A prediction model for flexural strength of corroded prestressed concrete beam using artificial neural network

Abstract

The prestressed concrete structures are taking the forefront in recent years due to the innovations in the construction industry. However, corrosion is one of the barriers to the serviceability of the prestressed structures. Therefore, a detailed investigation of the prestressed concrete structure under a corrosive environment is essential. This study uses Resilient Back Propagation with BackTracking Neural Network (RBPBTNN) to estimate the flexural strength of the corroded prestressed concrete beam. Three RBPBTNN-based prediction models are proposed to predict the ultimate load, ultimate moment and deflection. The datasets involving multiple influencing parameters are collected from experimentally verified literature. The best possible RMSE and R² values obtained during the training phase for ultimate load prediction are 3.2834 and 0.9964 and for ultimate moment prediction are 2.6128 and 0.9987 and for deflection prediction are 0.8252 and 0.9992 when K-fold cross-validation is three and training repetition is ten. The final performance measures (MAE, R², RMSE etc) of the prediction results are presented in comparison with other artificial neural network algorithms and it is found that the proposed models are the best fit for the collected datasets.

Keywords:

• Artificial neural network
• Prediction model
• Prestressed concrete
• Corrosion
• Flexural strength
• Parametric analysis
• Correlation
• Deflection
• Ultimate load

PUBLIC INTEREST STATEMENT

The prestressed concrete elements are most widely used in civil infrastructures in recent years. The high-strength steels are commonly used in these elements to transfer high-stress to the concrete. But, the steel corrosion is one of the barriers to the durability of reinforced and prestressed concrete structures in construction industry. This in turn reduces the load carrying capacity of the concrete structure. It is quite intuitive that such structures exposed to marine environments are more prone to the corrosion attacks and there is a high risk of sudden failure of structures without prior indication. Therefore, this study can be effectively used to know the strength of corroded prestressed beams in advance using Artificial Intelligence (AI) techniques and to adopt proper safety measures before the failure of the structure to improve its service life.

1. Introduction

Corrosion of steel is one of the major causes of deterioration of the reinforced concrete structures (Angst, 2018). The major consequences of corrosion deterioration have been identified as concrete spalling, delamination, cracking, reduced serviceability and ultimate resistance (Nasser et al., 2022; Pedrosa & Andrade, 2017). The internal pressure produced by the deposition of corrosion products breaks the bond between steel and concrete (Gao et al., 2021; Li & Yuan, 2013). This adverse effect would be severe in the prestressed concrete structure compared to the normal concrete since steel in the prestressed concrete has to handle very high stress. Researchers are extending their support to understand the reason behind corrosion-led structural degradation and reduce early-age sudden failure of the structure. The collapse of the structures with no prior indication causes economic and life losses. Therefore, it is necessary to properly understand the strength degradation with respect to time in prestressed structures for its safety and better serviceability. The effect of strand corrosion on bearing capacity, flexural capacity and bond strength should be thoroughly investigated.

The results obtained from the experiments to understand the mechanical properties of corroded prestressing steel and prestressed beam show that the corrosion directly reduces the cross-sectional area, ductility and tension strength of the steel (Jeon et al., 2019; Vecchi et al., 2021; Wang et al., 2020). The application of tensile stress in the prestressing steel creates more molecular gaps in the strand and increases the possibility of stress corrosion. Due to this inevitable stress, the stress corrosion cracking has been identified as the primary cause of the failure of prestressed steel. It was observed from the literature (Belletti et al., 2020; Dai et al., 2019) that brittle failure occurs due to the reduction in ductility of the steel. As the corrosion rate increases load carrying capacity, flexural strength, shear strength and deflection of the corroded prestressed beam were found to be reduced.

The practitioners have been directed towards the experimental evaluation of the corrosion in the prestressed beam in recent years. The experiment-based research (Rinaldi et al., 2010; Zhang, Wang, Zhang, Ma et al., 2017) was focused on the evaluation of corrosion behavior by artificially corroded prestressed beam through accelerated corrosion and a few case studies (Papãľ & Melchers, 2011; Vecchi et al., 2021) were focused on the degradation by the natural corrosion process. The corrosion of strands highly affects the overall behavior of simply supported beams in terms of load carrying capacity and failure mode when subjected to bending. The beam collapse in bending occurs due to the concrete crushing in the non-corroded beam. But, in the corroded beam, collapse in bending occurs due to the strand rupturing when the beam reaches 14%-20% mass loss. However, a mass loss of 7% can also reduce the ductility of the strand and change the failure mode of the beam (Rinaldi et al., 2010). Reduction of the ductile behavior fractures the strand due to the cross-sectional loss. Both corrosion degree and fracture location affect the flexural strength of the beam (Dai, Chen, Wang, Ma et al., 2021). Using experimental evidence, analytical (both simulation-based and prediction-based) and numerical studies (Dai et al., 2019, 2016) were conducted and proved to be equally important in understanding the corrosion behavior.

The analytical study is broadly categorized into simulation-based and prediction-based studies. The prediction-based advanced techniques focus on analyzing and predicting the strength of structures through experimental evidence. In particular, these advanced techniques were used to predict the behavior of structural elements such as columns, beams etc by exploring the behavior of corrosion via the rate of change of corrosion to give a hint and take precautionary measures to safeguard the structures. The soft computing techniques have been substituting the efforts and the time required to analyze the structural strength and durability. The Artificial Neural Network (ANN) is one of the prominent techniques that can be effectively used in such behavioral studies.

The ANN is originally inspired by the biological neural network. The Neural Network (NN) is a type of computational intelligence technique to simulate the human brain and nervous system. ANN is a simplified multi-layered self-adjustable weighted graph data structure. ANN consists of an interconnected network of processing elements or nodes or neurons which maps input to the desired output. The network is represented in terms of a series of layers such as input, hidden and output. The input information is fed into the input layer, then transferred to the output layer through the hidden layers. The interconnection of any two layers involves weights and these weights are randomly assigned to the nodes at the beginning of the prediction process. In addition to the weights, each layer involves a numerical bias value for individual neurons. The weighted average of the weight and bias values is calculated and passed through a special function called activation function to get the desired output.

The prediction-based analysis using ANN has now become a platform for the prediction of hardened properties of concrete, such as compression strength (Huang et al., 2022; Moradi et al., 2021; Ofuyatan et al., (2022), and durability of concrete (Amiri & Hatami, 2022). ANN was used in the prediction-based analysis of structural elements such as predicting the compression strength of axially loaded concrete-filled steel tube column (Bardhan et al., 2022), glass fibre reinforced concrete column (Karimipour et al., 2021) and flexural strength estimation of slabs (Congro et al., 2021). Collecting the pullout test results, the bond strength of concrete was predicted using the Levenberg-Marquardt algorithm for corroded deformed bars embedded in concrete (Ahmadi et al., 2021) and the bond strength of Fibre Reinforced Polymer (FRP) concrete at high temperature was predicted using Back Propagation Neural Network (BPNN; Huang et al., 2022). Erdem (Erdem, 2010) used the ANN to predict the flexural strength of the Reinforced Concrete (RC) slab in the fire. Cai et al. (Cai et al., 2020) proposed and compared the ANN models to predict the post-fire residual flexural strength of RC beam using BPNN and Genetic Algorithm BPNN (GA-BPNN) where GA-BPNN had shown the best fitting ability and reproducibility with less error. Mansour et al. (Mansour et al., 2004) used Multi-layer Back Propagation Neural Network (MBNN) to predict the ultimate shear strength of RC beam with lateral reinforcements. It was concluded that the predicted results from the ANN model are very much near to the actual value than the values obtained from equations of ACI and CSA building codes. Abuodeh et al. (Abuodeh et al., 2020) created a model to predict the shear strength of shear deficient RC beam strengthened using FRP sheets for shear by Resilient Back Propagation Neural Network (RBPNN). Fu and Feng (Fu & Feng, 2021) developed the machine learning model for predicting the residual shear capacity of corroded reinforced concrete beams at different times of service span using Gradient Boosting Regression Tree (GBRT).

On similar lines, Hung et al. (Nguyen et al., 2021) investigated to find the best ANN model to predict the shear strength of FRP reinforced concrete beams containing flexural and shear reinforcement. As a result, the best ANN model is finalized with the trial study with few records. This study confirmed model accuracy and stability by doing sensitivity analysis using 500 Monte Carlo simulations to predict the accurate shear strength. Alabduljabbar et al. (Alabduljabbar et al., 2020) formulated a model using BPNN with a single layer containing 12 neurons and conducted a non-linear finite element analysis of corroded beam using ABAQUS with ANN output. The need for prediction-based analysis has been confirmed again through a good agreement between the experiment-based and software-based analysis results.

Gene Expression Programming (GEP), one more promising and well-equipped counterpart of ANN was used by Murad et al. (Murad et al., 2021) to develop the empirical relation to find the flexural behavior of concrete beams reinforced with FRP bars. The validated results from both calculated values using the equation from ACI-440-17 and the CSA S806–12 guidelines and experimental results confirmed the superiority of prediction accuracy.

BPNN and its variants give promising results to the problems involving multiple influencing parameters. The summary of the most recent developments in prediction-based analysis studies has been given in Tables 1-2. However, most studies were concentrated on predicting the flexural capacity of RC beam. No substantial work on predicting the flexural capacity of the corroded prestressed concrete beams to date. This study mainly focuses on the prediction-based analysis of flexural strength of corroded prestressed concrete beams using ANN technique to fill this gap.

Table 1. Comparison of existing ANN models used in RC beam analysis

Scheme	Model	Input Size	Training,Testing Size (in %)	I-H-O Layers	I-H-O Neurons
Congro et al. (Congro et al., 2021)	^1BRNN	400	70,30	1-1-1	6–42-3
Fu and Feng (Fu & Feng, 2021)	GBRT	158	70,30	–	–
Murad et al. (Murad et al., 2021)	GEP	116	75,25	–	–
Hung et al. (Nguyen et al., 2021)	BPNN,BRNN	125	70,30	1-1-1	9–17–1
Cai et al. (Cai et al., 2020)	GA-BPNN	480	75,25	–	7-10-1
Abuodeh et al. (Abuodeh et al., 2020)	RBPNN	120	70,30	1-2-1	15–17,7-1
Alabduljabbar et al. (Alabduljabbar et al., 2020)	MBNN	107	90,10	1-1-1	6-12-1
Erdem (Erdem, 2010)	BPNN	294	70,30	1-1-1	7-8-1
Mansour et al. (Mansour et al., 2004)	MBNN	176	80,20	1-1-1	9-3-1

¹BRNN = Bayesian Regularization Neural Network

Table 2. Comparison of existing ANN models used in RC beam analysis (Continued)

Scheme	Epoch Count	Input Dataset	Normalization	Activation Function	Cost Function	Output Parameters
Congro et al. (Congro et al., 2021)	1000	literature	–	hyperbolic tangent, sigmoid	R^2 = 0.9372	three flexural strengths
Fu and Feng (Fu & Feng, 2021)	–	literature	–	—	R^2 = 0.955, RMSE = 19.19, MAE = 12.84	shear strength
Murad et al. (Murad et al., 2021)	–	literature	–	—	R^2 = 0.96, RMSE = 11.2%, MAE = 8.6%	flexural strength
Hung et al. (Nguyen et al., 2021)	1000	literature	–	sigmoid	R^2 = 0.9577, RMSE = 19.97, MAE = 14.98	shear strength
Cai et al. (Cai et al., 2020)	1000	ABAQUS	min-max	sigmoid	R^2 = 0.99617 RMSE = 4.7	post fire flexural capacity
Abuodeh et al. (Abuodeh et al., 2020)	–	–	min-max	sigmoid	R^2 = 0.885 RMSE = 8.1	shear strength
Alabduljabbar et al. (Alabduljabbar et al., 2020)	–	literature	–	sigmoid	R = 0.99934	compression strength, steel reinforcement area in corroded beam
Erdem (Erdem, 2010)	20	simulation	–	sigmoid	R ${ }^2$=0.9975, MSE = 0.00017	moment capacity of slab
Mansour et al. (Mansour et al., 2004)	45,000	ABAQUS	sigmoid	sigmoid	—	shear strength

R = correlation coefficient, R² = coefficient of determination

2. Methodology

The advent of sophisticated ANN-enabled tools such as OriginPro, TensorFlow, R etc has outreached the interdisciplinary applications, including modeling and predicting the compression strength, flexural strength, shear strength, structural crack and bond strength. It is evident that ANN provides an effective solution to complex structural problems and outperforms its counterparts. ANN is making a significant impact in terms of accuracy, speed and convergence (Mozaffari et al., 2019) that would not have been possible with experiment-based measurements.

The neural network generally consists of i) a single input layer that receives the input information, ii) one or more hidden layers which take the information from the input layer and undergo optimization, iii) one or more output layers that produce the desired output. The complete workflow of the proposed methodology is given in Figure 1. The input data is first fed to the data preprocessing phase, where undefined and outlier records are redefined through the process of normalization. The cleaned records are then split as training and testing datasets in the data splitting phase. The most crucial phases in the prediction process are model selection, training and testing. As there is no rule-of-thumb to deterministically choose a generic model that can best predict the output for any given input dataset, in the model selection phase, a random ANN model is selected and trained. A new model will be selected for training if the chosen model underperforms. This process repeats until finding the best fit model. After trial and error combinations of different ANN algorithms, Resilient Back Propagation with Back Tracking NN (RBPBTNN) has been outreached with its best predictions. Therefore, RBPBTNN is selected as the prediction algorithm in this work. During the RBPBTNN training phase, the proposed model fits the training data so that it should get the optimal prediction output. The proposed model repeatedly runs on top of the training data, learns the mistake and corrects it through the back-propagation technique. During the RBPBTNN testing phase, the validity of the predicted results is checked using testing data. If the model cannot give the best possible output, the parameters are varied and the training phase will be repeated, or an entirely different RBPBTNN model will be developed and proceeded with a new training process.

Figure 1. The workflow of the proposed methodology.

2.1. Data preprocessing and slicing

The accuracy of the prediction directly depends upon the volume and veracity of the input data. Therefore, a relatively large experimental dataset for the present study was collected from various literature (Belletti et al., 2020; Dai et al., 2020; Dai, Chen, Wang, Ma et al., 2021, 2021; Dai et al., 2016; ElBatanouny et al., 2015; Jeon et al., 2020; Liu & Fan, 2019; Lu et al., 2021; Menoufy & Soudki, 2014; Moawad, El-Karmoty et al., 2018; Moawad, Mahmoud et al., 2018; Papãľ & Melchers, 2011; Rinaldi et al., 2010; Saraswathy et al., 2017; Zeng et al., 2010; Zhang, Wang, Zhang, Liu et al., 2017). The primary reason for choosing the input dataset from the literature is that the collected data have strong experimental details with a four-point bending test on a corroded prestressed concrete beam.

The collected input data consists of three datasets. The first dataset ($\mathit{Datase}{t}_1$) consists of 185 rows with 15 input parameters (breadth ($\mathit{B}$) in $\mathit{mm}$, depth ($\mathit{D}$) in $\mathit{mm}$, total area ($A_c$) of the section in $\mathit{m}{m}^2$, cover for prestressing steel ($\mathit{c}$) in $\mathit{mm}$, effective depth of section ($d_p$) in $\mathit{mm}$, ratio of cover for prestressing steel to effective depth of section ($\mathit{c}/d_p$), span of beam ($\mathit{L}$) in $\mathit{mm}$, diameter of prestressing steel ($D_p$) in $\mathit{mm}$, total area of prestressing steel ($A_p$) in $\mathit{m}{m}^2$, ultimate strength of prestressing steel ($f_{\mathit{pu}}$) in $\mathit{N}/\mathit{m}{m}^2$, prestress in steel ($f_{\mathit{pr}}$) in $\mathit{N}/\mathit{m}{m}^2$, total area of non-prestressed normal steel ($A_s$) in $\mathit{m}{m}^2$, strength of normal steel ($f_y$) in $\mathit{N}/\mathit{m}{m}^2$, strength of concrete ($f_{\mathit{ck}}$) in $\mathit{N}/\mathit{m}{m}^2$ and percentage of corrosion (${\eta }_{\mathit{corr}}$)) and one output parameter (ultimate load ($P_u$) in $\mathit{kN}$). The second dataset ($\mathit{Datase}{t}_2$) consists of 111 rows with 16 input parameters that are taken from the first dataset and one output parameter (ultimate moment ($M_u$) in $\mathit{kNm}$). The third dataset ($\mathit{Datase}{t}_3$) consists of 111 rows with 16 input parameters that are taken from the first dataset and one output parameter (deflection ($\mathrm{\Delta }$) in $\mathit{mm}$). In addition, the corrosion percentage recorded in all three datasets is with respect to the mass loss of the prestressed steel embedded in the concrete beam. The sample dataset content of all the datasets is shown in Table 3.

Table 3. The input datasets taken from various literatures (Belletti et al., 2020; Dai et al., 2020; Dai, Chen, Wang, Ma et al., 2021, 2021; Dai et al., 2016; ElBatanouny et al., 2015; Jeon et al., 2020; Liu & Fan, 2019; Lu et al., 2021; Menoufy & Soudki, 2014; Moawad, El-Karmoty et al., 2018; Moawad, Mahmoud et al., 2018; Papãľ & Melchers, 2011; Rinaldi et al., 2010; Saraswathy et al., 2017; Zeng et al., 2010; Zhang, Wang, Zhang, Liu et al., 2017)

$\mathit{B}$	$\mathit{D}$	${\mathit{A}}_{\mathit{c}}$	$\mathit{c}$	${\mathit{d}}_{\mathit{p}}$	$\mathit{c}/{\mathit{d}}_{\mathit{p}}$	$\mathit{L}$	${\mathit{D}}_{\mathit{p}}$	${\mathit{A}}_{\mathit{p}}$	${\mathit{f}}_{\mathit{pu}}$	${\mathit{f}}_{\mathit{pr}}$	${\mathit{A}}_{\mathit{s}}$	${\mathit{f}}_{\mathit{y}}$	${\mathit{f}}_{\mathit{ck}}$	${\eta }_{\mathit{corr}}$	${\mathit{P}}_{\mathit{u}}$	Δ
600	680	194,000	85	595	0.14	12,660	50	3926.9	1670	1252.5	240.3	450	34	7	112	–
600	680	194,000	85	595	0.14	12,660	50	3926.9	1670	1252.5	240.3	450	34	76.5	57	–
150	150	22,500	55	95	0.57	1000	7	76.97	1470	1410	314.15	450	40	20	90.6	–
150	150	22,500	55	95	0.57	1000	7	76.97	1470	1410	314.15	450	40	25	80.9	–
200	300	60,000	45	255	0.18	3000	12	279	1976	1300	314.15	450	34	20	115	–
$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$
$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$
150	300	45,000	62	238	0.26	2600	15.2	139	1950	1014	628.3	435	44	5.8	276.7	–
150	300	45,000	62	238	0.26	2600	15.2	139	1950	994.5	628.3	435	44	0	211.5	–
130	150	19,500	42	108	0.39	2000	15.2	139	1910	1395	157.07	400	43.9	9.41	48	–
130	150	19,500	42	108	0.39	2000	15.2	139	1910	0	157.07	400	41.6	10.45	28.3	–
$\mathit{B}$	$\mathit{D}$	${\mathit{A}}_{\mathit{c}}$	$\mathit{c}$	${\mathit{d}}_{\mathit{p}}$	$\mathit{c}/{\mathit{d}}_{\mathit{p}}$	$\mathit{L}$	${\mathit{D}}_{\mathit{p}}$	${\mathit{A}}_{\mathit{p}}$	${\mathit{f}}_{\mathit{pu}}$	${\mathit{f}}_{\mathit{pr}}$	${\mathit{A}}_{\mathit{s}}$	${\mathit{f}}_{\mathit{y}}$	${\mathit{f}}_{\mathit{ck}}$	${\eta }_{\mathit{corr}}$	${\mathit{P}}_{\mathit{u}}$	${\mathit{M}}_{\mathit{u}}$
150	220	33,000	52	168	0.3	2000	15.2	139	1830	1395	326.7	335	34.1	0	126	37.8
150	220	33,000	52	168	0.3	2000	15.2	139	1830	1395	326.7	335	32.4	12.1	110	33
130	150	19,500	42	108	0.39	2000	15.2	139	1910	0	157.07	400	41.6	10.45	28.3	8.4
130	150	19,500	42	108	0.39	2000	15.2	139	1910	943	157.07	400	43.4	12.48	33.5	10.2
$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$
$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$
150	220	45,000	40	180	0.22	2500	15.2	139	1910	1200	327	290	31.8	0	65	37.8
150	220	45,000	40	180	0.22	2500	15.2	139	1910	1200	327	290	31.8	2.7	55	33
150	220	45,000	40	180	0.22	2500	15.2	139	1910	1200	327	290	31.8	6.2	52.5	31.5
150	220	45,000	40	180	0.22	2500	15.2	139	1910	1200	327	290	31.8	9.1	48	28.8
$\mathit{B}$	$\mathit{D}$	${\mathit{A}}_{\mathit{c}}$	$\mathit{c}$	${\mathit{d}}_{\mathit{p}}$	$\mathit{c}/{\mathit{d}}_{\mathit{p}}$	$\mathit{L}$	${\mathit{D}}_{\mathit{p}}$	${\mathit{A}}_{\mathit{p}}$	${\mathit{f}}_{\mathit{pu}}$	${\mathit{f}}_{\mathit{pr}}$	${\mathit{A}}_{\mathit{s}}$	${\mathit{f}}_{\mathit{y}}$	${\mathit{f}}_{\mathit{ck}}$	${\eta }_{\mathit{corr}}$	${\mathit{P}}_{\mathit{u}}$	$\mathrm{\Delta }$
600	680	194,000	85	595	0.14	12,660	50	3926.9	1670	1252.5	240.3	450	34	44.5	78	52
150	150	22,500	55	95	0.57	1000	7	76.97	1470	1410	314.15	450	40	0	120.6	4.8
150	150	22,500	55	95	0.57	1000	7	76.97	1470	1410	314.15	450	40	15	98.5	6.3
150	150	22,500	55	95	0.57	1000	7	76.97	1470	1410	314.15	450	40	25	80.9	5.2
200	300	60,000	45	255	0.18	3000	12	279	1976	1300	314.15	450	34	20	95	7
$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$
$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$	$\cdot$
400	300	45,000	50	250	0.2	3600	12.7	95.7	1860	1302	100.53	435	40	10	48.3	33.34
130	150	19,500	42	108	0.39	2000	15.2	139	1910	465	157.07	400	43.5	7.47	42	22
130	150	19,500	42	108	0.39	2000	15.2	139	1910	1395	157.07	400	43.9	9.41	48	18
610	381	92,720	35	346	0.1	4980	13	200	1860	1264.8	314.15	435	40.7	6.3	101	86.1

The correlation plots of the input v/s output parameters are shown in Figure 2. The ultimate load in $\mathit{Datase}{t}_1$ has a reasonable correlation with all the input parameters. The maximum positive correlation is 0.5 and the maximum negative correlation is −0.001. 75% of the input parameters are positively correlated and the remaining 25% are negatively correlated as shown in Figure 2a. The ultimate moment in $\mathit{Datase}{t}_2$ has a partial mix of correlation with the input parameters. The maximum positive and negative correlations are 0.9 and −0.07 respectively. 82% of the input parameters are positively correlated and the remaining 18% are negatively correlated as shown in Figure 2b. The ultimate deflection in $\mathit{Datase}{t}_3$ has a partial mix of correlation with the input parameters. The maximum positive and negative correlations are 0.7 and −0.08 respectively. 69% of the input parameters are positively correlated and the remaining 31% are negatively correlated as shown in Figure 2c. The positively correlated parameters highly influence each other and the relationship is always directly proportional. The negatively correlated parameters are less influencing each other and the relationship is always inversely proportional.

Figure 2. Correlation plots of input parameters v/s output parameters.

It is also observed from the marginal plot of Figure 3 that there are very few outliers in all three input datasets. Specifically, one outlier as shown in Figure 3a, four outliers as shown in Figure 3b, two outliers as shown in Figure 3c and three outliers as shown in Figure 3f, three outliers as shown in Figure 3d, three outliers as shown in Figure 3e. The input analysis through marginal plots confirms that the collected input positively contributes towards accurate predictions of ultimate load, ultimate moment and deflection as there are very few outliers. Even these outliers could be removed from the datasets to further increase the prediction accuracy through normalization. However, normalization could be helpful in scenarios where the input data ranges are highly scattered. But, it is intuitive from the statistical measurements (as shown in Tables 4 and 5) of the input datasets considered in this work that there is no input normalization required as the data ranges are well-organized. In addition to that, the standard deviation lights on the consistency of the input data i.e., a higher value implies that the input data is less consistent and a lower value implies that the input data is more consistent. Out of all the input datasets, the minimum standard deviation is seven and the maximum is 24.8, which is quite acceptable, as shown in Table 4.

Figure 3. Marginal histograms between corrosion, compressive strength and output parameters.

Table 4. The statistical measurements of input datasets

Parameter	Dataset1					Dataset2					Dataset3
	Min	Max	Mean	Median	SD	Min	Max	Mean	Median	SD	Min	Max	Mean	Median	SD
$\mathit{B}$	80	610	178.7	150	106.4	80	600	157.2	150	88.5	130	610	205.3	150	129.4
$\mathit{D}$	150	680	271.5	250	87.2	150	680	256.9	240	93	150	680	292.7	300	104.2
$A_c$	19,500	194,000	46,470.2	41,235	26,687.5	19,500	194,000	42,922.3	37,500	30,298.4	19,500	194,000	52,194.6	45,000	32,946.6
$\mathit{c}$	25	85	49.2	50	12.5	40	85	46.9	43	10	25	85	53.1	52	14.5
$d_p$	95	595	222.3	210	84.4	108	595	210	200	85.3	95	595	239.6	238	102.5
$\mathit{c}/d_p$	0.07	0.57	0.2	0.236	0.1	0.14	0.39	0.2	0.22	0.1	0.07	0.57	0.3	0.26	0.1
$\mathit{L}$	1000	12,660	2941.9	2500	1682.7	2000	12,660	2860	2500	1968.8	1000	12,660	3286.1	2600	2100.2
$D_p$	7	50	15	15.2	5.5	12.7	50	16	15.2	6.7	7	50	15.2	15.2	7
$A_p$	76.97	3926.9	231.5	139	552.5	95.7	3926.9	280.3	139	708.9	76.97	3926.9	292.7	139	707.5
$f_{\mathit{pu}}$	622.83	1976	1797.1	1910	308.8	622.83	1951.5	1743.2	1910	380.5	1470	1976	1853.7	1910	124.5
$f_{\mathit{pr}}$	0	1432.5	1136.2	1250	313.9	0	1408	1122.7	1250	314	0	1432.5	1147.4	1300	336.2
$A_s$	39.3	986	384.1	327	178.4	39.3	986	368.5	327	170.7	39.3	838.8	368.9	326.7	204.7
$f_y$	235	470	379.6	375	61.5	235	450	358.4	360	59.4	335	470	410.1	435	47.3
$f_{\mathit{ck}}$	31.8	87	42.4	43.5	8.6	31.8	55.7	41.9	44	7	31.8	87	42.4	41.7	9.1
${\eta }_{\mathit{corr}}$	0	100	15.9	7.47	22.1	0	100	16.7	7	24.8	0	100	16.6	9.41	20.8
$P_u$	16.4	276.7	99.6	93.4	54.5	16.5	276.7	82.4	79	45.2	16.4	276.7	118	111.36	59.8
$M_u$	—	—	—	—	—	8.4	490	54.7	42.03	71.3	—	—	—	—	—
$\mathrm{\Delta }$	—	—	—	—	—	—	—	—	—	—	4.66	120.7	31	20.2	28.6

SD = Standard Deviation

Table 5. The statistical measurements of input datasets (Continued …)

Parameter	Dataset1			Dataset2			Dataset3
	Variance	Skewness	Kurtosis	Variance	Skewness	Kurtosis	Variance	Skewness	Kurtosis
$\mathit{B}$	11,323.5	3.3	10.5	7833.4	4.6	21.0	15,453.0	2.5	5.1
$\mathit{D}$	7596.0	2.3	8.5	8646.9	3.4	3.6	10,849.7	1.7	4.9
$A_c$	712,223,310.3	3.8	18.3	917,993,658.1	4.5	6.0	1,085,480,806.9	2.9	10.4
$\mathit{c}$	156.6	0.7	0.4	99.5	2.2	2.1	208.8	0.0	−0.3
$d_p$	7116.3	2.0	6.5	7274.2	3.3	3.4	10,497.9	1.4	3.2
$\mathit{c}/d_p$	0.0	0.9	1.8	0.0	0.9	1.0	0.0	0.5	0.3
$\mathit{L}$	2,831,562.5	4.3	22.1	3,876,030.9	4.6	6.1	4,410,851.2	3.2	12.3
$D_p$	29.9	5.7	34.6	44.9	4.8	6.7	49.1	4.5	20.4
$A_p$	305,301.2	6.6	41.9	502,604.2	5.0	7.2	500,503.7	5.0	23.7
$f_{\mathit{pu}}$	95,368.4	−2.9	7.7	144,792.5	−2.2	−2.2	15,504.9	−1.8	2.5
$f_{\mathit{pr}}$	98,541.0	−1.5	1.5	98,603.8	−1.6	−1.6	113,000.0	−1.5	1.6
$A_s$	31,832.7	0.5	0.4	29,135.1	0.3	0.3	41,900.3	0.4	−0.5
$f_y$	3781.5	−0.6	0.0	3524.3	−0.6	−0.7	2240.0	−0.6	−1.0
$f_{\mathit{ck}}$	73.4	1.9	8.1	49.6	0.1	0.1	82.6	2.6	11.3
${\eta }_{\mathit{corr}}$	487.5	2.2	4.8	613.3	2.0	2.0	431.9	2.3	5.5
$P_u$	2970.6	1	0.8	2039.6	1.6	1.6	3578.0	0.6	−0.3
$M_u$	—	—	—	5088	4.8	4.3	—	—	–
$\mathrm{\Delta }$	—	—	—	—	—	–	820.6	1.7	2.1

2.2. ANN Model selection and parameter setting

After rigorous trial and error efforts, it is observed that one input layer, two hidden layers and one output layer model is giving the best possible result for the chosen datasets. The proposed RBPBTNN models are shown in Figure 4. There are 15–13-13-1 neurons in the first model as shown in Figure 4a. All fifteen input neurons (labeled as {$i_1,\cdot \cdot \cdot ,i_{15}$}) in Figure 4a takes the input from the corresponding input parameters and a single neuron (labeled as $o_1$) gives the output for the corresponding output parameter of $\mathit{Datase}{t}_1$. There are 16–15-15-1 neurons in the second model as shown in Figure 4b. All sixteen input neurons (labeled as {$i_1,\cdot \cdot \cdot ,i_{16}$}) in Figure 4b take the input from the corresponding input parameters and a single neuron (labeled as $o_2$) gives the output for the corresponding output parameter of $\mathit{Datase}{t}_2$. There are 16–16-16-1 neurons in the third model as shown in Figure 4c. All sixteen input neurons (labeled as {$i_1,\cdot \cdot \cdot ,i_{16}$}) in Figure 4c take the input from the corresponding input parameters and a single neuron (labeled as $o_3$) gives the output for the corresponding output parameter of $\mathit{Datase}{t}_3$.

Figure 4. The proposed RBPBTNN models for ultimate load, ultimate moment and deflection predictions.

(1) $\begin{array}{c}{o}_1=f_3\mathbf{(}\sum _{\mathit{m}=1}^{13}(f_2\mathbf{(}\sum _{l=1,n_2=1}^{\mathit{l}=13,n_2=13}(f_1\mathbf{(}\sum _{k=1,n_1=1}^{\mathit{k}=15,n_1=13}(i_k\cdot w_{k,n_1}^1+b_{n_1}^1)\mathbf{)}\cdot w_{l,n_2}^2+b_{n_2}^2)\mathbf{)}\cdot w_{m,1}^3+b_1^3)\mathbf{)}\end{array}$(1)

(2) $\begin{array}{c}{o}_2=f_3\mathbf{(}\sum _{\mathit{m}=1}^{15}(f_2\mathbf{(}\sum _{l=1,n_2=1}^{\mathit{l}=15,n_2=15}(f_1\mathbf{(}\sum _{k=1,n_1=1}^{\mathit{k}=16,n_1=15}(i_k\cdot w_{k,n_1}^1+b_{n_1}^1)\mathbf{)}\cdot w_{l,n_2}^2+b_{n_2}^2)\mathbf{)}\cdot w_{m,1}^3+b_1^3)\mathbf{)}\end{array}$(2)

(3) $\begin{array}{c}{o}_3=f_3\mathbf{(}\sum _{\mathit{m}=1}^{16}(f_2\mathbf{(}\sum _{l=1,n_2=1}^{\mathit{l}=16,n_2=16}(f_1\mathbf{(}\sum _{k=1,n_1=1}^{\mathit{k}=16,n_1=16}(i_k\cdot w_{k,n_1}^1+b_{n_1}^1)\mathbf{)}\cdot w_{l,n_2}^2+b_{n_2}^2)\mathbf{)}\cdot w_{m,1}^3+b_1^3)\mathbf{)}\end{array}$(3)

The final ANN model formula for predicting ultimate load ($P_u$), ultimate moment ($M_u$) and deflection ($\mathrm{\Delta }$) are given in equations (1)(1) $\begin{array}{c}{o}_1=f_3\mathbf{(}\sum _{\mathit{m}=1}^{13}(f_2\mathbf{(}\sum _{l=1,n_2=1}^{\mathit{l}=13,n_2=13}(f_1\mathbf{(}\sum _{k=1,n_1=1}^{\mathit{k}=15,n_1=13}(i_k\cdot w_{k,n_1}^1+b_{n_1}^1)\mathbf{)}\cdot w_{l,n_2}^2+b_{n_2}^2)\mathbf{)}\cdot w_{m,1}^3+b_1^3)\mathbf{)}\end{array}$(1) , (2(2) $\begin{array}{c}{o}_2=f_3\mathbf{(}\sum _{\mathit{m}=1}^{15}(f_2\mathbf{(}\sum _{l=1,n_2=1}^{\mathit{l}=15,n_2=15}(f_1\mathbf{(}\sum _{k=1,n_1=1}^{\mathit{k}=16,n_1=15}(i_k\cdot w_{k,n_1}^1+b_{n_1}^1)\mathbf{)}\cdot w_{l,n_2}^2+b_{n_2}^2)\mathbf{)}\cdot w_{m,1}^3+b_1^3)\mathbf{)}\end{array}$(2) ) and (3(3) $\begin{array}{c}{o}_3=f_3\mathbf{(}\sum _{\mathit{m}=1}^{16}(f_2\mathbf{(}\sum _{l=1,n_2=1}^{\mathit{l}=16,n_2=16}(f_1\mathbf{(}\sum _{k=1,n_1=1}^{\mathit{k}=16,n_1=16}(i_k\cdot w_{k,n_1}^1+b_{n_1}^1)\mathbf{)}\cdot w_{l,n_2}^2+b_{n_2}^2)\mathbf{)}\cdot w_{m,1}^3+b_1^3)\mathbf{)}\end{array}$(3) ) respectively where $i_{(\cdot )}$ is the neuron in the input layer, $w_{(\cdot ,\cdot )}^1$ is the input weight obtained for the neuron in the first hidden layer, $b_{(\cdot )}^1$ is the bias value obtained for the neuron in the first hidden layer, $w_{(\cdot ,\cdot )}^2$ is the input weight obtained for the neuron in the second hidden layer, $b_{(\cdot )}^2$ is the bias value obtained for the neuron in the second hidden layer, $w_{(\cdot ,\cdot )}^3$ is the input weight obtained for the neuron in the output layer, $b_1^3$ is the bias value obtained for the neuron in the output layer, $f_1$ is the activation function chosen for the first hidden layer, $f_2$ is the activation function chosen for the second hidden layer, $f_3$ is the activation function chosen for the output layer. After careful observation, the non-linear sigmoid function is chosen as the activation function for all the layers due to its superior performance over other activation functions. In this work, RBPBTNN is used to predict the output. The RBPBTNN uses the back propagation technique as its underlying architecture to approach the predefined error threshold set during the training phase.

2.3. Training and testing phases of RBPBTNN models

To fit a suitable ANN model that can predict the desired output, the model must undergo the training and testing phases to verify the result. It is customary to undergo training and testing phases on the same dataset before model deployment. Out of 185 records in $\mathit{Datase}{t}_1$ 155 records are selected for training and 30 records are selected for testing. Similarly, out of 111 records in $\mathit{Datase}{t}_2$ and $\mathit{Datase}{t}_3$, 91 records are selected for training and 20 records are selected for testing.

(4) $w_{(\cdot ,\cdot )}^1=\left(\begin{array}{ccccccc}1.30992& -1.5839& -0.39769& 0.48123& -1.1176& -1.05594& -0.54475\\ -1.90566& -3.93345& 0.34399& -2.03816& -0.18183& 1.23973& 1.3158\\ 5.66979& -4.70606& 2.97171& -3.72645& 5.83402& 6.7959& 15.72736\\ 0.77997& 0.34399& -2.5251& 1.40746& -0.15095& -1.87773& 0.24168\\ 9.17055& 3.26496& -0.32375& -1.90122& 1.67796& -0.2432& -12.2661\\ 1.12895& -0.4175& 2.01342& 1.29713& 0.62186& 0.45937& 0.39033\\ -1.28121& -1.00664& -1.37445& 1.35928& -1.34199& -0.192& 1.12198\\ 16.81078& -6.72747& -16.28797& 4.22142& -18.90289& 10.3331& -4.22435\\ 0.77584& 0.84723& -2.0819& 0.37399& -0.67302& -0.12092& 0.56637\\ -1.57615& -0.11786& -2.10989& -1.30529& -0.9048& -0.06708& -3.73422\\ 2.66313& 9.70148& 3.35554& -16.96176& 3.81534& -2.38273& -22.82482\\ 0.77064& -1.78709& 5.3207& -2.42165& -2.0367& -1.2342& -0.32596\\ 0.91391& 1.12046& 10.22896& 5.86671& -1.44571& -1.82679& 0.77484& \\ & \\ 2.20028& -9.37774& -1.40738& 1.40215& -0.31097& -1.05071& 1.93255& 0.17216\\ -2.00351& 6.97531& 0.32502& 1.52513& -3.99141& 0.71871& 4.66946& -3.54923\\ -66.04053& 25.67755& -0.11045& 2.43695& -10.23629& 0.90246& -11.33339& 235.09079\\ -2.47328& -1.0298& 1.0218& 0.24362& 0.11456& -0.25231& 4.22702& -0.72003\\ -6.71739& 32.52107& -0.36051& -1.12637& -0.88871& -5.06193& 3.53102& 46.64729\\ -1.56898& 18.6299& -1.11593& -0.52669& 1.02498& 0.39179& 0.92653& -1.82053\\ 2.40169& 2.48293& -0.79173& 1.64984& -0.50634& -0.57168& -5.7725& 0.24828\\ 5.92588& 64.89871& -4.38999& 1.96824& 0.70994& 1.24509& -39.29272& 199.34378\\ -1.75233& 4.80928& -0.35157& 2.05106& -1.85844& 2.85875& -5.70238& 1.41944\\ -1.32028& 1.62222& 0.82683& -0.93681& 2.79943& -0.88489& 5.60198& 3.64272\\ 17.96092& 56.77938& 2.34511& -1.08836& 0.2219& -3.80802& 12.94041& -41.09473\\ 0.17891& 18.89546& 1.72297& 1.35919& -6.68294& 2.68622& 3.92186& -0.91205\\ 1.04509& -16.89487& 0.09089& 2.27893& 5.47514& -0.95731& -30.58632& -17.04009\end{array}\right)$(4)

(5) $w_{(\cdot ,\cdot )}^2=\left(\begin{array}{cccccc}2.11069& 23.70965& -5.76198& -3.06317& 9.73949& 1.60436\\ -36.51321& 21.53749& 14.99468& -1.85766& 71.61841& -5.73144\\ 0.7547& -30.12395& -0.58873& -1.40339& 1.84403& -0.36363\\ 1.945& 31.60827& 2.11581& 1.80219& -1.86411& -0.46265\\ -0.46248& -0.71731& 1.07132& 2.05614& -0.78846& -1.07576\\ -1.28062& 5.08976& -1.87779& 1.17828& -2.07474& 1.61319\\ 2.1222& -17.20002& 12.85168& 1.5006& 0.6063& 5.49119\\ 1.56092& 0.4958& 0.38526& 0.09224& 1.01326& -1.96098\\ 4.65766& -4.96137& 1.36211& -2.73373& 1.62469& -0.09773\\ -0.4& 1.15281& 0.83194& 0.39147& -2.44754& -0.79563\\ -5.40278& 2.70667& -1.55204& 2.29849& -0.99888& 2.11014\\ 5.10934& 2.32425& -1.1329& 1.19736& -2.54067& 0.06161\\ -1.31719& 2.02427& 0.93841& -0.44607& -0.93408& -0.40481\\ & \\ -1.22808& -2.43016& 1.67529& 1.32585& 0.70034& 5.31192& -4.05278\\ 46.59451& 24.73337& -13.90927& 1.30693& -1.52389& -2.85341& 71.38682\\ 2.44984& -1.34388& -2.38671& 2.6656& -2.74437& 1.50519& 1.46934\\ 0.36913& 5.42838& 4.23289& 5.17908& -9.44664& -1.25863& -16.57862\\ 0.54252& -0.49786& -1.12254& 0.27869& -1.11998& 0.79762& -1.2724\\ -1.23879& 0.7644& 0.97791& -1.46245& 0.27955& -0.41602& -0.47034\\ 15.10773& -3.95178& 0.41674& 1.11819& -7.69363& 1.42879& -5.69715\\ 6.00488& -1.05436& -0.33356& 1.36382& -2.28949& -2.54949& -0.57757\\ 0.92576& -0.41635& -2.46074& 4.58575& 1.54803& 0.56819& -1.63445\\ 0.07506& -1.52416& -0.19431& -0.8608& -4.84294& 3.55029& -0.27515\\ -2.49591& -0.65393& -0.26597& 1.17861& -0.90914& -0.1847& 0.0866\\ 16.33326& 0.27861& -3.68937& 0.96461& 4.57526& 0.23389& -3.76296\\ -0.00267& -1.25438& 0.49692& -1.09028& 1.65491& 2.53344& -0.84461\end{array}\right)$(5)

(6) $w_{(\cdot ,\cdot )}^3=\left(\begin{array}{ccccccc}-1.33416& 0.86465& -5.14146& 1.49686& -1.88219& 2.01111& 1.77028\\ 3.21918& -2.54102& -4.53495& 3.35738& -1.29584& -1.40553\end{array}\right)$(6)

(7) $b_{(\cdot )}^1=\left(\begin{array}{ccccccc}0.27305& -1.41205& 0.1963& 0.33746& 1.07578& 0.09499& -0.81948\\ 1.15584& -1.27872& -0.6825& 3.35239& -0.37516& -0.29382\end{array}\right)$(7)

(8) $b_{(\cdot )}^2=\left(\begin{array}{ccccccc}0.77393& -1.00083& -0.46346& 1.74818& -0.62493& -1.10065& 0.8151\\ -0.92072& -0.02454& 0.95348& -0.17408& 0.55036& 0.86819\end{array}\right)$(8)

(9) $b_1^3=\left(\begin{array}{c}0.79104\end{array}\right)$(9)

(10) $w(⋅,⋅)1={−0.83904   −0.4668   −0.1275  −1.35966−0.17314   4.12816   1.08754  0.26599} { - 0.41077} { - 0.66629}  {0.14606}  {0.28521}  {0.14594}  {1.11745}  {1.19824}  {0.04723} {0.91755}  {1.05278}  { - 0.86482} {2.37025}  {1.57505}  {0.34128}  {0.47231}  { - 0.67525}  { - 0.04715}  { - 0.41875}  { - 0.60966}  { - 0.78333}  { - 0.00944}  {1.01826}  {0.06116}  { - 0.04913}  {0.4394}  { - 0.31717}  { - 0.30178}  {0.67699} {2.14725}  {0.55201}  {1.89373}  {1.51734}  {0.02051} { - 0.76306}  {1.00279}  { - 1.19664}  {0.92671} { - 3.63338}  {2.0601}  { - 0.66844}  { - 0.41424}  {0.69464} { - 2.34222} {0.33093}  {0.24612}  {0.29222}  { - 1.00437}  { - 0.78162} \cr {1.162}  {1.03436}  {0.10232}  { - 0.0958} { - 2.18586} {1.26314}  { - 0.0398}  {1.4391}  { - 0.21284}  {0.79627}  { - 1.43682}  {1.30004} {1.10916}  {0.38164}  {1.12503}  { - 0.28343}  { - 0.21533} {0.19939}  { - 1.47652}  { - 1.14073}  { - 0.98744}  {7.25014}  {0.28202}  { - 0.17928}  { - 5.24027}  { - 0.38823}  { - 1.83789}  { - 1.98681}  { - 1.106}  {0.48563}  { - 3.0643}  { - 1.2470}  {0.89903}  { - 1.89851}  { - 1.67625}  {0.71788}  { - 0.3604}  {3.45135}  { - 0.74641}  {0.52137}  {0.04466}  {0.14019}  {1.83167}  { - 0.43589}  {0.11807}  {2.12444}  { - 0.65322}  {0.13794}  {0.09839}  { - 0.56892}  { - 1.56063}  {0.24056}  { - 0.84421}  {0.61661}  {0.02872}  {0.15364}  {0.74405}  {0.2099}  {0.24994} { - 0.2644}  { - 0.48205}  { - 0.101}  { - 0.38999}  { - 5.05511}  { - 0.80105}  {1.77585}  {0.10866}  {1.99103}  {0.461}  {0.6449}  {2.73432}  {0.2076} { - 1.03126}  { - 0.58883}  { - 2.16194}  {1.13097} {0.09368}  {0.98256}  {3.40859} { - 0.6302}  { - 1.2133}  {0.9812}  {0.5811}  { - 1.82554} { - 0.75726}  { - 2.97865}  { - 2.21977}  { - 1.86608}  { - 0.17607}  {0.96013}  {1.17861}  {0.07406}  { - 0.70105}  {1.10303}  {1.57107}  { - 0.69183}  {0.35886}  { - 0.4054}  {0.88868}  {0.98643}  { - 0.01568}  {0.5568}  {0.62846}  {1.21517}  { - 1.13068}  { - 0.51091} { - 0.82153}  {2.05191}  { - 1.48851}  { - 1.30267}  {1.97931}  { - 1.0423}  {1.10984}  { - 1.3657}  { - 1.13177}  { - 0.08294} {0.6403} { - 0.36425} {1.80606}  { - 0.77355} {0.27444} { - 1.03036}  { - 0.40822}  { - 1.04999}  { - 1.17554}  {0.22025}  {1.22176}  { - 10.14312}  { - 1.05844}  {1.48761}  { - 0.01603} { - 2.23382}{ - 1.19395} 0.64841−2.71699−0.57154−2.18444​  1.4162  1.01739  0.2223  2.06442  −0.23303  2.20916  −2.78824  −3.00025  −0.45204  0.34514   −0.62596    0.58722   −0.47617   −0.62647  0.11437  −0.87395  0.66516   −0.35243 −1.16424  −1.00587  −2.30962  0.58455   0.20292  −0.03822  −0.1585  1.00775  −3.36478  1.60167  −0.58609   1.10717  8.88694  −0.54556  −0.07002  0.84144  −0.90605   −0.62703  7.21445   1.56045   0.57815  −2.479  −0.04445  −0.23385  −1.38231  2.24195  5.94E−05  −3.19447  −1.04849  $(10)

(11) $w_{(\cdot ,\cdot )}^2=\left(\begin{array}{ccccccc}0.77927& -0.93042& -1.61545& -2.23769& 2.05285& -1.90017& -0.98172\\ 1.43585& -1.66307& 0.81762& 0.55429& -1.33093& 0.51257& -2.74755\\ -1.23857& -0.7663& 1.09704& -1.06236& 1.5452& -1.77728& -2.64909\\ 1.61435& 0.0557& 0.68046& 0.51635& -0.89948& 2.53111& 4.77568\\ 1.13357& 0.66349& 0.2529& 0.22484& -0.36556& -1.08527& -0.41552\\ 0.89689& -0.81015& -0.97638& -0.41189& 0.81142& -1.55133& 0.58189\\ 1.17915& -3.03655& 6.2742& -1.20797& -0.51025& 14.91106& -2.27254\\ 0.41322& 1.94795& -0.51289& -0.37491& 1.51027& 11.03749& 2.716\\ -1.33221& 0.05741& -0.65416& 0.07573& 1.25784& -2.56009& -2.68572\\ 0.49487& -0.05527& -0.15964& -2.12137& 0.36101& -0.87975& -0.00735\\ -0.35502& -1.68303& 0.15328& -0.67232& 1.99617& -1.07191& 1.33101\\ 0.56645& 1.91273& -1.35198& 2.54815& -2.13396& 1.49682& 1.9449\\ 1.22696& 0.5422& 0.10388& 0.79584& -1.90703& 0.88708& 0.70909\\ 1.37981& 0.10391& -0.05018& -0.78432& -0.82698& -2.90594& 0.64166\\ 0.82895& 0.11146& 0.8172& -1.40912& 0.25884& 6.17555& 1.88983\\ \\ -0.09434& -1.36729& -0.3242& -0.56096& -1.54967& 0.53202& -0.00594& -0.47755\\ -11.57877& 0.66163& 0.20917& -2.04625& 6.11977& -1.03191& -1.9287& -0.55677\\ 0.32712& 0.73273& -2.00732& 1.52575& 0.18964& -0.59629& 0.49693& -0.07595\\ 1.01989& -0.72114& -0.50003& 0.01946& 2.40421& -0.92307& 1.54789& -0.94341\\ 2.2004& 1.21581& -1.29115& 2.19248& 5.39331& -1.13282& 0.4393& 1.84945\\ -1.59232& 0.95001& -0.29196& -2.38994& -1.45781& 1.74623& -2.13302& -1.63847\\ -14.91104& 19.19685& 1.70977& -20.47494& -12.16562& 2.42967& 1.41061& -6.14187\\ 28.38445& 1.50014& -0.12017& -0.41212& 2.46866& 1.94063& -0.47301& -1.54096\\ 0.08169& 0.95401& 0.05759& -1.29862& -1.37865& 0.0623& -1.19949& -0.38027\\ -0.01045& -0.15177& 1.28231& -1.2281& -2.46571& -0.07776& -1.36802& -1.37616\\ -2.64122& 2.84427& -0.58839& -2.29942& 1.05375& -1.21015& -0.4035& -0.11335\\ 0.48289& 0.71051& -1.04224& 0.09281& 1.92561& -1.99942& 0.14936& 1.45026\\ 0.62892& 1.03519& 1.20749& 0.97545& 1.86918& -0.57981& -0.35298& 1.09954\\ 2.86171& 0.4196& 1.03075& -0.55137& 1.25883& -0.01177& -1.13317& 0.49512\\ 21.0504& 0.61268& -0.95174& -0.30819& 2.46435& 0.37148& -0.52247& 1.11265\end{array}\right)$(11)

(12) $w_{(\cdot ,\cdot )}^3=\left(\begin{array}{cccccccc}1.01465& 0.31021& 2.38233& -0.46954& -1.31079& 1.98707& 0.13497& -0.47611\\ 1.9266& 0.48781& 1.78669& -0.4672& -0.63115& -1.21142& -0.27927\end{array}\right)$(12)

(13) $b_{(\cdot )}^1=\left(\begin{array}{ccccccc}-1.27214& 0.74293& 0.47722& -1.02678& -2.52935& -0.41163& 1.08434\\ 0.60001& -0.16287& 1.43962& -0.5693& -0.33109& -1.25747& 0.50508& -0.54384\end{array}\right)$(13)

(14) $b_{(\cdot )}^2=\left(\begin{array}{ccccccc}-1.02719& -0.19057& 0.51144& -0.68846& -1.24663& 0.16532& 2.12435\\ 1.20535& 0.0549& -0.96967& -0.65354& 2.33305& -1.11357& -0.12249& 1.97887\end{array}\right)$(14)

(15) $b_1^3=\left(\begin{array}{c}0.09641\end{array}\right)$(15)

(16) $\begin{array}{l}{w}_{(\cdot ,\cdot )}^1=-1.331790.10099-0.79290.138520.618940.099482.26380.\text{85991}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} - 3.32784\hspace{0.17em} - 1.43881\hspace{0.17em}\hspace{0.17em} - 0.81649\hspace{0.17em}\hspace{0.17em}1.95594\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}- 2.4815\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.97853\hspace{0.17em} - 1.94178\hspace{0.17em}\hspace{0.17em}1.13907}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.73303\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.81025\hspace{0.17em} 1.08221 0.65549 1.39029 - 0.12861 \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}4.1777 \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.22752}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} - 28.4072\hspace{0.17em} 0.90179\hspace{0.17em} - 1.17867\hspace{0.17em} - 1.92273\hspace{0.17em}\hspace{0.17em}1.73645\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.82859\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.04014\hspace{0.17em}\hspace{0.17em} - 0.11736 }\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.75134\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.90E - 04\hspace{0.17em} - 1.21556\hspace{0.17em}\hspace{0.17em} - 4.23925\hspace{0.17em} - 0.16477\hspace{0.17em} - 3.98891\hspace{0.17em} - 0.81004\hspace{0.17em} 0.21072}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}9.64008\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.40005\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}- 0.37421\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.3913\hspace{0.17em}\hspace{0.17em} - 0.42187\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.16283\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.90947\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.83533 }\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.97439\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.1409\hspace{0.17em}\hspace{0.17em} 0.89321 - 0.27058 0.23496 - 14.0501 - 5.04822 0.26384 }\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}- 1.10455\hspace{0.17em}\hspace{0.17em}1.11836\hspace{0.17em}\hspace{0.17em}3.04958\hspace{0.17em}\hspace{0.17em} - 0.64872\hspace{0.17em}\hspace{0.17em} - 0.21521\hspace{0.17em} - 3.43536\hspace{0.17em}\hspace{0.17em}4.85537\hspace{0.17em}\hspace{0.17em}2.06188}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.87054 1.65804 1.84487 2.18874 - 0.28649 1.00369 0.12168 - 1.47679}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.00043 - 0.89692 - 0.50515 - 0.22492 }0.\text{39301 0.71993 - 2.76691 - 0.72273}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}19.58302 - 2.71071 - 0.85304 - 3.83049 - 0.54049 8.61973 7.14381 - 18.47029 }\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.41535 - 0.03217 - 3.18815 - 0.67689 - 0.34534 2.54284 0.91757 - 0.25281 }\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}3.64583 0.42998 1.49889 0.15002 - 0.64925 1.03737 1.56292 - 0.26082}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.27386 - 0.39539 0.63753 - 2.65577 - 1.79411 - 1.04856 - 3.39794 2.80882}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.16864 - 0.98201 0.96485 - 1.48228 0.40666 - 3.09046 3.1948 1.4052}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.47523 2.92985 - 0.28276 0.03705 - 0.38112 - 0.01946 3.24538 - 1.11894}\\ \text{ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}- 8.76559 - 0.43392 0.848 - 0.28104 - 0.41763 - 1.04591 1.52538 - 0.14222}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} 6.44166 - 0.23373 0.019 1.19194 1.36906 0.67513 - 1.7939 0.59181}\\ \text{ - 5.68954\hspace{0.17em}\hspace{0.17em}0.07347 - 0.76518 0.76967 0.11834 1.61887 - 1.26802 - 1.5304}\\ \text{ - 32.71013 - 6.19851 0.33992 3.90874 - 3.9949 13.35755 26.54455 13.58262}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} 2.59871 0.70251 0.18247 - 8.40538 - 1.107 18.87626 5.12558 - 1.08194}\\ \text{ 9.23348 0.44141 0.33614 1.78588 - 0.33505 - 0.81128 - 12.21366 - 5.22149}\\ \text{ 8.85904 - 0.42293 0.83334 8.21911 1.34503 - 0.95163 11.54466 - 0.59667}\\ \text{ 0.98875 4.65484 2.61371 - 8.36332 - 2.64915 0.76087 - 8.38449 2.85161}\\ \text{ - 3.19318 - 0.22069 0.28269 2.89494 - 2.34362 - 0.0631 7.05684 - 8.13611}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} 0.67565 0.05591 - 0.76497 0.90107 - 0.13151 - 0.01567 8.29391 0.52898}\\ \text{ 4.0766 }-3.492270.4214317.272442.14845.68654-18.52478-1.62999-24.142640.96928-1.46442-3.46961.501330.6589-31.59711-0.606324.6557-0.216260.57747-1.28487-0.1956-2.54897-5.55506-1.260592.180590.097280.23798-4.46206-0.04128-5.641212.25751.44850.20147-0.36794-0.63426-2.969820.63273-1.61534-13.2529-0.7659-2.31739-0.10613-\text{1}.\text{64648 }-\text{0}.\text{21075 0}.\text{5386 }-\text{0}.\text{82701 }-\text{17}.\text{02842 0}.\text{4873 }\end{array}$(16)

(17) $w(⋅,⋅)2=1.59444   1.374410.08246−14.3686−11.803350.265251.53291−0.460640.071031.08321−0.9983−0.59131.32194 0.83937 −5.15232 0.41352 −0.02886 4.24844 0.44092 21.52419 1.78021 0.16504 1.35912  −0.24865 0.68712 −3.05583 1.85519 3.4193 42.09814 −16.10189 2.45281 9.18944 −2.18033 3.21015 −2.87406 −2.65924 −1.63166   2.11773  0.2259  0.22556  - 0.02443  - 0.23996  - 0.25933 2.50266 9.79031 0.35729 0.00453 0.74291  - 1.77344 1.43737 0.14838  - 4.38995  - 2.41929  - 1.08705  - 0.47383  - 0.3407 0.91235 6.42264  - 7.91324  2.67979  - 23.07227  - 2.38108  - 0.26728  - 6.6412  2.18116  - 1.97641  0.84271 1.1091  0.69688  - 0.4572  5.85349  0.80908 0.62809  - 0.56724  0.52351 - 0.08814  - 1.46343  - 0.4876  0.01357  1.77592 - 0.21627  0.26697  0.37209  0.00415  1.0432 1.55307 1.99866 - 2.32109  - 0.49835   - 2.08799 4.5515 0.06363 13.35438  - 0.40685 0.86924 0.83136 1.13519  - 0.33177  - 0.57808 5.2195 3.43116  - 0.18598 1.46334 - 0.3403  - 1.27676 1.46988  - 0.90317  - 2.88181 0.49047 1.72472 - 0.73717 0.09032 0.16506  - 3.73556 2.20811  - 2.09712  - 0.37874  - 0.84745 0.25325  - 1.89855  - 0.42056  - 1.73956  - 0.68133  - 3.82544  5.43908  - 0.10337 0.59736  - 0.10614  5.88872 1.16788 0.52967 2.04449 0.17697 1.6034  - 4.88021  - 3.67342  - 3.16443  - 3.80481 0.43883  - 2.77432 0.74847 0.34197 2.29412  - 0.53634 13.61469  - 3.50092 0.83081   - 17.00822  - 1.28632  - 30.07928  - 4.99074 6.17085  - 13.42477 1.11761 5.56685 6.92186  - 0.47691  - 6.472113.76509 6.25211 0.08923 1.92817 0.50941 0.33192 0.59373 14.89096 - 3.01504 0.32747 - 0.33903 0.87293  - 3.52739  - 8.5838 0.43991  - 7.50242  - 3.9938 - 2.14433 2.16659 2.4177 2.7793 4.99061  - 1.15627 8.16648 - 0.17308 1.58318 1.01933  - 0.62937 0.82288 5.49387  - 5.57786 0.8237 8.20826 5.50498 3.90456 0.02137  - 1.57529  - 17.03563  - 0.50824 17.0463 2.84456  - 3.99713  - 0.79834  - 0.64645  - 0.36926 - 0.18217 0.62207  - 0.53813  - 0.72676 0.55994  - 3.0044  - 0.53167 0.46937  - 0.32129   - 2.28853 - 0.02642 0.75698 - 0.13255 - 2.25147 0.43358 0.61629  - 3.17574 −1.51643  −6.78321  10.71132  1.96188  −4.96801  5.79232  −2.74981  −5.35273  −1.08634  0.3329  −2.6128 −1.51295 −6.62437 0.96287 0.349 0.89334 1.4128 0.68246 0.47649 −1.83139 −0.93862 −1.93173 −2.94282 2.47698 1.1675 −0.66479 9.5737 5.81911 70.31665 13.40772 −0.41635 −3.73189 −0.65804 4.74472 −12.32241 −1.64707 $(17)

(18) $w_{(\cdot ,\cdot )}^3=\left(\begin{array}{cccccccc}0.41998& 4.7676& 1.1575& 0.40345& -0.94073& -0.81895& -0.42949& -1.58796\\ -0.62729& 1.37791& -0.77688& 1.29469& -1.37013& -1.36542& 2.58703& -0.85273\end{array}\right)$(18)

(19) $b_{(\cdot )}^1=\left(\begin{array}{cccccccc}-0.68265& -0.28817& -0.91879& -1.28449& -0.1705& 1.71379& 1.14313& -0.28467\\ -1.3088& 1.11838& -0.2343& 1.28489& 1.13781& 1.69541& 0.64284& -1.21441\end{array}\right)$(19)

(20) $b_{(\cdot )}^2=\left(\begin{array}{cccccccc}-1.18978& -0.64785& 2.08001& 0.18852& -0.87108& -0.67752& 1.28692& -0.37449\\ 0.69863& 0.15144& 0.80098& 0.14935& 0.39817& 0.33561& 1.04919& 0.85973\end{array}\right)$(20)

(21) $b_1^3=\left(\begin{array}{c}0.23086\end{array}\right)$(21)

3. Results and discussion

To mitigate the complexity of the prediction process, three separate RBPBTNN models have been proposed as shown in Figure 4a, Figure 4b and Figure 4c. The proposed models are executed in two phases: training and testing. The learning process in the proposed models involve the cost associated with each phase and is measured through various cost functions such as Nash-Sutcliffe Efficiency (NSE), Mean Absolute Error (MAE), R-squared (R ${ }^2$), Mean Absolute Percentage Error (MAPE), Root Mean Square Error (RMSE) as shown in equation (22)(22) $\begin{array}{cc}{\mathrm{R}}^2& =1-\left(\frac{{\sum }_{k=1}^N{(T_k-P_k)}^2}{{\sum }_{k=1}^N{(P_k)}^2}\right)\\ \mathrm{RMSE}& =\sqrt{\frac{{\sum }_{k=1}^N{(T_k-P_k)}^2}{\mathit{N}}}\\ \mathrm{MAE}& =\frac{{\sum }_{k=1}^N|P_k-T_k|}{\mathit{N}}\\ \mathrm{MAPE}& =\frac{{\sum }_{k=1}^N(|\frac{P_k-T_k}{P_k}|\cdot 100)}{\mathit{N}}\\ \mathrm{NSE}& =1-\left(\frac{{\sum }_{k=1}^N{(T_k-P_k)}^2}{{\sum }_{k=1}^N{(T_k-\bar{T})}^2}\right)\\ \mathit{\hspace{0.17em}}& \mathit{\hspace{0.17em}}\end{array}$(22) .

(22) $\begin{array}{cc}{\mathrm{R}}^2& =1-\left(\frac{{\sum }_{k=1}^N{(T_k-P_k)}^2}{{\sum }_{k=1}^N{(P_k)}^2}\right)\\ \mathrm{RMSE}& =\sqrt{\frac{{\sum }_{k=1}^N{(T_k-P_k)}^2}{\mathit{N}}}\\ \mathrm{MAE}& =\frac{{\sum }_{k=1}^N|P_k-T_k|}{\mathit{N}}\\ \mathrm{MAPE}& =\frac{{\sum }_{k=1}^N(|\frac{P_k-T_k}{P_k}|\cdot 100)}{\mathit{N}}\\ \mathrm{NSE}& =1-\left(\frac{{\sum }_{k=1}^N{(T_k-P_k)}^2}{{\sum }_{k=1}^N{(T_k-\bar{T})}^2}\right)\\ \mathit{\hspace{0.17em}}& \mathit{\hspace{0.17em}}\end{array}$(22)

where $P_{(\cdot )}$ is the predicted output, $T_{(\cdot )}$ is the target output, $\bar{T}$ is the mean of target output values.

The proposed models are executed with K-fold cross-validation from 1 to 6 and the respective RMSE and R² values of the training datasets are given in Figure 5a and Figure 5b. One of the most significant performance measures of predictive analysis is RMSE, representing the statistical distance between the target and predicted output. The highest value of RMSE indicates underfit results and the lowest value of RMSE indicates overfit results. One more significant measure is R², which ranges from 0 to 1 and represents the degree of correctness of the predicted output. The highest value of R² indicates the perfect prediction results and the lowest value of R² indicates unfit results. Therefore, the idle prediction model should always exhibit reasonable RMSE and R² values.

Figure 5. The proposed RBPBTNN training performance comparison when K-fold and training repetitions are varied.

It is observed from Figure 5a and Figure 5b that the lowest RMSE and highest R² values are recorded when K-fold is three and training repetition is one. The K-fold cross-validation is a technique of running the training phase (K-fold)-1 times by randomly choosing (1-(1/K-fold)) records for training and remaining (1/K-fold) records for validating the training results. The K-fold cross-validation process will have a direct impact on the prediction results. The RMSE value has been decreasing till K-fold = 3 and starts increasing when K-fold $>3$ in all the proposed models. Conversely, the R² value has been increasing till K-fold = 3 and starts decreasing when K-fold $>3$ in all the proposed models. It is equally essential to run the model with a higher number of repetitions to fit the model. The training process of the proposed models involves 1 to 10 training repetitions.

The proposed models exhibit good performance (i.e., least RMSE and highest R²) when the number of repetitions is ten as shown in Figure 5c and Figure 5d. The parametric details of the execution of the training and validation phases of all the proposed models when K-fold = 3 and training repetition = 10 are shown in Table 6. The final performance details of the proposed models are shown in Table 7.

Table 6. Parametric details of the proposed ANN models during training and validation phases

Parameter	Details
	UltimateLoad	UltimateMoment	Deflection
Input neurons	15	16	16
Output neurons	1	1	1
Hidden layer count	2	2	2
Hidden neurons	13,13	15,15	16,16
Activation function	sigmoid	sigmoid	sigmoid
Training, testing data	155,30	91,20	91,20
RMSE	3.1373	2.1441	0.6795
R ${ }^2$	0.9967	0.9991	0.9995
K-fold	3	3	3
No. of repetition	10	10	10
No. of simulations	1	1	1

sigmoid $\Rightarrow \mathit{f}$ ($\mathit{z}$) = 1/(1 +$e^{-\mathit{z}}$)

Table 7. Performance comparison of the proposed RBPBTNN with other ANN models on testing data

Model	Cost	Ultimate	Ultimate	Deflection
	Function	Load	Moment
BPNN	R^2	0.9668	0.9760	0.8880
	RMSE	19.3695	15.0696	10.7425
	MAE	16.3099	8.7368	6.0040
	MAPE	19.22%	18.94%	32.86%
	NSE	0.8738	0.9523	0.7820
GCM-BPNN	R^2	0.9765	0.9466	0.9090
	RMSE	15.9672	24.8600	10.3240
	MAE	12.0127	9.2977	5.8680
	MAPE	15.82%	13.38%	23.31%
	NSE	0.9142	0.8712	0.7990
RBPWBTNN	R^2	0.9080	0.9272	0.8914
	RMSE	38.3460	30.4650	11.3150
	MAE	23.1595	9.3636	6.2180
	MAPE	39.45%	10.62%	24.10%
	NSE	0.5050	0.8066	0.7570
RBPBTNN	R^2	0.9860	0.9691	0.9180
	RMSE	13.2020	17.7800	10.3060
	MAE	10.5340	5.6900	5.8250
	MAPE	12.96%	7.79%	19.24%
	NSE	0.9414	0.9341	0.8060

GCM-BPNN = Globally Convergent Modification of BPNN

RBPWBTNN = Resilient Back Propogation Without Backtracking Neural Network

There are 15 $\times$ 13 = 195 weights in between the input and first hidden layer in the first proposed model as shown in Figure 4a. The corresponding predicted weight matrix $w_{(\cdot ,\cdot )}^1$ is listed in equation (4)(4) $w_{(\cdot ,\cdot )}^1=\left(\begin{array}{ccccccc}1.30992& -1.5839& -0.39769& 0.48123& -1.1176& -1.05594& -0.54475\\ -1.90566& -3.93345& 0.34399& -2.03816& -0.18183& 1.23973& 1.3158\\ 5.66979& -4.70606& 2.97171& -3.72645& 5.83402& 6.7959& 15.72736\\ 0.77997& 0.34399& -2.5251& 1.40746& -0.15095& -1.87773& 0.24168\\ 9.17055& 3.26496& -0.32375& -1.90122& 1.67796& -0.2432& -12.2661\\ 1.12895& -0.4175& 2.01342& 1.29713& 0.62186& 0.45937& 0.39033\\ -1.28121& -1.00664& -1.37445& 1.35928& -1.34199& -0.192& 1.12198\\ 16.81078& -6.72747& -16.28797& 4.22142& -18.90289& 10.3331& -4.22435\\ 0.77584& 0.84723& -2.0819& 0.37399& -0.67302& -0.12092& 0.56637\\ -1.57615& -0.11786& -2.10989& -1.30529& -0.9048& -0.06708& -3.73422\\ 2.66313& 9.70148& 3.35554& -16.96176& 3.81534& -2.38273& -22.82482\\ 0.77064& -1.78709& 5.3207& -2.42165& -2.0367& -1.2342& -0.32596\\ 0.91391& 1.12046& 10.22896& 5.86671& -1.44571& -1.82679& 0.77484& \\ & \\ 2.20028& -9.37774& -1.40738& 1.40215& -0.31097& -1.05071& 1.93255& 0.17216\\ -2.00351& 6.97531& 0.32502& 1.52513& -3.99141& 0.71871& 4.66946& -3.54923\\ -66.04053& 25.67755& -0.11045& 2.43695& -10.23629& 0.90246& -11.33339& 235.09079\\ -2.47328& -1.0298& 1.0218& 0.24362& 0.11456& -0.25231& 4.22702& -0.72003\\ -6.71739& 32.52107& -0.36051& -1.12637& -0.88871& -5.06193& 3.53102& 46.64729\\ -1.56898& 18.6299& -1.11593& -0.52669& 1.02498& 0.39179& 0.92653& -1.82053\\ 2.40169& 2.48293& -0.79173& 1.64984& -0.50634& -0.57168& -5.7725& 0.24828\\ 5.92588& 64.89871& -4.38999& 1.96824& 0.70994& 1.24509& -39.29272& 199.34378\\ -1.75233& 4.80928& -0.35157& 2.05106& -1.85844& 2.85875& -5.70238& 1.41944\\ -1.32028& 1.62222& 0.82683& -0.93681& 2.79943& -0.88489& 5.60198& 3.64272\\ 17.96092& 56.77938& 2.34511& -1.08836& 0.2219& -3.80802& 12.94041& -41.09473\\ 0.17891& 18.89546& 1.72297& 1.35919& -6.68294& 2.68622& 3.92186& -0.91205\\ 1.04509& -16.89487& 0.09089& 2.27893& 5.47514& -0.95731& -30.58632& -17.04009\end{array}\right)$(4) where each row represents the weights belonging to the input layer neuron. There are 13 $\times$ 13 = 169 weights in between hidden layers. The corresponding predicted weight matrix $w_{(\cdot ,\cdot )}^2$ is listed in equation (5)(5) $w_{(\cdot ,\cdot )}^2=\left(\begin{array}{cccccc}2.11069& 23.70965& -5.76198& -3.06317& 9.73949& 1.60436\\ -36.51321& 21.53749& 14.99468& -1.85766& 71.61841& -5.73144\\ 0.7547& -30.12395& -0.58873& -1.40339& 1.84403& -0.36363\\ 1.945& 31.60827& 2.11581& 1.80219& -1.86411& -0.46265\\ -0.46248& -0.71731& 1.07132& 2.05614& -0.78846& -1.07576\\ -1.28062& 5.08976& -1.87779& 1.17828& -2.07474& 1.61319\\ 2.1222& -17.20002& 12.85168& 1.5006& 0.6063& 5.49119\\ 1.56092& 0.4958& 0.38526& 0.09224& 1.01326& -1.96098\\ 4.65766& -4.96137& 1.36211& -2.73373& 1.62469& -0.09773\\ -0.4& 1.15281& 0.83194& 0.39147& -2.44754& -0.79563\\ -5.40278& 2.70667& -1.55204& 2.29849& -0.99888& 2.11014\\ 5.10934& 2.32425& -1.1329& 1.19736& -2.54067& 0.06161\\ -1.31719& 2.02427& 0.93841& -0.44607& -0.93408& -0.40481\\ & \\ -1.22808& -2.43016& 1.67529& 1.32585& 0.70034& 5.31192& -4.05278\\ 46.59451& 24.73337& -13.90927& 1.30693& -1.52389& -2.85341& 71.38682\\ 2.44984& -1.34388& -2.38671& 2.6656& -2.74437& 1.50519& 1.46934\\ 0.36913& 5.42838& 4.23289& 5.17908& -9.44664& -1.25863& -16.57862\\ 0.54252& -0.49786& -1.12254& 0.27869& -1.11998& 0.79762& -1.2724\\ -1.23879& 0.7644& 0.97791& -1.46245& 0.27955& -0.41602& -0.47034\\ 15.10773& -3.95178& 0.41674& 1.11819& -7.69363& 1.42879& -5.69715\\ 6.00488& -1.05436& -0.33356& 1.36382& -2.28949& -2.54949& -0.57757\\ 0.92576& -0.41635& -2.46074& 4.58575& 1.54803& 0.56819& -1.63445\\ 0.07506& -1.52416& -0.19431& -0.8608& -4.84294& 3.55029& -0.27515\\ -2.49591& -0.65393& -0.26597& 1.17861& -0.90914& -0.1847& 0.0866\\ 16.33326& 0.27861& -3.68937& 0.96461& 4.57526& 0.23389& -3.76296\\ -0.00267& -1.25438& 0.49692& -1.09028& 1.65491& 2.53344& -0.84461\end{array}\right)$(5) where each row represents the weights belonging to the neuron from the first hidden layer. There are 13 $\times$ 1 = 13 weights in between the second hidden layer and the output layer. The corresponding predicted weight matrix $w_{(\cdot ,\cdot )}^3$ is listed in equation (6)(6) $w_{(\cdot ,\cdot )}^3=\left(\begin{array}{ccccccc}-1.33416& 0.86465& -5.14146& 1.49686& -1.88219& 2.01111& 1.77028\\ 3.21918& -2.54102& -4.53495& 3.35738& -1.29584& -1.40553\end{array}\right)$(6) where each row represents the weights belonging to the neuron from the second hidden layer. The predicted bias matrices belongs to each layer are listed in equations (7)(7) $b_{(\cdot )}^1=\left(\begin{array}{ccccccc}0.27305& -1.41205& 0.1963& 0.33746& 1.07578& 0.09499& -0.81948\\ 1.15584& -1.27872& -0.6825& 3.35239& -0.37516& -0.29382\end{array}\right)$(7) , (8(8) $b_{(\cdot )}^2=\left(\begin{array}{ccccccc}0.77393& -1.00083& -0.46346& 1.74818& -0.62493& -1.10065& 0.8151\\ -0.92072& -0.02454& 0.95348& -0.17408& 0.55036& 0.86819\end{array}\right)$(8) ), (9(9) $b_1^3=\left(\begin{array}{c}0.79104\end{array}\right)$(9) ) respectively. Similarly, all the predicted weight matrices and bias matrices of the second proposed model (as shown in Figure 4b) are listed in equations (10)(10) $w(⋅,⋅)1={−0.83904   −0.4668   −0.1275  −1.35966−0.17314   4.12816   1.08754  0.26599} { - 0.41077} { - 0.66629}  {0.14606}  {0.28521}  {0.14594}  {1.11745}  {1.19824}  {0.04723} {0.91755}  {1.05278}  { - 0.86482} {2.37025}  {1.57505}  {0.34128}  {0.47231}  { - 0.67525}  { - 0.04715}  { - 0.41875}  { - 0.60966}  { - 0.78333}  { - 0.00944}  {1.01826}  {0.06116}  { - 0.04913}  {0.4394}  { - 0.31717}  { - 0.30178}  {0.67699} {2.14725}  {0.55201}  {1.89373}  {1.51734}  {0.02051} { - 0.76306}  {1.00279}  { - 1.19664}  {0.92671} { - 3.63338}  {2.0601}  { - 0.66844}  { - 0.41424}  {0.69464} { - 2.34222} {0.33093}  {0.24612}  {0.29222}  { - 1.00437}  { - 0.78162} \cr {1.162}  {1.03436}  {0.10232}  { - 0.0958} { - 2.18586} {1.26314}  { - 0.0398}  {1.4391}  { - 0.21284}  {0.79627}  { - 1.43682}  {1.30004} {1.10916}  {0.38164}  {1.12503}  { - 0.28343}  { - 0.21533} {0.19939}  { - 1.47652}  { - 1.14073}  { - 0.98744}  {7.25014}  {0.28202}  { - 0.17928}  { - 5.24027}  { - 0.38823}  { - 1.83789}  { - 1.98681}  { - 1.106}  {0.48563}  { - 3.0643}  { - 1.2470}  {0.89903}  { - 1.89851}  { - 1.67625}  {0.71788}  { - 0.3604}  {3.45135}  { - 0.74641}  {0.52137}  {0.04466}  {0.14019}  {1.83167}  { - 0.43589}  {0.11807}  {2.12444}  { - 0.65322}  {0.13794}  {0.09839}  { - 0.56892}  { - 1.56063}  {0.24056}  { - 0.84421}  {0.61661}  {0.02872}  {0.15364}  {0.74405}  {0.2099}  {0.24994} { - 0.2644}  { - 0.48205}  { - 0.101}  { - 0.38999}  { - 5.05511}  { - 0.80105}  {1.77585}  {0.10866}  {1.99103}  {0.461}  {0.6449}  {2.73432}  {0.2076} { - 1.03126}  { - 0.58883}  { - 2.16194}  {1.13097} {0.09368}  {0.98256}  {3.40859} { - 0.6302}  { - 1.2133}  {0.9812}  {0.5811}  { - 1.82554} { - 0.75726}  { - 2.97865}  { - 2.21977}  { - 1.86608}  { - 0.17607}  {0.96013}  {1.17861}  {0.07406}  { - 0.70105}  {1.10303}  {1.57107}  { - 0.69183}  {0.35886}  { - 0.4054}  {0.88868}  {0.98643}  { - 0.01568}  {0.5568}  {0.62846}  {1.21517}  { - 1.13068}  { - 0.51091} { - 0.82153}  {2.05191}  { - 1.48851}  { - 1.30267}  {1.97931}  { - 1.0423}  {1.10984}  { - 1.3657}  { - 1.13177}  { - 0.08294} {0.6403} { - 0.36425} {1.80606}  { - 0.77355} {0.27444} { - 1.03036}  { - 0.40822}  { - 1.04999}  { - 1.17554}  {0.22025}  {1.22176}  { - 10.14312}  { - 1.05844}  {1.48761}  { - 0.01603} { - 2.23382}{ - 1.19395} 0.64841−2.71699−0.57154−2.18444​  1.4162  1.01739  0.2223  2.06442  −0.23303  2.20916  −2.78824  −3.00025  −0.45204  0.34514   −0.62596    0.58722   −0.47617   −0.62647  0.11437  −0.87395  0.66516   −0.35243 −1.16424  −1.00587  −2.30962  0.58455   0.20292  −0.03822  −0.1585  1.00775  −3.36478  1.60167  −0.58609   1.10717  8.88694  −0.54556  −0.07002  0.84144  −0.90605   −0.62703  7.21445   1.56045   0.57815  −2.479  −0.04445  −0.23385  −1.38231  2.24195  5.94E−05  −3.19447  −1.04849  $(10) , (11(11) $w_{(\cdot ,\cdot )}^2=\left(\begin{array}{ccccccc}0.77927& -0.93042& -1.61545& -2.23769& 2.05285& -1.90017& -0.98172\\ 1.43585& -1.66307& 0.81762& 0.55429& -1.33093& 0.51257& -2.74755\\ -1.23857& -0.7663& 1.09704& -1.06236& 1.5452& -1.77728& -2.64909\\ 1.61435& 0.0557& 0.68046& 0.51635& -0.89948& 2.53111& 4.77568\\ 1.13357& 0.66349& 0.2529& 0.22484& -0.36556& -1.08527& -0.41552\\ 0.89689& -0.81015& -0.97638& -0.41189& 0.81142& -1.55133& 0.58189\\ 1.17915& -3.03655& 6.2742& -1.20797& -0.51025& 14.91106& -2.27254\\ 0.41322& 1.94795& -0.51289& -0.37491& 1.51027& 11.03749& 2.716\\ -1.33221& 0.05741& -0.65416& 0.07573& 1.25784& -2.56009& -2.68572\\ 0.49487& -0.05527& -0.15964& -2.12137& 0.36101& -0.87975& -0.00735\\ -0.35502& -1.68303& 0.15328& -0.67232& 1.99617& -1.07191& 1.33101\\ 0.56645& 1.91273& -1.35198& 2.54815& -2.13396& 1.49682& 1.9449\\ 1.22696& 0.5422& 0.10388& 0.79584& -1.90703& 0.88708& 0.70909\\ 1.37981& 0.10391& -0.05018& -0.78432& -0.82698& -2.90594& 0.64166\\ 0.82895& 0.11146& 0.8172& -1.40912& 0.25884& 6.17555& 1.88983\\ \\ -0.09434& -1.36729& -0.3242& -0.56096& -1.54967& 0.53202& -0.00594& -0.47755\\ -11.57877& 0.66163& 0.20917& -2.04625& 6.11977& -1.03191& -1.9287& -0.55677\\ 0.32712& 0.73273& -2.00732& 1.52575& 0.18964& -0.59629& 0.49693& -0.07595\\ 1.01989& -0.72114& -0.50003& 0.01946& 2.40421& -0.92307& 1.54789& -0.94341\\ 2.2004& 1.21581& -1.29115& 2.19248& 5.39331& -1.13282& 0.4393& 1.84945\\ -1.59232& 0.95001& -0.29196& -2.38994& -1.45781& 1.74623& -2.13302& -1.63847\\ -14.91104& 19.19685& 1.70977& -20.47494& -12.16562& 2.42967& 1.41061& -6.14187\\ 28.38445& 1.50014& -0.12017& -0.41212& 2.46866& 1.94063& -0.47301& -1.54096\\ 0.08169& 0.95401& 0.05759& -1.29862& -1.37865& 0.0623& -1.19949& -0.38027\\ -0.01045& -0.15177& 1.28231& -1.2281& -2.46571& -0.07776& -1.36802& -1.37616\\ -2.64122& 2.84427& -0.58839& -2.29942& 1.05375& -1.21015& -0.4035& -0.11335\\ 0.48289& 0.71051& -1.04224& 0.09281& 1.92561& -1.99942& 0.14936& 1.45026\\ 0.62892& 1.03519& 1.20749& 0.97545& 1.86918& -0.57981& -0.35298& 1.09954\\ 2.86171& 0.4196& 1.03075& -0.55137& 1.25883& -0.01177& -1.13317& 0.49512\\ 21.0504& 0.61268& -0.95174& -0.30819& 2.46435& 0.37148& -0.52247& 1.11265\end{array}\right)$(11) ), (12(12) $w_{(\cdot ,\cdot )}^3=\left(\begin{array}{cccccccc}1.01465& 0.31021& 2.38233& -0.46954& -1.31079& 1.98707& 0.13497& -0.47611\\ 1.9266& 0.48781& 1.78669& -0.4672& -0.63115& -1.21142& -0.27927\end{array}\right)$(12) ), (13(13) $b_{(\cdot )}^1=\left(\begin{array}{ccccccc}-1.27214& 0.74293& 0.47722& -1.02678& -2.52935& -0.41163& 1.08434\\ 0.60001& -0.16287& 1.43962& -0.5693& -0.33109& -1.25747& 0.50508& -0.54384\end{array}\right)$(13) ), (14(14) $b_{(\cdot )}^2=\left(\begin{array}{ccccccc}-1.02719& -0.19057& 0.51144& -0.68846& -1.24663& 0.16532& 2.12435\\ 1.20535& 0.0549& -0.96967& -0.65354& 2.33305& -1.11357& -0.12249& 1.97887\end{array}\right)$(14) ), (15(15) $b_1^3=\left(\begin{array}{c}0.09641\end{array}\right)$(15) ) respectively. All the predicted weight matrices and bias matrices of the proposed model (as shown in Figure 4c) are listed in equations (16)(16) $\begin{array}{l}{w}_{(\cdot ,\cdot )}^1=-1.331790.10099-0.79290.138520.618940.099482.26380.\text{85991}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} - 3.32784\hspace{0.17em} - 1.43881\hspace{0.17em}\hspace{0.17em} - 0.81649\hspace{0.17em}\hspace{0.17em}1.95594\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}- 2.4815\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.97853\hspace{0.17em} - 1.94178\hspace{0.17em}\hspace{0.17em}1.13907}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.73303\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.81025\hspace{0.17em} 1.08221 0.65549 1.39029 - 0.12861 \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}4.1777 \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.22752}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} - 28.4072\hspace{0.17em} 0.90179\hspace{0.17em} - 1.17867\hspace{0.17em} - 1.92273\hspace{0.17em}\hspace{0.17em}1.73645\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.82859\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.04014\hspace{0.17em}\hspace{0.17em} - 0.11736 }\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.75134\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.90E - 04\hspace{0.17em} - 1.21556\hspace{0.17em}\hspace{0.17em} - 4.23925\hspace{0.17em} - 0.16477\hspace{0.17em} - 3.98891\hspace{0.17em} - 0.81004\hspace{0.17em} 0.21072}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}9.64008\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.40005\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}- 0.37421\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.3913\hspace{0.17em}\hspace{0.17em} - 0.42187\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.16283\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.90947\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.83533 }\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.97439\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.1409\hspace{0.17em}\hspace{0.17em} 0.89321 - 0.27058 0.23496 - 14.0501 - 5.04822 0.26384 }\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}- 1.10455\hspace{0.17em}\hspace{0.17em}1.11836\hspace{0.17em}\hspace{0.17em}3.04958\hspace{0.17em}\hspace{0.17em} - 0.64872\hspace{0.17em}\hspace{0.17em} - 0.21521\hspace{0.17em} - 3.43536\hspace{0.17em}\hspace{0.17em}4.85537\hspace{0.17em}\hspace{0.17em}2.06188}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.87054 1.65804 1.84487 2.18874 - 0.28649 1.00369 0.12168 - 1.47679}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.00043 - 0.89692 - 0.50515 - 0.22492 }0.\text{39301 0.71993 - 2.76691 - 0.72273}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}19.58302 - 2.71071 - 0.85304 - 3.83049 - 0.54049 8.61973 7.14381 - 18.47029 }\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.41535 - 0.03217 - 3.18815 - 0.67689 - 0.34534 2.54284 0.91757 - 0.25281 }\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}3.64583 0.42998 1.49889 0.15002 - 0.64925 1.03737 1.56292 - 0.26082}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0.27386 - 0.39539 0.63753 - 2.65577 - 1.79411 - 1.04856 - 3.39794 2.80882}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.16864 - 0.98201 0.96485 - 1.48228 0.40666 - 3.09046 3.1948 1.4052}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1.47523 2.92985 - 0.28276 0.03705 - 0.38112 - 0.01946 3.24538 - 1.11894}\\ \text{ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}- 8.76559 - 0.43392 0.848 - 0.28104 - 0.41763 - 1.04591 1.52538 - 0.14222}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} 6.44166 - 0.23373 0.019 1.19194 1.36906 0.67513 - 1.7939 0.59181}\\ \text{ - 5.68954\hspace{0.17em}\hspace{0.17em}0.07347 - 0.76518 0.76967 0.11834 1.61887 - 1.26802 - 1.5304}\\ \text{ - 32.71013 - 6.19851 0.33992 3.90874 - 3.9949 13.35755 26.54455 13.58262}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} 2.59871 0.70251 0.18247 - 8.40538 - 1.107 18.87626 5.12558 - 1.08194}\\ \text{ 9.23348 0.44141 0.33614 1.78588 - 0.33505 - 0.81128 - 12.21366 - 5.22149}\\ \text{ 8.85904 - 0.42293 0.83334 8.21911 1.34503 - 0.95163 11.54466 - 0.59667}\\ \text{ 0.98875 4.65484 2.61371 - 8.36332 - 2.64915 0.76087 - 8.38449 2.85161}\\ \text{ - 3.19318 - 0.22069 0.28269 2.89494 - 2.34362 - 0.0631 7.05684 - 8.13611}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} 0.67565 0.05591 - 0.76497 0.90107 - 0.13151 - 0.01567 8.29391 0.52898}\\ \text{ 4.0766 }-3.492270.4214317.272442.14845.68654-18.52478-1.62999-24.142640.96928-1.46442-3.46961.501330.6589-31.59711-0.606324.6557-0.216260.57747-1.28487-0.1956-2.54897-5.55506-1.260592.180590.097280.23798-4.46206-0.04128-5.641212.25751.44850.20147-0.36794-0.63426-2.969820.63273-1.61534-13.2529-0.7659-2.31739-0.10613-\text{1}.\text{64648 }-\text{0}.\text{21075 0}.\text{5386 }-\text{0}.\text{82701 }-\text{17}.\text{02842 0}.\text{4873 }\end{array}$(16) , (17(17) $w(⋅,⋅)2=1.59444   1.374410.08246−14.3686−11.803350.265251.53291−0.460640.071031.08321−0.9983−0.59131.32194 0.83937 −5.15232 0.41352 −0.02886 4.24844 0.44092 21.52419 1.78021 0.16504 1.35912  −0.24865 0.68712 −3.05583 1.85519 3.4193 42.09814 −16.10189 2.45281 9.18944 −2.18033 3.21015 −2.87406 −2.65924 −1.63166   2.11773  0.2259  0.22556  - 0.02443  - 0.23996  - 0.25933 2.50266 9.79031 0.35729 0.00453 0.74291  - 1.77344 1.43737 0.14838  - 4.38995  - 2.41929  - 1.08705  - 0.47383  - 0.3407 0.91235 6.42264  - 7.91324  2.67979  - 23.07227  - 2.38108  - 0.26728  - 6.6412  2.18116  - 1.97641  0.84271 1.1091  0.69688  - 0.4572  5.85349  0.80908 0.62809  - 0.56724  0.52351 - 0.08814  - 1.46343  - 0.4876  0.01357  1.77592 - 0.21627  0.26697  0.37209  0.00415  1.0432 1.55307 1.99866 - 2.32109  - 0.49835   - 2.08799 4.5515 0.06363 13.35438  - 0.40685 0.86924 0.83136 1.13519  - 0.33177  - 0.57808 5.2195 3.43116  - 0.18598 1.46334 - 0.3403  - 1.27676 1.46988  - 0.90317  - 2.88181 0.49047 1.72472 - 0.73717 0.09032 0.16506  - 3.73556 2.20811  - 2.09712  - 0.37874  - 0.84745 0.25325  - 1.89855  - 0.42056  - 1.73956  - 0.68133  - 3.82544  5.43908  - 0.10337 0.59736  - 0.10614  5.88872 1.16788 0.52967 2.04449 0.17697 1.6034  - 4.88021  - 3.67342  - 3.16443  - 3.80481 0.43883  - 2.77432 0.74847 0.34197 2.29412  - 0.53634 13.61469  - 3.50092 0.83081   - 17.00822  - 1.28632  - 30.07928  - 4.99074 6.17085  - 13.42477 1.11761 5.56685 6.92186  - 0.47691  - 6.472113.76509 6.25211 0.08923 1.92817 0.50941 0.33192 0.59373 14.89096 - 3.01504 0.32747 - 0.33903 0.87293  - 3.52739  - 8.5838 0.43991  - 7.50242  - 3.9938 - 2.14433 2.16659 2.4177 2.7793 4.99061  - 1.15627 8.16648 - 0.17308 1.58318 1.01933  - 0.62937 0.82288 5.49387  - 5.57786 0.8237 8.20826 5.50498 3.90456 0.02137  - 1.57529  - 17.03563  - 0.50824 17.0463 2.84456  - 3.99713  - 0.79834  - 0.64645  - 0.36926 - 0.18217 0.62207  - 0.53813  - 0.72676 0.55994  - 3.0044  - 0.53167 0.46937  - 0.32129   - 2.28853 - 0.02642 0.75698 - 0.13255 - 2.25147 0.43358 0.61629  - 3.17574 −1.51643  −6.78321  10.71132  1.96188  −4.96801  5.79232  −2.74981  −5.35273  −1.08634  0.3329  −2.6128 −1.51295 −6.62437 0.96287 0.349 0.89334 1.4128 0.68246 0.47649 −1.83139 −0.93862 −1.93173 −2.94282 2.47698 1.1675 −0.66479 9.5737 5.81911 70.31665 13.40772 −0.41635 −3.73189 −0.65804 4.74472 −12.32241 −1.64707 $(17) ), (18(18) $w_{(\cdot ,\cdot )}^3=\left(\begin{array}{cccccccc}0.41998& 4.7676& 1.1575& 0.40345& -0.94073& -0.81895& -0.42949& -1.58796\\ -0.62729& 1.37791& -0.77688& 1.29469& -1.37013& -1.36542& 2.58703& -0.85273\end{array}\right)$(18) ), (19(19) $b_{(\cdot )}^1=\left(\begin{array}{cccccccc}-0.68265& -0.28817& -0.91879& -1.28449& -0.1705& 1.71379& 1.14313& -0.28467\\ -1.3088& 1.11838& -0.2343& 1.28489& 1.13781& 1.69541& 0.64284& -1.21441\end{array}\right)$(19) ), (20(20) $b_{(\cdot )}^2=\left(\begin{array}{cccccccc}-1.18978& -0.64785& 2.08001& 0.18852& -0.87108& -0.67752& 1.28692& -0.37449\\ 0.69863& 0.15144& 0.80098& 0.14935& 0.39817& 0.33561& 1.04919& 0.85973\end{array}\right)$(20) ), (21(21) $b_1^3=\left(\begin{array}{c}0.23086\end{array}\right)$(21) ) respectively.

3.1. Parametric analysis of input parameters on output prediction

The parametric analysis of the proposed models has been conducted using the Pandas library of Python in the Google Colaboratory platform. In particular, the “Series” function from Pandas is used to calculate the feature importance of each input parameter on the predicted output. The detailed parametric study is conducted mainly to evaluate the contribution of individual input parameters to the output prediction. The individual impact of input parameters on ultimate load, ultimate moment and deflection are shown in Figure 6. The analysis of input parameters of $\mathit{Datase}{t}_1$ shows that cover to prestressing steel, percentage of corrosion and total area of non-prestressed steel has a significant impact on the ultimate load prediction. The analysis of input parameters of $\mathit{Datase}{t}_2$ shows that depth of the beam, cover to prestressing steel, effective depth and the ratio of cover to effective depth influence the ultimate moment prediction. The analysis of input parameters of $\mathit{Datase}{t}_3$ shows that the span of the beam, later percentage of corrosion and cover to prestressing steel have a significant impact on the deflection prediction. Interestingly, rest of the input parameters influence less than 8% on the output parameter prediction in all the datasets.

Figure 6. Parametric analysis of input parameters on ultimate load, ultimate moment and deflection predictions.

The analysis results follow the same feature importance weightage mentioned in the literature. As the percentage of corrosion increases, the ultimate load, ultimate moment and deflection decreases (Moawad, El-Karmoty et al., 2018; Zhang, Wang, Zhang, Liu et al., 2017; Zhang, Wang, Zhang, Ma et al., 2017). With the increase in the corrosion rate, the ductility of the strand reduces; thus deflection value decreases. If the rate of corrosion increases then the failure mode of the concrete changes from concrete crushing to wire rupture and the brittle failure of the beam occurs (Hansapinyo et al., 2021; Zhu et al., 2017). As the cover thickness of the prestressing steel increases, corrosion initiation time increases and the thickness of the cover proportionally affects the strand corrosion with respect to time. As the cover thickness decreases, the possibility of moisture and chemical ingression into the beam increases, which intern increases the corrosion rate. Non-prestressed steel in the prestressed concrete beam is most likely to undergo deterioration earlier than prestressed steel. Therefore, the non-prestressed steel area affects the strength of the beam since prestressed steel is most likely to protect against corrosion in terms of high strength grouting (Moawad, Mahmoud et al., 2018). The moment capacity of the beam proportionally varies with the total depth, effective depth and cover to depth ratio of the beam. The spalling of the concrete due to the bond degradation by corrosion reduces the depth of the beam, thus reducing the moment capacity. The strength of concrete and strength of steel have a direct impact on the load carrying capacity of the beam. Since the prestressed steel has to carry high stress in prestressed concrete, the reduction in the area of steel causes an increase in the stress in the steel bar and affects the beam strength.

3.2. Comparison of the proposed RBPBTNN models with other ANN models

The back propagation neural network-based methods are the frontier techniques in soft computing. One of the fine features of BPNN is the back propagation of errors through multiple layers of the network to fit the output. In recent developments, Resilient Back Propagation Without Backtracking Neural Network (RBPWBTNN), RBPBTNN, and GCM-BPNN algorithms have been developed on top of BPNN. All of these try to minimize the error by adding a rate of learning to the weights going in the opposite direction of the partial derivatives. Unlike back propagation, RBPBTNN proved to be the fastest, chooses a different rate of learning for each weight and uses only sign of the partial derivates instead of magnitude. GCM-BPNN works on top of RBPBTNN by selecting a single learning rate in relation to other rates.

In general, the back propagation algorithm iteratively calculates partial derivatives of the error functions until all the absolute partial derivatives of the weights are smaller than the chosen threshold (Gãijnther & Fritsch, 2010). The reason for choosing an appropriate back propagation algorithm primarily lies in the provision for selecting an error threshold during the training phase for convergence. The proposed RBPBTNN and RBPWBTNN converge for a small error threshold of 0.0005, but the rest of the chosen models converge only after 0.01. To investigate further, BPNN, RBPWBTNN and GCM-BPNN are trained with the same number of layers as the proposed models for different K-fold values and the performance comparison of the most influencing cost function RMSE is shown in Figure 7. The trained BPNN, RBPWBTNN and GCM-BPNN models are tested with optimum K-fold and recorded the performance metrics in Table 7. The RBPWBTNN is showing poor performance and the proposed RBPBTNN is showing the best performance in ultimate load prediction when multiple performance metrics such as NSE, MAE, R ${ }^2$, and RMSE are considered. Along with these, MAPE should also be considered one of the measuring metrics. At K-fold = 3 and training repetition = 10, the MAPE values of RBPBTNN, BPNN, GCM-BPNN, and RBPWBTNN in ultimate load prediction are 12.9605, 19.224, 15.82, 39.452 respectively. Again, RBPBTNN shows the lowest value of MAPE in ultimate load prediction. All the model executions are conducted with the optimum K-fold and training repetition = 10.

Figure 7. RMSE comparison of different ANN algorithms when K-fold is varied during training phase.

The poor performance of RBPWBTNN is persistent in the ultimate moment and deflection predictions. The MAPE values of RBPBTNN, BPNN, GCM-BPNN, and RBPWBTNN in ultimate moment prediction are 7.793, 18.949,13.388, 10.62, and in deflection prediction are 19.244, 32.8695, 23.3186, 24.101 respectively. Even though BPNN shows better results in ultimate moment prediction, MAPE is very high, which degrades its performance. Therefore, the proposed RBPBTNN is showing consistently high performance in all three models in predicting ultimate load, ultimate moment and deflection and the respective linear fit of the predicted values are shown in Figure 8a, Figure 8b and Figure 8c. As a matter of fact, the prediction accuracy majorly depends on the size of the input dataset. Therefore, size of the dataset is always a limiting factor to accuracy.

Figure 8. Linear fit of predicted ultimate load, ultimate moment and deflection.

3.3. Multi-fold impact of proposed RBPBTNN models on flexural strength

The increasing use of artificial intelligence techniques alleviates the complexities of analyzing structures. The proposed neural network models help to estimate the capacity of the corroded prestressed concrete beam under bending. This study shows the dissection of the main prediction into sub-predictions and feed-forward the previous prediction to the next to increase the overall performance. The ultimate load prediction output from the proposed first model can be feed-forwarded as input to predict the ultimate moment in the proposed second model and deflection in the proposed third model for the prediction-based analysis of the corroded prestressed beam. The prediction through the proposed models is almost near to the actual values. Therefore, this study can be effectively used to know the strength of corroded prestressed beams in advance and adopt proper safety measures before the failure of the structure to improve its service life.

4. Conclusions and future scope

In this study, three resilient back propagation ANN models are developed to predict the corroded prestressed concrete beam’s flexural strength. The thorough assessment of the obtained results ensures that these models can be effectively used in the assessment of the prestressed concrete beam. The proposed models are the best fit for the present study and exhibit the following unique results.

• The analysis results show that the proposed models are outperforming with R² = 0.9860, RMSE = 13.2020, MAE = 10.5340, NSE = 0.9414 for ultimate load, R² = 0.9691, RMSE = 17.7800, MAE = 5.6900, NSE = 0.9341 for ultimate moment and R² =0.9180, RMSE = 10.3060, MAE = 5.8250, NSE = 0.8060 for deflection.

• The proposed resilient back propagation models converge early at the error threshold 0.0005 compared to other back propagation algorithms which are converging above 0.01.

• The parametric analysis guarantees that the feature importance weightage predicted is same as the actual feature importance weightage of corroded prestressed beam behavior. Cover for prestressed steel, depth and span of beam are the highly influencing features of ultimate load, ultimate moment and deflection respectively. Breadth, prestress in steel and ultimate strength of prestressing steel are the least influencing features of ultimate load, ultimate moment and deflection respectively.

Although the proposed models show significant performance, the prediction accuracy can be improved by gathering more data. All the proposed models use the most common sigmoid function in all the layers. The proposed models could be evaluated using other activation functions such as linear, tangent hyperbolic, ReLU etc. The proposed back propagation algorithm could be further executed and compared with recent counterpart prediction algorithms such as genetic programming in the future.

CRediT authorship contribution statement

Yamuna Bhagwat: Conceptualization, Methodology, Data collection, Formal analysis, Investigation, Writing original draft, Validation, Visualization, Project administration, Review and editing. Gopinatha Nayak: Supervision, Resources, Project administration, Validation, Writing - review and editing. Radhakrishna Bhat: Methodology, Formal analysis, Investigation, Writing - original draft, Validation, Visualization, Writing - review and editing. Muralidhar Kamath: Visualization, Validation, Writing - review and editing.

Disclosure statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

Yamuna Bhagwat

Yamuna Bhagwat received her B.E and M.Tech in civil engineering from Visveswaraya Technological University, Belagavi, India. She is currently pursuing her Ph.D in Department of Civil Engineering, Manipal Institute of Technology, MAHE, Manipal, India.

Gopinatha Nayak

Gopinatha Nayak received his M.Tech in Structural Engineering and Ph.D from National Institute of Technology, Surathkal, Karnataka, India. Presently he is working as a Professor in the Department of Civil Engineering, Manipal Institute of Technology, MAHE, Manipal, India.

Radhakrishna Bhat

Radhakrishna Bhat received his B.E and Integrated Ph.D degrees in Computer Science and Engineering from Visveswaraya Technological University, Belagavi, India. He is currently working as an Assistant Professor in the Department of Computer Science and Engineering, Manipal Institute of Technology, MAHE, Manipal, India.

Muralidhar Kamath

Muralidhar Kamath received his M.Tech in Construction Technology and Management from Visvesvaraya National Institute Of Technology, Nagpur, India and Ph.D from Manipal Institute of Technology, MAHE, Manipal, India.