Leone Corradi Lelio Luzzi e-mail: lelio.luzzi@polimi.it Fulvio Trudi Department of Nucelar Engineering, Politecnico di Milano, Via Ponzio 34/3, 20133 Milan, Italy Collapse of Thick Cylinders Under Radial Pressure and Axial Load This paper provides the theoretical collapse loads of thick, long cylindrical shells subject to pressure and axial forces. Tubes are made of isotropic, perfectly plastic von Mises’ material. Axial strains are assumed to be constant but possibly different from zero, so that elongation is permitted. This assumption, together with axial symmetry and the isochoric nature of plastic flow, unambiguously defines the set of possible collapse mechanisms, and collapse loads are computed on this basis. Results are contrasted to those presently available, based on thin-shell assumptions. Comparison shows that differences are of engineering significance, well worth considering for thick tubes, such as those envisaged in some nuclear power plant applications. DOI: 10.1115/1.1938204 1 Introduction The assessment of load-bearing capacity of shells can be con- sidered an issue satisfactorily settled when shells are thin enough to collapse because of elastic buckling, as typical of aeronautic or aerospace applications. Outside this context, however, shells of higher thickness often are required. Medium-thick shells are em- ployed, for instance, in the oil industry as pipes or casings with thickness increasing as the depth of the water in which the pipes operate, and recent proposals for innovative nuclear power plant design consider steam generator tubes of significant thickness pressurized from outside 1. When thick tubes are subject to external pressure, collapse is initiated and often dominated by yielding, but interaction with instability is meaningful, in that imperfections reduce the load bearing capacity by an amount of engineering significance also when thickness is considerable. At present, such an effect is ac- counted for by means of more or less empirical formulas, defining the reduction with respect to the plastic collapse load induced by coupling with instability 2–8. Independent of the adequacy of such formulas, often borrowed from problems, such as beam columns, only partially similar to thick tubes, the very definition of the reference value demands discussion. In general, the plastic collapse pressure is computed by exploiting thin-shell assumptions, which consider stresses con- stant throughout. Under uniform pressure the tube becomes stati- cally determinate, with the consequence that the elastic limit is overestimated and the collapse pressure underestimated. Discrep- ancies are negligible as long as the ratio between the radius of the cylinder and its wall thickness is large, but get more and more significant as this ratio decreases. The pressure values at the onset of yielding elastic limit are easily computed from the well-known elastic solutions 9, and the correct thick shell values are used by most codes see, e.g., 10. The analogous results for plastic collapse, on the contrary, are available only for tubes in plane strain 11,12, a situation of interest but by no means the only significance. To clarify this point, the kinematics of deformation of long cylinders is examined. The tube being subject to uniform pres- sures for completeness, internal pressure is also included and, possibly, to constant axial force, each cross section undergoes the same loading, and its response is essentially “plane,” in that stresses and strains are independent of the axial coordinate, say z. Classical plane solutions, however, are not adequate. A slice of the cylinder in plane stress conditions would experience in the elastic-plastic range nonuniform transverse strains  z , in general, conflicting with those of adjacent slices. A plane-strain assump- tion solves the conflict by imposing that  z be zero throughout, but this constraint appears excessively severe; in fact, continuity be- tween adjacent slices is merely expected to make axial strains uniform without preventing possible elongation. For long tubes the most realistic model seems that of generalized plane strain, which assumes that  z is constant, but not necessarily zero. In this paper long, thick cylinders subject to external and/or internal pressure and axial load are considered and the values of such loads bringing, individually or together, the cylinder to col- lapse are determined. To this purpose the kinematic theorem of limit analysis is employed in conjunction with the von Mises yield criterion. In spite of the upper-bound nature of the kinematic theo- rem, the result is exact; in fact, the assumptions of axial symme- try, generalized plane strain, and isochoric plastic flow unambigu- ously define the set of possible collapse mechanisms, governed by the ratio among two parameters, namely, the radius variation and axial elongation. Results are contrasted to the corresponding elastic limits and to predictions stemming from thin-shell assumptions. Comparison permits the assessment of some points, such as the resources with respect to the elastic limit provided by stress redistribution and the range of validity of thin-shell approximation. In particular, it ap- pears that the latter assumption is too restrictive for really thick tubes, as those required by some fourth-generation nuclear plant applications. Limit analysis is based inherently on the small strain hypothesis and results are unable to assess the influence of the plasticity- instability interaction on the collapse level—influence which is significant in medium-thick cylinders and plays some role even in the definitely thick cylinders. This aspect is presently investigated and some preliminary results are presented in 13. As was already mentioned, however, formulas evaluating such effects use the col- lapse load as a reference value and its correct definition seems to be a preliminary, but important, starting point toward a rational assessment of the load-bearing capacity of this structural typol- ogy. 2 General Relations The cylinder in Fig. 1 is considered. Loads consist of external pressure q, internal pressure p, and axial force F, all constant throughout. Pressures are supposed to be always positive, while F can assume either sign, with F  0 corresponding to tension. The Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the Applied Mechanics Division, May 27, 2004; final revi- sion, November 2, 2004. Associate Editor: A. Maniatty. Discussion on the paper should be addressed to the Editor, Prof. Robert M. McMeeking, Journal of Applied Mechanics, Department of Mechanical and Environmental Engineering, University of California—Santa Barbara, Santa Barbara, CA 93106-5070, and will be accepted until four months after final publication in the paper itself in the ASME JOURNAL OF APPLIED MECHANICS. 564 / Vol. 72, JULY 2005 Copyright © 2005 by ASME Transactions of the ASME