Risk-adjusted Design Basis Earthquake’s Adequacy for use in Seismic Design Codes

In most buildings’ seismic design codes design basis peak ground acceleration (PGA DBE ) is provided by employing a uniform-hazard approach. However, a new trend in updating seismic codes is to adopt a risk-informed method to estimate the PGA DBE so-called risk-adjusted design basis peak ground acceleration (PGA RDBE ). An attempt is made here to examine the adequacy of the PGA RDBE to fulfill the assumptions made in seismic codes for converting the maximum considered earthquake’s (MCE) intensity to PGA DBE . To this end, the performance of regular intermediate steel moment frames (IMF) is assessed in terms of collapse margin (CMR) and residual drift ratios in the event of MCE and design basis earthquake (DBE), respectively. The PGA RDBE s are computed for Karaj County, Iran. A set of 96 index archetypes of regular IMF are designed considering four design parameters, which include the number of stories (2, 3, 6, 9, 12, and 15), span lengths (4 and 8 meters), occupancies (residential and commercial), and seismic demands (0.15, 0.25, 0.35 and 0.45g). The PGA DBE prescribed by Standard No. 2800 for Karaj neither meets the assumed acceptance criteria nor stands on the safe side. Meanwhile, PGA RDBE fulfills the acceptance criteria but does not necessarily satisfy the implicit assumption made in codes that the code-conforming buildings have at least a CMR of 1.5 if the MCE occurs. This emphasizes that the PGA RDBE should not be used without examining the CMR fulfillment. The results recommend that a lower limit need to be set on PGA RDBE s, which is found to be 0.35g for Karaj. Outcomes also reveal that the code-conforming buildings designed with the proposed PGA RDBE can fulfill both repairability and life safety performances at the DBE and MCE, respectively. These buildings also have a high chance to be even considered as repairable ones at the seismic demand of MCE. Furthermore, regardless of the employed method for estimating PGA DBE , various relationships between design parameters with different performance indicators such as CMR, residual drift ratio, ductility demand, imposed drift ratio, and building’s normalized weight are presented. These relationships can be used to evaluate the buildings’ safety factor against collapse and repairability, justification of using IMF in regions with high seismicity, level of structural and nonstructural damage as well as the economic consequence of changes in PGA DBE . The presented relationships provide a multi-criteria decision-making tool to decide on the optimum PGA DBE leading to an affordable alternative and tolerable damage.


Introduction
Among different natural hazards, an earthquake could have major effects on a concentrated area within a short period leading to a significant number of deaths, especially in developing countries (Holzer and Savage 2013;Ibrion et al. 2015;Nadim et al. 2004). The majority of the death tolls in the event of a large earthquake are due to the collapse of buildings (Marano et al. 2010;Tobita et al. 2007;Yamazaki et al. 1996). Therefore, the authorities try to control and restrict earthquake consequences by establishing appropriate regulations and updating design codes based on new achievements. As a minimum acceptable building's performance in the event of a very rare earthquake, seismic design codes try to minimize the number of fatalities by limiting the probability of collapse through the prescriptive criteria. Most of the seismic design codes, such as Iranian Standard No. 2800 (BHRC 2014) and Eurocode 8 (CEN 2004) neglect the vulnerability of buildings and consider just a point of seismic hazard curve to estimate the design basis earthquake. For instance, in both Standard No. 2800 and Eurocode, a uniform-hazard approach is used to define the design basis earthquake as an earthquake with a 10% probability of exceedance in 50 years ignoring the occurrence probability of the other earthquakes. This approach leads to buildings with unknown reliability or probability of failure in their service life. To provide nearly uniform acceptable performance for buildings, both the earthquake hazard and building vulnerability need to be considered in estimating the design basis earthquake's intensity. The uniform-hazard approach fails to estimate the seismic risk by neglecting the vulnerability and earthquake hazard curves. To have buildings with certain and tolerable reliability, a new approach for estimating design basis peak ground accelerations (PGADBE) needs to be implemented.
A risk-informed approach is just implemented in ASCE/SEI 7 (2010, 2017) that takes into account both earthquake hazard and building vulnerability. In this approach, the occurrence probability of all the probable earthquakes is considered by employing the entire hazard curve. Furthermore, the buildings' vulnerability is also considered by utilizing the buildings' collapse fragility function. Many studies have been devoted to compute the building's collapse fragility function via different analytical, experimental, and empirical methods (Adam and Ibarra 2014;Black 2011;Gehl et al. 2013;Gkimprixis et al. 2018;Ibarra and Krawinkler 2005;Martins et al. 2018;Ulrich et al. 2014). A comprehensive review of developing building collapse fragility P a g e | 3 function for risk assessment can be found in Douglas and Gkimprixis (Gkimprixis et al. 2018).
Besides, a thorough review of different methods of the buildings' collapse capacity assessment is provided in Villaverde (Villaverde 2006). In the same fashion as ASCE /SEI 7 (2010/SEI 7 ( , 2017, the general building's collapse fragility function (Luco et al. 2007) is employed to compute riskadjusted design basis peak ground accelerations here (PGARDBE).
Since seismic risk assessment depends on the earthquake hazard curve, a specific region needs to be chosen to carry out the study. The authors previously studied buildings' seismic collapse risk in Tehran City (Zaman and Ghayamghamian 2019) to estimate PGARDBE. So, Karaj is selected here to conduct the study since it is located near the active faults and growing very fast with almost 2 million populations. Karaj is the second city with the most population density in Iran with about 1,200 people per square kilometer (Statistical Center of Iran 2015). This concentration of people near the active faults exposes their lives and assets to high seismic risk. Besides, it is located in Tehran's vicinity and has been chosen as a specific city to manage rescue operations when a large magnitude earthquake hits Tehran.
In this study, first, it is attempted to evaluate the adequacy of PGADBE for both design basis peak ground acceleration prescribed in Standard No. 2800 (PGA2800) and those PGARDBEs proposed here. Second, it is tried to evaluate the economic consequences made by PGADBE's variation regardless of the employed method to estimate it. To this end, buildings with regular intermediate steel moment frame as a lateral load-resisting system are selected. Then, a set of 96 index archetypes are constructed by considering four design parameters, namely number of stories ( ), span lengths ( ), occupancies ( ), and PGADBEs. The performance indicators including collapse margin ratio (CMR), imposed drift ratio (∆ ), ductility demand ( ), and residual drift ratio (∆ ) are calculated for all the building models. The adequacy of the PGA2800 and PGARDBE are explained based on their ability to provide an acceptable level of seismic collapse risk and the calculated CMRs as well. The relation between design parameters and seismic demand (PGAd) with those of normalized weight ( ), aspect ratio ( ), , ∆ , and ∆ are examined. These relationships together make a multi-criteria decision-making framework to ensure that the future proposed updates in seismic codes will be safe and affordable alternatives. Furthermore, this framework makes it also possible to provide desired performances at different considered hazard levels. P a g e | 4

Probabilistic Seismic Hazard Analysis
The first step in computing the building collapse risk in Karaj is to calculate the earthquake hazard curves. Probabilistic seismic hazard analysis (PSHA) is a tool to deal with the uncertainties (like magnitude and location) and to estimate different ground motion intensity and their probability of exceedance in a certain period (usually building's lifetime) by incorporating all the earthquake sources. Karaj is the capital of the Alborz Province, which is situated at the foot of the Alborz rocky mountain range and surrounded by many known active faults, such as Mosha (135 km), North Tehran (130 km), Robat Karim (82 km), Eshtehard (64 km), Taleghan (61 km) Kahrizak (35 km), and Ray (27 km) faults (Hesami et al. 2003).
To conduct PSHA, the studied region is divided into 99 square cells with each side measuring 3 km, and peak ground acceleration (PGA) hazard curves are estimated at the cells' center.
These grid points alongside the PGA hazard curves are depicted in Fig. 1. A detailed description of the composed earthquakes catalog, faults' location, seismicity parameters, and ground motion prediction equations (GMPEs) employed to estimate the PGA hazard curves can be found in Zaman and Ghayamghamian (Zaman and Ghayamghamian 2019). The peak ground accelerations of the maximum considered earthquake (PGAMCEs) also need to be extracted from the calculated PGA hazard curves. According to the Iranian National Building Code (INBC) part 6 (BHRC 2019), the maximum considered earthquake (MCE) is defined as an earthquake with a 2% probability of exceedance (PE) in 50 years. These PGAMCEs are depicted in Fig. 2 and will be used later to measure CMRs. P a g e | 5 The pseudo-spectral accelerations (PSAs) are also needed in computing the general building's collapse fragility functions. The PSAs can be directly estimated by given GMPEs

General Building's Collapse Fragility Function
Building collapse fragility functions can generally be obtained through four approaches, including empirical (Hisada et al. 2005;Jaiswal et al. 2011), experimental (Gardoni et al. 2002), analytical (Baker 2015;Ulrich et al. 2014), and expert opinion (Porter et al. 2007;Tobita et al. 2007). For code-conforming buildings, there is another approach to simulate the collapse fragility function which is so-called the general building's collapse fragility function (Luco et al. 2007). In this approach, the general building's collapse capacity ( ) is assumed to be a random variable whose logarithm has a normal distribution buildings with IMF as a lateral load-resisting system, which is used here to construct the general building's collapse fragility function.

Building Collapse Risk Assessment
General building's seismic collapse risk is evaluated through the risk integral in the following Where , ( ) , and ( ) stand for PSA, general building's collapse fragility function, and PSA hazard curve, respectively. The computed risk integral needs to be optimized to concur with two acceptance criteria proposed by FEMA P695 (ATC-63 2009). The first condition is the conditional collapse probability in the event of MCE be equal to 10%, and the second one is the probability of collapse be equal to 1% in 50 years. The first criterion is satisfied by replacing P a g e | 7 the μ with the + 1.28 ; the second one is also fulfilled by optimizing μ in such a way that the Eq. (1) be equal to 1%. So, the general building's collapse fragility function can be written in where, μ and σ are the mean and standard deviation, respectively.
The so-called risk-adjusted peak ground acceleration (PGAR) is the one that satisfies the above acceptance criteria. It is assumed that the well-designed buildings have at least a safety margin of 1.5 against collapse. Therefore, two-thirds of the calculated PGAR is introduced as a risk-adjusted design basis peak ground acceleration (PGARDBE) with values generally larger than 0.35g as depicted in Fig. 3(a). The calculated PGARDBE varies from 0.35g to 0.45g as moving from the southeast to the northwest of Karaj. When compared the calculated PGARDBE with the constant value of 0.35g prescribed by Standard No. 2800 for the entire Karaj, it is found that the PGA2800 neither maintains the uniform collapse risk nor stands on the safe side. This evidence suggests the PGA2800 needs to be modified by PGARDBEs. Furthermore, to easily convert the PGA2800 to PGARDBE in Karaj, a risk-adjustment conversion factor (RA) is defined here as the ratio of the calculated PGARDBEs to that of PGA2800. Fig. 3(b) displays the variation of calculated RA in Karaj.

IMFs' Performance Evaluation
In the previous section, PGARDBEs that satisfy the collapse risk acceptance criteria were presented. However, before adopting those values as PGADBE in seismic code, their adequacy to fulfill the other intended performance goals must be ensured. To this end, buildings' seismic performance is evaluated in terms of the CMR, ∆ , ∆ , , and . A set of 96 index archetypes with IMF as a lateral load-resisting system are constructed by considering several design parameters, namely, , , , , and PGADBE. Then, the effects of design parameters' variation on performance indicators are examined. It should be noted that the performances of the developed buildings are examined for a range of PGADBE values. Therefore, the results are relevant to be used for the performance assessment of IMFs regardless of the method used to estimate PGADBE and where the buildings are located.

Archetype Design Space
P a g e | 9 of 3.2 m for all stories, have three bays of equal length, and their floor system behaves like a rigid diaphragm (Fig. 4). In numerical models, 170 box-shape cross-sections for columns and 110 I-shape ones for beams are defined to be used in the design and optimization process. Structural members' cross-section size is determined through the equivalent lateral force procedure, which is the reference method for determining seismic demands (

Nonlinear Analysis
Static nonlinear (pushover) analysis is employed to evaluate the index archetypes' performances in terms of CMR, ∆ , ∆ , and . The advantages of pushover analysis can be summarized as follows: (1) allows easy visualization of system performance at different levels of demand; (2) leads to a unique solution at the considered level of ground motion intensity; (3) needs less computational effort and time than more sophisticated nonlinear history response analysis. Meanwhile, its disadvantages are: (1) ignores the record-to-record variability ( ); (2) is not suitable for buildings with various irregularities; (3) is not appropriate for buildings with significant higher mode effects. To avoid the above disadvantages, only buildings with regular IMFs are considered in compiling the archetype design space. A detailed discussion about the pushover analysis procedures and results accuracy can also be found in FEMA 440 (ATC-55 2005). A triangular load pattern is used in the pushover analysis to compute the global pushover curve (roof displacement against base shear), and then the idealized pushover curve is constructed through the procedure presented in ASCE/SEI 41 (2017).

Evaluation of CMR
In buildings' seismic design codes, it is assumed that the code-conforming buildings have at least a safety margin ratio of 1.5 against the collapse in the event of MCE. Therefore, CMR is employed here to examine the safety factor of buildings with IMF as a lateral load-resisting

Fig. 6 Illustration of collapse margin ratio (CMR) and terms used in performance assessment
Statistical regression analysis is used to find the relation between CMR and assumed design parameters. According to the statistical t-test analysis, the building's and have minor effects on the CMR and can be ignored from the regression analysis. Therefore, the relation between CMR with design parameters of PGADBE and is computed and given in Eq.
(3), which shows a strong dependence of CMR on PGADBE. Besides, CMRs' dispersion at each PGADBE follows a lognormal distribution, whose mean value and standard deviation (SDCMR) are given by Eqs. (3) and (4) where PGAd is the seismic demand (PGA at any desired intensity level) in fraction of g; PGADBE is the seismic intensity in fraction of g for which the considered building is designed; R 2 and RMSE are the coefficient of determination and root mean square error (residual standard deviation), respectively; is building occupancy (0.0 for residential and 1.0 for commercial buildings). Note that the value for occupancy with a live load intensity in the range of 200 to 500 kg/m 2 could be calculated by linear interpolation between 0.0 and 1.0. The positive sign of the 's coefficient indicates that the CMR increases as the (live-to-dead load ratio) increases. This is due to the overstrength created by increasing the live load intensity, which has a lower probability of existence during an earthquake. Eqs. (3) and (4)  CMR of 96 index archetypes designed for four levels of PGADBE easy-to-use tool to examine the impact of any seismic demand variation due to any future updates or modifications in design codes. Furthermore, Eq. (3) makes it possible to assess the effects of seismic importance ( ) and response modification ( ) factors variations on CMR due to their proportional relationship with the PGADBE.

IMFs Usage Justification
To justify the usage of IMFs in different seismic zones especially those that are to be constructed in regions with very high seismicity (PGADBE=0.35), ductility demand ( ) needs to be estimated. Next, the estimated ductility demand needs to be compared with the ductility capacity of IMFs stated in the design codes to ensure their stability. In the seismic portion of INBC part 10 (BHRC 2013), it is required that the IMFs have at least a rotation capacity of 2% radians at the PGADBE level of seismic demand, at least 1% of which need to be tolerated in the elastic range and the remaining can be tolerated in the plastic range of behavior. The justification of IMFs usage in regions with very high seismicity is made by providing the relationship between ductility demand as a performance indicator and design parameters via a multiple linear regression analysis as: where PGAd is the PGA of the desired level of earthquake hazard at which building performance needs to be evaluated; PGADBE is the earthquake intensity for which the considered building is designed; and are the building's span length in units of meter and aspect ratio, respectively.
The use of IMfs is allowed by Iranian Standard No. 2800 (BHRC 2014) while their usage is restricted by ASCE/SEI 7 (2017) in regions with very high seismicity. To ensure the ductility demand will not exceed the ductility capacity of IMFs, ductility demand is calculated at the PGARDBE as well as PGAMCE levels of demand for residential buildings designed with the PGARDBE by using Eq. (5) in Karaj. The distribution of calculated ductility demands is shown in Fig. 8.
Since all the resulting values are lower than 2.0%, it can be inferred that IMFs have sufficient ductility capacity to be used in the region with high seismicity. It also can be realized that P a g e | 15 putting a restriction on their usage as stipulated by ASCE/SEI 7 (2017) in regions with very high seismicity might be an unnecessary conservative requirement.

Damage and Repairability
The structural and many nonstructural (e.x. nonbearing masonry walls) damage could be related to the imposed drift ratio (∆ ), which is a function of the earthquake ground motion intensity and building's characteristics such as lateral strength, stiffness, geometric properties, and ductility. Moreover, the repairability of buildings at any desired levels of demand can be expressed in terms of residual drift ratio (∆ where PGAd is the seismic demand of desired hazard level at which ∆ and ∆ need to be evaluated; PGADBE is the seismic intensity used to design the considered building; , , and represent the occupancy (live-to-dead load ratio), span length, and building's aspect ratio, respectively. 's value should be set to 0.0 for residential and 1.0 for commercial occupancies.
For other occupancies with a live load intensity in the range of 200 to 500 kg/m 2 , the value of could be computed by linear interpolation between 0.0 and 1.0. Note that the unloading stiffness is assumed to be equal to the elastic stiffness in the derivation of Eq. (7).
According to the ATC-58 (2012), buildings with a residual drift ratio (∆ ) up to 0.5% could be repairable and those with 1%, although are repairable, may not be economically and practically feasible, meaning those buildings might be at a total economic loss. The repairability of residential buildings designed with the PGARDBEs proposed here and PGA2800 stipulated by Standard No. 2800 is examined at the MCE level of demand.
In Fig. 9, buildings designed for the PGARDBEs ( Fig. 9(a)) display a high chance of repairability as their ∆ s are in the mid-range of 0.5% to 1.0%. This means that the buildings designed by the PGARDBE although sustain structural damage, but could be considered as repairable ones if the MCE occurs. Meanwhile, the calculated ∆ s close to 1.0% ( Fig. 9(b)) show that the buildings designed by the PGA2800 are unrepairable. Furthermore, as a requirement by the seismic design codes, buildings should be repairable at the PGADBE level of demand. Since it is recommended here to replace the uniform-hazard PGADBEs in Standard No. 2800 with the uniform-risk PGARDBE, it is required to validate the buildings' repairability at this level of demand as well. So, ∆ s at PGARDBE demand level are also calculated in Karaj and shown in Fig. 10. The calculated values in the studied region are all lower than 0.5%, which confirms the repairability requirement. Furthermore, the comparison of ∆ 's distribution in Figs. 9(a) and 10 with those in Fig. 9(b) reveals a more uniform distribution of repairability's chance for buildings designed by the PGARDBE in comparison to those designed by PGA2800.

Economic Consequence of Changes in the PGADBE
An increase in the PGADBE improves the safety margin against collapse, however, it also causes an increase in construction cost that may render the safety enhancement an impossible choice.
To find the optimum affordable safety, the relationship between the PGADBE and building weight as a basic economic index needs to be known. Again, using the results obtained from developed 96 index archetypes, the PGADBE relation to the building normalized weight (WN) is investigated, which shows a linear relationship as shown in Fig. 11 where WN is the normalized weight (WN = Wi /W0.15g). This equation shows that changing the PGADBE from 0.35g to 0.40g leads to a 5% increase in the building structural weight, which has a minor economic impact on the individual scale.

Conclusion and Discussions
Many design codes such as the Iranian Standard No. 2800 use a uniform-hazard procedure to determine the design basis earthquake. These need to be updated by taking a risk-informed approach to make them compatible with the acceptable and tolerable level of risk.  (Fig. 3(b)). that the PGARDBE may fail to provide CMRs larger than 1.5. This emphasizes that the direct application of PGARDBE without ensuring its adequacy may provide misleading results. To prevent undesirable results (CMR<1.5), the calculated PGARDBEs are modified so that the CMRs larger than 1.5 are guaranteed for almost all buildings in the studied region. This is carried out by defining a lower limit for the calculated PGARDBEs in the studied area to be 0.35g (Fig. 7). The repairability requirement of buildings designed with the calculated PGARDBEs is also examined by computing residual drift ratios using Eq. (7) as shown in Figs. 11(a) and 12. Buildings designed for and evaluated at the PGARDBE are repairable by showing a residual drift ratio lower than 0.5%. Furthermore, they have a high chance to be considered as repairable at the PGAMCE demand level with drift ratios in the mid-range of 0.5% to 1.0%. Furthermore, ductility demand maps for residential buildings designed for PGARDBE and evaluated at the PGARDBE as well as the PGAMCE are produced for those located in the studied area (Fig. 8). The values of calculated P a g e | 20 ductility demand lower than 2% confirm that the IMFs could be utilized in regions with very high seismicity without concern about the imposed ductility demand.
From an economic point of view, although replacing the PGADBE with PGARDBEs in seismic codes enhances safety, it is necessary to check whether this increase of PGADBE is also affordable or not. So, by selecting the building's normalized weight as a basic economic index, the relationship between this parameter and the PGADBE is found and proposed to be used (Eq. (8)) as a tool for checking the economic consequences of an increase in PGADBE. From this relation, it is found that an increase of PGADBE from 0.35g to 0.40g leads to a 5% increase in the building's structural weight, which has a minor economic impact on the individual scale.
Finally, to design buildings beyond the intended performance goals assumed in the seismic design codes, the performance-based design procedure has been proposed as a suitable alternative. From the practical point of view, this approach needs iterative, complicated, and time-consuming nonlinear analyses that make it a difficult-to-use choice in traditional design.
Using the proposed relationships between performance indicators (CMR, ∆ , ∆ , ) and design parameters for buildings with IMF not only overcome those disadvantages but also provide a simple straightforward tool to achieve desired performances at required seismic demand levels in the preliminary design stages.

Declarations Funding
No funding was received to assist with the preparation of this manuscript.

Conflicts of interest/Competing interests
The authors declare that they have no conflict of interest.

Availability of data and material
Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request