Efficient Ensemble Learning-Based Models for Plastic Hinge Length Prediction of Reinforced Concrete Shear Walls

Journal of Soft Computing in Civil Engineering 8-3 (2024) 128-154
How to cite this article: Safaeian Hamzehkolaei N, Barkhordari MS. Efficient ensemble learning-based models for plastic hinge
length prediction of reinforced concrete shear walls. J Soft Comput Civ Eng 2024;8(3):128–154.
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Efficient Ensemble Learning-Based Models for Plastic Hinge
Length Prediction of Reinforced Concrete Shear Walls
Naser Safaeian Hamzekolaei 1,*
; Mohammad Sadegh Barkhordari 2
1. Assistant Professor, Department of Civil Engineering, Bozorgmehr University of Qaenat, Qaen, Iran
2. Ph.D., Department of Civil Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
* Corresponding author: nsafaeian@buqaen.ac.ir
https://doi.org/10.22115/SCCE.2023.408189.1688
ARTICLE INFO ABSTRACT
Article history:
Received: 22 July 2023
Revised: 27 October 2023
Accepted: 20 November 2023
Reinforced concrete shear wall (RCSW) significantly
improves the seismic performance of buildings. Accurate
estimation of the plastic hinge length (PHL) of RCSWs is
crucial as it significantly impacts the plastic deformation,
ultimate displacement, and ductility capacity of RCSWs.
This study aims to develop practical machine-learning (ML)
models for PHL prediction of RCSWs. For this purpose, 721
data of nonplanar and rectangular RCSWs were utilized.
Deep neural network-based ensemble learning models
including Simple Averaging Ensemble (SAE), Stacking
Ensemble (SE), Snapshot Ensemble (SSE), and Deep Forest
(DP), were leveraged. Meanwhile, inherently ensemble-
learning-based (IELB) algorithms including the XGBoost,
RandomForest, CatBoost, HistGradientBoosting, AdaBoost,
Bagging, ExtraTrees, and GradientBoosting regressor, and
data-driven empirical equations were considered for
comparison. The Taylor diagram and statistical comparison
of the results revealed that the proposed SE model with the
gradient boosting regressor (GBR) meta-learner
(MAE=0.043, MSE=0.012, R2
=0.916) outperformed all
employed deep-and IELB-based ensemble algorithms as well
as the empirical formulas for PHL of RCSWs. The SHapley
Additive exPlanations-based model interpretation together
with Sobol's sensitivity analysis results revealed that the wall
length is the most crucial input variable, followed by the
effective height and the axial load ratio.
Keywords:
Reinforced concrete wall
(RCSW);
Plastic hinge length (PHL);
Machine learning;
Ensemble algorithms;
Gradient boosting regressor
(GBR);
SHapley additive exPlanations
(SHAPE).

N. Safaeian Hamzekolaei; M.S. Barkhordari/ Journal of Soft Computing in Civil Engineering 8-3 (2024) 128-154 129
1. Introduction
In recent years, the application of machine learning (ML), as one of the most effective artificial
intelligence (AI) approaches, for effectively modeling and forecasting structural performance has
become popular [1,2]. Various ML algorithms have been established and employed to predict the
structural design of buildings [3,4], the strength of material [5–7], collapse and damage
identification [8,9], structural reliability analysis [10], interfacial-bond/axial/flexural/shear
strength [1,11,12], and so on [13–15].
Chou et al. [16] developed metaheuristics-optimized single and ensemble machine learning
models by using extreme gradient boosting (XGBoost), jellyfish search, and symbiotic
organisms search algorithms for estimating the nominal shear capacity of the shear wall systems.
Barkhordari and Massone [17] employed ensemble deep neural network models for predicting
the failure mode of the shear walls. Barkhordari and Es-haghi [18] investigated the predictive
capability of a hybrid approach combining artificial neural networks (ANN) and the Simulated
Annealing (SA) method for forecasting the response of the reinforced concrete shear walls
(RCSWs) subjected to strong ground vibrations. Zhang et al. [19] utilized various ML algorithms
to predict failure modes, shear, and deformation capacity of the shear walls. Barkhordari and
Massone [20] utilized ensemble techniques and hybrid intelligence algorithms to predict the
shear strength of squat-reinforced concrete walls. Parsa and Naderpour [21] investigated the ML-
based shear strength prediction of the RCSWs using a hybrid support vector regression algorithm
improved by, Particle Swarm, Teaching-learning, and Harris Hawks Optimization algorithms.
Table 1 summarizes the information on the application of the ML models on the RCSWs.
Table 1
Summary of the application of the ML models on the RCSWs.
Application Description Refs.
Numerical model
optimization
ML-based traditional layered shell model optimization procedure was
investigated for accurate finite element analysis of RCSW structures.
[4]
Predicting the shear
strength
ML models was employed to predict the shear strength of RCSWs based on the
geometry, material properties, and reinforcement details. This information can be
used to design safe and more efficient structures.
[16,18,21–
24]
Predicting ductility
and energy
dissipation
The ductility and energy dissipation capacity of RCSWs were predicted based on
the geometrical and mechanical characteristics. The model can be used to design
structures that are less likely to collapse suddenly under extreme events.
[19,25,26]
Failure mode
identification
A crucial component of structural engineering, particularly in the context of
seismic design and structural safety, is the identification of the failure mode for
RCSWs. Engineers can evaluate the structural performance and make the
appropriate design changes by accurately identifying the failure modes.
[17,27–29]
Seismic risk
assessments
An accurate ML-based seismic risk assessments framework was proposed to
predict a building’s post-earthquake damage state using structural properties and
ground motion intensity measures as model inputs.
[30]

130 N. Safaeian Hamzekolaei; M.S. Barkhordari/ Journal of Soft Computing in Civil Engineering 8-3 (2024) 128-154
Ensemble learning (EL) has gained increasing popularity in recent years [31,32]. Multiple
systems and committee-based learning are alternative names for ensemble learning, which
involves training and combining multiple learners to tackle a learning challenge. Therefore, it is
used in this study to develop various predictive models for plastic hinge length (PHL) of
reinforced concrete structural walls (RCSW). RCSWs are frequently used as lateral resisting
systems to withstand the earthquake load in seismic regions. In order to assess and create cost-
effective designs, it is crucial to understand the response and behavior of RCSWs, particularly
given the growing popularity of performance-based design approaches for new structures. For
this purpose, many numerical models have been presented to predict the response of the RCSWs
[33–35]. In this type of analysis, the common assumption is that the plastic hinge length ( ph
l ) is
related to wall length ( w
l ), for example 0.5
ph w
l l
 . PHL has an influence on determining ultimate
strength and drift capacity. Therefore, accurate prediction of ph
l is important. Although 0.5
ph w
l l

was employed in the UBC-97 and more recently in NZS 3101:2006, however, previous studies
indicate that approximating the PHL as 0.5 w
l can provide an overestimation of ph
l [36–38].
Given that the straightforward definition of 0.5
ph w
l l
 is being utilized for seismic assessment,
this overestimation of ph
l may be problematic. Lu et al. [39] explored slender RCSWs with two
layers of vertical reinforcement. The tested walls were utilized to investigate the impact of
reinforcement ties, aspect ratio, and axial load effect. The experimental findings proved that the
minimal vertical reinforcing restrictions in place are insufficient to cause the development of a
significant number of secondary flexural cracks in the plastic hinge. They reported that ph
l values
were discovered to be substantially lower than 0.5 w
l . Moreover, for non-rectangular (nonplanar)
RCSWs, is not appropriate to assume that ph
l is proportionate to the length of the wall [36]. This
issue is related to the significantly higher shear stress that occurs in non-rectangular RCSWs. It
has been determined that the shear stress demand is a significant factor in determining ph
l of the
RCSWs [38,40]. According to Massone and Alfaro [41], the distributions of curvature along the
height of the wall are non-linear because of variations in steel-related characteristics, such as
hardening and reinforcement ratio. They demonstrated how the final curvature is influenced by
the degree of hardening and reinforcing. They stated that the outcomes are conservative with a
significant dispersion of results when estimating the plastic hinge length using 0.5
ph w
l l
 , and the
hinge model concentrates all curvature at the base of the wall, often leading to an
underestimation of the curvature.
Various empirical equations have been proposed to estimate ph
l of RCSWs [36,38,42,43].
However, in general, empirical equations may have not an acceptable accuracy, since (1) the
limited number of samples are used, (2) the limited range of input design variables are
considered, (3) walls with non-rectangular sections may not considered for developing some
equations, and (4) some of the formulas are established as a result of numerical or experimental
research on beams or columns [36,42]. The literature review shows that there is a need for a new
model considering a comprehensive database of the wall types and also effective parameters
(e.g., shear stress) for the estimation of ph
l because PHL has a significant effect on the

calculation of plastic deformation, ultimate displacement, and ductility capacity of the RCSWs
[41,42].
The main objective of this study is to evaluate the efficiency of the ensemble learning method-
based ML frameworks for the PHL prediction of the RCSWs. For this purpose, ensemble
learning algorithms including simple averaging ensemble (SAE), stacking ensemble (SE), and
snap-shot ensemble (SSE) as well as deep forest (DP) algorithm are employed. The game theory-
based interpretation approach is also utilized to explain the impact of the input variables on the
outcomes of the best model. Meanwhile, inherently ensemble-learning-based (IELB) algorithms,
including XGBoost, RandomForest, CatBoost, HistGradientBoosting, AdaBoost, Bagging,
ExtraTrees, and GradientBoosting regressor, and existing mechanics-driven and data-driven
equations are also investigated for comparison of the results.
The rest of the paper is organized as follows. The structural parameters of RCSW systems and
the experimental dataset are presented in Section 2. The applied individual and ensemble models
are discussed in Section 3. Performance evaluation of the applied ML models, the SHAP analysis
and the sensitivity analysis to explain the impact of input variables are investigated in Section 4.
Finally, concluding remarks are presented in Section 5.
2. Dataset for PHL of the RCSW
Fig. 1 shows the geometry and effective structural parameters of the common RCSW. Hoult [42]
has created a brand-new dataset including 721 samples of the nonplanar and rectangular RCSWs,
which is used in this work. The target variable is the PHL ( ph
l ) of the RCSWs. The secondary
cracking ratio ( scr
r ), wall length ( w
l ), effective height ( /
e
H M V
 ), axial load ratio (ALR =
P/Pn), and shear stress variable (SSV) are the inputs. The secondary cracking ratio and the shear
stress variable can be calculated according to Eqs. (1) and (2), respectively.
,min
,min
1,
( )
,
w
scr
w
sl w t tr cts
w sl w
ult w
r
A t n d f
n
A f t


 
 

 
(1)
Max
g c
V
SSV
A f


(2)
where Max
V , A, c
f  , cts
f , sl
A , sl
n , w
t , tr
d , ult
f , and w
 are the maximum shear force, the section
area of the RCSWs, concrete compressive strength, concrete tensile strength, the section area of
the longitudinal bar, number of bars in the longitudinal direction, wall thickness, transverse bars
diameter, ultimate strength of the bars, and longitudinal reinforcement ratio, respectively.
Table 2 shows the characteristics of the database where Ave, Std, Min, and Max stand for
average, standard deviation, minimum, and maximum, respectively. The histogram and
correlation matrix of input variables are also presented in Fig. 2. It can be seen from Fig. 2 that

only the wall length ( w
l ) has a correlation coefficient higher than 0.5 with effective height (
/
e
H M V
 ). The shear stress variable (SSV) has a correlation coefficient of 0.45. All other input
variables have a correlation coefficient less than 0.5 which indicates a weak relationship between
them.
Table 2
The statistical data for input parameters.
Variable Unite Min. Max. Ave. Std.
Secondary cracking ratio  
scr
r - 0.15 1.00 0.95 0.13
Wall length  
w
l (m) 0.51 9.00 4.05 2.61
Effective height  
/
e
H M V
 (m) 0.50 54.90 12.54 11.05
Axial load ratio (ALR) - 0.00 0.43 0.09 0.07
Shear stress variable (SSV) - 0.02 2.22 0.29 0.23
Plastic hinge length (PHL) (m) 0.021 4.889 1.188 0.87
Fig. 1. Geometry, loadings, and sections of the RCSWs.
d)

Following that, the data are randomly split into training and testing sets using the conventional
split of 80% to 20%, respectively [5]. Then, the database is transformed by scaling each feature
to a range [-1,1], as follows:
min
max min
2( )
1
( )
n
f f
f
f f

 

(3)
where f , n
f , min
f , and max
f denote a feature of the sample, the normalized value, the minimum
and maximum values of the feature, respectively.
Fig. 2. Histogram and correlation matrix of inputs.
Fig. 3 shows the relationship between each input parameter and the ph
l . As can be seen, there is a
negative correlation between the ALR and SSV versus ph
l as well as a positive correlation
between the secondary cracking ratio ( scr
l ), and effective height ( /
e
H M V
 )
versus ph
l . Therefore, the calculation of the PHL based on the conventional equation ( 0.5
ph w
l l
 )
may not be enough accurate.

a) b)
c) d) e)
Fig. 3. Input and output parameters correlation.
3. Individual and ensemble models
In ensemble learning, firstly, a number of individual base models, called base learners, are
trained separately utilizing common/arbitrary algorithms. Then, the base learners are merged
with a unified strategy (Fig. 4). If all basic learners in an ensemble learning algorithm are of the
same type, the algorithm is called homogeneous. In this study, homogeneous ensemble learning
algorithms with either deep neural network (DNN) or decision trees (DT)-based base learners are
employed. The applied models in this study include simple averaging ensemble (SAE), stacking
ensemble (SE), snap-shot ensemble (SSE), and deep forest (DP) algorithm [44,45].

Fig. 4. General workflow of ensemble learning models.
3.1. Base learners
An artificial Neural Network (NN) [20] is composed of interconnected artificial neurons that
resemble the neurons in a biological brain. When a neural network has more than one hidden
layer with computational neurons, it is called a deep neural network (DNN) [20]. Co-adaptation
is more likely to occur when there are many neurons in a fully connected layer. Co-adaptation
happens when multiple neurons in a layer derive similar hidden characteristics from the inputs or
highly similar hidden features. To address this issue, dropout can be used during NN training.
Dropout algorithms randomly disable a portion of neurons in a layer during each training step.
The dropout rate represents the proportion of neurons that are deactivated.
Table 3
Characteristics of the applied DDNs.
Base-model number
1 2 3 4 5
Drop rate (DR) and layer ith
neurons number (LiNN)
L1NN 35 15 15 10 15
DR 0.01 0.01 - 0.01 0.01
L2NN 25 15 15 20 15
L3NN - 45 20 15 30
DR - - - 0.01 0.01
L4NN - - 35 20 25
DR - 0.01 - -
L5NN - - - 15 25
L6NN - - - - 35
Activation Tanh Tanh Tanh Tanh Tanh
Optimizer Adam Adam Adam Adam Adam
Coefficient of determination 0.87 0.85 0.87 0.89 0.90

In this study, the SAE, SE, and SSE utilize DNNs as base learners, while DP uses decision trees.
Therefore, the first step is to select several DNNs with different architectures to be used in
subsequent steps. For this purpose, five DNNs with different architectures are developed using
training data. These trained DNNs are then utilized as base learners. The GridSearchCV
algorithm [46] is employed to determine the activation function, optimizer, number of neurons,
and dropout rate values for each DNN. The characteristics of the employed base learners are
presented in Table 3.
3.2. Simple averaging ensemble (SAE) algorithm
During the prediction process, the deep learning model calculates each output value. An
ensemble model is formed by combining the predictions of multiple DNN models to make a
unified decision instead of relying on individual predictions [20]. This allows for simultaneous
forecasting of each value in the data using multiple models. The performance of an ensemble
model is measured by taking the average of the predicted values [20].
Fig. 5. Workflow of the SAE algorithm.
Fig. 5 shows the workflow of the SAE [20], where N represents the number of base learners and
Predi (i=1,..,N) represents the estimated value by base learners. It should be mentioned that the
output layer has a single neuron and a linear activation function (Fig. 5).
3.3. Stacking ensemble (SE) algorithm
Stacking [20], also known as stacked generalization, refers to the process of studying a learning
algorithm that combines the results of multiple learning algorithms. The main goal is to reduce
generalization error by utilizing a pool of base learners and employing another learner to
aggregate their estimates [20]. The stacking algorithm is an ensemble method with two operating
levels: level 0 consists of the base learners, while level 1 includes a meta-model. Specifically, in
this study, DNNs are used as the base learners at level 0, and their outputs serve as input data for
the meta-models. The meta-models employed in this study are support vector regressor (SVR),
decision tree regressor (DTR), gradient boosting regressor (GBR), random forest regressor
(RFR), Ada boost regressor (ABR), and bagging regressor (BR). Fig. 6 illustrates the workflow
of the SE algorithm, showing the interaction between the meta-model and the base learners.

Fig. 6. Workflow of the SE algorithm.
3.4. Snap-shot ensemble (SSE) algorithm
The methods used for tuning DNNs are stochastic gradient descent (SGD) and its variations [20].
While avoiding erroneous solutions due to fictitious local minima is generally considered
beneficial, some researchers argue that these local minima contain valuable data that can
improve the model performance [20]. The SSE algorithm was designed to train a group of DNNs
with an easy-to-implement training mechanism. It leverages the non-convex structure of DNNs
and the capability of the SGD to escape from local minima. Instead of training M independent
DNNs, the SSE algorithm allows SGD to converge M times to local minima along its
optimization path. Each time the model converges, its weights are saved and added to a pool. The
procedure then continues with a high learning rate to escape the current local minimum (Fig. 7).
Fig. 7. Learning schedule of the SSE algorithm.
Specifically, the SSE technique employs a cycling process where the learning rate is abruptly
increased and then rapidly decreased, following a cosine function. The resulting ensemble in the
SSE method comprises snapshots of the optimization loop. Based on the accuracy, the structure
of the base model 5 (Table 2) is selected to develop SSE. To study the impact of the number of
members on accuracy, a sensitivity analysis is conducted. Since models saved towards the end of
the session have undergone more training epochs compared to those saved earlier, it is possible

that they perform better. Therefore, the list of recorded models is reversed, and models from the
pool are chosen and added starting from the last, thereby increasing the number of SSE
members.
Fig. 8 illustrates the performance of SSE as the number of members increases from 2 to 20.
Generally, the performance remains relatively stable until reaching 10 model members. However,
as the number of members increases from 10 to 20, the performance of the proposed SSE model
declines. This may be attributed to the fact that the first models generate outlier predictions due
to their fewer training epochs and insufficient time to learn all the information. Furthermore, it is
worth noting that even with just two members, the SSE model outperforms individual models. In
this study, the SSE with 10 recorded models is utilized for PHL prediction of RCSW.
Fig. 8. The performance of the SSE model versus the size of the model.
3.5. Deep forest (DP) algorithm
Using the concept of Variation in Memory (VIM) of base learners, the DP algorithm aims to
provide a model with a strong ability to make precise estimations. The DP algorithm incorporates
various methods for VIM simultaneously, including sample manipulation, input feature
manipulation, and output representation manipulation [44]. Deep Forest is a relatively new
ensemble learning technique that utilizes decision trees and deep learning. Traditional decision
tree algorithms construct a single tree to model the data, which may result in overfitting or under
fitting. Deep forest can overcome this issue by constructing multiple levels of decision trees. The
initial layer of trees is built using original data, and the subsequent layers are constructed using
the results from the previous layer. This approach allows the creation of a hierarchical structure
of decision trees that can uncover more complex correlations in the data. The ability of Deep
Forest algorithm to handle noisy, high-dimensional data is one of its key advantages. High-
dimensional data can cause overfitting or under fitting in standard decision tree techniques. To
address this issue, Deep Forest employs a strategy where a random subset of features is chosen
for each tree in every layer. This process helps reduce the impact of irrelevant or distracting
features on model predictions. Additionally, each layer is utilized in a way similar to a random
forest to further mitigate overfitting.

Fig. 9. Workflow of deep forest algorithm.
As shown in Fig. 9, a sliding window in DP algorithm slides over the features of the samples.
Then, a set of decision trees are trained using the selected features. In this scenario, each tree is
randomly trained with a restricted number of samples. The sliding windows generate a large
number of outputs for each decision tree. The output of these decision trees serves as the input
for the subsequent decision trees. This process is repeated until there is no further improvement
in the accuracy of the model for the validation data. The key advantage of this algorithm is its
adaptive structural complexity. Instead of predefining the number of layers, the algorithm
dynamically increases the number of layers until the lowest error value is reached or no further
improvement in accuracy is observed. In this study, the recommended default values have been
used for the DP algorithm. These include 500 trees in each forest, a sliding window size of 1, and
eight forests.
3.6. Performance metrics
In this study, the coefficient of determination (R2
), mean-squared error (MSE), and median
absolute error (MAE) are utilized to evaluate the efficiency of the applied ML models. A higher
R2
value, along with lower MSE and MAE values, indicates a better fit.
2
1
( )
( )
N
vi vi
i
O E
MSE
N


  (4)
2
2
2 2
[ ( )( )]
( ) ( )
vi v vi v
vi v vi v
O O E E
R
O O E E
 

 


(5)
,1 ,1 , ,
,...
( )
,
v v v i v i
MAE O E O E
median
   (6)
where N is the number of samples, v
O denotes the observed value, v
O denotes the average of
recorded values, v
E represents the estimated value, and v
E represents the average of estimated
values.

4. Results and discussion
In this section, the efficiency and accuracy of the applied models (SAE, SSE, DF, SVR, DTR,
GBR, RFR, ABR, and BR) are evaluated.
Table 4 presents the MAE, MSE, and R² values for the training and testing phases. The boxplots
for the predicted PHL values are also presented in Fig. 10. Good agreement between the results
of the applied models and the observed PHL values are quite obvious from Fig. 10. Based on the
training results in Table 4, the SE-DTR is the best model with highest R² value and the lowest
MAE and MSE values. SE-RFR ranks second among all models. Both SE-RFR and SE-BR have
the same MSE and MAE values, with a slight difference in the R² values. Also, the SSE
algorithm had the worst performance in the training stage in terms of MSE, MAE, and R² values.
From the results of the testing phase, it is noticeable that there is inconsistency in the
performance assessment results across the three indices. For instance, SE-RFR performs the best
in terms of MAE, while SE-GBR exhibits the best performance in terms of MSE and R² among
all investigated models. However, this is not the case for MAE. Moreover, the MAE value for the
SE-GBR model is close to that of the SE-RFR model. Therefore, considering the overall
performance, the SE-GBR model is selected as the best model for the subsequent steps of this
study.
Table 4
Performance assessment of ensemble models.
Model
Training data Testing data
MSE (m2
) R2
MAE (m) MSE (m2
) R2
MAE (m2
)
SAE 0.011 0.911 0.043 0.015 0.898 0.046
SSE 0.012 0.898 0.047 0.021 0.858 0.068
DF 0.008 0.938 0.023 0.020 0.861 0.059
SE
SVR 0.009 0.926 0.045 0.014 0.900 0.053
DTR 0.0 1 0.0 0.018 0.877 0.058
GBR 0.003 0.978 0.028 0.012 0.916 0.043
RFR 0.002 0.984 0.014 0.013 0.907 0.042
ABR 0.008 0.934 0.056 0.017 0.883 0.079
BR 0.002 0.980 0.014 0.015 0.898 0.049
The variations of the observed and predicted PHL values for the best model (SE-GBR) are
presented in Fig. 11. It is obvious from Fig. 11 that the SE-based GBR model's predicted PHL
values thoroughly follow the corresponding observed PHL values.

Fig. 10. Box plots of predicted PHL values (testing dataset).
Fig. 11. Variation of the observed and predicted PHL values (testing dataset).
4.1. Inherently ensemble-learning algorithms
In this section, the performance of the proposed SE-GBR ensemble learning model is compared
with inherently ensemble-learning-based (IELB) algorithms. For this purpose, eight different
IELB algorithms including the XGBoost, RandomForest, CatBoost, HistGradientBoosting,
AdaBoost, Bagging, ExtraTrees, and GradientBoosting regressor are investigated.
In Table 5, the results of the applied IELB algorithms and the proposed SE-GBR model are
compared. As can be seen, the performance indices for the SE-GBR model are superior than
those for the IELB algorithms. However, the MAE value for SE-GBR and ExtraTrees models is
almost the same. It should be mentioned that the ExtraTrees algorithm (with MSE=0.016,
R2
=0.89, and MAE=0.046) performs best among other IELB algorithms. However, the MSE and
R2
of the RandomForest are the same as ExtraTrees model. Therefore, the RandomForest
algorithm (with MAE=0.5) has gained the second rank among all IELB algorithms. These
0
1000
2000
3000
4000
5000
0 10 20 30 40 50 60 70 80 90 100 110
Plastic
hinge
length
(mm)
Data Number
Measured PHL
SE-GBR Model

results, in general, verify that the proposed SE-GBR model outperformed the applied inherently
ensemble-learning-based algorithms for PHL prediction of the RCSWs.
Table 5
Comparison of IELB algorithms and SE-GBR model (testing data).
Model MSE (m2
) R2
MAE (m)
XGBoost 0.024 0.834 0.096
RandomForest 0.016 0.890 0.050
CatBoost 0.018 0.875 0.047
HistGradientBoosting 0.016 0.888 0.056
AdaBoost 0.028 0.809 0.101
Bagging 0.017 0.879 0.062
ExtraTrees 0.016 0.890 0.046
GradientBoost 0.017 0.880 0.081
SE-GBR 0.012 0.916 0.043
4.2. Comparison with empirical equations
This section provides a brief overview of the empirical equations derived from the literature for
comparison purpose. For this purpose, six existing empirical models proposed in
[36,38,42,43,47,48] are used. Bohl and Adebar [36] employed a nonlinear finite element (FE)
model to evaluate the flexural deformation capacity of RCSWs. Through testing various RCSWs
with different shear span ratios, lengths, and axial loads, they derived an equation that predicts
the equivalent plastic hinge length, as presented in Eq. (10). Priestley et al. [43] proposed a
plastic hinge formula to meet specific serviceability and damage limits. Kazaz [38], based on a
regression study, developed a model that considers variables such as wall length, shear span
ratio, axial load ratio, and reinforcement ratio. This model was calibrated using both the Turkish
seismic code requirements and an FE shear wall model. Two different methods were employed to
estimate the length of the plastic hinge zone. The first method involved evaluating the strain
distribution across the wall's cross-section to determine the ultimate curvature configuration,
whereas, the second method required calculating the curvature arrangement using strain data
from shell elements applied to the FE model at the opposing corners of the wall. Furthermore,
Hoult [42] introduced Eq. (12) for predicting the PHL of RCSWs. This equation was derived
using standard regression techniques and based on the same database utilized in this study.
A list of the empirical equations for calculating the PHL of RCSWs is provided in Eqs. (7)-(12).
NZS 3101.1:2006,
ASCE/SEI 41-06 [47]
0.5
ph w
l l
 (7)

Priestley et al. [43]
0.2 0.022
/ , 0.2( / 1) 0.08, 1.1
ye
ph e w bl
ult y ye y
e
l H l f d
H M V f f f f


  
    
(8)
Eurocode 8 [48] '
0.11
0.2 ( )
30
y
e
ph w bl
c
H
l l d f
f
   (9)
Bohl and Adebar [36] (0.2 0.05 )(1 1.5 ) 0.8
ph w e w
n
P
l l H l
P
    (10)
Kazaz [38]
0.45
'
0.27 (1 )(1 )( )
y
tr e
ph w
n c w
f H
P
l l
P f l

   (11)
Hoult [42]
0.6
(0.13 1.8 ) (1 0.3 ) (1 0.13log( ))
ph scr w e
n
P
l r l H SSV
P
      (12)
where P/Pn denotes the axial load ratio, bl
d represents the bars diameter, ult
f denotes the
ultimate strength of the bars, '
c
f denotes the compressive strength of the concrete, y
f represents
the yield strength of the bars, and tr
 represents the transverse reinforcement ratio.
The testing data was used for comparison between the accuracy of the empirical equations (7-12)
and the proposed SE-GBR model. The predictions from the SE-GBR model are adjusted based
on the range of the dataset using Eq. (3). The performance of the applied empirical equations is
presented in Table 6 and Fig. 12. The boxplots of the estimated PHL values are also shown in
Fig. 13. It is evident that the predictions of the PHL using empirical equations (7) and (9)
deviated significantly from the dashed line for Fig. 12(a) and Fig. 12(c), respectively. Figs. 12(a),
(b), and (e) demonstrate that Eqs. (7), (8), and (11) overestimates the PHL compared with
observed values. It should be mentioned that previous studies have also reported that
approximating the PHL by 0.5 w
l can overestimate the real values of the ph
l [42]. This
overestimation can be attributed to the neglect of other influential factors. Moreover, from Fig.
12(c), it can be concluded that Eq. (9) underestimates the PHL (Fig. 13).
Table 6
Indicators value for SE-GBR and empirical models.
Model
Performance indices
MSE (m2
) R2
MAE (m)
Eq. (7) 1.292 0.30 0.668
Eq. (8) 0.297 0.65 0.303
Eq. (9) 0.405 0.53 0.258
Eq. (10) 0.235 0.73 0.250
Eq. (11) 0.492 0.43 0.194
Eq. (12) 0.200 0.77 0.206
SE-GBR 0.012 0.916 0.043

Fig. 12. Comparison of the results of existing equations Vs. SE-GBR model.
(a) (b)
(c) (d)
(e) (f)
(g)

Fig. 13. Box plots of residual error (test data).
According to Table 6, Eq. (10) with R2
of 0.73 and Eq. (12) with R2
of 0.77 demonstrate better
agreement with the experimental results compared to other empirical equations. Among the
empirical equations, Eq. (12) with the MAE value of 0.206 m and the lowest MSE (0.2 m2
) has
the best performance. Nevertheless, the proposed SE-GBR model, with higher R2
values and
lower MSE and MAE values, performed better than all existing formulas. The good agreement
between the results of the SE-GBR and the observed PHL values is obvious from Fig. 13.
Fig. 14. Taylor diagram for the estimated PHL values.

The Taylor diagram for all applied models is depicted in Fig. 14. In this figure, the azimuth angle
denotes to the correlation (r) between the predicted values and the observed PHL values, and the
radial distance from the observed PHL values denotes the centered RMS error. As can be seen in
Fig. 14, among all applied algorithms, the SE-GBR model, having the highest correlation with
observations and the lowest RMS error (less than 0.1), agrees best with the observations values.
Fig. 14 also indicates that Eq. (12) has a higher correlation with observations and a lower
centered RMS error than other empirical equations. However, the SE-GBR has a lower distance
from the observed PHL values compared with all empirical models. Therefore, it can be
concluded that the proposed model provided the most accurate results.
4.3. Model explain-ability
It is crucial to understand the results of ML-based models and be able to interpret how and why
these models provide predictions in a human-readable manner [49]. In this study, the state-of-
the-art method known as SHAP (SHapley Additive exPlanations) [50] is employed to explain the
impact of input variables on the prediction capability of the SE-GBR model. SHAP combines
game theory concepts with linear surrogate modeling techniques and provides additional
mathematical guarantees concerning the accuracy of the explanations. In this method, the
average influence of a feature on the prediction through all possible subsets of features is
denoted as the Shapley value [49].
The SHAP-based relevance analysis results of the input variables affecting PHL of RCSWs are
presented in Fig. 15. This figure shows the global importance of each input feature on the output of
the SE-GBR model. It should be mentioned that the feature importance of input variables is
calculated as the mean value of the absolute Shapley values of the dataset. Fig. 15 demonstrates
that the wall length ( w
l ) is the most crucial input variable affecting the PHL of the RCSWs,
followed by the effective height (He=M/V) and the axial load ratio (ALR). Although SSV has an
impact on how plasticity develops, it seems that this factor is not as important as other
parameters.
Fig. 15. Global importance of the input features.
Fig. 16 shows the summary plot for the SE-GBR model’s prediction. In this figure, each point
represents a Shapley value for a specific input variable and a single observation of the test set.

The color in this figure denotes the parameters’ value, which varies from low (blue) to high (red).
This figure shows the order of significance of input parameters affecting the PHL of RCSWs,
i.e., whether the prediction is changed positively or negatively. As can be seen in Fig. 16, an
increase in the wall length ( w
l ) increases the plastic hinge length of the RCSWs. This is due to
the connection between tension shift effects and w
l . The tension shift action increases and the
plasticity spreads farther with a long wall length. Moreover, the model is more likely to estimate
larger w
l as the value of the effective height (M/V) increases, which is related to the lower
nonlinear shear deformations and a higher possibility of flexural yielding before reaching the
nominal shear strength of the wall. Moreover, it can be seen that the PHL has an inverse
relationship with the axial load level (ALR). In the other words, increasing the axial load reduces
PHL values which may be due to the simultaneous concrete crushing and reinforcement buckling
in the compression area, or concrete crushing in the compression area prior to reinforcement
yielding in the tension area. This caused a dramatic decrease of ph
l in axially loaded RCSWs by
closing the distance between the bending moment at the yielding point and the ultimate capacity.
Fig. 16. Summary plot for SHAP analysis using SE-GBR model.
In addition, this analysis reveals that the higher values of the shear stress variable (SSV) will
decrease ph
l . This issue can be related to (1) the interaction between nonlinear flexural and shear
modes in RCSWs, and (2) the effects of the nonlinear shear deformations on the axial
compressive strains even in slender RCSWs, which lead to reduced strength and ductility. It is
also revealed that increasing the secondary cracking ratio ( scr
r ) increases the ph
l . This is in line
with other studies that discovered a threshold amount of longitudinal steel is necessary to permit
secondary cracking in RCSW system. The PHL will be shortened if the longitudinal
reinforcement ratio of the wall is lower than the minimum value needed to permit secondary
cracking.
4.4. Sensitivity analysis using Sobol's technique
In this section, the Sensitivity Analysis (SA) is performed to assess how a model's outcomes
respond to variations in input variables. Sobol's SA method has gained considerable attention due

to its accuracy and simplicity [51,52]. This method is based on the decomposition of the variance
of the model output into summands of variances of the input variables in increasing
dimensionality. In this study, the variance-based Sobol's technique is employed for SA. Sobol's
method takes a probabilistic perspective, treating the model's inputs as a multidimensional vector
with a joint probability density function. The stochastic representation of the model can be
expressed as follows:
( )
Y MO x
 (13)
where MOrepresents a mathematical operator, x is a vector of random inputs, and Y denotes the
quantity of interest. The Sobol's index for a subset 'u' of the input vector and total Sobol indices
are defined as follows [51,52]:
1
( ( )) / ( ),
n
total
i i i i i
i
S Var MO x Var Y S S

   (14)
where  
1,..,
i
S i n
 denotes the Sobol's index, and total
i
S represents the total Sobol index.
Fig. 17. Summary plot for SHAP analysis using SE-GBR model.
Fig. 17 shows the Sobol's technique-based sensitivity analysis results for PHL of the RCSW
system. The SA results illustrate that the wall length and the wall effective height exhibit a higher
total-order sensitivity coefficient. Consequently, they exert a significant influence on the PHL of
the RCSWs. It is worth mentioning that, similar to the SHAP-based model explanations analysis,
discussed in section 4.3, the SA results indicates that the secondary cracking ratio ( scr
r ), axial
load ratio ( ALR ) have small effects and the shear stress variable ( SSV ) has negligible effects on
the PHL of RCSWs.

5. Conclusion
In this study, ensemble machine learning (ML) algorithms including the simple averaging
ensemble (SAE), stacking ensemble (SE), snap-shot ensemble (SSE), and deep forest (DP)
algorithm were employed for prediction of the plastic hinge length (PHL) of reinforced concrete
shear walls (RCSW) systems. For this purpose, a comprehensive dataset with 721 nonplanar and
rectangular RCSWs was used. The secondary cracking ratio ( scr
l ), effective
height ( /
e
H M V
 ), axial load ratio (ALR=P/Pn), and shear stress variable (SSV) are considered
as input variables. The result of the proposed ML-based models was also compared with
inherently ensemble-learning-based (IELB) algorithms including the XGBoost, RandomForest,
CatBoost, HistGradientBoosting, AdaBoost, Bagging, ExtraTrees, and GradientBoosting
regressor, as well as with available empirical equations for PHL of RCSWs. The performance of
the applied models was investigated in terms of the regression plot, boxplot, Taylor diagram, and
three statistical indices, i.e., median absolute error (MAE), mean-squared error (MSE), and
coefficient of determination (R2
). The SHAP analysis, an acronym for SHapley Additive
exPlanations, was also leveraged to interpret the degree of influence of each input variable on the
prediction. The following conclusions can be expressed in light of the results of comparisons
analysis:
 The stacking ensemble (SE)-based gradient boosting regressor (GBR) meta-learner with
the statistical values of MAE=0.043 m, MSE=0.012 m2
, and R2
=0.916 outperformed other
ensemble-based ML algorithms in the testing phase. Also, SE with a random forest
regressor meta-model achieved the second rank among all other applied models.
 SE-GBR model outperformed all employed IELB algorithms. Moreover, between IELB
algorithms, the ExtraTrees regressor with the statistical values of MAE=0.046 m,
MSE=0.016 m2
, and R2
=0.89 performed better than all applied IELB algorithms.
 The proposed stacking ensemble-based gradient boosting regressor model surpassed all
available empirical formulas for prediction of the PHL of RCSWs in terms of MSE, R2
,
and MAE.
 Among the six available empirical equations for PHL prediction of RCSWs, the Bohl and
Adebar's as well as Hoult's equations had better agreement with experimental results than
other equations.
 The results of sensitivity analysis and the SHAP interpretation-based feature importance
analysis revealed that the wall length ( w
l ) is the most influential input variable affecting the
PHL of the RCSWs, followed by the effective height ( /
e
H M V
 ) and the axial load ratio
(P/Pn).
 SHAP analysis results showed that an increase in the wall length, the effective height, and
the secondary cracking ratio increases the plastic hinge length of RCSWs. In contrast, an
increase in the axial load ration and shear stress decreases the plastic hinge length of the
RCSWs.

Results of this study verified that the developed SE-based GBR model outperform all empirical
equations and other compared IELB algorithms for estimating the PHL of RCSWs. However,
due to data limitations, like other machine learning algorithms, the robustness and accuracy of
the model output may be questionable for new big data. Expanding the data sources with new
high-quality data, employing advanced feature-extracting techniques to identify additional
relevant features, and combining machine learning models with advanced optimization
algorithms could enhance the robustness and generalization capability of the predictive model for
PHL of RCSWs by leveraging the strengths of different approaches.
Nomenclature
ph
l Plastic hinge length w
l Wall length
ALR Axial load ratio /
e
H M V
 effective height
ult
f bars ultimate strength '
c
f strength of the concrete
y
f bars yield strength tr
 transverse reinforcement ratio
sl
A
section area of longitudinal
bar
SSV shear stress variable
A section area of the wall w
t wall thickness
sl
n
bars number in longitudinal
direction ult
f ultimate strength of the bars
cts
f concrete tensile strength c
f  concrete compressive strength
Max
V maximum shear SE stacking ensemble
SAE simple averaging ensemble SSE snap-shot ensemble
DP deep forest
Data availability
The entire database is available on the online repository Zenodo
(https://zenodo.org/record/4321917).
Acknowledgments
The authors would like to acknowledge the financial support of Bozorgmehr University of
Qaenat for this research under contract number 39246.
Funding
This research received no external funding.
Conflicts of interest
The authors declare that they have no conflict of interest.

Authors contribution statement
NSH and MSB: Data creation, Methodology, Software, Validation, Writing–original draft. NSH:
Supervision, analysis of the results, Writing – review & editing.
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