Analytical Seismic Fragility Curves for Reinforced Concrete Wall pier using Shape Memory Alloys considering maximum drift

Fragility curves express the seismic vulnerability of bridge piers for different damage states at various earthquake intensities. A fragility curve typically gives a physical understanding of repair costs and functionally levels of a bridge pier. Shape memory alloys (SMAs) provide a promising alternative material in reducing the failure probability of a bridge pier. This study develops a family of seismic fragility curves for reinforced concrete (RC) wall piers reinforced with three different types of SMA rebar in plastic hinge regions. An incremental dynamic analysis (IDA) using a total of 20 earthquake ground motions was performed on a SMA-RC wall pier to evaluate its seismic performance. The maximum drift recorded for each ground motion was taken as the seismic performance demand parameter of interest in this study. The probabilistic seismic demand model (PSDM) was used to generate fragility curves for each RC-SMA wall pier. The results show that the different mechanical properties and type of SMAs affect the performance of RC-SMA wall pier.


Introduction
Performance-based seismic design of bridges is now more frequently reviewed as it is necessary to enhance the ductility and limit the residual deformation of bridge structures due to the widespread damage of structures caused by earthquakes activities. Current seismic design codes such as EC8-2 [1], JRA 2012 [2], CHBDC 2014 and AASHTO LRFD 2012 [3][4] have provisions for performance based seismic design that allows reinforced concrete (RC) bridges to be designed with the ability to dissipate energy without causing any residual deformation during a seismic event. However, past catastrophic earthquakes events such as the Loma Prieta earthquake in 1989, the Northridge earthquake in 1994 and the Kobe earthquake in 1995 demonstrated that RC bridges were still vulnerable to seismic loading with most affected structures suffered serious damage and some even collapsed.
The overall seismic response of a bridge is dependent on the response of its pier. Evidence from the 1989 Loma Prieta earthquake shows collapse of Cypress Street Viaduct bridge while field reports reveal bridge piers suffering most from the strong earthquake ground motions. Considering the importance of bridges in maintaining the post-earthquake functionality of bridges [5], it is necessary to evaluate the seismic vulnerability of bridges. Fragility curves provide an important decision making tool in assessing the seismic risk of structures being damaged by different ground motion levels. In principle, there are different methodologies and approaches that have been developed for evaluating the seismic fragility of bridges [5]. The methodologies for the development of fragility curves include expert judgments [6], empirical approaches [7][8], and experimental and analytical approaches [9][10][11][12].
Over the last few years, many researchers [13][14][15] have found the ability of shape memory alloys (SMAs) as one of the rival materials for reinforcement bars in RC structures and their findings are well-documented. SMAs belong to a class of shape memory materials which have many advantages including large resistance to corrosion and high recoverable residual deformation. Many studies concluded that SMAs have significant positive effects on mitigating deficiencies of seismic resistance of conventional bridges and enhancing the performance of bridges under earthquake ground motions [16][17][18].
Several studies attempted to investigate the seismic vulnerability of bridges reinforced with SMAs [19][20][21], expressing their seismic risks in the form of fragility curves. A recent study by Billah and Alam [22] has developed fragility curves representing the seismic risk of bridge piers reinforced with five different types of SMA. The results demonstrated that all SMA-reinforced bridge piers exhibited a low probability of collapse in terms of maximum drift, thus effectively reducing the overall seismic vulnerability of the bridge piers. In this paper, seismic fragility curves for concrete wall piers reinforced with three different types of shape memory alloy considering maximum drift as the single demand parameter. The evaluation of the seismic fragility of the SMA-reinforced concrete (SMA-RC) wall piers is based on an analytical approach. A total of 20 selected earthquakes scaled with peak ground acceleration ranging from 0.101g to 0.875g was used in a nonlinear incremental dynamic analysis (IDA) [23] on each SMA-RC wall pier and the maximum drift recorded for each ground motion at each levels of intensity up to an intensity of 2.0g. The probabilistic seismic demand model (PSDM) is used to generate the fragility curves. Finally, seismic fragility curves for three different SMA-RC wall piers are compared.

Fragility function methodology
Fragility curves are a seismic risk tool that represents the vulnerability of a bridge influenced by a specified range of ground motion intensity in terms of probability distribution functions. The development of fragility curves can be used as an effective strategy to increase the level of safety of highway bridges, and to improve the decision making process for the estimation of loss and risk mitigation. Fragility function can be expressed as [24] Fragility = P (LS│IM=y) (1) where P (LS│IM=y) is the probability of exceeding a damage state as a function of engineering demand parameters (EDPs) at a given ground motion with IM=y. The present study makes use of the PSDM to generate fragility curves from the IDA results. A power law function [25] was assumed to obtain the mean and standard deviation through a regression analysis for each limit state, which provides a logarithmic correlation between a median demand and a selected IM.
where, a and b are unknown coefficients which can be estimated from a regression analysis of the response data collected from the IDA. The EDPs are assumed to follow a lognormal distribution. Equation 4 [26] is used to estimate the dispersion of the demand (βEDP| IM), conditioned upon the IM.

N
where, N is the number of simulations. It is now possible to generate the fragilities using Equation 5 [27] with the PSDM and the limit states corresponding to various damage levels.
ln (IMn) is defined as the median value of the intensity measure for the damage state (i.e slight, moderate, extensive, collapse), a and b are the regression coefficients and the dispersion component is given in Equation 7. [27] where, Sc is the median and βc is the dispersion value for each damage state.

Wall pier configuration
The wall piers are seismically designed and assumed to be located in Vancouver, British Columbia Canada ( Fig.1). A 2% probability of exceedance in 50 years (2475 years) is selected for design spectrum according to the CHBDC-2014 [3].
Here, Lp is the member length in mm; db is the reinforcement diameter in mm and fy is the yield strength of the bar in MPa.

Finite Element Modeling
Seismostruct [29] was used to model the reinforced concrete wall piers reinforced with shape memory alloys. The SMA-RC wall piers were modelled using nonlinear beam-column elements with rectangular cross-sections. The uniaxial stress-strain concrete material constitutive law based on Mander et al. [30] was used to model both the unconfined and confined concrete in the appropriate regions of the SMA-RC wall pier cross-sections. The transverse and longitudinal steel reinforcements bars were modelled using the Menegotto Pinto model [31]. The Auricchio and Sacco model [32] was used to represent the SMA reinforcement using the parameters in Table 1. SMAs were only used as a vertical rebar at the bottom of the plastic hinge of the RC wall piers. In addition, a zero-length rotational spring element was introduced at the bottom of the column section (Fig. 1c) to represent the mechanical couplers used to connect the steel to the SMA reinforcements. A bond-slip model with modified Takeda hysteretic curve [33] was used to model the slippage of SMA reinforcement bar from the coupler following the work of Billah and Alam [13] who numerically validated the bond slip relation for a SMA rebar against the experimental study by Alam et al. [34].

Fragility assessment of SMA-RC wall pier
The fragility curves were developed based on the PSDM with the maximum drift taken as the engineering demand parameter. 20 different ground motions (Table 2) from the PEER strong motion database of different magnitudes and peak ground accelerations (PGAs) were selected for the IDA. SeismoMatch [34] was used to match all the 20 ground motion records to the Vancouver's target response spectrum with a 5% damping ratio as shown in Fig. 2.  The maximum drifts (MDs) of the wall piers which represent the different performance-based limit states were considered as the EDP. An IDA was performed on the wall piers under a set of scaled intensity measures (IMs) with a constant increment of 0.2g up to 2.0g of the selected earthquake records using SeismoStruct [30]. The maximum drift of RC-SMAs wall pier subjected to 20 scaled earthquakes were recorded for each PGA level. The IMs are the first mode spectral accelerations, Sa (T1). In the present study, a PGA was considered an IM due to its practically, sufficiency and efficiency to analyse structural seismic fragilities [35][36]. PSDMs are generated from the relationship between EDP and IM through a linear regression in a logtransformed space. Fig. 3 shows the PSDMs for the three different RC-SMA wall piers considering the maximum drift as the EDP. The slope, intercept and dispersion of the EDP|IM relationship were carried out from this linear regression model and shown in Table 4. A strong relationship between EDP and IM is evident from Fig.3 with R2 values higher than 0.8 for all the PSDMs. Table  3 summarizes the demand dispersion (βEDP| IM) for all the RC-SMA wall piers where RC-SMA1 wall pier yielded a higher dispersion in demand than the RC-SMA2, and both the RC-SMA1 and RC-SMA3 wall piers have almost a similar dispersion in demand (βEDP| IM) indicating the effectiveness of introducing the right SMAs in the right regions in reducing maximum drifts of a wall pier. Four damage levels namely slight, moderate, extensive and collapse were considered in accordance with HAZUS [37]. The damage states are quantified based on the performance limit states proposed by Hose et al [39]. The performance limit states considered here are the drift percentages for cracking of concrete, yielding of reinforcements, and spalling and crushing concrete, and they are based on a strain damage approach. Yielding of SMA rebar was defined by the yield strain of the SMA bar while spalling of confined concrete was assumed to take place at a strain of 0.004 as suggested by Hose et al. [39]. Crushing strain of core concrete was calculated based on the equation proposed by Paulay and Priestley [28]: where εcu is the ultimate compression strain, εsm is the steel strain at maximum tensile stress, fc is the concrete compressive strength in MPa, fyh is the yield strength of transverse steel in MPa, and ρs is the volumetric ratio of confining steel. Table 4 shows the four limit state capacities in terms of median (Sc) and dispersion (c). The dispersion of limit state models was calculated using the equation developed by Nielson and Pang [27]: A probabilistic distribution of limit states was used to calculate the coefficient of variation (COV) for each limit state and the COVs were found to be 0.21 for DS1, 0.27 for DS2, 0.41 for DS3 and 0.44 for DS4, yielding similar dispersion values Nielson and Pang [27].
The fragility curves developed for three different RC-SMA wall piers considering maximum drift as the EDP using the linear PSDMs and limit state models presented in Table 3 and Table 4 are plotted in Fig. 4. The probability of cracking damage is seen very high irrespective of the seismic intensity levels. A closer look into Fig. 4a reveals that all the wall piers possess similar level of vulnerability at the cracking damage state. The RC-SMA1 wall pier performs better than the other two wall piers with a probability of damage of 87% if built in a seismic zone with a PGA of 2g. The high yield strength of SMA3 could have contributed to the better performance of the wall pier. It is also observed from Fig. 4c that the probabilities associated spalling for RC-SMA2 is more closed to other compare to probability of yielding damage. This can be attributed to the limit state capacity considered in this study showed that the range of spalling capacity are 1.32 -1.35. On the other hand, the probability of crushing damage (Fig.4 d) are 20%, 35% and 18% by PGA=2g for RC-SMA1, RC-SMA2 and RC-SM3, respectively.  Table 5. and Fig. 4. shown the comparison of median values of PGA for different types RC-SMA wall piers in term of maximum drift. The median PGA are defined as 50% of the probability of RC-SMAs reaching a limit state. At DS1, the median PGA ranges from 0.03g up to 0.05g. This results consistent with finding from [21], which the authors found the similar ranges median values of PGA for slight damage state. When looking at higher damage state, the median PGA for RC-SMA2 was 0.58g and 1.31g for DS2 and DS3, which the lowest value than other two RC-SMAs. This results indicates that the RC-SMA2 is more fragile compare to others wall pier. On the other hand, RC-SMA1 and RC-SMA2 have almost the same median values of PGA around 3.15g and 3.19g at DS4.