Predicting shear strength of slender beams without reinforcement using hybrid gradient boosting trees and optimization algorithms

Abstract

Shear failure of slender reinforced concrete beams without stirrups has surely been a complicated occurrence that has proven challenging to adequately understand. The primary purpose of this work is to develop machine learning models capable of reliably predicting the shear strength of non-shear-reinforced slender beams (SB). A database encompassing 1118 experimental findings from the relevant literature was compiled, containing eight distinct factors. Gradient Boosting (GB) technique was developed and evaluated in combination with three different optimization algorithms, namely Particle Swarm Optimization (PSO), Random Annealing Optimization (RA), and Simulated Annealing Optimization (SA). The findings suggested that GB-SA could deliver strong prediction results and effectively generalizes the connection between the input and output variables. Shap values and two-dimensional PDP analysis were then carried out. Engineers may use the findings in this work to define beam's geometrical components and material used to achieve the desired shear strength of SB without reinforcement.

Graphical abstract

Keywords

slender beam / shear strength / gradient boosting / optimization algorithms

1 Introduction

It is difficult to predict the shear behavior of reinforced concrete (RC) beams having fractures that are developed on inclined sections. This is due to the fact that numerous factors affect the shear behavior of RC beams [1,2]. For calculating shear strength, one must model the interdependence of these several factors. Additionally, shear forces are always combined with other types of loads, such as bending, axial, and torsion forces, making calculations more difficult [3]. Because shear failure frequently happens abruptly and without warning, accurate shear capacity prediction is critical [4]. The shear strength of RC beams has been determined using a wide variety of techniques [5–8], in which shear strength tests were conducted on thin beams without girders. Simultaneously, key variables influencing the shear resistance of slender beams (SB) without reinforcement have been calculated, including the concrete compressive strength, the shear span-to-depth ratio, the proportion of longitudinal reinforcement, and the beam’s size [9–13]. However, owing to the complexity of the variables influencing the shear strength of unreinforced SB, many samples are required, making the experimental effort time-consuming and costly [14].

To address this issue, an empirical model based on regression analysis of the experimental data has been suggested [15–17]. Current design standards, such as Eurocode (EC2) [18] and ACI 318-14 [19], are all statistically justified on the basis of the best-fit empirical data. However, Russo and Puleri [20] examined several models for estimating the shear strength of SB in the absence of stirrups, demonstrating that they are only applicable within the constraints of the design parameters studied in a specific series of tests. This discrepancy may be explained by variations in the variables utilized and the experimental specifics, resulting in certain restrictions since these model-based analyses are restricted in their generalizability and application to a subset of unique thin beams [14]. Additionally, many theoretical explanations for the shear behavior of beams have been suggested, including the variable angle truss model [21], Kupferk et al.’s resistance hypothesis for RC beams [22], the modified compression field theory [23], and the strut-and-tie model [24]. However, these models are used in conjunction with certain assumptions to simplify the non-linear connection between shear strength and related factors. There are also robust computational approaches to solving complex problems, such as modeling cracks [25–27]. Regarding RC beams, scientists have also investigated models based on the finite element method for predicting the shear resistance [28,29]. Unfortunately, owing to the complexity of the problem in general or RC beams in particular, predicting shear behavior takes significant expertise and highly programmable computers.

Owing to the fast development of artificial intelligence methods in recent years, use of machine learning (ML) algorithms has grown, due to their efficiency in solving a wide variety of highly complex problems [30–37]. Several ML algorithms are worth noting, such as artificial neural network (ANN) [38], Kriging [39], polynomial Response Surface Method (RSM) [40], and Support Vector Machine (SVM) [41]. Among ML methods, the ANN model has been used by several scientists to estimate the shear strength of RC beams. In Oreta's study [42], the ANN model was developed using 155 data points and five input parameters to estimate the shear resistance of SB without reinforcement and simulate the impact of size on shear strength. In Mansour et al. [43], with a dataset of 176 data points and 9 input parameters, an ANN was utilized to estimate the shear strength of RC beams. The findings indicate that the ANN model is a viable tool for predicting the ultimate shear strength of RC beams. At the same time, the ANN model employed by Mansour et al. demonstrated that parameter studies may be used to assess the effect of certain input parameters on the desired output. To estimate the shear strength of deep and thin RC beams, Yang et al. [44] developed optimum multi-layer neural network models utilizing a robust backpropagation method. Amani and Moeini [45] created an ANN model and a neural fuzzy system to estimate the shear strength of RC beams using six influencing factors. The model used in this research outperformed the models developed in ACI 318. The ANN model is also used in a variety of other investigations, such as Cladera and Mari [46], Abdalla et al. [47], Chou et al. [48], Al-Shather and Redah [49], Dopico et al. [50], Seleemah [51], to predict the shear resistance of RC beams. Kaveh et al.’s study [52] offers a set of resistance prediction models for non-reinforced high-strength concrete (HSC) thin beams based on multilinear regression, the sophisticated ML technique of Multivariable Adaptive Regression (MARS), and Network Methods Data Processing (GMDH). This study’s database used 250 records with six input parameters to create models. Compared to the regression technique and the GMDH method, the MARS method gave the best estimates in terms of both accuracy and safety, and the prediction accuracy is considerably higher than that obtained with the most frequently constructed equations. Mohammed and Ismail [53] created a prediction model for RC beam shear resistance based on a random forest ML method, an SVM, and an Extreme Gradient Boosting (XGB) model. Tab.1 summarizes some of the findings obtained by use of ML methods to estimate shear resistance of RC beams.

The results shown in Tab.1 demonstrate that ML algorithms can accurately estimate the shear resistance of RC beams. The table also shows that many scientists have used and developed the ANN model in particular. However, the data set for these works is still small, and the range of input variables impact factors is not broad. Furthermore, these investigations only consider the prediction of cease predicting the shear resistance of RC beams when using different alternative ML methods. As a result, the impact of input parameters has not been considered.

One of the most powerful ML methods is the Gradient Boosting (GB) algorithm. GB is often used to solve supervised learning issues with high accuracy and has been used effectively in a variety of areas [56–58]. However, no research has been conducted too far on the efficacy of the GB method for predicting shear resistance issues in unreinforced SB. The authors, in this study, demonstrate use of the GB method to predict the shear resistance of unreinforced SB and to examine the variables that influence this resistance. The research illustrates its findings using a dataset of 1118 data points. The following sections comprise the paper’s content: the second section discusses the dataset’s basic information; the third section discusses the primary substance of the GB algorithm; the fourth section provides the ML simulation results along with sensitivity analysis; and lastly, the conclusion.

2 Database description

The database for this research was collected from published literature to develop an ML model for estimating the shear resistance of SB without reinforcing [3]. This dataset contains 1118 experimental findings from almost 80 studies conducted over a 60-year period.

Fig.1 shows a schematic of a beam test. When collecting data on beam samples, the following assumptions are made.

1) The rectangular cross-sectional shape of beams has no effect, on its own, on the size of the beams.

2) Beams with shear span to depth ratio, a/d ≥ 2, are called slender beams (denoted as SB).

3) The beams are strengthened longitudinally and do not have shear stirrups.

4) Shear tests are conducted with a focused force applied to one or two symmetrical locations on the beams.

5) Shear is the primary cause of specimen failure.

The database contains eight different parameters that affect the shear strength (designated by V) of SB, in cluding the overall section depth (h), the section width (b), the shear span to depth ratio (a/d), the characteristic concrete compressive strength (f_ck), the characteristic steel reinforcement yield strength (f_yk), the vertical shear reinforcement ratio, and the top plate width (w_T). Tab.2 summarizes the nomenclature, function, and statistical analysis of the input and output parameters (showing minimum, maximum, mean, median, and standard deviation). In addition, in order to comprehend the gathered dataset’s findings, the dataset’s variable distribution is examined. Fig.2 depicts the data distribution of input and output parameters, as well as the frequency with which they appear in the dataset. Finally, after examining Fig.2, certain conclusions are made.

1) The depth of the overall section ranges between 25 and 2200 mm (mean is 305 mm, and the standard deviation is 302.1 mm). In contrast, the majority (more than 85%) of data sets had have an h-value in the range (320 ± 20) mm, followed by 13.8% of data sets with an h-value in the range (240 ± 20) mm, and a few samples with a beam height greater than 1500 mm (1.6%).

2) The section width (b) value varies from 21 to 3000 mm (the mean is 154 mm, and the standard deviation is 202 mm). The majority of data (84.1%) have b values between 21 and 300 mm. 38.6% of beam samples have b in the range [140–180], followed by 10.9% with b between 180 and 220 mm. Only one sample has b value of 2000 mm, and one sample has a value of 3000 mm.

3) The range of shear span to depth ratio (a/d) ranges from 2 to 9.4 (The mean is 3.0, and the standard deviation is 1.1) (The mean is 3.0, and the standard deviation is 1.1). In the dataset, a significant quantity of data (91.4%) has a/d values in the range 2–5, while the least data are when a/d is between 6–9.4 (4.2%). When examining individual a/d values, the most frequent value for a/d is (3 ± 0.2) (37.8%).

4) The characteristic f_ck in the dataset varied from 6.1 to 127.5 MPa (the mean is 33.8 MPa and the standard deviation is 22.9 MPa). 4.7% of samples utilized lightweight concrete (f_ck ≤ 17 MPa), while 19.8% of beam samples used high strength concrete (f_ck ≥ 60 MPa).

5) The range of the primary reinforcement ratio (ρ) in beams ranges from 0.9% to 7.94%. However, most of the data (96.2%) have a value of ρ ≤ 4%, while approximately 0.5% of the samples have ρ ≥ 6%.

6) Characteristic yield strength of steel reinforcement (f_yk) varies from 283 to 1779 MPa (The mean is 425 MPa and the standard deviation is 164.5 MPa). Thus, the fluctuation range of f_yk is extensive. However, the f_yk value is typically in the range of 283–600 MPa (91.23%).

7) The size of the aggregate diameter (ag) ranges from 2 to 50 mm (the mean is 19 mm and the standard deviation is 7.7 mm), but the major aggregate diameters are generally around 10, 19, and 25 mm. The statistics for these sizes are 14%, 18.1%, and 21.8%, respectively.

8) The range of top plate width in this dataset (w_T) ranges from 10 mm to 400 mm (the mean is 100 mm, and the standard deviation is 71 mm). The sheet width is primarily concentrated in the range of 10–20 mm (28%), 100–110 mm (17%), and 150–160 mm (19%).

9) Finally, the value of the output parameter as the shear strength of the non-shear-reinforced SB varies across a broad range, from 1.9 to 2238.5 kN. However, most data (95.7%) indicate the shear strength of non-shear-reinforced SB to be up to 200 kN.

The correlation between input parameters and between input parameters and output parameters is also one of the key sources of evidence for evaluating the significance and relevance of input variables. Therefore, a correlation matrix between the characteristics has been studied and is shown in Fig.3, in which positive numbers indicate a positive connection, negative values imply a negative correlation. In addition, the intensity of the colors also indicates the correlation value between them. The study illustrated in Fig.3 indicates that the parameters in the obtained data set have a modest and medium correlation. This indicates that the eight input parameters of the dataset can be considered to be independent variables. Meanwhile, the shear resistance of SB is predicted based on these eight input parameters.

The data in this study are divided into two subsets: 70% (corresponding to 783 experimental samples) are used to develop ML models and are referred to as the training dataset; 30% (corresponding to 335 test samples) are used to test and evaluate the developed models’ accuracy and are referred to as the testing dataset. The partitioning method is chosen randomly to ensure that the sampled data is reflective of the actual data. The 335 experimental samples in the testing dataset were not used to build ML models. In other words, the test dataset’s 335 samples are seen as a novel dataset that the ML models have never encountered throughout their learning and development process. Additionally, some research indicates that a 7/3 data division ratio is acceptable for ensuring the reliability and representativeness of data used to train and validate ML models [59].

3 Machine learning methods

ML, which dates all the way back to the late 1950s, is the scientific research of algorithms and statistical models used by computer systems to execute a task without explicit instructions, relying instead on patterns and conclusions. The term “machine learning” refers to a subset of artificial intelligence (AI). ML algorithms construct a mathematical model using sample data referred to as “training data” to make predictions or choices without being subject to explicit programming for this. Computer systems learn to execute tasks such as categorization, prediction, and pattern recognition via ML. To store the learning process, the systems are trained by analyzing sample data using algorithms and different statistical models. Typically, sample data consists of quantifiable features, and ML algorithms seek to establish a relationship between the feature and an output value referred to as a label [60]. Following training on the sample data set, the acquired knowledge is utilized to detect patterns and make judgments about new data. Algorithms for ML are classified into two broad categories based on how they learn, namely supervised learning algorithms and unsupervised learning algorithms. Additionally, there are two types of learning: Semi-Supervised Learning and Reinforcement Learning. The distinction between these lies in the data used to train the model, the way the algorithms utilize the data, and the types of issues they address. The GB algorithm, for example, is described in depth below.

3.1 Gradient boosting

GB is a synthetic approach that develops an improved predictor via the use of boosting techniques. GB approaches the boosting technique issue as an optimization problem, using a loss function and attempting to reduce error. This is why it is named GB, since gradient descent technique is its inspiration. GB constructs a method for resolving the following optimization problem:

(1){min}_{c_n=1:N,{\omega }_n=1:N}L\left(y,\sum _{n=1}^{N}c{}_n{\omega }_n\right),

where L is the loss function value, y is the label, c_n is the confidence score of the nth weak learner (also known as the weight), and ω_n is the nth weak learner. Rather than attempting to discover the optimum global solution by scanning for all c_n and ω_n values, which is a process that consumes considerable time and resources, boosting’s strategy is to seek local solutions after adding each new model to the model chain with the aim of eventually arriving at a global solution.

{min}_{c_n,{\omega }_n}L\left(y,W_{n-1}+c{}_n{\omega }_n\right)

where

(2)W_{n-1}={\displaystyle \sum _{i=1}^{N-1}c{}_n{\omega }_n.}

Consider the sequence of boosting models as a function W and each learner function as a parameter m. Gradient descent is used here to reduce the loss function L(y, W),

(3)W_n=W_{n-1}-\eta \frac{\mathrm{\partial }}{\mathrm{\partial }\omega }L\left(W_{n-1}\right).

The following related relationship can be obtained.

(4)c_n{\omega }_n\approx -\eta \frac{\mathrm{\partial }}{\mathrm{\partial }\omega }L\left(W_{n-1}\right),

where ω_n is the model to be added next. Then, the new model must be “taught” to fit the -\eta {\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }\omega }}L(W_{n-1}) value, which is also known as pseudo-residuals.

In summary, the algorithm implementation is done as follows.

• Initialization of pseudo-residuals to be equal for each data point.

• At the ith loop.

1) Newly added train model is fitted to existing pseudo-residuals values.

2) Calculation of the confidence score ci of the model just trained.

3) Updating of the primary model.

4) Finally, calculation of the value of pseudo-residuals to label the next model.

• Then repeat with loop i + 1.

3.2 Optimization algorithms

3.2.1 Particle Swarm Optimization

Particle Swarm Optimization (PSO) was suggested by Eberhart and James Kennedy, and is well-known as a resilient optimization method [61] modeled on the behavior of a flock of birds or fish while on the lookout for food sources. PSO is capable of dealing with numerous optimization issues, each of which represents a potential solution. A PSO individual resembles a bird or a fish feeding in its foraging area. Individuals in the problem space comprise the population in PSO. Each individual’s movement is a mix of velocity and position. At any given moment, the location of each individual is determined by its optimal position and the optimal position of the herd as a whole. Individual efficiency is defined by an adaptive value that is decided by the issue at hand. Random initialization is used to create instances. Successive generations are generated. Each individual alters its velocity and position throughout time and has a fitness value, which will be decided by the generation’s optimum fitness function. Thereafter, each individual in the multidimensional search space finds a local optimum solution based on its fitness rating. Then, the locally optimum solution is compared to the swarm’s global optimal solution in order to update the globally optimal solution’s value. Finally, using the global optimum solution as a starting point, the system determines the optimal solution.

3.2.2 Simulated Annealing optimization

Simulated Annealing (SA) is a probabilistic method for determining a function’s global optimum. It is a meta-simulation, more precisely, whose primary characteristic is the ability to escape from local extremes by permitting backward movement with the aim of finding an optimum solution. It became renowned two decades ago because of its ease of installation, convergence, and the use of backward movement [62].

The method is named after the metallurgical process of annealing, in which a steel crystal is heated and then allowed to cool extremely slowly until it reaches its most regular crystal form. If the cooling process is slow enough, the resulting metal will have a good structure. Each iteration of the method applied to a combinatorial optimization problem takes the objective function for two solutions into account: the current solution and the new solution. They are compared, and the new answer is always selected if its accuracy is superior. On the other hand, even if the new solution’s accuracy is inferior, it is still acceptable with the aim of escaping the local extremes in pursuit of the global extreme. The chance of accepting the answer decreases as the temperature parameter decreases until the temperature nears zero. This process will be less frequent, and the solution distribution will have a greater chance of rapidly converging to the global extreme [62].

3.2.3 Random Search optimization

The Random Search (RS) algorithm is a kind of optimization method that is often utilized in practice. This method does not need the computation of the slope gradient to determine the parameter set’s optimum value. As a result, the RS method is often used to determine the optimality of non-zero continuous or non-differentiable functions. The RS method is simple to implement, and it operates by repeatedly traversing the search space at random locations surrounding the current position. Indeed, doing a numerical survey to thoroughly scan the SA hyper-parameter set and arrive at an optimum set is difficult. In this study, the RS method is used to determine the SA hyper-parameters, and simulations are conducted using the most optimum parameter settings. It is referred to in this section as Random Annealing, which is essentially a mixture of SA and RS that Simon introduced [63].

3.3 Repeated K-fold Cross-Validation

K-fold Cross-Validation (CV) is a technique for assessing an ML model’s performance using random samples of data collection. This guarantees that all information within the dataset is recorded. Additionally, CV enables us to get accurate estimates of the model’s generalization error or how well the model works on unknown data. However, a single run of the K-fold CV procedure may provide an imprecise assessment of the model’s performance. This is because different items of data may provide wildly divergent findings. Repeated K-fold CV is a variation of K-fold CV that enables the predictive performance of a ML model to be improved. This technique guarantees that no data points are lost during training and testing, even when data points are randomly selected for training and testing. The repeated K-fold method shuffles and randomly samples the dataset many times, resulting in a robust model with a high number of training and testing operations. Two factors affect the repeated K-fold CV method used to determine the predictive validity of the ML model. The first parameter K is a number indicating the data set randomly split into K subsets; one subset is chosen as the test dataset for evaluating the model’s performance, while the other subset is used as the training dataset. These stages will be repeated up to the number specified by the second parameter of this method, and therefore the process is called repeated K-fold, i.e., the repeated K-fold CV algorithm is performed a certain number of times. Because if K is too big, the training set will be considerably more significant than the test set, and evaluation results will not represent the fundamental nature of ML techniques, particularly with substantial data sets. This study uses repeated K-fold cross-evaluation with K=10 and repeats 10 times. This is also why many researchers choose a 10-fold cross-evaluation [64]. Fig.4 depicts the repeated 10-fold CV method.

3.4 Model performance assessment

To assess the performance and accuracy of ML models in predicting the shear strength of SB, the performance indicators utilized in this research are root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R²). These performance metrics are calculated according to the following formulas [65–68].

(7)R^2=1-\left[\frac{{\displaystyle \sum _{k=1}^N{\left(p_k-q_k\right)}^2}}{{\displaystyle \sum _{k=1}^N{\left(p_k\right)}^2}}\right],

(8)RMSE=\sqrt{\frac{1}{N}\sum _{k=1}^N{\left(p_k-q_k\right)}^2},

(9)MAE=\frac{1}{N}\sum _{k=1}^N\left|p_k-q_k\right|,

where p is the experimental value, q is the predicted value, calculated according to the model, and N is the number of samples in the database. Among the performance indicators used, RMSE and MAE achieved their optimum value when equal to 0, and R² reached the optimal value when equal to 1.

4 Methodology and flowchart

In this research, hybrid ML models are developed to forecast SB’ shear resistance, comprising four significant stages as follows (Fig.5).

1) Data preparation: 1118 experimental results are gathered to establish the dataset in this phase. Based on the dataset, hybrid ML models are constructed using nine input parameters and the shear resistance of SB as output variables. The data set is randomly split into two halves at a ratio of 7:3, in which the training data set accounts for 70% of the dataset and the testing data set accounts for 30% of the remaining dataset.

2) Model building: in this following phase, the training dataset is utilized for training the hybrid ML models. Three hybrid ML models are presented, namely GB-PSO, GB-RA, and GB-SA. The training procedure is iterative until the models are trained successfully (that is satisfying the stopping condition requirements).

3) Proposed Model Validation: The testing portion data is utilized to verify the proposed ML model in this third phase. Again, statistical measures like R², RMSE, and MAE are employed to assess the model’s performance.

4) Evaluate the impact of input parameters in ML models: in this final phase, an SHAP value graph is constructed to evaluate the significance and influence of the input parameters on the shear strength of non-shear-reinforced SB value.

5 Results and discussion

5.1 Performance evaluation of prediction machine learning technique

The block design shown in Fig.5 is used to create a model for predicting the shear resistance of RC beams. In the first phase, a basic GB model was created using the training dataset. The GB model’s accuracy is adjusted through six hyper-parameters, namely n-estimators, learning rate, max depth, max features, min samples split, and min samples leaf.

It is difficult to perform a fine-tuning process with many parameters in a model using ML methods, and the computing volume required is likewise greater. As a result, optimization methods are used in this research to determine the optimum values for the parameters in the GB model. The suggested PSO, RA, and SA layout optimization methods are used to determine the optimum parameter values for the GB model used to estimate the shear resistance of non-shear-reinforced SB RC beams. In addition, the coefficient of determination R² is used to assess the optimization process of the GB model.

Furthermore, the PSO hyperparameters have a considerable impact on the cost function of the optimization problem and the computational process, such as time-consumption. Population size is thought to have a significant impact on the PSO calculation procedure [69,70]. As a result, sensitivity analysis of population size is done in order to select the best population size for tuning the hyperparameter of the ML model by PSO. Fig.5 depicts the effect of population size on the convergence R² score and computational time. According to Fig.6, the highest R² score was obtained with population sizes of 20 and 60. Furthermore, the processing time is the shortest in the case of a population size of 20 (257 s). Consequently, it is possible to conclude that a population size of 20 is adequate for tuning ML hyperparameters by PSO.

The optimum GB model discovered by optimization techniques is the one with the most excellent R². The trial-and-error procedure is carried out using the values specified in Tab.3. Additionally, the performance of ML models is assessed in terms of the number of iterations required to fine-tune the model’s parameters. To avoid excessive optimization time, the maximum number of iterations is set to 500. It should be mentioned that the mean of the performance indices is crucial for assessing the suggested hybrid models’ accuracy and efficiency. The value of cost function R² and corresponding RMSE and MAE for three hybrid models with 500 repetitions are shown in Fig.7(a)–Fig.7(c), respectively. It could be observed that after about 250 iterations, the mean value of R² for the three suggested hybrid models reach relative convergence, and the R² values for all three models remain stable after 350 iterations. This demonstrates that using 500 iterations to assess the models’ performance is acceptable. According to the performance criteria R² and RMSE, the highest optimization performance process belongs to hybrid GB-SA model in which the R² score is the highest value and RMSE is the lowest value. Tab.4 summarizes the findings of the optimization algorithms’ discovery of the optimum parameters for the GB model.

The following paragraphs evaluate the performance of the three hybrid models in detail using statistical metrics such as R², RMSE and MAE over repeated 10-fold CV. The optimum GB hybrid model is the one with the lowest mean values of RMSE and MAE, and the highest mean value of R². The mean and standard deviation of these statistical criteria for the three hybrid ML models are detailed in Tab.5–Tab.7.

The mean values of R², RMSE, and MAE for the GB-PSO model are [0.8535; 0.8671], [55.1560; 57.0504], and [23.7524; 24.2966], respectively, while the standard deviation is [0.01; 0.026], [2.033; 6.123], and [0.467; 1.504]. For the GB-RA model, the mean values of the three criteria R², RMSE, and MAE are [0.8422; 0.8630], [55.1758; 59.44], and [23.9539; 24.6699], respectively, while the standard deviation has corresponding values of [0.009; 0.025], [2.173; 6.685], and [0.484; 1.651].

Finally, the range of mean values of the GB-SA model is [0.8564; 0.8782], [52.1567; 56.8385], and [22.2888; 23.2129], while for the standard deviation is [0.008; 0.023], [1.96; 5.974], and [0.439; 1.522], for R², RMSE, and MAE, respectively. Furthermore, the range of statistical criteria corresponding to 10-fold CV of three hybrid ML models is further shown in Fig.8–Fig.10.

The findings indicate that all three models, GB-PSO, GB-RA, and GB-SA, have excellent performance in predicting the shear strength of non-shear-reinforced SB (with low standard deviations). The GB-SA model, on the other hand, produced superior results with higher R² values and lower RMSE and MAE values than the other two models. This demonstrates that the GB-SA model is more effective in predicting the shear strength of non-shear-reinforced SB.

Fig.11 shows the performance comparison including (a) R², (b) RMSE, and (c) MAE between the best hybrid model GB-SA and 5 popular ML models such as XGB [71], SVM [72], Extreme Learning Machine (ELM) [73], Light Gradient Boosting (LightGBM) [74] and Gradient Boosting with Categorical features Support (CatBoost) [75]. The comparison is shown by boxplot of performance values after 10 repeats of 10-Fold CV. The boxplot in Fig.11 shows that the GB-SA has the lowest value of RMSE and the highest value of R² in comparison with the 5 ML models listed above. That confirms that the GB-SA model offers the best performance in predicting the shear strength of SB.

5.2 Prediction of typical machine learning algorithm

The findings of the 10-fold CV of the GB-SA model are shown in this section. There is a correlation between the shear strength of non-shear-reinforced SB matching the true value (green circle) and the predicted value (purple square) from the training and testing data sets, as shown in Fig.12. The horizontal axis on this graph shows the number of samples in the data sets, while the vertical axis depicts non-shear-reinforced SB shear strength in kN. The shear strength of 783 samples in the proposed model’s training database is quite similar to the experimental values (Fig.12(a)). In addition, 335 experimental outputs from the testing database are also predicted with minor errors (Fig.12(b)). The error values and the correlation between the experimental findings and the outputs using the GB-SA model are further quantified in the next section.

Fig.13(a) shows the GB-SA model’s distribution plot and cumulative distribution of errors for the training data, whereas Fig.13(b) shows the distribution and cumulative distribution of errors for the testing data. The error values corresponding to the training data set and the testing data set are modest, with approximately 350 data having an error of 0 for the training data set and 125 samples without error for the testing data set. There are just a few examples with substantial errors with a maximum error value of 200 kN for both the training and testing data sets. Additionally, it is simple to calculate the percentage inaccuracy of the samples inside a specific range using the cumulative distribution (red line). The proportion of samples having error values computed by GB-SA within the range [–20, 20] kN is 90% using the training database, for example. This result may be achieved using the testing database as well.

In Fig.14, the regression model depicts the correlation between the output value according to the GB-SA hybrid ML model and the actual value for the training database, testing database, and all data. The horizontal axis depicts the actual values of the gathered experiment, while the vertical axis depicts the output results of the suggested model.

Furthermore, the suggested model’s value for the training database (Fig.14(a)), testing database (Fig.14(b)), and total data (Fig.14(c)) is highly similar to the experimental findings. The GB-SA hybrid ML model can generalize across input and output characteristics and generate good predictions, according to these findings. In addition, the model’s performance is assessed using the statistical criteria mentioned earlier (Tab.4). These results demonstrate that the GB-SA hybrid ML model may be used to estimate non-shear-reinforced SB shear strength, saving time and money.

5.3 Sensitivity analysis

The hybrid ML model GB-SA predicts the shear strength value of non-shear-reinforced SB, and each input variable has a dependent connection with the others. Analysis of the effect of each input variable on the shear resistance value using a density scatter plot of SHAP data is shown in Fig.15. This graph uses all of the data, so that the rows, together, provide a record of all the data. The most significant features are listed first, followed by the least important. The color indicates the feature value, and the location on the x-axis is determined by the SHAP value; the lowest value corresponds to blue, while the greatest value corresponds to red. The color intensity indicates intermediate values. The number of points on the y-axis indicates the distribution of the SHAP value per feature, while the number of points on the x-axis reflects the number of features. The more positive the effect on the predicted shear strength grows as the feature’s SHAP value rises in tandem with the value of the corresponding feature, and vice versa. The size of the beam, main reinforcement ratio, shear span to depth ratio, and the compressive strength of the concrete are the fundamental factors that have the greatest impact on the shear strength of non-shear-reinforced SB, according to the results. Of the 8 input factors, the shear strength of SB was unaffected by three of them: the characteristic yield strength of steel reinforcement, aggregate diameter, and top plate width. Furthermore, the SHAP value does not necessarily rise or decrease as the feature value increases, demonstrating the complicated and non-linear connection between the input parameters and the shear resistance of SB. Some experimental investigations [9,10,13,76] in the literature have shown comparable findings. The beam dimensions, namely the cross-sectional height and breadth, had the greatest impact on the SB’s shear resistance in this study. Previous research using ML algorithms to estimate non-shear-reinforced SB’s shear strength backs this up. In Gandomi’s study [54], the contribution of each input parameter in the GEP models was evaluated through sensitivity analysis, and it was found that the shear strength of non-shear-reinforced SB increases as the size of the SB, concrete compressive strength, and longitudinal reinforcement ratio increase, but decreases as the shear span to depth ratio increases. The sequence of the impact of the factors, as determined by Gandomi, is consistent with the findings of this research. When utilizing sensitivity analysis for the ANN model to estimate the shear strength value of the non-shear-reinforced SB, Abdalla et al.’s research [47] makes the same finding. Zhang et al. [14] used the beetle antennae search method to build a random forest model to estimate the shear strength of non-shear-reinforced SB. The significance of factors that influence SB shear strength has been discussed. The most important variable is the shear span to depth ratio, followed by SB size, longitudinal reinforcement ratio, and concrete compressive strength. Because of the magnitude of the data set, the shear span to depth ratio has a different impact compared with the findings of this research. The dataset in Zhang’s research is based on shear span to depth ratios ranging from 0.25 to 3, while the shear span to depth ratio in our study is based on values ranging from 2 to 9.4. The size of the beam, including b and h, is the most complex of the factors influencing the shear strength of non-shear-reinforced SB. In certain experiments, the shear strength of the beam decreases as the size of the beam rises [9,76]. This may be owing to the limited number of beams examined, which means that generality cannot be ensured in all instances. Some computational models, such as the compression field-based model [77], or the ACI [78], Eurocode 2 [18], and CSA models [79], claim that the beam size is a parameter that positively affects the shear resistance of SB.

Finally, a partial dependence plot 2D (PDP-2D) is suggested for examining the connection between each pair of input parameters and the five most significant parameters discussed earlier. The shear strength value of non-shear-reinforced SB is shown by the color column. For example, using PDP-2D, it is feasible to calculate the appropriate beam size to produce the necessary shear strength in Fig.16(a). Precisely, when b and h exceed 500 mm, the value of the shear strength (V) progressively rises from 400 kN to its maximum value when the two indices exceed 700 mm. However, V stays constant in magnitude between 560–640 kN as b and h continue to rise to about 2500 mm. Considering Fig.14(b), when h is less than 500 mm, V is independent of ρ and ranges between 0 and 150 kN. As h rises, two spatial areas where distinct value of V could be observed determined. When ρ is less than 0.04, V is often less than 450 kN, whereas if ρ is higher than 0.04, V is typically greater than 900 kN. The transition domain of ρ must be extremely narrow in order to obtain a V value between 450 and 900 kN (i.e., only in the range of 0.04 to 0.042). Another input variable may have an effect on V within this range of ρ.

Fig.16(c) shows a similarity analysis relating to Fig.14(b), indicating that when h is smaller than 500, V will only reach a maximum value of 180 kN. When h is between 500 and 600, V will reach a value of 180–240 kN, and this value is practically independent of the ratio a/d (if a/d is more than 3, then V is in the 180–240 range, and if a/d is less than 3, then V will reach a more considerable value, ranging from 240 to 300 kN). There is a vast range of values as shown in Fig.14(c), when h is more than 700 and the ratio a/d is greater than 3, which offers a V value in the range 240–300. There is only a very narrow range for the two variables h and a/d that enables a maximum value of V (V between 560–640 with h around 700 and a/d between 2 and 3). As can be seen in Fig.16(d)–16(f), if a high value of V is desired, b must be more than 500 mm, ρ must be greater than 0.04, and the ratio a/d must be between 2–3. By combining the results shown in Fig.16(g)–16(j), it is possible to conclude that during the preliminary design phase, the aforementioned analysis assists the engineer in defining the geometrical components of the beam as well as the material to modify non-shear-reinforced SB’s shear strength.

6 Conclusions

The estimation of the shear strength of slender RC beams without stirrups has long been a difficult topic in structural engineering. The main purpose of this work is to propose ML models that can forecast the shear strength with high accuracy and consistency. A database of 1118 data points including eight separate components was compiled from the relevant literature. Three hybridized ML models, GB-PSO, GB-RA, and GB-SA, were built based on the original GB algorithm in conjunction with PSO, RA, and SA. For assessment of models, a 10-times repeated K-fold CV was employed, as well as additional statistical metrics. The findings reveal that the GB-SA model offers accurate prediction results with R² = 0.9471, RMSE = 29.8308, and MAE = 13.1549. Shap values show that the overall section depth, section width, vertical shear reinforcement ratio, shear span to depth ratio, and characteristic concrete compressive strength, are the most important factors that have the greatest impact on the shear strength of non-shear-reinforced SB. Finally, two-dimensional PDP graphs are presented, with the goal of assisting engineers in adjusting shear strength in relation to geometrical components and material properties.

References

[1] Collins M P, Mitchell D, Adebar P, Vecchio F J. A general shear design method. ACI Structural Journal, 1996, 93(1): 36–45

[2] Smith K N, Vantsiotis A S. Shear strength of deep beams. Journal Proceedings, 1982, 79(3): 201–213

[3] IsmailK S. Shear behaviour of reinforced concrete deep beams. Dissertation for the Doctoral Degree. Sheffield: University of Sheffield, 2016

[4] Kwak Y K, Eberhard M O, Kim W S, Kim J. Shear strength of steel fiber-reinforced concrete beams without stirrups. ACI Structural Journal, 2002, 99(4): 530–538

[5] Birkeland P W, Birkeland H W. Connections in precast concrete construction. Journal Proceedings, 1966, 63(3): 345–368

[6] ReganP E. Research on shear: a benefit to humanity or a waste of time? Structural Engineering, 1993, 71(19): 37–47

[7] Choi K K, Hong-Gun P, Wight J K. Unified shear strength model for reinforced concrete beams-Part I: Development. ACI Structural Journal, 2007, 104(2): 142

[8] NielsenM PBraestrupM WJensenB CBachF. Concrete Plasticity: Beam Shear—Shear in Joints—Punching Shear. Copenhagen: Danish Society for Structural Science and Engineering, 1978

[9] KaniG. How safe are our large reinforced concrete beams? Journal Proceedings, 1967, 64(3): 128–141

[10] CollinsM PKuchmaD. How safe are our large, lightly reinforced concrete beams, slabs, and footings? Structural Journal, 1999, 96(4): 482–490

[11] Bazant Z P, Kim J K. Size effect in shear failure of longitudinally reinforced beams. Journal of the American Concrete Institute, 1984, 81(5): 456–468

[12] Russo G, Zingone G. Flexure-shear interaction model for longitudinally reinforced beams. Structural Journal, 1991, 88(1): 60–68

[13] Shioya T, Iguro M, Nojiri Y, Akiyama H, Okada T. Shear strength of large reinforced concrete beams. Special Publication, 1990, 118: 259–280

[14] Zhang J, Sun Y, Li G, Wang Y, Sun J, Li J. Machine-learning-assisted shear strength prediction of reinforced concrete beams with and without stirrups. Engineering with Computers, 2020, 38: 1293–1307

[15] Zsutty T C. Beam shear strength prediction by analysis of existing data. Journal Proceedings, 1968, 65(11): 943–951

[16] Niwa J, Yamada K, Yokozawa K, Okamura H. Revaluation of the equation for shear strength of reinforced concrete beams without web reinforcement. Doboku Gakkai Rombunshuu, 1986, 1986(372): 167–176

[17] Sarsam K F, Al-Musawi J M. Shear design of high-and normal strength concrete beams with web reinforcement. Structural Journal, 1992, 89(6): 658–664

[18] BSEN 1992-1-1:2004. Eurocode 2: Design of Concrete Structures—Part 1-1: General Rules and Rules for Biddings. Brussels: European Committee for Standardization, 2015

[19] ACI318M-0. Building Code Requirements for Structural Concrete and Commentary. Farmington Hills: American Concrete Institute, 2008

[20] Russo G, Puleri G. Stirrup effectiveness in reinforced concrete beams under flexure and shear. Structural Journal, 1997, 94(3): 227–238

[21] Lampert P, Thürlimann B. Ultimate strength and design of reinforced concrete beams in torsion and bending. In: Ultimate Strength and Design of Reinforced Concrete Beams in Torsion and Bending. Basel: Springer, 1972, 107–131

[22] Kupfer H, Bulicek H. A consistent model for the design of shear reinforcement in slender beams with I- or Box-shaped cross section. Proceedings, Symposium on Concrete Shear in Earthquake, 1992, 256–265

[23] Vecchio F J, Collins M P. The modified compression-field theory for reinforced concrete elements subjected to shear. Journal of the American Concrete Institute, 1986, 83(2): 219–231

[24] Marti P. Basic tools of reinforced concrete beam design. Journal Proceedings, 1985, 82(1): 46–56

[25] RabczukTZiGBordasSNguyen-XuanH. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37−40): 2437−2455

[26] Rabczuk T, Belytschko T. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343

[27] RabczukTBelytschkoT. A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29−30): 2777−2799

[28] Gan D, Zhou Z, Yan F, Zhou X. Shear transfer capacity of composite sections in steel tubed-reinforced-concrete frames. Structures, 2017, 12: 54–63

[29] Xu T, Castel A, Gilbert R I. On the reliability of serviceability calculations for flexural cracked reinforced concrete beams. Structures, 2018, 13: 201–212

[30] Ly H B, Le L M, Duong H T, Nguyen T C, Pham T A, Le T T, Le V M, Nguyen-Ngoc L, Pham B T. Hybrid artificial intelligence approaches for predicting critical buckling load of structural members under compression considering the influence of initial geometric imperfections. Applied Sciences (Basel, Switzerland), 2019, 9(11): 2258

[31] Ly H B, Pham B, Dao D V, Le V M, Le L M, Le T T. Improvement of ANFIS model for prediction of compressive strength of manufactured sand concrete. Applied Sciences (Basel, Switzerland), 2019, 9(18): 3841

[32] Guo H, Zhuang X, Rabczuk T. A deep collocation method for the bending analysis of Kirchhoff plate. Computers, Materials & Continua, 2019, 59(2): 433–456

[33] Anitescu C, Atroshchenko E, Alajlan N, Rabczuk T. Artificial neural network methods for the solution of second order boundary value problems. Computers, Materials & Continua, 2019, 59(1): 345–359

[34] Samaniego E, Anitescu C, Goswami S, Nguyen-Thanh V M, Guo H, Hamdia K, Zhuang X, Rabczuk T. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 2020, 362: 112790

[35] Zhuang X, Guo H, Alajlan N, Zhu H, Rabczuk T. Deep autoencoder based energy method for the bending, vibration, and buckling analysis of Kirchhoff plates with transfer learning. European Journal of Mechanics. A, Solids, 2021, 87: 104225

[36] Guo H, Zhuang X, Chen P, Alajlan N, Rabczuk T. Stochastic deep collocation method based on neural architecture search and transfer learning for heterogeneous porous media. Engineering with Computers, 2022, 1–26

[37] Ly H B, Le T T, Vu H L T, Tran V Q, Le L M, Pham B T. Computational hybrid machine learning based prediction of shear capacity for steel fiber reinforced concrete beams. Sustainability (Basel), 2020, 12(7): 2709

[38] Lehký D, Slowik O, Novák D. Reliability-based design: Artificial neural networks and double-loop reliability-based optimization approaches. Advances in Engineering Software, 2018, 117: 123–135

[39] Zhang Y, Hu S, Wu J, Zhang Y, Chen L. Multi-objective optimization of double suction centrifugal pump using Kriging metamodels. Advances in Engineering Software, 2014, 74: 16–26

[40] Keshtegar B, Hao P, Wang Y, Hu Q. An adaptive response surface method and Gaussian global-best harmony search algorithm for optimization of aircraft stiffened panels. Applied Soft Computing, 2018, 66: 196–207

[41] Yang I T, Hsieh Y H. Reliability-based design optimization with cooperation between support vector machine and particle swarm optimization. Engineering with Computers, 2013, 29(2): 151–163

[42] Oreta A W C. Simulating size effect on shear strength of RC beams without stirrups using neural networks. Engineering Structures, 2004, 26(5): 681–691

[43] Mansour M Y, Dicleli M, Lee J Y, Zhang J. Predicting the shear strength of reinforced concrete beams using artificial neural networks. Engineering Structures, 2004, 26(6): 781–799

[44] YangK HAshourA FSongJ K. Shear capacity of reinforced concrete beams using neural network. International Journal of Concrete Structures and Materials, 2007, 1(1): 63–73

[45] Amani J, Moeini R. Prediction of shear strength of reinforced concrete beams using adaptive neuro-fuzzy inference system and artificial neural network. Scientia Iranica, 2012, 19(2): 242–248

[46] Cladera A, Mari A R. Shear design procedure for reinforced normal and high-strength concrete beams using artificial neural networks. Part II: Beams with stirrups. Engineering Structures, 2004, 26(7): 927–936

[47] Abdalla J A, Elsanosi A, Abdelwahab A. Modeling and simulation of shear resistance of R/C beams using artificial neural network. Journal of the Franklin Institute, 2007, 344(5): 741–756

[48] Chou J S, Pham T P T, Nguyen T K, Pham A D, Ngo N T. Shear strength prediction of reinforced concrete beams by baseline, ensemble, and hybrid machine learning models. Soft Computing, 2020, 24(5): 3393–3411

[49] Al-Shather L M, Redah S M A M. Prediction of shear strength of reinforced concrete beams using Artificial Neural Network and evaluated by Finite Element Software. International Journal of Scientific and Engineering Research, 2018, 9: 34–42

[50] DopicoJ ROrdóñezJ PBohigasA CGonzález-FonteboaBMartínez-AbellaF. Shear and bond analysis on structural concrete using artificial neural networks. In: The 5th International Engineering and Construction, Conference (IECC’5). Los Angeles: American Society of Civil Engineers, 2008

[51] Seleemah A A. A multilayer perceptron for predicting the ultimate shear strength of reinforced concrete beams. Journal of Civil Engineering and Construction Technology, 2012, 3(2): 64–79

[52] Kaveh A, Bakhshpoori T, Hamze-Ziabari S M. Development of predictive models for shear strength of HSC slender beams without web reinforcement using machine-learning based techniques. Scientia Iranica, 2019, 26(2): 709–725

[53] Mohammed H R M, Ismail S. Proposition of new computer artificial intelligence models for shear strength prediction of reinforced concrete beams. Engineering with Computers, 2021, 38(4): 3739–3757

[54] Gandomi A H, Alavi A H, Kazemi S, Gandomi M. Formulation of shear strength of slender RC beams using gene expression programming, part I: Without shear reinforcement. Automation in Construction, 2014, 42: 112–121

[55] Elsanadedy H M, Abbas H, Al-Salloum Y A, Almusallam T H. Shear strength prediction of HSC slender beams without web reinforcement. Materials and Structures, 2016, 49(9): 3749–3772

[56] Ahmad A, Ostrowski K A, Maślak M, Farooq F, Mehmood I, Nafees A. Comparative study of supervised machine learning algorithms for predicting the compressive strength of concrete at high temperature. Materials (Basel), 2021, 14(15): 4222

[57] Kaloop M R, Kumar D, Samui P, Hu J W, Kim D. Compressive strength prediction of high-performance concrete using gradient tree boosting machine. Construction & Building Materials, 2020, 264: 120198

[58] Vu Q V, Truong V H, Thai H T. Machine learning-based prediction of CFST columns using gradient tree boosting algorithm. Composite Structures, 2021, 259: 113505

[59] Khorsheed M S, Al-Thubaity A O. Comparative evaluation of text classification techniques using a large diverse Arabic dataset. Language Resources and Evaluation, 2013, 47(2): 513–538

[60] MohammedMKhanM BBashierE B M. Machine Learning: Algorithms and Applications. Boca Raton: CRC Press, 2016

[61] Kennedy J, Eberhart R. Particle swarm optimization. Proceedings of ICNN’95—International Conference on Neural Networks, 1995, 4: 1942–1948

[62] GloverF WKochenbergerG A. Handbook of Metaheuristics. New York: Springer Science & Business Media, 2006

[63] BlankeS. Hyperactive: An optimization and data collection toolbox for convenient and fast prototyping of computationally expensive models. Version 2.3.0. Available at GitHub

[64] Kohavi R. A study of cross-validation and bootstrap for accuracy estimation and model selection. International Joint Conference on Articial Intelligence (United States), 1995, 14(2): 1137–1145

[65] Nguyen Q H, Ly H B, Tran V Q, Nguyen T A, Phan V H, Le T T, Pham B T. A novel hybrid model based on a feedforward neural network and one step secant algorithm for prediction of load-bearing capacity of rectangular concrete-filled steel tube columns. Molecules (Basel, Switzerland), 2020, 25(15): 3486

[66] Ly H B, Nguyen T A, Thi Mai H V, Tran V Q. Development of deep neural network model to predict the compressive strength of rubber concrete. Construction & Building Materials, 2021, 301: 124081

[67] Ly H B, Nguyen M H, Pham B T. Metaheuristic optimization of Levenberg–Marquardt-based artificial neural network using particle swarm optimization for prediction of foamed concrete compressive strength. Neural Computing & Applications, 2021, 33(24): 17331

[68] Ly H B, Pham B T, Le L M, Le T T, Le V M, Asteris P G. Estimation of axial load-carrying capacity of concrete-filled steel tubes using surrogate models. Neural Computing & Applications, 2021, 33(8): 3437–3458

[69] Piotrowski A P, Napiorkowski J J, Piotrowska A E. Population size in particle swarm optimization. Swarm and Evolutionary Computation, 2020, 58: 100718

[70] EberhartR CShiY. Comparing inertia weights and constriction factors in particle swarm optimization. In: Proceedings of the 2000 Congress on Evolutionary Computation. CEC00 (Cat. No. 00TH8512). IEEE: 2000, 1: 84–88

[71] ChenTGuestrinC. XGBoost: A scalable tree boosting system. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. New York: Association for Computing Machinery, 2016

[72] Cortes C, Vapnik V. Support-vector networks. Machine Learning, 1995, 20(3): 273–297

[73] HuangG BZhuQ YSiewC K. Extreme learning machine: Theory and applications. Neurocomputing, 2006, 70(1−3): 489−501

[74] KeGMengQFinleyTWangTChenWMaWYeQLiuT. Lightgbm: A highly efficient gradient boosting decision tree. In: Advances in Neural Information Processing Systems 30 (NIPS 2017). Long Beach: Neural Information Processing Systems Foundation, Inc. (NeurIPS), 2017

[75] DorogushA VErshovVGulinA. CatBoost: gradient boosting with categorical features support. 2018, arXiv:1810.11363

[76] WalravenJ C. Division of Mechanics and Structures. Report 5-78-4. 1978

[77] Nadir W, Dhahir M K, Naser F H. A compression field based model to assess the shear strength of concrete slender beams without web reinforcement. Case Studies in Construction Materials, 2018, 9: e00210

[78] ACI318-11. Building Code Requirements for Structural Concrete and Commentary. Farmington Hills: American Concrete Institute, 2011

[79] CSAA23.3-14. Design of Concrete Structures. Mississauga: Canadian Standards Association, 2004

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