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Planar cellular lattice structures subject to axial compression may undergo elastic bending or buckling of the unit cells. If sufficient compression is applied, the columns of adjacent cells make contact. This changes the topology of the lattice by establishing new load paths. This topology change induces a corresponding shift in the effective stiffness characteristics of the lattice — in particular, the shear modulus undergoes a step-change. The ability to embed adaptive stiffness characteristics through a topology change allows structural reconfiguration to meet changing load/operational requirements efficiently. The concept, of topological reconfiguration, can be exploited across a range of length scales, from (meta-)materials to components. Here we focus on macroscopic behaviour presenting results obtained from finite element analysis that shows excellent correlation with the observed response of 3D-printed PLA lattices. Through a parametric study, we explore the role of key geometric and stiffness parameters and identify desirable regions of the design space. The non-linear responses demonstrated by this topology morphing lattice structure may offer designers a route to develop bespoke elastic systems.