A runflat radial ply tire (70) having a thread (72), a carcass (86) and a belt structure (74) radially located between the thread and the carcass has two sidewalls (80, 82) each reinforced with a circumferentially disposed wedge insert (50a, 50b) having a saw-tooth cross-sectional shape. The saw-tooth cross-sectional shape of each wedge insert (50a, 50b) is defined by a plurality of circumferentially disposed segments (52a-52e), each of which is separated from adjacent segments by circumferentially disposed grooves (54a-54d). During heavy loading of the tire, or during runflat operation, the radially loaded sidewall portions deform until the grooves (54a-54d) undergo closure.