Evaluation of seismic performance of as‐built and retrofitted reinforced concrete frame structures with LAP splice deficiencies

Several studies have looked at the development of hinge models to simulate the hysteretic response of flexure‐ and shear‐critical reinforced concrete (RC) beam‐column components from damage initiation to onset of gravity collapse. However, few studies have been conducted to develop similar models for older type bond‐critical beam‐column components. First, using existing experimental data, this paper describes the calibration of simple and efficient hinge models to simulate the inelastic hysteretic response of as‐built and retrofitted splice‐deficient columns. Subsequently, to demonstrate the applicability of the hinge models, the seismic performance of as‐built and fibre‐reinforced polymer (FRP) ‐retrofitted two‐ and four‐story nonductile RC frame buildings with splice‐deficient columns are assessed using nonlinear dynamic analysis procedures. The results show that buildings with bond‐critical columns may have a lower collapse potential than buildings with shear‐critical columns, implying that in certain cases a longer splice length may actually worsen performance. According to the analyses, local retrofitting of the columns can significantly improve the seismic performance of the buildings. Contrary to results presented in this study, a significant U.S. evaluation methodology for RC frame buildings, FEMA P‐2018, indicates that buildings with bond‐critical columns have similar collapse potential as buildings with shear‐critical columns. Modifications are proposed to improve the FEMA P‐2018 provisions. The modelling approach presented in this paper are recommended for incorporation into ASCE/SEI 41, engineering practice and future research.


INTRODUCTION
capacity of the poorly detailed RC columns in these buildings. Relative to other buildings, these structures are classified as collapse-prone and, therefore, a high priority for evaluation and retrofit. Despite the "collapse-prone" tag attached to pre-1980s RC structures, reconnaissance data have shown that only a relatively small proportion of these buildings have collapsed in past earthquakes. Otani 4 used data from four strong seismic events with magnitude M ≥ 6.7 to show that a significant number of pre-1980s buildings suffered operational (i.e., light to moderate) damage following strong earthquake shaking, while only about 2-10% of buildings suffered partial or total collapse. Likewise, despite the presence of several pre-1980s structures in the Christchurch (New Zealand) region, only one pre-1980s building collapsed following the 2010/2011 Canterbury earthquakes. 5 Instead of classifying all pre-1980s buildings as collapse-prone, it is important to identify the buildings with the highest collapse risk, such that they can be prioritised for risk mitigation efforts to improve overall community resilience.
As a step towards risk mitigation, seismic assessment procedures have been developed to assess the response of these older type structures at the component and global levels. Owing to a lack of sufficient knowledge on the seismic response of RC frame structures on the global and component level, codified seismic assessment provisions in seismically active regions, for example, the United States 6 and New Zealand, 7 can be inherently conservative leading to potentially unwarranted retrofits or potentially building closures 8 ; this conservatism can pose a significant financial burden on the global economy and lead to poor prioritisation of those buildings that most threaten community seismic resilience. With the aim of tackling these problems, FEMA developed guidelines for identifying older RC buildings with high collapse potential (see FEMA P-2018 9 ).
Prior to the 1980s, RC design codes allowed the presence of inadequate splices in the plastic hinge regions of RC columns; hence, it is not uncommon to have a significant number of buildings with inadequate column splice detailing. Experimental studies [10][11][12] have demonstrated that the response of RC columns with short splices (a term we use interchangeably with splice-deficient columns) may be significantly different from that of columns with continuous reinforcement. The response of columns with short splices is expected to have an influence on the global response of the structure, especially if a weak column-strong beam mechanism is prevalent. However, many efforts 13,14 focusing on modelling and collapse assessment of RC buildings have not accounted for these effects and there is uncertainty as to how best to represent the force-displacement response of components with short splices. 15 State-of-the-art seismic assessment procedures (e.g., ASCE/SEI 41-17, FEMA P-2018 9 , Eurocode 8-3 16 ) adopt varying approaches for the seismic assessment of columns with short splices. Following the ASCE/SEI 41-17 procedure, the majority of columns with short splices in the plastic hinge region will be classified as bond-critical (defined in ASCE/SEI 41-17 as columns controlled by inadequate development or splicing). ASCE/SEI 41-17 provides a formulation tailored for evaluating the response of columns classified as bond-critical. On the other hand, FEMA P-2018 provisions assume that all columns with short splices can be treated as columns with continuous reinforcement (in terms of deformation capacity). To account for a reduced deformation capacity at lateral failure in columns with provided splice length (l o ) shorter than the 'minimum required lap length for the column to develop its full ultimate deformation capacity as if its bars were continuous (l ou,min )', Eurocode 8-3 recommends that the full deformation capacity be multiplied l o /l ou,min . It is noteworthy that compared to ASCE/SEI 41-17, Eurocode 8-3 does not make provisions for evaluating the deformation capacity at the axial failure of concrete columns.
Experimental results have shown that columns with short splices in the plastic hinge region may experience bondcritical, flexure-shear or brittle shear-critical failure modes depending on detailing and boundary conditions. A review of these experimental studies 17 has shown that the seismic behaviour of RC columns with short splices is significantly influenced by the failure mode. Hence, fragility estimates based on ASCE/SEI 41-17, and FEMA P-2018 provisions, which do not appropriately account for failure mode effects, may not be representative of the actual fragility of RC frames with splice-deficient columns.
In this paper, using experimental data from various studies, simple and efficient hinge models are calibrated to simulate the inelastic hysteretic response of as-built and FRP-retrofitted columns with short splices. Subsequently, the proposed models are adopted in evaluating the seismic performance of two-and four-story archetype buildings designed to the 1967 Uniform Building Code 18 through nonlinear dynamic analyses. For each archetype building set, three different structures were considered -a structure with shear-critical columns, a structure with bond-critical columns and a structure with FRP-retrofitted columns -for a total of six frames. The seismic fragility of the structures at four limit states (immediate occupancy [IO], life safety [LS], collapse prevention [CP] and global collapse) were evaluated. Results show the impact of column detailing vulnerabilities and retrofit on the seismic performance of older type RC frame structures. Furthermore, the collapse potential of each building was evaluated using the FEMA P-2018 methodology. Subsequently, recommendations are proposed for improving the FEMA P-2018 methodology for assessing buildings with splice-deficient columns. Building code F I G U R E 1 Minimum lap-splice length requirements of pre-1980s building codes F I G U R E 2 Influence of failure mode and axial load ratio on the buffer between drifts at lateral failure (θ u ) and axial failure (θ a ) in columns with lap splice deficiencies (data available in Opabola and Elwood 17 )

RESPONSE OF COLUMNS WITH SHORT SPLICES IN THE PLASTIC HINGE REGION
Older RC design codes [19][20][21][22] allowed for splice length between 20d b and 30d b to be provided in the plastic hinge region of RC columns (see Figure 1). Modern design codes 23,24 recommend that splices should be located away from the regions of maximum moment demands. Aside from allowing short lap splices, older codes allowed the use of poorly anchored and widely spaced transverse reinforcement in the plastic hinge regions of RC columns. Hence, these columns are susceptible to brittle failure mechanisms.
To assess the response of columns with short splices in plastic hinge regions, a number of experimental programs 10,25 have been carried out on column specimens considering various governing parameters -axial load level, splice length, stirrup spacing and loading protocol. A summary of test data on columns with short straight splices is available in Opabola and Elwood. 17 Based on available experimental data, three distinct failure modes have been observed -brittle-shear, flexureshear and bond-critical failure modes. Similar to columns with continuous reinforcement, the response of brittle-shear and flexure-shear critical columns with short splices is characterised by failure through a diagonal plane (with the distinction depending on whether this occurs before or after flexural yielding). In these failure modes, following the development of the critical diagonal failure plane, an axial load is resisted through a shear-friction mechanism along the diagonal plane. 26 Experimental results in Figure 2 show that, as a result of the brittle nature of the shear-friction mechanism, loss of vertical load-carrying capacity typically occurs shortly after lateral failure -with a buffer of typically less than 1.5% drift. However, according to experimental data, 10 fixed-end rotation accounts for up to 90% of total rotation in bond-critical columns, i.e., bond-critical columns are dominated by a rocking mechanism. Due to the pseudo-ductile nature of the rocking mechanism of bond-critical columns, these columns tend to have a larger buffer between lateral failure (defined as 20% drop in lateral resistance) and axial failure (See Figure 2). In some of the reported test programs, no axial failure was in fact reported at drift levels above 7%. It is noteworthy that state-of-the-practice seismic assessment provisions 6,9,16 do not recognise the large buffer between lateral and axial failure in bond-critical columns. Hence, there is little or no information on how this large buffer affects the global seismic performance and collapse fragility of RC frame structures.

Existing approaches for modelling columns with short splices
The focus here is on macro-modelling approaches suitable for nonlinear response history analysis procedures in seismic assessment guidelines. Among these, Cho and Pincheira 27 proposed a macro-modelling approach for modelling the response of columns with short splices. The model consists of an elastic beam-column element with three zero-length springs -a nonlinear rotational spring to capture flexural deformation, a nonlinear shear spring to capture shear deformation and a nonlinear rotational bond-slip spring to capture the bond-slip deformation. The moment-slip rotation relationship of the bond-slip zero-length spring is calibrated by combining results from two analyses. Firstly, the uniaxial pull-out force versus uniaxial slip deformation is estimated using a one-dimensional analysis on an isolated reinforcing bar embedded in concrete. Then, a moment-curvature analysis is carried out to relate the pull-out force to the moment applied to the critical section of the column. Furthermore, the slip rotation is computed by dividing the estimated uniaxial slip deformation, obtained from the one-dimensional analysis on an isolated reinforcing bar embedded in concrete, by the difference between the effective depth and neutral axis depth obtained from the moment-curvature analysis.
The major disadvantage of the Cho and Pincheira 27 approach is that it requires multiple steps in the analysis of the column element which can be time-consuming. Furthermore, the moment-curvature analysis is based on an assumption of concrete-rebar deformation compatibility -an assumption that may not be valid once bond degradation has occurred. Also, as pointed out by Chowdhury and Orakcal, 28 using predefined loading and unloading rules for the bond-slip spring tends to introduce incompatibility between the bond-slip and flexural deformations, compromising the reliability of this type of approach.
ASCE/SEI 41-17 recommends modelling parameters for predicting the inelastic response of bond-critical columns. These modelling parameters for bond-critical columns were derived from regression analyses on a database of columns with short splices which included shear-critical columns with short splices. 15 Analytical studies 17 have shown that the ASCE/SEI 41-17 provisions are significantly conservative. Hence, hinge models using the ASCE/SEI 41-17 modelling parameters may not adequately capture the inelastic hysteretic response of bond-critical columns.
A number of other macro-and micro-modelling approaches have also been proposed by other researchers. 28,29 Aside from the complexities in the definition and implementation of these models, there are also difficulties in achieving numerical convergence under cyclic loading. There is also evidence that, despite the complexities of these models, they may be unable to capture the potential post-yield flexure-shear interaction as observed in experimental tests. 25,30

Proposed approach
A failure mode-based modelling approach is adopted in this section. The adopted approach is based on the well-established knowledge that the contribution of deformation components (i.e. flexure, shear and bar slip) to total deformation is dependent on failure mode; such that, for effective and efficient modelling, it is appropriate to first identify the probable failure mode of the column. The failure mode of a column with a short splice can be evaluated using the strength-based procedure. 17 In this strength-based procedure, the probable flexural strength, the lateral strength corresponding to maximum developable tensile stress in the splice, and the undegraded shear strength are compared. As presented in Opabola and Elwood, 17 the adequacy of the strength-based approach was validated using a database of 29 columns with short splices, achieving a 93% accuracy in predicting the observed failure mode. Interested readers are referred to Opabola and Elwood 17 for more information. Once the probable failure mode is determined, the column can be modelled as presented below. Based on experimental data, 10 the fixed-end rotation component contributes about 90% of the inelastic deformation of bond-critical columns. Hence, this study explores a lumped plasticity modelling approach for modelling the response of bond-critical columns. In the proposed approach, the column model consists of an elastic beam-column element and zero-length nonlinear rotational spring elements which models the bond degradation and rocking phases of the column (see Figure 3A). The moment-rotation relationship of the nonlinear rotational spring is modelled using the modified Ibarra-Medina-Krawinkler (IMK) uniaxial material model 31 incorporated in OpenSees, 32 the software platform adopted for this study. In this model, the response of the column is idealised as a trilinear moment-rotation backbone. The key backbone parameters are the effective stiffness (EI eff ), moment corresponding to the maximum developable tensile stress (M b ), peak moment strength (M max ), pre-capping rotation capacity (θ p ), post-capping rotation capacity (θ pc ), ultimate rotation capacity (θ u ) and the residual strength capacity (M res ) (see Figure 3B). In addition, a number of hysteretic parameters are used to capture cyclic strength and stiffness deterioration (Λ) and pinching behaviour (κ). The definitions of these parameters are described below.

Effective stiffness (EI eff )
The effective stiffness is defined as the secant stiffness to the strength corresponding to the maximum developable tensile stress in the splice (i.e., M b ). Various recommendations 6,16,[33][34][35] have been proposed to predict the effective stiffness of RC beam-column components. In this study, we adopt the Opabola and Elwood 35 formulation (Equation 1). To account for the contribution of bar slip and shear deformation components, Opabola and Elwood 35 proposed a modifier to the flexural rigidity coefficients of ASCE/SEI 41-17 as a function of aspect ratio (a/d). Equation (1) recognises that the contribution of bar slip and shear deformations to the elastic response is negligible in slender columns. For a database of 29 columns with short splices, Equation (1) provides an effective stiffness estimate with a mean ratio of observed to calculated of 1.0 and coefficient of variation (CoV) of 23.7%. 17 where the effective stiffness coefficient from ASCE/SEI 41 (α) is the flexural rigidity coefficient from ASCE/SEI 41-17 6 and taken as 0.3 for axial load ratio N/A g f' c ≤ 0.1, α = 0.7 N/A g f' c ≥ 0.5 and linear interpolation for 0.1 < N/A g f' c < 0.5; a/d is the aspect ratio of the column.

Peak strength
In bond-critical columns, bond failure is initiated once the maximum developable tensile stress f s is reached in the tensile bars. This corresponds to column moment capacity M b . Typically, M b is an output of the earlier mentioned strength-based failure mode classification procedure for columns with short splices. Given that longitudinal reinforcement yielding and strain hardening do not occur in bond-critical columns, the plateau in the moment-rotation can be assumed to be flat, i.e., M max /M b equals unity. As shown in Figure 4, the computed lateral strength corresponding to maximum developable tensile stress provides a good estimate of the measured peak strength of bond-critical columns with a mean measured-topredicted ratio of 0.95 and CoV of 14.5%.

Pre-capping rotation capacity (θ p )
Experimental results 10,11 have shown that, once bond failure is initiated, increased lateral drift demands will make the lap splices in the plastic hinge region progressively unzip along the provided splice length leading to significant concreterebar bond degradation. The progressive unzipping of the lap splice, together with the onset of damage to the concrete in the compression zone, leads to lateral failure of bond-critical columns. Opabola and Elwood 17 proposed a mechanistic formulation (Equation 2) for evaluating the inelastic rotation capacity at lateral failure (θ p ) of bond-critical columns with short splices. The key assumption of the formulation is that the plastic deformation capacity of a bond-critical column is related to the ability of the splices to continue transferring the maximum developed tensile stress (i.e. once the developable tensile stress starts dropping, lateral failure occurs). Assuming that tensile stress transfer along splices is through a shear-friction mechanism, the stress transfer capability of a splice is expressed in terms of a clamping coefficient which is defined as the ratio of tensile force in the splice to the clamping force from the transverse reinforcement, that is, ′ ∕ ( , ∕ ) (see Equation 3 and Figure 5A).
where λ ο accounts for the influence of axial load on the rotation capacity of the column and it is equal to 1 for N/A g f' c ≤ 0.2 and 0 for N/A g f' c ≥ 0.5. A linear interpolation is required for 0.2 < N/A g f' c < 0.5. a nl,o relates the plastic rotation capacity to the clamping coefficient and can be estimated as: where A' s is the area of tensile reinforcement, f s is the maximum allowable tensile stress in the longitudinal reinforcement (computed as in Elwood et al. 36 ), A vt is the transverse reinforcement area, f yt is the yield strength of the stirrups, s is the stirrup spacing, and l s,prov is the provided splice length. Equation (2) provides a plastic rotation capacity estimate with a mean ratio of observed to calculated of 1.05 and CoV of 11.4% (See Figure 5B).

(A) (B)
F I G U R E 5 Adequacy of formulation for predicting pre-capping rotation capacity (θ p ) (data available in Opabola and Elwood 17 ). (A) Clamping coefficient versus measured θ p and (B) measured θ p versus predicted θ p using Equation (2) Post-capping rotation capacity (θ pc ) The rocking model (Equation 4) proposed by Opabola et al. 37 for evaluating the plastic rotation capacity at the axial failure of columns with smooth reinforcement is adopted here for predicting the plastic rotation capacity at axial failure (b nl ) of bond-critical columns with short splices. The adoption of the rocking model is in line with experimental evidences 10,11 that the response of bond-critical columns, following splice failure, is similar to that of a rocking body.
where h is the column depth, a is the shear span the coefficient K accounts for the effect of aspect ratio on the rocking behaviour of RC columns: In this study, without loss in accuracy, we further simplify Equation 5 into 6 by assuming that the column effective depth is equal to 80% of the column depth.
We also further assume that concrete damage due to vertical bond-splitting cracks reduces the undamaged column depth to the depth of the concrete core (assumed to be 0.6 h); such that A g is replaced with A c (assumed to be 0.6A g ) in Equation (4).
Replacing A g with A c (assumed to be 0.6A g ) and replacing K with Equation (6), Equation (4) is simplified into Equation (7).
The adequacy of Equation (7) is shown in Figure 6. Equation (7) provides an estimate with a mean measured-tocalculated ratio of 1.1 and a CoV of 14.5%. Due to limited data, to avoid overestimating the plastic rotation capacity at the axial failure of bond-critical columns, it is recommended that in adopting Equation (7) N/A g f' c should not be taken less than 0.1.
The post-capping rotation capacity (θ pc ) of a bond-critical column with a short splice can be estimated as the difference between b nl , computed from Equation (7), and pre-capping rotation capacity (θ p ) as computed using Equation (2). The residual strength ratio, defined as the ratio of residual resistance (M res ) to the strength corresponding to the maximum developable tensile stress (M b ), of the bond-critical column dataset in the Opabola and Elwood 17 study, was measured. ASCE/SEI 41-17 adopts Equation (8), as a function of transverse reinforcement ratio (ρ t ), for predicting the residual strength ratio c of bond-critical columns. The adequacy of Equation (8), as well as a proposed recommendation, is presented in this section. Figure 7A shows that Equation (8) overestimates the residual strength ratio of bond-critical columns with a mean measured-to-calculated ratio of 0.77 and CoV of 26%. Hence, it was decided to develop a refined formulation to predict the residual strength ratio of bond-critical columns.
The statistical relationship between the measured residual strength ratio, axial load ratio and transverse reinforcement ratio is presented in Table 1. While Figure 7A suggests that the residual strength ratio is well-correlated with the transverse reinforcement ratio (i.e., the governing parameter, according to ASCE/SEI 41-17), Figure 7B and Table 1 suggest that axial load ratio is more correlated with residual strength ratio (correlation coefficient of 0.82). Table 1 also suggests that the correlation between residual strength ratio and transverse reinforcement ratio (correlation coefficient of 0.6) may be largely attributed to the correlation between axial load ratio and transverse reinforcement ratio (correlation coefficient of 0.48). TA B L E 1 Correlation coefficients between measured residual strength ratio (c), transverse reinforcement ratio (ρ t ) and axial load ratio Linear regression analyses were carried out to develop a formulation for predicting the residual strength ratio of bondcritical columns. Two equations were developed for predicting the residual strength ratio of bond-critical columns -the first considering axial load ratio and transverse reinforcement ratio (Equation (9)) and the second considering axial load ratio only (Equation (10)).
Equation (10) looks similar to the ASCE/SEI formulation for predicting the residual strength ratio provision for columns not controlled by inadequate development or splicing along with the clear height (See Equation (11)). As shown in Figure 7B, however, Equation (11) overestimates the residual strength ratio of bond-critical columns.
The adequacy of Equations (8-11) is presented in Table 2. As shown in Table 2, Equation (8) has the least accuracy. The inclusion of transverse reinforcement ratio does not make Equation (9) superior to Equation (10) (See Table 2); hence, Equation (10) is proposed for predicting the residual strength ratio of bond-critical columns. A comparison of Equation (10) with the measured residual strength ratio of bond-critical columns is presented in Figure 7B.

Cyclic strength and stiffness deterioration parameter (Λ) and pinching behaviour parameter (κ)
The hysteretic energy dissipation-based cyclic deterioration rules incorporated in the IMK were developed by Rahnama and Krawinkler. 38 As defined in Rahnama and Krawinkler, the cyclic deterioration in excursion i is defined by the parameter β i and expressed as: where E i is the hysteretic energy dissipated in excursion i, E t is the hysteretic energy dissipation capacity and E t = λM y θ p = Λθ p , ∑E j is the hysteretic energy dissipated in all previous excursions and c is an exponent defining the rate of deterioration (typically taken as 1.0 33 ). The parameter β can be used to incorporate various types of strength and stiffness deterioration phenomena into the hysteresis model. Using a procedure similar to that adopted in previous studies, 33,39 the hysteretic and pinching parameters (Λ and κ) were calibrated to the hysteretic force-displacement response for 24 bond-critical columns in the database compiled by Opabola and Elwood. 17 Existing literature 40 have shown that cyclic degradation is highly dependent on the axial load level and confinement of the concrete core. It is known that cyclic energy dissipation capacity reduces as axial load level increases or confinement level reduces. The influence of axial load ratio and transverse reinforcement ratio on the calibrated hysteretic and pinching parameters are shown in Figure 8. As shown in Figure 8A and B, Λ is not significantly influenced by the considered variables. The modest variation in Λ values between test specimens may be attributed to the fact that axial load ratio and transverse reinforcement ratio are implicitly included in E t through θ p (See Equations 2 and 12). For the dataset, the mean calibrated value of Λ equals 0.42 with a CoV of 19%. For the purpose of modelling bond-critical columns, a value of 0.4 is recommended for Λ. Given the scarcity of experimental data, additional test data are needed to further validate the recommendations provided here.
The pinching parameter κ is defined in Figure 3B. Figure 8C and D suggest that κ reduces as confinement level reduces. The relationship between κ and transverse reinforcement ratio can be expressed using Equation (13). Equation (13) A comparison of numerical and experimental results 10,11,41 from test specimens subjected to cyclic loading protocol is presented in Figure 9. As shown in Figure 9, the force-displacement response of bond-critical columns with short splices are well captured using the proposed approach. The proposed approach is recommended for adoption in ASCE/SEI 41.

3.2.2
Modelling shear-critical and flexure-shear critical columns with short splices A number of approaches have been proposed for modelling the response of shear-critical columns. Elwood 42 developed a shear limit-state material that can be combined with fibre elements or flexural rotation spring elements. In the Elwood approach, once the flexural force-displacement curve reaches the shear failure limit surface, a nonlinear shear spring is activated. The activated nonlinear spring controls strength and stiffness deterioration. Axial failure can be detected using the Elwood axial limit-state material. LeBorgne and Ghannoum 43 proposed a pinching limit-state material that works with a force-based and/or deformation-based limit surface, triggering a degrading behaviour when a force and/or deformation limit is reached.
F I G U R E 1 0 Measured backbone curves for an as-built column specimen with lap-splice deficiency, FRP-retrofitted and code-conforming column specimens (data from Harajli and Dagher 12 ) Similar to the rotation spring element proposed for bond-critical columns, Haselton et al. 33 have proposed approaches for modelling the cyclic response of flexure and flexure-shear critical columns using zero-length rotational spring elements controlled by trilinear backbone curves.
The three aforementioned approaches can be adopted for modelling flexure-shear critical and shear-critical columns with short splices. For the purpose of the current study, however, the Elwood 42 approach has been adopted. It is noteworthy that test data from shear-critical and flexure-shear critical columns with short splices were part of the dataset adopted in calibrating the modelling parameters for the Elwood 42 approach. 26,44

Modelling FRP-retrofitted columns with short splices
One significant challenge to evaluating the collapse fragility of retrofitted buildings is the fact that state-of-the-art seismic assessment procedures do not provide guidance on acceptance criteria and modelling parameters for retrofitted columns. Studies 45,46 have modelled retrofitted columns using fibre-element models with the confined concrete backbone properties modified to account for the enhanced confinement from the FRP wrapping or steel jacketing. In the aforementioned studies, 'additional' bond-slip deformation is typically modelled using a rotational bond-slip spring.
A key attribute of fibre-element models is their capability to capture inelastic deformations due to flexural curvature. Experimental results 12,47,48 on jacketed columns with short splices have, however, shown that fixed-end rotation accounts for up to 80% of total deformation during the inelastic phase. In fact, the main difference between as-built bond-critical columns and the retrofitted bond-critical columns is the enhanced deformation capacity and energy dissipation capacity, lesser pinching, lower strength and stiffness deterioration rates as a result of the improved concrete-rebar bond and external confinement to the concrete provided by the jackets. Hence, it may be more appropriate to explore the applicability of a lumped plasticity model to better capture the concentration of inelastic deformation at the column ends.
For the purpose of this study, experimental data from cyclic tests on FRP-retrofitted bond-critical columns with short splices were examined. 11,12,49 In some of the examined test programs, a code-conforming column specimen had also been tested in order to compare the retrofit effectiveness. Figure 10 compares the measured force-displacement backbone response of an as-built bond-critical critical with short splices with those of an FRP-retrofitted nominally identical column, and a code-conforming column (i.e., continuous longitudinal reinforcement and conforming stirrup detailing). As shown in Figure 10, the response of the FRP-retrofitted column specimen was nearly identical to that of a code-conforming column, with very similar flexural strength and deformation capacity. The authors 12 also noted that the energy absorption capacity of the FRP-retrofitted and code-conforming column specimens were similar.
In this study, it is recommended that the behaviour of the FRP-retrofitted column can be modelled using a hinge model (Figure 3). The recommended modelling parameters FRP-retrofitted columns are described below.
As reported in previous studies, 50,51 effective stiffness formulations developed for regular columns are generally applicable to FRP-retrofitted columns. Hence, Equation (1) is also adopted for FRP-retrofitted columns. As earlier mentioned, the spread of plasticity in FRP-retrofitted columns with short splices is mainly through strain penetration. The equivalent plastic length (L p ) 51 of an FRP-retrofitted column can be evaluated as g + 0.044f y d b ; where g is the gap between the jacket and the footing, 0.044f y d b is the strain penetration length, f y and d b are the yield strength and diameter of the longitudinal reinforcement respectively. The pre-capping rotation capacity of the FRP-retrofitted column θ p = (ϕ u -ϕ y )L p , 51 where ϕ u and ϕ y are the ultimate and yield curvatures of the column and are estimated through a moment-curvature analysis. In this study, however, we recognise that in certain cases (e.g. under low axial load), FRP-retrofitted columns may suffer bar fracture rather than concrete compression failure; 50 hence the ultimate curvature capacity is defined to correspond to a tensile strain of 0.06 52 or a concrete compressive strain accounting for the FRP-enhanced confinement level. 51 It is also noteworthy that the column yield strength (M y ) and peak strength (M max ) for defining the moment-rotation relationship of the retrofitted column are outputs of the moment-curvature analysis.
As described in experimental studies 12 and shown in Figure 10, the post-capping response of FRP-retrofitted and modern columns are similar. For lack of appropriate formulations specifically calibrated to a large database of FRP-retrofitted columns, it is recommended that the post-capping plastic rotation capacity (θ pc ) of an FRP-retrofitted column be computed using formulations proposed by Haselton et al. 33 using equivalent transverse reinforcement (stirrups) parameters and the axial load level, force-displacement nonlinear modelling parameters. The definition of the equivalent reinforcement parameters is defined as follows: The confinement level provided by the FRP wrapping can be expressed in terms of an equivalent volumetric transverse reinforcement ratio (ρ v,eff ): where f f is the strength of the FRP wrapping in the hoop direction, f yt is the yield strength of the transverse reinforcement in the column, and ρ f is the FRP volumetric reinforcement ratio, computed in accordance with ACI 440. 53 Similar to the procedure previously adopted by Alvarez and Brena, 54 assuming an area of transverse reinforcement A v,eff , the equivalent transverse reinforcement s eff corresponding to the equivalent volumetric transverse reinforcement ratio (evaluated using Equation (14)) can be computed, such that the provided FRP wrapping can be expressed as equivalent stirrups with yield strength of f yt , area of A v,eff and spacing of s eff . Figure 11 shows the adequacy of the proposed modelling approach for FRP-wrapped bond-critical columns tested by Ghosh and Sheikh. 11 As shown in Figure 11, the cyclic response of the test specimens is well captured using the proposed approach. Additional studies are needed to explore the applicability of this approach to other jacketed beam-column components with continuous deformed bars as well as with continuous and spliced plain bars.

As-built concrete frames
For the purpose of this study, a set of two-story and four-story space frame structures, designed according to the 1967 Uniform Building Code, 18 were considered. Each of the archetype frame structures is assumed to be situated at a Southern California site with coordinates 33.996 • N, -118.162 • W. Inventory studies 55 have shown that about 70% of pre-1980s RC structures in high seismic regions of the United States are between 1 and four stories. It is also noteworthy that the archetype frame structures, adopted in this study, are also representative of pre-1980s RC frame buildings in other seismically active regions, for example, New Zealand. A survey of RC frames constructed in Wellington, New Zealand between 1935 and 1975 showed that 52% of buildings were 2-3 stories and about 21% are 4-6 stories. 56 The archetype frame structures consist of three bays of equal widths (17 ft (5182 mm)), a ground story height of 13 ft (3965 mm) (see Figure 12). A constant story height of 11 ft (3353 mm) was adopted for the remaining stories in the building. As expected in structures from the design era, it was assumed that the floor slabs with a thickness of 8 in (∼200 mm) were monolithically cast with the beams. The expected combinations of dead load (DL) and live load (LL) under seismic demands, 57 on the structure (comprising dead load and live load) was taken to be equal to 209 psf (10 kPa).
All the structural components were designed using the working stress design method, as was typical of the era. For each structure, the column dimension (18 in by 18 in (457 mm x 457 mm)) and reinforcement detailing (longitudinal reinforcement ratio of 1.9% and transverse reinforcement ratio of 0.26%) were kept constant over the height of the building. Each beam is 30 in (762 mm) deep and 18 in (457 mm) wide. It is assumed that the splices in the beams are not located in their plastic hinge regions. Typical of pre-1970s structures, the beam-column joints were unreinforced.
After the design, the column and beam strengths were compared. For the internal and external subassemblies, the values of ΣMc/ΣMb correspond to 0.47 and 0.77, confirming that a column mechanism was the most probable mechanism. Also, the equivalent column moment corresponding to joint shear failure computed following the strength hierarchy approach, 58 was higher than the column yield strength -making joint failures unlikely. It is noteworthy that a number of experimental studies 59 have tested as-built and retrofitted RC soft-storey frame structures with splice-deficient columns, similar to case-study buildings in this paper.
For each building type, three variations were considered, making a total of six buildings. The first variation assumes that the columns have straight splices with a splice length corresponding to 30d b in the plastic hinge region of all columns in the building. Using the failure mode identification approach, 17 these columns are predicted to be bond-critical.
The second variation assumes the splice lengths in the columns were increased to 35d b , which is sufficient for the longitudinal reinforcement to reach its yield strength, and the columns able to attain their flexural yield capacity. As a result of the shear capacity ratio of the columns (V p /V n ≈ 0.8), however, they are susceptible to a flexure-shear critical mechanism as observed in tests. 25,30 Finally, for the third variation, the columns of the first variation were retrofitted to avoid a splice or shear failure. The design of this retrofit is described below.

Seismic evaluation and retrofit design in accordance with ASCE/SEI 41-17
In this study, retrofits of the splice-deficient columns are designed according to ASCE/SEI 41-17 Tier 3. Prior to retrofit design, the behaviour of each as-built frame was evaluated using the nonlinear dynamic procedure (NDP) in accordance with section 7.4.4 of ASCE/SEI 41-17. The adopted hinge-type models are as described previously in this paper. Nonlinear response history analyses were carried out to evaluate the structural performance of each as-built structure at BSE-2E hazard level (computed as S X1 = 0.87 g; S XS = 1.38 g). The site-specific seismic hazard was characterised based on the US Geological Survey (USGS) hazard maps. The structures are classified as Risk Category II and a site class D soil was assumed. For the two-and four-story structures, 11 ground motion records were selected using the conditional mean spectrum approach. 60 While according to ASCE/SEI 41-17, the building performance level is defined as the combination of the performance levels of nonstructural and structural systems, in the current study, however, only the structural performance levels are considered. Under the BSE-2E hazard level, a structure must fulfil the CP structural performance level. Analytical results showed that the as-built structures do not fulfil the CP structural performance level at the BSE-2E hazard level. Further discussions on the seismic fragility of the as-built structures at various performance objectives are presented subsequently in this paper. Here, we design the retrofit to achieve CP structural performance level at the BSE-2E hazard level for comparison with the as-built buildings.
The choice of retrofit technique is heavily influenced by economic considerations, desired continued functionality of the building during the retrofit, and desired global inelastic mechanism for the retrofitted structure. Retrofit strategies can be in form of component-level and subassembly-level modifications (e.g., selective strengthening using jacketing techniques or selective weakening techniques) and/or system-level modifications (including, but not limited to, structural strengthening, structural stiffening, mass reduction, or removal of potential structural weaknesses).
Three retrofit techniques were considered -concrete jacketing, steel jacketing and FRP wrapping. However, the FRPwrapped column was preferred as the column dimension remains almost the same, thereby keeping the useable floor area unchanged. Also, desirably, the natural period of the structure is unchanged. In addition, previous studies have shown that well-designed FRP-wrapped and steel-jacketed columns are able to achieve similar improvement in deformation capacities, 61 so results may be similar in both cases.
The ASCE/SEI 41-17 Tier 3 evaluation was adopted for the retrofit design. The retrofit was intended to improve the drift capacity and energy dissipation capacity of the columns. The FRP retrofit design entailed providing confinement to ensure sufficient clamping is provided to inhibit splice failure and also improve the shear strength of the columns. The FRP retrofit design was carried out in accordance with ACI 440 53 provisions. A shear capacity ratio of 0.6 is desirable to achieve a flexure-dominated mechanism. For the purpose of this study, however, a V p /V n of 0.4 was targeted as this value corresponds to the shear capacity ratio of a modern column with similar cross-section and transverse reinforcement conforming to ACI 318-19. 23

NONLINEAR SIMULATION MODELS OF ARCHETYPE STRUCTURES
The two archetype buildings described in the previous section are symmetric in the plan; hence, it is assumed that the structures respond to ground motion excitation without any torsional behaviour. Each of the six aforementioned variations was modelled using a two-dimensional frame that corresponds to one of the interior frames of the structure. The seismic mass of the frame was lumped at each beam-column joint and the ground floor columns were assumed to be fixed at the foundations and the floor diaphragms were assumed rigid. The beams were modelled using a combination of elastic beam-column elements and zero-length nonlinear rotational spring elements following the Haselton et al. 33 approach. The columns were modelled using the approaches described previously in Section 3. As noted previously, a joint mechanism was not expected to govern the behaviour, and, hence, no explicit joint modelling was incorporated in To account for nonlinear geometric effects, the P-delta transformation incorporated in OpenSees was employed. Damping was modelled using a value of 2% for Rayleigh damping in the first and third modes. 9,14,62 The fundamental period from eigenvalue analysis with the effective stiffness assumptions described above of the two-story building model is 0.71 s. The fundamental period of the four-story building model is 1.24 s.

Pushover analyses
Nonlinear pushover analyses, using inverted triangular load pattern was performed on each of the six building variations -two-and four-story buildings with bond-critical, flexure-shear critical (hereafter referred to as shear-critical for brevity) and retrofitted columns. Figure 13 plots the base shear versus roof drift relationship for all six variations. The base shear has been normalised by the seismic weight on the frame. As shown in Figure 13, the retrofitted structure has the highest deformation capacity. The buildings with shear-critical columns had a sharp drop in strength which is attributed to significant degradation in lateral resistance after lateral failure in the ground floor columns. In comparison, despite experiencing the earliest onset of lateral resistance degradation, the buildings with bond-critical columns were able to resist a significant amount of lateral shear demands at large drift ratios.

Incremental dynamic analysis
The collapse behaviour of each building variation was assessed using the incremental dynamic analysis (IDA) approach. 63 A suite of 22 pairs of recorded far-field ground motions representative of large seismic events were adopted from FEMA P-695. 64 The magnitude and distance for the ground motion records in the adopted suite were larger than M w = 6.5 and 10 km, respectively. For each model, the maximum story drift demand and spectral acceleration corresponding to the fundamental period (S a (T 1 )) were recorded. Global collapse capacity was defined to correspond to the smallest scaling of a ground motion sufficient to achieve either a maximum inter-story drift of 10% or when θ pc (see Figure 3B) of 50% of the columns at any story level is exceeded. This approach has previously been adopted by Galanis

Seismic fragilities at different performance objectives
Using results from the IDA analyses, the responses of all structures were analysed for four different limit states. The first three limit states are based on the ASCE/SEI 41-17 performance objectives, while the last limit state corresponds to the collapse capacity as defined in the previous section. ASCE/SEI 41-17 makes provisions for seismic assessment to satisfy three performance objectives -IO, LS and CP. For the sake of brevity, these performance objectives are not defined here and interested readers are referred to ASCE/SEI 41-17. The acceptance criteria for different RC components to satisfy each of the aforementioned ASCE/SEI 41-17 performance objectives, as fractions of inelastic rotation capacities at lateral (a) and axial failure (b), are provided in Chapter 10 of the standard. For reinforced concrete columns, the IO limit is defined to correspond to an inelastic rotation capacity of zero for bond-critical columns and 0.15a ≤ 0.5% for other failure modes. LS and CP performance objective limits correspond to 0.5b and 0.7b, respectively (See Table 10-8 of ASCE/SEI . For bond-critical columns, the corresponding drift ratios to these limits were computed using Equations (2) and (4). Fragility functions for each limit state were developed using results from IDA on the six buildings. Assuming a lognormal distribution for the demand and capacity, the probability of reaching or exceeding a specified damage state given an intensity measure (IM) can be estimated as: where D and C are the seismic demand and structural capacity, respectively; S D and S C are the median values of the demand and structural limit state, respectively; β D|IM is the record-to-record uncertainty as a function of S a (T 1 ); β C is the dispersion of all limit states and is taken as 0.3 65 ; and β M is the modelling uncertainty and is taken as 0.2. 66 The obtained fragility curves for the IO, LS and CP performance objectives for the two-and four-story buildings are presented in Figure 14. The collapse fragilities for the buildings are presented in Figure 15. As shown in Figure 14, the structures with the bond-critical columns have the lowest median S a (T 1 ) for the IO performance level. This is not surprising given that the IO drift limit for the bond-critical column corresponds to an inelastic rotation that equals zero. The buildings with bond-critical columns, however, have larger S a (T 1 ) values at the LS and CP limit states than the shearcritical buildings. This superior performance is attributed to the fact that both limit states are a function of inelastic rotation capacity at axial failure and the rocking mechanism in bond-critical columns is sufficient to delay axial failure following splice failure. For the LS, CP, and collapse limit states, the median S a (T 1 ) values for the bond-critical buildings are about 30% larger than that of the shear-critical buildings. It is worth recalling that the only difference between the bond-critical and shear-critical columns was a splice length of 30d b for bond-critical versus 35d b for shear-critical; hence, a shorter splice length is in fact resulting in an improved LS, CP, and collapse performance.
Regardless of the number of stories, the retrofitted buildings have the largest median S a (T 1 ) values at all limit states. At the collapse limit state, the median S a (T 1 ) values of the two-and four-story retrofitted buildings are 80 and 120% larger than the two-and four-story buildings with shear-critical columns. Likewise, the median S a (T 1 ) values of the two-and four-story retrofitted buildings are 36 and 70% larger than the two-and four-story buildings with bond-critical columns.
It is noted, however, that the two-and four-story case study buildings do not include columns with high axial loads (i.e. axial load ratio greater than 35%). At high axial loads, bond-critical columns with light transverse reinforcement may fail at similar drifts to that of shear-critical columns. A higher level of axial load may also trigger flexure-shear interaction. The likelihood of this interaction needs to be assessed on a case-by-case basis as it is also dependent on the geometric, material and reinforcement detailing. Also, the results presented in this study may not be valid for buildings dominated by a joint mechanism.

7.2
Influence of reinforcement detailing on collapse performance As previously discussed, columns in older type buildings typically have light transverse reinforcement with lap-spliced or continuous longitudinal reinforcement. The probable failure mode of these columns is dependent on this reinforcement detailing. Figure 16 presents the influence of longitudinal reinforcement detailing on the median collapse fragility of the original two-(2S-U) and four-story (4S-U) buildings with varying splice length in the columns. As shown in Figure 16, due to inadequate transverse reinforcement detailing, the seismic performance of buildings with provided splice length sufficient to attain flexural yield (prior to shear failure) perform worse than the bond-critical columns (i.e., l s,prov ≤ 30d b ). The axial capacity of shear-damaged columns is not significantly influenced by longitudinal reinforcement detailing (i.e. presence or absence of lap splices). 17,26 Hence, the median collapse fragility of all the buildings with shear-damaged columns (i.e. l s,prov ≥ 35d b ) is similar. As shown in Figure 16, older type buildings with bond-critical columns may perform better than buildings with shear-critical columns having continuous reinforcement (and sparsely spaced transverse reinforcement) when the response is dominated by a soft-story mechanism. As shown in Figure 16, for the two-(2S-R) and four-story (4S-R) buildings with retrofitted columns, the provision of additional lateral confinement (from the FRP wraps) in the poorly detailed columns prevents lap-splice and shear failure modes, allowing the column response to be flexure-dominated. For the retrofitted buildings, a similar median collapse fragility was attained (See Figure 16). Figure 16 shows that providing additional lateral confinement in the columns  switches the relationship between the median collapse S a and provided splice length from a step function to a linear function.

Evaluation of collapse potential using FEMA P-2018 methodology
FEMA P-2018 9 provides a simple methodology for evaluating the collapse potential of older type concrete buildings. This methodology first identifies the critical stories and components for various structural configurations. Then, given a target displacement demand, the collapse potential of a structure can be evaluated based on comparison of drift demands and drift capacities of critical components, as described in the FEMA P-2018 document. The component drift capacities adopted in the methodology are intended to represent the drift at which loss of gravity load-carrying capacity occurs. Following the FEMA P-2018 methodology, the collapse potentials of the four as-built and the two retrofitted case-study structures were evaluated for a seismic hazard level corresponding to ASCE/SEI 41-17's BSE-2E (i.e., 5% probability of exceedance in 50 years). According to FEMA P-2018, the deformation capacity of columns with short splices can be assessed by assuming the longitudinal reinforcement is continuous and based on the column's shear capacity ratio (V p /V n ), the failure mode may be categorised as flexure-critical, flexure-shear critical or brittle shear-critical -the bond-critical mechanism is not recognised in terms of its effect on drift demands or drift capacities.
The rating of each building, determined following the FEMA P-2018 methodology, is presented in Table 3. As shown in Table 3, the current FEMA P-2018 approach would assign the same Building Ratings (which is akin to a collapse probability, and higher values indicate greater collapse potential) for the buildings with shear-critical and bond-critical columns. This is because FEMA P-2018 does not currently recognise the higher drift at axial capacity associated with bond-critical columns. However, results from the fragility functions derived from the IDA here show that the buildings have different collapse probability at the selected hazard level (see Table 3). This discrepancy results in a different relative ranking and assessment of the bond-critical buildings than suggested by the collapse fragility analysis.
We proposed a modified approach for FEMA P-2018 for evaluating the collapse drift capacity of bond-critical columns. If Equation (3) is adopted in computing the plastic rotation capacity at axial failure for bond-critical columns, the influence of the column failure mode on the Building Rating becomes more consistent with the evaluated collapse risk. Considering this proposed modification in assessing the Building Rating (see Table 3), it is evident that buildings with shear-critical columns are more collapse-prone than buildings with bond-critical columns and should be prioritised for retrofit. The buildings with bond-critical columns are, however, still categorised as "High Risk" and, hence, should not be overlooked when developing risk mitigation policies.

CONCLUSIONS
In this study, the seismic performance of as-built and retrofitted pre-1970s RC frame structures with splice-deficient columns was evaluated using probabilistic nonlinear dynamic analysis procedures. In comparison with shear-critical columns (with or without short splices), bond-critical columns with splice deficiencies have a larger buffer between lateral and axial failure. For typical arrangements of transverse reinforcement found in older columns, it was identified that a column failure mode can switch from bond-critical to shear-critical by simply increasing the splice length from 30d b to 35d b . FRP wrapping of splice-deficient columns, to improve the bond behaviour of the splice region and the shear capacity of the plastic hinge region, was also considered to demonstrate the improved performance of retrofitted buildings. As a first step, using experimental data from various studies, simple and efficient hinge models are calibrated to simulate the inelastic hysteretic response of as-built and FRP-retrofitted columns with short splices. In the proposed approach for the as-built column, the probable failure mode was assessed using a strength-based procedure. Thereafter, the forcedisplacement backbone for the column model was developed using mechanistic formulations for predicting the precapping and post-capping rotation capacity of the columns considering the probable failure mode. The adequacy of the proposed approach was demonstrated using experimental data. A limitation in this study is the size of the experimental dataset used in validating the proposed approach. Additional test data are needed to further validate the recommendations provided in this study.
Using the proposed modelling approach, the seismic performance of a set of two-and four-story RC frames was evaluated at different limit states. The response of the considered frames is dominated by a soft-story mechanism. Incremental dynamic analyses, using a suite of 22 pairs of far-field ground motions, showed that the median collapse S a (T 1 ) for the retrofitted buildings was 40-70% larger than that of the bond-critical buildings and 80-120% larger than that of shearcritical buildings.
Due to the rocking behaviour of bond-critical columns after splice failure, the median S a (T 1 ) for the bond-critical buildings was about 30% larger than that of the shear-critical buildings at three limit states -LS, CP and collapse. This better behaviour of the buildings with bond-critical columns is because the columns were able to sustain their axial load-carrying capacity at large drift demands. However, the shear-critical buildings have a median S a (T 1 ) at IO about 33% larger than the bond-critical building because bond-critical columns experience the initiation of bond failure at low drift levels. It is noted, however, that the two-and four-story case study buildings do not include columns with high axial loads. At high axial loads, bond-critical columns with light transverse reinforcement may fail at similar drifts to that of shear-critical columns. Also, the results presented in this study may not be valid for buildings dominated by a joint mechanism.
The collapse potential of each considered building was assessed using FEMA P-2018 provisions. Results showed that, contrary to results from nonlinear dynamic analyses, the current FEMA P-2018 procedures produce ratings of buildings with bond-critical and shear-critical columns that indicate similar collapse potential. This is attributed to the fact that FEMA P-2018 does not account for the bond degradation effect on column deformation capacity. It is, therefore, proposed that future editions of FEMA P-2018 should account for the influence of failure mode on collapse capacity of columns with short splices using approaches proposed in this study.