How to use this calculator
- Find the working load. Use the total force at the checked extension, including initial tension.
- Enter spring body geometry. Enter wire diameter d and mean coil diameter D so the stress scale and body spring index are known.
- Enter hook radii. Enter the inside hook bend radius r1 and side/transition radius r2 from the drawing or catalog geometry.
- Enter allowables. Replace the default bending and shear allowables with your material, supplier or design-basis values.
- Read the governing result. Use the governing utilization and safety factor as a static screen before fatigue, catalog and drawing approval checks.
How it works
Extension spring ends usually govern before the body coils do. This page screens two common end-loop locations: hook bending at the loop radius and torsional stress at the side or transition radius.
The bend-radius indexes are
C1 = 2*r1/d
and
C2 = 2*r2/d
where d is wire diameter. Tight bend radii drive the
correction factors up quickly:
KA = (4*C1^2 - C1 - 1) / (4*C1*(C1 - 1)) KB = (4*C2 - 1) / (4*C2 - 4)
Hook bending and side torsion are then screened with
sigma_A = KA*16*F*D/(pi*d^3) tau_B = KB*8*F*D/(pi*d^3)
using total working force F, mean coil diameter
D and wire diameter d. For context, the body
coil is also checked with the Wahl-corrected shear stress at the same
load.
If you need the working force from extension and initial tension first, use the extension spring calculator. For body-coil rate only, compare the spring rate calculator; for wire sizing and index checks, use the spring wire size calculator and spring index calculator.
Worked example
Verified against the live calculator
A spring carries 50 N at the checked extension, with
d = 2 mm, D = 16 mm, and both hook radii
r1 = r2 = 3 mm. The bend indexes are
C1 = C2 = 3.
The factors are KA = 1.333 and KB = 1.375.
Hook bending stress is
1.333*16*50*16/(pi*2^3) = 679 MPa, and hook side torsion is
1.375*8*50*16/(pi*2^3) = 350 MPa. With allowables of
850 MPa bending and 480 MPa shear, the governing
utilization is about 0.80.
Spring material data
The calculator does not embed a material-specific hook allowable table. Enter the bending and shear allowables from your material, heat treatment, spring supplier, catalog rating or approved design basis.
| Quantity | Formula | Role |
|---|---|---|
| Hook bend index | C1 = 2*r1/d | Curvature index for the hook bending location. |
| Side bend index | C2 = 2*r2/d | Curvature index for the torsion/side location. |
| Hook bending factor | KA = (4*C1^2 - C1 - 1)/(4*C1*(C1 - 1)) | Curved-wire correction for hook bending stress. |
| Hook torsion factor | KB = (4*C2 - 1)/(4*C2 - 4) | Curved-wire correction for hook side torsion. |
| Hook bending stress | sigma_A = KA*16*F*D/(pi*d^3) | Normal stress at the hook/loop radius. |
| Hook side torsion stress | tau_B = KB*8*F*D/(pi*d^3) | Torsional shear stress at the side/transition radius. |
| Body coil stress | tau_body = Kw*8*F*D/(pi*d^3) | Wahl-corrected body-coil stress at the same load. |
Source: Standard machine-design extension-spring end-loop stress relations. Use supplier data and fatigue/design review for production springs.
Frequently asked questions
How do you calculate extension spring hook stress?
For a formed hook or loop, a common first-pass screen checks bending at the hook with sigma_A = KA*16*F*D/(pi*d^3) and torsion at the side with tau_B = KB*8*F*D/(pi*d^3). This calculator derives KA and KB from the bend-radius indexes C1 = 2*r1/d and C2 = 2*r2/d.
What load should I enter?
Enter the total working force at the extension you are checking. For an extension spring, that is the initial tension plus the body-coil rate times extension: F = Fi + k*x. The extension spring calculator can solve that force first.
What are r1 and r2?
r1 is the inside radius at the hook or loop section checked in bending. r2 is the inside radius at the side or transition section checked in torsion. Use the radii from the actual spring drawing or supplier geometry.
Is this an extension spring hook stress chart?
No. It is a calculator for the common analytical screen. Manufacturer hook charts, fatigue data, catalog load ratings and end-style limits still control final selection.
Can I use one allowable stress for every spring material?
No. The defaults are placeholders for a screen. Replace the bending and shear allowables with values from the material standard, spring supplier, catalog rating or your approved design basis.
Does this work for machine hooks and crossover loops?
Only as a first-pass check when the entered radii represent the actual end geometry. Machine hooks, crossover loops, swivel loops, open gaps, end orientation and forming residual stress can shift the real peak stress.
Method & assumptions
- Static screen only; fatigue life, shot peening, set, relaxation and residual forming stress are not modelled.
- Hook geometry is simplified to one hook bend radius and one side/transition radius.
- The entered allowables are user design inputs, not universal material limits.
- Machine hooks, crossover loops, open hooks, swivel loops and proprietary end forms can have different peak stresses.
- Final spring selection still needs the spring drawing, catalog limits, supplier data, load cycle, environment, surface finish, heat treatment and professional review where safety matters.