Design of geodesic dome structures.

Tofan, Mihai ; Tierean, Mircea ; Baltes, Liana 等

Abstract: This paper presents a MathCAD finite element application for Fuller's dome structures. The dome's structure is generated introducing

the radius of external sphere. For architectural reasons, some of the theoretical nodes of structures can be omitted. The introduction of material's properties and the geometrical parameters of the bars are the next step. After the assemblage of the stiffness matrix, the restraints and the forces are applied for several loading cases. The nodal displacements are calculated and the distorted structure is drawn, for each loading case. The structure's bars are checked at traction, bending and buckling. The welded joins are also calculated and the resonance frequencies of the dome.

Key words: FEA, simulation, structures, geodesic dome

1. INTRODUCTION

The geodesic dome structure was invented in 1947 by Richard Buckminster Fuller (July 12, 1895-July 1, 1983). Even if domes have existed for centuries, the geodesic domes are better than those because they combine the sphere, the most efficient container of volume per square meter, with the tetrahedron, which provides the greatest strength for the least weight, or using different words "the most economical momentary relationship among a plurality of points and events" (http://www.thirteen.org/bucky/dare.html).

This geodesic dome uses a pattern of self-bracing triangles in a pattern that gives maximum structural advantage, thus theoretically using the least material possible (a "geodesic" line on a sphere is the shortest distance between any two points).

The subject of this paper is the calculation program developed in MathCAD for geodesic dome structures.

2. GEODESIC DOME GEOMETRY

According to the number of struts' distinct dimensions it is possible to realize different geodesic geometries (http://www.desertdomes.com). The simplest geometry has the same length for all the struts and its name is "1v". The 1v dome is actually an icosahedron (solid figure with twenty faces) with the 5 struts bottom removed. The structure with two different struts' lengths (2v) is obtained removing 20 bars from the complete geodesic sphere.

The subject of this paper is the structure with three different struts' lengths (3v). The pattern for this structure contains nine triangles (Figure 1). The analyzed structure has the outer sphere radius of R=4.85 m.

[FIGURE 1 OMITTED]

The first step in geometry description of the structure is the numbering of nodes. The top of the structure was labeled as 0 and each level of triangles starts with the next number from the near right node in counterclockwise. The 3D coordinates of the nodal points are obtained applying the geodesic dome analytical relations into an Auto LISP program. These coordinates are imported in MathCAD using READPRN function and multiplied with the radius (Mathsoft, 2001).

Next, a nodal junction matrix was build, which defines the elements of the structure. In this matrix the first line shows the start node and the second line the arrival node. The structure defined in this way has 61 nodes and 150 elements. Now is time to introduce the struts' geometry, which includes the dimensions of the cross section and mechanical properties of the material: density, tensile and flexural Young modulus. Using these values the inertia characteristics are calculated for each bar.

3. NODAL DISPLACEMENTS COMPUTING

Because the struts are fully constrained in nodes, the typical element for this structure is the beam element. The relations are explained using a simple rectangular element. The linear relation between displacement and force can be explained using the relation:

[delta] = D x F, (1)

where: [delta]-displacements matrix;

D-compliance matrix;

F-forces matrix.

Considering the same cross sectional properties for all the elements, the local stiffness matrix (the inverse of compliance matrix) for each beam can be very easily built using the formula (Tofan et al., 1995):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where: [[alpha].sub.j] = A x [L.sup.2.sub.j]/Iz, [beta] = Iy/Iz, [tau] = G x It/2E x Iz;

A-cross sectional area;

Iz, Iy, It-inertia characteristics of bar;

E, G--tensile and flexural Young modulus;

[L.sub.j]-bar's length.

To transport this local stiffness in the global coordinate system (the center of the sphere) it is necessary to multiply them with the rotation and translation operators, proper for each bar. The assembling of the local stiffness into the global stiffness matrix can be realized using the positional incidence.

[FIGURE 2 OMITTED]

After this step, the fixing structure will be added and the outer forces are introduced in nodes. All the nodes from the lowest level are considered fixed into foundation. For the rest of the nodes, six loading values correspond: the first three refer to forces and the rest of them to moments. After that, the nodal displacements are calculated (six for each node) by multiplying the inverse of stiffness matrix with the outer forces matrix. These nodal displacements draw the distorted structure (Figure 2), amplified by [10.sup.3].

4. STRESSES AND VIBRATION CHECKING

Having the displacements of each node, the nodal forces are obtained, in the global coordinate system and also the forces for each element. Using these forces (six for each bar edge) the stresses that affect the struts are determined. For each bar the Von Mises stresses and buckling are verified.

The eigenfrequencies of the structure (Figure 3) that shows the ability of the structure to avoid collapses caused by earthquakes and winds are also calculated.

For the analyzed structure three types of loadings are considered:

* wind, 730 N/[mm.sup.2];

* snow, 2440 N/[mm.sup.2];

* gravitational loading of the bars and ceiling; applied for three kinds of structures realized from:

* rectangular pinewood bars (150x50 mm) with end steel connectors;

* aluminum tube ([phi]0x5 mm);

* stainless steel tube ([phi]60x5 mm), the last two welded on spheres connectors.

The elements are verified considering three loading hypothesis:

* gravitational loading plus wind force for all the free nodes;

* gravitational loading for all free nodes plus snow loading on upper two triangle levels;

* gravitational loading plus wind force for all free nodes plus snow loading on upper two triangle levels.

The values obtained for the worst loading case are presented in table 1.

[FIGURE 3 OMITTED]

If the struts are made of pinewood, special connectors at bars' ends are necessary. These connectors are made of steel tube ([phi]133x8 mm) welded with two steel strips for each beam. Using two bolts the struts are joined to node. Figure 4 presents the Von Mises stresses map obtained using Design Star F.E.A. software. Based on symmetry only half of the connector was modeled. For meshing, the 5 mm tetragonal solid element was used. The maximum stress is 9.364x[10.sup.7] Pa.

[FIGURE 4 OMITTED]

5. CONCLUSION

The presented programs show the deformed structures, the displacements' values, the bars stresses and the eigenfrequencies of the geodesic dome structure. Simply introducing the radius of the sphere, the material characteristics and bars' geometry, the problem is solved.

This program shows that local loads are distributed throughout the geodesic dome, using the entire structure. Geodesic domes get stronger, lighter and cheaper per unit of volume as their size increases, just the opposite of conventional building.

For the chosen example the top displacement is approximately 1 mm, the stresses are lower than admissible values and the eigenfrequencies are greater than the value proper to earthquakes and winds.

Due to the simplicity and this high velocity, the program is easy to implement in any type of calculation for geodesic dome structures, avoiding the laboriously methodology of the modeling, introducing data and calculation, claimed by commercial programs for F.E.A.

6. REFERENCES

Tofan, M.C., Goia, I., Tierean, M.H. & Ulea, M. (1995). Deformatele structurilor (Structure's strains), Editura Lux Libris, ISBN 973-96854-2-0, Brasov, Romania

Mathsoft, Inc. (2001) MathCAD User's Guide, Cambridge, MA Available from: http://www.desertdomes.com Accessed: 2007-07-17 Available from: http://www.thirteen.org/bucky/dare.html Accessed: 2007-07-17

Table 1. Values obtained for the worst loading case.

 Maximum Top
 Von Mises stress displacement

Material Pa mm

Rectangular pine bars 3.26 x [10.sup.5] 0.965
Aluminum tube 1.24 x [10.sup.6] 1.096
Stainless steel tube 1.45 x [10.sup.6] 0.557

 Minimum Buckling
 eigenfrequency safety factor

Material Hz

Rectangular pine bars 15.438 9.078
Aluminum tube 18.675 26.95
Stainless steel tube 19.478 29.17