aslani and miranda - fagility slab column connections

December 5, 2005 11:28 WSPC/124-JEE 00226

Journal of Earthquake Engineering, Vol. 9, No. 6 (2005) 777–804c© Imperial College Press

FRAGILITY ASSESSMENT OF SLAB-COLUMNCONNECTIONS IN EXISTING NON-DUCTILE

REINFORCED CONCRETE BUILDINGS

HESAMEDDIN ASLANI and EDUARDO MIRANDA

Dept. of Civil and Environmental EngineeringStanford University, Stanford, CA, USA

Received 12 November 2004Reviewed 11 April 2005Accepted 14 April 2005

Fragility functions that estimate the probability of exceeding different levels of damage inslab-column connections of existing non-ductile reinforced concrete buildings subjectedto earthquakes are presented. The proposed fragility functions are based on experimen-tal data from 16 investigations conducted in the last 36 years that include a total of82 specimens. Fragility functions corresponding to four damage states are presented asfunctions of the level of peak interstory drift imposed on the connection. For damagestates involving punching shear failure and loss of vertical carrying capacity, the fragilityfunctions are also a function of the vertical shear in the connection produced by gravityloads normalised by the nominal vertical shear strength in the absence of unbalancedmoments. Two sources of uncertainty in the estimation of damage as a function of lateraldeformation are studied and discussed. The first is the specimen-to-specimen variabilityof the drifts associated with a damage state, and the second the epistemic uncertaintyarising from using small samples of experimental data and from interpreting the exper-imental results. For a given peak interstorey drift ratio, the proposed fragility curvespermit the estimation of the probability of experiencing different levels of damage inslab-column connections.

Keywords: Performance evaluation; damage assessment; experimental data; uncertaintyanalysis; slab-column connections.

1. Introduction

The goal of performance-based seismic design (PBSD) is to design facilities withpredictable levels of seismic performance. An adequate implementation of PBSDrequires a relationship between the ground motion intensity and the damage in thestructure. Previous studies in damage assessment have typically provided globalestimates of damage in the facility in a single step through the use of damageprobability matrices relating a damage ratio with Modified Mercalli Intensity [ATC-13, 1985], empirical functions relating peak ground motion parameters to damageratios [Wald, 1999] and with functions relating damage ratios to spectral ordinates[NIBS, 1999]. More recent work has been aimed at improved estimates of damagein specific types of reinforced concrete buildings as a function of spectral ordinates.

777


778 H. Aslani & E. Miranda

For example, Singhal and Kiremidjian [1996] computed fragility of three classesof reinforced concrete frame buildings, namely low rise concrete frames that are1–3 storeys tall, mid rise frames that are 4–7 storeys tall, and high rise framesthat are 8 storeys or taller, to estimate the probability of reaching or exceedinga certain damage level in the building conditioned on the ground motion witha certain level of linear spectral acceleration. In another study, Hwang and Huo[1997] developed fragility functions for specific reinforced concrete frames as partof a loss assessment study of Memphis buildings. Similarly, Erberik and Elnashai[2004] computed fragility curves for 5-storey flat-slab structures.

In the probabilistic framework being developed at the Pacific Earthquake Engi-neering Research (PEER) Center, damage assessment is performed in two steps.In the first step, a probabilistic description of the structural response at increasinglevels of ground motion intensity is obtained through a series of response historyanalyses. Then, in a second step, damage to individual structural and nonstruc-tural components is estimated as a function of structural response parameters (e.g.peak interstory drift demands, peak floor accelerations, etc.) computed in the firststep. This approach requires fragility functions for various damage states for eachcomponent in the facility as a function of structural response parameters.

The objective of this work is to summarise research aimed at the development offragility functions that estimate damage in slab-column connections of non-ductilereinforced concrete buildings as a function of the peak interstorey drift imposed onthe connection. In particular, cast-in-place slab-column connections built prior to1976 which have no shear reinforcement have been studied. An important amountof research has been conducted on the seismic design and behaviour of slab-columnconnections. Moehle et al. [1988], ACI-352 [1988], Hueste and Wight [1999] andEnomoto and Roberston [2001] provide excellent summaries of previous research.These previous research has been aimed at improving design recommendations forslab-column connections in new buildings located in seismic regions, and in partic-ular design recommendations to avoid shear failures. However, results from previ-ous experimental studies also provide valuable information to estimate the level ofdamage that can occur in slab-column connections in existing reinforced concretebuildings as they are subjected to increasing levels of lateral deformation.

2. Definition of Damage States

In this study, four discrete damage states of slab-column connections have beenintroduced. These damage states have been defined based on specific actions thatwould have to be taken as a result of the observed damage. This approach facilitatesthe estimation of economic losses and other types of consequences (e.g. repair time,possible casualties, etc.) resulting from the occurrence of the damage.

The following damage states have been considered:

• DS 1 Light Cracking: Light cracking corresponds to crack widths smaller than0.3mm (0.013 in) which become visible at distances of about 2.0m (6.6 ft).


Fragility Assessment of Slab-Column Connections 779

DS1 DS2

DS3 DS4

Fig. 1. Schematic view of the four damage states considered for slab-column connections:(a) Light cracking; (b) severe cracking; (c) punching shear failure; and loss of vertical carryingcapacity.

Figure 1(a) shows a schematic view of a typical cracking pattern on the topface of the slab at this damage state. These crack widths can be tolerated inreinforced concrete in humid or moist air conditions [ACI 224R, 2001]. Hence,actions associated with this damage state typically consist of either “no repair”or a “light repair” by applying a coating on the concrete surface to conceal theprojection of cracks [FEMA 308, 1998]. The purpose of this repair action is toimprove the aesthetic appearance of the slab or to provide an additional bar-rier against water infiltration into the slab. This repair action is not intended toprovide any increase in strength and stiffness to the cracked slab.

• DS 2 Severe Cracking: This damage state involves extensive cracking with crackwidths between 0.3mm (0.013 in) and 2mm (0.08 in). Figure 1(b) shows a typicalcracking pattern on the top face of the slab at this damage state. For this levelof cracking most concrete repair guidelines suggest epoxy injection [ACI 224.1,1984; ACI 546, 1996; ACI 548, 1997]. The repair action associated with this dam-age state typically consists of crack epoxy injection which provides a structuralbinding agent to fill the crack and adhere to the substrate material [FEMA 308,



1998]. The main purpose of this repair action is to partially restore the originalstrength and stiffness of the connection [Krauss et al., 1995].

• DS 3 Punching Shear Failure: This damage state corresponds to severe crackingcharacterised by a roughly circular tangential cracking around the column areaof slab, radial cracks extending from that area, and considerable spalling of theconcrete cover [ASCE-ACI 426, 1974]. Figure 1(c) shows typical cracking patternon the top face of the slab at this damage state. Repair actions involve signifi-cant labour and cost, and consist of concrete spall repair and rebar replacement[FEMA 308, 1998]. Loose concrete is removed with chipping hammer. If reinforc-ing bars are exposed, concrete is removed to provide sufficient clearance aroundthe bar for the patch to bond to the full diameter. When repairing this type ofdamage, sometimes it is necessary to cut out the damaged length of reinforcingbars in order to replace it with new bars.

• DS 4 Loss of Vertical Carrying Capacity (LVCC ): At this damage state com-ponent loses its vertical carrying capacity, and collapses under its gravity load.Figure 1(d) shows a schematic view of a slab-column connection at this damagestate. Previous studies have shown that LVCC occurs at larger levels of defor-mation than those associated with punching shear failure [Hawkins and Mitchell,1979; Mitchell and Cook, 1984; Pan and Moehle, 1992]. If there is no possibility toredistribute the vertical load to other members, this damage state has disastrousconsequences, since it can lead to a local collapse. In many cases, local collapsesproduced by the LVCC have triggered a global progressive collapse in buildings[Rosenblueth and Meli, 1986; Hamburger, 1996].

3. Experimental Results Used in this Study

Estimation of the probability of experiencing various damage states in slab-columnconnections requires gathering information about the level of lateral deformation atwhich various damage states have been observed. Ideally, one would use informa-tion from actual buildings that have undergone various levels of earthquake damage.However, in most cases the level of lateral deformation that produced damage invarious floor levels is unknown. Therefore, results from experimental studies havebeen used as the basis for establishing levels of lateral deformations associated withdifferent damage states. Data considered in this study was limited to interior slab-column connections without shear reinforcement. Results of experimental researchconducted over the last 36 years in 10 different research universities were consid-ered. Information about the material properties and characteristics of all specimensconsidered in this study is summarised in Table 1, which includes 17 experimentalresearch investigations for a total of 82 specimens.

In most cases there were not enough information to establish the interstoreydrifts at which all four damage states took place. This was either because thedamage state did not occur (e.g. a punching shear failure or LVCC was not observedduring the test) or because the report did not document well enough information


Fragility Assessment of Slab-Column Connections 781Table

1.

Pro

per

ties

ofsl

ab-c

olu

mn

spec

imen

suse

din

this

study.

Spec

imen

pro

per

ties

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emen

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imen

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the

Sec

tion

f′c

fynum

ber

Ref

eren

ces

Label

(cm

)(M

pa)

(Mpa)

colu

mn

(cm

×cm

)(M

pa)

(Mpa)

Vg

V0

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

1H

anso

nand

P7

7.5

033.0

354.4

0.0

4—

30×

15

33.0

354.4

2H

anso

n[1

968]

C8

7.5

032.8

410.9

0.0

5—

30×

15

32.8

410.9

3Is

lam

and

Park

IP2

8.7

531.9

373.7

0.1

8Y

es22.5

×22.5

31.9

373.7

4[1

973]

IP3C

8.7

529.7

315.8

0.2

3Y

es22.5

×22.6

29.7

315.8

5IP

18.7

529.7

315.8

0.2

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×22.7

29.7

315.8

6H

awkin

set

al.

S1

15.0

034.8

460.0

0.3

3Y

es25×

25

34.8

460.0

7[1

974]

S2

15.0

023.4

462.6

0.4

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25

23.4

462.6

8S3

15.0

022.0

455.0

0.4

5Y

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25

22.0

455.0

9S4

15.0

032.3

460.0

0.4

0Y

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25

32.3

460.0

10

Sey

monds

etal.

S6

15.0

023.2

460.0

0.8

6Y

es25×

25

23.2

460.0

11

[1976]

S7

15.0

026.5

462.6

0.8

1Y

es25×

25

26.5

462.6

12

S8

15.0

026.5

462.6

—Y

es25×

25

26.5

462.6

13

Ghalliet

al.

SM

0.5

15.0

036.7

475.8

0.3

1—

25×

25

36.7

475.8

14

[1976]

SM

1.0

15.0

033.3

475.8

0.3

3—

25×

25

33.3

475.8

15

SM

1.5

15.0

039.9

475.8

0.3

0—

25×

25

39.9

475.8

16

Morr

ison

and

S1

7.5

045.8

322.7

0.0

2—

25×

25

45.8

322.7

17

Soze

n[1

981]

S2

7.5

035.1

330.3

0.0

2—

25×

25

35.1

330.3

18

S3

7.5

033.9

335.1

0.0

2—

25×

25

33.9

335.1

19

S4

7.5

034.9

317.2

0.0

5—

25×

25

34.9

317.2

20

S5

7.5

035.2

339.9

0.1

1—

25×

25

35.2

339.9

21

D1

7.5

036.3

290.0

0.0

4—

25×

25

36.3

290.0



Table

1.

(Continued

) Spec

imen

pro

per

ties

Sla

bC

olu

mn

Bott

om

rein

forc

emen

tSpec

imen

Thic

knes

sf′c

fyth

rough

the

Sec

tion

f′c

fynum

ber

Ref

eren

ces

Label

(cm

)(M

pa)

(Mpa)

colu

mn

(cm

×cm

)(M

pa)

(Mpa)

Vg

V0

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

22

D2

7.5

033.9

327.0

0.0

4—

25×

25

33.9

327.0

23

D3

7.5

036.5

355.0

0.0

4—

25×

25

36.5

355.0

24

Zee

and

Moeh

leIN

T6.0

037.2

435.1

0.2

1Y

es13.5

×13.5

37.2

435.1

[1984]

25

Pan

and

Moeh

leTes

t1

12.0

033.3

472.0

0.3

7Y

es27×

27

33.3

472.0

26

[1988]

Tes

t2

12.0

033.3

472.0

0.3

6Y

es27×

27

33.3

472.0

27

Tes

t3

12.0

031.4

472.0

0.1

8Y

es27×

27

31.4

472.0

28

Tes

t4

12.0

031.4

472.0

0.1

9Y

es27×

27

31.4

472.0

29

Haw

kin

set

al.

6A

H15.0

031.3

415.0

0.4

3Y

es30×

30

31.3

415.0

30

[1989]

9.6

AH

15.0

030.7

415.0

0.4

8Y

es30×

30

30.7

415.0

31

14A

H15.0

030.3

415.0

0.5

3Y

es30×

30

30.3

415.0

32

6A

L15.0

022.7

415.0

0.7

2Y

es30×

30

22.7

415.0

33

9.6

AL

15.0

028.9

415.0

0.6

8Y

es30×

30

28.9

415.0

34

14A

L15.0

027.0

415.0

1.3

8Y

es30×

30

27.0

415.0

35

7.3

BH

11.2

522.2

415.0

0.3

8Y

es30×

30

22.2

415.0

36

9.5

BH

11.2

519.8

415.0

0.4

8Y

es30×

30

19.8

415.0

37

14.2

BH

11.2

529.5

415.0

0.4

2Y

es30×

30

29.5

415.0

38

7.3

BL

11.2

518.1

415.0

0.6

9Y

es30×

30

18.1

415.0

39

9.5

BL

11.2

520.0

415.0

0.7

1Y

es30×

30

20.0

415.0

40

14.2

BL

11.2

520.5

415.0

0.8

0Y

es30×

30

20.5

415.0

41

6C

H15.0

052.4

415.0

0.3

7Y

es30×

30

52.4

415.0

42

Haw

kin

set

al.

9.6

CH

15.0

057.2

415.0

0.4

1Y

es30×

30

57.2

415.0

43

[1989]

14C

H15.0

054.7

415.0

0.4

8Y

es30×

30

54.7

415.0

44

6C

L15.0

049.5

415.0

0.5

5Y

es30×

30

49.5

415.0

45

14C

L15.0

047.7

415.0

0.7

4Y

es30×

30

47.7

415.0



Table

1.

(Continued

) Spec

imen

pro

per

ties

Sla

bC

olu

mn

Bott

om

rein

forc

emen

tSpec

imen

Thic

knes

sf′c

fyth

rough

the

Sec

tion

f′c

fynum

ber

Ref

eren

ces

Label

(cm

)(M

pa)

(Mpa)

colu

mn

(cm

×cm

)(M

pa)

(Mpa)

Vg

V0

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

46

6D

H15.0

030.0

415.0

0.3

5Y

es30×

30

30.0

415.0

47

14D

H15.0

031.7

415.0

0.4

9Y

es30×

30

31.7

415.0

48

6D

L15.0

028.7

415.0

0.6

1Y

es30×

30

28.7

415.0

49

14D

L15.0

024.3

415.0

0.8

1Y

es30×

30

24.3

415.0

50

10.2

FH

I15.0

025.9

415.0

0.4

3Y

es30×

30

25.9

415.0

51

10.2

FH

O15.0

033.8

415.0

0.4

5Y

es30×

30

33.8

415.0

52

14FH

15.0

031.2

415.0

0.5

2Y

es30×

30

31.2

415.0

53

6FLI

15.0

025.9

415.0

0.6

3Y

es30×

30

25.9

415.0

54

10.2

FLI

15.0

018.1

415.0

0.8

0Y

es30×

30

18.1

415.0

55

10.2

FLO

15.0

026.5

415.0

0.8

0Y

es30×

30

26.5

415.0

56

9G

H2

15.0

024.7

45.0

0.4

7Y

es20×

40

24.7

415.0

57

9.6

GH

0.5

15.0

026.3

415.0

0.5

0Y

es40×

20

26.3

415.0

58

9.6

GH

315.0

027.0

415.0

0.4

5Y

es45×

15

27.0

415.0

59

Rober

tson

and

2C

11.2

533.0

500.2

0.1

8Y

es25×

25

33.0

500.2

60

Durr

ani[1

990]

6LL

11.2

532.2

524.9

0.5

1Y

es25×

25

32.2

524.9

61

7L

11.2

530.8

524.9

0.3

7Y

es25×

25

30.8

524.9

62

8I

11.2

539.3

524.9

0.1

8Y

es25×

25

39.3

524.9

63

Dilger

and

Cao

CD

1—

——

0.8

5—

——

—64

[1991]

CD

2—

——

0.6

5—

——

—65

CD

8—

——

0.5

2—

——

—


784 H. Aslani & E. MirandaTable

1.

(Continued

)

Spec

imen

pro

per

ties

Sla

bC

olu

mn

Bott

om

rein

forc

emen

tSpec

imen

Thic

knes

sf′c

fyth

rough

the

Sec

tion

f′c

fynum

ber

Ref

eren

ces

Label

(cm

)(M

pa)

(Mpa)

colu

mn

(cm

×cm

)(M

pa)

(Mpa)

Vg

V0

66

Durr

aniand

Du

DN

Y1

11.2

535.3

372.3

0.2

6N

o25×

25

35.3

372.3

67

[1992]

DN

Y2

11.2

525.7

372.3

0.3

7N

o25×

25

25.7

372.3

68

DN

Y3

11.2

524.6

372.3

0.2

3N

o25×

25

24.6

372.3

69

DN

Y4

11.2

519.1

372.3

0.2

7N

o25×

25

19.1

372.3

70

Wey

and

Durr

ani

SC

O11.2

539.3

524.9

0.2

5Y

es25×

25

39.3

524.9

[1992]

71

Farh

eyet

al.

18.0

035.1

457.6

0.3

2Y

es30×

20

35.1

457.6

72

[1993]

28.0

035.1

457.6

0.3

2Y

es30×

20

35.1

457.6

73

38.0

015.0

457.6

0.4

9Y

es30×

20

15.0

457.6

74

48.0

015.0

457.6

0.5

6Y

es30×

12

15.0

457.6

75

Luo

etal.

II11.2

520.7

379.0

0.0

8N

o25×

25

20.7

379.0

[1994]

76

Johnso

nand

ND

1C

11.4

029.6

413.7

0.2

3N

o25×

25

29.6

413.7

77

Rober

tson

[2001],

ND

4LL

11.4

032.3

413.7

0.2

8N

o25×

25

32.3

413.7

78

from

Enom

oto

ND

5X

L11.4

024.1

413.7

0.4

7N

o25×

25

24.1

413.7

79

and

Rober

tson

ND

6H

R11.4

026.3

413.7

0.2

4N

o25×

25

26.3

413.7

80

[2001]

ND

7LR

11.4

018.9

413.7

0.2

8N

o25×

25

18.9

413.7

81

ND

8B

11.4

039.2

413.7

0.2

0N

o25×

25

39.2

413.7

82

Rober

tson

etal.

[2002]

1C

11.5

35.4

420.0

0.2

5Y

es25×

25

35.4

420.0

—In

form

ati

on

was

not

available



to properly establish the level of interstorey drift at which the damage state wasobserved. The later situation was particularly common for the first two damagestates (light cracking and severe cracking), primarily because until very recently,earthquake provisions have mainly been concerned with life safety and not damagecontrol.

Only five investigations included detailed information about the level of inter-storey drift at which light cracking was observed. For these specimens, the reportedinterstorey drift at which clearly visible cracking occurred, was accompanied with asignificant reduction in lateral stiffness. Therefore, in order to expand the number ofdata points associated with this damage state, the interstorey drift at which a 30%or more sudden reduction in lateral stiffness was observed in the hysteresis loop wasconsidered. Figure 2 presents a graphic representation of a point in loading historywhere this sudden reduction occurred. Table 2, summarises the interstorey driftsassociated with this first damage state. Interstorey drifts presented in Table 2 havebeen calculated using the slab centreline to slab centreline interstorey height. Theindirect way of determining the interstorey drift associated with first damage statewas only used in specimens where a good quality hysteresis loop at early loadingcycles was included in the report. Consequently, Table 2 reports interstorey driftsassociated with light cracking for only 43 specimens. It can be seen that the firstdamage state in slab-column connections occurs at very low deformations ragingfrom 0.19% to 0.8% drift ratios.

A few studies systematically reported cracking patterns and crack widths atvarious levels of lateral deformation to be used for the second damage state. Carefulstudy of such investigations [Pan and Moehle, 1988; Robertson and Durrani, 1990]suggests that damage state two, typically occurs when top steel reinforcement isat yield. Therefore, in order to gather more data points associated with the seconddamage state, it was assumed that this damage state occurs at peak interstorey

Lateral Load

IDRIDRDS1

Significant changein stiffness

Fig. 2. Typical drop in lateral stiffness observed in the hysteresis loop of slab-column specimensused to identify the first damage state, light cracking, in some of the specimens.



Table 2. Interstorey drift ratios∗ used to develop the fragility functions.

Specimen IDRDS1 IDRDS2 IDRDS3 IDRDS4 Specimen IDRDS1 IDRDS2 IDRDS3 IDRDS4

number (%) (%) (%) (%) number (%) (%) (%) (%)

(1) (2) (3) (4) (5) (1) (2) (3) (4) (5)

1 ** ** 3.80 42 ** ** 1.25

2 ** ** 5.80 43 ** ** 1.25

3 0.43 1.11 4.72 44 ** ** 1.93

4 0.30 1.11 4.22 45 ** ** 0.91

5 0.28 1.11 4.81 46 ** ** 2.39

6 0.65 0.76 3.75 47 ** ** 1.65

7 0.22 0.93 1.82 48 ** ** 1.48

8 0.32 0.76 2.18 49 ** ** 0.77

9 0.43 0.90 2.46 50 ** ** 0.80

10 0.42 0.69 1.39 51 ** ** 1.25

11 0.28 0.49 1.04 52 ** ** 0.91

12 0.35 0.69 *** 53 ** ** 1.08

13 ** ** 6.00 54 ** ** 0.79

14 ** ** 2.70 55 ** ** 1.09

15 ** ** 2.70 56 ** ** 0.80

16 0.31 ** 4.47 57 ** ** 1.48

17 0.31 ** 2.89 58 ** ** 0.57

18 0.28 ** 4.25 59 0.64 1.50 5.00∗∗∗∗

19 0.22 ** 4.30 4.90 60 0.54 ** 1.00 2.00

20 0.22 ** 4.70 4.90 61 0.40 1.00 1.50 3.00

21 0.31 0.89 3.11 62 0.80 1.00 4.00 5.00

22 0.40 0.89 2.89 63 ** ** 0.90

23 0.36 0.89 4.32 64 ** ** 1.20

24 0.28 0.98 3.30 65 ** ** 1.40

25 0.20 0.77 1.60 2.60 66 0.50 ** 4.50∗∗∗∗ 5.00

26 0.50 0.75 1.50 1.60 67 0.50 ** 2.00

27 0.25 0.80 3.70 5.00 68 0.50 ** 4.50∗∗∗∗ 5.00

28 0.50 0.79 3.50 3.50 69 0.50 ** 4.70 6.00

29 ** ** 2.27 70 ** ** 5.0

30 ** ** 1.70 71 0.19 1.63 3.48 4.70

31 ** ** 1.36 72 0.19 1.27 2.53 3.70

32 ** ** 1.48 73 0.28 1.20 2.40 3.30

33 ** ** 1.08 74 0.28 0.82 3.16 3.80

34 ** ** 0.97 75 0.40 1.00 5.00∗∗∗∗

35 ** ** 1.62 76 0.74 1.48 8.00

36 ** ** 1.97 77 0.54 0.76 5.00

37 ** ** 1.27 78 0.43 1.00 2.00

38 ** ** 1.62 79 0.71 1.07 4.00 5.00

39 ** ** 1.73 80 0.43 1.00 5.00

40 ** ** 1.48 81 0.50 1.14 5.00

41 ** ** 2.59 82 0.70 1.40 3.50 4.00

*Centreline dimensions were used to compute IDR**Information was not available***Punching shear failure did not occur****IDR corresponds to a flexural failure

drifts at which top steel reinforcement was reported to yield or at peak interstoreydrifts at which a residual drift in the hysteresis loop was observed after unloading.Figure 3 schematically illustrates the latter situation. The third column in Table 2lists the interstorey drifts that were identified or inferred corresponding to this



2DSIDR

Lateral Load

IDR2DSIDR

Permanent residualdisplacement

Fig. 3. The first occurrence of a residual lateral drift in the hysteresis loop of a slab-columnconnection specimen used to identify the second damage state, severe cracking, in some of thespecimens.

damage state. Interstorey drifts associated with severe cracking were obtained for33 specimens which range between 0.49% and 1.63% drift ratios.

With the exception of five specimens in which shear failure was not observed,all other specimens experienced a punching shear failure and the research reportscontained information about the interstorey drift at which this damage state wasobserved. The fourth column in Table 2 lists interstorey drift ratios at which punch-ing shear failures occurred in various specimens. It can be seen that drifts at whichthese failures have occurred exhibit a very large dispersion ranging from 0.57% toas large as 8% drift ratios.

Most experimental studies have investigated parameters that influence themoment capacity and deformation at which punching shear failure occurred. Hence,in most cases testing was stopped after punching shear failure was observed. Thereare no investigations in which testing continued until LVCC occurred. However,there are six studies representing a total of 18 cases in which the specimens weresubjected to one or more cycles or larger deformations after punching shear fail-ure was observed. The main goal of these subsequent cycles was the investigationof post-punching behaviour of the slab-column connections. In this study, thesespecimens have been used to obtain information of the post-punching deformationcapacity and of a conservative estimate of the probability of experiencing LVCCin the connection. Maximum drift ratios at which testing were stopped in differentspecimens range between 1.6% and 6.0%.

4. Fragility Functions

As shown in Table 2, the interstorey drift at which each damage state in slab-columnconnections was observed shows relatively large variations from one specimen to



another. This specimen-to-specimen variability has taken into account by devel-oping drift-based fragility functions that estimate the likelihood of experiencing aparticular damage state. In particular, a drift-based fragility function provides theprobability of experiencing or exceeding a particular damage state conditioned onthe peak interstorey drift.

At each damage state, a cumulative frequency distribution function was obtainedby plotting ascending-sorted interstorey drifts at which the damage state was exper-imentally observed against (i−0.5)/n, where i is the position of the peak interstoreydrift and n is the number of specimens. This cumulative frequency distribution func-tion provides information about the portion of the data set that does not exceed aparticular value of drift and represents an empirically derived cumulative distribu-tion function.

Figures 4(a) and 4(b) show cumulative distribution functions for the first andsecond damage states, respectively. As can be seen in Fig. 4(a), the first damagestate (light cracking) was observed in half of the specimens when an interstoreydrift ratio of about 0.4% was reached. The second damage state (severe cracking)occurred in half of the specimens when the interstorey drift ratio reached 0.9%,Figure 4(b). Also plotted in these figures are fitted lognormal cumulative distribu-tion functions of the interstorey drift ratios given by

P (DS ≥ dsi|IDR = idr) = Φ[Ln(idr) − Ln (IDR)

σLn IDR

], (1)

where P (DS ≥ dsi|IDR = idr) is the probability of experiencing or exceedingdamage state i, IDR is the median of the interstorey drift ratios, IDR’s, at which

(a) (b)

Fig. 4. Fragility functions fitted to interstorey drift ratios corresponding to the first and seconddamage states in slab-column connections.



damage state i was observed, σLn IDR is the standard deviation of the naturallogarithm of the IDR’s, and Φ is the cumulative standard normal distribution. Itcan be seen that the lognormal distribution fits the data relatively well. Parametersfor the fitted lognormal probability distribution are given in Table 3.

In order to verify if the cumulative distribution function could be assumed aslognormally distributed, a Kolmogorov-Smirnov goodness-of-fit test [Benjamin andCornell, 1970] was conducted. Also shown in Figs. 4(a) and 4(b) are graphicalrepresentations of this test for 10% significance levels. The hypothesis that theassumed cumulative probability distribution adequately fits the empirical data isaccepted if all data points lie between the two gray lines. It can be seen that forboth damage states this assumption is adequate.

Empirical and fitted cumulative distribution functions for the third damagestate are shown in Fig. 5. Also shown in the figure are the 10% significance levelsof the Kolmogorov-Smirnov goodness-of-fit test which indicate that the lognormalassumption is also valid for the third damage state. The corresponding parameters

Table 3. Statistical parameters estimated for interstorey drift ratios corre-sponding to the four damage states in slab-column connections.

Number of

Damage state IDR(%) σLn IDR specimens (n)

(1) (2) (3) (4)

DS1: Light cracking 0.40 0.39 43DS2: Severe cracking 0.95 0.25 33DS3: Punching shear failure 2.00 0.62 77DS4: Loss of vertical carrying capacity 4.28 0.36 18

P (DS≥ds3|IDR)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.00 0.02 0.04 0.06 0.08

IDR

Data

Lognormal fit

K-S test,10% significance

DS 3

σ ln IDR = 0.62

Fig. 5. Fragility function fitted to interstorey drift ratios corresponding to the third damagestate, punching shear failure, in a slab-column connection.



for the cumulative distribution of this damage sate are given in Table 3. The disper-sion parameter (σLn IDR) which is approximately comparable to the coefficient ofvariation of the data is shown in Column 3 of Table 3 for each damage state. It canbe seen that for the third damage state σLn IDR is 0.62 which is 1.6 times larger thanthe level of dispersion corresponding to the first damage state and 2.5 times largerthan the dispersion corresponding to the second damage state. This means that theestimation of whether a punching shear failure is likely to occur in a slab-columnconnection is very uncertain if only the peak interstorey drift ratio is used.

Experimental research has shown that the deformation capacity of slab-columnconnections is a function of the level of gravity load. In particular, Pan and Moehle[1988], Moehle et al. [1988], Robertson and Durrani [1990] indicated that the dis-placement ductility ratio at which punching shear failure occurs is a function of thegravity load ratio Vg/V0, where Vg is the vertical shear that acts on the slab criticalsection defined at a distance d/2 from the column face in which d is the average slabeffective depth, and the normalising shear force V0 is the theoretical punching shearstrength in the absence of moment transfer in the connection, which according toACI-352 [1988] is given by

V0 =(

16

+1

3βc

)√f ′

cb0d ≤ 13

√f ′

cb0d, (2)

where βc is the ratio of long-to-short cross-sectional dimensions of the supportingcolumn, f ′

c is the compressive strength of concrete in MPa and not to exceeding40MPa, b0 is the perimeter length of the slab critical section in mm, and d is theslab effective width in mm. For most columns encountered in practice the maximumlimit shown in Eq. (2) governs. More recently, Hueste and Wight [1999] proposeda trilinear equation to estimate the drift at which punching shear failure occurs.

On the basis of the above observations, a fragility surface was developed for thethird damage state, in which the probability of experiencing a punching shear failurewas computed as a function of the peak IDR and the vertical shear ratio. Columnseven in Table 1 lists the gravity shear ratio in all specimens where punching shearfailure was observed. Figure 6(a) shows the drift at which punching shear failureoccurred as a function of the vertical shear ratio, Vg/V0. Also shown in the figure isthe trilinear variation of the drift capacity at punching shear proposed by Huesteand Wight [1999]. This figure shows results from 77 specimens, which represent44 data points in addition to those originally gathered by Hueste and Wight. In theproposed fragility surface, the probability of experiencing punching shear failure iscomputed with Eq. (1), but the median interstorey drift capacity is computed as afunction of the level of gravity load present in the slab at the time of earthquakeas follows:

IDRDS3 = 0.014 + 0.03 exp[−4440

(Vg

V0

)8]. (3)



0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.0 0.2 0.4 0.6 0.8 1.0

Data

Hueste et al. (1999)

This study, Eq. (3)

0V

Vg

IDRDS3

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.0 0.2 0.4 0.6 0.8 1.0

Data

Hueste et al. (1999)

This study, Eq. (3)

0V

Vg

(a) (b)

Fig. 6. (a) Variations of interstorey drift ratio at punching shear failure, IDRDS3, with changesin the level of gravity shear, Vg/V0, in slab-column connections; (b) variations of the dispersionof the IDRDS3 with changes in Vg/V0.

Figure 6(b) shows the variations of the dispersion of IDR’s at which punchingshear failure occurred around IDRDS3, as a function of the gravity load ratio. Itcan be seen that the level of dispersion significantly increases as the level of gravityshear ratio increases. Points shown in this graph were obtained using a movingwindow analysis with a gravity load ratio width of 0.2 moving at increments of0.05. This variation of dispersion with changes with the level of gravity load ratiowas approximated as follows:

σLn IDRDS3 = 0.62 − 0.4 exp[−11.6

(Vg

V0

)2.9]. (4)

Figure 6(b) also shows the variation of dispersion computed with Eq. (4), where itcan be seen that this equation captures relatively well the variation of the dispersionparameter as a function of the gravity shear ratio.

Figure 7(a) shows the fragility surface resulting from the use of Eqs. (1), (3)and (4). Figure 7(b) shows fragility functions for the third damage state for slab-column connections with gravity shear ratios equal to 0.1, 0.3 and 0.5. Comparisonof Figs. 5 and 7(b) shows that computing the two parameters of the lognormaldistribution as a function of the gravity shear ratio using Eqs. (3) and (4) leads tomuch better estimations of the probability of experiencing or exceeding a damagecorresponding to a punching shear failure. For example, in a slab-column connec-tion with a vertical shear ratio of 0.1 subjected to an interstorey drift ratio of 0.02,the probability of experiencing or exceeding a punching shear failure is essentially



(a) (b)

Fig. 7. Proposed fragility surfce to account for high levels of specimen-to-specimen variabilityfor the third damage state: (a) 3-dimensional presentation; (b) 2-dimensional presentation.

zero, while if the effect of the gravity load ratio is neglected (as done in Fig. 5),this probability would be estimated as 46%, hence, significantly overestimating thefragility of the connection. Similarly, if a slab-column connection with a verticalshear ratio of 0.5 is subjected to the same level of drift, there is a very high prob-ability of experiencing or exceeding a punching shear failure, 75%, whereas if theeffect of the gravity load ratio is neglected, as done in Fig. 5, this probability wouldbe estimated as 46%, underestimated by 29%.

In order to obtain an estimate of the probability of experiencing LVCC, foreach specimen that was subjected to further cyclic loading after the punching shearfailure was observed, the following ratio was computed

Γ =IDRDS4

IDRDS3, (5)

where IDRDS4 is the interstorey drift ratio at which the test was stopped in spec-imens where the post-punching shear failure behaviour was studied. This ratio ofdrifts can be interpreted as an amplification factor acting on the drift correspond-ing to punching shear failure in order to provide a conservative estimate of thedrift at which LVCC occurs. Figure 8 shows a cumulative frequency distributionof Γ, P (Γ <γ), corresponding to the 18 specimens in which post-punching shearfailure was studied. Also shown in this figure is a lognormal fit of this parameterwith median equal to 1.23 and logarithmic standard deviation of 0.2, and 10% sig-nificance levels of the Kolmogorov-Smirnov goodness-of-fit test. As shown in thisfigure all data points lie within the confidence bands, indicating that it is adequateto assume that this ratio is also lognormally distributed.



Fig. 8. Lognormal cumulative distribution function fitted to the cumulative frequency distribu-tion of γ.

The interstorey drift ratio at which LVCC occurs is assumed to be lognormallydistributed with median and logarithmic standard deviations given by

IDRDS4 = 1.23 · IDRDS3, (6)

σLn IDRDS4 =√

σ2Ln IDRDS3

− 0.3σLn IDRDS3 + 0.04. (7)

The value of −0.3 in Eq. (7) corresponds to the correlation between the driftat which punching shear failure was reported and the amplification factor Γ.Figure 9(a) presents the fragility surface corresponding to the LVCC damage statecomputed with Eqs. (1), (6), and (7). Figure 9(b) shows fragility functions corre-sponding to LVCC for slab-column connections with vertical shear ratios of 0.1, 0.3and 0.5. It can be seen that if the vertical shear ratio has a low value such as 0.1,LVCC is not likely to occur provided that IDR’s are smaller than 3.8%. However,if one wants to avoid the possibility of having LVCC at high levels of gravity shearratio, such as 0.5, the IDR needs to be limited to 0.6%.

As can be seen in Fig. 9 estimating the two parameters of the lognormal dis-tribution for DS 4 as a function of the gravity shear ratio leads to much betterestimations of the probability of experiencing LVCC. Furthermore neglecting theeffect of gravity load can lead to results that in some cases are too conservativewhile in other are unconservative. For example, in a slab-column connection with avertical shear ratio of 0.1 subjected to an IDR of 0.04, the probability of experienc-ing a loss of vertical carrying capacity is approximately 2%. However, if the effectof the gravity load ratio is neglected this probability would be estimated as 55%which is significantly larger. Similarly, if a slab-column connection with a verticalshear ratio of 0.5 is subjected to the same level of IDR, there is a 97% probability



(a) (b)

Fig. 9. Fragility surface developed for the loss of vertical carrying capacity damage state inslab-column connections; (a) 3-dimensional presentation, (b) 2-dimensional presentation.

of experiencing LVCC, whereas if the effect of gravity load ratio is neglected thisprobability would be 55%, underestimated by 42%.

The drift level at which LVCC damage state occurs in a slab-column connectionis not only influenced by the level of gravity load, but also by whether or not theslab has longitudinal steel reinforcement that is continuous through the columnreinforcement cage. In particular, slab-column connections with continuous bottomreinforcement may resist significant vertical loads following initial punching failureof the slab which leads to a significant increase in the drift capacity associatedwith experiencing LVCC. Column 8 in Table 1 summarises the information onwhether the slab bottom reinforcement in the various specimens passed throughthe column reinforcement cage. It can be seen that of 11 specimens that did notcontain any continuous bottom reinforcement, post-punching behaviour was onlystudied in 4 specimens. Results from these 4 specimens indicate that, even whenthere is no continuous bottom reinforcement, the connection does not loose itsvertical carrying capacity immediately after experiencing punching shear failure,since top slab reinforcement, though less reliable and efficient than bottom slabreinforcement, is capable of resisting vertical load following initial punching [Panand Moehle, 1992]. For example, even though specimen 69 had no bottom slabreinforcement passing through the column, the test was stopped at 6% drift ratiowithout experiencing LVCC, which corresponds to a drift 28% larger than the oneat which the punching shear failure occurred. Since the maximum drift shown inTable 2 corresponds to the drift at which testing was stopped and not to the driftat which LVCC occurred, results of the four specimens with no continuous bottomreinforcement do not show a clear trend of decreased drift capacity compared to



that of the specimens with continuous bottom reinforcement. However, this shouldnot be interpreted as the bottom reinforcement not increasing the drift at whichLVCC occurs, and simply that the test setup, or loading protocols did not permitthis to be observed in the laboratory. As mentioned before, drifts associated to DS 4

provide a conservative estimate of the probability of experiencing a LVCC becausethis damage state was not observed in any of the specimens.

5. Probability of Being at Each Damage State

Fragility functions developed for a slab-column connection can be used to estimatethe probability that the connection is at a specific damage state when it is subjectedto a certain level of IDR. This probability is a primary input when performingloss estimation in buildings [Miranda and Aslani, 2003] and can be estimated asthe difference between fragility functions corresponding to two consequent damagestates as follows

P (DS = ds i|IDR = idr )

=

1 − P (DS ≥ dsi+1|IDR = idr) i = 0P (DS ≥ dsi|IDR = idr) − P (DS ≥ ds i+1|IDR = idr ) 1 ≤ i ≤ m

P (DS < dsi|IDR = idr) i = m

, (8)

where i = 0 corresponds to the state of no damage in the component, P (DS ≥dsi|IDR = idr) is the fragility function for the ith damage state of a componentwhich is computed form Eq. (1), and m is the number of damage states defined forthe component, and for the case of slab-column connections is 4.

In order to apply Eq. (8) to a specific slab-column connection, it is first requiredto estimate fragility curves corresponding to all the damage states defined in theconnection. Figures 10(a) and 10(c) present fragility functions corresponding to thefour damage states defined for slab-column connections at two levels of gravity shearratio, e.g. at two different locations in a building. Fragility functions presented inthese figures can then be used in Eq. (8) to estimate the probability of being ateach damage state as a function of the level of IDR in the slab-column connection.Figures 10(b) and 10(d) show the probability of being at each damage state for slab-column connections with gravity shear ratios equal to 0.2 and 0.5, respectively. Ascan be seen in Fig. 10(b), for a connection with a gravity load ratio of 0.2 and at 1%drift, the probability that the connection is in the first damage state is 40% whilethere is a 58% probability for the connection to be in the second damage state.The probability that the connection has not experienced any physical damage is,therefore, 2%. At 5% drift, however, the connection for sure has experienced physicaldamage and there is a 30% probability that the connection is in the second damagestate, 35% to be in the third damage state and 35% to be in the last damage state.As the level of gravity shear ratio increases, punching shear failure and LVCCdamage states occur at significantly lower levels of interstorey drift ratio, as can beseen in Fig. 10(d).



0.0

0.2

0.4

0.6

0.8

1.0

0.000 0.025 0.050 0.075 0.

IDR

DS1 DS2

DS3 DS4

P(DS ds i | IDR)

0.2V

V

0

g =

0.0

0.2

0.4

0.6

0.8

1.0

0.000 0.025 0.050 0.075 0.100

IDR

DS1 DS2

DS3 DS4

P(DS ds i | IDR)

0.2V

V

0

g =

0.0

0.2

0.4

0.6

0.8

1.0

0.000 0.025 0.050 0.075 0.100

IDR

DS1 DS2

DS3 DS4

P(DS≥ds i | IDR)

0.2V

V

0

g =

0.0

0.2

0.4

0.6

0.8

1.0

0.000 0.025 0.050 0.075 0.100

IDR

DS1

DS2

DS3

DS4

P(DS=ds i | IDR)

0.2V

V

0

g =

0.0

0.2

0.4

0.6

0.8

1.0

0.000 0.025 0.050 0.075 0.100

IDR

DS1

DS2

DS3

DS4

P(DS=ds i | IDR)

0.2V

V

0

g =

0.0

0.2

0.4

0.6

0.8

1.0

0.000 0.025 0.050 0.075 0.100

IDR

DS1

DS2

DS3

DS4

P(DS=ds i | IDR)

0.2V

V

0

g =

(a) (b)

0.0

0.2

0.4

0.6

0.8

1.0

0.000 0.025 0.050 0.075 0.100

IDR

DS1 DS2

DS3 DS4

P(DS ds i | IDR)

0.5V

V

0

g =

0.0

0.2

0.4

0.6

0.8

1.0

0.000 0.025 0.050 0.075 0.100

IDR

DS1 DS2

DS3 DS4

P(DS ds i | IDR)

0.5V

V

0

g =

0.0

0.2

0.4

0.6

0.8

1.0

0.000 0.025 0.050 0.075 0.100

IDR

DS1 DS2

DS3 DS4

P(DS≥ds i | IDR)

0.5V

V

0

g =

0.0

0.2

0.4

0.6

0.8

1.0

0.000 0.025 0.050 0.075 0.100

IDR

DS1

DS2

DS3

DS4

P(DS=ds i | IDR)

0.5V

V

0

g =

0.0

0.2

0.4

0.6

0.8

1.0

0.000 0.025 0.050 0.075 0.100

IDR

DS1

DS2

DS3

DS4

P( =ds i | R)

0.5V

V

0

g =

0.0

0.2

0.4

0.6

0.8

1.0

0.000 0.025 0.050 0.075 0.100

IDR

DS1

DS2

DS3

DS4

)

0.5V

V

0

g =

(c) (d)

Fig. 10. (a), (c) Component-specific fragility curves for slab-column connections with gravityshear ratios equal to 0.2 and 0.5, respectively; (b), (d) Probability of being at each damage statefor the two connections computed with Eq. (8).

6. Epistemic Uncertainty on Fragility Functions

In addition to specimen-to-specimen variability, another source of uncertainty thatneed to be considered in estimating the level of damage in slab-column connectionsis epistemic uncertainty. Two sources of epistemic uncertainty were incorporatedin this study. The first source is the uncertainty caused by using fragility functionswhose parameters have been obtained from a limited number of specimens. Thissource of epistemic uncertainty is referred here as finite-sample uncertainty. Thesecond source is the uncertainty produced by the fact that damage observations in



the specimen are only collected at peak values of the loading protocol. Therefore,the drifts reported at a certain damage state in a specimen will typically correspondto the peak drift that was imposed to the loading cycle in which the damage statewas observed. This source of uncertainty is associated with the drift increment inthe loading protocol used to test each specimen.

A quantitative measure for finite-sample uncertainty can be obtained by com-puting confidence intervals of the statistical parameters of the deformation levelscorresponding to each damage state. Different methods can be used to estimatethe confidence intervals of a statistical parameter. If the underlying probabilitydistribution is normal or lognormal, conventional statistical methods can be used[Crow et al., 1960; Benjamin and Cornell, 1970]. For cases in which the underlyingprobability distribution is not normal or lognormal, a generic method of findingthe confidence interval is to use re-sampling techniques, such as bootstrap statistics[Efron and Tibshirani, 1993].

Since it was verified that the fragility curves of a slab-column connection can beassumed lognormal, a conventional approach was used to estimate confidence inter-vals of the median and logarithmic standard deviation of the drifts correspondingto different damage states. The confidence intervals of the median of a lognormallydistributed sample can be approximated as follows [Crow et al., 1960]:

IDR · exp[± zα/2

σLn IDR√n

], (9)

where zα/2 is the value in the standard normal distribution such that the probabilityof a random deviation numerically greater than zα/2 is α, and n is the numberof data points. The number of slab-column specimens used to develop fragilityfunctions are given in the last column of Table 3.

Confidence intervals for the logarithmic standard deviation of the data are non-symmetric and can be computed as [Crow et al., 1960]:

[(n − 1)σ2

Ln IDR

χ2α/2,n−1

]1/2

and[(n − 1)σ2

Ln IDR

χ21−α/2,n−1

]1/2

, (10)

where χ2α/2,n−1 is the inverse of the χ2 distribution with n − 1 degrees of freedom

and a probability of occurrence of α/2, similarly, χ2α/2,n−1 is the inverse of the χ2

distribution with n−1 degrees of freedom and a probability of occurrence of 1−α/2.For each damage state, lower and upper confidence intervals of IDR and σLn IDR

were estimated using Eqs. (9) and (10). These confidence intervals were used tobuild a confidence band on the original fragility functions.

A quantitative measure of the second source of epistemic uncertainty wasobtained by identifying the drift increment used in the loading protocol in the cycleat which each damage state occurred in each specimen. Once these drift incrementswere collected, a bias and dispersion on the empirical median drift capacity wasestimated by assuming that the median capacity is a normally distributed random



variable. Finally, the two sources of epistemic uncertainty were combined assumingthat they are uncorrelated.

Figure 11 presents the effects of epistemic uncertainty on the fragility curvesfor each damage state. The black line in each graph corresponds to the fragilitycurve in the absence of epistemic uncertainty, while the grey bands correspond to a90% confidence band on the fragility curves when both sources of epistemic uncer-tainty have been considered. The corresponding parameters are listed in Table 4.Please note that the second source of uncertainty in addition to increasing the

P(DS≥ds1|IDR)

0.00.10.20.30.40.50.60.70.80.91.0

0.000 0.003 0.006 0.009 0.012 0.015IDR

y

envelope

Originalfragilit

Epistemicuncertainty

P(DS≥ds2|IDR)

0.00.10.20.30.40.50.60.70.80.91.0

0.000 0.005 0.010 0.015 0.020 0.025IDR

Originalfragility

Epistemicuncertaintyenvelope

(a) (b)

P(DS≥ds3|IDR)

0.00.10.20.30.40.50.60.70.80.91.0

0.00 0.02 0.04 0.06 0.08 0.10IDR

0.2V

V

0

g =

Originalfragility


P(DS≥ds3|IDR)

0.00.10.20.30.40.50.60.70.80.91.0

0.00 0.02 0.04 0.06 0.08 0.10IDR

0.5V

V

0

g =

Originalfragility


(c) (d)

Fig. 11. Incorporating epistemic uncertainty to fragility functions of slab-column connections;(a) Damage state 1, (b) Damage state 2, (c), (d) Damage state 3 With Vg/V0 = 0.2 and 0.5,(e), (f) Damage state 4 with Vg/V0 = 0.2 and 0.5.



P (DS4≥ds4| IDR)

0.00.10.20.30.40.50.60.70.80.91.0

0.00 0.02 0.04 0.06 0.08 0.10 0.12IDR

0.2V

V

0

g =

Originalfragility


P (DS4≥ds4| IDR)

0.00.10.20.30.40.50.60.70.80.91.0

0.00 0.02 0.04 0.06 0.08 0.10IDR

Originalfragility


0.5V

V

0

g =

(e) (f)

Fig. 11. (Continued)

confidence band on the median drift also decreases the drift capacity (i.e. shiftsthe fragility curve to the left) As shown in these graphs, the influence of epistemicuncertainty is significant and needs to be accounted for when performing sensitivitystudies in loss estimation [Aslani and Miranda, 2004]. For example, a 50% prob-ability of experiencing or exceeding the second damage state can occur anywherefrom 0.61% to 0.95% drift ratio, as can be seen in Fig. 11(b). Furthermore, epis-temic uncertainty can cause important changes in the probability of experiencingor exceeding a damage state. For example, for a slab-column connection with agravity shear ratio equal to 0.5, it can be seen in Fig. 11(f) that at 2% interstoreydrift ratio, probability of experiencing or exceeding LVCC varies anywhere from64% to 95%.

7. Conclusions

Fragility functions that provide a probabilistic estimation of the level of damageexperienced in slab-column connections of non-ductile reinforced concrete buildingshave been presented. In these fragility functions the damage is estimated as a func-tion of the peak interstorey drift imposed on the connection. These new fragilityinformation can be used in estimating damage at the component level and withdamage states associated to specific repair actions. Furthermore, by using resultsfrom experimental studies, the study has identified and quantified the uncertaintiesassociated with estimating damage in slab-column connections. Experimental datafrom 16 investigations conducted in the last 36 years, with a total of 82 slab-columnspecimens was used to develop fragility functions for slab-column connections.



Table

4.

Sta

tist

ical

para

met

ers

esti

mate

dto

inco

rpora

teep

iste

mic

unce

rtain

tyfo

rin

ters

tore

ydri

ftra

tios

corr

e-sp

ondin

gto

the

dam

age

state

sin

slab-c

olu

mn

connec

tions

at

90%

confiden

cein

terv

al.

Para

met

ers

consi

der

ing

only

finit

ePara

met

ers

consi

der

ing

both

sam

ple

epis

tem

icunce

rtain

tyso

urc

esofep

iste

mic

unce

rtain

ty

Dam

age

state

IDR

σL

nID

RID

Rσ

Ln

IDR

(1)

(2)

(3)

(4)

(5)

DS

1:Lig

ht

crack

ing

0.0

033×

1.1

00.3

9+

0.0

90.0

027×

1.3

50.3

9+

0.0

90.0

033×

0.9

00.3

9−

0.0

60.0

027×

0.7

40.3

9−

0.0

6

DS

2:Sev

ere

crack

ing

0.0

09×

1.0

70.2

5+

0.0

70.0

076×

1.2

50.2

5+

0.0

70.0

09×

0.9

30.2

5−

0.0

40.0

076×

0.8

00.2

5−

0.0

4

DS

3:P

unch

ing

shea

rID

R∗ D

S3×

1.1

2σ

Ln

IDR

DS3

+0.1

0(I

DR

DS3−

0.0

024∗∗

)×

1.2

1σ

Ln

IDR

DS3

+0.1

0

failure

IDR

∗ DS3×

0.8

9σ

Ln

IDR

DS3−

0.0

7(I

DR

DS3−

0.0

024∗∗

)×

0.8

2σ

Ln

IDR

DS3−

0.0

7

DS

4:Loss

ofver

tica

lID

R∗ D

S4×

1.1

5σLn

IDR

DS4

+0.1

4(I

DR

DS4−

0.0

032∗∗

)×

1.1

8σLn

IDR

DS4

+0.1

4

carr

yin

gca

paci

tyID

R∗ D

S4×

0.8

7σ

Ln

IDR

DS4−

0.0

8(I

DR

DS3−

0.0

024∗∗

)×

0.8

5σ

Ln

IDR

DS4−

0.0

8

*ID

RD

S3

and

IDR

DS4

are

com

pute

dfr

om

Eq.(3

)and

Eq.(6

),re

spec

tivel

y.**V

alu

es0.0

024

and

0.0

032

corr

espond

toep

iste

mic

unce

rtain

tyca

use

dby

dri

ftin

crem

ents

inth

elo

adin

gpro

toco

luse

dto

test

each

spec

imen

for

dam

age

state

s3

and

4,re

spec

tivel

y.



Two sources of uncertainty were incorporated in the fragility functions:Specimen-to-specimen variability and epistemic uncertainty. Specimen-to-specimenvariability corresponds to the fact that different specimens experience the vari-ous damage states at levels of deformation that in general are different for eachspecimen. The specimen-to-specimen variability was incorporated by developingdrift-based fragility functions at each damage state. It was observed that, for dam-age states involving punching shear failure and loss of vertical carrying capacity,developing fragility functions both as a function of interstorey drift ratio and as afunction of the gravity shear ratio leads to improved estimates of the probability ofexperiencing these damage states compared to damage estimates obtained only asa function of interstorey drift ratio.

Two kind of epistemic uncertainty were considered. The first one is associatedto the fact that the parameters of the fragility functions have been obtained from alimited number of specimens, and the second is associated to the fact that damageobservations in the various specimen were typically collected only at the end ofeach loading cycle and therefore the resulting fragility functions are influenced bythe drift increment used in the loading protocol that was employed to test of eachspecimen. Quantitative measures for each of these two kinds of epistemic uncer-tainty were developed using statistical procedures. Results indicate that in somecases the effects of epistemic uncertainty on the probability of experiencing eachdamage state are significant and therefore should be incorporated in probabilisticdamage assessment.

Acknowledgements

The authors would like to acknowledge the support by the Pacific Earthquake Engi-neering Research (PEER) Center with support from the Earthquake EngineeringResearch Centers Program of the National Science Foundation under Award Num-ber EEC-9701568. The authors also would like to express their gratitude for com-ments and suggestions from the director of the PEER Center, Professor Jack P.Moehle at University of California at Berkeley, and Professor Ian N. Robertson atUniversity of Hawaii at Manoa, during the course of this investigation.

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