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BIZOMAT MAKEOS · Springs

Extension Spring Designer (Body, Hooks, Initial Tension, Frequency)

Will this extension spring carry its working load without overstressing the body or the hooks, is the initial tension sensible, and how high is its natural frequency?

Design check of a helical extension spring with standard loop hooks: rate, load at the working extension, body shear stress with the Wahl factor (as the extension spring tool), the initial-tension stress against your range, hook bending stress at the loop and torsional stress at the bend radius (Wahl hook formulas), and the natural frequency of the spring fixed at both ends.

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How this tool works

Inputs, units and where to find them

InputUnitTypical source
Wire diametermmDiameter of the spring wire.
Mean coil diametermmMean diameter of the coils.
Active coilsNumber of active coils (for loop ends, about the body coils).
Shear modulusGPaShear modulus of the wire.
Initial tensionNInitial tension wound into the spring.
Working extension (beyond initial tension)mmExtension from the free position at the working load.
Allowable body shear stressMPaAllowable body shear stress at the working load (your value from the spring maker).
Hook loop mean radiusmmMean radius of the hook loop (usually half the mean coil diameter).
Hook bend mean radiusmmMean radius of the bend where the hook leaves the body.
Allowable hook bending stressMPaAllowable bending stress in the hook (your value).
Allowable hook torsional stressMPaAllowable torsional stress at the hook bend (your value).
Initial-tension stress, low endMPaLow end of the initial-tension stress the maker can wind for this index.
Initial-tension stress, high endMPaHigh end of the initial-tension stress the maker can wind for this index.
Wire densitykg/m³Density of the spring wire.

What each input means

Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.

What this means: Wire diameter
Illustration: Wire diameterA circle with a line straight through its centre from edge to edge, marking the diameter Wire diameter.dmeasured acrossthe centreExample: 1.6 mmWire diameter
What it is
The diameter of the round wire the spring is wound from, measured with a micrometer across the wire.
Units and format
Enter a number in mm: millimetres. The form accepts 0.000001 mm to 50 mm. Required: the tool does not calculate until every input is valid.
Example
The form opens with 1.6 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Rate rises with the fourth power of wire diameter, so small changes matter.
Get it right
Measure the wire, not the coil; a small error matters because the wire size enters the rate to the fourth power.
What this means: Mean coil diameter
Illustration: Mean coil diameterA circle with a line straight through its centre from edge to edge, marking the diameter Mean coil diameter.Dmeasured acrossthe centreExample: 12 mmMean coil diameter
What it is
The diameter of the coils measured to the centre of the wire: outside diameter minus one wire diameter.
Units and format
Enter a number in mm: millimetres. The form accepts 0.000001 mm to 500 mm. Required: the tool does not calculate until every input is valid.
Example
The form opens with 12 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Rate falls with the cube of the mean diameter.
Get it right
Do not enter the outside diameter; subtract one wire diameter from it.
What this means: Active coils
Illustration: Active coilsA coil spring between two plates; Active coils counts the coils.NₐExample: 20Active coils
What it is
The coils that deflect under load. For springs with ordinary loop ends it is about the number of body coils, as the loops add little.
Units and format
Enter a plain number with no unit. The form accepts 1 to 500. Required: the tool does not calculate until every input is valid.
Example
The form opens with 20 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Rate is inversely proportional to the active coils.
What this means: Shear modulus
Illustration: Shear modulusA stress against strain graph whose straight starting slope is the stiffness value Shear modulus.stressstrainGExample: 79.3 GPaShear modulus
What it is
The shear modulus of the spring wire material, from the wire supplier or a spring materials standard.
Units and format
Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 100 GPa. Required: the tool does not calculate until every input is valid.
Example
The form opens with 79.3 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Rate rises in proportion to the shear modulus.
Get it right
GPa, not MPa: a value a thousand times too small makes the spring look very soft.
What this means: Initial tension
Illustration: Initial tensionA block with an arrow pushing on it; the arrow is the force Initial tension.FᵢExample: 10 NInitial tension
What it is
The preload wound into a close-coiled spring: the pull needed before the coils start to separate.
Units and format
Enter a number in N: newtons. The form accepts 0 N to 100000 N. Required: the tool does not calculate until every input is valid.
Example
The form opens with 10 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
It adds directly to the working load and to every stress.
Get it right
It is not the working load; the tool adds the rate times extension on top of it.
What this means: Working extension (beyond initial tension)
Illustration: Working extension (beyond initial tension)A bar with a dimension line and arrows from one end to the other, marking the length Working extension (beyond initial tension).xExample: 25 mmWorking extension (beyond initial tension)
What it is
How far the spring stretches from its free (unloaded) length to the working position. The tool adds rate × this extension to the initial tension.
Units and format
Enter a number in mm: millimetres. The form accepts 0.000001 mm to 2000 mm. Required: the tool does not calculate until every input is valid.
Example
The form opens with 25 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Working load and stresses rise linearly with extension.
Get it right
Use the change in length, not the overall length at the working position.
What this means: Allowable body shear stress
Illustration: Allowable body shear stressA test bar pulled from both ends: stress Allowable body shear stress is the force carried by each square millimetre of its cross-section.σstress = force ÷ areaExample: 650 MPaAllowable body shear stress
What it is
The shear stress you allow in the spring body at the working load, Wahl-corrected, from the spring maker or a spring design standard.
Units and format
Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 3000 MPa. Required: the tool does not calculate until every input is valid.
Example
The form opens with 650 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Hook loop mean radius
Illustration: Hook loop mean radiusA circle with a line from its centre to its edge, marking the radius Hook loop mean radius; the radius is half the diameter.r₁centre to edge= half the diameterExample: 6 mmHook loop mean radius
What it is
The mean radius of the hook or loop, measured to the wire centre line. Bending is highest where the loop meets the axis of the spring.
Units and format
Enter a number in mm: millimetres. The form accepts 0.000001 mm to 250 mm. Required: the tool does not calculate until every input is valid.
Example
The form opens with 6 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A tighter loop (smaller radius) raises the stress concentration and the hook bending stress.
Get it right
Measure to the centre of the wire, not the inside of the loop.
What this means: Hook bend mean radius
Illustration: Hook bend mean radiusA circle with a line from its centre to its edge, marking the radius Hook bend mean radius; the radius is half the diameter.r₂centre to edge= half the diameterExample: 2.4 mmHook bend mean radius
What it is
The mean radius of the small bend where the wire turns from the body into the hook, measured to the wire centre line. Torsion is highest here.
Units and format
Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100 mm. Required: the tool does not calculate until every input is valid.
Example
The form opens with 2.4 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A smaller bend radius raises the stress concentration and the hook torsional stress.
Get it right
Measure to the wire centre; a sharp bend from tight tooling gives a small radius.
What this means: Allowable hook bending stress
Illustration: Allowable hook bending stressA test bar pulled from both ends: stress Allowable hook bending stress is the force carried by each square millimetre of its cross-section.σstress = force ÷ areaExample: 1200 MPaAllowable hook bending stress
What it is
The bending stress you allow in the hook loop at the working load, from the spring maker or a spring design standard.
Units and format
Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 4000 MPa. Required: the tool does not calculate until every input is valid.
Example
The form opens with 1200 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Allowable hook torsional stress
Illustration: Allowable hook torsional stressA test bar pulled from both ends: stress Allowable hook torsional stress is the force carried by each square millimetre of its cross-section.σstress = force ÷ areaExample: 650 MPaAllowable hook torsional stress
What it is
The torsional (shear) stress you allow at the bend where the hook leaves the body, from the spring maker or a spring design standard.
Units and format
Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 3000 MPa. Required: the tool does not calculate until every input is valid.
Example
The form opens with 650 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Initial-tension stress, low end
Illustration: Initial-tension stress, low endA test bar pulled from both ends: stress Initial-tension stress, low end is the force carried by each square millimetre of its cross-section.σstress = force ÷ areaExample: 40 MPaInitial-tension stress, low end
What it is
The lowest stress from initial tension that the spring maker can reliably wind for this spring index. The tool compares the uncorrected initial-tension stress with this range.
Units and format
Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0 MPa to 1000 MPa. Required: the tool does not calculate until every input is valid.
Example
The form opens with 40 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Get it right
Ask the maker for the range for your index; it only triggers a warning.
What this means: Initial-tension stress, high end
Illustration: Initial-tension stress, high endA test bar pulled from both ends: stress Initial-tension stress, high end is the force carried by each square millimetre of its cross-section.σstress = force ÷ areaExample: 120 MPaInitial-tension stress, high end
What it is
The highest stress from initial tension the spring maker can reliably wind for this spring index; above it the coils cannot be wound that tight.
Units and format
Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 1000 MPa. Required: the tool does not calculate until every input is valid.
Example
The form opens with 120 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Get it right
It must be higher than the low end, or the tool stops.
What this means: Wire density
Illustration: Wire densityA cube of material: density Wire density is its mass divided by the volume it fills.massvolumeρExample: 7850 kg/m³Wire density
What it is
The mass per unit volume of the spring wire material, used only for the active mass and the natural frequency.
Units and format
Enter a number in kg/m³: kilograms per cubic metre. The form accepts 0.000001 kg/m³ to 20000 kg/m³. Required: the tool does not calculate until every input is valid.
Example
The form opens with 7850 kg/m³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Natural frequency falls with the square root of density; stresses are not affected.

Method and formulas

  1. Rate k = G d⁴ ÷ (8 D³ N_a); working load F = F_i + k x; body stress τ = K_w 8 F D ÷ (π d³) with the Wahl factor (as the extension spring tool).
  2. Hook bending σ_A = F [K₁ 16 D ÷ (π d³) + 4 ÷ (π d²)], K₁ = (4C₁² − C₁ − 1) ÷ [4C₁(C₁ − 1)], C₁ = 2r₁ ÷ d.
  3. Hook torsion τ_B = K₂ 8 F D ÷ (π d³), K₂ = (4C₂ − 1) ÷ (4C₂ − 4), C₂ = 2r₂ ÷ d.
  4. Natural frequency (both ends fixed) f = ½ √(k ÷ m_a), m_a = ρ π² d² D N_a ÷ 4.
  5. Example: the spring carries 56.993 N at 25 mm; the body runs at 509.11 MPa and the hook at 972.48 MPa bending.

Limitations

  • Static design stresses; fatigue needs the stress range and the maker’s fatigue data.
  • Hook formulas are for standard full loops; other end types differ.
  • Example values are illustrations. Replace them with values for your actual task.

Typical uses

  • Design an extension spring for a return or counterbalance
  • Find why springs break at the hook
  • Check a spring for surge at a cycling frequency

Who it is for

  • Design engineers
  • Spring buyers
  • Maintenance engineers

What you get

  • Rate, working load and body stress
  • Hook bending and torsional stresses
  • Initial-tension stress and natural frequency

Interpreting the result

  • Hooks usually fail first: a larger bend radius r₂ lowers the hook torsional stress the most.

References

  • Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 325–340. Topic reference; no source tables or recommended settings are reproduced.

Learn the Concepts Behind This Tool

These BIZOMAT Library readings explain the method this tool uses, with worked examples.

BIZOMAT tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.

Open the Extension Spring Designer (Body, Hooks, Initial Tension, Frequency) calculator

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