Structural Optimization 8, 145-155 Q Springer-Verlag 1994 Optimal plastic shape design via the boundary perturbation method W. Egner, Z. Kordas and M. Zyczkowski Institute of Mechanics and Machine Design, Cracow University of Technology (Politechnika Krakowska), ul. Warszawska 24, PL-31-155 KrakSw, Poland Abstract The paper presents the boundary perturbation method as applied to optimal plastic shape design. Perfect plas- ticity is assumed. The procedure consists of two steps: deter- mination of a class of fully plastic solutions in the limit state (if possible), and then the choice of the optimal shape from among those solutions. Governing equations and boundary conditions for general perturbations of a circular cylinder under internal pressure are derived. Two examples of optimization concern non-circular shapes under plane strain conditions (heads of tension members) and pipes of variable diameter (transition zones). In both cases the minimal volume is the design objective under the constraint of given limit load-carrying capacity. 1 Introduction The boundary perturbation method (BPM) is now widely developed both in solid and fluid mechanics (although not always under this name). Older results are presented in the monograph by Morse and Feshbach (1953), and a recent sur- vey by Guz and Nemysh (1987) gives 310 references (mostly Ukrainian and Russian). The monograph by Guz and Ne- mysh (1989) presents in detail two variants of this method. Recently, much attention to BPM applied to elastic prob- lems has been paid by Parnes (1987, 1989) (eccentric load- ings) and Gao (1991) (inclusions). Fewer papers are devoted to BPM in the problems of plasticity; they were initiated by Ilyushin (1940), Ivlev (1957) and Spencer (1962). Applica- tions of BPM to optimal shape design are rather seldom; we mention here the paper by Schnack and Iancu (1989) who used numerically realized local perturbations in the elastic range. A series of papers by Kordas and her collaborators, started in 1970, used BPM in the investigation of fully plas- tic states at the stage of collapse of various perfectly plastic structural elements. Kordas and Zyczkowski (1970) consid- ered noncircular shapes of cylinders under pressure; Kordas (1973) discussed pipe-lines of variable diameter; Kordas and Skraba (1977) analysed cylinders under pressure with bend- ing; Kordas (1977) presented a general approach to the prob- lem under consideration; Kordas (1979) considered noncircu- lar shapes of disks under pressure; Dollar and Kordas (1980) discussed frame corners under bending, tension and shear; Kordas and Postrach (1990) analysed rotating disks. Full plastification at the stage of collapse is the first step towards optimization (in most cases the necessary condition), since the material in rigid or elastic zones at the stage of collapse is not properly utilized. In many cases, however, the above condition is not sufficient and then additional optimization is necessary. Examples of such additional optimization are given in the papers by Bochenek et al. (1983) and Egner et al. (1993). The first paper discussed plastic optimization of a doubly- connected cross-section of a bar under torsion with small bending; in this problem only one boundary condition along each contour holds, thus it is always possible to find a class of solutions satisfying that condition and then to perform the subsequent optimization. The second paper is devoted to the optimal design of yoke elements (ends of connecting rods, bolt joints, chain links), i.e. to plane problems of plasticity with two boundary conditions along each contour; it turns out that they may all be satisfied simultaneously and the optimal shape ensuring the maximal limit load-carrying ca- pacity may be found. In both problems the circular (annular) shape was subject to perturbations. If the thickness (of beams, plates or shells) is assumed as the design variable, then the Prager-Shield theorem on optimal plastic design is particularly useful (Rozvany 1976; Lellep 1991). On the other hand, if the shape of the bound- ary is subject to variations, its applications are less effective, although possible (Mr6z 1963). In such cases, BPM may suc- cessfully be employed proVided that the optimal shape is not too different from a certain classical shape with a simple per- fectly plastic solution. In the present paper we first consider general three-dimensional perturbations of a circular cylin- der under internal pressure and then give two examples of the application to optimal plastic shape design. 2 General perturbations for a plastic circular cylin- der under internal pressure 2.1 Assumptions, basic solulion We consider bodies with a shape close to a circular cylin- der under the following assumptions: the material is per- fectly plastic, isotropic, incompressible, subject to the Huber- Mises-Hencky (HMIt) yield condition. In the basic, unper- turbed state the cylinder with internal radius a 0 and external radius b0 is in a plane-strain condition. Small strains are as- sumed throughout the paper. Hencky-Ilyushin or Levy-Mises constitutive equations are employed; they lead to the same results only with strains replaced by strain rates in the second case. The cylindrical coordinates r, 6, z are used. Under the assumption of uniform internal pressure Pa =