  The equilibrium configuration of an engineering structure, able to withstand
a certain loading condition, is usually associated with a local minimum of the
underlying potential energy. However, in the nonlinear context, there may be
other equilibria present, and this brings with it the possibility of a
transition to an alternative (remote) minimum. That is, given a sufficient
disturbance, the structure might buckle, perhaps suddenly, to another shape.
This paper considers the dynamic mechanisms under which such transitions
(typically via saddle points) occur. A two-mode Hamiltonian is developed for a
shallow arch/buckled beam. The resulting form of the potential energy---two
stable wells connected by rank-1 saddle points---shows an analogy with
resonance transitions in celestial mechanics or molecular reconfigurations in
chemistry, whereas here the transition corresponds to switching between two
stable structural configurations. Then, from Hamilton's equations, the
analytical equilibria are determined and linearization of the equations of
motion about the saddle is obtained. After computing the eigenvalues and
eigenvectors of the coefficient matrix associated with the linearization, a
symplectic transformation is given which puts the Hamiltonian into normal form
and simplifies the equations, allowing us to use the conceptual framework known
as tube dynamics. The flow in the equilibrium region of phase space as well as
the invariant manifold tubes in position space are discussed. Also, we account
for the addition of damping in the tube dynamics framework, which leads to a
richer set of behaviors in transition dynamics than previously explored.
