Predicting torsional capacity of reinforced concrete members by data-driven machine learning models

Abstract

Due to the complicated three-dimensional behaviors and testing limitations of reinforced concrete (RC) members in torsion, torsional mechanism exploration and torsional performance prediction have always been difficult. In the present paper, several machine learning models were applied to predict the torsional capacity of RC members. Experimental results of a total of 287 torsional specimens were collected through an overall literature review. Algorithms of extreme gradient boosting machine (XGBM), random forest regression, back propagation artificial neural network and support vector machine, were trained and tested by 10-fold cross-validation method. Predictive performances of proposed machine learning models were evaluated and compared, both with each other and with the calculated results of existing design codes, i.e., GB 50010, ACI 318-19, and Eurocode 2. The results demonstrated that better predictive performance was achieved by machine learning models, whereas GB 50010 slightly overestimated the torsional capacity, and ACI 318-19 and Eurocode 2 underestimated it, especially in the case of ACI 318-19. The XGBM model gave the most favorable predictions with R² = 0.999, RMSE = 1.386, MAE = 0.86, and \overline{\lambda }= 0.976. Moreover, strength of concrete was the most sensitive input parameters affecting the reliability of the predictive model, followed by transverse-to-longitudinal reinforcement ratio and total reinforcement ratio.

Graphical abstract

Keywords

RC members / torsional capacity / machine learning models / design codes

1 Introduction

Torsion plays a crucial role in analyzing the mechanical performance of structures subjected to eccentric loads.

For common reinforced concrete (RC) specimens, dense diagonal cracks which spiral around the member and premature failure may appear due to torsional effects. Therefore, their torsional resisting capacities need to be evaluated during the design process. Complicated three-dimensional behaviors and testing limitations of members under torsion lead to the difficulties in understanding and exploring the torsional mechanism. Until now, some achievements in experimental and theoretical researches aiming at predicting the torsional performance of RC members have been accumulated. Two main accepted theories, i.e., skew bending theory [1,2] and space truss model theory, have been developed, of which the latter is more widely used. The specifications from the most widely practiced current design codes [3–5], i.e., GB 50010, ACI 318-19, and Eurocode 2 are all based on the space truss model. Fig.1 displays the general development of the space truss model. Rausch [6] first used the space truss model to calculate the torsional capacity of RC members. Diagonal compression field theory [7] and the developed version of variable angle truss model (VATM) [8,9] laid the theoretical foundation of torsional analysis of RC members. Three equilibrium equations, three compatibility equations and constitutive law of materials were adopted to compute the post-cracking torsional performance through iterations. subsequently, several modified versions of VAMT were created, such as modified VATM and generalized VATM by Bernardo et al. [10–12], soften membrane model for torsion by Jeng and Hsu [13–15].

Based on the theoretical models and experimental research, effects of key mechanisms of different variables on the torsional capacity of RC members have been previously studied. Variables such as cross-sectional area size, reinforcement ratio of transverse and longitudinal bars, strength of materials, thickness of concrete cover, and so on, greatly affect the torsional strength. Considering the effect of section size ratio and the amount of reinforcement, an empirical equation for torsional strength was proposed by Mcmullen and Rangan [16]. Hafez and Hassan [17] evaluated the effects of concrete strength and section size and found that increasing section dimensions and concrete strength enhanced torsional strength. However, Bernardo and Lopes [18] evaluated the torsional performance of beams with high strength concrete by theoretical models and found that concrete strength had little influence, and the most influential parameter was the reinforcement ratio. That’s why several codes of ACI 318-19 and Eurocode 2 do not consider the contribution of concrete. Lee et al. [19] explored the influence of maximum ratio of torsional reinforcements by experiments. They revealed that excessive torsional reinforcement could cause brittle failure. The maximum amount of torsional reinforcement should be limited to avoid over-reinforcement. Besides, codes of Eurocode 2 overestimate the maximum reinforcement limits. Recently, concrete and reinforcements with high strength tend to be widely applied in practice. Bernardo and Lopes [20,21], Joh et al. [22], and Fang and Shiau [23] experimentally demonstrated the enhancing effect of high strength concrete on torsional strength and cracking stiffness. Meanwhile, they found that such members had relatively lower torsional ductility and the range of reinforcement ratio to keep ductile failure was narrow. Twenty-nine RC members with high-strength torsional reinforcements were cast and experimentally investigated by Kim et al. [24] to study the applicability of high strength materials. It was revealed that most high-strength reinforcements did not reach yielding strength and a limitation of 550 MPa should be designed for torsional reinforcements. Torsional strength has also been found to be influenced by large concrete cover [25], steel fiber reinforcements [26,27] and prestress tendons [11,28].

In spite of these efforts, no easy-to-use, uniform and rational model for torsional capacity prediction of RC members has been suggested. Torsional models become more and more complicated and computation processes with huge iteration cannot be avoided. Furthermore, the most important point is that the stability and accuracy of the predictions are not convincing enough to be directly applied in actual practice. As a result, estimation of torsional strength is still a tough challenge.

Recent innovations in utilization of machine learning may provide better tools to solve the troublesome task [29–31]. Such methods can automatically search for the inherent relationship between RC members and their torsional capacity by learning from the experimental database, without relying on complicated equilibrium equations [32]. Generally, machine learning model development procedure consists of problem determination and data handling [33] (feature selection and feature extraction techniques), model development [34,35] (machine learning algorithm selection, training and validation of models by k-fold cross validation or ratio sampling) and performance evaluation [36,37] (performance metrics and fine-tuning). Different kinds of machine learning techniques including k-nearest neighbor, decision tree (DT), back propagation artificial neural network (BPANN), random forest regression (RFR), support vector regression (SVR), and so on, have been extensively applied in multiple fields of civil engineering such as manufacturing [30], construction industry [31], structural materials [38,39], design and performance evaluation of structures [40–43], and durability and service life prediction [44–46]. The perfect excellent predictive performances demonstrated that machine learning methods can efficiently overcome the disadvantages of the traditional calculation methods based on analysis of mechanical mechanisms.

It is evident from the literature that a lot of research has been conducted to predict ultimate shear capacity of different members [47–50] through use of machine learning methods. Based on the similar calculation process for members subjected to shear and torsion, the machine learning methods can be extended to predict the torsional capacity. However, few studies have been carried out to determine torsional strength of RC members using machine learning methods, compared with research related to the shear capacity. To the authors’ knowledge, the artificial neural network (ANN) model was trained and validated by Arslan [51], Haroon et al. [52] and Ilkhani et al. [53] based on limited experimental databases to predict torsional capacity of RC members. The results showed that ANN models can provide reasonable predictions. However, only the ANN model was tested in those studies and the number of tested specimens in the databases was limited (the number of collected data was around 100).

This paper aims at proposing an efficient predictive model for torsional capacity of RC members by means of machine learning algorithms. A throughout literature review was implemented to construct an abundant database consisting of 287 RC members under torsional failure. Four machine learning algorithms, namely BPANN, support vector machine (SVM), RFR, and extreme gradient boosting machine (XGBM), were trained and tested by 10-fold cross-validation method. Meanwhile, predicting results from different design code equations (GB 50010, ACI 318-19, and Eurocode 2) were calculated and compared with that from machine learning models. Besides, the best fitting model with XGBM algorithm was applied to implement sensitivity analysis to investigate the influence of input variables on the torsional capacity of RC members. This paper provides an overall description of the application of machine learning methods in solving the torsional problems and proposes new tools for exploration of torsional mechanisms.

2 Specification provisions for torsional capacity of RC members

The specification provisions for torsional capacity of RC members applied in different countries are summarized in this section. Most design guidelines were proposed based on different kinds of space truss models. All the predictive equations can be used to calculate the ultimate torque of tested specimens.

(1) Chinese code GB 50010 [4]

In China, torsional capacity of RC members has been considered by a combination of torsional resistance contributions from both concrete and steel bars [4]. The predictive equation was expressed by Eq. (1).

(1)T_{\mathrm{u}}=0.35f_{\mathrm{t}}{W}_{\mathrm{t}}+1.2\sqrt{\zeta }\frac{f_{\mathrm{y}\mathrm{v}}{A}_{\mathrm{s}\mathrm{t}1}{A}_{\mathrm{c}\mathrm{o}\mathrm{r}}}{s},

where f_{\mathrm{t}} is the concrete tensile strength; W_{\mathrm{t}} is the plastic moment of torsional resistance, which is calculated by b^2(3h- b)/6 for rectangular section, where b is the width and h is the height of the section; f_{\mathrm{y}\mathrm{v}} is the yield stress of stirrups; A_{\mathrm{s}\mathrm{t}1} is the area corresponding to a single leg of stirrups; s is the spacing between stirrups; A_{\mathrm{c}\mathrm{o}\mathrm{r}} is the area enclosed by the internal surface of stirrups, as shown in Fig.2; \zeta is the ratio defined by strength of longitudinal bars and stirrups, which is:

(2)\zeta =\frac{f_{\mathrm{y}\mathrm{l}}{A}_{\mathrm{s}\mathrm{t}\mathrm{l}}s}{f_{\mathrm{y}\mathrm{v}}{A}_{\mathrm{s}\mathrm{t}1}{u}_{\mathrm{c}\mathrm{o}\mathrm{r}}},(\zeta ⩽1.7 )

where u_{\mathrm{c}\mathrm{o}\mathrm{r}} is the perimeter of the path of shear flow; f_{\mathrm{y}\mathrm{l}}, A_{\mathrm{s}\mathrm{t}\mathrm{l}} are the yielding stress and gross area of longitudinal steel bars, respectively.

(2) American design code ACI 318-19 [3]

The torsional capacity predicted by ACI 318-19 only considers the contribution of steel bars. Based on the force equilibrium between longitudinal reinforcements and transverse ones, the derived equation for torsional capacity is

(3)T_{\mathrm{u},\mathrm{A}\mathrm{C}\mathrm{I}}=min\left(\frac{2A_0{A}_{\mathrm{s}\mathrm{t}1}{f}_{\mathrm{y}\mathrm{v}}}{s}\cot \theta ,\frac{2A_0{A}_{\mathrm{s}\mathrm{l}}{f}_{\mathrm{y}\mathrm{l}}}{s}\tan \theta \right)⩽\sqrt{f_{\mathrm{c}}^{\mathrm{\prime }}}\frac{A_{\mathrm{o}\mathrm{h}}^2}{p_{\mathrm{h}}},

where A_0 is the concrete area enclosed by shear flow zone, A_0=0.85A_{\mathrm{o}\mathrm{h}}; A_{\mathrm{o}\mathrm{h}},p_{\mathrm{h}} are the area and perimeter enclosed by the centerline of the outermost closed stirrups, respectively; f_{\mathrm{c}}^{\prime } is the compressive strength of concrete cylinder, \sqrt{f_{\mathrm{c}}^{\prime }} should be less than 8.3 MPa; f_{\mathrm{y}\mathrm{l}}, f_{\mathrm{y}\mathrm{v}} are respectively the yield stress of reinforcements, whose values should not exceed 420 MPa in ACI 318-19; \theta is the inclination of compressive strut in truss model. In the failure condition whereby both longitudinal bars and transverse bars happen to yield, \theta can be calculated as follows:

(4)\theta =\cot {}^{-1}\left(\sqrt{\frac{A_{\mathrm{s}\mathrm{l}}s{f}_{\mathrm{y}\mathrm{l}}}{A_{\mathrm{s}\mathrm{t}1}{p}_{\mathrm{h}}{f}_{\mathrm{y}\mathrm{v}}}}\right).

Generally, the value of \theta is limited to 30°–60°. For simplification in design, it is permitted to set \theta to be 45° for RC members.

(3) Code of Eurocode 2 [5]

The design equation provided in Eurocode 2 is similar to that in ACI 318-19. As is shown in Fig.2, the biggest difference is that the centerline of effective wall thickness (t_ef) is assumed as the shear flow path in Eurocode 2. As expressed in Eq. (5), torsional capacity of RC members can be predicted by:

(5)\begin{array}{rl}{T}_{\mathrm{u}}=& min\left(\frac{2A_{\mathrm{k}}{A}_{\mathrm{s}\mathrm{t}1}{f}_{\mathrm{y}\mathrm{v}}}{s}\mathrm{c}\mathrm{o}\mathrm{t}\theta ,\frac{2A_{\mathrm{k}}{A}_{\mathrm{s}\mathrm{t}\mathrm{l}}{f}_{\mathrm{y}\mathrm{l}}}{u_{\mathrm{k}}}\tan \theta \right)\\ & ⩽2v{f}_{\mathrm{c}}^{\prime }{A}_{\mathrm{k}}{t}_{\mathrm{e}\mathrm{f}}\sin \theta \cos \theta ,\end{array}

where f_{\mathrm{c}}^{\prime } should be less than 90 MPa; f_{\mathrm{y}\mathrm{l}} is limited within 600 MPa; \upsilon is the effective strength parameter of concrete, which is taken as 0.6 \left(1-f_{\mathrm{c}}^{\prime }/250\right) \ge 0.5; A_{\mathrm{k}},u_{\mathrm{k}} are the area and perimeter length enclosed by shear flow path, respectively. t_ef can be expressed by the following expression:

(6)t_{\mathrm{e}\mathrm{f}}=max\left(A_{\mathrm{c}}/u,2C\right) ,

where A_{\mathrm{c}} is the area of whole section; u is the perimeter of the section; C is the distance between the center of longitudinal bars and section edge. Similarly, inclination angle \theta can be calculated from the Eq. (4) by substituting p_{\mathrm{h}} with u_{\mathrm{k}}.

(7)\theta =\cot {}^{-1}\left(\sqrt{\frac{A_{\mathrm{s}\mathrm{l}}s{f}_{\mathrm{y}\mathrm{l}}}{A_{\mathrm{s}\mathrm{t}1}{u}_{\mathrm{k}}{f}_{\mathrm{y}\mathrm{v}}}}\right),

where \theta should be limited between 21.8° and 45°.

3 Mathematical algorithms of machine learning models

In this paper, four machine learning models, BPANN, SVM, RFR, and XGBM, are used to predict the torsional capacity of RC members. This section gives the brief overview of the mathematical algorithms of the different machine learning models.

3.1 Back propagation artificial neural network

The BPANN model based on the principle of neural systems is a widely used machine learning algorithm. As illustrated in Fig.3(a), a BPANN model is composed of three parts, i.e., input layer, hidden layer and output layer. Data can be transformed from neurons in the input layer to neurons in the output layer through hidden layers. Neurons exist in one layer have a relationship with ones in another layer through weight functions, as expressed by Eq. (8). Loss functions constructed by the actual values and predicted values are fed back through the neural network to optimize the weight function until the loss function satisfies the accuracy requirement.

(8){\mathit{\text{z}}}_i^l=f( {\sum }_{j=1}^m{W}_{ij}^l{\mathit{\text{z}}}_j^{l-1}+b_i^{l-1}),

where f(\cdot ) is the activation function; b_i^{l-1} is the bias; {\mathit{\text{z}}}_i^l is the value of ith neuron in lth layer; {\mathit{\text{z}}}_j^{l-1} is the value of jth neuron in (l−1)th layer; W_{ij}^l is the weights matrix.

3.2 Support vector machine

The SVM model is categorized as supervised learning method for regression problems. Dissanayake et al. [48] made great contributions to the development and applications of SVM in solving regression related projects. As shown in Fig.3(b), the fundamental concept behind the SVM is to construct a flat decision surface f(x) where the actual data y_i deviate from the decision surface in the range of [-\epsilon ,\epsilon ] for all the training data. Different kernel decision functions can map multiple dimensional feather space with linear or nonlinear characteristics. To determine the linear decision function, Cortes and Vapnik [54] proposed the \epsilon–insensitive loss function to achieve the formulation of the optimization problem.

3.3 Random forest regression

The RFR model is an ensemble learning algorithm for conducting regression projects, which adopts a bagging method [55]. RFR can improve its prediction performance by making a full use of multiple base learners. As shown in Fig.3(c), the database is randomly split into an n database by a Bootstrap sampling method to develop n regression tree models, which guarantees the RF model have more diversity of outputs. The final prediction values are acquired by taking the average of the prediction values of all regression tree models. The main hyperparameters that need to be set in RF model are the number of trees and tree depth. More trees mean that it is as much as possible to average out distractions, leading to a decrease of prediction error. However, the maximum allowable tree depth should be optimized to avoid the overfitting. Detailed information for RF model can be found in Ref. [55].

3.4 Extreme gradient boosting machine

As a developed model from DT, XGBM is used for regression and classification problems employing gradient boosting framework. Multiple simple regressor learners are combined in this ensemble method. Consequently, XGBM has been widely applied by researchers for bending or shear bearing capacity prediction [41,50,56,57]. Fig.3(d) shows the flowchart of XGBM. Several iterations exist in the training process. At each iteration step, another basic learner will be added to the model to reduce the overall pseudo residual errors. The final prediction can be determined by:

(9)P_m=P_{m-1}+\alpha (y-P_{m-1}),

where P_m, P_{m-1} are the predicting values at iteration m − 1 and m, respectively; y is the actual value, such that the pseudo residual error can be expressed by (y-P_{m-1}); \alpha is the learning rate to mitigate potential overfitting. More details on the XGBM can be found in Refs. [58,59].

4 Workflow of machine learning models

4.1 Experimental database of reinforced concrete members under torsion

The abundant database plays a significant role in successfully constructing effective machine learning models for torsional capacity of RC members. Hence, a thorough literature survey using multiple sources [11,12,15,16,19,20,23,60–65] was undertaken to collect existing experimental results of RC members subjected to torque. A total of 287 RC members loaded to torsional failure were collected, of which 37 were RC members with hollow section and 250 were RC members with solid section. Specific values of the parameters that may influence specimens’ torsional capacity are summarized in Tab.1, comprising width (b), height (h), and thickness (t) of the section, reinforcement ratio of longitudinal steel bars and stirrups ({\rho }_l,{\rho }_t), yielding strength of longitudinal bars and stirrups (f_{\mathrm{y}\mathrm{l}},f_{\mathrm{y}\mathrm{t}}), compressive stress of concrete (f_{\mathrm{c}}) and torsional capacity (T_{\mathrm{u}}). It should be noted that t is calculated by b/2 if the RC beam was with solid section. {\rho }_l= A_{\mathrm{s}\mathrm{t}\mathrm{l}}/\left(bh\right), where A_{\mathrm{s}\mathrm{t}\mathrm{l}} is the area of total longitudinal bars; {\rho }_{\mathrm{t}}=A_{\mathrm{s}\mathrm{v}}/\left(bs\right), where A_{\mathrm{s}\mathrm{v}} is the gross area of stirrups; s is the spacing.

Distribution of all collected influential parameters is shown in Fig.4. It can be seen that values of each parameter have an wide range to cover the different RC members that had been used in practice: section dimension ranged from 100 to 800 mm; reinforcement ratio ranged from 0.07% to 3.509%; concrete and steel bars ranged from common strength materials to high strength materials. Pearson correlation coefficient (r) between influential parameters and torsional capacity was used to evaluate the significance of the parameters. As depicted in Fig.5, parameters of b and h showed strong correlation with torsional capacity because values of r were respectively 0.84 and 0.71, thus more than 0.5. Parameters of f_{\mathrm{c}} and f_{\mathrm{y}\mathrm{v}} had the absolute value of r more than 0.4. Other parameters exhibited weak correlation with torsional capacity due to lower values of r, less than 0.3. As described by the truss model [61], the solid sections could be simplified to be hollow sections when the RC members were in ultimate bearing condition, which explains the lowest value of r between t and T_u. Consequently, parameter t was not selected as input variable. Although f_{\mathrm{y}\mathrm{l}}, {\rho }_l, {\rho }_t displayed weak correlation, experimental results by Bernardo and Lopes [18] and Lee et al. [19] confirmed that the total reinforcement ratio ({\rho }_{\mathrm{t}}+{\rho }_{\mathrm{l}}) and longitudinal reinforcement enhanced the torsional capacity of RC members. Generally, a comprehensive variable defined by transverse-to-longitudinal reinforcement ratio {\rho }_{\mathrm{t}} f_{\mathrm{y}\mathrm{v}}/{\rho }_{\mathrm{l}}{f}_{\mathrm{v}\mathrm{l}} can be used to evaluate the effect of total reinforcements. Above all, six variables of b, h, f_{\mathrm{y}\mathrm{v}}, f_{\mathrm{c}}, {\rho }_{\mathrm{t}}{f}_{\mathrm{y}\mathrm{v}}/{\rho }_{\mathrm{l}}{f}_{\mathrm{y}\mathrm{l}}, {\rho }_{\mathrm{t}}+{\rho }_{\mathrm{l}} were selected as the input variables to develop the machine learning models.

4.2 Model implementation

In this section, details of training and testing process of the four machine learning models, BPANN, SVM, RFM, and XGBM, are elaborated. The models were constructed in the Python environment.

4.2.1 Model development

Six parameters, b, h, f_{\mathrm{y}\mathrm{v}}, f_{\mathrm{c}}, {\rho }_{\mathrm{t}} f_{\mathrm{y}\mathrm{v}}/{\rho }_{\mathrm{l}}{f}_{\mathrm{y}\mathrm{l}}, {\rho }_{\mathrm{v}}+{\rho }_{\mathrm{l}}, were selected as input variables and torsional capacity T_u was taken as output for all machine learning models. Generally, the experimental data are statistically divided into two sets: a first set is applied for training the machine learning model and a second set is applied to validate the accuracy of trained models. To avoid the possible over-fitting problem, 10-fold cross-validation method, using 10 batches of data, was used in this study. This approach had been validated in several published works [34,35,48] with limited training data. As shown in Fig.6, the database was statistically separated ten batches, with equal size, giving ten training and testing rounds. In each round, nine batches (such as batches 1–9) were randomly selected as the training set and the remaining batch (batch 10) was used as testing set. After the finish of the training and testing round, a further round began: nine batches (such as batches 1–8 and 10) were applied as training set, while batch 9 was used as testing set. Thus each batch was tested once and trained nine times. Finally, the predictive results from the machine learning models were obtained by taking weighted mean of all training and testing rounds, which was expressed by:

(10)P={\sum }_{i=1}^{10}{W}_i{P}_i,

where P_i is the calculated results in each round; W_i is the weight of i round in the view of determination coefficient (R²), W_i=(R^2{)}_i/{\sum }_{j=1}^{10}(R^2{)}_j.

4.2.2 Optimization procedure

Hyperparameters in the machine learning models play the decisive role in predictive performance, so the hyperparameter tuning process is necessary. Tab.2 gives the sets of values predefined for each hyperparameter and its optimal value. Meanwhile, the default values were applied for those hyperparameters that are not listed in the table. The tuning process is achieved by the method of grid search tuning.

4.2.3 Evaluation indicators of prediction performance

A comprehensive application of performance metrics throughout the whole data range can be helpful in quantitatively evaluating the performance of developed machine learning models. Based on a bulk of reviewed works [29,36,37], a series of widely applied statistical parameters are adopted including coefficient of determination, R-Squared (R²), root mean squared error (RMSE), and mean absolute error (MAE). R² evaluates how well the predicted lines replicate the actual lines. IfR^2=1, a wonderful fitting is obtained between predicted values and actual values. The expression for R² can be expressed by:

(11)R^2=1-\frac{\sum _1^n{\left(T_{\mathrm{p}\mathrm{r}\mathrm{e},i}-T_{\mathrm{a}\mathrm{c}\mathrm{t},i}\right)}^2}{\sum _1^n{\left(T_{\mathrm{p}\mathrm{r}\mathrm{e},i}-{\overline{T}}_{\mathrm{a}\mathrm{c}\mathrm{t}}\right)}^2},

where T_{\mathrm{p}\mathrm{r}\mathrm{e},i} represents the predicted values and T_{\mathrm{a}\mathrm{c}\mathrm{t},i} stands for the actual values from the experiment; {\overline{T}}_{\mathrm{a}\mathrm{c}\mathrm{t}}={\displaystyle \frac{1}{n}}\sum _1^n{T}_{\mathrm{a}\mathrm{c}\mathrm{t},i} is the average of experimental values.

RMSE and MAE are parameters used to appraise the errors between predicted values and actual values. Better prediction performance will be achieved when the values of RMSE and MAE decrease to be closer to zero.

(12)RMSE=\sqrt{\frac{\sum _i^n{\left(T_{\mathrm{p}\mathrm{r}\mathrm{e},i}-T_{\mathrm{a}\mathrm{c}\mathrm{t},i}\right)}^2}{n}},

(13)MAE=\frac{\sum _i^n\left|T_{\mathrm{p}\mathrm{r}\mathrm{e},i}-T_{\mathrm{a}\mathrm{c}\mathrm{t},i}\right|}{n}.

In addition, a commonly used strength ratio between predicted results and experimental results is calculated to evaluate the different prediction methods for torsional capacity of RC members. The strength ratio (\lambda) can be expressed by:

(14)\lambda =\frac{T_{\mathrm{p}\mathrm{r}\mathrm{e}}}{T_{\mathrm{a}\mathrm{c}\mathrm{t}}},

where \lambda >1implies the predicted results overestimate the actual value of experimental results, and demonstrates the method predicts a lower value than the actual results. The mean value (\overline{\lambda }), Standard deviation value (S{D}_{\lambda }) and coefficient of variation (CO{V}_{\lambda }) of \lambda are calculated and provided in the results described below.

Consequently, the best prediction model is the one with the highest value of R², lowest values of RMSE, MAE, S D_{\lambda }, and CO{V}_{\lambda } (nearest to zero), and with \overline{\lambda } close to one.

5 Results and discussions

5.1 Performance evaluation of different machine learning models

Fig.7(a) and Fig.7(b) gives the results from training and testing process by 10-fold cross-validation method, respectively. The predictive performance of all ten rounds of training and testing process was recorded in terms of R². The circle mark in the boxplot indicates the potential outliers. The rectangular box represents the interquartile range, while the horizontal line inside the box represents the median value. For the training process in the Fig.7(a), values of R² were between 0.867 and 0.999. SVM and RFR showed the highest dispersion in the training process among these machine learning models. The generalized predicting capacity of the trained models can be further validated by the testing database because the testing data was not used for training. On the whole, testing performance is slightly worse than the training performance. For the testing process in Fig.7(b), R² varied from 0.568 to 0.995. Similarly, SVM showed the highest dispersion. For better comparisons, mean values of indicators including R², RMSE and MAE from all the models were calculated and are summarized in Tab.3. XGBM exhibited the highest predictive accuracy with the highest average values of R² (0.999 and 0.983) and lowest average values of RMSE (0.45, 10.28) and MAE (0.12, 5.63) both in training and testing process. The performance of RFR was not far behind. However, SVM provided the worst prediction with the lowest value of R² both in training (0.881) and testing processes (0.843).

The final predictive results of each machine learning method were based on weighted average of ten training and testing rounds. The final predictive performance for torsional capacity of RC members from four kinds of machine learning models, i.e., BPANN, SVM, RFR, and XGBM, are compared with each other in Fig.8. The abscissa of each figure is the actual torsional capacity obtained from experiments, while the ordinate displays the predicted results from different machine learning models. The brown diagonal line represents the fitting line where the predicted values are perfectly equal to the actual values. The boundaries of zones with tolerance of ±20% are shown by black dash lines. As illustrated in Fig.8, predicted points mostly distribute in the vicinity of the ideal fitting line, Y = X. Meanwhile, predictive errors (shown in Fig.9) from different machine learning models are concentrated close to zero. It can be seen that all machine learning models exhibit a reasonable accuracy in predicting torsional capacity. The XGBM model exhibited the best predictive ability with the highest R² = 0.999 and the minimal mismatch (RMSE = 1.386 kN·m, MAE = 0.86 kN·m), followed by the RFR model with R² = 0.979, RMSE = 12.82 kN·m, and MAE = 5.95 kN·m. On the other hand, the BPANN and SVM models showed the relatively poor predictive performance with lower values of R² and higher values of RMSE and MAE.

5.2 Evaluation of predictive performance of different design codes

The design equations for torsional capacity considered in this paper comprise Eq. (1) (GB 50010), Eq. (3) (ACI 318-19), and Eq. (5) (Eurocode 2). Torsional capacity of total 287 tested specimens was predicted by all the three design equations. Fig.10 gives a scatter plot of the predicting results from different design codes versus the experimental results. The abscissa of each figure is the experimental results, while the ordinate means the predicted results. An area within the range of ±40% of the ideal fitting line was marked by black dish lines in each figure. Performance indicators and strength ratio\lambdawere adopted to appraise the predictive performance of design codes. Specific values of the evaluation indicators are summarized in Tab.4. It can be seen in Fig.10 that the scatters basically locate in the area bounded by the black dash lines and the value of R² is more than 0.8, which means torsional capacity can be reasonably calculated by all the three design equations. The corresponding errors between equation predictions and experimental results are recorded in Fig.11. Smaller errors close to zero constitute more than 75% of the total errors. According to the evaluation indicators from all three design equations, GB 50010 generated the highest determination coefficient value (R² = 0.915), the lowest derivation errors (MAE = 14.48), and the closest value to one of strength ratio, \lambda =1.03, demonstrating the most favorable prediction performance, followed by Eurocode 2 and ACI 318-19.

To evaluate performance of design equations in more depth, Fig.12 was drawn to show the distribution characteristic of strength ratio\lambdaalong with various stresses. The average value of strength ratio from GB 50010 was 1.032 (> 1) and the mean of strength ratios from ACI 318-19 and Eurocode 2 were 0.726 and 0.954, respectively. This illustrates that predictions from GB 50010 slightly overestimated the torsional capacity, whereas those from ACI 318-19 and Eurocode 2 underestimated the ultimate torque (especially for ACI 318-19, where large underestimation happened). In code GB 50010, the additional concrete contribution to the torsional capacity is considered, leading to a higher prediction. ACI 318-19 and Eurocode 2 only take the contribution of reinforcement into account. Furthermore, allowable material strengths of concrete (f_{\mathrm{c}}) and reinforcements (f_{\mathrm{y}\mathrm{l}},f_{\mathrm{y}\mathrm{v}}) are strictly defined in ACI 318-19 and Eurocode 2 to avoid possible brittle failure. The maximum strengths of concrete allowed are 90 MPa in Eurocode 2 and 69 MPa in ACI 318-19, whereas those of reinforcements are 600 MPa in Eurocode 2 and 420 MPa in ACI 318-19. The results of specimens whose material strength exceeded the limitation are shown enclosed by the dotted box in Fig.12. These limitations may be the reasons for the underestimated predicting results. As illustrated, a relatively lower limitation for concrete and reinforcements surely leads to more conservative predictions by ACI 318-19. The different definitions of shear flow path in different design codes (displayed in Fig.2) are also factors that greatly influence the accuracy of the code predictions. The torsion-resisting mechanism of RC members is complicated and no authoritative theory is accepted and applied universally. Consequently, different design code equations should be applied with assessment of specific conditions, case by case.

5.3 Comparative evaluation of performance of developed machine learning models and design codes

The predictive performances of the trained machine learning models (BPANN, SVM, RFR, and XGBM) against the existing design codes (GB 50010, ACI 318-19, and Eurocode 2) for torsional capacity of RC members are comparatively assessed in this section. Statistical parameters and strength ratio were adopted as the evaluation indicators. The corresponding values of the indicators are summarized in Tab.4, which shows that the XGBM model and the RFR model greatly outperformed the existing design codes. A Taylor diagram, plotted in Fig.13, was used for further comparison, giving a visual illustration of how closely the predicted results match the actual results in terms of indicators of SD, r, and RMSD. Accordingly, the closer to the reference point (blue point in Fig.13), the better the predictive model, with minimal values of SD and RMSD. It can be seen that the XGBM model appears in the bottom part of Fig.13, which means the XGBM model achieved the minimum value of SD and RMSD, followed by the RFR model, the BPANN model and the SVM model. Moreover, codes of ACI 318-19 and Eurocode 2 had comparable predictive performance, while GB 50010 has the biggest RMSD value. Another observation is that all machine learning models possess similar or much better predictive ability than the existing design codes. Consequently, the developed XGBM model performed best among all the models and design codes in predicting the torsional capacity of RC members, which is also demonstrated by the ranking method introduced by Olalusi and Spyridis [66].

5.4 Sensitivity analysis of input variables

Six input variables, b, h, f_{\mathrm{y}\mathrm{v}}, f_{\mathrm{c}}, {\rho }_{\mathrm{t}}{f}_{\mathrm{y}\mathrm{v}}/{\rho }_{\mathrm{l}}{f}_{\mathrm{y}\mathrm{l}}, {\rho }_{\mathrm{t}}+{\rho }_{\mathrm{l}}, were considered in this paper, and these have significant impact on the accuracy of best model, i.e., XGBM. Feather importance analysis was conducted to evaluate the influence of input variables on the prediction. Fig.14 displays the relative importance of each variable in impacting the accuracy of the XGBM model. Variable f_{\mathrm{c}} is the most critical variable with the relative importance of 22.3%, which means f_{\mathrm{c}} is a critical factor affecting the torsional strength. Therefore, the contribution of concrete in bearing torsional load should be reasonably considered in the design code, as stipulated in Chinese code GB 50010. Variables of {\rho }_{\mathrm{t}}{f}_{\mathrm{y}\mathrm{v}}/{\rho }_{\mathrm{l}}{f}_{\mathrm{y}\mathrm{l}}, {\rho }_{\mathrm{t}}+{\rho }_{\mathrm{l}}, b, f_{\mathrm{y}\mathrm{v}}, whose relative importance ranged from 15.9% to 20.6% had less impact than f_{\mathrm{c}}. Both longitudinal and transverse reinforcements are necessary in resisting torque. However, the section height h has the least influence on the torsional strength.

The best fitting model, i.e., XGBM model, was selected for trend analysis to explore its consistent performance with different values of six input variables. The variations and trends of strength ratio\lambdaversus the input variables are depicted in Fig.15. Results from code of GB 50010 were also added to embody the superiority of XGBM model. As revealed from the figure, the accuracy of the predictions by the XGBM model exhibited no significant or apparent trend with the input variables of b, h, f_{\mathrm{y}\mathrm{v}}, f_{\mathrm{c}}, {\rho }_{\mathrm{t}}{f}_{\mathrm{y}\mathrm{v}}/{\rho }_{\mathrm{l}}{f}_{\mathrm{y}\mathrm{l}} and {\rho }_{\mathrm{t}}+{\rho }_{\mathrm{l}} for the range investigated in this paper, which demonstrates that the proposed model can be applied with acceptable confidence for the RC members with all values of torsional parameters. Besides, predicting results from GB 50010 showed great deviation from \lambda =1, although a horizontal regression line was achieved. A huge error appeared when the code was used to predict the torsional strength of a specific RC beam.

6 Conclusions

This paper developed four kinds of machine-learning-based approaches, based on BPANN, SVM, RFR, and XGBM, for prediction of torsional capacity of RC members. Data with a total of 287 torsional RC specimens with different sections were collected and applied as the training and testing databases. The predictive performance of all machine learning models were compared with each other and with calculated results from three kinds of design codes. The superiority of XGBM model for predicting RC members’ torsional capacity was demonstrated. The following findings can be drawn.

1) For the range of input variables considered in this paper, both XGBM model and RFR model have a reasonable accuracy in predicting the torsional capacity of RC members, as these models possess the value of R² higher than 0.979, the value of RMSE and MAE lower than 12.82 and 5.95, and the value of \overline{\lambda } close to 1. Between them, XGBM model achieved better overall predictive performance as evaluated by Taylor diagram.

2) Comparison with perfect fitting achieved by 10-fold cross-validation method and trend analysis demonstrated the generalization capacity of the machine learning models for torsional capacity of RC members. The XGBM model can be applied with acceptable confidence for torsional capacity of RC members with a full range of torsional parameters investigated in this paper.

3) Sensitivity analysis was conducted to explore the influence of input parameters on the performance of the XGBM model. The results showed that the concrete strength (f_{\mathrm{c}}) was the input parameter to which the precision of the predicting model was most sensitive, followed by transverse-to-longitudinal reinforcement ratio ({\rho }_{\mathrm{t}} f_{\mathrm{y}\mathrm{v}}/{\rho }_{\mathrm{l}}{f}_{\mathrm{y}\mathrm{l}}), total reinforcement ratio ({\rho }_{\mathrm{t}}+{\rho }_{\mathrm{l}}), section width (b) and strength of stirrups (f_{\mathrm{y}\mathrm{v}}). However, section height (h) had little influence on the prediction results by the XGBM model.

(4) Predictive performances of three state-of-practice design codes were poorer than those of machine learning models for higher errors and lower coefficient of determination. Among them, GB 50010 had the best predictive accuracy with R² = 0.9151, \overline{\lambda } \text{=1.032}, and MAE = 14.48.

(5) GB 50010 slightly overestimated the torsional capacity of RC members, whereas Eurocode 2 and especially ACI 318-19 underestimated it. Reasons for the underestimation by ACI 318-19 and Eurocode 2 may be limitations to the maximum of material strengths of concrete and steel bars used in the design code equations. A much lower allowable material strengths stipulated in ACI 318-19 leads to more underestimated results.

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