cost for overall service time versus per unit service time

Structural Safety 58 (2016) 40–51

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Structural Safety journal homepage: www.elsevier.com/locate/strusafe

Seismic design and importance factor: Benefit/cost for overall service time versus per unit service time A. Pozos-Estrada a, T.J. Liu b, R. Gomez a, H.P. Hong b,⇑ a b

Institute of Engineering, National Autonomous University of Mexico, Mexico D.F., Mexico Department of Civil and Environmental Engineering, University of Western Ontario, N6A 5B9, Canada

a r t i c l e

i n f o

Article history: Received 27 April 2015 Received in revised form 27 August 2015 Accepted 30 August 2015

Keywords: Benefit Cost Decision making Importance factor Optimization Seismic design Risk

a b s t r a c t The maximum expected monetary benefit or minimum expected cost rule are often adopted to assess the optimum structural design levels under infrequent large earthquakes. In the assessment, the monetary benefit or cost functions are frequently established by considering the overall benefit or lifecycle cost at present value for a given structural design life. The selection of the structural design life is somewhat arbitrary, and in many cases one is interested in maximizing the structural service time or the benefit per unit service time. The consideration of the benefit (or cost) at present value per lifecycle or per unit service time may lead to different optimum design levels for a given planning time horizon. Moreover, it is unknown if the recommended importance factor in design codes to increase the seismic design load for classes of buildings is optimum. These two issues are investigated through numerical analyses by placing a structure at several different locations in Mexico and considering assembled detailed seismic hazard model. The implication of the results for the codified designs and for selecting the importance factor is discussed. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The catastrophic losses due to seismic events for engineered systems are well-known and the losses are mostly due to severe structural and non-structural damages and business interruptions. Available resources and willingness to significantly further reduce the losses are limited. This perhaps is partly due to the fact or perception that the structures are already designed or strengthened to an optimum, efficient and/or acceptable level. If this is not the case, one should see more significant retrofitting activities and a greater increase in the seismic design load level in structural design codes, at least in affluent societies. The investigation of the optimum seismic design level for buildings is not new and has been investigated in Rosenblueth [1,2], Liu et al. [3], Rosenblueth and Jara [4], Kanda and Ellingwood [5], Ang and De Leon [6], Rackwitz [7], Kang and Wen [8], Ellingwood [9], Esteva et al. [10,11], Liu et al. [12], Ellingwood and Wen [13] and Goda and Hong [14,15]. In these studies, the selection of the seismic design load level takes into account the safety and the economic issues that balance benefit and cost for the structural lifecycle. Most of the studies adopt the maximum expected (monetary) benefit or

⇑ Corresponding author. Tel.: +1 (519) 661 2111x88315; fax: +1 (519) 661 3779. E-mail address: [email protected] (H.P. Hong). http://dx.doi.org/10.1016/j.strusafe.2015.08.005 0167-4730/Ó 2015 Elsevier Ltd. All rights reserved.

the minimum expected cost rule (i.e., maximum or minimum expected value (MEV) rule, where the value is associated with benefit or cost, respectively) to select the optimum design level, although the use of other decision theories has been considered in the literature [2,15–17]. Results in [15,16] indicated that the efficient or optimum seismic design level obtained based on the MEV rule represents the one that is preferred by a risk-neutral decision maker. This identified optimum design serves as the upper and lower bounds on the efficient seismic designs for risk-seeking and risk-averse decision makers, respectively. The failure probability for a reference period or the annual failure rate [6] corresponding to the optimum design can be used to aid the selection of the target reliability level for calibrating the design codes. It is noted that in the above-mentioned studies, the total benefit or the total cost per structural design life are employed. This can be adequate if the selected design life can be justified in a logical manner. However, such a justification, if any, is not discussed in the literature, and periods of 30-, 50- and 75-year structural design life have been adopted for calibrating design codes. Further, in some cases one is interested in maximizing the benefit per unit service time or minimizing the cost spent on the structure per unit service time [18]. Therefore, rather than focusing on maximizing the expected total benefit or minimizing the expected total cost per design life or planning time horizon, one could select the optimum seismic design level by maximizing the benefit or

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minimizing the cost per unit service time. For the optimization, the use of Poissonian model for earthquake occurrence [19] is usually considered for seismic source zones that do not have very clear identified faults, while use of non-Poissonian model could be considered adequate for subduction earthquakes such as characteristic earthquakes along Mexican subduction region [20,21]. To achieve economic efficiency for a class of structures where consequences of failure are extremely severe, an increased safety level and seismic design level is necessary. Quantification of this increase has been investigated in [1,13,22], and the design codes use an importance factor to cope with the design of a class of buildings of importance. To simplify the analysis and to assess the required importance factor, the structure under seismic loading is represented as failure/safe system in [1,13]. Moreover, the investigation of the adequacy of the importance factor for Mexican building code was presented in [23] by incorporating partial damage cost. For simplicity and computing efficiency, however, simplified seismic hazard models are used in these studies, and differences between the ground motion prediction equations (GMPEs) and between inelastic seismic demands for different earthquake types are not considered. In this study, we adopt the MEV rule to select the optimum seismic design level because it provides the optimum design level that is preferred by a risk-neutral decision maker and serves as a reference for decision makers of different risk attitudes. Both linear and nonlinear structural responses under seismic excitations are used to define the partial damage and collapse and to evaluate the expected damage cost and/or the expected annual average cost. For the analysis, a detailed seismic hazard model applicable to part of the Mexican Pacific coastal region and Mexico City is assembled and developed. The model is used to map seismic hazard and to estimate uniform hazard spectra (UHS). The model is used as the basis to investigate the differences between the optimum seismic design levels if the expected benefit (or cost) per life cycle or per unit service time is employed in the MEV rule. It is also used to investigate the optimum importance factor needed for an increased damage cost for a class of important structures. For the analyses, the same building is placed at several sites in Mexico. The effect of Poissonian and non-Poissonian earthquake occurrence modeling on the estimated optimum seismic design levels is also assessed.

2. Formulation of objective functions Consider that B(A, t) denotes the benefit at present value derived from the service and existence of an engineered structure up to the time t, where A is a set of design parameters. The construction of such a structure requires an initial capital investment C0(A). If it is damaged or collapsed due to a large earthquake at a time there would be a corresponding damage cost at the present value, that represents structural and non-structural damage cost, cost of lost life and limb and cost of demolition and removal. Consider that the structure is immediately repaired or reconstructed upon damage or collapse without modifying the design and construction rules (i.e., systematic reconstruction after failure), and that the total damage cost (including the repair and replacement cost) is denoted by CDT(A, t) for service until the end of a planning time horizon t. The optimum design dictated by the MEV rule is obtained by maximizing the following objective function O(A, t) [1],

OðA; tÞ ¼ BðA; tÞ  C 0 ðAÞ  C DT ðA; tÞ:

ð1Þ

O(A, t) at the optimum must be positive for the structure to be viewed as of a benefit.

By ignoring the possible failure at the completion of the structure, and considering that there are n seismic source zones that affect the structure, and that the earthquakes occur randomly in time sij, i = 1,   , Nj(t), j = 1,   , n, where Nj(t) denotes the total number of earthquakes originated from the j-th source zone in the time interval 0 to t, O(A, t) shown in Eq. (1) can be written as,

OðA; tÞ ¼ BðA; tÞ  C 0 ðAÞ 

N j ðtÞ n X X

  ðC D ðAxij Þ þ C R ðAxij ÞÞecsij ;

ð2Þ

j¼1 i¼1

where CD(A|xij) and CR(A|xij) represent the damage cost and repair/ reconstruction cost given that the damage state (or level) induced by the earthquake occurred at sij is xij, and c is a discount rate adjusted for inflation which is often set to 5%. If the annual average benefit (or cost) is of interest, the objective function presented in Eqs. (1) and (2), is replaced by Oa(A, t), which is defined by,

Oa ðA; tÞ ¼ OðA; tÞ=t:

ð3Þ

If the objective function O(A, t) is considered, the optimum seismic design is the one that maximizes the expected value of O(A, t), E(O(A, t)), where E() denotes the expectation. If the annual average benefit (or cost) rather than the benefit (or cost) for the planning time horizon is of interest, the optimum seismic design is obtained by maximizing EðOa ðA; tÞÞ ¼ EðOðA; tÞ=tÞ. In this case, the optimum seismic design is not only a function of the set of design parameters A but also a function of the planning time horizon t. This suggests that the maximization of EðOa ðA; tÞÞ for a given planning time horizon t, leads only to a suboptimum since it does not ensure that EðOa ðA; tÞÞ is minimum for all possible t values. The use of E(O(A, t)) or EðOa ðA; tÞÞ could lead to different optimum design level. In either case, one may need to treat t as a decision parameter as well to find the optimum seismic design level. Note that since the direct comparison of EðOðA; tÞÞ for two different t values is not meaningful, its use is not valuable for selecting the optimum design for all possible t values. However, the use of EðOa ðA; tÞÞ, that emphasizes the expected benefit per unit service time, for selecting the optimum seismic design level could overcome this problem. In general, if a limited planning time horizon is considered and the earthquake occurrence is non-Poissonian, simulation technique could be employed to evaluate EðOðA; tÞÞ and EðOa ðA; tÞÞ since no closed-form analytical solution is available. The analysis procedure includes the assessment of seismic hazard in terms of the UHS for a site of interest, the design of the structure for a considered seismic design level and, the evaluation of the objective function O(A, t) or Oa(A, t) defined in Eqs. (2) and (3) using the sampled seismic events and seismic demand for a considered planning time horizon. 3. Seismic hazard model The evaluation of EðOðA; tÞÞ and EðOa ðA; tÞÞ for a construction site of interest requires the information on the probabilistic characterizations of the seismic hazard and, on the initial construction cost that is a function of the seismic design level. The characterization of the seismic hazard is based on the earthquake occurrence modeling, magnitude-recurrence relation, seismic source zones and ground motion prediction equations (GMPEs) (i.e., attenuation relations). An often employed methodology to assess the seismic hazard is the one given in [19]. A computational implementation of this method based on simulation technique that is described in [24] is adopted in the present study. To characterize the seismic hazard for Mexico City and part of the Mexican Pacific coastal region, we adopt the seismic source zone model shown in Fig. 1 and Table 1 [25–29], where the source zones are classified into three groups, depending on the earthquake type and earthquake magnitude. The source zones for the first and second groups are located near the Mexican Pacific coastal

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Fig. 1. Mexican seismic source zones for: (a) small and moderate earthquakes with M 6 7 and (b) earthquakes with M > 7. The sites I–VI identified in plot b are used to represent or near Morelia, Acapulco, Oaxaca, Salina Cruz, Tuxtla Gutierrez and Puebla.

Table 1 Magnitude and recurrence relations for the source zones shown in Fig. 1 (from [26,27]). The parameters, k0, k7 y b are given in Fig. 1; the values given in Table 2 in [26] are interpreted as k7. Source zones

Magnitude and recurrence relation

M

Interplate earthquakes for zones 1S1, 1*, 1S2 and 2*

expðbMÞexpð7bÞ kðMÞ ¼ k0 expð4:5bÞexpð7bÞ

4.5–7.0

Interplate earthquakes for zones 1 to 14 Inslab earthquake: New west

kðMÞ ¼ k7 ð1  UððM  7:5Þ=0:3ÞÞ

M>7 4.5–7.9

Inslab earthquake: centre

expðbMÞexpð7:88bÞ kðMÞ ¼ k0 expð4:5bÞexpð7:88bÞ

region and are for interplate earthquakes. The first group is defined by 4 polygons for the interplate earthquakes with moment magnitude, M, less than or equal to 7. The second group is defined by 14 relatively small polygons for characteristic earthquakes [20] with M > 7. The third group for the inslab earthquakes is formed by

expðbMÞexpð7:9bÞ kðMÞ ¼ k0 expð4:5bÞexpð7:9bÞ

4.5–7.88

two large polygons. Part of Mexico City and the Valley of Mexico is directly above the source zones of this group. The local and continental earthquakes are not considered since they do not affect the seismic hazard significantly for Mexico City [30]. The annual occurrence rate k(M) for earthquakes with magnitude greater than

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Mexico City. The coefficients of the GMPEs considered in this study are those estimated based on the geometric mean of the ground motion measures along two orthogonal horizontal directions [35]. The regression coefficients for the GMPEs applicable to the CU station are given in [29]; they are estimated by considering records from 18 interplate earthquakes and records from 12 inslab earthquakes recorded at the CU station. For a few selected Tn values, the coefficients are included in Table 2. The coefficients for the GMPEs applicable to sites outside of the MVB are estimated in [36] by using 418 Mexican interplate records and 277 Mexican inslab records. However, it was subsequently noted that 5 out of 418 records and 2 out of 277 records lead to questionable inelastic responses [37]. Therefore, a re-analysis is carried out by neglecting these questionable records in this study following the same analysis procedure in [36]. The obtained regression coefficients are shown in Table 2 for a few selected Tn. Note that the standard deviation of the residual, r, for the GMPEs applicable to the CU station (see Eqs. (6) and (7)) is much smaller than that for the GMPEs applicable to sites outside of the MVB. This is consistent with the observation that r for a single site is smaller than that for a region [38]. Note also that in most studies the earthquake occurrence in each source zone is modeled as Poisson process. For some of the numerical analyses in the next section, this Poissonian assumption is relaxed for one or two zones for M > 7 which are associated with Michoacan segment and Guerrero Gap [20], where renewal process is considered. This is in agreement with the consideration made in [4].

or equal to M for each of these zones are shown in Fig. 1 and Table 1. The GMPEs for the considered source zones are,

log10 SðT n Þ ¼ c1 þ c2 M þ c3 R  c4 log10 ðR þ c5 10c6 M Þ þ c7 H þ e:

ð4Þ

log10 SðT n Þ ¼ c1 þ c2 M þ c3 R  c4 log10 R þ c5 H þ e:

ð5Þ

log10 SðT n Þ ¼ c1 þ c2 ðM  6Þ þ c3 ðM  6Þ2 þ c4 log10 R þ c5 R þ e: ð6Þ and,

log10 SðT n Þ ¼ c1 þ c2 M þ c3 R  c4 log10 R þ c5 H þ e:

ð7Þ

where S(Tn) (g) represents the spectral acceleration (SA) for the natural vibration period Tn (s) and a damping ratio of 5% (that is considered throughout this study), e is the residual, ci, for i within 1–7 are regression coefficients, and R is defined in Table 2. The functional forms shown in Eqs. (5) and (7) are suggested in [31], and the functional forms shown in Eq. (4) and Eq. (6) are suggested in [32] and [33], respectively. As indicated previously and shown in Table 2, for zones outside of the Mexican Volcanic Belt (MVB) (see Fig. 1) and the Valley of Mexico, the functional forms of the GMPEs given by [31–34] are considered. Only a single station located in Mexico City (i.e., Ciudad Universitaria (CU) station) is considered because the records at the CU station are rich and it is often used as the reference site for

Table 2 Ground motion prediction equations. This study

Interplate Earthquakes (for sites outside of MVB and Valley of Mexico) Eq. (4) (adopted from [32]) Tn (s)

c1

c2

c3

c5

c6

c7

r

0.5 1.0 2.0

1.612 0.788 0.300

0.2289 0.2934 0.3853

0.0017 0.0007 0.0004

0.003 0.002 0.002

0.515 0.509 0.489

0.0019 0.0038 0.0034

0.40 0.40 0.38

Inslab Earthquakes (for sites outside of MVB and Valley of Mexico) Eq. (5) (adopted from [31])

[29]

Tn (s)

c1

c2

c3

c5

r

0.5 1.0 2.0

0.907 1.934 2.927

0.686 0.781 0.869

0.0024 0.0016 0.0012

0.0034 0.0029 0.0014

0.29 0.31 0.30

Interplate Earthquakes for CU station Eq. (6) (adopted from [33]) Tn (s)

c1

c2

c3

c4

c5

r

0.5 1.0 2.0

3.058 2.975 2.685

0.0504 0.0069 0.0515

0.1775 0.1853 0.2187

0.500 0.500 0.500

0.0025 0.0021 0.0014

0.16 0.18 0.25

Inslab Earthquakes for CU station Eq. (7) (adopted from [31]) Tn (s)

c1

c2

c3

c5

r

0.5 1.0 2.0

1.238 0.229 1.060

0.4044 0.5332 0.4758

0.0013 0.0001 0.0018

0.0010 0.0046 0.0140

0.18 0.20 0.27

Notes: (1) e in the regression equations is a zero mean normal variate with standard deviation r for a random orientation. (2) For Eq. (4), R (km) is the closest distance to the fault surface for events with M > 6.0 or the hypocentral distance for the rest, H (km) is the focal depth; c4 = 1.82–0.16M. For the regression analysis, 413 records from 40 interplate events are considered. (3) For Eq. (5), c4 = 1; R = (R2cld + D0)1/2; Rcld (km) is the closest distance to the fault surface for events with M > 6.5, or the hypocentral distance for the rest; H (km) is already defined; D0 = 0.0075  100.507M is a near-source saturation term [34]. The model parameters are obtained using 275 records from 16 seismic events. (4) For Eq. (6), R (km) is the closest distance to the fault surface; ci, i = 1,. . ., 5 are model parameters. The values of the model parameters are obtained using records from 18 seismic events (see Table 3). This equation is applicable for M 6 8.1; if an event with M > 8.1, M = 8.1 is used in the equation (Reyes, personal communication 2007). The same is considered for Eq. (4). (5) For Eq. (7) see Note (3). The model parameters are obtained using records from 12 seismic events. (6) The average focal depth is 22.22 km for interplate earthquakes and 54.61 km for inslab earthquakes.

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A. Pozos-Estrada et al. / Structural Safety 58 (2016) 40–51

(a)

(b)

(d) 101

0

SA (g)

10

CU Site I Site II Site III Site IV Site V Site VI

10

-1

(c)

0

(e) 101

10

10

-1

0

10

0.5

1

1.5 Tn (s)

2

2.5

1.5 Tn (s)

2

2.5

3

3

CU Site I Site II Site III Site IV Site V Site VI

0

SA (g)

0

1

(f) 101

CU Site I Site II Site III Site IV Site V Site VI

SA (g)

10

0.5

-1

0

0.5

1

1.5 Tn (s)

2

2.5

3

Fig. 2. Contour maps of SA and UHS: (a) Map for SA at Tn = 1.0 s for T = 125 years, (b) Map for SA at Tn = 1.0 s for T = 475 years, (c) Map for SA at Tn = 1.0 s for T = 2475 years, (d) UHS for selected sites and T = 125 years, (e) UHS for selected sites and T = 475 years, (f) UHS for selected sites and T = 2475 years.

Based on the adopted seismic hazard model, we estimated seismic hazard contour maps for part of Mexico and UHS for a few selected locations identified in Fig. 1. We illustrated in Fig. 2 the contour map of SA for the return period (T) equal to 125, 475 and 2475 years. The UHS shown in Fig. 2 for the CU station has two peaks, one for Tn near 0.25 s and the other near 1.75 s. The first one is due to inslab earthquakes and the second one is caused by interplate earthquakes [29]. Moreover, the seismic hazard analysis results indicate that the seismic design level suggested in the current Mexico Federal District Code [39] for Tn = 1.0 (s) which equals 0.16 g corresponds approximately to the 1520-year return period value. This design return period will be discussed further in the subsequent sections.

The empirical equation to predict the expected seismic displacement ductility demand for bilinear single-degree-offreedom (SDOF), ml, is already developed in [37] by using the same sets of records that were employed to develop Eqs. (4) and (5). It is given by the following empirical expression, b

ml ¼ expðða1 ln /Þ Þ

ð8Þ

where / is the normalized yield strength which is defined as the ratio of the yield strength of the system to the strength that is required for the corresponding SDOF system to remain linear elastic under the same excitations, and a1 and b are model parameters.

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A. Pozos-Estrada et al. / Structural Safety 58 (2016) 40–51 Table 3 Parameters for estimating the mean seismic displacement ductility demand (ml). 

a1 ¼ a1 exp a2 =T an3

c



Model for b 0.2 < Tn < 3 (s)

Parameter

0.05 < Tn < 3 (s)

Interplate earthquakes (sites outside the valley of Mexico and the MVB) ([37]) 0 a1 0.068 a2 1.876 a3 0.399 0.05 a1 0.073 a2 2.063 a3 0.269

0.998 0.065 1.614 0.878 0.036 1.900

b1 b2 b3 b1 b2 b3

0.955 0.347 1.058 1.049 0.312 1.056

Inslab earthquakes (sites outside the valley of Mexico and the MVB) ([37]) 1.001 0 a1 a2 0.118 a3 1.063 0.05 a1 0.804 a2 0.101 0.955 a3

1.028 0.105 1.116 0.878 0.050 1.277

b1 b2 b3 b1 b2 b3

0.978 0.294 0.979 1.132 0.292 0.996

Interplate earthquakes CU station 0 a1 a2 a3 0.05 a1 a2 a3

1.968 0.759 1.269 0.312 3.288 0.240

b1 b2 b3 b1 b2 b3

0.701 0.375 1.218 0.609 0.112 1.223

Parameter

0.05 < Tn < 0.2 (s)

0.014 6.597 0.197 5.28E04 9.965 0.037

Inslab earthquakes CU station 0 a1 a2 a3 0.05 a1 a2 a3

0.010 1.096 b1 0.956 4.421 1.044 b2 0.465 0.287 0.259 b3 0.991 9.67E03 1.549 b1 0.810 6.362 0.883 b2 0.297 0.086 0.700 b3 1.479         Tn 15 < T and b ¼ b þ ðb  b Þ ln 6 T < 3, where T is set equal to 0.3 (s) for for 0:05 6 T = ln for T Notes: Model for b is given by b ¼ b1  b2 ln TT00n = ln T0:1 n n 00 00 1 3 1 00 T 00 T 00 00 Interplate and Inslab for sites outside the valley of Mexico and the MVB, and for Interplate at the CU station; T00 is set equal to 0.7 (s) for inslab at the CU station.

1

1

Tn (s) 0.05 0.1 0.2 0.3 0.4 0.5 1 2 3 1 COV of ductulity demand

Tn (s) 0.05 0.1 0.2 0.3 0.4 0.5 1 2 3 0.1

0.1 0.1

(b)

Normalized yield strength

Normalized yield strength

(a)

0.1

10

1

10

cov of ductulity demand

1

1 Normalized yield strength

Normalized yield strength

(c) Tn (s) 0.05 0.1 0.2 0.3 0.4 0.5 1 2 3

0.1 0.1

1 cov of ductulity demand

(d) Tn (s) 0.05 0.1 0.2 0.3 0.4 0.5 1 2 3

0.1

10

0.1

1 cov of ductulity demand

10

Fig. 3. Coefficient of variation of the ductility demand vl: (a) vl for interplate records obtained at CU station, and (b) vl for inslab records obtained at CU station, (c) vl for interplate records, (d) vl for interplate records (results shown in (c and d)) are from Hong et al. 2009 for records from a broad network of recording stations).

Their distribution fitting results indicate that the Frechet distribution is preferred for the displacement ductility demand l. Following the same procedure in [37], we assessed l using the records considered to develop Eqs. (6) and (7). The estimated a1

and b for selected Tn are summarized in Table 3 and the variation of the coefficient of variation of l, vl, for / ranging from 0.1 to 1.0 is illustrated in Fig. 3. For comparison purpose, Fig. 3 also included the plots of vl obtained using records from a broad net-

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A. Pozos-Estrada et al. / Structural Safety 58 (2016) 40–51

work of recording stations for interplate earthquakes and for inslab earthquakes. The comparison shows that there are differences between vl for records at the CU station and from a broad network of recording stations, and between vl for the interplate earthquakes and inslab earthquakes. Therefore, for consistency, the GMPE and the ductility demand model developed using the same records should be used as a set to reflect the differences in the characteristics of the ground motions from different earthquake types and for different local or regional conditions. Samples of l obtained from records at the CU station for each earthquake type are used in distribution fitting by considering the lognormal and Frechet distribution models. Results of the distribution fitting and the Akaike information criterion [40], again, indicate that the Frechet distribution is the preferred model for Tn > 0.3 s. 4. Consideration of structural design code requirements The current Mexico Federal District Code [39] recommends that the design base shear coefficient, Cs, equals the elastic seismic design coefficient a (g) divided by a force reduction factor Q0 . Q0 depends on Tn and the seismic behavior factor, Q, whose quantification and physical meaning are not entirely clear. Q is not the displacement ductility capacity and at most it is about 40% of the ductility capacity [41]. Therefore, it could be instructive to replace Q0 by the displacement ductility capacity related reduction factor Rl. This consideration is justified since in [42] it is considered that such a reduction factor stands for the approximate ductility deformation capacity of the selected structural system [43]. Furthermore, [39] recommends a period dependent over-strength reduction factor for some sites in Mexico City, while [42] suggests a structural system dependent over-strength factor, Ro. Therefore, it is desirable to incorporate the over-strength factor in assessing the optimum seismic design and the importance factor. In such a case, it is considered that Ro represents the ratio of the mean to the minimum required design base shear force; it includes the effect of the system behavior, the difference between mean material strength and its design strength, and the difference between the designed structural member sizes and their required sizes that are dictated by the minimum design requirements. Based on the above, and considering that the designed structure could be modeled as a bilinear hysteretical SDOF system whose design is governed by strength requirement (i.e., C s ¼ a=ðRo Rl Þ) and subjected to a seismic event with S(Tn) (g), the normalized yield strength / can be expressed as [14,15,22],

 / ¼ min

 a Rn ;1 : SðT n Þ Ro Rl

ð9Þ

where Rn denotes the ratio of the yield capacity of the designed structure to the minimum required design base shear force. If / equal to 1.0, it simply implies that the actual yield capacity of the structure is greater than or equal to the elastic seismic demand. The quantities such as the irregularity factor of the structure and the importance factor that could affect the design level are taken equal to one in writing Eq. (9). The ductility demand l for the simplified structural model can then be assessed using / and the probabilistic model shown in Table 3 and Fig. 3. It must be emphasized that the use of an equivalent bilinear SDOF system to represent the considered building is a crude approximation. The adequacy of such an approximation for steel frame systems was elaborated in [44]; it allows a simple and objective approximate quantitative assessment of the inelastic structural responses for a given ground motion measure. The use of nonlinear inelastic finite element model and considering time history analysis under seismic ground motions is an extremely computationally intensive task.

Since the seismic design level is controlled by Cs which is related to the selected seismic elastic design base shear coefficient a (i.e., the selected elastic design SA (g) value), for simplicity, it is considered that the set of design parameters A (see Eqs. (1–3)) that need to be considered for optimum seismic design is simply a. 5. Cost information and damage level The initial construction cost for structures to be located in Mexico City has been discussed in [4,5,10]. No general mathematical model was suggested in [5] for assessing the initial cost. The mathematical model developed in [10] depends on the fundamental natural vibration period of the designed structure Tn and the fundamental vibration period of the system if it were designed for gravity load only. Since these vibration periods are unknown a priori, the model is less convenient for a systematic parametric study. The initial cost function of the structural components C 0S ðC s Þ used in [4] is of the following form,

( C 0S ðC s Þ ¼

h i for C s P ac C 00  1 þ cc ðC s  ac Þbc for C s < ac

C 00

;

ð10Þ

where C00 represents the cost if it was not designed against earthquakes and, ac, bc and cc are model parameters. They suggested that ac = 0.05, bc = 1.1 and cc = 1.4 could be considered. Details of the designs used to derive these parameters are unclear and unavailable. Eq. (10) was adopted in [7], indicates that ac = 0.05, bc = 1.091 and cc = 6.547. Using the cost information given in [7] and convert them into Canadian practice, [14] indicated ac = 0.0082, bc = 1.08 and cc = 8.054 could be adequate for Vancouver. However, if only the designs with the design base shear coefficient Cs greater than 0.05 are considered, one concludes that ac = 0.05, bc = 1.04 and cc = 6.14, showing that cc depends on the considered lower bound value for Cs. The values ac = 0.05, bc = 1.05 and cc = 6.5 are used as the reference values for the following numerical analysis. By including the cost of the non-structural components, the initial total cost of the structure C 0 ðC s Þ is defined as,

C 0 ðC s Þ ¼ C 0S ðC s Þ þ C 0S ðC s;ref Þ=j;

ð11Þ

where Cs,ref is a reference earthquake design base shear coefficient and j represents the ratio of the cost of the structural to the nonstructural components. Eq. (11) implies that the non-structural cost is independent of Cs. The repair/reconstruction cost, CR(Cs|d), and the damage cost, CD(Cs|d), that consists of the loss of contents, relocation, rental and income, and the injury and fatality cost can be expressed as fractions or ratios of their values corresponding to collapse state (i.e., for d = 1), where the damage factor d is defined as,



d¼

0 minððl  1Þ=ðlC  1Þ; 1Þ

l<1 lP1

ð12Þ

and l and lC denote the seismic displacement ductility demand and the ductility capacity of the structure, respectively. Expressions for the above mentioned cost functions have been developed for a variety of structures [45] based on [46–48]. In particular, for concrete structures and commercial use category CD(Cs|d) and CR(Cs|d) are given by,

  C D ðC S jdÞ ¼ aI aBC d0:64 þ aBI d0:62 þ aF d9:9

ð13Þ

and,

C R ðC S jdÞ ¼ C 0 ðC s Þd0:77

ð14Þ

where aBC, aBI and aF are coefficients related to the damage cost with contents-related loss, business-interruption-related loss, and

47

A. Pozos-Estrada et al. / Structural Safety 58 (2016) 40–51

6. Analysis procedure and results 6.1. Procedure

(a) Expected value of [B(A,t) - O(A,t)]/C00

injury and fatality, respectively; and aI is a scaling factor. For normal buildings, aI is considered to be equal to one. As will be seen, the use of aI allows the investigation of the sensitivity of the optimum seismic design level to the increased damage cost or to the relative importance of damage cost. Eqs. (11), (13) and (14) are used for the numerical analyses in the following section. It is understood that the cost information considered in this study may not be applicable to other countries; an extensive discussion on the sources of failure costs focused on Japan and engineer’s role in structural safety is given in [49].

10 8 6

S

2

0 0.05

6.2. Numerical results on optimum design level For the numerical analysis, it is considered that there is a need for a ten story reinforced concrete structure, implying that the benefit derived from the structure is always higher than the expected total cost. Therefore, if the benefit per unit of time derived from the service and existence of the structure equals b, (i.e., B(A, t) = bt) one must select the value of b such that the expected values of O(A, t) and Oa(A, t) are positive at least when the design is near optimum. The cost information as well as the structural information for the considered structure are shown in Table 4. Note that the ductility capacity lC is assumed to be lognormally distributed with mean 2.5 and coefficient of variation (cov) of 0.3 based on the study given in [50] for a concrete structure of nine stories located in Mexico City. Rn is considered to be lognormally distributed with a mean

(b) Expected value of [B(A,t) - O(A,t)]/C00

The identification of the optimum design is carried out in this section based on the expected value of O(A, t) or Oa(A, t) calculated using simple simulation technique for the adopted seismic hazard model shown in Figs. 1 and 2 and Tables 1–3, and the cost model shown in Table 4. The analysis basically consists of [14]: (1) carrying out probabilistic seismic hazard analysis; (2) selecting the T-year return period value of SA, ST(Tn), for the elastic design base shear coefficient a (g) based on the results from the first step; (3) providing probabilistic characterization of capacity of the structure designed according to the applicable design code (except the seismic design level), and generating samples of O(A, t) and Oa(A, t) for the considered seismic hazard and evaluating their statistics. The results of Step (1) can be used to develop the UHS such as those shown in Fig. 2. Steps (2) and (3) are repeated for a range of considered design levels to identify the optimum design level. It must be noted that for the present study the designed structure is approximated by a bilinear hysteretic SDOF system with damping ratio of 5% and displacement ductility capacity lC. Since a is replaced by ST(Tn) for the design, a in Eq. (9) must be replaced by ST(Tn) as well to calculate /. This calculated / is then used together with Table 3 and Fig. 3 to calculate the mean and coefficient of variation of l and to define the probability distribution of the ductility demand.

Sopt = 0.400 (g)

4

= 0.276 (g)

Sopt = 0.211 (g)

CU Station Site I Site II 0.10

opt

0.2 SA (g)

0.4

0.6

0.8

1

10 8

Sopt = 0.377 (g)

6 Sopt = 0.318 (g)

4 2 0 0.05

S

Site III Site IV Site V Site VI

opt

= 0.253 (g)

Sopt = 0.217 (g)

0.10

0.2 SA (g)

0.4

0.6

0.8

1

Fig. 4. E[B(A, t)  O(A,t)]/C00 and optimum design level for t equal to 50 years for the considered sites: (a) at the CU station, Sites I and II, and (b) at Sites III to VI.

value equal to 4 as shown in Table 4 for the designed structure. The selection of this value is guided by the results obtained in [51] and the difference between Q and Rl discussed earlier and those used in the literature [14]. However, since the impact of the coefficient of variation of Rn on the estimated structural reliability and damage cost statistics is small [14], this cov is ignored for numerical analyses. For the analysis, first it is considered that the earthquake occurrence in each source zone shown in Fig. 1 can be modeled as Poisson process with parameters shown in Table 1, and that the optimum design is to be identified based on the objective function O(A, t). If the planning period t is considered to be equal to 50 years, the obtained expected value of (B(A, t)  O(A, t))/C00 is shown in Fig. 4 for a range of seismic design levels and the seven sites identified in Fig. 1, indicating that the expected objective function near the optimum design is relatively flat – a feature that is shared by the results given in [1]. Fig. 4a shows that the optimum design level, Sopt, for CU equals 0.211 g, and the corresponding optimum return period, Topt, is

Table 4 Information employed for numerical analyses. Damage cost

c = 0.05; C 0 ðC s Þ ¼ C 0S ðC s Þ þ C 0S ðC s;ref Þ=j where the structural cost is defined by C 0S ðC s Þ ¼ C 00  maxð1; 1 þ 6:5ðC s  0:05Þ1:05 Þ; the non-structural cost is defined as C 0S ðC s;125 Þ ¼ C 00  maxð1; 1 þ 6:5ðC s;125  0:05Þ1:05 Þ=0:3; C D ðdÞ ¼ C 00 ðaBC d0:64 þ aBI d0:62 þ aF d9:9 Þ where aBC =C 00 ¼ 3:05, aBI =C 00 ¼ 9:66 and aF =C 00 ¼ 31:82; C R ðd; C s Þ ¼ C 0 ðC s Þd0:77

Structural design

Structure is for commercial use; planning service period t is 30, 50 and 75 years; structure is modeled as an elastoplastic single-degree-of-freedom system; Tn is 1.0 s; damping ratio is 5%; lC is a lognormal variate with a mean of 2.5 and cov of 0.3; Rn is treated as a deterministic variable with a value equal to 4, and Ro is 1.5

A. Pozos-Estrada et al. / Structural Safety 58 (2016) 40–51

(a) 10

SA (g)

listed in Table 5. Note that this obtained Sopt is independent of the value of b since B(A, t) is considered to be independent of design level. The identified optimum seismic design level (for Tn = 1.0 (s)) is higher than that suggested in [39], which equals 0.16 g and corresponds approximately to 1520-year return period value, as mentioned previously. The expected values of (B(A, t)  O(A, t))/C00 for the remaining six sites are shown in Fig. 4a and b for a range of seismic design levels. Sopt for each site is identified in these plots, and the corresponding Topt and the minimum expected (B(A, t)  O(A, t))/C00 (i. e., minimum E½C 0 ðAÞ þ C DT ðA; tÞ=C 00 ) are summarized in Table 5. The results indicate that Topt is site dependent. The lower bound value is associated with Site II (near Acapulco) whose seismic hazard is greater than those of the rest of the sites analyzed, while the upper bound is associated with Site V (Tuxtla Gutierrez) (see Figs. 1 and 2). The large discrepancy between Topt for the CU station and for other sites could be explained by noting that r associated with the GMPEs applicable to the CU station is much smaller than that associated with the GMPEs applicable to sites outside of the MVB (see Table 2). To verify this, we repeated the analysis carried out for Fig. 4a and the UHS but by scaling upwards the r for the CU station (i.e., r for Eqs. (6) and (7)) by a factor fr equal to 1.4 or 1.8. The use of fr = 1.8 leads to the scaled r comparable to that applicable to sites outside of the MVB and shown in Table 2 (i.e., r for Eqs. (4) and (5) obtained using records from a broad network of stations). The UHS by considering the scaled r are compared in Fig. 5a for several return periods. The plot shows that the UHS can be very sensitive to the scaling of the standard deviation of GMPEs, especially as the return period increases. The optimum design levels are presented in Fig. 5b, indicating that as r increases Sopt increases but its corresponding Topt decreases. The latter can be explained by noting that the cov of SA as well as its probability distribution tail increase as r of GMPEs increases. Additional discussion of the effect of r on the UHS by using the site specific information can be found in [52]. It must be emphasized that the values of Sopt and Topt are sensitive to the adopted structural and cost information shown in Table 4. For example, an increase in Rn by a scaling factor will result Sopt to be scaled by the inverse of the same factor. Therefore, the conclusions drawn from the analysis results should be based on the differences among Topt (or Sopt) for different sites rather than the value of Topt for a single site. To investigate the impact of the structural design life t on the selected optimum seismic design level, we consider t equal to 30 and 75 years. The obtained results are included in Table 5, indicating that Sopt is slightly affected by the arbitrarily selected planning period. Sopt increases as the service period increases. However, the differences between the optimum design levels for different t

10

0

-1

fσ = 1

10

-2

0

0.5

fσ = 1.8 125 years 475 years 2475 years

fσ = 1.4 125 years 475 years 2475 years

125 years 475 years 2475 years

1

1.5 T (s)

2

2.5

3

n

(b) 10 Expected value of [B(A,t) - O(A,t)]/C00

48

CU Station S = 0.406 (g) opt Topt = 2786 years

8 6

Sopt = 0.314 (g) Topt = 6071 years

4 2 0 0.05

fσ = 1.0

S = 0.211 (g) opt T = 6261 years

fσ = 1.4 fσ = 1.8

opt

0.10

0.2 SA (g)

0.4

0.6

0.8

1

Fig. 5. Effect of standard deviation of the GMPEs on UHS and the expected value of [B(A, t)-O(A,t)]/C00 and optimum design level at the CU station.

values are marginal in some cases. This observation is consistent with the findings in [14]. 6.3. Optimum seismic design by maximizing the benefit per unit service time To assess the differences between the optimum seismic design levels obtained based on E(O(A, t)) and E(Oa(A, t)), we note that if the benefit per unit of time is considered to be a constant and without discounting, B(A, t) = bt, it can be shown that the maximization of E(Oa(A, t)) is equivalent to the minimization of E½C 0 ðAÞ þ C DT ðA; tÞ=ðtC 00 Þ. Therefore, Sopt in this case, is simply

Table 5 Identified optimum design spectral acceleration and its corresponding return period and expected total cost. Optimum

CU

Site I

Site II

Site III

Site IV

Site V

Site VI

t = 30 years Sopt (g) Topt (years) minfE½C 0 ðAÞ þ C DT ðA; tÞ=C 00 g

0.208 5539 4.864

0.254 748 5.211

0.398 603 5.899

0.310 701 5.435

0.340 694 5.590

0.195 1055 4.899

0.245 1033 5.042

t = 50 years Sopt (g) Topt (years) minfE½C 0 ðAÞ þ C DT ðA; tÞ=C 00 g

0.211 6261 4.877

0.276 941 5.243

0.400 613 5.945

0.318 746 5.477

0.377 919 5.631

0.217 1401 4.926

0.253 1106 5.057

t = 75 years Sopt (g) Topt (years) minfE½C 0 ðAÞ þ C DT ðA; tÞ=C 00 g

0.213 6685 4.886

0.295 1106 5.251

0.401 623 5.963

0.319 751 5.479

0.382 960 5.636

0.224 1496 4.928

0.262 1212 5.063

A. Pozos-Estrada et al. / Structural Safety 58 (2016) 40–51

equal to that for the case with minimum E½C 0 ðAÞ þ C DT ðA; tÞ=ðtC 00 Þ. Using the minimum E½C 0 ðAÞ þ C DT ðA; tÞ=C 00 tabulated in Table 5, it is concluded that Sopt is associated with the one with increased structural design life t. This implies the optimum t is beyond 75 years for the considered cases. It must be emphasized that this conclusion is based on the assumptions that there is no natural degradation of the structural strength and, the benefit derived from the service provided by the building per unit time (at present value) is a constant or the discounting of the benefit can be neglected. To relax the latter, we consider that the same discount rate c for the cost is equally applicable to the benefit, resulting in,

max½EðOa ðA; tÞ=C 00 ¼ fbð1  ect Þ=c  min½EðC 0 ðAÞ þ C DT ðA; tÞÞg=ðtC 00 Þ:

ð15Þ

By applying this equation and using the minimum expected cost value shown in Table 5, optimum structural design life t is illustrated in Fig. 6 for two selected values of b/C00. The results indicate that the optimum structural design life t depends on b/ C00. The optimum t is within 30–75 years (i.e., Sopt is within those shown in Table 5 for t = 30 and 75 years) for b/C00 = 0.4; it is less than or equal to 30 years for b/C00 = 0.55. It indicates that as b increases the consideration of discounted benefit results in a decreased optimum t, hence, in decreased Sopt. In short, a shorter structural design life is preferred to reduce the increased discounted benefit. Since the above simplifying assumptions on the degradation as well as on the discounted benefit must be improved before any convincing recommendations could be made, no further analysis based on max[E(Oa(A, t))] is considered in the following.

a

max[E(O (A,t))]/C00

(a)

0.06

0.04

0.02

CU Station Site I Site II Site III Site IV Site V Site VI

0.00 b/C00 = 0.4 -0.02 20

(b)

30

40

50 t (years)

60

70

80

0.14

a

max[E(O (A,t))]/C00

0.12 0.10 0.08 0.06 0.04 0.02 20

CU Station Site I Site II Site III Site IV Site V Site VI 30

b/C00 = 0.55 40

50 t (years)

60

70

Fig. 6. Maximum E(Oa(A, t)) versus structural design life t.

80

49

6.4. Effect of Poissonian and non-Poissonian occurrence modeling on the optimum design It should be noted that in the above analysis, the earthquake occurrences are considered to follow a Poisson Process. This may not be the case as discussed previously. To investigate the nonPoissonian occurrence model on the estimated optimum seismic design level, we repeat the above analysis for t = 50 years, but considering that the earthquake occurrences for source zone 11 (Michoacan) follows a renewal process with the interarrival time modeled as a lognormal variate with mean of 51.6 years, cov of 0.39 and the elapsed time since the last earthquake equal to 29 years [4]. Inspection of the obtained analysis results shows that the differences between the obtained optimum seismic design levels based on E(O(A, t)) and those shown in Table 5 are within 5%. This indicates that the consideration of non-Poissonian model does not affect significantly the optimum design level. This may be explained by noting that seismic source zones other than zone 11 (Michoacan) also contribute significantly to the seismic hazard for the sites of interest. 6.5. Importance factor In the previous section, an importance factor, I, equal to one and

aI = 1 are considered since the analyses are focused on ordinary or normal buildings. The importance factor is used to scale up the design base shear coefficient, Cs, (or the elastic seismic design coefficient) to design structures whose collapse could cause significant fatalities and extreme economic losses. Its use increases the safety level and reduces the risk of the considered class of buildings. An importance factor of 1.5 is recommended in [39]. The investigation of the adequacy of this importance factor was presented in [23]. Their study is focused on the possible differences in the importance factors for a site near and another far away from the seismic source. It provided a forward step towards the understanding of the rationale of the selected importance value I, although a very simplified treatment of seismic hazard model was considered. The analysis results presented in the following is focused on I versus the increased damage cost CD(d). More specifically, we consider that the damage cost of the structure is increased by the factor aI (greater than unity) shown in Eq. (13) for the structure to be considered important; we are interested in finding the optimum seismic design level I  Sopt, where Sopt are those obtained in the previous sections and for aI = 1 (i.e., normal building). By repeating the analysis carried out for the results shown in Fig. 4 and Table 5 but with aI > 1, the obtained relation between I and aI, is shown in Fig. 7 for a planning service period equal to 30 and 50 years. Fig. 7 indicates that optimum I is almost a linear function of logarithm of aI and this relation is insensitive to whether t is equal to 30 or 50 years. Fig. 7a and b indicate that for the CU station and using the single site r (i.e., for fr = 1.0) if the damage cost of the important structure is about 20 times of that of a normal structure, the optimum I is about 1.5. However, if the seismic design is to be carried out for sites near the CU station, the GMPEs applicable to CU station could be adequate but the single site r may not. In such a case, if an increase of the single site r is considered, the resulting optimum I equals 1.5 for aI about 10 if fr = 1.4 and for aI about 6 if fr = 1.8. Fig. 7c and d show that in all cases an importance factor of 1.5 could be considered to be optimum if aI is about 5. The above observations imply that if an existing seismic design code is calibrated to near optimum seismic design level. An importance factor of 1.5 is adequate provided that r for the GMPEs is similar in magnitude to those associated with Eqs. (4) and (5), and aI is about 5.

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A. Pozos-Estrada et al. / Structural Safety 58 (2016) 40–51

(b) 3.0

fσ = 1.0

fσ = 1.0

t = 30 years

2.5

2.5

2.0

2.0

I

I

(a) 3.0

t = 50 years

fσ = 1.4 fσ = 1.8

1.5

1.5

CU Station

CU Station 1.0

1

2

3

5

α

10

20

30

1.0

50

1

2

3

5

(c) 3.0

2.0

10

1.5

2.0

30

50

1.5

t = 30 years 1.0

20

Site I Site II Site III Site IV Site V Site VI

2.5 I

I

(d) 3.0

Site I Site II Site III Site IV Site V Site VI

2.5

α

I

I

1

2

3

5

α

10

20

30

50

t = 50 years 1.0

1

2

3

I

5

α

10

20

30

50

I

Fig. 7. Optimum importance factors (I) as a function of aI: (a) at the CU station and t = 30 years, (b) at the CU station and t = 50 years, (c) at six selected sites and t = 30 years, (d) at six selected sites and t = 50 years.

7. Discussion and conclusions The maximization of expected benefit or the minimization of expected cost rule is often used in assessing the optimum seismic design level. It is observed that such an optimum design level is selected based on expected overall benefit or lifecycle cost at present value for an arbitrarily selected time frame or design life. It is argued that instead of considering the expected overall benefit or cost for a specified service period, in many cases, one could be interested in maximizing the structural service time or the benefit per unit service time. Through numerical results presented in this study, it is shown that the consideration of the former and the latter can lead to different optimum seismic design level. However, it is emphasized that simple assumptions on the natural degradation of the structural strength and discounted benefit models must be improved and validated in order to make convincing recommendations. Furthermore, it must be emphasized that since the values of Topt (or Sopt) obtained are sensitive to the adopted structural and cost information, the conclusions drawn from the presented analysis results should be based on the differences among Topt (or Sopt) for different sites rather than the value of Topt for a single site. Specific conclusions include: (1) For the considered structure with Tn equal to 1.0 (s), Topt for Mexico City is significantly greater than that for the six considered sites outside of the MVB. It is shown that this is due to the use of single site r of the residual for the GMPEs. (2) The optimum seismic design level Sopt is not very sensitive to the considered structural design life t or the assumption of the Poissonian and non-Poissonian earthquake occurrence modeling for the assembled seismic hazard model. The latter may be due to the consideration of an elapsed time of 29 years since the last earthquake. (3) Sopt is influenced by whether the expected benefit (or cost) per life cycle or per unit service time is employed. The maximization of the expected (net) benefit per unit service time

(i.e., E(Oa(A, t) shown in Eq. (13)) eliminates the need for an arbitrarily selected value of t. If the future benefit is not discounted, the maximization of E(Oa(A, t) leads to the optimum t greater than or equal to the largest considered t ( = 75 years). If the future benefit is discounted, a shorter structural design life is preferred (so to reduce the discounted benefit) as the benefit per unit of time increases. (4) The optimum importance factor I is controlled by the ratio of the damage cost of an important structure to that of its corresponding normal structure, aI. I is almost a linear function of logarithm of aI. This relation is insensitive to the structural design life t. but sensitive to the applicable standard deviation of the residual for the GMPEs. (5) The adoption of an importance factor of 1.5 for seismic design can be optimum and justified if aI is about 5, and r of the residual for the applicable GMPEs is similar in magnitude to those obtained by using records from a broad network of recording stations.

Acknowledgments The financial supports of the National Council on Science and Technology (CONACYT) of Mexico and the Natural Sciences and Engineering Research Council of Canada are gratefully acknowledged. We also thank The Unidad de Instrumentacion Sismica of The Institute of Engineering for providing some of the records employed in this study. We thank two anonymous reviewers for their helpful comments and suggestions. References [1] Rosenblueth E. Optimum design for infrequent disturbances. J Struct Div ASCE 1976;102(9):1807–25. [2] Rosenblueth E. What should we do with structural reliabilities. In: Proc., ICASP 5, 1. Waterloo, Canada: Univ. of Waterloo; 1987. p. 24–34.

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