Engineering Structures 136 (2017) 406–419
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Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct
Evaluation of the economic benefits of using Buckling-Restrained Braces
in hospital structures located in very soft soils
Héctor Guerrero a,⇑, Amador Terán-Gilmore b, Tianjian Ji a, José A. Escobar c
a
School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Pariser Building, Manchester M13 9PL, UK
Universidad Autónoma Metropolitana, Department of Materials, México City, Mexico
c
Universidad Nacional Autónoma de México, Institute of Engineering, México City, Mexico
b
a r t i c l e
i n f o
Article history:
Received 11 February 2016
Revised 16 January 2017
Accepted 17 January 2017
Keywords:
Buckling-Restrained Braces (BRBs)
Hospital structures
Very soft soils
Lakebed zone of Mexico City
Economic benefits of BRBs
a b s t r a c t
Since economic quantities are more meaningful to decision makers than dynamic response parameters,
this paper examines the economic benefits of using Buckling-Restrained Braces (BRBs) in hospitals
located in the very soft soils of the lakebed zone of Mexico City. Since non-structural elements and contents are far more expensive than the structure itself, they are included in detail in the analyses. The
results of analyses on three-, six-, and nine-storey frames, which represent short-period structures, show
that the expected (or probable) losses and lifecycle costs are smaller when structures are equipped with
BRBs. Different cases (defined by the contribution of the BRBs to the strength capacity of the structure)
were analysed and compared. The best design options were identified from a comparison of lifecycle
costs. As an example, a comparison of cost-benefit analysis between a bare frame and a frame designed
under gravity loads and equipped with BRBs, shows that the latter is more economical; because, for the
same initial cost, the lifecycle costs are significantly smaller.
Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction
After the M8.1 19/09/1985 Michoacán Earthquake, significant
damage was observed in structures located in the lakebed zone
of Mexico City [1,2]. This zone is characterised by very soft soils
with low shear wave velocity <80 m/s [3]. It has been well documented that the structure of the soil deposits of the lakebed zone
of Mexico City is composed of randomly-organised pin-like solids;
and that the space among the solids is filled up with water –
achieving water contents of up to 400% (e.g. [4,5]). These types of
soils are of particular interest because they are linked with ground
motions which have long predominant periods of vibration with a
narrow-band frequency content and a long duration [6]. In particular, structures with a fundamental period of vibration similar to
the dominant period of the soil, or short-period structures with
stiffness degradation, can be vulnerable under such types of
ground motion [7]. As an example, more than 400 buildings suffered total, or partial, collapse and around 3200 were damaged
during the 19/09/1985 Michoacán Earthquake [8,9]. It is significant
to highlight that, among the collapses, over 240 buildings (including 11 hospitals) were less than 10 storeys high [10], which can be
classified as short-period structures.
⇑ Corresponding author.
E-mail address: hector.guerrerobobadilla@manchester.ac.uk (H. Guerrero).
http://dx.doi.org/10.1016/j.engstruct.2017.01.038
0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.
On the other hand, hospitals are regarded as one of the most
important structures in a modern society. This has been recognised
by seismic design codes, which classify hospitals as ‘‘important”
structures and require them to be designed with a higher load
capacity than conventional structures. In the pursuit of simplicity,
the codes impose a factor of importance; which is higher than
unity and is applied to the spectral ordinates of the design spectra.
For example, the Mexico City building code [11] has an importance
factor of 1.5, and Eurocode 8 [12] has a factor of 1.4. Unfortunately,
it has been observed that the use of importance factors has not
been sufficient to avoid interruptions of functionality, extensive
damage or collapse of these facilities [1,2]. Furthermore, providing
increased load capacity does not necessarily reduce the level of
damage in a building [13,14]. Therefore, a different approach,
based on energy dissipation and control of response, may be more
effective in achieving more efficient hospital structures with better
behaviour under earthquake loading. In other words, the use of
protective technologies (e.g. Buckling-Restrained Braces, BRBs)
combined with rational procedures of design (e.g. PerformanceBased Seismic Design, PBSD) may help to reduce economic losses
and loss of functionality in such important facilities.
BRBs represent a major development in the area of seismic protection because they are effective for dissipating energy (e.g. see
[15–17]). Although many variations have been proposed, they are
typically composed of two parts [18,19]: a core (which is
407
H. Guerrero et al. / Engineering Structures 136 (2017) 406–419
ZZZ
kðDVÞ ¼
GðDVjDMÞdGðDMjEDPÞdGðEDPjIMÞdkðIMÞ
ð1Þ
where k(DV) is the mean annual frequency of exceeding a given
decision variable (DV), for example, a repair cost threshold; k(IM)
is the mean annual frequency of exceeding a given intensity measure (IM), for example, the peak ground acceleration, pga; G(EDP|
IM) is the probability of exceeding a given engineering demand
parameter (EDP) such as inter-storey drift ratio, conditioned to a
given IM; G(DM|EDP) is the probability of exceeding a damage measure (DM), such as the number of cracks in a concrete element,
given a value of EDP; and G(DV|DM) is the probability of exceeding
a value of DV, given a value of DM.
The PEER framework was further developed by Yang et al. [29].
They proposed a methodology based on Monte Carlo simulations
to assess the expected losses in buildings. The methodology was
later adopted by the FEMA P58 Project [30].
In this paper, the FEMA P58 Project [30] methodology is used to
evaluate the economic benefits of using BRBs in hospital structures
located in very soft soils. For that purpose, three-, six-, and ninestorey frames in 2D, representative of short-period structures,
were analysed. For each frame, five cases (one without, and four
with, BRBs) were analysed. They are described in the next section
together with their design specification and calculated dynamic
response. Since the costs of non-structural elements and contents
are far more expensive than the cost of the structure itself, they
were included in detail in the analyses. Then, expected (or
probabilistic) losses, when subjected to earthquake ground
motions, were estimated. Cost-benefit analyses were also conducted to determine the value of using BRBs in structures over a
period of time (e.g. 50 years – which is commonly taken as the useful life of common structures).
It should be mentioned that the results presented in this paper
are valid for regular structures whose dynamic responses are dominated by the first mode. Also, the studied structures are designed
using the displacement-based procedure in [27], which estimates
the response demands of equivalent dual single-degree-offreedom (SDOF) oscillators and transforms them into the response
demands of multi-storey structures using simplified height-wise
distribution functions. In this study, it is assumed that, for regular,
first mode-dominated structures, the response demands of the
studied structures are reasonably well estimated using equivalent
dual SDOF oscillators. Therefore, it is expected that the results are
valid for comparison purposes in relative terms. For different types
of structural systems, such as irregular or high-rise buildings, different results might be obtained and specific studies may be
needed.
2. Design and dynamic response
2.1. Studied cases
Fig. 2 shows the layout of the structures studied in this paper.
They are three-, six- and nine-storey frames in 2D, and represent
typical hospitals located in the lakebed zone of Mexico City. For
comparison purposes, five cases are studied for each hospital. They
are represented in Fig. 3 and are described as follows:
Case 0. A bare frame is designed to resist the full seismic loads
alone (i.e. without BRBs). This serves as the reference case.
Case 1. The structure of Case 0 is upgraded using BRBs, which
increases the initial cost and capacity but reduces the lateral
displacement demands.
Case 2. The main frame is re-designed considering gravity loads
only, and then BRBs are provided to match the initial cost of
Case 0.
Three, six and nine storeys
commonly made of a steel plate) and a case (which is commonly
composed of a steel tube and a filling mortar). Further details
can be found elsewhere (e.g. [18–20]), where experimental tests
have demonstrated the great capacity of BRBs to dissipate energy
and their efficiency for reducing the seismic response of structures
(e.g. [21–23]). Furthermore, several methods have been proposed
to design structures equipped with BRBs (e.g. [20,24–27]). Most
of them are aimed at controlling lateral displacements. However,
lateral displacements and, more generally, dynamic response
may be meaningless to investors and decision makers, because
they prefer economic quantities for determining whether to
include BRBs in a structure. In this context, quantification of the
economic benefits of using BRBs in structures is required to provide understandable and practical information to decision makers
– especially for hospital structures, given their cost and importance
in modern societies.
In order to assess the economic benefits of using BRBs in hospitals, not only initial cost but also expected (or probable) losses due
to future earthquakes need to be considered. Since earthquakes
and the resulting ground motion intensities are uncertain, a probabilistic assessment needs to be conducted to include all possible
sources of uncertainty associated with future earthquakes. The
Pacific Earthquake Research Center (PEER) has established a framework for Performance-Based Assessment of structures, which is
founded on the analysis of four independent phases [28] (Fig. 1).
The phases are seismic hazard analysis, dynamic response analysis,
damage analysis and loss analysis. They are integrated using the
total probability theorem, namely
6 bays @ 8 m
Fig. 2. Layout of hospitals structures studied in this paper.
Seismic hazard
analysis
Dynamic response
analysis
Damage
analysis
Loss
analysis
Outcome: λ (IM)
Outcome: G(EDP|IM)
Outcome: G(DM|EDP)
Outcome: G(DV|DM)
IM = intensity
measure, e.g. pga
EDP = engineering
demand parameter,
e.g. inter-storey drift
DM = damage
measure, e.g. number
of cracks in a concrete
element
DV = decision
variable, e.g. total
repair cost after an
earthquake
Fig. 1. PEER framework for performance assessment [28].
0
1
2 3
Case
4
0
1
2
3
Case
4
No BRBs
Displacement demands
No BRBs
No BRBs
Load capacity
H. Guerrero et al. / Engineering Structures 136 (2017) 406–419
Initial cost
408
0
1
2 3
Case
4
Fig. 3. Schematic representation of the studied cases.
linear-elastic response demands. For Cases 1 to 4, n = 5%, 7%, 8%
and 10%, respectively, for the seismic intensities defined in
Table 1; while for Case 0, n = 3% was considered for pga = 0.05g
and n = 5% for the other intensities.
The design starts with an initial dimensioning of the frame
under gravity loads. Then, the designer has three options to
restrain the lateral displacement demands imposed by the earthquake ground motions: (1) increase the capacity of the frame, (2)
provide BRBs, or (3) a combination of both. For the cases represented in Fig. 3, option (1) was used for Case 0, option (2) for Case
3, and option (3) for Cases 1, 2 and 4. In all cases, the method considers capacity design principles, i.e. while the weak beam-strong
column mechanism is promoted in the frames, the fuse concept
is guaranteed in the BRBs by ensuring that they yield at smaller
displacements than the frames, in order to protect them.
It is important to highlight that the method of design considers
that the dynamic behaviour of a structure equipped with BRBs is
represented by a single-degree-of-freedom (SDOF) dual system.
Therefore, by applying earthquake ground motions to equivalent
SDOF dual systems, the method is capable of providing not only
maximum displacement demands but also velocities, accelerations
and residual displacements, which are useful for assessing the performance of the structures. More details of the design process can
be found in Ref. [31]. The resultant steel profiles, cross-sectional
areas of BRBs, fundamental periods of vibration, and statistics of
the dynamic response are shown in Tables 2 and 3, Table 4 and
Appendix A, respectively. Note that the areas of the BRBs (in Table 3)
correspond to those of the first storeys. The areas of those in other
storeys are proportional to the following vectors: (1.0, 0.614,
0.273)T, (1.0, 0.791, 0.626, 0.461, 0.296, 0.132)T and (1.0, 0.847,
0.738, 0.630, 0.521, 0.412, 0.304, 0.195, 0.087)T for the three-,
six- and nine-storey frames, respectively. Note that the BRBs of
the nine-storey frames are larger than those of the others and,
although they may seem rather large, they may be manufactured
using commercially available steel plates. For example, the 190
cm2 required in the first storey of Case 4 can be achieved using
steel plates with a thickness of 7.6 cm (3 in.) and a width of
25 cm. It should also be noted that these areas are of a similar order
to those presented by others (e.g. Montiel and Teran-Gilmore [32]).
Case 3. Similar to Case 2, the main frame is designed under
gravity loads, but the lateral capacity of the BRBs is smaller so
that the displacement demands are similar to those of Case 0.
This provides a structure with lower initial costs but a similar
response to that of Case 0.
Case 4. Again, similar to Case 2, the main frame is designed
under gravity loads. However, in this case the lateral capacity
of the BRBs is larger, so that the initial cost is larger than that
of Case 2. This increases the load capacity and reduces the displacement response.
The five cases are designed using the same methodology; which
was proposed by Guerrero et al. [27,31] and is based on control of
the lateral displacement demands. By doing this, only the effects of
the BRBs are compared, and the effects of the lack of control of the
lateral displacements are avoided.
2.2. Design
It was assumed that the studied structures were located in the
lakebed of Mexico City. The floor masses were 461 t for the top
floor and 576 t for the other floors. A rigid floor system was
assumed. The first storey had a height of 4 m and the others 3 m.
The materials used were steel ASTM A992 (fy = 350 MPa) in the
beams and columns and steel ASTM A36 (fy = 250 MPa) in the core
of the BRBs.
The method proposed by Guerrero et al. [27,31] was used to
design the structures. This method uses earthquake ground
motions as input, and is based on control of lateral displacements
for different levels of seismic intensity. Each displacement threshold is defined by the maximum inter-storey drift allowed at each
seismic intensity level. The pair seismic intensity-maximum
inter-storey drift is termed as objective of design. The design
objectives for the studied structures are given in Table 1. A set
of 30 earthquake ground motions, with dominant periods of
vibration around 2 s and recorded in the lakebed zone of Mexico
City, were used. They were scaled to account for the different
levels of seismic intensity defined in Table 1. It should be noted
that peak-ground-acceleration (pga) was selected as seismic
intensity, which may be considered adequate for the studied
structures because they are short-period structures located in
the acceleration-sensitive region of the spectrum (this will be evident in Section 4.2). Regarding the damping ratio, the following
values were considered, which are in agreement with shaking
table experiments that suggests that BRBs increase damping significantly [31] even for low intensity ground motions and
2.3. Dynamic response
In the previous subsection, statistics related to the engineering
demand parameters (EDPs) were generated during the design.
They are shown in detail in Appendix A. The EDPs are: peak displacements, residual displacement ratios (defined as the ratio of
Table 1
Objectives of design of the hospitals studied in this chapter.
Objective of design
1
2
3
4
Performance
Seismic intensity, pga
Max. drift
Fully operability
0.05g
0.0025
Operability
0.10g
0.005
Life safety
0.20g
0.010
Collapse prevention
0.30g
0.020
409
H. Guerrero et al. / Engineering Structures 136 (2017) 406–419
Table 2
Steel profiles of the studied structures.
Structure
Storey
Columns (HSS profiles, mm)
Cases 0 & 1
Cases 2, 3, 4
Cases 0 & 1
Cases 2,3,4
3-storeys
1
2
3
500 25
500 25
500 19
500 19
500 13
500 13
W21 68
W21 68
W21 62
W21 68
W21 68
W21 62
6-storeys
1 to 3
4 to 6
600 38
600 19
500 16
500 13
W27 94
W27 84
W24 68
W24 68
9-storeys
1 to 3
4 to 6
7 to 9
900 38
900 25
800 19
500 25
500 16
500 13
W27 129
W27 102
W27 84
W24 68
W24 68
W24 68
Table 3
Cross-sectional areas, in cm2, of the BRBs of the first storey.
Structure
Case 1
Case 2
Case 3
Case 4
3-storeys
6-storeys
9-storeys
13.5
46.9
64.0
18.0
105.0
127.0
6.84
60.0
93.0
32.0
152.0
190.0
Table 4
Fundamental periods of the studied structures, in seconds.
Storeys
Case 0
Case 1
Case 2
Case 3
Case 4
Three
Six
Nine
0.85
1.17
1.41
0.73
0.97
1.20
0.74
0.97
1.26
0.84
1.10
1.35
0.66
0.88
1.15
the residual to the peak displacements), absolute velocities and
accelerations. Since these EDPs correspond to SDOF dual systems,
to assess their performance they need to be converted into vectors
of demands for equivalent MDOF structures. This provides acceptable results [27,31] and was conducted as follows:
1. Estimation of vectors of inter-storey drift demands. First, the mean
displacement demand at the top floor of each frame (for each
case) was determined. Each mean displacement demand (in
Appendix A), which corresponds to SDOF dual systems, was multiplied by the ratio dN/dmax (where dN and dmax were determined
during the design process [31], dN is the displacement threshold
at the top floor of the structure and dmax is the displacement
threshold of the equivalent SDOF dual system). For the studied
cases, dN/dmax = 1.39, 1.41 and 1.44 for the three-, six- and
nine-storey frames, respectively. Then, with the mean
displacement demands at the top floor of each structure, the
corresponding inter-storey drift profile was selected from the
pushover curve (which was also calculated during the design
process).
2. Estimation of the residual drift demands. The mean residual displacement ratios in Appendix A, defined as the ratio of the residual to peak displacement demands, were multiplied by the drift
demands estimated in the previous step. The maximum value,
among all the storeys, was selected.
Cases:
1
Examples of the vectors of mean demands are shown in Figs. 4–
6 for a pga = 0.20g. The inter-storey drifts, floor velocities and floor
accelerations for the studied structures are presented. It can be
appreciated that the response demands are consistently larger in
Case 0 (i.e. frames designed without BRBs). The inter-storey drifts
of Cases 0 and 3 are similar but slightly smaller in Case 3. Cases
1, 2 and 4 consistently have smaller inter-storey drifts. The floor
velocities and accelerations are smaller when BRBs are used (i.e.
Cases 1 to 4). The improvement of the response due to the BRBs
is evident and is attributed to diverse aspects, such as: (1) the good
dissipation capacity of the BRBs; (2) the fact that structures with
BRBs tend to be stiffer than moment resisting frames – this is beneficial for short-period structures like those analysed here because
their period of vibration becomes shorter and farther from the resonance region of the spectrum; and (3) higher levels of damping –
which is attributed to the friction between a BRBs core and its case,
as found in the shaking table experiments reported in Ref. [31].
3. Initial cost
3.1. Definitions
For convenience three costs are defined:
(a) Total initial cost (CT). This includes the total cost of structural
elements, non-structural elements and contents; so the hospital would be fully functional. In this paper, the total cost of
Case 0 is referred to as C0 and is taken as the reference value
for comparison purposes.
3
Storey
2
3. Estimation of vectors of floor velocities and accelerations. The mean
of the absolute velocities and accelerations in Appendix A were
multiplied by the ratio dN/dmax to obtain the values at the top
floor. Then, these were linearly distributed to find the
peak- ground velocity and peak-ground acceleration. These
distributions are based on modal analysis and the following
assumptions: (1) the mode shape of the fundamental mode is linear – which is reasonable for low-rise structures; and (2) the
dynamic response of the studied structures is dominated by the
fundamental mode, which is the case for the studied structures.
3
4
1
2
3
0
Storey
Storey
3
Beams
2
1
0
0.005
0.01
a) Inter-storey drifts, m/m
1
0
0
0
2
0
1
b) Floor velocities, m/s
0
0.2
0.4
0.6
c) Floor accelerations, g
Fig. 4. Response estimated for pga = 0.20g: three-storey frame.
H. Guerrero et al. / Engineering Structures 136 (2017) 406–419
0
0.005
0.01
6
5
4
3
2
1
0
6
5
4
3
2
1
0
Storey
6
5
4
3
2
1
0
Storey
Storey
410
0
0.2
0.4
0.6
c) Floor accelerations, g
0
1
b) Floor velocities, m/s
a) Inter-storey drifts, m/m
Fig. 5. Response estimated for pga = 0.20g: six-storey frame.
0
0.005
0.01
a) Inter-storey drifts, m/m
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
Storey
Storey
Storey
9
8
7
6
5
4
3
2
1
0
0
0.5
1
0
1.5
b) Floor velocities, m/s
0.2
0.4
0.6
c) Floor accelerations, g
Fig. 6. Response estimated for pga = 0.20g: nine-storey frame.
weight alone – however, after consulting a few contractors, it
was found that the steel weight is a very good indicator of CS,
which simplifies the estimation of the initial cost. The total cost,
CT, shown in the last column of the table, was estimated by dividing the structure cost by 0.2. To be consistent with the statistics in
[33] and [34], the non-structural and contents cost, Cn, was calculated as 0.8C0, which is shown in the fourth column for illustration
purposes.
The structural steel weight (ws) and the weight of the BRBs
(wBRBs) for Cases 1 to 4 are shown in Table 6. It can be appreciated
that, for each structure, the steel weight of Case 1 is the same as
that of Case 0 (see Table 5). On the other hand, the steel weight
of Cases 2 to 4 is smaller, because these cases were designed for
gravity loads only. The weight of the BRBs is least for Case 3 and
most for Case 4.
Table 7 shows the estimated structure and total costs for Cases
1 to 4. For convenience, they are presented in terms of C0. They
were estimated as follows:
(b) Structure cost (CS). This is only the cost of the structural elements and their connections. In this study, and to be consistent with [33] and [34], the structure cost for Case 0 is
considered to be 20% of the total cost, i.e. CS = 0.2C0.
(c) Non-structural and contents cost (Cn). This only includes the
cost of the non-structural elements and the contents of the
building, i.e. Cn = CT CS. To be consistent, the five cases have
the same Cn – which is given from Case 0 as Cn = 0.8C0.
In order to compare the economic benefits of using BRBs in
hospital structures, the total cost of Case 0, C0, is estimated first;
then the costs corresponding to Cases 1 to 4 are determined
relative to C0.
3.2. Estimation
Table 5 shows the estimated total costs for Case 0 of the three-,
six- and nine-storey frames. The weight of the structural steel is
shown in the second column of the table. The structure cost is estimated using a cost for steel (in US Dollars) of $3/kg and an additional cost of 5% due to beam-to-column connections; this is
shown in the third column. Here, it is recognised that the structure
cost, CS, may be affected by several factors other than the steel
(a) The structure cost is CS = (3ws + 9wBRBS)(1 + 0.05), i.e. ws and
wBRBS, from the previous table, and this was multiplied by
$3/kg and $9/kg, respectively. An additional cost of 5%, due
to connections, was included. CS was then normalised by C0.
Table 5
Estimation of initial cost for Case 0.
Structure
Steel weight (ws), kg
Structure cost, CS = 3(1 + 0.05)ws
Non-struct. & contents cost, Cn = 0.8C0
Total cost, C0 = CT = CS/0.2
3-storeys
6-storeys
9-storeys
38,495
107,018
213,063
$121,261
$337,106
$671,178
$485,042
$1,348,424
$2,684,591
$606,303
$1,685,531
$3,355,739
Table 6
Steel weight, in kg, for Cases 1 to 4.
Structure
3-storeys
6-storeys
9-storeys
Case 1
Case 2
Case 3
Case 4
wS
wBRBs
wS
wBRBs
wS
wBRBs
wS
wBRBs
38,495
107,018
213,063
1972
6290
12,236
30,406
64,478
140,375
2632
14,076
24,280
30,406
64,478
140,375
999
8044
17,780
30,406
64,478
140,375
4674
20,377
36,325
411
H. Guerrero et al. / Engineering Structures 136 (2017) 406–419
Table 7
Estimates of initial cost for Cases 1 to 4.
Structure
Case 1
3-storeys
6-storeys
9-storeys
Case 2
Case 3
CS
CT
CS
CT
CS
CT
CS
CT
0.23C0
0.24C0
0.23C0
1.03C0
1.04C0
1.03C0
0.20C0
0.20C0
0.20C0
1.0C0
1.0C0
1.0C0
0.17C0
0.17C0
0.18C0
0.97C0
0.97C0
0.98C0
0.23C0
0.23C0
0.23C0
1.03C0
1.03C0
1.03C0
(b) The total initial cost is CT = CS + Cn; where CS was estimated
in the previous step and Cn = 0.8C0 is shown in Table 5. It
should be noted that Cn = 0.8C0 means that, consistently,
the five cases have the same costs for non-structural components and contents. However, the total and structure costs
are different.
By analysing Tables 5–7, it is apparent that, even though the
steel weight in the structures may be significantly different, the
impact on the total initial cost, CT, can be very small (i.e. differences
smaller than 5% are observed).
4. Expected losses
To evaluate the expected (or probabilistic) losses in the studied
hospital structures, the assessment methodology proposed in the
FEMA P58 Project [30] and described previously in the introduction
was used here. The procedure consists of four analyses; which are
described in the next subsections.
4.1. Seismic hazard analysis
This analysis is normally conducted using a Probabilistic Seismic Hazard Analysis (PSHA) whose outcome is a seismic hazard
curve (see [35]). In PSHA, all the possible source regions that
may generate potentially damaging earthquakes are included,
along with their associated uncertainties. In this paper, a PSHA is
conducted using the computer program CRISIS [36] – which
includes information on the seismicity of the Mexican Republic
and ground motion attenuation laws. The resulting seismic hazard
curves for rock and soft soil sites in Mexico City are shown in Fig. 7.
These curves provide the Mean Annual Frequency (MAF) of exceeding a given value of pga. For example, for a pga = 0.10g, the MAF is
0.004 in rock and 0.02 in soft soils; which are equal to return periods of 250 years and 50 years, respectively. Since the studied structures are considered to be located in the soft soil of Mexico City,
the soft soil curve, which contains site effects, is used in this study.
4.2. Response analysis
Mean Anual Frequency
In Ref. [31], it was found that the use of equivalent SDOF dual
oscillators to estimate the response of low- and medium-rise,
10
1
0.1
0.01
0.001
0.001
Case 4
0.01
0.1
1
peak ground acceleration, g
Fig. 7. Seismic hazard curves for Mexico City.
10
regular, multi-storey structures equipped with BRBs provides reasonable results, while the time of analysis is significantly reduced.
Therefore, the vectors of the responses obtained in Section 2.3
were used to approximate the responses of the studied structures.
Because the probability of collapse has a significant impact on
the estimated losses, this parameter is obtained using Incremental
Dynamic Analysis (IDA) [37]. For that purpose, equivalent SDOF
dual oscillators and the 30 ground motions used during the design
process (see Appendix D in [31]) were used. The ground motions
were increasingly scaled between pga = 0.025g and 1.0g with increments of 0.025g. Collapse was considered to occur when: (1) a
small increment of seismic intensity generates a very large
increase of displacement; (2) the computer program shows numerical instability; or (3) the displacement demands are larger than
the corresponding collapse displacement threshold, i.e. the displacement at which the lateral load capacity of the structure (as
obtained from pushover analysis) is reduced by more than 20%.
For the equivalent SDOF oscillators (corresponding to the three-,
six- and nine-storey frames), the displacement thresholds were
0.15 m, 0.27 m and 0.39 m, respectively.
Fig. 8 shows the results for Case 0 of the six-storey frame. Fig. 8a
shows the IDA curve, where the horizontal axis shows the peak displacements and the vertical axis the seismic intensity, or pga. The
mean, and the mean plus and minus one standard deviation, are
indicated in the figure by dark lines. It can be observed from the
figure that, as compared to some example results presented by
others (e.g. Fig. 9 in Ref. [37]) where the record-to-record dispersion is very large, the record-to-record dispersion in Fig. 8a is considerably smaller (with a coefficient of variation around 0.17). This
may be attributed to the fact that the period of the structure is
located within the acceleration-sensitive region, as observed in
the 5%-damped pseudo-velocity spectra of Fig. 9. This observation
suggests that the pga is a reasonable parameter to represent the
seismic intensity for the structures studied in this paper.
Fig. 8b shows a collapse fragility function estimated by counting
the number of ground motions that predicted collapse (for a given
pga) divided by the total number of analyses, i.e. 30. The observed
data were fitted to a log-normal distributed function with a mean
of pga = 0.53g and a dispersion (or record-to-record variability) of
ba = 0.17. Then, the dispersion was increased to include other
sources of uncertainty using the following equation [30]:
b¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b2a þ b2c þ b2m
ð2Þ
where bc is the uncertainty associated with construction quality and
bm is the uncertainty associated with the completeness of the
numerical model. Since the construction quality of hospitals is
expected to be rigorous, a value of bc = 0.10 was selected; while
bm = 0.40 was chosen reflecting the assumption that a dual SDOF
system would not represent the true behaviour of a MDOF structure
fitted with BRBs. In this way, the total dispersion is b = 0.45. Similarly, the total dispersions of all the other response parameters (or
EDPs) of Appendix A were determined using Eq. (2).
Fig. 10 shows the log-normal fitted collapse fragility functions
for each case for each frame. For all the studied structures, Cases
1 and 4 had the smallest probability of collapse (conditioned to a
given pga); then, in order, Case 2, Case 0 and Case 3.
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H. Guerrero et al. / Engineering Structures 136 (2017) 406–419
Fig. 8. Results of IDA corresponding to Case 0 for the six-storey frame.
Once the components and quantities were defined, the corresponding DS were defined in the form of fragility functions. An
example of these functions is given in Fig. 11 for a typical partition wall made of gypsum with metal studs. Three damage states
(DSi) were defined. If the wall was subjected to an inter-storey
drift demand of 0.005, it would have a probability of: 0.93 of
being in damage state DS1 or worse; 0.2 of being in DS2 or worse;
and 0.03 of being in DS3 or worse. Fig. 11b shows the repair costs
and actions for the damage state DS1 of the example partition
wall. It is appreciated that the unit repair cost may reduce as
the quantity increases, i.e. the efficiency of scale is considered.
Uncertainty was also considered by defining mean, dispersion
and distribution of the costs. In this study, the PACT package
[30] was used, in which a vast database has been compiled and
contains information of many fragility functions with the definitions of damage states, repair actions and mean costs, along with
their dispersions and types of distributions (e.g. normal, lognormal, etc.). The estimation of repair costs using this database is
reliable as it contains the contributions from many engineers,
researchers and contractors.
Fig. 9. Pseudo-velocity spectra for the 30 earthquake ground motions.
4.4. Loss analysis
4.3. Damage state analysis
This analysis requires detailed definition of the damage states
(DS), their corresponding consequence actions, and repair costs
for each component. Since hypothetical hospitals are analysed, neither the components nor their quantities are known. Therefore,
normative components and quantities, which are considered sufficient for preliminary assessment [30] and for comparison purposes, are selected from the Normative Quantity Estimation Tool
provided by the FEMA P58 Project [30] for healthcare occupancy.
They are shown elsewhere (see Appendix C in [31]).
0.5
1
0.25
0
0 0.25 0.5 0.75
1
pga
a) Three-storey frame
1
Collapse probability
0.75
0
1
2
3
4
Collapse probability
Collapse probability
1
The total repair cost was estimated using statistics from the
structural response and the fragility data given in the previous sections. For that purpose, the Monte Carlo procedure proposed by
Yang et al. [29] and adapted by the FEMA P58 Project [30] was
used. It is important to highlight that the repair cost estimated
herein only includes the typical cost associated with repair of the
components of the hospitals. No costs were associated with sophisticated equipment, or compensation due to injuries or loss of
human lives, because of the difficulties associated with their
estimation.
0.75
0.5
0.25
0
0 0.25 0.5 0.75
1
pga
b) Six-storey frame
Fig. 10. Collapse fragility functions.
0.75
0.5
0.25
0
0 0.25 0.5 0.75
1
pga
c) Nine-storey frame
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H. Guerrero et al. / Engineering Structures 136 (2017) 406–419
Repair cost (USD)
Probability
P(DM > DSi | Drift)
1
0.75
0.5
DSi:
state
0.25
ith
damage
0
0
0.01
0.02
0.03
2800
DS1: Screws pop-out,
minor cracking of wall
board, warping or cracks.
- action: repair 15 m of
wall in both sides
2750
2700
Uncertainty:
Distr: Normal
Dispersion: 0.44
2650
2600
0
Inter-storey drift ratio
a) Fragility functions
5
10
15
20
Quantity
b) repair cost and actions for DS1
Fig. 11. Fragility functions and repair actions of a typical partition wall (data taken from the PACT database [30]). DSi refers to the ith damage state, in this case, of the
partition wall.
4.4.1. Intensity-based assessment
First, various intensity-based assessments were conducted for
intensities between pga = 0.05g and 0.8g, with intervals of 0.05g.
For simplicity, Figs. 12–14 show only the cumulative distribution
functions of the repair costs for intensities of pga = 0.10g, 0.20g
and 0.30g. The repair costs have been normalised by 1.2C0; which
includes the initial cost of Case 0, C0, plus 20% for demolition and
clearance of the site. It was found that the repair costs, from the
lowest to the higest, are in the following order: Case 4, Case 1, Case
2, Case 3 and Case 0.
4.4.2. Time-based assessment
To estimate the average annual value of the repair costs, timebased analyses were conducted by integrating the intensitybased cumulative distribution functions over all the hazard levels
– which are defined by the hazard curve of Section 4.1. Further
guidance can be found in [29,30]. Figs. 15–17 show the average
annual repair costs and times for each structure studied. It was
Case 1
1
5. Cost-benefit analysis
Although the estimation of initial costs and annualised losses
help to provide a good understanding of the convenience of a given
case, or design option, cost-benefit analysis provides a further
Case 2
Case 3
Case 0
0.5
pga=
0.10g
0.25
0
0
0.75
P(Repair cost
0.75
1
c)
c)
1
P(Repair cost
P(Repair cost
c)
Case 4
found that the highest annualised losses are those of Case 0, while
the lowest are those of Cases 1 to 4, i.e. with BRBs. The annualised
losses of Case 4 are always the smallest; followed by Cases 1, 2 and
3, respectively.
Similarly, the probabilities of collapse and of loss of functionality during the lifetime of the hospitals (50 years) were estimated.
They are shown in Figs. 18–20. It is appreciated that, compared
to the limit of 10% established by FEMA P695 [38], the probability
of collapse is small in each case. However, in relative terms, Cases 1
and 4 consistently present the lowest probabilities of collapse.
Also, the probability of loss of functionality of Cases 1 and 4 is consistently the lowest.
0.5
pga=
0.20g
0.25
0.5
0.25
0
0
0.15 0. 3 0.45
pga=
0.30g
0.75
1
0 0.25 0.5 0.75 1
Repair Cost / 1.2C0
b) pga=0.20g
Repair Cost / 1.2C0
0 0.25 0.5 0.75
Repair Cost / 1.2C0
a) pga=0.10g
c) pga=0.30g
Fig. 12. Cumulative distribution functions of repair cost: three-storey frame.
Case 1
Case 2
Case 3
1
0.5
pga=
0.10g
0.25
0
0
0.1
0.2
0.3
Repair Cost / 1.2C0
a) pga=0.10g
0.75
0.5
0.25
pga=
0.20g
0
0
0.25 0.5 0.75
Repair Cost / 1.2C0
b) pga=0.20g
P(Repair cost
c)
0.75
Case 0
c)
1
1
P(Repair cost
P(Repair cost
c)
Case 4
0.75
0.5
0.25
pga=
0.30g
0
0 0.25 0.5 0.75
1
Repair Cost / 1.2C0
c) pga=0.30g
Fig. 13. Cumulative distribution functions of repair cost: six-storey frame.
H. Guerrero et al. / Engineering Structures 136 (2017) 406–419
0.5
0.25
0
0
0.1
Case 3
Case 0
1
c)
pga=
0.10g
Case 2
1
c)
0.75
Case 1
P(Repair cost
c)
1
P(Repair cost
Case 4
0.75
0.5
pga=
0.20g
0.25
0.75
P(Repair cost
414
0.5
0
0.2
0
0 0.2 0.4 0.6
Repair Cost / 1.2C0
Repair Cost / 1.2C0
a) pga=0.10g
pga=
0.30g
0.25
0 0.25 0.5 0.75 1
Repair Cost / 1.2C0
b) pga=0.20g
c) pga=0.30g
Fig. 14. Cumulative distribution functions of repair cost: nine-storey frame.
3.0
15,000
10,000
5,000
0
0
1
2
3
4
Repair time, days
Repair cost, $
20,000
2.5
2.0
1.5
1.0
0.5
0
1
2
3
4
0.0
Case
Case
a) Repair costs
b) Repair times
Fig. 15. Average annual repair costs and times: three-storey frame.
5.0
40,000
30,000
0
1
2
3
4
20,000
Repair time, days
Repair cost, $
50,000
4.5
4.0
3.5
3.0
2.5
0
2.0
1
2
3
Case
Case
a) Repair costs
b) Repair times
4
Fig. 16. Average annual repair costs and times: six-storey frame.
6.5
70,000
60,000
50,000
40,000
0 1
2 3
4
30,000
Repair time, days
Repair cost, $
80,000
6.0
5.5
5.0
4.5
4.0
Case
a) Repair costs
0
1 2
3 4
Case
b) Repair times
Fig. 17. Average annual repair costs and times: nine-storey frame.
comparison helpful for choosing which option may be the most
convenient over a period of time (e.g. 50 years). Herein, the present
value of annualised losses, associated with future damage,
was added to the corresponding initial cost of each studied case.
In this way, not only initial costs but also lifecycle costs can be
compared, to help decide which case or design option is the most
convenient.
First, the net present value (NPV) of the stream of annualised
losses was estimated as:
t
NPV ¼ An ½ð1 1=ð1 þ in Þ Þ=in ð3Þ
where t is the period of time in years, which is considered to be
50 years in this study; in is the interest rate, considered to be 7%;
and An is the value of the annualised losses, which includes the
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H. Guerrero et al. / Engineering Structures 136 (2017) 406–419
1.0%
0.6%
0.4%
0.2%
0.0%
0
1
2
3
4
Probability of loss of
functionality
Probability of
collapse
0.8%
0.8%
0.6%
0.4%
0.2%
0.0%
0
1
2
3
4
Case
Case
a) Collapse in 50 years
b) Loss of functionality in 50 years
Fig. 18. Probabilities of collapse and of loss of functionality: three-storey frame.
Probability of
collapse
1.0%
0.5%
0.0%
0
1
2
3
4
Probability of loss of
functionality
2.0%
1.5%
1.5%
1.0%
0.5%
0
1
2
3
4
0.0%
Case
Case
a) Collapse in 50 years
b) Loss of functionality in 50 years
Fig. 19. Probabilities of collapse and of loss of functionality: six-storey frame.
2.0%
1.5%
1.0%
0.5%
0.0%
0
1
2
3
4
Probability of loss of
functionality
Probability of
collapse
2.0%
1.5%
1.0%
0.5%
0.0%
Case
a) Collapse in 50 years
0
1
2
3
4
Case
b) Loss of functionality in 50 years
Fig. 20. Probabilities of collapse and of loss of functionality: nine-storey frame.
repair costs and the costs due to loss of functionality. In this study,
it is assumed that each day of downtime has a cost of 0.01C0. As an
example, for Case 0 of the three-storey frame, the annualised repair
cost is $18,185 and the annualised repair time is 2.71 days (see
Fig. 15); thus, A = $18,185 x (C0/$606,303) + 2.71 x 0.01C0 = 0.057C0.
Therefore, NPV = 0.057C0[(1–1/(1 + 0.07)50)/0.07] = 0.79C0.
Figs. 21–23 show the initial costs and initial costs plus NPV of
the annualised losses for all the studied structures. They are presented in terms of C0. It can be appreciated that:
In terms of initial costs, Case 3 is the cheapest, while Cases 1
and 4 are the most expensive. However, it has to be recognised
that the differences of the initial costs are less than 5% when
compared to Case 0. In this context, the differences may be
regarded as insignificant.
In terms of the total lifecycle costs (i.e. Initial costs + NPV), the
cheapest case for the three-, six- and nine-storey frames is Case
4, while Case 0 (i.e. frames without BRBs) is consistently the
most expensive.
Comparison between Cases 0 and 2 shows that, even when they
have the same initial costs, the lifecycle cost is significantly
lower for Case 2 (i.e. with BRBs).
Comparison between Cases 1 and 4 shows that they have similar initial costs but Case 4 has smaller lifecycle costs. Since
the contribution of BRBs is higher in Case 4 (see Tables 2 and
3), it can be said that the higher the contribution of the BRBs
the lower the lifecycle costs.
6. Discussion
Benefits of BRBs. By analysing Figs. 15–23, it can be appreciated
that, for the studied frames, the BRBs help to significantly
reduce the expected losses, lifecycle costs, probability of collapse, and probability of loss of functionally.
The best options for design. Since Case 4 consistently had the
lowest repair costs, repair times, and probabilities of collapse
and loss of functionality, it is regarded as the best option. Cases
2 and 3 may be also seen as good options because they had better behaviour than the bare frame counterpart, at similar or
smaller initial cost.
Smaller cross-sectional profiles. By analysing Table 2, it can be
observed that the steel profiles are smaller in Cases 2, 3 and
4. There are additional benefits from using BRBs because: (a)
smaller cross-sectional depths of beams provides a higher
H. Guerrero et al. / Engineering Structures 136 (2017) 406–419
Initial cost, C0
1.05
1.00
0.95
0
1
2
3
4
0.90
Case
a) Initial costs
Initial cost + NPV, C0
416
1.8
1.6
1.4
0
1
2
3
4
1.2
Case
b) Expected total costs in 50 years
Initial cost, C0
1.05
1.00
0.95
0.90
0
1
2
3
4
Initial cost+NPV, C0
Fig. 21. Initial and lifecycle costs: three-storey frame.
2.0
1.9
1.8
0
1
2
3
4
1.7
Case
Case
b) Expected total costs in 50 years
a) Initial costs
Initial cost, C0
1.05
1.00
0.95
0
1
2
3
4
0.90
Initial cost+NPV, C0
Fig. 22. Initial and lifecycle costs: six-storey frame.
2.2
2.1
2.0
1.9
Case
a) Initial costs
0
1
2
3
4
Case
b) Expected total costs in 50 years
Fig. 23. Initial and lifecycle costs for the nine-storey frame.
inter-storey clearance, which is significant from an architectural
point of view; and (b) lighter beams and columns allow further
reductions of cost and time during construction and demolition
of the frames.
Variability of costs. It should be noted that this study is based on
fixed costs of structural steel, BRBs, components and downtime.
However, with significantly different costs, different conclusions may be found.
Limitation of this study. The dynamic response of the frames is
estimated using equivalent dual SDOF systems. The response
is then converted to vectors of response for MDOF structures.
This may have an impact in the estimation of the expected
losses. However, it is considered that this should affect all the
studied cases proportionally; therefore, relative comparisons
should still be valid.
Implications of this study. The results of this paper suggest that
decision makers (such as the Minister of Health of Mexico)
should consider constructing hospitals protected with BRBs
because, with similar initial costs, better response and smaller
losses due to future earthquakes are to be expected. On the
other hand, by comparing Case 0 and 1 it is apparent that
upgrading a hospital with BRBs could cost less than 5% of its
total cost; while the benefits might be substantially higher
because losses due to future earthquakes will be reduced
significantly.
BRBs versus other stiffening options. As discussed before, the
improvement of the response due to the use of BRBs can be
attributed to diverse aspects, such as: (1) plastic dissipation
capacity; (2) high lateral stiffness; and (3) higher levels of damping. Because an in depth analysis, that explains how these individual structural properties reduce the lateral response of the
studied frames, is lacking; it can be argued that other stiffening
options may provide similar benefits than those discussed for
the BRBs. Although detailed studies are underway to establish
a comparison of the life cycle benefits of different stiffening
options, at this point some observations can be made. Firstly
and as discussed by Guerrero et al. [25,29], the lateral stiffness
of the BRBs can be fine-tuned in such a manner as to achieve
an optimum stiffness-based design. Secondly, and as discussed
in detail by Teran-Gilmore et al. [39], in the soft soils of Mexico
City, stiffness-degrading systems located in the accelerationsensitive region of the spectra develop larger seismic demands
417
H. Guerrero et al. / Engineering Structures 136 (2017) 406–419
than those with elasto-plastic behaviour. Within this context,
the use of reinforced concrete walls does not allow for the
fine-tuning of the structural system, fact that results in systems
having much larger lateral strength and stiffness than those that
result from the use BRBs. Because this usually represents very
large costs in terms of the foundation system, reinforced concrete walls are being used less and less in low-rise systems
located in the soft soils of Mexico City. In terms of the use of
concentric steel braces, the buckling of the braces result in
stiffness-degrading behaviour and important limitations in
terms of fine-tuning the lateral stiffness (the need to control
lateral deformations and buckling usually results in large
cross-sectional areas). The use of concentric steel braces usually
requires much larger connection plates and stronger foundation
systems. Because of this, several recent rehabilitation projects
for low-rise schools and hospitals in Mexico City have used BRBs
as an alternative to concentric steel braces and reinforced concrete walls.
7. Conclusions
Three-, six-, and nine-storey framed structures, representative
of hospitals located in the lakebed zone of Mexico City, were
designed with, and without, BRBs. For comparison purposes, five
cases were studied, namely: Case (0) which serves as reference,
consists of a frame designed without BRBs to fully support the seismic loads; Case (1) the structure of Case 0 is upgraded with BRBs;
Case (2) the main frame is re-designed under gravity loads only,
then BRBs are provided so that the initial cost matches that of Case
0; Case (3) the capacity of the BRBs of Case 2 is reduced so that the
lateral displacements are similar to those of Case 0; and Case (4)
the capacity of the BRBs of Case 2 is increased so that the initial
cost and load capacity are similar to Case 1. The following conclusions can be drawn:
When BRBs are introduced in structures, representative of hospitals located in very soft soils (such as those in the lakebed
zone of Mexico City), the expected losses and lifecycle costs
are reduced significantly, as can be appreciated in Figs. 21–23.
In particular, Case 4 is regarded as the most convenient option
for design, because it consistently had the lowest repair costs,
lifecycle costs, repair times, probability of collapse and probability of loss of functionality.
Comparison between Cases 0 and 2 shows that, even with similar initial costs, the lifecycle costs were significantly lower for
Case 2 (i.e. with BRBs).
Comparison between Cases 0 and 3 shows that, even when they
were designed for similar displacement demands, the initial and
lifecycle costs of the latter were smaller.
Comparison between Cases 1 and 4 shows that, even when they
had similar initial costs and capacity, Case 4 had significantly
smaller lifecycle costs than Case 1, it is therefore suggested that
the higher the contribution of the BRBs the better.
Comparison of Cases 0 and 1 shows that upgrading a hospital
with BRBs could cost less than 5% of its total cost; while the
benefits might be substantially higher.
An additional exploitable benefit of BRBs is that they provide
smaller structural profiles allowing higher inter-storey clearances and lighter profiles.
Finally, the reader should be aware of the fact that the response
demands of the studied structures were estimated using equivalent dual SDOF oscillators and then transformed into the demands
of the multi-storey hospitals using simplified height-wise distribution functions. Although small variations may be expected, it is
assumed that for regular, first mode-dominated structures, the
response demands of the studied structures are reasonably well
estimated using equivalent dual SDOF oscillators. Therefore, it is
expected that the results are valid for comparison purposes in relative terms. For different types of structural systems, such as irregular or high-rise buildings, different results might be observed and
specific studies may be needed.
Acknowledgements
The first author acknowledges the sponsorship provided by the
National Council for Science and Technology (CONACyT) and the
Institute of Engineering at the UNAM in Mexico. We acknowledge
to Miguel A. Jaimes for kindly providing the hazard curves for rock
and a soft soil sites in Mexico City. Finally, the kind revision of the
English of this manuscript by Brian Ellis is recognised.
Appendix A
Statistics of Engineer demand parameters of the equivalent dual
SDOF systems (see Tables A.1–A.3).
Table A.1
Three-storey hospital.
Case
pga
Displ., cm
bdispl
cr
Standard dev. of cr
Velocity, m/s
bvel
Accel., g
baccel
0
0.05g
0.10g
0.20g
0.30g
1.32
2.32
4.63
7.00
0.30
0.25
0.25
0.27
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.080
0.20
0.37
0.74
1.09
0.13
0.12
0.12
0.12
0.11
0.19
0.38
0.55
0.30
0.25
0.25
0.18
1
0.05g
0.10g
0.20g
0.30g
0.85
1.53
3.01
4.57
0.31
0.25
0.23
0.18
0.000
0.000
0.012
0.010
0.000
0.009
0.013
0.013
0.18
0.34
0.67
0.98
0.14
0.14
0.14
0.15
0.09
0.17
0.30
0.42
0.31
0.24
0.18
0.15
2
0.05g
0.10g
0.20g
0.30g
0.90
1.63
3.19
5.01
0.33
0.27
0.23
0.16
0.000
0.000
0.017
0.013
0.000
0.017
0.020
0.019
0.18
0.34
0.67
0.98
0.14
0.14
0.14
0.15
0.09
0.17
0.29
0.40
0.33
0.24
0.17
0.13
3
0.05g
0.10g
0.20g
0.30g
1.15
2.09
4.10
6.05
0.25
0.21
0.17
0.15
0.000
0.000
0.005
0.004
0.000
0.008
0.006
0.021
0.18
0.35
0.68
1.01
0.12
0.12
0.12
0.13
0.10
0.17
0.31
0.44
0.25
0.19
0.16
0.13
4
0.05g
0.10g
0.20g
0.30g
0.61
1.16
2.27
3.56
0.24
0.20
0.20
0.18
0.000
0.000
0.000
0.035
0.000
0.000
0.048
0.034
0.16
0.32
0.64
0.96
0.16
0.16
0.16
0.15
0.08
0.15
0.28
0.37
0.24
0.20
0.13
0.11
418
H. Guerrero et al. / Engineering Structures 136 (2017) 406–419
Table A.2
Six-storey hospital.
Case
pga
Displ., cm
bdispl
cr
Standard dev. of cr
Velocity, m/s
bvel
Accel., g
baccel
0
0.05g
0.10g
0.20g
0.30g
2.24
3.95
7.88
11.90
0.32
0.27
0.27
0.27
0.000
0.000
0.000
0.001
0.001
0.000
0.026
0.104
0.22
0.42
0.83
1.20
0.16
0.13
0.13
0.11
0.11
0.19
0.37
0.49
0.32
0.27
0.25
0.11
1
0.05g
0.10g
0.20g
0.30g
1.48
2.70
5.18
7.46
0.21
0.18
0.16
0.15
0.000
0.000
0.003
0.010
0.000
0.000
0.013
0.013
0.19
0.36
0.72
1.05
0.12
0.11
0.12
0.12
0.10
0.17
0.32
0.43
0.21
0.19
0.13
0.13
2
0.05g
0.10g
0.20g
0.30g
1.58
2.87
5.52
8.24
0.26
0.21
0.18
0.17
0.000
0.000
0.004
0.029
0.000
0.003
0.032
0.032
0.19
0.37
0.72
1.07
0.14
0.12
0.12
0.13
0.10
0.17
0.31
0.40
0.26
0.21
0.10
0.10
3
0.05g
0.10g
0.20g
0.30g
1.98
3.62
7.10
10.79
0.27
0.23
0.22
0.21
0.000
0.000
0.019
0.034
0.000
0.012
0.014
0.066
0.21
0.40
0.77
1.14
0.13
0.12
0.12
0.12
0.09
0.17
0.30
0.41
0.27
0.22
0.16
0.13
4
0.05g
0.10g
0.20g
0.30g
1.32
2.39
4.61
6.62
0.23
0.19
0.18
0.18
0.000
0.000
0.000
0.018
0.000
0.000
0.041
0.045
0.18
0.36
0.70
1.03
0.12
0.12
0.12
0.13
0.10
0.17
0.33
0.43
0.23
0.19
0.15
0.08
Table A.3
Nine-storey hospital.
Case
pga
Displ., cm
bdispl
cr
Standard dev. of cr
Velocity, m/s
bvel
Accel., g
baccel
0
0.05g
0.10g
0.20g
0.30g
3.67
6.32
12.40
19.67
0.36
0.32
0.25
0.30
0.000
0.000
0.000
0.065
0.004
0.001
0.059
0.148
0.26
0.48
0.92
1.29
0.22
0.20
0.15
0.13
0.11
0.20
0.37
0.44
0.36
0.32
0.17
0.05
1
0.05g
0.10g
0.20g
0.30g
2.31
4.24
8.16
11.97
0.28
0.24
0.23
0.21
0.000
0.000
0.007
0.014
0.002
0.004
0.013
0.048
0.22
0.42
0.81
1.19
0.16
0.14
0.13
0.13
0.09
0.17
0.32
0.43
0.28
0.24
0.18
0.15
2
0.05g
0.10g
0.20g
0.30g
2.65
4.89
9.56
15.15
0.29
0.26
0.23
0.22
0.000
0.000
0.011
0.045
0.001
0.017
0.043
0.064
0.23
0.43
0.83
1.20
0.15
0.14
0.12
0.14
0.09
0.17
0.29
0.38
0.29
0.24
0.13
0.11
3
0.05g
0.10g
0.20g
0.30g
3.13
5.71
11.29
19.41
0.32
0.25
0.24
0.27
0.000
0.000
0.030
0.112
0.001
0.015
0.040
0.117
0.24
0.45
0.85
1.23
0.20
0.15
0.14
0.13
0.10
0.18
0.29
0.37
0.32
0.21
0.14
0.08
4
0.05g
0.10g
0.20g
0.30g
2.26
4.11
7.86
11.91
0.28
0.24
0.22
0.21
0.000
0.000
0.000
0.022
0.000
0.000
0.064
0.054
0.22
0.41
0.81
1.16
0.16
0.14
0.13
0.12
0.10
0.17
0.32
0.40
0.28
0.24
0.15
0.10
Symbols in Tables A.1–A.3:
pga = peak ground acceleration.
g = acceleration of the gravity.
b = Coefficient of variation.
cr = Residual to peak displacement ratio.
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