Supplementary File S1 — Central-axis RB bending steady-demand estimator

Input parameters

RB diameter, dmm

Elastic modulus, E_eMPa

Fulcrum roller radius, R₁mm

Yield stress, σ_yMPa

Rotating bending roller radius, R₂mm

Bilinear tangent modulus, E_HMPa

Enable bilinear outputyes/no

Bending-roller rotation radius, R₃mm

Power-law strength coefficient, KMPa

Torque unitoutput

Power-law exponent, n–

Enable power-law outputyes/no

Reference input ranges: d = 8–16 mm, R₁ = 10–35 mm, R₂ = 10–35 mm, and R₃ = 55–110 mm. Out-of-range inputs are flagged, but calculations remain available when the input geometry satisfies the model requirements.

When the required power-law parameters are available, the power-law formulation is recommended because it generally provided the closest overall agreement, while the bilinear formulation remains a simpler alternative when only a tangent hardening modulus is available.

Material parameters should be determined from true stress–strain data, not fitted directly from engineering stress–strain data.

Enter parameters and calculate the steady demand.

Geometry schematic

Loading schematic of the central-axis RB bending configuration showing d, R1, R2, R3, the fulcrum roller, rotating bending roller, and rotation direction.

Schematic of the central-axis bending configuration and the geometric parameters used by the estimator.

Estimated steady demand

Bilinear hardening

Steady bending force

—N

Steady shaft torque

—N·m

Model statusNot calculated

Power-law hardening

Steady bending force

—N

Steady shaft torque

—N·m

Model statusNot calculated

Model scope and implemented relations

Scope of use

This estimator applies the analytical model within the assumptions adopted in the manuscript, representing the RB as an equivalent smooth round bar. Local transverse-rib effects on tool–bar contact, surface deformation, and force or torque fluctuations are not captured.

Implemented relations

r = d/2 r_n = R1 + r β = r_n σ_y / (E_e r) δ_r = 0.5 mm A(β) = arcsin(β) − β sqrt(1 − β²)(1 − 2β²) M_bilinear = [r⁴/(2r_n)](E_e − E_H)A(β) + [π E_H r⁴/(4r_n)] + [4 σ_y r³/3](1 − E_H/E_e)(1 − β²)^(3/2) M_power = [E_e r⁴/(2r_n)]A(β) + [2 K r^(n+3)/r_n^n] × [B(n/2 + 1, 3/2) − B_{β²}(n/2 + 1, 3/2)] B(a,b) is the complete Beta function; B_x(a,b) is the unregularized incomplete Beta function. L_N = R3 sin[acos((R1 + 2r + R2)/R3)] L_r = R1 + 2r L_eff = L_N + δ_r L_r / R2 F_k = (M_k / L_eff) sqrt[1 + (δ_r/R2)^2] T_k = L_eff F_k / sqrt[1 + (δ_r/R2)^2] k ∈ {bilinear, power}

Input and output units

Quantity	Unit	Variables/output
Geometry	mm	d, R1, R2, and R3
Stress/modulus	MPa = N/mm²	E_e, σ_y, E_H, and K
Power-law exponent	–	n
Force	N	Steady bending force
Torque	N·mm or N·m	User-selectable