5. Deployable Ring

The deployable ring problem was proposed by Goto et al.[12]and it has been studied, among others, by Rebel[13], although with shell elements. It possesses the remarkable property that the ring can considerably be reduced in size by twisting it. The accompanying finite rotations in space make the deployable ring a severe test for the finite rotations capabilities of non-linear beam and shell elements.

The ring has a radius R = 60 and a rectangular cross section shape with a width of 0.6 a height of 6.0 and a modulus of elasticity of 2.0 105 [no units indicated]. It is clamped in one point and a rotation in the radial direction of the point opposite to the clamped point is applied. The rotation ranges from 0 to 2π.

The test is performed with B2 and B3 elements, the B3 elements obviously giving more accurate results because they approximate the circle better than the B2 elements. The FE mesh consist of a variable number of B2.S.RS or B3.S.RS elements. The figures below display the geometry and mesh points as well as results for a mesh with 36 B2.S.RS elements.

Deployable ring: Mesh (36 B2.S.RS elements).

Figure 27. Deployable ring: Mesh (36 B2.S.RS elements).

The analysis results with 36 B2.S.RS elements are displayed in the two figures below. The graph displays the applied moment Mx due to the applied rotation in the global x-direction, i.e the reaction to the applied rotation, as a function of the applied rotation. The deformation process is illustrated with steps corresponding to 0.5π, π, 1.5π, and 2π.

Deployable ring: Mxvs applied rotation (36 B2.S.RS elements).

Figure 28. Deployable ring: Mxvs applied rotation (36 B2.S.RS elements).

Deployable ring: Deformation shapes at steps 0π, 0.5π, π, 1.5π, and 2π. (36 B2.S.RS elements).

Figure 29. Deployable ring: Deformation shapes at steps 0π, 0.5π, π, 1.5π, and 2π. (36 B2.S.RS elements).

[12] Goto Y., Watanabe Y., Kasugai T., Obata M., Elastic buckling phenomenon applicable to deployable rings, International Journal of Solids and Structures, Vol. 29, No. 7, pp. 893-909 (1992).

[13] G. Rebel; Finite Rotation Shell Theory including Drill Rotations and its Finite Element Implementation; PhD Thesis, Delft University Press (1998).