The opportunities of umbrella-type shells

Relevance. The necessity of division of umbrella surfaces and surfaces of umbrella type into two separated classes is explained in introduction. Earlier, umbrella surfaces and surfaces of umbrella type were in the same class of surfaces because they consist of the identical fragments lying on the surfaces of revolution. Umbrella surfaces are compound surfaces on the base surface of revolution but umbrella-type surfaces are kinematic surfaces formed by continuous movement of a changing curve and that is why taking into account the methods of construction of these surfaces they were divided in two separate classes. The aim of the work is a collection of main publications on all areas of the investigation of umbrella-type shells. Methods. For the determination of principal results of investigation of umbrella-type shells, it is necessary to know differential geometry of surfaces, structural mechanics of thin shells, and approaches used in architecture of spatial structures. Results. In this article, the principal scientific papers on geometry, strength analysis, and offers of applications of thin-walled shells of umbrella type in building and of reflectors of umbrella type for space apparatuses. The accurate parametric equations of some determined surfaces are presented. The approximated computer models of middle surfaces of the real umbrella shells but in the form of umbrella-type surfaces are given. The examples of determination of stress-strain state of thin-walled shells of umbrella type without dividing of the whole shell in identical fragments are shown. New information and materials already known about shells of umbrella type give reasons to suppose that the shells of this type will be claimed by engineers and architects.