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# Stiffness

[stif]
Stiffness is the resistance of an elastic body to deformation by an applied force. It is an extensive material property.

## Definition

The stiffness, k, of a body is a measure of the resistance offered by an elastic body to deformation (bending, stretching or compression).

$k=frac \left\{P\right\} \left\{delta\right\}$

where

P is a steady force applied on the body
δ is the displacement produced by the force (for instance, the deflection of a beam, or the change in length of a stretched spring)

In the International System of Units, stiffness is typically measured in newtons per metre.

As both the applied force and displacement are vectors (respectively P and δ), in general their relationship is characterised by a stiffness matrix, k, where:

$P=k delta ,$

The displacement can, in general, refer to a point distinct from that where the force is applied and a complicated structure will not deflect purely in the same direction as an applied force. The stiffness matrix enables such systems to be characterised in straightforward terms.

The inverse of stiffness is compliance, typically measured in units of metres per newton. In rheology it may be defined as the ratio of strain to stress , and so take the units of reciprocal stress, e.g. 1/Pa.

## Rotational stiffness

A body may also have a rotational stiffness, k, given by

$k=frac \left\{M\right\} \left\{theta\right\}$

where

M is the applied moment
θ is the rotation

In the SI system, rotational stiffness is typically measured in newton-metres per radian.

In the SAE system, rotational stiffness is typically measured in inch-pounds per degree.

Further measures of stiffness are derived on a similar basis, including:

• shear stiffness - ratio of applied shear force to shear deformation
• torsional stiffness - ratio of applied torsion moment to angle of twist

## Relationship to elasticity

In general, elastic modulus is not the same as stiffness. Elastic modulus is a property of the constituent material; stiffness is a property of a solid body. That is, the modulus is an intensive property of the material; stiffness, on the other hand, is an extensive property of the solid body dependent on the material and the shape and boundary conditions. For example, for an element in tension or compression, the axial stiffness is

$k=frac \left\{AE\right\} \left\{L\right\}$

where

A is the cross-sectional area,
E is the (tensile) elastic modulus (or Young's modulus),
L is the length of the element.

For the special case of unconstrained uniaxial tension or compression, Young's modulus can be thought of as a measure of the stiffness of a material.

## Use in engineering

The stiffness of a structure is of principal importance in many engineering applications, so the modulus of elasticity is often one of the primary properties considered when selecting a material. A high modulus of elasticity is sought when deflections are undesirable, while a low modulus of elasticity is required when flexibility is needed.