Helical Spring Design
An EngineersToolbox Calculation Module

ETB Modules | Solid Mechanics | Dynamics and Controls | Fatigue and Fracture Mechanics | Numeric Analysis | Utilities

Background Information
The Helical Spring Design module calculates spring design parameters for close-coiled round wire helical compression springs, including spring rate, maximum force, maximum displacement, and maximum shear stress. It requires the following basic input:
  • Geometry - Length, Wire Diameter, Coil Diameter
  • Material Data - Elastic Modulus, Poisson's Ratio, Density
  • End Treatment - Plain, Ground, Closed, Closed and Ground


Spring Geometry
Figure 1 illustrates the basic geometric parameters defining the helical compression spring.

Figure 1. Helical spring geometry


The primary spring geometric design parameters are:

  • Free Length (Lo) - The length of the unloaded spring.
  • Wire Diameter (d) - The diameter of the wire that is wound into a helix.
  • Coil Diameter (D) - The mean diameter of the helix, i.e., (Douter + Dinner)/2.
  • Total Coils (Nt)- The number of coils or turns in the spring.


End Treatment
The four most common types of end treatment are shown in Table 1. The following geometric parameters are derived by the module based on the specified end type:


  • Active Coils (Na) - The number of coils which actually deform when the spring is loaded, as opposed to the inactive turns at each end which are in contact with the spring seat or base.
  • Solid Length (Ls) - The minimum length of the spring, when the load is sufficiently large to close all the gaps between the coils.
  • Pitch (p) - The distance from center to center of the wire in adjacent active coils.
  • Pitch Angle (a) - The angle between the coils and the base of the spring.  The pitch angle is calculated from the equation:

Typically, either closed ends or closed and ground ends are specified due to the greater area of contact between the spring and its base.

Table 1. Effect of end treatment.

  Plain Ends Closed Ends Plain Ends
Closed Ends

Active Coils, Na


Nt-2 Nt-1 Nt-2
Free Length, Lo Nap+d Nap+3d (Na+1)p Nap+2d
Solid Length, Ls (Na+1)d (Na+1)d (Na+1)d (Na+2)d
Pitch, p (Lo-d)/Na (Lo-3d)/Na Lo/(Na+1) (Lo-2d)/Na
* The equations shown for the Closed and Ground end type assume one inactive coil at each end of the spring.  The Helical Spring Design module allows the user to specify any number of inactive coils for a spring with Closed and Ground ends.


Spring Materials
The selection of the spring material is usually the first step in parametric spring design. Material selection can be based on a number of factors, including temperature range, tensile strength, elastic modulus, fatigue life, corrosion resistance, electrical properties, cost, etc. The Helical Spring Design module requires the following material properties as input:
  • Elastic Modulus (E)
  • Poisson's Ratio (n)
  • Material Mass Density (r)

Nominal properties for materials commonly used in spring design can be accessed using the ETB Materials Database.  A short description of common spring materials is given in the following paragraphs.

High-carbon spring steels are the most commonly used of all springs materials.  They are least expensive, readily available, easily worked, and most popular. These materials are not satisfactory for high or low temperatures or for shock or impact loading.  Examples include:

  • Music Wire (ASTM A228)
  • Hard Drawn (ASTM A227)
  • High Tensile Hard Drawn (ASTM A679)
  • Oil Tempered (ASTM A229)
  • Carbon Valve (ASTM A230)

Alloy spring steels have a definite place in the field of spring materials, particularly for conditions involving high stress and for applications where shock or impact loading occurs. Alloy spring steels also can withstand higher and lower temperatures than the high-carbon steels.  Examples include:

  • Chrome Vanadium (ASTM A231)
  • Chrome Silicon (ASTM A401)

Stainless spring steels have seen increased use in recent years. Several new compositions are now available to withstand corrosion. All of these materials can be used for high temperatures up to 650°F.  Examples include:

  • AISI 302/304 - ASTM A313
  • AISI 316 - ASTM A313
  • 17-7 PH - ASTM A313(631)

Copper-base alloys are important spring materials because of their good electrical properties combined with their excellent resistance to corrosion. Although these materials are more expensive than the high-carbon and the alloy steels, they nevertheless are frequently used in electrical components and in subzero temperatures. All copper-base alloys are nonmagnetic.  Examples include:

  • Phosphor Bronze (Grade A) - ASTM B159
  • Beryllium Copper - ASTM B197
  • Monel 400 (AMS 7233)
  • Monel K500 (QQ-N-286)

Nickel-based alloys are especially useful spring materials to combat corrosion and to withstand both elevated and below-zero temperature application. Their nonmagnetic characteristic is important for such devices as gyroscopes, chronoscopes, and indicating instruments. These materials have high electrical resistance and should not be used for conductors of electrical current.  Examples include:

  • A286 Alloy
  • Inconel 600 (QQ-W-390)
  • Inconel 718
  • Inconel X-750 (AMS 5698, 5699)



Module Input
The Helical Spring Design module input form is shown in Figure 1. The module will accept either the Spring Rate (k) or Number of Active Coils (Na) as input.  If k is specified, Na will be calculated by the module; if Na is specified, k will be calculated.

Spring end types supported by the module are: plain ends, closed ends, plain ends ground, and closed ends ground.  For the Closed Ends Ground end type, the user can specify the total number of inactive coils (Nia).


Figure 1. Module input form



Module Output

The Helical Spring Design module follows the standard ETB convention for tabular output as shown in Figure 3.


Figure 3. Module tabulated results.


The module calculates the following design parameters:

  • Outer Diameter (Do) - The outer diameter of the spring coil,


  • Inner Diameter (Di) - The inner diameter of the spring coil,


  • Spring Index (C) - The ratio of mean coil diameter to wire diameter. A low index indicates a tightly wound spring (a relatively large wire size wound around a relatively small diameter mandrel giving a high rate).


  • Shear Modulus (G) - A material property calculated from the material's elastic modulus E and Poisson ratio n,


  • Slenderness Ratio (Lo / D) - The ratio of spring length to mean coil diameter.
  • Spring Rate (k) - The force required to produce a unit deflection, F/d. For close-coiled helical springs the F-d characteristic is approximately linear and can be calculated from the geometry and shear modulus of the spring:


  • Maximum Deflection (dmax) - The deflection required to go from the free length to the solid length of the spring,


  • Maximum Load (Pmax) - The maximum force the spring can take occurs when the spring is deformed all the way to its solid height,

  • Uncorrected Maximum Shear Stress (tmax) - The formula for uncorrected stress is obtained by dividing the torsion moment WD/2 acting on the wire, by the section modulus in torsion, p d3/16, giving:


  • Wahl Correction Factor (Kw) - In addition to shear stress due to torque, there are two other sources of stress: the direct shear stress and the curvature stress. The stress due to curvature comes from the spring being unable to twist as its load is applied, thus contributing to the shear stress in the wire. To account for these additional sources of stress, the uncorrected shear stress t is multiplied by the Wahl correction factor K, which is a function of the spring index. The Wahl correction factor can be calculated from the following: 


  • Corrected Maximum Shear Stress (tmax') - The corrected stress is calculated by multiplying the Wahl correction factor K by the initial uncorrected stress t:


  • Wire Length (Lw) - The length of wire needed to make the spring.


  • Spring Mass (M) - The mass of the spring.


  • Natural Frequency (fn) - The lowest natural frequency of the spring in the axial direction.



Faires, V.M. (1965) Design of Machine Elements, 4th Edition, The Macmillan Company (Toronto, Ontario).
Society of Automotive Engineers (1996) Spring Design Manual, 2nd Edition, Society of Automotive Engineers, Inc. (Warrendale, PA).
Wahl, A.M. (1963) Mechanical Springs, 2nd Edition, McGraw-Hill, Inc. (New York).
Young, W.C. (1989) Roark's Formulas for Stress and Strain, 6th Edition, McGraw-Hill, Inc. (New York).

Roark's Formulas for Stress and Strain
by Warren C. Young, Richard G. Budynas

Mechanical Engineering Design
by Joseph Edward Shigley, et al

Marks' Standard Handbook for Mechanical Engineers
by Eugene A. Avallone (Editor), Theodore, III Baumeister (Editor)

Advanced Mechanics of Materials
by Robert Davis Cook, Warren C. Young

This page was last updated on 03/25/03.

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