{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T10:57:42Z","timestamp":1774436262796,"version":"3.50.1"},"reference-count":60,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,20]],"date-time":"2024-09-20T00:00:00Z","timestamp":1726790400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico (CNPq\u2014Ministry of Science, Technology and Innovation of Brazil)","award":["307640\/2022-1"],"award-info":[{"award-number":["307640\/2022-1"]}]},{"name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico (CNPq\u2014Ministry of Science, Technology and Innovation of Brazil)","award":["UIDB\/04625\/2020"],"award-info":[{"award-number":["UIDB\/04625\/2020"]}]},{"name":"FCT (\u201cFunda\u00e7\u00e3o para a Ci\u00eancia e Tecnologia\u201d\u2014Portugal)","award":["307640\/2022-1"],"award-info":[{"award-number":["307640\/2022-1"]}]},{"name":"FCT (\u201cFunda\u00e7\u00e3o para a Ci\u00eancia e Tecnologia\u201d\u2014Portugal)","award":["UIDB\/04625\/2020"],"award-info":[{"award-number":["UIDB\/04625\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Buildings"],"abstract":"<jats:p>Based on a systematic literature review, no design guidelines addressing practical issues related to steel circular hollow section (CHS) members under hydrostatic pressure have been documented thus far, except for the design recommendations proposed by the European standard EN 1993-1-6 concerning the strength and stability of shell structures. Considering that the existing codes are often outdated and subject to improvement, there is a growing demand for modern solutions that can accurately replicate the stability loss response of equipment subject to external pressure. The development of a more direct and simple method, augmented by reliable computational techniques, is advisable to avoid the numerous challenges associated with manual calculations. This approach will streamline the design process and mitigate the complexity of current practical or technical approaches. Thus, this research has formulated a method to predict design strength through the direct application of the critical elastic buckling stress, determined from Generalised Beam Theory (GBT) analyses, induced by hydrostatic pressure in custom-crafted equations, finely tuned for this specific objective. The authors present a formulation of GBT to analyze the elastic buckling behavior of CHS subject to both combined axial compression and external pressure. This work provides an extensive study of the behavior of circular cylindrical shells under external loading conditions. It first introduces the analyses of GBT and the theoretical and empirical formulas applicable to this objective. It then describes the steps involved in the development and validation of a numerical model that simulates the response of circular cylindrical shells under hydrostatic pressure, conducting an extensive parametric study using finite element analyses (FEAs). The results from available tests validate the finite element (FE) model. Upon achieving adequate agreement between the generated numerical and experimental results, new resistance reduction curves for cylindrical shell members are developed based on the results of the parametric study. Additionally, the study compares the existing provisions of EN 1993-1-6 for the design of cylindrical shells under hydrostatic pressure. Finally, a statistical approach is adopted to compare the responses of cylindrical shells calculated according to the EN 1993-1-6 code recommendations with those obtained through the newly proposed method.<\/jats:p>","DOI":"10.3390\/buildings14092992","type":"journal-article","created":{"date-parts":[[2024,9,23]],"date-time":"2024-09-23T04:45:28Z","timestamp":1727066728000},"page":"2992","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Direct Strength Design of Circular Cylindrical Steel Shells under Uniform External Pressure"],"prefix":"10.3390","volume":"14","author":[{"given":"Rodrigo da Silva","family":"Manera","sequence":"first","affiliation":[{"name":"Institute of Geology and Engineering, Federal University of Southern and Southeast of Para, Folha 17, Quadra 4, Lote Especial, Maraba 68050-080, Brazil"},{"name":"Departamento de Estruturas, Faculdade de Engenharia Civil, Arquitetura e Urbanismo, Universidade Estadual de Campinas (UNICAMP), Av. Albert Einstein, 951, Campinas 13083-852, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3852-9420","authenticated-orcid":false,"given":"Cilmar","family":"Basaglia","sequence":"additional","affiliation":[{"name":"Departamento de Estruturas, Faculdade de Engenharia Civil, Arquitetura e Urbanismo, Universidade Estadual de Campinas (UNICAMP), Av. Albert Einstein, 951, Campinas 13083-852, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7994-2687","authenticated-orcid":false,"given":"Dinar","family":"Camotim","sequence":"additional","affiliation":[{"name":"CERIS, DECivil, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,20]]},"reference":[{"key":"ref_1","unstructured":"(2007). Eurocode 3 Design of Steel Structures\u2014Part 1-6: Strength and Stability of Shell Structures (Standard No. CEN. EN 1993-1-6)."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Annaratone, D. (2007). Pressure Vessel Design, Springer.","DOI":"10.1007\/978-3-540-49144-6"},{"key":"ref_3","unstructured":"AIChE (2024, June 06). Process Safety Beacon Archives. Available online: https:\/\/www.aiche.org\/ccps\/resources\/process-safety-beacon\/archives."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.ijpvp.2013.03.008","article-title":"Fourier series analysis of a cylindrical pressure vessel subjected to axial end load and external pressure","volume":"107","author":"Brar","year":"2013","journal-title":"Int. J. Press. Vessel. Pip."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"105221","DOI":"10.1016\/j.ijpvp.2024.105221","article-title":"A new analytical tool for the elastic\/plastic buckling design of pressure vessels subjected to external pressure","volume":"210","author":"Do","year":"2024","journal-title":"Int. J. Press. Vessel. Pip."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"110011","DOI":"10.1016\/j.tws.2022.110011","article-title":"Buckling behavior of steel and steel\u2013composite cylinders under external pressure","volume":"181","author":"Zuo","year":"2022","journal-title":"Thin Walled Struct."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/j.tws.2018.04.016","article-title":"Buckling of externally pressurized cylindrical shell: A comparison of theoretical and experimental data","volume":"129","author":"Zhu","year":"2018","journal-title":"Thin Walled Struct."},{"key":"ref_8","unstructured":"De Campos, E.F. (2017). An\u00e1lise Num\u00e9rica e Experimental de Flambagem em Cilindros de Paredes Finas. [Master\u2019s Thesis, Universidade Estadual Paulista\u2014FEIS]."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1017\/jmech.2016.59","article-title":"Buckling of cylindrical shells with arbitrary circumferential thickness variations under external pressure","volume":"33","author":"Feng","year":"2017","journal-title":"J. Mech."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.tws.2012.03.001","article-title":"Prediction of vacuum-induced buckling pressures of thin-walled cylinders","volume":"55","author":"Kelliher","year":"2012","journal-title":"Thin Walled Struct."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4163","DOI":"10.1016\/j.ijsolstr.2009.07.025","article-title":"Buckling under the external pressure of cylindrical shells with variable thickness","volume":"46","author":"Nguyen","year":"2009","journal-title":"Int. J. Solids Struct."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"904","DOI":"10.1016\/j.tws.2006.08.015","article-title":"Buckling and post-buckling behavior of thin-walled cylindrical steel shells with varying thickness subjected to Uniform External Pressure","volume":"44","author":"Aghajari","year":"2006","journal-title":"Thin Walled Struct."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/0045-7949(90)90300-Q","article-title":"Imperfection sensitivity of stiffened cylinders subjected to external pressure","volume":"34","author":"Seleim","year":"1990","journal-title":"Comput. Struct."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"883","DOI":"10.1016\/0020-7683(71)90070-9","article-title":"Buckling under external pressure of cylindrical shells with dimple shaped initial imperfections","volume":"7","author":"Amazigo","year":"1971","journal-title":"Int. J. Solids Struct."},{"key":"ref_15","unstructured":"ASME\u2014American Society of Mechanical Engineers (2023). Boiler and Pressure Vessel Code, Section A-4, American Society of Mechanical Engineers."},{"key":"ref_16","unstructured":"CODAP (2022). Code de Construction des Appareils \u00e0 Pression non Soumis \u00e0 l\u2019action de la Flamme, SNCT Publications. (In French)."},{"key":"ref_17","unstructured":"Maraveas, C., and Miamis, K. (2013, January 16\u201320). Shell buckling evaluation of thin-walled steel tanks filled at low liquid level according to current design codes. Proceedings of the Structural Stability Research Council, St. Louis, MO, USA."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1467","DOI":"10.1016\/j.jcsr.2007.01.003","article-title":"Buckling strength verification of cantilevered cylindrical shells subjected to transverse load using Eurocode 3","volume":"63","author":"Gettel","year":"2007","journal-title":"J. Constr. Steel Res."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/S0143-974X(99)00084-X","article-title":"Stability of steel shell structures: General report","volume":"55","author":"Schmidt","year":"2000","journal-title":"J. Constr. Steel Res."},{"key":"ref_20","unstructured":"ECCS (2008). Buckling of Steel Shells\u2014European Design Recommendations, European Convention for Constructional Steelwork."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"104861","DOI":"10.1016\/j.euromechsol.2022.104861","article-title":"Closed-Form Solutions for the Elastic\u2013Plastic Buckling Design of Shell Structures under External Pressure","volume":"98","author":"Do","year":"2023","journal-title":"Eur. J. Mech. A Solids"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"106274","DOI":"10.1016\/j.tws.2019.106274","article-title":"GBT-based buckling analysis of steel cylindrical shells under combinations of compression and external pressure","volume":"144","author":"Basaglia","year":"2019","journal-title":"Thin Walled Struct."},{"key":"ref_23","first-page":"819","article-title":"Collapse by instability of thin cylindrical shells under external pressure","volume":"11","author":"Windenburg","year":"1934","journal-title":"Trans. ASME"},{"key":"ref_24","unstructured":"ANSYS (2020). Structural Analysis, Mechanical APDL Structural Analysis Guide, Release 20, ANSYS."},{"key":"ref_25","unstructured":"De Freitas, A.S.N. (2017). An\u00e1lise Estrutural e de Estabilidade do Vaso de Press\u00e3o de um AUV. [Master\u2019s Thesis, Escola Polit\u00e9cnica da Universidade de S\u00e3o Paulo]."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"107485","DOI":"10.1016\/j.tws.2021.107485","article-title":"Buckling Behaviour of Double-Layer and Single-Layer Stainless Steel Cylinders under External Pressure","volume":"161","author":"Zhang","year":"2021","journal-title":"Thin Walled Struct."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"360","DOI":"10.1016\/j.joes.2021.03.002","article-title":"Buckling Analysis of Ring Stiffened Thin Cylindrical Shell under External Pressure","volume":"6","author":"Rathinam","year":"2021","journal-title":"J. Ocean Eng. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1002\/stco.201610031","article-title":"Comparative Assessment of the Design of Tubular Elements According to Offshore Design Standards and Eurocode 3","volume":"9","author":"Manco","year":"2016","journal-title":"Steel Constr."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.tws.2007.01.014","article-title":"Buckling of Long Steel Cylindrical Shells Subjected to External Pressure","volume":"45","author":"Albiez","year":"2007","journal-title":"Thin Walled Struct."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1016\/S0020-7462(01)00087-7","article-title":"Buckling Loads of Tank Shells with Imperfections","volume":"37","author":"Hornung","year":"2002","journal-title":"Int. J. Non Linear Mech."},{"key":"ref_31","unstructured":"Timoshenko, S.P., and Gere, J.M. (1961). Theory of Elastic Stability, McGraw-Hill."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Rotter, J. (2016). Advances in understanding shell buckling phenomena and their characterisation for practical design. Recent Progress in Steel and Composite Structures, Proceedings of the XIII International Conference on Metal Structures, Zielona G\u00f3ra, Poland, 15\u201317 June 2016, CRC Press.","DOI":"10.1201\/b21417-3"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1540004","DOI":"10.1142\/S0219455415400040","article-title":"Buckling analysis of thin-walled steel structural systems using generalized beam theory (GBT)","volume":"15","author":"Basaglia","year":"2015","journal-title":"Int. J. Struct. Stab. Dyn."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/j.tws.2007.02.001","article-title":"Generalised beam theory to analyse the buckling behaviour of circular cylindrical shells and tubes","volume":"45","author":"Silvestre","year":"2007","journal-title":"Thin Walled Struct."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1540021","DOI":"10.1142\/S0219455415400210","article-title":"Buckling and vibration analysis of cold-formed steel CHS members and frames using generalized beam theory","volume":"15","author":"Basaglia","year":"2015","journal-title":"Int. J. Str. Stab. Dyn."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Schardt, R. (1989). Verallgemeinerte Technische Biegetheorie: Lineare Probleme, Springer.","DOI":"10.1007\/978-3-642-52330-4"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/0263-8223(92)90055-H","article-title":"Failure of composite cylinders under combined external pressure and axial loading","volume":"22","author":"Mistry","year":"1992","journal-title":"Compos. Struct."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.ijpvp.2018.02.003","article-title":"Strength and buckling of an untypical dished head of a cylindrical pressure vessel","volume":"161","author":"Magnucki","year":"2018","journal-title":"Int. J. Press. Vessel. Pip."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/j.compstruct.2017.03.109","article-title":"Buckling of composite cylindrical shells with rigid end disks under hydrostatic pressure","volume":"173","author":"Lopatin","year":"2017","journal-title":"Compos. Struct."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.ijpvp.2016.05.002","article-title":"Buckling of externally pressurized steel toriconical shells","volume":"144","year":"2016","journal-title":"Int. J. Press. Vessel. Pip."},{"key":"ref_41","unstructured":"Simpson, B. (1997). A Concept for the Classification of Steel Containments Due to Safety Considerations. Containment Structures: Risk, Safety and Reliability, Proceedings of the IABSE Henderson Colloquium, Cambridge, UK, 11\u201312 July 1995, CRC Press."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Simpson, B. (1997). Challenges for the Future in the Design of Bulk Solid Storages. Containment Structures. Containment Structures: Risk, Safety and Reliability, Proceedings of the IABSE Henderson Colloquium, Cambridge, UK, 11\u201312 July 1995, CRC Press.","DOI":"10.1201\/9781482271751"},{"key":"ref_43","unstructured":"Dunphy, E. (2013). Study and Comparison of Shell Design Codes. [Master\u2019s Thesis, Delft University of Technology]."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1243\/PIME_PROC_1985_199_123_02","article-title":"Interactive buckling tests on cylindrical shells subjected to axial compression and external pressure\u2014A comparison of experiment, theory and various codes","volume":"199","author":"Galletly","year":"1985","journal-title":"Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1243\/JMES_JOUR_1969_011_007_02","article-title":"Experimental behaviour of thin-walled cylindrical shells subjected to external pressure","volume":"11","author":"Montague","year":"1969","journal-title":"J. Mech. Eng. Sci."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"109122","DOI":"10.1016\/j.tws.2022.109122","article-title":"Experiments on buckling behavior of thin-walled steel pipes subjected to axial compression and external pressure","volume":"174","author":"Badamchi","year":"2022","journal-title":"Thin Walled Struct."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.engstruct.2018.02.012","article-title":"Design of cold-formed stainless steel circular hollow section columns using direct strength method","volume":"163","author":"Huang","year":"2018","journal-title":"Eng. Struct."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1016\/j.engstruct.2015.12.020","article-title":"Local buckling and ultimate strength of slender elliptical hollow sections in compression","volume":"111","author":"McCann","year":"2016","journal-title":"Eng. Struct."},{"key":"ref_49","unstructured":"ANSYS (2020). Mechanical APDL Theory Reference, ANSYS, Inc."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"117855","DOI":"10.1016\/j.oceaneng.2024.117855","article-title":"Buckling Failure of Winding Composite Cylindrical Hulls under Hydrostatic Pressure","volume":"303","author":"Yuan","year":"2024","journal-title":"Ocean Eng."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"111583","DOI":"10.1016\/j.tws.2024.111583","article-title":"S890 hot-rolled ultra-high strength steel (UHSS) seamless circular hollow section beam\u2013columns: Testing, modelling and design","volume":"197","author":"Wang","year":"2024","journal-title":"Thin Walled Struct."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"111102","DOI":"10.1016\/j.tws.2023.111102","article-title":"Testing, numerical modelling and design of S890 and S960 ultra-high strength steel circular hollow sections under combined loading","volume":"192","author":"Wang","year":"2023","journal-title":"Thin Walled Struct."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"110871","DOI":"10.1016\/j.oceaneng.2022.110871","article-title":"Experimental and Numerical Analyses on Buckling and Strength Failure of Composite Cylindrical Shells under Hydrostatic Pressure","volume":"249","author":"Zhang","year":"2022","journal-title":"Ocean Eng."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"108708","DOI":"10.1016\/j.tws.2021.108708","article-title":"Nonlinear Buckling Analysis of a Semi-Elliptical Dome: Numerical and Experimental Investigations","volume":"171","author":"Barathan","year":"2022","journal-title":"Thin Walled Struct."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.ast.2016.05.018","article-title":"Nonlinear Buckling Analysis of the Conical and Cylindrical Shells Using the SGL Strain Based Reduced Order Model and the PHC Method","volume":"55","author":"Liang","year":"2016","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1016\/j.tws.2015.06.013","article-title":"On Nonlinear Thermal Buckling Analysis of Cylindrical Shells","volume":"95","author":"Alijani","year":"2015","journal-title":"Thin Walled Struct."},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Nassiraei, H. (2024). Probabilistic Analysis of Strength in Retrofitted X-Joints under Tensile Loading and Fire Conditions. Buildings, 14.","DOI":"10.3390\/buildings14072105"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"118143","DOI":"10.1016\/j.engstruct.2024.118143","article-title":"Experimental and numerical investigations of S890 and S960 ultra-high strength steel circular hollow section columns","volume":"311","author":"Zhang","year":"2024","journal-title":"Eng. Struct."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"1586","DOI":"10.1061\/(ASCE)0733-9445(2004)130:10(1586)","article-title":"Numerical modeling of stainless steel structural components\u2014A consistent approach","volume":"130","author":"Gardner","year":"2004","journal-title":"J. Struct. Eng."},{"key":"ref_60","unstructured":"OriginLab (2020). OriginLab\u2014Origin and OriginPro-Data Analysis and Graphing, version 8.1, OriginLab."}],"container-title":["Buildings"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-5309\/14\/9\/2992\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:01:20Z","timestamp":1760112080000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-5309\/14\/9\/2992"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,20]]},"references-count":60,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["buildings14092992"],"URL":"https:\/\/doi.org\/10.3390\/buildings14092992","relation":{},"ISSN":["2075-5309"],"issn-type":[{"value":"2075-5309","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,9,20]]}}}