Seismic damage classification for axially-loaded well-detailed reinforced concrete frame members based on compressive strain

Quantification of seismic damage to reinforced concrete (RC) members in terms of damages states is important for condition assessment and performance-based seismic design. This paper presents a classification for axially-loaded flexural-dominant well-detailed RC members into clearly-differentiated damage states based on change in failure mode, principles of mechanics, and loss in lateral load capacity. The paper considers extreme fiber compressive strain as the Engineering Response Parameter (ERP) for classification of the damage states. The identification of member damage states based on compressive strain can be very useful for seismic assessment of the existing buildings and performance-based design of new buildings. Published experimentally observed cyclic force–displacement relationships of RC columns have been numerically simulated and are used to develop statistics of compressive strain. These are further used to develop semi-empirical expressions for extreme fiber compressive strain in terms of RC member properties, viz. axial load ratio, longitudinal reinforcement ratio, and confinement reinforcement ratio. It is shown that the proposed expressions accurately predict experimentally observed compressive strain at different damage states. The fragility functions of damage states, that provide the probability of occurrence of the strain limits, are also presented. It is seen that the proposed expressions for strain and their limits accurately identify the damage states on the monotonic stress–strain relationship of confined concrete and validate the use of strain as an ERP. The present study also provides strain limits for different damage states with consistent level of conservatism.


Introduction
An axially loaded flexure-critical RC frame member, such as those present in frame buildings, when experience increasing lateral deformations due to seismic excitations, sequentially undergoes cracking of cover concrete, tensile yielding of reinforcement, crushing and spalling of concrete cover, initiation of core crushing, buckling of longitudinal reinforcement, and fracture of longitudinal or transverse reinforcement. The continuous progression of damage in RC members is typically divided into discrete damage classifications or member performance levels. The damage classifications and their relationship with engineering response parameters (ERP) are required for seismic design of the new buildings (ACI-318 2019), assessment of existing buildings (FEMA P-58 2018), and damage assessment of buildings after an earthquake based on residual load-carrying capacity (ATC-20 1989;Ohkubo 1992). For illustration, the description of damage states of an RC column given by FEMA P-58 (FEMA P-58 2018) is shown in Table 1. ERP is a generic term indicating measurable responses used to represent damage states of member or of a complete building. Some examples of ERPs are strain, deformation, drift, and rotation of a member, and total drift and maximum interstory drift of a building. Definition of damage states or criteria for damage classification and the corresponding limiting values of ERPs is an important requirement for seismic design, evaluation of existing structures, and seismic health monitoring (Shan et al. 2022).
Numerous proposals have been made in the literature to relate damage in RC structural members with ERPs. Panagiotakos and Fardis (2001) proposed expressions for deformation (chord rotations) of beams, columns, and walls at the yielding of reinforcement and ultimate (failure) damage states. Matamoros and Sozen (2003) proposed drift limits at collapse state for high-strength columns. Similarly, Pujol et al. (2006), Elwood and Moehle (2005a, b), Elwood and Moehle (2005a, b), and Kato and Ohnishi (2002) expressed drift capacity of shear-critical columns in terms of cross-sectional properties and axial load ratio. Watson et al. (1994), Sheikh and Khoury (1997), and Saatcioglu and Razvi (1992) developed procedures to determine amount of confinement reinforcement to achieve specific ultimate drift capacity of RC columns. Zhu et al. (2007) proposed probabilistic drift capacity models for lateral strength degradation and axial load at failure. Berry and Concrete Spalling: Spalling of cover concrete that exposes transverse but not longitudinal reinforcing steel 3 Concrete crushing: Spalling of cover concrete exposes longitudinal reinforcement. Strength loss initiates in laboratory 4 Steel yielding, buckling, and fracture: reinforcing steel experiences severe inelastic deformation and requires replacement. Longitudinal steel exhibits severe inelastic strain, buckling, or fracture 1 3 Eberhard (2005) proposed a probabilistic model to predict the deformation demands at onset of bar buckling of RC columns. Haselton et al. (2008) related buckling of longitudinal reinforcement in RC column to inter-story drift ratio (IDR %). Seismic damage indices are also developed as another approach to classify damage progression in RC members. Examples of seismic damage index include Park and Ang (1985), Roufaiel and Meyer (1987), Wang and Shah (1987), Powell and Allahabadi (1988), Niu and Ren (1996), Mehanny and Deierlein (2001), and Colombo and Negro (2005) indices. These damage indices are generally calibrated from observations at ultimate or collapse damage of the member.
The gap in the available literature related to damage classification in RC members for seismic design and assessment is twofold. First, available ERPs, including seismic damage indices in published literature, while accurately identifying the ultimate or failure/collapse damage state of an RC member, do not address the initial and intermediate damage states with similar accuracy. Second, the ERP values at damage states of RC members in literature are obtained from limited analytical or test results of the models considered in that study. As a result, the limits of ERP are only applicable to RC members with similar structural characteristics (Ahmed and Dasgputa 2022).
These issues can be addressed by defining all damage states on the basis of mechanics of load transfer and relating them to the ERP. Further, generalized expressions to determine the value of ERP at various damage states structural characteristics are required. In the present study, a new seismic damage state definition is presented for axially loaded flexural-critical well-detailed RC frame members. The damage is defined in terms of concrete compressive strain as the Engineering Response Parameter (ERP) ensuring that the definition covers the full range of seismic damage, viz. initial, intermediate, and collapse damage states. The use of strain as ERP also permits its seamless use for the design of new buildings as well as for assessment of existing buildings. The important damage states are defined on the basis of change in failure mode, principle of mechanics, and extent of loss in load capacity of the member.
The compressive strains at the damage states are estimated using calibrated numerical simulations of published experimental inelastic cyclic responses of RC members. A semiempirical approach is adopted to develop expressions for compressive strains at important damage states. Fragility functions are developed at the important damage states to estimate their probability of occurrence. The strain expressions and strain limits proposed in the study allow for distinct identification of important damage states from the stress-strain relationship of confined concrete. The study improves the widely used subjective definition of important damage states for RC members. The present study also provides a framework to determine strain limits at a different probability of exceedance and thus ensures consistent levels of conservatism in strain limits of different damage states.

Mechanics of damage progression, important damage states, and ERPs
Flexural failure of axially-loaded RC members when subjected to lateral loading is governed by its confinement reinforcement (Berry and Eberhard 2005;Bae and Bayrak 2008). Figure 1 shows a representative stress-strain behavior of both unconfined and confined concrete sections of an RC member (Penelis and Kappos 1997). The stress-strain behavior of concrete comprises ascending and descending branches. The confinement reinforcement significantly affects the descending part of the stress-strain curve. As seen in Fig. 1, the stress-strain curves of both unconfined and confined concrete are identical at low strain levels and separate at higher strain towards the end of the ascending branch. The peak stress for confined concrete (f cc ) is higher than unconfined concrete (f c ). However, the strain (ε cc ) at peak stress of confined concrete remains nearly the same as strain at peak stress of unconfined concrete (ε c ).
In the present study, the progression of damage for flexural-dominant axially loaded well-detailed RC members has been studied based on the stress-strain behavior at their critical section. RC members having axial load ratios up to 0.4 have been considered, which typically occurs in members of RC frame buildings of up to around 20 stories. When the axial load ratio is less than around 0.4, yielding of longitudinal reinforcement generally occurs in the high strain region of the ascending portion of the stress-strain curve (Paulay and Priestley 1992). The strain (ε cc ) at peak stress (f cc ) generally lies at 0.002k (with k ~ 1.0-1.3) (Sakai and Sheikh 1989). At this stage, the RC section also experiences minor but irreversible flexural cracking in the cover concrete. However, the concrete in compression zone, transverse reinforcement in inclined crack zones, and aggregate interlocking provide sufficient shear resistance to avoid brittle failure. Thus, yielding of longitudinal reinforcement (ε yld ) which occurs in the ascending portion of the curve has been designated as Light damage state of the member.
As flexural strain increases, the onset of spalling (initiation of spalling) of cover concrete occurs, and the resulting loss of capacity represents start of the descending portion of the stress-strain curve in confined member (Fig. 1). Initially, the spalling reduces the shear resistance of concrete by destroying the bond between reinforcement steel and concrete. At this stage, aggregate interlocking and transverse reinforcement provide sufficient shear resistance and prevent brittle shear failure of the member (Penelis and Kappos 1997). Thus, onset of spalling, which occurs at ε osp , represents the start of progressive loss of section capacity, and is designated as Moderate damage state.
Significant spalling of cover concrete, which occurs subsequently at higher flexural strain, indicates a complete separation of longitudinal reinforcement from surrounding concrete and penetration of cracks into the core concrete at the section. Grinding of cracked concrete under cyclic seismic loading smoothens the crack interface and reduces the 1 3 contribution of shear resistance from aggregate interlocking. This leads to a rapid reduction in strength and stiffness of the member (Penelis and Kappos 1997). Hence, significant spalling of cover concrete, which occurs at ε ssp , is designated as Severe damage state. The damage states, their description and the corresponding concrete strain are summarized in Table 2.
The shear resistance at this stage is mainly maintained by the dowel action of the transverse reinforcement across cracks. As the flexural strain increases, the transverse reinforcement yields, and even the unsupported longitudinal reinforcement under compressive load can buckle. Crushing of concrete core is initiated with buckling of the reinforcement. Buckling of significant number of longitudinal reinforcement bars due to further spalling causes complete crushing of core concrete (at strain ε ccu ) and leads to a significant loss of gravity load-carrying capacity of the RC member. Buckling of longitudinal reinforcement or crushing of core concrete, which occurs at ε ibb , is designated as Collapse damage state of the member.
Several studies have related compressive strain to damage. Priestley (2000) proposed compressive strain value of 0.0040 at the onset of crushing of concrete cover in axiallyloaded RC members. Kowalsky (2000) proposed similar compressive strain value of 0.0040 at cover concrete crushing in circular bridge piers. Goodnight et al. (2016) reported mean and standard deviation of compressive strains as 0.0049 and 0.0014 at crushing of cover concrete. Babazedeh et al. (2015) reported a compressive strain of 0.005 for initial crushing of concrete. Lehman et al. (2004) and Chen et al. (2009) reported mean and standard deviation of compressive strains as 0.0066 and 0.0022 at initial spalling of cover concrete. Lehman et al. (2004) also reported mean compressive strain of 0.0188 with a standard deviation of 0.01 at crushing of core concrete. The initial spalling of cover concrete, reported by Lehman et al. (2004) and Chen et al. (2009), occurs after onset of spalling (initiation of crushing) but before significant spalling. They reported the mean minus one standard deviation values of compressive strain at initial spalling and core crushing as 0.0044 and 0.0088, respectively. These values are closer to strain limits at crushing of cover concrete reported by Goodnight et al. (2016) and Babazedeh et al. (2015). Lehman et al. (2004) also proposed strain limits corresponding to initiation of bar buckling (similar to crushing of core concrete). The compressive strain limits proposed in the literature at important damage states proposed in this paper are summarized in Table 3.
The damage (or inelastic response) in a RC member is concentrated over plastic hinge region. The length of plastic hinge increases with progression of damage in the member and stabilizes at collapse or ultimate damage state of the member (Bachmann 1971). The length of plastic hinge region is generally longer than the length over which the bending moment (M) under seismic forces is greater than the yield moment capacity of RC section (M y ). The presence of shear causes an increase in the tensile strain in the reinforcement and causes reinforcement bars to yield over a wider area (Priestley 2000). The spread of damage in RC members is closely associated with strain in the damaged region. Also, unlike other response quantities such as rotation, drift, etc., compressive strain is inherently related to damage. Thus damage states defined in terms of compressive strain are applicable to all RC frame members of different lengths and with different end conditions. The strain-based definition also allows classification of damage in flexural-dominant well-detailed RC members of various lateral load-resisting systems such as RC bare frames, RC frame with shear walls, and infill-RC frames with In-plane (IP) and out-of-plane (OOP) interactions (Mazza and Donnici 2021). Therefore, the extreme fiber strain has been selected as the ERP to represent damage in the RC member.

Selection of experimental datasets
The objective of the study is to investigate strain-based damage state definition for axially loaded flexure dominant well-detailed RC frame members. Therefore, experimental datasets of RC frame members are considered in the present study. Two experimental datasets viz. PEER structural performance database (Berry et al. 2004) and Chen et al. (2009), are considered in the present study. The PEER database (Berry et al. 2004) used in this study includes data from cyclic lateral load tests on 274 rectangular and 160 spiral RC columns. For each column test, the database provides material, geometry properties, reinforcement details, axial load ratio, and force-displacement histories. The experimental studies reported displacement at one or multiple damage stages. The damage states reported in the experimental studies are at (1) onset of spalling (first observation of spalling), (2) onset of significant spalling (minor cracks in core concrete), (3) onset of bar buckling, and (4) fracture of transverse/longitudinal reinforcement. The displacement ratios at occurrences of the above-listed damage states are included in the observations. The axial load ratio varies between 0 to 0.8 and span to depth ratio between 2 to 6 for experimental datasets of 179 rectangular columns in the database. Of these, the axial load ratio varies between the range of 0 to 0.4 for 150 datasets. However, only 51 of these experimental studies have reported displacements corresponding to (1) or (2) or (3) damage states. Furthermore, 6 out of the 51 experimental studies from PEER database do not provide complete details of confinement reinforcement. Hence, the present study has considered observations from the remaining 45 experimental studies from PEER database. Crushing of core concrete 0.0188 0.0100 Lehman et al. (2004) Another important source of experimental data used in this study is from Chen et al. (2009). They presented data from ten columns and three RC beam specimens that were quasi-statically tested to study the effects of member parameters on ductility and damage progress. This study recorded displacements at (1) first yield of reinforcement, (2) onset of significant spalling, and (3) onset of bar buckling. In the present study, 2 of 7 column specimens that had axial load ratio less than 0.4 and all 3 beam specimens from Chen et al. (2009) are considered. Therefore, a total of 50 specimens, consisting of 45 rectangular columns from PEER database and 5 rectangular beam/columns, that exhibited flexural failure mode, from Chen et al. dataset, were used. The range of parameter values of selected RC specimens is shown in Table 4.

Analytical modelling scheme
Member-type macro modelling scheme is used in the present study for numerical simulation of force-displacement relationship in above listed experimental datasets. The membertype macro modeling scheme is used to reproduce non-linear force-displacement relationship of RC member. In member-type macro modeling scheme, the RC member is modelled as line element, and the inelastic behavior of RC section is represented using spread plastic hinge. The inelastic behavior of RC section is represented using moment-curvature relationship.
The use of moment-curvature relationship with spread plasticity models captures yield penetration. Thus, in member-type modeling spread plasticity model is most suitable to capture initial progression of damage in RC members. The spread plasticity model simulates inelastic behavior in RC members by a combination of concentrated plasticity at the end sections of the member and distributed flexibility rule. The spread plasticity model also accounts for the variation of the point of contra-flexure during incremental nonlinear inelastic response. The length of plastic hinge is updated at each step of the analysis based on the instantaneous moment distribution diagram and yield penetration model. Therefore, in the present study spread plasticity model with moment-curvature relationship is used for reproducing inelastic response in RC member. The details of the mathematical formulation of spread plasticity model can be found in Reinhorn et al. (2009).
The characteristics of hysteretic behavior RC member are degradation in strength, stiffness, and pinching under cyclic loading. The strength degradation occurs due to cyclic degradation and in-cycle deterioration at high deformation. Haselton et al. (2016), Pelliciari (2020), Kenawy (2020), and Di Domenico (2022) highlighted the importance of calibrating strength degradation for cyclic degradation and in-cycle deterioration. The characteristics of inelastic cyclic behavior of RC member has been modelled using non-symmetric threeparameter hysteretic models proposed by Park et al. (1987). In this model, the characteristics of hysteretic behavior of RC members viz. reduction in stiffness, strength, and pinching are controlled using parameters α, β, and γ, respectively. The strength degradation parameter β is a combination of ductility-based strength decay parameter (β 1 ) and hysteretic energy-based strength decay parameter (β 2 ). The reduction in strength with increase in ductility (displacement) demand has been captured with parameter β 1 . While strength reduction due to cyclic loading (in-cycle strength reduction) has been captured with parameter β 2 . The model permits tracing the variability of the hysteretic loop at different deformation levels under repeated loading reversals from a combination of the values of these parameters. In the present study, IDARC 2D 7.0 (Reinhorn et al. 2009) is used for numerical simulation of experimental force-displacement relationship.
The simulated force-displacement relationship is estimated by statistically minimizing the difference in initial stiffness, yield displacement, backbone curve, and hysteresis characteristics with the experimental observation. The calibration of analytical hysteretic behavior is carried out to match average deteriorations that occur over displacement history.
The moment-curvature relationship of the section has been developed from section geometry, reinforcement details, and stress-strain relationships of concrete and steel. The stress-strain properties of the confined concrete are determined based on the arrangement of transverse reinforcement in the sections. In the present study, confinement models considered are Park et al. (1982), Kappos (1991), and EC-8 (2004). For estimation of moment-curvature relationship, it is assumed that the plane section remains plane during bending. The contribution of concrete under tension in the overall tensile resistance is very marginal and does not affect moment (or curvature) values. Therefore, tensile strength of concrete (i.e. portion below neutral axis) is not considered in the development of moment-curvature relationships. The neutral axis, curvature, and moments are estimated for all compressive strain values to develop the moment-curvature relationship of the RC section.
The ERP of interest, extreme fiber compressive strain, can be determined from its relationship with curvature (ϕ) of an RC section. The curvature of an RC section is the ratio of strain at the extreme fiber of the RC section (ε) to its distance from the neutral axis (x) (Park and Paulay 1975).
Thus, compressive strain is directly associated with each curvature value from moment-curvature relationship.
For each experimental data, quasi-cyclic analysis of its analytical model is carried out using the displacement time history used during experiments. For each numerically simulated experiment response, the initial stiffness of the tri-linear moment-curvature relationship and the parameters of the hysteretic model are determined by minimizing the error (difference) between simulated and experimentally observed force-displacement response at yield, reduction in stiffness, and reduction in strength, respectively. The force-displacement relationship from the analytical simulation is verified with the experimentally observed force-displacement relationship before further use.
The overall methodology followed in the present study is summarized in Fig. 2. The time history of strain is obtained from the numerical simulation of the experimentally observed response. The proposed methodology maps the damage progression in RC member under cyclic loading on the monotonic stress-strain relationship of the concrete. In Fig. 2, the path shown with Black arrows, (a)- , is forward path. The forward path is followed for numerical simulation of experimental force-displacement. The path shown with Green arrows, (e)-(d)-(c)-(b), is backward path. The backward path is followed for the estimation of time-history of strain from displacement time history and identification of strain values from the displacements recorded at critical damage states. This path includes estimation of curvature and compressive strain time histories from the force-displacement and moment-curvature relationships, respectively. The steps in forward path represent modeling member section properties (Fig. 2a), development of confined concrete stress-strain relationship (Fig. 2b), estimation of moment-curvature relationship (Fig. 2c), construction of analytical member model for time-history simulations (Fig. 2d), and determination of its force-displacement relationship (Fig. 2e). For estimation of moment-curvature relationship, the stress-strain relationship of reinforcement steel is idealized as elastoplastic (bilinear) as shown in Fig. 2b. A moment-curvature relationship for an RC section is determined by considering the locations of the reinforcing bars in the RC section and stress-strain relationships of both the concrete and the reinforcing steel, resulting in the typical continuously varying (or smooth) relationship. In IDARC-2D, the continuous moment-curvature relationship is approximated as a tri-linear function for further analysis to determine the inelastic response of the RC member. The tri-linear key points represent moment and curvatures at cracking, first yield of longitudinal reinforcement, and ultimate state, as shown in Fig. 2c.
The backward path provides curvature and compressive strain time history, and their values at observed critical damage states. The curvature time history and values at observed damage states are determined from exclusive relationship maintained by analytical model (Fig. 2d) between simulated force-displacement relationship (Fig. 2e) and continuous moment-curvature relationship (Fig. 2c). The compressive strain time history and values at observed damage states are determined from exclusive relationship continuous  (Fig. 2c) and stress-strain relationship (Fig. 2b) represented by Eq. 1.

Observations from numerical simulations
For simulated experimental datasets, when the member axial load ratio varies between 0.1 and 0.4, the initial (effective) stiffness is found to vary between 0.35 and 0.85 of gross stiffness (EI g ). These effective stiffness factors match those reported by Elwood and Eberhard (2006). Figure 3 shows the experimentally observed and simulated force-deflection relationship from tri-linear moment-curvature approximation for Gill et al.  2, 148.6, and 84.3, 148.7, respectively. The relative error in total hysteretic energy from numerical simulations with reference to that from experimental studies is 0.07% and 0.03%, respectively. Thus, it can be seen that macro modeling scheme with line elements and spread plasticity considered in this study accurately predicts the experimental force-displacement relationship.

Determination of compressive strain at important damage states
Equation 1 is used to determine curvature from the force-displacement and moment-curvature relationships. The compressive strains at observed important damage states are obtained thereafter. The important damage states that have been considered, as discussed previously, are (1) first yield of reinforcement, (2) onset of spalling, (3) significant spalling, and (4) bar buckling or core crushing. Details of experimental specimens and strain values at observed damage states are summarized in "Appendix". The strain values at observed damage states are determined at first occurrence of recorded displacement from calibrated analytical model. The strength deterioration occurring under same magnitude displacement cycle does not affect the strain determined at observed damage state, i.e., the strain identified at first occurrence of recorded displacement is the lower bound strain at which the observed damage state can commence.
The statistics of compressive strain at critical damage states obtained for numerical simulations at the identified damage states are given in Table 5. The mean minus and the mean plus standard deviation (16% and 84% percentile) limits of compressive strain are also shown. The statistics shown in Table 5 are based on strain values of all the specimens and collectively represent observed strains in all RC sections (summarized in Table 4). The statistics are useful to investigate the conservatism available for strain limits at important damage states. The strain limits shown in Table 5 have been used to identify all damage states for an RC section.
The mean minus and plus standard deviation range of compressive strain at onset of spalling is found between 0.0040 and 0.0078 in the present study. The mean minus standard deviation and mean strain limits proposed by Chen et al. (2009) for initial spalling (i.e., 0.0044 and 0.0063, respectively) and by Lehman et al. (2004) (i.e., 0.0043 and 0.0066, respectively) are found to be within this range. The mean minus standard deviation strain  318 2019) is also found to be close to the mean minus standard deviation strain at the onset of spalling (0.0040) in the present study. This implies that the compressive strains specified in the design standards are conservative. Priestley (2000) and Kowalsky (2000) have specified only a single value of mean compressive strain (0.004) at initiation of concrete crushing. The mean minus standard deviation strain from the present study also has the same value. The results obtained in this study thus help to evaluate the recommendation in these standards and in published literature with respect to the extent of conservatism.
The mean strain at onset of spalling and significant spalling obtained in the present study is 0.0059 and 0.0095. On the other hand, mean compressive strain proposed by Goodnight et al. (2016) and Babazedeh et al. (2015) for initial crushing of concrete (i.e., 0.0049) is lower than the mean strain at onset of spalling and mean minus standard deviation strain at significant spalling obtained in the present study. This implies that these recommendations of Goodnight et al. (2016) and Babazedeh et al. (2015) have high conservatism.
At collapse damage state, which corresponds to bar buckling, the mean compressive strains of 0.018 suggested by Lehman et al. (2004) is found to be close to mean minus standard deviation strain range (0.016) from the present study. The strain limits proposed by Priestley and Lehman et al. are intended for development of seismic design procedures. Thus, the strain limits proposed by Priestley and Lehman et al. for moderate and collapse damage states are found to be conservative with reference to the mean strains and close to mean minus standard deviation strains obtained in the present study.
It can also be observed from Table 5 that the bounds of compressive strain (in terms of mean minus and plus standard deviation) at Light damage state are well separate from the bounds of other damage states. However, the bounds of Moderate, Severe, and Collapse damage states are overlapping. This implies that the occurrence of more than one damage state is possible for a particular value of compressive strain. For a given RC section, the occurrence of important damage states is sequential but the strain at which a damage state occurs depends on the section properties. As a result, simply stating the bounds of compressive strain for these damage states from Table 5 or other cited studies is not sufficient to identify their occurrences for a specific RC section. In order to have an accurate assessment of damage, there is thus a need to accurately predict occurrence of a damage state from compressive strain, and is presented below.

Sensitivity of compressive strain with sectional details
The sensitivity study of compressive strains at important damage states has been carried out with cross sectional parameters to develop semi-empirical prediction expressions. These parameters, which are related to compressive strains, and are important in the mechanics of load transfer, are used to develop semi-empirical prediction expressions. The member properties of RC section considered to develop the prediction expressions are (1) shear span to depth ratio (L/D), (2) axial load ratio (n 0 ), (3) longitudinal reinforcement ratio (ρ l ), (4) confinement reinforcement ratio (ρ w ), (5) bar diameter to depth ratio, and (6) concrete cover to depth ratio.
All the listed parameters, except span to depth ratio, can be estimated from RC section geometry and reinforcement details. Estimation of shear span to depth ratio (L/D) requires shear span which is the distance between points of contra flexure in flexural deformation of the member. The estimation of shear span is straightforward in determinate structural configurations. However, when a member of an indeterminate RC building frame is subjected to seismic forces, the location of point of contra-flexure, which depends on magnitude of load, changes as a function of time as the magnitude of seismic force changes. Thus, for practical applications, the exact estimation of shear span at different damage states is not possible. In this study, the variation of span to depth ratio (L/D) are not considered. Figure 4 shows the variation of compressive strain at light damage (i.e., first yield of reinforcement) against the following RC member sectional properties: (a) concrete cover to depth ratio, (b) axial load ratio (n 0 ), (c) confinement reinforcement ratio (ρ w ), and (d) longitudinal reinforcement ratio (ρ l ), respectively. In order to broadly understand the variation in compressive strain at important damage states with changes in sectional properties, trend lines from linear correlation with these section properties are also shown in Fig. 4. The correlation is useful to evaluate the effect of variation of section properties on the compressive strain. The coefficient of regression between compressive strain and RC sectional properties are also shown in Fig. 4.
It can be seen from Fig. 4 that the concrete cover to depth ratio has high scatter, and is therefore not correlated with compressive strains at the important damage states. The reason can be explained from the mechanics of load transfer, as the cover to depth ratio of RC section does not influence load transfer, strain distribution, and progression of damage. Thus, cover to depth ratio is not considered for the development of semi-empirical prediction expression. Fig. 4 Variation of compressive strain at first yield of reinforcement bar with sectional parameters As expected, no specific trend is observed between the compressive strains at first yield of reinforcement and confinement reinforcement ratio (ρ w ) (Fig. 4d). The confinement reinforcement ratio increases strength of confined concrete in post-yield region and mainly delays buckling of longitudinal reinforcement. The confinement reinforcement ratio (ρ w ) is not found to be correlated with compressive strains up to severe damage state, and the parameter is not considered for semi-empirical prediction expressions of these damage states. Figure 5 shows variations of compressive strain at onset of spalling against other sectional properties, i.e., axial load ratio (n 0 ) and longitudinal reinforcement ratio (ρ l ). The results of variation of compressive strain at significant spalling and bar buckling against these cross-sectional properties were investigated and found to be similar. For brevity, the variation of compressive strain at significant spalling and bar buckling against cross-sectional properties are not presented.
The axial load causes initial prestressing of the member cross-section and delays the incidence of initial cracking. Thus, an increase in axial load ratio delays the occurrence of first yield of reinforcement. This is seen in Fig. 4a, which compares compressive strain of first yield of reinforcement and axial load ratio. However, the increase in axial load ratio of RC member accelerates the occurrence of subsequent damage states. This effect is observed in Fig. 5a, which shows a decrease in compressive strain of onset of spalling with increase in axial load ratio. Similarly, the comparison of compressive strain at significant spalling with axial load ratio show decrease in compressive strain with an increase in axial load ratio. A similar trend is expected but not observed for compressive strain at bar buckling with increase in axial load ratio for complete dataset. This is because in the complete dataset, the confinement reinforcement is relatively lesser for specimens with low axial load ratio (i.e. ≤ 0.1) when compared to range of confinement reinforcement ratios available for specimens with axial load ratio more than 0.15 (i.e., for axial load ratio between 0.15 and 0.4). This leads to relatively lower strain values at bar buckling for data points with low axial load ratios. The dataset with axial load ratio of more than 0.15 exhibits the expected trend.
The other parameter, which influences all damage states, is the longitudinal reinforcement ratio. As is well known, during design, the longitudinal reinforcement ratio is estimated from axial force and moment demands. Thus, the compressive strain at first yield of reinforcement remains nearly constant with longitudinal reinforcement ratio, as observed  Fig. 4c. For the important damage states occurring after first yield of reinforcement, the increase in the longitudinal reinforcement ratio in an under-reinforced RC section increases its moment of resistance. This delays the occurrence of subsequent important damage states. This can be observed in Fig. 5b, which shows an increase in compressive strain with the increase in longitudinal reinforcement ratio at strain values corresponding to onset of spalling. A similar trend is also observed when the compressive strains at significant spalling and bar buckling are compared with longitudinal reinforcement ratios.
Based on the above discussions, the axial load ratio and longitudinal reinforcement ratio are found to be the most important parameters affecting the occurrence of important damage states. Both these parameters are used to develop semi-empirical prediction expressions for all important damage states.
The confinement reinforcement ratio (ρ w ) significantly influences the properties of the confined core concrete and occurrence of bar buckling (Penelis and Kappos 1997). The increase in confinement reinforcement is expected to delay the occurrence of bar buckling due to a reduction in the effective length of the compression reinforcement. Therefore, confinement reinforcement ratio is included for developing the prediction expression for strains at bar buckling.
To summarize, the axial load ratio (n 0 ) and longitudinal reinforcement ratio (ρ l ) have strong influence at all damage states and are therefore considered for development of prediction expressions. Additionally, confinement reinforcement ratio (ρ w ) is considered only for development of prediction expression for bar buckling damage state.

Proposed semi-empirical expressions
The mathematical form of semi-empirical prediction expressions for compressive strain is based on the product of capacity and demand parameters. The axial load on the member is due to forces applied to the building and represents the demand on the member. The longitudinal and confinement reinforcement ratios are properties of RC section and are associated with the capacity of RC section. The longitudinal and confinement reinforcement ratios are summed together to represent the capacity term of the section.
Axial load ratio is the most important parameter in all damage states which delays occurrence of first yielding but accelerates occurrence of subsequent damage states. In addition, the occurrence of all damage states including bar buckling is prolonged with an increase in longitudinal reinforcement ratio. Taking all these into consideration, the following non-linear functional form is considered for the semi-empirical prediction expressions for compressive strains: Parameters A, B, C, and D in Eq. 2 are constants and are determined from nonlinear regression analysis by minimizing the square of error between experimentally observed and numerically simulated compressive strain. The important damage states, identified on the basis of mechanics of load transfer, are found to be statistically independent. Therefore, the constants of Eq. 2 are determined from experimentally-observed strain corresponding to each damage state. Since the confinement ratio does not affect strain at first yield, onset of spalling, and significant spalling, the exponent, C, of confinement ratio is taken as equal to 0 for these damage states. Table 6 presents the coefficients of best-fitted expression, number of data points, values of constants, regressions coefficient, and residual standard error for all-important damage  states. The statistical significance of resultant coefficients has been evaluated for the data set of each damage state. The P value from the statistical significance tests is also shown in Table 6. It can be seen that the null hypotheses, which are defined as "observed compressive strain is not related to combinations of sectional properties," are rejected at a 2 percent level of significance for every damage state. Thus, the proposed expression (Eq. 2) and estimated coefficients meet the statistical significance test. Table 6 also presents the mean and standard deviation of the ratios of compressive strain determined from Eq. 2 to the observed experimental data. It can be seen that the mean values are closer to unity and Coefficient of Variance (COV) is less than 0.35 for all four damage states. Thus, the proposed expression accurately predicts the compressive strain at important damage states for various sectional properties. Based on the above discussions, it is concluded that the proposed semi-empirical expressions accurately predict the concrete strain for the whole range of member damage from Light damage to Collapse damage state.

Fragility functions of important damage states
Seismic fragility functions have been developed to study the sequence of damage and the likelihood of their occurrence. The fragility functions are also used for the selection of strain limits of important damage states. The seismic fragility function has been defined as the normal cumulative distribution function (CDF) (Singhal and Kiremidjian 1996). It is formulated as, where F ds (ε) is fragility function for damage state ds evaluated at ε and Φ is the normal cumulative distribution function. DS is uncertain damage state, ds is particular damage state, and Ε is the uncertain demand parameter (i.e., compressive strain). The particular value of compressive strain is denoted as ε. The mean and the standard deviation of damage state ds are denoted as ε ds are β ds , respectively. Figure 6 shows the cumulative probability of occurrence of important damage states as a function of extreme compressive strain for the database along with the fitted normal cumulative distribution function (CDF). It can be seen that the fragility functions of important damage states are non-overlapping. This confirms the sequential nature of predicted damage states, for example, the spalling in RC members occurs only after yielding of longitudinal reinforcement.
The mean and standard deviation of the fragility functions for important damage states are also shown in Fig. 6. The standard deviations at onset of spalling (moderate damage state) and significant spalling (severe damage states) are found close to uncertainties in the thresholds of all damage states provided by HAZUS (1999) (β M = 0.4). The standard deviations are lower for Light and Collapse damages states. This implies larger uncertainties associated with the prediction of intermediate damage states compared to relatively undamaged and failure states of the member.
Fragility functions are used in this study to determine probability of occurrence of damage states for different limiting strain values. In order to ensure that the damage state is identified with reasonable conservatism in practical applications, this study proposes that the damage categorization should be based on the concrete strain corresponding to mean minus one standard deviation level. Table 7 shows the corresponding probability of exceedance of the damage states. From the table, it can be seen that when the damage is categorized as Moderate, there is 6.33% probability of actual damage being Severe, and only 0.49% probability of the member being in Collapse damage state. This implies that in only 6.33% of cases, a higher damage state has been underestimated as Moderate damage state. Similarly, the other damage states are also categorized with consistent conservatism, as shown in Table 7. The present study shows that the collapse damage state of a member occurs due to bar buckling (crushing of core concrete). The fracture of longitudinal or transverse reinforcement occurs after bar buckling. Thus, on the basis of experimental observations used in the present study, it is concluded that the probability of occurrence of fracture of longitudinal or transverse reinforcement for mean minus standard deviation (0.0116) strain limit of bar buckling is negligible. The strain limits at any other probability of exceedance (e.g. 50% i.e. mean value) can also be determined from Table 5.

Applications of proposed strain prediction expressions and recommendations
The proposed strain prediction expression can be used to determine the compressive strain at important damage states for seismic assessment of the existing buildings. This information can be used to estimate available capacity and the extent of expected damage for various seismic demands. For the seismic design of new buildings, the inverse of the proposed expression can be used to finalize member properties (specifically member dimensions) for the chosen limiting value of compressive strain under specified loads. In performance-based design, the proposed expression can be used to determine reinforcement areas for chosen limiting values of compressive strain at damage states of interest for given member dimensions. Design parameters such as the maximum allowable longitudinal reinforcement ratio and the minimum required confinement reinforcement ratio can be determined using the proposed expression for a specified probability of occurrence, target damage state, and axial load ratio of the member. The target damage state for the estimation of member properties can be determined from the objectives of single or multi-performance seismic design.
The following recommendations are made on the basis of strain values at important damage states against section properties (summarized in Sect. 4.2) and proposed strain prediction equations: 1. The study shows that the occurrence of initial and intermediate damage states can be delayed by reducing the axial load ratio. This can be achieved by increasing the member size. When larger cross-section is used to reduce axial load ratio, it makes a building stiffer, and thus also reduces its displacement demand. The initial and intermediate damage states are related to Enhanced Performance Objectives in Performance Based Design (ASCE-41 2017) and the adoption of larger member sizes can help in achieving these performance objectives. The extent of increase in member size for satisfying enhanced performance objectives can be estimated from the proposed strain prediction equations. 2. The study also shows that confinement reinforcement plays a crucial role in controlling the ultimate or collapse damage states. The collapse performance of an RC building can be enhanced by increasing the confinement ratio along with a reduction in the axial load ratio. The required increase in confinement ratio that is required for improving the collapse performance level can be estimated using the proposed strain prediction equations.

Summary and conclusions
This paper discusses the progression of seismic damage to axially-loaded well-detailed RC members dominated by the flexural response. Important damage states from light damage to extensive damage for axially-loaded well-detailed RC members have been defined on the basis of the mechanics of load transfer. The limiting strain values for identifying these damage states and their statistical bounds are also quantified. A new member damage classification based on the mechanics of the RC section and loss in force capacity is proposed. Four important member damage states have been defined, corresponding to (1) first yield of reinforcement, (2) onset of spalling, (3) significant spalling, and (4) bar buckling or core crushing. It is shown that the progression of damage is well represented by extreme fiber compressive strain as the ERP. The compressive strain at important member damage states is determined by analytically reproducing experimentally observed force-displacement relationships. The semi-empirical expressions to determine compressive strain at important damage states are developed using data from a number of experimental observations. The fragility functions of important damage states in terms of compressive strain are also developed to quantify the conservatism associated with the identified damage states. Based on the investigations presented in the paper, the following major conclusions are drawn.
1. The axial load ratio has an important role to govern member damage. As is well known, this study also shows that the presence of axial load delays the occurrence of first yield of reinforcement. However, the presence of high axial load ratio is found to advance the occurrence of subsequent damage states. 2. The compressive strains at important damage states are found to be dependent on axial load ratio, longitudinal reinforcement ratio, and confinement reinforcement ratio. 3. The proposed semi-empirical expression accurately predicts the compressive strain at important damage states for a wide range of section properties. 4. The strain bounds for moderate, severe, and extensive damage states are found to be overlapping, showing that the damage state is not uniquely related to the strain levels.
The study shows that the damage state at a particular strain value must be determined based on sectional properties of the member. 5. The proposed expression is useful for seismic assessment of existing RC buildings.
The proposed expression also provides the basis to estimate member properties that are required for limiting damage to a particular state during the seismic design of new buildings. 6. The study shows that increasing member size helps in achieving Enhanced Performance Objectives, which are related to the occurrence of initial and intermediate damage states. However, the collapse performance level can be enhanced only by increasing confinement reinforcement. 7. The mean and standard deviation of compressive strains at important damage states presented in the study provide the benchmark to evaluate conservatism of strain limits specified in published literature and the design standards. 8. The fragility functions show that the standard deviations of compressive strains at first yield of reinforcement (slight damage) and bar buckling (severe damage) are lower as compared to other damage states. This implies that quantification of moderate damage has higher uncertainty than the quantification of light or severe damage. Author contributions All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by SRS. The first draft of the manuscript was written by SRS and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Funding The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Conflict of interest
The authors have no relevant financial or non-financial interest to disclose.