Thin-Walled Structures 34 (1999) 295–330 www.elsevier.com/locate/tws Analytical modelling of tube hydroforming Nader Asnafi * Volvo Car Components Corporation/Industrial Development Centre, Olofstro ¨m, Sweden Abstract The automotive industry has shown a growing interest in tube hydroforming during the past years. The advantages of hydroforming (less thinning, a more efficient manufacturing process etc.) can, for instance, be combined with the high strength of extra high strength steels, which are usually less formable, to produce structural automotive components which exhibit lower weight and improved service performance. Design and production of tubular components require knowledge about tube material behaviour and tribological effects during hydroforming and how the hydroforming operation itself should be controlled. These issues are studied ana- lytically in the present paper. Hydroforming consists of free forming and calibration. Only the so-called free forming is treated here. The analytical models constructed in this paper are used to show what the limits are during the free forming, how different material and process para- meters influence the loading path and the forming result, and what an experimental investi- gation into hydroforming should focus on. The present study was a part of a larger investi- gation, in which finite-element simulations and experiments were also conducted. The results of these simulations and experiments will be accounted for in coming papers.  1999 Elsevier Science Ltd. All rights reserved. Keywords: Fracture; Hydroforming; Modelling; Tube; Yielding Notation C Length of the expansion zone, Fig. 13 d 0 Initial (outer) tube diameter, Fig. 10 d fr Tube diameter at fracture * Present address: Gra ¨nges Technology, SE-612 81 Finspång, Sweden. Tel: + 46-122-170-00; fax: + 46-122-124-87; e-mail: nader.asnafi@techno.graenges.se 0263-8231/99/$ - see front matter  1999 Elsevier Science Ltd. All rights reserved. PII:S0263-8231(99)00018-X