4. Hexadome (Healey problem)

The buckling load of the Hexadome model are calculated. The Hexadome was first analysed by Tim Healey[26]. Later, Hexadome problem was solved by J.C. Wohlever in order to test his group theory. In this test the Hexadome is analysed in the conventional way.

Hexadome (Healey problem): Geometry

Figure 58. Hexadome (Healey problem): Geometry

All rods are of equal length L=9m. The center node height i 2h, with h=1.5m. The cross section area of a rod is A=1cm2. Material is linear isotropic, with the modulus of elasticity E=1·108N/m2.

On each of the intersection nodes (where more than two bars are attached to each other), an external force F=10N is applied. All six points (where two rods are connected) of the Hexadome are simply supported.

Every bar in the structure is modelled with one rod element.

The nodes of the rod elements have three degrees of freedom, the translations in the three directions. For this reason the rod element can only be simply supported, as a hinge or, in this case, as a ball-in-socket joint. (The bending stiffness of a rod element is zero.) Therefore, a rod element on its own can't be used for buckling analysis. The dome only buckles as a result of the instability of the construction, not as a result of Euler buckling of the rod. Due to the forces, the rods shorten and at a certain point the dome gets instable and the dome buckles. The rods have only stiffness in one direction. This is the reason that only one element is used over the length l. In the case two or more elements are used one gets singularities because 3 DOFs per node are present. This problem can be solved by defining node local coordinate systems and suppressing the DOFs perpendicular to the rod.

The results of the predictions obtained with this model are summarised in the following table:

Table 4. Hexadome (Healey problem): Linear buckling analysis results.

Mode number Eigenvalue
6 0.341
7 0.561
8 0.561
9 1.277

[26] Tim Healey; A group theoretic approach to computational bifurcation problems with symmetry; Computer Methods in Applied Mechanics and Engineering, Vol. 67 pp. 257-295, 1988.