  In this paper, we discuss reduced order modelling approaches to bifurcating
systems arising from continuum mechanics benchmarks. The investigation of the
beam's deflection is a relevant topic of investigation with fundamental
implications on their design for structural analysis and health. When the beams
are exposed to external forces, their equilibrium state can undergo to a sudden
variation. This happens when a compression, acting along the axial boundaries,
exceeds a certain critical value. Linear elasticity models are not complex
enough to capture the so-called beam's buckling, and nonlinear constitutive
relations, as the hyperelastic laws, are required to investigate this behavior,
whose mathematical counterpart is represented by bifurcating phenomena. The
numerical analysis of the bifurcating modes and the post-buckling behavior, is
usually unaffordable by means of standard high-fidelity techniques such (as the
Finite Element method) and the efficiency of Reduced Order Models (ROMs), e.g.\
based on Proper Orthogonal Decomposition (POD), are necessary to obtain
consistent speed-up in the reconstruction of the bifurcation diagram. The aim
of this work is to provide insights regarding the application of POD-based ROMs
for buckling phenomena occurring for 2-D and 3-D beams governed by different
constitutive relations. The benchmarks will involve multi-parametric settings
with geometrically parametrized domains, where the buckling's location depends
on the material and geometrical properties induced by the parameter. Finally,
we exploit the acquired notions from these toy problems, to simulate a real
case scenario coming from the Norwegian petroleum industry.
