Cross-Spring Flexure Hinge Calculator
Estimate small-angle rotational stiffness for a symmetric orthogonal crossed-leaf flexure hinge.
Engineering screening tool—verify inputs and results independently. Use at your own risk.
Hinge geometry and material
Enter rectangular-leaf dimensions in millimetres and material modulus in gigapascals.
Stiffness result
Linear elastic moment per radian for the entered hinge.
Stress caution: maximum local stress depends strongly on cross-section and root rounding. This stiffness relation does not calculate stress or fatigue life.
How to use
- Confirm that the hinge is a symmetric crossed-leaf flexure operating at small rotation; this calculator has one mode.
- Enter effective leaf length L, bending thickness t, and out-of-plane width w in mm.
- Enter Young’s modulus E in GPa and Poisson’s ratio ν, keeping −1 < ν < 0.5.
- Enter the finite configuration ratio λ established for the selected hinge geometry.
- Select Calculate, or edit a valid input, to update kθ and Eps.
- Select Reset example to restore the aluminium demonstration values.
- Read kθ as small-angle moment per radian; the mN·m/rad metric preserves nearest-integer display rounding.
- Correct any red warning. Do not use this screen when large rotation, yielding, fatigue, buckling, root shape, attachment compliance, or safety qualification governs.
Equations used
Entered dimensions are converted from mm to m and E from GPa to Pa before calculation.
Symbols: Eps is plane-strain modulus in Pa; E is Young’s modulus in Pa; ν is Poisson’s ratio; I is rectangular-section second moment in m⁴; w, t, and L are width, thickness, and effective length in m; λ is the dimensionless configuration ratio; and kθ is torsional stiffness in N·m/rad. Positive kθ is a stiffness magnitude; the equation does not set torque or rotation direction.
Worked example
1. Set L = 10 mm, t = 0.1 mm, w = 10 mm, E = 69 GPa, ν = 0.34, and λ = 1.
2. The calculator gives Eps = 78.02 GPa and kθ = 0.084521 N·m/rad, equivalent to 85 mN·m/rad after nearest-integer display rounding.
3. Under the linear model, a 0.01-rad rotation corresponds to approximately 0.000845 N·m. Verify stress, fatigue, root radii, and geometric applicability separately.
Input definitions
- L: effective deforming leaf length, positive, in mm.
- t: rectangular bending thickness, positive, in mm; cubic stiffness sensitivity.
- w: out-of-plane leaf width, positive, in mm.
- E: positive isotropic Young’s modulus, in GPa.
- ν: signed isotropic Poisson ratio, strictly between −1 and 0.5.
- λ: finite dimensionless configuration ratio. Use a geometry-validated value; only λ² affects this relation.
Theory and method
The model treats each compliant member as a slender rectangular elastic leaf and combines its bending section property I = wt³/12 with a plane-strain elastic modulus. The factor 12λ² + 1 represents the implemented crossed-leaf configuration relation. The result is a linear rotational stiffness about the ideal pivot; multiplying kθ by a sufficiently small rotation in radians estimates elastic reaction moment.
The method assumes identical symmetric orthogonal leaves, homogeneous isotropic linear elasticity, small rotation and strain, ideal rigid attachments, and a configuration ratio already validated for the geometry. It excludes local notch/root stress, yielding, fatigue, creep, residual stress, buckling, parasitic translation, load-axis error, attachment compliance, and large-deflection geometric nonlinearity. Thickness has cubic influence, so tolerance and unit errors can dominate the estimate.
Reference: Anton van Beek, Advanced Engineering Design, crossed-spring hinge case relation. The implemented relation uses E/(1−ν²) and (12λ²+1)Epswt³/(12L).
Continue in TriboSolver
Use TriboSolver to define the actual flexure geometry and refine stress, deformation, and nonlinear behavior before design release.