Fragmented structures are discontinuous structures consisting of fragments not connected together by any binder. These structures and solids are encountered in a number of engineering and natural systems. The integrity of a fragmented structure is preserved by a combination of kinematic boundary constraints and compression applied at the boundaries. These compressive stresses, though essential for maintaining the continuity, may cause the loss of structural stability (buckling). In this study, we investigate this dual role of the compressive stresses using an example of fragmented beams. First we develop a continuum model for the large deflection analysis of fragmented beams, which includes the destabilising effect of the axial forces. To account for the discontinuous nature of a fragmented beam, we introduce delamination-controlled stiffnesses that depend upon the contact area between the fragments. We apply this model for the stability analysis of fragmented beams with various boundary conditions. The numerical simulations capture two failure mechanisms characterising fragmented structures: full delamination due to bending and buckling under compressive loads.