Roymech engineering encyclopedia

Leaf Spring Design


Leaf springs are beam-type springs used to support loads, store strain energy and provide controlled deflection. They are widely associated with vehicle suspension systems, particularly road and railway vehicles, but the same engineering principles apply to many mechanical spring applications.

This page provides engineering reference information for single and laminated leaf springs, including bending stress, deflection, spring rate and the principles behind constant-strength multi-leaf spring design.

Not sure whether a leaf spring is the right spring type?
See the Spring Selection and Design Guide for guidance on selecting between leaf, compression, extension, torsion, disc and other spring types.

Contents

Leaf Spring Principles

A leaf spring stores energy by elastic bending. The simplest form is a single beam or plate spring. Higher loads can be accommodated using laminated multi-leaf arrangements in which several leaves act together.

Traditional laminated leaf springs are particularly useful in suspension systems because the spring can perform more than one mechanical function. Depending on the arrangement, the spring may support the vehicle load, provide suspension compliance and contribute to locating or controlling the axle.

Leaf Spring Characteristics

Single Leaf Springs

A single leaf spring is essentially a beam designed to operate elastically under repeated bending. Typical characteristics include:

  • Suitable for low to moderate loads, depending on material and geometry.
  • Relatively simple construction.
  • Approximately linear force-deflection behaviour within the elastic range for simple constant-section geometries.
  • Spring rate determined principally by material modulus, section geometry and effective length.
  • Can provide a relatively long working length with a small cross-sectional depth.
  • Generally straightforward to manufacture.

Laminated Leaf Springs

A laminated leaf spring consists of two or more leaves arranged so that they share the applied load. Typical characteristics include:

  • Suitable for relatively high loads.
  • Load capacity and stiffness can be varied by changing the number, length and section of the leaves.
  • Interleaf friction can introduce hysteresis and damping into the force-deflection characteristic.
  • The leaves can be proportioned to obtain a more nearly uniform bending-stress distribution.
  • Traditional multi-leaf designs may require attention to interleaf friction, corrosion, wear and lubrication.
  • Modern designs may use tapered or parabolic leaves to reduce mass and interleaf contact.
Interleaf friction: An ideal beam calculation does not include friction between leaves. In an actual laminated spring, friction can alter the measured spring characteristic and produce hysteresis between loading and unloading.

Leaf Spring Nomenclature

The following notation is used in the equations on this page. SI units are shown, although millimetres and N/mm2 are often more convenient for practical spring calculations provided units are kept consistent.

Symbol Description SI Unit
E Young's modulus Pa (N/m2)
F Applied force N
t Leaf thickness m
b Leaf width m
L Effective spring length m
x Distance from the load point or section reference distance as defined in the relevant diagram m
n Number of effective leaves dimensionless
δ Spring deflection m
σ Bending stress Pa (N/m2)
k Spring rate (stiffness), F/δ N/m

Single Leaf Springs

Two basic beam arrangements are useful when considering the behaviour of single leaf springs: the cantilever spring and the simply supported spring.

Cantilever and simply supported single leaf spring arrangements

Cantilever Leaf Spring

A cantilever leaf spring is fixed at one end and loaded at or towards the free end. The maximum bending moment normally occurs at the fixed end.

Cantilever Leaf Spring

A cantilever leaf spring is fixed at one end and loaded at or towards the free end. The maximum bending moment normally occurs at the fixed end.

Stress (max) σ max = 6 F L b t 2
Stiffness k = F δ = 3 E I L 3 = E b t 3 4 L 3

Simply Supported Leaf Spring

A simply supported leaf spring is supported at both ends and subjected to a transverse load. For the conventional centrally loaded arrangement, maximum bending moment and maximum deflection occur at the centre of the span.

Simply Supported Leaf Spring

A simply supported leaf spring is supported at both ends and subjected to a transverse load. For the conventional centrally loaded arrangement, maximum bending moment and maximum deflection occur at the centre of the span.

Stress (max) σ max = 3 F L 2 b t 2
Stiffness k = F δ = 48 E I L 3 = 4 E b t 3 L 3

Multiple Leaf Springs

For a cantilever-type leaf spring, bending moment and therefore bending stress vary with distance from the load point. For a rectangular section, the bending stress at a section may be written:

σ = 6 F x b t 2

For constant thickness t, a constant bending stress can theoretically be obtained by varying the width b so that the ratio x/b remains constant.

This produces the idealised triangular constant-strength spring. A traditional laminated leaf spring approximates this condition by dividing the triangular form into a number of individual leaves of different lengths.

Development of a laminated multi-leaf spring from an ideal triangular constant-strength leaf spring

For the idealised laminated spring, the maximum bending stress and spring stiffness are given by the following relationships.

For the idealised laminated spring, the maximum bending stress and spring stiffness are given by the following relationships.

Stress (max) σ max = 6 F L n b t 2
Stiffness k = F δ = E n b t 3 6 L 3

Quarter-Elliptic Leaf Spring

A quarter-elliptic spring operates essentially as a cantilever, with one end fixed and the load applied towards the free end. The preceding laminated cantilever relationships apply to the idealised arrangement shown below.

Quarter-elliptic laminated leaf spring arrangement

Semi-Elliptic Leaf Spring

A semi-elliptic leaf spring is supported towards both ends and is commonly loaded near its centre. This arrangement has been extensively used for vehicle suspension systems.

The relevant stress, deflection and stiffness relationships for the idealised semi-elliptic spring are:

Semi-Elliptic Leaf Spring

A semi-elliptic leaf spring is supported towards both ends and is commonly loaded near its centre. This arrangement has been extensively used for vehicle suspension systems.

The relevant stress, deflection and stiffness relationships for the idealised semi-elliptic spring are:

Stress (max) σ max = 3 F L 2 n b t 2
Stiffness k = F δ = 8 E n b t 3 3 L 3

Semi-elliptic laminated leaf spring with central load F and effective span L

Leaf Spring Design Considerations

The elementary equations on this page provide useful relationships between load, stress, deflection and stiffness, but practical leaf-spring design requires additional considerations.

  • Fatigue: suspension springs are normally subjected to large numbers of fluctuating load cycles, so fatigue stress and surface condition can govern the design.
  • Stress concentration: eyes, clamps, holes, changes of section and other geometric details can produce local stresses not represented by simple beam equations.
  • Interleaf friction: friction in a laminated spring produces hysteresis and affects the actual force-deflection characteristic.
  • Pre-stressing: traditional multi-leaf springs may use pre-stressing or nipping so that the leaves share working stresses more effectively after assembly.
  • Material and heat treatment: allowable stress depends on material grade, heat treatment, surface treatment and manufacturing process.
  • Corrosion and wear: surface damage is particularly important where the spring operates under cyclic bending stress.
  • Large deflection: simple beam equations become progressively less representative where geometry changes substantially during spring movement.

For cyclic applications also see Fatigue in Springs and Spring Materials.

Standards and Design References

Leaf springs used in vehicle suspension are manufactured and specified for particular applications. Relevant standards depend on the vehicle, spring construction, units and procurement requirements.

  • SAE J1123_201604 – Leaf Springs for Motor Vehicle Suspension – Made to Metric Units. SAE lists this standard with a status of Stabilized.
  • SAE J510_201604 – Leaf Springs for Motor Vehicle Suspension – Made to Customary U.S. Units. SAE likewise lists this standard with a status of Stabilized.

These SAE standards principally address practical requirements for finished motor-vehicle suspension leaf springs rather than replacing the elementary beam relationships presented on this page. SAE identifies its leaf-spring design manual HS-J788 as the source in which the basic design concepts were addressed.

Engineering reference: The equations on this page are useful for understanding and preliminary calculation of idealised leaf springs. Final suspension-spring design should also consider fatigue, material condition, heat treatment, surface treatment, manufacturing tolerances, interleaf effects, attachment details and the applicable vehicle or equipment specification.
See Also