Large web openings introduce complex structural behaviors and additional fail- ure modes of steel cellular beams, which must be considered in the design using laborious calculations (e.g., exercising SCI P355). This paper presents seven machine learning (ML) models, including decision tree (DT), random forest (RF), k-nearest neighbor (KNN), gradient boosting regressor (GBR), extreme gradient boosting (XGBoost), light gradient boosting machine (LightGBM), and gradient boosting with categorical features support (CatBoost), for predicting the elastic buckling and ultimate loads of steel cellular beams. Large datasets of finite element (FE) simulation results, validated against experimental data, were used to develop the models. The ML models were fine-tuned via an extensive hyperparameter search to obtain their best performance. The elastic buckling and ultimate loads predicted by the optimized ML models demonstrated ex- cellent agreement with the numerical data. The accuracy of the ultimate load predictions by the ML models exceeded the accuracy provided by the existing design provisions for steel cellular beams published in SCI P355 and AISC De- ∗Corresponding author Email addresses: vitaliy.degtyarev@newmill.com, vitdegtyarev@yahoo.com (Vitaliy V. Degtyarev), konstantinos.tsavdaridis@city.ac.uk (Konstantinos Daniel Tsavdaridis) December 2, 2021 sign Guide 31. The relative feature importance and feature dependence of the models were evaluated and discussed in the paper. An interactive Python-based notebook and a user-friendly web application for predicting the elastic buckling and ultimate loads of steel cellular beams using the developed optimized ML models were created and made publicly available. The web application deployed to the cloud allows for making predictions in any web browser on any device, including mobile. The source code of the application available on GitHub allows running the application locally and independently from the cloud service. Keywords: Cellular beams, Perforated web, Elastic buckling, Ultimate strength, Predictive models, Machine learning 1. Introduction 1 Perforated steel beams with repeating web openings have been used in con- 2 struction for more than a century [1]. They offer several advantages over steel 3 beams with solid webs, including weight reduction, higher strength-to-weight 4 ratio, integration of utilities, and improved aesthetics. Castellated beams with 5 hexagonal openings, which were the first type of beams with perforated web sec- 6 tions, have practically been replaced in modern construction by cellular beams 7 with circular openings [2]. Multiple large openings cause a significant reduc- 8 tion in the beam shear strength and introduce additional possible failure modes 9 of the beams, which makes the flexural behavior and design of cellular beams 10 complicated. A cellular beam may exhibit one of the following failure modes: 11 global bending, lateral-torsional buckling, vertical shear, local Vierendeel bend- 12 ing, web post horizontal shear, web post bending, web post buckling, and shear 13 buckling. Many researchers have contributed to the body of knowledge about 14 the strength and structural behavior of steel cellular beams. Several research 15 publications describe numerical studies on the lateral-torsional buckling of cel- 16 lular beams [3–8], which allowed for determining the effects of different design 17 2 parameters on the beam strength governed by the elastic and inelastic lateral- 18 torsional buckling. It was found in particular that the cellular beam geometry 19 affected the moment-gradient coefficient, which is not the case for the solid-web 20 beams [4]. T-shaped stiffeners were proposed to improve the flexural stiffness of 21 cellular beams and reduce the lateral-torsional buckling occurrence [5]. Modi- 22 fied calculations of the cross-sectional properties and a modified buckling curve 23 selection were developed based on the existing European guidelines [6]. 24 Web post buckling of cellular beams and beams with web openings of dif- 25 ferent shapes was studied in [9–11]. Tsavdaridis and D’Mello [9] demonstrated 26 that particular non-standard opening shapes improved the beam structural per- 27 formance compared with the beams with standard circular, hexagonal, and elon- 28 gated web openings. They also proposed an empirical formula for predicting the 29 ultimate vertical shear strength of web posts formed by the different opening 30 shapes. Panedpojaman et al. [10] proposed design equations for predicting the 31 shear strength of local web post buckling in symmetric and asymmetric cellular 32 beams, which demonstrated improved accuracy in predicting the shear strength 33 compared with BS EN 1993-1-1 [12] and AISC 360 [13]. 34 Chung et al. [14] investigated the Vierendeel mechanism in cellular steel 35 beams and found that shear yielding is more critical in steel beams with cir- 36 cular openings than in beams with rectangular openings. They proposed an 37 empirical shear moment interaction curve at the perforated sections. Kang et 38 al. [15] studied the shear behavior and strength of cellular beams and proposed 39 a rational design model for predicting the beam shear strength, which showed a 40 good agreement with the numerical and experimental results. Ellobody [16] in- 41 vestigated combined buckling modes of steel cellular beams and found that the 42 failure load could be significantly reduced when the beams failed in combined 43 web distortional and web post-buckling. 44 3 Several research papers have been dedicated to the optimal design of cellular 45 beams [9, 17–24]. The studies demonstrated that the strength and weight of 46 the beams with web openings could be significantly improved by using non- 47 standard opening shapes [9, 17, 19, 21], applying special optimization techniques 48 [18, 20, 22], and selecting specific sizes and spacing of web openings [21, 23, 24]. 49 Akrami and Erfani [25] assessed design methodologies for perforated steel 50 beams presented in ASCE 23-97 [26], SCI P100 [27], SCI P355 [2], Chung et 51 al. [28], and Tsavdaridis and D’Mello [17]. The two latter methods were found 52 least restrictive and produced the lowest errors. The authors proposed ASCE 53 23-97 modifications, which showed a good agreement with experimental and 54 numerical data. 55 The presented literature review shows that the published research concen- 56 trated mainly on studying specific failure modes of cellular beams. To fill the 57 gap in the information about the global response of such members, Rajana et al. 58 [29] performed an extensive numerical parametric study of the elastic and inelas- 59 tic buckling of cellular beams subjected to strong axis bending. The effects of 60 various parameters on the elastic buckling and ultimate loads of cellular beams 61 were investigated, and an extensive database of the FE simulation results was 62 generated. The study showed that the elastic buckling was affected mainly by 63 the web thickness and the flange geometry. The diameter of web openings, their 64 spacing, flange geometry, and web thickness were the most critical parameters 65 affecting the beam strength. It was also determined that the initial geometric 66 imperfections had an insignificant effect on the predicted beam strength. 67 Artificial intelligence (AI) and machine learning (ML) are emerging fields 68 of computer science that allow for developing machines with simulated human 69 intelligence and creating data-based descriptive models capable of handling very 70 complex problems efficiently. A properly developed ML model for engineering 71 4 applications reveals hidden relations between the predicted variable and input 72 parameters based on the underlying physics. Many industries have successfully 73 adopted AI and ML [30–33], whereas their deployment in structural engineering 74 is still somewhat limited despite many research publications demonstrating the 75 accuracy and effectiveness of the AI and ML methods. 76 The number of research publications on ML applications in civil and struc- 77 tural engineering had increased exponentially since the second half of the 1980s, 78 when the first papers on this topic were published [34–39]. Many publica- 79 tions described ML models considered in this study for predicting properties 80 of concrete and reinforced concrete structures [40–65]. Fewer papers have been 81 published on ML applications to steel structures, including buckling analysis of 82 beam-columns [66], cold-formed steel (CFS) space structure optimization [67], 83 web crippling strength prediction [68], elastic distortional buckling stress de- 84 termination [69, 70], rotation capacity prediction [71], strength prediction of 85 concrete-filled steel tubular columns [72], failure mode identification of column 86 base plate connection [73], capacity prediction of cold-formed stainless steel 87 tubular columns [74], seismic drift demand estimation for steel moment frame 88 buildings [75], and shear strength of CFS channels with staggered perforated 89 webs [76–79]. ML techniques were previously applied to steel cellular beams. 90 Sharifi et al. [80] developed an artificial neural network (ANN) to predict the 91 flexural strength of steel cellular beams governed by lateral-torsional buckling 92 using a relatively small dataset with 99 samples. The predicting abilities of 93 the developed ANN were superior to those by the Australian Standard [81]. 94 Abambres et al. [82] developed an ANN model and an ANN-based formula for 95 predicting the elastic buckling load of cellular beams using a large dataset of 96 numerical results described in [29]. The ANN and the proposed formula showed 97 an excellent agreement with the FE simulation results. An ANN and ANN- 98 5 based formula for predicting the lateral-torsional buckling resistance of slender 99 steel cellular beams were presented by Ferreira et al. [83]. 100 The presented review indicates that ML has excellent potential for develop- 101 ing structural engineering expert tools, which can help designers make informed 102 decisions during pre-design and design checking. ML models built on top of the 103 available test or numerical data are computationally efficient and often more 104 accurate than the existing design methods based on the traditional approaches, 105 which often rely on fewer data points and engineering intuition. They can 106 replace computationally intensive finite element simulations when the design 107 parameters are within the ranges of the data used for the ML model training. It 108 should also be noted that accurate finite element simulations require advanced 109 software resources, which are not always available to designers, and advanced 110 techniques, thus skills and knowledge. Even when the appropriate software is 111 available, and the designers possess the required expertise, it is impractical to 112 perform advanced finite element simulations during the project design phase due 113 to time constraints. Because of that, engineers are often using simplified FEA 114 to perform stress analyses. The downside is that these are not very accurate 115 due to the number of assumptions, leading to the very same initial problem: the 116 need of higher safety factors and the excessive use of material where it is not 117 needed. 118 This study aims to explore various ML algorithms for predicting the elastic 119 buckling and ultimate loads of steel cellular beams. Considering the complexity 120 of the structural behavior and design of such members, ML models are deemed to 121 be a promising alternative to the existing design guidelines and computationally 122 expensive FE modeling. The objectives of the study were as follows: 123 1. Develop and optimize ML models for predicting the elastic buckling and ul- 124 timate loads of steel cellular beams based on seven popular ML regressors, 125 6 including decision tree (DT), random forest (RF), k-nearest neighbors 126 (KNN), gradient boosting regressor (GBR), extreme gradient boosting 127 (XGBoost), light gradient boosting machine (LightGBM), and gradient 128 boosting with categorical features support (CatBoost). 129 2. Interpret and explain the developed models using the permutations and 130 SHapley Additive exPlanations (SHAP) [84] methods. 131 3. Compare predictions by the developed ML models with those per SCI 132 P355 [2] and AISC Design Guide 31 [85]. 133 ML models were trained using FE simulation results of steel cellular beams 134 published in [29]. The elastic buckling load dataset included 3645 samples. The 135 ultimate load (inelastic buckling load) dataset consisted of 78390 samples. All 136 models were implemented in open-source Python-based frameworks, and their 137 hyperparameters were optimized via an extensive tuning process. The ten-fold 138 cross-validation method was employed for the model training and performance 139 evaluation. The final evaluation of the models was performed on the data unseen 140 by the models during training. The ML model predictions showed an excellent 141 agreement with the FE simulation results. The ultimate loads of the cellular 142 beams predicted by the models compared with the FE analysis data considerably 143 better than the ultimate loads predicted by SCI P355 [2] and AISC Design Guide 144 31 [85]. 145 A web application for predicting the elastic buckling and ultimate loads of 146 steel cellular beams was created in Streamlit. A light version of the application 147 was deployed to the cloud at https://scba-cb.herokuapp.com/ . It allows for 148 making predictions in any web browser on any device, including mobile. The 149 source codes of the full and lite application versions are available on GitHub 150 athttps://github.com/vitdegtyarev/SCBA-Streamlit and https://gi 151 thub.com/vitdegtyarev/SCBA-Streamlit-CB , respectively. They allow for 152 7 running the application on a local machine. 153 The novelty of the presented work consists of the development of new opti- 154 mized ML models for accurate and computationally efficient predictions of the 155 elastic buckling and ultimate loads of steel cellular beams, interpretation and ex- 156 planation of the developed models using the permutations and SHAP methods, 157 comparison of the performance of seven different ML models, and development 158 of a web application based on the optimized ML models for the ease of use by 159 engineers in practice. 160 2. Datasets 161 The elastic buckling load, wcr, and ultimate load (inelastic buckling load), 162 wmax, datasets of FE simulation results from [29] were used for training and per- 163 formance evaluation of the ML models. The FE models were validated against 164 the experimental data, as described in [29]. The wcrandwmaxdatasets consist 165 of 3645 and 78390 samples, respectively. Fig. 1 shows dimensional parameters 166 of the cellular beams considered in the numerical parametric study, including 167 beam span length, L; beam height, H; flange width, bf; flange thickness, tf; 168 web thickness, tw; opening diameter, Do; web post width, WP; and opening 169 end distance, Led. 170 Figure 1: Dimensional parameters of steel cellular beams In addition to the dimensional parameters of the beams, the ultimate load 171 dataset included steel yield stress, Fy; steel ultimate stress, Fu; steel yield strain, 172 8 ϵy; steel ultimate strain, ϵu; and initial geometric imperfections considered in 173 the FE models. The dimensional beam characteristics shown in Fig. 1 and Fy 174 (in the wmaxmodels only) were considered the ML models’ input parameters. 175 The initial geometric imperfections were excluded from the input parameters 176 because they have an insignificant effect on wmax[29], and their exact shape 177 and magnitude are not usually known to the designer. 178 Distributions of the parameters in the elastic buckling and ultimate load 179 datasets presented in Figs. 2 and 3 demonstrate that the datasets cover a wide 180 range of the beams and represent the steel cellular beams used in construction. 181 Figure 2: Distributions of variables of the elastic buckling load dataset Fig. 4 and 5 show correlation matrices for the dataset variables. The beam 182 span length, L, has the highest negative correlation with wcrandwmax, char- 183 acterized by moderate coefficients of correlation of -0.67 and -0.60, respectively. 184 All other variables have weak correlations with wcrandwmax, with coefficients 185 of correlation not exceeding 0.37. It is interesting to note that wmaxhas a con- 186 siderably stronger correlation with WP than wcr, which indirectly highlights 187 the positive contribution of the web post plastic behavior to the ultimate load 188 9 Figure 3: Distributions of variables of the ultimate load dataset of the cellular beams. Dohas a strong positive correlation with Hbecause the 189 Dovalues were set as fractions of the Hvalues in the numerical parametric 190 study. All other dataset variables have weak correlations between themselves. 191 Figure 4: Correlation matrix for the elastic buckling load dataset 10 Figure 5: Correlation matrix for the ultimate load dataset The datasets used in this study can be found at the following link: https: 192 //www.kaggle.com/vitdegtyarev/buckling-and-ultimate-loads-of-cel 193 lular-beams . 194 3. Review of machine learning algorithms 195 The abilities of seven popular supervised ML algorithms to predict the elas- 196 tic buckling load, wcr, and ultimate load, wmax, of steel cellular beams were 197 evaluated. Supervised ML algorithms learn by example using labeled train- 198 ing data, which consist of input parameters (also known as features) and one 199 or more output values (also known as targets). The evaluated ML algorithms 200 included decision tree (DT), random forest (RF), k-nearest neighbors (KNN), 201 gradient boosting regressor (GBR), extreme gradient boosting (XGBoost), light 202 gradient boosting machine (LightGBM), and gradient boosting with categorical 203 features support (CatBoost). These algorithms are commonly employed to de- 204 11 velop predictive ML models in civil/structural engineering (as was discussed in 205 the Introduction section) and other domains. They are based on different princi- 206 ples and may result in different performances when used for different problems. 207 One algorithm may demonstrate a better predictive accuracy than others on 208 one problem and inferior performance on a different problem. Therefore, it is 209 important to find an algorithm and its optimal hyperparameters that works the 210 best for a given problem. 211 Fig. 6 demonstrates the schematic architecture of the considered ML models 212 for predicting wcrandwmax. The models consisted of features, ML algorithms, 213 and targets. The features of the models for predicting wcrwere L,H,bf,tf, 214 tw,Do,WP, and Led. The wmaxmodels also included Fyas a feature. 215 All ML algorithms have hyperparameters, or the parameters specified before 216 the model training to control the learning process and avoid overfitting or un- 217 derfitting. Overfitting is characterized by the ability of a model to make good 218 predictions for the samples used in training while making poor predictions on 219 the new samples of data unseen by the model before. An underfitted model 220 produces poor predictions on the seen and unseen data. The ability of an ML 221 model to make good predictions for previously unseen data is referred to as gen- 222 eralization. Finding optimal hyperparameters is essential for obtaining a model 223 with the best performance and generalization [53, 76]. It is equivalent to finding 224 the form and coefficients of a regression equation that gives the best prediction 225 accuracy for a given problem. 226 The following sections present a brief overview of each ML algorithm con- 227 sidered in the study. Detailed information about the ML algorithms and their 228 practical implementation can be found in published literature, including [86] 229 and [87]. 230 12 Figure 6: Architecture of ML models 3.1. Decision tree 231 The DT algorithm bears its name from its tree structure incrementally de- 232 veloped by splitting the dataset into smaller subsets. DT models have three 233 types of nodes: root node, decision nodes, and terminal nodes (also known as 234 leaves). The learning starts at the root node, which includes all training data. 235 The root node splits into two or more decision nodes, which include subsets of 236 the original training data. The splitting occurs based on a series of questions 237 determined by the algorithm. It continues for the subsequent levels until a pre- 238 defined maximum depth of the tree is reached or when the nodes have only one 239 sample of the training data. The algorithm stops at the terminal nodes, which 240 do not split. 241 Various algorithms for growing a DT exist. They differ by the possible tree structure, the split finding criteria, the splitting stoppage criteria, and the model estimation within the terminal nodes. The classification and regression trees (CART) algorithm [86] was used in this study. In this algorithm, a dataset (xi, yi) fori= 1,2, ..., N , with xi= (xi1, xi2, ..., x ip) is considered, where xiand yiare features and targets, Nis the number of samples, and pis the number of features. The original dataset is split into Mregions R1,R2,...,RM. The model prediction in each region is a constant cmdescribed by Eq. 1. f(x) =MX m=1cmI(x∈Rm) (1) 13 where I(x∈Rm) is the identity function that returns 1 if xis in the subset Rm 242 and 0 otherwise. 243 The best ˆ cmis the average of yiin region Rmwhen the sum of squared errors P(yi−f(xi))2is used as the criterion of minimization: ˆcm= ave( yi|xi∈Rm) (2) The following greedy algorithm is employed to find the best binary partition 244 of each node in terms of the minimum sum of squared errors. The pair of 245 half-planes partitioned by a splitting variable jand a split point sis defined as 246 follows. 247 R1(j, s) ={X|Xj≤s}andR2(j, s) ={X|Xj> s} (3) The splitting variable jand the split point sthat solve Eq. (4) are sought. 248 min j,s min c1X xi∈R1(j,s)(yi−c1)2+ min c2X xi∈R2(j,s)(yi−c2)2  (4) The inner minimization is solved by 249 ˆc1= ave( yi|xi∈R1(j, s)) and ˆ c2= ave( yi|xi∈R2(j, s)) (5) Once the best split is found, the dataset is partitioned into two resulting 250 subsets, after which the splitting process is repeated for each subset and each 251 node in the subsequent levels. 252 The DT advantages consist of the relative ease of data preparation, the ease 253 of understanding and interpretation, and robustness against missing values. One 254 of the main disadvantages of DT is their proneness to overfitting when the tree 255 is very large [88, 89]. To avoid overfitting, the DT model should not be very 256 14 large. At the same time, the model should be large enough to capture the 257 important relationships between the features and targets to avoid underfitting. 258 DT hyperparameters include the maximum depth of the tree, the minimum 259 number of samples required to split an internal node, the minimum number of 260 samples required to be at a leaf node, and others. 261 3.2. Random forest 262 RF is an ensemble of DTs generally trained via bagging , which stands for 263 bootstrap aggregating [86]. In this method, the same algorithm (DT) is trained 264 many times on different random subsets of the entire training set. The sampling 265 is performed with replacement, meaning that the same sample may appear 266 in different subsets. Predictions from multiple randomly generated DTs are 267 averaged to obtain the final output value of the RF algorithm. 268 The RF regression algorithm consists of the following steps [86]. 269 1. For b=1 to B, where bis an individual DT and Bis the total number of 270 DTs (estimators): 271 (a) A bootstrap sample of size Nis drawn from the training data. 272 (b) A tree Tbis grown to the bootstrapped data by repeating the fol- 273 lowing substeps for each node until the maximum tree depth or the 274 minimum node size is reached: 275 i.mvariables are randomly selected from pvariables. 276 ii. The best variable among mand the best split point is found. 277 iii. The node is split into two nodes. 278 2. The ensemble of trees {Tb}B 1is output. 279 3. The final prediction is made as ˆfB RF(x) =1 BPB b=1Tb(x). 280 The RF advantages include those listed in Subsection 3.1 for DT and its 281 robustness against overfitting due to the presence of multiple independent DTs 282 15 making predictions. On the negative side, RF requires more computational 283 power and resources to build numerous trees and combine their outputs com- 284 pared with DT. The RF hyperparameters include those for DTs plus the number 285 of trees in the forest. 286 3.3. K-nearest neighbors 287 The KNN regression algorithm predicts the output value by interpolating 288 the output values of knearest neighbors in the training set. The number of 289 neighbors kis a hyperparameter set before training. The distance between the 290 neighbors is defined by the distance function in the form of the Minkowski metric 291 described by Eq. (6). The Euclidean and Manhattan distances, which are other 292 typical distance metrics, can be obtained from the Minkowski metric by setting 293 the power parameter, p, equal to 1 and 2, respectively. 294 D(X, Y ) = kX i=1|xi−yi|p!1 p (6) The output values are obtained by taking either an average (Eq.(7)) or an inverse distance weighted average (Eq.(8)) of the knearest neighbors with sim- ilar features. In the latter approach, closer neighbors have a more significant influence on the target than the more distant neighbors. ˆf(x) =1 kX xi∈Nk(x)yi (7) ˆf(x) =P xi∈Nk(x)1 diyiP xi∈Nk(x)1 di(8) where Nk(x) is the neighborhood of xdefined by the kclosest points xiin the 295 training data, diis the distance from the ithpoint to the estimated point. 296 The KNN advantages include the ease of implementation, the ability to 297 16 add new data without the effect on the algorithm’s accuracy, and the training 298 period absence, which makes the KNN algorithm significantly faster than other 299 ML algorithms when the dataset size and the number of input variables are 300 relatively small. The KNN disadvantages consist of sensitivity to noisy data, 301 missing values and outliers, and slow predictions for large datasets and datasets 302 with a large number of features. The KNN hyperparameters are the number of 303 neighbors, the weight function (uniform or inverse distance weighted), and the 304 distance metric. 305 3.4. Gradient boosting 306 Boosting algorithms, or boosting machines, are ensemble methods that com- 307 bine several weak learners (usually DTs) to produce a strong learner. Boosting 308 machine predictors are trained sequentially, with each subsequent learner im- 309 proving the predecessor’s predictions. The algorithm stops when a predefined 310 number of predictors is reached or when the perfect fit is achieved. The two 311 common boosting algorithms are gradient boosting and adaptive boosting. 312 In gradient boosting, the boosting algorithm is combined with gradient de- 313 scent, which is an iterative optimization algorithm for finding a local minimum 314 of a function. New predictors are fitted to the residual errors from the previous 315 predictors. The gradient boosting algorithm includes the following steps [90]: 316 1. For a training set ( xi, yi)n i=1, the model is initialized with a constant value 317 of 318 F0(x) = argmin γnX i=1L(yi, γ) (9) where iandndenote the ithsample and the total number of samples 319 2. For m=1 to M, where mandMare the mthiteration and the total number 320 of iterations: 321 17 (a) Pseudo-residuals are computed as follows: 322 rim=−∂L(yi, F(xi)) ∂F(xi) F(x)=Fm−1(x)(10) (b) The training set ( xi, rim)n i=1is used to fit a predictor hm(x) to 323 pseudo-residuals. 324 (c) Multiplier γmis computed by solving the following optimization prob- 325 lem: 326 γm= argmin γnX i=1L[yi, Fm−1(xi) +γhm(xi)] (11) (d) The model is updated using the following equation: 327 Fm(x) =Fm−1(x) +γmhm(x) (12) 3.FM(x) is obtained. 328 The gradient boosting algorithm’s advantages include high accuracy, flexi- 329 bility, and the ability to handle missing data. It is generally considered resistant 330 to overfitting due to many weak learners involved in the prediction. However, 331 the algorithm may overfit when its hyperparameters are poorly selected. The 332 disadvantages of gradient boosting are computation cost, the number of hyper- 333 parameters that require proper tuning, and limited interpretability. 334 The gradient boosting hyperparameters include learning rate, the number of 335 boosting iterations, maximum depth of the individual regression estimators, the 336 minimum number of samples required to split an internal node, the minimum 337 number of samples required to be at a leaf node, and others. 338 The gradient boosting algorithms have been implemented in several frame- 339 works: GBR [91], XGBoost [92], LightGBM [93], and CatBoost [94]. XGBoost, 340 LightGBM, and CatBoost are improved implementations of GBR. XGBoost was 341 18 optimized for more accurate and faster predictions via regularization, custom 342 loss functions, parallel processing, and other algorithm improvements. Light- 343 GBM offers improved training speed, higher efficiency, better accuracy, lower 344 memory use, and the ability to process large datasets by applying the Gradient- 345 based One-Side Sampling (GOSS) method and parallel learning. CatBoost can 346 process categorical features to improve accuracy for datasets with categorical 347 features, ordered boosting to fight overfitting, missing value support, and others. 348 4. Implementation and results 349 The ML algorithms were implemented in the following Python-based open- 350 source libraries: scikit-learn (DT, RF, KNN, and GBR) [91], XGBoost [92], 351 LightGBM [93], and CatBoost [94]. The models were optimized, validated, and 352 tested using the ten-fold cross-validation method. The wcrandwmaxdatasets 353 were randomly divided into training and test sets in the 80/20 proportion. The 354 training set of each dataset was partitioned into ten groups. The models were 355 trained on nine groups of the training set and validated on the remaining group. 356 The process was repeated for the remaining groups of the training set until each 357 group had served as the validation set. The final performance of the models was 358 evaluated on the test data unseen by the models in training. Compared with the 359 hold-out method, where the dataset is divided into training, validation, and test 360 sets, with each set used for its purpose only, the ten-fold cross-validation method 361 makes more samples available for model training and excludes model dependence 362 on a particular random choice of the samples selected for the training, validation, 363 and test sets. As a result, the ten-fold cross-validation method usually produces 364 more accurate models with better generalization performance. 365 Figs. 2 and 3 demonstrate that the numerical ranges of the features ranged widely in the datasets, which is not ideal for ML models, as it might cause 19 difficulties for the algorithms in finding optimal model parameters. Each feature value in the training set was standardized using Eq. (13) to make the features’ scales uniform. Each feature in the test set was also standardized using the mean and standard deviation values of the feature obtained for the training set. x′=x−µ σ(13) where x′is the standardized value of the input parameter, xis the original 366 (non-standardized) value of the input parameter, µis the mean of the original 367 values of the input parameter, and σis the standard deviation of the original 368 values of the input parameter. 369 Performance of the ML learning models was evaluated based on the mean squared error (MSE) values obtained for the test set calculated as follows. MSE =1 nnX i=1(y−ˆy)2(14) where nis the number of samples, yis the output value, and ˆ yis the predicted 370 output value. 371 Mean absolute error (MAE), mean absolute percentage error (MAPE), the coefficient of determination ( R2), the minimum, maximum, mean, and coef- ficient of variation values of the prediction-to-FEA ratios, which are metrics commonly used for performance evaluation of ML models [95], calculated using the following equations were also determined for the training and test sets. MAE =1 nnX i=1|y−ˆy| (15) MAPE =100 nnX i=1 y−ˆy y (16) 20 R2= 1−Pn i=1(y−ˆy)2 Pn i=1(y−¯y)2(17) where ¯ yis the mean of the yvalues. 372 Extensive hyperparameter tuning was carried out for each ML model us- 373 ing the grid and random searches to find optimal hyperparameter values that 374 give the best model performance. The obtained optimal hyperparameters for 375 each ML model are listed below. The hyperparameter designations used in the 376 Python libraries [91–94] are shown in parentheses. The hyperparameters not 377 presented below had default values. 378 •DT: 379 –the maximum depth of the tree ( maxdepth ): None for wcrandwmax, 380 –the minimum number of samples required to split an internal node 381 (minsamples split ): 4 for wcrand 2 for wmax, 382 –the minimum number of samples at a leaf node ( minsamples leaf ): 383 2 for wcrandwmax. 384 •RF: 385 –the number of trees in the forest ( nestimators ): 80 for wcrand 200 386 forwmax, 387 –the maximum depth of the tree ( maxdepth ): None for wcrandwmax, 388 –the minimum number of samples required to split an internal node 389 (minsamples split ): 2 for wcrandwmax, 390 –the minimum number of samples at a leaf node ( minsamples leaf ): 391 1 for wcrandwmax. 392 •KNN 393 21 –the number of neighbors ( nneighbors ): 5 for wcrand 4 for wmax, 394 –weight function ( weights ): uniform for wcrandwmax, 395 –the power parameter for the Minkowski metric ( p): 1 for wcrand 396 wmax, 397 –leaf size ( leaf size ): 20 for wcrand 30 for wmax. 398 •GBR: 399 –learning rate ( learning rate ): 0.1 for wcrandwmax, 400 –the number of boosting stages ( nestimators ): 200 for wcrand 1300 401 forwmax, 402 –maximum depth of individual regression estimators ( maxdepth ): 5 403 forwcrand 9 for wmax, 404 –the minimum number of samples required to split an internal node 405 (minsamples split ): 2 for wcrandwmax, 406 –the minimum number of samples at a leaf node ( minsamples leaf ): 407 3 for wcrand 4 for wmax. 408 •XGBoost: 409 –learning rate ( eta): 0.2 for wcrandwmax, 410 –minimum loss reduction required to make a further partition on a 411 leaf node of the tree ( gamma ): 1 for wcrandwmax, 412 –the maximum tree depth of base learners ( maxdepth ): 5 for wcrand 413 12 for wmax, 414 –the minimum sum of instance weight (hessian) needed in a child 415 (minchild weight ): 3 for wcrand 6 for wmax. 416 •LightGBM: 417 22 –learning rate ( learning rate ): 0.1 for wcrandwmax, 418 –the number of boosting iterations ( numiterations ): 100 for wcrand 419 3800 for wmax, 420 –maximum tree leaves for base learners ( numleaves ): 50 for wcrand 421 wmax, 422 –the minimum number of observations that must fall into a tree node 423 for it to be added ( mindata inleaf ): 10 for wcrandwmax, 424 –maximum tree depth for base learners ( maxdepth ): -1 (None) for 425 wcrandwmax, 426 –the maximum number of bins ( maxbin): 100 for wcrandwmax. 427 •CatBoost 428 –learning rate ( learning rate ): 0.03 for wcrandwmax, 429 –the number of iterations ( iterations ): 850 for wcrand 4000 for 430 wmax, 431 –tree depth ( depth ): 6 for wcrand 11 for wmax, 432 –L2 regularization term coefficient of the cost function ( l2leaf reg): 433 3 for wcrand 1 for wmax, 434 –the amount of randomness to use for scoring splits when the tree 435 structure is selected ( random strength ): 1 for wcrand 2 for wmax. 436 Figs. 7 and 8 show the performance of the developed ML models with the op- 437 timal hyperparameters for predicting wcrandwmax, respectively. The compar- 438 isons of the ML model predictions with FE simulation results are demonstrated 439 for the training and test datasets in each figure. The values of R2, minimum, 440 maximum, mean, and coefficient of variation of the prediction-to-FEA ratios 441 23 Figure 7: Performance of ML models for predicting elastic buckling load of steel cellular beams Table 1: Performance metrics of ML models for predicting elastic buckling loads of steel cellular beams, wcr(Train/Test) Model MSE ((kN/m)2) MAE (kN/m) MAPE (%) DT 155.94/1538.26 6.04/19.07 2.35/7.05 RF 83.79/682.24 4.44/13.00 1.83/5.27 KNN 1320.87/2617.14 21.09/28.99 10.22/12.66 GBR 39.68/319.27 3.85/7.66 1.99/3.24 XGBoost 48.44/294.91 4.51/8.59 2.51/3.95 LightGBM 41.88/366.81 3.57/7.56 1.82/3.06 CatBoost 33.90/295.43 3.16/6.15 1.52/2.48 are presented in Figs. 7 and 8.The MSE, MAE, and MAPE values are given in 442 Tables 1 and 2. 443 24 Figure 8: Performance of ML models for predicting ultimate load of steel cellular beams Table 2: Performance metrics of ML models for predicting ultimate loads of steel cellular beams, wmax (Train/Test) Model MSE ((kN/m)2) MAE (kN/m) MAPE (%) DT 15.99/20.25 2.08/2.37 1.90/2.18 RF 16.01/20.17 2.08/2.38 1.90/2.18 KNN 18.43/22.61 2.18/2.49 1.99/2.27 GBR 16.01/20.24 2.08/2.38 1.90/2.18 XGBoost 16.03/20.13 2.09/2.39 1.92/2.19 LightGBM 16.10/20.10 2.11/2.40 1.94/2.20 CatBoost 16.05/20.17 2.10/2.39 1.92/2.19 As can be seen from Fig. 7 and Table 1, CatBoost, XGBoost, and GBR 444 demonstrated comparable performances in predicting wcrfor the test set. The 445 performance metrics for LightGBM were slightly worse than those for CatBoost, 446 25 XGBoost, and GBR. KNN provided inferior performance compared with other 447 considered ML models. Fig. 8 and Table 2 show that all models performed 448 well in predicting wmax, with KNN providing slightly worse metrics than other 449 algorithms. It is worth reminding that the ultimate load dataset included 78390 450 samples and was significantly larger than the elastic buckling load dataset with 451 3645 samples. The good predictions of the ultimate load by all considered 452 ML models, which was not the case for the elastic buckling dataset, highlight 453 that good data with a very large number of samples contributes more to the 454 accuracy of ML models than the ML algorithm differences. It can also be seen 455 from comparisons of the performance metrics for the training and test sets that 456 the created ML models with the optimal hyperparameters have a reasonably 457 good generalization performance. 458 Developed ML models can be accessed at the following link: https://ww 459 w.kaggle.com/vitdegtyarev/cellular-beams-ml-models . An interactive 460 notebook for predicting the elastic buckling and ultimate loads of steel cellular 461 beams with the developed ML models can be found at the following link: https: 462 //www.kaggle.com/vitdegtyarev/ml-models-for-cellular-beams?scrip 463 tVersionId=63075739 . 464 5. Relative feature importance and feature dependence 465 Structural engineers often perceive ML methods as black boxes because hu- 466 mans cannot easily explain and interpret ML predictions. To remove this barrier 467 to adopting ML methods, several ML explainability and interpretability tech- 468 niques are available, including relative feature importance, partial dependence, 469 feature interactions, and SHAP [96]. These techniques shed light on why and 470 how an ML model made its predictions and expose how ML model predictions 471 compare with mechanics-based knowledge. The application of the explainability 472 26 and interpretability methods to the developed ML models is described in this 473 section. 474 Relative effects of the features on the wcrandwmaxpredictions by each con- 475 sidered ML model were analyzed using the permutation and SHAP methods. 476 The permutation feature importance is a decrease in a model score when the 477 feature values are randomly shuffled (permuted). A feature with a more sig- 478 nificant score decrease is more important than others. The model score in the 479 form of coefficient of determination, R2, was used in this study. The random 480 shuffling of values is repeated several times for each feature to obtain the mean 481 and the standard deviation of the permutation importance score. 482 The SHAP method [84] aims to explain a prediction for a sample by deter- 483 mining the contribution of each feature to the prediction by computing Shapley 484 values from coalitional game theory [97]. The Shapley value represents the av- 485 erage contribution of one player, which is a model feature in our case, to the 486 model predictions taken for all possible combinations, which may consist of all 487 dataset samples or a predefined portion of them. SHAP uses an additive feature 488 attribution method – a linear explanation model of the summation of present 489 features. The feature importance is determined based on the absolute average 490 Shapley values. Features with larger Shapley values are more important than 491 others. SHAP feature importance is based on the magnitude of feature attri- 492 butions, while permutation feature importance is based on the decrease in the 493 model performance. Thus, the relative feature importance predicted by these 494 two methods might be different. 495 The relative feature importance was determined for all considered ML models 496 using both methods. The relative feature importance plots were similar for all 497 models. Therefore, the relative feature importance for the CatBoost models, 498 which are ones of the most accurate models for predicting wcrandwmax, is 499 27 presented and discussed hereafter. Fig. 9 shows permutation and SHAP feature 500 importance plots for the optimized CatBoost models for predicting wcrand 501 wmax. 502 Figure 9: Permutation and SHAP feature importance for CatBoost models The relative feature importance in predicting the elastic buckling load, wcr, 503 of steel cellular beams is discussed first. The span length, L, has the most 504 significant importance according to both methods, which was expected. The 505 next important beam parameters are the flange width, bf, the web thickness, 506 tw, and the flange thickness, tf. The permutation method indicated that tw507 is more important than bfin predicting wcr, while the SHAP method showed 508 bfabove tw. However, the difference in the importance scores for bfandtwis 509 small, especially per the permutation method. These results compare well with 510 the conclusions made in [29], indicating that the CatBoost model can capture 511 the mechanics of the cellular beam behavior. The relative importance of the 512 web post width, WP, the opening diameter, Do, the beam height, H, and the 513 28 opening end distance, Led, have relatively small importance in predicting wcr 514 according to both methods. 515 The permutation and SHAP relative feature importance plots for wmax516 demonstrate that the span length, L, is the most important feature, followed 517 byWP,tw,bf,Fy, and tf. It should be noted that WP has a more significant 518 impact on wmaxthan on wcr. These results align with the conclusion made 519 in [29] and confirm the positive contribution of the web post plastic behavior 520 to the strength of steel cellular beams mentioned in Section 2. The relative 521 importance of H,Do, and Ledfor predicting wmaxis minor. 522 SHAP feature importance plots provide useful information, which is, how- 523 ever, somewhat limited. SHAP summary plots shown in Fig. 10 are more 524 informative as they combine feature importance and feature effects. Each point 525 on the summary plots represents a Shapley value for a dataset sample. The 526 color shows the feature value from low (blue) to high (red). Points with the 527 same Shapley values are scattered vertically to demonstrate their distribution 528 for each feature. The order of the features follows their importance; so, it is 529 the same as shown in Fig. 9. The SHAP summary plots presented in Fig. 530 10 indicate that wcrandwmaxincrease when the beam span reduces and vice 531 versa. Wide web posts have higher wcrandwmax, which decrease when the 532 web post width reduces. Greater values of tw,bf,tf, and Ledproduce higher 533 wcrandwmax, whereas an increase in the opening diameter Doresults in wcr 534 andwmaxreductions. The beam height Haffects wcrandwmaxdifferently: 535 wcrgoes down when Hincreases, while wmaxincreases when Hgoes up. The 536 reduction of wcrwith an increase in Hcan be explained by an increase in the 537 web post slenderness, which results in the elastic buckling load reduction. 538 SHAP dependence plots given in Figs. 11 and 12 for wcrandwmaxillus- 539 trate exact relationships between feature values and predictions. Each point 540 29 Figure 10: SHAP summary plots for CatBoost models represents a prediction for a dataset sample. Feature values are shown on the 541 horizontal axes, while SHAP values are given on the vertical axes. The SHAP 542 values demonstrate the magnitude of change in wcrandwmaxwhen the feature’s 543 value is known. The color of each point corresponds to the second feature, which 544 was determined by the algorithm to have the highest interaction with the con- 545 sidered feature shown on the horizontal axis. 546 The SHAP dependence plots for wcrshow that Landbfhave the highest 547 interactions with tw, while other features interact with Lmost frequently. An 548 increase in Lresults in an exponential decrease in wcr, which is more pronounced 549 for the cellular beams with thicker webs. An increase in Hresults in reductions 550 ofwcrfor the beams with short spans and smaller reductions or no reduction 551 for the beams with long spans, which can be seen from the comparison of the 552 SHAP values for the beams with long spans (red dots) with the SHAP values for 553 the beams with short spans (blue dots). These results indicate that the elastic 554 buckling load of the beams with short spans was likely governed by the local web 555 buckling. In contrast, the elastic buckling load of the beams with long spans was 556 likely governed by the global lateral-torsional buckling of the beams, which was 557 less sensitive to the changes in Hfor the beams considered in this study. The 558 elastic buckling loads increase when bfincreases, especially for the beams with 559 thicker webs. The elastic buckling load, wcr, increases when tfandtwincrease, 560 30 Figure 11: SHAP dependence plots for CatBoost model for predicting elastic buckling load especially in the beams with short spans. The elastic buckling loads reduce 561 when Dogoes up. The wcrreduction is more significant due to the Doincrease 562 for the cellular beams with short spans. When WP increases, wcrincreases, 563 especially for the beams with short spans. The plot also shows that web posts 564 with widths of 56 mm and narrower contribute to wcrreductions, indicated by 565 the negative SHAP values, while web posts of 72 mm wide and wider contribute 566 31 towcrpositively. An increase in Ledresults in a more significant increase in wcr 567 for the beams with short spans and a smaller increase in wcrfor those with long 568 spans. It should be noted that the beam elastic buckling loads in the dataset 569 were obtained for different buckling modes, including global lateral-torsional 570 buckling of the beams, local buckling of the web posts, and their interaction. 571 Therefore, the effects of the ML model features discussed above reflect possible 572 changes in the buckling modes when the beam geometry changes. 573 The SHAP dependence plots for wmaxdemonstrate that the strongest in- 574 teractions are between Landtw,HandDo,bf/tfandWP, and Ledandbf. 575 All other features of the ultimate load model interact with Lthe most. An 576 increase in Lresults in a wmaxreduction, which is more pronounced in the cel- 577 lular beams with thicker webs. Increases in H,bf,tf, and twcause an increase 578 inwmax, which goes down when Doincreases. An increase in WP makes the 579 beam ultimate load higher, which is more pronounced for the beams with short 580 spans. Similar to the observed effect of WP onwcr, web post widths up to 56 581 mm have a negative contribution to the beam ultimate load, while web posts 582 of 72 mm wide and wider contribute to the beam ultimate load positively. It 583 implies that the beam ultimate load in the dataset was governed by the web 584 post strength when WP was 56 mm or lower. The beam ultimate load becomes 585 higher when LedandFyincrease. The positive effect of Fyonwmaxis more 586 significant in the beams with short spans and when Fyincreases from 235 to 587 355 MPa compared with the Fyincrease from 355 to 440 MPa. 588 Fig. 13 shows contour plots of wcrpredicted by the developed CatBoost 589 model as functions of H/DoandSo/Do(where Sois the center-to-center spacing 590 of the web openings) for the beams with different span lengths and cross-section 591 dimensions. The beam designations are presented in the L-H-tw-bf-tfformat, 592 with all dimensions in mm. Fig. 13 demonstrates that So/Dohas a greater 593 32 influence on wcrthan H/Dofor the beams with short spans. For many short- 594 span beams, an increase in So/Dofrom 1.1 to 1.3 results in a greater increase 595 inwcrthan a further increase in So/Dofrom 1.3 to 1.49. It indicates that 596 web opening spacing of approximately 1.3 Dois optimal for many short-span 597 beams. Only short-span beams with H=420 mm and tw=9 mm demonstrate 598 an approximately uniform wcrincrease when So/Doincreases from 1.1 to 1.49. 599 The long-span beams show a wider variety of the wcrcontour shapes. For 600 example, H/Dohas a more significant effect on wcrthan So/Dofor the beam 601 with H=420 mm, tw=9 mm, bf=162 mm, and tf=15 mm compared with other 602 analyzed beams. The wcrvalues for the 8000-420-9-162-15 and 8000-700-15- 603 162-15 beams reduce slightly when So/Doincrease from 1.1 to 1.3 and increase 604 with the further increase in So/Dofrom 1.3 to 1.49. However, it should be noted 605 that the absolute magnitude of the wcrchange is relatively small in those cases. 606 The contour plots also show the effects of the cross-section dimensions on the 607 wcrvalues of the cellular beams discussed earlier in the paper. 608 Figs. 14 and 15 present contour plots of wmaxpredicted by the CatBoost 609 model as functions of H/DoandSo/Dofor the cellular beams made from steel 610 with Fyof 235 and 440 MPa, respectively. The beam designation format is as 611 described previously, with the steel yield strength added at the end. Similar 612 towcr,So/Doshows a more significant effect on wmaxthan H/Dofor most of 613 the considered beams. The effect of the opening diameter on wmaxis more pro- 614 nounced in the H/Dorange from 1.25 to approximately 1.45 for many beams. 615 A further increase in H/Doat a constant So/Dovalue changes wmaxinsignif- 616 icantly. It can also be seen from the contour plots that for some beams (see 617 4000-420-15-162-15-235, 7000-420-15-162-15-235, 4000-420-15-270-15-440, and 618 7000-700-9-270-15-440, for example), the effects of H/Dovarying in the range 619 from 1.25 to 1.45 on wmaxare relatively small when So/Dois low. They be- 620 33 come more pronounced as So/Doincreases. The So/Doof approximately 1.3 is 621 optimal for many considered beams. The effects of the cross-section dimensions 622 onwmaxdiscussed earlier in the paper can also be seen from Figs. 14 and 15. 623 34 Figure 12: SHAP dependence plots for CatBoost model for predicting ultimate load 35 Figure 13: Contour plots of wcr(kN/m) as functions of H/DoandSo/Do 36 Figure 14: Contour plots of wmax (kN/m) as functions of H/DoandSo/Dofor beams made from 235 MPa steel 37 Figure 15: Contour plots of wmax (kN/m) as functions of H/DoandSo/Dofor beams made from 440 MPa steel 38 6. Comparisons of ultimate loads of cellular beams predicted by ML 624 models, SCI P355, and AISC Design Guide 31 625 The ultimate loads of cellular beams predicted by the developed ML models 626 were compared with the nominal beam strengths determined per SCI P355 [2] 627 and AISC Design Guide 31 [85]. According to SCI P355 and AISC Design 628 Guide 31, the cellular beam strength may be governed by shear resistance of 629 perforated beam section, shear resistance of solid beam section, shear buckling 630 resistance of perforated web, bending resistance of beam at the centerline of 631 opening, bending resistance of tees, web post shear resistance, and web post 632 buckling resistance. The beam, tee, and web post resistances are computed 633 per EN 1993-1-1 [12] and EN 1993-1-5 [98] in SCI P355 and per AISC 360 634 [99] in AISC Design Guide 31. The most significant differences between the 635 SCI P355 and AISC Design Guide 31 provisions are in the web post buckling 636 resistance and lateral-torsional buckling calculations [100]. In SCI P355, the 637 web post buckling resistance is calculated using analytical equations, which 638 account for the web post slenderness, while AISC Design Guide 31 adopted 639 empirical equations from SCI P100 [27]. SCI P355 also requires checking web 640 shear buckling near openings, whereas AISC Design Guide 31 does not include 641 such a requirement. 642 The SCI P355 provisions apply to cellular beams with the following geo- 643 metric limits: H/D o≥1.25, 2 .0≥So/Do≥1.3,Led/Do≥0.5, and the 644 depth of tees not less than tf+30 mm. The beams considered in the study 645 had the following parameters: 1 .70≥H/D o≥1.25, 1 .49≥So/Do≥1.10, 646 1.49≥Led/Do≥0.04, and the depth of tees between tf+29.5 mm and tf+136.5 647 mm. So/Do,Led/Do, and the depth of tees of some beams did not comply 648 with the SCI P355 limits. Therefore, the SCI P355 predictions were com- 649 pared with the FE simulation results for all beams and 17,982 beams that met 650 39 the geometric limits. AISC Design Guide 31 applies to cellular beams with 651 1.75≥H/D o≥1.25 and 1 .50≥So/Do≥1.08. All beams considered in the 652 present study complied with the AISC Design Guide 31 limits. 653 Fig. 16 compares the ultimate loads of the cellular beams from the FE 654 simulations with those predicted by the developed ML models, SCI P355, and 655 AISC Design Guide 31. Fig. 16 clearly shows that the developed ML models 656 predict the ultimate loads of the cellular beams considerably better than SCI 657 P355 and AISC Design Guide 31. For the best models, the mean ratio and the 658 coefficient of variation of the ML predictions to the FE simulation results are 659 1.00 and 0.034, respectively. The coefficient of determination, R2, is 0.997. The 660 corresponding metrics for SCI P355 are 0.75, 0.253, and 0.638 for all beams 661 and 0.75, 0.298, and 0.535 for the beams meeting the geometric limits. It is 662 interesting to note that the SCI P355 provisions demonstrate better comparison 663 with the FE simulation results when all beams are considered neglecting the 664 geometric limits. AISC Design Guide 31 showed even worse accuracy than SCI 665 P355, characterized by the mean ratio and the coefficient of variation of the 666 prediction-to-FEA ratios of 0.69 and 0.429, and the coefficient of determination 667 of 0.416. 668 40 Figure 16: Comparisons of ultimate loads of steel cellular beams predicted by ML models, SCI P355, and AISC Design Guide 31 with FE simulation results 7. Web application 669 A user-friendly web application was created in the Streamlit framework ( ht 670 tps://streamlit.io ) to predict the elastic buckling and ultimate loads of 671 41 steel cellular beams with the ML models developed in the present work. Fig. 672 17 demonstrates a flow chart of the web application. 673 Figure 17: Flow chart of the web application In the beginning, the user specifies the following parameters via the web 674 application sliders and radio buttons: L,H,bf,tf,tw,H/D o,So/Do, and Fy. 675 Ranges of the parameters available in the application correspond to the feature 676 ranges in the datasets used for the ML training. At the next step, the following 677 parameters are computed: Do,So,Led, the number of openings evenly spaced 678 along the beam length, the cellular beam weight, and the percentage of the 679 beam weight reduction due to the openings compared with the identical solid- 680 web beam. The parameters specified by the user and the computed ones are 681 displayed on the screen. Next, the developed ML models and scalers are loaded; 682 the features are standardized, and predictions by all ML models considered 683 42 in this study are made and displayed. The code runs automatically after any 684 change of input variables. The prediction process takes only several seconds. 685 The graphical user interface of the web application is presented in Fig. 18. 686 Figure 18: Graphical user interface of the developed web application It was found challenging to deploy the web application with all consid- 687 ered ML to cloud platforms due to the application size and required compu- 688 tational resources, which exceeded the limits of free cloud accounts. There- 689 fore, a lite version of the application based on CatBoost predictions was cre- 690 ated and successfully deployed on Heroku at the following address: https: 691 //scba-cb.herokuapp.com/ . The deployed lite version of the application 692 opens and runs in any web browser on any device, including mobile. 693 It should be noted that the computational resources provided by the free 694 Heroku account are sufficient for running the application by one user at a time. 695 Multiple users can open the application, but it crashes when two or more users 696 run the computations simultaneously. If that happens, it is recommended to 697 close the application and use it later. The use of a paid Heroku account, which 698 offers more powerful computational resources, would resolve this issue. 699 The source codes of the full and lite application versions can be accessed 700 43 on GitHub at https://github.com/vitdegtyarev/SCBA-Streamlit and 701 https://github.com/vitdegtyarev/SCBA-Streamlit-CB , respectively. The 702 GitHub pages include instructions on how the web applications can be used 703 independently from the cloud services on a local machine. 704 8. Conclusions 705 ML models for predicting the elastic buckling and ultimate loads of steel 706 cellular beams were developed and optimized using the following algorithms: 707 decision tree (DT), random forest (RF), k-nearest neighbor (KNN), gradient 708 boosting regressor (GBR), extreme gradient boosting (XGBoost), light gradient 709 boosting machine (LightGBM), and gradient boosting with categorical features 710 support (CatBoost). Large datasets of FE simulation results from the literature 711 [29], validated against experimental data, were employed to train and evaluate 712 the ML models implemented in open-source Python-based libraries. 713 The ML models were optimized by tuning their hyperparameters via ex- 714 tensive grid and random searches and validated through the ten-fold cross- 715 validation method. The final evaluation of the ML models was performed on 716 the test sets unseen by the models during training. The elastic buckling and 717 ultimate loads predicted by the optimized ML models demonstrated excellent 718 agreements with the numerical data. The accuracy of the ultimate load predic- 719 tions by the ML models exceeded the accuracy provided by the existing design 720 provisions for steel cellular beams. An interactive Python-based notebook for 721 predicting the elastic buckling and ultimate loads of steel cellular beams using 722 the developed optimized ML models was created and made publicly available 723 at the following link: https://www.kaggle.com/vitdegtyarev/ml-models- 724 for-cellular-beams?scriptVersionId=63075739 . 725 The developed ML models were explained and interpreted by evaluating the 726 44 relative feature importance using the permutations and SHAP methods. SHAP 727 feature dependence was also determined and discussed. It was demonstrated 728 that the beam span length, beam flange width, and beam web thickness are 729 the most important features in predicting the elastic buckling by the developed 730 models, with the opening end distance and beam height being the least im- 731 portant parameters. The most important features in predicting the ultimate 732 load are the beam span length, web post width, beam web thickness, and beam 733 flange width. The opening end distance and opening diameter are the least 734 important characteristics. These results align well with the mechanics-based 735 knowledge demonstrating that the developed ML models can capture the web 736 opening effects from the data used for their training. Contour plots of wcrand 737 wmaxpredicted by the CatBoost model as functions of H/DoandSo/Dowere 738 presented and discussed. For most beams, So/Doaffects wcrandwmaxmore 739 significantly than H/Do, with So/Do=1.3 being the optimal value. 740 A web application for predicting the elastic buckling and ultimate loads was 741 created in Streamlit. The lite version of the application has been deployed to 742 the cloud at: https://scba-cb.herokuapp.com/ . It can be opened and 743 run in any web browser on any device, including mobile. The source codes 744 of the full and lite application versions can be accessed on GitHub at https: 745 //github.com/vitdegtyarev/SCBA-Streamlit and https://github.com/v 746 itdegtyarev/SCBA-Streamlit-CB , respectively. 747 The presented study demonstrates the opportunities for using ML methods 748 for predicting the elastic buckling and ultimate loads of cellular beams. However, 749 it should be noted that the developed models are based on the data for cellular 750 beams with relatively short spans, not exceeding 8 m in the elastic buckling 751 load dataset and 7 m in the ultimate load dataset. Therefore, the developed 752 models are limited to beams with such spans. In modern construction, cellular 753 45 beams are often used for spans ranging from 9 to 18 m [101]. Future work 754 should concentrate on extending the datasets to the beams with longer spans 755 and retraining the ML models using the extended data. The reliability of the 756 ultimate load predictions by the ML models should also be evaluated, and an 757 appropriate safety factor determined. 758