EESD_2005_Crowley_Bommer_A Probabilistic Displacement Based Vulnerability Assessment Procedure

8/3/2019 EESD_2005_Crowley_Bommer_A Probabilistic Displacement Based Vulnerability Assessment Procedure

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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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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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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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(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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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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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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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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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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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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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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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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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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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


16/47


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)


17/47


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


18/47


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)


19/47


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:


20/47


=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


21/47


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.


22/47


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)


23/47


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:


24/47


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.


25/47


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


26/47


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


27/47


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


28/47


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)


29/47


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 (%)

PDF

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

PDF

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.


30/47


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


31/47


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


32/47


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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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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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.


35/47


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


36/47


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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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


38/47


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

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EESD_2005_Crowley_Bommer_A Probabilistic Displacement Based Vulnerability Assessment Procedure