Conical monolithic thin-walled reinforced concrete blocks (shells) can be used as foundations for frame structures and buildings with load-bearing walls. As an analogue of columnar foundations, this type has the advantage of the speed of installation. It almost wholly excludes earthwork and formwork from the technological process, provided that the physical and mechanical properties of the soil base are improved. Shells are immersed in the supporting soil base with the help of dynamic or static loading. Additional soil base compaction occurs due to interaction with the conical outer surface of the shells. The work aims to develop the theoretical foundations for modelling, mathematical description and the possibility of optimizing the process of sinking monolithic thin-walled spatial reinforced concrete shells that model foundations for low-rise technological or residential facilities. Based on the analysis of the ”hammer-head-shell-soil” system, an elastic-viscous-plastic model with an attached soil mass was developed and presented, which is used as a basis for developing a system of nonlinear differential equations of the second order. The article provides recommendations on the impact of structural and technological factors on the failure rate and the contact stresses in the hammer-shell zone. The following factors have a significant influence on the failure of the shell: the height of the hammer lift, soil resistance, and the ratio of the masses of the hammer and the shell. The effect of shock-absorbing elements is insignificant. The greatest influence on the magnitude of the contact forces in the zone of the hammer-shell has: the height of the fall of the hammer and the stiffness of the resilient gasket.