Flexural performance tests and machine learning analysis of prestressed high strength steel reinforced concrete beams

ABSTRACT

This paper reports a study on the flexural performance of prestressed high strength steel reinforced concrete (PHSSRC) beams and proposes machine learning (ML) models for predicting the flexural bearing capacity of steel reinforced concrete (SRC) beams. Seven prestressed steel reinforced concrete (PSRC) beams were tested under a four-point bending load to investigate the effect of steel strength grade, steel ratio, and longitudinal reinforcement ratio on the flexural load capacity of beams. The test results indicated that increasing the steel strength grade and steel ratio significantly enhanced the bearing capacity of PSRC beams. A database consisting of 112 sets of experimental data on the flexural performance of SRC beams, including the test data in this study, was established for training and validating the ML models. Artificial Neural Network (ANN) model and eXtreme Gradient Boosting (XGBoost) model were respectively developed to predict the flexural load capacity of SRC beams, and the accuracy of the flexural load capacity values predicted by the design specification, the ANN model and the XGBoost model were compared. The results show that the XGBoost model has the best prediction performance. Finally, the sensitivity analysis of the parameters was performed using the Shapley Additivity (SHAP) interpretation method.

KEYWORDS:

• Steel reinforced concrete beam
• flexural bearing capacity
• machine learning
• prediction model
• SHAP

1. Introduction

Steel reinforced concrete constructions are widely used in high-rise buildings (Wang et al. 2019, 2020; Wang, Shi, and Zhou 2019; Wang, Zhou, and Shi 2023a) and large-span structures (Wang, Wang, and Shi 2021; Wang et al. 2020, 2021, 2023) because of their excellent mechanical, seismic and fire resistance properties (Wang et al. 2020, 2022, 2023; Wang, Shi, and Zhou 2022; Wang, Zhou, and Shi 2023b). In recent years, high strength steel (HSS) has been widely researched and applied in composite structures due to its high yield strength (F, L, and L 2007; Gang, Jin-qing, and Wei-qing 2014; L et al. 2020; T et al. 2022). However, there are few reports on the application of HSS in steel reinforced concrete (SRC) beams. Kabir (Kabir et al. 2020) conducted flexural performance tests on Engineered Cementitious Composites and Lightweight Concrete (ECC-LWC), and the test results showed that the load carrying capacity and ductility properties of HSS composite beams wrapped by ECC-LWC are higher than those of HSS composite beams wrapped by ordinary concrete. Liu (Zu-qiang, Beng-you, and Jian-yang 2023) designed and fabricated six Q460 HSS ultra-high-performance concrete composite beams to study their bending performance, using steel ratio, section location, and steel fiber volume as test parameters. The test results indicated that all specimens had reached ductile failure, and the yielding of the tensile reinforcement and the lower flange of the section occurred before the concrete was crushed in the compression zone. The above studies did not use conventional stirrups or longitudinal reinforcement, were designed with fewer study parameters, and did not incorporate prestressing tendons, which hinder the development of HSS. The application of HSS to SRC beams can meet the demand for the bearing capacity of large-span structures and reduce steel consumption, which has good economic benefits. In addition, the application of prestress improves the cracking resistance and deformation resistance of the members.

There are two main methods for calculating the flexural load capacity of SRC beams. The calculation accuracy of the Superposition Method is low, and the calculation process of the Deformation Synergy Method is cumbersome. Machine Learning (ML) has been applied in the field of structural engineering to solve complex engineering computational problems. Wakjira (Wakjira et al. 2022) developed a total of 11 ML models for predicting the shear capacity of FRP-RC beams, where the extreme gradient boosting (XGBoost) model has superior predictive accuracy over the available codes and the guiding equations. Golafshani (Golafshani et al. 2012) developed an ANN model using test data from 179 test beams to predict the bond strength of reinforced steel in concrete structures. The above studies show that for specific engineering computational problems, different ML methods have an important impact on the accuracy of the computational results, and the selection of appropriate ML models can help to reduce the amount of computation and improve the prediction accuracy.

Therefore, tests on the flexural performance of seven prestressed steel reinforced concrete (PSRC) beams were carried out to investigate the effect of steel strength grade, steel ratio, and longitudinal reinforcement ratio on the flexural performance of the specimen. 112 sets of test data investigating the bending performance of SRC beams were collected and two ML models were developed with a database of test data from this paper and the published literature covering a wide range of geometrical and material properties. A comparative study between the proposed model and the canonical computational methods was carried out to obtain a load capacity prediction model with relatively high accuracy and good generalization performance, which provides a reference for engineering practice. Finally, the contribution of the input features to the bending capacity of SRC beams was ranked using the Shapley Additive Properties (SHAP) method of interpretation. By addressing the limitations of previous research, incorporating conventional reinforcement and prestressing tendons, and utilizing ML techniques, this study provides novel insights into the flexural performance of PSRC beams and offers a valuable reference for engineering practice. The workflow diagram of this study is shown in Figure 1. This article is overloaded with noun abbreviations, and to improve the readability of the article, the abbreviations, and their meanings appearing in the text are summarized in Table 1.

Figure 1. Workflow of this study.

Table 1. Meanings of nouns.

Nouns	Meaning
PHSSRC	prestressed high strength steel reinforced concrete
ML	machine learning
SRC	steel reinforced concrete
PSRC	prestressed steel reinforced concrete
ANN	Artificial Neural Network
XGBoost	extreme gradient boosting
ECC	Engineered Cementitious Composites （ECC-LWC)
LWC	Lightweight Concrete
MAPE	mean absolute percentage error
RMSE	root mean squared error
R^2	coefficient of determination
BP	backpropagation
LM	Levenberg-Marquardt
SHAP	Shapley Additivity interpretation method

2. Experimental study

2.1. Specimens and material properties

For the test, seven beams were made. The beam length was 3100 mm, and the concrete cover depth was 20 mm. The cross-sectional sections of the different specimens are shown in Figure 2. The spacing of the stirrups at the support was 50 mm, the encryption length was 300 mm, and the spacing of the rest of the stirrups was 100 mm. The section steels were made of welded H-beams. The application of prestress was completed by the unbonded post-tensioning process. The design parameters of the tested beams are shown in Table 2. Table 3 shows the mechanical parameters of the steel used in welded H-beams and bars. The specimens were selected from C50 commercial concrete. Concrete specimens had an average cubic compressive strength of 55.3 MPa.

Figure 2. Cross-sectional reinforcement diagram.

Table 2. Parameter of specimens.

Specimen	Steel strength grade	Steel ratio (%)	Steel shape(mm) bf×h×t1×t2	Longitudinal reinforcement ratio (%)
B1Q3S3L1	Q355	3.03%	100×150×8×5	1.67%
B2Q5S3L1	Q550	3.03%	100×150×8×5	1.67%
B3Q6S2L1	Q690	2.27%	100×150×5×5	1.67%
B4Q6S3L1	Q690	3.03%	100×150×8×5	1.67%
B5Q6S3L2	Q690	3.03%	100×150×8×5	2.29%
B6Q6S3L0	Q690	3.03%	100×150×8×5	1.09%
B7Q6S4L1	Q690	4.4%	100×150×10×10	1.67%

Table 3. Mechanical properties of steel.

Grade	Thicknesses (Diameters)/mm	Yield strength fy/MPa	Ultimate strength fu/MPa	Elongation at break δ/%
Q355	5	397	527	42.9
Q355	8	391	540	22.6
Q550	5	626	713	26.5
Q550	8	633	756	19.4
Q690	5	740	819	22.6
Q690	8	787	845	17.7
Q690	10	690	766	21.7
Prestressed wire	15.2	1820	1910	20.5
HRB400	14	453	591	25.6
HRB400	18	415	586	28.7
HRB400	20	433	566	29.5
HPB300	8	358	537	23.9

2.2. Loading method and measurement point arrangement

The test used a four-point bending graded loading method, and the loading device is shown in Figure 3. The test procedure began with the preloading of the specimen, followed by the commissioning of the test equipment, and subsequently initiating normal loading. The load increment for each stage of normal loading was 10% of the estimated ultimate load. Upon reaching 90% of the estimated ultimate load of the tested beam, the loading approach was altered to displacement-controlled loading at a 2 mm/min loading speed until the specimen was destroyed. The crushing of the concrete in the compression zone indicated damage to the tested beams. The loading ceased when the load on the specimens decreased to 0.85 times the ultimate load.

Figure 3. Test device and instrumentation.

The vertical deformation of the specimen was measured by displacement meters. Three displacement meters were evenly arranged in the span section of the specimen, and two displacement meters were arranged at the support. To quantify the strain variations over the whole test, strain gauges were positioned at 150 mm intervals along the length of the pure bending section’s longitudinal reinforcement. The arrangement of the strain gauges for the section steel is shown in Figure 4(a), and the position of the displacement gauges is shown in Figure 4(b).

Figure 4. Arrangement of strain gauge and displacement meter.

3. Experimental results and analysis

3.1. Failure mode

As shown in Figure 5, all specimens exhibited bending damage mode. At the initial stage of loading, no early cracking occurred in tested beams due to the applied prestressing load. The specimens cracked when the load was increased to 0.22 ~ 0.3 times the ultimate load (P_u). The tensile reinforcement yielded when loaded to (0.49 ~ 0.6) P_u. The load when the bottom flange of the steel at the pure bend section was completely yielded is taken as the yield load (P_y) (J, M, and Z 2014). When the specimen was subjected to (0.74 ~ 0.91) Pu, yielding occurred in the bottom flange of the steel. This led to an increased rate of deflection, as well as a rapid extension of cracks from the tensile zone towards the position of the neutral axis. When the web yielded to (0.37–0.75) times the height of the web, the concrete in the compression zone crushed and the specimens reached the ultimate bearing capacity. After that, the bearing capacity of the beams started to drop slowly, and when the load dropped to 0.85 P_u, the unloading was completed. The ultimate load P_u and yield load P_y of each specimen are listed in Table 4.

Figure 5. Damage pattern of some test beams.

Table 4. Yield load, ultimate load of the test.

Specimen	$P_{\mathrm{y}}^{\hspace{0.17em}}$/kN	$P_{\mathrm{u}}^{\hspace{0.17em}}$/kN
B1Q3S3L1	140	190
B2Q5S3L1	170	223
B3Q6S2L1	160	210
B4Q6S3L1	190	240
B5Q6S3L2	240	264
B6Q6S3L0	200	224
B7Q6S4L1	240	277

3.2. Load-deflection curves

Figure 6 shows the comparison of the load-deflection curves of the seven specimens. Comparing specimens B1 and B4, the bearing capacity of the test beams was increased by 26.3% when the steel strength grade was increased from Q355 to Q690. In comparison to specimens B3 and B7, when the steel ratio was increased from 2.27% to 4.4%, the bearing capacity was increased by 31.9%. The carrying capacity of the members rose by 17.9% when the longitudinal reinforcement ratio was raised from 1.09% to 2.29%. The above analysis shows that increasing the steel strength grade, steel ratio and longitudinal reinforcement ratio can improve the bearing capacity of PSRC beams.

Figure 6. Load-deflection curves for different beams.

3.3. Strain distribution of cross-section in the span

The strain distribution of steel and concrete are shown in Figures 7 and 8.

Figure 7. Strain distribution along the depth of the steels.

Figure 8. Strain distribution along the depth of the beams.

From the figure, it can be seen that:(1) the in-span strain shows a good linear relationship along the direction of beam height, which satisfies the assumption of flat cross-section;(2) the bottom flange of each steel beam is fully yielded under tension, and the steel web is highly yielded, which fully utilizes the tensile strength of the steel. (3) The beams with lower steel ratio (B3) and more compression reinforcement (B4) have a higher utilization of steel strength.

4. Research methodology

The ANN model and the XGBoost model are two commonly used ML models that have demonstrated success in previous research (Li et al. 2022; Wakjira et al. 2022, 2022, 2022). The ANN model is composed of multiple basic units called neurons, which transfer information through weighted connections. It possesses the ability to automatically extract and learn complex patterns and features from data. The XGBoost model is an ensemble learning model that combines multiple weak classifiers based on decision trees. This model offers better tunability and requires smaller amounts of data. In this study, both models were employed to predict the bending capacity of SRC beams. Based on the calculated results, the model exhibiting superior performance and more accurate predictions was selected.

4.1. Database for ML models

In this paper, a total of 112 test data of SRC beams including this test were collected, and a database was established (Wang, Wu, and Zheng 2009; Zhang, Sima, and Zhang 2005; Jia et al. 2015; Jiang et al. 2022; F, L, and L 2007; Li 2007, 2005; Liu 2012; Shao et al. 2001; Shen 2005; Tang, Zhou, and Zhang 2018; Wang 2006; Wang et al. 2015; Yang 2007; Zhang 2013). The database covered a total of 11 input parameters, including the width of section b (180-260 mm), the height of section h (200-400 mm), the eccentricity distance of section E (−25 ± 75 mm, downward eccentricity is positive), effective prestress f_p0 (0–312.5kN), concrete compressive strength f_c (20.3–86.2Mpa), yield strength of steel f_a (235-840Mpa), yield strength of reinforcement f_y (320-552Mpa), modulus of elasticity of concrete E_c (18200–48700Mpa), the cross-sectional area of reinforcement A_s (144-1718 mm²), the cross-sectional area of steel A_a (1255.9-6208 mm²), and cross-sectional area of prestressing tendon A_p (0-280 mm²). The output parameter chosen was the flexural bearing capacity M. Details of all specimens are listed in Appendix Table A1. 80% of the samples in the database were used to train the ML model, while the other 20% were used to assess the model’s performance (Li et al. 2022; Tang et al. 2015; Wakjira et al. 2022).

4.2. The preprocessing method of the model

4.2.1. Indicators to evaluate performance

To assess the performance and accuracy of the ANN model, this section and the following sections used several statistical metrics. These include mean absolute percentage error (MAPE), root mean squared error (RMSE), and coefficient of determination (R²), which are expressed in equations (1)-(3), respectively.

(1) $\mathit{MAPE}=\left((\sum _{\mathit{i}=1}^n\left|(T_{\mathrm{i}}-P_{\mathrm{i}})/T_{\mathrm{i}}\right|)/\mathit{n}\right)\times 100\mathrm{\%}$(1)

(2) $\mathit{RMSE}=\sqrt{\frac{1}{n}\sum _{\mathit{i}=1}^n{({\mathit{T}}_{\mathrm{i}}-P_{\mathrm{i}})}^2}$(2)

(3) $R^2=1-\left(\sum _{\mathit{i}=1}^n{({\mathit{T}}_{\mathrm{i}}-P_i)}^2\right)/\left(\sum _{\mathit{i}=1}^n{({\mathit{T}}_{\mathrm{i}}-T_{\mathrm{mean}})}^2\right)$(3)

Where, $T_{\mathrm{i}}$, $P_{\mathrm{i}}$, and $T_{\mathrm{mean}}$are the average of the true, predicted, and real values, respectively; n is the number of samples.

Mean absolute percentage error (MAPE) and root mean squared error (RMSE) reflect the deviation between the test and predicted values, and the coefficient of determination R² indicates the degree of linear correlation between the test and predicted values. The lower the values of MAPE and RMSE, the closer the R² is to 1, which proves the better the prediction performance of the model.

4.2.2. Normalization of models

To control the influence of weights and bias values on the ANN model and to improve the efficiency of the training algorithm, the input parameters were normalized in the training process, and the normalization equation was given in Equation (4)(4) $x^{\prime }=\frac{({\mathit{y}}_{max}-y_{min})(\mathit{x}-x_{min})}{x_{max}-x_{min}}+y_{min}$(4) :

(4) $x^{\prime }=\frac{({\mathit{y}}_{max}-y_{min})(\mathit{x}-x_{min})}{x_{max}-x_{min}}+y_{min}$(4)

Where x^’ is the normalized data; $y_{min}$and $y_{max}$default to 0 and 1, respectively; $x_{min}$and $x_{max}$are the maximum and minimum values of the input data.

4.2.3. Optimized models

Hyperparameter optimization significantly enhances the performance and generalization ability of ML models, resulting in improved predictions and more dependable results. In this study, the grid search method was employed to conduct hyperparameter tuning, drawing from successful experiences documented in previous research (Wakjira et al. 2022). To address the issue of overfitting, a 10 times 10-fold cross-validation approach was utilized for hyperparameter optimization. The training dataset was randomly divided into 10 groups, each including nine training sets and one validation set. After completing the training process, the validation set was replaced, and this procedure was repeated 10 times.

4.3. Building ML models

4.3.1. ANN model

The artificial neural network proposes weights and deviations of each neuron to connect the layers. Equation (5)(5) $\mathit{y}=\mathit{f}(\mathit{v})=\mathit{f}(\sum _{\mathit{i}=1}^n{w}_i{x}_i+\mathit{b})$(5) may be used to compute the connection between the values of input and output.

(5) $\mathit{y}=\mathit{f}(\mathit{v})=\mathit{f}(\sum _{\mathit{i}=1}^n{w}_i{x}_i+\mathit{b})$(5)

where x_i is the value of input; w_i is the weight value of the nerve; b is the bias value of the model; f is the activation function; and y is the value of output.

The ANN model used a back propagation (BP) algorithm to predict the flexural bearing capacity of beams, which can minimize the error of a specific training model. The Levenberg-Marquardt (LM) algorithm has the advantages of fast convergence, high accuracy, and efficient handling of redundant parameters. Therefore, the LM algorithm was used in the modeling phase. The ANN model established in this paper sets up an input layer, a hidden layer, and an output layer, as shown in Figure 9.

Figure 9. Architecture of the ANN model.

To determine the optimal number of neurons in the hidden layer, the hyperparameter optimization of the model was performed using the methodology provided in Section 4.2.2, which allows multiple attempts at different configurations. Two metrics, MAPE and R², were chosen to assess the performance of the model with different numbers of neurons. The number of neurons in the hidden layer was increased from 2 to 20, and the calculation results are shown in Figure 10. The training results showed that different numbers of neurons had a large impact on the model performance. ANN 11-10-1 (11 input parameters, 10 hidden layer neurons, and 1 output parameter) had the smallest MAPE value and high R² value, which proved the excellent performance of the model. The ANN model utilized a sigmoid transfer function within the hidden layer, which transforms the input signal to produce the encoded output. Meanwhile, a linear transfer function was used in the output layer to provide a mapping of the input to corresponding output values.

Figure 10. Performance of models with different numbers of neurons.

4.3.2. XGBoost model

XGBoost effectively controls the complexity of the model through regularization methods. It uses two regularization techniques: weight decay and leaf-wise minimum constraints on leaf nodes. These techniques help prevent overfitting and improve the generalization ability of the model. The objective function of XGBoost consists of a loss function and a regularization term as shown in Equation (6)(6) $\mathit{Obj}(\mathit{\theta })=\sum _{\mathit{i}=1}^N\mathit{L}(Y_i,\widehat{Y_i})+\sum _{\mathit{t}=1}^T\mathrm{\Omega }(f_t)$(6) (Chen and Guestrin 2016). The regularization term is shown in Equation (7)(7) $\mathrm{\Omega }(f_t)=\mathit{\gamma K}+\frac{1}{2}\mathit{\lambda }{∥{\omega }_t∥}^2$(7) .

(6) $\mathit{Obj}(\mathit{\theta })=\sum _{\mathit{i}=1}^N\mathit{L}(Y_i,\widehat{Y_i})+\sum _{\mathit{t}=1}^T\mathrm{\Omega }(f_t)$(6)

(7) $\mathrm{\Omega }(f_t)=\mathit{\gamma K}+\frac{1}{2}\mathit{\lambda }{∥{\omega }_t∥}^2$(7)

where K represents the leaf count; $\mathit{\gamma }$and $\mathit{\lambda }$ denote the regularization hyperparameters; ${\omega }_t$ corresponds to the leaf weights.

Combining the grid search method with the K-fold cross-validation method, the hyperparameter optimization of the XGBoost model was carried out using the training set data. The optimization results of hyperparameters such as the number of estimators, maximum depth, and learning rate are shown in Table 5.

Table 5. Optimized values of the hyperparameters for the XGBoost model.

Hyperparameter	Optimal value
Maximum number of estimators (CARTs)	100
Maximum depth of tree	5
Learning rate	0.1

5. Predictive performances of models

5.1. Calculation methods of the current codes

At present, there are two main methods for calculating the flexural bearing capacity of SRC beams: one is the Superposition Method, such as the Japanese Standard for Calculation of Steel and Bone Concrete Structures (AIJ- SRC (2001) 2001); the other is the Deformation Synergy Method, such as the Code for Design of Prestressed Concrete Structure (JGJ 369–2016) in China.

(a) AIJ-SRC 2001

(8) $\mathit{M}\le M_{\mathrm{b}\mathrm{y}}^{\mathrm{s}\mathrm{s}}+M_{\mathrm{b}\mathrm{u}}^{\mathrm{r}\mathrm{c}}$(8)

(9) $M_{\mathrm{b}\mathrm{y}}^{\mathrm{s}\mathrm{s}}\le {\gamma }_{\mathrm{s}}{W}_{\mathrm{s}}{f}_{\mathrm{a}}$(9)

(10) $\begin{array}{rl}& M_{\mathrm{b}\mathrm{u}}^{\mathrm{r}\mathrm{c}}=A_{\mathrm{s}}{f}_{\mathrm{y}}(\mathit{h}-a_{\mathrm{s}}-\frac{x}{2})+A_{\mathrm{p}}{f}_{\mathrm{p}}(\mathit{h}-a_{\mathrm{p}}-\frac{x}{2})\\ & +{\mathit{A}{\mathit{ }}^{\mathrm{\prime }}}_{\mathrm{s}}{\mathit{f}{\mathit{ }}^{\mathrm{\prime }}}_{\mathrm{y}}(\frac{x}{2}-{\mathit{a}{\mathit{ }}^{\mathrm{\prime }}}_{\mathrm{s}})\mathit{\hspace{0.17em}}\end{array}$(10)

where, $M_{\mathrm{b}\mathrm{u}}^{\mathrm{r}\mathrm{c}}$ is the bearing capacity of prestressed reinforced concrete; $M_{\mathrm{b}\mathrm{y}}^{ss}$ is the bearing capacity of the section steel; x is the height of the concrete compressive zone; ${\gamma }_{\mathrm{s}}$ is the plastic development coefficient of cross-section, 1.05 for H-beams; $W_{\mathrm{s}}$ is the resistance moment of the section steel; $\mathit{\hspace{0.17em}}\mathit{f}{{\mathit{ }}^{\mathrm{\prime }}}_{\mathrm{y}}$, $\mathit{\hspace{0.17em}}{f}_{\mathrm{y}}$, $\mathit{\hspace{0.17em}}{f}_{\mathrm{a}}$ and $\mathit{\hspace{0.17em}}{f}_{\mathrm{p}}$are the yield strength of the compression reinforcement, tensile reinforcement, section steel, and steel strands respectively; $a_{\mathrm{s}}$, $a_{\mathrm{s}}^{\prime }$ and $a_{\mathrm{p}}$are the distance from the center of the tensile and compression reinforcement, and steel strands, respectively, to the edge of the compressive zone;$A_{\mathrm{s}}$, $A_{\mathrm{s}}^{\prime }$ and $A_{\mathrm{p}}$are the cross-sectional areas of tensile reinforcement, compression reinforcement, and steel strands respectively.

（b）JGJ369–2016

(11) $\begin{array}{rl}& \mathit{M}\le {\alpha }_1{f}_c\mathit{bx}(h_0-\frac{x}{2})+{\mathit{f}{\mathit{ }}^{\mathrm{\prime }}}_{\mathrm{y}}{\mathit{A}{\mathit{ }}^{\mathrm{\prime }}}_{\mathrm{s}}(h_0-{\mathit{a}{\mathit{ }}^{\mathrm{\prime }}}_{\mathrm{s}})\\ & -f_{\mathrm{a}}{A}_{\mathrm{af}}(a_{\mathrm{a}}-a_0)-f_{\mathrm{p}}{A}_{\mathrm{p}}(a_{\mathrm{p}}-a_0)\\ & +{\mathit{f}{\mathit{ }}^{\mathrm{\prime }}}_{\mathrm{a}}{\mathit{A}{\mathit{ }}^{\mathrm{\prime }}}_{\mathrm{af}}(h_0-{\mathit{a}{\mathit{ }}^{\mathrm{\prime }}}_{\mathrm{a}})+M_{\mathrm{aw}}\end{array}$(11)

(12) $\begin{array}{rl}& {\alpha }_1{f}_{\mathrm{c}}\mathit{bx}=f_{\mathrm{y}}{A}_{\mathrm{s}}+f_{\mathrm{p}}{A}_{\mathrm{p}}\\ & +f_{\mathrm{a}}{A}_{\mathrm{af}}-{\mathit{f}{\mathit{ }}^{\mathrm{\prime }}}_{\mathrm{y}}{\mathit{A}{\mathit{ }}^{\mathrm{\prime }}}_{\mathrm{s}}-{\mathit{f}{\mathit{ }}^{\mathrm{\prime }}}_{\mathrm{a}}{\mathit{A}{\mathit{ }}^{\mathrm{\prime }}}_{\mathrm{af}}-N_{\mathrm{aw}}\end{array}$(12)

The specific section stress distribution is shown in Figure 11. The specific meaning of symbols in the formula is detailed in the specification JGJ 369–2016.

Figure 11. Stress distribution diagram of beam section of JGJ369–2016.

5.2. Comparison of the predicted results

Figure 12 and Figure 13 show the comparison between the predicted and tested values in the training and test sets computed by the ANN model and XGBoost model. The R² of the ANN model was 0.998 and 0.969 for the training and test sets, respectively; the MAPE was 0.0297 and 0.067, respectively; and the RMSE was 10 kN·m and 12.66 kN·m, respectively. The R² of the XGBoost model was 0.999 and 0.962 for the training and test sets, respectively; the MAPE was 0.0055 and 0.044, respectively; and the RMSE was 1.072 kN·m and 11 kN·m, respectively. The values of RMSE and MAPE for the XGBoost model are less than the ANN model, which indicates that the model is more predictive than the ANN model and the model generalizes better.

Figure 12. Comparison of the predicted and experimental values of the ANN model.

Figure 13. Comparison of the predicted and experimental values of the XGBoost model.

For a more visual comparison, the calculated results of the ANN model, XGBoost model, JGJ 369–2016, and AIJ-SRC2001 were compared with the test values, as shown in Figure 14. The XGBoost model calculated R², MAPE, and RMSE of 0.994, 0.013 and 4.97 kN·m, respectively, which are within 10% error from the experimental values. The errors between the calculated results of the ML models and the test values are smaller than those of JGJ 369–2016 and AIJ-SRC 2001, indicating that the flexural capacity of SRC beams calculated according to the code is on the conservative side. The calculated results of the ML models are closer to the test values and have higher prediction accuracy, while the XGBoost model has better prediction performance than the ANN model.

Figure 14. Comparison of models and codes.

5.3. Sensitivity analysis of the XGBoost model

XGBoost model can be regarded as a black box model, even though the model has good performance on the training set, we still cannot explain the decision-making process behind the model. Shapley interpretation method provides an interpretable approach that allows us to understand how each feature is ranked and decided based on its impact on the model prediction, and this methodology has been used with previous machine learning research with success (Gao and Lin 2021; Mangalathu, Hwang, and Jeon 2020; Wakjira et al. 2022). With Shapley’s Interpretive Approach, we can understand the extent to which each feature contributes to the model’s predictions, which allows us to understand which features are most critical to the model’s predictions and how they affect the model’s output.

Figure 15(a) illustrates the distribution of Shap values for each input parameter. The position of each point on the x-axis indicates the Shap value of each factor indicating the effect of each factor on the flexural capacity of the SRC beams, and the y-axis gives the factors in order of importance. The colors in Figure 13(a) indicate the values of the factors. The average of the absolute Shap values for each input variable is plotted in Figure 15(b) in descending order of its importance. As can be seen from the figure, the four most influential factors are the yield strength of section (f_a), the section area of section (A_a), the cross-sectional area of the reinforcement (A_a), and the section height (h). In addition, Figure 16 shows the results of the analysis based on the correlation of SHAP features. From Figure 16, it can be seen that there are negative interactions between section and reinforcement, and section and concrete strength.

Figure 15. Shap values of input parameters, (a) each point (b) average value.

Figure 16. Interaction plot of factors.

6. Conclusion

In this paper, flexural performance tests of seven types of PSRC beams were completed. After collecting and building a database containing 112 sets of credible test data, two ML models were developed to predict the flexural capacity of SRC beams. The computational results of the ML models were compared with the specifications. The main result of this study is that the machine learning model suitable for predicting the load bearing capacity of SRC beams is selected through comparison, avoiding the tedious manual calculation process, but the main problem of the study is that fewer experimental data is collected, and more experimental data on the flexural capacity of SRC beams will be collected in the future, so that the model’s generalization ability and accuracy can be further improved. The main conclusions of this paper are as follows:

1. Increasing the steel strength grade, steel ratio and longitudinal reinforcement ratio can increase the flexural bearing capacity of PSRC beams.

2. Compared with the calculation results of JGJ 369-2016 and AIJ-SRC 2001 for the flexural capacity of SRC beams, the prediction results of ANN model and XGBoost model are in good agreement with the experimental results, and the error is within 10%, which can predict the flexural capacity of SRC beams more accurately. The generalization ability and stability of the XGBoost model are better than that of the ANN model.

3. The sensitivity analysis of the model shows that the correlation between the input parameters and the flexural capacity is in the following order: steel yield strength > steel cross-sectional area > steel cross-sectional area, and there is a negative interaction between the strength of the steel section and the reinforcement, and between the strength of the steel section and the strength of the concrete.

Acknowledgements

This research was financially supported by the National College Student Innovation and Entrepreneurship Training Program (202210225271) and the Fundamental Research Funds for the Central Universities (2572019CT01).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by the National College Student Innovation and Entrepreneurship Training Program [2572019CT01]; the Fundamental Research Funds for the Central Universities [2572019CT01].

Notes on contributors

Zhibo Bao

Zhibo Bao is an undergraduate student at Northeast Forestry University, majoring in Civil Engineering, and is the leader of the China Student Innovation and Entrepreneurship Program and a recipient of the China National Scholarship. is an undergraduate student at Northeast Forestry University, majoring in Civil Engineering, and is the leader of the China Student Innovation and Entrepreneurship Program and a recipient of the China National Scholarship.

Jun Wang

Jun Wang is a professor and doctoral supervisor at the School of Civil Engineering, Northeast Forestry University. He is mainly engaged in the research on the theory of combined structures and the basic theory and structure of concrete composites.

Yurong Jiao

Jiao Yurong is a master's student at the School of Civil Engineering, Northeast Forestry University, with research interests in modular structures.

Appendix

Table A1. Details of all specimens

number	b (mm)	h (mm)	E (mm)	fp0 (kN)	fc (Mpa)	fa (Mpa)	fy (Mpa)	Ea (Mpa)	As (mm^2)	Aa ((mm^2)	Ap ((mm^2)	M (kN⋅m)	reference
1	210	280	0	270	24.396	275	347.2	27440	610	3910	280	152.04	(Wang, Wu, and Zheng 2009)
2	235	310	0	270	24.396	275	347.2	27440	610	4997.5	280	177.52
3	235	310	0	270	24.396	275	35.2	27440	864	4997.5	280	200.32
4	260	330	0	270	24.396	275	347.2	27440	610	6208	280	255.31
5	260	330	0	270	24.396	275	351.9	27440	1118	6208	280	278.11
6	200	300	0	0	26.5	611	200	35000	252	2328	0	147.2	(Li 2005)
7	200	300	0	0	26	611	200	35000	396	2328	0	168.4
8	200	300	0	0	24.1	611	200	35000	252	2328	0	147.6
9	200	300	0	0	26.4	611	200	35000	396	2328	0	166.4
10	200	300	0	0	3.8	600	300	35000	252	2892	0	175.6
11	200	300	0	0	31.4	600	300	35000	396	2892	0	195.6
12	200	300	0	0	31.7	600	300	35000	252	2892	0	176.8
13	200	300	0	0	31.4	600	300	35000	396	2892	0	192.8
14	200	300	0	0	32.2	300	300	35000	396	1728	0	130.4
15	200	300	0	0	32.4	815	300	35000	396	1728	0	214.8
16	200	300	0	0	31	600	300	35000	144	2892	0	156.4
17	200	300	0	0	3.6	611	300	35000	396	2328	0	160.8
18	200	300	0	0	34.1	600	300	35000	396	2892	0	181.6
19	200	300	0	0	35.6	605	300	35000	396	3456	0	219.6
20	200	300	0	0	37.1	600	300	35000	540	2892	0	186
21	200	300	0	0	3.9	300	300	35000	396	2892	0	146.4
22	200	300	0	0	36.4	500	300	35000	252	2892	0	184
23	200	300	0	0	37	840	300	35000	396	2892	0	256
24	200	300	0	107.8	34.3	284	434	35000	308	2580	98	124.8	(Tang, Zhou, and Zhang 2018)
25	200	300	0	154	32	284	434	35000	308	2580	140	132.6
26	200	300	0	215.6	35.9	284	434	35000	308	2580	196	147.6
27	200	300	0	261.8	32.1	284	434	35000	308	2580	238	151.2
28	200	300	0	308	35.4	284	434	35000	308	2580	280	166.2
29	220	400	40	0	31.17	325	365	35000	1531.2	2612	0	245	(J, M, and Z 2014)
30	220	400	40	262.5	36.55	325	365	35000	1531.2	2612	240	320
31	220	400	40	281.2	3.23	325	365	35000	1531.2	2612	240	324
32	220	320	0	0	35.05	325	365	35000	1070.1	2612	0	155
33	220	320	0	262.5	35.21	325	365	35000	1070.1	2612	240	205
34	220	320	0	281.2	31.1	325	365	35000	1070.1	2612	240	203
35	220	320	0	0	38.5	331	320	35000	1013.8	2612	0	151.9	(L et al. 2020)
36	220	320	0	22.3	38.5	331	320	35000	1013.8	2612	240	200.9
37	220	320	0	22.3	39.6	331	340	35000	1013.8	2612	240	199.9
38	220	400	40	0	39.93	331	320	35000	1531.2	2612	0	240.1
39	220	400	40	237.1	39.93	331	320	35000	1531.2	2612	240	313.6
40	220	400	40	237.1	39.6	331	340	35000	1531.2	2612	240	312.6
41	200	300	0	99.6	34.3	283.7	434	34500	615	2580	98	124.8	(T et al. 2022)
42	200	300	0	142.2	32	283.7	434	34500	615	2580	140	132.6
43	200	300	0	199.1	35.9	283.7	434	34500	615	2580	196	147.6
44	200	300	0	241.8	32.1	283.7	434	34500	615	2580	238	151.2
45	200	300	0	284.5	35.4	283.7	434	34500	615	2580	280	166.2
46	200	300	0	142.2	33.8	283.7	434	34500	615	2580	140	100.8
47	200	400	0	0	31.6	315	365	35000	628	1840	0	140	(Kabir et al. 2020)
48	200	400	0	0	31.6	315	365	35000	989	1840	0	158.8
49	200	400	0	0	31.6	315	365	35000	628	4240	0	185.2
50	200	400	0	0	31.6	315	365	35000	989	4240	0	227.5
51	200	260	0	0	26.8	235	400	32500	603	1789.2	0	75	(Zu-qiang, Beng-you, and Jian-yang 2023)
52	180	300	0	0	28.3	335.4	362.7	32500	314	2428	0	168	(Wakjira et al. 2022)
53	200	300	50	0	4.6	365	521	18200	452	1255.9	0	97	(Golafshani et al. 2012)
54	200	300	50	0	4.6	365	521	18200	452	1255.9	0	96
55	180	250	0	0	22.1	270	552	32000	452	2104	0	58.9	(Wakjira et al. 2022)
56	180	250	0	0	22.1	270	552	32000	452	2104	0	58.9
57	180	250	0	0	22.1	270	552	32000	452	2104	0	58.1
58	180	250	0	0	22.1	270	552	32000	452	2104	0	58.8
59	180	280	0	0	26.8	325	400	32500	628	2354	0	106.5	(Li et al. 2022)
60	200	200	0	0	2.3	272	320	30000	452	1297.4	0	47.18	(Wakjira et al. 2022)
61	200	200	0	0	2.8	272	320	30000	452	1297.4	0	47.32
62	200	250	25	0	2.3	272	320	30000	452	1297.4	0	63.35
63	200	250	25	0	24.2	272	320	30000	452	1297.4	0	66.5
64	200	300	50	0	2.3	272	320	30000	452	1297.4	0	96.6
65	200	300	50	0	25	272	320	30000	452	1297.4	0	103.14
66	200	300	0	0	24.3	272	320	30000	452	1297.4	0	75.78
67	200	300	0	0	2.3	272	320	30000	452	1297.4	0	71.91
68	200	350	75	0	2.3	272	320	30000	452	1297.4	0	119.88
69	200	350	75	0	27	272	320	30000	452	1297.4	0	133.2
70	200	350	25	0	2.3	272	320	30000	452	1297.4	0	93.78
71	200	350	25	0	23	272	320	30000	452	1297.4	0	96.3
72	250	300	0	28.6	37.1	400	430	35500	1250	2270	280	190	This paper
73	250	300	0	223.4	37.1	630	430	35500	1250	2270	280	223
74	250	300	0	268.3	37.1	740	430	35500	1250	1450	280	210
75	250	300	0	26.5	37.1	780	430	35500	1250	2270	280	240
76	250	300	0	282.9	37.1	780	430	35500	1718	2270	280	264
77	250	300	0	29.7	37.1	780	430	35500	817	2270	280	224
78	250	300	0	27.8	37.1	690	430	35500	1250	3300	280	277
79	200	300	0	249.2	86.18	254	410	48700	465	2125.9	280	157.3	(Wang, Wu, and Zheng 2009)
80	200	300	0	289.8	86.18	254	410	48700	465	2125.9	280	165.1
81	200	300	0	312.5	86.18	254	410	48700	465	2125.9	280	165.1
82	200	300	0	292.9	86.18	254	410	48700	666	2125.9	280	183.3
83	200	300	0	282.2	86.18	254	410	48700	920	2125.9	280	222.3
84	200	300	0	203.3	86.18	254	410	48700	920	2125.9	197	184
85	200	300	30	295.7	86.18	254	410	48700	465	2125.9	280	204.8
86	200	300	30	287	86.18	254	410	48700	666	2125.9	280	224.3
87	200	300	30	297.1	86.18	254	410	48700	920	2125.9	280	243.8
88	200	300	0	285.3	86.18	254	410	48700	465	2613.1	280	171.6
89	200	300	0	29.9	86.18	254	410	48700	465	2796.4	280	180.7
90	200	350	−5	0	37.21	272	365	35100	534	2612	0	115.35	(Wang, Shi, and Zhou 2019)
91	200	350	−25	0	37.21	272	365	35100	534	2612	0	108.61
92	200	350	−5	0	37.21	272	390	35100	854	2612	0	147.61
93	200	350	−5	0	4.99	272	365	36300	534	2612	0	127.29
94	200	350	−5	0	47.97	272	365	37100	534	2612	0	122.31
95	200	350	−5	279.61	37.21	272	365	35100	534	2612	280	172.58
96	200	350	−5	222.15	37.21	272	365	35100	534	2612	280	184.72
97	200	350	−5	269.53	37.21	272	365	35100	534	2612	280	174.31
98	200	350	−5	261.77	37.21	272	470	35100	854	2612	280	202.48
99	200	350	−5	283.88	4.99	272	365	36500	534	2612	280	189.79
100	200	350	−5	293.77	47.97	272	365	36500	534	2612	280	196.65
101	200	350	−5	271.66	37.21	272	365	36500	534	2612	280	172.58
102	180	250	0	0	22.1	270	552	32000	452	2104	0	53.9	(Wakjira et al. 2022)
103	180	250	0	0	22.1	270	552	32000	452	2104	0	54.6
104	180	250	0	0	22.1	270	552	32000	452	2104	0	59.4
105	180	250	0	0	22.1	270	552	32000	452	2104	0	58.5
106	180	240	0	0	26.8	325	410	32500	452	2354	0	96.6	(Li et al. 2022)
107	180	240	0	0	26.8	325	400	32500	628	2354	0	91.6
108	180	240	0	0	26.8	325	400	32500	628	2354	0	96.5
109	180	240	0	0	26.8	325	400	32500	628	2354	0	91.9
110	180	240	0	0	26.8	325	400	32500	628	2354	0	92.2
111	180	280	0	0	26.8	325	400	32500	628	2354	0	114.9
112	180	280	0	0	26.8	325	400	32500	628	2354	0	106.5