## FEM Analysis of geodesic domes

Modern types of geodesic shells are complex single-contour and double-contour structures. Geodesic shells that perform the function of the building envelope are composed of flat and/or curved elements that form the total surface of the dome. This complicates the analysis of geodesic shells for strength and stability. An even greater complication of computations is the analysis of double-contour structures combining an asymmetric and non-smooth shell with a complex multi-rod system.
For automated analysis of the strength and stability of geodesic shells, the finite element method and the Patran/Nastran FEM software were used. Patran is a preprocessor/postprocessor by MSCSoftware, used to prepare the analysis scheme and visualize the results of analysis in various solvers (Nastran/Marc/Dytran).

Computing complex Nastran allows to determine the elastic stress-strain state of the shell by the finite element method

[K]e{u}e = {F}e - {F}ee0 - {F}ep - {F}eb

where [K]e - element stiffness matrix,
{u}e - element displacement vector,
{F}e - nodal force vector,
{F}ee0 - equivalent nodal forces of temperature distortions,
{F}ep - equivalent nodal forces of surface distributed loads (pressures),
{F}eb - equivalent nodal forces of inertial loads.

At the same time, the accuracy of the solution essentially depends on the type and parameters of the user-selected finite element. To perform analysis from the Nastran library, the CQUAD element of a standard membrane with the topology Tria3, Quad4 was adopted.
As an example, consider the analysis of three types of geodesic single-contour domes of class I1; 6 of dead load and the analysis of a similar smooth hemispherical dome. Three types of geodesic single-contour shells, approaching a hemisphere with a diameter of D = 10 m, thickness δ = 0.005m from aluminum (ρ = 2700kg / m3, E = 70000MPa, μ = 0.3).

Breakdown of the geodesic dome into plates: First variant. Second variant. Third variant.

Patran preprocessor allow to appoint of the characteristic size of the finite element by hand or perform its automatic selection. The following figure shows the results of analysis. Maximum stress diagram in smooth and geodesic domes calculated numerically for three types of shells and a smooth spherical dome

To assess the accuracy of the results obtained, a comparison of numerical values was made. σm, calculated for smooth domes and for geodesic domes for two values of the characteristic size of a finite element: with l = 0.05m and with l chosen automatically by the Patran program itself. It is clear from the previous figure that the numerical results for the third variant of the geodesic shell (l = 0.05m) and the smooth dome are quite close.

Also, analysis of geodesic domes of class I1; 6.4 on the effect dead load were made. Depending on the number of breakdown of class I1; 6.4 into stress concentration elements σm regarding σma spherical dome (λ = σm / σma) made up λ = 35.1 for 130 elements, λ = 17.5 for 630 elements, λ = 2.03 for 2373 elements. With the increase in the number of breakdown elements of the geodesic shell, the global stress concentrator gradually shifts from the top of the dome to its base. Von Mises stress distributions in a class I1; 6.4 geodesic dome according to the first breakdown variant (top view).

Also, the analysis of geodesic domes on the dead load and snow load were made. The function of application of snow load during the development of a analysis scheme in Patran was developed in the form of a PCL program.
Numerical analysis of stresses and deformations arising in double-contour geodesic domes under the dead load and wind load were performed. The function of wind load application in the form of PCL program has been developed also.
A technology has been developed for analysis geodesic domes for stability using the FEM in the Patran/Nastran system. The presence of buckling type loss in the plates of the geodesic dome is demonstrated. Demonstrated the presence of loss of stability of the type of bifurcation in the rods of the secondary contour of the geodesic dome.
Analysis of rigid, deformable and destructible geodesic domes for explosive impact were performed using the Dytran solver.