Keywords: buckling, external pressure, pressure hull, simulated annealing, shell.

Various optimisation techniques would normally be needed to deal with problems encountered in practice and these techniques may be classified in a variety of ways. A useful approach could be based on the type of information that must be provided in order to find the optimum, e.g. whether the information on gradients of the objective and constraints is required. A class of methods, requiring only function values, is collectively known as zero-order search methods. The emphasis has recently been directed toward new zero-order techniques, which have arisen, at least in part, from a study of nature and they include, amongst others, genetic algorithms, simulated annealing and tabu search. This paper applies one of these methods, i.e. simulated annealing, SA, to structural optimisation of a pressure hull component.

The paper begins with an overview of the current use of the simulated annealing technique in structural optimisation. After a brief discussion of generic parameters, the paper highlights a method, which allows the SA to be used for continuous domain problems. Next, the shape optimisation of metallic, doubly curved shells subjected to static external pressure is attempted. The maximum load carrying capacity is sought within a class of shells with the positive Gaussian curvature in which the meridional profile is described by generalised ellipses. Both geometrical non-linearity and material non-linearity are taken into account. It is assumed that shells are made from mild steel with Young's modulus E = 217 GPa, Poisson's ratio = 0.27 and the yield point of material = 330 MPa. Structural analyses are based on linear elastic, perfectly-plastic modelling of material (with isotropic strain hardening). The structural responses were obtained from self-contained codes with both bifurcation buckling and collapse load being considered as possible modes of structural failure.

Standard SA algorithm with a geometric schedule for lowering temperature has been adopted. A variable step length was used for generation of a candidate configuration with the average percentage of accepted moves at about 50% per temperature level. The epoch length of 20 was assumed and it was kept constant at all temperature levels. The stopping criterion was based on no change of the cost function for 10 consecutive iterations. Computational effort of a single re-analysis is such that a predominantly sequential optimisation tool, such as SA, was capable to navigate through the non-convex design space in an affordable amount of time. Whilst it is not certain whether the obtained solution is a global maximum, it offers some 40% increase over the reference geometry's load bearing capacity. Yielding of the optimal geometry starts on the inner surface of the barrel at both top and bottom edges and the maximum, effective, plastic strain at the collapse reaches 3.2%. Calculations have shown that 90% of the cross section has yielded at the collapse load. The optimal solution has been verified experimentally by collapsing two, nominally identical shells. Models, designated here as E1 and E1a, were CNC- machined from thick, mild steel tube with 180 mm and 310 mm inner and outer diameters, respectively. Shells were machined with integral top and bottom rings. After machining to over-sized dimensions both shells were stress relieved at 600C. There has been no stress relieving after the final machining. The wall thickness of the optimal shells was about 2.6 mm, their height was about 100 mm and their diameter was 200 mm. Their experimental collapse pressure was about 17 MPa. Calculations for E1 and E1a were carried out for both the lower and upper yield of the material. The ratio varies from 0.91 to 1.01 for barrel E1 and 0.90 1.02 for barrel E1a. The load bearing capacity of some shell components is known to be sensitive to initial geometric imperfections. The type of shell geometry and type of loading considered in this paper belong to this type of structural elements. The initial deviations from perfect geometry can take various shapes and magnitudes. Traditionally, the most critical shape deviations are considered in order to ascertain the possible loss of the load carrying capacity. Imperfections in the form affine to the eigenmode are used for this purpose. Numerical results show that for the optimal barrel, the eigenmode corresponding to 13 circumferential waves is most detrimental. When one compares the loss of the load carrying capacity of cylinder and that of optimal barrel it is clear that the latter profile is more sensitive to initial shape deviations from perfect geometry. The differences are however not significant. It is therefore safe to say that shape optimisation has not created the solution which would be dismissed from a practical point of view because of greatly enhanced imperfection sensitivity to initial geometric imperfections. It is seen from results presented in the paper that both the initial and optimal designs have a comparable sensitivity of load carrying capacity to initial shape deviations from perfect geometry.

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