Risk-based seismic performance assessment of Yielding Shear Panel Device

Engineering Structures 56 (2013) 1570–1579

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Risk-based seismic performance assessment of Yielding Shear Panel Device Md Raquibul Hossain a, Mahmud Ashraf b,⇑, Jamie E. Padgett c a

School of Civil Engineering, The University of Queensland, St Lucia, QLD 4072, Australia School of Engineering and Information Technology, The University of New South Wales, Canberra, ACT 2610, Australia c Department of Civil and Environmental Engineering, Rice University, 6100 Main Street, MS-318, Houston, TX 77005, USA b

a r t i c l e

i n f o

Article history: Received 31 January 2012 Revised 21 July 2013 Accepted 23 July 2013

Keywords: Bouc–Wen–Baber–Noori (BWBN) model Earthquake Seismic energy dissipation Probabilistic seismic performance Seismic fragility Yielding Shear Panel Device

a b s t r a c t Yielding Shear Panel Devices (YSPDs) have recently been proposed to facilitate passive energy dissipation of building frames during seismic activity and hence protect major structural components from excessive stress. A YSPD is composed of a thin steel diaphragm plate encapsulated within a square hollow steel tube, which is bolted to the structure to utilize the inelastic shear deformation capability of the steel diaphragm plate for energy dissipation and for consequent modification to the structural response. This paper conducts probabilistic performance evaluation to assess the appropriateness of YSPDs given uncertainty in the occurrence and intensity of earthquakes, material strength, stiffness, structural response, etc., and evaluate performance based on size, number and configuration of YSPDs. A fragility analysis is conducted, which identifies the probability of exceeding a structural damage level depending on ground motion intensity, along with a limit state probability analysis to quantify the annual exceedance probability of a specified damage level. A mathematical model to represent YSPDs in a finite element code is developed and the model is used for analyzing a case study steel moment frame with alternative YSPD designs. The study reveals the potential for considerable reduction in the median fragility and annual limit state exceedance probability due to the inclusion of YSPDs through a V-brace system. However, the effectiveness of different YSPD orientations is varied and their relative performance levels are discussed in detail. Overall, the study shows the suitability of YSPD as a passive energy dissipation device and the potential to utilize this device to help achieve performance-based objectives for buildings in seismic zones. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Recent seismic activities around the globe have revealed the need to identify sustainable solutions for reducing the catastrophic impacts of earthquakes. Researchers have introduced a variety of active, semi-active and passive energy dissipation devices to diminish the damaging seismic effects. Passive control mechanisms received attention due to their simplicity in design as well as some additional unique advantages over other control mechanisms. For example, passive control devices are generally more reliable due to their independence of external power, are easy to rehabilitate, economical and less complex. Some commonly used metal yielding passive energy dissipating devices include the

⇑ Corresponding author. Address: Room 127, Building 20, Australian Defence Forces Academy, School of Engineering and Information Technology, The University of New South Wales, Canberra, ACT 2610, Australia. Tel.: +61 2 6268 8334; fax: +61 2 6268 8450. E-mail addresses: [email protected] (M.R. Hossain), mahmud.ashraf@unsw. edu.au (M. Ashraf), [email protected] (J.E. Padgett). 0141-0296/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2013.07.032

added damping and stiffness (ADAS) device [1–3], triangular added damping and stiffness (TADAS) device [4,5], steel plate shear wall (SPSW) [6,7], among others. These metal yielding devices utilise the stable hysteretic response of the constituent materials to dissipate energy. The Yielding Shear Panel Device (YSPD) [8,9] is another recently proposed metal yielding energy dissipating device as shown in Fig. 1. YSPD dissipates energy by taking advantage of its stable hysteretic in-plane shear deformation of the steel diaphragm plate. YSPD is composed of a steel diaphragm plate welded inside a steel square hollow section (SHS). SHS provides the supporting boundary for the diaphragm plate as well as connectivity with the Vbrace and the beam through bolted connection. Previous studies have focused on the device development, numerical modelling and response assessment in a deterministic fashion [9–12]. Uncertainties arising from the occurrence and intensity of earthquakes, material strength and stiffness, structural response and consequences were not considered for the performance evaluation of YSPDs. Probabilistic seismic performance evaluation techniques, such as seismic fragility analysis and limit state probability

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M.R. Hossain et al. / Engineering Structures 56 (2013) 1570–1579 Thickness = T

Thickness = t

S = 50mm

d

D

Elevation

Top View

Fig. 1. Yielding Shear Panel Device (YSPD) [5].

analysis, consider these uncertainties and have been used to evaluate the performance of various seismic retrofit techniques for buildings and bridges [13,14]. This paper uses a probabilistic approach to assess the performance of YSPDs using a benchmark structure to provide insights into the relative performance of an as-built and YSPD retrofitted structure. Uncertainties associated with seismic intensity and structural responses are considered for probabilistic performance evaluation of YSPDs. Probabilistic tools such as fragility curves and point estimates of risk of damage are used to uncover the impacts of these uncertainties and subsequently identify the performance of YSPDs for different design configurations. Obtained results provide some useful insight into changes in the probabilistic performance of a structure as a result of change in the device size and their relative positioning within the structure. It is worth noting that for other building configurations the probabilistic performance may vary based on the building design and the arrangement of YSPDs. 2. Structural model for seismic performance assessment The moment resisting frame of the Los Angeles (LA) three storey SAC model structure, designed for the SAC Phase II Steel Project [15,16] has been used by many researchers for performance evaluation of various seismic control devices [17–20]. The four bayed

North–South lateral load-bearing moment resisting frame of this benchmark structure is chosen to evaluate the performance of YSPD. Fig. 2 shows the floor plan and the moment resisting frame of the three storey SAC model structure. Ohtori and Spencer [16] provided a detailed description of the structural design, loadings and evaluation model for this benchmark structure. The fundamental frequency of the first mode was found to be 0.99 Hz and spectral acceleration (Sa) for a time period of T1 = 1.0 s is used as the earthquake intensity measure. The North–South lateral load-bearing frame of the three storey Los Angeles SAC structure is modelled as a 2-dimensional frame using Opensees [21]. The slab system is assumed sufficiently stiff for preventing the lateral movement in the normal direction of the frame. The columns and the far end simply supported beams are modelled as elastic beam–column elements. Zero-length springs with negligible stiffness are used as the connecting elements between the column joints and simply supported beams. Fixed supported beams are modelled as elastic beams with hinges. The plastic hinge is modelled by concentrated plasticity over a hinge length of 10% beam length at the end of all beams [20]. A uni-axial bilinear strain hardening material (1% kinematic hardening with an yield strength of 345 MPa) is used to model the plastic hinge as a fibre section [10]. The inherent damping is modelled as Rayleigh damping with a damping ratio of 2% and has been

4 @ 30ft

Rigid Link

Hinge Connection

6 @ 30ft

13ft

13ft

13ft

(a)

(b)

Fig. 2. (a) Floor plan of the Los Angeles three storey SAC model structure and (b) analytical model of the North–South lateral load-bearing moment resisting frame [11,12].

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element with large axial and bending stiffness and a hinge with negligible stiffness is introduced at the beam ends to simulate similar moment release as rigid links. The stiffness contributions of the gravity columns are neglected whilst the P–D effects are considered by following the above mentioned procedure.

Table 1 Member sizes of the North–South lateral load-bearing frame [12]. Storey

1 2 3

Columns

Beams

With one fixed beam

With two fixed beams

Other column (weak axis)

Fixed ends

Hinged ends

W14  257 W14  257 W14  257

W14  311 W14  311 W14  311

W14  68 W14  68 W14  68

W33  118 W30  116 W24  68

W21  44 W21  44 W21  44

assigned to the first and the third mode of the frame [22]. Table 1 summarizes the beam and the column sections of the North–South lateral load-bearing frame. Gravity load of the entire structure is 10,142 kN for the top floor whilst 9376 kN for the other floors [15]. Each lateral load bearing frame carries half of this seismic load as the interior frames are considered as gravity frames. The seismic load is applied to the frame as a combination of distributed load and point load. A distributed load is applied to the beams calculated from the tributary area of the adjacent East–West Bay. Remaining seismic load is applied to a dummy gravity column (known as ‘leaning column’) as point load for considering the P–D effect. The dummy column is pin connected with the lateral force resisting frame using rigid links. A rigid link considers the translational constrains between the two connected nodes and it is moment released at its ends. The dummy column is modelled as an elastic beam–column

3. YSPDs within the moment resisting frame Hossain and Ashraf [12] used the Bouc–Wen–Baber–Noori (BWBN) model [23] to represent the pinching hysteretic force deformation relationship of YSPDs and provided closed-form relationships among the physical parameters and model parameters. Haukaas and Der Kiureghian [24] implemented the BWBN model in Opensees [21] as a uni-axial material model excluding the pinching effect. This existing code is modified to incorporate the pinching effect herein for probabilistic seismic performance assessment of YSPDs. A detailed explanation of the BWBN model and incremental algorithm for Opensees implementation is presented in Appendix A. YSPDs are modelled as spring elements connected between the beam and the V-brace. The BWBN material model is used as the constituent material model for these spring elements. YSPDs are installed in the North–South lateral load-bearing moment resisting frame of the SAC three storey building for improving its seismic performance. YSPDs are installed in the each storey of one interior bay as shown in Fig. 3 (Case 1) and all moment resisting bays as shown in Fig. 3 (Case 2). Three different

Rigid Link

Hinge Connection

YSPD

13ft

13ft

13ft

30ft

30ft

30ft

30ft

Case 1 YSPD

Rigid Link

Hinge Connection

13ft

13ft

13ft

30ft

30ft

30ft

30ft

Case 2 Fig. 3. Analytical model of the North–South lateral load-bearing moment resisting frame equipped with YSPDs.

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M.R. Hossain et al. / Engineering Structures 56 (2013) 1570–1579 Table 2 BWBN model parameters for the YSPDs [7]. YDPD (D  T  t)

fy (MPa)

kt (kN/mm)

Fi (kN)

A

b

c

n

q

f1o

p

w0

dw

k

100  4  2 110  5  3 120  6  4

250 300 350

0.33 0.42 0.49

26.76 54.22 93.51

1.0 1.0 1.0

0.5 0.5 0.5

0.5 0.5 0.5

1.213 0.544 0.300

0.52 0.38 0.30

0.96 0.95 0.95

0.018 0.015 0.012

0.41 0.27 0.22

0.00001 0.00001 0.00001

0.0300 0.0014 0.0002

D is the Size of YSPD (mm). T is the thickness of SHS plate (mm). t is the thickness of diaphragm plate (mm). fy is the yield strength of SHS and diaphragm plates (MPa). kt is the tangential stiffness of YSPD after tension field formation (kN/mm). Fi, A, b, c, n are hysteretic parameters and q, f1o, p, w0, dw, k are pinching parameters (See Appendix A for detail description).

120

4. Probabilistic seismic performance evaluation Probabilistic seismic performance evaluation identifies the response of a structure considering the uncertainties associated with the seismic events and the subsequent structural responses. The limit state, which is denoted by damage state (DS), probability of the seismic risk assessment is defined by Eq. (1) assuming seismic intensity demand (Q) and structural capacity (R) as random variables [25].

Force (kN)

60

0

PLS ¼

-60 YSPD 100X4X2

YSPD 120X6X4

-120 -12.5

0

12.5

ð1Þ

x

YSPD 110X5X3

-25

X P½DSjQ ¼ x  P½Q ¼ x

25

Displacement (mm) Fig. 4. Cyclic force displacement relationship of the YSPDs generated using the BWBN models proposed by Hossain et al. [7].

Table 3 Size of the braces used in the current study. YSPD

Steel Brace

YSPD 100  4  2 YSPD 110  5  3 YSPD 120  6  4

HSS4  4  1/8 HSS4  4  1/4 HSS4  4  1/2

sizes of YSPDs are considered for the evaluation of seismic performance. Table 2 summarizes the BWBN model parameters calculated from the closed form equations recently proposed by Hossain and Ashraf [12] and Fig. 4 shows the force displacement relationships for the corresponding YSPDs. Cold-formed welded hollow structural section (HSS) made of ASTM A500 Grade B steel is used as the V-braces for connecting the YSPDs at the middle of the beam. The braces are designed to remain elastic by considering a design force of 2Fi and the brace sizes are summarized in Table 3. The inclusion of the YSPD will alter the structural response of the parent frame since the device will introduce hysteretic damping and some stiffness [11]. The seismic energy will be dissipated through the inelastic shear deformation of the diaphragm plate. The added stiffness by the V-brace system will increase the storey elastic stiffness until the YSPD response becomes inelastic. As YSPD’s response becomes inelastic quickly after experiencing small deformation, the added initial stiffness will not significantly alter the structural response. After becoming inelastic, YSPD dissipates energy through hysteretic damping connected between the beam and the V-brace system as they provide necessary support condition. This hysteretic damping will significantly reduce the seismic response of the structure.

where P[DS|Q = x] represents the seismic fragility of the structure and P[Q = x] represents the seismic hazard. Seismic fragility incorporates uncertainties associated with structural response, which is affected by the response modification device or retrofit. Seismic hazard estimates, on the other hand, are characterized from probabilistic seismic hazard analyses and are often represented on the basis of mean earthquake occurrence rates. Combination of this information yields estimates of probabilities of different levels of damage state risks. Appropriate definition of damage state measure and earthquake intensity measure is one of the vital steps for probabilistic seismic assessment. Researchers proposed several displacement-based, energy-based and hybrid damage measures for structures subjected to earthquake but there is no specific guideline to choose the most appropriate damage measure [26]. FEMA 356 [27] suggests using the maximum drift ratio for assessing the structural performance levels and the corresponding damage to structural components. The inter-storey drift ratio (h) is used as the damage measure in the present study by following the FEMA guideline. h is measured as the ratio of the relative displacement between the adjacent floors and the storey height. FEMA 356 [27] proposed three structural performance levels, i.e. Collapse Prevention (CP), Life Safety (LS) and Immediate Occupancy (IO) performance levels with corresponding maximum allowable drift ratios of 5%, 2.5% and 0.7%. Spectral acceleration (Sa) at the fundamental period of structure (T1 = 1.0 s for SAC moment frame) is considered as the earthquake intensity measure for the current study. The fragility Fr(x) is estimated as a lognormal distribution indicating probability of exceedance of different damage stages by Eq. (2) and the seismic hazard H(x) is approximated using Eq. (3), which offers the annual exceedance probability of a specific level of earthquake intensity [25].

h  i S a =bR F r ðxÞ ¼ U ln x  ln b

ð2Þ

HðxÞ ¼ k0 xk

ð3Þ

where b S a is the median value of the fragility of the structure in units of Sa, bR is the lognormal standard deviation of the system fragility, U is the standard normal cumulative distribution function, k0 and k

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Table 4 Selected ground motion records used in the current study. PGMD record no.

Earthquake name

Year

1838

Hector Mine

1999

1153

Kocaeli, Turkey Chi-Chi, Taiwan Denali, Alaska Imperial Valley-06 Loma Prieta

2112 169 804

Earthquake magnitude

Distance to rupture (km)

7.13

62.9

1999

Whitewater Trout Farm Botas

7.51

127.1

1999

TCU076

7.62

2.8

2002

TAPS Pump Station #08 Delta

7.90

104.9

6.53

22.0

So. San Francisco, Sierra Pt. Mission Creek Fault Tabas Brawley Airport Auletta

6.93

63.1

1979 1989

880

Landers

1992

143 719

Tabas, Iran Superstition Hills-02 Irpinia, Italy-01

1978 1987

284

1980

Spectra Acceleration (g)

1

0.5

0 0

7.28

27.0

7.35 6.54

2.1 17.0

6.90

9.6

0.1

0.2

0.3

θmax Fig. 6. IDA curve of YSPD 100  4  2 (Case 2) for Tabas Earthquake (PGMD No. 143).

5 Mean Spectrum (Geometric)

4

1.5

Sa (g)

Station name

Normalized Median θ max (%)

1511

2

Design Spectrum (ASCE/SEI-7-05)

3

-3.3% (0.32)

100.0 -11.0% (0.30)

75.0

-6.0% (0.32) -18.0% (0.29)

-9.1% (0.31) -23.6% (0.29)

50.0 25.0

Case 1 0.0 Case 2

YSPD 100X4X2

2

YSPD 110X5X3 Posted Values on Bar Chart: Drift demand reduction% (βD|Sa)

1

YSPD 120X6X4

Fig. 7. Reduction in inter storey drift demands (%) and dispersion of seismic demand (bD|Sa) for ground motions with design hazard level of LA.

0 0

1

2

3

Time Period (s) Fig. 5. Response spectrum of the scaled ground motion records and the design response spectrum at downtown Los Angeles for the site class D (stiff soil).

are constants that depend on the site of the building. The dispersion parameter bR reflecting uncertainties associated with seismic demand and structural capacity is calculated as following equation [25],

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bR ¼ b2DjSa þ b2c

ð4Þ

where uncertainty in seismic demand bDjSa is represented by the dispersion in hmax and uncertainty in structural capacity bc depends on different structural damage states. bc is set equal to 0.25 for IO and LS limit states, whilst 0.15 for CP limit state [25,28]. The limit state annual exceedance probability is approximately defined by integrating the fragility and seismic hazard as shown in following equation [25].

" PLS ¼ k0 b S k a exp

ðkbR Þ 2

2

# ð5Þ

where k0 b S k a represents the seismic hazard with zero dispersion and the exponential term is a correction factor for considering the variability of seismic demand and structural capacity. Seismic fragility analysis and limit state probability analysis of the SAC three storey building with and without installing YSPDs will identify the reduction in fragility and limit state probability

Table 5 Seismic demand statistics and median fragility values for frames with and without YSPDs. YSPDs

bDjSa

No YSPD YSPD 100  4  2 YSPD 110  5  3 YSPD 120  6  4 YSPD 100  4  2 YSPD 110  5  3 YSPD 120  6  4

0.34 0.35 0.34 0.34 0.32 0.30 0.29

(Case (Case (Case (Case (Case (Case

1) 1) 1) 2) 2) 2)

b S a ðgÞ

bR IO

LS

CP

IO

LS

CP

0.42 0.43 0.42 0.42 0.41 0.39 0.38

0.42 0.43 0.42 0.42 0.41 0.39 0.38

0.37 0.38 0.37 0.37 0.35 0.34 0.33

0.12 0.13 0.15 0.16 0.17 0.19 0.21

0.49 0.52 0.53 0.55 0.55 0.60 0.64

0.90 0.92 0.94 0.97 1.02 1.03 1.08

by considering aforementioned uncertainties. YSPDs of different sizes and combinations are considered for the risk-based seismic performance evaluation. The methodology provides necessary guidelines for risk-based retrofit design for structures. A series of nonlinear time history analysis (NTHA) known as incremental dynamic analysis (IDA) [29] is carried out with a range of ground motion records for the probabilistic performance evaluation. Ten scaled earthquake records are chosen from the PEER Ground Motion Database (PGMD) [30] based on spectral matching with the design spectrum provided by ASCE/SEI-7-05 design code [31]. PEER Ground Motion Database searches and identifies historical earthquake records that match with the seismic design spectrum for a

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1

1 Life Safety

Probability of Exceedance

Probability of Exceedance

Immediate Occupancy 0.75

0.5

0.25

0

0.75

0.5

0.25

0 0

0.1

0.2

0.3

0.4

0.5

0

0.3

0.6

Sa (g)

0.9

1.2

1.5

Sa (g)

1

Probability of Exceedance

Collapse Prevention 0.75

0.5

0.25

0 0

0.5

1

1.5

2

Sa (g) Fig. 8. Fragility curves for the North–South lateral load-bearing frame of the SAC three storey LA building with and without YSPDs.

Annual Probability of Exceedance, H(Sa)

1.0E-01

Table 6 Annual exceedance probability of different performance limit states (PLS) with and without YSPDs.

-k H(Sa) = k0 Sa

1.0E-02

k0 = 3.03×10-4

PLS for different performance limit states

k = 2.69

IO

LS

CP

1/6 1/7 1/10 1/13 1/15 1/22 1/29

1/250 1/290 1/310 1/350 1/360 1/480 1/585

1/1550 1/1560 1/1680 1/1860 1/2206 1/2360 1/2760

No YSPD YSPD 100  4  2 YSPD 110  5  3 YSPD 120  6  4 YSPD 100  4  2 YSPD 110  5  3 YSPD 120  6  4

1.0E-03

1.0E-04

1.0E-05 0.0

0.5

1.0

1.5

(Case (Case (Case (Case (Case (Case

1) 1) 1) 2) 2) 2)

the design spectrum). The IDA curve of YSPD 100  4  2 (Case 2) for Tabas Earthquake (PGMD No. 143) is presented in Fig. 6.

Sa (g) Fig. 9. Annual spectral hazard curve of LA Downtown (T1 = 1.0 s and 2% damping).

5. Seismic performance evaluation of YSPDs 5.1. Performance based on seismic drift demand assessment

structure. Ten earthquake records identified by PGMD are scaled through scalar multiplication to match the target spectrum at the time period of T1 = 1.0 s. The building is assumed to be located on stiff soil (site class D based on ASCE/SEI-7-05). Table 4 summaries the list of earthquake records used for IDA and Fig. 5 shows the fault normal and the fault parallel spectral acceleration spectrum of these earthquakes (scaled to match the mean spectrum with

The drift demand for design ground motion intensity for LA (Sa = 0.72 for T1 = 1 s) are compared to identify the reduction for various size and configurations of YSPDs and plotted in Fig. 7. A significant reduction in the seismic drift demand is observed after introducing YSPDs. The dispersion of seismic demand (bDjSa ) is also shown for these configurations. The minimum median demand reduction of 3.3% is attained by YSPD 100  4  2 (Case 1) and a maximum reduction of 23.6% of the seismic drift demand is

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achieved by installing YSPD 120  6  4 (Case 2). The dispersion of seismic demand (0.33 for the frame without YSPDs) is also reduced to 0.32 and 0.29 respectively, although this impact on dispersion is rather minimal. The reductions in the demands in particular suggest the applicability of YSPD as a passive seismic control device that can improve the probabilistic structural performance. It is worth mentioning that larger YSPDs have shown greater reduction

Table 7 Seismic hazard parameters k and k0 (T1 = 1 s and 2% damping) [21]. Site

k

k0

Charleston, SC Memphis, TN Seattle, WA Los Angeles, CA

0.81 1.00 2.14 2.69

2.66  104 1.85  104 1.44  104 3.03  104

Annual Probability of Exceedance, H(ds)

Immediate Occupancy Limit State 2.00E-03

2.00E-03

2.50E-02

2.00E-01

1.00E-03

1.00E-03

1.25E-02

1.00E-01

0.00E+00

0.00E+00

0.00E+00

Charleston, SC

Memphis, TN

0.00E+00

S ea t t l e , W A

L o s A n g el e s , C A

Annual Probability of Exceedance, H(ds)

Life Safety Limit State 6.00E-04

5.00E-04

1.25E-03

5.00E-03

3.00E-04

2.50E-04

6.25E-04

2.50E-03

0.00E+00

0.00E+00

0.00E+00

Charleston, SC

Memphis, TN

0.00E+00

Seattle, WA

Los Angeles, CA

Annual Probability of Exceedance, H(ds)

Collapse Prevention Limit State 3.50E-04

3.50E-04

3.50E-04

7.00E-04

1.75E-04

1.75E-04

1.75E-04

3.50E-04

0.00E+00

0.00E+00

Charleston, SC

0.00E+00

Memphis, TN

0.00E+00

Seattle, WA

Los Angeles, CA

Fig. 10. Annual performance limit state exceeding probability (PLS) for regions of moderate seismicity and high seismicity. (Note: scales vary per limit state and site.)

M.R. Hossain et al. / Engineering Structures 56 (2013) 1570–1579

due to the higher elastic stiffness and larger energy dissipation capability. The configuration of YSPDs applied to all bays produce more favourable results than the configuration of applying YSPDs in a single bay. 5.2. Performance based on fragility analysis Seismic fragility analysis determines the exceedance probability of a damage state for a structure against earthquakes of a particular intensity. hmax is chosen as the damage state indicator for fragility analysis and three drift limits (i.e., 0.7%, 2.5% and 5.0%) are defined based on FEMA performance levels, i.e. IO, LS and CP levels. The uncertainties associated seismic demand and structural capacity is considered for the fragility analysis, whilst uncertainties of material strength and stiffness is ignored for their insignificant role [25]. Table 5 summarizes the values of bDjSa and bR for frames considered with and without YSPDs. The median fragility ð Sba ) in terms of spectral acceleration for frames with and without YSPDs is calculated based on Eq. (2). Table 5 summarizes the median fragility values whilst the fragility curves are shown in Fig. 8 for each performance levels. Significant reduction in fragility is observed after introducing YSPDs. One measure of the improvement of the performance is revealed by the increase of the median value of the frame’s fragility. The median value of fragility is increased from 0.49 g to 0.52 g by introducing YSPD 100  4  2 (Case 1), whilst the value is further increase to 0.64 g for YSPD 120  6  4 (Case 2) for the life safety performance level. Similar increments are also observed for immediate occupancy and collapse prevention levels. The improved performance of the North–South lateral load-bearing frame of the SAC three storey LA building in terms of fragility after introducing YSPDs is a clear evidence of the appropriateness of YSPDs as seismic energy dissipating device. YSPDs of larger size and thickness have a higher initial stiffness, yield strength and larger energy dissipation capability. As a result, bigger YSPDs show better seismic performance (Fig. 4). The larger sized YSPD has more energy dissipation potential which is well demonstrated as the installation YSPD 120  6  4 increases the median value of fragility more compared with same number of YSPD 100  4  2. Regardless of YSPD size, The dispersion in the frame fragility is also reduced for YSPDs in all moment resisting bays by limiting the drift demand but the reduction is insignificant when YSPDs are installed in a single bay. 5.3. Performance based on limit state exceedance probability A seismic hazard curve provides the annual exceedance probability of earthquakes with a particular intensity. Hazard curves for fundamental period of 0, 0.2 and 1.0 s and 5% damping are readily available from USGS [32]; and the seismic hazard curve for 2% damping may be easily generated using available information. The hazard curve (T1 = 1.0 s and 2% damping) for LA downtown is shown in Fig. 9 by fitting a line (defined by the parameters k and k0) to the points representing annual probability of exceedance for different spectral accelerations (Sa). The annual probability of exceedance of drift demand corresponding to different performance limit states for the SAC frame equipped with or without YSPDs is calculated based on Eq. (5) and summarized in Table 6. The annual exceedance probability of life safety limit state, or damage state, decreased from 1/250 to 1/290 for YSPD 100  4  2 (Case 1) and to 1/585 for YSPD 120  6  4 (Case 2). Similar reductions are also observed for immediate occupancy and collapse prevention limit states. This reduction in damage state exceedance probability indicates the capability of YSPDs for use as an effective seismic control device.

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Seismic hazard characteristics significantly vary between highseismic zones and regions of moderate seismicity as the hazard curves in moderate-seismic zone are relatively flat with a lower k value compared with steeper hazard curves of high-seismic zones [25]. Four sites are chosen in the current study to identify the effect of seismic hazard on the performance of YSPDs; two of them are from moderate-seismic zone (Charleston, SC and Memphis, TN) and the other two are from high-seismic zones (Seattle, WA and Los Angeles, CA). Seismic hazard parameters for these cities are summarized in Table 7. Fig. 10 shows the limit state annual exceedance probability of SAC frame for different performance limits of these sites. Reduction of the limit state probability is higher for high-seismic zones when compared with moderate-seismic zones for increasing size and number of YSPDs. The higher value of k indicates a higher reduction in annual exceedance probability of all performance limit states for the same YSPD configuration. This phenomenon indicates a better performance of YSPDs in the high-seismic zones compared with regions of moderate seismicity.

6. Conclusion Probabilistic seismic performance evaluation techniques are employed herein to assess the suitability of Yielding Shear Panel Device as a passive control device. An incremental dynamic analysis has been conducted using a suite of available earthquake records and a finite element model of the SAC building [16] with a recently calibrated BWBN model [23,33] introduced in the OpenSees platform to reflect the behaviour of the Yielding Shear Panel Device. Results obtained from the nonlinear dynamic analyses are used for probabilistic assessment of the effectiveness of YSPD in terms of seismic fragility and limit state probability analysis. The fragility analysis evaluates the exceedance probability of various levels of damage, defined on the basis of inter storey drift, whilst the limit state probability analysis calculates the annual probabilities of damage state exceedance at a given site. The presented case study evaluates six different YSPD configurations including three different YSPD sizes; obtained results reveal that the performance varies with change in size, number and configuration of YSPDs. Median fragility and limit state annual exceedance probability are used for evaluating the probabilistic performance. Increasing the size of YSPDs reduced the median fragility, limit state annual exceedance probability and caused a small reduction of seismic demand dispersion for both single bay and all moment resisting bays YSPD installations. Due to their higher initial stiffness and yield strength as well as larger energy dissipation capability, larger YSPDs showed better seismic performance. The effect of seismic zone on the retrofitted building in different hazard conditions has also been evaluated by considering moderate and high seismic zones for limit state probability analysis. The results reveal that YSPDs show better damage state risk reduction in the high-seismic zone when compared against its performance at the moderate-seismic zones. Two different configurations are used for the probabilistic performance evaluation; single bay installation (Case 1) and all moment resisting bay installation (Case 2). For the same size of YSPD, the increasing number of YSPDs in the all moment bay installation significantly reduced the median fragility and limit state annual exceedance probability compared with those for single bay installation. A small reduction in the seismic demand desperation is also observed. A comparison of single bay and all moment bay YSPD installation shows that YSPD 120  6  4 (Case 2) has a similar performance in terms of median fragility and limit state probability as the YSPD 100  4  2 (Case 1) configuration with a small reduction of the seismic demand dispersion, which indicates all moment resisting bays has lower uncertainties for

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seismic demand when compared with single bay installation. Although this study explores alternative YSPD configurations for a benchmark structure, the results suggest the prospect of utilizing YSPDs for performance based seismic design. Future research is required to explore alternative structural geometries and to develop appropriate performance based design guidelines for YSPDs.

where p controls the rate of initial drop in the slope, f1o is the total slip, w0 contributes to the amount of pinching, dw controls the rate of pinching spread, k controls the rate of change of f2 as f1 changes. The rate of hysteretic energy is given by following equation,

Appendix A. Pinching hysteretic model for YSPDs

A.2. Incremental algorithm for finite element implementation

ð12Þ

Using Eq. (6), the force at time tn+1 becomes,

A.1. Hysteretic model for YSPDs The restoring force F produced in the YSPD as shown in Fig. 11 is expressed according to BWBN model [23,33] is given in following equation,

F ¼ F e þ F h ¼ aF i d þ ð1  aÞF i z

e_ ¼ ð1  aÞF i zd_

ð6Þ

F n þ 1 ¼ aF i dnþ1 þ ð1  aÞF i znþ1

ð13Þ

The rate equation of z is discredited using the Backward Euler method as follows,

znþ1 ¼zn

    ðdnþ1  dn Þ ðdnþ1  dn Þ þ Dthnþ1 A  c þ bsgn znþ1 jznþ1 jn Dt Dt

    ðdnþ1  dn Þ n ¼ zn þ hnþ1 A  c þ bsgn znþ1 jznþ1 j ðdnþ1  dn Þ Dt

where Fe and Fh are the elastic and hysteretic component of the restoring force respectively; Eq. (6) provides a nonlinear force displacement (F–d) relationship based on the parameters Fi, a and z. Fi is defined here as the force representing the intersecting point of the bilinear envelope of the force displacement relationship at a unit displacement of di. kt is the tangential stiffness of YSPD whilst a = kt/Fi. The non-degrading pinching hysteretic response depends on the hysteretic displacement z, which is defined by the following first order nonlinear differential equation [23,34],

en+1 is found by discrediting Eq. (15) with the Backward Euler method as follows,

n h i o _ _ jzjn z_ ¼ dhðzÞ A  c þ bsgnðdzÞ

enþ1 ¼ en þ Dtð1  aÞF i znþ1

ð7Þ

where A, b, c and n are hysteretic model parameters which control _ depends on the sign of dz _ the shape of the curve. Value of sgnðdzÞ _ _ and the value becomes +1 if ðdzÞ is positive or becomes 1 if ðdzÞ is negative. The a pinching inducing function h(z) is expressed by following equations as [34,35],

  2 2 _ hðzÞ ¼ 1:0  f1 e ðzsgnðdÞqzu Þ =f2 zu ¼



1 1 n bþc

ð8Þ ð9Þ

where 0 6 f1 < 1 controls the severity of the pinching, f2 causes the pinching region to spread, zu is the ultimate value of z and q are model parameters. Depending on the hysteretic energy (e), f1, f2 are expressed by following equations as,



f1 ¼ f1o 1:0  eðpeÞ

ð10Þ

f2 ¼ ðw0 þ dw eÞðk þ f1 Þ

ð11Þ

ð14Þ

ðdnþ1  dn Þ Dt ¼ en þ ð1  aÞF i znþ1 ðdnþ1  dn Þ

ð15Þ

where zn, dn and en are the history variables and these values should be stored for the next step after each incremental step. The above incremental equations is solved by Newton–Raphson method for incremental strain (dn+1  dn). The incremental algorithm of the pinching hysteretic model is presented below, new 1. While ðjzold nþ1  znþ1 j > tolaranceÞ (a) Evaluation function f(zn+1):

enþ1 ¼ en þ ð1  aÞF i znþ1 ðdnþ1  dn Þ

f1nþ1 ¼ f1o 1:0  eðpenþ1 Þ f2nþ1 ¼ ðw0 þ dw enþ1 Þðk þ f1nþ1 Þ ððznþ1 sgnðdnþ1 dn Þqzu Þ2 =f22

hnþ1 ¼ 1:0  f1nþ1 e

nþ1

Þ

a1 ¼ c þ bsgnððdnþ1  dn Þznþ1 Þ a2 ¼ A  a1 jznþ1 jn

Force (F)

f ðznþ1 Þ ¼ znþ1  zn  hnþ1 a2 ðdnþ1  dn Þ (b) Evaluation of function derivatives with respect to zn+1:

e0nþ1 ¼ ð1  aÞF i ðdnþ1  dn Þ f01nþ1 ¼ f1o peðpenþ1 Þ e0nþ1

f02nþ1 ¼ w0 f1nþ1 þ kdw e0nþ1 þ dw enþ1 f01nþ1 þ dw f1nþ1 e0nþ1 ððznþ1 sgnðdnþ1 dn Þqzu Þ2 =f22

a3 ¼ e Displacement (δ) Fig. 11. Nonlinear force displacement (F–d) relationship of YSPD.

a4 ¼

nþ1

Þ

2f1nþ1 ðznþ1  sgnðdnþ1  dn Þ  qzu Þsgnðdnþ1  dn Þ f22nþ1

M.R. Hossain et al. / Engineering Structures 56 (2013) 1570–1579

a5 ¼ 0 hnþ1

2f1nþ1 ðznþ1  sgnðdnþ1  dn Þ  qzu Þ2

References

f32nþ1 ¼

a3 ðf01nþ1

 a4 þ

f02nþ1 a5 Þ

a02 ¼ na1 jznþ1 jn1 sgnðznþ1 Þ 0

F 0 ðznþ1 Þ ¼ 1:0  ðhnþ1 a2 þ hnþ1 a02 Þðdnþ1  dn Þ (c) Trial value in the Newton–Raphson method:

znew nþ1

¼ znþ1 

f ðznþ1 Þ f 0 ðznþ1 Þ

(d) Update zn+1: new zold nþ1 ¼ znþ1 and znþ1 ¼ znþ1

2. Compute force:

F nþ1 ¼ aF i dnþ1 þ ð1  aÞF i znþ1 3. Update parameters:

enþ1 ¼ en þ ð1  aÞF i znþ1 ðdnþ1  dn Þ

f1nþ1 ¼ f1o 1:0  eðpenþ1 Þ f2nþ1 ¼ ðw0 þ dw enþ1 Þðk þ f1nþ1 Þ hnþ1 ¼ 1:0  f1nþ1 e

ððznþ1 sgnðdnþ1 dn Þqzu Þ2 =f22

nþ1

Þ

4. Compute algorithmically consistent tangent:

a1 ¼ c þ bsgnððdnþ1  dn Þznþ1 Þ a2 ¼ A  a1 jznþ1 jn ððznþ1 sgnðdnþ1 dn Þqzu Þ2 =f22

a3 ¼ e a4 ¼

a5 ¼

nþ1

Þ

2f1nþ1 ðznþ1  sgnðdnþ1  dn Þ  qzu Þsgnðdnþ1  dn Þ f22nþ1 2f1nþ1 ðznþ1  sgnðdnþ1  dn Þ  qzu Þ2 f32nþ1

b1 ¼ ð1  aÞF i zn þ 1 b2 ¼ f1o peðpenþ1 Þ b1 b3 ¼ w0 b2 þ kdw b1 þ dw enþ1 b2 þ dw f1nþ1 b1 b4 ¼ a3 ðb2 þ b3 a5 Þ b5 ¼ ð1  aÞF i ðdnþ1  dn Þ b6 ¼ f1o peðpenþ1 Þ b5 b7 ¼ w0 b6 þ kdw b5 þ dw enþ1 b6 þ dw f1nþ1 b5 b8 ¼ a3 ðb6  a4 þ b7 a5 Þ b9 ¼ na1 jznþ1 jn1 sgnðznþ1 Þ @znþ1 a2 ðh  b4 Þ ¼ @dnþ1 1:0  ðb8 a2 þ hnþ1 b9 Þðdnþ1  dn Þ Consistent Tangent;

1579

@ rnþ1 @znþ1 ¼ aF i þ ð1  aÞF i @dnþ1 @dnþ1

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