Limit states for post-earthquake assessment and recovery analysis of ductile concrete components

For post-EQ residual capacity assessment, it is important to understand the effect of the cyclic loading history imposed on concrete components on the performance of these components in future EQ events. This section reviews experimental results from nominally identical components tested under a varying number of cycles at each peak deformation/load demand to evaluate the effect of loading history.

Kawashima and Koyama (1988) tested three nominally identical, flexure-controlled columns with an aspect ratio of 5.4 subjected to different numbers of cycles at each drift demand to assess the influence of the number of cycles on damage progression and hysteretic response of RC columns. Figure 2 shows the damage accumulation history for two nominally identical columns subjected to 3 and 10 repeated cycles at each drift demand. This figure indicates that the influence of the number of cycles per drift level (N_cyc) on the component damage level only became significant when the drift demand exceeded 2.1% (corresponding to a ductility demand of 4). Prior to reaching this ductility demand, the damage progression of both specimens was similar irrespective of the number of repeated cycles at each drift level. As indicated in Figure 2 with the red dashed line, this 2.1% drift demand corresponds to the initiation of LSL. As shown in Figure 2, the adopted definition of the point of initiation of LSL throughout this article is the point beyond which a lower peak strength corresponds to the subsequent larger peak deformation demand. It is should be noted that this point is different from the 20% LSL typically adopted in defining the ultimate deformation capacity of components. Similar conclusions were reached when examining the results from another set of tests reported by the same authors on flexure-controlled columns with an aspect ratio of 3.8 (with similar transverse reinforcement detailing as the specimens with an aspect ratio of 5.4).

Similar results have also been observed for RC structural walls. For example, Oesterle et al., (1979) tested two identical flexure-shear-controlled walls with barbell-shaped cross-sections under two different reversed cyclic loading protocols to investigate the significance of loading history on the behavior of the walls, as shown in Figure 3. As can be noted, one reversed cycle at a rotational ductility of five (i.e. roughly drift at the initiation of LSL) resulted in response of Wall B9 being comparable to that of Wall B7, which had sustained three complete reversed cycles at ductility demands of 1 to 5. Thus, the authors concluded that structural wall behavior under load reversals was not dependent on the entire previous load history but instead was dependent on whether the wall was previously loaded to initiation of LSL or not.

These limited experimental results suggest that the number of cycles applied prior to the initiation of LSL may not have a significant influence on the damage progression and hysteretic response of a concrete component, provided that low-cycle fatigue limit state is not triggered.

Experimental results (Chiu et al., 2021; Colmenares and Santa María, 2021; Maeda et al., 2017; Marder, 2018; Moscoso et al., 2021; Opabola and Elwood, 2023) are reviewed to understand the limit state at which demands from a past seismic event have compromised the residual capacity of a component in future seismic events. Each test program includes two or more tests on nominally identical components—one of which is subjected to a standard cyclic loading protocol (Figure 4a), and the other specimen(s) subjected to an initial loading protocol (an EQ or a cyclic loading) prior to applying a standard cyclic loading protocol (Figure 4b).

First, six nominally-identical ductile beam specimens reported by Marder et al. (2018) were examined. The test specimens included: (1) Specimen CYC subjected to a quasi-static standard cyclic protocol; (2) Specimen CYC-DYN subjected to a pseudo-dynamic standard cyclic protocol; (3) Specimen LD-1 subjected to an initial pseudo-dynamic long-duration displacement history (with a peak drift demand of 1.36%) is applied and followed by a quasi-static standard cyclic protocol; (4) Specimen LD-2 subjected to an initial pseudo-dynamic long-duration displacement history (with a peak drift demand of 2.17%) followed by a quasi-static standard cyclic protocol; (5) Specimen P-1 subjected to an initial pseudo-dynamic pulse-type displacement history (with a peak drift demand of 1.36%) followed by a quasi-static standard cyclic protocol; (6) Specimen P-2 subjected to an initial pseudo-dynamic pulse-type displacement history (with a peak drift demand of 2.17%) followed by a quasi-static standard cyclic protocol. The measured drift at the initiation of LSL for specimen CYC-DYN was 1.9%, which means that specimens LD-1 and P-1 were subjected to an initial pseudo-dynamic loading protocol with a peak drift (1.36%) smaller than the drift at the initiation of LSL from a nominally identical specimen subjected to a standard cyclic protocol. On the other hand, specimens LD-2 and P-2 were subjected to initial dynamic loading protocols with a peak drift (2.17%) larger than the drift at the initiation of LSL from a nominally identical specimen subjected to a standard cyclic protocol. A summary of the test results is presented in Table 1. The force–displacement plots of the beam specimens are presented in Figure 5. It is noted that Figure 5 presents only the hysteretic behavior during the standard cyclic loading protocol of specimens LD-1, LD-2, P-1, and P-2. The tests showed that the initial EQ protocol influenced the initial stiffness of the damaged specimens (the residual displacements in the EQ-damaged specimens are also noteworthy—see Figure 5e). This article focuses on the influence of initial EQ protocol on residual strength and deformation capacity of components. Studies on the influence of initial EQ protocol on the initial stiffness of the EQ-damaged frame components are available in existing literature (Abdullah et al., 2020; Di Ludovico et al., 2013; Marder, 2018; Opabola and Elwood, 2023).

As shown in Figure 5a and b, irrespective of the type of initial EQ protocol (i.e. long duration or pulse-type), lower initial stiffness and residual displacement in the EQ-damaged specimens LD-1 and P-1, the force–displacement plots of both specimens were similar to that of specimen CYC once the EQ-damaged LD-1 and P-1 were pushed to drifts larger than 1.36%. Furthermore, the deformation capacities of CYC, LD-1, and P-1 were similar (see Table 1). The similarities in the force–displacement plots of specimens CYC, EQ-damaged LD-1, and P-1 at drift demands > 1.36% can be attributed to the fact that LSL was not initiated during the initial EQ protocol (i.e. the peak drift demand of 1.36% is less than the measured drift at LSL in specimen CYC – 1.9%).

These experimental observations also suggest that the hysteretic behavior of an EQ-damaged component would be similar to that of a pristine nominally identical component in a subsequent event at drift demands larger than the initial peak drift demand the EQ-damaged component was subjected to during the initial EQ.

On the other hand, LSL was reached during the initial EQ protocol in specimens LD-2 and P-2, resulting in reduced residual capacity—as observed by the lower residual strength and deformation capacity in EQ-damaged LD-2 and lower deformation capacity in EQ-damaged P-2 (see Table 1 and Figure 5c and d). The difference in the force-displacement response of EQ-damaged LD-2 and P-2 relative to specimen CYC is attributed to the fact that the initial EQ-protocol on pristine LD-2 and P-2 did not initiate LSL.

It is noteworthy that similar conclusions have been reached in a system-level study on two nominally identical frame structures subjected to different loading histories by Cecen (1979). Cecen (1979) concluded that two nominally identical EQ-damaged frames, subjected to different displacement histories, would have similar responses under a similar subsequent ground motion provided that the intensities of all the preceding displacement histories are not greater than that of the subsequent ground motion.

Results from tests on ductile beam specimens reported by Opabola and Elwood (2023) and Chiu et al. (2021) led to conclusions similar to those by Marder et al. (2018). For the sake of brevity, no detailed information of the experimental programs by Opabola and Elwood (2023) and Chiu et al. (2021) is provided here.

Similar to frame members, experimental results from studies on the residual (reserve) capacity of RC walls are reviewed (i.e. tests involving the application of two sequential loading protocols). It is important to note that although the results reviewed herein are from tests on non-ductile (ordinary) flexure-controlled walls and shear-controlled walls, the concepts and conclusions are equally applicable to flexure-controlled walls.

Colmenares and Santa María (2021) conducted an experimental program on four flexure-controlled walls to understand the effect of number of cycles on the residual capacity of typical Chilean RC walls with unconfined boundaries. The baseline wall was tested under a standard cyclic loading protocol, while the other three walls were subjected to a cyclic loading protocol with a constant amplitude at a target drift of ∼1.0% prior to applying the standard cyclic protocol. The varied parameter between the three tests was the number of cycles applied during the initial constant-amplitude protocol, which ranged from 30 to 120 cycles. Since the target drift was smaller than the drift capacity of the baseline wall at the initiation of LSL (roughly 2.0%), the number of cycles in the initial protocol did not have a noticeable effect on the strength and deformation capacity of the wall tests.

Furthermore, Moscoso et al. (2021) conducted an experimental study on walls with details similar to those tested by Colmenares and Santa María (2021), except that Moscoso et al. (2021) used a standard cyclic loading protocol for the initial loading. Wall RW2 S2.5 was tested initially under a standard cyclic loading protocol with maximum drifts of + 1.76% and –1.57%, which were close to drift capacity at the initiation of LSL of a companion specimen (RW1 S2.5) tested to failure under only a standard cyclic protocol. As a result, during the subsequent standard cyclic protocol, RW2 S2.5 was able to sustain only one additional cycle with a drift demand greater than the maximum drift reached during the initial protocol. However, for the other two wall tests (RW4 S2.5 and RW6 S1.75), since the maximum drift reached during the initial loading protocol was only about 50% of drift at the initiation of LSL, no noticeable effect on the strength and deformation capacity of the walls was observed.

The results above suggest that the behavior of an EQ-damaged component follows a “peak-oriented” assumption—that is, provided the drift at LSL of an EQ-damaged component was not reached in any preceding displacement histories (and no low-cycle fatigue issues), the residual capacity of the component is uncompromised.

Figure 6 is a graphical representation of the peak-oriented assumption using an arbitrary force-displacement plot for an arbitrary component that has been subjected to three arbitrary cyclic events, with each event imposing larger a peak deformation demand than the preceding one. Figure 6 shows that a damaged component will only be able to reach its original peak strength if the original component was subjected to initial cyclic demands lower than the drift capacity at the initiation of LSL (i.e. point 8). As shown in Figure 6, ductility demand histories reduce the effective stiffness (taken as the slope of the first force-displacement curve each event) of a damaged component (from K₁ to K₂ to K₃), irrespective of whether the drift at the initiation of LSL is reached or not.

As previously concluded, provided that the low-cycle fatigue limit state is not triggered, the influence of prior loading history on the residual capacity of ductile concrete components becomes significant only once the deformation at the initiation of LSL (θ_LSL) is reached. State-of-practice standards for seismic evaluation and retrofit, such as ASCE/SEI 41, provide formulations or tables for evaluating component deformation capacities, typically at 20% LSL and at axial failure (AF). In this article, the deformation at 20% LSL is referred to as “lateral failure” or “LF.” ASCE/SEI 41 provides component deformation capacities at LF in terms of either plastic hinge rotation (i.e. modeling parameter a) or total plastic hinge rotation (i.e. modeling parameter d) (see Figure 7 for the definition of these modeling parameters).

For structural walls, ASCE/SEI 41-17 (2017) nonlinear deformation-based modeling parameters are given as plastic hinge rotations (i.e. Parameters a and b). However, approved updates of these provisions (Abdullah and Wallace, 2021b) for ACI 369.1-23 (2023) (which will be incorporated into the next ASCE/SEI 41) use total hinge rotation capacities for the deformation-based modeling parameters of flexure-controlled walls (Figure 7b), which include both the elastic and plastic deformations contributed by the hinge region (Abdullah, 2019). For this study, these updated modeling parameters are used for walls, as they represent the most accurate and up-to-date provisions and recommendations for flexure-controlled walls.

There are currently no codified formulations (in ASCE/SEI 41-17 (2017) or other documents) for predicting θ_LSL of concrete components. In this study, rather than developing new formulations for predicting θ_LSL, it was considered more efficient to adopt a simple methodology of calibrating a multiplier, x, to be applied to the ASCE/SEI 41 deformation capacity at LF (θ_LF) to obtain θ_LSL (see Figure 8a and Equation 1). If the ASCE/SEI 41 model for θ_LF is a reasonable estimate of mean deformation capacity at LF, then x will be less than unity.

Using information from collated databases of experimental tests (described later) on concrete components (frame elements and walls), the statistical distribution (median and dispersion) of multiplier x can be evaluated by fitting a distribution to the calculated ratio of deformation capacity at LSL to deformation capacity at LF (see Figure 8b). The subsequent section describes the methodology proposed to develop the detailed visual inspection and repair limits, which are expressed as fractions of the ultimate component deformation limit.

In this section, a general methodology is proposed for defining the detailed visual inspection limits (θ_IT) of structural components, which is defined such that there is a low likelihood that a location of damage will be missed. A reliability analysis, where all relevant sources of uncertainty are considered, is needed to accomplish this. For this study, a value of p ≤ 10% is recommended based on engineering judgment. In addition to considering the uncertainty β_LSL in predicting θ_LSL (see Figure 8b), uncertainties in demand estimates from ground motion (β_gm) and modeling (β_model) need to be incorporated. The adopted approach in this study accounts for the fact that there could be a significant level of uncertainties in the demand estimates for various reasons, that is, the proximity of the structure to a ground motion recording station, availability and reliability of data obtained from building instrumentation, knowledgeability of material and structural properties, and complexity of adopted modeling techniques. For example, it is reasonable to assume that β_gm and β_model would be low for instrumented buildings and high for non-instrumented buildings. β_gm would be low for a building with a ground motion station or a building situated near a ground motion station. The argument for β_model follows the assumption that numerical models of instrumented buildings can be properly calibrated using data obtained from the instrumentation. From a post-EQ assessment perspective, β_model could be further reduced if the engineer is able to improve their structural models using post-event observations on the expected behavior and failure modes of the components.

The authors are unaware of available studies with recommended values for β_gm and β_model for adoption in post-EQ probabilistic modeling and analysis. FEMA P-58 (FEMA, 2018) (in Table 5–2 of FEMA P-58) recommends values for β_model depending on the quality and completeness of the analytical model, where β_model values of 0.10, 0.25, and 0.40 are recommended for superior, average, and limited quality models. In this study, it is assumed that the β_model for instrumented and non-instrumented buildings would be in the range of 0.10 to 0.40, respectively.

On the other hand, the record-to record variability values (β_RTR) provided in Table 5–3 of FEMA P-58 are deemed too conservative for adoption as β_gm because it is expected that there would be a fair amount of available information from ground motion stations at or near the building, and the post-EQ assessment would likely not involve analyses using a suite of ground motion records. This study assumes that β_gm ranges from zero for buildings with ground motion stations on-site to 0.70 for buildings without any ground motion stations on a similar site class within approximately 50 km (Abrahamson, 2021).

In this study, it is assumed that the random variables representing the deformation parameters (demand and capacity) are statistically independent and lognormally distributed. In structural reliability terms, the limit state function for triggering the median ${\tilde{\theta }}_{\mathrm{LSL}}$ can be written as Equation 2, where ${\tilde{\theta }}_D$ is the median component deformation demand:

The reliability index Z, accounting for the uncertainties in capacity and demand, can be computed as:

The reliability index Z defines probability p of ${\tilde{\theta }}_D$ exceeding ${\tilde{\theta }}_{\mathrm{LSL}}$. The relationship between Z and p is defined as:

where Φ(.) is standard normal distribution function.

In this section, the limit state of interest is the detailed visual inspection limit, which is activated when ${\tilde{\theta }}_D$ ≥ θ_IT. Replacing ${\tilde{\theta }}_D$ with θ_IT in Equation 3, θ_IT can be expressed as:

where 1/$e^{Z\sqrt{{\beta }_{\mathrm{gm}}^2+{\beta }_{\mathrm{model}}^2+{\beta }_{\mathrm{LSL}}^2}}$ is the multiplier with respect to the median value of ${\tilde{\theta }}_{\mathrm{LSL}}$ and defines the fraction of ${\tilde{\theta }}_{\mathrm{LSL}}$ that θ_IT needs to be equal to, or to exceed, in order to trigger the limit state of interest. As previously stated, θ_LSL can be defined as a fraction of the deformation capacity at LF (θ_LF), that is, θ_LSL = x·θ_LF. Hence, θ_IT can also be expressed as:

As such, the detailed visual inspection limit multiplier x_IT is defined using the probability p of exceeding θ_LSL (defined in terms of a reliability index Z), considering various sources of uncertainty. In Equation 6, β_LSL and x are the only parameters influencing x_IT which are dependent on the component type.

The purpose of the component repair limit is to identify structural damage that needs safety-critical repair, which is defined as the deformation limit beyond which the lateral strength and deformation capacity of the component is compromised, and that safety-critical repair may be required to restore the structural characteristics of the component. A component below repair limit is still likely to see repairs for durability, aesthetics, and maybe serviceability.

In addition, the repair limit is used to fine-tune the analytical model of the building to accurately predict demands and damage, that is, to reconcile model results and observed damage. Therefore, the repair limit should not be as conservatively defined, as was the case for the detailed visual inspection limit, because the need for repair has to be confirmed through detailed visual inspection of the structural components by a professional engineer. For this study, the proposed repair limit is defined as the median estimate of θ_LSL. As is demonstrated later using experimental data, the repair limit typically corresponds to the deformation at the initiation of longitudinal bar buckling in flexure-controlled components.

As part of this study, a database of cyclic tests on ductile beams and columns was collated, where ductile components are defined as components where no brittle response was observed, ratio of transverse reinforcement spacing to effective depth (s/d) is not greater than 0.5, the shear capacity ratio (defined as the ratio of flexural strength to undegraded shear strength as defined by Sezen and Moehle, 2004) of the component is less than 0.6, and the transverse reinforcement is properly anchored with 135 degree hooks. This definition was adopted from ASCE/SEI 41-13 (2014). It is noted that all considered specimens were subjected to significant drift demands beyond peak strength. Furthermore, only test specimens with N_cyc of 2 to 4 were considered.

The collated frame component database consists of 48 beams and 61 columns. The distribution of key parameters is presented in Table 2. Four of the beam specimens have concrete compressive strength greater than 100 MPa (14.5ksi). These four data points were not excluded from the database because it was of interest to extend the applicability of the framework over a wide range of steel and concrete properties.

For each component in the database, four drift capacities were extracted from the measured force-displacement backbones, as the drift at:

Furthermore, wherever reported, information on deformation demands at bar yielding, concrete cover spalling, longitudinal bar buckling, longitudinal bar fracture, and transverse bar fracture were collected. An example of a force-displacement curve with reported damage states is presented in Figure 9. It is noted that P-delta effects were consistently corrected for the backbones of the column specimens as described in Berry et al. (2004).

The reported damage states from the frame component database were reviewed to identify the drift/rotation capacities at the initiation of LSL, which are used to propose detailed visual inspection and repair limits in the subsequent subsections. The typical damage sequence includes the formation of horizontal flexural cracks (in some components, diagonal-tension shear cracks are also observed), tensile yielding of longitudinal reinforcement, delamination and spalling of concrete in the extreme compression fiber, buckling of longitudinal reinforcement followed by fracture of longitudinal reinforcement with or without unhooking of transverse reinforcement.

It was of interest to identify the component damage state that influences the initiation of LSL in beam-column components. For the tests in the database, relevant information on drift corresponding to the reported observed damage were collected, which indicated that the drift at the initiation of LSL corresponds to the initiation of bar buckling (Figure 10). The ratio of measured drifts at the initiation of LSL to the drifts at the initiation of bar buckling has a mean of 1.0 (with a coefficient of variation of 0.10) for beams and a mean of 1.0 (with a coefficient of variation of 0.17) for columns. The high correlation between drift at the initiation of LSL and drift at bar buckling can be explained by the fact that bar buckling is associated with high localized strains, leading to high likelihood of fracture in subsequent cycles.

The observation that drift at the initiation of LSL corresponds to drift at the initiation of bar buckling further buttresses the choice of drift at the initiation of LSL as a threshold beyond which the component capacity is compromised. For components that experience bar buckling, it is expected that significant safety repair (typically in form of bar replacement) would be required to restore the component capacity. Prior to the initiation of bar buckling, only minor repair works related to concrete patching and epoxy injection may be required.

The authors note there may be significant uncertainty in accurately identifying the initiation of bar buckling (or kinking) by visual inspection, especially in cases where the concrete cover has not spalled yet. This was one of the main reasons this study decided to develop the proposed procedure around the deformation at LSL rather than bar buckling. This approach also allowed this study to consider other test specimens where bar buckling information was not reported.

Figure 11 presents the variation of measured drift at the initiation of LSL against the ratio of transverse reinforcement spacing to bar diameter (s/d_b) for the beam tests and the axial load ratio (N/A_gf’_c) for the column tests. As shown in Figure 11b, there is a strong correlation between the drifts at the initiation of LSL and N/A_gf’_c.

The lateral failure of beam-column components in ACI 369.1-23 is defined by modeling parameter a, which represents the plastic rotation capacity at 20% drop in lateral strength from peak strength. Following the methodology outlined above, this section seeks to define a multiplier x that needs to be applied to ACI 369.1-23 modeling parameter a for defining the component deformation limit corresponding to LSL (θ_LSL) for beams and columns. ACI 369.1-23 modeling parameter a for beams and columns are presented in Equations 7 and 8, respectively.

where N is the axial load, A_g is the gross cross-sectional area, f’_c is concrete strength, ρ_t is the transverse reinforcement ratio, V_y is the flexural yield strength, and V_o is the undegraded shear strength as provided in section 10.4.2.3.1 of ASCE/SEI 41-17.

where M/Vd is the shear span ratio, f’_c is concrete strength, ρ_t is the transverse reinforcement ratio, and f_yt is the yield strength of the transverse reinforcement.

Following the outlined methodology, the ratio of plastic rotation at the initiation of LSL to the computed modeling parameter a was calculated for each beam–column component. Figure 12 represents the distribution of the ratio of plastic rotation at LSL to the computed a-value for beams and columns, respectively. For the beams, the median ratio equals 0.75a with a logarithmic distribution of 0.46. For the column specimens, the median ratio equals 1.2a with a logarithmic distribution of 0.38. The larger multiplier for the column specimens is attributed to the conservatism of modeling parameter a for ductile columns (See Ghannoum and Matamoros, 2014). It is noted that the ASCE/SEI 41-17 (2017) modeling parameter a model was fitted to a wide range of conforming and non-conforming column specimens with different failure modes (i.e. flexure, flexure-shear, and shear). This model achieves a median estimate across all column specimens in the database. However, this resulted in a bias for certain subset of columns (e.g. flexure-controlled columns with s/d < 0.4). For the entire column database, the median ratio of measured plastic rotation at the initiation of LSL to the measured plastic rotation at LF is 0.74 with a logarithmic dispersion of 0.3.

To address the over-conservatism of the ASCE/SEI 41 formulation for ductile columns, a linear regression model was proposed to predict the drift at initiation of LSL for ductile columns (Equation 9). Equation 9 provides an estimate of the drift at initiation of LSL for ductile columns with a median ratio of 1.0 and logarithmic distribution of 0.29.

Hence, if a future update to the ASCE/SEI 41 modeling parameter a provides a good median estimate, the plastic rotation at the initiation of LSL can be updated to 0.74a.

As previously mentioned, to avoid false negatives, the detailed visual inspection limit needs to be a lower-bound estimate with a high confidence level that there is a low probability of exceeding the component deformation limit at the initiation of LSL. Using the adopted methodology presented in this article, Tables 3 and 4 present the results of a parametric study, considering ground motion (β_gm), modeling (β_model) uncertainties, and values of unit-normal distribution parameter Z (a function of probability p%), to define a range of multipliers x_IT to the computed a-values for the detailed visual inspection limit.

As shown in Table 3, adopting a Z-value of 1.28 (i.e. probability of exceeding θ_LSL p = 0.10) results in a multiplier to modeling parameter a with a range of 0.24 to 0.41 and an average value of 0.32 for ductile beams. Notably, the immediate occupancy (IO) limit in ASCE/SEI 41 is defined as 0.15a. Hence, depending on the considered modeling and ground motion uncertainties, the detailed visual inspection limit for ductile beams ranges from 1 to 2 times ASCE/SEI 41 IO limits.

As shown in Table 4 adopting a Z-value of 1.28 (i.e. probability of exceeding θ_LSL p = 0.10) results in a multiplier to modeling parameter a with a range of 0.4 to 0.7 and an average value of 0.54 for ductile columns. The detailed visual inspection limit for ductile columns ranges from 2 to 4.6 times ASCE-41 IO values.

The component deformation limits corresponding to the repair limit are proposed to correspond to the median estimate of the deformation at the initiation of LSL (i.e. initiation of bar buckling), which corresponds to a plastic rotation capacity of 0.75a for ductile beams and 1.0θ_LSL for ductile columns.

A comprehensive database (Abdullah, 2019; Abdullah and Wallace, 2019), which stores data on more than 1100 RC wall tests reported in the literature, was utilized to obtain a dataset of ductile (conforming) flexure-controlled walls. The reported information includes three major clusters of data: (1) information about the test specimen, test setup, and axial and lateral loading protocols; (2) computed data, for example, moment-curvature relationships and wall shear strength according to ACI 318-19 (2019); and (3) test results, for example, backbone relations and failure modes. Database information related to the objectives of this study are briefly presented below; however, detailed information about the database can be found elsewhere (Abdullah, 2019).

For buildings assigned to Seismic Design Category D, E, and F, design of RC structural walls is currently governed by the requirements of ASCE/SEI 7-16 (2016) and ACI 318-19 (2019), which includes provisions for structural walls with special boundary elements (SBE) that satisfy ACI 318-19 §18.10.6.4. Since the recommendations provided herein are applied for post-EQ assessment of buildings designed based on prior versions of ACI 318, as well as the current version, the database was filtered using the following criteria, some of which are less restrictive than the detailing requirements of ACI 318-19 §18.10.6.4 to cover walls with SBEs that are complaint to previous version of ACI 318 since 1983. It is noted that for simplicity, these walls are termed as ductile walls:

Based on the above-selected criteria, a total of 188 wall tests (hereafter referred to as ductile dataset) were identified. Histograms for various dataset parameters for the 188 tests are shown in Figure 13, where P/A_gf’_c is the compressive axial load normalized by the measured concrete compressive strength (f’_c) and gross concrete area (A_g), and M/Vl_w is the ratio of base moment-to-base shear normalized by wall length (l_w). Analysis of reported failure modes of about 1000 wall tests indicated that the flexure- and shear-controlled walls have a nominal shear-to-flexure strength ratio (V_n/V_@Mn) > 1.0 and < 1.0, respectively, whereas walls with failure modes reported as flexure-shear are mainly scattered between 0.7 < V_n/V_@Mn < 1.3 (Abdullah, 2019). Walls with t_w less than 88 mm [3.5 in.] were excluded because use of two curtains of web reinforcement along with realistic concrete cover is not practical in such thin walls. The limit on ratio f_u/f_y is slightly less restrictive than the limit of 1.25 specified in ACI 318-19 §20.2.2.5. The specified limits on s/d_b ≤ 8.0 and A_{sh, provided}/A_{sh, required} (A_sh required by ACI 318-19 Eq 18.10.6.4b) ≥ 0.7 are slightly less restrictive than the current limits in ACI 318-19 §18.10.6.4 of 6.0 and 1.0, respectively. Finally, the cyclic loading protocols used for these 62 walls consisted for either one repeated cycle (1 wall), two repeated cycles (23 walls) or three repeated cycles (38 walls) at each load/deformation demand.

It is noted this dataset is the same dataset used by the authors to develop the updated modeling parameters and acceptance criteria for ACI 369-1.23 (see Abdullah, 2019).

The database includes reported failure mode and backbone relations (e.g. base shear-total top displacement), consisting of seven points (origin, cracking, general yielding, peak, ultimate, residual, and axial failure), as shown in Figure 14. In addition, the database includes reported deformation at key damage states such as initiation of concrete cover spalling, initiation longitudinal bar buckling, and longitudinal bar fracture, if such information is reported in the reference document. Out of the 188 wall tests in the ductile dataset, 62 walls had at least reported deformation corresponding to initiations of cover spalling and bar buckling. This smaller subset (62 walls) was utilized for the recommendations proposed herein, especially identifying hinge deformation capacity at initiation of LSL (θ_LSL). It is noted that only half of the walls in this subset had reported bar fracture because this information was either not reported or the wall did not experience longitudinal bar fracture and failed due to out-of-plane instability or, in a rare case, due to crushing of concrete in the web region next to the boundary elements (common in barbell or flanged walls). The statistics (medians and lognormal standard deviations) and fragility curves for total hinge rotation capacities at each key damage state normalized by d from ACI 369-1.23 are presented in Table 5 and Figure 15, respectively.

In this subsection, results and damage states typical of ductile flexure-controlled walls are reviewed to identify the hinge rotation capacity at initiation of LSL, which is used to propose detailed visual inspection and repair limits later. For ductile flexure-controlled walls subjected to gradual increase of cyclic demands, the damage sequence typically involves:

Review of damage states of walls in the dataset (e.g. as shown in Figure 16) suggested that initiation of LSL of ductile flexure-controlled walls is typically associated with buckling of longitudinal bars in the boundary elements. Once longitudinal bars buckle, the concrete core loses part of the confining pressure and begins to crush, and, as a result, the depth of neutral axis shifts inward, leading to a smaller moment arm and thus reduction in moment capacity. Thus, longitudinal bar buckling is considered as the damage state beyond which the residual capacity of the wall (in terms of strength and deformation capacity) is compromised, and that structural repair might be needed to restore its structural characteristic for future events.

Figure 17 compares the hinge rotation capacities of the reduced dataset of 62 wall tests at the three key damage states and indicates that the rotation capacities for all three damage states vary significantly (by a factor of 2.0–4.0), although all the walls are fully or nearly code-compliant. In addition, there is a significant buffer zone (on average about 0.01 hinge rotation) between occurrence of initiation of cover spalling and initiation of bar buckling, indicating that it is unlikely for cover spalling and bar buckling damage states to happen simultaneously in a ductile wall. As was shown in Table 5, the median values of rotation at cover spalling and bar buckling normalized by Parameter d are 0.35 and 0.86, respectively, indicating that median values of hinge rotation capacity at bar buckling are more than twice the rotation capacity at cover spalling. This suggests that the use of cover spalling as the component deformation limit for repair might be overly conservative. Figure 17 and Table 5 also indicate that there is only a slight reserve rotation capacity between bar buckling and bar fracture (median value of ratios of rotation at bar buckling to bar fracture ≈ 0.86), and that fracture of the extreme layer of longitudinal bars results in roughly 20% LSL, which is typically defined as LF (median value of ratios of rotation at bar fracture to LF ≈ 1.0). These results support the approach of using the damage state of longitudinal bar buckling as the component deformation limit for ductile flexure-controlled walls.

Finally, Figure 17 indicates that there is a significant correlation between hinge rotation capacity and a slenderness parameter, (l_w/b)(c/b) = l_wc/b², where l_w is the length of the wall, c is the depth of neutral axial corresponding to a concrete compressive strain of 0.003, and b is the width of the flexural compression zone. This parameter provides an efficient means to account for the slenderness of the cross-section (l_w/b) and the slenderness of the flexural compression zone of the cross-section (c/b) on the deformation capacity of ductile flexure-controlled walls (Abdullah and Wallace, 2019). Walls with values of l_wc/b² lower than 10 tend to be flexure-tension controlled and generally have large deformation capacities (e.g. Figure 16), whereas walls with values of l_wc/b² exceeding 70 (slender cross-sections and large compression zones) tend to be flexure-compression-controlled and generally have low deformation capacities and simultaneous occurrence of lateral and axial failures (Abdullah and Wallace, 2021a), suggesting that these walls could be considered as components always requiring inspection. Further review of the data revealed that rotation capacity at the initiation of bar buckling is moderately impacted by the longitudinal bar slenderness ratio, s/d_b, which ranges from 2.5 to 8.0 in the reduced dataset.

As discussed previously, the purpose of the component deformation limit for detailed visual inspection limit state is to help the engineer identify locations that may have potentially sustained structural damage for visual inspection. Using the proposed methodology described previously and the uncertainty in the capacity reported in Table 5, a parametric study was carried out to determine the range of multipliers on Parameter d for component deformation limits for inspection of ductile flexure-controlled walls. The results are presented in Table 6, which indicates that, depending on the variability associated with modeling and the ground motion, the multipliers range from 0.29 to 0.6, with average values of 0.44. Figure 15 indicates that a detailed visual inspection limit of 0.4d roughly corresponds to a median value of rotation at cover spalling, which confirms the rationale of the approach used to select the detailed visual inspection limit. That is, when there is 50% probability that cover spalling has occurred, visual inspection should be conducted to confirm whether the element has sustained structural damage in the form of bar buckling and/or fracture.

In the updated nonlinear acceptance criteria for flexure-controlled walls in ACI 369-1.23, IO performance objective is defined as the sum of elastic hinge rotation and 10% of the inelastic hinge rotation, that is, IO = ${\theta }_y+0.1(d-{\theta }_y)$. For ductile flexure-controlled walls, ${\theta }_y$ typically ranges from 0.3% to 0.4%. Assuming ${\theta }_y$≈ 0.35%, IO ≈ 0.4% to 0.64% rotation and ≈ 0.2 to 0.4 d. Thus, detailed visual inspection limit of 0.4d is roughly 1 to 2 times IO, as can be seen in Figure 18.

It should be recognized that there are vertical load-carrying components whose probable residual capacity may not be reliably assessed, or their post-LSL behavior is relatively brittle, that is, once the lateral strength degradation initiates, the component possesses little or no additional deformation capacity before axial failure occurs (Abdullah and Wallace, 2019, 2021a); and thus failure of such components in a subsequent event may result in catastrophic consequences. It is recommended that such components be inspected regardless of whether the inspection trigger is exceeded or not. Also, careful engineering judgment should be exercised when classifying such components as those not requiring repair. Examples of walls described in this study and are falling in this category include flexure-controlled walls with values of l_wc/b² exceeding 70.

As discussed previously, this limit is defined as the median value of hinge rotation capacity at bar buckling, that is, 50% probability of reaching initiation of LSL. Table 5 shows that the median value of hinge rotation capacity at bar buckling corresponds to a repair limit of 0.86d. To account for the possibility of bar buckling occurring prior to the reported observation (i.e. buckled bars being concealed by split but un-spalled concrete cover), a slightly smaller multiplier for repair limit, that is, 0.8 d, is proposed. Figure 15 shows that median value of rotation at bar buckling corresponds to roughly 25% probability of bars fracturing and almost 100% probability of cover spalling.

The LS performance objective in ACI 369-1.23 is defined as 0.75 Parameter e (rotation at AF), which is roughly equal to 0.9 d, resulting in the repair limit (0.8 d) being equal to about 0.9 times LS. Figure 18 compares the proposed repair limit with the hinge rotation capacity data of the reduced dataset and LS performance objective. Recommended component deformation limits.

As earlier mentioned, visual inspection limits are needed to identify locations requiring detailed inspection. For defining these limits, recommended lognormal dispersions in the demand (i.e. ground motion, β_gm, and modeling, β_model) are presented Table 7, where β_gm is defined as a function of the building distance to the closest ground motion station on the same site class. This study assumes β_gm = zero for instrumented buildings and buildings with a ground motion station on site. It is noteworthy that β_gm has been defined to account for within-event spatial correlation in ground shaking. Furthermore, β_model is taken as 0.1 and 0.2 for instrumented and non-instrumented buildings. Based on these dispersions, component deformation limits for visual inspection are selected from Tables 3, 4, and 6, and presented in Table 8. Furthermore, Table 8 presents the component deformation limits for safety-critical repair, which are defined to correspond to the median estimate of the deformation at the initiation of LSL, as noted previously.