
Stiffness is the extent to which an object resists
deformation in response to an applied
force
In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
.
The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
Calculations
The stiffness,
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 (DOF) (for example, stretching or compression of a rod), the stiffness is defined as
where,
*
is the force on the body
*
is the
displacement produced by the force along the same degree of freedom (for instance, the change in length of a stretched spring)
Stiffness is usually defined under
quasi-static conditions, but sometimes under dynamic loading.
In the
International System of Units
The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official s ...
, stiffness is typically measured in
newtons per meter (
). In Imperial units, stiffness is typically measured in
pounds (lbs) per inch.
Generally speaking,
deflections (or motions) of an infinitesimal element (which is viewed as a point) in an elastic body can occur along multiple DOF (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
degrees of freedom a
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 direction (or degree of freedom) but also those along with other directions.
For a body with multiple DOF, 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 obtain the direct-related stiffness for the degree of unconstrained freedom. The ratios between the reaction forces (or moments) and the produced deflection are the coupling stiffnesses.
The
elasticity tensor is a generalization that describes all possible stretch and shear parameters.
A single spring may intentionally be designed to have variable (non-linear) stiffness throughout its displacement.
Compliance
The
inverse of stiffness is or , 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, for example, 1/
Pa.
Rotational stiffness

A body may also have a rotational stiffness,
given by
where
*
is the applied
moment
*
is the rotation angle
In the SI system, rotational stiffness is typically measured in
newton-metres per
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
.
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 - the ratio of applied
shear force to shear deformation
* torsional stiffness - the ratio of applied
torsion moment to the angle of twist
Relationship to elasticity
The
elastic modulus of a material is not the same as the stiffness of a component made from that material. Elastic modulus is a property of the constituent material; stiffness is a property of a structure or component of a structure, and hence it is dependent upon various physical dimensions that describe that component. That is, the modulus is an
intensive property of the material; stiffness, on the other hand, is an
extensive property of the solid body that is dependent on the material its shape and boundary conditions. For example, for an element in
tension or
compression, the axial stiffness is
where
*
is the (tensile) elastic modulus (or
Young's modulus),
*
is the
cross-sectional area,
*
is the
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...
of the element.
Similarly, the torsional stiffness of a straight section is
where
*
is the
rigidity modulus of the material,
*
is the
torsion constant for the section.
Note that the torsional stiffness has dimensions
orce*
ength/
ngle so that its SI units are N*m/rad.
For the special case of unconstrained uniaxial tension or compression,
Young's modulus be thought of as a measure of the stiffness of a structure.
Applications
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.
In biology, the stiffness of the
extracellular matrix is important for guiding the migration of cells in a phenomenon called
durotaxis.
Another application of stiffness finds itself in
skin biology. The skin maintains its structure due to its intrinsic tension, contributed to by
collagen, an extracellular protein that accounts for approximately 75% of its dry weight. The pliability of skin is a parameter of interest that represents its firmness and extensibility, encompassing characteristics such as elasticity, stiffness, and adherence. These factors are of functional significance to patients. This is of significance to patients with traumatic injuries to the skin, whereby the pliability can be reduced due to the formation and replacement of healthy skin tissue by a pathological
scar. This can be evaluated both subjectively, or objectively using a device such as the Cutometer. The Cutometer applies a vacuum to the skin and measures the extent to which it can be vertically distended. These measurements are able to distinguish between healthy skin, normal scarring, and pathological scarring,
and the method has been applied within clinical and industrial settings to monitor both pathophysiological sequelae, and the effects of treatments on skin.
See also
References
{{Authority control
Physical quantities
Continuum mechanics
Structural analysis