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
10^{5} [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.

The analysis results with 36 B2.S.RS elements are displayed in the two
figures below. The graph displays the applied moment
M_{x} 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.

**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).