Optimized neural network-based model to predict the shear strength of trapezoidal-corrugated steel webs

Abstract

Beam-like members use corrugated webs to increase their shear strength, stability, and efficiency. The corrugation positively affects the members' structural characteristics, especially those governed by the web parameters, such as the shear strength, while reducing the total weight. Existing code and analytical models for predicting the shear strength of trapezoidal corrugated steel webs (TCSWs) are summarized. This paper presents an optimized Artificial Neural Network (ANN)-based model to estimate the shear strength of steel girders with a TCSW subjected to a concentrated force. A database of 206 experimental results from the literature is used to feed the ANNs. Six geometrical and material parameters were identified as input variables, and the experimental shear strength at failure was considered the output variable. Four hyperparameter optimization techniques are applied to refine the ANN models: Bayesian Optimization (BO), Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS), Firefly Algorithm (FA), and African Buffalo Optimization (ABO). The performance metrics indicate that the ABO-ANN model is the most effective among these. The predictions of the developed ML model were also compared with those of existing code and analytical models. The comparisons illustrated that the ANN-based model outperforms the other existing models. The sensitivity analysis using the proposed ANN-based model captured the relationships and interactions among the geometric and material parameters and their impact on shear strength. One main finding is that the corrugation angle in the 35–45° range maximized the TCSW shear strength.

Highlights

• •A dataset of 206 experimental shear strength tests of TCWBs was compiled.
• •The African Buffalo Optimization (ABO) resulted in the best-performed ANN model.
• •The proposed ANN model outperformed existing design codes for TCWBs as EN 1993-1-5.
• •A sensitivity study indicates that the corrugation angle has the greatest influence among the modeling parameters.

1. Introduction

Corrugated web steel beams (CWSBs) are considered a significant advancement in structural engineering, offering superior performance to traditional welded plate structures. By incorporating thin corrugated webs, CWSBs eliminate the need for transverse stiffeners while enhancing shear strength capabilities [1]. The corrugated web design allows for more efficient shear stress distribution, improving shear capacity and overall structural performance. The superior load-bearing and flexural strength of CWSBs enhance structural integrity and safety in building constructions, making them an attractive choice for designers and engineers. Additionally, the inherent resistance of CWSBs to buckling allows for innovative geometrical designs, enabling architects and engineers to explore new possibilities in building aesthetics and functionality [[2], [3], [4]].

Furthermore, CWSBs contribute to sustainability in the construction industry by reducing material usage and the associated carbon footprint. The corrugated web design allows for the use of thinner steel plates, significantly reducing the structure's overall weight without compromising its strength and stability. This reduction in material use leads to cost savings and minimizes the environmental impact of the construction process, aligning with the growing global emphasis on sustainable building practices.

2. Literature review

This section presents the main equations used to estimate the shear strength of trapezoidal corrugated steel webs (TCSWs). While various methods exist for determining shear stresses, they follow similar procedures. Fig. 1 and Table 9 illustrate the essential dimensional parameters for calculating Local, Global, and Interaction shear buckling stresses. Although Local shear buckling stress is usually lower than Global buckling stress, empirical evidence by Nie et al. (2013) [4] shows that most specimen failures are due to Global buckling stress. The calculation of Local, Global, and Interaction buckling slenderness ratios is consistent across different models. However, these models show distinct variations in normalized interaction shear buckling strengths. The shear yield strength (τ_y) is determined by applying the von Mises yield criterion shown in Equations (1), (2), where f_y represents the uniaxial yield strength of the steel material.

$$\tau =\frac{V}{\rho \times h_w\times t_w}$$ (1)

$${\tau }_y=\frac{f_y}{\sqrt{3}}$$ (2)

Fig. 1.

Profile of TCSW. T. Wang et al. (2021) [5].

Table 9.

Notation.

a	Flat panel width
b = a1	Horizontal projection of the inclined panel width
c	Inclined panel width
d	Longitudinal width of inclined fold panel
m	Number of corrugation
hw	Web depth
hr = a3	Corrugation depth
L	Web length
α	Corrugation angle
t	Web thickness
w	Larger values of b and c
w2	Smaller values of b and c
w2/w	Fold ratio
Ix	Moment of inertia about the axis along web height
Iy	Moment of inertia about the neutral axis along web height
E	Young's modulus of elasticity
$\nu$	Poisson's ratio
fy	Yield tensile stress
τy	Yield shear stress
τu	Ultimate shear stress, τu = Qu/(ht)
τLel	Local elastic shear buckling stress
τGel	Global elastic shear buckling stress
τIel	Interactive elastic shear buckling stress
τcr,L	Critical local shear buckling stress
τcr,G	Critical global shear buckling stress
τcr,I	Critical interactive shear buckling stress
kLel	Local shear buckling stress buckling coefficient
kG	Global shear buckling stress buckling coefficient
λL	Local buckling slenderness ratio
λG	Global buckling slenderness ratio
λI,n	Interactive buckling slenderness ratio
ρL	Normalized local elastic shear buckling strength
ρG	Normalized global elastic shear buckling strength
ρI,n,el	Normalized interaction elastic shear buckling strength

2.1. Local shear buckling

The determination of elastic local shear buckling stress is fundamentally based on the plate buckling theory principles, which refers to predicting the buckling behavior of thin-walled plates under various stress conditions. These analytical formulas estimate critical buckling stresses by considering the effects of geometric and material properties on buckling behavior. Plate buckling theories have gained recognition in published literature. Ziemian (2010) [6] analytically applied a formula for local shear buckling stress, emphasizing the importance of these theories in understanding plate buckling phenomena, as presented in Equation (3):

$${\tau }_l=\frac{k_L{\pi }^{2}E}{12\left(1-v^2\right)}{\left(\frac{t_w}{b}\right)}^2$$ (3)

The computation of the local shear buckling coefficient, k_L, is derived for various boundary conditions, as outlined in Equations (4), (5), (6):

• a)For configurations where long edges are simply supported and short edges are clamped:

$$k_L=5.34+2.31\left(\frac{b}{h}\right)-3.44{\left(\frac{b}{h}\right)}^2+8.39{\left(\frac{b}{h}\right)}^3$$ (4)

• b)In cases where all edges are clamped:

$$k_L=8.98+5.6{\left(\frac{b}{h}\right)}^2$$ (5)

• c)When all edges are simply supported:

$$k_L=5.34+4.0{\left(\frac{b}{h}\right)}^2$$ (6)

Driver et al. (2006) [3] conducted a comprehensive analysis of existing data, concluding that the shear strength of corrugated webs is frequently overestimated by conventional equations derived from plate buckling theories. The study calculated the elastic local shear buckling stress, utilizing the standard plate buckling theory, expressed as:

$${\left({\tau }_{cr}\right)}_{el}=k_L\frac{{\pi }^{2}E}{12\left(1-v^2\right){\left(\frac{w}{t_w}\right)}^2}$$ (7)

In Equation (7), k_L represents values of 5.34 and 8.98 for scenarios where the fold is considered to have simply supported edges and fixed edges, respectively. The term w denotes the maximum fold width, which is the larger of the longitudinal fold width b and the inclined fold width c. Further expanding on this concept, Yi et al. (2008) [7] suggested that with all panel sides simply supported, the local buckling strength presented in Equation (8) can be reformulated as:

$${\left({\tau }_{cr}\right)}_{el}=\frac{5.34{\pi }^{2}E}{12\left(1-v^2\right)}{\left(\frac{t_w}{b}\right)}^2$$ (8)

2.2. Global buckling stress

Global shear buckling failure is characterized by diagonal buckles extending over the web's depth, impacting numerous panels. Easley and McFarland (1969) [8] established a mathematical formula to estimate the elastic global buckling stress, employing orthotropic theory. This formulation is represented as:

$${\tau }_G=k_G\frac{{\left(D_x{D}_y^3\right)}^{0.25}}{t_w{h}^2}$$ (9)

Where k_G denotes the global shear buckling coefficient, and D_x and D_y are the bending stiffnesses per unit length of the corrugated web about the longitudinal and vertical axes, respectively. The expressions for D_x and D_y are given by Equations (10), (11), integrating parameters such as bending stiffness E, web thickness t_w, and angles of corrugation α.

$$D_x=\frac{E}{b+d}\left(\frac{b{t}_w{\left(d\mathit{tan}\alpha \right)}^2}{4}+\frac{t_w{\left(d\mathit{tan}\alpha \right)}^3}{12\mathit{sin}\alpha }\right)$$ (10)

$$D_y=\frac{b+d}{b+d\mathit{sec}\alpha }\left(\frac{E{t}_w^3}{12}\right)$$ (11)

Subsequently, Driver et al. (2006) [3] approached global buckling by considering the corrugated web as an orthotropic flat web, leading to the expression of the elastic shear buckling stress ${\left({\tau }_{cr}\right)}_{Gl}$ in Equation (12). F(α,β) represents a non-dimensional coefficient, calculated in Equation (13), characterizing the web corrugation geometry, where β is the ratio of longitudinal (b) to inclined fold width (c).

$${\left({\tau }_{cr}\right)}_{Gl}=k_G\frac{E{t}_w^{\frac{1}{2}}{b}^{\frac{3}{2}}}{12h_w^2}F\left(\alpha ,\beta \right)$$ (12)

$$F\left(\alpha ,\beta \right)=\sqrt{\frac{\left(1+\beta \right){\mathit{sin}}^3\alpha }{\beta +\mathit{cos}\alpha }}.{\left\{\frac{3\beta +1}{{\beta }^2\left(\beta +1\right)}\right\}}^{3/4}$$ (13)

Furthermore, Yi et al. (2008) [7] presented an alternative expression for global buckling strength in Equation (14), which involves parameters such as material elasticity modulus E, Poisson's ratio v, and web dimensions.

$${\left({\tau }_{cr}\right)}_{Gl}=36\beta E\frac{1}{{\left[12\left(1-v^2\right)\right]}^{\frac{1}{4}}}{\left[\frac{{\left(\frac{d}{t}\right)}^2+1}{6\eta }\right]}^{3/4}{\left(\frac{t}{h}\right)}^2$$ (14)

Where $\eta =\frac{d+b}{d+c}$.

Most recently, Wang et al. (2021) [5] proposed a refined model considering simply supported edges of TCSWs and a coefficient k_g = 36. Their formulation for global buckling stress, ${\tau }_{Gl}$ , is detailed in Equation (15), integrating factors like web thickness t and height h_w and dimensions of corrugation folds b and c.

$${\tau }_{Gl}=6.839\frac{E}{h^2}{\left[\frac{t^2{h}_w^6}{\left(b+c\right){\left(b+d\right)}^2}{\left(\frac{c}{3}+b\right)}^3\right]}^{1/4}$$ (15)

2.3. Interactive shear buckling strengths

Shear buckling strength in steel beams with corrugated webs has been a significant research focus in structural engineering. In 2005, Sayed-Ahmed [9] proposed an interaction equation (Equation (16)) that considers the relationship between buckling and yielding. This research highlighted the complex failure modes essential when designing web structures. The study used numerical simulations and analytical evaluations to investigate the buckling behavior of corrugated webs and confirmed the accuracy of the proposed interaction equations. One key finding from this research is the distribution of load-bearing responsibilities in steel beams with corrugated webs. It was revealed that while the flanges primarily contribute to flexural strength, the contribution of the webs was minimal, displaying negligible interaction between bending and shearing forces. The study also emphasized the resilience of these beams, which can still bear loads even after web buckling occurs. Since this foundational work, increasing efforts have been made to develop equations that more accurately describe the relationship between local and global shear buckling. These equations usually follow a basic format, Equation (16):

$${\left(\frac{1}{{\tau }_I}\right)}^n={\left(\frac{1}{{\tau }_L}\right)}^n+{\left(\frac{1}{{\tau }_G}\right)}^n+{\left(\frac{a}{{\tau }_y}\right)}^n$$ (16)

Where τ_I represents the interactive shear buckling strength, τ_L and τ_G represent the local and global shear buckling strengths, respectively, and τ_y represents the shear yield strength. The coefficients a and n are used to adjust the contribution of each term to the overall interactive shear buckling strength. Multiple design models have been proposed, assigning specific values to a and n. Table 1 provides a comprehensive overview of these models, delineating their approaches to determining interactive shear buckling strength.

Table 1.

Comparative analysis of models for determining interactive shear buckling strength: a and n values.

Researcher	a and n values
Bergfelt & L. Leiva-Aravena., 1984 [10]	$n=1,a=0$
El Metwally, 1998 [11]	$n=2,a=1$
Sayed-Ahmed, 2005 [9]	$n=3,a=1$
Abbas et al., 2002 [12]	$n=2,a=0$
Shiratani et al., 2003 [13]	$n=4,a=0$
Driver et al., 2006 [3]	$n=2,a=0$
Sause & Braxtan, 2011 [14]	$n=3,a=2$
Leblouba et al., 2017, 2019 [15,16]	$n=4,a=1$
Wang et al., 2021 [5]	$n=1.25,a=$ 1

The slenderness ratios relevant to local (L), global (G), and interactive (I) buckling of TCSWs are defined in Equations (17), (18), (19) as follows:

$$\text{Local}\text{Slenderness}\text{Ratio}\left({\mathrm{\lambda }}_{\mathrm{L}}\right){\lambda }_L=\sqrt{\frac{{\tau }_y}{{\tau }_{el,L}}}$$ (17)

$$\text{Global}\text{Slenderness}\text{Ratio}\left({\mathrm{\lambda }}_{\mathrm{G}}\right){\lambda }_G=\sqrt{\frac{{\tau }_y}{{\tau }_{el,G}}}$$ (18)

$$\text{Interactive}\text{Slenderness}\text{Ratio}\left({\mathrm{\lambda }}_{\mathrm{I},\mathrm{n}}\right){\lambda }_{I,n}=\sqrt{\frac{{\tau }_y}{{\tau }_{I,n,el}}}$$ (19)

Recent studies have significantly advanced the understanding of TCSWs, focusing on shear strength, buckling modes, and the role of flanges in steel beams [3,14,17]. Leblouba et al. (2017, 2019) [15,16] further refined the mathematical models for determining the shear strength of TCSWs by integrating experimental data from 144 beam tests, leading to the development of a more accurate three-parameter model validated through statistical comparison with experimental data. While the study provided advanced modeling, there is a lack of research on the application of machine learning techniques, particularly in predicting TCSW shear strength, despite recent literature suggesting that machine learning models can improve predictive accuracy and computational efficiency in structural engineering problems [[18], [19], [20], [21], [22], [23], [24], [25], [26]].

This study aims to address this research gap by applying Artificial Neural Network (ANN) algorithms to a dataset of 206 experimental results, leveraging the capabilities of ANNs to forecast the shear strength of TCSWs while implementing various optimization methodologies to enhance the robustness and precision of the ANN model. The novelty of this work lies in its ability to surpass traditional modeling techniques, such as established models like EN 1993-1-5 [27], Leblouba et al. (2017, 2019) [15,16], and Wang et al. (2021) [5], in precision and adaptability to complex structural behaviors while offering insights into the impact of geometric and material variables on the shear strength of TCSWs.

3. Data collection and methodology

This study addresses the lack of research on applying machine learning techniques to predict shear strength in TCSWs. The methodology, outlined in Fig. 2, follows a systematic approach that begins with data collection, focusing on datasets relevant to the study's objectives. The collected data is then prepared and optimized for computational analysis. The methodology's core is Machine Learning Analysis, which employs Artificial Neural Network (ANN) algorithms to identify patterns within the data. The Comparative Analysis phase is a crucial component of the study, where various prediction models are compared using metrics such as R squared, Root Mean Square Error (RMSE), and Mean Absolute Error (MAE). This phase is further validated through importance and sensitivity analysis, which ensures the model's robustness. These phases' results present the optimal model characterized by its precision, computational efficiency, and accuracy.

Fig. 2.

Research Methodology Flowchart for Shear Strength Prediction using Machine Learning.

The study relies on a dataset comprising 206 experimental shear strength test results of TCWBs, collected from various published studies and summarized in Table 2. Despite the dataset's comprehensive nature, its relatively small size and variability due to different experimental setups may introduce biases or skew the model's predictions. To mitigate these effects, robust data preprocessing techniques, including normalization and outlier detection, were implemented to ensure data quality. Additionally, a sensitivity analysis was conducted to evaluate the influence of each variable on the model outputs, providing the model's robustness against variations in input data. The design of TCWBs requires considering a diverse array of critical variables. Each dataset includes eight geometrical and material input variables: web depth (h_w), web thickness (t_w), width of longitudinal fold panel (b), longitudinal width of inclined fold panel (d), corrugation depth (h_r), width of the inclined fold panel (c), steel yield strength (τ_y), and normalized elastic shear buckling strength (ρ). The experimental shear strength (V_test) of the TCWBs is the sole output variable in this study.

Table 2.

Shear strength of TCWB - Experimental Data from Literature.

Reference	No. Specimens	Reference	No. Specimens
Shimada [28]	15	Nie et al. [4]	8
Leiva-Aravena and Edlund [29]	2	Ibrahim [30],Usman [31]	12
Lindner [32]	25	Leblouba et al. [15,16,33]	21
Elgaaly [34]	42	Wang et al. [35]	1
Johnson et al. [36]	3	Zhang et al. [37]	1
Peil [38]	20	Elamary and Taha [39]	3
Shiratani et al. [13]	1	Elamary et al. [40]	5
Lee et al. [41]	9	Kadhim and Ammash [42]	3
Abbas [43,44]	3	Ammash and Al-Bader [45]	9
Watanabe et al. [46]	3	Deng et al. [47]	3
Architektur [48]	2	Abdullah and Muhaisin [49]	6
Moon et al. [50]	3	Abdullah and Almayah [51]	6
Total		206

3.1. Descriptive data analysis

Before developing the proposed model, a descriptive analysis was conducted to examine the statistical characteristics of the variables under investigation. Key statistical properties, such as variability and distribution patterns, were thoroughly analyzed, and the results are summarized in Table 3. A notable observation from the analysis is the mean shear strength (V_test) of 295.9 kN, with a high standard deviation of 469.2 kN, indicating considerable variability within the dataset.

Table 3.

Descriptive statistics of trapezoidal corrugated web beam properties.

Descriptive Statistics	Mean	Std. Deviation	Minimum	Maximum
Web depth hw (mm)	662.1	477.6	250.0	2210.0
Web Thickness tw (mm)	2.2	1.5	0.6	8.0
Steel yield strength fy (MPa)	378.6	148.1	189.0	714.0
Angle $\mathit{\theta }°$ (degree)	41.0	13.1	9.4	90.0
Longitudinal fold panel a1 (mm)	106.6	102.2	0.0	550.0
Corrugation depth a3 (mm)	50.9	33.6	10.0	200.1
Shear strength Vu (kN)	295.9	469.2	25.3	3861.0

Fig. 3(a–g) illustrates the distribution of all the studied variables, including the output variable, providing a better understanding of the data variability. The dataset comprises 206 experimental results from various studies, each with unique design parameters, test conditions, and loading scenarios, leading to different parameter distribution patterns. TCWB girders have been designed with diverse shapes and material properties based on specific needs, resulting in wide variations in geometric parameters such as web depth (h_w), web thickness (t_w), and corrugation angle. Moreover, researchers and engineers have employed different design practices and specifications, contributing to inconsistencies in the geometric and material properties of the dataset. Most variables tended to have positive distribution skewness. The distribution plot for the V_test shows more pronounced positive skewness and more dispersion. The analysis of beam thickness reveals a unique distribution profile, with most data points clustered around the median thickness value and fewer occurrences towards the extreme thickness values. This pattern is observed in all parameters except for the experimental shear strength analysis, where numerous outliers, primarily from extensive beam studies and supporting literature, contribute to the observed difference.

Fig. 3.

Distribution Plots - Geometrical, Material, and Shear Strength Characteristics of Trapezoidal Corrugated Webs: (a) Web depth (b) Web thickness (c) Steel yield strength (d) Angle (e) Longitudinal fold panel (f) Corrugation depth (g) Shear strength.

3.2. Model development

In the initial phase of implementing ANN models, the dataset was randomly divided into two subsets: a training and validation set containing 80 % of the data and a testing set containing the remaining 20 %. Before this division, Min-Max scaling was applied to normalize the range of independent variables. This scaling process was essential for enhancing neural network performance by initially standardizing features to a [0,1] range and experimenting with a [−1,1] range to assess the impact on model convergence and generalization.

To ensure a robust evaluation and mitigate the risk of overfitting, a 10-fold cross-validation technique was employed during model development. This technique involved partitioning the training set into 10 subsets, training the model on 9 subsets, and validating it on the remaining subset. The process was repeated ten times, using a different subset for validation in each iteration. Scaling the data to [−1,1], as recommended by Obaid et al. (2019) [52], demonstrated improved convergence speeds, facilitating the gradient descent process during backpropagation.

The overall model performance metrics, including R², Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Relative Error (MRE), were calculated as the average of these 10 iterations, providing a comprehensive assessment of model efficacy. The testing subset, consisting of data points not used in the training phase, served as the benchmark for evaluating the generalizability and predictive power of the developed models.

3.3. Performance metrics

The evaluation of the machine learning models in this study employed a set of widely recognized and effective performance metrics for comparing different machine learning methodologies. These metrics include:

• •Coefficient of Determination (R²): This metric measures the proportion of variance in the dependent variable that the independent variables can explain. It indicates the model's ability to fit the data effectively, with higher values suggesting a better fit.
• •Root Mean Square Error (RMSE): RMSE calculates the average magnitude of the predictive errors by measuring the difference between the predicted and actual values within the dataset. It provides a single metric for assessing the accuracy of predictions, with lower values indicating better performance.
• •Mean Absolute Error (MAE): MAE computes the average absolute differences between the predicted and actual values. This metric offers a straightforward and intuitive measure of predictive accuracy, with lower values indicating better performance.
• •Mean Relative Error (MRE): MRE determines the average percentage difference between the predicted and actual values. This metric provides a relative scale for evaluating the precision of the predictions, with lower values indicating better performance.

3.4. Artificial Neural Network

ANNs are a sophisticated computational model inspired by the mathematical abstraction of learning processes in human cognition. These networks comprise a series of interconnected processing units, each designed to replicate essential cognitive functions observed in the human brain, such as learning, memory formation, knowledge acquisition, generalization, and exploratory behavior. The operation of ANNs relies on implementing various learning algorithms during their training phase, with the Multilayer Perceptron (MLP) learning rule being particularly notable. MLPs are feed-forward networks that facilitate the sequential data flow from input nodes through intermediate hidden layers to output nodes, ultimately yielding the desired results. Extensive literature has been dedicated to this subject, with numerous studies providing empirical evidence supporting the assertion that MLPs can serve as universal predictors, particularly when configured with a single hidden layer. One of the most significant attributes of ANNs is their ability to model complex non-linear relationships between input and output variables with remarkable accuracy. The underlying mathematical framework of an individual neuron and its activation function application can be defined by Equations (20), (21):

$$z=\sum _{i=1}^n{w}_i{x}_i+b$$ (20)

$$a=f\left(z\right)$$ (21)

Where:

• • z is the aggregated weighted sum of the inputs.

• • x_i represents the input variables.

• • w_i denotes the respective weights assigned to each input.

• • b signifies the bias component.

• • f represents the activation function (sigmoid, ReLU, or tanh functions).

• • a is the output of the neuron.

3.4.1. ANN architecture

Fig. 4 illustrates the architecture of the ANNs employed in this study. The selection of ANN architecture is crucial in determining the models' precision, necessitating the identification of the most effective ANN characteristics. The following are the architectural features of the ANNs developed for this research:

• •Topology: The arrangement of neurons across various layers and the types of inter-neuronal connections significantly influence model performance. This study employs a multilayer feed-forward perceptron topology.
• •Input Parameters: The study incorporates eight distinct input variables to capture the relevant factors influencing the shear strength of trapezoidal corrugated steel web beams (TCWBs).
• •Hidden Layers: The architecture includes one to three hidden layers to investigate their impact on model efficacy and to identify the optimal number of hidden layers for accurate predictions.
• •Neurons per Hidden Layer: The study examines the effect of different neuron densities in hidden layers, specifically considering 10 (narrow), 25 (medium), and 100 (wide) neurons per layer. This analysis aims to determine the optimal number of neurons for achieving the best predictive performance.
• •Activation/Transfer Function: The hyperbolic tangent function is utilized for activation and transfer processes within the network, enabling the modeling of non-linear relationships between input and output variables.
• •Training Algorithm: The Levenberg–Marquardt backpropagation learning algorithm is employed to train the ANNs, ensuring efficient and effective learning from the available data.
• •Output Variable: The primary output of interest is the shear strength (V_test) of the TCWB, which the ANN models aim to predict accurately based on the given input variables.

Fig. 4.

General neural network topology for shear strength estimation.

3.5. Optimizable neural network

Improving the performance of neural networks requires careful hyperparameter optimization. Studies have shown that specific hyperparameters, such as the number of neurons per hidden layer and the total number of hidden layers, significantly impact model performance [53]. Traditionally, this process involves manually adjusting hyperparameters within a defined range to optimize model accuracy. This research adopts a more balanced approach by integrating advanced optimization techniques within a Python-based framework. These techniques aim to minimize the Mean Squared Error (MSE), which serves as the primary objective function, resulting in a model with optimally tuned hyperparameters. This strategy allows for a more systematic exploration of the hyperparameter space, ultimately leading to a model representing the optimal balance of these parameters.

The study employs the following optimization methodologies:

• •Bayesian Optimization: Based on the principles of probabilistic modeling, Bayesian optimization provides a structured approach to exploring the hyperparameter space of ANN models [54]. It uses a substitute probabilistic model, typically a Gaussian process, to map the complex relationship between hyperparameters and the MSE. This model offers valuable insights into uncertainty, guiding the identification of promising hyperparameter configurations and facilitating faster convergence toward optimal settings. The computational overhead and the sensitivity to the acquisition function choice are this method's main limitations.
• •Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS): L-BFGS is a quasi-Newtonian algorithm that excels at optimizing smooth or nearly convex objective functions [55]. In ANN hyperparameter optimization, L-BFGS utilizes an efficient approximation of the Hessian matrix, enabling a guided and effective minimization of the MSE. It is beneficial in scenarios with continuous hyperparameter spaces and relatively uniform objective functions. Requires reasonable initial parameter estimates to avoid local minima.
• •Firefly Algorithm: Inspired by the bioluminescence of fireflies, this nature-inspired algorithm can handle both continuous and discrete optimization problems [56]. When applied to ANN hyperparameter optimization, it iteratively explores different hyperparameter combinations, demonstrating a solid ability to search complex hyperparameter landscapes globally. It requires careful tuning of parameters, and it has slower convergence in high-dimensional spaces.
• •African Buffalo Optimization (ABO): Mimicking the social dynamics of African buffaloes, ABO is a population-based strategy [57]. It alternates between exploration and exploitation to iteratively search for optimal solutions. In ANN hyperparameter tuning, ABO iteratively refines a set of hyperparameters, assisting in the minimization of the MSE. This method is particularly effective in navigating complex hyperparameter spaces, ensuring a balanced approach to achieving optimal ANN model performance. ABO is computationally expensive due to multiple communication strategies and is sensitive to the choice of communication parameters.

4. Established models in literature

4.1. Model according to EN 1993-1-5

This study employs the framework outlined in EN 1993-1-5 [27] to model the shear strength of steel beams with corrugated webs. This approach requires a comprehensive consideration of both the geometric and material properties of the web, as well as the intricate interaction between local and global buckling phenomena and their influence on post-buckling strength. Adhering to this rigorous methodology is crucial to ensure that the design of steel beams with corrugated webs meets the strict safety and performance requirements specified in the Eurocode.

For beams with trapezoidal web configurations, the model identifies two main shear buckling modes: local (${\tau }_{cl,L})$ and global (${\tau }_{cl,G})$. The local buckling mode is primarily influenced by the dimensions of the largest flat panel, while the global buckling mode encompasses one or more corrugations. The critical stress values for these buckling modes are expressed in Equations (22), (23), (24), (25), (26), (27), (28) as follows:

$${\tau }_{cl,L}=4.83E{\left[\frac{t_w}{a_{\mathit{max}}}\right]}^2$$ (22)

$${\tau }_{cl,G}=\frac{32.4}{t_w{h}_w^2}\sqrt[4]{D_x{D}_z^3}$$ (23)

Where D_x and D_y are defined as:

$$D_x=\frac{E{t}_w^3}{12\left(1-v^2\right)}\frac{a_1+a_4}{a_1+a_2}$$ (24)

$$D_z=\frac{E{t_{w}a}_3^2}{12}\frac{3a_1+a_2}{a_1+a_4}$$ (25)

The slenderness parameters, crucial for assessing stability, are denoted by

$${\lambda }_i=\sqrt{\frac{f_y}{{\tau }_{cr,i}\sqrt{3}}}$$ (26)

The interaction buckling, represented as the combined critical stress τ_cr3, is computed using the Equation:

$$\frac{1}{{\tau }_{cr3}^n}=\frac{1}{{\tau }_{cr1}^n}+\frac{1}{{\tau }_{cr2}^n},wheren=1$$ (27)

Finally, the characteristic shear resistance is expressed by:

$$V_R=\rho \frac{f_y}{\sqrt{3}}{h}_w{t}_w$$ (28)

In this Equation, $\rho$ is the minimum of the reduction values ${\rho }_i$ calculated for λ₁ and λ₂. The abovementioned parameters are universal and will be consistently utilized throughout the analysis.

4.2. Model according to Leblouba et al. (2017, 2019)

Leblouba et al. (2017) [15] conducted a comprehensive study on the shear strength of TCSWs to refine predictive models for normalized shear strength. They developed an analytical model that integrates local and global shear buckling and interaction modes based on an extensive collection of existing experimental and numerical studies. This led to a more refined equation that accurately captures the shear strength of TCSWs, considering the complex interaction between local and global buckling modes. In a subsequent study, Leblouba et al. (2019) [16] further improved this model by incorporating geometric imperfections and material uncertainties, significantly enhancing the model's precision and reliability, especially when dealing with variable input parameters.

Their study's methodology for calculating local and global elastic shear buckling stresses remains consistent with previously established equations (Equations (7), (9)). For the interaction regression model, the authors introduced two coefficients for the interaction section, specifically for n = 3 and n = 4, while maintaining a value of a = 1 for both models. The normalized interaction shear buckling strength Equations (29), (30) are as follows:

$${\rho }_{M-1}=\frac{1}{{\left[1+{\left(\frac{{\lambda }_{I,4}}{1.58}\right)}^{1.6}\right]}^{1.15}}$$ (29)

$${\rho }_{M-2}=\frac{1}{1+{\left(\frac{{\lambda }_{I,3}}{1.4}\right)}^{1.7}}$$ (30)

4.3. Model according to Wang et al. (2021)

In the study conducted by Wang et al. (2021) [5], a new equation is proposed for predicting the normalized shear strength of TCSWs that are primarily influenced by local buckling phenomena. This innovative approach stands out due to its ability to incorporate the complex interaction between local and global buckling modes, providing a more accurate prediction of shear strength compared to existing models. Additionally, the study investigates the influence of various geometrical and material parameters on the shear strength of TCSWs. This comprehensive analysis contributes to a deeper understanding of the characteristics of TCSWs under different conditions. The local and global buckling stresses are mathematically expressed through Equations (8), (15)). Equations (31), (32) form the foundation for the normalized interaction shear buckling strength equations used in the study:

$${\lambda }_I={\left[{\left(\frac{{\tau }_L}{{\tau }_y}\right)}^{-1.25}+{\left(\frac{{\tau }_G}{{\tau }_y}\right)}^{-1.25}\right]}^{0.4}$$ (31)

$${\rho }_S={\left({\lambda }_I^{2.5}+1\right)}^{-0.8}$$ (32)

5. Results and discussion

5.1. Results of Optimizable neural network performance

This study presents a detailed analysis of the performance of trained and tested ANN models. The models' comprehensive evaluation is summarized in Table 4. Several performance metrics were employed, including R², RMSE, MAE, and MRE. Notably, performance varied across different models. The narrow ANN model exhibited the lowest performance, evidenced by an R² of 0.82, an RMSE of 179.22 kN, an MAE of 62.55 kN, and an MRE of 23.1 %. On the other hand, the Trilayered ANN model demonstrated superior performance metrics, with an R² of 0.90, an RMSE of 119.83 kN, an MAE of 59.02 kN, and an MRE of 13.6 %.

Table 4.

Results of Performance measures of ANN models.

Model	Trilayered Neural Network	Bilayered Neural Network	Medium Neural Network	Narrow Neural Network
R2	0.90	0.88	0.86	0.82
RMSE	119.83	114.37	143.25	179.22
MAE	59.02	52.12	63.95	62.55
MRE	13.6 %	15.7 %	18.0 %	23.1 %

Furthermore, the study applied four hyperparameter optimization techniques (BO, L-BFGS, FA, and ABO) to minimize the MSE and refine the ANN models. Quantitative and visual summaries of the optimization outcomes are presented in Table 5 and Fig. 5. Among these, the ABO-ANN model emerged as the most effective, as indicated by its R² value of 0.98, an RMSE of 40.01 kN, an MAE of 24.80 kN, and an MRE of 7.9 %. The optimal architecture of this model comprised a single hidden layer with 30 neurons. The ReLU function was identified as the most effective activation function. The optimization process concluded after 832 iterations upon achieving the minimum MSE (refer to Fig. 6). Consequently, this model was established as the most accurate in predicting the shear strength of the TCSW, depending upon the input parameters aligning with the trained value range.

Table 5.

Results of Performance measures of optimized ANN models.

Model	Bayesian Optimization	L-BFGS Optimization	African Buffalo Optimization	Firefly Optimization
R2	0.95	0.98	0.98	0.98
RMSE	51.53	53.61	40.01	41.40
MAE	30.15	36.32	24.80	26.70
MRE	10.3 %	15.1 %	7.9 %	11.4 %
No. neurons	Layer 1: 50 Layer 2: 93 Layer 3: 292	15	30	13

Fig. 5.

Comparative analysis of accuracy across optimized models.

Fig. 6.

Analysis of minimum mean squared error (MSE).

5.2. Model performance comparison

A detailed comparison of the results of the optimized ANN models developed in this study is summarized in Table 6. Based on the results, the ABO method is recommended for optimizing neural networks in this context due to its superior accuracy and global optimization capability. The combination of high R², low RMSE, and minimal MRE emphasizes the method's efficiency in predicting the shear strength of TCSW.

Table 6.

Comparison of the results of the optimized ANN models.

Optimization Method	Accuracy	Computational Efficiency
ABO	-Highest accuracy: R2 = 0.98, MRE = 7.9 % -Balanced exploration-exploitation approach effectively navigates complex hyperparameter spaces	-Balances computational efficiency with global optimization capabilities -Requires 30 neurons in a single hidden layer, striking a balance between model complexity and computational time -Exploration-exploitation strategy ensures the lowest MRE of 7.9 %
L-BFGS Optimization	-High accuracy: R2 ≥ 0.95 -Relatively high MRE of 15.1 %, indicating potential overfitting or less effective generalization compared to ABO	-Moderate computational efficiency -Requires only 15 neurons in a single hidden layer while achieving relatively high accuracy -Higher MRE suggests potential sensitivity to local minima
Firefly Optimization	-High accuracy: R2 = 0.98 with 13 neurons -Strong global optimization capabilities	-Stochastic nature allows it to escape local minima effectively -Converges slower than other methods due to extensive exploration
Bayesian Optimization	-High accuracy: R2 ≥ 0.95	-Complex architecture (435 neurons), rapid convergence, high computational overhead

5.3. Comparative analysis of models for predicting shear strength of TCSW

A comparative analysis of four models for predicting the shear strength of TCSW is presented in Table 7 and Fig. 7. The models considered are the Eurocode-based EN 1993-1-5 [27], the model by Leblouba et al. (2017, 2019) [15,16], the model proposed by Wang et al. (2021) [5], and the ABO-optimized ANN with a 30-neuron hidden layer. The EN 1993-1-5 model, despite its theoretical solid foundation, exhibits limitations with an R² value of 0.86, an RMSE of 120.4, and an MRE of 21 %, suggesting potential areas for improvement in complex TCSW modeling scenarios. The model of Leblouba et al. (2017) achieves an R² value of 0.96, an RMSE of 81.82, and an MRE of 19 %, indicating substantial accuracy. However, its reliance on a limited dataset may restrict its applicability to diverse TCSW designs. The Wang et al. (2021) model demonstrates reasonable consistency with an R² of 0.94 and a coefficient of variation (CoV) of 0.36. Still, its RMSE of 111.67 and MRE of 20 % raise questions about its universal applicability. The ABO-optimized ANN model exhibits superior performance across all metrics, with an R² of 0.98, an RMSE of 40.01, an MRE of 8 %, and a CoV of 0.11, indicating exceptional predictive accuracy.

Table 7.

Comparative evaluation of predictive models for TCSW shear strength using key performance metrics.

Model	EN 1993-1-5 [27]	Leblouba et al. (2017, 2019) [15,16]	T. Wang et al. (2021) [5]	ANN- Optimized by African Buffalo Algorithm
R2	0.86	0.96	0.94	0.98
RMSE (kN)	120.40	81.82	111.67	40.01
MRE	0.21	0.19	0.20	0.08
MAE (kN)	237.05	44.92	51.62	24.80
Mean (Vexp/Vpred)	1.06	1.04	1.02	1.00
CoV (Vexp/Vpred)	0.38	0.38	0.36	0.11

Fig. 7.

Correlation between experimental results and theoretical predictions of shear load in TCSW.

Fig. 8 presents box plots that illustrate the models' performance by displaying the spread and central tendencies of error metrics. A narrower interquartile range in the box plot suggests consistent performance, which is crucial in engineering applications. These visual representations complement the quantitative analysis, providing a comprehensive understanding of each model's strengths and weaknesses.

Fig. 8.

Variability in model performance for TCSW shear strength: Box plot analysis.

6. Analysis of variable importance and sensitivity in TCSW

Understanding the importance of different variables and their sensitivity in predicting the shear strength of TCSWs is crucial for model validation and practical design insights. Accurate models should predict the shear strength precisely and represent the expected conceptual effect of each input parameter on the output. This study employed importance and sensitivity analysis to evaluate the model's accuracy in predicting the shear strength and understanding the relationship between input and output parameters. The key objectives of this analysis are to identify the critical input variables, such as web geometry and material properties, that have the most significant influence on shear strength, to quantify the relative impact of each parameter on shear strength prediction, and to validate that the optimized ANN model accurately captures the influence of these different variables. This analysis provides valuable information for optimizing the design of TCSW girders and demonstrates the ANN model's robustness in representing the intricate relationships between input parameters and shear strength.

6.1. Importance analysis

The importance method applied to TCSW reveals critical insights, as in Table 8 and Fig. 9. The web depth (h_w), with a mean of approximately 1.65, is a significant variable contributing to the shear strength (V_u). This effect is closely followed by the web thickness (t_w), with an importance mean of around 0.93. Both h_w and t_w demonstrate a linear relationship with V_u, emphasizing their influence on the beam's shear capacity. An increment in either dimension correlates positively with enhanced shear strength. It is compatible with established structural engineering principles that associate increased section depth and thickness with increased strength and resistance to deformation.

Table 8.

Importance analysis of feature variables in TCSW shear strength prediction.

Feature	Importance Mean
hw	1.65
tw	0.93
fy	0.67
Angle	0.42
a1	0.06
a3	0.01

Fig. 9.

Neural network assessment of shear strength variation with Configuration parameters in TCSW.

6.2. Steel yield strength

The importance analysis reveals that steel yield strength (f_y), with a mean importance value of 0.67, is more complex in determining shear strength than geometric parameters. Although f_y is a fundamental material property that contributes to overall shear strength, its influence is less significant than the geometric attributes of the web in TCSW scenarios. However, when material yielding is the primary failure mode, f_y may play a more critical role.

6.3. Corrugation depth and longitudinal fold panel

The longitudinal fold panel (a₁) and corrugation depth (a₃) have lower importance means of 0.057 and 0.008, respectively, in determining the shear strength (V_u) of trapezoidal corrugated steel web beams (TCSWs). While a₃ shows a mild positive correlation with V_u, suggesting that deeper corrugations might slightly improve the structural performance of TCSW designs, a₁ exhibits an inverse relationship with V_u. This finding, which aligns with previous studies [51,58] indicates that larger longitudinal fold panels could reduce shear strength, potentially due to increased flexibility or stress concentrations. The positive correlation between a₃ and V_u was confirmed, but this relationship does not turn negative even when a₃ approaches or exceeds a₁. Instead, the correlation shows diminishing returns at lower values of a₃. A balance between a₃ and other geometric parameters should be considered for optimal design to maximize shear strength. While the importance analysis shows that a₃ remains a critical parameter, its effect highly depends on its proportion to other geometric factors. Future research can explore a parametric study to determine the precise ranges where a₃ positively or negatively influences V_u.

6.4. Corrugation angle

The corrugation angle significantly impacts the shear strength (V_u) of trapezoidal corrugated steel web beams (TCSWs), with an optimal range falling between 20 and 60°. Several studies have consistently shown that angles around 35–40° correspond to a peak in shear strength [3,4,33,59,60]. The angle distinguishes between three types of corrugation: a 0-degree angle resembles a flat web, while a 90-degree angle represents a rectangular corrugation. The shear strength reaches its maximum at approximately 45° and decreases as the angle approaches that of rectangular corrugations [45,61,62] The optimal corrugation angle for maximizing shear strength in TCSWs is 35–45°.

For beams with shear strengths exceeding 600 kN, carefully considering geometric and material properties is essential. The increased effects at such scales require meticulous design to prevent localized failures. When designing TCSWs for high-load applications, prioritizing web geometry (depth and thickness) is crucial, while material properties like yield strength remain essential but secondary.

7. Summary and conclusions

This study explored the use of Artificial Neural Network (ANN) models to predict the shear strength of trapezoidal corrugated steel webs (TCSWs) using a dataset of 206 experimental results from the literature. The ANN model's hyperparameters were optimized using several techniques, namely, Bayesian Optimization (BO), Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS), Firefly Algorithm (FA), and African Buffalo Optimization (ABO). The following conclusions were drawn:

• 1.The African Buffalo Optimization (ABO) algorithm yields the best optimized ANN model.
• 2.The optimized ABO-ANN model yielded an MRE of 8.0 % for the 206 points from the database, outperforming the EN 1993-1-5 code model and the existing calculation models assessed in this work. The latter models exhibit mean errors of about 20 %.
• 3.The sensitivity analysis using the proposed ANN-based model captured the impact of the geometric and material parameters on TCSW shear strength. It was found that the corrugation angle significantly impacts TCSW shear strength, with optimal performance observed between 35 and 45°.
• 4.The study highlights the ANN model's potential but also acknowledges its limitations. The dataset, though substantial, may not fully capture the wide range of beam configurations and loading scenarios. Future research should expand the dataset and include additional geometric and material variables to improve the ANN model's predictive accuracy.
• 5.This conclusion offers valuable insights into predicting the shear strength of TCSWs using optimized ANN models, providing a basis for further research and practical applications in structural engineering.

Funding

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

Data availability statement

The data used in this study will be available upon request.

CRediT authorship contribution statement

Mazen Shrif: Writing – original draft, Visualization, Software, Methodology, Investigation, Data curation, Conceptualization. Samer Barakat: Writing – review & editing, Validation, Project administration, Methodology, Investigation, Data curation, Conceptualization. Zaid Al-Sadoon: Writing – original draft, Validation, Supervision, Investigation. Omar Mostafa: Writing – original draft, Visualization. Raghad Awad: Writing – review & editing, Resources, Data curation.

Declaration of competing interest

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