Transfer-Learning Prediction Model for Low-Cycle Fatigue Life of Bimetallic Steel Bars

Abstract

The prediction of the low-cycle fatigue life of bimetallic steel bars (BSBs) is essential to promote the engineering application of BSBs. However, research on the low-cycle fatigue properties of BSB is limited, and fatigue experiments are time-consuming. Moreover, considering that sufficient data are needed for model training, the lack of data hinders the leverage of typical data-driven machine learning, which is widely used in fatigue life prediction. To address this issue, a transfer learning framework was suggested to accurately predict the low-cycle fatigue life of BSBs with limited data. To achieve this goal, 54 data points obtained from low-cycle fatigue tests on BSBs and 264 data points of other metallic bars were collected. Source models based on artificial neural networks (ANNs) were first constructed using the collected source dataset. Then, the learned knowledge stored in the source models was transferred to the transfer models. After that, transfer models were further fine-tuned and then tested using the target dataset of BSBs. The ANN models, which were of the same structure as the transfer models but only trained with the target dataset without transferring deep features from the source models, were set as baseline models. Compared with baseline models, the constructed transfer models could be used to accurately predict the fatigue life of BSBs. Moreover, the influence of hidden layers of ANNs on accuracy was examined by comparing one-layer and two-layer transfer models. Furthermore, the influence of key parameters on fatigue life of metallic bars was evaluated by feature analysis.

1. Introduction

Longitudinal steel bars in reinforced concrete (RC) members are often damaged by low-cycle fatigue loading during earthquakes [1,2,3,4]. Thus, it is important to determine the low-cycle fatigue life of steel bars to evaluate the hysteretic behavior of RC structures. Several researchers have performed comprehensive investigations on the low-cycle fatigue properties of steel bars. Recent research indicated that the low-cycle fatigue properties of steel bars were influenced by many factors, such as the slenderness ratio (λ), mechanical properties of steel, and fatigue strain amplitude (ε_a). Aldabagh and Alam [5] investigated the influence of ε_a and λ on the low-cycle fatigue performance of ASTMA1035 Grade 690 bars. Apostolopoulos [6] conducted low-cycle fatigue tests on corroded reinforcing steel bars (S500s). The experimental results illustrated that the corroded steel bars exhibited a gradual reduction in both their load bearing ability and available energy. Hawileh et al. [7] studied the low-cycle fatigue behavior of ASTM A706 and A615 Grade 60 steel bars with different ε_a values and established an equation relating dissipated energy to ε_a and fatigue life. Kashani et al. [8] quantified the combined effects of inelastic buckling and chloride-induced corrosion damage on the low-cycle fatigue life of embedded reinforcing bars in concrete. By utilizing the fatigue life data of existing structures, the fatigue life of newly constructed structures can be predicted, providing a scientific basis for structural design and ensuring the safety and economy of the structure. It should be noted that the dimensions and mechanical properties of steel bar specimens were different in different experimental investigations. Furthermore, the applied fatigue load in previous studies also varied. Therefore, it is difficult to directly compare different experimental results. Although many empirical regression formulas have been proposed by different scholars to clarify the S–N curve of steel bars under low-cycle fatigue loads, the generality of these prediction formulas is questionable. Traditional prediction methods are based on numerical regression, which makes it difficult to capture the complex features hidden in the data, necessitating that they be calculated separately for each situation. The lack of a general quantitative method for determining the low-cycle fatigue properties of steel bars also led to difficulties in engineering design and evaluation. Therefore, it is necessary to establish a unified method to effectively analyze the low-cycle fatigue properties of different types of steel bars. Some investigations were conducted in this study to address this issue.

Bimetallic steel bars (BSBs) are novel steel reinforcements that exhibit outstanding corrosion resistance and good mechanical properties. BSBs consist of stainless steel cladding, carbon steel substrate, and a metallurgical bonding layer between the stainless-steel cladding and carbon steel substrate, which is produced through the hot-rolling process (Figure 1) and ensures cooperative deformation of the stainless-steel cladding and carbon steel substrate in the BSB [9,10]. Research conducted by Shi et al. [11,12] indicated that the metallurgical bonding layer from the hot-rolling process had outstanding connection performance, especially for shear resistance. Stainless-steel cladding enhances the corrosion resistance of BSB by isolating the corrosion factor from the carbon steel substrate, which blocks the process of carbon steel corrosion [13]. Wang et al. [14] performed a series of investigations into the monotonic mechanical properties of BSB. The experimental results indicated that the BSB exhibited good strength properties, and its stress–strain curve was influenced by the cladding ratio. The ductile properties of the bimetallic steel bars (BSBs) met the requirements of RC structures. Li et al. [3] conducted an experimental study on the fatigue properties of BSBs, where the fatigue life and fracture behavior were investigated. The BSB also performs well in terms of residual mechanical properties after exposure to fire [15,16,17]. Based on the experimental results in [18], the bonding properties of BSBs in seawater sea sand concrete were similar to those of the traditional carbon steel bar, where the predictive bond-slip model was suggested. The above investigations indicate that the BSB, which is a substitute for traditional carbon steel bars, can be reasonably applied to RC structures to enhance their durability. Furthermore, a comprehensive economic analysis of BSB was conducted by Liu et al. [19], which indicated that for RC structures in a corrosive environment, the BSB had the lowest life-cycle cost compared with stainless steel bars and carbon steel bars. In general, recent studies on BSB have demonstrated satisfactory engineering application potential.

As stated previously, longitudinal steel bars in concrete members are commonly damaged by low-cycle fatigue loading during earthquakes. Hence, it is necessary to conduct a comprehensive investigation of the low-cycle fatigue behavior of BSB to promote its engineering application. Although there have been studies on the low-cycle fatigue properties of BSB [15,20,21,22], the key parameters considered in previous investigations were very limited, and the dataset is insufficient for a comprehensive predictive method for the low-cycle fatigue properties of BSB. The material properties of bimetallic steel bars are more complex than those of single metals, and their fatigue performance is not only affected by the characteristics of each metal but also by the interaction between the two metal interfaces. Therefore, it is necessary to consider more factors when predicting BSB fatigue life, which increases the difficulty of prediction. Conventional regression prediction methods cannot meet these requirements, so it is necessary to use better models to predict BSB fatigue performance.

Owing to its capability to describe the nonlinear relationship between high-dimensional data, machine learning (ML) has been applied in many fields of structural engineering [23,24,25,26]. Machine learning can consider multiple independent variables, thereby achieving higher accuracy in predicting material properties [27]. Recently, machine learning has been widely selected to quantify fatigue life of various materials [28,29,30,31,32,33,34]. For instance, various ML models have been used to predict the creep, fatigue, and creep fatigue of stainless steel [35]. Yang et al. [36] proposed a novel ML-based method to handle multiaxial fatigue life prediction by utilizing features extracted from a long short-term memory network as the input of a neural network. Kamiyama et al. [37] predicted the low-cycle fatigue crack development of metal thin films using a deep convolutional neural network with microscopic images as the input. By constraining the biases and weights of the neural network, Chen and Liu [38] developed a physics-guided neural network for the estimation of fatigue S–N curves. Guo et al. [39] trained a model using the adaptive boosting algorithm to explore the quantitative influence of corrosion degree, stress ranges, loading ratio, loading frequency and other factors on the fatigue life of corroded high-strength steel wires. The literature mentioned above indicates that ML achieves satisfactory applicability and accuracy for fatigue life prediction. However, the aforementioned studies were all based on the assumption of a sufficient dataset, because ML is mostly a data-driven method. Therefore, despite the high-dimensional regression capacity of typical data-driven ML models, the time-consuming fatigue experiments that are unfavorable for data acquisition hinder the application of typical ML models in fatigue life prediction.

Transfer learning has proven to be one of the most effective approaches, where the small-sample learning needs to be addressed [40,41], and can be used to improve a model from one domain by transferring information from a related domain. The two common steps in transfer learning are deep feature extraction and fine-tuning [42]. In the deep feature extraction, a source model is constructed with the source dataset, and activation values of hidden layers are stored as features. In fine-tuning, the source model is fine-tuned with the target dataset for a similar task by fixing the first several layers and fine-tuning the final layers. It has been widely used in many applications, such as predicting the remaining useful life of machines [43], machine fault diagnosis [44], and plant classification [42]. By combining LSTM and transfer learning, Wei et al. [45] suggested a method to reveal S–N curves. A comprehensive review of the literature on transfer learning can be found in [40,41].

In this paper, a framework based on machine learning and transfer learning is proposed, aimed at achieving efficient prediction of low-cycle fatigue life of BSBs with limited data. This framework is also suitable for fatigue behavior research on other materials. The basic idea is to construct a source model using artificial neural networks (ANNs) based on a comprehensive source dataset collected from existing research on the low-cycle fatigue life of other metallic materials. Then, by leveraging the prior knowledge learned from the source dataset, transfer models fine-tuned using the target dataset of BSBs with very limited data can predict the low-cycle fatigue life of BSBs. Thus, the problem of predicting the low-cycle fatigue life of BSBs with insufficient BSB samples will be efficiently solved. Based on our knowledge, the transfer learning approach to low-cycle fatigue life prediction still remains very limited.

The main contributions of this paper are as follows:

(1) A novel framework based on neural networks and transfer learning is proposed, which significantly reduces the cost and simplifies the quantification of the low-cycle fatigue life of BSBs.

(2) An ML model is trained to efficiently predict the low-cycle fatigue life of various types of metallic bars with high accuracy, which could provide a practical analysis method in engineering.

(3) The relative importance of several key parameters to the low-cycle fatigue life of steel bars is evaluated.

Experimental results from existing research are introduced in Section 2. The framework and model training process are proposed in Section 3. The prediction results are discussed in Section 4. The efficiency of transfer learning is discussed, and the parametric influence is analyzed in Section 5. The main conclusions are presented in Section 6.

2. Experimental Results from Fatigue Experiment

2.1. Test Approach

Although concrete and stirrups were designed to provide an embedding effect on the longitudinal steel bar in RC structures, the concrete between stirrups often cracked and separated (Figure 2) under an earthquake because of the poor tensile properties of concrete. Then, without the embedded effect of concrete, the slenderness ratios of the longitudinal steel bars between the transverse stirrups were relatively large. Thus, the longitudinal steel bars were prone to overall buckling during earthquakes and exhibited noteworthy transverse deformations, as shown in Figure 2. To accurately reveal the low-cycle fatigue properties of steel bars under hysteretic loading, it is necessary to properly simulate the above boundary conditions and steel bar deformations in a laboratory environment. The experimental methods and MTS system shown in Figure 3 were widely employed in previous research [5,7,8,22] to clarify the low-cycle fatigue performance of steel bars. The test specimen shown in Figure 3 consists of a clamping section and a test section. To simulate the embedding effect of concrete and transverse stirrups on the longitudinal steel bar, the clamping section’s in-plane translation and rotation degrees of freedom were limited [21,22,46,47,48,49]. During fatigue loading, the vertical deformation of the testing section, the fatigue load, and the loading cycles were recorded and employed to determine the strain, stress, and fatigue life of specimens, respectively.

2.2. Variables

Cycles to failure (N_f) was selected as the performance index to quantify the low-cycle fatigue properties of steel bars in this study, which has been employed by many researchers in their studies [3,5,22,50,51]. The experimental results indicated that N_f was influenced by many parameters. The mechanical properties of steel bars might cause differences in the accumulation of fatigue damage [52], and the influences of mechanical properties on N_f were reflected by the yield strength f_y, ultimate strength f_u, and ratio of f_y/f_u. With respect to the geometric parameters, the diameter d and slenderness ratio λ = l/d significantly affected the failure mode of the steel bars, where l is the length of the test segment. When d is relatively small and λ is relatively large, the specimen under fatigue loading is prone to inelastic buckling, as shown in Figure 4. Large plastic deformation occurred at the inflection points of inelastic buckling deformation, which accelerated the accumulation of fatigue damage and reduced the fatigue life. The effects of fatigue load on N_f are reflected by the fatigue strain amplitude ε_a. With an increase in ε_a, more obvious plastic deformation occurred in the steel bars, which accelerated the accumulation of fatigue damage. In addition, the increase in ε_a led to a greater compressive stress and compression deformation in the steel bars, which aggravated the generation of inelastic buckling. With the increase in ε_a, the fatigue life N_f of the steel bars decreased gradually. Therefore, the yield strength f_y, ultimate strength f_u, ratio f_y/f_u, slenderness ratio λ, diameter d, and fatigue strain amplitude ε_a are included in the following research. It should be noted that the corrosion effect, which changes the properties of steel bars, was not considered in this study.

2.3. Dataset of Experimental Data

The objective of this study is to quantify the low-cycle fatigue life of BSBs by leveraging existing experimental results on different metallic bars. To achieve this objective, a source dataset consisting of seven types of other metallic bars (264 samples in total) was collected from existing research [3,5,22,50,51], as summarized in

Table 1. Experimental results from existing research [3,5,22,50,51].

Group	Type		Number of Samples	Nf
1	Other metallic bars (264 samples in total)	ASTM A1035 Grade 690	64	1~49
2		B460 smooth bar	29	17~1226
3		HRB400E/316L stainless steel clad bar	24	3~319
4		B500B bar	92	6~171
5		B500A bar	8	8~20
6		B400C bar	15	7~20
7		B450C bar	32	7~20
8	BSB	304-HRB400 stainless-clad BSB	54	12~2637

. To avoid a decrease in model accuracy caused by redundancy or noise, the complexity of the dataset should be minimized as much as possible [53,54]. In this study, 7 variables from the dataset were selected based on test results, including f_y, f_u, f_y/f_u, d, ε_a, λ, and N_f. The collected target dataset of BSBs is also listed in Table 1. As stated above, studies on the low-cycle fatigue behavior of BSBs are very limited. Therefore, only 54 samples were collected. It is worth mentioning that among the 54 samples, every three formed a group with the same geometric parameters and loading conditions. The average test result for each group was adopted in the model training. Hence, there were only 18 efficient data samples for BSB, which was insufficient for directly constructing an ML model. The transfer framework in this study, however, will be able to capture the fatigue behavior of BSBs using the dataset of other metallic bars.

The details on the experimental results are listed in Appendix A,

Table A1. Experimental results from current investigations.

Number	Type	λ	d (mm)	εa	fy (MPa)	fu (MPa)	Nf
T-6-0.006	304-HRB400 stainless-clad bimetallic steel bar [22]	6	18	0.60%	486.15	652.52	410
T-6-0.006		6	18	0.60%	486.15	652.52	387
T-6-0.006		6	18	0.60%	486.15	652.52	415
T-6-0.015		6	18	1.50%	486.15	652.52	50
T-6-0.015		6	18	1.50%	486.15	652.52	42
T-6-0.015		6	18	1.50%	486.15	652.52	44
T-6-0.024		6	18	2.40%	486.15	652.52	17
T-6-0.024		6	18	2.40%	486.15	652.52	18
T-6-0.024		6	18	2.40%	486.15	652.52	16
T-3-0.01		3	18	1.00%	486.15	652.52	2390
T-3-0.01		3	18	1.00%	486.15	652.52	2001
T-3-0.01		3	18	1.00%	486.15	652.52	2637
T-6-0.01		6	18	1.00%	486.15	652.52	370
T-6-0.01		6	18	1.00%	486.15	652.52	297
T-6-0.01		6	18	1.00%	486.15	652.52	325
T-9-0.01		9	18	1.00%	486.15	652.52	164
T-9-0.01		9	18	1.00%	486.15	652.52	144
T-9-0.01		9	18	1.00%	486.15	652.52	136
T-12-0.01		12	18	1.00%	486.15	652.52	76
T-12-0.01		12	18	1.00%	486.15	652.52	73
T-12-0.01		12	18	1.00%	486.15	652.52	72
T-15-0.01		15	18	1.00%	486.15	652.52	76
T-15-0.01		15	18	1.00%	486.15	652.52	71
T-15-0.01		15	18	1.00%	486.15	652.52	80
T-3-0.02		3	18	2.00%	486.15	652.52	269
T-3-0.02		3	18	2.00%	486.15	652.52	300
T-3-0.02		3	18	2.00%	486.15	652.52	339
T-6-0.02		6	18	2.00%	486.15	652.52	47
T-6-0.02		6	18	2.00%	486.15	652.52	46
T-6-0.02		6	18	2.00%	486.15	652.52	48
T-9-0.02		9	18	2.00%	486.15	652.52	31
T-9-0.02		9	18	2.00%	486.15	652.52	23
T-9-0.02		9	18	2.00%	486.15	652.52	22
T-12-0.02		12	18	2.00%	486.15	652.52	28
T-12-0.02		12	18	2.00%	486.15	652.52	25
T-12-0.02		12	18	2.00%	486.15	652.52	23
T-15-0.02		15	18	2.00%	486.15	652.52	26
T-15-0.02		15	18	2.00%	486.15	652.52	34
T-15-0.02		15	18	2.00%	486.15	652.52	29
T-3-0.03		3	18	3.00%	486.15	652.52	71
T-3-0.03		3	18	3.00%	486.15	652.52	70
T-3-0.03		3	18	3.00%	486.15	652.52	76
T-6-0.03		6	18	3.00%	486.15	652.52	14
T-6-0.03		6	18	3.00%	486.15	652.52	17
T-6-0.03		6	18	3.00%	486.15	652.52	17
T-9-0.03		9	18	3.00%	486.15	652.52	13
T-9-0.03		9	18	3.00%	486.15	652.52	13
T-9-0.03		9	18	3.00%	486.15	652.52	13
T-12-0.03		12	18	3.00%	486.15	652.52	12
T-12-0.03		12	18	3.00%	486.15	652.52	14
T-12-0.03		12	18	3.00%	486.15	652.52	17
T-15-0.03		15	18	3.00%	486.15	652.52	18
T-15-0.03		15	18	3.00%	486.15	652.52	15
T-15-0.03		15	18	3.00%	486.15	652.52	14
12.7-6db-0.01 (1)	ASTM A1035 Grade 690 [5]	6	12.7	1.00%	795.5	1054.65	49
12.7-6db-0.01 (2)		6	12.7	1.00%	795.5	1054.65	44
12.7-6db-0.02 (1)		6	12.7	2.00%	795.5	1054.65	4
12.7-6db-0.02 (2)		6	12.7	2.00%	795.5	1054.65	4
12.7-6db-0.03 (1)		6	12.7	3.00%	795.5	1054.65	1
12.7-6db-0.03 (2)		6	12.7	3.00%	795.5	1054.65	2
12.7-6db-0.04 (1)		6	12.7	4.00%	795.5	1054.65	1
12.7-6db-0.04 (2)		6	12.7	4.00%	795.5	1054.65	1
12.7-9db-0.01 (1)		9	12.7	1.00%	795.5	1054.65	30
12.7-9db-0.01 (2)		9	12.7	1.00%	795.5	1054.65	28
12.7-9db-0.02 (1)		9	12.7	2.00%	795.5	1054.65	4
12.7-9db-0.02 (2)		9	12.7	2.00%	795.5	1054.65	3
12.7-9db-0.03 (1)		9	12.7	3.00%	795.5	1054.65	2
12.7-9db-0.03 (2)		9	12.7	3.00%	795.5	1054.65	2
12.7-9db-0.04 (1)		9	12.7	4.00%	795.5	1054.65	2
12.7-9db-0.04 (2)		9	12.7	4.00%	795.5	1054.65	2
12.7-12db-0.01 (1)		12	12.7	1.00%	795.5	1054.65	15
12.7-12db-0.01 (2)		12	12.7	1.00%	795.5	1054.65	17
12.7-12db-0.02 (1)		12	12.7	2.00%	795.5	1054.65	4
12.7-12db-0.02 (2)		12	12.7	2.00%	795.5	1054.65	4
12.7-12db-0.03 (1)		12	12.7	3.00%	795.5	1054.65	3
12.7-12db-0.03 (2)		12	12.7	3.00%	795.5	1054.65	2
12.7-12db-0.04 (1)		12	12.7	4.00%	795.5	1054.65	2
12.7-12db-0.04 (2)		12	12.7	4.00%	795.5	1054.65	2
12.7-15db-0.01 (1)		15	12.7	1.00%	795.5	1054.65	15
12.7-15db-0.01 (2)		15	12.7	1.00%	795.5	1054.65	14
12.7-15db-0.02 (1)		15	12.7	2.00%	795.5	1054.65	5
12.7-15db-0.02 (2)		15	12.7	2.00%	795.5	1054.65	5
12.7-15db-0.03 (1)		15	12.7	3.00%	795.5	1054.65	3
12.7-15db-0.03 (2)		15	12.7	3.00%	795.5	1054.65	3
12.7-15db-0.04 (1)		15	12.7	4.00%	795.5	1054.65	3
12.7-15db-0.04 (2)		15	12.7	4.00%	795.5	1054.65	3
15.88-6db-0.01 (1)		6	15.88	1.00%	774	1017.3	48
15.88-6db-0.01 (2)		6	15.88	1.00%	774	1017.3	31
15.88-6db-0.02 (1)		6	15.88	2.00%	774	1017.3	3
15.88-6db-0.02 (2)		6	15.88	2.00%	774	1017.3	5
15.88-6db-0.03 (1)		6	15.88	3.00%	774	1017.3	2
15.88-6db-0.03 (2)		6	15.88	3.00%	774	1017.3	1
15.88-6db-0.04 (1)		6	15.88	4.00%	774	1017.3	1
15.88-6db-0.04 (2)		6	15.88	4.00%	774	1017.3	1
15.88-9db-0.01 (1)		9	15.88	1.00%	774	1017.3	27
15.88-9db-0.01 (2)		9	15.88	1.00%	774	1017.3	23
15.88-9db-0.02 (1)		9	15.88	2.00%	774	1017.3	3
15.88-9db-0.02 (2)		9	15.88	2.00%	774	1017.3	4
15.88-9db-0.03 (1)		9	15.88	3.00%	774	1017.3	2
15.88-9db-0.03 (2)		9	15.88	3.00%	774	1017.3	2
15.88-9db-0.04 (1)		9	15.88	4.00%	774	1017.3	2
15.88-9db-0.04 (2)		9	15.88	4.00%	774	1017.3	1
15.88-12db-0.01 (1)		12	15.88	1.00%	774	1017.3	15
15.88-12db-0.01 (2)		12	15.88	1.00%	774	1017.3	17
15.88-12db-0.02 (1)		12	15.88	2.00%	774	1017.3	6
15.88-12db-0.02 (2)		12	15.88	2.00%	774	1017.3	4
15.88-12db-0.03 (1)		12	15.88	3.00%	774	1017.3	3
15.88-12db-0.03 (2)		12	15.88	3.00%	774	1017.3	2
15.88-12db-0.04 (1)		12	15.88	4.00%	774	1017.3	2
15.88-12db-0.04 (2)		12	15.88	4.00%	774	1017.3	2
15.88-15db-0.01 (1)		15	15.88	1.00%	774	1017.3	14
15.88-15db-0.01 (2)		15	15.88	1.00%	774	1017.3	14
15.88-15db-0.02 (1)		15	15.88	2.00%	774	1017.3	4
15.88-15db-0.02 (2)		15	15.88	2.00%	774	1017.3	5
15.88-15db-0.03 (1)		15	15.88	3.00%	774	1017.3	3
15.88-15db-0.03 (2)		15	15.88	3.00%	774	1017.3	3
15.88-15db-0.04 (1)		15	15.88	4.00%	774	1017.3	2
15.88-15db-0.04 (2)		15	15.88	4.00%	774	1017.3	3
B460-5d-0.01	B460 smooth bar [50]	5	12	1.00%	474.5	510.564	1226
B460-10d-0.01		10	12	1.00%	474.5	510.564	243
B460-12d-0.01		12	12	1.00%	474.5	510.564	192
B460-15d-0.01		15	12	1.00%	474.5	510.564	140
B460-5d-0.015		5	12	1.50%	474.5	510.564	367
B460-8d-0.015		8	12	1.50%	474.5	510.564	115
B460-10d-0.015		10	12	1.50%	474.5	510.564	100
B460-12d-0.015		12	12	1.50%	474.5	510.564	94
B460-15d-0.015		15	12	1.50%	474.5	510.564	85
B460-5d-0.02		5	12	2.00%	474.5	510.564	184
B460-8d-0.02		8	12	2.00%	474.5	510.564	69
B460-10d-0.02		10	12	2.00%	474.5	510.564	64
B460-12d-0.02		12	12	2.00%	474.5	510.564	58
B460-15d-0.02		15	12	2.00%	474.5	510.564	52
B460-5d-0.03		5	12	3.00%	474.5	510.564	69
B460-8d-0.03		8	12	3.00%	474.5	510.564	38
B460-10d-0.03		10	12	3.00%	474.5	510.564	37
B460-12d-0.03		12	12	3.00%	474.5	510.564	32
B460-15d-0.03		15	12	3.00%	474.5	510.564	32
B460-5d-0.04		5	12	4.00%	474.5	510.564	40
B460-8d-0.04		8	12	4.00%	474.5	510.564	23
B460-10d-0.04		10	12	4.00%	474.5	510.564	22
B460-12d-0.04		12	12	4.00%	474.5	510.564	22
B460-15d-0.04		15	12	4.00%	474.5	510.564	24
B460-5d-0.05		5	12	5.00%	474.5	510.564	26
B460-8d-0.05		8	12	5.00%	474.5	510.564	17
B460-10d-0.05		10	12	5.00%	474.5	510.564	17
B460-12d-0.05		12	12	5.00%	474.5	510.564	17
B460-15d-0.05		15	12	5.00%	474.5	510.564	18
FSC14S2.0R0.1(1)	HRB400E/316L stainless steel clad bar [3]	6	14	1.00%	466.5	661	175
FSC14S2.0R0.1(2)		6	14	1.00%	466.5	661	159
FSC14S4.0R0.1(1)		6	14	2.00%	466.5	661	24
FSC14S4.0R0.1(2)		6	14	2.00%	466.5	661	23
FSC14S6.0R0.1(1)		6	14	3.00%	466.5	661	6
FSC14S6.0R0.1(2)		6	14	3.00%	466.5	661	6
FSC14S8.0R0.1(1)		6	14	4.00%	466.5	661	3
FSC14S8.0R0.1(2)		6	14	4.00%	466.5	661	3
FSC18S2.0R0.1(1)		6	18	1.00%	474.6	645	181
FSC18S2.0R0.1(2)		6	18	1.00%	474.6	645	183
FSC18S4.0R0.1(1)		6	18	2.00%	474.6	645	22
FSC18S4.0R0.1(2)		6	18	2.00%	474.6	645	18
FSC18S6.0R0.1(1)		6	18	3.00%	474.6	645	9
FSC18S6.0R0.1(2)		6	18	3.00%	474.6	645	7
FSC18S8.0R0.1(1)		6	18	4.00%	474.6	645	6
FSC18S8.0R0.1(2)		6	18	4.00%	474.6	645	5
FSC25S2.0R0.1(1)		6	25	1.00%	495.5	662.1	292
FSC25S2.0R0.1(2)		6	25	1.00%	495.5	662.1	319
FSC25S4.0R0.1(1)		6	25	2.00%	495.5	662.1	31
FSC25S4.0R0.1(2)		6	25	2.00%	495.5	662.1	35
FSC25S6.0R0.1(1)		6	25	3.00%	495.5	662.1	13
FSC25S6.0R0.1(2)		6	25	3.00%	495.5	662.1	18
FSC25S8.0R0.1(1)		6	25	4.00%	495.5	662.1	6
FSC25S8.0R0.1(2)		6	25	4.00%	495.5	662.1	6
B500B-5d-0.01	B500B bar [50]	5	16	1.00%	535.67	633.75	609
B500B-8d-0.01		8	16	1.00%	535.67	633.75	171
B500B-10d-0.01		10	16	1.00%	535.67	633.75	92
B500B-12d-0.01		12	16	1.00%	535.67	633.75	68
B500B-15d-0.01		15	16	1.00%	535.67	633.75	62
B500B-5d-0.015		5	16	1.50%	535.67	633.75	162
B500B-8d-0.015		8	16	1.50%	535.67	633.75	57
B500B-10d-0.015		10	16	1.50%	535.67	633.75	33
B500B-12d-0.015		12	16	1.50%	535.67	633.75	32
B500B-15d-0.015		15	16	1.50%	535.67	633.75	31
B500B-5d-0.02		5	16	2.00%	535.67	633.75	66
B500B-8d-0.02		8	16	2.00%	535.67	633.75	23
B500B-10d-0.02		10	16	2.00%	535.67	633.75	22
B500B-12d-0.02		12	16	2.00%	535.67	633.75	25
B500B-15d-0.02		15	16	2.00%	535.67	633.75	19
B500B-5d-0.025		5	16	2.50%	535.67	633.75	32
B500B-8d-0.025		8	16	2.50%	535.67	633.75	15
B500B-10d-0.025		10	16	2.50%	535.67	633.75	14
B500B-12d-0.025		12	16	2.50%	535.67	633.75	14
B500B-15d-0.025		15	16	2.50%	535.67	633.75	16
B500B-5d-0.03		5	16	3.00%	535.67	633.75	24
B500B-8d-0.03		8	16	3.00%	535.67	633.75	11
B500B-10d-0.03		10	16	3.00%	535.67	633.75	11
B500B-12d-0.03		12	16	3.00%	535.67	633.75	11
B500B-15d-0.03		15	16	3.00%	535.67	633.75	13
B500B-5d-0.04		5	16	4.00%	535.67	633.75	13
B500B-8d-0.04		8	16	4.00%	535.67	633.75	7
B500B-10d-0.04		10	16	4.00%	535.67	633.75	7
B500B-12d-0.04		12	16	4.00%	535.67	633.75	9
B500B-15d-0.04		15	16	4.00%	535.67	633.75	9
B500B-5d-0.01		5	12	1.00%	544.33	640.67	467
B500B-8d-0.01		8	12	1.00%	544.33	640.67	167
B500B-10d-0.01		10	12	1.00%	544.33	640.67	111
B500B-12d-0.01		12	12	1.00%	544.33	640.67	74
B500B-15d-0.01		15	12	1.00%	544.33	640.67	72
B500B-5d-0.015		5	12	1.50%	544.33	640.67	166
B500B-8d-0.015		8	12	1.50%	544.33	640.67	64
B500B-10d-0.015		10	12	1.50%	544.33	640.67	50
B500B-12d-0.015		12	12	1.50%	544.33	640.67	41
B500B-15d-0.015		15	12	1.50%	544.33	640.67	36
B500B-5d-0.02		5	12	2.00%	544.33	640.67	71
B500B-8d-0.02		8	12	2.00%	544.33	640.67	32
B500B-10d-0.02		10	12	2.00%	544.33	640.67	26
B500B-12d-0.02		12	12	2.00%	544.33	640.67	23
B500B-15d-0.02		15	12	2.00%	544.33	640.67	27
B500B-5d-0.03		5	12	3.00%	544.33	640.67	26
B500B-8d-0.03		8	12	3.00%	544.33	640.67	15
B500B-10d-0.03		10	12	3.00%	544.33	640.67	13
B500B-12d-0.03		12	12	3.00%	544.33	640.67	19
B500B-15d-0.03		15	12	3.00%	544.33	640.67	17
B500B-5d-0.04		5	12	4.00%	544.33	640.67	17
B500B-8d-0.04		8	12	4.00%	544.33	640.67	12
B500B-10d-0.04		10	12	4.00%	544.33	640.67	9
B500B-12d-0.04		12	12	4.00%	544.33	640.67	14
B500B-15d-0.04		15	12	4.00%	544.33	640.67	12
B500B-5d-0.05		5	12	5.00%	544.33	640.67	10
B500B-8d-0.05		8	12	5.00%	544.33	640.67	8
B500B-10d-0.05		10	12	5.00%	544.33	640.67	8
B500B-12d-0.05		12	12	5.00%	544.33	640.67	8
B500B-15d-0.05		15	12	5.00%	544.33	640.67	9
B500B-16-TEMP Prod.1.1	B500B bar [51]	6	16	2.50%	596.6	671.4	19
B500B-16-TEMP Prod.1.1		6	16	4.00%	596.6	671.4	8
B500B-16-TEMP Prod.1.1		8	16	2.50%	596.6	671.4	15
B500B-16-TEMP Prod.1.1		8	16	4.00%	596.6	671.4	10
B500B-16-TEMP Prod.1.2		6	16	2.50%	572	668.3	19
B500B-16-TEMP Prod.1.2		6	16	4.00%	572	668.3	8
B500B-16-TEMP Prod.1.2		8	16	2.50%	572	668.3	15
B500B-16-TEMP Prod.1.2		8	16	4.00%	572	668.3	6
B500B-16-TEMP Prod.1.3		6	16	2.50%	513.1	616.3	19
B500B-16-TEMP Prod.1.3		6	16	4.00%	513.1	616.3	9
B500B-16-TEMP Prod.1.3		8	16	2.50%	513.1	616.3	13
B500B-16-TEMP Prod.1.3		8	16	4.00%	513.1	616.3	8
B500B-16-TEMP Prod.2		6	16	2.50%	516.9	635	20
B500B-16-TEMP Prod.2		6	16	4.00%	516.9	635	11
B500B-16-TEMP Prod.2		8	16	2.50%	516.9	635	14
B500B-16-TEMP Prod.2		8	16	4.00%	516.9	635	7
B500B-20-TEMP Prod.2		6	20	2.50%	515.3	621.8	20
B500B-20-TEMP Prod.2		6	20	4.00%	515.3	621.8	9
B500B-20-TEMP Prod.2		8	20	2.50%	515.3	621.8	16
B500B-20-TEMP Prod.2		8	20	4.00%	515.3	621.8	7
B500B-8-STR Prod.1		6	8	2.50%	565.6	619.3	20
B500B-8-STR Prod.1		6	8	4.00%	565.6	619.3	11
B500B-8-STR Prod.1		8	8	2.50%	565.6	619.3	20
B500B-8-STR Prod.1		8	8	4.00%	565.6	619.3	10
B500B-8-TEMP Prod.2		6	8	2.50%	584.7	671.5	20
B500B-8-TEMP Prod.2		6	8	4.00%	584.7	671.5	11
B500B-8-TEMP Prod.2		8	8	2.50%	584.7	671.5	17
B500B-8-TEMP Prod.2		8	8	4.00%	584.7	671.5	9
B500B-12-STR Prod.1		6	12	2.50%	538.4	627	25
B500B-12-STR Prod.1		6	12	4.00%	538.4	627	8
B500B-12-STR Prod.1		8	12	2.50%	538.4	627	15
B500B-12-STR Prod.1		8	12	4.00%	538.4	627	7
B500A-8-CW Prod.2	B500A bar [51]	6	8	2.50%	526.4	546.8	20
B500A-8-CW Prod.2		6	8	4.00%	526.4	546.8	14
B500A-8-CW Prod.2		8	8	2.50%	526.4	546.8	19
B500A-8-CW Prod.2		8	8	4.00%	526.4	546.8	12
B500A-12-CW Prod.2		6	12	2.50%	567.7	589	20
B500A-12-CW Prod.2		6	12	4.00%	567.7	589	8
B500A-12-CW Prod.2		8	12	2.50%	567.7	589	17
B500A-12-CW Prod.2		8	12	4.00%	567.7	589	8
B400C-8-TEMP Prod.1	B400C bar [51]	6	8	2.50%	442.9	567.3	20
B400C-8-TEMP Prod.1		6	8	4.00%	442.9	567.3	12
B400C-8-TEMP Prod.1		8	8	2.50%	442.9	567.3	20
B400C-8-TEMP Prod.1		8	8	4.00%	442.9	567.3	12
B400C-16-MA Prod.2		6	16	2.50%	434.5	565.3	20
B400C-16-MA Prod.2		6	16	4.00%	434.5	565.3	12
B400C-16-MA Prod.2		8	16	2.50%	434.5	565.3	17
B400C-16-MA Prod.2		8	16	4.00%	434.5	565.3	8
B400C-20-MA Prod.2		6	20	2.50%	416	563.3	20
B400C-20-MA Prod.2		6	20	4.00%	416	563.3	9
B400C-20-MA Prod.2		8	20	2.50%	416	563.3	18
B400C-20-MA Prod.2		8	20	4.00%	416	563.3	9
B400C-20-TEMP Prod.2		6	20	2.50%	436.2	557.2	20
B400C-20-TEMP Prod.2		6	20	4.00%	436.2	557.2	7
B400C-20-TEMP Prod.2		8	20	2.50%	436.2	557.2	7
B450C-8-STR Prod.1	B450C bar [51]	6	8	2.50%	530.2	624.8	20
B450C-8-STR Prod.1		6	8	4.00%	530.2	624.8	15
B450C-8-STR Prod.1		8	8	2.50%	530.2	624.8	20
B450C-8-STR Prod.1		8	8	4.00%	530.2	624.8	16
B450C-12-STR Prod.1		6	12	2.50%	530.2	619.6	28
B450C-12-STR Prod.1		6	12	4.00%	530.2	619.6	9
B450C-12-STR Prod.1		8	12	2.50%	530.2	619.6	18
B450C-12-STR Prod.1		8	12	4.00%	530.2	619.6	8
B450C-12-STR Prod.2		6	12	2.50%	513.8	599.7	20
B450C-12-STR Prod.2		6	12	4.00%	513.8	599.7	14
B450C-12-STR Prod.2		8	12	2.50%	513.8	599.7	20
B450C-12-STR Prod.2		8	12	4.00%	513.8	599.7	12
B450C-16-TEMP Prod.1.1		6	16	2.50%	537.3	640.5	19
B450C-16-TEMP Prod.1.1		6	16	4.00%	537.3	640.5	9
B450C-16-TEMP Prod.1.1		8	16	2.50%	537.3	640.5	13
B450C-16-TEMP Prod.1.1		8	16	4.00%	537.3	640.5	11
B450C-16-TEMP Prod.1.2		6	16	2.50%	446.7	542.7	18
B450C-16-TEMP Prod.1.2		6	16	4.00%	446.7	542.7	18
B450C-16-TEMP Prod.1.2		8	16	2.50%	446.7	542.7	18
B450C-16-TEMP Prod.1.2		8	16	4.00%	446.7	542.7	9
B450C-16-TEMP Prod.1.3		6	16	2.50%	517.8	615.4	19
B450C-16-TEMP Prod.1.3		6	16	4.00%	517.8	615.4	14
B450C-16-TEMP Prod.1.3		8	16	2.50%	517.8	615.4	15
B450C-16-TEMP Prod.1.3		8	16	4.00%	517.8	615.4	9
B450C-16-TEMP Prod.2		6	16	2.50%	479.3	601.1	18
B450C-16-TEMP Prod.2		6	16	4.00%	479.3	601.1	8
B450C-16-TEMP Prod.2		8	16	2.50%	479.3	601.1	18
B450C-16-TEMP Prod.2		8	16	4.00%	479.3	601.1	7
B450C-20-TEMP Prod.2		6	20	2.50%	492.9	591.4	19
B450C-20-TEMP Prod.2		6	20	4.00%	492.9	591.4	7
B450C-20-TEMP Prod.2		8	20	2.50%	492.9	591.4	19
B450C-20-TEMP Prod.2		8	20	4.00%	492.9	591.4	7

. The statistical distributions of all variables are shown in Figure 5, which has been normalized within the range between zero and one to avoid the scale effect when training the ML models.

3. Framework

The goal of this framework is to realize low-cycle fatigue life prediction of BSBs by leveraging the source dataset of other metallic bars. As shown in Figure 6, the framework consists of two parts. A source model to predict the fatigue life of other metallic bars is constructed based on the collected source dataset with sufficient data. Subsequently, the learned knowledge in the source model is transferred to the transfer model to reveal the fatigue life of BSBs by retraining only a few model parameters with limited BSB data. To predict low-cycle fatigue life, a typical machine learning model, an artificial neural network (ANN), was used.

3.1. Performance Metrics

To assess the constructed model's accuracy, the predicted results are studied in terms of three performance metrics: the coefficient of determination (${\mathrm{R}}^2$), root mean squared error (RMSE), and mean absolute error (MAE), which are expressed as

$${\mathrm{R}}^2=1-{\sum }_{i=1}^N{\left({\widehat{y}}_i-y_i\right)}^2/{\sum }_{i=1}^N{\left(y_i-\overline{y}\right)}^2$$

(1)

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \raisebox{1ex}{${{\sum }_{i=1}^N({\widehat{y}}_i-y_i)}^2$}\!\left/ \!\raisebox{-1ex}{$N$}\right.}}$$

(2)

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \raisebox{1ex}{${\sum }_{i=1}^N{|\widehat{y}}_i-y_i|$}\!\left/ \!\raisebox{-1ex}{$N$}\right.}$$

(3)

where $y_i$ denotes the test result for the normalized low-cycle fatigue life of the $i^{th}$ sample, ${\widehat{y}}_i$ is the fatigue life predicted by the ML models, $\overline{y}$ denotes the average value of the test results in the training or testing set, and N denotes the total number of samples in the corresponding set.

3.2. Construction of Source Model

3.2.1. Neural Network Architecture

The ANN model processes input data through a series of layers, with the most critical component being the hidden layer, which is responsible for extracting features from the data to improve model performance. The typical structure of an ANN is illustrated in Figure 7 using an ANN with two hidden layers. The network starts with an input layer. Then, there are two sequential hidden layers. In the end, there is a regression output layer. The behavior of the$1^{st}$ and $2^{nd}$ hidden layer can be determined as

$$y_i=f_i\left(\sum _{k=1}^6{w}_k^i{x}_k+b^i\right)$$

(4)

$$z_i=f_i\left(\sum _{k=1}^n{w}_k^i{y}_k+b^i\right)$$

(5)

Then, the output is

$$o=f_i\left(\sum _{k=1}^n{w}_k^i{z}_k+b^i\right)$$

(6)

where $x$ is the input feature; $w$, $b$, and $f$ denote the weights, bias, and activation functions, respectively; and $n$ is the number of neurons in the hidden layer. The activation function is utilized to ensure that all values passed to the next layer are within the range between zero and one. The initial weights and biases were assigned randomly. The gradient descent algorithm was then used to minimize the loss function in order to update the weights and biases in the training process.

3.2.2. Model Training and Hyperparameter Tuning

The performance of ML models is highly dependent on hyperparameters, and appropriate hyperparameters help improve the model prediction accuracy. In this paper, the hyperparameters were selected based on the grid search methods and K-fold cross-validation [23,24]. Grid search is a methodical approach to tuning the hyperparameters of a machine learning model. The K-fold cross-validation method can fully utilize the dataset by splitting the training set into K folds, and each fold serves for training and validation sequentially. The model performance on different data subsets was evaluated by repeating the training and testing process, so as to ensure that the grouping does not affect the generalization ability of the model. Then, the optimal hyperparameters were determined according to the best average performance score (such as R²) of the training. The grid search method extensively searches for all hyperparameters within specified ranges.

The dataset for the other metallic bars was first randomly split into training (85%) and testing (15%) sets. The training set was then split ten-fold. The model was trained 10 times for the given hyperparameters, wherein in each iteration, nine out of the 10 folds were used for training and the remaining one was selected for validation. The performance scores of all 10 folds were averaged to evaluate the performance of the hyperparameters. After the grid search tuning phase, the best performing model was the one with the highest score. Five hyperparameters which significantly affected the model’s performance were tuned, i.e., the number of neurons in the hidden layers, activation function, learning rate, batch size, and epochs. Both one-layer and two-layer ANNs were investigated. The inspected and optimal values for each ANN are listed in

Table 2. Hyperparameters in source model.

Hyperparameter	Inspected Values	Optimal Value for One-Layer ANN	Optimal Value for Two-Layer ANN
Number of neurons in hidden layers	7, 9, 11, 13, and 15	15	15
Activation function	Relu, Tanh and Sigmoid	Relu	Relu
Learning rate	0.0001, 0.001, and 0.01	0.01	0.001
Batch size	16 and 32	16	16
Epochs	100, 200 and 300	300	300

. To ensure the generalization ability of the model, the epoch was determined to be 300 based on the analyses with different epochs. In addition, the number of neurons in the two hidden layers was set the same for the one-layer ANN to reduce the computational cost. After optimal hyperparameters were obtained, the final models were trained using the entire training set.

3.3. Construction of Transfer Model

As stated above, the transfer learning approach is based on feature extraction and fine-tuning. A source model was first trained to extract meaningful features from a large dataset to achieve a specific task. Then, in fine-tuning, the learned knowledge was transferred by fixing the first several layers of the source model and fine-tuning the final layers of the model to learn the properties of a new smaller dataset for similar tasks. In this paper, the one-layer and two-layer transfer models for the low-cycle fatigue life prediction for BSBs were trained as follows: (1) the hidden layers of the corresponding source models were fixed and were not adjusted in fine training; (2) the weights and biases of the output layer (16 parameters in total) were trained with the training set of BSBs. To fully validate the accuracy of the model, the BSBs dataset was split into a training set (30%) and a testing set (70%). Usually, in typical ML model construction, the size of the training set is larger than that of the testing set to fully extract features. But in the transfer framework, the feature extraction has been completed in the construction of source models. In the construction of transfer models, only a few parameters need to be fine-tuned. Moreover, the reliability of transfer models should be fully validated. Thus, a ratio of 30:70 was adopted in this work. Baseline models are the corresponding one-layer and two-layer ANNs trained directly with the dataset of BSBs without transfer from source models.

4. Results

4.1. Prediction Results for Source Models

Figure 8a,b show the variations in the predicted low-cycle fatigue life of other metallic bars by the one-layer and two-layer ANN source models against the test results, respectively, wherein the results of the training and testing sets are both indicated. From Figure 8, it can be seen that both the one-layer and two-layer ANNs predict the low-cycle fatigue life reasonably well. The three metrics defined in Section 3.1 are furthermore utilized to evaluate the accuracy of predictions.

Table 3. Performance metrics of one-layer and two-layer source models, transfer models and corresponding baseline models.

Model	Set	One-Layer Model			Two-Layer Model
		${\mathbf{R}}^2$	RMSE	MAE	${\mathbf{R}}^2$	RMSE	MAE
Source model	Training	0.959	21.130	6.784	0.979	15.101	5.060
	Testing	0.911	32.231	11.700	0.906	33.023	13.882
Transfer model	Training	1.000	5.121	2.290	1.000	0.278	0.125
	Testing	0.861	38.321	19.820	0.392	80.143	39.832
Baseline model	Training	0.919	253.032	163.764	0.831	365.041	214.744
	Testing	−0.976	144.467	99.245	−0.113	108.389	74.993

shows the ${\mathrm{R}}^2$, RMSE, and MAE values for the two models. From the validation results on the testing set, these two models were capable of predicting the fatigue life with an ${\mathrm{R}}^2$ value higher than 0.90. The one-layer model gave a slightly better coefficient of determination and smaller values of RMSE and MAE on the testing set, compared with the two-layer model. The above phenomenon is due to the noise introduced by the double-layer model with too many parameters when the dataset is small. The RMSE of the test set for the one-layer model was 38.321, which was due to the dimensionality of the dataset. By far, the ability of the source models to extract the features of low-cycle fatigue life has been validated, which will be leveraged by the transfer model to predict the fatigue life of BSBs based on the limited target dataset.

4.2. Prediction Results for Transfer Models

Figure 9 compares the predicted results for the low-cycle fatigue life of BSBs to the test results for both the one-layer and two-layer transfer models. For the testing set, these two models achieved considerable accuracy, as almost all of the dots were spotted on the line of y = x. Furthermore, the coefficients of determination were both 1.00, as shown in Table 3. For the testing set, the one-layer transfer model attained a coefficient of determination of 0.861, which indicates that the one-layered transfer model can predict the low-cycle fatigue life of BSB with satisfactory accuracy. However, the accuracy of the two-layer transfer model deteriorated significantly on the testing set compared to that of the training set. The value of ${\mathrm{R}}^2$ decreases from 1.00 to 0.392, which is smaller than the results for the one-layer transfer model. The results in Figure 9b show a relatively higher scatter on the testing set than the results in Figure 9a. Thus, the one-layer ANN model possesses a higher representative ability under the current BSBs dataset.

5. Discussion

5.1. Efficiency of the Transfer Framework

The objective of this paper is to validate the benefit of leveraging the dataset of other metallic bars to predict the low-cycle fatigue life of BSBs, which will remarkably reduce the cost by decreasing the need for data samples of BSBs. To highlight the efficiency and accuracy of the constructed transfer models, as stated above, one- and two-layer baseline models based solely on the dataset of BSBs without transferring the features learned in the source models were also constructed. The low-cycle fatigue life of BSBs predicted by the one-layer and two-layer baseline models against the test results is shown in Figure 10a,b, respectively. Compared with the corresponding results for the transfer models in Figure 9, the results for both the one-layer and two-layer baseline models display significantly larger dispersion. It is noteworthy that the coefficients of determination of these two baseline models were negative, as listed in Table 3, which indicates that the BSB dataset is far from sufficient for fatigue life prediction. However, by leveraging the dataset of other metallic bars, with the same dataset of BSBs, a satisfactory coefficient of determination of 0.861 was achieved using the one-layered transfer model. Thus, the advantage of the transfer model was validated.

5.2. Influence of Key Parameters

Based on the one-layer transfer model, the influences of key parameters on the low-cycle fatigue life of BSBs were studied. Figure 11 shows the evolution of the low-cycle fatigue life of BSBs with variations in the fatigue strain amplitude ε_a and slenderness ratio λ. It can be seen that as ε_a and λ increase, the fatigue life decreases significantly. The colored dots in Figure 11 denote test results for the BSBs, which again indicate the accuracy of the transfer model. It should be stated that the low-cycle fatigue tests in the existing literature [15,20,21,22,55] only considered the value of ε_a within the range of 0.00 and 0.03. By utilizing the constructed model, the fatigue life with ε_a in the range of 0.03 and 0.05 can be predicted, as shown in Figure 11.

5.3. The Relative Importance of Features on the Fatigue Life

As stated above, the low-cycle fatigue life of metallic bars is influenced by several parameters, and the influence is complicated owing to the coupling effect among different parameters. Figure 12 depicts the influence of ε_a on N_f based on the collected dataset. In general, with an increase in ε_a, N_f of different steel bars decreased gradually. However, even for the same ε_a, there were clear differences in N_f among different steel bars. Regarding the coupling effect, it is difficult to evaluate the relative importance of each parameter using conventional methods. By contrast, ML provides an easier way to evaluate the relative importance of each parameter. Based on ML regression models, the impact of the input parameters on the fatigue life was evaluated using the SHapley Additive exPlanations (SHAP) framework [23,56,57,58,59]. The SHAP value, which is determined by comparing model predictions with and without each feature, is central to the method.

Figure 13 displays the relative importance of the input features on the low-cycle fatigue life of metallic bars. According to the SHAP value, the fatigue strain amplitude ε_a has the greatest impact on fatigue life, and as the value of ε_a increases, N_f decreases. The other two parameters that have relatively more significant impact are f_y and λ. Similarly, increases in f_y and λ tend to decrease N_f.

The interaction between two features, that is, how one feature affects the impact of another feature on the low-cycle fatigue life, can be revealed from the SHAP feature dependence plot. The dependence of λ on ε_a is shown in Figure 14. The effect of ε_a on the contribution of λ depends on the value of λ. When λ is relatively large, increasing ε_a leads to an increase in the SHAP value for λ. However, when λ is comparatively small, increasing ε_a generally decreases its SHAP value.

5.4. Limitations and Prospects

In order to facilitate the operation and data analysis of the low-cycle fatigue test, some hypotheses were set up. One of the important assumptions is that the load is a tensile and compressive equal amplitude load. Another important assumption is that the reinforcement is consolidated at both ends. In the concrete structure, the vertical reinforcement is constrained by stirrups and concrete, and its two ends can be basically fixed. In some extreme situations, the restraining effect of concrete and stirrups may fail due to concrete crushing or stirrup fracture. Therefore, a variety of boundary conditions can be considered in future studies to fully grasp the low-cycle fatigue properties of BSBs.

The prediction model proposed in this study was based on 54 BSB data points and 264 other steel bar data points. Due to the limited research on low-cycle fatigue of BSBs, there were only 54 BSB data points in the dataset. It is worth noting that strict model testing and validation are crucial to ensure the accuracy and reliability of transfer learning frameworks in fatigue life prediction, which requires sufficient datasets. Therefore, more experimental research should be conducted in the future to enrich the BSB dataset. In addition, a framework based on machine learning and transfer learning is proposed in this study, aiming at achieving efficient prediction on low-cycle fatigue life of BSBs with limited data. When the dataset is too large, the computational cost of transfer learning should not be ignored. Many methods can be attempted in future studies to reduce computational costs, such as removing unimportant neurons or connections to reduce the size and complexity of the model, thereby lowering computational costs. In addition to transfer learning frameworks, traditional regression models and other machine learning models also play important roles in the fields of machine learning and artificial intelligence, each with its unique advantages and disadvantages. The choice of model depends on the specific type of problem, the characteristics of the data, and the requirements of the model. Therefore, it is still necessary to make choices based on the specific situation when solving similar issues.

The applicability of the results in this study may be limited when applied to other types of metal materials or fatigue loading conditions not included in the dataset. Especially for metal materials with significantly different material properties from ordinary steel bars, such as nickel-based alloys, titanium alloys, aluminum alloys, etc. In order to adapt the model to other materials and fatigue loading conditions, it is necessary to supplement it with more fatigue test datasets.

6. Conclusions

A transfer learning framework was proposed in this study, in which a dataset of 264 other metallic bars was leveraged to predict the low-cycle fatigue life of BSBs with a very limited experimental dataset (48 samples). The main conclusions of this study are as follows:

(1) The transfer models accurately predicted the low-cycle fatigue life of BSBs using a very small number of samples. The one-layer transfer model achieved a coefficient of determination of 0.861, which was significantly better than the performance of baseline models that did not transfer knowledge learned from the dataset of other metallic bars.

(2) The constructed source model was capable of accurately predicting the low-cycle fatigue life of seven types of metallic bars collected, thus achieving a unified prediction among various metallic materials.

(3) The influence of the hidden layers of the ANN was considered by comparing the prediction performance of one-layer and two-layer transfer models. The results indicated that the one-layer model outperformed the two-layer model on the current BSB dataset.

(4) Feature analysis revealed that the three parameters of relatively high importance for the low-cycle fatigue life of metallic bars are the fatigue strain amplitude ε_a, yield strength f_y, and slenderness ratio λ.

The proposed transfer framework was validated as an efficient method to predict the low-cycle fatigue life of BSBs quickly and at low cost. In future studies, more experiments will be considered to improve the predictive performance of the model.