= Axial stiffness
{title2=$K_{\rm ax}$}
<Axial stiffness> measures resistance to extension or shortening along a specified loading axis. For a proportional response, $F=K_{\rm ax}\Delta L$; 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> has
$$
K_{\rm ax}=\frac{EA}{L},
$$
where $E$ is <Young's modulus>, $A$ is the load-bearing <cross-sectional area> and $L$ the deforming length.<Structural Stiffness Concept by TU Delft OpenCourseWare>[Axial force-extension relation and structural stiffness]{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:
$$
\Delta L=F\left(\frac{L_1}{E_1A_1}+\frac{L_2}{E_2A_2}\right),\qquad
K_{\rm ax}=\left(\frac{L_1}{E_1A_1}+\frac{L_2}{E_2A_2}\right)^{-1}.
$$
Choose $E_1=E_2=70\,\mathrm{GPa}$, $L_1=20\,\mathrm{mm}$, $L_2=30\,\mathrm{mm}$, $A_1=1\,\mathrm{mm^2}$ and $A_2=2\,\mathrm{mm^2}$. The total <mechanical compliance> is $0.0005\,\mathrm{mm\,N^{-1}}$, giving $K_{\rm ax}=2000\,\mathrm{N\,mm^{-1}}$. 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>.
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