Keywords: earthquake resistance, shell, revolution, finite element method, probabilistic analysis.

In the calculation of the seismic stability of structures, the dynamic character of seismic effect should be taken into account by means of various factors. To determine accurately the reactions of building structures it is recommended that calculations be done for real or synthesized accelerograms. The numerical methods of calculation presented in this paper are based on the application of the theory of random processes combined with the finite element method. Equations of the discrete model displacement are given within the framework of the finite element method. They connect matrices of mass, damping and stiffness; the vector of structure joint displacements; the vector of space load distribution in the joints; the determined envelope; and stationary random process. The determined envelope is assumed in the form which is most frequent when analyzing reactions of structures exposed to non-stationary random effect. To illustrate the methods of probabilistic analysis of seismic stability of structures, two real projects and a projected one are considered. Seismic action in the form of non-stationary random oscillations of acceleration of the foundation is assumed as the product of the determined envelope by a stationary random process, the chart of its spectral density taking the form of a rectangle. Displacements and tensions resulting from the action of seismic load have been determined. The projects are:

• An ancient dome of a mosque in Akhaltsikhe:
  The building is situated in a seismic area in Georgia. Its dome was built of locally manufactured thin bricks using a lime-clay mortar. The dome dimensions are as follows: its span is 16.2m, with a rise of 8.0m and a shell thickness of 0.6m. Seismic stability has been calculated in accordance the norms using a determinist approach.
• The building of the museum of the Defenders of the Caucasian Passes near the city of Karachayevsk in the North Caucasia:
  The museum building is constructed as a reinforced-concrete shell of revolution. The walls are built in the form of a truncated cone with a height of 4.2m, a lower base diameter of 21.74m and an upper base diameter of 19.65m. The construction coating has the form of a sphere segment with the rise being 1.6m.
• Water tower project:
  The water tower construction is a coupled reinforced-concrete shell of revolution consisting of a cylindrical (pillar) and toroid (water tower) shells. The height of a cylindrical shell is 25m and its external diameter is 7.7m; the external diameter of the toroid shell is 13.7m and its internal diameter is 7.7m. The height is 11.8m. The bottom of the reservoir is at the mark of 25m.

The numerical studies demonstrate a low level of tension stress and displacement when affected by most severe earthquakes which proves that shells of revolution are highly effective where their seismic stability is concerned.

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