Stiffness is the rigidity of an object — the extent to which it resists deformation in response to an applied force. The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
The stiffness, k, of a body is a measure of the resistance offered by an elastic body to deformation. For an elastic body with a single degree of freedom (for example, stretching or compression of a rod), the stiffness is defined as
- F is the force applied on the body
- δ is the displacement produced by the force along the same degree of freedom (for instance, the change in length of a stretched spring)
Generally speaking, deflections (or motions) of an infinitesimal element (which is viewed as a point) in an elastic body can occur along multiple degrees of freedom (maximum of six DOF at a point). For example, a point on a horizontal beam can undergo both a vertical displacement and a rotation relative to its undeformed axis. When there are M degrees of freedom a M x M matrix must be used to describe the stiffness at the point. The diagonal terms in the matrix are the direct-related stiffnesses (or simply stiffnesses) along the same degree of freedom and the off-diagonal terms are the coupling stiffnesses between two different degrees of freedom (either at the same or different points) or the same degree of freedom at two different points. In industry, the term influence coefficient is sometimes used to refer to the coupling stiffness.
It is noted that for a body with multiple DOF, the equation above generally does not apply since the applied force generates not only the deflection along its own direction (or degree of freedom), but also those along other directions.
For a body with multiple DOF, in order to calculate a particular direct-related stiffness (the diagonal terms), the corresponding DOF is left free while the remaining should be constrained. Under such a condition, the above equation can be used to obtain the direct-related stiffness for the degree of freedom which is unconstrained. The ratios between the reaction forces (or moments) and the produced deflection are the coupling stiffnesses.
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.
A body may also have a rotational stiffness, k, given by
- M is the applied moment
- θ is the rotation
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 structure. 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
- A is the cross-sectional area,
- E is the (tensile) elastic modulus (or Young's modulus),
- L is the length of the element.
Similarly, the rotational stiffness is
- "J" is the second moment of area about the rotation axis,
- "n" is an integer depending on the boundary condition (=4 for fixed ends)
- "G" is the rigidity modulus of the material
Note that in SI, these units yield . 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 deflection is undesirable, while a low modulus of elasticity is required when flexibility is needed.
- Elasticity (physics)
- Elastic modulus
- Mechanical impedance
- Hooke's law
- Moment of inertia
- Stiffness (mathematics)
- Young's modulus
- Compliant mechanism
- Baumgart F. (2000). "Stiffness--an unknown world of mechanical science?". Injury (Elsevier) 31. doi:10.1016/S0020-1383(00)80040-6. Retrieved 2012-05-04. "“Stiffness” = “Load” divided by “Deformation”"
- Martin Wenham (2001), "Stiffness and flexibility", 200 science investigations for young students, p. 126, ISBN 978-0-7619-6349-3
- V. GOPALAKRISHNAN and CHARLES F. ZUKOSKI; "Delayed flow in thermo-reversible colloidal gels"; Journal of Rheology; Society of Rheology, U.S.A.; July/August 2007; 51 (4): pp. 623–644.