Multidimensional functionality limit states for seismic resilience analysis of urban buildings

Currently, most of methodologies available to evaluate the seismic performance of buildings use as index maximum interstory drifts. However, recent earthquakes have evidenced the need to develop performance levels that incorporate seismic resilience concepts to evaluate the level of post-seismic functionality of buildings and their capacity to recover functionality. Furthermore, such performance levels should explicitly consider the performance of structural, non-structural elements and contents. For this purpose, this paper proposes a set of six performance limit states for office-type buildings, in which the seismic performance of structural, non-structural elements and contents is explicitly considered. Each of these limit states is associated with a set of probable events that generically determine its recovery of functionality (e.g., post-seismic inspection and management of financial resources). To exemplify the proposed scheme a seven-story reinforced concrete building with unreinforced infill masonry walls and located in Mexico City is evaluated. The results obtained suggest that the building has a significant probability of experiencing loss of functionality due to the damage suffered mainly by the non-structural elements and contents. This indicates that modern seismic design codes, as that used for this study, accomplish their main objective, which is to reduce the probability of collapse and to prevent the loss of human lives. However, these results also demonstrate that the main objective of decision makers when designing such buildings, which is to be functional for one or several needs, is not achieved.


Introduction
The last two decades have witnessed the development of probabilistic methodologies to evaluate the seismic performance of individual buildings (Porter 2000;Moehle and Deierlein 2004;Mitrani-Reiser 2007;Günay and Mosalam 2013, among others). A major aspect characterizing these formulations is the performance limit states, used to estimate the probability of reaching or exceeding an undesired performance level, e.g., leading to demolition or collapse. Most of the existing methodologies use interstory drifts (maximum or residual) Extended author information available on the last page of the article as an Engineering Demand Parameter ( EDP ). With this criterion it is possible to evaluate the structural safety given a seismic hazard level; however, it is not feasible to evaluate the loss of functionality of a building under study from a resilient perspective. Therefore, it is convenient to formulate multidimensional performance limit states (MPLS) based on seismic resilience concepts that explicitly and jointly relate the structural response (e.g., drifts, velocities, and accelerations) to the damage that structural ( S ), non-structural ( NS ) elements and contents ( CS ) are susceptible to experience.
In this regard, Cimellaro et al. (2010) proposed a unified analytical framework to quantify the seismic resilience of individual structures, e.g., bridges, hospitals, etc. They defined resilience as a function that indicates the capability of a given building to sustain a level of performance functionality over the recovery time t REC defined as the time necessary to return a damaged building to a level of operation and comfort equal or better than that offered before it was affected by the disruptive event. Under these conditions, this research group proposed as a building resilience index R(t) the area under the recovery curve normalized by the recovery time (see Fig. 1), as shown by the following equation: where defines the functionality of the system as a function of time t and t 0 corresponds to the instant when a potentially destructive earthquake occurs; L T IM, t REC is the loss of functionality caused by a seismic intensity IM ; H(⋅) is the Heaviside step function and f REC t, t 0 , t REC is the recovery function.
According to this formulation, the loss function considers structural, and non-structural economic losses. Non-structural losses are classified into direct and indirect losses. The former is due to the damage caused to the contents and to the people injured and/ or casualties. Indirect economic losses are generated by business interruption (highly dependent on the recovery time). The total losses are obtained by summing the fraction of each type of loss L i multiplied by the probability of exceeding one or more MPLS, R ≥ r lim , conditioned to an intensity measure, IM , e.g., Sa T 1 , PGA , etc., Fig. 1 Schematic representation of the resilience of a building proposed by Bruneau et al. (2003) and extended by Cimellaro et al. (2010) The MPLSs are characterized by a set of fragility functions, each one representing the probability of reaching or exceeding the corresponding limit state associated with the structural response of the system, e.g., interstory drifts. This framework is of a general nature, so it is necessary to perform a particular formulation of the MPLS for the type of building under study.
Due to the limitations and the uncertainties that characterize seismic demands and structural response, in the last two decades several research groups have been developing probabilistic methodologies to estimate structural performance in terms of decision variables, DV , e.g., repair costs and repair time. In such methodologies the performance levels are not controlled by a physical variable, but by the probability of exceedance of the DV s associated with an IM with a specific return period, e.g., 250 years.
The first methodology oriented to the probabilistic evaluation of buildings subject to the influence of earthquakes was the scheme developed by the Pacific Earthquake Engineering Research Center (PEER) (Cornell and Krawinkler 2000;Moehle and Deierlein 2004;Günay and Mosalam 2013). This methodology primarily aims to estimate the consequences of seismic damage experienced by individual buildings in terms of a DV as a performance indicator. This scheme represented an important step towards the quantification of the seismic risk of individual buildings. Nevertheless, it does not explicitly consider the fundamental aspects of seismic resilience, i.e., loss of functionality and recovery time (Krawinkler and Deierlein 2013).
Along this line of thought, Porter (2000) developed a methodology denominated Assembly-Based Vulnerability (ABV) of buildings, which follows the same structure of the PBEE-PEER framework. This methodology uses fragility functions to infer the damage that structural and non-structural elements may develop as a function of the physical variable to which they are sensitive. The DV s obtained with this approach are repair costs and repair times, which require a large number of non-linear dynamic analyses (NLDA) for each IM of interest to determine their probability distribution.
The methodology developed by Porter (2000) was extended by Mitrani-Reiser (2007), who considered the delay time for the start of repairs within the seismic performance evaluation scheme. The delay time is composed by the post-seismic inspection, the bidding process, the process to obtain financial resources and the governmental permitting stage (Comerio 2006). The most innovative aspect of Mitrani-Reiser's work was the probabilistic methodology called Virtual Inspector. This approximation consists of estimating the probability of tagging the building in study as safe, restricted use or unsafe, according to the post-seismic inspection criteria developed by the ATC 20 (1989). The assessment of the probability of the building being tagged as unsafe, and therefore non-usable, considers only the structural damage. Therefore, since the influence of non-structural elements and contents on the loss of functionality is not considered, these probabilities correspond to the lower limit of the tagging probability.
The FEMA P-58-1 (2012a), based on the performance-assessment framework developed by PEER, adopted the probabilistic post-seismic inspection approach proposed by Mitrani-Reiser (2007). This methodology was modified to infer the number of people who may lose their lives given that the building is tagged as unsafe. This methodology allows for probabilistic estimates of repair costs and times; however, it does not consider the fraction of recovery time associated with delays, which have been shown to have a significant impact on indirect economic losses associated with business interruption (Almufti and Willford 2014). Mieler et al. (2016) developed a conceptual framework for estimating the probability of loss of functionality in buildings. This scheme uses fault tree analysis to estimate such probabilities. This tool is characterized by considering the performance of each component that can affect the functionality of the building in case of damage. The methodology was designed to evaluate performance objectives considering structural and non-structural elements and relate them to global levels of resilience. The implementation of this methodology requires the development of a robust algorithm of the performance of structural and non-structural components and prescribing their interdependence, therefore, it requires a detailed inventory of the elements that are essential for the operation of the facility. The accuracy of this methodology depends on the level of detail with which the fault tree is elaborated. However, it has been observed that this type of approximation is very sensitive, and it is possible that the model under or overestimates the failure probabilities (Porter and Ramer 2012).
The previous models only consider the assessment of postearthquake damage to evaluate the recovery of a building. In recent years, Burton et al. (2016) developed five functionality limit states of the shelter in-place housing capacity of a residential community in terms of seismic resilience. Each one of them is characterized not only by the prescription of a specific damage configuration, but also considers a set of events involved in the recovery of functionality (e.g., post-seismic inspection, structural redesign, and execution of repairs). Due to the scarcity of systematically collected information on the events involved in the post-seismic recovery of buildings, these limit states are based on heuristic reasoning, i.e., using deduction by analogy (Polya 1954). This means that these limit states must be reviewed in future earthquakes to verify their veracity, or if necessary, improve them or even propose new ones. Despite this limitation, and the current state of the art of seismic resilience, the limit states developed by this research group can be of great help to identify damage configurations potentially harmful to building performance and to simulate scenarios of recovery of functionality at different levels of seismic hazard. This paper proposes a MPLS formulation for the seismic performance assessment of urban buildings which explicitly considers the damage of non-structural elements and contents. This formulation is an extension of the methodology developed by Mitrani-Reiser (2007) for estimating the tagging probability of the post-seismic inspection process in adaptation of the LS s for residential buildings used by Burton et al. (2016). The formulation of the LS s is consistent with the MPLS of the seismic resilience assessment framework developed by Cimellaro et al. (2010). To exemplify the proposed formulation, the probabilistic evaluation of the LS s in a seven-story RC office-type building with URM infill-walls designed with a modern seismic design code is presented. The results obtained indicate that the loss of functionality in this type of buildings is mainly due to the damage experienced by the non-structural elements and contents, proving that although modern seismic design guidelines are aimed to prevent collapse under severe earthquakes to minimize loss of life, they are not oriented to prevent the loss of functionality for low and moderate seismic intensities.

Assessment of the structural safety of a building using the inspection criteria proposed by ATC 20
When a potentially destructive earthquake affects a building, an immediate inspection is required to assess the potential damage and determine whether the building can be partially or fully operational, or in the worst case, demolition is required (Gutiérrez et al. 2020).
To restore the functionality of the buildings damaged by an earthquake, which affect their occupants, communities or an entire city, the post-seismic inspection of all these constructions must be done in the shortest possible time. Therefore, to reduce and optimize the inspection time, a significant number of specialized workers and engineers must be available to perform the post-seismic assessments. However, the experience acquired from past destructive events show that it is unlikely that the affected society will have enough qualified personnel to carry out this task due to the possible large number of buildings affected and their dispersed location within a city (Gallagher 1989). Due to this problem, the ATC 20 (1989) developed a procedure for inspecting conventional buildings affected by an earthquake with the objective of minimize the time involved in this activity and for issuing a report that defines the general activities required to mitigate the effects of the damage produced and to recover their functionality. This procedure is divided into the three levels of inspection: rapid evaluation ( RE ), detailed evaluation ( DE ), and engineering evaluation ( EE ). The first two only require visual inspections, while the third requires an engineering calculation process to determine the residual load capacity of the affected building. The objective of the evaluation is to describe post-seismic structural safety using one of three types of tagging: inspected, restrained entry, or unsafe. These tags are identified with the green ( G ), yellow ( Y ), or red ( R ) colors, respectively. A fourth category involves the inspection of NS components, that although their damage does not affect the stability of the structure, it can be a cause of injury to people in and outside the affected building. The tag assigned to the building due to this type of damaged elements is unsafe area ( R a ), and the red color is used to identify it. The details of each type of evaluation are described below.

Numerical simulation of building safety using the ATC 20 inspection criteria
Mitrani-Reiser (2007) developed a probabilistic methodology to estimate the post-seismic structural safety based on the ATC 20 (1989) inspection criteria. This methodology, called Virtual Inspector, reproduces numerically the rapid and detailed evaluation. For the first one, in case of assigning a Y tag, i.e. , a restricted entry tag, the process of a detailed inspection is then simulated. Finally, the probabilities of each type of inspection are combined to obtain the probability that the building receives a G or R tagged given an IM.
The main limitation of this approach is that the evaluation process only considers S elements; therefore, the probability of tagging the building corresponds to the lower limit of such probabilities (Gutiérrez 2021). To address this limitation, this paper explicitly considers the simulation of the post-seismic inspection of NS elements and CS . With this consideration, the probability of tagging the post-seismic inspection corresponds to the upper limit of probabilities. The development of NS − CS damage combined with null structural damage is not a reason to post the building as structurally unsafe; however, it can be considered as a potential risk due to the possibility of the NS − CS falling ( R a tag), and naturally, the possibility to induce injury to people and partial loss of functionality.
The probabilities that the building being R , Y or G tagged given an IM , are calculated with the following proposed equations: where the subscript T refers to the type of performance group (PG) , i.e., S , NS elements or CS . Examples of these types of PG s are, to mention a few, columns, windows, and furniture, respectively. The terms Pr S D |IM, NC , Pr M D |IM, NC and Pr N D , L D |IM, NC , are the probability of experiencing severe ( S D ), moderate ( M D ), null ( N D ) or light ( L D ) damage given that the building does not collapse. The term Pr(C|IM) indicates the probability of collapse and Pr(NC|IM) = 1 − Pr(C|IM) the probability of survival. Note that only the severe (overturning) and null (rest) states are considered in the evaluation of the contents, as is described in Sect. 2.3.2. Thus, for this case the Eqs. 4a and 4c are used, which consider the R and G tags.
The G , Y or R color tagging that governs the post-seismic inspection of a building in evaluation is based on the prescription of hierarchy levels among the types of PG s, the most important are the S , followed by the NS elements and finally the CS . Likewise, in each of these types of PG s, a hierarchy is prescribed for the level of damage that can develop. This second hierarchy starts with severe damage, followed by moderate damage and ending with light damage. Additionally, in the approach proposed here only the detailed inspection is performed, instead of doing the rapid and detailed virtual inspection. This is because the rapid inspection is performed due to the lack of enough personnel specialized to inspect many buildings potentially damaged in a short time. From a computational point of view, it is more efficient to perform one numerical inspection instead of two.

Functionality limit states formulation
The notion of the functionality limit state was introduced by Burton et al. (2016). The LS s explicitly relate a series of damage configurations in a building caused by an earthquake to a set of generic activities that must be performed to recover its functionality. In this work, the idea proposed by Burton et al. (2016) is used and adapted in such a way that the configuration of each LS is derived from tagging of the post-seismic evaluation of structural safety, and from the evaluation of the unsafe areas, corresponding to the NS elements and CS damaged. Due to the great uncertainties in the inference of the DS s that characterize the different PG s that integrate a building, the LS s allow to visualize generic configurations of the damage combination patterns for different levels of IM s. Likewise, since there are few real events that can be used as a basis for generating a robust and reliable recovery model, currently, the only alternative so far is to propose generic recovery events RQ(t) formulated via deduction by analogy, i.e., from the perspective of a heuristic point of view (Polya 1954). The configuration of each LS and RQ(t) , is described in the Table 1.
Due to the complex interaction of the events that define each LS , to evaluate them, concepts of probability and set theory are used. For example, the probability for the LS 3 conditioned to an IM , is The events in this equation are obtained from the random realizations that match with the events functionality limit states, defined in Table 1. The probability of the LS 3 is obtained recurring to the frequentist definition of probability. In general, the canonical expression for estimating the probability for each LS is calculated by the following expression: The LS s defined in Table 1 are similar to those proposed by Burton et al. (2016). The main difference is that here they are obtained from the combination of the events associated with the virtual inspection corresponding to S , NS elements and CS , deducted under a heuristic criterion. This not only gives a logical character to the deduction of the LS s, but also allows the calculation of the probability of each one of them in a less ambiguous way. Therefore, its numerical implementation is straightforward.
Likewise, the assumption of the events that integrate the recovery of each of these LS s has been deduced from a heuristic reasoning. This means that the events contained in the recovery process can be refined and improved with the availability of more and better databases of S , NS and CS damage produced by potentially destructive seismic events in a wide range of buildings, and of the activities involved in the recovery of its original functioning level. In general, LS s can be interpreted as a series of combinations of discrete damage ( DS ) levels experienced by the various PG s that comprise a building, together with the likely generic events involved in their recovery of functionality.

Damage analysis
The loss of functionality of a building that has experienced damage from an earthquake is mainly influenced by the damage developed by the NS and CS elements. These elements can experience damage even at lower seismic intensities than the structural system was designed for, mainly because of their low deformation capacity (Almufti and Willford 2014;Filitrault and Sullivan 2014). In this section it is described the criteria to infer the damage experienced by S and NS elements.

Inference of the damage in structural and non-structural elements
The PBEE-PEER methodology does not establish a general criterion for determining whether damage is inferred by element or by group of elements. However, several research groups, e.g., Mitrani-Reiser (2007), Goulet et al. (2007), Yang et al. (2009), and guidelines as FEMA P-58-1 (2012a), to name a few, evaluate the damage by grouping several elements of the same PG assuming that they can experience the same level of EDP given an IM , and therefore the same level of DS . To exemplify this, consider that on a given floor of a building, a set of buckling restrained braces are installed on the same frame. With this approach, it is accepted that all these elements develop the same level of maximum drift and therefore the same level of DS.
This criterion may be appropriate if the structural models are bidimensional, if the uncertainties associated with the mechanical properties of the materials used and their mass distribution are small enough to disregard their variability. However, for 3D structural analysis, in-plan torsional effects can cause important differences in the EDP s . Therefore, although the building is safe, a tag indicative that it is partially non-functional is assigned because some usable spaces cannot be occupiable by the users. This state could occur in buildings whose contents may experience accelerations of great magnitude causing them to fall The generic recovery process for this limit state, RQ consist of the following events: (1) an inspection of the building due to the visual impact of the damage to the contents and furnishings, an after the completion of the inspection, the owners/stakeholders could apply for (2) an insurance to pay for the replacement of damaged elements, if the amount and severity require it.  The recovery path, RQ , is assumed to be described by the following events: (1) post-seismic inspection, where the building is declared restricted entry, i.e., it receives a yellow tag; (2) obtaining the financing to pay for the repairs; (3) execution of the structural repairs. Once the structural repairs are completed the building is declared safety to be reoccupied, but with restrict use. The next step (4) The recovery process, RQ , is assumed to be integrated of the following events: (1) the post-seismic inspection is performed, in which it is determined that the building is unsafe, i.e., it receives a red tag. Due to the quantity and severity of damage it is determined that the building can be repaired, i.e., it did not experience significant residual drifts and the repair cost/time are acceptable to the stakeholders. After carrying out these activities, the building condition becomes restricted entry. During this state the building could be occupied by authorized personnel and the following events are performed: (2) obtain financial resources to pay for repairs, (3) repair process planning and (4) bidding process. Once the structural retrofit has been determined and the financing is obtained, (5) the structural repairs are carried out. After this step, the building is safe to be reoccupied, i.e., with restricted use. However, to be fully functional, The recovery process, RQ 6 (t) , is given by the cleaning of debris and the reconstruction of the building. In this limit state, it is considered, under a conservative judgment, that the number of fatalities is equal to the number of occupants who were in the building at the time of the seismic event corresponding with a same set of elements located in a same floor. Therefore, the level of DS in each element of the set could be substantially different. Furthermore, the damage level of an element varies, even for the same values of EDP due to the differences in the pattern and history of the structural response (Günay and Mosalam 2013).
In the methodology presented here, three-dimensional structural models are used and both EDP s and DS s are evaluated individually, i.e., element by element, so that each of them has its own probability density function (PDF) of structural response p(EDP|IM) , and therefore an individual PDF discrete damage state p(DS|EDP) . To infer the damage that each S and NS element may experience, the methodology proposed by Gutiérrez (2021) and Gutiérrez and Ayala (2022) is used, which corresponds to an extension of the approaches carried out by Porter (2000) and Mitrani-Reiser (2007). This method uses the fragility functions developed by FEMA P-58-3 (2012b), which have been calibrated and tested in laboratories with acceptable results (Beck et al. 2002).

Inference of the damage in contents
To infer the damage in a content element, the FEMA P-58-1 (2012a) adopted the procedure proposed in ASCE/SEI 43-05 (2005) assuming that this damage can be estimated from the dynamic response of a rigid block subjected to a history of accelerations at its base. The response of a rigid block to a horizontal excitation at its base can be classified in four types: rest, slide, rocking and overturning. This methodology uses fragility functions, similarly to those used to evaluate damage in the PG s described in Sect. 2.3.1. Nonetheless, the results of a comparative study performed by Dar et al. (2016) between the predictions obtained from the fragility functions proposed in ASCE/SEI 43-05 (2005) and FEMA P-58-1 (2012a) and a series of NLDAs of a set of rigid blocks subjected to various levels of excitation concluded that, the former offer unreliable, and in many cases, non-conservative results. Due to this inconsistency, in this paper the theory of rigid block dynamics is used to estimate the performance of typical office furniture, e.g., bookcases and shelves. Details of the methodology used to estimate the damage to the contents can be found in Gutiérrez (2021).

Numerical example
This section presents the probabilistic evaluation of the functional limit states for a hypothetical office-type reinforced concrete building. The architectural floor plan is shown in Fig. 2. The building has seven floors and three bays in each orthogonal direction. The story height of all floors is 4.0 m and the width of all bays is 6.5 m. The ground floor is composed of a reception, a lobby, a restaurant, a couple of bathrooms, and the stairs that give access to the upper floors. The rest of the floors are integrated of six offices, two bathrooms and a meeting room. Fifteen particular types of PG s are considered for the damage analysis: (1) columns, (2) beams, (3) column-beam connections, (4) exterior windows, (5) ceiling system, (6) sprinkler system, (7) gypsum interior partitions, (8) unreinforced masonry exterior walls, (9) interior windows, (10) unreinforced masonry interior walls, (11) concrete stairs, (12) piping for cold and hot water system, (13) ducting HVAC (Heating, Ventilating and Air Conditioning) system, (14) central unit HVAC and (15) contents (bookcases and shelves for typical use). Table 2

Structural design and location of the building
The building was designed in accordance with the Mexico City Building Code (NTC-DS) (GCDMX 2004). It was assumed that the building is located at the accelerometer station of the Secretary of Communications and Transportation (SCT), which is in Zone IIIb, corresponding to a soft soil, according to the geotechnical zonation of the valley of Mexico City. The structural system chosen for the building was a reinforced concrete moment resisting frame with low ductility as it is the most common system used in Mexico City for such type of buildings. Accordingly, a seismic behavior factor Q = 2 was considered in its design. The dimensions of the cross section of all columns and beams are 100 × 100 cm and 90 × 70 cm, respectively . The fundamental period of the designed building is T 1 = 0.61s. The design of this building is shown in detail in Gutiérrez (2021).

Seismic hazard analysis
The evaluation of the damage and functionality limit states of the hypothetical building was performed for seven levels of seismic hazard, with return periods of 50, 125, 250, 500, 1000, 1500 and 2500 years, with a probability of exceedance of 63, 33, 18, 10, 5, 3 and 2%, respectively, in a 50-year interval. Spectral acceleration Sa(T 1 ) was used as the IM . The target mean spectral accelerations for each IM level considered were: 0.15g, 0.22g, 0.29g, 0.36g, 0.47g, 0.54g and 0.63g, obtained from the seismic hazard curve for the SCT site and a fundamental period of vibration of T 1 =0.6s (see Fig. 4). The intensities associated with earthquakes with return periods of 50 years correspond to service seismic events, that is, of immediate occupation, and the events corresponding to a 250-year return period are associated with the collapse prevention (GCDMX 2004). To represent the variability of the seismic demand, sets of 20 pairs of accelerograms were simulated for each IM using the   Kohrs-Sansorny et al. (2005). Details of this analysis can be found in Gutiérrez (2021). It should be noted that the building studied in this research has limited ductility, so it is not appropriate to estimate the probability of collapse using Incremental Dynamic Analysis (Vamvatsikos and Cornell 2002) since this methodology is more suitable for buildings that can develop significant ductility demands (Shoraka et al. 2013).

Structural analysis
The NLDA was performed using the open-source software OpenSees (Mazzoni et al. 2007). The columns and beams were modeled using inelastic fiber elements with five integration points. The behavior of concrete was modeled using the constitutive model developed by Chang and Mander (1994), and the longitudinal reinforcement steel with the model proposed by Giuffrè and Pinto (1970). To consider the stiffness contribution of the   Kadysiewski and Mosalam (2009) was used. This approach considers the in-and out-of-plan behavior of an infill wall. The parameters of the constitutive model for masonry were calculated in accordance with FEMA 356 (2000). The evaluation of the demolition limit state LS 5 associated with significant residual drifts was performed using the methodology developed by Ramirez and Miranda (2012).
Structural collapse was assumed to occur when at least one building column exceeded a prescribed maximum drift (first component failure) in a seismic random realization given a seismic intensity. This approach is consistent with the criterion proposed by ASCE 41-06 (2006) for the limit state of collapse prevention. A value of 0.025 was used as the limiting collapse drift, as suggested by Xue et al. (2008) and Nazri (2018) for buildings with low ductile capacity. The probability of collapse is estimated using the frequentist approach to estimate probabilities, i.e., by dividing the number of times the structure collapsed by the total number of seismic realizations associated with a specific seismic hazard level.
Regarding content damage analysis, to estimate the history of accelerations at the base of each block located on each floor, a bilinear interpolation of the accelerations calculated at the corners of the panels forming the slab was carried out. To determine if the block under analysis experienced overturning, the maximum angular displacement of the rotation point the block was used as EDP , with the maximum displacement being max (t) = 1.40rad(≈ 80 • ) . For values of (t) ≥ max (t) the numerical solution diverges, an indication that the block was overturning. The results obtained are consistent with those observed in recent earthquakes (e.g., Chile 2010 and México 2017).

Analysis of the results
For the sake of brevity, only the damage analysis associated with the functional limit states of the event LS 4 , IM 6 , is presented. More details about the damage analysis can be found in Gutiérrez (2021). This event is the one that contributes most to the probability distribution of the seismic hazard level IM 6 ( Sa(T 1 ) = 0.54g ), which correspond to recurrence period of 1500-years. Figure 5 shows the visual damage simulation of each of the PG s corresponding to this event, generated by an orthogonal pair of synthetic accelerograms. In this seismic intensity the S elements experienced light, moderate and severe damage (see Fig. 5a, c). It is observed that a significant number of columns of the first two floors experienced light damage, and one column, located on the second floor, developed moderate damage. Likewise, an important number of column-beam connections located in the first three levels experienced severe, moderate, and light damage, a condition sufficient for the building to receive a red tag, indicating that the structural system is not safe for this level of seismic hazard. In this regard, the beam-type elements did not experience damage of any type in any seismic realization (Fig. 5b). It can also be seen that a facade window experienced moderate damage (Fig. 5d), which according to its pathology would imply its fall, and possibly the cause of injuries on the users of the building, or on passers-by. On the other hand, it can be observed that many partition wall elements experienced light and severe damage in most of the floors (Fig. 5h). Also, most of the interior and exterior masonry walls located in the first three floors experienced severe damage (Fig. 5i, j), while in the rest of the floors it is observed that this type of elements experienced light and moderate damage. The concentration of damage in this type of elements in the first three floors is a product of the demand of shear forces accumulated gradually from the top floor to the base of the building. In this case, the stairways also experienced damage on all floors, especially moderate and severe damage on the first and second floors (Fig. 5k). This condition automatically implies loss of functionality because these elements are the transportation routes within the building.
An interesting result is due to the performance of CS . The Fig. 5o shows that this type of element suffered severe damage, i.e., overturning, in the last three floors of the facility. This is since floor accelerations increase gradually from the base to the top floor of the building. This physical aspect is typical of short-period buildings, such as the one under evaluation. It can also be observed that the periphery of the last three floors is where the greatest number of CS that experienced overturning is concentrated. This is because to the fact that the points furthest from the center of mass of the building-floors experienced the largest intensity due to plant rotational accelerations (see Fig. 6).
The estimation of the DS state element-by-element can help to identify vulnerable areas and, thus, which may generate loss of functionality, and even the total or partial collapse of a building. At the same time, it can be useful to identify the most vulnerable elements to propose structural retrofits that mitigate as much as possible future earthquake induced damage.

Post-seismic evaluation analysis and functionality limit states
The fundamental aspect of the methodology proposed in this paper is the probabilistic evaluation of the post-seismic inspection, in which not only S elements are considered, but also NS elements and CS . This process is essential to determine which seismic realizations, associated with each IM produce the damage patterns of the LS s, and therefore the configuration of the events that allow the estimation of the functionality recovery of the system (see Table 1).
In Sect. 2.1, the importance of evaluating the performance of NS elements and CS was exposed because, although they are not a cause of structural unsafety, they are the main cause of loss of functionality since the spaces where these elements are installed can become non-usable during the period in which they are repaired or replaced. Figure 7a shows the probability of tagging the post-seismic structural inspection for each IM considered in the study. This figure shows that the building is safe, from the structural point of view (green color line), up to intensity IM 5 (0.47g) , that is, the probability of the building receiving a green tag is Pr TAG = R s |0.47g = 0.95 , of being tagged with restricted use is Pr TAG = Y s |0.47g = 0.05 , and of being tagged as unsafe is Pr TAG = R s |0.47g = 0.0 . However, at IM 6 (0.54g) , the probability that the building is assigned a red tag is Pr TAG = R s |0.54g = 0.5 . For this same level of seismic intensity, the probability of the structure being tagged with restricted use is Pr TAG = Y s |0.54g = 0.35 , and of being tagged as safe Pr TAG = G s |0.54g = 0.15 . For seismic intensity with return period of 2500-years the probability that the building is structurally unsafe is Pr TAG = R s |0.63g = 1.0. Figure 7b presents the probability of tagging of NS safety, analogous to S inspection; however, for this case the tagging refers to the safety of a certain area of the building. For this event, it is observed that the performance of the NS elements completely change the system performance, not of the structural safety, but of the functionality itself. The red line represents the probability that at least one element of the non-structural PG s will develop severe damage given a seismic intensity Pr TAG = R NS , IM . This probability starts to become relevant at IM 3 (0.29g) , which corresponds to the performance level of collapse prevention (GCDMX 2004). This probability is Pr TAG = R NS |0.29g = 0.5 . Comparing this result with the one obtained for the same intensity but with respect to structural safety tagging, is evident that NS damage controls the loss of functionality. This is a clear indication that the design regulations are not developed to take care of the integrity of the NS elements nor to avoid injury to users, but only in minimizing the damage that may be experienced by the S elements to reduce the probability of collapse and avoid loss of life. In Fig. 7c, which corresponds to the post-seismic inspection simulation of CS , content overturning starts to occur at IM 4 (0.36g) . The probability that this type of element experiences severe damage is Pr TAG = R CS |0.36g = 0.05 . For this same IM level the probability of the building being tagged as unsafe is Pr TAG = R S |0.36g = 0.0 . Again, this is a clear indication that loss of functionality is caused at seismic intensities lower than those that cause unsafety in the structural system, or even restricted use (according to the tagging criteria proposed by the ATC 20 (1989), in which only structural safety, but not functionality, is evaluated). Figure 7d shows the probability of each LS as a function of each IM . This figure shows that for the immediate occupancy seismic demand IM 1 (0.15g) the limit state controlling the loss of functionality is LS 1 , therefore, the building is structurally safe, as expected. However, for this case, it is possible that some NS elements may develop light damage, which can be repaired in a short period of time, without rendering the building non-functional, i.e., only cosmetic repairs are required on a small number of elements.
With respect to the collapse prevention IM 3 (0.29g) it is observed that the limit states LS 1 and LS 3 have the same probability, that is, Pr LS 1 |0.29g = Pr LS 3 |0.29g = 0.5 . Under a conservative criterion, it is concluded that the building has a high probability of being yellow tagged, i.e., with restricted use. According to the shape of the probability curves Pr(LS|IM) it is appreciated that such probabilities are governed by the performance of the NS elements (see Fig. 7b) up to intensity IM 5 (Tr = 1000 years, Sa(T 1 ) = 0.47g).
The LS 4 reflects severe structural damage with a Pr LS 4 |0.54g = 0.4 . It should also be noted that for this same seismic intensity the probability that the building will experience structural collapse is Pr LS 5 |0.54g = 0.1 , which is an indication that the structural system is not safe at this level of IM since this probability is not negligible according to seismic design standards against collapse.
For the IM 7 (0.63g) , it is observed that the LS s that govern the performance of the system are LS 4 and LS 6 , with probability Pr LS 4 |0.63g = 0.5 and Pr LS 6 |0.63g = 0.5 , respectively. Taking a conservative criterion, that is, accepting that the governing limit state of functionality is LS 6 , which corresponds to structural collapse, it is concluded that the building is not resilient to this level of seismic demand. The LS s that allow the building to recover are LS 0 ...LS 4 . The LS 5 also does not have the capacity to be resilient because this state corresponds to the demolition of the facility; however, for this application example the probability that the building will have to be demolished is zero for all the seismic intensities considered in the study.
In general, it can be concluded that the building has the capacity to recover from seismic intensities less than or equal to Sa(T 1 ) = 0.47g , that is, with a recurrence period of approximately 1000-years. However, the resilience at these levels of IM can be costly, not only from the point of view of S , NS and CS repairs, but also in terms of business interruption and the equivalent economic value associated with people injured or losing their lives (Gutiérrez 2021).

Conclusions
This paper presents the formulation and evaluation of a set of six MPLS, referred as functionality limit states, LS s. These limit states are consistent with the seismic resilience evaluation scheme of individual buildings proposed by Cimellaro et al. (2010). The limit states formulation for office-type buildings corresponds to an adaptation of the formulation developed by Burton et al. (2016). The main difference consists in that the formulation presented here is based on the results of post-seismic simulation following the criteria proposed by ATC 20 (1989). In this regard, this research work explicitly considers the numerical simulation of the inspection of NS elements and CS , which results in the upper limit of the tagging probability of a building, whereas in the approach proposed by Mitrani-Reiser (2007) provides the lower bound of the tagging probability is obtained since only the inspection of S elements are considered. Therefore, it can be concluded that the methodology proposed in this paper is an extension of the approach developed by Mitrani-Reiser (2007) and adapted later by the FEMA P-58-1 (2012a) guidelines.
The formulation of the LS s presented in this work is based on a heuristic criterion, developed using empirical knowledge obtained from the observation of the damage experienced by the components that integrate a building for office use and its impact on the loss of functionality. Likewise, this formulation considers in general terms the social and technical events that occur during the recovery of the functionality of a damaged building. Therefore, this approximation should be refined and extended, to consider the damage generated by future potentially destructive earthquakes, their impact on the functionality of the buildings, and the factors that prevent the initiation of repairs. In this regard, it is desirable that post-seismic emergency and mitigation plans be created, from which systematically collected information will be extracted to develop LS s such as those presented in this paper. Also, using a strategy like the one used here, it is feasible to develop MPLS for other types of facilities, e.g., schools, hospitals or government buildings.
To illustrate the application of the methodology proposed, this paper described the probabilistic evaluation of the LS s for a seven-story RC building with URM infill-walls along its entire height and the architectural components that have been observed to be more susceptible to experience damage. The dynamic behavior of CS , i.e., bookcases and shelves, was also modeled using rigid body dynamics; however, electronic type contents, e.g., desktop computers, were not considered due to the conditions employed in the solution of the dynamic equilibrium equation. The consideration of this elements in the LS s analysis can be carried out by means of the more refined approach, such as the one developed by Jaimes and Reinoso (2013). The use of this type of elements in the analysis can substantially increase the probability that the building loses functionality in the face of seismic intensities that do not necessarily cause structural damage.
The construction system studied in the application is typical of office-type buildings located in Mexico City. The results of the evaluation suggest, as expected, that the building may lose functionality mainly due to the damage experienced by the NS elements and CS under seismic intensities comparable to those used in the limit state of collapse prevention. This is a clear indication that the design regulations are formulated to provide structures with sufficient strength and stiffness to reduce the probability of collapse, and, therefore, to reduce the loss of life; however, they are not focused on reducing the probability of loss of functionality at the same level of IM s. These results also suggest that, in addition to the direct economic losses associated with the repair of damaged elements, there may also be indirect economic losses derived from business interruptions.
Another innovative aspect of the methodology proposed in this paper is the probabilistic assessment of the damage that each element may experience individually, and not in sets of elements, as suggested by the most popular methodologies for assessing structural performance (e.g., Porter 2000;Mitrani-Reiser 2007;FEMA 2012a). Through element-byelement damage assessment it is possible to identify the most vulnerable elements, and therefore, the areas of a building that are most likely to experience either partial or total loss of functionality. At the same time, this information can be useful to propose reinforcement systems not only for S elements, but also for NS ones and CS . For example, if the probability of experiencing irreparable damage to the CS of a laboratory is not negligible compared to the probability of the S elements experiencing severe damage given a seismic hazard level, then under this condition it is possible to establish a mitigation strategy, e.g., by providing a special anchoring system at the base of the most vulnerable elements, or by establishing a strategy to replace the damaged elements in the shortest possible time.
Finally, the formulation proposed in this paper can be introduced into any of the seismic performance evaluation formulations, e.g., Porter (2000), Mitrani-Reiser (2007), etc., to predict the expected value of the DV s associated with each LS , e.g., direct economic losses due to repairs, or the recovery time of a building, and at the same time identify more clearly the origin of such losses.
Author contributions All authors contributed to the study conception and design. The development of codes was performed by JG. Outputs were reviewed by GA and SL. The first draft of the manuscript was written by JG. All authors read and approved the final manuscript.