Experimental and Transformer-Based Study on Seismic Behavior and Plastic Hinge Length of RC Columns Reinforced with End-Fixed Ultra-High Strength Rebars

Abstract

The application of machine learning (ML) in structural engineering is receiving increasing attention recently. This paper experimentally studies three self-restoring reinforced concrete (SRRC) columns reinforced with low-bond ultra-high strength rebars, to first discuss the reliability and evaluation of the SRRC columns under multiple reversed cyclic (MRC) loads induced by strong earthquakes, and to also first introduce the Transformer method into the analysis and discussion of structural tests. The tests confirmed the superior seismic behavior and high self-centering performance of the columns and presented how MRC loads affect the seismic performance of SRRC columns in terms of the lateral load-carrying capacity and energy dissipation capacity. Superior to conventional methods, a high-accuracy Transformer-based model is proposed to evaluate the plastic hinge height (PHL) of the tested SRRC columns compared with the other three algorithms (MLP, KNN, and XGBoost). Furthermore, the Shapley Additive exPlanations (SHAP) approach is adopted to explain the insight relationship between the structural parameters and PHL of the columns.

1. Introduction

Recently, mega earthquakes such as the Great East Japan Earthquake, the Kumamoto Earthquake [1], and the Northern Osaka Earthquake [2] have caused major damage to human society, physically and mentally. According to the literature [3], as the next possible mega earthquakes, the Nankai Trough Earthquake and Tokyo direct-type earthquake will probably occur in the next 20 years, which is an earthquake that occurs directly beneath the Tokyo area (there is no clear definition in seismology, and earthquakes that cause major damage when their epicenter is in inland areas or areas where people live, such as urban areas, are called direct-type earthquakes). The long-period ground motion (LPGM) earthquake is another type of strong earthquake that has received attention since the Tokachi-Oki earthquake (Japan, Hokkaido) in 2003 [4]. It can cause long-period reciprocating horizontal vibrations in building structures, especially long-period buildings. Specifically, the column components inside the structures will be subjected to multiple reversed cyclic (MRC) loads. Faced with such strong earthquakes, in the current seismic design codes, the protection of human life is highly prioritized against extremely rare earthquakes, which means that structures are allowed to be damaged but have to be prevented from collapsing. RC structures are widely used as one of the main earthquake-resistant structure types in Japan. To ensure that these structures can effectively resist collapse under strong earthquakes, ductile RC (DRC) structures have been widely accepted in the past 50 years. The earthquake disaster lessons prove that DRC structures can resist earthquakes effectively even at the large deformation stages, ensuring that the structures do not collapse during earthquakes. However, when DRC structures are subjected to repeated earthquake loads induced by LPGM, it is expected that they will repeatedly undergo large deformations in the plastic region. More and more researchers are interested in different structural element tests under LPGM, such as concrete-filled steel tubular (CFT) columns [5,6,7,8], and reinforced concrete elements (columns, beams, and beam-column joints) [9,10,11]. These studies show that MRC loading has a non-negligible influence on the structural performance of DRC elements.

On the other hand, recent research trends show that not only is the collapse resistance of the structures concerned important, but also the fast repairability of the structures post-earthquake. This is because it involves the post-earthquake restoration of the structures and the reconstruction of the earthquake-affected areas, not only in terms of time but also in terms of costs. The structures and all built environments should be better prepared for new conditions, which promote the sustainable and resilient development of RC structures in earthquake-prone zones. According to the previous research [12,13,14,15], it is clear that the use of low-bond ultra-high-strength rebars as the longitudinal rebars of reinforced concrete (RC) columns can effectively and easily realize SRRC columns. This way, we can reduce residual deformation and keep a high load-carrying capacity even at a large deformation stage. As a result, the effects of the bond strength of the bars in such columns have revealed that the rebars are highly restored in the large deformation stage. Combining the high capacity and the self-centering capacity of SRRC columns, suppressing the damage to the structural members, keeping the residual deformation smaller, and improving the reuse and repairability after the earthquake are being considered in order to develop a structural form with better use and repairability. However, the previous research on the seismic behavior of the columns under multiple positive- and negative-load-induced strong earthquakes such as long-period ground motion earthquakes has not progressed.

Moreover, the damage of RC elements is caused by the energy absorption of the structures during the earthquakes, usually including crack damage and concrete spalling at the plastic hinge zones. Since the seismic performance of the RC columns depends significantly on the deformation capacity, many empirical models for predicting the plastic hinge length (PHL) of the columns have been proposed in the past decades. The PHL-affecting factors mainly include the sectional depth, compressive strength of unconfined concrete, longitudinal steel ratios, axial load ratio, transverse steel ratio, and shear span-depth ratio [16]. With the development and propriety of AI technology, we can apply machine learning (ML) methods to solve challenges in structural engineering [17,18,19,20]. Wakjira et al. [21] adopted ensemble ML algorithms to predict and discuss the RC column PHL based on a dataset collected from the literature.

Based on the above brief literature review, the paper, for the first time, experimentally investigates the structural performance of SRRC columns under MRC loads without axial force as a preliminary exploratory study. The method of not applying axial force was also used to study the influence of internal and external damage accumulation caused by simple multiple cycles on the horizontal deformation and resistance mechanism of the columns. The focus of the paper is to study the effects of MRC loads on the seismic performance of SRRC columns, including the seismic mechanism, seismic capacity, and deformation capacity. At the same time, different from Wakjira’s work [21], this paper selects the Transformer to train ML models to predict the PHL of the investigated SRRC columns. Furthermore, regarding the “black box” in ML prediction, the Shapley Additive exPlanations (SHAP) approach is used for its apparent explainability.

2. Test Programs

2.1. Details of Specimen

Table 1. Outline of tested specimens.

Specimen #	Cross-Section			Longitudinal Rebars		Steel Stirrups			Load Types
	B	D	a/D	Type	pg	pw	ph	S
	(mm)	(mm)			(%)	(%)	(%)	(mm)
F60S3N33U-NRC	250	250	3.0	U bars	2.4	0.914	1.13	70	NRC
F60S3N33SB-NRC				SB bars	2.43		2.26	35	NRC
F60S3N33SB-MRC				SB bars	2.43		2.26	35	MRC

NRC: normal reversed cyclic, MRC: multiple reversed cyclic, p_g: area ratio of longitudinal rebar, p_w: area ratio of transverse reinforcement, p_h: volumetric ratio of transverse reinforcements, S: stirrup spacing, a/D: shear span ratio.

shows the outline of the tested specimens. Three 1/3-scaled square column specimens with a cross-section of 250 mm × 250 mm and a shear span ratio of 3.0 were produced. The longitudinal rebars were fixed at their ends in a mechanical method by strong steel nuts and steel plates. In addition, the below end of the columns was fixed with a force stub and was considered to be joined to a rigid column position. The distance from the stub surface to the lateral loading position was 750.0 mm. The reinforcement arrangement of the specimens is plotted in Figure 1. The control column was longitudinally reinforced by low-bond high-strength bars called SBPDN1275/1420 (U rebars, Neturen company, Tokyo, Japan) (here, S: Steel, B: bar, P: prestressed concrete, D: deformed, N means the steel bar is not needed in order to consider the relaxation, 1275/1420 means that the yield strength is over 1275 N/mm², and the ultimate strength is over 1420 N/mm²), while the other two specimens were reinforced by normal-bond high-strength steel bars called SBPD1080/1430 (SB rebars, Neturen company, Tokyo, Japan). The loading methods and type of longitudinal reinforcements were the main experimental variables, as shown in Table 1.

2.2. Properties of Used Materials

2.2.1. Steel Reinforcements

The reinforcements SBPDN1275/1420 used for the longitudinal rebars of the control column are low-bond ultra-high strength steel rebars with a diameter of 12.6 mm and spiral grooves on their surface, as shown in Figure 2a. The stirrups were also SBPDN1275/1420 rebars for all tested specimens with a diameter of 7.1 mm. The properties of the SBPDN1275/1420 rebars were measured based on standard tensile tests as per the Japanese test code [22]. The yielding strength (f_y) and yield strain (ε_y) of the rebars were calculated by the 0.2% offset method, which was 1435 N/mm² and 0.0087 for SBPDN1275/1420 and 1106 N/mm² and 0.0078 for SBPDN1080/1430 rebars, respectively. The mechanical properties of the rebars are listed in

Table 2. Mechanical properties of steel bars.

#	E (kN/mm2)	fy (N/mm2)	εy (×10−2)	fm (N/mm2)
U rebar	213	1435	0.87	1512
SB rebar	191	1106	0.78	-

, including the elastic modulus (E), and the maximum tensile strength at the end of the test (f_m).

2.2.2. Concrete

A ready-mixed concrete was used in this study. In all specimens, the maximum particle size of the coarse aggregates was 20.0 mm, while the used cement is common Portland cement. A standard compressive test using three standard concrete cylinders (diameter of 100 mm, and height of 200 mm) was conducted to measure the compressive strength of the concrete, as per current Japanese test methods [22]. The average measured compressive strength of the concrete at the test was 60.1 N/mm².

2.3. Test Setup and Loading Protocols

2.3.1. Test Setups

The lateral load of the specimens was placed at the point of contra-flexure of the two cantilever columns, which is under antisymmetric bending during an earthquake. The distance from the lateral force to the column stub was 750.0 mm to ensure the shear span ratio of the columns was 3.0. Figure 3 shows the test setup used in the experiments. A 1000 kN hydraulic jack was installed in a strong steel frame to provide the lateral force in the positive and negative directions. The position of strain gauges in longitudinal rebars are showed in Figure 4(a). The drift ratio of the column (R) was calculated by excluding the relative lateral displacement of the stub and the column measured at the loading point position by LVDT. In terms of collecting the axial deformation within the range of 250 mm and 500 mm from the stub, four LVDTs were set near the surface of the tested columns, as presented in Figure 4(b). A lateral LVDT was placed 40 mm away from the ground to measure the possible displacement of the base stub.

2.3.2. Loading Protocols

As shown in Figure 5, two types of loading methods were applied for the tests, that is, normal reversed cyclic (NRC) load and multiple reversed cyclic (MRC) load, respectively. The detailed program of the loads is as follows: NRC loading protocol—Referring to the literature [10], the target drift ratios of each cycle were first 0.0025 rad, 0.0050 rad, 0.0075 rad, 0.010 rad, and 0.020 rad. Each cycle was performed twice. After that, 0.030 rad, 0.040 rad, and 0.050 rad were conducted for one cycle, respectively. MRC loading protocol— As a preliminary study, the number of repetitions in the MRC loading in this study was developed concerning the seismic response analysis of a 43-story RC building [23]. Two simulated seismic motions designed for the long-period ground motion and the Building Center of Japan simulated wave (BCJ-L2) were adopted. The drift ratios of each member were recorded, and classified into three amplitude categories (high, medium, and low). Each level includes 10 cycles of small amplitude deformation, 10 cycles of medium amplitude deformation, and then 5 cycles of large amplitude loads, which was repeated 3 times for a total of 105 cycles of loads. The maximum inter-story drift ratio of the elastic level is assumed as 1/100 when receiving a very rare occurrence of earthquake movements. After repeating the elastic level 105 times, a total of 3 sets of plastic level 3 sets were set 105 times. Different from the literature [23], at the last level of the loads, if it is possible to add force, the drift ratios of 1/33 and 1/25 were repeated 5 times each to study the failure state of the columns under repeated large deformation.

3. Test Results

3.1. Hysteretic Response

Figure 6 shows the lateral force-drift ratio curves of the specimens obtained in the experiment, in which △ indicates the yield stress at 75% on both sides which is important for the safety design of structures, ○ means the maximum or minimum lateral force, and □ represents the stiffness when the first crack occurred on both sides. Comparing specimen U-NRC with SB-NRC, there is no clear degradation of U-NRC with the increase in drift ratio while the lateral force began to decrease after the drift ratio of 4%. The residual drift ratio of the U-NRC specimen is smaller than that of SB-NRC specimens, especially after the drift ratio of 3%. These will be discussed more in Section 3.3.

Regarding the specimen SB-NRC and SB-MRC, it can be realized that the maximum lateral force of the specimens under the NRC load was 10% higher than the column subjected to the MRC load, while there is little difference in stiffness for the two columns before the drift ratio of 2%. Moreover, the residual drift ratio of specimen SB-NRC and SB-MRC both stay at a relatively small level before the drift ratio of 3%.

3.2. Crack Observation

According to Figure 7, for specimen U-NRC, during the initial loading cycle with a positive target member angle of R = 0.25%, the formation of flexural cracks was first observed at locations 100 mm and 225 mm from the base of the column on the east face of the south side. Subsequently, during the initial loading cycle with R = −0.25%, flexural cracks were observed at a location 200 mm from the base of the column on the west face of the south side. From the cycle of R = 2.0% to the cycle with R = 3.0% rad, numerous shear cracks developed within a 400 mm range from the base of the column. In the cycle of R = 4.0%, both sides within a 100 mm range from the base of the column experienced concrete spalling.

For specimen SB-NRC, a flexural crack was observed at 375 mm from the column base on the east side during the first loading cycle at R = 0.25%. This crack had a width of 0.05 mm at the peak of the R = 0.25% cycle and completely closed upon unloading, becoming almost invisible. At the peak of the first cycle at R = −0.25%, a flexural crack with a width of 0.35 mm appeared at 225 mm from the column base on the west side. In addition, another flexural crack with a width of 0.10 mm was observed at 500 mm from the column base. In the R = 2.0% loading cycle, many new cracks appeared and the crack width increased significantly within 250 mm from the column base.

However, many residual cracks remained small, with sizes of approximately 0.05 mm to 0.10 mm, indicating a reduction. During the loading cycle R = 3.0%, the crack width increased further, with many cracks approaching 1.0 mm in width. Severe spalling occurred near the base of the columns on both the east and west sides. In the loading cycle with R = −3.0%, large vertical cracks appeared at the base of the columns. In the loading cycle with R = 4.0%, large areas of spalling occurred near the base of the columns on both sides. In addition, serious damage was also observed on the north side of the west facade within 250 mm of the base of the column. In addition, serious damage was also observed on the north side of the west facade within 250 mm of the base of the column. Previous flexural cracks developed severely, resulting in more spalling of the cover concrete. At the end of the last loading cycle at R = 5.0%, severe spalling occurred in most areas within 250 mm of the base of the west facade column, resulting in significant material loss. The SB-MRC specimen showed a similar failure progression as SB-NRC, but with more severe damage in the 250 mm height range, larger crack widths, higher concrete spalling heights, and more cracks.

3.3. Residual Deformation

Figure 8 presents the relation between residual drift ratios and target drift ratios of the specimens reinforced by different types of rebars under the same NRC loading methods. Upon comparison, it is evident that the residual drift ratio of Specimen U-NRC remained comparatively lower throughout the entire loading process. In addition, the residual drift rate of the SB-NRC specimens shows a significant increase after the drift rate of 3%, while, before this threshold, the curve remains relatively stable. This is mainly because a large amount of concrete cover begins to peel off from 3.0%, the steel bars begin to enter the pre-yield stage, and the mechanical bonding of the steel bar surface begins to hinder the self-recovery of the concrete column during unloading. In contrast, for the U-NRC specimens, the residual drift rate did not exceed 0.6 even when the drift rate reached 5%, which is about one-third of that of the SB-series specimens. This is due to the spiral grooves on the surface of the steel bars and the strong yield strength of the steel bars to provide a strong restoring force during the unloading stage. These all reflect the excellent resilience of the U-shaped-steel-bar-reinforced column.

3.4. Energy Dissipation Capacity

The equivalent viscous damping factor (h_eq), proposed by Jacobsen [24], presented in Figure 9, was applied to discuss the energy dissipation capacity of the columns. Figure 10 presents the comparison of the damping factor between different specimens within this experiment. During the entire loading process, although the h_eq values of all columns hover around 0.08, the damping factors of the U-NRC specimen at various target drift ratios were consistently higher than those for the SB-NRC specimen, achieving its peak value at a drift ratio (R) of 2.5% that was 1.74 times greater than the SB-NRC. This is different from our previous study [12,13,14,15]; that is, the small bond strength of the steel bars will be beneficial for controlling the residual deformation, but, to maintain the same longitudinal reinforcement ratio as in the U column, only four SB steel bars were used in the column in this study, which greatly reduced the bonding area of the longitudinal reinforcement in the hinge zone, thereby reducing the effect of bonding on the residual deformation.

3.5. Effect of MRC Loads

Figure 11 illustrates the relation between the number of cycles and the lateral force ratio. The maximum lateral forces in the positive and negative positions of the SB-MRC were 181 kN and −193.5 kN under MRC loads, respectively, as depicted in Figure 5. Here, V/V_max represents the ratio of the maximum lateral force of each cycle to the overall maximum lateral force throughout the loading process. Figure 11a–c delineate three distinct states based on the loading plan: the low drift ratio state, medium drift ratio state, and high drift ratio state. Moreover, due to malfunctions in the experimental equipment, there are only four cycles of each maximum drift ratio within the range of R = 2% to R = 4% in the high drift ratio state.

The V/V_max ratios maintain relative stability for each load loop with the escalation in cycle numbers before reaching the drift ratio of 1.0%. However, beyond this threshold, the repeated cycles significantly degraded the lateral forces, particularly around the peak lateral forces. This is mainly because, after the column exceeds this drift ratio, serious cracks and concrete spalling usually begin to occur in the plastic hinge deformation zone. Notably, in the high drift ratio state, the positive and negative positions achieve maximum lateral force at drift ratios of 4% and 3%, with repeated cycling resulting in a 17.13% and 20.30% reduction in lateral force, respectively. At this time, the damage to the concrete is further increased, and the steel bars are close to their yield level, and serious damage such as deformation and buckling will occur.

In addition, contrary to the trend of the V/Vmax ratio with loading cycles, the effective viscous damping factor (h_eq) has a decreasing trend under the low and medium deformation stages, but it always hovers around 0.1, even when the drift rate exceeds 4%. This is mainly because the mechanical bite produced by the internal concrete and the ribs on the surface of the steel bars during small deformations hinders the free recovery of the column. Thereafter, the reduction in this factor is also small with the increase in the drift ratios. This is because, during large deformations, unlike the traditional ordinary steel RC structure, the strong mechanical anchoring at the ends of the ultra-high-strength steel bar can provide a strong restoring force to the steel bars during the unloading period to easily overcome the bond force between the deformed ribs on the steel bar surface and the internal concrete, thereby controlling the area of the hysteresis curves to be small–flat. This makes the area of the hysteresis curve increase in proportion to the equivalent potential area (±W_e), resulting in little change in h_eq values.

4. Discussion

4.1. Residual Deformation Control Mechanism of the U-NRC Specimen

The control mechanism of residual deformation of the SRRC columns tested in the study is elucidated in Figure 12. Initially, as depicted in Figure 12a, the steel rebars are characterized by six spiral shallow grooves. During the period of small deformation, this is an important factor in supporting the joint resistance of concrete and steel rebars against external loads, because the two ends of these steel bars are fixed by two steel plates. The specific performance is reflected in the hysteretic curves. The initial stiffness of the columns proposed here usually is greater than that using smooth steel bars [17]. In the stage of large deformation, as the concrete around the steel bars is destroyed by the repeated pull-out bonding of the steel bars, these grooves are filled with concrete, and the bonds of these deformed steel bars disappear completely, just like round steel bars.

Moreover, Figure 12d details the mechanism through which the steel rebar generates restoring forces during large deformation phases. Due to the high performance of the steel rebars, the restoring forces were large for the strain history data of the steel bars, supporting the notion that the steel bars do not reach their yielding strength even at the end loading cycles of the test. This adaptation of rebars with reduced bond strength contributes to minimizing the obstacles to the free recovery of columns, and then controlling the residual deformation in the specimens under conditions of large deformation. In summary, the transformation of deformed rebars with spiral grooves into plain rebars (unbonded rebars) at the large displacement stages and the end fixation of ultra-high-strength steel bars are the keys to controlling residual deformation of the columns under strong earthquakes.

4.2. PHL Evaluation

4.2.1. Experimental Observation

According to the literature [16], the PHL usually is calculated by (a) using a predefined curvature profile with the displacements and curvatures measured from tests, or (b) by experimental observation or the measured curvature profile along the height of the columns. To more closely approximate real physical phenomena, this paper adopted Method (b) to determine the PHL of the tested specimens. Based on the experimental observation presented in Figure 7 and the experimental results of the strain and curvature contribution, Figure 13 shows the strain and curvature contribution of the SB-NRC specimen along height. The PHL of SB specimens is about 375.0 mm regardless of the load methods, while the PHL of the U specimen is about 150–250 mm.

4.2.2. Existing PHL Model

In the past decades, many empirical models were proposed to estimate the PHL of RC columns. As shown in

Table 3. Comparison between empirical models and experimental observations.

Ref.	Authors	Plastic Hinge Length (Lp) Expressions	Lp,cal (mm)
			U	SB
[26]	Sheikh and Khoury	${\mathrm{L}}_{\mathrm{p}}=1.0h$	250	250
[27]	Priestley and Park	${\mathrm{L}}_{\mathrm{p}}=0.08L+6d_b$	136	198
[28]	Paulay and Priestly	${\mathrm{L}}_{\mathrm{p}}=0.08L+0.022{f_{y}d}_b$	458	620
[29]	Priestley et al.	${\mathrm{L}}_{\mathrm{p}}=0.08L+0.022{f_{y}d}_b\le 0.044{f_{y}d}_b$	458	620
[30]	Panagiotakos and Fardis	${\mathrm{L}}_{\mathrm{p}}=0.12L+0.014a_{sl}{f_{y}d}_b$	90	90
[31]	Lu et al.	${\mathrm{L}}_{\mathrm{p}}=0.077L+8.16d_b$	161	245
[32]	Berry and Eberhard	${\mathrm{L}}_{\mathrm{p}}=0.05L+0.1d_b{\displaystyle \frac{f_y}{\sqrt{f_c^{\prime }}}}(\mathrm{m}\mathrm{m},\mathrm{M}\mathrm{P}\mathrm{a})\le 0.25L$	188	188
[33]	EN 1998-3 (2005) EC8	${\mathrm{L}}_{\mathrm{p}}={\displaystyle \frac{L}{30}}+0.2h+0.11d_b{\displaystyle \frac{f_y}{\sqrt{f_c^{\prime }}}}(\mathrm{m}\mathrm{m},\mathrm{M}\mathrm{P}\mathrm{a})$	333	438
[34]	Bae and Bayrak	${\mathrm{L}}_{\mathrm{p}}=\left(0.3{\displaystyle \frac{P}{P_0}}+3{\rho }_S-0.1\right)L+0.25h\ge 0.25h$ $P_0=0.85f_c^{\prime }\left(A_g-A_s\right)+A_s{f}_y,{\rho }_s=A_S/A_g$	73	73
[16]	Ning and Li	${\mathrm{L}}_{\mathrm{P}}=\left(0.042+0.072{\displaystyle \frac{P}{P_0}}\right)L+0.298h+6.407d_b$	187	253
[35]	Ho	${\mathrm{L}}_{\mathrm{P}}=\left(20\sqrt{P/P_0}{\left({\displaystyle \frac{f_c^{\prime }}{f_y}}\right)}^{1.5}{\left({\displaystyle \frac{{\rho }_s}{{\rho }_{st}}}\right)}^{0.5}+0.6\right)h$	150	150
[33]	Biskinis and Fardis	${\mathrm{L}}_{\mathrm{P}}=0.2h\left(1+{\displaystyle \frac{1}{3}}minmum\left(9,{\displaystyle \frac{L}{h}}\right)\right)$	100	100
		Experimental results (Lp,exp)	120	120

$d_b$: diameter of longitudinal bars; $L$: shear span, $f_y$: yield strength of longitudinal bars; $a_{sl}$: coefficient of the fixed-end rotation due to the slippage of longitudinal bars; $h$: overall cross-sectional depth; $f_c^{\prime }$: compressive strength of concrete; ${\rho }_s$: longitudinal reinforcement ratio; $A_s$: area of longitudinal reinforcements; $A_g$: cross area of concrete section; $P$: applied axial load; $P_0$: nominal axial capacity; ${\rho }_{st}$: transverse reinforcement ratio; L_p,exp and L_p,cal: experimental and calculative L_p, respectively.

, these models are used to calculate the PHL of the resilient columns investigated in this paper. The results show that non-negligible errors occur between the calculations based on empirical models and experimental observations. The calculating results closest to the experimental observations of the RC columns are based on EN 1998-3 (2005) EC8 [25], but an 11.2% error still appears in the U-series column and a 16.8% error for the SB-series column. An effective model for estimating the PHL of the resilient RC columns reinforced by high-strength reinforcements is required.

5. ML-Based PHL Model

In this study, an ML-based approach is applied to develop an effective and simple model for predicting the PHL of the RC columns including the test RRC columns. As for the ML methods, it is very crucial for the dataset used to train the ML models. According to the literature [16], 133 RC columns are selected to discuss the PHL of the columns. To make the ML model more accurate, only square columns from the 133 RC columns are used to comprise a new dataset for machine learning. As presented in

Table 4. Dataset for ML training.

Ref.	B (mm)	L (mm)	${\mathit{f}}_{\mathit{c}}^{\prime }$ (MPa)	Longitudinal Rebars			Transverse Hoops			$\mathit{P}/{\mathit{P}}_0$	Observed PHL Ratio
				${\mathit{d}}_{\mathit{b}}$ (mm)	${\mathit{\rho }}_{\mathit{s}}$ (%)	${\mathit{f}}_{\mathit{y}}$ (MPa)	s (mm)	${\mathit{\rho }}_{\mathit{s}\mathit{t}}$ (%)	${\mathit{f}}_{\mathit{v}\mathit{y}}$ (MPa)
[36]	610	3050	43.3	22	1.25	503	95	2.04	427	0.43	0.66
	440	3050	43.3	16	1.25	496	86	1.76	496	0.43	0.91
	610	3050	36.5	22	1.25	400	152	0.72	455	0.17	0.49
	610	3050	41.4	22	1.25	400	152	1.3	434	0.17	0.47
[24]	325	1895	50	32	6.1	531	70	2.1	531	0.61	1.47
	325	1895	83.3	20	2.4	531	70	2.1	531	0.33	1.42
	325	1895	77.8	12	0.9	339	85	1.73	339	0.12	0.92
	325	1895	80.6	20	2.4	531	105	2.1	531	0.31	1.48
	325	1895	56.1	32	6.1	531	110	2	531	0.59	1.2
	325	1895	96.4	20	2.4	531	90	2.45	531	0.34	1.47
	325	1895	94.7	32	6.1	531	100	2.2	531	0.35	1.69
	325	1895	85	32	6.1	572	120	3.2	572	0.63	2.05
[37]	305	1841	32.4	19	2.44	508	95	1.68	508	0.76	0.85
	305	1841	32.5	19	2.44	508	114	1.69	464	0.76	1.1
	305	1841	33.2	19	2.44	508	108	1.68	508	0.6	0.96
	305	1841	31.3	19	2.44	508	108	1.68	464	0.77	1.05
	305	1841	32.8	19	2.44	508	108	3.06	508	0.77	0.99
	305	1841	32.3	19	2.44	508	108	1.3	508	0.47	1.16
[38]	305	1841	54.1	19	2.44	508	108	1.68	508	0.62	1.05
	305	1841	54.7	19	2.44	508	108	3.06	464	0.64	0.89
	305	1841	53.6	19	2.44	508	76	4.3	464	0.64	1.08
[39]	300	1200	72	16	2.98	483	50	1.92	952	0.57	1.5
	300	1200	74	16	2.98	483	80	1.2	952	0.57	1
	300	1200	80.5	16	2.98	483	50	1.92	952	0.45	1
	300	1200	86.4	16	2.98	483	80	1.2	952	0.45	1.2
	300	1200	72.6	16	2.98	483	50	2.48	952	0.57	0.84
	300	1200	80	16	2.98	483	80	1.55	952	0.55	1.06
	300	1200	76	16	2.98	483	50	2.48	952	0.5	1.17
	300	1200	88	16	2.98	483	80	1.55	952	0.5	1.61
	300	1200	76.2	16	2.98	483	50	2.48	952	0.5	0.93
	300	1200	75.1	16	2.98	483	50	2.48	952	0.56	1.55
[40]	400	1800	25.6	20	1.57	474	80	2.55	333	0.2	0.43
[41]	550	1650	32	20	1.25	511	110	1.7	325	0.1	0.39
	550	1650	32	20	1.25	511	110	1.7	325	0.1	0.69
	550	1650	32.1	20	1.25	511	90	2.8	325	0.3	0.67
	550	1650	32.1	20	1.25	511	90	2.8	325	0.3	0.86
[42]	400	1600	24.8	19	0.82	362	100	0.32	325	0.03	1
	400	1600	24.8	19	0.82	362	100	0.32	325	0.03	1
	400	1600	24.8	19	0.82	362	100	0.32	325	0.08	1
	400	1600	24.8	19	0.82	362	100	0.32	325	0.03	1
[43]	325	1895	89.6	12	0.86	556	100	0.38	378	0.11	0.83
	325	1895	85.4	12	0.86	556	175	0.38	362	0.11	0.68
	325	1895	83.2	12	0.86	556	220	0.47	344	0.11	0.8
	325	1895	85.9	12	0.86	556	85	1.73	339	0.11	0.77
[44]	305	2010	72.1	20	2.58	454	95	3.15	463	0.5	1.15
	305	2010	71.7	20	2.58	454	90	2.84	542	0.36	0.98
	305	2010	71.8	20	2.58	454	90	2.84	542	0.5	0.91
	305	2010	71.9	20	2.58	454	100	5.12	463	0.5	0.89
	305	2010	101.8	20	2.58	454	90	4.83	542	0.45	0.86
	305	2010	101.9	20	2.58	454	76	6.72	463	0.46	1.05
	305	2010	102	20	2.58	454	94	2.72	542	0.45	0.88
	305	2010	102.2	20	2.58	454	70	4.29	463	0.47	1.26

, B (mm) is the sectional width ranging from 300 mm to 610 mm, L (mm) is the column length ranging from 1200 mm to 3050 mm, $f_c^{\prime }$(MPa) is the concrete compressive strength varying from 24.8 to 102.2, $d_b$ (mm) is the diameter of longitudinal reinforcement changing from 12 mm to 32 mm, ${\rho }_s$(%) is the longitudinal reinforcement ratio in percent ranging from 0.82 to 6.1, $f_y$ (MPa) is the reinforcing steel yield strength changing from 339 MPa to 572 MPa, s (mm) is the spacing of transverse reinforcement varying from 50 mm to 220 mm, ${\rho }_{st}$ (%) is the stirrup ratio ranging from 0.32 to 6.72, $f_{vy}$ (MPa) is the yield strength of transverse reinforcement changing from 325 MPa to 952 MPa, and $P/P_0$ is the axial load ratio ranging from 0.03 to 0.07, respectively. The PHL ratio is defined as follows:

$$PHLratio={\displaystyle \frac{L_P}{B}}$$

(1)

where L_p is the plastic hinge length, and B is the sectional width.

5.1. Introduction of Machine Learning Methods

5.1.1. MLP

The multi-layer perceptron (MLP) [45] is a fundamental ML algorithm that belongs to the class of feedforward artificial neural networks. Its basic structure consists of three primary components: the input layer, hidden layers, and the output layer. These layers are made up of neurons, which are the essential building blocks of the network. The primary function of an MLP is to map input data to corresponding output predictions. This mapping is accomplished through forward propagation, a process where input data pass through each layer of the network. At each neuron, a weighted sum of the inputs is computed, followed by the application of an activation function. Training an MLP involves the backpropagation algorithm, a supervised learning technique. During backpropagation, the model adjusts the weights and biases of the neurons to minimize the error between the predicted output and the actual target values. This process iteratively reduces the prediction error by propagating the error back through the network and updating the parameters accordingly. In conclusion, MLP is a powerful ML tool for solving regression problems, as a result of its layered architecture and the effectiveness of the backpropagation training method.

5.1.2. XGBoost

XGBoost, short for extreme gradient boosting [46], is a powerful ensemble machine learning algorithm. Ensemble techniques, such as boosting, enhance robustness and accuracy by combining the predictions of multiple base learners. In XGBoost, decision trees serve as the base learners. A key concept in XGBoost is gradient boosting. This approach iteratively corrects the residual errors of previous models by minimizing a loss function through gradient descent. XGBoost improves upon traditional decision tree algorithms by providing a scalable, distributed framework. The training process of XGBoost starts with initializing predictions to the mean value for regression tasks. During each boosting round, a new decision tree is added to the model. The gradients of the loss function relative to the current predictions are calculated, and a new decision tree is fitted to these gradients. The model is then updated using the predictions from the new tree, scaled by a learning rate. This process optimizes a regularized objective function, incorporating both the loss function and a regularization term. XGBoost’s efficiency and high performance come from its advanced algorithms and features, including regularization, sparsity awareness, and parallelization. Even for small datasets, XGBoost can deliver a superior performance due to these sophisticated mechanisms.

5.1.3. KNN

KNN, short for the k-nearest neighbors algorithm [47], is a simple, instance-based machine learning method used for both classification and regression tasks. It makes predictions by considering the characteristics of the k most similar data points, known as neighbors. Two critical parameters for the performance of the KNN model are the selection of k, and the number of neighbors. The KNN algorithm requires intensive computation during the prediction phase due to the necessity of calculating the distance between the query point and all points in the training dataset. Despite this, KNN is powerful and effective for many applications, particularly when the decision boundary is irregular. Its intuitiveness and ease of implementation make it a popular choice for various pattern recognition and data mining tasks.

5.1.4. Transformer

Transformer [48], originally designed for natural language processing tasks and renowned for powering models like ChatGPT, consists of an encoder–decoder structure. In the context of the aforementioned regression task, only the encoder part is utilized. The Transformer employs self-attention mechanisms to weigh the influence of each input feature, enabling it to capture complex dependencies and relationships within the data. This attention mechanism computes a weighted sum of input features, with the weights determined by the relevance of each feature to the task at hand. For regression tasks, the Transformer model processes input sequences through multiple layers of self-attention and feedforward neural networks. Each layer refines the representation of the input data, capturing intricate patterns and interactions. The final output layer generates continuous values as predictions, making the Transformer suitable for regression. A key advantage of using Transformers for regression is their ability to handle large and complex datasets, capturing long-range dependencies without losing information over distance. Additionally, Transformers are highly parallelizable, resulting in efficient training and inference.

5.1.5. Training Process

As shown in Figure 14, in total, 52 datasets are divided into two parts, including all the inputs: the structural parameters (B, L, $f_c^{\prime }$, $P/P_0$, ${\rho }_s$, $d_b$, $f_y$, s, ${\rho }_{st}$, and $f_{vy}$) and the output: the corresponding PHL; 90% of the dataset is used as the training set and the other 10% is used as the test set. The four ML algorithms introduced above are trained and well-tuned. To select the optimal model, the performance of the models was evaluated based on the following four statistical parameters.

Mean absolute error (MAE):

$$MAE={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\left|y_i-{\widehat{y}}_i\right|$$

(2)

Root mean squared error (RMSE):

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{(y_i-{\widehat{y}}_i)}^2}$$

(3)

Coefficient of determination (R²):

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}{{\sum }_{i=1}^n{\left(y_i-{\overline{y}}_i\right)}^2}}$$

(4)

Index of agreement (d) [48]:

$$d=1-{\displaystyle \frac{{\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}{{\sum }_{i=1}^n{\left(\left|y_i-{\overline{y}}_i\right|+\left|{\widehat{y}}_i-{\overline{y}}_i\right|\right)}^2}}$$

(5)

where n is the number of observations; ${\mathrm{y}}_{\mathrm{i}}$ is the actual value; ${\widehat{\mathrm{y}}}_{\mathrm{i}}$ is the predicted value; and ${\overline{\mathrm{y}}}_{\mathrm{i}}$ is the mean of the actual values.

These metrics are comprehensively assessed on both the training and testing datasets. Once the optimal model is identified, we will utilize SHapley Additive exPlanations (SHAP) to visualize the contribution of different input parameters to the prediction process. SHAP values provide detailed insights into how each feature influences the model’s predictions, enhancing our understanding of the model’s behavior and feature importance.

5.2. Comparison and Validation

The relationship between the experimental PHL and predicted PHL based on four different ML algorithms is presented in Figure 15. The four models are all well-tuned by the optimal hyperparameters. The models not only predict in the test set but also in the training set, which is that 90% of predictions are within the 20% to −20% error range (the green dashed lines). According to Figure 16b, the MLP is the optimal model in the training set with the following metrics: MAE = 0.318; RMSE = 0.0762; R² = 0.9448; and D = 0.9856. The optimal model in the test set is the Transformer with the following metrics: MAE = 0.1097; RMSE = 0.1262; R² = 0.7163; and d = 0.9244.

In addition, in terms of validating the usability of these ML models in the resilient columns, which is investigated in the tests mentioned in Section 3, the structural parameters of the columns are regarded as input to conduct two more predictions of all models. The predicting results are presented in

Table 5. Comparison between ML models and experimental observations.

ML Models	Prediction of Plastic Hinge Height (×λ) (mm)
	U-Series	SB-Series
KNN	108.2	95.0
XGBoost	104.9	106.6
KNN	95.7	95.7
Transformer	118.8	114.8
Experimental observations	120.0	120.0

, with the best PHL prediction model of the U-series RC column which is KNN with a result of 290 mm, and SB-series RC columns with a result of 348 mm for the Transformer model, which had 1.6% and 7.2% errors compared to the PHL from experimental observation.

Due to the different surface textures of U- and SB-series reinforcement, as shown in Figure 2, the grooves on the surface of U-series reinforcement (low-bond rebar) contribute more resilient capacity and a shorter PHL than the SB series. Moreover, as presented in Table 4, the data of columns that are selected are reinforced by normal-bond rebars that are similar to SB-series reinforcement. Due to this significant reason which is that the shear span ratio of the investigated RC columns is smaller than that of most columns of the dataset range, and the use of the ultra-high strength bars fixed by the ends made the restoring force of the columns much higher than the level of ductile concrete columns in the dataset, the predictions of ML models are not ideal enough. As a starting study here, the authors proposed a correction parameter of 0.33 because (1) the ratio of the average length in the dataset to the investigated RC columns’ length and (2) the yield strength of the used steel bars in this study is three times that of the traditional deformed rebars, to modify the models. A significant contribution is still required in the future to confirm the relationship of the PHL between low-bond rebars and normal-bond rebars. Moreover, a dataset consisting of the PHL of low-bond-rebar-reinforced columns is required for the better evolution of ML models.

5.3. Explanation of the Optimal ML Model and Feature Importance

5.3.1. SHAP Method

SHAP (SHapley Additive exPlanations) [49] is a game-theory-based approach to explaining the output of machine learning models. The method attributes the output of a model to its input features by computing the SHAP value for each feature, which represents the average contribution of a feature across all possible subsets of features. The SHAP value is derived from cooperative game theory and ensures a fair distribution of the prediction among the features, considering their interactions. The key idea behind SHAP is to treat the prediction task as a game where each feature is a player contributing to the final prediction, as shown in Figure 17. By evaluating all possible combinations of features, SHAP calculates the marginal contribution of each feature to the prediction. This approach provides a unique solution that satisfies properties such as efficiency, symmetry, and additivity, making it a robust and consistent method for feature attribution. SHAP values are computed by training the model on different subsets of data and observing the changes in the prediction when a particular feature is added or removed. This process involves a considerable computational cost but offers deep insights into the model’s behavior. The SHAP framework can be applied to any machine learning model, including black-box models like deep neural networks and tree-based methods, making it a versatile tool for model interpretation.

5.3.2. Explanation of Transformer on Test Set

The SHAP method was applied to the optimal ML model Transformer in the test set. As presented in Figure 18, the mean absolute values of the SHAP values of each input variable are measured, and, the global importance of the input variables is shown based on the mean value. The blue bar means there is a negative impact of the presentive structural parameter to the output PHL value while the red bar means a positive impact. The parameter s has the most negative impact on the PHL while the most positive impact is from the $f_c^{\prime }$.

5.3.3. Explanation of Prediction on Investigated Columns

As for the two varieties of resilient RC columns—the U series and SB series which are investigated in the experiment—the SHAP method was also applied in the validating prediction to show a deep inside relationship between the input structural parameters and output PHL values. According to Figure 19, for the prediction of the U-series columns, the yield strength of longitudinal reinforcement bars $f_y$ plays the most positive role in the output, followed by the cross-sectional depth B. Moreover, the most negative impact input parameter is the diameter of longitudinal bars $d_b$ followed by the axial force ratio P/P₀. As presented in Figure 20, for the prediction of the SB−series RC columns, the same as the prediction of the U-series columns, $f_y$ and B are the two most positive input parameters and $d_b$ and P/P₀ are the most negative input parameters, respectively.

6. Conclusions and Remark

In this study, a comprehensive experimental analysis was carried out to elucidate the seismic performance of resilient reinforced concrete (RC) columns reinforced with ultra-high-strength materials, emphasizing the impact of various reinforcement and loading types. In terms of the effective evaluation of the plastic hinge height of the resilient RC columns, four ML models are trained and compared. In addition, the SHAP method is adopted to investigate the deep connection inside the ML model. The main findings and results are as follows:

(1)

Comparing the F60S3N33U−NRC specimen to the F60S3N33SB−NRC specimen, the F60S3N33SB−NRC specimen exhibits a markedly reduced residual drift ratio and equivalent viscous damping factors, which can be explained by the large bond of the rebar or the deformed ribs on the surface of the bars, which hinders the free recovery of the column during large deformations, although these high-strength steel bars provide a strong restoring force by fixing their ends.

(2)

Regarding the impact of multiple reversed cyclic loads, it significantly induces degradation around the drift ratio corresponding to the peak lateral force of each loop, with the lateral force experiencing reductions of 17.13% and 20.30% in the positive and negative directions, respectively. However, this is not the case for the equivalent viscous damping factors, which remain stably close to 0.1.

(3)

An effective ML model based on the Transformer algorithm is introduced to evaluate the PHL of RC square columns including the RRC tested by the study, which is compared to the KNN, MLP, and XGBoost algorithms. The ML model predicts the PHL of the columns with relatively minor errors of 1.6% and 7.2% compared to the experimental observation. Moreover, the optimal Transformer model in the test set is with the following metrics: MAE = 0.1097; RMSE = 0.1262; R² = 0.7163; and d = 0.9244.

(4)

According to the analysis based on the SHAP method of the Transformer model, the yield strength of the longitudinal reinforcement bars and the cross-sectional depth make the most positive contribution while the $d_b$ and P/P₀ are the most negative input parameters to the prediction of the investigated columns.

To exclude the influence of the axial force and only study the degradation under the MRC load, and as part of the preliminary study, this paper studies the seismic performance of the columns without the axial force. Currently, the tests on concrete columns with axial forces applied are ongoing. Based on similar experimental studies that have been completed in our lab, it is expected that the maximum lateral resistance of concrete columns increases in the presence of axial forces and begins to decrease at around R = 5%. These studies will be analyzed in detail in the next article.