  We present a general approach to the bifurcation analysis of elastic
frameworks with symmetry. While group-theoretic methods for bifurcation
problems with symmetry are well known, their actual implementation in the
context of elastic frameworks is not straightforward. We consider frames
comprising assemblages of Cosserat rods, and the main difficulty arises from
the nonlinear configuration space, due to the presence of (cross-sectional)
rotation fields. We avoid this via a single-rod formulation, developed earlier
by one of the authors, whereby the governing equations are embedded in a linear
space. The field equations comprise the assembly of all rod equations,
supplemented by compatibility and equilibrium conditions at the joints. We
demonstrate their equivariance under the symmetry-group action, and the
implementation of group-theoretic methods is now natural within the
linear-space context. All potential generic, symmetry-breaking bifurcations are
predicted apriori. We then employ an open-source path-following code, which can
detect and compute simple, onedimensional bifurcations; multiple bifurcation
points are beyond its capabilities. For the latter, we construct
symmetry-reduced problems implemented by appropriate substructures. Multiple
bifurcations are rendered simple, and the path-following code is again
applicable. We first analyze a simple tripod framework, providing all details
of our methodology. We then treat a hexagonal space frame via the same
approach. The tripod and the hexagonal dome both exhibit simple and double
bifurcation points.
