Evaluation of the Economic Benefits of using Buckling-Restrained Braces in Hospital Structures Located in Very Soft Soils

Héctor Guerrero1,*, Amador Terán-Gilmore2, Tianjian Ji1 and José A. Escobar3

1 School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Pariser Building, Manchester, M13 9PL, UK

2 Universidad Autónoma Metropolitana, Department of Materials, México City, México

3 Universidad Nacional Autónoma de México, Institute of Engineering, México City, México

Abstract

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.

Keywords: Buckling-Restrained Braces (BRBs), Hospital Structures, Very Soft Soils, Lakebed Zone of Mexico City, Economic Benefits of BRBs

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

1

hospitals) were less than 10 stories high [10], which can be classified as short-period structures.

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. Performance-Based 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 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] (Figure 1). The phases are seismic hazard analysis, dynamic response analysis, damage analysis and loss analysis. They are integrated using the total probability theorem, namely

(1)

where (DV) is the mean annual frequency of exceeding a given decision variable (DV), for example, a repair cost threshold; (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

2

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.

Seismic hazard analysis

Outcome: (IM)

IM = intensity measure, e.g. pga

Dynamic response analysis

Outcome: G(EDP|IM)

EDP = engineering demand parameter,

e.g. inter-storey drift

Damage analysis

Outcome: G(DM|EDP)

DM = damage measure, e.g. number

of cracks in a concrete element

Lossanalysis

Outcome: G(DV|DM)

DV = decision variable, e.g. total repair cost after an

earthquake

Figure 1. PEER framework for performance assessment [28]

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 nine-storey 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-of-freedom (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 response2.1 Studied CasesFigure 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 Figure 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.

3

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. 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.

Thre

e, si

x an

d ni

ne st

orey

s

6 bays @ 8 m

Figure 2. Layout of hospitals structures studied in this paper

No

BR

Bs

0 1 2 3 4Case

Initi

al co

st

0 1 2 3 4Case

Load

cap

acity

0 1 2 3 4Case

Dis

plac

emen

t dem

ands

No

BR

Bs

No

BR

Bs

Figure 3. Schematic representation of the studied cases

2.2 DesignIt 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

4

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 linear-elastic response demands. For Cases 1 to 4, =5%, 7%, 8% and 10%, respectively, for the seismic intensities defined in Table 1; while for Case 0, =3% was considered for pga=0.05g and =5% for the other intensities.

Table 1. Objectives of design of the hospitals studied in this chapterObjective of design 1 2 3 4

Performance Fully Operability Operability Life Safety Collapse

PreventionSeismic intensity, pga 0.05g 0.10g 0.20g 0.30gMax. drift 0.0025 0.005 0.010 0.020

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 Figure 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 Table 2, Table 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

5

steel plates with a thickness of 7.6 cm (3 inches) 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]).

Table 2. Steel profiles of the studied structuresColumns (HSS profiles,

mm)Beams

Structure Storey Cases 0 & 1 Cases 2, 3, 4 Cases 0 & 1 Cases 2,3,43-storeys 1 500x25 500x19 W21x68 W21x68

2 500x25 500x13 W21x68 W21x683 500x19 500x13 W21x62 W21x62

6-storeys 1 to 3 600x38 500x16 W27x94 W24x684 to 6 600x19 500x13 W27x84 W24x68

9-storeys 1 to 3 900x38 500x25 W27x129 W24x684 to 6 900x25 500x16 W27x102 W24x687 to 9 800x19 500x13 W27x84 W24x68

Table 3. Cross-sectional areas, in cm2, of the BRBs of the first storeyStructure Case 1 Case 2 Case 3 Case 43-storeys 13.5 18.0 6.84 32.06-storeys 46.9 105.0 60.0 152.09-storeys 64.0 127.0 93.0 190.0

Table 4. Fundamental periods of the studied structures, in secondsStoreys Case 0 Case 1 Case 2 Case 3 Case 4Three 0.85 0.73 0.74 0.84 0.66Six 1.17 0.97 0.97 1.10 0.88Nine 1.41 1.20 1.26 1.35 1.15

2.3 Dynamic ResponseIn 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 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).

6

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.

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.

Examples of the vectors of mean demands are shown in Figures 4 to 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].

0

1

2

3

0 0.005 0.01

Stor

ey

Inter-storey drifts

41230

Cases:

0

1

2

3

0 1

Stor

ey

Velocity, m/s

0

1

2

3

0 0.2 0.4 0.6

Stor

ey

Acceleration, ga) Inter-storey drifts, m/m b) Floor velocities, m/s c) Floor accelerations, gFigure 4. Response estimated for pga=0.20g: three-storey frame

0123456

0 0.005 0.01

Stor

ey

Inter-storey drifts0123456

0 1

Stor

ey

Velocity

0123456

0 0.2 0.4 0.6

Stor

ey

Accelerationa) Inter-storey drifts, m/m b) Floor velocities, m/s c) Floor accelerations, gFigure 5. Response estimated for pga=0.20g: six-storey frame

7

0123456789

0 0.005 0.01

Stor

ey

Inter-storey drifts

0123456789

0 0.5 1 1.5

Stor

ey

Floor velocities

0123456789

0 0.2 0.4 0.6

Stor

ey

Floor accelerationsa) Inter-storey drifts, m/m b) Floor velocities, m/s c) Floor accelerations, gFigure 6. Response estimated for pga=0.20g: nine-storey frame

3 Initial cost3.1 DefinitionsFor 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.

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 EstimationTable 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 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.

Table 5. Estimation of initial cost for Case 0Structur

eSteel

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 38,495 $121,261 $485,042 $606,3036-storeys 107,018 $337,106 $1,348,424 $1,685,5319-storeys 213,063 $671,178 $2,684,591 $3,355,739

8

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 6. Steel weight, in kg, for Cases 1 to 4Case 1 Case 2 Case 3 Case 4

Structure wS wBRBs wS wBRBs wS wBRBs wS wBRBs

3-storeys 38,495 1,972 30,406 2,632 30,406 999 30,406 4,6746-storeys 107,018 6,290 64,478 14,076 64,478 8,044 64,478 20,3779-storeys 213,063 12,236 140,375 24,280 140,37

517,780 140,375 36,325

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: 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.

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.

Table 7. Estimates of initial cost for Cases 1 to 4Case 1 Case 2 Case 3 Case 4

Structure CS CT CS CT CS CT CS CT

3-storeys 0.23C0 1.03C0 0.20C0 1.0C0 0.17C0 0.97C0 0.23C0 1.03C0

6-storeys 0.24C0 1.04C0 0.20C0 1.0C0 0.17C0 0.97C0 0.23C0 1.03C0

9-storeys 0.23C0 1.03C0 0.20C0 1.0C0 0.18C0 0.98C0 0.23C0 1.03C0

By analysing tables 5 to 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 analysisThis 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

9

City are shown in Figure 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.

0.001

0.01

0.1

1

10

0.001 0.01 0.1 1 10

Mea

n A

nual

Fre

quen

cy

peak ground acceleration, g

Figure 7. Seismic hazard curves for Mexico City

4.2 Response analysisIn ref. [31], it was found that the use of equivalent SDOF dual oscillators to estimate the response of low- and medium-rise, 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.

Figure 8 shows the results for Case 0 of the six-storey frame. Figure 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. Figure 9 in ref. [37]) where the record-to-record dispersion is very large, the record-to-record dispersion in Figure 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 Figure 9. This observation suggests that the pga is a reasonable parameter to represent the seismic intensity for the structures studied in this paper.

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Figure 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 a=0.17. Then, the dispersion was increased to include other sources of uncertainty using the following equation [30]:

(2)

where c is the uncertainty associated with construction quality and m 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 c=0.10 was selected; while m=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 =0.45. Similarly, the total dispersions of all the other response parameters (or EDPs) of Appendix A were determined using equation (2).

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8

pga,

g

displacement demand, m

0.0

0.3

0.5

0.8

1.0

0 0.2 0.4 0.6 0.8 1

Col

laps

e pr

obab

ility

pga, g

ObservationsFitted =0.45

a) IDA curve b) Collapse fragility functionFigure 8. Results of IDA corresponding to Case 0 for the six-storey frame

0.001

0.01

0.1

1

10

0.1 1 10 100

Pseu

do-v

eloc

ity, m

/s

Period, sec

Accel.sensitive

Displ.sensitive

Vel.sensitive

Spectral regions

Stud

ied

stru

ctur

es

Figure 9. Pseudo-velocity spectra for the 30 earthquake ground motions

11

Figure 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.

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1

Col

laps

e pr

obab

ility

pga

01234

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1C

olla

pse

prob

abili

typga

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1

Col

laps

e pr

obab

ility

pgaa) Three-storey frame b) Six-storey frame c) Nine-storey frame

Figure 10. Collapse fragility functions

4.3 Damage State AnalysisThis 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]).

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 Figure 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. Figure 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.

0

0.25

0.5

0.75

1

0 0.01 0.02 0.03

Prob

abili

ty

P(D

M >

DS i

| Drif

t)

Inter-storey drift ratio

2600

2650

2700

2750

2800

0 5 10 15 20

Rep

air c

ost (

USD

)

Quantity

DS1: Screws pop-out, minor cracking of wall board, warping or cracks.- action: repair 15 m of

wall in both sides

Uncertainty:Distr: NormalDispersion: 0.44

a) Fragility functions b) repair cost and actions for DS1

12

DSi: ith damage state

Figure 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 Loss AnalysisThe 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.

4.4.1 Intensity-based assessmentFirst, various intensity-based assessments were conducted for intensities between pga=0.05g and 0.8g, with intervals of 0.05g. For simplicity, Figures 12 to 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.

0

0.25

0.5

0.75

1

0 0.05 0.1 0.15 0.2

P(R

epai

r cos

t ≤ c

)

Repair Cost / 1.2C0

Case 4 Case 1 Case 2 Case 3 Case 0

pga=0.10g0

0.25

0.5

0.75

1

0 0.15 0.3 0.45

P(R

epai

r cos

t ≤ c

)

Repair Cost / 1.2C0

pga=0.10g

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1

P(R

epai

r cos

t ≤ c

)

Repair Cost / 1.2C0

pga=0.20g 0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1

P(R

epai

r cos

t ≤ c

)

Repair Cost / 1.2C0

pga=0.30g

a) pga=0.10g b) pga=0.20g c) pga=0.30gFigure 12. Cumulative distribution functions of repair cost: three-storey frame

0

0.25

0.5

0.75

1

0 0.05 0.1 0.15 0.2

P(R

epai

r cos

t ≤ c

)

Repair Cost / 1.2C0

Case 4 Case 1 Case 2 Case 3 Case 0

pga=0.10g0

0.25

0.5

0.75

1

0 0.1 0.2 0.3

P(R

epai

r cos

t ≤ c

)

Repair Cost / 1.2C0

pga=0.10g

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75

P(R

epai

r cos

t ≤ c

)

Repair Cost / 1.2C0

pga=0.20g

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1

P(R

epai

r cos

t ≤ c

)

Repair Cost / 1.2C0

pga=0.30g

a) pga=0.10g b) pga=0.20g c) pga=0.30g

13

Figure 13. Cumulative distribution functions of repair cost: six-storey frame

0

0.25

0.5

0.75

1

0 0.05 0.1 0.15 0.2

P(R

epai

r cos

t ≤ c

)

Repair Cost / 1.2C0

Case 4 Case 1 Case 2 Case 3 Case 0

pga=0.10g0

0.25

0.5

0.75

1

0 0.1 0.2

P(R

epai

r cos

t ≤ c

)

Repair Cost / 1.2C0

pga=0.10g

0

0.25

0.5

0.75

1

0 0.2 0.4 0.6

P(R

epai

r cos

t ≤ c

)

Repair Cost / 1.2C0

pga=0.20g

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1

P(R

epai

r cos

t ≤ c

)

Repair Cost / 1.2C0

pga=0.30g

a) pga=0.10g b) pga=0.20g c) pga=0.30gFigure 14. Cumulative distribution functions of repair cost: nine-storey frame

4.4.2 Time-based assessmentTo estimate the average annual value of the repair costs, time-based analyses were conducted by integrating the intensity-based 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]. Figures 15 to 17 show the average annual repair costs and times for each structure studied. It was 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.

0

5,000

10,000

15,000

20,000

1

Rep

air c

ost,

$

Case

0 1 2 3 40.00.51.01.52.02.53.0

1

Rep

air t

ime,

day

s

Case

0 1 2 3 4

a) Repair costs b) Repair timesFigure 15. Average annual repair costs and times: three-storey frame

20,000

30,000

40,000

50,000

1

Rep

air c

ost,

$

Case

0 1 2 3 42.02.53.03.54.04.55.0

1

Rep

air t

ime,

day

s

Case

0 1 2 3 4

a) Repair costs b) Repair timesFigure 16. Average annual repair costs and times: six-storey frame

14

30,000

40,000

50,000

60,000

70,000

80,000

1

Rep

air c

ost,

$

Case

0 1 2 3 44.0

4.5

5.0

5.5

6.0

6.5

1

Rep

air t

ime,

day

s

Case

0 1 2 3 4

a) Repair costs b) Repair timesFigure 17. Average annual repair costs and times: nine-storey frame

Similarly, the probabilities of collapse and of loss of functionality during the lifetime of the hospitals (50 years) were estimated. They are shown in Figures 18 to 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.

0.0%

0.2%

0.4%

0.6%

0.8%

1

Prob

abili

ty o

f co

llaps

e

Case

0 1 2 3 40.0%

0.2%

0.4%

0.6%

0.8%

1.0%

1

Prob

abili

ty o

f los

s of

func

tiona

lity

Case

0 1 2 3 4

a) Collapse in 50 years b) Loss of functionality in 50 yearsFigure 18. Probabilities of collapse and of loss of functionality: three-storey frame

0.0%

0.5%

1.0%

1.5%

1

Prob

abili

ty o

f co

llaps

e

Case

0 1 2 3 40.0%

0.5%

1.0%

1.5%

2.0%

1

Prob

abili

ty o

f los

s of

func

tiona

lity

Case

0 1 2 3 4

a) Collapse in 50 years b) Loss of functionality in 50 yearsFigure 19. Probabilities of collapse and of loss of functionality: six-storey frame

15

0.0%

0.5%

1.0%

1.5%

2.0%

1

Prob

abili

ty o

f co

llaps

e

Case

0 1 2 3 40.0%

0.5%

1.0%

1.5%

2.0%

1

Prob

abili

ty o

f los

s of

func

tiona

lity

Case

0 1 2 3 4

a) Collapse in 50 years b) Loss of functionality in 50 yearsFigure 20. Probabilities of collapse and of loss of functionality: nine-storey frame

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 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:

(3)

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 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 Figure15); 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.

Figures 21 to 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

16

2 and 3), it can be said that the higher the contribution of the BRBs the lower the lifecycle costs.

0.90

0.95

1.00

1.05

1

Initi

al c

ost,

C0

Case

0 1 2 3 41.2

1.4

1.6

1.8

1

Initi

al c

ost +

NPV

, C0

Case

0 1 2 3 4

a) Initial costs b) Expected total costs in 50 yearsFigure 21. Initial and lifecycle costs: three-storey frame

0.90

0.95

1.00

1.05

1

Initi

al c

ost,

C0

Case

0 1 2 3 41.7

1.8

1.9

2.0

1

Initi

al c

ost+

NPV

, C0

Case

0 1 2 3 4

a) Initial costs b) Expected total costs in 50 yearsFigure 22. Initial and lifecycle costs: six-storey frame

0.90

0.95

1.00

1.05

1

Initi

al c

ost,

C0

Case

0 1 2 3 41.9

2.0

2.1

2.2

1

Initi

al c

ost+

NPV

, C0

Case

0 1 2 3 4

a) Initial costs b) Expected total costs in 50 years

Figure 23. Initial and lifecycle costs for the nine-storey frame

6 Discussion Benefits of BRBs. By analysing Figures 15 to 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

17

because: a) smaller cross-sectional depths of beams provides a higher 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 acceleration-sensitive region of the spectra develop larger seismic demands 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 ConclusionsThree-, 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

18

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 Figures 21 to 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.

AcknowledgementsThe 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.

References

[1] Meli R. Evaluation of Performance of Concrete Buildings Damaged by the September 19, 1985 Mexico Earthquake. In: Casssaro MA, Martinez-Romero E, editors. International Conference on the Mexico Earthquakes - 1985: Factors Involved and Lessons Learned. Mexico City: American Society of Civil Engineers; 1986. p. 308-27.[2] Rosenblueth E, Meli R. The 1985 Mexico earthquake: causes and effects in Mexico City. Concrete International. 1986;8:23-34.

19

[3] Singh SK, Suarez G. Review of the Seismicity of Mexico with Emphasis on the September 1985, Michoacan Earthquakes. Intenational Conference on the Mexico Earthquakes of 1985. Mexico City1986.[4] Ovando-Shelley E, Ossa A, Romo MP. The sinking of Mexico City: Its effects on soil properties and seismic response. Soil Dynamics and Earthquake Engineering. 2007;27:333-43.[5] Díaz-Rodríguez A. Dynamic properties of Mexico City clay for wide strain range. 10th WCEE. Rotterdam1992.[6] Bojórquez E, Ruiz-García J. Residual drift demands in moment-resisting steel frames subjected to narrow-band earthquake ground motions. Earthquake Engineering & Structural Dynamics. 2013;42:1583-98.[7] Meli R, Avila J. The Mexico earthquake of September 19, 1985—analysis of building response. Earthquake Spectra. 1989;5:1-18.[8] USGS. Historic Earthquakes - Michoacan, Mexico - 1985 September 19 13:17:47 UTC - Magnitude 8.0. 2013. p. http://earthquake.usgs.gov/earthquakes/world/events/1985_09_19.php.[9] Cassaro M, Martinez-Romero E. The Mexico Earthquakes of 1985: Factors Involved and Lessons Learned. International Conference on the Mexico Earthquakes of 1985. Mexico City1986.[10] Borja-Navarrete G, Diaz-Canales M, Vazquez-Vera A, Valle-Calderon Ed. Damage Statistics of the September 19, 1985 Earthquake in Mexico City. International Conference on the Mexico Earthquakes of 1985. Mexico City1986.[11] RCDF. Mexico City Building Code and its Complementary Specifications. Mexico City, Mexico2004.[12] Eurocode-8. Design of structures for Earthquake Resistance, Part1: General rules, seismic actions and rules for buildings. British Standard; 2004.[13] Priestley M. Myths and Fallacies in Earthquake Engineering. The Ninth Mallet Milne Lecture. The ROSE School, Italy2003.[14] Priestley MJN, Calvi GM, Kowalsky MJ. Displacement-based design of structures. Istituto Universitario di Studi Superiori di Pavia, Pavia, Italy. 2007.[15] Merrit S, Uang C-M, Benzoni G. Subassemblage testing of Corebrace Buckling-Restrained Braces. La Jolla, California: University of California, San Diego; 2003.[16] Black CJ, Makris N, Aiken ID. Component testing, seismic evaluation and characterization of buckling-restrained braces. J Struct Eng-ASCE. 2004;130:880–94.[17] Kersting RA, Fahnestock LA, López WA. Seismic Design of Steel Buckling-Restrained Braced Frames. NIST GCR 15-917-342015.[18] Uang C-M, Nakashima M. Steel Buckling-Restrained Braced Frames. In: Bozorgnia Y, Bertero VV, editors. Earthquake Engineering from Engineering Seismology to Performance-Based Engineering: CRC Press; 2004.[19] Tremblay R, Bolduc P, Neville R, DeVall R. Seismic testing and performance of buckling-restrained bracing systems. Canadian Journal of Civil Engineering. 2006;33:183-98.[20] Teran-Gilmore A, Virto N. Preliminary design of low-rise buildings stiffened with buckling restrained braces by a displacement-based approach. Earthquake Spectra. 2009;25:185-211.[21] Fahnestock L, Ricles J, Sause R. Experimental Evaluation of a Large-Scale Buckling-Restrained Braced Frame. Journal of Structural Engineering. 2007;133:1205-14.[22] Vargas R, Bruneau M. Experimental Response of Buildings Designed with Metallic Structural Fuses. II. J Struct Eng-ASCE. 2009;135:394-403.

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[23] Guerrero H, Ji T, Escobar JA, Teran-Gilmore A. Effects of Buckling-Restrained Braces on Reinforced Concrete Precast Models Subjected to Shaking Table Movements. (In revision, submitted in October 2015 to) Earthquake Engineering and Structural Dynamics 2015.[24] Kim J, Seo Y. Seismic design of low-rise steel frames with buckling-restrained braces. Engineering Structures. 2004;26:543-51.[25] Vargas R, Bruneau M. Analytical Response and Design of Buildings with Metallic Structural Fuses. I. J Struct Eng-ASCE. 2009;135:386–93.[26] Maley TJ, Sullivan TJ, Della Corte G. Development of a Displacement-Based Design Method for Steel Dual Systems With Buckling-Restrained Braces and Moment-Resisting Frames. J Earthqu Eng. 2010;14:106-40.[27] Guerrero H, Ji T, Teran-Gilmore A, Escobar JA. A Method for Preliminary Seismic Design and Assessment of Low-Rise Structures Protected with Buckling-Restrained Braces. Engineering Structures. 2016;123:141-54.[28] Moehle J, Deierlein GG. A Framework Methodology for Performance-Based Earthquake Engineering. 13th WCEE. Vancouver, Canada2004.[29] Yang T, Moehle J, Stojadinovic B, Der Kiureghian A. Seismic Performance Evaluation of Facilities: Methodology and Implementation. Journal of Structural Engineering. 2009;135:1146-54.[30] FEMA-P58. Seismic Performance Assessment of Buildings. Washington, D.C.: Federal Emergency Management Agency; 2012.[31] Guerrero H. Seismic Design and Performance of Hospital Structures Equipped with Buckling-Restrained Braces in the Lakebed Zone of Mexico City. PhD Thesis, The University of Manchester, UK. 2016.[32] Montiel M, Teran-Gilmore A. Comparative reliability of two 24-story braced buildings: traditional versus innovative. The structural design of tall and special buildings. 2011.[33] UN-ISDR. 2008-2009 World Disaster Reduction Campaign. United Nations, https://www.unisdr.org/2009/campaign/pdf/wdrc-2008-2009-information-kit.pdf; 2013.[34] Taghavi S, Miranda E. Response Assessment of Nonstructural Building Elements. In: Center P, editor. Berkeley, California: Pacific Earthquake Engineering Research Center, Univesity of California at Berkeley; 2003.[35] McGuire RK. Seismic Hazard and Risk Analysis. California, USA: Earthquake Engineering Research Institute; 2004.[36] CRISIS. Institute of Engineering, UNAM. Mexico2007.[37] Vamvatsikos D, Cornell CA. Incremental dynamic analysis. Earthquake Engineering & Structural Dynamics. 2002;31:491-514.[38] FEMA-P695. Quantification of Building Seismic Performance Factors. In: Agency FEM, editor. Washington, D.C.2009.[39] Teran-Gilmore A, Sanchez-Badillo A, Espinosa-Johnson M. Performance-based seismic design of reinforced concrete ductile buildings subjected to large energy demands. Earthquakes and Structures. 2010;1:69-91.

Appendix A. Statistics of Engineer Demand Parameters of the Equivalent Dual SDOF systems

Table A.1. Three-storey hospitalCase pga Displ.,

cmdispl cr Standard

dev. of cr

Velocity, m/s

vel Accel., g

accel

0 0.05g 1.32 0.30 0.000 0.000 0.20 0.13 0.11 0.300.10g 2.32 0.25 0.000 0.000 0.37 0.12 0.19 0.25

21

0.20g 4.63 0.25 0.000 0.000 0.74 0.12 0.38 0.250.30g 7.00 0.27 0.000 0.080 1.09 0.12 0.55 0.18

1

0.05g 0.85 0.31 0.000 0.000 0.18 0.14 0.09 0.310.10g 1.53 0.25 0.000 0.009 0.34 0.14 0.17 0.240.20g 3.01 0.23 0.012 0.013 0.67 0.14 0.30 0.180.30g 4.57 0.18 0.010 0.013 0.98 0.15 0.42 0.15

2

0.05g 0.90 0.33 0.000 0.000 0.18 0.14 0.09 0.330.10g 1.63 0.27 0.000 0.017 0.34 0.14 0.17 0.240.20g 3.19 0.23 0.017 0.020 0.67 0.14 0.29 0.170.30g 5.01 0.16 0.013 0.019 0.98 0.15 0.40 0.13

3

0.05g 1.15 0.25 0.000 0.000 0.18 0.12 0.10 0.250.10g 2.09 0.21 0.000 0.008 0.35 0.12 0.17 0.190.20g 4.10 0.17 0.005 0.006 0.68 0.12 0.31 0.160.30g 6.05 0.15 0.004 0.021 1.01 0.13 0.44 0.13

4

0.05g 0.61 0.24 0.000 0.000 0.16 0.16 0.08 0.240.10g 1.16 0.20 0.000 0.000 0.32 0.16 0.15 0.200.20g 2.27 0.20 0.000 0.048 0.64 0.16 0.28 0.130.30g 3.56 0.18 0.035 0.034 0.96 0.15 0.37 0.11

Table A.2. Six-storey hospitalCase pga Displ.,

cmdispl cr Standard

dev. of cr

Velocity, m/s

vel Accel., g

accel

0

0.05g 2.24 0.32 0.000 0.001 0.22 0.16 0.11 0.320.10g 3.95 0.27 0.000 0.000 0.42 0.13 0.19 0.270.20g 7.88 0.27 0.000 0.026 0.83 0.13 0.37 0.250.30g 11.90 0.27 0.001 0.104 1.20 0.11 0.49 0.11

1

0.05g 1.48 0.21 0.000 0.000 0.19 0.12 0.10 0.210.10g 2.70 0.18 0.000 0.000 0.36 0.11 0.17 0.190.20g 5.18 0.16 0.003 0.013 0.72 0.12 0.32 0.130.30g 7.46 0.15 0.010 0.013 1.05 0.12 0.43 0.13

2

0.05g 1.58 0.26 0.000 0.000 0.19 0.14 0.10 0.260.10g 2.87 0.21 0.000 0.003 0.37 0.12 0.17 0.210.20g 5.52 0.18 0.004 0.032 0.72 0.12 0.31 0.100.30g 8.24 0.17 0.029 0.032 1.07 0.13 0.40 0.10

3

0.05g 1.98 0.27 0.000 0.000 0.21 0.13 0.09 0.270.10g 3.62 0.23 0.000 0.012 0.40 0.12 0.17 0.220.20g 7.10 0.22 0.019 0.014 0.77 0.12 0.30 0.160.30g 10.79 0.21 0.034 0.066 1.14 0.12 0.41 0.13

4

0.05g 1.32 0.23 0.000 0.000 0.18 0.12 0.10 0.230.10g 2.39 0.19 0.000 0.000 0.36 0.12 0.17 0.190.20g 4.61 0.18 0.000 0.041 0.70 0.12 0.33 0.150.30g 6.62 0.18 0.018 0.045 1.03 0.13 0.43 0.08

Table A.3. Nine-storey hospitalCase pga Displ.,

cmdispl cr Standard

dev. of cr

Velocity, m/s

vel Accel., g

accel

0 0.05g 3.67 0.36 0.000 0.004 0.26 0.22 0.11 0.360.10g 6.32 0.32 0.000 0.001 0.48 0.20 0.20 0.320.20g 12.40 0.25 0.000 0.059 0.92 0.15 0.37 0.17

22

0.30g 19.67 0.30 0.065 0.148 1.29 0.13 0.44 0.05

1

0.05g 2.31 0.28 0.000 0.002 0.22 0.16 0.09 0.280.10g 4.24 0.24 0.000 0.004 0.42 0.14 0.17 0.240.20g 8.16 0.23 0.007 0.013 0.81 0.13 0.32 0.180.30g 11.97 0.21 0.014 0.048 1.19 0.13 0.43 0.15

2

0.05g 2.65 0.29 0.000 0.001 0.23 0.15 0.09 0.290.10g 4.89 0.26 0.000 0.017 0.43 0.14 0.17 0.240.20g 9.56 0.23 0.011 0.043 0.83 0.12 0.29 0.130.30g 15.15 0.22 0.045 0.064 1.20 0.14 0.38 0.11

3

0.05g 3.13 0.32 0.000 0.001 0.24 0.20 0.10 0.320.10g 5.71 0.25 0.000 0.015 0.45 0.15 0.18 0.210.20g 11.29 0.24 0.030 0.040 0.85 0.14 0.29 0.140.30g 19.41 0.27 0.112 0.117 1.23 0.13 0.37 0.08

4

0.05g 2.26 0.28 0.000 0.000 0.22 0.16 0.10 0.280.10g 4.11 0.24 0.000 0.000 0.41 0.14 0.17 0.240.20g 7.86 0.22 0.000 0.064 0.81 0.13 0.32 0.150.30g 11.91 0.21 0.022 0.054 1.16 0.12 0.40 0.10

Symbols in Tables A.1 to A.3:pga = peak ground accelerationg = acceleration of the gravity= Coefficient of variationcr = Residual to peak displacement ratio

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