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Axial stiffness measures resistance to extension or shortening along a specified loading axis. For a proportional response, $F=K_{\mathrm{ax}}\Delta L$; for a nonlinear response, specify whether the reported value is a secant stiffness or tangent stiffness. A straight uniform bar in small-strain linear elasticity haswhere $E$ is Young's modulus, $A$ is the load-bearing cross-sectional area and $L$ the deforming length.^[ref] Axial rigidity $EA$ has units of force; axial stiffness $EA/L$ has units of force per length.

An original stepped-bar example shows why a single area is sometimes insufficient. Join two ideal straight segments end to end, with no distributed axial loading. Both carry the same force $F$, so their extensions add:Choose $E_1=E_2=70\mathrm{GPa}$, $L_1=20\mathrm{mm}$, $L_2=30\mathrm{mm}$, $A_1=1\mathrm{m}{\mathrm{m}}^2$ and $A_2=2\mathrm{m}{\mathrm{m}}^2$. The total mechanical compliance is $0.0005\mathrm{mm}{\mathrm{N}}^{-1}$, giving $K_{\mathrm{ax}}=2000\mathrm{N}\mathrm{m}{\mathrm{m}}^{-1}$. A $100\mathrm{N}$ force extends the bar by $0.050\mathrm{mm}$, divided into $0.0286$ and $0.0214\mathrm{mm}$ in the two segments. The thinner segment extends more despite being shorter.

In parallel, components sharing the same extension instead add their axial stiffnesses, as in a straight unbonded material-fibre bundle with every strand engaged. These are different constraints, described by series and parallel spring stiffness.

A physical test should measure extension between defined specimen points. Crosshead displacement can include machine compliance and grip motion, so the slope of the complete apparatus is not automatically the specimen's axial stiffness. Structural geometry and loading constraints must be separated from the material's Young's modulus.

Measure displacement from each arrangement's gravitational equilibrium. The series and parallel spring stiffness follows by adding forces or extensions. In the parallel arrangement the two spring forces change by $kx$ each, giving equivalent stiffness $2k$. In series the common force $F$ extends each spring by $F/k$, so total extension is $2F/k$ and the equivalent stiffness is $k/2$. HenceThe corresponding angular frequencies are $2\pi f$. gravity shifts the equilibrium extensions but does not alter the small-oscillation stiffness of these ideal linear, massless springs.

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,the stiffness $k$ is constant and has units $\mathrm{N}{\mathrm{m}}^{-1}$. 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,where $E$ is Young's modulus, $A$ is cross-sectional area and $L$ is length.^[ref] Unlike the structural stiffness, Young's modulus relates stress to strain and is expressed in units of pressure.

For an original example, take $E=200\mathrm{GPa}$, $A=10\mathrm{m}{\mathrm{m}}^2$ and $L=1\mathrm{m}$. ThenA $1000\mathrm{N}$ 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.