Springs are elastic members that exert forces, or torques, and absorb energy, which is usually stored and later released. The force produced by a spring can be compressive or tensile and linear or radial.

Springs are usually, but not necessarily, made of metal. Plastics can be used when loads are light. Modern structural composites are being introduced for some applications requiring minimum spring mass. Blocks of rubber often constitute springs, as in bumpers and in vibration isolation mountings of various machines such as electric motors and internal combustion engines. Pneumatic springs of various types take advantage of the elastic compressibility of gases, as compressed air in automotive "air shocks" and as hermetic ally sealed high-pressure nitrogen gas in the hydro-pneumatic sus­pensions of French Citroen automobiles. For applications requiring compact springs providing very large forces with small deflections, hydraulic springs have proved effective. These work on the basis of the slight compressibility of liquids, as indicated by their bulk modulus of elasticity. Product cost can sometimes be reduced by designing the required elasticity into other parts, rather than making those parts rigid and adding a separate spring.

Springs can be classified by the direction and nature of the force exerted when they are deflected. Several types of spring are listed in Table 1 according to the nature of force or torque exerted. The principal characteristics of various classes of springs are summarized in Table 2.

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image001.jpg



Table 1 Classification of springs according to the nature of force or torque exerted


Type of spring


Helical compression springs
Belleville springs
Flat springs, e.g. cantilever or leaf springs


Helical extension springs
Flat springs, e.g. cantilever of leaf springs
Drawbar springs
Constant force springs


Garter springs
Elastomeric bands
Spring clamps


Torsion springs
Power springs


Table 2 Principle characteristics of a variety of types of spring

Type of spring

Principle characteristics

Helical compression springs

These are usually made from round wire wrapped into a straight cylindrical form with a constant pitch between adjacent coils

Helical extension springs

These are usually made from round wire wrapped into a straight cylindrical form but with the coils closely spaced in the no -load condition. As an axial load is applied the spring will extend but resisting the motion

Drawbar springs

A helical spring is incorporate into an assembly with two loops of wire. As a load is applied the spring is compressed in the assembly resisting the motion

Torsion springs

These exert a torque as the spring id deflected by rotation about their axis. A common example of the application of a torsion spring is the clothes peg

Leaf springs

Leaf springs are made from flat strips of material and loaded as cantilever beams. They can produce a tensile or compressive force depending on the mode of loading applied

Belleville springs

These comprise shallow conical discs with a central hole

Garter spring

These consist of coiled wire formed into a continuous ring so that they can exert radial inward force when stretched

Volute springs

These consist of a spiral wound strip, that functions in compression. They are subject to significant friction and hysteresis

Compression Spring:

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image002.jpg

Extension Spring

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image003.jpg

Torsion Spring

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image004.jpg


Description: C:\My Site\subjects\machine_elements_design\Springs_files\image005.jpg

Spring design involves the relationship between force, torque, deflection, and stress. Springs have many uses in connection with machine design, such as to cushion impact and shock loading, to store energy, to maintain contact between machine members, for force measuring devices, to control vibration, and other related functions.

Helical springs

Figure 1(a) shows a helical coil spring with ends adapted to support a compressive load. At (b) the spring has been deflected by the axial load P, which may be assumed to compress the spring as between two parallel plates. The notation applies:


          P       = axial load, N

          D       = mean diameter of coil, mm

                    = (Do + Di) / 2

          d        = diameter of wire, mm

                    = (Do – Di) / 2

          p       = pitch of coils, mm

          δ        = deflection of spring, mm

          n        = number of active coils

          C       = spring index = D/d = (Do-d)/d 

= (4 : 12); less that 4 it is difficult to manufacture, more than 12 is likely to buckle.

          G       = torsional modulus of     

                     elasticity, N/mm2

          τs            = shearing stress, N/mm2

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image006.jpg

  The number of active coils, n, represents the coils in the spring with the exception of those which lie flat against the compression plats. The inactive coils do not contribute to the deflection of the spring.


Description: C:\My Site\subjects\machine_elements_design\Springs_files\image007.jpg


Springs (DIN ISO 2167)

Representation of a cylindrical helical compression spring made out of round wire.

A- in view

B- in section and

C- as name plate drawing

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image008.jpg


Description: C:\My Site\subjects\machine_elements_design\Springs_files\image009.jpgStress in a helical spring made of round wire


   Figure 2 shows a part of a compression spring that supports a compressive spring that supports a compressive load P and a section of the wire cut by axial plane. The part of spring shown in the figure is in equilibrium under the action of the two forces P and resisting torsional moment T. The torque equals

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image010.gif

The shearing stress due to the torque T is

          Description: C:\My Site\subjects\machine_elements_design\Springs_files\image011.gif

and the direct shearing stress is

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image012.gif


Description: C:\My Site\subjects\machine_elements_design\Springs_files\image013.gif and Description: C:\My Site\subjects\machine_elements_design\Springs_files\image014.gif


for solid circle cross section


   The maximum shearing stress, which is located at the inner side of the curved wire, may determined by

          Description: C:\My Site\subjects\machine_elements_design\Springs_files\image015.gif


          Description: C:\My Site\subjects\machine_elements_design\Springs_files\image016.gif is the direct shear factor


 It is evident form the above equation that for springs of small index C the effect of direct shear (1/2C) is appreciable.

   The effect of curvature of the wire as it forms the coil should be considered also.

   In order to include the effects of both direct shear and wire curvature, a stress factor had been determined by the use of approximate analytical methods developed by A. M. Wahl which may be used in the above equation to determine the maximum shearing stress in the wire as follows:


          Description: C:\My Site\subjects\machine_elements_design\Springs_files\image017.gif


Description: C:\My Site\subjects\machine_elements_design\Springs_files\image018.gif



Description: C:\My Site\subjects\machine_elements_design\Springs_files\image019.gif


Deflection of helical spring


   An equation for the axial deflection of a helical spring in terms of the axial load, spring dimensions, and a materials constant may be conveniently determined by equating the work re3fquired to deflect the spring to the strain energy in the twisted wire. For close-coiled springs the bending of the wire is small and the strain energy of bending may be neglected.

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image020.gif   The axial load as shown in Fig. 2 increases linearly from zero to P, and hence the work required to compress the spring is the average force P/2 times the deflection, or /2.


   The strain energy in a bar twisted by a torsional moment T through a total angle θ is /2. The total angle of twist θ = TL/GJ, where L equals the length of the twisted wire and J equals the polar moment of inertia of the wire section. The active length of the wire in the helical spring equals (πDn/cos α), where α is the lead angle of the helix, which for close-coiled spring is of the order of 5o; hence cos α is unity.

   For springs of indexes as used in machinery, namely, 2.5 or 3 and over, the strain energy in the curved wire forming the coils will be closed to the strain energy in a straight wire; hence from the above relations the following equation for δ may be obtain

          Description: C:\My Site\subjects\machine_elements_design\Springs_files\image021.gif


          T = P D/2

          L = π D n

          J = π d4/32

          Description: C:\My Site\subjects\machine_elements_design\Springs_files\image022.gif

From the above equation

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image023.gif

where (P/δ) is known as the spring rate. The spring rate represents the slope of the load-deflection line as shown in Fig. 3. Any difference between the ideal line and the actual curve is due principally to the change in number of active coils as the spring is deflected, to mechanical hysteresis of the material, and the friction between the end coils in contact with each other or with the loading plate.

   Compression springs in which the free length is more than four times the mean diameter of the coils may fail by sidewise buckling.   

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image024.jpg

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image025.jpg

The spring rate for springs in parallel:

For springs having rates as shown in figure (a)

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image026.gif


The spring rate for springs in series:

For springs having rates as shown in figure (b)

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image027.gif







Nomogram for helical spring calculation:  (G = 81,400 MPa)

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image028.jpg

Explanation of the nomogram:

The Nomogram applies to cylindrical helical extension and compression springs made of round steel wire (shear modulus G=81,400 N/mm2). In the case of materials with a different shear modulus G spring deflection must be multiplied by G/G.

The nomogram indicates the deflection s of one coil. Total deflection s is obtained by multiplying (s) by the number of active coils n: s = n . s.


Spring ends:

   Helical spring ends may be either plain, plain ground, squared, or squared and ground as shown in Fig. 4 below. This results in a decrease of the number of active coils and affects the free length and solid length of the spring as shown below.

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image029.jpg

p=(D/3 : D/4), n= ( 3: 15)



Buckling may occur in compression springs if the free length is over 4 times the mean diameter unless the spring is properly guided. The critical axial load that will cause buckling may be approximated by

          Fcr = k Lf KL


          Fcr = axial load to produce buckling, N

          k = spring rate, N/m, of axial deflection

          Lf = free length of the spring, m

          kL = a factor depending on the ratio Lf/D (slenderness factor).


Description: C:\My Site\subjects\machine_elements_design\Springs_files\image030.jpg


or Lf/D < 3 for no buckling

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image031.jpgBuckling depends on

Lf/D and δ/Lf











Buckling can be avoided by providing a spring guide

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image032.gif









Allowable stresses:

For helical chrome-vanadium steel springs, oil-tempered, hot wound and heat-treated after forming, are tabulated below.

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image033.jpg


   Sever service includes rapid continuous loading where the ratio of minimum to maximum stress is one half or less. Average service is the same as severe except for intermittent operation. Light service includes springs subjected to static loads or to infrequently varied loads.


Spring design approach

·        Specific material

·        Guess a trail diameter (space available)

·        Check spring rate and free length


-         spring rate: k = (Fo-Fi) / (Li-Lo)

-         spring free length Lf = Li + (Fi/k)


          Fo is the operating force, and Lo is the operating length

          Fi is the installed force, and Li is the installed length 


Leaf Springs



Description: C:\My Site\subjects\machine_elements_design\Springs_files\image034.jpgActs as:

-         an elastic element

-         guide

-         damper

+ they are easy to manufacturer and convincement to repair



-         large metal content (energy stored in unit volume in spring or torsion bar is 4 times more than a leaf spring; that lead to increase metal content).

-         sufficient unsprung mass

-         short service life.


If it is desired to maintain uniform bending stresses over the length of the beam then the width of the cantilever needs to vary linearly with the location as illustrated in Figure i, (σb = M y/I = 12 F L y / b h3). The concept used in producing compact cantilever springs of uniform bending stress is to chop the triangular form illustrated in Figure i, into a number of strips and recombine them as illustrated in Figure ii. The multi-leaf spring shown and the single triangular section beam both have the same stress and deflection characteristics with the exceptions that the multi-leaf spring is subject to additional damping due to friction between the leaves and that the multi-leaf spring can carry a full load in only one direction due to the tendency for the leaves to separate. Leaf separation can be partially overcome by the provision of clips around the leaves.

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image035.jpg

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image036.jpg


The deflection of a triangular leaf spring is given by

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image037.gif


          F is force (N)

          L is the length (m)

          E is the Young’s modulus (N/m2)

          I is the second moment of area (m4)

For a rectangular cross-section,

          Description: C:\My Site\subjects\machine_elements_design\Springs_files\image038.gif

where b is width (m); and h is thickness (m).


The spring rate is given by

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image039.jpg          Description: C:\My Site\subjects\machine_elements_design\Springs_files\image040.gif

The corresponding bending stress (for cantilever, semi-elliptic and full-elliptic) is given by

          Description: C:\My Site\subjects\machine_elements_design\Springs_files\image041.gif

For semi-elliptic beam the maximum deflection at the center is given by

          Description: C:\My Site\subjects\machine_elements_design\Springs_files\image042.gif

For a full-elliptic beam the maximum deflection at the center is given by

          Description: C:\My Site\subjects\machine_elements_design\Springs_files\image043.gif


* in the above equation the length b can be changed by nb where n is the number of leafs and b is the width of single leaf. The leaf width b is chosen from the available range of rolled products. It is desired that the following condition should hold

          6 < b/h < 10

          6 <   n   < 14

The spring length can be determined using the equations for spring stiffness and stress to obtain

          Description: C:\My Site\subjects\machine_elements_design\Springs_files\image044.gif


          δ = is the total deflection (δst + δd)


- Effect of adding an extra full length number of leaves (ne) to the graduate length number of leaves (ng):

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image045.gif

That means the extra leaves will have more stress than graduate ones.

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image046.gif


Torsion beam suspension (torsion bars)


Definition: A steel bar that is twisted to support the weight of the vehicle. Torsion bars are used in place of coil or leaf springs on some vehicles, and allow ride height to be adjusted to compensate for sage that occurs over time.

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image047.jpg

Description: C:\My Site\subjects\machine_elements_design\Springs_files\image048.jpg

The bars are usually solid of circular cross section although hollow tubes and rectangular bars are used.

          Description: C:\My Site\subjects\machine_elements_design\Springs_files\image049.gif

τall = 1000- 1050 MPa

The angle of twist and stiffness of a torsion bar are expressed as

          Description: C:\My Site\subjects\machine_elements_design\Springs_files\image050.gif

The torsion-bar working length L (without spline ends) is determined by the θ equation. It is recommended to choose the diameters and lengths of spline ends depending on the torsion-bar diameter

          dsp = (1.2 -1.3)d; and Lsp= (0.6-1.2)d




Suspension Formulae


          Description: C:\My Site\subjects\machine_elements_design\Springs_files\image051.gif


where b is the width of spring blade (m), L is the distance between the eyes of the spring when laden (m), t is the thickness of the blade (m), n is the number of blades, and E is the modulus of elasticity, which (modified to allow for internal friction) is 159 x 106 kN/m2.

    For a torsion bar, the spring rate is given as the twisting moment per angular deflection. When a lever is added, this can be converted into a rate for the vertical deflection of the end of the lever.


    Spring rate (torsion bar, for deflection at end of lever)


                   Description: C:\My Site\subjects\machine_elements_design\Springs_files\image052.gif


where G is the modulus of rigidity, which is 78.5 x 106 kN/m2 in this case, d is the diameter of the torsion bar (m), l is the effective length of the torsion bar (m), i.e. half the length of the bar for an anti-roll bar, and e is the length of the lever (m).



 Spring rate (coil spring)


                   Description: C:\My Site\subjects\machine_elements_design\Springs_files\image053.gif


where G is the modulus of rigidity, which is 81.5 x 106 kN/m2 in this case, d is the wire diameter (m), n is the number of free coils, and D is the mean coil diameter (m). To find the number of free coils it is necessary to subtract the number of dead coils form the total number of coils. The dead coils are those that provide the abutment and so cannot be deflected, usually 1.5 to 2 coils.






Helical Compression

Round and Rectangular Wire

Description: Helical Compression

Push - wide load and deflection range - constant rate.

Push - wide load and deflection range.

Conical springs can be made with minimum solid height and with constant or increasing rate.

Barrel, hourglass, and variable pitch springs used to minimize resonant surging and vibration.

Spring Washer

Column 1:


Column 2:


Description: spring washerDescription: spring washer

1) Push - high loads, low deflections - choice of rates (constant, increasing or decreasing)

2) Push - Light Loads, low deflection-uses limited radial space.

3) Push - Higher deflections than bellevilles.

4) Push - for axial loading of bearings.

5) Push - uesd to absorb axial end play.


Description: volute

Push - may have inherently high friction damping.




1) Cantilever, Retangular Section

2) Cantilever, Trapezoidal Section

3) Simple Beam

1Description: cantilever section beam
2Description: trapezoid beam
3Description: simple beam

Push or pull - wide range of loads, low dflection range.



Helical Torsion

Round or Rectangular Wire

Description: Helical Torsion

Twist - constant rate.




1) Hairspring

2) Brush

1Description: Hairspring Spiral
2Description: brush sprial


Twist or Push



Constant Force Spring Motor

Level Torque

Description: Constant Force Spring Motor

Twist - exerts close-to-constant torque over many turns.



Helical Extension

Description: Helical Extension

Pull - wide load and deflection range - constant rate.




Description: Drawbar

Pull - extension to a solid stop.



Constant Force

Description: Constant Force

Pull - very long deflection at constant load or low rate.



Retaining Rings

Round or Rectangular Wire

Description: Reatining Rings

Pull or push - to resist axial loads.



Garter Springs

1) Extension

2) Compression

1Description: Garter Springs
2Description: Garter Springs

1) Pull with radial pressure.

2) Push with radial pressure.