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.