Machine learning based prediction model for plastic hinge length calculation of reinforced concrete structural walls

Abstract

Reinforced concrete structural walls (RCSWs) are integral part of buildings and other structures and are used to carry in-plane and out-plane loads. For assessment purposes and to ensure safety and resilience of the structure, the curvature, capacity, and strain demands of RCSWs need to be estimated. Nonlinear numerical models are increasingly used in earthquake engineering design and assessment, where it is critical to develop high fidelity simulation tools to precisely forecast the global and local behavior of RCSWs. Plastic deformations concentrated in small regions, i.e. plastic hinges, and characterized through the concept of plastic hinge length (PHL), are conventionally used to define the inelastic response of RCSWs. In this study machine learning (ML) algorithms were leveraged in evaluating the PHL of RCSWs. A database containing 721 planar and nonplanar RCSW samples were utilized for training and testing ML models. Four different algorithms were employed, namely: XGBoost, CatBoost, Random Forest (RF), and genetic programming. The RF model outperformed other ML counterparts. The results of the best ML model were also compared against empirical equations available in literature and nonlinear regression analysis. The proposed ML model provided much better predictions than the existing relationships. Predictions of most empirical models were found to be extremely over (or under) conservative. The proposed ML model was also presented via an online tool where users can obtain the PHL of any RCSW simply by inputting wall parameters and clicking a run button.

Keywords

reinforced concrete wall plastic hinge length machine learning empirical equations planar and nonplanar wall

Highlights

Developed a machine learning (ML) model to predict the plastic hinge length (PHL) of reinforced concrete structural walls (RCSWs).

Four ML models examined: XGBoost, CatBoost, Random Forest (RF), and genetic programming.

RF model outperformed other ML models, existing expressions, and regression models.

SHapley Additive exPlanation (SHAP) framework was used to identify the significant wall parameters on PHL.

A user-friendly web application is presented for the ML model, allowing practitioners to determine the PHL easily.

Introduction

Reinforced concrete structural walls (RCSWs) are particularly advantageous at bearing lateral loads like wind or seismic loads due to their high strength and rigidity (Cao et al., 2021). In regions of high seismic activities, using structural walls to counteract lateral loads is a widely common practice (Barkhordari, 2022; Cao et al., 2022). It is desirable to design a structural wall to behave inelastically during a devastating earthquake (Wang et al., 2021), where it can maintain large deformations without failure (Barkhordari and Tehranizadeh, 2021). The characteristics of the structural system affect the pattern and size of the inelastic deformations inside the structure (Barkhordari, 2022). To effectively assess structural walls, it is essential to follow well-established guidelines that consider a complete understanding of the wall’s behavior for the geometry, materials, and loading (Massone et al., 2021; López et al., 2022).

Although the inelastic curvature will vary significantly as the wall height increases, it is conventional to assume it is uniform above the base (Figure 1) at a height corresponding to the length of the plastic hinge (Nasr et al., 2022). Tall buildings have several floor slabs connecting the walls, causing complicated interactions between walls and floor slabs (Barkhordari and Es-Haghi, 2021). To ensure consistency of the overall deformations at the story level, slender walls exhibit larger flexural deformations around the wall base (Beiraghi and Siahpolo, 2017; Beiraghi, 2018a, 2018b; Barkhordari et al., 2021). Furthermore, a large plastic hinge is more likely to occur at higher levels of tall buildings due to higher-mode effects (Panagiotou and Restrepo, 2009; Beiraghi, 2018c, 2019a, 2019b; Beiraghi and Kheyroddin, 2021). As a result, the assumed value of the plastic hinge length (PHL) has a significant impact on the plastic deformation and associated maximum displacement capacity of RCWs (Pozo et al., 2021).

Figure 1.

Curvature distribution of shear reinforced concrete wall.

Many studies initially advocated using a PHL equal to half of the wall length ( $0.5 \times l_{w}$ ) as a reasonable, lower-bound estimate when computing a conservative value of the displacement capacity of RCWs (Hoult, 2022; Hoult et al., 2018). This recommendation was mostly based on beam experiments (Hoult 2022; Hoult et al., 2018). Numerous studies demonstrated that assuming PHL to be equal to $0.5 l_{w}$ may result in an overestimation of PHL (Bohl and Adebar, 2011a). Lu et al. (2017) used experimental campaigns and proved that the minimal vertical reinforcing restrictions are insufficient to prevent a significant number of secondary flexural cracks developing in the region of the plastic hinge. The response of the walls was governed by the main sizable flexural cracks at the base of the walls. This tendency significantly slowed the propagation of plasticity and led to a number of possible problems, including a lower drift limit and early reinforcement fracture or buckling. They suggested that PHL of the tested walls was less than $0.5 l_{w}$ . Kazaz (2013) used the findings of nonlinear finite element studies to establish an analytical formula for the determination of PHL for RCWs. The study findings of the parametric investigation rely on several variables, such as wall length, wall height, and axial load ratio, to name but a few, to calculate the change of plastic zone length at the base of the cantilever shear wall models. Additionally, Kazaz (2013) demonstrated that it is unrealistic to have a PHL that corresponds to a fixed percentage of the wall length. When examining PHL of nonplanar RCSWs, this finding becomes especially clear since the shear stress level is identified as a critical parameter in the determination of PHL for RCSWs (Constantin, 2016; Hoult et al., 2020).

Although different PHL formulations are available for design or evaluation purposes, many are based on limited databases (Massone and Alfaro, 2016; Hoult, 2022) or are developed for other structural members like columns (Wakjira et al., 2021). In addition, many of these empirical equations are derived from experimental or computational studies on beams or columns (Panagiotakos and Fardis, 2001). Moreover, recently developed equations were found to extremely under or overestimate PHL (Hoult, 2022). Therefore, an accurate and reliable model is needed to estimate PHL of RCSWs. With the emergence of machine learning (ML) methodologies, popular tools such as artificial neural networks (ANNs) and ensemble models have become increasingly used in material science and structural engineering research and applications. As an example, Barkhordari and Massone (2022a) developed ensemble deep neural network algorithms to predict the failure mode of RCSWs. They suggested that the weighted average ensemble model, which has the highest accuracy of the competing models, be used for determining the failure mechanism. Barkhordari and Tehranizadeh (2021) established a hybrid algorithm using ANN and Simulated Annealing (SA) algorithm to estimate stresses, strains, and curvature of RCSWs. Zhang et al. (2022) utilized various algorithms to compute failure modes, strength, and deformation of RCSWs. The results showed that the XGBoost algorithm properly identified the RCSWs failure modes. Furthermore, the lateral strength and ultimate drift were well predicted by the gradient boosting and Random Forest (RF) methods. Table 1 provides a summary of various ML models available in literature for evaluating RCSWs, their inputs, algorithms, and result capabilities (Barkhordari and Es-Haghi, 2021; Keshtegar et al., 2021; Nguyen et al., 2021; Pizarro et al., 2021; Topaloglu et al., 2022; Xiong et al., 2022; Barkhordari and Massone 2023; Yaghoubi et al., 2023).

Table 1.

Applications of various ML algorithms in RCSWs.

Ref.	Dataset size	Algorithm used	Output
Barkhordari (2022)	393	Averaging ensemble, weighted average ensemble, and integrated stacking ensemble	Failure mode
Barkhordari and Tehranizadeh (2021)	150	Hybrid ANN-SA	Stresses, strains and curvature
Zhang et al. (2022)	429	K-nearest neighbors, decision tree, ANN, light GBM, CatBoost, XGBoost, gradient boosting, and random forest	Failure modes, strength, and deformation capacity
Keshtegar et al. (2021)	434	Honey badger algorithm with ANN, hunger games search with ANN, fitness-distance balance coyote optimization algorithm with ANN, averaging ensemble, snapshot ensemble, and stacked generalization ensemble	Shear strength
Nguyen et al. (2021)	369	ANN	Shear capacity
Pizarro et al. (2021)	209	Support vector regression and response surface model	Shear capacity
Barkhordari and Es-Haghi (2021)	600	Hybrid ANN-SA	Displacement, drift, and base shear
Topaloglu et al. (2022)	161	Robust linear regression, K-nearest neighbor regression, kernel ridge regression, support vector regression, and Gaussian process regression	Equivalent damping ratio
Xiong et al. (2022)	181	Naïve bayes, nearest neighbors, logistic regression, SVC, decision tree, random forest, AdaBoost, gradient boosting	Failure mode
Barkhordari and Massone (2023)	312	Gaussian process regression	Energy dissipation
Yaghoubi et al., (2023)	165	Convolutional networks	Conceptual structural design

The main goal of this study is to fill a gap in the literature by developing a comprehensive and useful ML based model that processes large amounts of data using different ML algorithms with a database of nonplanar and rectangular RCWs t predict PHL for RCSWs. Four different algorithms are considered including XGBoost, CatBoost, RF, and Genetic Programming (GP). The root-mean-squared error, coefficient of determination, mean absolute percentage error, and median absolute error are utilized as assessment metrics to assess the efficiency of ML models. Results showed the RF algorithm outperformed other ML models in terms of assessment metrics. The results of the best model are compared against previous equations developed for PHL calculation. The performance of the proposed model was much better than the existing relationships. Also, the SHapley Additive exPlanation (SHAP) method is employed to describe the outputs of the best model and identify the most notable variables in determining the PHL.

Dataset description

The accuracy and robustness of prediction in ML models is significantly influenced by the inputs. This study takes into account a brand-new database of nonplanar and rectangular RCSWs which is compiled by (Hoult, 2022). The database contains 721 planar and nonplanar reinforced concrete structural walls (RCSW) samples. The PHL of RCSWs is the desired output variable. The input parameters of this database are the ratio of the secondary cracking ( $r_{s c}$ , Equation (1)), wall length ( $l_{w}$ ), wall effective height ( $H_{e} = M / V$ ), the ratio of axial load ( $A L R$ ), and shear stress variable ( $S S P$ , Equation (2)).

r_{s c} = \frac{ρ^{w}}{ρ^{w, \min}} \leq 1, ρ^{w} = n_{l} \frac{A_{s}}{A_{g}}, ρ^{w, \min} = \frac{(t_{w} - n_{t} d_{b t}) f_{c t f}}{f_{u} t_{w}}

(1)

S S P = \frac{V_{M}}{A_{g} \sqrt{f_{c}}}

(2)

where

V_{M}

A_{g}

f_{c}

A_{s}

n_{l}

t_{w}

d_{b t}

f_{c t f}

, and

f_{u}

are the maximum shear, section area of RCSW, compressive strength of concrete, longitudinal bar area, number of longitudinal bars, wall thickness, transverse reinforcement diameter, tensile strength of concrete, and the ultimate strength of bars, respectively. Figure 2 shows a typical wall with illustrations on loading conditions and cross section details. The database considers four wall sections, namely: rectangular, H-shaped, Barbell, and U-shaped. Table 2 shows the statistical properties of input and output parameters. It should be noted that 70% of RCSW samples were utilized for the training set, 15% for testing set, and the same for validation set.

Figure 2.

Cross-sections of RCSWs in database compiled by Hoult (2022).

Table 2.

Statistical properties of input and output variables.

Statistical properties	$r_{s c}$	$l_{w}$ (mm)	$M / V$ (mm)	ALR	SSP	PHL (mm)
SD	0.13	2610	11,048	0.07	0.23	879
Minimum	0.15	507	500	0.00	0.02	21
Maximum	1.00	9000	54,900	0.43	2.22	4889

A correlation matrix (Figure 3) was used to show how the input parameters relate to one another. As can be seen, all input parameters had very low correlations (<0.5). The ratio of shear span ( $M / V$ ) did seem to correlate with the wall length ( $l_{w}$ ). The ratio of the secondary cracking ( $r_{s c}$ ), did not seem to significantly correlate with other input parameters. Figure 4 displays inputs and the output (PHL) correlation.

Figure 3.

Correlation matrix of the inputs.

Figure 4.

Inputs and output correlation.

Description of ML models

Among many algorithms, some have been reported to perform well in previous studies (Barkhordari and Es-Haghi, 2021; Feng et al., 2021; Sarir et al., 2021; Asteris et al., 2022; Liu et al., 2022), have been used in this research. Four types of ML models, namely Extreme Gradient Boosting (XGBoost), RF, Categorical features-support gradient Boosting (CatBoost), and GP, are developed in this study to forecast the PHL of RCSWs. For XGBoost, RF, and CatBoost models, Bayesian optimization is employed to efficiently tune their hypermeters (Abdollahi et al., 2022).

XGBoost model

Chen and Guestrin (2016) created the decision-tree-based ensemble ML technique known as XGBoost. This approach, which is a more sophisticated variant of the gradient boosting algorithm, was created to produce results with both high accuracy and little chance of overfitting (Asteris et al., 2022). Three key components of gradient boosting are a loss component to be minimized, a weak learner to produce estimates, and an additive model to append more weak learners. Regularization is appended to the target function with a loss function as an upgrade from a straightforward gradient boosting approach to get rid of overfitting. The model’s objective function (J) and general formula for the prediction at step t are written as follows:

\begin{array}{l} J (Θ) = L (Θ) + Ω (Θ) \\ f_{i}^{t} = \sum_{k = 1}^{t} f_{k} (x_{i}) = f_{i}^{(t - 1)} + f_{t} (x_{i}) \\ J (Θ) = \sum_{i = 1}^{N} L (y_{i}, {\overset{⌢}{y}}_{i}) + \sum_{k = 1}^{K} Ω (f_{k}) \end{array}

(3)

where

Θ

is the vector of parameters,

L (Θ)

is the loss function, and

Ω (Θ)

is the regularization term.

x_{i}

f_{i}^{t}

and

f_{t} (x_{i})

y_{i}

{\overset{⌢}{y}}_{i}

f_{k}

, and

k

denotes the input parameter, the predictions, learner, real value, model output, score function of an independent tree, and number of trees, respectively. The hypermeters of XGBoost are tuned using Bayesian optimization. The hyperparameters of XGBoost and their optimized value are listed in Table 3. Bayesian optimization techniques are considered one of the effective methods to evaluate the quantity of functions. The capacity of Bayesian optimization to consider prior knowledge of the issue at hand and guide sampling and assist in balancing exploration and exploitation of the search area accounts for a significant portion of its efficiency. This is likely because the method is based on the well-known "Bayes' theorem,” which can be simplified to show that the posterior probability of a theory or hypothesis M, given information E, is related to the likelihood of E given M multiplied by the prior probability of M (Barkhordari and Billah, 2022; Fu et al., 2022).

P (M | E) \propto P (E | M) \cdot P (M)

(4)

Table 3.

Hyperparameter of XGBoost model.

Hyperparameter	Subsample ratio (level)	Subsample ratio (tree)	Learning rate	Max. Tree depth	Number of trees	Training instance
Value	0.8	0.7	0.1	3	100	0.89

The posterior probability serves as a stand-in for an objective function in this way. The surrogate function can be utilized to approximate the objective function and use that information to guide future sample decisions. While sampling, the posterior is carefully used in what is known as the “acquisition” function, for example when acquiring extra samples. The acquisition will improve the conditional probability of positions in the search to create the next instance, whereas the model leverages the knowledge of the objective function to sample the portion of the search space that is most likely to be profitable. The posterior is modified once fresh examples have been added to the dataset and evaluated using the objective function. This procedure is continued until the extrema of the target function are found, a satisfactory outcome is found, or the available resources are depleted (Fu et al., 2022).

Random forest model

Regression trees (RTs) have a very high variation, and a small change in the input data can have a big impact on the output because of the training process’ inherent hierarchy, which causes an error in the top splitting to ripple down to all the splits below (Liu et al., 2022). As a result, overfitting frequently occurs, which lowers the RT model’s capacity for applicability. To address the instability of a standalone RT, RF is offered as a simple ML model. Following the continuous learning approach, RF builds many RT models based on subsamples arbitrarily chosen from the training set using a bagging strategy. Additionally, each RT model’s input feature count is randomized in relation to the total features. Finally, the results from all RT models are integrated to generate an inference based on a particular input by using averaging. In Figure 5, the RF procedure is depicted. The optimized hyperparameters’ values for the tunning process are shown in Table 4.

Figure 5.

RF model procedure.

Table 4.

Hyperparameter of RF model.

Hyperparameter	Max. Tree depth	Number of features	Min. Samples at a leaf node	Min. Samples for an internal node	Number of trees
Value	12	5	1	2	180

CatBoost model

XGBoost is comparable to CatBoost, an updated Gradient Boosting Decision Tree framework (Prokhorenkova et al., 2018). Traditional boosting techniques reduce errors by changing a training set’s weights (Alzubi et al., 2022). One of CatBoost’s main advantages is its capacity to combine many data kinds, like photos, audio, and text features. In contrast to most ML algorithms, which are unable to manage non-numeric values, CatBoost provides an unconventional method of managing categorical data, needing only a minimal amount of categorical feature transformation. The trees are instead generated by CatBoost by enforcing the requirement that all nodes at the same grade test the same predictor under the same conditions (symmetric trees). As a result, an index of a leaf can be determined using bitwise operations. The tree process enables a straightforward fitting strategy, and the tree architecture functions as a regularization to identify the best solution and prevent overfitting. In other words, symmetric trees, also known as balanced trees, in CatBoost refer to trees where the separating criterion holds true for all nodes at the same level of the tree. On the other hand, XGBoost produces asymmetric trees, which means the separation criteria for every node within the same level can vary. Figure 6 displays CatBoost structure versus XGBoost structure. The hyperparameters of CatBoost and their optimized value are listed in Table 5.

Figure 6.

CatBoost structure versus XGBoost structure.

Table 5.

Hyperparameter of CatBoost model.

Hyperparameter	Tree depth	Max. Number of trees	L2 regularization	Learning rate	Training samples in a leaf
Value	3	359	2.2	0.46	29

Genetic programming model

GP, an evolutionary computing algorithm, finds extremely realistic near-optimal expressions for complex systems using heuristic optimization techniques (Naderpour et al., 2022). The following explanations provide a summary of the primary GP evolutionary cycle processes:

1. Initialization: immediately after the analysis starts, GP model initializes a predetermined number of arbitrary candidates. The candidates are sometimes referred to as a model or a solution. Trees expressions are frequently employed to represent GP individuals. Figure 7 depicts a straightforward illustration of a GP model that is similar to equation (5). The population is the group of candidates with which GP begins.

x_{1}^{3} + (x_{2} \times \frac{x_{4}}{x_{3}})

(5)

2. Fitness evaluation: following initialization, a preset assessment metric is used to investigate the performance of the produced individuals. The correlation between the predicted values acquired by the candidate and the required real values is measured using the root mean squared error. Through this procedure, the created candidates' predictability quality is assessed.

3. Reproduction: at this point, a series of GP operators are executed to candidates that were probabilistically chosen based on their fitness value. The selection strategy refers to the manner in which this procedure is carried out. The more weight a candidate possesses, the more likely it is that they will be chosen. Some genetic operations are: (1) Crossover: Crossover is the process by which two candidates interchange some of their structural components. Crossover operators come in a variety of forms, from simple to complicated. In the straightforward crossover scenario shown in Figure 8, two candidates swap randomly chosen subtrees from their models; (2) Mutation: This operation is performed on a single individual by replacing a random location in the tree model of the individual with a different randomly created subtree. The rate at which the mutation operator is used is typically lower than the crossover rate. That is done to prevent the mutation operator’s substantial alterations from significantly destroying quality among individuals. The mutation operator, on the other hand, aids in preserving variation over the period of iterations. The newly created populations substitute the old ones after executing genetic operations repeatedly; (3) Elitism: Elitism is a straightforward strategy frequently used to keep the best candidates in the population and pass them unchanged to the following generation.

Figure 7.

A simple GP candidate.

Figure 8.

GP crossover procedure.

Evaluation metrics

The root-mean-squared error (RMSE, Equation (6)), coefficient of determination (R², Equation (7)), mean absolute percentage error (MAPE, Equation (8)), and median absolute error (MAE, Equation (9)) are employed to assess the efficiency of four ML models. A better fit is indicated by a higher R² value. Lower RMSE, MAPE, and MAE values are preferred and signify better model performance.

R M S E = \sqrt{\sum (\frac{{(A_{t} - F_{t})}^{2}}{N})}

(6)

R^{2} = \frac{{[\sum (A_{t} - {\bar{A}}_{t}) (F_{t} - {\bar{F}}_{t})]}^{2}}{{\sum (A_{t} - {\bar{A}}_{t})}^{2} {(F_{t} - {\bar{F}}_{t})}^{2}}

(7)

M A P E = \frac{1}{N} \sum_{t = 1}^{N} | \frac{A_{t} - F_{t}}{A_{t}} |

(8)

M A E = m e d i a n (| A_{1} - F_{1} |, . . ., | A_{N} - F_{N} |)

(9)

where

A_{t}

is the observed value,

{\bar{A}}_{t}

is the average of the observed value,

F_{t}

is the predicted value,

{\bar{F}}_{t}

is the average of the predicted value, and

N

is the number of samples in the set.

Results and discussion

The ML models are created using the chosen dataset and the identified optimal hyper-parameters. Comparison of the predicted and observed PHL for training and testing subsets are used to assess the performance of the models. Figure 9 compares the predicted versus real values for four ML models in the training phase. The diagonal line (y = x) denotes that the estimation is equal to the real value. Performance indices of four models are listed in Table 6 for the training stage.

Figure 9.

Predicted values versus real values for training phase: (a) XGBoost; (b) RF; (c) CatBoost; (d) GP.

Table 6.

Performance indices of four models – training phase.

Models	Indices
Models	RMSE (mm)	R²	MAPE	MAE (mm)
XGBoost	138.40	0.974	0.186	79.376
RF	118.19	0.981	0.115	42.572
CatBoost	96.89	0.987	0.108	51.324
GP	300.20	0.880	0.442	166.701

It can be observed from Figure 9 that all the ML models obtain good outcomes as the dots (predicted values) are very close to the diagonal line except for the GP model. Overall, the CatBoost model achieves better performance than other models with RMSE, R², MAPE, and MAE values of 96.89 mm, 0.987, 0.108, and 51.324 mm, respectively. The RF model obtains the second rank among the models in the learning phase. In addition, the GP model has the worst performance among all ML models in the training stage. In the training stage, the R² values of the RF, CatBoost, and XGBoost models are all greater than 0.95, demonstrating the high level of estimation accuracy of these three models in the training stage. The RF model contains a single best quantitative metric in terms of the MAE with the lowest value of 42.57 mm.

The predictions of the models in the testing stage are compared with the experimental results in Figure 10. Also, Table 7 presents the performance indices of four models in the testing stage. By comparing the four quantitative metrics, the RF model has the four quantitative indicators with the best values in the testing stage than other models, namely the MAPE with the lowest value of 0.201, R² with the highest value of 0.889, MAE with the lowest value of 100.974 mm, and the RMSE with the lowest value of 308.68 mm. The CatBoost model attains the third rank in the testing phase in terms of RMSE, R², and MAPE. The RF model outperforms other models in the testing phase; therefore, it is selected as the best model.

Figure 10.

Predicted values versus real values for testing phase: (a) XGBoost; (b) RF; (c) CatBoost; (d) GP.

Table 7.

Performance indices of four models – testing phase.

Models	Indices
Models	RMSE (mm)	R²	MAPE	MAE (mm)
XGBoost	314.32	0.885	0.281	126.175
RF	308.68	0.889	0.201	100.974
CatBoost	328.03	0.874	0.263	119.313
GP	340.29	0.865	0.390	191.425

Existing PHL prediction models

Six existing empirical equations for the PHL of the RCSWs are also employed as a comparison to further support the suggested prediction model’s validity and show its efficiency. Bohl and Adebar (2011a) evaluated the pattern of inelastic curvatures that should be utilized to approximate the flexural displacement capacity of RCSWs using a nonlinear finite element model. A model that is able of estimating the equivalent PHL, as expressed in equation (13), was consequently established based on the range of RCSWs with different shear span ratios, wall lengths, and axial compression levels. Priestley et al. (2007) suggested a plastic hinge formula to satisfy serviceability and damage levels. The tension shift brought on by diagonal shear cracks and the strain penetration into the foundation are both taken into consideration by the model as factors of the bar diameter and the ultimate-to-yield strengths ratio of the rebars. Kazaz (2013) developed a model based on the regression analysis considering wall length, shear span ratio, axial load ratio, and the ratio of the reinforcement. The finite element shear wall model and Turkish seismic code regulations were also used to calibrate the model. In order to determine the length of the plastic hinge zone, two distinct approaches were used (11). The first approach used an analysis of the wall-cross sectional strain pattern to determine the ultimate curvature pattern when the outside tension steel achieved a strain of 0.06; while the second technique involved calculating the curvature pattern employing the shell element strains added to the finite element model at the wall’s opposite edges. Hoult (2022) used the same database used in this paper along with conventional regression methods and presented a new relationship for PHL.

The following is a concise summary of the empirical equations:

NZS 3101.1:2006, ASCE/SEI 41-06 (2006)

P H L = 0.5 l_{w}

(10)

Priestley et al. (2007)

\begin{array}{l} P H L = k H_{e} + 0.2 l_{w} + 0.022 f_{y e} d_{b l} \\ k = 0.2 (f_{u} / f_{y} - 1) \leq 0.08, f_{y e} = 1.1 f_{y} \end{array}

(11)

Eurocode 8 (2005)

P H L = \frac{H_{e}}{30} + 0.2 l_{w} + 0.11 (\frac{d_{b l} f_{y}}{\sqrt{f_{c}^{'}}})

(12)

Bohl and Adebar (2011a)

P H L = (0.2 l_{w} + 0.05 H_{e}) (1 - 1.5 A L R) \leq 0.8 l_{w}

(13)

Kazaz (2013)

P H L = 0.27 l_{w} (1 - A L R) (1 - \frac{f_{y} ρ_{T R R}}{f_{c}^{'}}) {(\frac{H_{e}}{l_{w}})}^{0.45}

(14)

Hoult (2022)

P H L = r_{s c} (0.13 l_{w} + 1.8 H_{e}^{0.6}) (1 - 0.3 A L R) (1 - 0.13 \log (S S P))

(15)

Where $H_{e}$ is the wall effective height, $d_{b l}$ is the bars’ diameter, $f_{u}$ is the bars’ ultimate strength, $f_{y}$ is the bars’ yield strength, $f_{c}^{'}$ is the strength of the concrete, and $ρ_{T R R}$ is the transverse reinforcement ratio, other parameters are previously defined. Figure 11 and Table 8 compare the calculated PHL of the various empirical equations with the experiments using the test set. It can be observed that the PHL calculated using equation (10) was mostly higher than the observed results. It is consistent with previous studies since numerous investigations on the length of the plastic hinge of RCSWs have found that a length of $0.5 l_{w}$ can overestimate the real plastic hinge length (Hoult et al., 2018; Bohl and Adebar, 2011b; Hoult, 2022). This overestimation can be attributed to the influence of some parameters that were neglected. On the other hand, the performance measures of the empirical equations show that Equation (13) and Equation (15) are close to the experimental results than other empirical equations. Equation (15) has the best performance with the MAPE value of 0.471, the R² value of 0.766, the MAE value of 206.895 mm, and the RMSE with the lowest value of 447.71 mm. However, by comparing the results of Tables 7 and 8, it is observed that the proposed ML model (RF model) in this study outperforms the empirical equations.

Figure 11.

Scatter diagrams of the existing equations results. (a) Eq.10; (b) Eq.11; (c) Eq.12; (d) Eq.13; (e) Eq.14; (f) Eq.15.

Table 8.

Performance measures of empirical equations.

Models	Indices
Models	RMSE (mm)	R²	MAPE	MAE (mm)
Equation 10	1137.08	0.302	0.465	668.117
Equation 11	545.13	0.653	0.732	303.411
Equation 12	636.65	0.527	0.503	258.198
Equation 13	485.43	0.725	0.474	250.990
Equation 14	702.07	0.425	0.593	194.564
Equation 15	447.71	0.766	0.471	206.895

Interpretation of best ML model

The majority of ML-based models typically do not produce mathematical expressions that clearly connect the predicted outcome to the input parameters. This study utilizes a state-of-the-art method (SHAP (Lundberg and Lee 2017) method) to describe the outputs of the RF model and identify the most notable variables in determining the PHL. In order to create an understandable model, SHAP employs an additive characteristic assignment method, where an outcome framework is described as the linear sum of input parameters. The explaining model $E m (x^{'})$ with a modified input $x^{'}$ for an original model $f_{o r g} (x)$ is calculated as:

f_{o r g} (x) = E m (x^{'}) = φ_{0} + \sum_{i = 1}^{M} φ_{i} x_{i}^{'}

(16)

where

x

is input variable (

x_{1}, x_{2}, . . ., x_{n i v}

n i v

is the number of input variables,

M

is the number of samples features.

φ_{0}

reflects the fixed value when all the input variables are absent and

φ_{i}

are SHAP values.

x^{'}

have relationships to

x

through a mapping function. Lundberg and Lee (2017) showed the Shapley values are the solution to find

φ_{i}

. Further information about the formulation and application of SHAP algorithm can be viewed in (Lundberg and Lee 2017).

Figure 12(a) displays the importance of each input variable. It can be observed from Figure 12(a) that the most important input variable is the wall effective height ( $H_{e} = M / V$ ), followed by the wall length ( $l_{w}$ ) and the axial load ratio ( $A L R$ ). Stress parameter (SSP) also seems to impact the development of plasticity even it doesn’t appear to be as significant as some other variables. Figure 12(b) shows the summary plot for the prediction of the PHL. The input variables are ranked in order of relevance in Figure 12(b). The hue in this diagram denotes the variable value, which ranges from low (blue) to high (red), and the dots in it stand for the SHAP values of the specimens (in the test set) for each input variable. Figure 12(b) displays that the model is more likely to anticipate larger PHL as the value of the ratio of shear span rises, which relates to a higher possibility of RCSW yielding in flexure before meeting the RCSW’s nominal shear strength (Barkhordari and Massone 2022a, 2022b). Also, the higher values of wall length ( $l_{w}$ ) will increase the length of the plastic hinge. This issue can be related to the relationship between tension shift effects and $l_{w}$ . With a larger wall length, the tension shift effect grows and the plasticity spreads more widely (Pauley et al., 1992; Hoult, 2022). The higher values of SSP will reduce PHL because the shear behavior can have a significant influence on axial compressive strains even in slender RCSWs governed by flexure and can cause diminished lateral stiffness, strength, and ductility (Barkhordari et al., 2021; Massone et al., 2021; Kolozvari et al., 2019). According to Figure 12(b), the length of the plastic hinge has an inverse connection with the axial load ratio ( $A L R$ ) as opposed to the ratio of shear span. In other words, the PHL of the RCSWs is shortened by the axial load level. This is related to the concurrent occurrence of concrete crushing and reinforcement buckling in the compression zone, or concrete crushing in the compression zone before reinforcement yielding in the tension zone. As a result, the gap between the bending moment at the yielding point and the bending moment capacity gets shorter and shorter and the PHL reduction in axially loaded RCSWs occurs (Ren et al., 2022; Hashim et al., 2020). The impact of the ratio of the secondary cracking ( $r_{s c}$ ) on the PHL prediction cannot be interpreted using Figure 12(b) because the color points are intermingled and do not accurately depict the variation in the SHAP value with the alteration of the secondary cracking ( $r_{s c}$ ).

Figure 12.

SHAP method plots. (a) Relative importance; (b) Summary plot.

Regression models

In this section, the regression method is used to develop the model. Two cases are considered. The first case (C-1) is to consider samples that only have a rectangular section (because there are many of them in the database). The second case (C-2) considers all the data but with four input parameters (wall effective height, the wall length, ALR, and SSP) because based on the results of the SHAP method, the ratio of the secondary cracking has less impact than other parameters. Figure 13 compares the output of the regression models with the experiments. Table 9 presents performance measures of regression models. The performance metrics RMSE, R², and MAE of 449.83, 0.766, and 247.13 are achieved in C-1 which are 17.12%, 44.8%, and 36.0% better than C-2. In comparison with Table 7, ML models outperform the regression models in terms of RMSE, R², and MAE. The results show that the RF model could be employed in PHL approximation application with superior performance.

Figure 13.

Scatter diagrams of the models. (a) case 1 (C-1); (b) case 2 (C-2).

Table 9.

Performance measures of regression models.

Models	Indices
Models	RMSE (mm)	R²	MAPE	MAE (mm)
C-1	449.83	0.766	0.534	247.13
C-2	542.81	0.529	0.362	386.61

Web application

The excellent potential of the RF model for PHL calculation of RCSWs has been validated by prior comparisons. However, a significant obstacle to the practical application of ML techniques within routine design work has been the complexity of the processes involved in database development, model training, validation, and so on. To solve these problems, a user-friendly web application in Python has been developed that incorporates the model and its tuned hyperparameters. The web address provided in Appendix A, as seen in Figure 14, can be utilized to view the graphical user interface (GUI) of the RF model. Engineers only need to specify numerical inputs. When the “Run” button is pressed, the inputs are sent to the model that is running on a web server, and a forecast is displayed right away. Anyone can visit it using the provided URL without the need to install any additional software.

Figure 14.

GUI of the RF model.

Sensitivity analysis

In this section, a sensitivity analysis method (SAM) is conducted to quantify the sensitivity of a model’s state or outcome variations to fluctuations in each random parameter over the full range of the variable. The Sobol’s technique has received a lot of attention due to its simplicity and accuracy (Cacuci and Ionescu-Bujor, 2017; Ionescu-Bujor and Cacuci, 2017). The variance-based method, namely Sobol’s technique, is used as the SAM. This approach analyzes the model from a probabilistic point of view, treating the model’s inputs as an arbitrary vector using a joint probability density function. The model’s stochastic representation is shown as (Cacuci and Ionescu-Bujor, 2017; Ionescu-Bujor and Cacuci, 2017):

Y = M (X)

(17)

where

M

is a mathematical operator,

X

is a random input vector, and

Y

is a quantity of interest. The definition of Sobol’s index for the subset u of the input vector and the total Sobol’ indices are (Cacuci and Ionescu-Bujor, 2017; Ionescu-Bujor and Cacuci, 2017):

S_{u} = \frac{V a r (M_{u} (X_{u}))}{V a r (Y)}

(18)

S_{i}^{T} = \sum_{\frac{u \subset {1, . . ., n}}{i \subset u}} S_{u}

(19)

where

S_{u}

is Sobol’ index and

S_{i}^{T}

total Sobol’ index.

Figure 15 displays the results of the sensitivity analysis and shows that the wall length ( $l_{w}$ ) and wall effective height ( $H_{e} = M / V$ ), have a high total-order sensitivity coefficient, correspondingly large effect on the PHL of RCSWs. Contrary to SHAP method discussed in section of the interpretation of best ML model, the SAM shows the ratio of the secondary cracking ( $r_{s c}$ ), the axial load ratio ( $A L R$ ), and shear stress variable ( $S S P$ ) have negligible effects on the PHL of RCSWs.

Figure 15.

Sobol’ Total order results.

Conclusions

Due to their considerable stiffness, strength, and ductility, reinforced concrete structural walls (RCSW) are typically employed in structures to resist seismic loads. In addition, nonlinear numerical models are increasingly being used in earthquake engineering design. Therefore, it is crucial to create a practical model that can forecast the length of the plastic hinge of the RCSWs with acceptable accuracy. This paper assessed the application of some machine learning (ML) models in predicting the plastic hinge length (PHL) of RCSWs. Four ML models, namely Extreme Gradient Boosting (XGBoost), RF, Categorical features-support gradient Boosting (CatBoost), and Genetic Programming (GP), are used in this study. The following conclusions were drawn:

1. The results show the RF model outperformed other ML models with the lowest value of the MAPE, MAE, and RMSE as well as the highest value of R². In the training phase, the CatBoost model had the best performance but in the testing phase, the RF model outperformed.

2. The most effective ML model was explained using the SHapley Additive exPlanation (SHAP) framework. The outcomes of the SHAP method showed that the effective parameters in the PHL were the ratio of shear span ( $M / V$ ), followed by the wall length ( $l_{w}$ ) and the axial load ratio ( $A L R$ ).

3. The SHAP method revealed that the ratio of the shear span as well as the wall length both positively influence the PHL. However, the shear stress parameter (SSP) and the axial load ratio (ALR) showed a negative influence on the plastic hinge length.

4. Of the empirical equations available in literature for estimating the PHL of RCSWs, Hoult’s model was the most accurate.

5. Comparing the predictions of empirical equations and the RF based ML model, the latter provided much better predictions for the PHL of RCSWs.

6. A multilinear regression analysis was also performed to determine the PHL, using the existing wall database and the significant parameters identified by the SHAP method. The regression models yielded a large scatter in predictions of the PHL and were less precise than the RF ML-based model.

7. A user-friendly web application in Python has been developed that incorporates the model and its tuned hyperparameters. Users can obtain the PHL of any RCSW simply by inputting parameters and clicking a run button.

The presented ML models tend to perform significantly better than existing analytical models, for the database implemented in the study. However, a larger database with more parameters and datapoints per parameter might be needed to fully evaluate the model’s predictability. The model predictions could also be tested for hybrid walls such as concrete walls with stay-in-place fiber reinforced polymer (FRP) skins, and sandwich concrete walls. Future research can also concentrate on reducing the influence of unstructured and potentially abnormal data sources using abnormality detection methods. Novel, creative ML models must also be developed and evaluated. The bagging and bootstrap approach, as well as hybrid neural networks with metaheuristic algorithms, which are often used for creating ML models, can be used for this purpose.

Footnotes

Authors’ contributions

Mohammad Sadegh Barkhordari: formal analysis, investigation, resources, conceptualization, methodology, data curation, and writing—original draft. Akram Jawdhari: conceptualization, methodology, and writing—review. All authors have read and approved the final manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

ORCID iDs

Mohammad Sadegh Barkhordari

Akram Jawdhari

Appendix

Web application can be found online at: https://msbearthquake-plastichingelengthwalls1-streamphl-vb82yd.streamlitapp.com/

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