System‐reliability‐based disaster resilience analysis for structures considering aleatory uncertainties in external loads

The concept of disaster resilience is getting more prominent in the era of climate change due to the increase in the intensities and uncertainties of disaster events. To effectively assess the holistic capacity of structural systems, a disaster resilience analysis framework has been developed from a system‐reliability‐based perspective. The framework evaluates resilience in terms of reliability, redundancy, and recoverability and provides quantitative indices of reliability and redundancy for structures with a resilience threshold. Although this framework enables the comprehensive evaluation of disaster resilience performance, practical applications of such concepts to the structures subjected to dynamic excitations with large aleatory uncertainty, such as earthquakes, remain challenging. This study develops a framework to assess the resilience performance of structures by taking into account the aleatory uncertainties in external forces. Along with the development of reliability and redundancy curves that can effectively accommodate such excitations, a new resilience threshold representation is proposed to incorporate recoverability in the decision‐making process. Moreover, we provide efficient procedures for calculating the reliability and redundancy curves to alleviate the computational complexity during the resilience analysis. Two earthquake application examples are presented targeting a nine‐story building and a cable‐stayed bridge system to demonstrate the enhanced practical applicability of the proposed framework.


INTRODUCTION
Civil infrastructures are designed to limit the extent of damages for frequent hazardous events, and further, to ensure life safety under extreme hazardous events.The growing complexity of urban communities complicates the prediction of disaster performance and challenges the associated design decisions.Furthermore, the effects of such mispredictions are often extended to the recovery stages, demanding significant time and resources to regain the pre-event condition.As a result, there is a growing emphasis on incorporating long-term outcomes in the disaster risk management framework, such as technical, environmental, economic, and social consequences.To account for the broader impact of disasters, the risk management paradigm is being shifted from "fail-safe" to "safe-to-fail" 1 -motivating the introduction of resilient infrastructure.
While disciplines such as physics, psychology, and economics, have been using different definitions for resilience, in the context of structural engineering, resilience is defined as the holistic ability or capacity of a structure to "sustain internal and external disruptions without discontinuity of the original functionality or, if discontinued, to recover fully and rapidly. 2 " Based on the concept, a series of studies were undertaken to develop the criteria and methodologies for evaluating the resilience performance of civil structural systems. 3,4A framework consisting of four attributes-robustness, redundancy, resourcefulness, and rapidity-is the most widely used resilience concept in the structural engineering field. 36][7][8] Although the framework enables quantitative assessment of the initial loss and the recovery process considering various uncertainties, Lim et al. 4 identified its three critical limitations.First, the restoration curve models are often arbitrarily chosen by the modelers, which may lead to different resilience performance evaluations.Second, although the underlying structural functionality is determined by an intertwined relationship between components-and system-level performances, most of the research efforts based on the resilience triangle focused on estimating the resilience performance of either structural components or systems only.Third, it may not be straightforward to employ the resilience triangle framework in post-disaster decisionmaking because the component/system performances are often aggregated into a single measure, such as the area of the triangle.
To address such issues, Lim et al. 4 proposed a new concept of disaster resilience from a system-reliability perspective.In their work, disaster resilience is characterized by three criteria, that is, reliability, redundancy, and recoverability.In the analysis, the resilience performance of a structure is described by inspecting possible sequences of the progressive system failure scenarios.For each initial disruption scenario, reliability (β), redundancy (π), and recoverability indices are computed and presented in a single plot as shown in Figure 1B.Note that the recoverability index is visualized by a color.Such a two-dimensional scattered plot is termed the β-π diagram in Lim et al., 4 and is used to visualize the resilience performance of the structure.Moreover, those reliability and redundancy indices are capable of not only describing the likelihood of each disruption scenario but also identifying the fatal disruption cases by introducing a resilience performance limit-in terms of per-hazard de minimis level of risk.The de minimis risk stands for a threshold value of the annual system failure probability given a hazard below which a society normally does not impose any regulations. 9This allows the β-π diagram to provide an instantaneous intuition on the likelihood of each disruption scenario (as a coordinate of β) and its impact (as a coordinate of π), as well as to define the system-level resilience limit, in terms of the two indices.
Although Lim et al. 4 effectively addressed the limitations of the resilience triangle model, the practical application of the concept to structures under realistic loading conditions, for example, earthquakes, remains challenging because of the following three reasons.First, since Lim et al. 4 proposed the resilience indices focusing on the structural systems subjected to static loads, it is not straightforward to calculate such indices under the presence of high stochastic aleatory uncertainties.In other words, new formulations need to be derived for the reliability and redundancy indices that consider the aleatoric uncertainty characteristics of external loads (i.e., the inherent randomness that cannot be explained by feature variables) and their impacts on the systems.Second, the initial disruption scenarios are defined as mutually exclusive and collectively exhaustive (MECE) events, but a procedure to obtain the resilience metrics for each MECE event was hardly addressed in Lim et al. 4 limiting the widespread adoption of the method in real-world applications.Third, it is computationally demanding to evaluate a set of β and π when a large number of structural components are considered; yet no efficient methods have been proposed.
To address these research needs within the system-reliability-based resilience assessment framework, we aim to develop new formulations and algorithms that can accommodate earthquakes or earthquake-like dynamic excitations, which we refer to as "stochastic" excitations in this context.Note that the term "stochasticity" pertains to the "aleatoric characteristic" of the hazards and is not related to certain excitation (ground motion or wind load) models.In other words, the application of the proposed method is not limited to stochastic ground motion models, as demonstrated in the examples.
After a literature review of the system-reliability-based disaster resilience framework, we newly formulate reliability and redundancy indices for structures exposed to stochastic excitations in Section 2. Motivated by the traditional performancebased engineering formulations which utilize fragility curves and total probability theorem, the new indices are built upon the concepts of reliability and redundancy curves.Furthermore, an improved resilience performance limit is proposed from the concept of factored de minimis risk.In addition to the definitions, the essential pieces of information required in resilience assessment are listed to show the framework at a glance and promote its practical applications.With an example of a three-story building structure exposed to earthquake excitations, the relationship between component failure and initial disruption scenarios represented by MECE events is thoroughly investigated.Section 3 proposes several efficient methods to reduce the computational demands in estimating the reliability and redundancy curves.To demonstrate the applicability and merits of the proposed method, the framework is applied to two earthquake engineering problems in Section 4. The paper concludes with a summary and discussion in Section 5.

SYSTEM-RELIABILITY-BASED RESILIENCE ASSESSMENT OF STRUCTURES UNDER DYNAMIC EXCITATIONS WITH HIGH ALEATORY UNCERTAINTY
The resilience performance of structural systems should be defined by joint states of statistically dependent components and their interrelationship. 10To consider such characteristics in the resilience performance assessment, Lim et al. 4 characterized disaster resilience using three criteria and developed reliability (β) and redundancy (π) indices for individual structures.The indices have limitations to a direct application to structures under earthquake ground motions or wind forces, which is characterized by high aleatory uncertainties.Thus, in this section, after reviewing the resilience indices in Lim et al., 4 a new disaster resilience assessment framework is proposed to embrace the aleatoric characteristics of external forces.It is remarked that since the recoverability should be evaluated considering various factors of socioeconomic impacts, this study focuses on proposing reliability and redundancy indices and their relationship, while the recoverability index will be discussed more conceptually.

Review of system-reliability-based resilience indices
In Lim et al., 4 the resilience criteria-reliability, redundancy, and recoverability-of a structure are proposed to be evaluated considering multiple progressive failure scenarios.In particular, given a system failure path or an "initial disruption scenario," the initial component failures triggered by the external loads represent the lack of "reliability" of the system, and the subsequent system failure induced by both the external loads and the initial component disruptions are represented by the lack of "redundancy."In other words, the reliability index represents the capability of structural elements, such as columns, joints, or cables, to avoid significant initial disruptions while the redundancy index requires to reflect the system's capability in preventing a system-level failure after some structural members' disruption.The third resilience criterion, recoverability, on the other hand, is associated with the repair time and costs of structural elements to recover the original (or desired) level of safety or functionality of the structure.Thus, the three criteria are evaluated for different "initial disruption scenarios" (and different hazard types), and the collection of those values determines the overall system-level resilience.
Let us consider -th initial disruption scenario   and -th hazard event   .The initial disruption scenarios are defined as different possible combinations of structural component failure events that occur immediately after an extreme hazard event, and the hazard is an event that induces external forces on structural systems.The reliability index for   is formulated in terms of the probability of   given   , that is, where Φ −1 (⋅) denotes the inverse cumulative distribution function (CDF) of the standard Gaussian distribution.On the other hand, the redundancy index is defined in terms of the probability of system-level failure given   and   , that is, where   denotes the system-level failure of the given structure.For the recoverability index, Lim et al. 4 employed the economic losses of the system for the given component disruption scenario.The reliability, redundancy, and recoverability indices estimated for each disruption scenario () and hazard type () are then used to plot β−π diagram (see Figure 1B), which is a two-dimensional scatter plot between  , and  , with the colors representing the recoverability.
Given that the initial disruption scenarios are mutually exclusive and collectively exhaustive (MECE), the unconditional annual failure probability of structural systems associated with the hazard   , ( , ), can be expressed by the two resilience indices as follows: where ( ,, ) stands for the annual probability of the system failure event originated from the i-th initial disruption scenario under   , and    represents the annual mean occurrence rate of   .The upper threshold of ( , ) or ( ,, ) should be decided based on social consensus.To this aim, Lim et al. 4 employed the concept of de minimis risk, 11 the highest tolerable risk level in society, for which often the order of 10 −7 ∕ is adopted for the civil structural systems. 9Using the de minimis risk level   as the threshold, the following resilience constraint was obtained as Dividing Equation (4) by    , the inequality can be written as where   ∕   stands for the per-hazard de minimis risk.By intertwining with the β−π diagram, it is possible to quantitatively assess the resilience performance of the structure, and further identify the critical components associated with the risky scenarios.Nonetheless, the original resilience threshold in Equation ( 5) reveals two limitations.First, recoverability is not explicitly considered in Equation ( 5), while it is important to incorporate recoverability into the resilience assessment to obtain a more comprehensive understanding of the system's ability to withstand and recover from disruptions.The second limitation pertains to the use of per-hazard de minimis risk as a resilience limit.While the framework allows for versatile choices of initial disruption scenario definitions, imposing the same per-hazard de minimis risk resilience threshold for different possible granularity of initial scenarios can potentially lead to over/under-estimation of the resilience performance.As an example on the extreme end, when the initial disruption scenarios are decomposed into a large number of sub-scenarios with extremely small occurrence probabilities, it is likely that all scenarios will satisfy the resilience threshold regardless of design details.This may not accurately reflect the resilience performance of the system, and to avoid this, the resilience threshold should be defined such that it is (approximately) inversely proportional to the total number of alternative failure paths considered in the analysis.

Disaster resilience assessment framework to consider aleatory uncertainties
Two contributions are made in this section to develop a system-reliability-based disaster resilience assessment framework of structures under stochastic excitations.First, a new concept of reliability and redundancy curves is proposed to deal with the variabilities of stochastic excitations.Second, a new resilience limit-state surface that accounts for the recoverability and the different granularity of the initial disruption scenarios is proposed.

Reliability and redundancy indices for stochastic excitations
The resilience indices in Equations ( 1) and ( 2) are applicable to general types of individual structures.However, it is challenging to apply the current formulation of such indices to structures subjected to stochastic excitations because of the high-dimensional nature in its randomness.To consider the variabilities in stochastic excitations in the systemreliability-based resilience framework, the conditional probability expression of the structural response is introduced, which is widely adopted in the traditional performance-based engineering formulation represented as the fragility analysis.Intensity measure(s) (IM or im) are introduced to characterize the stochastic excitations, and the failure probability of components and system (reliability and redundancy analysis, respectively) are evaluated conditional to .Using this concept, the probability of the -th failure scenario given a hazard event  can be written through the total probability theorem as follows: where (  |, ) is the scenario-level fragility given the hazard event , termed as the "reliability curve," and   (|) is the probability density function (PDF) of  given the hazard event .For example, the hazard event  for an earthquake event can be characterized by various features such as source and site conditions.On the other hand, the seismic event, which represents the site-specific realizations of ground motions for a given hazard event, is featured by intensity measures that inherently involve significant amount of aleatory uncertainty.The term (  |im, ) captures the effect of the aleatory uncertainty of the latter.For the sake of notational simplicity, we hereafter omit the subscript  in  (i.e.,   in Equations ( 1) and ( 2)) to consider only a single hazard scenario.In a similar manner, the probability of a system-level failure given the -th disruption scenario caused by the hazard event  is in which ( sys |  , im, ) represents the system-level fragility induced by the -th initial disruption scenario given the hazard event , termed as the "redundancy curve," and   (|  , ) is the PDF of  given   and .Note that, not only the redundancy curve is conditioned on   but the distribution of  is also updated after   has occurred.This is because, an unlikely failure of a strong component or simultaneous failure of multiple components may indicate that the applied intensity of the stochastic excitation was high, and such a strong excitation is likely to cause the subsequent system failure.  (|  , ) can be obtained through Bayes' theorem as follows: By substituting Equation (8) into Equation (7), ( sys |  , ) becomes which involves both the reliability and redundancy curves as well as (  |).Following Equations ( 1) and (2), generalized reliability and redundancy indices are written as follows: F I G U R E 2 Probabilistic relationship between hazard/structural random variables.
The dependency between the random variables of hazard and structural system in the reliability and redundancy analyses are graphically summarized in Figure 2A,B, respectively.Figure 2B indicates that   and  sys are dependent on the same .This implicitly assumes that the hazard event that causes the initial disruption (in the reliability analysis context) is the event that triggers the system failure (in the redundancy analysis context).In such a case, the observation of   changes the distribution of  as in Equation ( 8), and the redundancy index should consider this as in Equation (11).However, when one wants to consider a case where each of the initial disruptions and the system failure occurs due to a sequence of independent hazard realizations, unconditional  IM (im|) should be used instead of Equation ( 8) to estimate the redundancy index as instead of Equation ( 11).Under such assumptions, no arrow exists between  and   in Figure 2B, indicating that the s in Figure 2A,B are treated as independent variables.In short, Equations ( 10) and ( 11) are employed for the resilience assessment of structures under a single event, while Equations ( 10) and ( 12) assume sequential events.The focus of this paper lies on the former.

New resilience limit-state to account for the recoverability and granularity of the initial disruption scenarios
In the work of Lim et al., 4 the per-hazard de minimis risk   ∕  was employed as the resilience threshold (Equation ( 5)) in the disaster resilience assessment framework.While   ∕  incorporates the reliability and redundancy performance taking into account the annual occurrence rate of hazard, it does not explicitly consider the recoverability characteristics of each initial disruption scenario nor the number of MECE initial disruption scenarios.
To address these limitations, we propose a factored de minimis risk, denoted as  * , , by multiplying the original de minimis risk to the recoverability index and dividing it by the number of MECE events: where   is a recoverability index given the -th component disruption scenario   , which should always be positive, and   represents the number of MECE events.The recoverability index in Equation ( 13) plays a role as a scenario-specific reduction/amplification factor and its values are determined considering various socioeconomic parameters (e.g., importance of structure, social and economic factors, availability of engineers, and community capital).Meanwhile,  * , decreases as the granularity of the MECE events increases.Note that     represents a system-level resilience threshold, i.e., maximum allowable annual failure probability of a structural system, where all possible failure paths are aggregated (consider the case of   = 1 in Equation ( 13)).
Using the factored de minimis risk, Equation ( 5) can be rewritten as This enables the comprehensive assessment of the resilience performance incorporating all three criteria, and accounting for the level of granularity in the selected initial disruption scenarios.For instance, if an investigated disruption scenario

F I G U R E 4
Five critical features for the system-reliability-based resilience assessment.
does not have enough recoverability performance (i.e., low   ), the resilience threshold becomes more stringent (i.e., low  *  ∕  ) requiring higher values of reliability and redundancy indices to satisfy Equation (14).Furthermore, if a large number of MECE events is considered, the resilience threshold again becomes more stringent.Such adjustment allows the framework to be less affected by the arbitrary selection of MECE events.The relationship between the three indices with the resilience limit surface is visually illustrated in Figure 3.We refer to this three-dimensional scatter plot as a "β−π−γ diagram." Finally, it is remarked that one notable merit of the system-reliability resilience analysis framework is the clear separation of the recoverability index from the other two indices.This is attributed to the fact that each of the three resilience indices is directly conditioned on the initial disruption scenarios.This facilitates interdisciplinary communications and collaborations by allowing engineers to focus on assessing the "structural" performance only, while social scientists only aim at evaluating the recoverability performance for each initial disruption scenario without demanding onerous efforts to understand complex structural failure mechanisms.Ongoing research is being conducted to further demonstrate this, and the numerical examples in this study focus on the reliability and redundancy indices only by assuming  = 1.

Required information in the resilience assessment framework
The assessment of resilience performance for structures subjected to stochastic excitations requires five essential pieces of information: (1) hazard model, (2) initial disruption scenarios, (3) component-level limit-state, (4) component damage model and system-level limit-state, and (5) socioeconomic information.Figure 4 depicts the roles of each feature adopting the illustrational analogy in Lim et al. 4 The detailed descriptions associated with the five features are illustrated in the following paragraph with an example of a three-story building structure under seismic hazard environments to facilitate a comprehensive understanding.• Target structure A numerical model of the target structure is required to estimate the reliability and resilience curves used in Equations (10) and (11), respectively.As an example, Figure 5 shows a three-story, four-bay SAC building structure which is designed by Brandow & Johnston Associates as a benchmark structure in the SAC joint venture project.The design meets the seismic code of typical low-and medium-rise buildings located in Los Angeles, California.A numerical simulation model is constructed in OpenSees 12 utilizing a bilinear material (Steel 01) and a fiber section for both beams and columns.Each story consists of a weak column on the rightmost side of the building, and a rigid diaphragm assumption has been made.The first mode period of the structure is estimated as 1.01 s, and further details of modeling parameters including material properties are found in Refs.[13, 14].

• Hazard model
Hazard discerption is used twice in the analysis framework.The first is to get the site-specific IM distribution,   (|) used in Equations ( 10) and (11), and the second is to select/generate a site-specific events, for example, ground motions, when estimating the reliability and redundancy curves.Recall that the main goal of the hazard analysis is to produce an explicit description of the distribution of future hazardous events considering various uncertainties.As such, the relationship between IM and its annual mean rate of occurrence is the main outcome of the hazard analysis, in general.IM could be either a scalar value or a combination of various IMs depending on the problem.For example, in the earthquake engineering field, spectral acceleration at the first mode period, ( 1 ), which shows a strong correlation with typical engineering demand parameters (EDP) is widely used.][17] In the demonstration examples, we used the response spectrum estimated from a ground motion prediction equation (GMPE) by Boore and Atkinson 18 as a design spectrum.The annual mean occurrence rate of the hazard,   , is set to 10 −3 .With a series of assumptions-unspecified fault type, moment magnitude 7, 30 km of the Joyner-Boore distance, and 700 m/s of the shear-wave velocity over the top 30 m-the seismic hazard curve for Sa( 1 = 1.01) and the PDF of Sa( 1 = 1.01) are respectively determined as shown in Figure 6A,B.Note that the seismic hazard curve in Figure 6A is the multiplication of   to the complementary cumulative distribution function (CCDF) of the PDF in Figure 6B.F I G U R E 7 An example of MECE events () and non-MECE events () of the three-story building.

• Initial disruption scenarios
In order to express the system failure probability in terms of β and π following Equation (3), it is important to ensure that the initial disruption scenarios   ,  = 1, 2, ⋯,   are MECE events, in which   is the number of initial disruption scenarios.One may be tempted to select the initial disruption scenarios in terms of the failure of structural components,   ,  = 1, 2, ⋯,   , where   is the number of components of interest, but such a set, in most cases (if not always), violates the MECE combination.To illustrate the difference between   and   , let us consider the three-story building model.The failure of -th story weak column is considered as the component failure events of the building,   ,  = 1, 2, 3. Figure 7 shows that  1 ,  2 , and  3 are not mutually exclusive due to the intersection of multiple events, for example, joint failure of 1st and 2nd stories.
Using the set theory, however, the MECE initial disruption scenarios can be defined in terms of the component failure events: where   denotes the survival of member  and   is the complement set of .For example, in the three-story building,  6 =  1  2  3 (intersection notation ∩ is omitted here) represents 6-th disruption scenario of which 2nd and 3rd floors have failed ( = {2, 3}) while the first floor has survived (   = {1}).According to Equation ( 15), the number of disruption scenarios increases exponentially as the number of components increases, that is,   = 2   .However, as will be discussed in Section 3, many scenarios in fact are significantly rare (i.e., have extremely low (  |)) and can be disregard in the resilience analysis.Note that the choice of components for defining the MECE is not unique and the number of MECE failure scenarios can be flexibly chosen based on engineering judgment and the computational costs.For instance, in the building example, it is possible to further divide weak columns or beams into several sections and treat these sections as individual components.This finer granularity allows for a more detailed analysis of the resilience performance of specific structural elements.Note that, as mentioned in Section 2.2.2, the resilience threshold is adjusted based on the number of MECE events to minimize the effect of different MECE choices on the final evaluation of the structural resilience status.
• Component-level limit-state A numerical definition of component failure is essential in obtaining the reliability curve in Equation (10).Given that the disruption scenarios are defined as Equation ( 15), the limit-state functions of each   ,  = 1, 2, ⋯,   can be defined in terms of those of the component failure event   ,  = 1, 2, ⋯,   .For example, in the previous building model, the limit-state for the component failure can be established by excessive tensile stress at the weak column (rightmost column) of each story: where   is the maximum tensile stress computed at -th story's weak column, and  tr, is its maximum allowable threshold level.Using Equation ( 16), the limit-state function of   is then defined as the joint occurrence of   and C as defined in Equation ( 15).For explanation purpose, in the three-story building,  tr, = 350 Mpa is assumed.The estimated reliability curves (  |, ) and indices   will be investigated in Section 3.
• Component damage model and system-level limit-state The redundancy analysis starts by numerically modeling the degraded performance originating from the given disruption scenarios.Given the fact that the performance degradation stems from the component-level (or scenario-level) disruptions, one of the convenient options to represent the performance degradation is to replace the material properties, for example, stiffness and strength, or geometric area with those of the damaged ones.Figure 8 shows an illustrative example in which the bilinear envelope (solid line) of the material model of the damaged weak columns is replaced by a new bilinear envelope (dashed line).The stiffness of the original material property,  1 , is reduced by multiplying   , while the yield strength   is reduced to     , in which   = 0.4 and   = 0.2 are used in this example following Li. 19he degraded numerical model can describe the load redistribution initiated by the disruption scenario and represent the performance degradation of the structure.
In addition to the updated numerical model, a proper system-level limit-state needs to be defined to estimate the redundancy curve in Equation (11).The system failure event in our example is defined in terms of the global response of the system following the common practice 20,21 given by where   stands for the maximum allowable peak roof drift, and  roof , represents the peak roof drift of the structure obtained from the dynamic analysis with taking into account the initial disruption   .In our example,   = 0.07 is assumed.

• Socioeconomic information
Since recoverability stands for the ability to quickly respond to disaster impacts and rapidly recover the damaged structural components to the original state or the desired performance level, it should be determined not only as a direct repair cost but by a comprehensive analysis of the structure and social science aspects.Furthermore, the recoverability index should incorporate enough information to help engineers or stakeholders determine whether the structure needs to be retrofitted or not.Based on the desired properties, proper socioeconomic information is required to estimate recoverability.Many research efforts have been made to incorporate social science aspects in the recoverability index, 5,22,23 nevertheless no index is available to estimate the recoverability index for each initial disruption scenario.Thus, further study is currently underway to quantitatively define the recoverability index and investigate its relationship with the resilience limit-state.

ESTIMATION OF RELIABILITY AND REDUNDANCY CURVES FOR EACH DISRUPTION SCENARIO
The estimation of reliability and redundancy curves is the most computationally intensive step in the proposed resilience assessment framework.This section provides computationally efficient and practically feasible methods to estimate those curves.For the sake of notational brevity, we use the followings to represent the reliability and redundancy curves, respectively.
, (im) =  (  |im, ) Unlike conventional fragility curves often defined as non-decreasing functions, the reliability curves typically have a non-monotonic shape because the initial disruption scenario   describes a mixed state of failed and survived components instead of only the failed components.In fact, the MECE condition of the initial disruption scenarios constrains the sum of the reliability curves to always be 1, regardless of  values.For instance, when im = 0 (no external forces are applied to the structure) the probability of the "no components failure" scenario should be 1, while the other scenarios take the probability of 0. On the other hand, considering another extreme case where  → ∞, only the "all components failure" scenario or a practical equivalent will have the probability of 1, which implies that the reliability curves of the other scenarios will decay to 0. In other words, all except these two special cases has skewed bell-shape curves along with IM.This implies that the reliability curves cannot (1) be assumed to have a simple functional form, such as a lognormal CDF, and (2) be calibrated independently for each   because of the constraint that all the reliability curves should sum up to 1.
After a high-level overview of the existing fragility analysis following Yi et al., 24 three methods are proposed to estimate the reliability curves to consider the aforementioned characteristics, followed by a discussion on the redundancy analysis.To provide a comprehensive overview, we present Table 1 to summarize the computational aspects of the proposed three methods for estimating reliability curves.The methods are illustrated using the three-story building example.

High-level overview of fragility analysis methods
The fragility curve is defined as the conditional failure probability given IM of a hazard: where  is a binary damage state index that takes 1 if the component or system is damaged and 0 otherwise.In practice,  is represented as the demand being greater than capacity, i.e.,

DS = 𝕀 {𝛿
where (⋅) is the indicator function,   represents the response threshold (capacity), and  stands for the response of the component/system due to hazard loads (demand), which is often referred to as an EDP.Among various fragility analysis methods, incremental dynamic analysis (IDA), cloud analysis, maximum likelihood estimation of the binary classification model, and extended fragility analysis are summarized in the subsequent paragraphs.IDA gained popularity in light of intuitive analysis steps and the easiness of calibrating the parameters of a fragility function. 25IDA creates multiple splines on {, } space, each obtained by running multiple dynamic structural analyses for varying scales of ground motion time histories.The uncertainty in the capacity of the system is represented in terms of IM values at which the splines cross the response threshold   .The fragility curve of typical IDA procedure takes the form of lognormal CDF The parameters  and  are respectively log-mean and log-standard deviation of the collected IM capacity samples during the IDA analysis.
The cloud analysis predicts the mean response by introducing the power law assumption between IM and  26

𝐸 [ln 𝑑] = 𝑎 + 𝑏 ln im + 𝜀
where  follows a normal distribution, whose mean is 0 and the standard deviation is , that is, (0,  2 ).By minimizing the squared error of the linear regression under homoscedasticity assumption, {, , } are estimated.Using the estimated parameters, the following fragility curve is obtained.
Next, a method by Shinozuka et al. 27 treats the fragility analysis as a binary classification task.Using the lognormal CDF in Equation ( 22) as the form of the fragility function, parameters  and  are obtained by maximizing the following Bernoulli likelihood function where  sim represents the number of samples obtained from dynamic analyses, and the superscript () stands for the -th analysis data.Once  and  are calibrated, the fragility can be described using Equation (22).Lastly, as an alternative to the lognormal CDF, a log-logistic distribution is used as a fragility function in the extended fragility analysis method 28   (im) = 1 1 + exp − (  +  1 ln im) (26)   where   and  1 are coefficients calculated again by maximizing Equation (25).A merit of introducing the Bernoulli likelihood function is that the parameters of the fragility function are estimated in terms of  instead of the actual response quantity .This is useful particularly when the system failure is defined as a combination of multiple response quantities, for example, F I G U R E 9 MECE events () and their supersets (red) of the three-story building example.

Method 1: Subtraction method
To address the challenges discussed in the beginning of Section 3, a new method termed the "subtraction method" is introduced.This method allows us to apply conventional fragility methods for the reliability tasks by drawing a relationship between the probability of   ,  = 1, 2, … ,   and those of joint   ,  = 1, 2, … ,   in Equation (15).For an initial disruption scenario   =   S , the reliability curve can be reformulated using the subtraction method as in which and where    is the number of elements in   .The subtraction method converts the task of the reliability curve estimation (lefthand side term of Equation ( 28)) to the fragility analysis of joint component failures (righthand side terms of Equation ( 28)).Thereby, no care needs to be made to consider the constraints discussed previously.Since these joint component failures do not condition on survival events,   in Equation ( 29), the conventional fragility analysis methods, for example, under lognormal assumption, can be adopted in the reliability analysis.For example, in the three-story building example, we can represent the reliability curve of  1 =  1 2 3 using Equation (28) as follows: In the same manner as above, the followings are the expressions of other MECE events using the subtraction method A graphical illustration of the subtraction method used in the three-story building example is shown in Figure 9.The following is the summary of the procedure when applying the subtraction method to the three-story budling.},  = 1, 2, … ,   , where   = 50 is the total number of model evaluations,   = 2   = 8, and   is the binary occurrence index that takes 1 if    has occurred, and 0 otherwise.(ii) For  = 1, 2, … ,   , using { (𝑛) ,  ()  }, calibrate the fragility function parameters in Equation ( 22) by maximizing the likelihood defined in Equation ( 25) to obtain the fragility curves (   |).2. Calculate the reliability curves of   =    S  using Equation (28), that is,  , () = (   S  |).
Figure 10 describes the estimated reliability curves using the above procedure.A total of 50 ground motions are used in the dynamic analysis which are spectrum-matched or spectrum-compatible to a design spectrum presented in Section 2.3 (See Figure 14B).The ground motion time histories are selected from the NGA-West database. 29It is remarked that one may get a negative  , () using the subtraction method.To the authors' observation, the effect of negativity was not significant as it was apparent only at the improbable range of hazard magnitude, for example, beyond 4g in the numerical example, where g is the gravitational acceleration.Thus, we decided to enforce the negative values to zero in the calculation.However, it is possible to strictly prevent the negative probability density by applying a constraint such that a common dispersion parameter,  in Equation (22), is assigned to all (   |), that is,  1 =  2 = ⋯ =    = .Note that similar tricks are often introduced in the traditional fragility analysis to prevent crossings between multiple damage states, for example, as used in Shinozuka et al. 30 Furthermore, an approximation approach is proposed to facilitate the efficient estimation of the joint component fragility function (  |) (and all the righthand side terms in Equation ( 28)) using the fragility functions of the single components (  |),  ∈ , and their correlation information.By substituting the component failure definition in Equation ( 21) into Equation (30) after applying the natural logarithm, the joint component failure is written as a series system reliability problem Assuming that log(  ) are joint normal distribution, the below can be derived 31,32  (  |im) = Φ  (− (im) ;  (im)) (34)   where Φ  (⋅; ()) is the -dimensional multivariate standard Gaussian CDF with correlation matrix of (), (im) is a vector of reliability indices whose element is defined as   = Φ −1 ((  |im)) which is different from the reliability index in Section 2, and () is constructed by the inner product of the normalized negative gradient vector of each components' limit-state function at the design point.Using Equation (33) and (  |), (  |) can be approximated with a small computational cost, which facilitates the efficient computation of the subtraction method.However, one should be cautious about the fact that since it relies on the normality assumption, this error can be accumulated in the calculation of Equation ( 28).Therefore, for example, one may want to perform a goodness-of-fit test to measure how well log(  ) follows the joint normal distribution.This effect of error accumulation is alleviated when the scenario screening, which will be discussed in Section 3.2.3, is introduced.

Method 2: Multinomial logistic regression
Alternatively, the task of estimating reliability curves can be formulated into a multi-class classification problem of which the input is IM and the categorical outcomes are   .Then the membership probability (i.e., the probability that a given sample belongs to a particular category) is in nature equivalent to the definition of reliability curve.In particular, the membership probability of the logistic regression model takes the form of 33  , (im) = exp ( oi +   ln im) for  = 1, … ,   − 1, and Therefore, the formulation naturally satisfies ∑   =1  , (im) = 1.The coefficients   and  1 are calibrated by maximizing the following likelihood function where  (𝑛) is the -th sample of the categorical outcome as the index of the disruption scenario.Once {  ,  1 } for  = 1, … ,   − 1 are obtained by maximizing the likelihood function of Equation ( 37), the reliability curve for  =   can be automatically determined from Equation (36).A merit of this procedure is that the reliability curves for all disruption scenarios are obtained simultaneously with attaining the condition ∑   =1  , (im) = 1.The following is the application of the multinomial logistic regression to estimate the reliability curves of the three-story building.Procedure 1. Perform structural dynamic analysis using a set of ground motions to obtain a cloud of data samples { (𝑛) , z () },  = 1, 2, ⋯,  sim , where  sim = 50.2. Find {  ,   }, where  = 1, 2, ⋯, 7, by maximizing the likelihood function in Equation (37).3. Following the definition, the reliability curves are equivalent to the calibrated logistic regression model in Equations ( 35) and (36).
Figure 11 shows the results of the reliability curve estimated using the multinomial logistic regression.As shown in Figure 11A, the summation of  , () for all MECE events is always 1 for every IM. Figure 11B plots the reliability curves of each MECE event, which shows a good agreement with the results using the subtraction method in Figure 10B.Note that  2 ,  3 , and  6 are not observed in the dataset {z () }, thus assumed to have zero probability.The underlying assumption is that the reliability indices of those scenarios are smaller than those observed.Therefore, if deemed needed, one needs to revisit this assumption, and run more simulations to make sure all the critical cases are taken into account in the resilience assessment.The additional simulations are not needed if at least one scenario in the β−π diagram satisfies the screening condition that will be discussed in Section 3.2.3.

F I G U R E 1 1
Reliability curves obtained by multinomial logistic regression.

3.2.3
Method 3: Screening of force majeure scenarios One critical challenge in the resilience assessment is that the number of initial disruption scenarios increases exponentially as that of the structural components increases.Since the previously introduced methods should check whether the reliability index satisfies the resilience limit-state for every MECE event, it is still limited to applying the reliability-based resilience assessment framework to a structure having a large number of structural components such as a cable-stayed bridge.However, since there are lots of force majeure MECE events, which have extremely small occurrence probability, the screening method can exclude those from the resilience analysis.
In particular, for a scenario of   ⊂   , if one can show that the below is satisfied no more reliability analysis is required for   because   will always satisfy the resilience threshold in Equation (5) ( (  |) = Φ(−  ) < (  |) always holds).For instance, in Figure 9, if the failure probability of C 1 satisfies the resilience performance threshold of Equation (38), we can infer that  1 ,  4 ,  5 , and  7 (or  1 2 3,  12 3, 1 23 , and  123 , respectively) meet the disaster resilience goal without further analysis.Similarly, if C 23 satisfies the resilience limit-state, the analysis of F 7 and  6 ( 123 and ̄1 23 ) can be disregarded in the resilience analysis.Using this property, it is possible to drastically reduce the number of MECE events considered in the resilience assessment framework.Moreover, the screening method enables not only to efficiently assess the resilience performance of the existing structures but also to quickly check whether a candidate structure is within the resilience-safe domain with the β−π−γ diagram during the design phase.

Estimation of the redundancy curves
The redundancy curve in Equation ( 19) can be straightforwardly obtained from a fragility analysis described in Section 3.1 after considering the component damage scenarios in the numerical model.In the analysis, the same stochastic excitation set used in the reliability analysis is employed.As already discussed in Lim et al., 4 the force majeure scenarios with sufficiently low occurrence probability can be omitted in the redundancy analysis.For example, if is satisfied, Equation ( 14) is already satisfied for the scenario   regardless of the redundancy index   .Furthermore, by extending the discussion in Section 3.2.3, it can be shown that if F I G U R E 1 2 Results of the redundancy analysis.
is satisfied, any reliability and redundancy analyses associated with all   ⊂   can be omitted.A procedure to estimate the redundancy curves for the three-story building example is provided in the following.

Procedure
Repeat below for  = 1, 2, ⋯, 8: 1.If Equation (39) or Equation ( 40) is satisfied, label   as safe and exclude   from further analysis.In other words, neglect Steps 2 and 3 and move on to  + 1, else move on to Step 2. 2. Update the structural model in accordance with the damage scenario   .3. Perform fragility analysis with a predefined system-level limit-state using the damaged structure to obtain  , () Figure 12A provides an example of the IDA results to evaluate the system performance given  1 =  1 2 3 (failure of the first story only), while Figure 12B illustrates the estimated redundancy curves.Note that among the failure scenarios,  1 ,  4 ,  5 ,  7 , and  8 are inspected in accordance with the discussion in Section 3.2.2.By comparing the curves of  8 and  7 , one can notice that, in this example, only a minor performance decay is observed even when many components failed, indicating that the component damages do not in fact have a critical influence on the global structural response.This is attributed to the assumption of the damage model we introduced.A summary of the reliability and redundancy analyses is presented in Table 2 with the traditional fragility analysis in performance-based engineering.

NUMERICAL INVESTIGATIONS
The proposed seismic resilience assessment framework is demonstrated using a mid-rise building and a bridge model.For the reliability analysis, the multinomial logistic regression method (Section 3.2.2) and screening approach (Section 3.2.3)are respectively applied in the examples.

Target structure and hazard
The first example considers a benchmark nine-story building model shown in Figure 13 adopted from the SAC Phase II Steel project report.This building is designed to meet the design standard of the mid-rise building located in Los Angeles, California, region.The model has a basement level as shown in Figure 13, and the horizontal displacement at the ground level is restrained to be zero.The building is modeled using OpenSees 12 using bilinear material model (Steel 01) for both TA B L E 2 Summary of reliability and redundancy curves in comparison with traditional fragility curves.

Category
Performance-based Engineering Resilience-based Engineering X X X X X X X X X X X X

Initial disruption scenarios and limit-states
The component failure events are defined as an occurrence of an excessive drift ratio at each story: where   is the peak inter-story drift ratio at the -th story and   = 0.02 is its maximum allowable threshold.Note that the response at the basement level is indexed with  = 0.The ten components lead to 1024 (2 10 ) initial disruption scenarios.
The system-level limit-state is represented in terms of the maximum roof drift ratio as in Equation ( 17) with   = 0.07.that the probability of "no component failure" case decreases as the IM increases.In the range of high IM values, the event of C1-9 (failure of all components except for the basement level) dominates the response followed by C1-8 (failure of all components except for the basement and the top story).Figure 15C,D summarizes the results in terms of the number of failed components.Among different cases, the "no component failure" case dominates under relatively small IM values, but the increase has been observed for the probability of 8~10 components failure cases as IM increases.The redundancy analysis is performed for the 84 scenarios and the results are presented in Figure 16.It is shown in Figure 16A that the most critical scenarios in terms of redundancy curves are the "all components failure" case and C0-8.On the other hand, the "no component failure" case and several scenarios with a few members failure such as C3 and C8 appear to have relatively high redundancy, which agrees well with the general intuition-a larger number of remaining load-resisting members leads to a higher redundancy.Meanwhile, the updated distribution of IM (as defined in Equation ( 8)) used for redundancy analysis is presented in Figure 16B and the scenarios with the five largest and five smallest mean IM of the updated distribution are listed in Table 3, where [⋅] represents the mathematical expectation.It can be seen that different scenarios lead to various ranges of updated IM.
The β−π diagram is shown in Figure 17A.The color represents the number of failed components, which can be used as a recoverability indicator.From the decaying trend of the scatter plot, one can draw insight into the complementary nature of the reliability and redundancy across the scenarios.The event C1,2,4-9, for example, has high reliability (i.e., it is rare to have the combination of components 1,2,4-9 failed) and low redundancy (i.e., the failure of the components 1,2,4-9 is associated with high IM values as shown in Table 3, which is likely to trigger the progressive system failure).On the contrary, C3 has a low reliability but a high redundancy level.
To investigate the effect of IM updating in the redundancy assessment, the β−π diagram without updating the IM (i.e., using Equation ( 12)) is presented in Figure 17B.While the reliability indices remain the same as Figure 17A, the redundancy characteristics are significantly different from those with updating.In this case, π directly follows the trend observed in the redundancy curves in Figure 16A.Meanwhile, the reason that some single-member failures have higher reliability than multiple-member failures can be explained by the high correlation between the member failure events.In other words, it is likely to have multiple member failures than only a single member failure in this example.

Target structure
A cable-stayed bridge is introduced to attest to the applicability and effectiveness of the proposed framework to a more complex civil structure.A nonlinear three-dimensional finite element model is constructed using OpenSees 12 as shown in Figure 18.The bridge consists of two pylons, girder, and 128 cable elements, and its total length is 1,069 m.Note that no soil-structure interaction is considered in this study.
A bilinear tension-only material with a yield stress of 1,770 MPa and 1% of the post-yield stiffness ratio is introduced to model the cable elements.The sagging of each cable element is considered from Ernst 34 with Young's modulus of the cable strand of 195 GPa.The initial tension force of the cable element is converted to the initial strain in the truss model.On the other hand, linear elastic frame elements are employed to model the girder and pylons.In addition, linear springs are used to model the bridge bearings for simplicity.Because no nonlinear element except the cables is introduced in the numerical model, a limitation exists in describing the local collapse of structural elements and seismic behaviors after the yield point.][37] Dynamic characteristics of the numerical model are investigated by performing the eigenvalue analysis.The estimated modal periods are tabulated in Table 4, while Figure 19 illustrates the corresponding mode shapes.Note that the eigenvalue analysis is performed after applying the dead load and pretension force of the cables.

Hazard analysis
In the same manner as the three-and nine-story building examples, we assume a point source earthquake event with moment magnitude of  = 7.The distance between the epicenter and the cable-stayed bridge and the shear wave velocity are set as 20 km and 750 m/s, respectively.Under these assumptions, the PDF of the IM given the hazard is obtained by using the GMPE by Boore and Atkinson. 182.3Initial disruption scenarios and limit-states Among various system damage scenarios, this study considers those induced by initial cable disruptions, as the cable elements are the main medium of the load transfer from the superstructure to the pylon.Note that while other structural elements or combinations of various structural elements could be selected to define the initial disruption scenarios, this study only employs the cable elements for the purpose of explaining the proposed framework.The limit-state of the cable elements used to derive reliability curves,  , (), is defined as the seismic demand exceeding 50% of the yield stress (i.e., 885 MPa).Since there are 128 cable elements in the model, the total number of MECE initial disruption scenarios is 2 128 , including the "no element failure" scenario.Since the cable-stayed bridge in the system-level has multiple failure modes, the system failure limit-state function, in this research, is defined as the presence of at least one failure mode.Thus, the redundancy analysis is considered as a series system reliability problem following the approach summarized by Der Kiureghian. 31Based on the literature survey, [38][39][40][41] four critical system failure scenarios are identified, and the corresponding limit-states are summarized in Table 5.When computing the redundancy curves,  , (), dynamic analyses are conducted after removing the failed cable elements of the bridge.

Resilience performance
As discussed earlier, a huge number of structural components in the cable-stayed bridge may result in numerous initial disruption scenarios.However, it may not be necessary to evaluate all the reliability and redundancy indices for each scenario, if many scenarios conservatively satisfy the resilience threshold as discussed in Sections 3.2.3 and 3.3.In this example, we illustrate a case where it is sufficient to assess the resilience performance for individual component failure events instead of all initial disruption scenarios.In other words, as described in Section 3.2.3, a set of β and π is first estimated for   (128 cases) and is shown that we do not need to estimate them for all   (2 128 cases) because they are guaranteed to be safe.However, note that if some scenarios do not secure the resilience criteria, further steps are needed to estimate β and π for initial disruption scenarios   .
To test the applicability of the proposed framework to general stochastic excitations, spectrum-compatible, bidirectional artificial ground motions are generated by following an algorithm and parameter sets provided in Kim  et al. (2021). 35Although the algorithm enables to simulate multi-variate ground motions, in this research, the same set of orthogonal ground motion time histories is used for each support.By assuming the mean of a response spectrum obtained using the assumptions in Section 4.2.2 as the target spectrum, 30 sets of ground motion time histories are generated.When generating the spectrum-compatible orthogonal ground motion time histories, we scale the target spectrum to capture the seismic behavior of the structural system for a broad range of ground motion intensities.Thirty different scale factors are introduced to make peak ground acceleration (PGA) of the target spectrum ranging from 0.16-1.0g.Using the cloud analysis in Section 3.1, both the reliability and redundancy curves are estimated for the component failure scenarios.The scalar IM is established as the geometric mean of PGA of the two orthogonal ground motions.
Figure 20 shows the β−π diagram of 32 component failure cases with the resilience limit-state surface corresponding to  dm ∕(    ) = 10 −4 .Note that because of the bidirectional symmetry of the bridge system, only a quarter of the elements are considered.In the figure, we disregard the component failure scenario having a reliability index greater than 12, which is considered as force majeure.Because all the β values already exceed the resilience criterion, no redundancy analysis is required.However, for visualization purposes, the redundancy is evaluated where the conditioning scenario is "every component survives but member ."As shown in the figure, even though we conservatively assess the reliability performance of the bridge, all cases of the β−π are located outside the resilience limit-state surface (i.e., satisfy the socially-accepted criteria).The estimated reliability and redundancy values are well-matched with the characteristics of the cable bridge, in that scatter points indicated by blue solid and red dashed boxes in Figure 20 are respectively the failure scenario of the first and second outermost cables in which the highest tension forces are measured during the seismic excitations.Furthermore, a typical inverse proportional relationship between reliability and redundancy, where higher reliability corresponds to lower redundancy, is observed in the numerical example.

CONCLUSIONS
This study newly established a resilience assessment framework for structures subjected to external forces having high aleatory uncertainties from a system-reliability-based perspective.The framework leveraged the concept of reliability and redundancy curves to accommodate the aleatoric variabilities in excitation.Using these curves, a pair of reliability and redundancy indices were estimated for each mutually exclusive and collectively exhaustive (MECE) initial disruption scenario, which was then evaluated by the factored de minimis level of risk that considers the recoverability of each failure scenario and the number of MECE events.To facilitate a comprehensive understanding of the proposed concept, we presented and summarized five core elements needed to successfully assess the resilience performance of structures subjected to stochastic excitations.Furthermore, to increase the applicability of the proposed framework, efficient and effective computational procedures for calculating the reliability and redundancy curves were provided.
After describing the developed procedure using a three-story building structure, two more sophisticated structural systems were studied with an example of earthquake excitations to demonstrate the ideas and potential benefits of the proposed framework.The numerical investigation confirmed that the proposed framework can systemically assess the disaster resilience performance of structures subjected to stochastic excitations by efficiently dealing with MECE initial failure disruption scenarios.Although the numerical investigations focused on evaluating the seismic performance, the concept can be applied to other types of hazards such as winds, waves, or vibrations from vehicles.Currently, two further studies are underway to extend the framework and enhance the applicability of the assessment procedure.First, a mathematical expression is being developed to define and quantify the recoverability index in Equation (14).Second, the framework is being extended to consider aging infrastructure under varying environmental conditions associated with climate change.Furthermore, it is desirable to investigate the results of resilience analysis for different scales/granularities of initial disruption scenarios.A sequential decomposition approach can be employed to systematically explore the resilience of the system and provide insights into the hierarchical nature of different components to the overall system resilience.Another interesting research topic would be to further extend the proposed methods to accommodate uncertain structural properties.
The proposed resilience assessment methodology and computational procedure are expected to enhance the applicability of the framework to more complex civil engineering systems and realistic hazards, further bridging the gap between advanced reliability theories and current performance-based engineering practices.

A C K N O W L E D G E M E N T S
This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (RS-2023-00242859).

D ATA AVA I L A B I L I T Y S TAT E M E N T
The data that support the findings of this study are available from the corresponding author upon reasonable request.

F I G U R E 1
Illustrative comparison of resilience triangle (left) with four resilience attributes (red), and system-reliability-based resilience diagram (right) with three resilience criteria (black).

F I G U R E 5
Configuration of the three-story steel building.F I G U R E 6 Hazard curve and the corresponding hazard frequency function.

F I G U R E 8
Properties of damaged components.

F I G U R E 1 0
Reliability curves obained through the subtraction method.Procedure 1. Estimate reliability curves of each joint component failure events    ,   ⊂ {1, 2, 3} using fragility analysis methods described in Section 3.1.The Bernoulli model-based fragility method is used in this example.(i)Collect/generate the multiple ground motion time histories for a specific region of interest, and run structural dynamic analysis to collect a cloud of data samples {()  , Hazard typesDynamic excitations with high aleatory uncertainty (i.e., stochastic excitations)a  in the conditioning term is omitted following the convention in fragility analysisF I G U R E1 3 Nine-story steel building model.beams and columns, and the Rayleigh damping with damping ratio of 0.03 is introduced.The first mode period of the structure is  1 = 2.27 s.The hazard description in Section 2.3 is employed, which is characterized by the PDF of spectral acceleration Sa(T 1 = 2.27) shown in Figure 14A.Moreover, a set of spectrum-compatible 50 ground motion time histories is shown in Figure 14B.

F I G U R E 1 4 F I G U R E 1 5
Hazard description.Reliability curves of nine-story building ("C" represents the failure of components in  and survival of all the other components).

F I G U R E 1 6
motions with different scaling factors, a total of 485 simulations are performed, and 84 among possible 1,024 scenarios are observed.The framework assumes that only 84 scenarios are plausible, while other scenarios are considered to have an occurrence probability (near) zero.The 485 data points are used to estimate the logistic regression parameters in Equation(35), and the results are shown in Figure15.Figure15A,B is equivalent figures that show the probability of the system lying in a certain initial disruption scenario given IM, where the significant scenarios are labeled as "C" meaning that members in  fail while all the other members are safe (i.e., equivalent to   S in Equation (29)).As expected, the probabilities of all MECE events always sum up to one because of the MECE condition.The figure shows Redundancy curves of the nine-story building ("C" represents the failure of components in  and survival of all the other components).

F
I G U R E 1 7 β−π diagram of the nine-story building structure.F I G U R E 1 8 Configuration of the example structural system.TA B L E 4 Modal periods of the cable-stayed bridge.

F I G U R E 1 9
Modal shapes of the long-span bridge.TA B L E 5 System-level limit-states of the cable-stayed bridge.peak displacement of pylon to the height of pylon >1% Girder Ratio of the peak transverse displacement of girder to the length of girder >1% Cable Cable tension force >885 Mpa

F
I G U R E 2 0 β−π diagram of the cable-stayed bridge.
Summary of proposed methods to compute reliability curves.
TA B L E 1 Mean of IM conditioned on each disruption scenario.
TA B L E 3