a two-story CBF tested by Uriz and Mahin [31] at University of California, Berkeley (UCB). This two-story CBF, referred to as UCB hereafter, was the first full-size frame designed using the post-Northridge era seismic design provisions for special CBFs. As shown in Fig. 12(a), the frame has the HSS 6 × 6 × 3/8, W24 × 117, and W10 × 45 for braces, beams, and columns, respectively, used on both floors, and was laterally loaded on the roof with cyclic roof displacement history shown in Fig. 12(b). The first (lower) floor (south and north) braces, named LSB and LNB, demonstrated the typical cyclic response of global buckling first, followed by fracture initiation after local buckling, and fracture propagation till terminal fracture. The OpenSees [50] model of the frame is shown in Fig. 12(c). The proposed fracture process procedure

implemented in OpenSees [50] was followed to simulate the cyclic response of the frame, and the predicted response of the first floor braces is compared with the tested responses in Fig. 12(e) and (f). These two figures show that the fracture initiation in both LSB and LNB, Stage B, predicted by Eq. (13) was about μf = Δc Δcr = 9.0 , earlier than the fracture initiations observed from the experiment for this particular specimen. The predicted fracture propagation (Stage B) and terminal point (Stage D) appear to be comparable with the tested results. In particular, the predicted tension deformation and tensile strength at the terminal fracture point match well with the observations from the tests of both braces. It should be noted that identical parameters were used in OpenSees [50] or modeling the braces in the UCB frame, shown in Fig. 12, and for modeling HSS 1–1 and HSS 2–1 specimens of Fell et al. [34], except for the initial camber, the strain hardening ratio, and material model parameters that were 0.07%, 0.003, RO = 15, cR1 = 0.93, and cR2 = 0.93.

15

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(a)

(b)

Te rminal stage Stage D Fracture propagation Stage C

Fracture initiation Stage B

(c)

(d)

Fig. 13. Cyclic fracture prediction of HSS1-1 tested by Fell et al. [45]: (a) Uriz and Mahin [31] prediction, (b) Tirca and Chen [62] prediction, (c) Karamanci and Lignos [63] prediction, and (d) the prediction by the fracturing process model proposed in this study.

and 1 failure. The rain-flow cycle counting procedure recommended by ASTM E1049-85 [61], was used to convert irregular seismic loading history into elementary ranges and cycles that are equivalent to that of constant amplitudes. The fatigue ductility properties ε0 and m are coefficients dependent on materials and structural members. Uriz and Mahin [31] recommended that ε0 = 0.095 and m = −0.5 be used specifically for HSS bracing members, based on the experimental results of five SHS specimens tested by Yang et al. [25]. Tirca and Chen [62] evaluated the coefficient ε0 and m values proposed by Uriz and Mahin [31] for Eq. (14a) using experimental results of a group of HSS braces, and reported that ε0 of 0.095 and m of − 0.5 failed to predict the fracture experienced in the physical tests of the specimens. Subsequently, with m = −0.5 unchanged, Tirca and Chen [62] proposed a regression empirical equation to estimate ε0 for HSS braces with different geometric and material parameters (b t , KL r , and E Fy ), as expressed in Eq. (15) that was calibrated based on experimental results of 14 HSS brace specimens. Eq. (15) is applicable only for HSS braces with 50 ⩽ KL r ⩽ 150 .

5.3. Comparing the proposed fracturing process concept with the existing fracture models Many of the analysis-based studies on CBFs [7,31,44,52–56] included brace fracture in their seismic analyses. As more and more analytical studies on the seismic performance of CBFs are conducted using software-specific numerical models (such as the fiber-based models in OpenSees [50]) to simulate brace fracture, a fracture simulation model in OpenSees [50] that has been proven to be reliable is needed as an improved alternative to the existing fiber-based fracture models currently avaliable in OpenSees [50]. For that purpose, a comparison between the fracturing process concept using the proposed fracture initiation model in Eq. (13) and the fiber-based fracture models in OpenSees [50] has been made as follows. Uriz and Mahin [31] introduced cumulative damage theory and its associated assumptions to consider low-cycle fatigue-induced fracture [57–60] in the form shown in Eqs. (14a) and (14b).

εi = ε0 (Nf )m k

DI =

KL 0.859 ⎛ b ⎞−0.6 ⎛ E ⎞ ⎞ ε0 = 0.006 ⎛ ⎜ ⎟ ⎝ r ⎠ ⎝ t ⎠ ⎝ Fy ⎠

n

∑ Ni i=1

(14a)

fi

(14b)

0.1

(15)

Karamanci and Lignos [63] also evaluated the coefficient ε0 and m values proposed by Uriz and Mahin [31] using another group of tested HSS braces, and proposed their own empirical equation, Eq. (16), for estimating ε0 , and selected m to be − 0.3. Eq. (16) was based on the regression analysis of experimental results of 65 SHS braces with the 4.2 ⩽ b t ⩽ 30.4, 27 ⩽ KL r ⩽ following design properties: 85, and 32 ⩽ Fy ⩽ 77 ksi .

In Eq. (14a), εi is the plastic strain amplitude, Nf is the number of cycles of constant amplitude to fatigue failure, ε0 is the fatigue ductility coefficient, and m is the fatigue ductility exponent. In Eq. (14b), damage index, DI , is used to quantify accumulative damage through the number of cycles at a constant amplitude of the ith segment in ak series of segments, divided by Nf of the ith segment with 0 indicating no damage 16

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(a)

(b) Terminal stage Stage D Fracture propagation Stage C

Fracture initiation Stage B

(c)

(d)

Fig. 14. Cyclic fracture prediction of HSS2-1 tested by Fell et al. [45]: (a) Uriz and Mahin [31] prediction, (b) Tirca and Chen [62] prediction, (c) Karamanci and Lignos [63] prediction, and (d) the prediction by the fracturing process model proposed in this study.

KL −0.484 ⎛ b ⎞−0.613 ⎛ E ⎞ ⎞ ε0 = 0.291 ⎛ ⎜ ⎟ ⎝ r ⎠ ⎝t ⎠ ⎝ Fy ⎠

0.303

strain increases. In their study, they suggested a minimum of 16 elements for strain-related analysis. Thus, when the fatigue model of Uriz and Mahin [31] was evaluated, the braces were modeled with 16 elements. Likewise, Karamanci and Lignos [63] recommended square HSS braces to be modeled using 8 elements with each having 5 integration points and fibers with 10 × 4 mesh, strain hardening ratio of 0.001, RO = 22 , cR1 = 0.925, and cR2 = 0.25. Accordingly, the modeling parameters were adopted as recommended [63]. A similar approach was followed for Tirca and Chen [62] model. In other words, the modeling parameters suggested by each researcher were employed when the fatigue prediction of the associated model was assessed in Figs. 13 and 14.

(16)

These three strain-based fracture models have been implemented in OpenSees [50] for the structural engineering community to use, and here they are compared with the fracturing process-based model proposed in this study using the tests conducted by Fell et al. [45]. The test setup and specimens HSS1-1 and HSS2-1 are shown in Fig. 11. Figs. 13 and 14 show the the predicted results of two specimens, HSS1-1 and HSS2-1, by (a) Uriz and Mahin, (b) Tirca and Chen, and (c) Karamanci and Lignos, and (d) the prediction by the proposed fracturing process model in this study in comparison with the test results obtained by Fell et al. [45]. The comparisons shown in Figs. 13 and 14 clearly indicate that these three existing models (a, b, and c) fail to simulate the fracture propagation and terminal fracture to varying degrees. The fracturing process simulation with the proposed fracture initiation model, on the other hand, successfully predicts both specimens throughout cyclic response, from global buckling, tension yielding, and fracture initiation and propagation, to terminal fracture stages, as shown in plot (d) of Figs. 13 and 14. It is important to emphasize that Uriz and Mahin [31], Tirca and Chen [62], and Karamanci and Lignos [63] conducted sensitivity analyses related to the hysteretic behavior of HSS braces to optimize the input modeling parameters with respect to their proposed low-cycle fatigue predictions. Therefore, when HSS 1–1 and HSS 2–1 were evaluated in Figs. 13 and 14, the braces were modeled based on the input parameters for simulating the brace response proposed by each group of researchers. For example, Uriz and Mahin [31] had shown that strains reported from sections defined by fibers are mesh dependent. In other words, as the number of elements increases, the

6. Conclusions The paper developed a fracture initiation model of steel braces with square hollow structural sections for seismic evaluation of the postfracture response of concentrically braced frames. The following conclusions can be drawn: (1) Existing ductility-related empirical models, based on either accumulative or maximum deformation, cannot be relied upon to simulate experimentally observed brace fracture. (2) The strain-related fiber models that utilize the available fatigue material model in OpenSees by proposing constant fatigue parameters or through empirical predictions regarding the braces with different sizes and proportions failed to predict the fracture process observed from physical tests. (3) A model to predict fracture initiation is developed based on 17

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multiple regression analysis of an inclusive experimental data collected from different tests. The algorithm of the fracturing process using the fracture initiation prediction can be easily implemented in any simulation program such as OpenSees for seismic evaluation of post-fracture performance of concentrically braced frames. (4) The fracturing process concept with the proposed fracture initiation model is able to predict the cyclic response of the braces, tested as single brace members or in actual braced frames, throughout inelastic deformation, fracture initiation, fracture propagation, and terminal fracture stages. It provides a much-improved tool for the numerical simulation of post-fracture response of concentrically braced frames.

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Behaviour of square hss braces with end connections under reversed cyclic axial loading MSc Thesis Calgary, Alberta: Department of Civil Engineering, University of Calgary; 2003. [25] Yang F, Mahin SA, Uriz P. Limiting net section failure in slotted HSS braces. Steel TIPS Report, Structural Steel Education Council, Moraga, CA. 2005. [26] Goggins JM, Broderick BM, Elghazouli AY, Lucas AS. Behaviour of tubular steel members under cyclic axial loading. J Constr Steel Res 2006;62. https://doi.org/10. 1016/j.jcsr.2005.04.012. [27] Johnson SM. Improved seismic performance of special concentrically braced frames MSc Thesis Seattle: Department of Civil and Environmental Engineering, University of Washington; 2005. [28] Han SW, Kim WT, Foutch DA, ASCE M. Seismic behavior of hss bracing members according to width – thickness ratio under symmetric cyclic loading. J Struct Eng 2007;133(2):264–73. https://doi.org/10.1061/(ASCE)0733-9445(2007) 133:2(264). [29] Herman DJ. 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