The stiffness, k, of a body i s a measure of the resistance offered by an elastic body to deformation.

**Bars of Solid and hollow circular section**

**The stiffness**,** ***k*,** of a body **i s a
measure of the resistance offered by an elastic body to** **deformation. For
an elastic body with a single Degree of Freedom (for e xample, stretching or
compression of a rod), the stiffness is defined as

where

*F *is
the force applied on the body

d is the displacement
produced by the force along the same degree of freedom (for instance, the
change in length of a stretched spring)

In the International System
of Units, stiffness is typically measured in newtons per metre. In English
Units, stiffness is typically measured in pound force (lbf) per inch.

Generally speaking, deflections
(or motions) of an infinitesimal element (which is viewed as a point) in an
elastic body can occur along multipledegrees 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,^{[1]} and so take the units of
reciprocal stress, *e.g*. 1/Pa.

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