Simplified deep-learning approach for estimating the ultimate axial load of circular composite columns

Composite columns were preferred over reinforced concrete columns in modern-day construction techniques due to their confinement effect. Different materials were utilized as the outer confining tube and are mainly characterized by their mechanical properties. The main objective of this research is to develop a novel simplified artificial neural network model for the determination of the ultimate axial load of the circular composite columns irrespective of the type of confining tube. A database had been created with the existing experimental results of the composite columns and is employed for training, testing, and validation of the model. A set of composite columns were selected from the real-time experimental study and the ultimate axial load of the columns was determined and validated against the developed model. A user-friendly graphical user interface is created from the proposed model which can help the researchers for anticipating the ultimate axial load of the circular composite columns easily and efficiently.


Introduction
The type of materials used in the buildings was a significant aspect of the structural design of the structures. Reinforced concrete (RC) members were employed as conservative structural members for a long time. Nowadays, composite structural members were gaining more attention in the construction sectors than traditional RC members which offer greater structural efficiency, strength, durability, aesthetic appearance, and faster construction (Wang et al., 2017). Instead of employing the concrete and other components as individual members, utilizing them in the composite form will be more efficient (Han et al., 2014;Liew et al., 2016). The combination of these compressive and tensile members together enhances the structural efficiency concerning strength and stiffness. The confinement can resist compressive failure of the concrete and improves the ductility and resilience of the concrete members. The composite structure is economical as it does not require any type of formwork. Composite members can bear a greater load with a small area of cross-section and it leads to increased floor space. In recent decades, steel and fiber reinforced polymer (FRP) tubes were highly used for confining the concrete and enhancing the overall performance of the members. As these components were provided in the periphery of the cross-section, which enhances the compressive performance significantly. Over the past two decades, numerous experimental studies were reported on concrete-filled composite columns (O'Shea & Bridge, 1994;Schneider, 1998;O'Shea & Bridge, 2000). These composite columns can be employed as effective structural members in high-rise structures, bridge piers, seismic-prone areas, etc., (Kitada, 1998;Sakino & Sun, 2000;Zhou & Liu, 2019). It is reported that composite columns with different cross-sections, different types and grades of concrete were employed (O'Shea & Bridge 1998;Uy, 2001;Han & Yao, 2004;Liu & Gho, 2005;Lue et al., 2007;Yu et al., 2008;Ibañez et al., 2018). It was mentioned that various codal provisions determine the anticipated axial load of composite columns. However, the accuracy of these various codal provisions was different and each codal provision has certain limitations for different materials and their properties.

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FRP tubes were used to prevent both corrosion and local buckling. Compared to the steel tube, the FRP tube has a lighter weight and higher strength-to-weight ratio which makes it an efficient component to be used in seismic-prone regions. Recently, several studies were reported on the different behaviour of the FRP composite columns (Du et al., 2022;Li et al., 2018;Mohamed & Masmoudi, 2010).
Even though experimental studies were the best approach, it is performed with a very limited number of parameter variations and requires more time and cost. To save time and cost, Artificial Intelligence (AI) based prediction models can be employed for getting the required results. In 1943, McCulloch and Pitts were the first to develop simplified neurons for designing neural networks. In recent years, soft computing methods have been gaining attention and have applications in many fields. Instead of spending time, money, and effort on experimental studies, there are several statistical tools for predicting the results of the experiments. The Artificial Neural Network (ANN) is a developing technology in AI used for solving various engineering problems. Among the statistical tools, ANN is highly employed due to its extreme accuracy than the other tools. The ANN method can be used for analyzing the relationships where traditional mathematical methods find difficulties. The ANN works like the process of a biological neural network where the neurons are interlinked and are processed by training, testing, and validation of the data. In the ANN, the learning process can be characterized into (a) Supervised learning which involves the comparison of the target with the output directly, and (b) Unsupervised learning which involves the correlation of the inputs. The ANN method predicts the results (Output) by training the neural networks with the help of the available experimental data (Input). There are some works performed using ANN on structural concrete, structural steel, reinforced concrete (RC) members, shear connectors, seismic performance, fire resistance, the vibration of structures, and structural health monitoring (SHM) (Kaveh & Khaleghi, 2000;Alli et al., 2003;Mansour et al., 2004;Duan et al., 2013;Chojaczyk et al., 2015;Kotsovou et al., 2017;Morfidis & Kostinakis, 2017;Tran-Ngoc et al., 2019;Abambres & Lantsoght, 2020;Charalampakis & Papanikolaou, 2021;Moradi et al., 2021). However, researchers employed ANN in the design and analysis of double-layer grids, transmission lines, and large-scale space structures with back-propagation, counter-propagation, and genetic algorithms, and it was reported that counter-propagation functions faster than back-propagation (Kaveh & Iranmanesh, 1998;Kaveh & Servati, 2001;Kaveh et al., 2008). Some of the recent deep learning approaches employed were Gaussian process regression, support vector regression, grey-wolf optimization, multivariate adaptive regression splines, relevance vector machines, etc., (Yuvaraj et al., 2014;Chithra et al., 2016;Mansouri et al., 2016;Behnood & Golafshani, 2018;Prasanna et al., 2018;Avci-Karatas, 2019;Ngo et al., 2021;Le, 2022).
More research studies on the rectangular composite sections were reported. But, studies on the circular composite sections were a lack in number. Generally, the circular crosssection is preferred over any other cross sections as it bears the greater axial load and the circular confinement offers higher post-yield ductility. Axial compression is a significant behaviour that is involved with crucial parameters such as geometrical and mechanical aspects. Although several works using the ANN were reported, studies on the prediction of the axial load capacity of composite columns were very limited. The research works on ANNs with these circular composite columns was rarely reported. The primary aim of this study is to create a simplified ANN model from which the ultimate axial load of circular composite columns can be predicted. Several experimental data were extracted from the reported literature and are trained, tested, and validated. A novel Graphical User Interface (GUI) model can be developed for computing the ultimate axial load of the composite columns irrespective of the parameters such as type, diameter, height, thickness, and tensile strength of composite material and compressive strength of concrete. Real-time experiments were conducted with the composite columns under axial compressive loading and the results were validated with the developed model. Further, this model can be employed for estimating the ultimate axial load of the circular composite columns.

Development of database
Several research works were reported in investigating the axial compressive performance of the circular composite columns, especially concrete-filled steel tubular (CFST) and concrete-filled FRP tubular (CFFT) columns. A database of 255 composite columns is developed with these two types of composite columns. Out of which, 135 numbers were steel composite columns and 120 numbers were FRP composite columns. The selected columns were circular composite columns with different diameters and heights. The typical sectional view of the composite column is presented in Fig. 1. The crucial parameters include the outer diameter (D) and height (H) of the column, the thickness (t) and tensile strength of the outer confinement tube (f u ), the compressive strength of the in-filled concrete (f c ), and axial load capacity of the columns (P u ) were extracted from various experiments. The statistical data of these parameters were provided in Table 1 and the histogram for the distribution of statistical data is presented in Fig. 2.

ANN and their architecture
The ANN and its architecture, as shown in Fig. 3, are brain-inspired algorithms used to identify problems and depict complex patterns. The concept of neural networks made up of neurons in the human brain facilitated the emergence of the ANN, a computational intelligence approach. ANN was formulated to replicate the functionality of the brain in humans and the ANN algorithm only accepts structured and mathematical data. Despite their distinctive characteristics, the functions of ANN and biological neural networks are functionally equivalent.

Input, output, and elements of ANN
The input parameters used are outer diameter (D) and height (H) of the column, thickness (t), and tensile strength (f u ) of the outer confinement tube, column, and compressive strength of unconfined concrete (f c ) and output parameter is the axial load (P u ) of the composite columns.
In the input layer, the domain provides unprocessed inputs and the layer does not perform any calculations. The nodes in the input layer automatically transfer data to the hidden layer. The nodes of the hidden layer are not visible and operate as an interpretation for the neural network. The hidden layer processes the variables received through the input layer in different methods, and the results are obtained at the output layer. As a result, the output layer of the network resembles the information gathered through the hidden layer and offers the ultimate target value.
The activation function employed by the hidden layer is often the same however, in contrast to the hidden layers, the output layer often employs a distinct activation function. The decision is based on the objective or nature of the computation of the model. The Tangent-Sigmoidal (tansig) function is employed as the activation function of the hidden layer as shown in Fig. 4a, which is identical to the sigmoid or logistic function, and includes the same S-shape with the variation in the output range of − 1 to 1. The tangential value Tanh's reaches 1 if the input is higher (+), and − 1 if the input is lower (−). As illustrated in Fig. 4b, the purelin function is implemented as the activation function of the output layer and is corresponding to the input as a linear activation function, also known as "no activation". The function returns the given data and does not have any reflections on the weighted sum of the input.

Development of ANN model
The fundamental objective of this study is to use ANN approaches to create a model that anticipates the ultimate axial load of composite columns. An ANN-based model is created using the general-purpose soft-computing tool, MATLAB R2018a software, which is described in more detail. For modeling ANN, a total of 255 experimental datasets were used, which consist of 135 datasets of steel composite columns and 120 datasets of FRP composite columns.
The Levenberg-Marquardt (LM) algorithm was applied to train the network. The LM Algorithm uses Gauss-Newton and Gradient descent methods for curve fitting, which could be more accurate and faster than other algorithms.   This hybrid technique is frequently used to operate over the ideal characteristics of various algorithms to address a multidimensional complex problem. The LM algorithm has comparatively enhanced optimization performance and incorporates the principle of the neural neighborhood to improve the behaviour of both memory and time limitations. It can handle systems with a large number of unknown model parameters and hardly generate an optimal answer, even if the initial assumption is unreliable. The data were divided randomly into three sections by default using the algorithm 70, 15, and 15% for training, validation, and testing. The hidden layer activation function is tansig, which produces data values between [-1, + 1]. The activation function of the output layer is thought to be purelin, a linear transfer function that preserves the input, and both layers are characterized by weight and bias as shown in Fig. 5. The design parameters for the proposed ANN model are given in Table 2.

Training, validation, and testing of ANN model
About 75% of the overall data are utilized to train the ANN, with the remaining being utilized to test and validate the created model. The training phase gets converged after about 74 epochs or iterations for the axial load capacity of composite columns (Pu). In this study, the "Mean Squared Error" (MSE) criterion was taken into consideration to assess correlations of training, testing, and validation data to assess the accuracy of a neural network structure. The performance of the proposed model throughout the training, testing, and validation process is presented in Fig. 6. It shows during further training epochs, the error often decreases, but when the network starts to overfit the training data, the error may begin to rise on the validation part. The greatest performance is extracted from the epoch with the minimum validation error when using the default configuration, which terminates training after reaching six continuous rises in a validation error. The greatest validation performance is 2312.8013 at epoch 68. The mean square  The regression plots of training, validation, testing, and overall process are presented in Fig. 8 and it illustrates an ideal model. The Regression (R) results obtained for training, validation, and testing were 0.99575, 0.99416, and 0.98081 correspondingly. The optimum regression for the overall process is 0.99379, which shows good performance in training, validating, and testing the proposed ANN model.

ANN-based formulation
In considering the results from the preceding section, the proposed ANN model can determine the ultimate axial load of composite columns extremely well. The developed ANN model can be utilized to create an explicit empirical formulation that simplifies the process to employ it in the practical design due to the difficulties of analytical solutions. By utilizing their activation functions and parameters (weights and biases), the created ANN model immediately yielded the explicit formulation of the ultimate axial load. The target value of the ultimate axial load (P u ), which was based on the suggested ANN model, was a function of the parameters taken into consideration and is given by the expression where n = 7 (hidden layer neuron); W 0 , b 0 = 0.0446 ; are the weights (7 x 1) and bias of the output layer; W h , b h are the weights (7 x 5) and bias (7 x 1) of the hidden layer; x i are the   Fig. 6 Performance of the proposed model inputs (5 x 1) parameter and those data were represented by the following

Interactive graphical user interface
Nowadays, structural designers prioritize the creation of software that is more reliable and user-friendly, expanding its scope of use. The system created for this study has undergone a lot of work to make sure it is helpful and applicable. Moreover, for ease of use, a GUI tool has been implemented in MATLAB has been included for convenience for users who have preinstalled MATLAB software and a standalone GUI application has been created for users who don't have MATLAB software. The primary user interface, shown in Fig. 9 is straightforward to use. Good computer software encourages the user to request the necessary parameters, as illustrated in Fig. 9. The outer diameter (D) and height (H) of the composite column, thickness (t) and tensile strength (f y ) of the outer confinement tube, and compressive strength of unconfined concrete (f c ) can all be entered numerically by users. Finally, the ultimate axial load of composite columns can be obtained as the output immediately by clicking the PREDICT option.

Real-time experimental validation
The developed ANN model is validated with real-time experiments for better accuracy. Apart from the database created with the existing literature, some of the circular composite columns were selected from the real-time experiments from which the axial load capacity is calculated. The geometry and parameters extracted from the real-time experimental study were employed only for the validation of the developed model which does not require any consideration in selecting the parameters. The circular composite columns were prepared and tested subjected to axial compressive loading and the ultimate axial load of the specimens was determined. The real-time experimental data were analyzed and validated with the results obtained from the proposed model.

Test specimens, setup, and loading
Twenty-one composite columns were fabricated and tested under axial compressive load. Two types of composite columns viz., steel and fiber reinforced polymer (FRP) composite columns were employed in the experimental study. The columns used for the real-time experimental study were presented in Fig. 10. The axial compressive load is applied to the columns with the help of the hydraulic jack placed with the help of the loading frame and the experimental test setup is shown in Fig. 11. The axial load capacity of the composite columns under axial compression loading is calculated. The geometry and mechanical properties of the columns were given as the inputs in the developed GUI and the ultimate axial load of the columns was predicted. The geometry and mechanical properties  Table 3. The ultimate axial load of the columns obtained from the real-time experiments and predicted values were compared as shown in Fig. 12. The graph shows that the predicted load capacity was quite comparable with the experimental results with a linear regression R 2 value of 0.9885. It shows that the developed ANN along with the GUI model is also in good agreement with the real-time experimental data. Therefore, the model can be employed for anticipating the ultimate axial load of the composite columns effectively.

Conclusion
The primary aim of the research is to design a simplified ANN model to determine the ultimate axial load capacity of the composite columns. The designed model is based on the experimental database created with the reported literature. An empirical model along with the ANN model is proposed to obtain the best fit with the database. From this presented study, the following research outcomes can be derived. • The proposed model is trained with various numbers of trials, however, the ANN model with 7 neurons in  with the existing and real-time experimental results and can be used for determining the axial load of the composite columns. • The developed model is analyzed and validated with the real-time experimental study with the composite columns with a percentage error of less than 1%. Hence, the accuracy of the model is good enough for anticipating the axial load of the composite columns with negligible errors. • A novel GUI model is created to simplify the determination of the axial load of the circular composite columns. The efficiency of the developed GUI model in anticipating the axial load for both reported and real-time experimental studies are significant with R 2 = 0.9885. • This developed ANN model, empirical equation and GUI platform can estimate the axial load of the circular composite columns irrespective of the type, diameter, height, and thickness of the confining material and compressive strength of the unconfined concrete.
Correspondingly, the proposed model has some limitations, which are addressed. The developed model is limited as it can be employed only for composite columns with a circular cross-section. Future studies are required for the development of similar models for composite columns with different cross-sections. The model can perform better if the new input data and values which are within the limits of the trained input data. However, the performance of the developed model is acceptable; the accuracy can still be improved by training the model with larger data. Training larger datasets may be time-consuming and requires more memory. More attention is needed in optimizing the design parameters, as these neural network models were computationally more intensive. In further studies, a simplified and reliable software system can be developed from the GUI model which can be used by engineers and researchers for practical applications.