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Bulletin of Earthquake Engineering 2: 173219, 2004.
2004 Kluwer Academic Publishers. Printed in the Netherlands.
A Probabilistic Displacement-based
Vulnerability Assessment Procedure
for Earthquake Loss Estimation
HELEN CROWLEY1, RUI PINHO1, and JULIAN J. BOMMER21European School for Advanced Studies in Reduction of Seismic Risk (ROSE School),
University of Pavia, Via Ferrata, 27100 Pavia, Italy, 2Department of Civil and Environmental
Engineering, Imperial College London, South Kensington campus, London SW7 2AZ, UK
*Corresponding author. Tel: +39-0382-505859, Fax: +39-0382-528422, E-mail:[email protected]
Abstract. Earthquake loss estimation studies require predictions to be made of the propor-
tion of a building class falling within discrete damage bands from a specified earthquake
demand. These predictions should be made using methods that incorporate both computa-
tional efficiency and accuracy such that studies on regional or national levels can be effec-
tively carried out, even when the triggering of multiple earthquake scenarios, as opposed
to the use of probabilistic hazard maps and uniform hazard spectra, is employed to real-
istically assess seismic demand and its consequences on the built environment. Earthquake
actions should be represented by a parameter that shows good correlation to damage and
that accounts for the relationship between the frequency content of the ground motion and
the fundamental period of the building; hence recent proposals to use displacement responsespectra. A rational method is proposed herein that defines the capacity of a building class
by relating its deformation potential to its fundamental period of vibration at different limit
states and comparing this with a displacement response spectrum. The uncertainty in the
geometrical, material and limit state properties of a building class is considered and the first-
order reliability method, FORM, is used to produce an approximate joint probability density
function (JPDF) of displacement capacity and period. The JPDF of capacity may be used
in conjunction with the lognormal cumulative distribution function of demand in the classi-
cal reliability formula to calculate the probability of failing a given limit state. Vulnerability
curves may be produced which, although not directly used in the methodology, serve to illus-
trate the conceptual soundness of the method and make comparisons with other methods.
Key words: displacement-based assessment, earthquake loss estimation, RC structures,
reliability, vulnerability curves
1. Introduction
The principal requirement of an earthquake loss model is an estimation of
the proportion of buildings in an urban environment which will fall within
discrete damage bands, both structural and non-structural, when subject
to a specified earthquake demand. Currently available methods include a
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174 H. CROWLEY ET AL.
number of features which may limit their accuracy and computational effi-ciency, as described in what follows. A method that attempts to meet, in
a harmonised fashion, the two fundamental requirements of accuracy and
computational efficiency for reliable loss assessments, is proposed herein.
1.1. Limitations of the current methods for earthquake loss
estimation
Traditionally, the assessment of damage for loss estimation studies has been
based on macroseismic intensity or peak ground acceleration (PGA). Both
parameters, however, have their shortcomings: intensity, although directly
related to building damage (Musson, 2000), is erroneously treated as a con-
tinuous variable in predictive relationships when in fact it is a discreteindex with non-uniform intervals, whilst PGA shows almost no correla-
tion with the damage potential of the ground motion. In addition, neither
parameter accounts for the relationship between the frequency content of
the ground motion and the dominant period of the buildings. Nonetheless,
these parameters are typically applied in damage matrix methods such as
that developed by the Applied Technology Council (ATC, 1985) wherein
damage ratios or factors, defined as the ratio between the cost of repair
and the replacement value of the structure, are related to the intensity of
shaking through the post-processing of field data collected following dam-
aging earthquakes. The development of the damage matrices is subjective,
however, since the determination of the intensity of shaking, as well as thelevel of observed damage in a structure, are based on expert opinion and
thus cannot be judged as exact procedures. Another pitfall in this approach
is that changing practices in construction may make observations of past
events of little relevance to the prediction of damage in future earthquakes.
Furthermore, the validity of applying statistics gathered from events that
may be fundamentally distinct from the area under assessment, both in
terms of seismic demand and supply, is debatable.
In order to compensate for the aforementioned shortcomings in tra-
ditional loss estimation procedures, recent proposals (e.g., Calvi, 1999;
FEMA, 1999) have made use of response spectra, in particular the
displacement (or accelerationdisplacement) spectrum, to represent the
destructive capacity of the ground motion. The rationale for using dis-placement spectra in assessment arises from the movement towards defor-
mation-based philosophy in seismic design, which reflects the much closer
correlation of displacements, as opposed to transient forces, with structural
damage.
In the HAZUS methodology (Kircher et al., 1997; FEMA, 1999), the
performance point of a building type under a particular ground shaking
scenario is found from the intersection of an accelerationdisplacement
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PROBABILISTIC DISPLACEMENT-BASED VULNERABILITY ASSESSMENT 175
spectrum, representing the ground motion, and a capacity spectrum(pushover curve), representing the horizontal displacement of the structure
under increasing lateral load. This performance point provides the displace-
ment input into limit state vulnerability curves to give the probability of
exceeding the given damage band. A potential weakness in the approach is
the difficulty in obtaining a physically realistic representation of the inelas-
tic response of the structure using pushover analysis. Although this aspect
can be somewhat improved using displacement-based adaptive pushover
techniques (Antoniou and Pinho, 2004), a faithful representation of the real
structural behaviour requires a great deal of information about the struc-
ture, including reinforcement details, which are unlikely to be well known
for a large building stock. Another feature of the method is that the capac-
ity curves published in the HAZUS manual are only available for buildingsin the USA having a limited range of storey heights, thus the application
of this method to other parts of the world requires additional research to
be carried out, although, of course, any method requires the gathering of
local data (e.g. Bommer et al., 2002).
Loss estimation methods are generally demanding in terms of time,
computing power and required input data. The HAZUS methodology was
originally derived not for probabilistic loss estimation but as a tool for
estimating the impact of individual earthquake scenarios. The method has
been adapted to use with models of earthquakes derived from probabilistic
seismic hazard assessment (PSHA), as in FEMA 366 (FEMA, 2001), but
it is preferable, as discussed by Bommer et al. (2002), to represent the seis-mic demand by triggering a large number of earthquake scenarios that are
compatible in magnitude, location and associated frequency of occurrence
with the regional seismicity. However, Bommer et al. (2002) also demon-
strated that this approach becomes extremely demanding in terms of com-
putational effort: the earthquake loss model developed for Turkey using an
adaptation of the HAZUS approach had to be limited to just over 1000
scenarios for the entire country in order to reduce computer run times to
acceptable levels.
Following the long-established tradition in earthquake loss modelling for
insurance purposes initiated 30 years ago at UNAM, in Mexico City, by
Emilio Rosenbleuth and Luis Esteva, Ordaz et al. (2000) present a prob-
abilistic method for earthquake loss estimation that uses both accelera-tion response spectra and a drift-based damage parameter. The method
uses both analytical and empirical relationships to define the vulnerabil-
ity of realistic structural models and can account for the flexibility of
foundations. In addition, the authors have extended the method to full loss,
rather than just damage, calculations. In the method of Ordaz et al. (2000)
the seismic demand is obtained using hazard maps derived from PSHA as
opposed to the use of scenario earthquakes.
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176 H. CROWLEY ET AL.
In this paper a proposal for a displacement-based vulnerability assess-ment procedure is presented, which is particularly suitable for an earth-
quake loss model owing to its computational efficiency, without loss of
accuracy. The more physical model underlying the new approach is also
likely to represent an additional improvement with respect to existing
methodologies.
1.2. Proposed methodology
The most up-to-date version of a displacement-based method for seismic
vulnerability assessment, first proposed by Pinho et al. (2002) and subse-
quently developed by Glaister and Pinho (2003), is presented herein. Fur-
thermore, an implementation strategy, as well as further developments, arealso provided, thus bringing the method one step closer to practical appli-
cation.
The procedure uses mechanically derived formulae to describe the dis-
placement capacity of classes of buildings at three different limit states.
These equations are given in terms of material and geometrical proper-
ties, including the average height of buildings in the class. By substitu-
tion of this height through a formula relating height to the limit state
period, displacement capacity functions in terms of period are attained; the
advantage being that a direct comparison can now be made at any period
between the displacement capacity of a building class and the displacement
demand predicted from a response spectrum. The original concept is illus-trated in Figure 1, whereby the range of periods with displacement capacity
below the displacement demand is obtained and transformed into a range
of heights using the aforementioned relationship between limit state period
and height. This range of heights is then superimposed onto the cumula-
tive distribution function (CDF) of building stock to find the proportion
of buildings failing the given limit state.
The inclusion of a probabilistic framework into the method that was
lacking in the original proposal (Pinho et al., 2002) has allowed for a con-
sideration of the uncertainty in the displacement demand spectrum and the
uncertainty in the displacement capacity that arises when a group of build-
ings, which may have different geometrical and material properties, is con-
sidered together. The addition of this probabilistic framework, however, hasmeant that the simple graphical procedure outlined in Figure 1 that treated
the beam- or column-sway RC building stock as single building classes can
no longer be directly implemented, but instead, separate building classes
based on the number of storeys need to be defined; this issue is addressed
further in Section 3.4.
The aleatory variability in the demand is modelled using the widely
accepted assumption of a lognormal distribution of residuals (e.g.,
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PROBABILISTIC DISPLACEMENT-BASED VULNERABILITY ASSESSMENT 177
Height
cumulative
frequency
HLS1HLS2HLS3
PLS3
0
PLS2
PLS1
PLSi percentage of
buildings failing LSi
effective
period
displacement
LS1
LS2
LS3
Demand
Spectra
TLS1TLS2TLS3
HLSi=f(TLsi , LSi)
LS1
LS2
LS3
Figure 1. A deformation-based seismic vulnerability assessment procedure (Glaister
and Pinho, 2003). LS stands for limit state.
Restrepo-Velez and Bommer, 2003), whilst modelling of the displacement
capacity uncertainty requires a slightly more sophisticated approach: the
use of a first-order reliability method (FORM). FORM can be used to cal-
culate the approximate CDF of a non-linear function of correlated random
variables. Once the CDF of the demand and the capacity have been found,
the calculation of the probability of exceedance of a specified limit state
can be obtained using the standard time-invariant reliability formulation
(e.g. Pinto et al., 2004). The probability of being in a particular damageband may then be found from the difference between the bordering limit
state exceedance probabilities.
The authors believe that the use of the method described in this paper
leads not only to a more computationally efficient process of earthquake
loss estimation, with the possibility to calculate the losses from multiple-
scenario earthquakes, but also to a method that can be easily adapted to
suit the varied construction and design practices around the world, owing
to its transparent means of building class vulnerability assessment.
2. Deterministic Implementation of Proposed Methodology
2.1. Classification of buildings
The initial step required in this method is the division of the building pop-
ulation into separate building classes. A building class is to be considered
as a group of buildings which share the same construction material, failure
mechanism and number of storeys. The building classes currently consid-
ered within this methodology comprise the following structural types:
(1) reinforced concrete beam-sway moment resisting frames,
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178 H. CROWLEY ET AL.
(2) reinforced concrete column-sway moment resisting frames,(3) reinforced concrete structural wall buildings,
(4) un-reinforced masonry buildings exhibiting an out-of-plane failure
mechanism,
(5) un-reinforced masonry buildings exhibiting an in-plane failure mechanism.Within each structural type, further building classes may be defined to
separate, for example, buildings with different number of storeys, buildings
designed with distinct steel grades or those built without adequate confin-
ing reinforcement. A decision regarding whether a moment resisting frame
will exhibit a beam-sway (class 1) or a column-sway (class 2) mechanism
may be made considering the construction type, construction year and
evidence of a weak ground floor storey. Many buildings built before the
inclusion of sound seismic design philosophy (i.e. capacity design) into acountrys seismic design code and those with a weak ground floor storey
will generally adopt a soft-storey (column-sway) mechanism. The treatment
of classes 4 and 5, relating to un-reinforced masonry structures, have been
dealt with by Restrepo-Velez and Magenes (2004) in an independent effort
and will not be considered further in this study.
2.2. Structural and non-structural limit states
Damage to the structural (load-bearing) system of the building class is esti-
mated using three limit states of the displacement capacity. The building
class may thus fall within one of four discrete bands of structural damage:none to slight, moderate, extensive or complete. A qualitative description of
each damage band for reinforced concrete frames is given in Table I along
with quantitative suggestions for the definition of the mechanical material
properties for each limit state taken from the work of Priestley (1997) and
Calvi (1999). The first structural limit state is defined as the yield point of
the structure and the second and third structural limit states are attained
when the sectional steel and concrete strains reach the limits suggested in
Table I. Two alternative pairs of limit state 3 sectional strains have been
reported because the ultimate sectional strains that can be reached depend
on the level of confinement of the structural members. Nevertheless, it
should be noted that one is not constrained to employ these limit state steel
and concrete strains and has the ability to control these, and other, param-eters used in the building class capacity calculations.
Damage to non-structural components within a building can be con-
sidered to be either drift- or acceleration-sensitive (Freeman et al., 1985;
Kircher et al., 1997). Drift-sensitive non-structural components such as
partition walls can become hazardous through tiles and plaster spalling
off the walls, doors becoming jammed and windows breaking. Acceler-
ation-sensitive non-structural components include suspended ceilings and
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PROBABILISTIC DISPLACEMENT-BASED VULNERABILITY ASSESSMENT 179
Table I. Description of RC frame structural discrete damage bands
Structural damage band Description
None to slight Linear elastic response, flexural or shear type hairline cracks
(
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180 H. CROWLEY ET AL.
Table II. Description of non-structural discrete damage bands
Non-structural damage band Description
Undamaged No damage to any non-structural element, damage
assumed to initiate at drift ratios between 0.1% and 0.3%,
but may depend on quality of partitions
Moderate To maintain moderate, easily repairable damage to non-
structural elements, drift ratios should not exceed 0.3%
0.5%
Extensive Extensive damage to non-structural elements, to ensure
damage is reasonably repairable, drift ratios should not
exceed the range of 0.51.0%
Complete Repair of non-structural elements not feasible, exceedance
of extensive damage drift ratio limits
different limit states, is the basis of this methodology. Structural displace-
ment capacity formulae for all of the building classes described in Section
2.1 have been, or are in the process of being, derived, but only the beam-
sway and column-sway failure mechanisms of reinforced concrete frames
(classes 1 and 2) shall be presented herein. The derivation of displacement
capacity formulae for structural wall buildings (class 3) is currently under-
way. Whilst a more thorough description of the origin of the structural dis-
placement capacity formulae for classes 1 and 2 can be found in Glaisterand Pinho (2003), important developments have been carried out since the
original derivation of these equations, such as the inclusion of a robust for-
mula to relate the yield period of a RC frame to its height, and the deriva-
tion of non-structural displacement capacity formulae, as will be discussed
presently.
2.3.1. Displacement capacity at the centre of seismic force
(i) Beam-sway frames
As stated previously, the demand in this methodology is represented by a
displacement spectrum which can be described as providing the expected
displacement induced by an earthquake on a single degree of freedom
(SDOF) oscillator of given period and damping. Therefore, the displace-
ment capacity equations that are derived must describe the capacity of a
SDOF substitute structure and hence must give the displacement capac-
ity, both structural and non-structural, at the centre of seismic force of the
original structure.
The displacement capacity at the centre of seismic force is dealt with
in two different ways in this method depending on whether it is the limit
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PROBABILISTIC DISPLACEMENT-BASED VULNERABILITY ASSESSMENT 181
state base rotation/drift or the roof deformation of the original structurethat needs to be predicted.
In the structural displacement capacity equations, presented in Section
2.3.2, a base rotation can be mechanically derived for both beam- and
column-sway frames and the displacement at the centre of seismic force
is given by multiplying this rotation by an effective height. The effective
height is calculated by multiplying the total height of the structure by an
effective height coefficient (efh), defined as the ratio of the height to the
centre of mass of a SDOF substitute structure (HSDOF), that has the same
displacement capacity as the original structure at its centre of seismic force
(HCSF), and the total height of the original structure (HT), as schematically
shown in Figure 2.
For beam-sway frames, the ratio of HCSF to HT varies with the height,independently of ductility, from 0.67 for frames less than 4 storeys high to
0.61 for frames with more than 20 storeys; however, it has been suggested
by Priestley (1997) that, for regular structures, an average ratio of 0.64 may
be taken, irrespective of building height. The effective height coefficient can
then in turn be defined as a function of the number of storeys n using the
following equations, as suggested by Priestley (1997):
efh=0.64 n4 (1)
efh=0.640.0125(n4) 4
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182 H. CROWLEY ET AL.
In the derivation of the non-structural displacement capacity equa-tions for beam-sway frames, the effective height coefficient cannot be used
directly because, rather than mechanically deriving a base rotation capacity,
as in the structural displacement capacity formulation, it is the roof defor-
mation capacity that is directly obtained, as will be described in Section
2.3.3.
Hence a relationship between the deformation at the roof and the defor-
mation at the centre of seismic force is required. The factor relating these
two displacements shall be named a shape factor (S) and it can be found
from the displacement profiles suggested by Priestley (2003) for beam-sway
frames of various heights (Figure 3), where, as above, the elastic and inelas-
tic profiles are assumed to be equivalent.
The shape factor at the centre of seismic force can be found directlyfrom Figure 3 using an assumed ratio of the height to the centre of seis-
mic force (HCSF) to the total height (HT) of 0.64, as suggested previously.
Thus it can be seen in Figure 3 that the displacement at HCSF varies from
around 0.64 to 0.85 times the roof displacement depending on the number
of storeys.
(ii) Column-sway frames
As stated previously, the structural displacement capacity formulae are
derived by multiplying a base rotation by an effective height coefficient.
For column-sway frames, the elastic and inelastic deformed shapes vary
from a linear profile for elastic (pre-yield) limit states to a non-linear
profile at inelastic (post-yield) limit states (Figure 4). As suggested by
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
Shape Factor
He
ightratio
(H
i/H
n)
n < 4
n = 8
n = 12
n = 16
n > 20
efh = 0.64
Figure 3. Displacement profiles for beam-sway frames for varying number of
storeys, n.
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PROBABILISTIC DISPLACEMENT-BASED VULNERABILITY ASSESSMENT 183
0 0.2 0.4 0.6 0.8 1
Displacement ratio
elastic
inelasticHeight
of
ground
floor
Height to
centre of
seismic
forcesy1st p
psy
i
Height
Figure 4. Elastic and inelastic deformed shapes of column-sway frames with ground
floor drift capacity 1.
Priestley (1997), the linear profile at pre-yield limit states means that the
ratio ofHCSF to HT can be assumed to be 0.67 and so this is to be taken
as the effective height coefficient.
At post-yield limit states, the height to the centre of seismic force of a
column-sway frame is dependent on the ductility (Lsi ) and decreases from
a low ductility value of 0.67 to a high ductility value of 0.5, as inferredfrom Figure 4 and captured in the following equation, first proposed by
Priestley (1997) and then adapted by Glaister and Pinho (2003):
efh=0.0670.17Lsi 1Lsi
(4)
The ductility cannot be calculated, however, unless the yield displace-
ment at the effective height is known, thus leading to an iterative proce-
dure to find the effective height. Glaister and Pinho (2003) proposed that,
for the sake of simplicity, a formula similar to Eq. (4) could be used where,
instead of ductility, the steel strain s(Lsi) corresponding to a given limit
state is used, as presented in Eq. (5).
efh=0.0670.17s(Lsi)ys(Lsi )
(5)
For the derivation of the non-structural capacity, the inter-storey drift
capacity of the ground floor, i , is equated to a base rotation, as will be
described in Section 2.3.3, and so the effective height coefficient is required
to find the displacement capacity at the centre of seismic force. For pre-
yield limit states, this coefficient will be equivalent to that used in the
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184 H. CROWLEY ET AL.
structural displacement capacity formulae described above (i.e. 0.67HT). Atpost-yield limit states, (that is, when the non-structural limit state exceeds
the structural yield limit state), it is proposed that an initial effective height
of 0.6HT is assumed in order to estimate the structural yield displace-
ment and corresponding ductility. This resulting ductility is then input into
Eq. (4) to obtain a better estimate of the effective height coefficient; only
one iteration is required to arrive at a stable converged solution.
2.3.2. Structural displacement capacity vs height
By considering the yield strain of the reinforcing steel and the geometry of
the beam and column sections used in a building class, yield section curva-
tures can be defined using the relationships suggested by Priestley (2003).These beam and column yield curvatures are then multiplied by empirical
coefficients to account for shear and joint deformation to obtain a formula
for the yield chord rotation. This chord rotation is equated to base rotation
and multiplied by the total building height and an effective height coeffi-
cient, as introduced in Section 2.3.1, to produce the yield displacement
capacity of a SDOF substitute structure. Sound, rational and deformation-
based equations of displacement capacity can thus be derived through first
principles and mechanical considerations.
The yield displacement capacity formulae for beam- and column-sway
frames are presented in Eqs. (6) and (7), respectively; these are used to
define the first structural limit state.
Sy=0.5efhHTylb
hb(6)
Sy=0.43efhHTyhs
hc(7)
The parameters employed in these and subsequent equations are
described below:
Sy structural yield (limit state 1) displacement capacity,
efh effective height coefficient, as defined in Section 2.3.1,
HT total height of the original structure,
y yield strain of the reinforcement,
lb length of beam,hb depth of beam section,
hs height of storey,
hc depth of column section,
Post-yield displacement capacity formulae are obtained by adding a
plastic displacement component to the yield displacement, calculated by
multiplying together the limit state plastic section curvature, the plas-
tic hinge length, and the height or length of the yielding member. The
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PROBABILISTIC DISPLACEMENT-BASED VULNERABILITY ASSESSMENT 185
post-yield displacement capacity formulae for RC beam- and column-swayframes are presented here in Eqs. (8) and (9), respectively. In this formu-
lation, the soft-storey of the column-sway mechanism is assumed to form
at the ground floor. Straightforward adaptation of the equations could eas-
ily be introduced in the cases where the soft-storey is expected to form at
storeys other than the ground floor, but this is not dealt with herein.
SLsi = 0.5efhHTylb
hb+0.5
C(Lsi)+ S(Lsi )1.7y
efhHT (8)
SLsi = 0.43efhHTyhs
hc+0.5
C(Lsi)+ S(Lsi )2.14y
hs (9)
where, SLsi is the structural limit state i (2 or 3) displacement capacity,
C(Lsi ), maximum allowable concrete strain for limit state i, S(Lsi ), maxi-
mum allowable steel strain for limit state i.
Formulae for the ductility (SLsi) of beam- and column-sway frames are
shown in Eqs. (10) and (11), respectively. A detailed account of the deriva-
tion of Eqs. (6)(11) can be obtained from the work of Glaister and Pinho
(2003).
SLsi = 1+C(Lsi)+ S(Lsi )1.7y
hb
ylb(10)
SLsi = 1+ C(Lsi)+ S(Lsi )2.14y
hc
0.86efhHTy(11)
An important development that will need to be included in the meth-
odology is the calculation of the shear capacity of the structure, to ensure
that shear failure does not occur before the flexural displacement capacity
is reached. Within the purely displacement-based framework of the method
it would be most convenient for such a shear capacity check to be car-
ried out through comparison between the displacement demand and shear
capacity of reinforced concrete members. The recent work of Miranda
(2004), where formulae for the shear displacement capacity of members
have been derived by relating their shear force capacity to a displace-
ment capacity, will be used in the future developments of this proposed
vulnerability assessment method.
2.3.3. Non-structural displacement capacity vs height
Non-structural displacement capacity is found from the inter-storey drift
capacity of the non-structural components, such as partition walls. Exam-
ples of the limit state drift ratios have been described previously in Table II.
For beam-sway frames, the non-linear displaced shape leads to a var-
iation in inter-storey drift from the ground floor to the roof. However,
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186 H. CROWLEY ET AL.
by multiplying the drift ratio capacity by the total height of the build-ing, a roof displacement capacity corresponding to the average inter-sto-
rey drift capacity is attained. The non-structural displacement capacity of
the SDOF substitute structure, as introduced in Section 2.3.1, can thus be
found by multiplying the roof displacement by the shape factor to give the
displacement at the centre of seismic force of the structure, as presented in
Eq. (12).
NSLsi =SiHT (12)
where NSLsi is the non-structural limit state i displacement capacity, S is
the shape factor giving the ratio of the deformation at the effective height
to the roof deformation, described in Section 2.3.1, i the limit state i driftratio capacity.
For column-sway frames, the potential for concentration of non-
structural damage at the ground floor should be considered, as illustrated
previously in Figure 4. Thus it is assumed that once the first floor reaches
the limit state inter-storey drift capacity then the non-structural damage
limit state has been attained. Therefore it should be ascertained whether
the displacement at the first floor (NS1st), given in Eq. (13) by multiplying
the inter-storey drift with the storey height, is greater than the first floor
structural yield displacement (Sy1st), found by multiplying the yield base
rotation by the height of the first storey.
NS1st=ihs (13)If NS1st is lower than Sy1st, the non-structural displacement capa-
city at the centre of seismic force at this pre-yield limit state can sim-
ply be given by Eq. (12) with S= 0.67 due to the linear deformed shape,defined in Figure 4. If NS1st is higher than Sy1st, then the post-yield
non-structural displacement capacity of the SDOF substitute structure can
be found by the following steps. The plastic component of the displacement
(p) may be calculated by subtracting the yield displacement at the first
storey (Sy1st) from NS1st.
p=NS1st
Sy1st
=ihs
0.43hsy
hs
hc(14)
This plastic component (p) may then be added to the yield displace-
ment at the centre of seismic force (Sy) to obtain the non-structural limit
state displacement capacity of the SDOF substitute structure (NSLsi), as
illustrated in Figure 4. As has been discussed in Section 2.3.1, it is sug-
gested that an effective height coefficient be calculated using Eq. (4), where
the ductility may be first estimated for an initial guess of the yield dis-
placement at the centre of seismic force found with an effective height of
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PROBABILISTIC DISPLACEMENT-BASED VULNERABILITY ASSESSMENT 187
0.6HT, and then iterated once for a final solution. The formula for thenon-structural limit state displacement capacity of the SDOF substitute
structure for a column-sway frame, failing in the first storey and having
entered the non-linear range, is thus presented in Eq. (15).
NSLsi =p+Sy=ihs+0.43(efhHThs)yhs
hc(15)
To summarise, the non-structural displacement capacity of the SDOF
substitute structure may be calculated for beam-sway frames using Eq. (16)
where S can be found from Figure 2, assuming a HCSF to HT ratio of 0.64.
The non-structural displacement capacity of column-sway frames for limit
states before structural yielding, ascertained at the first floor, may be foundusing Eq. (17) and for limit states after structural yielding at the first floor,
using Eq. (18).
NSLsi =SiHT (16)
NSLsi =0.67iHT (17)
NSLsi =ihs+0.43(efhHThs)yhs
hc(18)
2.4. Period of vibration of buildings as a function of height
Simple empirical relationships are available in many design codes to relate
the fundamental period of vibration of a building to its height. However,
these relationships have been realised for force-based design and so produce
lower bound estimates of period such that the base shear force will be con-
servatively predicted. Hence the displacement demand on a structure needs
to be accurately estimated; however with a conservative periodheight rela-
tionship the displacement demand would generally be under-predicted. The
use of a reliable relationship between period and height is a fundamental
requirement in this methodology, so that the displacement capacity formu-
lae can be accurately defined in terms of period and directly compared with
the displacement demand.
Glaister and Pinho (2003) recognised the need for a sound relation-
ship between period and height that would be valid throughout the entire
displacement range. However, in the absence of such a relationship, they
used a modified version of the suggested formula given in EC8 (CEN,
2003). A suitable relationship between yield period and height has since
been derived by Crowley and Pinho (2004), which can be easily related
to inelastic period as will be shown in the subsequent sections. The
pre- and post-yield structural displacement capacity formulae given in
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188 H. CROWLEY ET AL.
Glaister and Pinho (2003) in terms of period have thus been updated, aswill be presented in Section 2.5.
2.4.1. Yield period
Crowley and Pinho (2004) describe how analytical procedures have been
used to obtain the yield period of European RC buildings designed before
the inclusion of capacity design in the design codes. Eigenvalue, pushover
and dynamic analyses have all been employed in the yield period determi-
nation for many buildings of various heights. Regression analysis of the
data has led to a group of best-fit yield periodheight curves that are in
general agreement despite having been derived from different theoretical
backgrounds. Hence there is a high degree of confidence in the resultsobtained which then lead to a straightforward choice of a linear yield
period vs. height (HT in metres) formula for European RC moment resist-
ing frames, presented in Eq. (19):
Ty=0.1HT (19)
2.4.2. Post-yield period
For post-yield limit states, the limit state period of the substitute structure,
as introduced in Section 2.3.2, can be obtained from the secant stiff-
ness to the point of maximum deflection on an idealised bi-linear force
displacement curve as described already in Glaister and Pinho (2003) butrepeated here for the sake of clarity. Assuming an elasto-plastic force
displacement relationship, the secant stiffness to the point of maximum
deflection (kLsi ) can be shown to be a geometric function of the elastic
stiffness (ky) and ductility (Lsi ) only. Since the elastic period (Ty) is also
a function of elastic stiffness, it can be assumed that the effective period
(TLSi ) of the inelastic structure is a function of elastic period and ductil-
ity alone. Eqs. (20)(23) show the working through of these premises and
the resulting equation relating effective period at a limit state i with the
corresponding ductility level and the elastic period, independent of the fail-
ure mechanism:
fy=kyy=kLsiLsi (20)
kLsi =kyy
Lsi= kyLsi
(21)
T k1/21/2 (22)
TLsi =TyLsi (23)
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PROBABILISTIC DISPLACEMENT-BASED VULNERABILITY ASSESSMENT 189
2.5. Structural and non-structural displacement capacity as afunction of period
2.5.1. Structural displacement capacity vs period
(i) Pre-yield
The derivation of a relationship between period and height that is valid
for all limit states allows the previous displacement capacity formulae pre-
sented in Glaister and Pinho (2003) to be developed into conceptually
sound functions of period. For the first (yield) limit state, the building
height may be simply defined in terms of the yield period by rearranging
Eq. (19) as follows:
HT=10Ty (24)In the case of beam-sway RC frames, the yield capacity equations can be
obtained by substituting the height in Eq. (6) (the formula for the yield dis-
placement capacity in terms of height) with the formula in Eq. (24) above:
Sy=5efhTyylb
hb(25)
For column-sway RC frames, the yield displacement equation is also
simply transformed into a function of period by substituting Eq. (24) into
Eq. (7).
Sy=4.3efhTyy hshc
(26)
(ii) Post-yield
For the post-yield structural limit states (2 and 3), the height of the build-
ing needs to be written in terms of the post-yield period. For beam-sway
frames the height is simply given by rearranging Eq. (23) to give the
formula shown in Eq. (27):
HT=10TLsiSLsi
(27)
The post-yield displacement capacity in terms of post-yield period is
then found by replacing the height in Eq. (8) (the formula for the post-yield displacement capacity in terms of height) with Eq. (27), to give the
following formula:
SLsi =5efhTLsiylb
hb
Lsi (28)
For the post-yield limit states of column-sway frames, the resulting for-
mula for the height has a slightly more complicated form as compared to
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190 H. CROWLEY ET AL.
beam-sway frames due to the dependence of the ductility on the height,(see Eq. (11)):
HT=1
2
Cl+ (C2l +400T2Lsi )1/2
(29)
where
Cl=c+s2.14y
0.86 y
hc
efh
The post-yield displacement capacity in terms of post-yield period, pre-
sented in Eq. (30), is thus obtained by replacing the height in Eq. (9) with
Eq. (29).
SLsi =0.215efhyhs
hc(C2l +400T2Lsi )1/2+0.25(c+s2.14y)hs (30)
2.5.2. Non-structural displacement capacity vs period
(i) Pre-yield
The initiation of non-structural damage can be confidently assumed to
occur before structural yielding, at a drift ratio 1. The relationship
between the height and yield period of Eq. (24) is also used in the substitu-
tion of height for period in the non-structural displacement capacity equa-
tions. The first limit state non-structural displacement capacity in terms
of period is thus presented in Eq. (31), where S can be obtained fromFigure 3 for beam-sway frames and may be taken as 0.67 for column-sway
frames, as has been described in Section 2.3.1.
NS=S1(10Ty) (31)
(ii) Post-yieldThe moderate and significant non-structural damage drift limit ratios,
2 and 3 respectively, may or may not occur before structural yield-
ing and so this check needs to be carried out. For beam-sway frames,
if the moderate or significant non-structural damage displacement capac-
ity is less than the structural yield displacement capacity at the centre of
seismic force, then Eq. (31) above can be used. However, if these displace-ments are higher than the yield displacement, then the yield period can no
longer be applied. Instead, the height should be substituted using Eq. (27),
where the ductility (Lsi) of the beam-sway frames can be calculated from
the ratio between the moderate/significant non-structural damage displace-
ment capacity and the structural yield displacement:
NSLsi =NSLsi
Sy= SiHT
Sy= Si
0.5efhylb/hbi=2,3 (32)
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PROBABILISTIC DISPLACEMENT-BASED VULNERABILITY ASSESSMENT 191
The final equation for the non-structural displacement capacity ofbeam-sway frames in terms of the inelastic period is found by replacing HT,
as defined in Eq. (27), in Eq. (12) to give:
NSLsi =Si
10TLsiNSLsi
=Si(10TLsi )
0.5yefhlb
ihbSi=2,3 (33)
For column-sway frames, if the non-structural displacement at the first
storey is greater than the yield displacement, then the height of the struc-
ture should be calculated using the post-yield period, as presented previ-
ously in Eq. (27), where the ductility can be found using Eq. (34), using
p computed from Eq. (14). The effective height coefficient in Eq. (34) is
initially taken as 0.6 to find the ductility and then Eq. (4) is used to finda better estimate of the effective height coefficient for further calculations.
NSLsi =NSLsi
Sy= p+Sy
Sy=1+ p
Sy=1+ p
0.43(efhHT)yhs/hc(34)
The height can then be represented in terms of inelastic period, using
the formula shown below, which again is slightly more complicated
than the formula for beam-sway frames due to the dependence of the
ductility on the height:
HT= 12C2
p+ (2p+400C22T2Lsi )1/2
(35)
where,
C2=0.43efhyhs/hc
The resulting formula for the non-structural displacement capacity in
terms of inelastic period is then found by substituting Eq. (35) into Eq.
(18):
NSLsi=
1
2p+ (
2p
+400C22T
2Lsi )
1/2 (36)2.6. Displacement demand
Displacement response spectra are used in this method to represent the
input from the earthquake to the building class under consideration. The
relationship between equivalent viscous damping ( ) and ductility (), used
to account for the energy dissipated through hysteretic action at a given
level of ductility demand is presented in the following equation:
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192 H. CROWLEY ET AL.
=a1 1bi
+ E (37)where a and b are calibrating parameters which vary according to the char-
acteristics of the energy dissipation mechanisms, whilst E represents the
equivalent viscous damping when the structure is within the elastic, or pre-
yield, response range. It is recognised, however, that the level of energy dis-
sipation of a given structural system may depend on the characteristics of
the input such as duration and phase content, for which reason research is
currently underway to assess the manner in which Eq. (37) can be adjusted
or improved to include this influence. In the meantime, values of a=25 andb=
0.5, as suggested by Calvi (1999), are adopted in Eq. (37), together with
an E=5%.The equivalent viscous damping values obtained through Eq. (37), for
different ductility levels, can then be combined with Eq. (38), proposed by
Bommer et al. (2000) and currently implemented in EC8 (CEN, 2003), to
compute a reduction factor to be applied to the 5% damped spectra at
periods from the beginning of the acceleration plateau to the end of the
displacement plateau:
=
10
5+ (38)
Bommer and Mendis (2004) have investigated the dependence of the
ratio of displacement spectral ordinates for higher damping levels to the
ordinates at 5% of critical damping on features of the earthquake motion.
The ratios are shown to decrease with increasing magnitude and with
increasing distance, both observations being consistent with the ratios
decreasing as the duration of the ground shaking increases. In the proposed
procedure of using earthquake scenarios rather than probabilistic hazard
maps to model the demand, this refinement of the prediction of the spec-
tral ordinates at higher damping levels can be easily incorporated.
2.7. Illustrative example of deterministic implementation
Many of the existing buildings in Europe have not been designed with
sound seismic design philosophy, hence, as has been discussed in Section
2.1, a large proportion may be assumed to behave with a column-sway fail-
ure mechanism. A deterministic example is provided herein to show how
the yield displacement capacity of column-sway frames varies with period
and how the failure of this building class can be ascertained through com-
parison with a displacement demand spectrum. The aim of this example is
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PROBABILISTIC DISPLACEMENT-BASED VULNERABILITY ASSESSMENT 193
Table III. Values used for the parameters
in the limit state 1 (yield) displacement and
period capacity equations for column-sway
frames
Parameter Value
Column depth, hc 0.38m
Storey height, hs 3.22m
Steel grade 275 yield strain, y 0.165 %
merely to illustrate the workings of the deterministic method described thus
far.Table III shows the values that have been assigned to the parameters
required to define the yield displacement capacity of column-sway frames,
presented previously in Eq. (26). The geometrical data has been taken from
the mean values obtained from a study of European building stock data;
this is discussed further in Section 3.3.1. The reinforcing steel in this exam-
ple has a 5% characteristic strength of 275 MPa; the calculation of the
mean yield strain shown in Table III is described in Section 3.3.2. The dis-
placement demand spectrum used in this example is based on the 1992
Erzincan (Turkey) earthquake record, but the ordinates have been scaled to
20% of their original value, for the convenience of providing a clearer dem-
onstration of the intersection between the demand and capacity curves.In Figure 5, the yield displacement capacity/demand curves for a
column-sway mechanism are given; the circles correspond to the
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
Period (s)
Disp
lacement(m)
T = 0.90 to 3.15 seconds
H = 9.0 to 31.5 metres
T
0 1 2 3 4 5 6 7
Figure 5. Column-sway yield capacity and demand curves.
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194 H. CROWLEY ET AL.
displacement capacity at a distinct number of storeys. As has been intro-duced in Section 1.2, failure of the limit state is assumed to occur when the
displacement capacity curve falls below the displacement demand curve;
hence a probability of failure of unity when the capacity is below the
demand and zero when the capacity is above the demand. Thus it is appar-
ent from Figure 5 that with a deterministic approach, all column-sway
buildings responding at a yield period between 0.9 and 3.15 s would be pre-
dicted to fail the first limit state. By using the relationship between yield
period and height described in Section 2.4.1, the height range of the build-
ings failing the limit state can be found to be between 9.0 and 31.5 m,
which corresponds to buildings between 3 and 10 storeys.
3. Probabilistic Framework
3.1. Overview
A large number of geometrical and material parameters can vary among
buildings within a given class. A fully probabilistic framework is thus
necessary, and has been applied to this method to account for the fol-
lowing sources of epistemic (knowledge-based) and aleatory (random)
uncertainty:
(1) The uncertainty concerning the geometrical and material properties of
a building class.
(2) The uncertainty regarding the definition of steel and concrete strains
reached at each limit state of structural damage.
(3) The uncertainty as to the drift rotations required to define each limit
state of non-structural damage.
(4) The model uncertainty caused by the dispersion of the empirical coeffi-
cients used in the derivation of the displacement capacity formulae,
such as those used to define the yield curvature, plastic hinge length
and yield period.
(5) The aleatory uncertainty in the estimation of the 5% damped response
spectrum. (It should be noted that the mean ductility is used to reduce
the 5% damped demand spectrum for higher limit states, using the
reduction factor that has been discussed previously. This assumption
has been made to simplify the method as otherwise the variability inthe demand would be dependent on the variability in the capacity).
The probability that the earthquake demand is greater than the capac-
ity of a building, and thus failure occurs, is given by the classical time-
invariant reliability formula (e.g. Pinto et al., 2004):
Pf=
0
[1FD()]fSC()d (39)
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PROBABILISTIC DISPLACEMENT-BASED VULNERABILITY ASSESSMENT 195
where FD() is the CDF of the demand and fSC() is the probabilitydensity function of the capacity, defined in terms of a particular damage
parameter (). The adaptation of this reliability formulation initially car-
ried out by Restrepo-Velez and Magenes (2004) to suit the methodology
described herein is shown in (40):
Pf=x
y
[1FD(x/TLsi =y)]fLSiTLSi (x,y)dxdy (40)
where FD(x/TLSi = y) is the CDF of the displacement demand, x, givena period, TLsi and fLSiTLSi(x,y), is the joint probability density func-
tion (JPDF) of the limit state displacement capacity, Lsi, and limit stateperiod, TLsi .
The JPDF, fLSiTLSi (x,y), may be defined as the product of the probabil-
ity density function of Lsi , conditioned to TLsi , and the probability den-
sity function of TLsi :
fLsiTLsi (x,y)=fLsi (x/y)fTLsi (y) (41)
Thus the final formulation for the calculation of the probability that the
displacement demand is greater than the displacement capacity of a build-
ing class, for a given limit state, is given by Eq. (42).
Pf=y
x
[1FD(x/TLsi =y)]fLSi /TLSi (x/TLsi =y)fTLSi dxdy (42)
The inner integral in the above equation gives the probability that the
displacement demand is greater than the displacement capacity, condi-
tioned to a period, and so may be referred to as the conditional probability
of failure. Thus it may be read that Eq. (42) is the integral of the product
of the conditional probabilities of failure by the probabilities of the con-
ditioning events, over the full range of their possible intensities (Franchin
et al., 2002).
The JPDH can be used in conjunction with the demand CDF through
the use of the reliability formulation of Eq. (42) to find the probability
of exceeding each of the three limit states described in Section 2.2. The
probability of a building class being in each of the four structural damage
bands, outlined in Table I, can then simply be found from the difference
between the exceedance probabilities of the bordering limit states to the
damage band in question. This probability is equated to the proportion of
buildings (P) falling within each damage band:
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196 H. CROWLEY ET AL.
Pnone/slight=1Pf1 (43)Pmoderate=Pf1Pf2 (44)
Pextensive =Pf2Pf3 (45)
Pcomplete =Pf3 (46)
The same process is also applied to find the proportion of a building class
that falls within one of the four non-structural damage bands in Table II.
3.2. Probabilistic treatment of the demand
The CDF of the displacement demand can be found using the median dis-placement demand values and their associated logarithmic standard devia-
tion at each period. The CDF gives the probability that the displacement
demand exceeds a certain value (x), given a response period (TLsi ) for a
given M-D scenario.
The displacement demand spectrum that might be used in a loss estima-
tion study could take the form of a code spectrum or else a uniform hazard
spectrum derived from PSHA for one or more annual frequencies of excee-
dance. Both of these options have drawbacks in being obtained from PSHA
wherein the contributions from all relevant sources of seismicity are com-
bined into a single rate of occurrence for each level of a particular ground-
motion parameter. The consequence is that if the hazard is calculatedin terms of a range of parameters, such as spectral ordinates at several
periods, the resulting spectrum will sometimes not be compatible with any
physically feasible earthquake scenario (Bommer, 2002). Furthermore, if
additional ground-motion parameters, such as duration of shaking, are to
be incorporated as they are in HAZUS, in the definition of the inelas-
tic demand spectrum then it is more rational not to combine all sources
of seismicity into a single response spectrum but rather to treat individ-
ual earthquakes separately, notwithstanding the computational penalty that
this entails.
The approach recommended therefore is to use multiple earthquake
scenarios, each with an annual frequency of occurrence determined from
recurrence relationships. For each triggered scenario, the resulting spectra
are found from a ground-motion prediction equation. In this way, the ale-
atory uncertainty, as represented by the standard deviation of the lognor-
mal residuals, can be directly accounted for in each spectrum. The CDF
of the displacement demand can then be compared with the JPDFs of
displacement capacity, using Eq. (42), and the annual probability of failure
for a class of buildings can be found by integrating the failure probabilities
for all the earthquake scenarios.
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PROBABILISTIC DISPLACEMENT-BASED VULNERABILITY ASSESSMENT 197
The method proposed herein for vulnerability assessment can equally beemployed in conjunction with seismic demand obtained from probabilistic
hazard maps, provided that the aleatory variability of the ground motion
is then removed from the calculation of the probability of exceeding the
limit states. The authors also acknowledge that such an approach is sig-
nificantly more efficient in terms of computational effort. However, there
are many benefits in using a multiple earthquake scenario approach, not
least amongst which is the facility to obtain clear and reliable disaggre-
gations of the calculated losses. The probabilistic implementation of the
method enables scenario-based loss calculations, which take full account of
the ground-motion variability, to be performed efficiently.
3.3. Probabilistic treatment of the capacity
The probability density functions of the limit state displacement capacity
and period are found using the FORM. The reader is referred for example
to Pinto et al., (2004) for a description of the theory of FORM, as well
as Restrepo-Velez (2004) for a detailed description of the application of
FORM to the displacement capacity equations for un-reinforced masonry
structures. Essentially, FORM can be used to compute the approximate
CDF of a non-linear function of correlated parameters, such as the limit
state displacement capacity function and limit state period function.
As has been presented previously, the limit state displacement capac-
ity Lsi ) of each building class can be defined as a function of the fun-damental period (TLsi ), the geometrical properties of the building, and
the mechanical properties of the construction materials. Similarly, the limit
state period (TLsi ) of each building class can be defined as a function of the
height (or number of storeys), the geometrical properties of the building,
and the mechanical properties of the construction materials. The uncer-
tainty in Lsi and in TLsi is accounted for by constructing a vector of
parameters that collects their mean values and standard deviations. By
assigning probability distributions to each parameter, FORM can be used
to find both the CDF of the limit state displacement capacity, conditioned
to a period, and the CDF of the limit state period.
In the following section, the probability distributions suggested for each
parameter in the capacity equations are discussed. In the absence of datafrom which the definition of the probabilistic distributions for the param-
eters could be obtained, the work of other researchers has been con-
sulted, as indicated below. Sufficient data to fully construct the matrix of
correlation coefficients between parameters are not available at present and
so the parameters are currently assumed to be uncorrelated. Where exten-
sive data are not available, it is apparent that statistical properties are often
based on engineering judgement. This identifies an area where additional
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198 H. CROWLEY ET AL.
research could be focused, but the authors believe that systematic andcomprehensive sensitivity studies should first be carried out in order to
establish a hierarchy of priorities for refinement of input parameters to
earthquake loss models.
3.3.1. Probabilistic modelling of geometrical properties
A given building class within a selected urban area may comprise a large
number of structures that present the same number of storeys and failure
mode, but that feature varying geometrical properties (e.g., beam height,
beam length, column depth, column/storey height), due to the diverse
architectural and loading constraints that drove their original design andconstruction. Since such uncertainty does affect in a significant manner
the results of loss assessment studies (see Glaister and Pinho, 2003), it
is duly accounted for in the current method by means of the probabilis-
tic modelling described below. Clearly, one could argue that by carrying
out a detailed inspection of the building stock, such variability could be
significantly reduced (in the limit, if all buildings were to be examined,
it could be wholly eliminated), however at a prohibitive cost in terms of
necessary field surveys and modelling requirements (vulnerability would
then be effectively assessed on a case-by-case basis).
The geometrical properties of buildings present also a random variabil-
ity, due to imperfections introduced at the construction phase, which
affects nominally identical structures. This aleatory variability in the geo-metrical properties of reinforced concrete structural members, documented
by Mirza and MacGregor (1979a), amongst others, is much smaller in
magnitude than its epistemic counterpart described above (up to 20 times
smaller), for which reason its influence in a loss assessment outcome is of
reduced importance. In addition, the inclusion of geometrical random var-
iability in the proposed methodology, although feasible, would increase sig-
nificantly the computation efforts involved. Therefore, only the epistemic
component of the geometrical variability of reinforced concrete members
has been modelled in the present work, as described in what follows.
Preliminary studies have been carried out to aid the somewhat demon-
strative scope of this presentation. The probability distribution functionsto describe the variability of geometrical properties in an urban environ-
ment have been studied using a database of 21 European buildings from
the following countries: Portugal, Italy, Greece, Romania, and Yugoslavia
(see Crowley, 2003). The recently designed buildings have been separated
from buildings designed before 1980; it is assumed that the latter have been
designed before the advent of sound seismic design philosophy and so can
be used to describe the parameters of column-sway frames. The geometric
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PROBABILISTIC DISPLACEMENT-BASED VULNERABILITY ASSESSMENT 199
Figure 6. Histogram to show the proportions of beam length found in a population of
recently designed (i.e. post-1980) European buildings and a normal distribution fitted
to the data.
properties that have been obtained from this population of buildings com-
prise the following: beam length, beam depth, storey height and column
depth.
Normal or lognormal probability distributions have been fitted to the
histograms produced from the data; an example is given in Figure 6 wherea normal distribution can be seen to describe fairly well the distribution of
beam length in recently designed structures.
The data used in this brief study is by no means extensive and fur-
ther data will be added as it becomes available to this ongoing research.
Nevertheless, the current values and probability distributions for the
geometric properties, which have been obtained from the aforementioned
European database, are presented in Tables IV and V, respectively for old
and recent buildings.
Table IV. Mean and standard deviation values and probability distribution
for the geometrical parameters from a database of old (i.e. pre-1980) Euro-pean RC buildings
Parameter Mean (m) Standard deviation (m) Distribution
Beam length, lb 4.02 1.14 Normal
Beam depth, hb 0.44 0.06 Normal
Storey height, hs 3.22 0.59 Lognormal
Column depth, hc 0.38 0.14 Lognormal
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200 H. CROWLEY ET AL.
Table V. Mean and standard deviation values and probability distribution for
the geometrical parameters from a database of recent (i.e. post-1980) RC
European buildings
Parameter Mean (m) Standard deviation (m) Distribution
Beam length, lb 4.57 0.62 Normal
Beam depth, hb 0.56 0.06 Normal
Storey height, hs 3.00 0.12 Normal
Column depth, hc 0.51 0.09 Lognormal
The values in Tables IV and V seem rational; for example the mean
beam length of older structures is shorter than newer structures (expected
since recent years have witnessed an increase in adopted spans) which then
accounts for the higher mean beam depth found in the newer structures
category. The mean column depth of older structures is lower than that
in newer structures due to the lack of consideration of capacity design in
the former. The standard deviations of the geometric properties in older
structures are generally higher than in newer structures, which would also
be expected as structures built to more recent design codes are more likely
to comply with prevalent dimension standards.
The mean values found for the older buildings in Table IV havebeen used in the deterministic example application in Section 2.7 whilst
the mean values, standard deviations and probabilistic distributions in
Table IV are used in a probabilistic example application to be presented in
Section 3.4.
3.3.2. Probabilistic modelling of reinforcing bar yield strain
It will be assumed that once a probabilistic distribution for yield strength
has been found, it can be divided by a deterministic value of the modulus
of elasticity of 200 GPa to find the distribution of the yield strains due to
the low coefficient of variation (CV) of this property in reinforcing steel
(Mirza and MacGregor, 1979b). Mirza and MacGregor (1979b) studied the
variability of the material properties of Grade 40 and Grade 60 reinforc-
ing bars using the test data available in North America. They concluded
that for the yield strength of the bars, a normal distribution correlated well
in the vicinity of the mean whilst a beta distribution correlated well over
the whole range of data. The CV in the yield strength was found to be
between 8% and 12% when data were taken from different bar sizes from
many sources. More recently, the Probabilistic Model Code (JCSS, 2001)
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PROBABILISTIC DISPLACEMENT-BASED VULNERABILITY ASSESSMENT 201
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24
Yield strain (%)
Yield strain (%)
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.1 0.15 0.2 0.25 0.3 0.35
275 MPa
325 MPa
400 MPa
(a)
(b)
Figure 7. The normal distribution of yield strain for reinforcing steel with an assumed
CV of 10% for (a) a 5% characteristic strength of 275 MPa alone and (b) 5% charac-
teristic strengths of 275, 325 and 400 MPa compared together.
has suggested that a normal distribution can be adopted to model the yield
strength of steel. A normal distribution for the steel yield strength (and
subsequently yield strain) will be used in this method.
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202 H. CROWLEY ET AL.
Figure 7a shows an example of the normal probability density func-tions of yield strain for reinforcing steel with a characteristic strength
of 275 MPa, defined as the strain that has a 95% probability of being
exceeded. The CV has been assumed to be 10% using the aforementioned
suggestions by Mirza and Macgregor (1979b) to account for the variability
in the strength of bars of different sizes and from different manufacturers.
Figure 7b shows the probability density functions of yield strain for
three different characteristic yield strengths, each with an assumed CV
of 10%. The mean yield strain obviously increases with the mean yield
strength, and as the CV is assumed equal for all steel strengths, the stan-
dard deviation (equal to the CV multiplied by the mean) thus increases
with increased strength. The shape of the three functions shown in
Figure 7b can be explained by considering that the dispersion increaseswith strength but the area under the probability density function must
always be equal to 1.
The main difficulty in assigning a probability distribution to the yield
strength of the steel used in a group of buildings, however, is the possibility
that different grades have been used which would lead to a distribution
with multiple peaks and troughs, as illustrated in the example in Figure 7b.
One approach to solve this problem could be to calculate the probabil-
ity of failure for the building class given each possible steel grade, using
the normal distribution to model the dispersion for each grade such as in
Figure 7a, and then a weighted average of failure can be found, knowing
or judging the use of each steel grade within the building class. The validityof such an approach would become questionable, however, if different steel
grades were often used within individual buildings.
3.3.3. Probabilistic modelling of limit states threshold parameters
Dymiotis et al. (1999) have studied the seismic reliability of RC frames
using inter-storey drift to define the serviceability and ultimate structural
limit states. They have found that a lognormal distribution may be used to
describe the variability in inter-storey drift for both limit states; the drift
ratios found from test specimens for the serviceability limit state are plot-
ted in Figure 8a as a histogram with the corresponding lognormal distribu-tion superimposed. Kappos et al. (1999) report the ultimate concrete strain
reached in 48 tests of very well-confined RC members. A simple statistical
analysis of this data shows that it would appear that in the case of limit
state sectional strains a lognormal distribution is also able to describe the
variability (Figure 8b).
The non-structural limit states are defined in this method using inter-
storey drift. Considering it has been found by Dymiotis et al. (1999) that
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PROBABILISTIC DISPLACEMENT-BASED VULNERABILITY ASSESSMENT 203
Figure 8. (a) Histogram and suggested lognormal distribution of maximum experi-
mental inter-storey drifts at the structural serviceability limit state, reproduced from
Dymiotis et al. (1999) and (b) histogram and suggested lognormal distribution for the
ultimate concrete strain of very well-confined test specimens using data taken fromKappos et al. (1999).
a lognormal distribution defines well the variability in the limit state inter-
storey drift for test specimens, a lognormal distribution will be assumed
herein, using the mean drift ratios that have been provided previously in
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204 H. CROWLEY ET AL.
Table II. For the structural limit states, it is the sectional steel and concretestrains that are used to define the limit states in this method and it would
appear from Figure 8b that a lognormal distribution may also be applied
to describe the variability in these limit state parameters. The CV of the
limit state parameters can be seen from Figure 8a and b to be high and so
a value of 50% will be currently assumed until further data is available to
substantiate this assumption.
3.3.4. Probabilistic modelling of scatter in empirical relationships
A number of empirical relationships have been used to derive the functions
of displacement capacity and period that have been presented in Section 4.
These include expressions for the plastic hinge length members and theyield curvature of RC members, all of which are discussed in Glaister and
Pinho (2003). An additional empirical relationship has since been added to
the methodology and that is the formula derived by Crowley and Pinho
(2004) to relate the height of the building to its yield period of vibration
that has been discussed in Section 2.4. All of the aforementioned relation-
ships rely on a given coefficient to relate one set of structural properties
to another, as for example the coefficient of 0.1 in the yield period ver-
sus height equation, Ty = 0.1HT. The mean value and standard deviationof these coefficients have been taken from the studies carried out to derive
these formulae and a normal distribution is used to model the dispersion
in the coefficient.As has been frequently noted in the literature, the use of a normal dis-
tribution for quantities that are non-negative is inappropriate; however, it is
also claimed that when the variability in the parameter is small in relation
to its mean, the probability of obtaining a negative quantity would be vir-
tually zero (Sasani and Der Kiureghian, 2001). Thus it has been decided
that a normal distribution will be used to model the dispersion in the
empirical coefficients in the example given in the following section.
3.4. Illustrative example of probabilistic implementation
The illustrative example introduced in Section 2.7 is considered again here
in a probabilistic sense and so the uncertainty in the displacement demandspectrum as well as the dispersion in each of the parameters used to calcu-
late the yield limit state displacement capacity are considered. Readers are
referred to the work of Iaccino (2004) for an example of application of the
current vulnerability assessment method, in its probabilistic version, to a
real case study: the province of Imperia in Liguria, Italy.
For comparative purposes, the median displacement demand spectrum is
assumed to be that used in the deterministic example in Section 2.7, which
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PROBABILISTIC DISPLACEMENT-BASED VULNERABILITY ASSESSMENT 205
was taken from the 1992 Erzincan record. In order to obtain a measure ofthe variability, it is assumed that this spectrum represents that produced by
a specific scenario and that the aleatory variability of the spectral ordinates
would be represented by the logarithmic standard deviations of the ground-
motion prediction equations. For the purpose of this exercise, the standard
deviations associated with the predictive equations for displacement ordi-
nates derived by Bommer et al. (1998) are employed; these only cover peri-
ods up to 3.0 s, but are stable from about 0.8 s and are therefore assumed
to remain constant for longer response periods. The CDF of the demand
displacement at each period is then easily obtained assuming a lognormal
distribution, leading to the three-dimensional surface shown in Figure 9a.
At 3 s, the 50-percentile (median), 16-percentile (median minus 1 standard
deviation) and 84-percentile (median plus 1 standard deviation) values ofthe displacement spectrum are indicated. The median displacement spec-
trum, that has been presented previously in Figure 5, is again shown here
in Figure 9b, along with the 16-percentile and 84-percentile spectra. The
response ordinates obtained at 3 s for each spectrum have been highlighted;
these correspond to those values indicated in the three-dimensional CDF
plot in Figure 9a.
The parameters required in the definition of the displacement capac-
ity in this example are presented in Table VI. The origin of the values of
mean and standard deviation and the chosen probabilistic distributions has
been discussed in Section 3.3, however it is recalled here that these are
merely indicative, obtained from a preliminary analysis of a limited sampleof European buildings (Crowley, 2003). These parameters are assumed to
be uncorrelated at present until extensive data is available to calculate the
correlation coefficients between pairs of parameters. For the coefficients in
Table VI. Mean and standard deviation values and assumed distributions used for
the parameters in the limit state 1 (yield) displacement and period capacity equa-
tions for column-sway frames
Parameter Mean Standard deviation Distribution
Storey height, hs 3.22 m 0.59 m Lognormal
Column depth, hc 0.38 m 0.14 m Lognormal
Steel grade 275 yield strain, y 0.165% 0.0165% Normal
Coefficients
Periodheight 0.1 0.015 Normal
Column-sway yield rotation 0.43 0.09 Normal
Plastic hinge length 0.5 0.15 Normal
Column yield curvature 2.14 0.214 Normal
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206 H. CROWLEY ET AL.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0 1 2 3 4 5
Period (s)
SpectralDispla
cement(m)
50-percentile
16-percentile
84-percentile
(a)
(b)
76
Figure 9. (a) CDFs of the demand displacement at each period, with median, 16-per-
centile and 84-percentile values of displacement response indicated at 3 s and (b) the
median, 16-percentile and 84-percentile displacement demand spectra.
the empirical equations, the mean values have been taken from the liter-
ature where the derivation of these relationships is presented, introduced
earlier. The standard deviation has been found from either the CV pub-
lished with the associated empirical formula or, where this was not avail-
able, by assuming a tentative CV from the degree of scatter.
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PROBABILISTIC DISPLACEMENT-BASED VULNERABILITY ASSESSMENT 207
0.00
0.05
0.10
0.15
0.20
0.25
0 0.5 1 1.5 2
Period (s)
Displacement(m
)
(a)
(b)
Figure 10. (a) Probability density functions of yield displacement capacity, condi-tioned to period and (b) two-dimensional deterministic curve of yield displacement
capacity vs period.
Figure 10 shows an example of the probability density functions, for
a range of periods, of the first limit state (yield) displacement capac-
ity of a column-sway RC building class. If there were no uncertainty in
the calculation of displacement capacity then this graph would be the
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208 H. CROWLEY ET AL.
two-dimensional curve simply relating displacement capacity to period thathas been shown previously in the deterministic example in Figure 5. This
is shown by the line on the plot in Figure 10a and is repeated, for clarity,
in two-dimensions in Figure 10b.
The conditional probability of failure, introduced in Section 3.1, can
be calculated using the inner integral of Eq. (42) now that the condi-
tional CDF of the displacement demand (Figure 9a) and the conditional
probability density function of the displacement capacity (Figure 10a) have
been found. In this probabilistic framework, the aforementioned condi-
tional probabilities of failure can be unconditioned using the probability
density function of period corresponding to a given number of storeys.
Figure 11 shows the probability density functions of yield period for
various heights of column-sway RC frames. The increased dispersion in theprobability density function with increased number of storeys is notable
and can be explained when one considers that the mean storey height and
its associated standard deviation of all frames has been assumed to be the
same; however, the dispersion in the total height will be much higher when
the building contains more storeys. Considering that the period has been
shown to be related to the total height of a building, as discussed in Sec-
tion 2.4, it is expected that there will be more dispersion in the period of
vibration of buildings as the number of storeys increases.
For a given number of storeys, at a given period, the probability den-
sity of that period (from Figure 11) may be multiplied by the probability
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
0 0.5 1 1.5 2 2.5 3
Period (s)
PDF(T)
1 storeys
2 storeys
3 storeys
4 storeys
5 storeys
Figure 11. Example probability density functions of yield period for column-sway
frames of varying number of storeys.
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PROBABILISTIC DISPLACEMENT-BASED VULNERABILITY ASSESSMENT 209
Figure 12. Example JPDF of capacity for a four storey column-sway RC building
class.
density function of the corresponding displacement capacity (from Fig-
ure 10) so that the JPDF of period and displacement can be obtained, as
illustrated in Figure 12 for a four storey column-sway building class. As
should be expected, the volume below this surface is 100%; if there were no
uncertainty in the period or displacement capacity then this figure would
show a single spike with a probability equal to unity, such as the single
points of displacement capacity shown for each number of storeys in the
deterministic example in Figure 5.
The JPDF can be used in conjunction with the demand CDF through
the use of the reliability formulation of Eq. (42) to find the probability of
exceeding the yield limit state for each number of storeys. Table VII shows the
results of this probabilistic example and compares them with those obtained
in the deterministic example. It is observed that although earthquake loss
estimation studies based on a deterministic procedure of vulnerability assess-
ment would differ greatly from those based on a fully probabilistic method,
the higher vulnerability of the building stock between 3 and 10 storeys is
identified with both methods. The application of a fully probabilistic method
is recommended as this allows for a systematic and rational treatment of the
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210 H. CROWLEY ET AL.
Table VII. Comparison of yield limit state exceedance probabilities (Pf) for column-
sway frames obtained using a deterministic and a fully probabilistic procedure
Number of storeys Pf in deterministic example Pf in probabilistic example
0 0 0.00
1 0 0.06
2 0 0.24
3 1 0.44
4 1 0.54
5 1 0.60
6 1 0.62
7 1 0.60
8 1 0.55
9 1 0.49
10 1 0.42
11 0 0.35
12 0 0.29
uncertainties that exist when trying to predict the actions from future earth-
quakes and the resulting response of groups of buildings.
4. Brief Comparison with Existing Methodologies
4.1. Preamble
The methodology described in this paper allows the proportion of build-
ings falling within defined damage bands to be calculated for loss estima-
tion studies. The method of HAZUS (FEMA, 1999) has been discussed
in Section 1 wherein it was mentioned that the proportion of buildings
exceeding a given damage band is found in HAZUS using vulnerability
curves. A vulnerability curve gives the probability of failing a limit state,
given a value of displacement demand. In order to make a brief compar-
ison between HAZUS and the method outlined in this paper, the implied
vulnerability curves associated with this method are presented, even though
they are neither derived nor required for the application of the proposed
new approach. By making this comparison only in terms of the vulnera-
bility functions, it is possible to compare the new approach with HAZUS
without also considering the diffe