Thin steel shells in the form of spherical segments, loaded by external pressure, have a tendency to sudden buckling under a lower pressure than their plastic resistance. The buckling phenomenon of spherical shells was the subject of research done by many authors and the previous results of theoretical and experimental research were used in the European Design Recommendations published in 2008 by European Convention for Constructional Steelwork. This article presents experimental research done on a specially designed and manufactured stand. Specimens were manufactured with a metal spinning technique from steel sheets usually used for deep drawing and stamping of metal objects. The actual shape of every specimen was determined with the three-dimensional scanning technique. Material research was also done on the steel sheets. Pressure was exerted with an air compressor and displacement was measured with the optical measurement system ARAMIS. The deformation modes of the whole specimen were registered precisely at every step of pressure exertion till the final postbuckling stage. All the data measured were registered automatically by two computers. The result of every experiment is presented in the form of an equilibrium path as the relationship of the pressure p as a function of a chosen displacement parameter. This article also presents a comparison between the critical pressures obtained during the experiments and the results obtained from the numerical simulations done with COSMOS/M. The experimental and the numerical results were also compared with the buckling resistances proposed by Rotter and Schmidt and Błażejewski and Marcinowski.

The shape of the optimal rod determined in the work meets the condition of mass conservation in relation to the reference rod. At the same time, this rod shows a significant increase in resistance to axial force. In the examples presented, this increase was 80% and 117%, respectively, for rods with slenderness of 125 and 175. A practical benefit from the use of compression rods of the proposed shapes is clearly visible. The example presented in this publication shows how great the utility in the structural mechanics can be, resulting from the applications of complex analysis (complex numbers). This approach to many problems can find its solutions, while they are lacking in the real numbers domains. What is more, although these are operations on complex numbers, these solutions have often their real representations, as the numerical example shows. There are too few applications of complex numbers in the technique and science, therefore it is obvious that the use of complex analysis should have an increasing range. One of the first people to use complex numbers was Girolamo Cardano. Cardano, using complex numbers, was solving cubic equations, unsolvable to his times – as the famous Franciscan and professor of mathematics Luca Pacioli put it in his paper Summa de arithmetica, geometria, proportioni et proportionalita (1494). It is worth mentioning that history has given Cardano priority in the use of complex numbers, but most probably they were discovered by another professor of mathematics – Scipione del Ferro (cf. [1]). We can see, that already then, they were definitely important (complex numbers).