= Stiffness
{title2=$k$}
\b[Stiffness] describes the resistance of a specified mechanical system to <displacement> under an applied <force>. For a linear one-dimensional response measured from an unloaded reference,
$$
F=kx,
$$
the stiffness $k$ is constant and has units $\mathrm{N\,m^{-1}}$. A nonlinear <force-displacement curve> requires a more specific definition: <secant stiffness> compares a finite change from a chosen reference, while <tangent stiffness> is the local derivative $dF/dx$.
Stiffness depends on geometry as well as material. For a straight, uniform bar under axial <linear elasticity>,
$$
x=\frac{FL}{EA},\qquad k=\frac{EA}{L},
$$
where $E$ is <Young's modulus>, $A$ is cross-sectional area and $L$ is length.<Structural Stiffness Concept by TU Delft OpenCourseWare>[Axial relation delta=PL/(EA), force-deflection slope and geometrical contributions]{ref} Unlike the structural stiffness, <Young's modulus> relates <stress> to <strain> and is expressed in units of pressure.
For an original example, take $E=200\,\mathrm{GPa}$, $A=10\,\mathrm{mm^2}$ and $L=1\,\mathrm m$. Then
$$
k=\frac{(200\times10^9)(10\times10^{-6})}{1}
=2.0\times10^6\,\mathrm{N\,m^{-1}}.
$$
A $1000\,\mathrm N$ axial load gives $x=0.0005\,\mathrm m=0.5\,\mathrm{mm}$. Doubling only the length halves the stiffness and doubles this extension. The material's <Young's modulus> has remained the same.
<Series and parallel spring stiffness> depends on how the components are connected. Two identical ideal <springs> of stiffness $k$ carrying the same force in series have total extension $2F/k$, so their combined stiffness is $k/2$. In parallel, equal displacement gives total force $2kx$ and combined stiffness $2k$. These relationships assume the stated force and displacement constraints.
The reciprocal $C=1/k$ is <mechanical compliance> in the scalar linear case. For a response involving several coupled displacements and forces, a <stiffness matrix> records the additional relationships; one scalar cannot describe every loading direction.
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