Keywords

1 Introduction

During the past few decades, it has been observed that a substantial number of building structures existing in seismically active zones undergo extensive damage, which is quite disturbing for the long-term structural performance. Reinforced concrete Frame-shear wall buildings are extensively used because of their good seismic resistance. In the present study such buildings have been focused at. Vulnerability assessment of such buildings in their inelastic range due to high seismicity can be suitably performed through nonlinear dynamic analysis, which necessitates detailed nonlinear modeling of the shear wall building. Fragility analysis (FA) is an essential tool used for damage assessment of buildings (Kappos et al. 2006; Gogus and Wallace 2015; Nazari and Saatcioglu 2017). Design codes focus on the site conditions and design practices adopted for designing buildings. Design codes are prescriptive in nature (ATC 2012; FEMA 2010). Several available literature have primarily focused on collapse damage states of the structures to reduce seismic risk of the building during earthquakes (FEMA P-695 2010; Galanis and Moehle 2015; Attar and Liel 2016). Many researchers conducted a dynamic analysis of buildings using different model types (Ji et al. 2007; Ji et al. 2009; Zhai et al. 2019). Fardis and Krawinkler (2010) worked on assessing the structural performance in the natural disaster for both old and new shear wall buildings designed as per EC-8 and derived to see the performance of the structures through fragility curves. Another study had been done by Pejovic and Jankovic (2015) using Perform 3D software to find the vulnerability of tall Reinforced concrete (RC) structures with core wall structural systems. Other recent works have focused on mid-rise frame-shear wall buildings conforming EC-8 for medium to high-class ductility (Antoniou et al. 2015) Kappos et al. (2006) used the hybrid approach, which combines an experimental and analytical approach. Many researchers are striving towards damage assessment of the structures such that there is no unexpected failure of the buildings during an earthquake. Investigation can be done by conducting seismic fragility analysis of the buildings to get expected damage of the existing structures (Gogus and Wallace 2015; Nazari and Saatcioglu 2017; Kolozvari et al. 2017). Several available literature primarily focused on collapse damage states (Park and Ang 1985; Galanis and Moehle 2015, Elkady and Lignos 2014, Sattar and Liel 2016) to reduce seismic risk building during an earthquake. Initially, the seismic fragility analysis has been developed for the nuclear plant by Kennedy and Ravindra (1984), where the fragility curve of various dangerous equipment has been plotted. Hwang improved this method in 1990, and it stretched its effect over the evaluation of normal buildings. Many researchers conducted the dynamic analysis with different levels of buildings using different model types (Ji et al. 2007; Ji et al. 2009; Zareian and Krawinkler, 2010; Nazari and Saatcioglu 2017). For obtaining the fragility curves several time history analyses have to be performed under specified set of ground motions (GM). Limited research work had been conducted so far, considering the requirement for probabilistic risk evaluation. In this regard, fragility analysis is used to estimate the disproportionate collapse of any structures. Several methods are available for FA. The most often used methods are: (i) the Incremental dynamic analysis (IDA) method, (ii) Multiple Stripes Analysis (MSA), (iii) Maximum likelihood estimation (MLE), and, (iv) Approximate Approach (AA). In earthquake engineering, the IDA method is a well-established approach. This clearly accounts for the demand and capacity uncertainties of the structures and provides the failure probability of a given intensity measure (IM). In this method, a set of GM is continuously scaled to get the level of the IM where the GM induces collapse (FEMA 2010; Vamvatsikos and Cornell 2002). In MSA, a set of IMs are to be chosen for the execution of the analysis. In the case of MLE, parameters having maximum probabilistic influence are used. AA is becoming popular nowadays. Limited works have been reported using AA (Korkmaz 2008; Réveillère et al. 2012; Borele and Datta 2015; Sil et al. 2019). The FA is less time taking yet gives satisfactory results. In this paper, different existing methods of fragility analysis have been reviewed for Reinforced concrete (RC) buildings. The suggestion has been given for best approach.

2 Classification of Fragility Analysis

The fragility assessment of structures can be done by applying many approaches which is used to find the vulnerability of the structures. Classifications of fragility analysis for structures are shown in Table 1.

Table 1. Classification of fragility analysis with their works and findings.
Full size table

3 Limitations of the Available Methods

Researchers have developed many methods to define the failure probability of a structure for a given IM of structures, both numerically and experimentally in the last few decades. Approaches are being developed to improve accuracy. Table 1 shows the benefits and drawbacks of the various approaches for determining a structure’s vulnerability. However, structural damage can occurs in the structure due to some of the loads like seismic load, wind load, any accidental load, etc. There are some of the important parameters like IDR, joint rotation, stiffness degradation etc. which plays an important role to measure the reliability of structure. Many works have been done so far by many researchers considering those parameters to find the vulnerability of the buildings by using several approaches. However, in Table 1 several methods of fragility assessment have been discussed, and the most preferable method has been suggested so that it takes lesser time to found out the vulnerability of the structures.

4 Analysis Based on Fragility

Seismic fragility is another form of reliability that expresses the exceedance probability of a damage limit state (DLS) for a structure under seismic excitation. Fragility may be defined as the probability of exceedance of the demand acting on the structure over the structure’s capacity for a specified intensity measure (IM). Kennedy and Ravindra (1984) have introduced analysis on Seismic fragility to measure the safety and risk assessment of mechanical assemblies and components of structures in nuclear power plants. The output of the investigation showed the risk of seismic exposure fragility, which is in the form of curve and represents the exceedance of probability of the performance level of structure. The Input-output relationship for fragility analysis and the fragility curve has been shown by Korkmaz, 2004. The seismic damage limit is used in fragility analysis to represent the performance levels of the structures. Thus, every fragility curve denotes the exceeding probability of damage limits when earthquake intensity increases. Fragility is nothing but the exceedance probability of the DLS of a structure that has been exposed to seismic excitation. Though, the fragility curve is a numerical statistical measure that reflects the exceedance probability of DLS at a specific IM. Here, ground motion is indicated as an IM, and it could be in terms of pseudo-spectral acceleration (PSA), peak ground acceleration (PGA), Spectral Displacement (SD), spectral acceleration (SA), spectral velocity (SV). Though, from all the IMs, the most common IM is PGA (Kennedy and Ravindra 1984; Hwang and Huo 1994; Hwang and Jaw 1990; Shinozuka et al. 2000).

5 Methods for Seismic Fragility Analysis (FA)

There are numerous ways to calculate the values of the parameters used in determining for a fragility function with observed data. It fully depends on the procedure used to get the data from the structural analysis.

The cumulative lognormal distribution function generally defines fragility function:

$$ P\left( {C\backslash IM = x} \right) = \Phi \left( {\frac{{\ln \left( {\frac{x}{\theta }} \right)}}{\beta }} \right) $$
(1)

where the probability of the structure is designated by \(P\left( {C\backslash IM = x} \right)\) Where the GM intensity measure (IM = x) will cause collapse; Ф () is the function of cumulative distribution (CDF); Median of ln IM is designated by θ, and standard deviation of ln is designated by β. The above Eq. (1) indicates the value of IM of ground motion will cause structure collapse are lognormal distributed. This assumption has been long-established by many researchers (Eads et al. 2013; Porter et al. 2007; Ibarra and Krawinkler 2005; Ghafory-Ashtiany et al. 2011; Bradley and Dhakal 2008). The response of the structures under various ground motions is determined through probabilistic Seismic Demand Analysis (PSDA).

5.1 Incremental Dynamic Analysis (IDA)

In earthquake engineering, the IDA method is a well-established approach. This accounts for the demand and capacity uncertainties of the structures and provides the failure probability of a given intensity measure (IM). In this method, a set of GMs is continuously scaled to get the level of the IM where the ground motion induces collapse (FEMA 2010; Vamvatsikos and Cornell 2002). Here, ground motions are in the form of IM and have to be increased incrementally in every analysis. The extreme values are plotted against the IM values for every intensity level. In this analysis, a group of IM values produces which is linked with the beginning of collapse for every ground motion shown by Baker and Eeri (2005.a). The collapse probability of the structure can be evaluated at which level Collapse happens. Visualization of this probability has been shown by Baker and Eeri (2005.b) and its denoted as observed CDF and Parameters of the fragility functions can be calculated from the same data by considering logarithms of every ground motion which is connected with the beginning of collapse at IM level ‘x’ and computed standard deviation as well as mean values (Ibarra and Krawinkler 2005). Equation 1 has to be calibrated for a specified structure which requires assessing of θ and β based on structural analysis findings. Those parameters have been terms as \(\hat{\theta }\) and \(\hat{\beta }\).

$$ ln\hat{\theta } = \frac{1}{n}\mathop \sum \nolimits_{i = 1}^n \ln IM_i $$
(2)
$$ \hat{\beta } = \sqrt {\frac{1}{n - 1}} \mathop \sum \nolimits_{i = 1}^n ln\left( {IM_i /\hat{\theta } } \right))^2 $$
(3)

where a number of considered ground motions is designated by n and IMi is the IM value at ith ground motion. Here, the mean of the normal distribution signifying the values of ln IM is designated by ln θ, and the standard deviation of the normal distribution signifying the values of ln IM is designated by ln β.

5.2 Maximum Likelihood Method (MLM)

In this method, find the parameters in such a way that the distribution of the result has the maximum likelihood from the observed data. In this method, ‘m’ is the number of observed ground motions to bring collapse, where the values of IM at collapse level (IMi) are known. In this method, a random ground motion can cause collapse at IMi, specified a fragility function in Eq. (1).

$$ {\text{MLM}}\, = \,\Phi \left( {\frac{{ln\left( {\frac{IM_i }{\theta }} \right)}}{\beta }} \right) $$
(4)

where, standard normal distribution is denoted by Ф, here the specified ground motion is scaled to IMmax without causing the building to collapse where the probability of IMi is more than IMmax (Klugman et al. 2012).

$$ {\text{MLM}}\, = \,1 - \Phi \left( {\frac{{{\text{ln}}(IM_i /\theta }}{\beta }} \right) $$
(5)

assume, IMi value for every ground motion data is not reliant on whole observed data of the likelihood method.

$$ {\text{MLM}}\, = \,\left[ {\mathop \prod \nolimits_{i = 1}^m \emptyset (\frac{{{\text{ln}}(IM_i /\theta }}{\beta })} \right]\left[ {1 - \Phi \left( {\frac{{{\text{ln}}(IM_{max} /\theta }}{\beta }} \right){ }^{n - m} } \right] $$
(6)

here, Π signifies that ‘i’ is the product value which starts from 1 to m. Where, ‘m’ denotes the ground motion values of collapse at IMs inferior to IMmax. Using this expression, the parameters of fragility functions are determined by changing the parameters until the function reaches the maximum. It can be maximized by using the logarithm of the likelihood function:

$$ \left\{ {\hat{\theta } , \hat{\beta }} \right\} = \begin{array}{*{20}c} {argmax} \\ {\theta , \beta } \\ \end{array} \mathop \sum \nolimits_{j - 1}^m \left[ {ln\phi \left( {\frac{{{\text{ln}}(IM_i /\theta }}{\beta }} \right)} \right] + \left[ {n - m} \right] ln\left[ {1 - \Phi \left( {\frac{{\ln (IM_{max} /\theta }}{\beta }} \right)} \right] $$
(7)

The Observed collapse as a function of IM and a fragility function calculated using Eq. 7 can be seen the figure (Baker and Eeri 2005.a).

5.3 Multiple Stripes Analysis (MSA)

This method is a special case of IDA. Here, only one IM level is chosen, and the data of EDP has been obtained. This method is also known as a unique method because more than one hazard level can be chosen. In this method, it is not necessary to perform nonlinear analysis till all the IM amplitudes get the collapse, and Picturing of a number of collapses causing plotted at IDR of 0.08 (Example MSA results) and Collapse observed in terms of IM and an estimated fragility function by implementing Eq. (11) is shown by (Baker and Cornell 2005.a). Pictorial is shown by (Baker and Cornell 2005.b). Here, the IM level changes of every ground motion target properties (Lin et al. 2013; Iervolino et al. 2010; Bradley 2010). Several ground motions have been used in every IM level, and because of the several data used in the analysis may not show the increasing fraction of collapse with the increase IM, but in spite, it is assumed that failure probability will increase with IM. On the other hand, the results of the structural analysis show the ground motions percentage that causes collapse at every IM level. Assuming that, from every ground motion’s data, the observation of failure or no failure is independent of various ground motions, the binomial distribution describes how the probability of observed collapses zj out of GM nj with IM = xj.

$$ {\text{P}}(z_j \;{\text{collapses}}\,{\text{in}}\,n_j \,{\text{ground}}\,{\text{motions}})\, = \,\left( {\begin{array}{*{20}c} {n_j } \\ {z_j } \\ \end{array} } \right)P_j^{zj} \left( {1 - p_j } \right)nj - zj $$
(8)

where probability is designated by pj when the ground movement IM = xj will cause collapse, fragility function can be found by the maximum probability, which gives the maximum probability of observing collapse data found from the structural investigation. The investigation data are found from various IM levels; at every level of IM, the binomial probabilities product is used to find the probability for the whole data set.

$$ {\text{Probability}}\, = \,\mathop \prod \nolimits_{j = 1}^m \left( {\begin{array}{*{20}c} {n_j } \\ {z_j } \\ \end{array} } \right)p_j^{zj} \left( {1 - p_j^{nj - zj} } \right) $$
(9)

where the number of IM levels is denoted by m and whole product levels are denoted by Π. Therefore, substitute the Eq. 1 in place of pj, so that the parameters of fragility are clear in the probability function.

$$ {\text{Probability}}\, = \,\mathop \prod \nolimits_{j = 1}^m \left( {\begin{array}{*{20}c} {n_j } \\ {z_j } \\ \end{array} } \right)\Phi \left( {\frac{{{\text{ln}}(xj/\theta )^{zj} }}{\beta }} \right)\left[ {1 - \Phi \left( {\frac{{{\text{ln}}(x_j /\theta }}{\beta }} \right)} \right]{\text{nj}} - {\text{zj }} $$
(10)

The probability function is maximized to estimates and gets the fragility function parameters. It is statistically and equivalently easier to increase the logarithm probability function.

$$ \left\{ {\hat{\theta } , \hat{\beta }} \right\} = \begin{array}{*{20}c} {argmax} \\ {\theta , \beta } \\ \end{array} \mathop \sum \nolimits_{j - 1}^m \left\{ {ln\left( {\begin{array}{*{20}c} {n_i } \\ {z_i } \\ \end{array} } \right) + z_j ln\Phi \left( {\frac{{{\text{ln}}(x_i /\theta }}{\beta }} \right) + \left[ {nj - zj} \right] ln\left[ {1 - \Phi \left( {\frac{\ln (x_j /\theta }{\beta }} \right)} \right]} \right\} $$
(11)

5.4 Approximate Approach

Firstly, numbers of finite element models (FEM) of any structures have been generated. The generated buildings are then matched with the selected ground motions that have been considered and then carried out by nonlinear dynamic analyses to find the structural response corresponding to each building model. Fragility may be defined as the probability of the demand acting on the structure exceeding the structure’s capacity for a specified intensity measure (IM).

Therefore, the expression of seismic fragility is

$$ {\text{Fragility }}\left( {\text{F}} \right) = \Pr \left[ {D \ge C/IM} \right] = {\text{Pr}}\left[ {C - D \le 0.0/IM} \right] $$
(12)

here, seismic demand is designated by D, and capacity is designated by C; IM is the ground motion IM. Considering the time-variant effect on the seismic fragility of the RC buildings, by Eq. (13) (Sudret and Mai 2013).

$$ {\text{F}} = {\text{Pr}}[{\text{D}}\left( {{\text{Demand}}} \right)\left( t \right) \ge C\left( {Capacity} \right)\left( t \right)/IM = Pr\left[ {C\left( t \right) - D\left( t \right) \le 0.0/IM} \right] $$
(13)

assuming that the demand and capacity of the structure follow a lognormal distribution, Eq. (13) takes the form Eq. (14).

$$ {\text{Pr}}[{\text{D }}\left( {{\text{Demand}}} \right)\left( t \right) \ge \frac{{C \left( {capacity} \right)\left( t \right)}}{IM} = \Phi \left[ {\frac{{\ln \left( {\frac{N_d \left( t \right)}{{N_c \left( t \right)}}} \right)}}{{\sqrt {\beta_{D\backslash IM}^2 \left( t \right) + \beta_C^2 } \left( t \right)}}} \right] $$
(14)

where, \( N_d \left( t \right)\) is the estimated median of the demand acting on the structure at the time, t; \( N_c \left( t \right)\) is the estimated median of the capacity of the structure at the time, t. \(\beta_{D\backslash IM}^2 \left( t \right)\) and \(\beta_{C\backslash IM}^2 \left( t \right)\) are the dispersion of demand and capacity of the structure at time, t.

The PGA levels have to be chosen based on the structure’s vulnerability in various seismic zones across the globe. Further, PSDM at different PGA levels and storey heights have been generated based on Eq. (15).

The equation of the probabilistic seismic demand model (PSDM) is expressed in Eq. (15).

$$ {\text{ln}}(N_d \left( t \right) = y\left( t \right) + z\left( t \right){\text{ln}}\left( {IM} \right) $$
(15)

here, \(y\left( t \right)\) and \(z\left( t \right)\) are the parameters of regression estimated during the time, t, and it can be found by conducting regressing analysis considering the building’s demands at various times.

6 Ground Motion Intensity Measure (IM)

It is the utmost important index to define the characteristics of the earthquake from the perspective of structural engineering. By considering the formulation of fragility analysis, it is well understood that the IM is the function of the seismic fragility curve. Therefore, the use of more appropriate IM gives more precise and accurate results in the fragility curve. To get the characteristic of maximum IM, sums of GM records have to be considered for scaling and also for the purpose of nonlinear analysis, which influences both the accuracy of the results as well as computational time. In this case, a minimum of 3 ground motion records is required according to ASCE/SEI 7-10 (2010). Reyes and Kalkan (2012) have reported that the ground motion records should not be less than 7 so that it can be able to reach to the optimum accuracy.

7 Summary

Seismic fragility is another form of reliability that expresses the exceedance probability of DLS for any given type of structure under seismic excitation. Fragility may be defined as the probability that the demand acting on the structure exceeds the structure’s capacity for a specified IM. In seismic fragility analysis, the PGA (g) has been considered as an IM, and all the ground motion considered has been scaled to the expected level. The incorporation of scaled records captures the worst scenario of the structural degradation that could be identified. This paper presents a review on seismic fragility analysis. A wide range of research on seismic reliability and risk has bought an important advancement for recovery, mitigation, preparedness, and a response against the Seismic risk. Several methods of fragility assessment have been discussed, and the most preferable method has been suggested so that it takes lesser time to found out the vulnerability of the structures. From the literature survey, it has been found that implementing an approximate approach over the other available approaches of seismic fragility analysis is efficient in accurately estimating the exceedance probability of the damage limit states without tedious computational requirements.

8 Conclusion

In this article some of the approaches that are available for finding out the vulnerability of the structure along with their mathematical expressions have been discussed. Benefits and drawbacks of these approaches have been discussed and also the possibilities of further study are explored. Investigation on seismic vulnerability is an active field of research. The fragility curves can be used in mitigation planning and risk assessment for a long-term strategy of the community to decrease seismic losses and damages. In this paper, the available methods of conducting fragility analysis for risk assessment and seismic reliability structures have been reviewed. It is found that scanty work has been done to assess fragility of RC shear wall buildings using the approximate approach. The approximate approach gives good outcome at the same time saves computational time. This approach is also applicable for all types of structures, such as RC buildings, bridges, and other structures.