A detailed design study is presented analyzing the influence of transverse clamping flexibility on the critical speeds and transverse stiffness of centrally clamped, centripetally tensioned, rotating circular disks. Tensioning techniques, which introduce in-plane stresses into thin plates, are commonly used to improve the critical speed and transverse stiffness of industrial circular saws. In centripetal tensioning, the magnitude of tensioning increases with the rotation speed of the disk, an objective that requires the disk to expand radially within the clamp as it rotates. Clamp designs that allow this radial expansion are necessarily more flexible in the transverse direction than traditional clamps, an effect that can be detrimental to the performance of high speed circular saws. The analysis here provides quantitative design guidelines for centripetal tensioning clamps to ensure that they improve circular saw performance.