Gradient beam radius profile r_beam (s) along the radial coordinate s of the spring cross-section. The outer coil fiber corresponds to s = R + r_out, and the inner coil fiber to s = R − r_out. The beam radius increases linearly from r_min to r_max = f ⋅ r_min to redistribute material toward higher stress regions. The red beam illustrates how the radius is assigned based on its midpoint position along s: the beam's endpoints are marked with red circles and the midpoint with a red plus symbol.