Notched Flexure Hinge Calculator

How to use

1. Confirm that the component is a single cylindrical-hole notched flexure operating at small rotation; this calculator has one mode.
2. Enter hole diameter D, minimum ridge height h, and out-of-plane plate thickness t in mm.
3. Enter Young’s modulus E in GPa and Poisson’s ratio ν, keeping −1 < ν < 0.5.
4. Enter positive 0.2% proof strength R_p0.2 in MPa.
5. Select Calculate, or edit any valid input, to update all results.
6. Select Reset example to restore D = 10 mm, h = 1 mm, t = 10 mm, E = 69 GPa, ν = 0.34, and R_p0.2 = 275 MPa.
7. Interpret k as small-angle torque per radian and the torque/rotation limits as first-yield screening magnitudes. Correct any red warning before use.
8. Do not use this model when fatigue, plasticity, fracture, buckling, detailed notch stress, large rotation, tolerances, or safety qualification governs.

Worked example

For D = 10 mm, h = 1 mm, t = 10 mm, E = 69 GPa, ν = 0.34, and R_p0.2 = 275 MPa, the result is E_ps = 78.02 GPa, c_xx = 118.42 MN/m, c_zz/c_xx = 0.10, k = 23.06 N·m/rad, T_max = 0.443 N·m, and ψ_max = 19.2 mrad = 1.101°. The correlation therefore reaches its proof-strength torque at about 1.10°; fatigue and safety factors still require separate evaluation.

Equations used

Convert D_m = D × 10⁻³, h_m = h × 10⁻³, and t_m = t × 10⁻³ from mm to m; E_Pa = E × 10⁹; and R_p0.2,Pa = R_p0.2 × 10⁶.

E_ps = E_Pa / (1 − ν²)
c_xx = 0.48 E_ps t_m √(h_m/D_m)
c_zz/c_xx = (0.56/0.48) / (1.2 + D_m/h_m)
k = √(h_m/D_m) E_ps t_m h_m² / 10.7
W_b = t_m h_m² / 6
T_max = W_b R_p0.2,Pa / 1.034
ψ_max = T_max / k
α_max = (180/π) ψ_max

E_ps is plane-strain modulus in Pa; c_xx and c_zz are normal and shear stiffnesses in N/m; k is rotational stiffness in N·m/rad; W_b is ligament section modulus in m³; T_max is torque in N·m; ψ_max is rotation in rad; and α_max is the same angle in degrees. A radian is dimensionless in SI. Display rounding is two decimals for E_ps, c_xx, c_zz/c_xx, and k; three decimals for T_max and α_max; and one decimal for milliradians.

Theory and method

The cylindrical relief hole leaves a narrow ligament that behaves as a compliant rotational joint. The design-chart correlation combines the dimensionless notch ratio h/D with a plane-strain elastic modulus. Normal stiffness scales with t√(h/D), while rotational stiffness scales strongly with t h²√(h/D). The rectangular ligament section modulus and 0.2% proof strength provide a first-yield torque; dividing that torque by rotational stiffness gives the elastic rotation screen.

Assumptions are a symmetric cylindrical-hole notch, homogeneous isotropic linear elasticity, plane-strain correction, small displacement/rotation, and rigid attachments. The model excludes fatigue, fracture, buckling, plastic redistribution, detailed peak stress, surface condition, residual stress, manufacturing tolerances, and attachment compliance. The proof-strength estimate is not a safety-rated allowable.

Reference: Anton van Beek, Advanced Engineering Design, cylindrical-hole flexure hinge design chart (Figure 14.6).

Continue in TriboSolver to refine material, loading, stiffness, and stress assumptions with a more detailed engineering workflow.