Developing Soft-Computing Models for Simulating the Maximum Moment of Circular Reinforced Concrete Columns

Journal of Soft Computing in Civil Engineering 8-2 (2024) 46-66
How to cite this article: Al-kharabsheh B, Alqawasmeh H. Developing soft-computing models for simulating the maximum
moment of circular reinforced concrete columns. J Soft Comput Civ Eng 2024;8(2):46–66.
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Journal of Soft Computing in Civil Engineering
Journal homepage: www.jsoftcivil.com
Developing Soft-Computing Models for Simulating the Maximum
Moment of Circular Reinforced Concrete Columns
Buthainah Al-kharabsheh1*
, Hasan Alqawasmeh2
1. Ph.D., Civil Engineering Department, Faculty of Engineering, Al Albayt University, 25113 Mafraq, Jordan
2. Ph.D., Civil Engineering Department, Faculty of Engineering Technology Al-Balqa Applied University, 11134
Amman, Jordan
Corresponding author: alkharabsheh@aabu.edu.jo
https://doi.org/10.22115/SCCE.2023.369096.1556
ARTICLE INFO ABSTRACT
Article history:
Received: 08 November 2022
Revised: 24 June 2023
Accepted: 17 July 2023
There has been a significant rise in research using soft-computing
techniques to predict critical structural engineering parameters. A
variety of models have been designed and implemented to predict
crucial elements such as the load-bearing capacity and the mode of
failure in reinforced concrete columns. These advancements have
made significant contributions to the field of structural engineering,
aiding in more accurate and reliable design processes. Despite this
progress, a noticeable gap remains in literature. There's a notable
lack of comprehensive studies that evaluate and compare the
capabilities of various machine learning models in predicting the
maximum moment capacity of circular reinforced concrete
columns. The present study addresses a gap in the literature by
examining and comparing the capabilities of various machine
learning models in predicting the ultimate moment capacity of
spiral reinforced concrete columns. The main models explored
include AdaBoost, Gradient Boosting, and Extreme Gradient
Boosting. The R2
value for Histogram-Based Gradient Boosting,
Random Forest, and Extremely Randomized Trees models
demonstrated high accuracy for testing data at 0.95, 0.96, and 0.95,
respectively, indicating their robust performance. Furthermore, the
Mean Absolute Error of Gradient Boosting and Extremely
Randomized Trees on testing data was the lowest at 36.81 and
35.88 respectively, indicating their precision. This comparative
analysis presents a benchmark for understanding the strengths and
limitations of each method. These machine learning models have
shown the potential to significantly outperform empirical
formulations currently used in practice, offering a pathway to more
reliable predictions of the ultimate moment capacity of spiral RC
columns.
Keywords:
Circular reinforced concrete
columns;
Moment capacity;
Machine learning models;
Soft-computing techniques.

B. Al-kharabsheh, H. Alqawasmeh/ Journal of Soft Computing in Civil Engineering 8-2 (2024) 46-66 47
1. Introduction
Designing reinforced concrete (RC) structures requires a thorough understanding and precise
estimation of the moment capacity of the columns. These columns, particularly those with
spirals, are believed to possess a higher lateral capacity than their counterparts with ties, due to
the greater confinement pressure they can offer [1,2]. Over the years, this hypothesis has been
tested and verified through a number of experimental studies, building a robust body of literature
on the behavior of these integral structural components [3–6]. Complementing this experimental
approach, the last few years have seen an upsurge in the application of machine learning-based
methods to predict the behavior of RC columns with interconnecting spirals. This surge has been
motivated by the capacity of machine learning techniques to process vast amounts of data, adapt
to new information, and provide robust predictions. For instance, studies conducted by Ahmadi
et al. [7] and Chang & Zheng [8] have utilized artificial neural network models to evaluate the
compressive strength of circular columns. In a similar vein, Koçer et al. [9]examined the ability
of an artificial neural network model to predict the moment capacity of spirally reinforced
concrete columns. More comprehensive research on this topic was carried out by Naderpour et
al.[10], focusing on the accuracy of artificial neural networks in estimating the moment capacity
of such columns. Going a step further, other research investigated the effectiveness of several
machine learning approaches, including the artificial neural network and ensemble machine
learning techniques, in solving other structural engineering problems including design, modeling,
assessment, and evaluation of structural behavior and resistance to external forces [11–16]. For
instance, Noori and Varaee [14]utilized artificial neural networks to approximate the nonlinear
seismic response of steel moment frames, demonstrating the potential of AI-based methods in
complex structural analysis. Furthermore, an investigation by Shishegaran et al. [15] offered
computational predictions of steel panel shear wall performance under explosive loads, revealing
crucial insights into the impacts of extreme conditions. The same research group developed a
high correlated variables creator machine to predict concrete's compressive strength, showcasing
the power of computational tools in materials testing [16]. Compressive strength, a key
parameter of concrete, has also been the focus of numerous AI-based studies. Zhang et al. [17]
proposed a novel method combining the Multivariate Adaptive Regression Splines (MARS), the
Grasshopper Optimization Algorithm (GOA), and the Monte Carlo Simulation (MCS) to assess
the reliability of compressive and splitting tensile strength prediction of roller compacted
concrete pavement. Ashrafian et al. [18]introduced an evolutionary Neuro-Fuzzy-Based
approach to estimate the compressive strength of eco-friendly concrete containing recycled
construction wastes. This trend extends to the estimation of seismic retrofit costs, where
Hamzehkolaei and Alizamir [19] evaluated machine learning algorithms' performance using
structural parameters. On the optimization front, a hybrid generalized reduced gradient-based
particle swarm optimizer was proposed by Varaee et al. [20] for addressing constrained
engineering optimization problems. Pour et al. [21] investigated the performance of composite
concrete-filled steel tube columns with different cross-section shapes and steel fibers, further
highlighting the need for accurate computational modeling in modern construction. Despite this
progress, it is apparent that most of the preceding research has been predominantly confined to
the use of artificial neural networks, with limited exploration of the potential of other machine
learning models. This research project aims to fill this gap by examining the performance of a

48 B. Al-kharabsheh, H. Alqawasmeh/ Journal of Soft Computing in Civil Engineering 8-2 (2024) 46-66
variety of machine learning models in determining the ultimate moment capacity of spirally
reinforced concrete columns. The study will delve into each machine learning model's
performance, analyzing and contrasting them to ascertain the most effective technique.
Following this, the outcomes of the selected machine learning models will be juxtaposed with
the methodology delineated in the ACI 318 standard by the American Concrete Institute. This
comparative analysis will provide valuable insights into the predictive accuracy of the machine
learning models relative to conventional structural engineering approaches. In addition to the
comparative analysis, the study will conduct an in-depth parametric analysis using the feature
importance to underscore the impact of each input variable on the model output. Feature
importance is a powerful tool that enables us to identify which variables have the most
significant impact on the predictions of the model, thereby shedding light on the most crucial
factors that influence the moment capacity of spirally reinforced concrete columns. Accordingly,
this article presents novel research focused on using soft-computing techniques to predict the
maximum moment capacity of circular RC columns. To address a gap in existing literature, the
researchers developed and evaluated various machine learning models for this purpose. By
benchmarking these models against experimental and conventional model results, they were able
to demonstrate that machine learning can provide highly reliable predictions for the ultimate
moment capacity of spiral RC columns. This study further suggests that the developed machine
learning models can significantly outperform the empirical formulation introduced by the codes
of practice. Thus, the research not only pioneers the application of soft-computing models to this
specific area of structural engineering but also highlights their potential for improving accuracy
and reliability in predicting the behavior of reinforced concrete structures. This article is
structured as follows: Section 2 discusses the materials and methods of the study; Section 3
provides the results and discussions of the study; Section 4 highlights the main conclusions of
the paper.
2. Materials and methods
According to Koçer et al. [9], evaluating the moment capacity of RC columns with spirals
exposed to simultaneous bending and axial movements is estimated in particles using
mathematical expressions typically supplied by the local standards of each nation. Such
empirical methods have the disadvantage of being limited in ability to suit a variety of column
attributes. In order to solve this problem, prior studies advise employing the ANN technique to
calculate the moment capacity of RC columns. Nevertheless, the research has been presented in
the literature did not highlight the capability and effectiveness of various machine learning
models in the estimation. Therefore, this study compares the accuracy of the three most popular
neural network models to the empirical methods used in ACI 318 to solve regression problems.
2.1. Utilized database
In fact, the Pacific Earthquake Engineering Research Center (PEER) provides an open-source
respiratory for experimental data of many circular spirally-reinforced concrete columns. This
database is used in this study to develop the selected machine learning models. Table 1 and
Figure 1 provides the descriptive statistics for this dataset.

Table 1
Descriptive statistics of the utilized database.
Variable
Sample
Size
Mean
Standard
Deviation
Variance
Coefficient
of variation
Minimum Q1 Median Q3 Maximum
Diameter (m) 104 0.390 0.131 0.017 33.440 0.150 0.280 0.400 0.490 0.610
Cross-section Circular sections or octagonal ones
Concrete cover (m) 104 0.018 0.006 0.000 34.400 0.010 0.010 0.020 0.020 0.040
Element length (m) 104 1.422 1.154 1.331 81.160 0.300 0.600 1.200 1.650 6.100
Concrete
compressive
strength (MPa)
104 35.5 13.9 192.1 39.0 18.9 29.4 32.5 35.5 90.0
Concrete tensile
strength (MPa)
104 3.676 0.609 0.370 16.560 2.720 3.388 3.560 3.720 5.930
Yield stress of
longitudinal
reinforcement
(MPa)
104 418.2 59.3 3519.9 14.2 294.0 366.0 441.3 453.3 565.4
Longitudinal
reinforcement ratio
104 2.796 1.077 1.160 38.510 0.520 2.040 2.570 3.793 5.580
Yield stress of
transverse
reinforcement
(MPa)
104 422.3 152.1 23127.9 36.0 0.0 361.0 401.5 444.0 1000.0
Volumetric
transverse
reinforcement ratio
104 0.910 0.546 0.298 60.020 0.000 0.630 0.920 1.135 3.040
Axial load applied
(kN)
104 798.7 846.4 716313.5 106.0 26.4 220.5 430.5 921.7 4300.0
Moment capacity
(kN.m)
104 342.5 313.1 98050.3 91.4 22.0 85.8 295.0 479.3 1300.0
2.2. Machine learning models
The present era witnesses a prevalent use of the multiple linear regression (MLR) method in
determining the linear relationship between a single dependent parameter and multiple
independent factors. This mathematical model provides a compelling way of understanding the
interplay between several variables and their impact on an outcome of interest. Currently, the
method of stochastic gradient descent (SGD) is widely employed in large-scale MLR modeling.
This method shows remarkable efficiency and delivers superior results when dealing with large
datasets. SGD essentially operates by computing the gradient of the loss function for each
sample in the dataset and subsequently updates the model parameters. Generally, there are a
variety of options available when choosing a loss function for SGD, including the squared
Euclidean norm, the absolute norm of ElasticNet, or even a combination of these two. These are
all techniques aimed at reducing the coefficients of the model to a zero vector, thus minimizing
the error and ensuring the best possible model fit. Another significant method utilized in
engineering, particularly in regression problems, is the support vector regressor (SVR). This
supervised learning algorithm was developed to handle cases where the nature of the problem
made traditional methods unsuitable. SVR allows for a variety of kernel functions, which could
be linear, polynomial, radial basis function (RBF), or sigmoid, each with distinct advantages.
This research aims to evaluate multiple kernels to pinpoint the one that yields the most optimal
results. Furthermore, the decision tree (DT) and SVR techniques have gained considerable
recognition in the data mining industry. Both are extensively used for solving regression and

classification problems. DTs come with certain unique advantages such as providing an
exhaustive analysis of all potential outcomes and the ability to track the path leading to each
solution. The DT method is known for conducting an in-depth examination and evaluation of
each outcome and its path. This feature is particularly crucial for conducting additional analysis
on decision nodes. Also, the DT method operates by dividing the dataset iteratively. Each
partition serves as the basis for generating the estimating model, and the decision tree itself is
ultimately produced from the integration of these predictive models. In recent times, a myriad of
DT methods have been proposed and developed. This study will employ the classification and
regression tree (CART) technique under the DT model. This approach is known for its efficiency
in handling numerical target parameters, making it a robust choice for various applications.
Fig. 1. Visualized descriptive statistics of the utilized database (Parameters' units are the same as that
defined in Table 1).

Random forest (RF) has emerged as one of the most sought-after machine learning algorithms in
the literature. Known for its precision and adaptability with various datasets, RF has gained
significant traction in the field of civil engineering, wherein it has been utilized to construct
functional models that help in decision-making and prediction. An RF model comprises
numerous decision trees (DT), each controlled by a random subset of the training data, distinct
from the singular tree structure of the DT model. RF employs the concept of bootstrapping and
aggregation, creating multiple Classification And Regression Trees (CARTs), contributing to the
robustness of the model. A key distinction arises when comparing the RF and DT methodologies.
While RF works based on critical thresholds, an extremely randomized trees (ETR) approach
hinges on random determination of thresholds for each potential feature. In the ETR model, the
splits within the tree's nodes are also randomly determined to identify the most suitable candidate
for the splitting criterion. The ETR method significantly reduces the model's variance but tends
to increase the bias. Unlike the RF, which uses bootstrap replicas (substituting input data through
down-sampling), ETR examines the entire original sample. Adaptive boosting (AdaBoost or
Ada) is a versatile meta-algorithm designed to enhance prediction performance. It has been
implemented in a myriad of learning methodologies cited in literature. This algorithm is typically
based on an iterative process, adjusting the weights when a previous experiment doesn't yield the
desired results. AdaBoost uses the real training dataset and regressor to fit multiple instances of
the regression model, adhering to Drucker's rules for handling complex situations. In this study,
the Ada weak model is a single-level DT regressor. Stochastic gradient boosting (GB) is an
enhancement of the traditional gradient boosting technique, used extensively for regression and
classification tasks. The primary difference between Ada and GB lies in their weak learner
models. While Ada's weak learner is a single-level regressor, GB's weak learner has a deeper
decision tree. Another distinctive feature of GB is that it does not require training on the entire
dataset, thereby reducing the risk of overfitting and lowering the correlation of the trees.
Histogram-based gradient boosting (HGB) distinguishes itself from other machine learning
techniques by assigning fixed data points into bins to create a histogram during the training
phase. This strategy facilitates faster training stages, rapidly improves model quality, and reduces
the model's memory requirements. Therefore, the HGB approach promises superior machine
learning applications in shorter timeframes. Extreme gradient boosting (XGB), a highly flexible
and effective machine learning technique, produces a series of decision trees by allocating a
weight classification to each independent parameter. These weight classifications are then
assigned to the decision tree to predict outcomes. In subsequent decision trees, classifications are
performed for the incorrectly predicted parameters, but with higher weight allocations. Primarily,
the XGB method relies on the effect of the weight, merging several predictions and classifiers to
create an accurate and robust model. The XGB model is particularly effective at mitigating
overfitting problems and consistently delivering superior results. A key similarity between XGB
and gradient boosting is their reliance on the gradient boosting concept, which contributes to
unique modeling features. These various methodologies are each powerful tools within the
machine learning repertoire. However, they all have their unique characteristics and suitable

applications. Understanding these distinctions is crucial in selecting the appropriate model for a
given task. Random forests, for example, excel in dealing with high-dimensional data and are
immune to overfitting due to their ensemble nature. Extremely randomized trees, on the other
hand, offer an even more randomized variant of the decision tree, which can prove beneficial in
certain cases where an additional level of randomness can help improve generalization. Adaptive
boosting, in contrast, works by adjusting weights of the instances in the data set based on the
previous classification. If an instance was incorrectly classified, the model increases the weight
of this instance and the next classifier in the sequence is forced to focus on this instance. This
approach makes AdaBoost very responsive to changes and can therefore quickly adapt to shifting
data patterns. On the other hand, stochastic gradient boosting and histogram-based gradient
boosting are both extensions of the traditional gradient boosting method. Both methods
iteratively add new models into the ensemble and use gradient descent to minimize the loss
function. However, the way they handle data and train models is fundamentally different, which
results in different strengths and weaknesses for these two methods. Lastly, extreme gradient
boosting, as its name suggests, pushes the boosting technique to its limit. It is known for its
efficiency and performance, often outperforming other methods on many machine learning
benchmarks. While it is similar to gradient boosting in that it relies on the concept of boosting, it
comes with several enhancements that reduce overfitting, improve computational speed, and
allow for better model interpretation. It represents one of the most advanced machine learning
techniques available today, capable of handling a wide variety of tasks with high accuracy and
robustness.
2.3. Model development and hyperparameters tunning
During the training process, the hyperparameters are optimized using the grid search approach
with 10 folds cross-validation. As a result, each model was trained extensively to reach an
optimal parameter that provides the best accuracy. Figure 2 illustrates the flowchart and the
models created for this method. Overall, the method starts by defining a set of search parameters
for each model. After that, the dataset is split into training and testing at 70% and 30% ratios,
respectively. Then the training dataset is used to train the model using the k-fold cross-validation
method to overcome the overfitting issue. Once many setups are built, the best model is used to
predict the testing dataset the model performance is captured for later comparison in this study.
The coefficient of determination (𝑅2
) was used to assess the goodness of fit for the linear
regression model, where the numerator of the 𝑅2
fraction depends on the unidentified
dissimilarities by the response of the independent parameters and the denominator of the 𝑅2
fraction depends on the total dissimilarities in the response, Eq. 1 [12]. The range of 𝑅2
where 1
represents the strongest linear relationship and values are between 0 and 1. Therefore, the
measurement tool used for the variations between evaluated and observed values is root-mean-
square error (RMSE), an error analysis method, Eq. 2. Furthermore, the variations between the
absolute evaluated and observed values are the mean absolute error (MAE) a type of error
analysis, Eq. 3.

Fig. 2. Flowchart of the approach used for developing the machine learning models' performance
assessment.
𝑅2
= 1 −
∑ (𝑥𝑖−𝑦𝑖)2
𝑛
𝑖=1
∑ (𝑥𝑖−𝑥̅𝑖)2
𝑛
𝑖=1
(1)
RMSE = √
∑ (𝑥𝑖−𝑦𝑖)2
𝑛
𝑖=1
𝑛
(2)
MAE =
1
𝑛
∑ |𝑦𝑖 − 𝑦
̅𝑖|
𝑛
𝑖=1 (3)
where 𝑥𝑖 is the measured value, 𝑥̅𝑖 is the mean of the measured values, 𝑦𝑖 is the predicted
value, 𝑦
̅𝑖 is the mean of the predicted values, and 𝑛 is the number of observations.

2.4. Feature importance analysis
The inference of the model is an essential element for attaining the areas of enhancing various
models, developing trust among trained algorithms, and formulating strategies for decision-
making. The use of machine learning models has advanced in several ways, including the ability
to do parametric evaluations by assessing the importance of each input characteristic on the
prediction outcomes. Therefore, the parametric evaluation using machine learning features the
importance of analysis, which has been used in many articles. Feature selection and feature
extraction are two key techniques employed in machine learning to identify the most relevant
features and transform them into meaningful representations, respectively. These techniques have
a significant impact on feature importance analysis, which aims to understand the relative
contribution of different features to the predictive power of the model. By incorporating feature
selection or feature extraction into the analysis, it becomes possible to identify the most
influential features that significantly impact the model's performance. This knowledge not only
improves model interpretability but also guides feature engineering efforts and enables better
decision-making regarding feature inclusion or exclusion. In the present study, feature
importance analysis is performed to assess the effect of each input parameter on the moment
capacity of circular spirally reinforced concrete columns. By understanding the importance of
different input variables, it becomes possible to determine their impact on the performance of the
anticipated model, specifically the dependent variable. This analysis provides valuable insights
for developing more accurate and reliable models, enhancing their predictive capabilities, and
formulating effective strategies for decision-making.
3. Results and discussions
The ERT, Ada, GB, and XGB models were found to be the top-performing machine learning
models for predicting maximum moment capacity in RC structural circular columns. It should be
highlighted that these models were exceptionally robust and reliable, showcasing promising
precision in their predictions compared to the rest. They displayed lesser deviations from the
equity line, representing a minimal disparity between their predicted and actual values. In
contrast, the SGD case was the underperformer amongst all the models examined. Its predictions
demonstrated a substantial deviation from the equity line, thus indicating a lack of precision and
reliability.
Table 2 presents the optimized hyperparameters for seven models utilized in the study, namely
DT, RF, ETR, Ada, GB, HGB, and XGB. For the DT model, the 'criterion' hyperparameter is
optimized to 'absolute_error', suggesting that the model minimizes the total absolute error when
making splits. The 'max_depth' is unrestricted, allowing the tree to expand freely, and 'splitter' is
set to 'best', meaning the decision tree utilizes the optimal split at each node. The RF model
utilizes 'squared_error' as the 'criterion' to minimize total squared error when creating splits. The
'max_depth' hyperparameter is capped at 10, limiting the tree's depth to prevent overfitting.
Moreover, the 'n_estimators' hyperparameter is optimized to 250, building a random forest
comprised of 250 decision trees. The ETR model employs similar hyperpar ameters to the RF
model, but the 'max_depth' remains unrestricted and the model includes 100 estimators
('n_estimators': 100). For the AdaBoost model, a DecisionTreeRegressor serves as the

'base_estimator' and the 'learning_rate' is established at 1.0. The 'loss' function employed is
'linear', and the number of weak learners ('n_estimators') is set to 100. The GB model uses 'least
squares' ('ls') as the loss function and a learning rate of 0.05. It also incorporates an elaborate
configuration with 'max_depth': 100, 'min_samples_leaf': 1, 'min_samples_split': 10,
'n_estimators': 200. The 'subsample' is set to 1.0, implying that the entire dataset is utilized for
training each tree. The HGB model applies 'least_squares' as the loss function with an
'l2_regularization' of 0.25 to prevent overfitting. The 'max_bins' hyperparameter is set to 180,
and 'max_depth' remains unrestricted. Lastly, for the XGB model, a 'dart' booster is applied, and
'reg:squarederror' serves as the objective. The 'max_depth' is set to 1, and the 'n_estimators' to
1000. All 'colsample_by*' parameters are set to 1.0, meaning all columns are utilized at every
level, node, and tree. Furthermore, specific regularization parameters, 'estimator__gamma': 2 and
'estimator__lambda': 0, are part of the optimized setup.
Table 2
Optimized hyperparameters for the models developed in this study.
Model Optimized Hyperparameters
DT {'criterion': 'absolute_error', 'max_depth': None, 'random_state': 0, 'splitter': 'best'}
RF {'criterion': 'squared_error', 'max_depth': 10, 'n_estimators': 250, 'random_state': 0}
ETR {'criterion': 'squared_error', 'max_depth': None, 'n_estimators': 100, 'random_state': 0}
Ada
{'base_estimator': DecisionTreeRegressor(), 'learning_rate': 1.0, 'loss': 'linear', 'n_estimators': 100,
'random_state': 0}
GB
{'learning_rate': 0.05, 'loss': 'ls', 'max_depth': 100, 'max_features': None, 'min_samples_leaf': 1,
'min_samples_split': 10, 'n_estimators': 200, 'random_state': 0, 'subsample': 1.0}
HGB
{'l2_regularization': 0.25, 'learning_rate': 0.17, 'loss': 'least_squares', 'max_bins': 180, 'max_depth':
None, 'random_state': 0}
XGB
{'booster': 'dart', 'colsample_bylevel': 1.0, 'colsample_bynode': 1.0, 'colsample_bytree': 1.0,
'estimator__gamma': 2, 'estimator__lambda': 0, 'learning_rate': 1.0, 'max_depth': 1, 'n_estimators': 1000,
'objective': 'reg:squarederror', 'random_state': 0}
Furthermore, an essential observation derived from Figure 3 was the performance difference
between the training and testing datasets. As anticipated, all the models performed better on the
training data, likely due to the direct exposure and adaptation to this dataset during the training
phase. However, the difference between these performances was not substantial enough to
suggest the presence of overfitting, which is a common concern in machine learning practices.
This indicates that all models, including the top performers, were able to generalize their learning
effectively to unseen data, an essential attribute for real-world application.
Although this analysis presents a clear comparative view of the models, it should be noted that
model selection is often dependent on specific criteria or requirements of a given scenario.
Certain situations might favor less computationally expensive models or models that are more
interpretable, despite a small trade-off in prediction accuracy. Therefore, while the ERT, Ada,
GB, and XGB models outperformed others in this study, the choice of machine learning model
for practical application should be a nuanced decision. Further study and validation might also be
required to verify these results in different contexts and for a more diverse range of datasets. As
defined in Eq. 4, the residual analysis is used in this study to calculate the difference between the
observed and predicted values and further investigate the models' performances.
𝑒𝑖 = (𝑦𝑖 − 𝑦
̂𝑖) (4)

Fig. 3. Performance of the developed machine learning models.

Figure 4 displays the residuals of several machine learning models as they correspond to each
observation in the training dataset. The term 'residual' here represents the difference between the
observed and predicted values. From the figure, it's apparent that most residuals are concentrated
around zero, implying a high level of accuracy in predictions for most of the models, excluding
SGD, RF, and HGB. This finding suggests that these models may not perform as well as others
in the given task, as the residuals are larger. Conversely, Figure 5 represents the residual plot for
the test dataset. Unlike the training dataset, the residuals of the test dataset show considerable
scatter for all the models. This is an expected phenomenon as testing data usually contains
unseen scenarios that can be challenging to predict precisely. The DT model shows the highest
negative residual at approximately -300 kN.m, which implies it has the highest under-prediction,
while the XGB model recorded the maximum positive residual, around 300 kN.m, showing the
highest over-prediction. It's crucial to note that both under-predictions and over-predictions can
be problematic depending on the application, and a balance is often desirable. Interestingly, the
HGB model demonstrated a consistent range of residuals between the training and testing
datasets. This is an important observation as it suggests that the HGB model is controlling
overfitting issues effectively. Overfitting occurs when a model learns the training data too well,
leading to poor performance on unseen data.
Fig. 4. Residuals of the developed machine learning models for the training dataset.
Fig. 5. Residuals of the developed machine learning models for the testing dataset.

The ability to produce a similar range of residuals in both datasets indicates that the HGB model
has achieved a good balance between learning the data and generalizing to unseen scenarios.
This is an important quality in a machine learning model and shows the robustness of HGB in
this context. However, these observations should be considered in conjunction with other
performance metrics to get a comprehensive understanding of model performance. The
importance of residual magnitude can vary based on the context and practical implications of
prediction errors in a particular application.
Figures 6 and 7 present the comparison between the predicted values from the machine learning
models and the actual measurements for both the training and testing datasets, respectively. This
comparison aims to give us a deeper understanding of the predictive capabilities of these models
and how well they align with the real-world findings. Continuing the trend from previous
observations, the SGD, RF, and HGB models did not perform satisfactorily in the training
dataset. This was highlighted by the mismatch in the statistical distributions of the predicted and
measured values. Such a divergence suggests a poor fitting of these models to the training data,
leading to unsatisfactory predictions. It could be due to the models not being able to capture the
complexities of the data, or perhaps the models are not suitable for this particular task. Further
investigation would be required to diagnose the exact issue. In the case of the testing dataset, the
XGB model, which previously displayed strong performance, was noted to have two significant
outliers that adversely affected its performance. An 'outlier' in this context means a data point
that deviates significantly from the expected range or pattern. These outliers can be problematic
as they can cause the model to produce predictions that are inconsistent with the overall pattern
of the data. In other words, the presence of such outliers may distort the model's ability to
accurately predict maximum moment capacity in RC structural circular columns. However, it's
worth noting that the presence of outliers is not always an indication of poor model performance.
Sometimes, outliers can arise due to exceptional cases in the data that are not representative of
the typical scenario. Thus, it would be beneficial to investigate the source of these outliers in the
XGB model's predictions and whether they are due to unusual data points or some characteristic
of the model itself.
Fig. 6. Boxplots for the developed machine learning models in the training dataset.

Fig. 7. Boxplots for the developed machine learning models in the testing dataset.
Figure 8 presents a quantitative evaluation of the machine learning models using three metrics
for goodness-of-fit and error assessment. These metrics provide a more objective measure of
model performance, as they capture the degree of agreement between the predicted and actual
values, as well as the magnitude of errors made by the models. As per this evaluation, the XGB
model demonstrated poor performance in the testing dataset due to the presence of two
significant outliers that adversely affected its results. This supports the earlier observation of the
model's vulnerability to extreme cases, which could potentially limit its effectiveness in making
precise predictions in certain situations. Conversely, the SGD and RF models recorded the
poorest overall performance, combining both training and testing datasets. This finding further
substantiates the previous indications of these models' deficiencies in adequately capturing the
intricacies of the data and the task at hand. In contrast, the Histogram-based HGB model was
found to have the most robust performance. It demonstrated exceptional control over overfitting
issues, as evidenced by the consistency in the range of residuals between training and testing
datasets and the corresponding goodness-of-fit metrics. This illustrates the HGB model's
potential as a suitable choice for predicting the maximum moment capacity of RC structural
circular columns.
In general, the ACI 318 provides engineers with the theoretical frameworks and practical design
procedures necessary for creating safe, durable, and efficient structures. One of the core
components of ACI 318 is the approach it offers for estimating the maximum moment capacity
of spiral RC columns. These types of columns, typically characterized by a continuous helical
reinforcement wrapped around the longitudinal reinforcement, are widely recognized for their
high load-bearing capacity and ductility. The ACI 318 approach to estimating the maximum
moment capacity of these columns combines various mechanical properties of the constituent
materials and geometrical characteristics of the column. The calculation procedure is initiated by
determining the axial load capacity (Pn) of the column. This is done by adding the contribution
from the concrete (0.85 f'c Ag) and the steel (fy As), where f'c is the specified compressive
strength of the concrete, Ag is the gross area of the column, fy is the yield strength of the steel,
and As is the area of the steel.

Fig. 8. Goodness of fit and errors of the developed machine learning models.

For spiral RC columns, ACI 318 specifies an enhancement factor due to the confinement
provided by the spiral reinforcement. The concrete strength, f'c, is multiplied by this
enhancement factor. This factor is higher for spirally reinforced columns, acknowledging the
additional confinement and subsequent strength the spiral provides to the concrete core. Next,
the nominal moment capacity (Mn) is calculated using the equation Mn=Pn*(h/2-d'), where h is
the overall dimension of the column, and d' is the distance from the extreme compression fiber to
the centroid of the longitudinal reinforcement. This equation reflects the understanding that
moment is the product of force and distance (lever arm). To ensure the structure's safety under
service loads, ACI 318 requires that the design strength (φPn for axial force and φMn for
moment) must be greater than the factored loads (Pu, Mu), where φ is the strength reduction
factor reflecting the variability of material strengths, workmanship, and the level of strain in the
failure mode. The values of φ are specified in the code based on the type of failure mode, with
lower values for brittle failure modes and higher for ductile modes. To consider the biaxial
bending, ACI 318 introduces the interaction diagram for the column, which is a graphical
representation of the axial load capacity and the moment capacity in two orthogonal directions. It
is a plot of the axial force versus bending moment for various eccentricities. For a specific
combination of axial force and bending moment, the design is considered safe if it falls within
the boundary of the interaction diagram.
Figures 9 and 10 further delve into a performance comparison, where the machine learning
models are benchmarked against the conventional ACI 318 approach. The ACI 318 is a widely
used method for estimating the maximum moment capacity of spiral RC columns.
Comparatively, the machine learning models, with the exception of SGD, achieved superior
performance, indicating their potential for enhanced precision in structural capacity predictions.
The SGD model, however, was the outlier with the highest residual value, reiterating its
unsatisfactory performance in this context. The comparative superiority of machine learning
models over the traditional ACI 318 equation demonstrates the significant potential of machine
learning in enhancing the precision of maximum moment capacity prediction in spiral RC
columns. However, model selection and calibration remain pivotal in realizing this potential, as
the performance can significantly vary based on the characteristics of the model and the data.
Fig. 9. Overall performance of the developed machine learning models against the ACI 318.

Fig. 10. Overall residuals of the developed machine learning models against the ACI 318.
Figure 11 gives an interesting perspective by comparing the statistical distribution of the results
from the machine learning models and the ACI 318 approach. The statistical distribution is an
essential tool for understanding data as it provides information on the variability, central
tendency, and skewness in the data, which are important characteristics for interpreting and
predicting outcomes. The American Concrete Institute's ACI 318 approach, a traditionally
employed method for estimating the maximum moment capacity of RC columns, shows a
significant underestimation of the moment capacity when compared to the measured values.
Such a tendency towards underestimation can be problematic as it might lead to overly
conservative designs and higher costs. Contrarily, the ERT and Ada machine learning models
showed promising results, effectively replicating the overall distribution of the observed values.
This is a desirable quality in predictive models, as it implies these models can better capture the
variability in the data and, therefore, can provide more accurate and reliable predictions. These
findings serve to underline the potential of machine learning models in enhancing the accuracy
of structural capacity predictions. By mimicking the actual distribution of the outcomes more
closely, these models can provide more reliable and cost-effective solutions. However, as always
in machine learning, a single measure such as the statistical distribution should not be the only
criterion for evaluating model performance. Other aspects such as the magnitude of residuals,
handling of outliers, and computational efficiency should also be considered to ensure a
comprehensive assessment.
In general, the success of employed algorithms such as XGB can be attributed to several factors.
Firstly, XGB is known for its strong performance due to its ensemble approach, which combines
multiple weak learners to create a powerful model. XGB leverages gradient boosting, which
sequentially adds new models that focus on the errors of the previous models, leading to better
overall predictions. Moreover, XGB handles complex datasets effectively, thanks to its ability to
handle missing values, feature interactions, and nonlinear relationships. On the other hand, the
weak results observed with SGD could be due to various reasons. SGD is a powerful
optimization algorithm for large-scale datasets, but it may struggle with certain characteristics.

For instance, SGD's performance can suffer when dealing with sparse data or when the learning
rate is not properly tuned. Additionally, SGD might require careful hyperparameter tuning and
regularization to prevent overfitting. Overall, the success of XGB lies in its ensemble approach
and ability to handle complex data, while the weaker results with SGD could be due to issues
related to the dataset or parameter settings.
Fig. 11. Overall reliability of the developed machine learning models against the ACI 318.
Figure 12 delivers an essential perspective on the machine learning models used in this study by
assessing the influence of input parameters on the outcomes of the models. This parameter
evaluation, often referred to as feature importance, is vital as it provides insight into which
factors significantly contribute to the predictive capabilities of the models. From the evaluation,
it appears that concrete compressive strength is the most impactful parameter in the prediction
decision across the machine learning models, indicating its central role in determining the
maximum moment capacity of RC structural circular columns. This is followed by the element
length, which is also deemed significantly important in driving the predictions. On the contrary,
some parameters such as the concrete cover, concrete tensile strength, and the cross-section of
the element exhibited lower relative importance in the prediction process. Although these
parameters are less influential in isolation, it's important to note that they still contribute to the
overall predictive power of the models. In machine learning, even seemingly less important
features can add value by enhancing the models' ability to capture the complexity and diversity
of the data. They may interact with other features in ways that improve the models' accuracy and
ability to generalize to new data. It's also worth mentioning that the importance of a feature can
depend on the specific model being used. Different models might assign different levels of
importance to the same feature, depending on how they learn from the data and make
predictions.

Fig. 12. Relative importance of each input variable on the machine learning model evaluated using the
XGB method.
4. Conclusions
In summary, this study aimed to evaluate the effectiveness and potential of various machine
learning models for predicting the maximum moment capacity of spirally RC columns. The
performance of the developed machine learning models was compared to the conventional ACI
318 method for estimating the maximum moment capacity. Further, a parametric evaluation was
carried out to assess the impact of different input variables on the machine learning models'
outputs. Based on the analysis and observations, the following conclusions can be drawn:
1. The machine learning models demonstrated high accuracy in predicting the maximum
moment capacity of spiral RC columns, as reflected in the R2 testing values. Particularly,
the HGB, RF, and ETR models achieved R2
values of 0.95, 0.96, and 0.95, respectively,
indicating their superior accuracy.
2. The HGB model was identified as the most robust, showing excellent control over
overfitting, high goodness-of-fit, and lower RMSE (62.37) and MAE (46.05) on testing
data compared to other models.
3. The performance of the machine learning models, in general, surpassed that of the
conventional ACI 318 approach. The ACI 318 method tends to underestimate the

maximum moment capacity of spiral RC columns, which could potentially lead to overly
conservative and costly designs.
4. From the feature importance analysis, it was determined that the concrete compressive
strength and element length have the highest influence on the predictions of the machine
learning models. This highlights the significance of these factors in the structural capacity
of RC columns and validates the role of these parameters in the design process.
It's important to note that while these findings provide promising indications of the potential of
machine learning in structural engineering, further research could be beneficial to enhance the
predictive capabilities of these models and to investigate their performance in other scenarios or
with different datasets. This could include exploring different machine learning techniques,
optimizing model parameters, and considering additional input features that could improve the
models' accuracy.
Funding
This research received no external funding.
Conflicts of interest
The authors declare no conflict of interest.
Authors contribution statement
BA: Conceptualization; BA: Formal analysis; HA: Investigation; BA and HA: Methodology;
BA: Writing – original draft; HA: Reviewing/editing –final draft.
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