Rapid Estimation Method of Allowable Axial Load for Existing RC Building Structures to Improve Sustainability Performance

Abstract

Existing reinforced concrete building structures have low lateral resistance capacities due to seismically deficient details. Since these building structures suffer an increase in axial loads to the main structural elements due to green retrofits (e.g., installation of energy equipment/devices, roof gardens) as one of the sustainable building solutions and/or vertical extensions, their capacities can be reduced. This paper aims to propose a rapid estimation method incorporating a previously developed machine-learning model to find an allowable range of axial loads for reinforced concrete columns using simple structural details for enhancement in the sustainability performance of existing buildings in structural and energy fields. The methodology consists of two steps: (1) a machine-learning-based failure detection model, and (2) column damage limits proposed by previous researchers. To demonstrate this proposed method, an existing building structure built in the 1990s was selected, and the allowable range for the target structure was computed for both exterior and interior columns. A machine-learning-based method showed that axial loading could be increased by a factor of 1.35. Additionally, nonlinear time-history analysis for the target structure was performed to compare the seismic responses before and after applying the maximum allowable axial load. Based on the dynamic responses, the increased axial loads from green retrofits and/or vertical extensions could degrade structural performance and change its failure mode. The proposed methodology can rapidly estimate the allowable axial load range for existing reinforced concrete buildings without repeated modeling and computing processes. In addition, nonlinear time-history analysis is needed to accurately evaluate the impact of the increased axial loads from green retrofits/vertical extensions on structural performance.

1. Introduction

Policies for carbon emission reduction are becoming more stringent. At the 26th Conference of the Parties (COP26) to the United Nations Framework Convention on Climate Change (UNFCCC), nations pledged to reduce their carbon emissions and outlined mid- to long-term plans for achieving these goals [1]. With the Nationally Determined Contributions (NDCs) being updated to be more stringent by 2030 compared to those in 2018, there is a pressing global need for industry-wide improvements. In the construction sector, various activities are ongoing to this effect, such as recycling and reusing building materials, minimizing construction supply chains, and implementing energy-efficient equipment/devices as sustainable building solutions. Furthermore, green remodeling research is being promoted for existing buildings (over 30 years old) that focuses on reviewing and improving structural performance and energy efficiency. Green remodeling involves minimizing the environmental footprint of existing buildings through the installation of rooftop gardens and equipment/devices that improve energy performance [2,3] for sustainable building maintenance. It is particularly applicable to existing reinforced concrete (RC) building structures with high energy consumption, including those designed before the 1970s, considering only gravity loads. The columns in these structures have properties such as low shear-reinforcement ratios, 90° L-shaped corner hooks for rectangular column ties, and lap-splices located at points of maximum moment development [4,5,6]. In past cases, such as the Tangshan earthquake (1976, China) and the İzmit earthquake (1999, Turkey), the lack of lateral resistance of RC buildings that were not seismically detailed resulted [7,8]. Damage accumulates in buildings as the frequency of exposure to natural hazards increases, leading to a degradation of the strength and stiffness performance intended by the original design [9]. Shin et al. (2014) [10] confirmed that piloti-type reinforced concrete structures can experience significant additional damage or collapse due to successive earthquakes if damage accumulates after the first earthquake. Furthermore, Shin et al. (2018) [11] found that initial damage induced by the mainshock amplifies the seismic vulnerability of structures due to aftershocks. Thus, such performance degradation increases the vulnerability of buildings, potentially causing significant damage in the event of future hazards.

Along with green remodeling, vertical extensions can also be considered to enhance the sustainable conditions of buildings and reduce carbon emissions by improving the utilization and functionality of old and seismically vulnerable buildings. Vertical extensions can be recycled capitalize on existing infrastructures, creating new living space with a simpler process and lower cost compared to reconstruction. While most countries do not prohibit vertical extensions, they are only allowed under strict procedures ensuring that the physical environment, urban aesthetics, future urban planning, and structural stability are not compromised. In some countries where rooftop floors are not included in the building footprint, vertical extensions on rooftop floors are frequently carried out without appropriate permits [12]. Illegal vertical extensions may cause material and structural stability issues and affect the failure modes of columns, which are the primary structural members of buildings. Such vertical extensions increase the loads on existing RC structures and increase the axial loads on the columns. In past experimental studies, increased axial loads on RC columns resulted in a decrease in ductility [13,14,15]. This may be attributed to the original shear capacity of the column being exceeded by the increased shear demand caused by the P-Δ effect.

The number of existing buildings with poor energy performance is increasing worldwide. As of 2023, 41% of all buildings in South Korea were over 30 years old [16]. Prior to green remodeling or vertical extension efforts, it is essential to review the structural safety of existing buildings due to increased gravity loads. However, determining the structural safety of an existing building using traditional structural performance assessment methodologies (computerized structural analysis-based assessment methods) when the axial loads of vertical members are increased by green remodeling or vertical extension requires a lot of workforce, cost, and time. Due to these characteristics, there are difficulties in implementing green remodeling or vertical extension in practice. Additionally, the lack of rapid structural assessment methodologies limits their application for green remodeling or vertical extension of existing structures at an early-stage design [17,18,19]. Therefore, this study aims to improve the efficiency of the structural performance evaluation process under increasing axial load by incorporating a machine-learning (ML) model that can determine the failure mode of RC columns using simple structural details [20].

This study proposes a methodology incorporating a previously developed ML model for the determination of column failure modes to rapidly estimate the allowable axial load using simple structural information. This is intended to maintain sustainable building performance for the existing building structures in terms of structural safety and energy consumption. After that, the estimated axial loads were applied to the existing RC building structure built before more than 30 years. To compare the seismic responses of the RC structures before and after application of the maximum axial loads computed from the proposed methodology, nonlinear time-history analyses (NTHAs) of the selected building structure were performed under 2%/50-year earthquake records (2400-year return period earthquake). To develop a methodology for rapid estimation of the allowable axial load with simple structural details, the final allowable axial load is calculated by comparing the failure mode determined by the following three considerations: (1) comparing the shear demand (force–displacement relationship) and the current code-defined shear capacity of the interior and exterior columns of the first floor of the target reinforced concrete building through macro analysis; (2) the failure mode with increasing axial load determined by using the ML model; and (3) the damage limit proposed in previous studies.

2. Numerical Analysis Modeling Methodology

2.1. Column Failure Modes

Green remodeling or vertical extension increases the loads in existing RC buildings by increasing the axial loads on columns, which are the main structural members. Increased axial loads can affect the failure modes of RC columns. When a column is simultaneously subjected to lateral and axial loads, horizontal displacement occurs as the member rotates about the central axis of the column. This horizontal displacement triggers a chain where an additional moment is created, thereby causing additional horizontal displacement, which continues. This P-Δ effect increases the bending moment of RC columns with an increase in axial load [21]. Figure 1 shows the relationship between the shear demand curve and the shear capacity curve as the axial load increases. The shear demand curve is derived from the nonlinear static pushover analysis, while the shear capacity curve is calculated according to the standard equations. Section 3 discusses this in further detail. When the axial load is increased in an RC column that has not previously experienced shear failure because the shear demand ($V_D$) is lower than the shear capacity ($V_C$), the shear demand curve increases due to the increase in strength of the column caused by the P-Δ effect [13,14,15]. If the shear demand ($V_D$) exceeds the shear capacity ($V_C$) owing to the changed axial load, shear failure occurs, consequently changing the failure mode.

2.2. Column Modeling Methodology

2.2.1. Development of Numerical Model

The failure modes of RC columns under lateral loads are classified into flexure failure, shear failure, and flexure-shear failure modes, depending on the type of crack [22,23]. The shear demand curve is derived from nonlinear static pushover analysis by developing a model to describe the flexure behavior of RC columns without considering shear failure. A model that describes the shear and flexure-shear behaviors of RC columns is used to analyze the effect of changes in axial loads due to green remodeling or vertical extension on the dynamic behavior of existing RC buildings. Therefore, an analysis model for failure mode is developed, and the modeling methodology is validated based on the results of past experimental studies [24,25,26]. In this process, OpenSees [27], which involves a macroscopic approach, is used. The RC column model basically reflects the nonlinear behavior of the material model, the effect of confining pressure from transverse reinforcement, and the bond–slip effect. In addition, axial loads equal to those applied in past experimental studies are used, and the reversed cyclic pushover loadings are calculated and applied based on the resulting data.

Figure 2 illustrates the details of the flexure behavior model. Fixed and sliding boundary conditions are set for the double bending model, and fixed and free boundary conditions are set for the single bending model. The columns are constructed with elastic beam–column elements, while the plastic hinge region uses displacement-based nonlinear beam–column elements with four integration points. The length of the plastic hinge region is set equal to the cross-sectional height of the column based on the analysis model of Berry et al. [28]. Constructing a plastic hinge to reflect the deformation in the column is the focus of the analysis. The model based on Concrete02 material is used to construct the fiber section of the nonlinear beam–column element. The model by Mander et al. (1988) [29] is used to consider the enhancement of compressive strength and ductility of concrete in the region confined by shear rebars in the cross section of RC columns. The model for the main rebar is based on Steel02 material, where strain hardening can be applied.

Bond slip is a phenomenon in RC columns where the bond stress between the rebar and concrete exceeds the allowable bond stress, causing the member to fail, usually at the bottom and top of the column where a lap splice is located. To reflect the bond–slip effect in the flexure failure model, zero-length section elements are applied to the bottom and top of the columns. The fiber section in the zero-length section element has the same composition as the fiber section in the column; however, the model of the material changes from a material strength–strain relationship to a material strength–slip relationship. Equations (1) and (2) presented in the study by Sezen et al. (2002) [30] are applied for the material strength–slip relationship as follows:

$${slip}_y={\displaystyle \frac{{\epsilon }_y{f}_y{\varphi }_c}{{8u}_e}}$$

(1)

$${SF}_{slip}={\displaystyle \frac{{slip}_y}{{\epsilon }_y}}$$

(2)

where, ${slip}_y$: bar slip at yield (mm), ${\epsilon }_y$: longitudinal reinforcement yield strain, $f_y$: longitudinal reinforcement yield stress (MPa), ${\varphi }_c$: diameter of longitudinal reinforcement (mm), ue: elastic bond stress (=$0.5\sqrt{{f^{\prime }}_c}$) (MPa), $f_c^{\prime }$: concrete compressive strength (MPa), and ${SF}_{slip}$: bond–slip scale factor. Material models with stress–slip relationships can be used in zero-length fiber sections by multiplying the strains in the rebar and concrete material models by the bond–slip variable (${SF}_{slip}$).

Figure 3 shows the details of the model that reflects flexure-shear failure. The elements of the model are constructed the same as the flexure-failure model, with zero-length shear springs applied at the bottom or top to depict shear behavior. The shear limit curve presented by Elwood (2004) [31] is used as the limit-state material model for this shear. For shear failure, the member exhibits flexure-failure behavior, as shown in Figure 3c. However, upon reaching a certain shear strength ($V_n$), the shear limit curve is applied, resulting in shear-failure behavior. The shear strength ($V_n$) is calculated using Equation (3) given by ASCE 41-23 [32], taking into account that the shear strength decreases as the displacement ductility (${\mu }_{∆}$) increases, as follows:

$$V_n=k({\displaystyle \frac{A_v{f}_{y}d}{s}}+\lambda ({\displaystyle \frac{0.5\sqrt{{f^{\prime }}_c}}{M/Vd}}\sqrt{1+{\displaystyle \frac{P}{0.5\sqrt{{f^{\prime }}_c}{A}_g}}}){0.85A}_g)$$

(3)

where, $k$: strength degradation coefficient that depends on displacement ductility (${\mu }_{∆}$) ($k=1.0$ for ${\mu }_{∆}\le 2$, $k=0.7$ for ${\mu }_{∆}\ge 6$, and linear interpolation is used for $2<{\mu }_{∆}<6$: ratio of the ultimate displacement to yield displacement, $A_v$: area of transverse reinforcement (mm²), $f_y$: yield stress of transverse reinforcement (MPa), $d$: effective depth (=0.8 h) (mm), $s$: spacing of transverse reinforcement (mm), $\lambda$: 0.75 and 1.0 for light and normal weight aggregate concrete, respectively, ${f^{\prime }}_c$: compressive strength of concrete, $M/Vd$: largest ratio of moment to shear times effective depth (2 ≤ $M/Vd$ ≤ 4), $P$: axial compressive load for previous converged solution state (N), $A_g$: gross cross-sectional area (mm²).

In Elwood (2004) [31], the unloading stiffness ($K_{deg}$) of a shear spring is defined by the unloading stiffness in shear behavior ($K_{deg}^t$) and the unloading stiffness in flexure behavior ($K_{unload}$) of the member, as given by Equation (4). For the load stiffness ($K_{deg}^t$), Equation (5) proposed by Baradaran (2013) [33] is used. Here, $A_v$: area of transverse reinforcement (mm²), $f_{yv}$: yield stress of transverse reinforcement (MPa), $d_c$: depth of the column core from centerline to centerline of transverse reinforcement (mm), and $L$: column length (mm). For the residual shear strength ($V_{res}$), 80% of the shear strength ($V_n$) is applied.

$${\displaystyle \frac{1}{K_{deg}^t}}={\displaystyle \frac{1}{K_{unload}}}+{\displaystyle \frac{1}{K_{deg}}}$$

(4)

$$K_{deg}^t={-4.5(4.6{\displaystyle \frac{A_v{f}_{yv}{d}_e}{Ps}}+1)}^{2}L$$

(5)

Table 1. Material information of the column used in past experiments.

Specimens	${{\mathit{f}}^{\prime }}_{\mathit{c}}$ (MPa)	${\mathit{f}}_{\mathit{y}\mathit{l}}$ (MPa)	${\mathit{f}}_{\mathit{y}\mathit{t}}$ (MPa)	$\mathit{L}/\mathit{D}$	$\mathit{P}/{\mathit{A}}_{\mathit{g}}{{\mathit{f}}^{\prime }}_{\mathit{c}}$	${\mathit{\rho }}_{\mathit{l}}$	${\mathit{\rho }}_{\mathit{t}}$	$\mathit{s}$ (mm)
C3-1 [24]	26.4	497	460	3.93	0.107	0.0214	0.0126	54
3SLH18 [25]	26.9	333	403	3.65	0.089	0.0303	0.0015	457
Specimen 1 [26]	21.8	438	476	3.90	0.151	0.0247	0.004	305

lists the details of the specimens from past experimental studies used to develop the modeling methodology for RC columns in this study. C3-1 by Mo and Wang (2000) [24], which exhibits flexure-fracture behavior, 3SLH18 by Lynn et al. (1996) [25], which exhibits shear-fracture behavior, and Specimen 1 by Sezen and Moehle (2006) [26], which exhibits flexure-shear-fracture behavior were selected. The compressive strength of concrete (${f^{\prime }}_c$), yield strength of the main rebar ($f_{yl}$), yield strength of the shear rebar ($f_{yt}$), axial load ratio ($P/A_g{f^{\prime }}_c$), aspect ratio ($L/D$), main reinforcement ratio (${\rho }_l$), shear reinforcement ratio (${\rho }_t$), and shear reinforcement spacing ($s$) of each specimen are listed.

2.2.2. Validation of Numerical Model

The results of the modeling methodology and past experimental studies [24,25,26] are compared using nonlinear static hysteresis analysis in fracture mode, as shown in Figure 4. A comparison of the analytical results with the flexural-failure behavior of C3-1 by Mo and Wang (2000) [24] reveals a gradual strength reduction in the overall geometry. On the contrary, a comparison of the shear-failure behavior of 3SLH18 by Lynn et al. (1996) [25] with the analytical results shows a sharp strength reduction in the overall geometry and a pinching effect. A comparison of the flexure-shear-failure behavior of Specimen 1 by Sezen and Moehle (2006) [26] with the analytical results shows that the strength decreases gradually and then rapidly over the entire geometry. The significant strength degradation observed in specimen 3SLH18 and specimen 1 is a behavior that occurs when the shear strength exceeds the shear capacity limit. This is attributed to features such as low shear-reinforcement ratios, 90° L-shaped corner hooks in rectangular column ties, and lap splices located at points of maximum moment, which result in a reduction in shear capacity limits.

Table 2. Variation of column modeling method.

Specimens		Initial Stiffness (kN/mm)	Maximum Strength (kN)	Strength Reduction Ratio
C3-1 (Mo and Wang 2000)	Experiment	16	235	0.70
	Simulation	16	229	0.81
	Error (%)	0.00%	2.55%	15.71%
3SLH18 (Lynn et al. 1996)	Experiment	14	270	0.26
	Simulation	15	260	0.27
	Error (%)	7.14%	3.70%	3.85%
Specimen 1 (Sezen et al. 2006)	Experiment	19	303	0.22
	Simulation	20	309	0.25
	Error (%)	5.26%	1.98%	13.64%

lists the initial stiffness, maximum strength, and strength reduction rate of the experimental and analytical results for each failure mode. The error rates from the analytical and experimental results are calculated and found to be relatively consistent with each other. The initial stiffness is calculated from the initial data after loading. The maximum strength is the maximum shear force in the resulting data, and the strength reduction is calculated as the ratio of the residual shear strength ($V_{res}$) to the maximum shear strength ($V_{max}$). C3-1 by Mo and Wang (2000) [24] and Specimen 1 by Sezen and Moehle (2006) [26] have the largest error rates for strength reduction rates at 16% and 14%, respectively. As the drop in strength decreases with the adjustment of the bond–slip parameter (${SF}_{slip}$) to match the initial stiffness, the drop in strength is depicted by tuning to strain hardening. 3SLH18 by Lynn et al. (1996) [25] has a maximum error of 7% for the initial stiffness. As the initial stiffness increases with the strain hardening adjustment to depict the rapid strength reduction due to shear failure, the bond–slip variable (${SF}_{slip}$) is adjusted to reduce the error rate. Overall, the numerical analysis model shows high accuracy in predicting the behavior specific to each failure mode. In this study, the modeling methodology of flexure-dominated columns is used to estimate the shear demands for individual columns presented in Section 4 through nonlinear pushover analysis, and the modeling methodology of shear-dominated columns is applied to the target existing building for NTHA before and after green remodeling.

3. Column Performance Evaluation

This study uses the following three methods to evaluate the performance of existing RC buildings by changing the axial load ratio of columns arising owing to green remodeling or vertical extension: a code-defined capacity curve [32], a machine-learning-based model to predict the failure mode of columns [34], and a drift-based damage limit metric for columns proposed in past studies [35].

The failure mode can be distinguished by the relationship between the shear demand and the shear capacity curves. The shear demand curve is estimated by nonlinear static pushover analysis of a model of RC column capable of depicting flexure behavior. For the shear capacity curve, Equation (3) given by ASCE 41-23 [32] is used, where the shear strength decreases as the displacement ductility (${\mu }_{∆}$) increases. In this equation, the displacement ductility ($k$) is calculated as the yield displacement of the shear demand curve. The relationship between the two curves distinguishes three modes of failure, namely, flexure, shear, and flexure-shear failures [22,23]. When the shear demand and shear capacity curves are plotted on a single coordinate plane, the flexure failure is determined to have occurred if there was no intersection between the two curves, as shown in Figure 5a [36]. If an intersection occurs, as shown in Figure 5b, it is determined that shear failure has occurred at that displacement ratio.

In order to estimate the allowable axial load of columns in existing RC buildings, a machine learning-based model by Kim et al. (2023) [34] is used to predict the failure mode of columns. Based on the database of experimental results for 330 RC columns, a classification technique is used to predict the failure mode. The main input variables are the compressive strength of concrete (${f^{\prime }}_c$), yield strength of the rebar ($f_y$), axial load ratio ($P/A_g{f^{\prime }}_c$), aspect ratio ($L/D$), main reinforcement ratio (${\rho }_l$), and shear reinforcement ratio (${\rho }_t$), which can affect the failure behavior of the column. The minimum, maximum, and nominal values of the input variables are shown in

Table 3. Summary of experimental database.

Parameter	Model Range	Extreme Values		Nominal
		Minimum	Maximum
Concrete compressive strength (MPa)	13.10 to 48.30	13.10	48.30	34.11
Axial load ratio	0.03 to 0.9	0.03	0.90	0.14
Slenderness ratio	1.12 to 8.67	1.12	8.67	2.43
Transverse reinforcement ratio	0.0068 to 0.0694	0.0068	0.0694	0.0174
Longitudinal reinforcement ratio	0.0009 to 0.0615	0.0009	0.0615	0.0073

. The input variables are used to predict the failure mode of the column, among the possible flexure, flexure-shear, or shear failures. Previous studies have applied classification techniques such as artificial neural network (ANN), k-nearest neighbor (KNN), decision tree (DT), and random forest (RF) to train the data. The hyperparameters of each model were optimized through iterative learning, and then the most appropriate machine-learning model for classifying failure modes was developed and validated.

To evaluate the performance indicators of the classification models, accuracy, precision, recall, F1-score, and area under the curve (AUC) were determined based on the confusion matrix results. Accuracy is the ratio of correctly predicted cases; precision is the ratio of actual flexure failures among those predicted as flexure failures; and recall is the ratio of actual flexure failures predicted as such. F1-score is the harmonic mean of recall and precision, and AUC represents the area under the ROC curve, with an AUC value close to 1 indicating a superior classification model. While the KNN model showed the highest accuracy, other performance metrics (precision, recall, and AUC) were considered due to data imbalance. All models demonstrated precision, recall, F1-score, and AUC values above 0.9 for the classification of the flexure failure modes. For the shear failure modes, the RF model showed the highest precision (0.71) and recall (0.83) among the considered methodologies. For the flexure-shear failure modes, both the ANN and RF models showed the highest precision (0.83), while the KNN model showed the highest recall (0.46). Based on the F1-score, the performance of the machine-learning models decreases in the order of RF, DT, KNN, and ANN. AUC values for shear and flexure-shear failure modes were above 0.85 for all models except KNN. Consequently, the DT and RF models demonstrated the best performance. Considering that the machine learning-based model aims to predict column failure modes using simple information before experimental or analytical processes, a conservative model with a high recall for shear failure modes was deemed most appropriate. Therefore, the RF model, which showed high recall for shear failure modes, was selected for this study. More detailed information can be found in Kim et al. [34]. In this study, the allowable axial load was estimated by determining the limit axial load ratio at which shear failure can occur by gradually increasing the axial load ratio in the input variables.

The axial load ratio, which initiates shear failure, is predicted using machine learning. At this axial load ratio, the results of a previous study by Catherine and Laura (2006) [35] are used to assess the damage state of the member based on the damage limit.

Table 4. Damage limit defined in the current study [35].

$\mathit{D}\mathit{S}$	Repair Method	Damage Description	Maximum Drift (%)
			Median	Dispersion
${DS}_1$	Cosmetic repair finishes	■ Initial hairline cracking at beam–column interface and within the joint area ■ Maximum crack width within the joint <0.5 mm	0.6	0.37
${DS}_2$	Epoxy inject concrete cracks	■ Wider maximum crack width within the joint ■ Yield of beam longitudinal rebar	1.2	0.50
${DS}_3$	Patch spalled concrete	■ Spalling of joint surface concrete ■ Initiation of deterioration of joint shear strength	2.4	0.39
${DS}_4$	Replace RC	■ Cracks extend into the beam and/or column ■ Spalling over 80% of joint surface concrete	3.0	0.28
${DS}_5$	Replace rebars	■ Buckling of column longitudinal rebar ■ Loss of beam rebar anchorage within joint ■ Pull-out of discontinuous beam longitudinal rebar	3.6	0.26

lists the damage limit based on the story drift ratio. In existing RC buildings, the behavior of the beam–column joints directly affects the behavior of the entire existing building. Moreover, because the damage state of all beam–column joints in a story is the same, the maximum story drift ratio for all columns in any story is the same [37]. Therefore, the close relationship between the beam–column joint and the column allows the damage limit of the joint to be used as the damage limit of the column. The failure mode is verified for the axial load ratio after green remodeling or vertical extension by designating ${DS}_3$ as the primary damage limit where the shear strength degradation starts.

4. Methodology for Calculating the Allowable Axial Load

4.1. Process for Calculating the Allowable Axial Load

Figure 6 illustrates the steps to calculate the allowable axial load for green remodeling or vertical extension of an existing RC building proposed in this study. To identify the range of allowable axial loads, a two-step evaluation procedure is applied to the RC columns as follows: step 1: machine learning-based failure mode determination; and step 2: damage limit evaluation.

A column is selected for analysis in an existing RC building, and a model of the flexure behavior of an RC column is developed using the column length, cross-sectional size, material properties, and axial load ratio. A pushover analysis is performed on this model to derive the shear demand curve, and the shear capacity curve is calculated using Equation (3) given by ASCE 41-23 [32]. When comparing the shear forces of the shear demand ($V_D$) and shear capacity ($V_C$) curves, an intersection event ($V_D\ge$ $V_C$) or a ductility less than 1.0 determines that shear failure can occur, preventing the green remodeling and vertical extension from proceeding. Without an intersection occurring ($V_D$ < $V_C$) or a ductility greater than 1.0, flexure failure can occur, and steps are taken to estimate the allowable axial load.

In the first step, if shear failure of the existing building columns is determined not to occur based on the shear capacity curve, the allowable axial load is calculated using a model by Kim et al. (2023) [34] that predicts the failure mode of RC columns based on machine learning. The compressive strength of concrete (${f^{\prime }}_c$), axial load ratio ($P/A_g{f^{\prime }}_c$), aspect ratio ($L/D$), main reinforcement ratio (${\rho }_l$), and shear reinforcement ratio (${\rho }_t$) in RC columns are used as the input variables. The input variables are used to predict the failure mode of the column out of flexure, flexure-shear, and shear failures. While increasing the axial load ratio from the input variables at a constant rate, the limit axial load ratio at which the failure mode changes from flexure failure to shear failure is determined. The axial load causing shear failure is considered the limit axial load ($P_{limits}$). The limit axial load ($P_{limits}$) of the column is used to estimate the allowable axial load of the existing RC building. As shown in Equation (6), it can be calculated by dividing the first-story columns into external and internal columns and multiplying the limit axial load ($P_{limits}$) by the number of external columns ($N_{ex}$) or the number of internal columns ($N_{in}$). For the external columns, the supporting section is considered half the size of the internal columns.

$$\left\{\begin{array}{c}{P}_{allow}=P_{limits}\times {\displaystyle \frac{1}{2}}\times N_{ex} for exterior column\\ P_{allow}=P_{limits}\times N_{in} for interior column \end{array}\right.$$

(6)

In the second step, the damage state of the column is evaluated based on the damage limits from a previous study by Catherine and Laura (2006) [35] for the shear demand and shear capacity curves redefined by the limit axial load ratio. Among the damage limit indicators, ${DS}_3$, where the shear strength started to degrade, is designated as the main damage limit indicator. If the displacement (${\Delta }_s$) at the intersection of the shear demand and the shear capacity curves is calculated to be less than or equal to ${DS}_3$, shear failure is determined to occur, preventing the green remodeling or vertical extension from proceeding. On the contrary, if there is no intersection of the shear demand and the shear capacity curves or the location of the intersection calculated is greater than ${DS}_3$ of the damage limit, flexure failure is determined to occur, and the axial load ratio can be increased further.

4.2. Allowable Axial Load for the Target Building

The methodology for calculating the allowable axial load is applied to an existing RC building to estimate the allowable range for green remodeling or vertical extension efforts. In the target existing RC building, the first-story columns in the central part of the structure where the loads were concentrated are selected to apply the flexure behavior modeling methodology. Using the flexure behavior model in Section 2.1, shear demand curves are derived for the first-story exterior and interior columns. The details of the frame to be analyzed are shown in Figure 7. The length, cross-sectional size, material properties, and axial load ratios of the first-story columns are considered in the modeling. A column on the first story of an existing building has a length of 3800 mm and a cross-section area of 600 × 600 mm. The cross-section consists of 16 main rebars with a cover thickness of 40 mm and diameter of 22 mm, and the shear rebars with a diameter of 10 mm are evenly spaced at 150 mm intervals.

Table 5. Summary of axial load to column for existing RC structure.

$\mathbf{Axial} \mathbf{Load}$ $(\mathit{P}/{\mathit{A}}_{\mathit{g}}{{\mathit{f}}^{\prime }}_{\mathit{c}}$)	1st Story Column	2nd Story Column	3rd Story Column	4th Story Column	5th Story Column
Exterior	0.08	0.06	0.05	0.03	0.01
Interior	0.21	0.16	0.12	0.07	0.03

lists the axial load ratios of interior and exterior columns for each story of the target building. The representative axial load values for the first-story interior and exterior columns are assumed to be 1361 kN (axial load ratio of interior first-story columns, 0.21) and 518 kN (axial load ratio of exterior first-story columns, 0.08), respectively.

The shear demand and shear capacity curves of the first-story exterior and interior columns of the existing building are derived and shown in Figure 8. For both the exterior and interior first-story columns, $V_D$ < $V_C$, meaning no intersection between the shear curves occurs. Therefore, the machine learning-based failure mode prediction model by Kim et al. (2023) [34] is used to identify the axial load ratio for which shear failure is predicted by increasing the axial load ratio from the input variables, as depicted in

Table 6. ML-based determination of allowable axial loading ratio.

1st Story Exterior Column
Increment ratio	$P/A_g{f^{\prime }}_c$	${f^{\prime }}_c$ (MPa)	$L/D$	${\rho }_l$	${\rho }_t$	Failure mode
1.00	0.08	18	3.52	0.017	0.004	Flexure
2.00	0.16					Flexure
3.00	0.24					Flexure
3.50	0.28					Flexure
3.54	0.28					Shear
1st Story Interior Column
Increment ratio	$P/A_g{f^{\prime }}_c$	${f^{\prime }}_c$ (MPa)	$L/D$	${\rho }_l$	${\rho }_t$	Failure mode
1.00	0.21	18	3.52	0.017	0.004	Flexure
1.10	0.23					Flexure
1.20	0.25					Flexure
1.30	0.27					Flexure
1.35	0.28					Shear

. To determine the limit axial load ratio at which shear failure is predicted, the failure modes were reviewed by incrementally increasing the axial load ratio at intervals of 0.01. It is found that the existing axial load ratio can be increased by 1.35 and 3.54 times for the first-story interior and exterior columns, respectively. The shear demand and shear capacity curves are plotted again with the limit axial load ratio to determine the damage status of the first-story exterior and interior columns when the limit axial load ratio is applied owing to green remodeling or vertical extension, as shown in Figure 9. The intersection between the shear demand and shear capacity curves does not occur after the axial load ratio is increased, indicating the occurrence of flexure failure. This suggests that the axial load ratio can be further increased.

The allowable axial load is calculated using the limit axial load ratio. In the target existing building, the first story of the frame to be analyzed consists of 12 columns. Therefore, it is calculated to withstand 1.35 times 1361 kN, the axial load from aging on the internal columns, and hence a total of 18,374 kN, considering that there are 10 internal columns. Considering that the outer columns have an axial load of 518 kN from aging and half the support area of the inner columns, a factor of 1.77 times (half of the 3.54) is applied to the outer columns. Calculation for two external columns shows that the axial load can be increased up to 1834 kN. Finally, the allowable axial load is calculated to be 20,208 kN (20,208 kN/14,646 kN, 1.38), which is the sum of 18,374 kN and 1834 kN.

The shear demand and shear capacity curves before and after green remodeling or vertical extension are shown in Figure 10. As the axial load ratio increases, the shear demand increases due to the P-Δ effect. The shear demand increases with increasing axial load, according to Equation (3). The shear demand increases due to the application of the limit axial load ratio, but the shear capacity increases due to the increase in axial load, resulting in no intersection between the two curves.

5. Nonlinear Time-History Analysis

In the process of calculating the seismic response with dynamic analysis, the loads on the building are converted to mass. Consequently, changes in axial loads due to green remodeling or vertical extension can affect the mass distribution and seismic response of RC buildings. In this study, the difference in dynamic behavior of existing RC buildings under changing axial loads is investigated using nonlinear time-history analysis (NTHA).

5.1. Modeling of the Target Existing RC Building

The details of the target existing RC building are illustrated in Figure 7 in Section 4.2. The frame has 5 stories and 11 spans, where the height of the first story is 4500 mm and the height of all other stories is 4000 mm. The six outer beams are 6300 mm long each, and the five center beams are 7500 mm long each. The representative axial load values for the exterior and interior of each story are listed in Table 5. The boundary conditions of the analysis model are assumed to be fixed for all directions. To consider the possibility of shear failure in the columns due to increased loads, a model capable of depicting the flexure-shear behavior from Section 2.1 is used. To improve analysis accuracy at the building scale, the nonlinear beam–column element is changed to the displacement-based beam–column element. The beam is constructed with an elastic beam–column element, and the joint area is set to rigid offset. The existing RC building designed under gravity load (1.2 DL + 1.6 LL, where DL = dead load, LL = live load) is considered in the analysis model. The allowable axial load calculated in Section 4.2 is used to calculate the value of the additional axial load and is applied to the roof of the target existing RC building.

5.2. Ground Motion

For this study, ground acceleration data with a 2% occurrence probability over 50 years (recurrence interval of approximately 2400 years) provided by past studies are selected to perform the NTHA [38]. Four seismic waves, as shown in

Table 7. Selected records for the 2% in 50-year hazard level.

Name	Record	Eq. Magnitude	Distance (km)	Scale Factor	Number of Points	DT (sec)	Duration (sec)	PGA (cm/sec2)
EQ1	1989 Loma Prieta (earthquake)	7	3.5	0.82	2500	0.01	24.99	409.95
EQ2	1989 Loma Prieta (earthquake)	7	3.5	0.82	2500	0.01	24.99	463.76
EQ3	Elysian Park (simulated)	7.1	10.7	0.97	3000	0.01	29.99	667.59
EQ4	Palos Verdes (simulated)	7.1	1.5	0.88	3000	0.02	59.98	613.28

, are selected for the analysis: EQ1 (Loma Prieta earthquake, 1989), EQ2 (Loma Prieta earthquake, 1989), EQ3 (Elysian Park simulated), and EQ4 (Palos Verdes simulated). Seismic waves with different distances from the epicenter and different magnitudes are selected to reflect various conditions, and the selected input seismic waves are shown in Figure 11.

5.3. Dynamic Response

Eigenvalue analysis is performed to understand the dynamic characteristics of the model before and after green remodeling or vertical extension. The natural periods of the target existing RC building are listed in

Table 8. Natural periods by mode before and after green retrofitting.

Period	Before Green Retrofit	After Green Retrofit
Mode 1	0.40	0.45
Mode 2	0.14	0.15
Mode 3	0.09	0.09

. Overall, there is no significant change in natural periods, but the period in Mode 1 increases by 12.5% after the green remodeling. The change in eigen vectors due to green remodeling or vertical extension is found to be insignificant.

Dynamic analysis is used to analyze the effect of mass change due to green remodeling or vertical extension on the structural safety of an existing RC building. The capacity limits for RC framing proposed by FEMA356 [39] are listed in

Table 9. Performance limits for RC frame [39].

Performance Level	Maximum Story Drift Limits	Residual Drift Limits	Description
IO	≤1.0%	Negligible	Minor hairline cracking. Limited yielding possible at a few locations. No crushing (strains below 0.003).
LS	≤2.0%	1.0%	Extensive damage to beams. Spalling of cover and shear cracking (<1/8″ width) for ductile columns. Minor spalling in nonductile columns. Joint cracks <1/8″ wide.
CP	≤4.0%	>1.0%	Extensive cracking and hinge formation in ductile elements. Limited cracking and/or splice failure in some nonductile columns. Severe damage in short columns.
Collapse	>4.0%

for immediate occupancy (IO), life safety (LS), and collapse prevention (CP) cases with a story drift ratio of 1.0%, 2.0%, and 4.0%, respectively. The residual deformation of RC frames is defined as IO for less than 1.0%, LS for 1.0%, and CP or higher for more than 1.0%. In addition, the responses of the building columns before and after the green remodeling or vertical extension to the expected drift of story failure for the first-story interior columns presented in Section 4.2 are compared.

Figure 12 shows the change in behavior of the first-story column of an existing RC building before and after green remodeling or vertical extension using selected earthquake records. As green remodeling or vertical extension increases the axial loads in the existing RC building, the period of the first-story columns increases. In all cases except EQ4, the change in the period results in an amplified displacement at the last coordinate of the graph. The change in the period is due to a decrease in stiffness caused by damage to the structure. The displacements of the last coordinates of EQ1, EQ2, and EQ3 exceed 1.0%, indicating that the residual deformation is above the collapse prevention (CP) level. It is also confirmed that the shear failure and permanent deformation of the primary structural member, that is, the first-story column, occur owing to the displacement exceeding the drift of story failure.

Based on the selected seismic waves, the maximum story drift ratios of the target existing RC building before and after green remodeling or vertical extension are presented in Figure 13. The maximum story drift ratio before the green remodeling shows the largest displacement change on the first story for all seismic waves, with negligible changes on other stories. After green remodeling, a similar geometrical change is observed, and the maximum story drift ratio on the first story particularly increases by up to four times (10.4% from 2.6%). This suggests that a soft-story mechanism occurs where damage is concentrated on certain stories of the building, amplified by increasing axial loads. The impact of this increase in the story drift ratio on the performance level of the building is then examined. In the case of EQ4, the life safety (LS) level remains the same before and after green remodeling or vertical extension, but the damage is amplified on the first story as the axial load increases. For the other seismic waves, the performance level of the building degrades from the collapse prevention (CP) level to the collapse level as the axial loads increase. The seismic performance target for ground acceleration with a 2% probability of occurrence over 50 years (recurrence interval of approximately 2400 years) is CP, which stands exceeded. Therefore, existing buildings that have undergone green remodeling or vertical extension are found to be more susceptible to severe structural damage in disasters such as earthquakes.

In Section 4.2, the application of the limit axial load ratio, calculated using the allowable axial load range estimation methodology, concluded that green remodeling/vertical extension is feasible. However, verification through time history analysis, which can reflect changes in the additional masses of the structure, indicated that applying the limit axial load ratio could result in significant structural damage. To enhance the validity of the estimation methodology for the allowable axial load range, it is necessary to establish the methodology based on nonlinear time history analyses.

6. Conclusions

This study aimed to propose a methodology that can estimate the allowable axial load for existing RC buildings with green remodeling or vertical extension based on simple information and analysis of column members in such buildings. The allowable axial load was calculated by evaluating the column performance using methods involving shear demand and shear capacity curves, predicting failure modes based on machine learning, and predicting failure modes based on damage limit. To derive the shear demand curve, a model describing the flexure behavior of RC columns was developed. An existing RC building was selected, and the allowable axial load was calculated to verify the applicability of the methodology. NTHA was performed to compare the seismic response of the building as the maximum allowable axial load was applied, and the following conclusions were drawn:

(1)

The proposed process for computing the allowable axial load for the column simply consists of the following four steps. First of all, shear demand and shear capacity curves were calculated using the length, cross-sectional size, material properties, and axial load ratio of existing RC building columns to determine the shear capacity of the columns. By applying a model that predicts the failure mode of the column based on machine learning, the limit axial load ratio between flexure failure and shear failure was identified, and the allowable axial load was calculated. The shear demand and shear capacity curves were established again by applying the calculated limit axial load ratio, and the allowable axial load was evaluated until reaching the damage limit, representing shear strength degradation.

(2)

The limit axial load ratios were derived for the interior and exterior columns of the first story of the target existing RC building, revealing that the axial load ratio can be increased by 3.54 times for the first-story exterior columns and 1.35 times for the first-story interior columns. The total allowable axial load for green remodeling or vertical extension was calculated as 20,208 kN. The final shear curves of the first-story interior and exterior columns showed that shear failure did not occur because of the increase in the shear capacity as the axial load ratio increased.

(3)

The NTHA results of the existing RC building with green remodeling or vertical extension without structural reinforcement showed that the soft-story phenomenon was amplified, in which damage was concentrated on a specific story (first story) because of the increased axial loads. This increased the story drift ratio by up to four times and exceeded the target seismic performance level (CP in the 2%/50-year return period). It also led to permanent deformation, amplifying the damage risk from subsequent earthquakes. Based on these observations, green remodeling efforts are suggested to consider seismic performance evaluation along with energy performance improvements.