A one-dimensional stationary problem of the dynamics of a console circular panel with applied normal pressure has been discussed. The boundary value problem has been considered for various shells, such as the Kirchhoff–Love shell and the Donnel–Mushtari shell with extensible and non-extensible middle plane. The possibility of finding an analytical solution for the above-said shells has been considered as well. The solution of the problem has been sought in the form of using integral expressions with Green's functions as their core. Green's functions would be the boundary value problem's solution with pressure in a form of Dirac delta function. An analytical algorithm has been developed to calculate Green's functions value for any fluctuation frequency. This algorithm is based on the numerical (in case of the Kirchhoff–Love shell and the Donnel–Mushtari shell with extensible middle plane) and precise (in any other cases) solution of the characteristic equation and also on developing a common solution in a matrix form for the boundary value problem. Examples of the calculations of dynamics of a console circular panel with applied concentrated force (Green's function) and normal pressure have been given. Comparison of various shells with the applied concentrated and distributed loads has been performed.