{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T07:00:11Z","timestamp":1777359611338,"version":"3.51.4"},"reference-count":65,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2018,12,26]],"date-time":"2018-12-26T00:00:00Z","timestamp":1545782400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computers"],"abstract":"<jats:p>Cellular beams are an attractive option for the steel construction industry due to their versatility in terms of strength, size, and weight. Further benefits are the integration of services thereby reducing ceiling-to-floor depth (thus, building\u2019s height), which has a great economic impact. Moreover, the complex localized and global failures characterizing those members have led several scientists to focus their research on the development of more efficient design guidelines. This paper aims to propose an artificial neural network (ANN)-based formula to precisely compute the critical elastic buckling load of simply supported cellular beams under uniformly distributed vertical loads. The 3645-point dataset used in ANN design was obtained from an extensive parametric finite element analysis performed in ABAQUS. The independent variables adopted as ANN inputs are the following: beam\u2019s length, opening diameter, web-post width, cross-section height, web thickness, flange width, flange thickness, and the distance between the last opening edge and the end support. The proposed model shows a strong potential as an effective design tool. The maximum and average relative errors among the 3645 data points were found to be 3.7% and 0.4%, respectively, whereas the average computing time per data point is smaller than a millisecond for any current personal computer.<\/jats:p>","DOI":"10.3390\/computers8010002","type":"journal-article","created":{"date-parts":[[2018,12,26]],"date-time":"2018-12-26T11:31:21Z","timestamp":1545823881000},"page":"2","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":52,"title":["Neural Network-Based Formula for the Buckling Load Prediction of I-Section Cellular Steel Beams"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4107-8501","authenticated-orcid":false,"given":"Miguel","family":"Abambres","sequence":"first","affiliation":[{"name":"Research &amp; Development, Abambres\u2019 Lab, 1600-275 Lisbon, Portugal"}]},{"given":"Komal","family":"Rajana","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, University of Leeds, LS2 9JT, Leeds, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8349-3979","authenticated-orcid":false,"given":"Konstantinos Daniel","family":"Tsavdaridis","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, University of Leeds, LS2 9JT, Leeds, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5967-0864","authenticated-orcid":false,"given":"Tiago Pinto","family":"Ribeiro","sequence":"additional","affiliation":[{"name":"Tal Projecto, 1350-252 Lisboa, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2018,12,26]]},"reference":[{"key":"ref_1","unstructured":"Tsavdaridis, K.D. (2010). Structural Performance of Perforated Steel Beams with Novel Web Openings and with Partial Concrete Encasement. [Ph.D. Thesis, City University of London]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/s40091-015-0095-4","article-title":"An experimental and parametric study on steel beams with web openings","volume":"7","author":"Morkhade","year":"2015","journal-title":"Int. J. Adv. Struct. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1061\/(ASCE)ST.1943-541X.0001421","article-title":"Review and assessment of design methodologies for perforated steel beams","volume":"142","author":"Akrami","year":"2016","journal-title":"J. Struct. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/0045-7949(78)90138-4","article-title":"Buckling of webs with openings","volume":"9","author":"Uenoya","year":"1978","journal-title":"Comput. Struct."},{"key":"ref_5","unstructured":"Lucas, W.K., and Darwin, D. (1990). Steel and Composite Beams with Web Openings, The American Iron and Steel Institute."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Darwin, D. (1990). Steel and Composite Beams with Web Opening, American Institute of Steel Construction (AISC).","DOI":"10.1061\/(ASCE)0733-9445(1990)116:6(1579)"},{"key":"ref_7","unstructured":"SEI\/ASCE (1998). Specifications for Structural Steel Beams with Openings, ASCE. SEI\/ASCE 23-97."},{"key":"ref_8","unstructured":"Ward, J.K. (1990). Design of Composite and Non-Composite Cellular Beams SCI P100, Steel Construction Institute."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1016\/S0143-974X(00)00035-3","article-title":"Investigation on vierendeel mechanism in steel beams with circular web openings","volume":"57","author":"Chung","year":"2001","journal-title":"J. Constr. Steel Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1177","DOI":"10.1016\/S0143-974X(03)00029-4","article-title":"Steel beams with large web openings of various shapes and sizes: An empirical design method using a generalized moment-shear interaction curve","volume":"59","author":"Chung","year":"2003","journal-title":"J. Constr. Steel Res."},{"key":"ref_11","unstructured":"Tsavdaridis, K.D., and D\u2019Mello, C. (2009, January 16\u201318). Finite Element Investigation of Perforated Beams with Different Web Opening Configurations. Proceedings of the 6th International Conference on Advances is Steel Structures (ICASS 2009), Hong Kong, China."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1605","DOI":"10.1016\/j.jcsr.2011.04.004","article-title":"Web buckling study of the behaviour and strength of perforated steel beams with different novel web opening shapes","volume":"67","author":"Tsavdaridis","year":"2011","journal-title":"J. Constr. Steel Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1214","DOI":"10.1061\/(ASCE)ST.1943-541X.0000562","article-title":"Vierendeel bending study of perforated steel beams with various novel web opening shapes through non-linear finite element analyses","volume":"138","author":"Tsavdaridis","year":"2012","journal-title":"J. Struct. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.compstruc.2015.05.004","article-title":"Application of Structural Topology Optimisation to Perforated Steel Beams","volume":"158","author":"Tsavdaridis","year":"2015","journal-title":"Comput. Struct."},{"key":"ref_15","unstructured":"Lawson, R.M., and Hicks, S.J. (2011). Design of Composite Beams with Large Openings SCI P355, Steel Construction Institute."},{"key":"ref_16","unstructured":"Lawson, R.M. (1987). Design for Openings in the Webs of Composite Beams SCI P068, Steel Construction Institute."},{"key":"ref_17","unstructured":"Verweij, J.G. (2010). Cellular Beam-Columns in Portal Frame Structures. [Master\u2019s Thesis, Delft University of Technology]."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1016\/j.jcsr.2011.01.001","article-title":"Assessment of load carrying capacity of castellated steel beams by neural networks","volume":"67","author":"Gholizadeh","year":"2011","journal-title":"J. Constr. Steel Res."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/s11709-014-0236-z","article-title":"Lateral-torsional buckling capacity assessment of web opening steel girders by artificial neural networks\u2014Elastic investigation","volume":"8","author":"Sharifi","year":"2014","journal-title":"Front. Struct. Civ. Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1080\/19373260.2014.955139","article-title":"Inelastic lateral-torsional buckling capacity of corroded web opening steel beams using artificial neural networks","volume":"8","author":"Tohidi","year":"2014","journal-title":"IES J. Part A Civ. Struct. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.tws.2015.12.007","article-title":"Load-carrying capacity of locally corroded steel plate girder ends using artificial neural network","volume":"100","author":"Tohidi","year":"2016","journal-title":"Thin-Walled Struct."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"S102","DOI":"10.1080\/19648189.2016.1246693","article-title":"Prediction of self-compacting concrete strength using artificial neural networks","volume":"20","author":"Asteris","year":"2016","journal-title":"Eur. J. Environ. Civ. Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/j.engstruct.2017.02.047","article-title":"An Artificial Neural Networks model for the prediction of the compressive strength of FRP-confined concrete circular columns","volume":"140","author":"Cascardi","year":"2017","journal-title":"Eng. Struct."},{"key":"ref_24","first-page":"1","article-title":"Urban Road Infrastructure Maintenance Planning with Application of Neural Networks","volume":"2018","author":"Jajac","year":"2018","journal-title":"Complexity"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1013","DOI":"10.1051\/matecconf\/201820301013","article-title":"Development of ANN Model for the Prediction of VIV Fatigue Damage of Top-tensioned Riser","volume":"203","author":"Wong","year":"2018","journal-title":"MATEC Web Conf."},{"key":"ref_26","unstructured":"Dassault Syst\u00e8mes (2011). ABAQUS 6.11, Abaqus\/CAE User\u2019s Manual, Dassault Systemes."},{"key":"ref_27","unstructured":"Dassault Syst\u00e8mes Simulia Corp (2017). ABAQUS CAE (2017), Dassault Syst\u00e8mes Simulia Corp.. Software."},{"key":"ref_28","unstructured":"Surtees, J.O., and Lui, Z. (1995). Report of Loading Tests on Cellform Beams, University of Leeds. Research Report."},{"key":"ref_29","unstructured":"Rajana, K. (2018). Advanced Computational Parametric Study of the Linear Elastic and Non-Linear Post Buckling Behaviour of Non-Composite Cellular Steel Beams. [Master\u2019s Thesis, University of Leeds]."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.jcsr.2014.02.007","article-title":"Moment gradient factor of cellular steel beams under inelastic flexure","volume":"98","author":"Sweedan","year":"2014","journal-title":"J. Constr. Steel Res."},{"key":"ref_31","unstructured":"(2018, November 29). Developer. Dataset ANN [Data Set]. Zenodo. Available online: http:\/\/doi.org\/10.5281\/zenodo.1486181."},{"key":"ref_32","unstructured":"Hertzmann, A., and Fleet, D. (2012). Machine Learning and Data Mining, Computer Science Department, University of Toronto. Lecture Notes CSC 411\/D11."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1007\/BF02478259","article-title":"A logical calculus of the ideas immanent in nervous activity","volume":"5","author":"McCulloch","year":"1943","journal-title":"Bull. Math. Biophys."},{"key":"ref_34","unstructured":"Hern, A. (2016, November 02). Google Says Machine Learning Is the Future. So I Tried It Myself. Available online: www.theguardian.com\/technology\/2016\/jun\/28\/all."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1016\/j.neucom.2016.06.014","article-title":"Neural networks: An overview of early research, current frameworks and new challenges","volume":"214","author":"Prieto","year":"2016","journal-title":"Neurocomputing"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Wilamowski, B.M., and Irwin, J.D. (2011). The Industrial Electronics Handbook: Intelligent Systems, CRC Press.","DOI":"10.1201\/NOE1439802892"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1016\/j.aei.2007.07.001","article-title":"Towards the next generation of artificial neural networks for civil engineering","volume":"228","author":"Flood","year":"2008","journal-title":"Adv. Eng. Inform."},{"key":"ref_38","unstructured":"Haykin, S.S. (2009). Neural Networks and Learning Machines, Prentice Hall\/Pearson."},{"key":"ref_39","unstructured":"The Mathworks, Inc. (2017). MATLAB R2017a, User\u2019s Guide, The Mathworks, Inc."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/0004-3702(90)90038-2","article-title":"Qualitative physics using dimensional analysis","volume":"45","author":"Bhaskar","year":"1990","journal-title":"Artif. Intell."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"3906","DOI":"10.1109\/TIP.2016.2570569","article-title":"Dimension reduction with extreme learning machine","volume":"25","author":"Kasun","year":"2016","journal-title":"IEEE Trans. Image Process."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1002\/for.3980140405","article-title":"Backpropagation in time-series forecasting","volume":"14","author":"Lachtermacher","year":"1995","journal-title":"J. Forecast."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1190","DOI":"10.1016\/j.engstruct.2005.12.009","article-title":"Application of artificial neural networks to evaluation of ultimate strength of steel panels","volume":"28","author":"Pu","year":"2006","journal-title":"Eng. Struct."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1061\/(ASCE)0887-3801(1994)8:2(131)","article-title":"Neural Networks in Civil Engineering: I-Principals and Understanding","volume":"8","author":"Flood","year":"1994","journal-title":"J. Comput. Civ. Eng."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1385","DOI":"10.1061\/(ASCE)0733-9445(1996)122:11(1385)","article-title":"Prediction of buckling load of columns using artificial neural networks","volume":"122","author":"Mukherjee","year":"1996","journal-title":"J. Struct. Eng."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1109\/MIE.2009.934790","article-title":"Neural Network Architectures and Learning algorithms","volume":"3","author":"Wilamowski","year":"2009","journal-title":"IEEE Ind. Electron. Mag."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Xie, T., Yu, H., and Wilamowski, B. (2011, January 27\u201330). Comparison between traditional neural networks and radial basis function networks. Proceedings of the 2011 IEEE International Symposium on Industrial Electronics (ISIE), Gdansk University of Technology Gdansk, Gdansk, Poland.","DOI":"10.1109\/ISIE.2011.5984328"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"231","DOI":"10.4028\/www.scientific.net\/KEM.144.231","article-title":"Prediction of fatigue strength of composite laminates by means of neural networks","volume":"144","author":"Aymerich","year":"1998","journal-title":"Key Eng. Mater."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1541","DOI":"10.1016\/S0045-7949(01)00039-6","article-title":"Neural network design for engineering applications","volume":"79","author":"Rafiq","year":"2001","journal-title":"Comput. Struct."},{"key":"ref_50","unstructured":"Xu, S., and Chen, L. (2008, January 23\u201326). Novel approach for determining the optimal number of hidden layer neurons for FNN\u2019s and its application in data mining. Proceedings of the International Conference on Information Technology and Applications (ICITA), Cairns, Australia."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1111\/j.1467-8667.1994.tb00365.x","article-title":"Effect of representation on the performance of neural networks in structural engineering applications","volume":"9","author":"Gunaratnam","year":"1994","journal-title":"Comput.-Aided Civ. Infrastruct. Eng."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"3265","DOI":"10.1016\/S0045-7825(03)00350-5","article-title":"Artificial neural network as an incremental non-linear constitutive model for a finite element code","volume":"192","author":"Lefik","year":"2003","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1858","DOI":"10.1109\/TCYB.2014.2298235","article-title":"Sparse extreme learning machine for classification","volume":"44","author":"Bai","year":"2014","journal-title":"IEEE Trans. Cybern."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1016\/S0893-6080(01)00027-2","article-title":"Three learning phases for radial-basis-function networks","volume":"14","author":"Schwenker","year":"2001","journal-title":"Neural Netw."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Waszczyszyn, Z. (1999). Neural Networks in the Analysis and Design of Structures, Springer. CISM Courses and Lectures No. 404.","DOI":"10.1007\/978-3-7091-2484-0"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.neunet.2015.09.003","article-title":"A fast SVD-Hidden-nodes based extreme learning machine for large-scale data Analytics","volume":"77","author":"Deng","year":"2016","journal-title":"Neural Netw."},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Wilamowski, B.M. (2011, January 14\u201316). How to not get frustrated with neural networks. Proceedings of the 2011 IEEE International Conference on Industrial Technology (ICIT), Auburn University, Auburn, AL, USA.","DOI":"10.1109\/ICIT.2011.5754336"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1016\/j.neucom.2005.12.126","article-title":"Extreme learning machine: Theory and applications","volume":"70","author":"Huang","year":"2006","journal-title":"Neurocomputing"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"1411","DOI":"10.1109\/TNN.2006.880583","article-title":"A fast and accurate online Sequential learning algorithm for Feedforward networks","volume":"17","author":"Liang","year":"2006","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1109\/TNN.2006.875977","article-title":"Universal approximation using incremental constructive feedforward networks with random hidden nodes","volume":"17","author":"Huang","year":"2006","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"3056","DOI":"10.1016\/j.neucom.2007.02.009","article-title":"Convex incremental extreme learning machine","volume":"70","author":"Huang","year":"2007","journal-title":"Neurocomputing"},{"key":"ref_62","unstructured":"Beyer, W., Liebscher, M., Beer, M., and Graf, W. (2006, January 12\u201313). Neural Network Based Response Surface Methods\u2014A Comparative Study. Proceedings of the 5th German LS-DYNA Forum, Ulm, Germany."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"1429","DOI":"10.1016\/S0893-6080(03)00138-2","article-title":"The general inefficiency of batch training for gradient descent learning","volume":"16","author":"Wilson","year":"2003","journal-title":"Neural Netw."},{"key":"ref_64","unstructured":"The Researcher (2018, November 29). ANNSoftwareValidation-Report.pdf. Available online: https:\/\/www.researchgate.net\/profile\/Abambres_M\/project\/Applied-Artificial-Intelligence\/attachment\/5aff6a82b53d2f63c3ccbaa0\/AS:627790747541504@1526688386824\/download\/ANN+Software+Validation+-+Report.pdf?context=ProjectUpdatesLog."},{"key":"ref_65","unstructured":"(2018, November 29). Developer. W and b Arrays [Data Set]. Zenodo. Available online: http:\/\/doi.org\/10.5281\/zenodo.1486268."}],"container-title":["Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-431X\/8\/1\/2\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:36:14Z","timestamp":1760196974000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-431X\/8\/1\/2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,12,26]]},"references-count":65,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2019,3]]}},"alternative-id":["computers8010002"],"URL":"https:\/\/doi.org\/10.3390\/computers8010002","relation":{"has-preprint":[{"id-type":"doi","id":"10.31224\/osf.io\/wg7hd","asserted-by":"object"}]},"ISSN":["2073-431X"],"issn-type":[{"value":"2073-431X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,12,26]]}}}