A numerical assessment of the load bearing capacity of externally pressurized moderately thick tubes Leone Corradi * , Valentino Di Marcello, Lelio Luzzi, Fulvio Trudi Politecnico di Milano – Department of Energy, Enrico Fermi Center for Nuclear Studies (CeSNEF), via Ponzio 34/3 – 20133 Milano, Italy article info Article history: Received 8 June 2007 Received in revised form 3 February 2009 Accepted 10 February 2009 Keywords: Moderately thick tubes Pressure vessels Collapse Plasticity–instability interaction Imperfection sensitivity abstract The collapse behavior of cylindrical shells pressurized from outside is examined. Attention is focused on tubes of moderate thickness, as required by very deep water pipelines or some innovative nuclear power plant proposals. Their collapse is expected to be dominated by yielding but, because of the decreasing nature of the post-collapse evolution, interaction with instability is likely to be significant enough to demand consideration. At present, no quantitative assessment of such effect is available, because little study has been devoted to tubes in this thickness range. Plasticity–instability interaction is activated by imperfections and to assess their influence on a systematic numerical study is undertaken. Computations produce a meaningful measure of the collapse pressure and it is proposed that the allowable pressure be determined on its basis, by introducing a suitable safety factor. This is chosen so that results reproduce those provided by presently accepted procedures in the well explored and reliable range of medium-thin tubes. When the same factor is applied to thicker tubes, the resulting allowable pressure is significantly higher than the values suggested by codes, which apparently react to the present lack of knowledge by assuming an extremely conservative attitude. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Buckling of shells under external pressure is a satisfactorily settled subject when shells are thin enough to collapse in the elastic range. In some situations, however, shells of higher thickness are required: medium-thin cylinders are presently employed in oil industry as pipes or casings and, since pipe laying in increasingly deeper water is envisaged, increasingly thicker tubes become of interest; also, recent proposals for innovative nuclear power plant design consider tubes of moderate thickness pressurized from outside [1]. Medium-thin tubes, typical of oil industry applications, are considered in a number of papers [2–9], mostly providing numer- ical computations of the collapse pressure. Attempts at reproducing results with empirical design formulas are also made [2,3,9] and the approximations obtained are adequate in the thickness range explored. However, such formulas, often borrowed from the only partially similar problem of beam-columns, contribute marginally to the understanding of the behavior and their use can be justified only on numerical ground. This makes at least questionable their direct application to thicker tubes, whose collapse behavior has been little explored both from the experimental and the numerical points of view. When employed in this range existing design formulas turn out to be extremely conservative and accepted design procedures, such as those based on ASME Boiler & Pressure Vessel code [10], produce surprisingly high values for the thickness, reflecting a substantial lack of knowledge on the phenomena involved. This situation has significant consequences for the IRIS (International Reactor Inno- vative and Secure) project. In its design, steam generator tubes are contained inside the vessel and pressurized from outside [1]. Present ASME Section III rules require an external diameter to thickness ratio less than 8.5: this entails a major contribution of the thermal conduction through the wall thickness to the thermal resistance in the heat exchange process between primary and secondary fluids, with detrimental consequences on the dimen- sioning of the heat transfer surface. It is felt that ASME code requirements are exceedingly conser- vative in this thickness range. Reason is that imperfections are not explicitly considered when defining a reference failure pressure. Provided that their entities are below given values, imperfections are accounted for by means of a safety factor that is essentially slenderness independent. However, at different slenderness imperfections have different effects on the load bearing capacity. The most detrimental ones are experienced when the interaction * Corresponding author. Tel.: þ39 02 2399 6343; fax: þ39 02 2399 6309. E-mail address: leone.corradi@polimi.it (L. Corradi). Contents lists available at ScienceDirect International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp 0308-0161/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2009.02.001 International Journal of Pressure Vessels and Piping 86 (2009) 525–532