Extension spring calculator to calculate fatigue resistance of ordinary extension springs with full twisted end. The points at extension spring hook where maximum stresses occurred (tensile stress at point A and shear stress at pont B) are shown in the following figure. In addition to these points, high shear stresses are occured at the body of the extension springs. This calculator can be used to check these critical points fatigue resistance against dynamic loading. Gerber and Goodman failure criterias are used in this calculator for the fatigue evaluation of the spring.

For the fatigue resistance calculations of the points where shear stresses occur (body of the spring and point B at extension spring end), Zimmerli's data [Ref 2] are used to calculate shear endurance limit value.  Zimmerli's data is based on torsion in spring so it has not been used for the points where tensile stresses occur. See the "Definitions" section for more information about the Zimmerli's Data.

Mechanical springs - Extension spring terminology Mechanical springs - Extension spring terminology
Location of Maximum Bending and Torsion Stresses in Twisted Loops

For the point where maximum tensile stresses occur (at point-A) due to bending, fatigue calculations are done by using maximum allowable tensile stress value , which is entered as an input parameter in the calculator.  Values given in the "Supplement" section can be used as a reference for maximum allowable tensile stress.  

The calculator is valid for dynamic loading case, un-peened spring steel material and ordinary extension spring with full twisted end as shown in the figure.

Mechanical springs - Extension spring terminology

Extension Spring Design with Twisted End

For the extension spring design which works under dynamic loading, first define the design parameters with the "Dimensional Design of Extension Spring". Then use "Stress Analysis of Extension Spring for Static Loading" calculator to check spring against yielding and use "Stress Analysis of Extension Spring for Fatigue Loading" calculator to check spring against fatigue.

Extension Spring Calculator for Fatigue Loading:

Parameter Value
Wire diameter [d]
Radius-1 [R1]
Radius-2 [R2]
Maximum working load [Fmax]
Minimum working load [Fmin]
Parameter Value
Material selectionx
Material tensile strength [Sut]
Allowable bending strength for the spring end for cycling loading  (% of Sut) + %
Design factor for dynamic loading [nd]o ---

Note 1 : x Material properties are from Ref-2 except "User defined" selection.

Note 2 : +  See supplements for reference values.

Note 3 : o  The design factor value that used for all of the points of interest ( Tensile stress at point-A, shear stress at point-B and shear stress at spring body).

Parameter Value
Factor of safety (According to Gerber) [fosgerber] + --- ---
Factor of safety (Acc.to Goodman) [fosgoodman]+ ---
Shear stress amplitude [τa]
Midrange shear stress [τm]
Factor of safety @ B (According to Gerber) [fosgerber]+ --- ---
Factor of safety @ B (According to Goodman) [fosgoodman]+ ---
Shear stress amplitude @ B [τa] ---
Midrange shear stress @ B [τm] ---
Factor of safety @ A (According to Gerber) [fosgerber] + --- ---
Factor of safety @ A (According to Goodman) [fosgoodman] + ---
Tensile stress amplitude @ A [σa] ---
Midrange tensile stress @ A [σm] ---  
Ultimate tensile strength of material [Sut] ---
Shearing ultimate strength [Ssu] ---
Material ASTM No. ---

Note 1 : + Green color means, fos ≥ nd, red color means fos ≤ nd


Design factor (nd):  The ratio of failure stress to allowable stress. The design factor is what the item is required to withstand .The design factor is defined for an application (generally provided in advance and often set by regulatory code or policy) and is not an actual calculation.

Dynamic Loading: A loading which varies with time with a number of load cycles over 104 and torsional stress range greater than 10 % of  fatigue strength (or endurance strength) at:

  • Constant torsional stress range
  • Variable torsional stress range

Extension spring: Extension / tension springs are coil springs which work under tensile loading. In order to carry and transfer tensile loads, extension springs require special ends in the form of hooks or loops. These special ends are generally produced by using the last coil of the spring or a separate component like screwed inserts. Generally, extension springs are connected to other component via these ends. If there is a motion to extend extension spring, it exerts force to component to move it back.

Extension springs are usually coiled with an initial tension which keeps the extension spring coils closed. Due to initial tension incorporated into spring, spring can’t be extended theoretically until a force that is greater than initial tension. In practice, extension springs extends slightly with smaller forces than initial tension due to deflection of end loops.

Tension springs are generally used to return back the component to its default position by providing return force.

Gerber fatigue criteria

Factor of Safety (Safety Factor): The ratio of failure stress to actual/expected stress. The difference between the factor of safety (safety factor) and design factor is: The factor of safety gives the safety margin of designed part against failure. The design factor gives the requirement value for the design. Safety factor shall be greater than or equal to design factor.

Gerber fatigue criteria: A fatigue failure criteria with characteristics shown in the figure.

Goodman fatigue criteria

Goodman fatigue criteria: A fatigue  failure criteria with characteristics shown in the figure.

Spring index: Ratio of spring mean diameter to coil diameter.

Spring rate: Parameter which shows relation between applied force and deflection. In other words, reaction force per unit deflection or spring resistance to length change.

Static/Quasistatic Loading: Following loading cases are defined as Static/Quasistatic loading:

  • A constant loading
  • A cycling loading with torsional shear stress range up to 10 % of fatigue strength (or endurance strength)
  • A cycling loading with torsional shear stress range more than 10 % of fatigue strength (or endurance strength) up to 10

Zimmerli's Data: Data reported in Ref-2 about the torsional endurance limits of spring steels. According to these data, spring steel material and its tensile strength has no effect on the torsional endurance limit for the wire size under 3/8 in (10mm). The endurace strength components for infinite life are reported as follows:

Shot Peended Ssa Ssm
Unpeened 35 kpsi (241 MPa) 55 kpsi (379 MPa)
Peened 57.5 kpsi (398 MPa) 77.5 kpsi (534 MPa)


Link Usage
Spring Steels for Coil Springs List of spring steel materials given in the calculator.
Formulas For Extension Spring Fatigue Design List of formulas used in the calculator.
Allowable Stresses for Extension Springs Supplemantary tables about the material strength properties of helical extension springs.


  • Budynas.R , Nisbett.K . (2014) . Shigley's Mechanical Engineering Design . 10th edition.  McGraw-Hill
  • F. P. Zimmerli, “Human Failures in Spring Applications,” The Mainspring, no. 17, Associated Spring Corporation, Bristol, Conn., August–September 1957.
  • EN 13906-1: 2002 - Cylindrical helical springs made from round wire and bar – Calculation and design – Part 1: Compression springs