Resilience evaluation of elastic-plastic high-rise buildings under resonant long-duration ground motion

A simple evaluation method of building resilience is proposed, and the resilience of elastic-plastic high-rise buildings is evaluated under resonant long-duration ground motions. The concept of availability is incorporated into the evaluation model of the building resilience, which was proposed in the previous paper. Not only in-building components but also external factors such as lifelines and redundancy of facility systems are considered in the evaluation model. Based on the evaluation model, the following probabilistic indices are formulated: (1) the expected value of the total recovery time, (2) the expected values of the availability of the structural frame, facilities, and non-structural components just after an earthquake event, and (3) the maximum likelihood recovery curves (MLRCs) of the structural frame, facilities, and non-structural components. Numerical examples are presented for evaluating the resilience of elastic-plastic high-rise building models under a pseudo-multi impulse (PMI), which substitutes for resonant long-duration ground motions. Finally, the inﬂuences of the passive viscous dampers, the redundancy of facility systems, and the uncertainty of manpower to repair the components on the building resilience are clariﬁed through the results of numerical examples.


Introduction
Rapid recovery and business continuity management after earthquake events are important in order to minimize social and economic losses.2][3] Recently, many researchers have focused on the resilience of buildings, lifelines, and communities.Comerio 4 proposed the concept of rational and irrational components of downtime.Mitrani-Reiser 5 investigated the performance-based design method and treated downtime as one of the indirect indices of the performance.The influence of the irrational components on downtime was also investigated.Lantada et al. 6 used a capacity spectrum-based method to conduct the seismic risk analysis in urban areas.Cimellaro et al. 7 evaluated the post-earthquake resilience of hospital systems.Hutt et al. 8 demonstrated that the downtime and the economic loss of existing tall buildings can be largely reduced by the seismic improvements of structural and non-structural systems and mitigation.Cimellaro et al. 9 proposed a comprehensive framework PEOPLES for measuring community resilience.Burton et al. 10 and Lin and Wang 11,12 treated recovery processes of communities at the level of an individual building.Cai et al. 13 presented an availabilitybased engineering resilience metric.De Iuliis et al. 14 proposed a fuzzy logic-based method for estimating the downtime of buildings.Cremen et al. 15 developed an analytical framework for modeling a post-earthquake business recovery time.Nakamura et al. 16 investigated influences of the damage correlation between in-building components on the recovery time of buildings.Yamagishi and Senna 17 proposed an event tree analysis-based prediction method of the restoration time of productive facilities.Kinugasa and Mukai 18 presented an index for evaluating the severity of the damage from the viewpoint of post-seismic functional recovery of buildings.Okano et al. 19 extended a critical path method to assess the total repair time of damaged buildings.
1][22][23][24][25][26][27][28] Especially in Japan, long-duration ground motions caused by big earthquakes may bring great losses to large cities on soft sedimentary basin.Therefore, it is necessary to clarify what is effective for improving the resilience of high-rise buildings.
In this paper, a simple evaluation method of building resilience is proposed, and the resilience of elastic-plastic highrise buildings is evaluated under resonant long-duration ground motions.In Section 2, the concept of availability is incorporated into the evaluation model of the building resilience, which was proposed by Akehashi and Takewaki. 29In Section 3, the influences of external factors such as lifelines and redundancy of facility systems on the building resilience are investigated.In Section 4, the probabilistic indices are formulated based on the evaluation model.In Section 5, numerical examples are demonstrated for evaluating the resilience of elastic-plastic high-rise building models under a pseudo-multi impulse (PMI), 28 which substitutes for resonant long-duration ground motions.Through the numerical examples, the influences of the passive viscous dampers, the redundancy of facility systems, and the uncertainty of manpower to repair the components on the building resilience are investigated.

Evaluation Model of Building Resilience
Akehashi and Takewaki 29 proposed a simple evaluation model of the building resilience.This model mainly treats the recovery time.In this section, the concept of availability is incorporated into the model.In Section 2.1, the basic concept of the evaluation method is briefly explained.In Section 2.2, the method for assessing the recovery times of facilities, nonstructural components, and structural frames is explained.In Sections 2.3 and 2.4, the concept of the availability of facilities, non-structural components, and structural frames are introduced, and the method for assessing the availability is explained.

Basic concept of evaluation model
Let T S total , T F total denote the total recovery time of the structural frame and that of the facilities and non-structural components.Assuming that the facilities and non-structural components start to be repaired after the structural frame is totally repaired, the total recovery time T total of a building is expressed as follows. 29 It is noted that some factors (eg, damage of transportation systems and post-earthquake inspection) may delay the start of the repair work.Such kind of delay is easily incorporated into the evaluation of T total , although it is not treated in this paper.
In the evaluation model proposed by Akehashi and Takewaki, 29 each facility (or non-structural component) in a building is regarded as a component contributing to the functionality of the building.When a function of a component links up with that of another component, they are classified into the same functional system.Let T i,j , T i denote the recovery time of component j which belongs to a functional system i and the recovery time of system i.Then, a relation between T i and T i,j can be obtained as where M i denotes the number of components in system i.Equation ( 2) means that T i depends on the manpower to repair the components.When the manpower is fully available and all the components in system i simultaneously begin to be repaired, T i is equal to max j T i,j È É . When the manpower is limited and the components are sequentially repaired, T i is equal to ∑ M i j¼1 T i,j .By expanding Equation (2) into the relation between T i,j and T F total , the following equation can be obtained.
where n is the number of functional systems in the building.When all the components in the building simultaneously begin to be repaired, T F total is equal to max i, j T i,j È É .When the components are sequentially repaired, T F total is equal to ∑ n i¼1 ∑ M i j¼1 T i,j .The former recovery scenario corresponds to Scenario FA (full ability to recover), and the latter recovery scenario corresponds to Scenario LA (limited ability to recover).As shown in Figure 1A, both the scenarios give the upper and lower limits of the recovery time of the probable recovery scenarios. 29Figure 1A means that the increase of the earthquake level (or the increase of the number of damaged components) enlarges the difference in the recovery time between Scenario FA and Scenario LA. Figure 1B illustrates an example of repairing three components by Scenario FA and Scenario LA.The arrows in the figure correspond to the repair time of each component.In Scenario LA, the arrows line up in series.In Scenario FA, the arrows line up in parallel.
It is noted that the evaluation of the recovery time by Scenario LA and Scenario FA aims for a simple expression of the uncertainty with regard to the manpower to repair the components.Although stochastic models of the manpower may work well, it requires much computational load for conducting Monte Carlo methods.In addition, the stochastic modeling itself is not easy.It is also pointed out that the recovery time can be calculated in consideration of the floor space. 30owever, such consideration does not always provide an accurate evaluation of the recovery time because the manpower is highly uncertain and difficult to be predicted.For example, when an earthquake causes severe damage in wide area, the manpower may be even smaller than expected.
The total recovery time T S total of a structural frame can be calculated in the same manner as T F total .

Evaluation of availability of facilities and non-structural components
In this paper, the availability of facilities and non-structural components indicates whether they are workable or not.The total availability r F of facilities and non-structural components and the availability r F i of system i can be expressed as Equation ( 4) means that r F is expressed as the sum of r F i and the function of system i fails when any one of the components in system i is damaged.This means that the systems line up in parallel and the components in system i line up in series.When all the components in the building are undamaged, the value of r F becomes one.r F i takes zero or 1=n ð Þ depending on the damage state of each component in system i, although there is still room for consideration of the weighting coefficients.In other words, the validation of the value 1=n ð Þ can be further considered.It is noted that the influences of the external factors such as lifelines will be discussed in Section 3.
Figure 2A,B illustrates examples of recovery of electrical equipment systems of a 4-story building.The function of the power supply at each floor is regarded as one functional system.Then, four functional systems exist in total.In these cases, the distribution panels at the 2nd and 3rd floors and the switchgear at the 1st floor fail.The failure of the switchgear leads to the failure of all of the systems because the switchgear links up with all of the systems.It is assumed that the damage of the switchgear is smaller than those of the distribution panels, and the switchgear recoveries faster.In Scenario FA (Figure 2A), the functions at the 1st and 3rd floors become available after the repair of the switchgear.Then, the functions at the 2nd and 4th floors simultaneously become available after the repair of the distribution panels because both distribution panels simultaneously begin to be repaired.In Scenario LA (Figure 2B), the components are sequentially repaired.Thus, the function at the 4th floor becomes available after the function at the 2nd floor becomes available.

Evaluation of availability of structural frame
In this paper, the availability of a structural frame indicates whether it is able to be occupied or not.The availability r S of the structural frame can be expressed as Examples of recovery process, (A) recovery of power availability by Scenario FA, (B) recovery of power availability by Scenario LA r S becomes zero or one depending on the damage state.After the repair, the value of r S changes from zero to one.Figure 3 illustrates an example of recovery of a structural frame and electrical equipment systems of a 4-story building by Scenario FA.In this case, the structural frame, the distribution panels at the 2nd and 3rd floors and the switchgear at the 1st floor fail.The red dashed line and the blue solid line in the figure correspond to the recovery curve of the structural frame and that of the electrical equipment system, respectively.The electrical equipment system begins to be repaired after the repair of the structural frame.

Influences of Lifelines and Redundancy of Facility System on Building Resilience
In this section, the influences of external factors such as lifelines and redundancy of facility systems on the building resilience are investigated.
Consider a 24-story building model and the electrical equipment system shown in Figure 4.When the power supply by the lifeline is interrupted, the power is not available in the building regardless of the damage states of the in-building facility components.However, when the emergency power system is available, the power will become available in the building for a certain period regardless of the state of the lifeline.From this perspective, the emergency power system provides redundancy for the facility system.By considering the availability r ll , r eps of the lifeline and the emergency power system, r F in Equation ( 4) can be rewritten in the following manner.
Equation ( 6) means that the external factors such as lifelines may decrease the building resilience and the redundancy of facility systems counteracts the influence of the external factors.It is noted that components which provide redundancy for facility systems should be strongly designed so that they surely work.

Probabilistic Evaluation of Buildings' Resilience
The recovery time and the availability can be probabilistically evaluated.In this section, the following probabilistic indices are formulated: (1) the expected value (mean value) T avg total of the total recovery time, (2) the expected values r S,avg , r F,avg of the availability of the structural frame and the facilities/non-structural components just after the earthquake event, and (3) the maximum likelihood recovery curves (MLRCs) of the structural frame and the facilities/nonstructural components.The 24-story building model and the electrical equipment system shown in Figure 4 are treated again.

Expected value T avg total of total recovery time
The expected value of Equation ( 1) can be expressed as where D k , τ i,j , τ S denote the damage state "k," the recovery time of component j of system i, and the recovery time of the structural frame.P is the symbol which expresses the probability.It is noted that, in Scenario FA, T avg total corresponds to an approximate value of the expected value of the recovery time because Scenario FA requires the function max Á f g.

4.
2 Expected values r S,avg , r F,avg of availability of structural frame and facilities/non-structural components just after earthquake event By the damage states of the lifeline and the emergency power system, the power availability at each floor can be classified into the following four cases (see Figure 5): (A) both of the lifeline and the emergency power system are undamaged r ll ¼ 1, r eps ¼ 1 À Á , (B) only the lifeline is undamaged r ll ¼ 1, r eps ¼ 0 À Á , (C) only the emergency power system is undamaged r ll ¼ 0, r eps ¼ 1 À Á , and (D) both are damaged r ll ¼ 0, r eps ¼ 0 À Á .In the cases A-C, the value of U r ll þ r eps À Á in Equation (6a) becomes one.In the case D, the value becomes zero.As explained in Section 2.3, r F i takes zero or 1=n ð Þdepending on the damage state of each component in the system i.When the emergency power system is equipped, all the cases A-D are considered.When it is not equipped, only the cases B and D are considered.
Consider the expectation of Equation (6a).When the emergency power system is not equipped, the expectation r F,avg of Equation (6a) can be expressed as where D 0 denotes the undamaged state.Q Mi j¼1 P i,j D 0 ð Þ is the probability of the case where all the components in system i are undamaged.In the case of Figure 5, M i ¼ 3 (conduit cable, distribution panel, switchgear), and Q M i j¼1 P i,j D 0 ð Þ corresponds to the area in green.When the emergency power system is equipped, r F,avg can be expressed as follows.
It should be pointed out that when the emergency power system is designed strongly enough, P r eps ¼ 1 ð Þbecomes almost one, and it leads to P r ll ¼ 1 Thus, the value of r F,avg becomes almost 1=P r ll ¼ 1 À Á È É times more than that in the case where the emergency power system is not equipped.
It is noted that r S,avg can be calculated in the same manner as r F,avg .

MLRCs of structural frame, facilities, and non-structural components
MLRC is the recovery curve in the case where each component belongs to the state with the highest probability.It should be pointed out that the probability of MLRC is independent of the selection of the recovery scenario.
It is noted that, although the Monte Carlo simulation is one of probabilistic methods for expressing the recovery process, it is not treated in this paper.

Resilience Evaluation of Elastic-plastic High-rise Buildings Under Resonant Long-duration Ground Motion
In this section, some numerical examples of the resilience evaluation of an elastic-plastic high-rise building model under a resonant long-duration ground motion are dealt with.In Section 5.1, the building model and the fragility functions are explained.In Section 5.2, a pseudo-multi impulse (PMI) as a representative for resonant long-duration ground motions is explained.PMI can treat not only the resonant response for the fundamental mode but also that for the higher modes.In Section 5.3, the influences of the passive viscous dampers, the redundancy of the facility systems, and the uncertainty of the manpower on the building resilience are investigated.

Models
Consider a 24-story shear building model.All the floor masses have the same value (= 400 × 10 3 kg).The model has a trapezoidal distribution of story stiffnesses (k 1 =k 24 ¼ 2:5), and the structural damping ratio is 0.01 (stiffness proportional type).
The undamped fundamental natural period is 2.4 s.The common story height is 4 m.The common yield interstory drift is d y ¼ 4=150 m and the bilinear hysteresis (kinematic hardening) in the story shear-interstory drift relation is assumed.The postyield stiffness ratio is set to 0.1.Figure 6 shows the 1st-4th undamped participation vectors and undamped natural periods.
Figure 7A and Table 1 show the fragility curves of the structural frame and the parameters (log-normal distributions).Although the maximum interstory drift d max,i at each story and the cumulative plastic deformation ductility demand may be related to the structural damage, it is assumed that the fragility curves of the structural frame depend only on Only the electrical equipment system is considered for the facility systems.The fragility curves (log-normal distributions) and the parameters of them are shown in Figure 7B-E and Tables 2, 3.It is assumed that the switchgear, the distribution panels, and the emergency power system are susceptible to the maximum floor accelerations a max,i at each story, and the conduit cables are susceptible to d max,i .It is also assumed that the lifeline is susceptible to the velocity amplitude of the input, and the emergency power system can provide power with the building for 0.1 month.When the emergency power system is undamaged and the damage state of the lifeline is slight or undamaged, the power supply to the building is kept up without interruption.This is because the recovery time of the lifeline with slight damage is 0.1 month.
It is noted that the values of the recovery time and parameters of the fragility curves are set with reference to the documents from FEMA 2 and Akehashi and Takewaki. 29or clarifying the influences of the passive viscous dampers, the redundancy of the facility systems, and the It is widely known that a multi-cycle sine wave (or a harmonic excitation) substitutes well for a long-duration ground motion.However, an iterative procedure with much computational load is required to obtain the resonant (critical) responses of the elastic-plastic structures by parametrically changing the input  period.On the contrary, a pseudo-multi impulse (PMI) 28 can precisely simulate the resonant responses without such iterative procedure (Figure 8).Although the conventional multi-impulse (MI) proposed by Kojima and Takewaki 31 is treated as a ground acceleration, PMI is treated as a multitude of impulsive lateral forces whose influence coefficient vector ι corresponds to the undamped fundamental participation vector α 1 φ 1 (see Figure 9).It should be noted that the resonant responses for the higher modes can also be treated by the appropriate settings of ι.

Numerical examples
Figures 10-13 represent the results of the resilience evaluation under PMI with V ¼ 0:4, 0:7 m=s.As shown in Figure 7E, the damage state of the lifeline is most likely to be undamaged in the cases of V ¼ 0:4 m=s, and it is most likely to be slight in the figures express the expected values T avg total of the total recovery time and the expected values r F,avg of the availability of the facilities just after the earthquake events.The expected values r S,avg of the availability of the structural frame are omitted for visibility.
Figure 10 presents the case of V ¼ 0:4 m=s which is resonant for the fundamental mode.The damage state of the structural frame is most likely to be slight for all the models.Although the viscous dampers do not improve MLRC of the structural frame, the values of T avg total , r F,avg and the probability of MLRCs of the facilities are improved.The emergency power system also improves the values of r F,avg .
Figure 11 illustrates the case of V ¼ 0:7 m=s which is resonant for the fundamental mode.The damage state of the structural frame is most likely to be moderate for the bare model.This delays the recovery of the facility components in the upper stories.The viscous dampers improve the damage state of the structural frame and the failure probabilities of the facility components.On the contrary, the effectiveness of the emergency power system is small because the structural frame and the lifeline are most likely to recover at the same time.In other words, the structural frame with slight damage and the lifeline with slight damage recover in 0.1 month.In this case, the emergency power system will be much more effective if the damage of the structural frame can be reduced.Figure 12 indicates the case of V ¼ 0:4 m=s which is resonant for the second mode.The damage state of the structural frame is most likely to be slight for the bare model.The failure probabilities of the facility components in the upper and middle stories are high because the maximum floor acceleration responses are large in those stories of the bare model.Thus, the recovery time becomes longer than the case of Figure 10 (resonant for the fundamental mode), and the availability of the facility components is decreased much more.On the contrary, the viscous dampers make the structural frame undamaged and largely decrease the failure probabilities of the facility components.In addition, the emergency power system improves the value of r F,avg .
Figure 13 shows the case of V ¼ 0:7 m=s which is resonant for the second mode.The damage state of the structural frame is most likely to be slight for the bare model.The failure probabilities of the facility components in the upper and middle stories are also high.The viscous dampers effectively reduce the failure probabilities of the structural frame and the facility components.Moreover, the emergency power system improves the value of r F,avg and MLRC of the facility because it counteracts the interruption of the power supply by the lifeline.
For all the cases in Figures 10-13, the bare model has much uncertainty with regard to the recovery time (or uncertainty with regard to the manpower to repair the components).In other words, the difference between the values of T avg total by Scenario FA and that by Scenario LA is large.The structural responses of the bare model are not reduced enough, and the recovery time is susceptible to the uncertainty with regard to the manpower.In contrast, the controlled model has small uncertainty because the viscous dampers reduce the structural responses and the number of the damaged components, and improve the damage states of the components.

Conclusions
A simple evaluation method of building resilience was proposed, and the resilience of elastic-plastic high-rise buildings under resonant long-duration ground motions was evaluated.The main conclusions can be summarized as follows.
1 The concept of availability was incorporated into the evaluation model of building resilience, which was proposed by Akehashi and Takewaki. 29The availability of facilities and non-structural components indicates whether they are workable or not.The availability of a structural frame indicates whether it is able to be occupied or not. 2 Not only in-building components but also external factors such as lifelines and redundancy of facility systems are considered in the evaluation model.Based on the evaluation model, the following probabilistic indices were formulated: (1) the expected value of the total recovery time, (2) the expected values of the availability of the structural frame, the facilities, and the non-structural components just after the earthquake event, and (3) the maximum likelihood recovery curves (MLRCs) of the structural frame and the facilities/non-structural components.3 Numerical examples were presented for evaluating the resilience of elastic-plastic high-rise building models under a pseudo-multi impulse (PMI), which substitutes for a representative for resonant long-duration ground motions.PMI can precisely simulate the resonant responses without any iterative procedure.Moreover, PMI can treat not only the resonant response for the fundamental mode but also that for the higher modes.The results of numerical examples can be summarized as follows: (1) the damage tendencies of structural frames and facility components are different from the resonant case for the fundamental mode to that for the second mode, (2) the control by passive viscous dampers largely improves the resilience and decreases the uncertainty with regard to the manpower to repair the components, (3) when the amplitudes of the inputs are large and the lifelines are likely to be damaged, the redundancy of facility systems is effective for improving the building resilience, and (4) when the damage of structural frame, facilities, and non-structural components are not reduced sufficiently, the effectiveness of the redundancy of facility systems is small.

FIGURE 1 .
FIGURE 1. Evaluation of recovery time based on Scenario LA and Scenario FA, (A) relation between Scenario LA and Scenario FA, (B) examples of recovery process

FIGURE 3 .
FIGURE 3. Examples of recovery process of structural frame and electrical equipment system

FIGURE 6 .FIGURE 7 .
FIGURE 6. Participation vectors and natural periods of 24-story shear building model

FIGURE 11 .
FIGURE 11.Resilience evaluation under PMI (V = 0.7 m/s) which is resonant for fundamental mode, (A) bare model, (B) controlled model, (C) controlled model with emergency power system, (D) maximum interstory drifts, and (E) maximum floor accelerations

FIGURE 12 .
FIGURE 12. Resilience evaluation under PMI (V = 0.4 m/s) which is resonant for second mode, (A) bare model, (B) controlled model, (C) controlled model with emergency power system, (D) maximum interstory drifts, and (E) maximum floor accelerations

FIGURE 13 .
FIGURE 13.Resilience evaluation under PMI (V = 0.7 m/s) which is resonant for second mode, (A) bare model, (B) controlled model, (C) controlled model with emergency power system, (D) maximum interstory drifts, and (E) maximum floor accelerations

TABLE 1 .
Parameters of fragility functions for structural frame

TABLE 2 .
Parameters of fragility functions for facilities