# Comparison of Two Mechanics-Based Methods for Simplified Structural Analysis in Vulnerability Assessment

Authors: H. Crowley; B. Borzi; R. Pinho; M. Colombi; M. Onida
Journal: Advances in Civil Engineering
 (2008)

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## Abstract

Analytical vulnerability assessment methods should ideally be validated or verified by comparing their damage predictions with actual observed damage data. However, there are a number of difficulties related to the comparison of analytical damage predictions with observed damage; for example, there are large uncertainties related to the prediction of the ground motions to which the damaged buildings have been subjected. Until such problems can be resolved, it is worthwhile considering the mechanics of simplified analytical vulnerability assessment methods and validating this part of the methodology through comparisons with detailed structural models. This paper looks at two mechanics-based vulnerability assessment methods (DBELA and SP-BELA) and compares the nonlinear static response predicted with these methods with finite elements-based nonlinear analyses of prototype buildings. A comparison of the predicted response of urban populations of buildings using the two methods is then carried out, and the influence of these differences on vulnerability curves is studied.


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## Body

## 1. Introduction

The vulnerability of reinforced concrete buildings has been the focus of
much research over the past 30 years (e.g., [1]), with methods varying from those based on the
observed damage of buildings in past earthquakes to detailed analytical studies
of prototype buildings under seismic action. The advantage of empirical
vulnerability assessment methods, based on observed damage data for a given
type of construction, is that they convey a high level of confidence of what
might be the damage to that building type under similar levels of future ground
motions. However, there are a number of disadvantages of using observed damage
data to produce vulnerability functions; these shortcomings are mainly related
to errors in the compilation of damage survey forms, the incompleteness of the
surveys, and the difficulties in predicting the ground motion to which the
buildings were subjected (see, e.g., [2]).Analytical vulnerability methods are becoming ever more popular, but one
of the main criticisms assigned to these methods relates to a lack of
validation or verification of the damage predictions. Comparisons of analytical
damage predictions with observed damage data is one way in which these
criticisms can be addressed; however, there are a large number of difficulties with
such comparisons. Even with access to detailed exposure and damage data, the
main difficulty in comparing the predictions from an analytical method with the
actual damage data resides in the uncertainty in the prediction of the ground
motion (see, e.g., [3, 4]).
Without recordings of the ground motion at the locations of the damaged
buildings, it is hard to know whether the errors in the predictions are due to
an incorrect estimation of the vulnerability or of the ground motion. Whilst
studies will continue to be carried out with the aim of verifying or validating
analytical vulnerability assessment methodologies, the authors of this paper
have decided to focus their efforts on checking the conceptual aspects of two
mechanics-based methods which they have proposed (DBELA and SP-BELA). In fact,
one of the major advantages of analytical vulnerability assessment methods is
that even without the possibility to check the methods with field data, the
patterns in the behaviour of different building classes can be physically
explained.The premise for the verification of the structural capacity defined in
the two simplified vulnerability assessment methods considered herein is that
the nonlinear response of a reinforced concrete structure can be obtained from
a nonlinear static analysis of the structure. This requires the variability in
the dynamic response of the building to be ignored, but it is noted that this
source of variability could nevertheless be subsequently included in the methods
by carrying out a number of nonlinear dynamic analyses. In this paper, finite
element nonlinear static analyses of gravity-load designed and
seismically-designed 4 storey reinforced concrete buildings will be compared
with the lateral strength and initial stiffness predicted from two simplified vulnerability
methods, DBELA and SP-BELA. The aim of this comparison is to verify and compare
the definition of the structural capacity within these two methods for
individual structures. The vulnerability predictions of these two methods in
terms of vulnerability curves, describing the probability of exceeding a damage
state given a level of severity of the ground motion, will also be compared and
the differences between the two methods will be discussed.


## 2. Mechanics-Based Methodologies

In the following sections,
the two vulnerability methodologies considered herein are briefly described.
More details on the derivation and implementation of the methods are given in
Crowley et al. [5] for the displacement-based
earthquake loss assessment (DBELA) method and Borzi et al. [6] for the simplified pushover-based earthquake loss
assessment (SP-BELA) method.


### 2.1. DBELA

The displacement-based
earthquake loss assessment (DBELA) method is based on a procedure proposed by
Calvi [7] which utilises the principles of the direct displacement-based
design method (e.g., [8]). In DBELA, as in the method proposed by
Calvi [7], the period of vibration of a building is defined from an
empirical period-height relationship. The multidegree-of-freedom (MDOF) building
is modelled as a single DOF system and the displacement capacity of the SDOF
system is compared with the displacement demand from a displacement response
spectrum, as described further in the following.For reinforced
concrete frames, the elastic period of vibration is related to the building
height through a relationship that has been derived from the periods of
vibration calculated analytically for a number of existing European RC
buildings [9, 10]. For bare reinforced concrete frames,
the following equation has been derived: (1)TLSy=0.1H, where TLSy is the period of vibration at yield and H is the
height of the building, in metres. Buildings designed using design codes based
on modern seismic design philosophy were also briefly studied by Crowley
[11], and it was
found that the period of vibration of these frames was much lower than that of
the older pre-1980 buildings. The yield period of vibration for a limited
number of frames from 4 to 12 storeys designed with the modern codes was found
to be represented by the following formula: (2)TLSy=0.48H0.3. In DBELA, elastic perfectly-plastic behaviour is
assumed and, therefore, the relationship between the elastic period of
vibration and the period corresponding to a limit state i(TLSi) is given by the following relationship, which
is also illustrated in Figure 1: (3)TLSi=TLSyμLSi. where μLSi is the ductility at the limit state under consideration. The
collapse multiplier shown in Figure 1 is the base shear capacity of
the building divided by its seismic weight (which is 100% dead load plus 30% of the live load).Figure 1
Capacity curve illustrating elastic perfectly-plastic
structural behaviour and the displacement capacity at different
limit states (ΔLSi), the collapse multiplier
(λ), and secant stiffness (Ki).The displacement
capacity of the building is related to different damage states which are
identifiable through limit states (see Figure1). Three limit state
conditions have currently been taken into account: light damage, significant
damage, and collapse. The light damage limit condition refers to the situation
where the building can be used after the earthquake without the need for repair
and/or strengthening. If a building deforms beyond the significant damage limit
state, it cannot be used after the earthquake without strengthening.
Furthermore, at this level of damage it might not be economically advantageous
to repair the building. If the collapse limit condition is achieved, the
building becomes unsafe for its occupants as it is no longer capable of
sustaining any further lateral force nor the gravity loads for which it has
been designed.The limit state conditions are related to the rotation requirements imposed on
the plastic hinges within the building that lead to the formation of a
mechanism. Figure2 shows the possible response mechanisms of a
reinforced concrete frame: a beam-sway mechanism is caused by plastic hinges
forming in all the beams above the first floor and in all of the columns at the
base of the building whilst the column-sway mechanism forms when plastic hinges
form at both ends of the columns in the ground floor. Based on the shape of the
displaced profile, the displacement capacity of the equivalent SDOF system can
be calculated using the elastic displacement and the post-elastic displacement
at the height of the SDOF system, as described below. In DBELA, without detailed information on the
strength of the building, the collapse failure mechanism is an assumption that
has to be made a priori. For
seismically designed buildings, a beam sway collapse mechanism is considered
with the activation of the mechanism at the foundation level (see Figure 2(a)). For nonseismically
designed RC buildings, a column-sway collapse mechanism is generally assumed
with a soft storey at the first floor (see Figure 2(b)). Future developments of
the method will consider the possibility of attaining a mechanism at any storey
within the building.Response mechanisms for a frame and the associated deformed shapes (a) beam-sway mechanism, (b) column-sway
mechanism.
(a)(b)The equations used
to calculate the displacement capacity in DBELA are described in detail in
Crowley et al. [5]. The
yield displacement capacity, ΔLSy, is calculated by the yield chord rotation of a single degree of freedom
system (Θy) multiplied by the effective height of the building (κ1HT), as shown in (4). The yield rotation for the
beam-sway frames is shown in (5) whilst that of the column-sway frames is given
in (6), where lb is the length of the beam, hb is the depth
of the beam, hs is the height of the storey, and hc is
the depth of the column: (4)ΔLSy=κ1HTΘy,(5)Θy=0.5εylbhb,(6)Θy=0.43εyhshc. The post-yield
displacement capacity is calculated using (7) and (8) for beam and column-sway
frames, respectively, where Hk is the equivalent height of the part
of the building above the activation of the global collapse failure mechanism.
The limit state chord rotation (ΘLSi) is given by (9), where ϕLSi and ϕy
represent the limit state and yield
curvatures, respectively, and Lpl is the plastic hinge length: (7)ΔLSi=ΔLSy+(ΘLSi−Θy)Hk,(8)ΔLSi=ΔLSy+(ΘLSi−Θy)hs,(9)ΘLSi=Θy+(ϕLSi−ϕy)Lpl. The limit state
and yield curvatures are proportional to the strains in the material and
inversely proportional to the section depth (e.g., [12]) whilst the
plastic hinge length is proportional to the section depth. Hence, it is
apparent that knowledge of a number of geometrical and material properties
allows the displacement capacity of reinforced concrete buildings to be
estimated.


### 2.2. SP-BELA

The simplified pushover-based earthquake loss assessment
(SP-BELA) method makes use of a simplified pushover methodology to evaluate the
capacity of the building stock. Pushover curves are calculated for
buildings with a given layout in plan, such as that shown in Figure3; this is a typical plan for Southern European and Italian
buildings (see, e.g., [13–15]). The prototype structure
defined for nonseismically designed buildings has the frames oriented in one
direction only (the x direction), and in the orthogonal direction the frame
effect is guaranteed by an effective width of the floor slab. For seismically
designed frames, the frame action is ensured in both orthogonal directions of
the building. A simulated design procedure is first used to design each
prototype building and assign beam and column dimensions and reinforcement
details. In SP-BELA, to account for seismically designed buildings, the method
of equivalent static lateral distribution of forces has been adopted during the
simulated design procedure by assuming a lateral load coefficient, c, corresponding to 5%, 7.5%, 10%, and 12.5% of the seismic weight (which is the
dead weight plus 30% of the live load). These values of lateral load
coefficient have been chosen considering the evolution of seismic design
practice in Italy
[16, 17]. In the 1930s in Italy,
the lateral load coefficient was defined as 10% of the weight in the highest
seismic zone and 7% in the second seismic zone. A third seismic zone was
introduced in the 1980s with a lateral load coefficient of 4%. Considering
these values as a basis, and the possibility that importance factors would
increase these values of c for certain structures, the aforementioned range of
lateral load coefficients was decided upon for the current study. In the design
procedure, capacity design is not currently integrated, since seismic design
codes in the Mediterranean area have only recently begun to take a hierarchy of
resistance into account.Figure 3
Plan view of the RC frame building assumed as
representative of the building structural type for seismically
designed buildings in Mediterranean countries.Once
the prototype building has been designed, a simplified pushover curve is
produced considering simple principles of mechanics in order to define the
collapse multiplier(λ) (which represents the base shear capacity divided
by the seismic weight). This multiplier, together with the yield displacement
capacity (Δy), is used to calculate the yield stiffness and subsequently the
elastic period of vibration (TLSy) based on the elastoplastic
behaviour shown previously in Figure 1: (10)TLSy=2πmKy=2πΔygλ. The post-yield
period of vibration is calculated in the same way as described previously for
DBELA. The limit state definition and displacement-based framework used in
SP-BELA is also based on (4), (7), and (8); however the yield and ultimate
chord rotations are taken from the proposals by Panagiotakos & Fardis (2001) [18]. The chord rotation corresponding to yielding Θy is thus given by (11)Θy=ϕyLV3+0.0013(1+1,5hLV)+0.13ϕydbfyfc, where ϕy
is the yield curvature of the columns,
h is the section height, db is the longitudinal bar diameter, fy and fc are the resistance of steel and concrete in MPa,
respectively, and LV
is the shear span of the columns (equal to the ratio between bending moment and
shear). A double bending distribution is commonly assumed for columns, and
hence LV
is taken as half of the interstorey height. The ultimate limit state chord
rotation is calculated using the following formula: (12)ΘLSi=1γel(Θy+(ϕLSi−ϕy)Lpl(1−0.5LplLV)), where γel is 1.5 for the primary structural elements and 1 for all others, ϕu
is the ultimate curvature, and Lpl is the plastic hinge length. According to the new
Italian regulations OPCM N° 3274 [19], the plastic hinge length can be
calculated as (13)Lpl=0,1LV+0,17h+0,24dbfyfc, whilst the
ultimate curvature is assumed to be (14)ϕu=εcu+εsuh, where
εcu and εsu are the ultimate concrete and steel strains, respectively.In
SP-BELA no a priori assumptions
are necessary to define the collapse mechanism, as this is determined within
the procedure by checking which elements of the frame will fail first considering
the relative strength between the beams and the columns; thus, the mechanism
can be found to form on any floor of the building. The definition of the
failure mechanism in SP-BELA is discussed further in Section3.2.


## 2.1. DBELA

The displacement-based
earthquake loss assessment (DBELA) method is based on a procedure proposed by
Calvi [7] which utilises the principles of the direct displacement-based
design method (e.g., [8]). In DBELA, as in the method proposed by
Calvi [7], the period of vibration of a building is defined from an
empirical period-height relationship. The multidegree-of-freedom (MDOF) building
is modelled as a single DOF system and the displacement capacity of the SDOF
system is compared with the displacement demand from a displacement response
spectrum, as described further in the following.For reinforced
concrete frames, the elastic period of vibration is related to the building
height through a relationship that has been derived from the periods of
vibration calculated analytically for a number of existing European RC
buildings [9, 10]. For bare reinforced concrete frames,
the following equation has been derived: (1)TLSy=0.1H, where TLSy is the period of vibration at yield and H is the
height of the building, in metres. Buildings designed using design codes based
on modern seismic design philosophy were also briefly studied by Crowley
[11], and it was
found that the period of vibration of these frames was much lower than that of
the older pre-1980 buildings. The yield period of vibration for a limited
number of frames from 4 to 12 storeys designed with the modern codes was found
to be represented by the following formula: (2)TLSy=0.48H0.3. In DBELA, elastic perfectly-plastic behaviour is
assumed and, therefore, the relationship between the elastic period of
vibration and the period corresponding to a limit state i(TLSi) is given by the following relationship, which
is also illustrated in Figure 1: (3)TLSi=TLSyμLSi. where μLSi is the ductility at the limit state under consideration. The
collapse multiplier shown in Figure 1 is the base shear capacity of
the building divided by its seismic weight (which is 100% dead load plus 30% of the live load).Figure 1
Capacity curve illustrating elastic perfectly-plastic
structural behaviour and the displacement capacity at different
limit states (ΔLSi), the collapse multiplier
(λ), and secant stiffness (Ki).The displacement
capacity of the building is related to different damage states which are
identifiable through limit states (see Figure1). Three limit state
conditions have currently been taken into account: light damage, significant
damage, and collapse. The light damage limit condition refers to the situation
where the building can be used after the earthquake without the need for repair
and/or strengthening. If a building deforms beyond the significant damage limit
state, it cannot be used after the earthquake without strengthening.
Furthermore, at this level of damage it might not be economically advantageous
to repair the building. If the collapse limit condition is achieved, the
building becomes unsafe for its occupants as it is no longer capable of
sustaining any further lateral force nor the gravity loads for which it has
been designed.The limit state conditions are related to the rotation requirements imposed on
the plastic hinges within the building that lead to the formation of a
mechanism. Figure2 shows the possible response mechanisms of a
reinforced concrete frame: a beam-sway mechanism is caused by plastic hinges
forming in all the beams above the first floor and in all of the columns at the
base of the building whilst the column-sway mechanism forms when plastic hinges
form at both ends of the columns in the ground floor. Based on the shape of the
displaced profile, the displacement capacity of the equivalent SDOF system can
be calculated using the elastic displacement and the post-elastic displacement
at the height of the SDOF system, as described below. In DBELA, without detailed information on the
strength of the building, the collapse failure mechanism is an assumption that
has to be made a priori. For
seismically designed buildings, a beam sway collapse mechanism is considered
with the activation of the mechanism at the foundation level (see Figure 2(a)). For nonseismically
designed RC buildings, a column-sway collapse mechanism is generally assumed
with a soft storey at the first floor (see Figure 2(b)). Future developments of
the method will consider the possibility of attaining a mechanism at any storey
within the building.Response mechanisms for a frame and the associated deformed shapes (a) beam-sway mechanism, (b) column-sway
mechanism.
(a)(b)The equations used
to calculate the displacement capacity in DBELA are described in detail in
Crowley et al. [5]. The
yield displacement capacity, ΔLSy, is calculated by the yield chord rotation of a single degree of freedom
system (Θy) multiplied by the effective height of the building (κ1HT), as shown in (4). The yield rotation for the
beam-sway frames is shown in (5) whilst that of the column-sway frames is given
in (6), where lb is the length of the beam, hb is the depth
of the beam, hs is the height of the storey, and hc is
the depth of the column: (4)ΔLSy=κ1HTΘy,(5)Θy=0.5εylbhb,(6)Θy=0.43εyhshc. The post-yield
displacement capacity is calculated using (7) and (8) for beam and column-sway
frames, respectively, where Hk is the equivalent height of the part
of the building above the activation of the global collapse failure mechanism.
The limit state chord rotation (ΘLSi) is given by (9), where ϕLSi and ϕy
represent the limit state and yield
curvatures, respectively, and Lpl is the plastic hinge length: (7)ΔLSi=ΔLSy+(ΘLSi−Θy)Hk,(8)ΔLSi=ΔLSy+(ΘLSi−Θy)hs,(9)ΘLSi=Θy+(ϕLSi−ϕy)Lpl. The limit state
and yield curvatures are proportional to the strains in the material and
inversely proportional to the section depth (e.g., [12]) whilst the
plastic hinge length is proportional to the section depth. Hence, it is
apparent that knowledge of a number of geometrical and material properties
allows the displacement capacity of reinforced concrete buildings to be
estimated.


## 2.2. SP-BELA

The simplified pushover-based earthquake loss assessment
(SP-BELA) method makes use of a simplified pushover methodology to evaluate the
capacity of the building stock. Pushover curves are calculated for
buildings with a given layout in plan, such as that shown in Figure3; this is a typical plan for Southern European and Italian
buildings (see, e.g., [13–15]). The prototype structure
defined for nonseismically designed buildings has the frames oriented in one
direction only (the x direction), and in the orthogonal direction the frame
effect is guaranteed by an effective width of the floor slab. For seismically
designed frames, the frame action is ensured in both orthogonal directions of
the building. A simulated design procedure is first used to design each
prototype building and assign beam and column dimensions and reinforcement
details. In SP-BELA, to account for seismically designed buildings, the method
of equivalent static lateral distribution of forces has been adopted during the
simulated design procedure by assuming a lateral load coefficient, c, corresponding to 5%, 7.5%, 10%, and 12.5% of the seismic weight (which is the
dead weight plus 30% of the live load). These values of lateral load
coefficient have been chosen considering the evolution of seismic design
practice in Italy
[16, 17]. In the 1930s in Italy,
the lateral load coefficient was defined as 10% of the weight in the highest
seismic zone and 7% in the second seismic zone. A third seismic zone was
introduced in the 1980s with a lateral load coefficient of 4%. Considering
these values as a basis, and the possibility that importance factors would
increase these values of c for certain structures, the aforementioned range of
lateral load coefficients was decided upon for the current study. In the design
procedure, capacity design is not currently integrated, since seismic design
codes in the Mediterranean area have only recently begun to take a hierarchy of
resistance into account.Figure 3
Plan view of the RC frame building assumed as
representative of the building structural type for seismically
designed buildings in Mediterranean countries.Once
the prototype building has been designed, a simplified pushover curve is
produced considering simple principles of mechanics in order to define the
collapse multiplier(λ) (which represents the base shear capacity divided
by the seismic weight). This multiplier, together with the yield displacement
capacity (Δy), is used to calculate the yield stiffness and subsequently the
elastic period of vibration (TLSy) based on the elastoplastic
behaviour shown previously in Figure 1: (10)TLSy=2πmKy=2πΔygλ. The post-yield
period of vibration is calculated in the same way as described previously for
DBELA. The limit state definition and displacement-based framework used in
SP-BELA is also based on (4), (7), and (8); however the yield and ultimate
chord rotations are taken from the proposals by Panagiotakos & Fardis (2001) [18]. The chord rotation corresponding to yielding Θy is thus given by (11)Θy=ϕyLV3+0.0013(1+1,5hLV)+0.13ϕydbfyfc, where ϕy
is the yield curvature of the columns,
h is the section height, db is the longitudinal bar diameter, fy and fc are the resistance of steel and concrete in MPa,
respectively, and LV
is the shear span of the columns (equal to the ratio between bending moment and
shear). A double bending distribution is commonly assumed for columns, and
hence LV
is taken as half of the interstorey height. The ultimate limit state chord
rotation is calculated using the following formula: (12)ΘLSi=1γel(Θy+(ϕLSi−ϕy)Lpl(1−0.5LplLV)), where γel is 1.5 for the primary structural elements and 1 for all others, ϕu
is the ultimate curvature, and Lpl is the plastic hinge length. According to the new
Italian regulations OPCM N° 3274 [19], the plastic hinge length can be
calculated as (13)Lpl=0,1LV+0,17h+0,24dbfyfc, whilst the
ultimate curvature is assumed to be (14)ϕu=εcu+εsuh, where
εcu and εsu are the ultimate concrete and steel strains, respectively.In
SP-BELA no a priori assumptions
are necessary to define the collapse mechanism, as this is determined within
the procedure by checking which elements of the frame will fail first considering
the relative strength between the beams and the columns; thus, the mechanism
can be found to form on any floor of the building. The definition of the
failure mechanism in SP-BELA is discussed further in Section3.2.


## 3. Verification through Finite Elements Analysis

### 3.1. Case-Studies Considered and Modelling

The prototype structure
presented in Figure3 has been designed with 4 storeys according to the
1992 Italian reinforced concrete design code [20],
first considering gravity loads
alone and then assuming lateral loads of 5%, 7.5%, 10%,
and 12.5% of the seismic weight. The characteristic concrete compressive
strength has been taken as 30 MPa and the characteristic yield strength of the
steel has been taken as 440 MPa. The main characteristics of the central frame
in the x direction are presented for each design case in Tables 1 and 2.Table 1
Geometrical properties of the central frame of the
prototype building in Figure3
designed to
different proportions of lateral load (c).
SectionDepth*/width (mm)c=0%c=5%c=7.5%c=10%c=12.5%Central beam320/1300370/1500370/1800420/1600420/1900Col P41st Fl300/400300/300300/400300/400300/6002nd Fl300/3003rd Fl4th FlCol P51st Fl700/300600/300900/300800/400800/4002nd Fl600/3003rd Fl400/3004th Fl300/300*The depth of the column is the section dimension in the x direction.Table 2
Reinforcing steel in the design of the central frame
of the prototype building in Figure3 designed to different proportions of lateral load (c).
Section% area of tensile reinforcing steelc=0%c=5%c=7.5%c=10%c=12.5%Central beam0.30/0.68**0.40/0.740.38/0.750.42/0.790.39/0.75Col P41st Fl0.671.41.22.11.52nd Fl0.893rd Fl0.454th Fl0.45Col
P51st Fl1.201.51.10.90.92nd Fl0.813rd Fl0.674th Fl0.45**The first number refers to the edge supports whilst the second number refers to
the central supports of the building.Nonlinear finite
element models of the designed structures have been modelled using the program
SeismoStruct [21]. This program is capable
of predicting the large displacement behaviour and the collapse load of framed
structures under static or dynamic loading, duly accounting for geometric
nonlinearities and material inelasticity. Its accuracy in predicting the
seismic response of reinforced concrete structures has been demonstrated
through comparisons with experimental results derived from pseudodynamic tests
carried out on large-scale models (e.g., [22, 23]).A static pushover
analysis of each building has been carried out in thex direction of the
building as this is the direction of the building where the frame action is
guaranteed for both nonseismically and seismically designed buildings. For the
pushover analysis, the mean compressive strength of the concrete has been
assumed to be 37.35 MPa whilst the mean yield strength of the steel has been
taken as 473 MPa.


### 3.2. Failure Mechanism

One first
comparison that can be made regards the failure mechanism obtained with both
methods. In SP-BELA, the mechanism is calculated by considering the following criteria.(i)
If there is a shear failure
mechanism detected in at least one column, the capacity curve will be
interrupted at the lateral force that produces this failure. This choice
is consistent with the fact that the shear failure mechanism is brittle
and does not have significant associated dissipative capacity. Therefore,
the structure cannot enter the nonlinear range.(ii)
If all the columns within a certain
storey fail in bending, than a column-sway collapse mechanism will be
activated.(iii)
If after the development of plastic
hinges in all beams, plastic hinges form in all columns at a certain
level, a beam-sway collapse mechanism will be activated.There could be a
situation in which at the storey where the mechanism is activated, some of the
columns are stronger than the beams, or vice versa. Therefore, it cannot be
clearly identified whether a beam or a column-sway mechanism will be activated.
The analyses undertaken to validate the SP-BELA procedure have shown that in
such cases the collapse mechanism can be assumed as a mixed mechanism defined using
the average displacement capacities of the column and beam-sway mechanisms.If the reinforcing steel is known in the building then a
sway potential index can be estimated in DBELA in order to define the
mechanism, using the following simplified formula based on the proposal by
Priestley et al. [24]: (15)SI=∑(Mbl+Mbr)∑(Mca+Mcb)∝∑(hbAsbfy)l+(hbAsbfy)r∑(hcAscfy)c+(hcAscfy)b=∑(hbAsb)l+(hbAsb)r∑(hcAsc)a+(hcAsc)b, where Mbl and Mbr are the flexural strengths at either side of the joint (where
l is left and r is right), and Mca and Mcb are the
flexural column strengths (a)
above and (b) below the joint, all of which are summed for all joints of
a given floor. This formula can be simplified using just the area of tensile
reinforcing steel Asb in each beam left and right of the joint, the
area of tensile reinforcing steel in each column, Asc above and
below the joint, the beam section dimensions, hb, and the column
section dimensions, hc, either side of the joint and fy which
is the yield strength of the steel, as shown in (15). If the sway index is
higher than 1, a column-sway mechanism is assumed to form (see
Figure 2(b)), below 0.85 a beam sway mechanism (see Figure 2(a)) will form whilst between
0.85 and 1 it is conservative to assume a column-sway mechanism. It is
clear from this formula that when the
reinforcement is unknown, in DBELA the collapse mechanism needs to be selected a priori assuming that seismically
and nonseismically designed buildings are more prone to a beam- and column-sway
mechanism, respectively.The calculated
sway potential indices from DBELA for the 5 different designs of the 4 storey
building are reported in Table3. These calculations show that a column-sway mechanism
in the third floor of the nonseismically designed building is predicted whilst
all seismically designed buildings demonstrate a beam-sway or global mechanism.
SP-BELA also predicts a column-sway at the third floor of the building designed
to gravity loads (i.e., c=0%) and global mechanisms for all other buildings.Table 3
Sway potential index
calculated using the formula in (15) for the 4-storey building from Figure 3 designed to different proportions of lateral load (c).
Floorc=0%c=5%c=7.5%c=10%c=12.5%1st0.50.60.50.40.42nd1.00.60.50.40.43rd*2.60.60.50.40.4*The
storey mechanism for the fourth floor is not calculated as plastic hinges are
likely to form in the top of the columns at the roof, even for a global
mechanism.The
mechanisms observed by plotting the deformed shapes from the finite elements
analyses were in full agreement with the predictions by both DBELA and SP-BELA,
as can be seen from the post-yield interstorey drift profiles for the
nonseismically designed and the seismically-designed (c=5%) design cases in Figure 4. The third storey sway mechanism is clearly present
in the nonseismically designed frame whilst the seismically designed building
has a global mechanism.Post-yield interstorey drift profiles inx direction of the building in Figure 3(a) nonseismically designed (b) c=5%.
(a)(b)


### 3.3. Yield Displacement Capacity, Collapse Multiplier, and Initial Stiffness

The yield displacement
capacity of the building in thex direction has been calculated in both
methodologies using the section properties, the yield strain of the steel, and
the appropriate yield rotation equations. The value of the collapse multiplier
is explicitly calculated in SP-BELA based on the strength of the structural
members. On the other hand, in DBELA the collapse multiplier (λ) has been calculated using
the results of the displacement capacity (Δy) and the period of vibration at yield (Ty) using the
following formula, which is simply a rearrangement of (10): (16)λ=4π2ΔygT2=4π2Δyg(0.1H)2. The pushover plots of collapse multiplier (which is equal to the base shear
force divided by the seismic weight) versus displacement at the effective
height of the building for the nonseismically designed and the 5% laterally
loaded building are plotted in Figures 5(a) and  5(b), respectively. These figures show the nonlinear
pushover plot from the finite elements model compared with the simplified
bilinear pushover plots of SP-BELA and DBELA. A bilinear pushover curve based
on the method described in Annex B of Eurocode 8 [25] has also been
added to the plots to ease comparison between the results of the simplified
methods and the finite element analyses. The comparison with the finite elements
pushover analysis is seen to be acceptable for both DBELA and SP-BELA in terms
of the collapse multiplier. However, it is apparent that SP-BELA underestimates
and DBELA slightly overestimates the stiffness of the nonseismically designed
building, which responds with a column-sway mechanism. In SP-BELA the yield
stiffness is obtained from the calculated base shear capacity and the yield
displacement. The yield displacement is based on the yield rotation in (11)
which includes additional shear and joint deformations which are not accounted
for in the finite elements package, thus explaining the lower stiffness
prediction with SP-BELA. Both SP-BELA and DBELA underestimate the stiffness of
the seismically designed building (see Figure
5(b)) which responds with a beam-sway mechanism; again
this is probably due to the additional shear and joint deformations explicitly accounted
for in the beam-sway rotation in DBELA (see (5) )and the equivalent equation used in SP-BELA (see (11)). It
is noted that the equation for beam-sway rotation used in DBELA (see (5)) has been specifically calibrated to
account for shear and joint deformations, whereas the associated increase
assumed in the column-sway rotation equation (see (6)) has not yet been verified (see [26]).Comparison of pushover analyses inx direction of the
building in Figure 3(a) nonseismically designed (b) c=5%.
(a)(b)Figure6(a) presents the comparison for the building designed
to a lateral load of 7.5% of the seismic weight. SP-BELA is seen to correctly
predict the collapse multiplier of this frame, whilst DBELA greatly underpredicts
the strength, though the displacement capacity is similar for both methods. The
reason for this difference is due to the imposed period of vibration and thus
initial stiffness used in DBELA. This formula, presented previously in (1), has
been derived using the regression analysis of the period of vibration of
nonseismically or weakly seismically designed frames. Under conventional
force-based design, the consideration of lateral loads in the design of a frame
or building will undoubtedly increase the lateral stiffness of the building
through the need to use larger section dimensions for the columns. This can be
seen by studying the section dimensions of the frame shown in Table 1. A lower period of vibration should thus be used
for seismically designed buildings. A modified period-height relationship
(T=0.085⋅H) which is found
to give good results for the building designed to the 7.5% lateral force is
shown in Figure
6(b),
whereas even lower periods of vibration are required for the buildings designed
to lateral forces of 10% and 12.5% of the seismic weight, as shown in Figures 7(a) and 7(b),
respectively. SP-BELA is seen in these figures to consistently match the
collapse multiplier for all seismic designs as the strength of the building is
explicitly calculated as part of this methodology.Comparison of pushover analyses for seismically
designed building,c=7.5%, (a) original period-height relationship in DBELA (b)
modified period-height relationship in DBELA.
(a)(b)Comparison of pushover analyses for seismically
designed building (a)c=10% (b) c=12.5%.
(a)(b)It appears from the analyses
presented herein that the SP-BELA methodology defines well the
single-degree-of-freedom structural characteristics of individual buildings,
whilst further calibration of the period-height relationship used in the DBELA
methodology would be required for the Southern European/Italian building stock
in order to match the different levels of seismic design.


## 3.1. Case-Studies Considered and Modelling

The prototype structure
presented in Figure3 has been designed with 4 storeys according to the
1992 Italian reinforced concrete design code [20],
first considering gravity loads
alone and then assuming lateral loads of 5%, 7.5%, 10%,
and 12.5% of the seismic weight. The characteristic concrete compressive
strength has been taken as 30 MPa and the characteristic yield strength of the
steel has been taken as 440 MPa. The main characteristics of the central frame
in the x direction are presented for each design case in Tables 1 and 2.Table 1
Geometrical properties of the central frame of the
prototype building in Figure3
designed to
different proportions of lateral load (c).
SectionDepth*/width (mm)c=0%c=5%c=7.5%c=10%c=12.5%Central beam320/1300370/1500370/1800420/1600420/1900Col P41st Fl300/400300/300300/400300/400300/6002nd Fl300/3003rd Fl4th FlCol P51st Fl700/300600/300900/300800/400800/4002nd Fl600/3003rd Fl400/3004th Fl300/300*The depth of the column is the section dimension in the x direction.Table 2
Reinforcing steel in the design of the central frame
of the prototype building in Figure3 designed to different proportions of lateral load (c).
Section% area of tensile reinforcing steelc=0%c=5%c=7.5%c=10%c=12.5%Central beam0.30/0.68**0.40/0.740.38/0.750.42/0.790.39/0.75Col P41st Fl0.671.41.22.11.52nd Fl0.893rd Fl0.454th Fl0.45Col
P51st Fl1.201.51.10.90.92nd Fl0.813rd Fl0.674th Fl0.45**The first number refers to the edge supports whilst the second number refers to
the central supports of the building.Nonlinear finite
element models of the designed structures have been modelled using the program
SeismoStruct [21]. This program is capable
of predicting the large displacement behaviour and the collapse load of framed
structures under static or dynamic loading, duly accounting for geometric
nonlinearities and material inelasticity. Its accuracy in predicting the
seismic response of reinforced concrete structures has been demonstrated
through comparisons with experimental results derived from pseudodynamic tests
carried out on large-scale models (e.g., [22, 23]).A static pushover
analysis of each building has been carried out in thex direction of the
building as this is the direction of the building where the frame action is
guaranteed for both nonseismically and seismically designed buildings. For the
pushover analysis, the mean compressive strength of the concrete has been
assumed to be 37.35 MPa whilst the mean yield strength of the steel has been
taken as 473 MPa.


## 3.2. Failure Mechanism

One first
comparison that can be made regards the failure mechanism obtained with both
methods. In SP-BELA, the mechanism is calculated by considering the following criteria.(i)
If there is a shear failure
mechanism detected in at least one column, the capacity curve will be
interrupted at the lateral force that produces this failure. This choice
is consistent with the fact that the shear failure mechanism is brittle
and does not have significant associated dissipative capacity. Therefore,
the structure cannot enter the nonlinear range.(ii)
If all the columns within a certain
storey fail in bending, than a column-sway collapse mechanism will be
activated.(iii)
If after the development of plastic
hinges in all beams, plastic hinges form in all columns at a certain
level, a beam-sway collapse mechanism will be activated.There could be a
situation in which at the storey where the mechanism is activated, some of the
columns are stronger than the beams, or vice versa. Therefore, it cannot be
clearly identified whether a beam or a column-sway mechanism will be activated.
The analyses undertaken to validate the SP-BELA procedure have shown that in
such cases the collapse mechanism can be assumed as a mixed mechanism defined using
the average displacement capacities of the column and beam-sway mechanisms.If the reinforcing steel is known in the building then a
sway potential index can be estimated in DBELA in order to define the
mechanism, using the following simplified formula based on the proposal by
Priestley et al. [24]: (15)SI=∑(Mbl+Mbr)∑(Mca+Mcb)∝∑(hbAsbfy)l+(hbAsbfy)r∑(hcAscfy)c+(hcAscfy)b=∑(hbAsb)l+(hbAsb)r∑(hcAsc)a+(hcAsc)b, where Mbl and Mbr are the flexural strengths at either side of the joint (where
l is left and r is right), and Mca and Mcb are the
flexural column strengths (a)
above and (b) below the joint, all of which are summed for all joints of
a given floor. This formula can be simplified using just the area of tensile
reinforcing steel Asb in each beam left and right of the joint, the
area of tensile reinforcing steel in each column, Asc above and
below the joint, the beam section dimensions, hb, and the column
section dimensions, hc, either side of the joint and fy which
is the yield strength of the steel, as shown in (15). If the sway index is
higher than 1, a column-sway mechanism is assumed to form (see
Figure 2(b)), below 0.85 a beam sway mechanism (see Figure 2(a)) will form whilst between
0.85 and 1 it is conservative to assume a column-sway mechanism. It is
clear from this formula that when the
reinforcement is unknown, in DBELA the collapse mechanism needs to be selected a priori assuming that seismically
and nonseismically designed buildings are more prone to a beam- and column-sway
mechanism, respectively.The calculated
sway potential indices from DBELA for the 5 different designs of the 4 storey
building are reported in Table3. These calculations show that a column-sway mechanism
in the third floor of the nonseismically designed building is predicted whilst
all seismically designed buildings demonstrate a beam-sway or global mechanism.
SP-BELA also predicts a column-sway at the third floor of the building designed
to gravity loads (i.e., c=0%) and global mechanisms for all other buildings.Table 3
Sway potential index
calculated using the formula in (15) for the 4-storey building from Figure 3 designed to different proportions of lateral load (c).
Floorc=0%c=5%c=7.5%c=10%c=12.5%1st0.50.60.50.40.42nd1.00.60.50.40.43rd*2.60.60.50.40.4*The
storey mechanism for the fourth floor is not calculated as plastic hinges are
likely to form in the top of the columns at the roof, even for a global
mechanism.The
mechanisms observed by plotting the deformed shapes from the finite elements
analyses were in full agreement with the predictions by both DBELA and SP-BELA,
as can be seen from the post-yield interstorey drift profiles for the
nonseismically designed and the seismically-designed (c=5%) design cases in Figure 4. The third storey sway mechanism is clearly present
in the nonseismically designed frame whilst the seismically designed building
has a global mechanism.Post-yield interstorey drift profiles inx direction of the building in Figure 3(a) nonseismically designed (b) c=5%.
(a)(b)


## 3.3. Yield Displacement Capacity, Collapse Multiplier, and Initial Stiffness

The yield displacement
capacity of the building in thex direction has been calculated in both
methodologies using the section properties, the yield strain of the steel, and
the appropriate yield rotation equations. The value of the collapse multiplier
is explicitly calculated in SP-BELA based on the strength of the structural
members. On the other hand, in DBELA the collapse multiplier (λ) has been calculated using
the results of the displacement capacity (Δy) and the period of vibration at yield (Ty) using the
following formula, which is simply a rearrangement of (10): (16)λ=4π2ΔygT2=4π2Δyg(0.1H)2. The pushover plots of collapse multiplier (which is equal to the base shear
force divided by the seismic weight) versus displacement at the effective
height of the building for the nonseismically designed and the 5% laterally
loaded building are plotted in Figures 5(a) and  5(b), respectively. These figures show the nonlinear
pushover plot from the finite elements model compared with the simplified
bilinear pushover plots of SP-BELA and DBELA. A bilinear pushover curve based
on the method described in Annex B of Eurocode 8 [25] has also been
added to the plots to ease comparison between the results of the simplified
methods and the finite element analyses. The comparison with the finite elements
pushover analysis is seen to be acceptable for both DBELA and SP-BELA in terms
of the collapse multiplier. However, it is apparent that SP-BELA underestimates
and DBELA slightly overestimates the stiffness of the nonseismically designed
building, which responds with a column-sway mechanism. In SP-BELA the yield
stiffness is obtained from the calculated base shear capacity and the yield
displacement. The yield displacement is based on the yield rotation in (11)
which includes additional shear and joint deformations which are not accounted
for in the finite elements package, thus explaining the lower stiffness
prediction with SP-BELA. Both SP-BELA and DBELA underestimate the stiffness of
the seismically designed building (see Figure
5(b)) which responds with a beam-sway mechanism; again
this is probably due to the additional shear and joint deformations explicitly accounted
for in the beam-sway rotation in DBELA (see (5) )and the equivalent equation used in SP-BELA (see (11)). It
is noted that the equation for beam-sway rotation used in DBELA (see (5)) has been specifically calibrated to
account for shear and joint deformations, whereas the associated increase
assumed in the column-sway rotation equation (see (6)) has not yet been verified (see [26]).Comparison of pushover analyses inx direction of the
building in Figure 3(a) nonseismically designed (b) c=5%.
(a)(b)Figure6(a) presents the comparison for the building designed
to a lateral load of 7.5% of the seismic weight. SP-BELA is seen to correctly
predict the collapse multiplier of this frame, whilst DBELA greatly underpredicts
the strength, though the displacement capacity is similar for both methods. The
reason for this difference is due to the imposed period of vibration and thus
initial stiffness used in DBELA. This formula, presented previously in (1), has
been derived using the regression analysis of the period of vibration of
nonseismically or weakly seismically designed frames. Under conventional
force-based design, the consideration of lateral loads in the design of a frame
or building will undoubtedly increase the lateral stiffness of the building
through the need to use larger section dimensions for the columns. This can be
seen by studying the section dimensions of the frame shown in Table 1. A lower period of vibration should thus be used
for seismically designed buildings. A modified period-height relationship
(T=0.085⋅H) which is found
to give good results for the building designed to the 7.5% lateral force is
shown in Figure
6(b),
whereas even lower periods of vibration are required for the buildings designed
to lateral forces of 10% and 12.5% of the seismic weight, as shown in Figures 7(a) and 7(b),
respectively. SP-BELA is seen in these figures to consistently match the
collapse multiplier for all seismic designs as the strength of the building is
explicitly calculated as part of this methodology.Comparison of pushover analyses for seismically
designed building,c=7.5%, (a) original period-height relationship in DBELA (b)
modified period-height relationship in DBELA.
(a)(b)Comparison of pushover analyses for seismically
designed building (a)c=10% (b) c=12.5%.
(a)(b)It appears from the analyses
presented herein that the SP-BELA methodology defines well the
single-degree-of-freedom structural characteristics of individual buildings,
whilst further calibration of the period-height relationship used in the DBELA
methodology would be required for the Southern European/Italian building stock
in order to match the different levels of seismic design.


## 4. Pushover Analyses of Urban Populations

The use of DBELA and SP-BELA in the
vulnerability assessment of urban areas requires many buildings with varying
characteristics to be considered. In SP-BELA, this simply translates to the
variation of input parameters such as design loads and bay lengths which are
then used in the simulated design of hundreds of different buildings, whether
they are designed to lateral loads and/or gravity loads. A simplified pushover
is then carried out for each designed building to ascertain the lateral
strength and displacement capacity based on a random definition of the material
properties. DBELA requires input parameters based on just the geometrical
properties and the material properties, whilst no information on the loads is
needed. As the sections in the building are not designed in DBELA, additional statistics
on the variation of each structural property are necessary and assumptions on
the proportion of buildings with each response mechanism need to be made a priori. In order to study the
influence of the differences in the way urban populations are generated in the
two methods, a comparison is carried out in the next section in terms of the
mean and variance of the SDOF characteristics, where the only prior information
of the building stock is based on collected statistics.


### 4.1. Random Variables to Describe Uncertainty in Building Characteristics

The two
mechanics-based vulnerability assessment methodologies treated herein use the
generation of a random population of buildings to model the variation of the
structural characteristics of a given building class within an urban
environment. Once a random population has been derived, in DBELA the period of
vibration and limit state capacity of each building are directly calculated. In
SP-BELA, the design of each random building is carried out (to gravity load or
lateral load design), and the simplified pushover curve is generated to obtain
the period of vibration, and then the limit state displacement capacities are
calculated. Further details on the random variables used in each method are
provided below.SP-BELA
In order to derive vulnerability curves using this type of
analytical procedure, it is necessary to randomly characterise the structural
parameters and produce pushover curves for each set of randomly generated buildings.
In the current application, the building characteristics that are described
through random variables have been defined as follows.
(i) Geometrical dimensions: the span lengthslx and ly are in the x and y directions, respectively, and the interstorey
height are considered as random variables. Following typical construction practice, the span length was
assumed as a random variable with a uniform distribution between 4 and 6 m. The interstorey
height was, on the other hand, assumed to be deterministic and equal to
3 m. This choice is due to the fact that the interstorey height does not generally
vary significantly, and this value is usually the minimum allowed from
construction standards.
(ii) Material properties: for steel, a
random choice between a characteristic yielding value of 380 MPa and 440 MPa was performed. A randomly chosen cubic characteristic resistance of 25 MPa, 30 MPa, and 35 MPa was, on the other hand, associated to the concrete
for each building of the generated dataset. The characteristic strengths (used
in the simulated design) are multiplied by parameters to obtain the mean values
of resistance (employed in the assessment); these parameters have a mean
value of 1.1 and 1.5 for steel and concrete, respectively [28], and a coefficient
of variation of 10%. In both cases, a normal distribution has been assumed
for these parameters; although this may lead to negative values in a very
small number of cases, this only happens when a very large number of
standard deviations are
considered.
(iii) Design loads: a normal distribution
has been adopted to represent gravity and wind loads, with the former
featuring a mean value of 3 kN/m2 (2.5 kN/m2 on the
roof) and a standard deviation of 0.5 kN/m2, whilst the latter has
a mean value of 0.9 kN/m2 (with a standard deviation 0.2 kN/m2).
Snow and vertical variable loads have instead been considered in a deterministic
manner, with values of 1.6 and 2.0 kN/m2, respectively.
The dimensions and
the reinforcement of the structural elements such as beams, columns, and floor
slabs are not taken as random variables, since the details of structural
elements are defined during the design of each randomly generated building
using the chosen design regulations. The Monte Carlo generation of the random variables with a normal distribution has been
undertaken using the Latin Hypercube algorithm, a technique that allows a
better representation of the sample in the low probability region of the distribution
(e.g., [29]).DBELA
The same random
variables described above for the geometric and material properties have been
used, but additional random variables for the beam depth and column depth are
required. As stated before, in SP-BELA these dimensions are assigned during the
design of each random building in the dataset, whilst with DBELA the
distribution of these parameters needs to be defined a priori; this can be carried out by a statistical analysis of a
sample of buildings. For the comparison carried our herein, the variability of
the beam depth and column depth of Italian buildings has been taken from a
study by Marino [30]:
(i)
mean beam depth = 0.34 m, standard
deviation = 0.07 m, lognormal distribution,(ii)
mean column depth = 0.37, standard
deviation = 0.11, lognormal distribution.


### 4.2. Building Characteristics of Urban Populations

An urban
population of reinforced concrete buildings has been derived for both DBELA and
SP-BELA using the random variables described above. The mean period of
vibration in thex and y direction of the prototype building (see Figure 3) for each design type has
been calculated using the SP-BELA methodology for each number of storeys from 2
to 8. The variation of the mean periods of vibration with storey number is presented in Figure 8 for the x direction and in Figure 9 for the y direction along with the period of vibration calculated using (1) (as used in DBELA for the older pre-1980 buildings). The
coefficient of variation of the period of vibration in SP-BELA is approximately
6% in all cases whilst in DBELA it has been found to be 15%. The higher
coefficient of variation in DBELA for the period of vibration is likely to be
due to the wide range of frames with different numbers of bays which were considered
in the regression analysis, whilst the SP-BELA calculations have been based on
a single prototype structural system (see Figure
3).Figure 8
Mean yield period of vibration versus number of
storeys for DBELA compared with each of the SP-BELA design
categories in thex direction.Figure 9
Mean yield period of vibration versus number of
storeys for DBELA compared with each of the SP-BELA design
categories in they direction.The comparison of the periods of vibration obtained
with the two methods in thex direction follows the conclusions obtained for
the single building in Section 3; the period of vibration from the DBELA
equation is similar to the period of vibration of the nonseismically designed
buildings but is much higher than the period of vibration of the seismically
designed buildings. In the y direction of the building, the period of the
nonseismically designed buildings in SP-BELA is high due to the lack of beams
in this direction of the building. On the other hand, the introduction of beams
in this direction (which occurs in the seismically designed frames) leads to
much higher stiffness and lower periods of vibration.As mentioned in
Section2, the period of vibration of post-1980 frames has been studied as part
of the development of DBELA, and it was found that due to the larger sections
used in these frames, the period of vibration versus height formula was lower
than that found for pre-1980 frames. The periods predicted with (2) for
different number of storeys (assuming a storey height of 3 metres) are presented in Figure 10 together with the periods of vibration found in SP-BELA for the seismically designed buildings. The close match between the
seismically designed frames with c=5% and the formula found for DBELA using
newer buildings can be observed in Figure
10, whereas the frames designed to higher seismic loads
in SP-BELA are seen to have even lower periods of vibration. Hence, it appears that (2) could be used for
weakly seismically designed buildings in Italy. However, this formula has
been based on a limited number of buildings, and further analyses are necessary
to calculate the variability in the equation; for this reason, this
period-height formula will not be considered in the calculations presented
subsequently in this paper.Figure 10
Mean yield period of
vibration versus number of storeys for new buildings (DBELA)
compared with the seismically designed buildings (SP-BELA) in thex
direction.Verderame et al. [27] have derived
period-height equations for the x and y directions of nonseismically designed Italian
reinforced concrete frames with a very similar layout to that considered
herein, however, with the inclusion of a staircase. The interstorey height was
taken as 3 m, the mean yield strength of the steel was assumed to be 360 MPa, and
the mean compressive strength of the concrete was taken as 25 MPa. The mean
period of vibration versus number of storeys found using the equations proposed
by Verderame et al. [27] in
both the x and y directions is
presented in Figure 11 along with the periods of vibration from SP-BELA and DBELA.Mean yield period of vibration versus number of
storeys for DBELA compared with nonseismically designed (c=0%) SP-BELA frames
and the equations proposed by Verderame et al.
[27], (a) the x
direction and (b) the y direction of the buildings.
(a)(b)The variation in period with height in both thex and
y directions is seen to match the SP-BELA results closely, whilst the formula
for DBELA is closer to the longitudinal (x) direction of the building because
this formula has been derived considering 2D frames, and in the nonseismically
designed buildings the frames are only included in the x direction. The
variation in the period of vibration of the random buildings generated by
Verderame et al. [27] was
found to be larger than that from SP-BELA; this increase in variability is most
likely due to the changes in plan dimensions considered by Verderame et al. [27], whilst in the SP-BELA
calculations herein the number of bays has been kept constant in each
direction.The mean yield displacement capacity of the randomly generated buildings has
been calculated using DBELA and SP-BELA for thex and y directions of the
building, as presented in Figures 12 and 13, respectively. The coefficient of variation of the
displacement capacity was found to be 36% for DBELA and just 13% for SP-BELA. The
larger variability in DBELA is due to the larger number of random variables
used to describe the geometry of the building.Figure 12
Mean yield displacement capacity versus number of
storeys for the beam- and column-sway cases of DBELA compared with
each of the SP-BELA design categories in thex direction.Figure 13
Mean yield displacement capacity versus number of
storeys for the beam- and column-sway cases of DBELA compared with each of the
SP-BELA design categories in they direction.For the DBELA calculations, the beam- and column-sway
displacement capacities are shown separately whilst the calculations for
SP-BELA implicitly take into account the proportion of buildings which have
each of these mechanisms, hence it is difficult to
compare the results of the two methods without considering first the mechanism.
The proportion of beam and column-sway mechanisms obtained for the random
population of buildings in the SP-BELA calculations is presented in Figure14. The low-rise frames are seen to have a predominantly
column-sway (i.e., strong beam-weak column) mechanism in both the x and y
directions of the building. The mid-to-high rise buildings have a mixture of
beam- and column-sway mechanisms, with a higher proportion of beam-sway
mechanisms in the x direction of the building. For low
rise buildings, the ratio of beam-sway to column-sway mechanisms increases in
the x direction with the lateral design force because an increased lateral load
means the buildings are designed to have larger column dimensions and thus are
less likely to fail in a column-sway mechanism. This behaviour is less apparent
in the y direction as often the columns are placed with the longer dimension in
the x direction of the building (see Table 1). On the other hand, this
trend is inverted in the high-rise buildings. As discussed in Borzi et al. [6], this effect is due to the fact that by increasing the
lateral load coefficient “c,” the required increase of the beam resistance leads
to emergent beams (as opposed to embedded beams which are used in the design of
the low-rise buildings) and this increase occurs at a higher rate than the
increase of the column resistance. Hence, the high-rise buildings have stiff,
strong beams, and a column-sway mechanism is thus more likely to occur.Percentage of beam-sway and column-sway mechanisms in
SP-BELA for the prototype building.
(a)(b)(c)(d)(e)(f)(g)(h)


### 4.3. Comparison between SP-BELA and DBELA

Considering that
100% of the 2-storey building class with 5% lateral force design have a
column-sway mechanism, and almost 100% of the 8-storey building class with 5%
lateral force design have a beam-sway mechanism in thex direction, the
capacities employed in DBELA for a column- and beam-sway mechanism,
respectively, have been compared with those of SP-BELA for these classes only (see
Table 4).Table 4
Comparison of periods of vibration, displacement capacity,
and collapse multipliers in DBELA and SP-BELA (inx direction).
SP-BELA,x directionDBELA(Seismically designed, 5%)Building classParameterμCoVμCoV2 storeysTLS1 (s)0.866%0.615%TLS2 (s)1.149%0.7825%TLS3 (s)1.2910%0.8627%ΔLS1 (m)0.04512%0.03437%ΔLS2 (m)0.07817%0.05328%ΔLS3 (m)0.10117%0.06428%λ0.2514%0.3855%8 storeysTLS1 (s)1.187%2.3815%TLS2 (s)1.6510%2.9925%TLS3 (s)1.8910%3.1927%ΔLS1 (m)0.1313%0.2431%ΔLS2 (m)0.2519%0.3826%ΔLS3 (m)0.3319%0.4326%λ0.3712%0.1850%Such comparison is valid since the
influence due to the failure mechanism has been removed, and thus some thought
can be given to the differences in the two methodologies related to structural
properties such as the displacement capacity and the period of vibration. The
higher limit state properties in terms of displacement capacity and period of
vibration are also considered here for the first time.Table4
shows that for the 2-storey
building class, the values of displacement capacity and period of vibration are
larger for SP-BELA whilst the collapse multiplier is lower. The difference in
displacement capacity is due to the different displacement capacity predicted
for column-sway frames in the two methods (see Section 3.3) but this is further
amplified by the difference in the column dimensions assumed in DBELA and those
assigned during the design for SP-BELA. In DBELA, due to a lack of data, the
column dimensions have not been assumed to vary for each number of storeys, and
instead a large coefficient of variation has been applied. In SP-BELA, each
random building is designed and thus the column dimension changes depending on
the numbers of storeys and the proportion of lateral force. The yield period of
vibration predicted with DBELA has already been seen to be underestimated for low-rise
seismically designed buildings (see Figure 8). As expected, the higher
yield stiffness and lower yield displacement capacity predicted with DBELA lead to a higher
strength.For the 8-storey building, DBELA predicts
higher yield displacement capacity and yield period of vibration, and thus
lower lateral strength. The limit state displacement capacity for SP-BELA uses the
column yield curvature, as shown in (11), whilst the DBELA equation for this
building class uses the beam yield curvature.
The yield curvature is inversely proportional to the section depth (e.g.,
[12]) and thus with larger column dimensions (that are used in this
high, seismically designed building), the column yield curvature will be lower
than the beam curvature; this explains the lower displacement capacities in
SP-BELA. It is clear that further comparisons with nonlinear finite elements models
of high-rise buildings are necessary to study whether the displacement capacity
of beam-sway frames matches better the rotation capacity used in DBELA or in
SP-BELA. The higher yield period of vibration predicted with DBELA for the 8-storey
building follows the conclusions presented in the previous section (see Figure 8).Another difference
which is apparent from Table4 is
the larger variability in the DBELA results compared to the SP-BELA results. As mentioned
previously, this variability is due to the larger number of variables to
describe the geometry of the building (i.e., beam and column dimensions) and
the variance assigned to each of these variables in DBELA.As mentioned in the Introduction, the focus of this
paper is not on verifying the vulnerability estimates themselves, but the structural
characteristics predicted with the two methodologies. So far, it has been shown
that the differences between the two analytical methods considered herein are
limited when considering individual buildings for which the structural
characteristics are well known. The application of the two methods to an urban
population, however, leads to larger variations between the two methods caused
by the limited knowledge in DBELA and the need to use input data based on
statistical studies. The variability in the two methods could have been reduced
if the
statistics of the buildings designed with SP-BELA had been used in DBELA, however the
idea herein has been to apply the two methods blindly, and not use one method to calibrate the other. The next section of the paper explores the
influence of the differences between the two methods for an urban population,
in terms of vulnerability curves. Both methods predict the proportion of buildings
which exceed the limit state for a range of values of peak ground acceleration,
with the most vulnerable direction of the building being considered in SP-BELA.


## 4.1. Random Variables to Describe Uncertainty in Building Characteristics

The two
mechanics-based vulnerability assessment methodologies treated herein use the
generation of a random population of buildings to model the variation of the
structural characteristics of a given building class within an urban
environment. Once a random population has been derived, in DBELA the period of
vibration and limit state capacity of each building are directly calculated. In
SP-BELA, the design of each random building is carried out (to gravity load or
lateral load design), and the simplified pushover curve is generated to obtain
the period of vibration, and then the limit state displacement capacities are
calculated. Further details on the random variables used in each method are
provided below.SP-BELA
In order to derive vulnerability curves using this type of
analytical procedure, it is necessary to randomly characterise the structural
parameters and produce pushover curves for each set of randomly generated buildings.
In the current application, the building characteristics that are described
through random variables have been defined as follows.
(i) Geometrical dimensions: the span lengthslx and ly are in the x and y directions, respectively, and the interstorey
height are considered as random variables. Following typical construction practice, the span length was
assumed as a random variable with a uniform distribution between 4 and 6 m. The interstorey
height was, on the other hand, assumed to be deterministic and equal to
3 m. This choice is due to the fact that the interstorey height does not generally
vary significantly, and this value is usually the minimum allowed from
construction standards.
(ii) Material properties: for steel, a
random choice between a characteristic yielding value of 380 MPa and 440 MPa was performed. A randomly chosen cubic characteristic resistance of 25 MPa, 30 MPa, and 35 MPa was, on the other hand, associated to the concrete
for each building of the generated dataset. The characteristic strengths (used
in the simulated design) are multiplied by parameters to obtain the mean values
of resistance (employed in the assessment); these parameters have a mean
value of 1.1 and 1.5 for steel and concrete, respectively [28], and a coefficient
of variation of 10%. In both cases, a normal distribution has been assumed
for these parameters; although this may lead to negative values in a very
small number of cases, this only happens when a very large number of
standard deviations are
considered.
(iii) Design loads: a normal distribution
has been adopted to represent gravity and wind loads, with the former
featuring a mean value of 3 kN/m2 (2.5 kN/m2 on the
roof) and a standard deviation of 0.5 kN/m2, whilst the latter has
a mean value of 0.9 kN/m2 (with a standard deviation 0.2 kN/m2).
Snow and vertical variable loads have instead been considered in a deterministic
manner, with values of 1.6 and 2.0 kN/m2, respectively.
The dimensions and
the reinforcement of the structural elements such as beams, columns, and floor
slabs are not taken as random variables, since the details of structural
elements are defined during the design of each randomly generated building
using the chosen design regulations. The Monte Carlo generation of the random variables with a normal distribution has been
undertaken using the Latin Hypercube algorithm, a technique that allows a
better representation of the sample in the low probability region of the distribution
(e.g., [29]).DBELA
The same random
variables described above for the geometric and material properties have been
used, but additional random variables for the beam depth and column depth are
required. As stated before, in SP-BELA these dimensions are assigned during the
design of each random building in the dataset, whilst with DBELA the
distribution of these parameters needs to be defined a priori; this can be carried out by a statistical analysis of a
sample of buildings. For the comparison carried our herein, the variability of
the beam depth and column depth of Italian buildings has been taken from a
study by Marino [30]:
(i)
mean beam depth = 0.34 m, standard
deviation = 0.07 m, lognormal distribution,(ii)
mean column depth = 0.37, standard
deviation = 0.11, lognormal distribution.


## 4.2. Building Characteristics of Urban Populations

An urban
population of reinforced concrete buildings has been derived for both DBELA and
SP-BELA using the random variables described above. The mean period of
vibration in thex and y direction of the prototype building (see Figure 3) for each design type has
been calculated using the SP-BELA methodology for each number of storeys from 2
to 8. The variation of the mean periods of vibration with storey number is presented in Figure 8 for the x direction and in Figure 9 for the y direction along with the period of vibration calculated using (1) (as used in DBELA for the older pre-1980 buildings). The
coefficient of variation of the period of vibration in SP-BELA is approximately
6% in all cases whilst in DBELA it has been found to be 15%. The higher
coefficient of variation in DBELA for the period of vibration is likely to be
due to the wide range of frames with different numbers of bays which were considered
in the regression analysis, whilst the SP-BELA calculations have been based on
a single prototype structural system (see Figure
3).Figure 8
Mean yield period of vibration versus number of
storeys for DBELA compared with each of the SP-BELA design
categories in thex direction.Figure 9
Mean yield period of vibration versus number of
storeys for DBELA compared with each of the SP-BELA design
categories in they direction.The comparison of the periods of vibration obtained
with the two methods in thex direction follows the conclusions obtained for
the single building in Section 3; the period of vibration from the DBELA
equation is similar to the period of vibration of the nonseismically designed
buildings but is much higher than the period of vibration of the seismically
designed buildings. In the y direction of the building, the period of the
nonseismically designed buildings in SP-BELA is high due to the lack of beams
in this direction of the building. On the other hand, the introduction of beams
in this direction (which occurs in the seismically designed frames) leads to
much higher stiffness and lower periods of vibration.As mentioned in
Section2, the period of vibration of post-1980 frames has been studied as part
of the development of DBELA, and it was found that due to the larger sections
used in these frames, the period of vibration versus height formula was lower
than that found for pre-1980 frames. The periods predicted with (2) for
different number of storeys (assuming a storey height of 3 metres) are presented in Figure 10 together with the periods of vibration found in SP-BELA for the seismically designed buildings. The close match between the
seismically designed frames with c=5% and the formula found for DBELA using
newer buildings can be observed in Figure
10, whereas the frames designed to higher seismic loads
in SP-BELA are seen to have even lower periods of vibration. Hence, it appears that (2) could be used for
weakly seismically designed buildings in Italy. However, this formula has
been based on a limited number of buildings, and further analyses are necessary
to calculate the variability in the equation; for this reason, this
period-height formula will not be considered in the calculations presented
subsequently in this paper.Figure 10
Mean yield period of
vibration versus number of storeys for new buildings (DBELA)
compared with the seismically designed buildings (SP-BELA) in thex
direction.Verderame et al. [27] have derived
period-height equations for the x and y directions of nonseismically designed Italian
reinforced concrete frames with a very similar layout to that considered
herein, however, with the inclusion of a staircase. The interstorey height was
taken as 3 m, the mean yield strength of the steel was assumed to be 360 MPa, and
the mean compressive strength of the concrete was taken as 25 MPa. The mean
period of vibration versus number of storeys found using the equations proposed
by Verderame et al. [27] in
both the x and y directions is
presented in Figure 11 along with the periods of vibration from SP-BELA and DBELA.Mean yield period of vibration versus number of
storeys for DBELA compared with nonseismically designed (c=0%) SP-BELA frames
and the equations proposed by Verderame et al.
[27], (a) the x
direction and (b) the y direction of the buildings.
(a)(b)The variation in period with height in both thex and
y directions is seen to match the SP-BELA results closely, whilst the formula
for DBELA is closer to the longitudinal (x) direction of the building because
this formula has been derived considering 2D frames, and in the nonseismically
designed buildings the frames are only included in the x direction. The
variation in the period of vibration of the random buildings generated by
Verderame et al. [27] was
found to be larger than that from SP-BELA; this increase in variability is most
likely due to the changes in plan dimensions considered by Verderame et al. [27], whilst in the SP-BELA
calculations herein the number of bays has been kept constant in each
direction.The mean yield displacement capacity of the randomly generated buildings has
been calculated using DBELA and SP-BELA for thex and y directions of the
building, as presented in Figures 12 and 13, respectively. The coefficient of variation of the
displacement capacity was found to be 36% for DBELA and just 13% for SP-BELA. The
larger variability in DBELA is due to the larger number of random variables
used to describe the geometry of the building.Figure 12
Mean yield displacement capacity versus number of
storeys for the beam- and column-sway cases of DBELA compared with
each of the SP-BELA design categories in thex direction.Figure 13
Mean yield displacement capacity versus number of
storeys for the beam- and column-sway cases of DBELA compared with each of the
SP-BELA design categories in they direction.For the DBELA calculations, the beam- and column-sway
displacement capacities are shown separately whilst the calculations for
SP-BELA implicitly take into account the proportion of buildings which have
each of these mechanisms, hence it is difficult to
compare the results of the two methods without considering first the mechanism.
The proportion of beam and column-sway mechanisms obtained for the random
population of buildings in the SP-BELA calculations is presented in Figure14. The low-rise frames are seen to have a predominantly
column-sway (i.e., strong beam-weak column) mechanism in both the x and y
directions of the building. The mid-to-high rise buildings have a mixture of
beam- and column-sway mechanisms, with a higher proportion of beam-sway
mechanisms in the x direction of the building. For low
rise buildings, the ratio of beam-sway to column-sway mechanisms increases in
the x direction with the lateral design force because an increased lateral load
means the buildings are designed to have larger column dimensions and thus are
less likely to fail in a column-sway mechanism. This behaviour is less apparent
in the y direction as often the columns are placed with the longer dimension in
the x direction of the building (see Table 1). On the other hand, this
trend is inverted in the high-rise buildings. As discussed in Borzi et al. [6], this effect is due to the fact that by increasing the
lateral load coefficient “c,” the required increase of the beam resistance leads
to emergent beams (as opposed to embedded beams which are used in the design of
the low-rise buildings) and this increase occurs at a higher rate than the
increase of the column resistance. Hence, the high-rise buildings have stiff,
strong beams, and a column-sway mechanism is thus more likely to occur.Percentage of beam-sway and column-sway mechanisms in
SP-BELA for the prototype building.
(a)(b)(c)(d)(e)(f)(g)(h)


## 4.3. Comparison between SP-BELA and DBELA

Considering that
100% of the 2-storey building class with 5% lateral force design have a
column-sway mechanism, and almost 100% of the 8-storey building class with 5%
lateral force design have a beam-sway mechanism in thex direction, the
capacities employed in DBELA for a column- and beam-sway mechanism,
respectively, have been compared with those of SP-BELA for these classes only (see
Table 4).Table 4
Comparison of periods of vibration, displacement capacity,
and collapse multipliers in DBELA and SP-BELA (inx direction).
SP-BELA,x directionDBELA(Seismically designed, 5%)Building classParameterμCoVμCoV2 storeysTLS1 (s)0.866%0.615%TLS2 (s)1.149%0.7825%TLS3 (s)1.2910%0.8627%ΔLS1 (m)0.04512%0.03437%ΔLS2 (m)0.07817%0.05328%ΔLS3 (m)0.10117%0.06428%λ0.2514%0.3855%8 storeysTLS1 (s)1.187%2.3815%TLS2 (s)1.6510%2.9925%TLS3 (s)1.8910%3.1927%ΔLS1 (m)0.1313%0.2431%ΔLS2 (m)0.2519%0.3826%ΔLS3 (m)0.3319%0.4326%λ0.3712%0.1850%Such comparison is valid since the
influence due to the failure mechanism has been removed, and thus some thought
can be given to the differences in the two methodologies related to structural
properties such as the displacement capacity and the period of vibration. The
higher limit state properties in terms of displacement capacity and period of
vibration are also considered here for the first time.Table4
shows that for the 2-storey
building class, the values of displacement capacity and period of vibration are
larger for SP-BELA whilst the collapse multiplier is lower. The difference in
displacement capacity is due to the different displacement capacity predicted
for column-sway frames in the two methods (see Section 3.3) but this is further
amplified by the difference in the column dimensions assumed in DBELA and those
assigned during the design for SP-BELA. In DBELA, due to a lack of data, the
column dimensions have not been assumed to vary for each number of storeys, and
instead a large coefficient of variation has been applied. In SP-BELA, each
random building is designed and thus the column dimension changes depending on
the numbers of storeys and the proportion of lateral force. The yield period of
vibration predicted with DBELA has already been seen to be underestimated for low-rise
seismically designed buildings (see Figure 8). As expected, the higher
yield stiffness and lower yield displacement capacity predicted with DBELA lead to a higher
strength.For the 8-storey building, DBELA predicts
higher yield displacement capacity and yield period of vibration, and thus
lower lateral strength. The limit state displacement capacity for SP-BELA uses the
column yield curvature, as shown in (11), whilst the DBELA equation for this
building class uses the beam yield curvature.
The yield curvature is inversely proportional to the section depth (e.g.,
[12]) and thus with larger column dimensions (that are used in this
high, seismically designed building), the column yield curvature will be lower
than the beam curvature; this explains the lower displacement capacities in
SP-BELA. It is clear that further comparisons with nonlinear finite elements models
of high-rise buildings are necessary to study whether the displacement capacity
of beam-sway frames matches better the rotation capacity used in DBELA or in
SP-BELA. The higher yield period of vibration predicted with DBELA for the 8-storey
building follows the conclusions presented in the previous section (see Figure 8).Another difference
which is apparent from Table4 is
the larger variability in the DBELA results compared to the SP-BELA results. As mentioned
previously, this variability is due to the larger number of variables to
describe the geometry of the building (i.e., beam and column dimensions) and
the variance assigned to each of these variables in DBELA.As mentioned in the Introduction, the focus of this
paper is not on verifying the vulnerability estimates themselves, but the structural
characteristics predicted with the two methodologies. So far, it has been shown
that the differences between the two analytical methods considered herein are
limited when considering individual buildings for which the structural
characteristics are well known. The application of the two methods to an urban
population, however, leads to larger variations between the two methods caused
by the limited knowledge in DBELA and the need to use input data based on
statistical studies. The variability in the two methods could have been reduced
if the
statistics of the buildings designed with SP-BELA had been used in DBELA, however the
idea herein has been to apply the two methods blindly, and not use one method to calibrate the other. The next section of the paper explores the
influence of the differences between the two methods for an urban population,
in terms of vulnerability curves. Both methods predict the proportion of buildings
which exceed the limit state for a range of values of peak ground acceleration,
with the most vulnerable direction of the building being considered in SP-BELA.


## 5. Vulnerability Curves for Urban Populations

Figure15 presents the procedure considered
herein for the comparison of the displacement capacity of the randomly
generated buildings with the displacement demand using a displacement response
spectrum: the random variables used to describe the variability of the spectrum
are presented in Section 5.2. The differences in DBELA and SP-BELA for what
concerns the generation of the random population of buildings and the
calculation of the pushover curve are highlighted in Figure 15. It is noted that a
preliminary comparison of the two methods based on vulnerability curves has
been made in Borzi et al.
[6].Figure 15
Flowchart of the DBELA and SP-BELA methodologies for
the derivation of vulnerability curves for RC buildings.


### 5.1. Displacement Spectral Shape

A displacement spectral shape has been anchored to the
PGA values for which the vulnerability is to be calculated. The spectral shape
has been defined using the acceleration spectrum proposed by recent Italian
regulations [19], multiplied by (T/2π)2 where T is the period of
vibration, to give the displacement spectrum for 5% damping. This displacement
spectra differs from that proposed in EC8 [25] because the displacement
response does not reduce to the peak ground displacement at long periods, but
remains constant after 2 seconds. The choice of a codified
spectral shape has been made for the comparison of the two methodologies, but
it would be just as feasible to use a displacement spectrum from a ground-motion
prediction equation for a given scenario, or a uniform hazard spectrum from a
probabilistic seismic hazard assessment.Equivalent linearisation is currently applied to the
structural model in both DBELA and SP-BELA, and thus the amount of energy that
the system can dissipate is taken into account through the use of over-damped
elastic response spectra (alternatively, and if preferred, it would be possible
to employ constant-ductility spectra instead). The equivalent viscous dampingξLSi which quantifies the energy dissipated when the limit state i is achieved, was calculated using the
expression proposed by Calvi [7], though a wealth of other similar
expressions available in the literature could also have been used (see, e.g.,
[31]): (17)ξLSi=a(1+1μLSib)+ξel, where ξel is the damping ratio characterising the elastic response, commonly
assumed as 5%, and a and b are two coefficients that vary between 20 and 30 and
0.5 and 1, respectively. In this application, a=25 and b=0.5.The spectral reduction factorηLSi, which is multiplied by the 5%
damped displacement spectrum, can then be expressed as a function of the equivalent
viscous damping defined above, as suggested in EC8 [25], though it is again
noted that several other alternatives do exist in the literature (see, e.g.,
[31]): (18)ηLSi=105+ξLSi.


### 5.2. Random Variables to Describe Uncertainty in Displacement Demand

In the probabilistic procedure used to calculate the vulnerability
curves, the uncertainty in the definition of the demand is accounted for by
assuming the following.(i)
The corner
periods of the spectrum are random variables having a constant distribution
between the maximum and the minimum values (forTB the distribution
is between 0.15 and 0.20 seconds and for TC the distribution is
between 0.4 and 0.8 seconds).(ii)
The coefficientα (spectral amplification coefficient) has
been taken as a random variable with a mean value of 2.5, a standard
deviation of 1.1, and a logarithmic distribution.
The sampling of
random variables with a normal distribution to describe the uncertainty in the
displacement demand has been generated using the Latin Hypercube algorithm [29], as mentioned previously.The upper and lower bound mean displacement spectral
shapes due to the variation in the characteristic periods of the spectrum are
shown in bold in Figure16, for a PGA=1 g and α=2.5. The shape of the
spectrum is assumed to vary uniformly between these two bounds. Further
variability is incorporated due to the uncertainty in the spectral
amplification factor; the upper bound plus 1 standard deviation and the lower
bound spectrum minus 1 standard deviation are also shown in Figure 16 (thin lines).Figure 16
Upper and lower bound mean displacement spectral
shapes due to variation of the characteristic periods (in bold). The thin lines
show the upper bound spectrum +1σ and the lower bound spectrum −1σ
that represents the variability in
the coefficient α.


### 5.3. Vulnerability Curves

Vulnerability curves have been produced for the 2-storey
and 8-storey buildings designed to 5% lateral loads using both DBELA and
SP-BELA, as presented in Figure17.Vulnerability curves calculated with DBELA and SP-BELA
for (a) 2-storey and (b) 8-storey buildings.
(a)
2 storeys(b)
8 storeysFor SP-BELA, thex
direction was found to be the most vulnerable direction of the building for
both building classes. The DBELA vulnerability curves are presented for the
column-sway mechanism for the 2-storey building class and the beam-sway
mechanism for the 8-storey building class as these were the predominant
mechanisms found by SP-BELA in this direction (see Figure 14). Hence, the only differences
in the two methods are due to the different period and displacement capacity
predictions, as studied for these two particular building classes in Section
4.2. Although rather different periods of vibration and displacement capacities
were predicted by the two methods, it can be seen in Figure 17 that the vulnerability curves
are rather similar.The first
comparison that can be made from Figure17(a) is that DBELA vulnerability curves are flatter
than the SP-BELA curves. This is caused by the larger variability in DBELA due to the increased number of random variables; additional
geometrical properties are included as random variables and the coefficient of
variation of the period of vibration is higher as it is based on a wider range
of types of frames.As has been
described in the previous section, higher periods and higher displacement
capacities have been predicted in SP-BELA for the 2 storey building class (see Table4). Considering that the
displacement demand increases with period up to 2 seconds (as shown in Figure 16), higher demands are predicted
when the period of vibration is higher; however these higher demands are
counterbalanced by the higher displacement capacities. Thus, it is not
surprising that different predictions of the displacement capacity and period
lead to similar predictions of vulnerability. For the 8-storey buildings,
higher periods of vibration and displacement capacities have been predicted
with DBELA. Hence, again the higher displacement demands in DBELA are
compensated by the higher displacement capacities; however considering that the
demand is constant after 2 seconds, the vulnerability is lower as can be
clearly seen in Figure
17(b).


## 5.1. Displacement Spectral Shape

A displacement spectral shape has been anchored to the
PGA values for which the vulnerability is to be calculated. The spectral shape
has been defined using the acceleration spectrum proposed by recent Italian
regulations [19], multiplied by (T/2π)2 where T is the period of
vibration, to give the displacement spectrum for 5% damping. This displacement
spectra differs from that proposed in EC8 [25] because the displacement
response does not reduce to the peak ground displacement at long periods, but
remains constant after 2 seconds. The choice of a codified
spectral shape has been made for the comparison of the two methodologies, but
it would be just as feasible to use a displacement spectrum from a ground-motion
prediction equation for a given scenario, or a uniform hazard spectrum from a
probabilistic seismic hazard assessment.Equivalent linearisation is currently applied to the
structural model in both DBELA and SP-BELA, and thus the amount of energy that
the system can dissipate is taken into account through the use of over-damped
elastic response spectra (alternatively, and if preferred, it would be possible
to employ constant-ductility spectra instead). The equivalent viscous dampingξLSi which quantifies the energy dissipated when the limit state i is achieved, was calculated using the
expression proposed by Calvi [7], though a wealth of other similar
expressions available in the literature could also have been used (see, e.g.,
[31]): (17)ξLSi=a(1+1μLSib)+ξel, where ξel is the damping ratio characterising the elastic response, commonly
assumed as 5%, and a and b are two coefficients that vary between 20 and 30 and
0.5 and 1, respectively. In this application, a=25 and b=0.5.The spectral reduction factorηLSi, which is multiplied by the 5%
damped displacement spectrum, can then be expressed as a function of the equivalent
viscous damping defined above, as suggested in EC8 [25], though it is again
noted that several other alternatives do exist in the literature (see, e.g.,
[31]): (18)ηLSi=105+ξLSi.


## 5.2. Random Variables to Describe Uncertainty in Displacement Demand

In the probabilistic procedure used to calculate the vulnerability
curves, the uncertainty in the definition of the demand is accounted for by
assuming the following.(i)
The corner
periods of the spectrum are random variables having a constant distribution
between the maximum and the minimum values (forTB the distribution
is between 0.15 and 0.20 seconds and for TC the distribution is
between 0.4 and 0.8 seconds).(ii)
The coefficientα (spectral amplification coefficient) has
been taken as a random variable with a mean value of 2.5, a standard
deviation of 1.1, and a logarithmic distribution.
The sampling of
random variables with a normal distribution to describe the uncertainty in the
displacement demand has been generated using the Latin Hypercube algorithm [29], as mentioned previously.The upper and lower bound mean displacement spectral
shapes due to the variation in the characteristic periods of the spectrum are
shown in bold in Figure16, for a PGA=1 g and α=2.5. The shape of the
spectrum is assumed to vary uniformly between these two bounds. Further
variability is incorporated due to the uncertainty in the spectral
amplification factor; the upper bound plus 1 standard deviation and the lower
bound spectrum minus 1 standard deviation are also shown in Figure 16 (thin lines).Figure 16
Upper and lower bound mean displacement spectral
shapes due to variation of the characteristic periods (in bold). The thin lines
show the upper bound spectrum +1σ and the lower bound spectrum −1σ
that represents the variability in
the coefficient α.


## 5.3. Vulnerability Curves

Vulnerability curves have been produced for the 2-storey
and 8-storey buildings designed to 5% lateral loads using both DBELA and
SP-BELA, as presented in Figure17.Vulnerability curves calculated with DBELA and SP-BELA
for (a) 2-storey and (b) 8-storey buildings.
(a)
2 storeys(b)
8 storeysFor SP-BELA, thex
direction was found to be the most vulnerable direction of the building for
both building classes. The DBELA vulnerability curves are presented for the
column-sway mechanism for the 2-storey building class and the beam-sway
mechanism for the 8-storey building class as these were the predominant
mechanisms found by SP-BELA in this direction (see Figure 14). Hence, the only differences
in the two methods are due to the different period and displacement capacity
predictions, as studied for these two particular building classes in Section
4.2. Although rather different periods of vibration and displacement capacities
were predicted by the two methods, it can be seen in Figure 17 that the vulnerability curves
are rather similar.The first
comparison that can be made from Figure17(a) is that DBELA vulnerability curves are flatter
than the SP-BELA curves. This is caused by the larger variability in DBELA due to the increased number of random variables; additional
geometrical properties are included as random variables and the coefficient of
variation of the period of vibration is higher as it is based on a wider range
of types of frames.As has been
described in the previous section, higher periods and higher displacement
capacities have been predicted in SP-BELA for the 2 storey building class (see Table4). Considering that the
displacement demand increases with period up to 2 seconds (as shown in Figure 16), higher demands are predicted
when the period of vibration is higher; however these higher demands are
counterbalanced by the higher displacement capacities. Thus, it is not
surprising that different predictions of the displacement capacity and period
lead to similar predictions of vulnerability. For the 8-storey buildings,
higher periods of vibration and displacement capacities have been predicted
with DBELA. Hence, again the higher displacement demands in DBELA are
compensated by the higher displacement capacities; however considering that the
demand is constant after 2 seconds, the vulnerability is lower as can be
clearly seen in Figure
17(b).


## 6. Conclusions

The mechanics of two analytical vulnerability assessment methods (DBELA and SP-BELA)
have been compared in this paper in terms of periods of vibration/base shear
capacities and displacement capacities. Both methods have been seen to match
well the nonlinear static pushover of nonseismically and weakly seismically
designed buildings carried out with a finite elements-based structural analysis
package. SP-BELA was seen to be able to capture the nonlinear static response
of a number of buildings designed to different lateral load forces due to the
implicit consideration of the strength in the method. On the other hand, the
DBELA method was seen to underestimate the strength of seismically designed
buildings due to the use of an initial stiffness based on the regression
analysis of nonseismically designed frames. Buildings designed to a lateral
force-based procedure tend to have larger column and beam dimensions and are
thus stiffer and stronger; it appears that further calibration of the DBELA
equations for initial stiffness/period is required for Southern European seismically
designed reinforced concrete frames.The initial studies
on the individual building have shown how SP-BELA defines with a higher degree
of accuracy the building behaviour, though further analyses are necessary to
ascertain which method leads to a closer prediction of the yield displacement
capacity of buildings with different heights and section dimensions. However,
it should be pointed out that the use of SP-BELA requires the building stock
under consideration to have been legally designed following particular code provisions,
though in many countries such an assumption may not be valid. Nevertheless,
when one is confident in the use of a code-based approach another advantage of
the SP-BELA procedure is that it implicitly accounts for the proportion of the
existing building stock which will have either a beam- or column-sway mechanism
(or a combination of the two). Considering the higher accuracy of the SP-BELA
methodology, comparisons in terms of the period-height equation, the displacement
capacity, and the response mechanism have been made at an urban scale with the
more simplified method, DBELA.The aforementioned
comparisons have shown how there is a variation in the period of vibration in
the two directions of the prototype building, and the DBELA period-height equation
matches quite well the longitudinal (x) direction of the nonseismically
designed prototype building, as this is the direction in which the frames are
placed. As mentioned previously, further calibration of the equation to match
the seismically designed frames, as well as the transverse (y) direction of the
building, is necessary. The displacement capacity equations of the two methods
are based on similar mechanics principles but on different definitions of the
rotation capacity. Differences in the displacement capacity were thus seen to
be quite large due to both the different equations used and the different
section dimensions employed in the calculation of yield and ultimate
curvatures. Furthermore, the displacement capacity in SP-BELA takes into account
the proportion of each response mechanism whilst the DBELA equations are
considered for each mechanism separately. Despite the differences in terms of
displacement capacity and period of vibration, for building classes with a
single dominant response mechanism, the two methods were seen to produce
similar vulnerability curves. The similarities in the vulnerability predictions
can be easily explained by comparing the relationship between the displacement
demand in the two methods and the predicted displacement capacities.To conclude, it can be stated that the results presented in this work are
an initial step towards validating the adequacy and accuracy of the analytical
vulnerability assessment methods discussed herein. This paper has highlighted how
the difficulties in validating the latter through a comparison with actual
observed damage data can begin to be overcome by comparing the results with
those obtained from detailed nonlinear numerical structural models. Nonlinear
dynamic analyses are currently being carried out such that further comparisons
between the dynamic response of MDOF structures and the static response of the SDOF
systems predicted by the two methods can be made, and the influence of higher
mode effects, cyclic degradation of members, and so forth can be explicitly
accounted for.



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*Source: 438379-2008-08-18.xml*