Hinge zones accommodate the differential tidal movement between floating ice shelves, which move with the sea tide, and grounded ice sheets, which do not. Observations on Rutford Ice Stream and Ronne Ice Shelf, Antarctica, using the kinematic method of Global Positioning System, have yielded the first continuous tidal displacement profiles from an ice sheet hinge zone. The form of these tidal displacement profiles indicates that the flexure can be fit to an elastic beam model using a parameter space optimization technique. The model matches the observed tidal deflection profile to within 5 cm which is similar to the observational uncertainty. The same model and parameter fitting technique is then successfully applied to various other published and unpublished tidal displacement data from Doake Ice Rumples, Bach Ice Shelf, and Ekström Ice Shelf, Antarctica, and Jakobshavns Glacier, Greenland. I conclude that the elastic beam model with a single value of the elastic modulus (0.88±0.35 GPa) adequately describes almost all the data and so can usefully be applied elsewhere. Earlier studies of tidal dissipation in ice shelf hinge zones, based on a viscous or transient creep rheology, showed that perhaps 30% of global tidal energy dissipation could be occurring in ice shelf hinge zones. This study suggests, however, that an elastic rheology may be equally appropriate, in which case this estimation of the tidal energy dissipation is excessively high.