Holistic design of pre-tensioned concrete beams based on Artificial Intelligence

ABSTRACT This research demonstrates how pre-tensioned concrete beams (PT beams) are designed holistically using artificial neural networks (ANNs). To establish reverse design scenarios, large input and output data are generated using the mechanics-based software AutoPTbeam. ANN-trained reverse-forward networks are proposed to solve reverse designs with 15 input and 18 output parameters for engineers. ANNs for reverse designs pre-tensioned concrete beams are formulated based on 15 input structural parameters to investigate the performances of PT beams with pin-pin boundaries. Useful reverse designs based on neural networks can be established by relocating preferable control parameters on an input-side, such as when four output parameters ( ) (reverse scenario) are reversely pre-assigned on an input-side, all associated design parameters, including crack width, rebar strains at transfer load stage, rebar strains, and displacement ductility ratio at ultimate load stage are computed on an output-side. Deep neural networks trained by chained training scheme with revised sequence (CRS) for the reverse network of Step 1 show the better design accuracies when compared to those obtained based on ANNs trained by parallel training method (PTM) and based on shallow neural networks trained by CRS when the deflection ductility ratios (μΔ ) within generated big datasets are reversely pre-assigned on an input-side. GRAPHICAL ABSTRACT


Previous studies
Several artificial neural network-based studies of prestressed beams have been conducted. Torky and Aburawwash (Torky and Aburawwash 2018) published a simple prestressed beam to demonstrate the viability of neural networks over traditional approaches. They proposed a deep learning approach to optimizing prestressed beams. Their data, however, are limited to beam depth, beam width, bending moment, eccentricity, and a number of strands.
Sumangala and Jeyasehar (Sumangala and Antony Jeyasehar 2011) also studied a damage assessment procedure using an artificial neural network (ANN) for prestressed concrete beams. They formulated the methodology using the results obtained from an experimental study conducted in the laboratory. The measured output from both static and dynamic tests was taken as input to train the neural network based on MATLAB. The quantitative evaluation of the degree of damage was possible by ANN using the natural frequency and stiffness in the post-cracking range. Their damage assessment procedure was validated using the data available in the literature, and the outcome is presented in Sumangala and Antony Jeyasehar (2011). Lee et al. (2019) proposed a novelty detection approach for tendons of PSC bridges based on a convolutional autoencoder (CAE). The proposed method used simulation data from nine different accelerometers. According to their research, the accuracies of CAEs for multi-vehicle are 79.5%-85.8% for 100% and 75% damage severities. Sanqiang Yang and Yong Huang also (Yang and Huang 2021) identified damages of prestressed concrete beam bridge based on a convolutional neural network. An additional study has been performed by (Alhassan, Ababneh, and Betoush 2020) who studied an innovative model for accurate prediction of the transfer length of prestressing strands based on artificial neural networks. Tam et al. (2004) also diagnosed prestressed concrete pile defects using probabilistic neural networks. Mo and Han (1995) investigated prestressed concrete frame behavior with neural networks. Slowik, Lehký, and Novák (2021) investigated reliability-based optimization of a prestressed concrete roof girder using a surrogate model and the double-loop approach, while Khandel et al. (2021) used a statistical damage detection and localization approach to evaluate the performance of prestressed concrete bridge girders using fiber Bragg grating sensors based on artificial neural networks. Roya (Solhmirzaei et al. 2020) investigated 360 test results ultra high performance concrete beams using different machine learning algorithms, resulting in good accuracies in predicting beam failure mode and shear capacity.
However, these studies are not dedicated to a holistic design of prestressed beams for practical applications. In the current study, both forward and reverse designs for prestressed beams are possible, with real-world big datasets of 50,000 generated using the full scale of structural mechanics-based software and trained using their previous training methods shown below. Hong, Pham, and Nguyen (2021) proposed training methods based on a feature selection for a reverse design of doubly reinforced concrete beams. They trained large datasets using TED, PTM, CTS, and CRS that they developed. Hong et al. also presented artificial intelligence-based novel design charts for doubly reinforced concrete beams  and novel design methods for reinforced concrete columns (Hong, Nguyen, and Pham 2022).  performed reverse designs of doubly reinforced concrete beams using Gaussian process regression models enhanced by sequence training/designing techniques based on feature selection algorithms. The training methods developed in Hong, Pham, and Nguyen (2021),   Hong, Nguyen, and Pham (2022), and  are used to map 15 input parameters to 18 output parameters for a holistic design of pre-tensioned concrete beams. The large datasets, which contain 33 parameters, including 15 input and 18 output parameters, aid in the accurate training of ANNs that represent prestressed beams.

Research significance
There are numerous available computer-aided engineering tools, including CAD packages, FEM software, self-written calculation codes, that are used to study the performance of the PT structures. In this study, ANNsprogramed based on MATLAB Deep Learning Toolbox (MathWorks 2022a), MATLAB Parallel Computing Toolbox (MathWorks 2022b), MATLAB Statistics and Machine Learning Toolbox (MathWorks 2022c), and MATLAB R2022a (MathWorks) are used to design PT structures based on EC2 2004 (CEN (European Committee for Standardization) 2004) in both forward and reverse directions. When big data is good enough to show a tendency of PT beam behaviors, a reverse solution is generalizable by using ANNs trained on big data. Based on large datasets, forward and reverse designs using ANNs are possible, with no need for primary engineering knowledge to effectively determine design parameters for practices. The proposed networks allow for both forward and reverse designs for PT beams, which is difficult to achieve with classic techniques. AI-based PT beam design with sufficient training accuracy can completely replace classical design software while exhibiting excellent productivity for both forward and reverse designs. To better demonstrate the design steps, numerical examples are provided. Design charts are also constructed for design scenarios. Design charts can be extended as further as possible to meet the requirements of engineers.
The proposed method is advantageous in that it is less dependent on problem types such as column, beam, frame, seismic design, and so on, and instead relies on characteristics of large datasets of the considered problem. As a result, the proposed method's applications are not limited to designing PT beams but can also be extended to other structures.

Description of input parameters
ANNs are implemented in designing bonded pretensioned beams with pin-pin ends shown in Figure 1, illustrating beam width (mm) b, beam depth (mm) h, and beam length (mm) L. Table 1 presents fifteen input parameters to generate big datasets using AutoPTbeam for an AI-based reverse design. Concrete properties include compressive cylinder strength of concrete at 28 days (MPa, f ck ), compressive cylinder strength of concrete at transfer stage (MPa, f cki ), and long-term deflection factor considering creep effect, (normally 2.5 ~ 3.5, K c ). Reinforcing bar properties include top, bottom rebar ratios (ρ st and ρ sc ), yield stress of rebar (MPa, f sy ), yield stress of stirrup (MPa, f sw ) , and preferred diameter (mm, ϕ s ). Tendon properties include tendon ratios (ρ p ), yield stress (MPa, f py ), and preferred diameter (mm, ϕ p ). Long-term pre-stress losses (PTLoss lt ) is also one of fifteen input parameters.
Tensile strength of tendon f pu is set as 1860 MPa, and hence, f pu is not included in inputs of big datasets.

Description of output parameters
There are three load stages considered in prestressed designs, as shown in Figure 2. The present study investigates 18 output parameters, including seven output parameters at the transfer load stage, five output parameters at the service load limit stages, and six outputs at the ultimate load limit stage, respectively, as shown in Table 2.
Table 2(a) illustrates seven outputs of forward designs at a transverse stage, where EC 2 (CEN (European Committee for Standardization) 2004) requires concrete compressive stress (σ c ) to be smaller than 0:6f cki , crack widths to be under a limitation depending on environmental condititons, and top rebar stress (σ sc ) to be less than a smaller number between 0:8f su andf sy . It is noted that, EC2 (CEN (European Committee for Standardization) 2004) determines crack limitations using six classes of environmental exposures, which are classified based on a risk of being attacked by carbonation-induced corrosion, chloride-induced corrosion, freeze thaw, or chemicals. ). Thirdly, concrete compressive stresses (σ c ) under characteristic load combinations should be smaller than 0:6f ck to avoid longitudinal cracks. Finally, rebar stresses are limited at a minimum of 0:8f su and f sy , whereas tendon stresses (σ P ) should not exceed a smaller number between 0:75f pu and f py to avoid unacceptable cracks or deformations at characteristic load combinations.   Tables 1 and 2 summarizes all the 33 input and output parameters used for generating large datasets for training ANNs. The large datasets are used for AIbased design for both forward (Section 4) and reverse designs (Section 5) of simply supported bonded pretensioned beams for all stages. Table 3 and Figures 3-6 contain a list of random variables, their corresponding ranges, and data distributions from the large structural datasets. The algorithm of AutoPTbeam shown in Figure 7(a) was developed for designing the forward design of bonded pretensioned beams by Cuong and Hong . Figure 7(b) shows flow charts for generating large structural datasets for training ANNs. Table 7(b) displays large structural datasets with selected non-normalized fifteen inputs and eighteen outputs generated based on Figure 7(b).

Generation of large structural datasets and network training
Regression models are often evaluated using several common metrics, namely Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Root Mean Squared Error (RMSE), and Coefficient of Determination (R 2 ) (Naser and Alavi 2020). The present paper employs MAE and correlation coefficients (R) calculated according to (1) and (2) to preliminarily benchmark performances of ANN models. However, it is noted that reliabilities of ANN models shall only be concluded based on errors in practical designs. Engineers must always check differences between results provided by ANN predictions and structure mechanic calculations. The present paper also suggests several methods, such as adjusting numbers of layers, numbers of neurons, or CRS training scheme, to enhance design accuracies.
Where:  Figure 8 shows the investigation of one forward design and one reverse design. An ANN predicts eighteen outputs for a set of fifteen inputs. Figure 8(b) depicts a reverse scenario. In a reverse scenario, four design parameters (b, ρ st ,ρ sc , and ρ p ) are calculated on an output-side when nominal capacities of pretensioned beams (q L/250 ,q 0.2 mm, q str , and μ Δ ) are prescribed on an input-side. Q 0.2 mm and μ Δ represent applied loads reaching crack width of 0.2 mm and deflection ductility ratios, respectively. Nominal capacities resisting applied distributed load (q L/250 ) of 70 kN/m at SLL (service load level) reaching a deflection of L/250 and resisting applied distributed load (q str ) of 150 kN/m at ULL (ultimate load level) are, then, calculated.    Table 4 where training accuracies in mapping  (Hong, Pham, and Nguyen 2021) and (Hong 2021) can be used to improve design accuracy further. Lagrange-based optimizations were also developed by  that demonstrates acceptable design accuracies for use in design applications.

Design charts corresponding to ρ p for forward design
Design charts shown in Figure 11 are established based on the following parameters; pre-assigned inputs are parameters b = 500 mm, h = 700 mm, f ck = 40 MPa, f cki = 20 MPa, ρ st = 0.005, ρ sc = 0.005, f sy = 500 MPa, f sw = 400 MPa, ϕ s = 16 mm, f py = 1650 MPa, ϕ p = 12.7 mm, PTLoss lt = 0.15, L = 10,000 mm, and K c = 3; output parameter is σ c /f cki at transfer stage. In Figure 11, design parameters (σ c /f cki ,ε st ,ε sc, ε p , and σ ct /f cki ) are plotted in Y-axis by varying tendon ratio (ρ p ) which is shown with X-axis. In Figure 11 (a), σ c /f cki (Output # 16) represents upper concrete stress per concrete strength at transfer stage. Upper concrete is usually in tension during the transfer stage because the load is upward. In actual concrete sections, tension concrete stresses (σ c of Output # 16 and σ ct Output # 20) immediately falls to 0 when reaching its tensile concrete strength at the transfer stage (f cki ) by an assumption that tensile concrete strength is lost after cracking. A discontinuity is shown in tensile concrete sections due to this assumption, whereas an ANN with a feed-forward network approximates a discontinuity based on real-valued continuous functions. A significant difference in data tendency between tensile concrete sections and ANNs is observed at the vicinity of initial cracking.

Formulation of ANNs based on back-substitution (BS) applicable to reverse designs
In a reverse scenario shown in Figure 8(b) and Table 5, design parameters (b, ρ st ,ρ sc , and ρ p in green cell) targeting nominal capacities (q L/250 , q 0.2 mm ,q str , and μ Δ in the yellow cell) are calculated on an output-side the reverse network of Step 1. Table 5 summarizes the training accuracy based on PTM. An ANN trained by R3.PTM.a based on 20 layers-20 neurons is used to calculate design parameters (b, ρ st ,ρ sc , and ρ p ) as shown in Boxes 1, 5, 6, and 10 on an output-side the Step 1 of Figure 12.
An ANN with FW.PTM.c based on 40 layers and 40 neurons is implemented in training the forward network of Step 2, when 18 outputs including reverse input parameters (q L/250 ,q 0.2 mm ,q str , and μ Δ which were already prescribed in Boxes 23, 24, 28, and 33 in Step 1) are calculated in Boxes 16 to 33 on an output-side of Step 2. Fifteen input design parameters including b, ρ st ,ρ sc , and ρ p pre-calculated on an output-side the reverse network of Step 1 are used in Boxes 1 to 15 of the forward network of Step 2. When a deflection ductility ratio (μ Δ ) of 1.50 is used, the reverse network of Step 1 produces a beam width (b) of 1252.12 mm, a lower tensile rebar ratio (ρ st ) of  Generation of large datasets of bonded pre-tensioned beams for AI-based design  0.00564, a compressive rebar ratio (ρ sc ) of 0.00122, and a tendon ratio (ρ p ) of 0.00197 in Boxes 1, 5, 6, and 10 on the output side of Figure 12(a). Significant errors are caused in a design as shown in Step 2 forward network, with the largest errors reaching −41.94% (0.062 mm vs. 0.088 mm) and −21.38% (−7.055 mm vs. −8.564 mm) for crack width and deflection (camber) at transfer load limit state, respectively, as shown in Figure 12(b). This is due to the reverse input parameter such as deflection ductility ratio (μ Δ ) of 1.50 being incorrectly preassigned out of data range at ULL in Step 1 of Figure 12. Errors of crack width and deflection (camber) at transfer load limit state reach −8.00% (0.050 mm vs. 0.054 mm) and 15.58% (−10.985 mm vs. −12.696 mm), respectively, indicating that errors decrease when deflection ductility ratio (μ Δ ) of 1.50 is adjusted to 1.95 as shown in Figure 13  Step 1 of Figures 14-16, respectively, where the deflection ductility ratio (μ Δ ) of 2.00 is adjusted to 2.50. An ANN with FW.PTM.c based on 40 layers and 40 neurons is trained in the forward network of Step 2 to obtain 18 outputs in Boxes 16 to 33. The best design accuracies are found with ANNs using R3.PTM.c, which is based on 40 layers-40 neurons and based on a deflection ductility ratio (μ Δ ) of 2.50. The network parameters depend on a number of hidden layers and neurons. Input conflicts should also be removed by selecting reversely pre-assigned input parameters that satisfy structural mechanics.

Formulation of ANNs based on back-substitution (BS) applicable to reverse designs
Readers are referred to the book (Hong 2021) to review the CRS method which is used in training the reverse network. A sequence of training networks should be determined based on the feature scores shown in Table 6(a). Reverse inputs (q L/250 , q 0.2 mm ,q str , and μ Δ ) are pre-assigned in the reverse network of Step 1 to calculate reverse outputs (b, ρ st , ρ sc , and ρ p ) as shown in Reverse Scenario 3 of Table 6(b), where training accuracies based on CRS with deep neural networks (DNN) are presented. Training a reverse ANN in Step 1 using the CRS method, rather than the PTM described in Section 5.1, begins with beam width (b) because beam width (b) can be used to train networks without other output parameters, such as ρ st ,ρ sc , and ρ p , as feature indexes as shown in Table 6. Important feature indexes affecting beam width (b) include h (5.77), L (10.92), q L/250 (4.14), q 0.2 mm (19.05), and q str (3.04) without ρ st and ρ p as shown in Table 6.  Table 6(a) indicate feature scores selected using the NCA method . As shown in Table 6(b), beam width (b) is, then, used as a feature index to train ρ st , resulting in higher training accuracy for ρ st . Similarly, beam width (b) and ρ st are used as feature indexes to train ρ p , followed by training ρ sc using all output parameters, beam width (b), ρ st , and ρ p , as feature indexes. The output parameters located on an output-side cannot be used to train other output parameters when using the TED training method (Hong 2021).

Design accuracies
Lower rebar ratio (ρ st ) has a feature score of 18.70 on ρ p whereas the upper rebar ratio (ρ sc ) is influenced by lower tensile rebar ratio (ρ st ) and tendon ratio (ρ p ) as much as 11.35 and 3.74, respectively. In the backsubstitution method, ANNs with R3.CRS.aare trained based on 20 layers-20 neurons, respectively, for the reverse networks of Step 1 as shown in Figure 17 whereas an ANN with FW.PTM.c based on 40 layers and 40 neurons is implemented in training the forward network of Step 2 when 18 outputs including reverse input parameters (q L/250 ,q 0.2 mm ,q str , and μ Δ which are    In designs, q L/250 = 80 kN/m at a deflection of L/250 at SLL and q str = 150 kN/m at ULL are reversely preassigned on an input side. Design accuracies based on the reversely pre-assigned deflection ductility ratio (μ Δ ) of 2.00 shown in Figure 17(c) are acceptable. Design accuracies based on reversely pre-assigned deflection ductility ratios (μ Δ ) of 1.75 shown in Figures 18(b) are acceptable whereas those based on pre-assigned deflection ductility ratios (μ Δ ) of 1.50 shown in Figures 18(a) yield errors that are unacceptable for use in design practice. Input conflicts occur when the deflection ductility ratios (μ Δ ) is 1.5. Deflection ductility ratios (μ Δ ) are adjusted from 1.50 to 1.75 as shown in Figures 18(a,b) to be within large data ranges. Figures 19-21 investigates the influence of a number of layers-neurons used for CRS in Step 1 on design accuracies of ANNs for a reverse design. For the reverse networks of Step 1, ANNs with R3.CRS.a, R3.CRS.b, and R3.CRS.c are trained using 20 layers-20 neurons, 30 layers-30 neurons, and 40 layers-40 neurons, respectively, as shown in Figures 19-21 for reversely preassigned deflection ductility ratios (μ Δ ) of 2.00 and 2.50. When FW.PTM.c based on R3.CRS.a, R3.CRS.b, and R3.CRS.c is used, design accuracies similar to those of the three networks are found. For the reverse network of Step 1 trained based on 40 layers-40 neurons shown in Figure 21(b), the largest error is found with −9.32% (91.98 kN/m vs.100.56 kN/m when compared to those calculated based on structural mechanics) for q 0.75fpu that represents nominal strength when tendons reach stresses smaller of 0.75 f pu and f py . The next largest error of deflection (camber) at transfer load limit state reaches −7.27% (−9.896 mm vs. −10.616 mm) compared with those calculated using structural mechanics. Good accuracies among all three designs shown in Figures 19-21 are obtained when deflection ductility ratios (μ Δ ) within a range between 1.75 and 2.75 are trained. Fewer input conflicts occur with a deflection ductility ratio (μ Δ ) of 2.50 than with 2.00.
It should be noted that networks used in the first reverse networks of Step 1 can be replaced by CRS to improve design accuracy.

Over-fitting
Because over-fitting occurs during training ANNs, the design accuracies are obtained using R3.CRS.c (40 layers-40 neurons) in Figure 21 are slightly lower than those obtained using R3.CRS.a (20 layers-20 neurons)

Design charts
Figures 22(a-d) present reciprocal design charts for design parameters of ρ sc ,ρ p ,ρ st , and b concerning ductility (μ Δ ). Design parameters of ρ sc ,ρ p ,ρ st , and b are determined accurately based on deflection ductility ratios (μ Δ ) within ranges between 1.75 and 2.75 as shown in Figures 22(a-d), indicating that upper and lower bounds of deflection ductility ratios (μ Δ ) are governed by of ρ sc ,ρ p ,ρ st , and b as can be seen in Figure 22. Design charts to determine ρ sc ,ρ p ,ρ st , and b are obtained using Figures 14-21 obtained based on the boundary of deflection ductility ratios (μ Δ ) between 1.75 and 2.75. Errors increase rapidly outside this  Step 1 neurons for Step 1; q L/250 = 80 kN/m at SLL (at a deflection of L/250), q str = 150 kN/m at ULL, deflection ductility ratio (μ Δ ) of 2.00 and 2.50 at an ultimate load limit state (ULL). Table 6. Training sequence of a reverse scenario based on CRS (DNN).   region in which AutoPTbeam results also diverge. Figures 22(a-d) are useful for calculating design parameters, ρ sc ,ρ p ,ρ st , and b, for specified deflection ductility ratios (μ Δ ) between 1.75 and 2.75. Step 2, when 18 outputs including reverse input parameters (q L/250 ,q 0.2 mm ,q str , and μ Δ which are already reversely prescribed in Boxes 23, 24, 28, and 33 in Step 1) are calculated in Boxes 16 to 33 as shown in Figures 23 and 24. However, design accuracies are not acceptable because the preassigned deflection ductility ratios (μ Δ ) are not with the range between 1.75 and 2.75 as shown in Figure 22.

Design examples
In Figures 25 and 26, ANNs with R3.CRS.b and R3. CRS.c are trained using 30 layers-30 neurons and 40 layers-40 neurons, respectively, for the reverse networks of Step 1 when deflection ductility ratios (μ Δ ) of 2.50 and 3.00 at ULL are reversely pre-assigned based on q L/250 = 80 kN/m at SLL and q str = 150 kN/m at ULL. ANNs with FW.PTM.c based on 40 layers and 40 neurons are used to train the forward network of Step 2, when 18 outputs including reverse input parameters (q L/250 ,q 0.2 mm ,q str , and μ Δ which are already reversely prescribed in Boxes 23, 24, 28, and 33 in Step 1) are calculated in Boxes 16 to 33 as shown in Figures 25 and  26. Design accuracies based on the pre-assigned deflection ductility ratio (μ Δ ) of 2.50 are improved because the pre-assigned deflection ductility ratio (μ Δ ) is within the range between 1.75 and 2.75 whereas design accuracies based on the pre-assigned deflection ductility ratio (μ Δ ) of 3.00 are not improved as much as those with deflection ductility ratio (μ Δ ) of 2.50. After all, the ductility ratio (μ Δ ) of 3.00 falls outside the range between 1.75 and 2.75 as shown in Figure 22.
In Figures 27(a,b), ANNs with R3.CRS.c and R3.CRS.b are trained based on 40 layers-40 neurons and 30 layers-30 neurons for the reverse networks of Step 1,  respectively. The deflection ductility ratios (μ Δ ) of 1.75 at ULL for Figure 27(a) and 2.00 at ULL and for Figure 27(b) are reversely pre-assigned based on q L/250 = 80 kN/m at SLL and q str = 150 kN/m at ULL. ANNs with FW.PTM.c based on 40 layers and 40 neurons are implemented in training the forward network of Step 2 when 18 outputs including reverse input parameters (q L/250 ,q 0.2 mm ,q str , and μ Δ which are already reversely prescribed in Boxes 23, 24, 28, and 33 in Step 1) are calculated in Boxes 16 to 33 as shown in Figures 27(a,b). Design accuracies based on the preassigned deflection ductility ratio (μ Δ ) of 2.00 shown in Figures 27(b) are improved because the ductility ratio (μ Δ ) of 2.00 is within the range between 1.75 and 2.75 whereas design accuracies based on the ductility ratio (μ Δ ) of 1.75 shown in Figures 27(a) are not improved as much as those with the ductility ratio (μ Δ ) of 2.00. This is due to the ductility ratio (μ Δ ) of 1.75 being on the data region's boundary, as shown in Figure 22.

Design based on CRS using shallow neural networks (SNN)
Table 7 presents training accuracies of a reverse scenario based on CRS using three types of deep and six types of shallow layers implemented in obtaining reverse outputs (b, ρ st ,ρ sc , and ρ p ) in the reverse networks (based on DNN and SNN) of Step 1. The sequence of the training networks determined for deep layers is also used for shallow networks as can be seen in Table 7. The reverse output parameter, beam width b, is firstly obtained in Box 1 of Figure 28 (a) based on R3.CRS.d. The rest of the reverse output parameters, ρ st ,ρ sc , and ρ p , based on R3.CRS.d are also obtained in Boxes 5, 6, and 10 of Figure 28(a), respectively. R3.CRS.a to R3.CRS.i using three types of deep and six types of shallow layers shown in Table 7 are   Similarly, deep ANNs trained by R3.CRS.a to R3. CRS.c based on 20 to 40 hidden layers -20 to 40 neurons and shallow ANNs trained by R3.CRS.d to R3.CRS.i based on 1 and 2 hidden layers -20 to 40 neurons are used to map input parameters to the rest of the reverse output parameters, ρ st ,ρ sc , and ρ p , respectively, For example, R3.CRS.10e represents an ANN trained based on 1 hidden layer -30 neurons to obtain ρ p in the reverse network of Step 1 as shown in Table 7. ANNs with FW.CRS.c are implemented in determining 18 outputs in the forward network of Step 2 as shown in Boxes 16 to 33 of Figure 28.
A dataset representing 15% of the total large datasets that are not used to train networks is used to verify training accuracies. Users can, however, provide the dataset ratio for verifying training accuracies. Figures 28(a-f) investigate the impact of a number of hidden layers and neurons implemented in Step 1    reverse networks on design accuracies obtained in Step 2 forward networks. When using shallow ANNs trained by R3.CRS.d to R3.CRS.i, design accuracies are similar to those obtained by deep ANNs trained by R3. CRS.a to R3.CRS.c, implying that ANNs trained with the CRS method in the reverse forward network of Step 1 can serve to determine accurate reverse outputs (b, ρ st , ρ sc , and ρ p ) with both deep and shallow layers as long as enough neurons are used. However, design accuracies heavily depend on the ranges of datasets. Preassigning a deflection ductility ratio (μ Δ ) out of the ranges between 1.7 to 2.75 causes significant errors as shown in Figure 28. The reversely pre-assigned input parameters should be adjusted to reduce design accuracies within the ranges between 1.7 to 2.75.
In Figure 28, design accuracies are compared with those based on structural calculations. Acceptable design accuracies are found as shown in Figures 28(a-f) where the a deflection ductility ratio (μ Δ ) of 2.00 is reversely pre-assigned on an input side. Errors of upper rebar strains (ε sc ) and deflection (camber) at transfer load limit state range from −10.77% to −11.68% and from −9.46% to −16.62%, respectively, when a reverse scenario is designed based on ANNs trained by R3.PTM. d, R3.CRS.e, R3.CRS.f, R3.CRS.g, R3.CRS.h, R3.CRS.i with 1 layer -20 neurons as shown in Figures 28(a-f), respectively, where the deflection ductility ratios (μ Δ ) of 2.00. Step 1) shows higher design accuracies compared with those obtained in Figure 16(a) based on ANNs trained by R3.PTM.c (40 layers-40 neurons) and Figure 28(e) based on shallow neural networks trained by R3.CRS.h (2 layers-30 neurons) when the deflection ductility ratios (μ Δ ) of 2.00 is reversely pre-assigned on an input-side.

Conclusions
Based on ANNs, this study demonstrates how to design pre-tensioned concrete beams. To establish reverse design scenarios, large amounts of input and output parameters are generated. For engineers, reverse designs with 15 input and 18 output parameters are proposed using ANN-trained reverse-forward networks. Here are some of the study's findings.
(1) Two-step networks are formulated to solve reverse design scenarios. In the reverse network of Step 1, reverse output parameters are determined which are, then, used as input parameters in the forward network of Step 2 to determine the design parameters.
(2) TED, PTM, and CRS can be used for both the reverse and forward networks with selected training parameters such as a number of hidden layers, neurons, and validation checks. Extra input features can be selected from the output-side when training ANNs based on CRS which is not possible with TED, in which all inputs are mapped to entire outputs simultaneously. For additional explanations of training methods such as CRS and TED, readers are referred to the book (Hong 2021) and (Hong 2019).
(3) Selection of training networks depends on many parameters such as feature scores, the volume of datasets, and types of big datasets. In particular, when the volume of datasets is insufficient, deep neural networks (DNN) provide less accurate design accuracies than shallow neural networks (SNN). Deep neural networks trained with CRS outperform shallow neural networks trained with CRS and ANNs trained with PTM for the reverse network in Step 1.
(4) ANNs formulated to train reverses datasets generated from pre-tensioned concrete beams are adequate to produce acceptable design accuracies for use in practical designs. Design tables derived from reverse designs can be extended to plot design charts that connect all design parameters to achieve entire designs in a streak. The reverse design facilitates the rapid identification of design parameters, helping engineers with fast decisions based on acceptable accuracies. engineering with hybrid composite structures. He has provided many useful solutions to issues in current structural design and construction technologies as a result of his research that combines structural engineering with construction technologies. He is the author of numerous papers and patents both in Korea and the USA. Currently, Dr. Hong is developing new connections that can be used with various types of frames including hybrid steel-concrete precast composite frames, precast frames and steel frames. These connections would help enable the modular construction of heavy plant structures and buildings. He recently published a book titled as "Hybrid Composite Precast Systems: Numerical Investigation to Construction" (Elsevier).