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Frontiers of Structural and Civil Engineering

Front. Struct. Civ. Eng.    2018, Vol. 12 Issue (1) : 163-182     https://doi.org/10.1007/s11709-017-0384-z
RESEARCH ARTICLE |
Probabilistic safety assessment of self-centering steel braced frame
Navid RAHGOZAR1(), Nima RAHGOZAR1, Abdolreza S. MOGHADAM2
1. Department of Structural Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran
2. Structural Engineering Research Center, International Institute of Earthquake Engineering and Seismology (IIEES), Tehran, Iran
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Abstract

The main drawback of conventional braced frames is implicitly accepting structural damage under the design earthquake load, which leads to considerable economic losses. Controlled rocking self-centering system as a modern low-damage system is capable of minimizing the drawbacks of conventional braced frames. This paper quantifies main limit states and investigates the seismic performance of self-centering braced frame using a Probabilistic Safety Assessment procedure. Margin of safety, confidence level, and mean annual frequency of the self-centering archetypes for their main limit states, including PT yield, fuse fracture, and global collapse, are established and are compared with their acceptance criteria. Considering incorporating aleatory and epistemic uncertainties, the efficiency of the system is examined. Results of the investigation indicate that the design of low- and mid-rise self-centering archetypes could provide the adequate margin of safety against exceeding the undesirable limit-states.

Keywords self-centering steel braced frame      mean annual frequency      safety assessment      confidence level      margin of safety     
Corresponding Authors: Navid RAHGOZAR   
Online First Date: 19 June 2017    Issue Date: 08 March 2018
 Cite this article:   
Navid RAHGOZAR,Nima RAHGOZAR,Abdolreza S. MOGHADAM. Probabilistic safety assessment of self-centering steel braced frame[J]. Front. Struct. Civ. Eng., 2018, 12(1): 163-182.
 URL:  
http://journal.hep.com.cn/fsce/EN/10.1007/s11709-017-0384-z
http://journal.hep.com.cn/fsce/EN/Y2018/V12/I1/163
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Navid RAHGOZAR
Nima RAHGOZAR
Abdolreza S. MOGHADAM
Fig.1  (a) Configuration of the self-centering system; (b) push and hysteretic curves of the self-centering system, fuse and PT strand components
Fig.2  Outline procedure for PSA of the self-centering steel braced frame
Fig.3  Schematic plan of the PSA procedure: correlation between (a) seismic hazard curve and (b) structural demand and capacity parameters
performance archetype stories number of archetype in each direction design load level
group ID design ID gravity frame type seismic loads seismic design category
PG1 A1 3 2 perimeter low Dmax
A2 6 4 perimeter low Dmax
A3 9 4 perimeter low Dmax
PG2 A4 3 2 perimeter low Dmax
A5 6 4 perimeter low Dmax
A6 9 4 perimeter low Dmax
PG3 A7 3 2 space high Dmax
A8 6 4 space high Dmax
A9 9 4 space high Dmax
PG4 A10 3 2 space high Dmax
A11 6 4 space high Dmax
A12 9 4 space high Dmax
Tab.1   Performance groups along with considered archetypes
Fig.4  Configurations of self-centering braced frame archetypes (a) space SC-CR; (b) perimeter SC-CR; (c) perimeter and space SC-CR
Fig.5  (a) Configuration and (b) material of post-tensioning strand. (c) Configuration, (d) modeling, (e) and hysteretic behavior of butterfly-shaped fuse [41]
Fig.6  Three design load cases. (a) Inverted triangular; (b) upward triangular; (c) reverse triangular profiles [13]
PG1 PG2 PG3 PG4
Dmax/perimeter Dmin/perimeter Dmax/space Dmin/space
3-st. 6-st. 9-st. 3-st. 6-st. 9-st. 3-st. 6-st. 9-st. 3-st. 6-st. 9-st.
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12
FPti (kN) 2062 3689 7256 768 743 1345 1543 2652 3305 249 300 425
APT (cm2) 22 28 52 8.3 5.7 9.7 16.8 20 41 2.7 2.6 3.1
NPT 16 20 30 6 4 7 12 15 30 2 2 2
VfP(kN) 280 256 318 139 96 105 280 256 318 139 96 105
Nfs - Nlfs 3-8 6-8 9-10 3-4 3-4 3-6 3-8 6-8 9-10 3-4 3-4 3-6
Tab.2   Design properties of self-centering braced frame archetypes
story no. seismic design category (SDC)
brace section beam section middle column section side column section
Dmax Dmin Dmax Dmin Dmax Dmin Dmax Dmin
1 W14×74 W14×61 W14×34 W14×30 W14×233 W14×176 W14×90 W14×74
2 W14×82 W14×68 W14×30 W14×30 W14×233 W14×176 W14×90 W14×74
3 W14×90 W14×74 W14×38 W14×34 W14×233 W14×176 W14×90 W14×74
1 W14×68 W14×53 W14×30 W14×30 W14×311 W14×257 W14×159 W14×132
2 W14×68 W14×53 W14×30 W14×30 W14×311 W14×257 W14×159 W14×132
3 W14×68 W14×53 W14×30 W14×30 W14×311 W14×257 W14×159 W14×132
4 W14×68 W14×53 W14×30 W14×30 W14×211 W14×176 W14×120 W14×99
5 W14×68 W14×53 W14×34 W14×30 W14×211 W14×176 W14×120 W14×99
6 W14×90 W14×74 W14×34 W14×30 W14×211 W14×176 W14×120 W14×99
1 W14×90 W14×74 W14×30 W14×30 W14×500 W14×370 W14×311 W14×257
2 W14×90 W14×74 W14×34 W14×30 W14×500 W14×370 W14×311 W14×257
3 W14×90 W14×74 W14×34 W14×30 W14×500 W14×370 W14×311 W14×257
4 W14×90 W14×74 W14×34 W14×30 W14×398 W14×370 W14×257 W14×233
5 W14×90 W14×74 W14×34 W14×30 W14×398 W14×370 W14×257 W14×233
6 W14×90 W14×74 W14×34 W14×30 W14×398 W14×370 W14×257 W14×233
7 W14×90 W14×74 W14×34 W14×30 W14×311 W14×257 W14×145 W14×120
8 W14×99 W14×82 W14×34 W14×30 W14×311 W14×257 W14×145 W14×120
9 W14×145 W14×99 W14×34 W14×30 W14×311 W14×257 W14×145 W14×120
Tab.3  Brace, beam, and column sections of self-centering archetypes
Fig.7  Nonlinear simulation of self-centering archetypes in OpenSees
Fig.8  (a) Brace with gusset-plate model; (b) details of gusset plate connection
Fig.9  (a) Ground motion hazard curves for the archetypes of PG1 (A1, A2, and A3); (b) spectral accelerations and median spectrum of far-field FEMA P695 ground motion set.
Fig.10  (a) IDAs of (1) A1, (2) A2, and (3) A3 archetypes under a set of selected ground motions; (b) (1) roof drift ratio, (2) PT axial force versus (2) time and (3) roof drift ratio for the A3 archetype under the scaled horizontal component of Hector Mine earthquake to PGA of 0.27 and 1.91 g
Fig.11  (a) Idealized schematic lateral force-drift ratio of self-centering systems; (b) definition of performance levels and limit-states
Fig.12  Quantified PT yield (LS(1); red plus), fuse fracture (LS(2); green plus), and global collapse (LS(3); black plus) limit-state points (IDR and Sa) for (a) A1, (b) A2, and (c) A3 archetypes
archetype ID design configuration S?LS (g) mean IDRLS (%)
number of stories gravity loads seismic design category LS(1) a) LS(2) b) LS(3) c) LS(1) a) LS(2) b) LS(3) c)
performance group number PG1: SDC Dmax and low gravity loading (perimeter frame)?????????
A1 3 low Dmax 1.13 1.44 2.01 3.48 4.48 8.59
A2 6 low Dmax 1.10 1.35 1.86 4.03 4.47 9.02
A3 9 low Dmax 0.89 0.97 1.37 3.89 4.62 8.67
mean of performance group 1.04 1.25 1.75 3.80 4.52 8.76
performance group number PG2: SDC Dmin and low gravity loading (perimeter frame)?????????
A4 3 low Dmin 0.75 0.86 1.14 3.40 4.33 5.56
A5 6 low Dmin 0.71 0.82 0.96 3.62 4.81 5.49
A6 9 low Dmin 0.61 0.64 0.70 3.17 4.15 5.23
mean of performance group 0.69 0.77 0.93 3.34 4.43 5.43
performance group number PG3: SDC Dmax and high gravity loading (space frame)?????????
A7 3 high Dmax 0.91 1.02 1.33 3.44 4.41 7.35
A8 6 high Dmax 0.88 0.97 1.27 3.87 4.73 7.80
A9 9 high Dmax 0.78 0.86 0.97 3.76 4.56 7.52
mean of performance group 0.86 0.95 1.19 3.69 4.57 7.56
performance group number PG4: SDC Dmin and high gravity loading (space frame)?????????
A10 3 high Dmin 0.73 0.81 1.05 3.29 4.13 5.22
A11 6 high Dmin 0.69 0.76 0.81 3.12 4.42 5.17
A12 9 high Dmin 0.61 0.63 0.69 3.22 4.21 5.41
mean of performance group 0.67 0.73 0.85 3.21 4.25 5.29
Tab.4  Summary of the pushover and IDA results for self-centering archetypes
Fig.13  (a) Fractile IDA curves and (b) record-to-record uncertainty of (b) maximum inter-story drift (IDRmax) versus Sa (bDR(IDRmax |Sa)) and (c) Sa versus IDRmax (bDR(Sa| IDRmax)) for (1) A1, (2) A2, and (3) A3 archetypes
Fig.14  Lognormal and adjusted fragility curves for PT yield, fuse fracture, and global collapse limit states of (a) A1, (b) A2, and (c) A3 archetypes.
archetype ID design configuration S?LS (g) safety margin ratio confidence level of x (%)
number of stories gravity/SDC loads SMT (g) LS(1)a) 10%/50 LS(2)b) 2%/50 LS(3)c) 2%/50 LS(1) a) LS(2) b) LS(3) c) LS(1) a) LS(2) b) LS(3) c)
performance group number PG1: SDC Dmax and low gravity loading (perimeter frame)???????
A1 3 low/Dmax 0.87 1.13 1.44 2.01 2.31 1.66 1.30 96.08 83.87 77.08
A2 6 low/Dmax 0.69 1.10 1.35 1.86 2.70 1.96 1.59 99.98 94.87 79.57
A3 9 low/Dmax 0.53 0.89 0.97 1.37 2.59 1.83 1.68 98.32 91.22 81.69
mean of performance group 0.69 1.04 1.25 1.75 2.54 1.82 1.51 98.12 89.98 79.44
performance group number PG2: SDC Dmin and low gravity loading (perimeter frame)???????
A4 3 low/Dmin 0.37 0.75 0.86 1.14 3.08 2.32 2.03 99.98 91.61 85.69
A5 6 low/Dmin 0.23 0.71 0.82 0.96 4.17 3.57 3.09 99.18 99.01 79.00
A6 9 low/Dmin 0.17 0.61 0.64 0.70 4.12 3.76 3.59 98.22 99.74 85.60
mean of performance group 0.26 0.69 0.77 0.93 3.58 2.96 2.65 99.12 96.78 84.76
performance group number PG3: SDC Dmax and high gravity loading (space frame)???????
A7 3 high/Dmax 0.87 0.91 1.02 1.33 1.52 1.17 1.05 93.98 85.53 78.88
A8 6 high/Dmax 0.69 0.88 0.97 1.27 1.80 1.40 1.27 96.12 97.19 70.00
A9 9 high/Dmax 0.53 0.78 0.86 0.97 1.83 1.62 1.47 97.37 93.63 74.85
mean of performance group 0.69 0.86 0.95 1.19 1.70 1.35 1.22 95.82 92.11 74.57
performance group number PG4: SDC Dmin and high gravity loading (space frame)???????
A10 3 high/Dmin 0.37 0.73 0.81 1.05 2.84 2.19 1.97 97.35 92.15 83.33
A11 6 high/Dmin 0.23 0.69 0.76 0.81 3.52 3.30 3.00 99.38 98.02 79.42
A12 9 high/Dmin 0.17 0.61 0.63 0.69 4.06 3.71 3.59 99.05 99.31 68.13
mean of performance group 0.26 0.67 0.73 0.85 3.27 2.81 2.58 98.59 96.49 76.96
Tab.5  Summary of IDA results, safety margin ratios, and confidence levels of self-centering archetypes at defined limit states.
Fig.15  Derivative MAF spectra of PT yield, fuse fracture, and global collapse limit states respecting Sa(d(lLS)/d(Sa)) for (a) A1, (b) A2, and (c) A3 archetypes
archetype ID design configuration λLS( × 10−4) λP0( × 10−4) acceptance check
number of stories gravity loads SDC LS(1)a)10%/50 LS(2)b)2%/50 LS(3)c)2%/50 10%/50 2%/50 LS(1)a) LS(2)b) LS(3)c)
performance group number PG1: SDC Dmax and low gravity loading (perimeter frame)
A1 3 low Dmax 3.02 1.90 1.11 21 4 pass pass pass
A2 6 low Dmax 1.64 1.71 0.67 21 4 pass pass pass
A3 9 low Dmax 1.90 1.23 0.80 21 4 pass pass pass
mean of performance group 2.19 1.61 0.86 21 4 pass pass pass
performance group number PG2: SDC Dmin and low gravity loading (perimeter frame)
A4 3 low Dmin 3.41 2.34 1.45 21 4 pass pass pass
A5 6 low Dmin 2.82 2.01 1.28 21 4 pass pass pass
A6 9 low Dmin 3.08 2.78 1.83 21 4 pass pass pass
mean of performance group 3.10 2.38 1.52 21 4 Pass Pass pass
performance group number PG2: SDC Dmin and low gravity loading (perimeter frame)
A7 3 high Dmax 3.27 2.76 1.96 21 4 pass pass pass
A8 6 high Dmax 2.21 1.83 1.12 21 4 pass pass pass
A9 9 high Dmax 2.60 2.35 1.50 21 4 pass pass pass
mean of performance group 2.69 2.31 1.53 21 4 pass pass pass
performance group number PG4: SDC Dmin and high gravity loading (space frame)
A10 3 high Dmin 2.88 2.25 1.30 21 4 pass pass pass
A11 6 high Dmin 1.84 1.77 0.97 21 4 pass pass pass
A12 9 high Dmin 2.11 1.96 1.38 21 4 pass pass pass
mean of performance group 1.22 1.99 2.28 21 4 Pass pass pass
Tab.6  Performance evaluation of self-centering archetypes
ID event year station fault type Mwa) Tg (s)
Comp.1-2
distance (km) Vs30(m/s2)
Clst.b) Epi.c)
1 Northridge 1994 Beverly Hills Mulhol blind thrust 6.7 0.91-0.55 17.2 13.3 356
2 Northridge 1994 Canyon W Lost Cany blind thrust 6.7 0.59-0.71 12.4 26.5 309
3 Duzce, Turkey 1999 Bolu strike-slip 7.1 0.56-0.99 12.0 41.3 326
4 Hector Mine 1999 Hector strike-slip 7.1 1.23-0.52 11.7 26.5 685
5 Imperial Valley 1979 Delta strike-slip 6.5 3.28-1.66 22.0 33.7 275
6 Imperial Valley 1979 El Centro Array#11 strike-slip 6.5 1.76-0.74 12.5 29.4 196
7 Kobe, Japan 1995 Nishi-Akashi strike-slip 6.9 0.47-0.72 7.1 8.7 609
8 Kobe, Japan 1995 Shin-Osaka strike-slip 6.9 0.67-1.23 19.2 46 256
9 Kocaeli, Turkey 1999 Duzce strike-slip 7.5 3.86-0.51 15.4 98.2 276
10 Kocaeli, Turkey 1999 Arcelik strike-slip 7.5 1.24-9.28 13.5 53.7 523
11 Landers 1992 Yermo Fire Station strike-slip 7.3 6.38-1.36 23.6 86 354
12 Landers 1992 Coolwater strike-slip 7.3 1.42-0.61 19.7 82.1 271
13 Loma Prieta 1989 Capitola strike-slip 6.9 0.85-1.49 15.2 9.8 289
14 Loma Prieta 1989 Gilroy Array #3 strike-slip 6.9 0.67-0.46 12.8 31.4 350
15 Manjil, Iran 1990 Abbar strike-slip 7.4 2.16-0.82 12.6 40.4 724
16 Superstition Hills 1990 El Centro Imp. Cent strike-slip 6.5 2.67-1.42 18.2 35.8 192
17 Superstition Hills 1987 Poe Road (temp) strike-slip 6.5 2.45-0.46 11.2 11.2 208
18 Cape Mendocino 1992 Rio Dell Overpass FF thrust 7.0 2.37-1.29 14.3 22.7 312
19 Chi-Chi, Taiwan, China 1999 CHY101 thrust 7.6 0.47-3.4 10.0 32.0 259
20 Chi-Chi, Taiwan, China 1999 TCU045 thrust 7.6 4.88-0.48 26.0 77.5 705
21 San Fernando 1971 LA- Hollywood Stor thrust 6.6 3.74-2.24 22.8 39.5 316
22 Friuli, Italy 1976 Tolmezzo thrust 6.5 0.51-0.66 15.8 20.2 425
  Table A1Far-field ground motion record set [44]
The following symbols are used in this paper:
APT = cross-sectional area of post-tensioning strands;
afs/bfs = link width ratio;
bfs = link depth at the end;
bfs/tfs = width-to-thickness ratio of fuse link ends;
Dmax = maximum spectral acceleration intensity associated with SDC D;
Dmin = minimum spectral acceleration intensity associated with SDC D;
EPT = PT modulus of elasticity;
fyfs = yield strength of the fuse;
fyPT = yield strength of the post-tensioning strand;
fPTi = initial post-tensioning stress;
FPTi = initial post-tensioning force;
fufs = fracturing strength of the fuse;
fUPT = ultimate strength of the post-tensioning strand;
f50 = median IDA curve;
f16 = 16% fractile curve;
f84 = 84% fractile curve;
k= logarithmic slope of the hazard curve;
k0 = real and positive constant of the prediction of the site seismicity;
KAB = hardening stiffness;
Kfs = initial fuse stiffness;
KOA = initial stiffness of the archetype;
KX = standard normal value of the inverse cumulative distribution index of x;
Leff = unbraced length of the brace;
Lfs = effective width of the fuse;
Lfs/tfs = slenderness ratio;
LPT = length of the post-tensioning strand;
Mfsy = yield moment of the fuse;
Mu = overturning moment;
MUP = uplift moment of the system;
Mw = moment magnitudes;
My = yield moment of the system;
Nfs = number of fuses;
Nlfs = number of links per fuse;
NPT = number of post-tensioning strands;
PD = total gravity load;
R= response modification coefficient;
Sa = spectral intensity at the fundamental period of the archetype;
SC= self-centering ratio;
SDS = design, 5% damped spectral response acceleration parameter at short period;
SD1 = design, 5% damped spectral response acceleration parameter at a period of 1 s;
SMT = spectra intensity at maximum considered earthquake ground motion;
S?LS = median limit state spectral intensity for the entire ground motion record set;
t= fundamental period of the system;
tfs = plate thickness of the fuse;
Vfp = shear strength of the fuse;
Vmax = maximum lateral strength;
V= design base shear of the system;
X= confidence level;
α= hardening ratio;
β= energy dissipation ratio;
βC,U = capacity uncertainty;
βC,R = capacity aleatory randomness;
βEDP,R = demand aleatory randomness;
βEDP,R|16 = demand randomness calculated by 16% fractile;
βEDP,R|84 = demand randomness calculated by 84% fractile;
βEDP,R|s = aleatory randomness of demand at each seismic intensity level;
βDes = design requirement-related uncertainty;
βMDL = model uncertainty;
βtotal,D = test data-related uncertainty;
βTOT = total uncertainty;
βtotal,R = record to record uncertainty;
βtotal,U = total epistemic uncertainty;
βUD = demand uncertainty;
δu = ultimate roof drift ratio;
δup = system uplift drift;
δy,eff = effective yield roof drift ratio;
δy = yield drift of the system;
εi = initial strain of the post-tensioning strand;
εlimit = strain limit of the post-tensioning strand;
εm = fracture strain of the post-tensioning strand;
εtarget = target strain of the post-tensioning strand;
εu = ultimate strain of the post-tensioning strand;
εy = yield strain of the post-tensioning strand;
φ= capacity factor;
φ-1(x) = inverse cumulative distribution function of the normal variable of x;
γ= demand factor;
γm = shear strain of the fuse;
λLS = mean annual frequency of exceeding a given limit-state;
λP0 = acceptable risk of occurring a limit state at a given seismic intensity;
λx = confidence factor.
  
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