The Role of Seismic Structural Health Monitoring (S2HM) in the Assessment of the Delay Time after Earthquakes

Abstract

The concept of seismic resilience has been introduced in the design of buildings in the last decade. In this regard, the delay time may be defined as the time that occurs between the event and the moment the repair process begins. In the literature, only a few contributions have considered delay time, and even its definition is still under discussion. However, it is a key parameter in the assessment of resilience after earthquakes since it may significantly increase the total time after which a structure may be considered recovered. The principle at the base of the paper is that seismic structural health monitoring (S2HM) may play a significant role in reducing the delay time. Therefore, delay time needs to be considered since it may significantly reduce the seismic resilience of structural systems. The paper aims to consider this important issue demonstrating the relationship between S2HM and the assessment of the seismic resilience of buildings. In particular, the assumption herein is that the accuracy of the S2HM may be described with different levels, and in correspondence with these levels, certain values of the delay time may be considered. In addition, the delay time is considered as a percentage of the total repair time. A multidimensional definition that includes the accuracy of S2HM in the description of the delay time is herein proposed to be included in methodologies that aim to assess seismic resilience.

1. Introduction

Seismic resilience has become an important issue in recent decades for earthquake engineering and has been defined since 2010 by [1], which proposed a formulation that may quantify resilience by considering several contributions. A more general definition was proposed in 2003 by [2] that defined resilience as the velocity of a system to recover from the consequences of an event. More recently, [3] proposed another definition that considers the role of the community. In particular, the computation of the repair time has been the object of the most recent publications in the literature. For example, the resilience-based earthquake design initiative (REDi, [4]) developed a framework that was applied to study the seismic resilience of several structural configurations. This approach proposes a loss evaluation methodology for evaluating the success of the pre-earthquake measures. Another contribution considered the seismic loss and downtime assessment of existing tall steel-framed buildings and strategies for increased resilience [5].

Among this literature, downtime was defined as the time between the occurrence of an earthquake and the re-occupancy of a building [6]. This range of time depends on two components [7]: the delay time (necessary to assess the state of the building, take decisions, and, consequently, mobilize economic and human resources) and repair time (consisting of the process of returning to the original functionality). In 2006, Comerio [8] introduced the concept of downtime in modeling structural losses. In 2015, [9] proposed to consider the actual mobilization and repair times together with different building limit states, including functionality limit state (FLS), detailed inspection limit state (DILS), and repairability limit state (RLS). The procedure proposed by FEMA P-85 is based on the estimation of the total repair time without considering the building’s functional recovery process [10]. Another study [11] named this delay time as impeding time, which is due to indirect and external factors that delay the beginning of the recovery process.

In this regard, several contributions considered these external factors (e.g., [8,12,13,14]. REDi [4] considered impeding factors as those that consist of construction delays proposing to consider impeding times ranging from 5 days for inspection to around 50 weeks for some contractor, design, and finance delays (more details can be found in [4]).

Marquis et al. [15] considered that the characteristics of human infrastructures and post-disaster decision making might considerably affect the recovery process. In addition, [16] considered that the decisions to repair and rebuild depend on unpredictable characteristics of the owners (e.g., income, ownership, time at current residence, earthquake insurance, etc.). Moreover, [17] demonstrated that implementing strategies to recover business is a key factor in reducing delay time by considering 22 businesses affected by the 2011 Christchurch Earthquake. Ref. [3] also considered the Christchurch Earthquake as a case study to support the definition of resilience as a process and particularly focused on how the prioritization of needs changes as time moves on following a disaster. Recently, [10] named mobilization time the range between the time of the earthquake and the time at which the repair process begins. This mobilization time consists of the time for building inspections, relocations of occupants and activities, and decision-making procedures (more details are found in [8]).

In this paper, the time that runs between the time of the event and the beginning of the repair process is called delay time, considering a negative connotation to this time since it significantly reduces the seismic resilience of the system. The duration of delay time depends on several aspects, such as the width of the affected region, the vulnerability of the existing buildings, and the availability of resources for seismic structural health monitoring (S2HM) or, more specifically, inspections. In particular, emergency decisions are strictly connected to the uncertainties of the structural state. In this regard, the decisions to evacuate the building, suspend ordinary activities, or preserve its functionality (total or partisan) have been investigated in many contributions (e.g., [18,19]).

This paper proposes several novelties. Firstly, it aims to calculate seismic resilience by considering the role of S2HM on the delay time. In particular, the approach adopted in [20,21] is herein extended to consider the delay time with a simplified formulation that concentrates all the various uncertainties inside a comprehensive parameter that may consider the different typologies of S2HM. The quantification of the delay time was herein based on the definition of the level of S2HM that may realistically be conducted on the structure. This may be considered a second novel contribution of the paper. Finally, a new formulation that includes the delay time was proposed herein in order to develop the previous definition of resilience with multiple dimensions [22].

The paper is divided into seven sections. In Section 2, the concept of resilience is described by considering the historical development of the terms during the last 30 years in various fields. In Section 3, the concept of seismic resilience (SR) is defined in the specific field of earthquake engineering by considering the introduction of the delay time inside the traditional definition of SR. Section 4 investigates the relationship of S2HM to SR by assuming that the level of pre-earthquake knowledge is a key parameter in the definition of the delay. Section 5 consists of the core of the paper: a multidimensional definition of SR is described, and the role of S2HM on the delay time is introduced by assuming that the delay time can be quantified with a percentage of total repair time. Section 6 discusses the assumptions that have been made in the paper by comparing with the similar approach described in the Eurocode EC3-8 in the definition of several levels (visual, limited, extended, extensive, and full) of the knowledge of a building.

2. Resilience

This section summarizes the definitions of resilience during the last 35 years. In 1988, Wildavsky [23] defined resilience as “the capacity to cope with unanticipated dangers after they have become manifest, learning to bounce back”. It is worth noting that two base principles were considered: 1. prevention and 2. learning from the paper for future events. Later, Hoiling et al. [24] expanded the previous definition by considering resilience as “the buffer capacity or the ability to a system to absorb perturbation, or magnitude of disturbance that can be absorbed before a system changes its structure by changing the variables”. Moreover, [25] described resilience by considering “individuals, group and organizations, and systems”. In 1998, Mallak [26] applied the concept to healthcare systems by defining it as “the ability of an individual or organization to expeditiously design and implement positive adaptive behaviors matched to the immediate situation, while enduring minimal stress”. In 1999, [27] considered inside resilience the possibility of an “amount of assistance from outside the community”, meaning (1) the eventuality of help from the outside and (2) the application of resilience to the community scale. Furthermore, [28] developed resilience as “the capacity to adapt existing resources and skills to new systems and operating conditions”, by including (1) the capacity and (2) adaptation to external events. Twenty-three years ago, [29] proposed a modern definition of resilience as “an active process of self-righting, learned resourcefulness and growth the ability to function psychologically at a level far greater than expected given the individual’s capabilities and previous experiences”, which considered (1) resilience a process, (2) the psychological contribution to resilience, and (3) the active role of individuals. Moreover, [30,31,32] considered resilience based on the mutual correlation of technical, organizational, social, and economic dimensions. In particular, they developed analytical formulations to quantitatively calculate resilience by including recovery time (the period necessary to restore the functionality of the system to an acceptable level) but neglecting the role of the delay time inside the formulation. A comprehensive definition of resilience was proposed by the Hyogo Framework, [33] which proposed to consider resilience “the capacity of a system, community or society potentially exposed to hazards to adapt, by resisting or changing in order to reach and maintain an acceptable level of functioning and structure. This is determined by the degree to which the social system is capable of organizing itself to increase this capacity for learning from past disasters for better future protection and to improve risk reduction measures”. This definition considers several aspects: the social dimension of resilience, the importance of learning from the past, community exposure, the possibility of change, the need to define some levels of performance, and the importance of future protection actions. These contributions are herein considered to extend the state-of-the-art methodologies to assess resilience in order to account for two fundamental aspects in the computation of resilience. Firstly, it is important to include delay time that may considerably reduce the resilience of the system. Second, the paper considers the accuracy of pre-earthquake S2HM important, even if it has been investigated extensively in the literature; the quantification of them inside resilience is a gap that this paper aims to cover.

3. Seismic Resilience (SR)

Seismic resilience (SR) applies this general content to study earthquake events. In particular, many contributions have been proposed during the ages, from the first application by [1]. In particular, SR was described as the ability to maintain or restore the flows in an effort toward community resilience [2]. In [34], a framework for modeling post-earthquake functional recovery of buildings was proposed by modeling the repair process of the recovery. In addition, Molina Hutt et al. [5] calculated the seismic loss and downtime on existing tall steel-framed buildings while [35] performed several studies on two tall buildings. In addition, [36] proposed a simplified model to analyze a super-tall building equipped with mega-braced frames, and [37] proposed to use SR to study several case studies of acute-care facilities.

Following this background, this paper includes the application of the delay time that is significantly important in the calculation of the SR. In particular, as shown in [10], seismic resilience (Figure 1) may be defined as:

$$\mathrm{SR}={{\displaystyle \int }}_{t_{0E}}^{t_{0E}+\mathrm{RT}}\frac{Q\left(t\right)}{RT}dt$$

(1)

where:

• $t_{0E}$ is the time of occurrence of the earthquake E;
• $t_{0R}$ is the time at which the repair process starts;
• $Q_0$ is the functionality just after the earthquake;
• L are the losses (calculated as L $=$ 1 $-Q_0)$;
• RT is the repair time necessary to recover the original functionality (expressed in Crew Working Days, CWD);
• $Q \left(t\right)$ is the recovery function describing the recovery process.
• In this regard, the delay time is calculated as $DT=t_{0R}-t_{0E}$.

It is worth noting that RT includes DT and that the calculation of SR significantly depends on this term since it is the extreme of the integral, at the RT is the upper limit of the integral, and also at the denominator of the integrand function. Therefore, the estimation of DT is crucial in the reduction in SR that is graphically represented by the normalized area underneath the recovery function Q(t). This area significantly reduces when the delay time increases since the system remains functional, which occurs at the time of occurrence of the earthquake. Therefore, the definition of the delay time is herein presented on the basis of the level of the pre-earthquake S2HM. It is worth noting that this formulation is valid for one dimension only, and it was discussed herein in order to be applied in Section 5 after the description of the role of the seismic structural health monitoring (S2HM) inside a formulation that considered multiple dimensions that are described in Section 5.

4. The Role of S2HM on SR

Seismic structural health monitoring (S2HM) plays an important role in the description of the state of a structure in pre-earthquake circumstances. In particular, the S2HM has been significantly improved over the last decades, as shown in [38], by considering different techniques, such as novel algorithms that consider structural damage-sensitive features (e.g., inter-story drifts [39], frequencies [40], and nonlinear mode shapes [41]) that may be applied to detect damage due to earthquakes. These techniques may be implemented in numerical models, as shown in [42]. In particular, the principal stakeholders may apply such techniques to organize the emergency procedures and recovery process. In this regard, [43] investigated the value of information (VoI) in the case of long-term structural health monitoring, providing a rational basis to rank measuring systems, including the decision-making process affected by the monitoring campaign. Moreover, [44] proposed a framework that allows the optimization of structural risk and integrity management by considering fatigue deteriorating structural systems and their characteristics in order to demonstrate how inspection strategies can be performed. In addition, [45] considered the VoI framework to assess the use of visual inspection to support decision-making procedures in managing buildings. In addition, [46] considered the possibility of applying mutually VoI and structural health monitoring. Recently, [47,48] considered the application of the probabilistic-based methodology of fragility curves to the assessment of seismic emergency procedures. In particular, [47] considered proposed to apply the VoI from Bayesian decision theory to convince stakeholders of the importance of S2HM. In [48], the importance of inspections to improve the prediction of gradual and shock deterioration processes was considered.

In this background, the accuracy of S2HM is considerably important for the definition of seismic resilience. Therefore, DT is assumed herein to depend on the S2HM, and thus several levels of accuracy were proposed herein. This principle consists of considering S2HM fundamental tools for the definition of the procedures that follow the immediate occurrence of the earthquake. However, considering that the various uncertainties due to the extreme variability of structural typologies cannot be represented by factors, DT was herein considered proportional to RT:

$$\mathrm{DT}=\lambda ·RT$$

(2)

The coefficient λ may define several levels of accuracy in the general assessment of the conditions of the building due to the S2HM. For example,

Table 1. Levels of S2HM.

Level	Accuracy	$\mathit{\lambda }$
L1	Slight	0.50
L2	Moderate	0.35
L3	Extensive	0.20
L4	Complete	0.05

quantifies the coefficient λ on the basis of four levels of S2HM accuracy (L1, L2, L3, and L4), named Slight, Moderate, Extensive, and Complete, proposing to consider DT as 50%, 35%, 20% and 5% of the RT. In particular, these four limit states were assumed to be similar to the approach proposed by [49] for the definition of loss models. It is worth noting that this formulation is a first attempt to assess the role of S2HM in the definition of the delay time by considering that DT depends on several contributions, such as inspections, site preparation, eventual relocation of people and activities, engineering services, obtaining permits, securing financing and recovery activities, as in [8,10].

Case Study

A case study has been considered herein in order to assess the definition of DT with the proposed values in Table 1. In particular, SR has been calculated by applying Equation (1) and thus considering Equation (2) to calculate the values of DT. Therefore, including Equation (2) inside Equation (1) leads to the following:

$$\mathrm{SR}={\displaystyle {\int }_{t_{0E}}^{t_{0E}+\mathrm{DT}+RT}}\frac{Q\left(t\right)}{RT}dt$$

(3)

Considering a linear representation of the repair process, the calculation of SR depends on the definition of RT. By adding the hypothesis that the final functionality is the same as the original ones, RT may be calculated directly, and thus, it is possible to derive SR for the four values of DT and shown in

Table 2. Case Study (Hp: linear repair function).

Level	Accuracy	$\mathbf{S}\mathbf{R}$
L1	Slight	0.502
L2	Moderate	0.531
L3	Extensive	0.644
L4	Complete	0.951

. It is worth noting that considering DT is fundamental in decreasing the value of SR. In particular, S2HM accuracy is shown to be extremely important in the assessment of SR: the obtained value for Slight accuracy (0.502) is almost half of the value obtained with a Complete accuracy (0.951)

5. Multidimensional Definition of SR

Following the previous [33], a multidimensional formulation of SR needs to be defined. In particular, the calculation of SR needs the application of the loss model and the recovery model:

(1)

The loss model describes the reduction in functionality due to the initial impact of an event on a system. The quantification of the losses that are at the base of this model may be a challenging issue since several typologies of losses need to be considered, such as direct and indirect losses, as described in [20].

(2)

The recovery model aims to assess the ability of the system to recover from the impact by considering the process of recovery that is generally represented with an analytical formulation. Such formulation needs to describe several variables that are challenging to be defined and generally depend on the preparedness of a community, the level of technological know-how, and the distribution of economic funds for the recovery process.

In this background, herein is extended what is described in Section 3 to assess the resilience to earthquake hazard by considering the presence of DT. SR is formulated on the basis of the recovery function Q_N(t) that describes the recovery process to return an estimated level of functionality:

$$Q=Q\left(t, \lambda ,Q_1,\dots ., Q_N\right)$$

(4)

which is defined in the R^N space and is a function of time (t), of the coefficient $\lambda$ and all the various dimensions Q_i of the problem/system. Consequently, SR may be computed by the following [24]:

$$\mathrm{SR}={{\displaystyle \int }}_{t0F}^{t0F+RT}\frac{Q\left(t,Q_1,\dots ., Q_N\right)}{RT}dt$$

(5)

It is worth noticing that this formulation has the same shape as that proposed above, and it expresses the dependency of SR not only on time but also on the N dimensions with that the system may be described.

Graphically, Equation (4) represents the normalized volume underneath the recovery function QN(t) and the plane t = RT (Figure 2). It is worth noting that in Figure 2, the values of RT could be different for the various dimensions.

It is worth noting that for every time t = k, with k < RT, it is possible to consider the plane:

$$Q\left(t=k,Q_1,\dots ., Q_N\right)$$

(6)

which represents a plain surface connecting the points on the recovery function corresponding at t = k for all the variable Qi. Therefore, Equation (5) describes the section of the cone whose perimeter represents the sum of the functionalities Qi reached by the total system at the selected time t = k. Figure 3 shows the contour lines (lines joining points at the same time t = k_i).

Following [1], in the present paper, the simplification of the linear repair function has been considered realistic when there is not sufficient information regarding the preparedness of the community or the consistency of the resources. Therefore, the solid assumes a conical shape with different angles with the t-axis, depending on the coefficient ci:

$$\mathrm{ci}=\frac{Q_{F,i}}{R{T}_i}$$

(7)

where $Q_{F,i}$ is the value of the functionality after the complete recovery of the system i. Such value may vary from 1% to 100% with respect to the original functionality but may also be bigger than 100% when the repair function allows improvements to the original functionality. Herein it was assumed that the loss is 100%, meaning that the value of functionality just after the earthquake is 0. It is worth noting that because of the presence of the DT, the hypothesis of a common vertex of the solid, as proposed in [22], is still not possible, and the solid shape may be more complicated than that shown in Figure 2. It is worth noting that the novelty of the present formulation consists of the fact that it may consider the effect of DT and, in particular, the role of S2HM in the definition of seismic resilience in a multidimensional space.

6. Discussion

The formulation presented herein may combine two important issues in structural engineering. The importance of pre-earthquake S2HM has been extensively discussed in the literature, and it is one of the most significant areas of research in terms of new applications and methodologies. The second important research question is the ability of the proposed approach to considering delay time inside the SR quantification. This parameter needs to be considered together inside the framework since it may significantly reduce the total repair cost of the system subjected to earthquakes. Until now, only a few contributions have investigated this issue, which may be considered a hole in the literature. The presented formulation aims to quantify the DT with the accuracy of S2HM by proposing several limits that need to be set up with expertise and case studies. The principle to connect S2HM to DT in a multi-dimension definition of SR is the core of the paper. The proposed values are only a first attempt, and they need to be calibrated with more specific information. However, they are based on the principle that the knowledge of the state of the structure is fundamental to saving precious time when the earthquake occurs, as included inside the Eurocode EC8-3 that considers several levels: visual, limited, extended, extensive, and full of the knowledge of a building. In particular, three categories of knowledge are considered: geometry, details, and material. In particular, the code prescribes the levels of inspection and testing to reach the recommended requirements of accuracy. Therefore, these prescriptions aim to carry out the pre-earthquake structural assessments with certain rigor and objectivity, considering the various uncertainties that affect the definition of these three quantities: geometry, details, and materials.

However, the principle applied in the present paper is similar to that of the EC8-3 to condense all types of uncertainties into a single factor, the so-called confidence factor (CF), which can be strictly applied only to the mechanical properties of the materials. This factor may be used to calculate the capacities of the materials obtained from in situ tests, inspections, and S2HM. Additionally, the values of CF are proposed by the EC3-8 on the base of extensive activities and debates between researchers; the herein proposed table needs to be discussed and proposed by experts and stakeholders. The aim of the paper is only to propose the approach, not to establish the quantification of the relationship between S2HM and DT. In this regard, several observations need to be considered. Firstly, the proposed table is a simplified approach to cover the various uncertainties, and many of them cannot be represented by factors since every structure has its own peculiarities in terms of dimensions, characteristics, and arrangements, and sometimes even ascertaining the real situation may become challenging. In addition, DT depends on various issues, as defined in [8,10], and calculating it only on the basis of the level of accuracy of S2HM may be considered a simplification. Other factors need to be considered, for example, the allocation of human resources, level of knowledge, and availability of economic funds for recovery procedures. Such limitations will be developed in future studies.

7. Conclusions

The paper investigates the role of delay time in the assessment of seismic resilience. Only a few contributions have considered delay time in the calculation of seismic resilience, even if it may be considered a key parameter that may significantly reduce the resilience of the systems. The paper proposes the definition of the delay time considering the existing ones that are still not unique and introduces this parameter inside the original formulation of seismic resilience. The delay time is assumed to depend on the S2HM, and thus several levels of accuracy were proposed herein. This principle consists of considering S2HM fundamental tools for the definition of the procedures that follow the immediate occurrence of the earthquake. The seismic resilience was then assessed by considering a loss model and a recovery model. The first allows for the losses occurring at the various infrastructures to be assessed, and the latter is applied to quantify the repair time (RT). The formulation herein proposed was simplified with linear functions that may be considered realistic because not much information is generally available due to the extreme uncertainties of seismic events. In particular, these kinds of data regard the preparedness of the community or the consistency of the resources. A multidimensional formulation was then proposed to consider the different dimensions that earthquakes may affect by proposing a simplified (but realistic) formulation of seismic resilience. Such an approach may be considered a first attempt to catch the extremely challenging issue of modeling delay time. In particular, the main limitation of the presented approach consists of including all the uncertainties regarding the definition of S2HM inside a unique parameter. In this paper, no data were available on the various typologies of S2HM methods, and thus only a simplified estimation of the parameter was proposed. Future studies will develop more advanced formulations of the role of S2HM on the delay time by including the proposed formulation inside a framework that may assess seismic resilience.