Axial rigidity ($EA$)

For a uniform bar in uniaxial linear elasticity, axial rigidity is the product $EA$ of Young's modulus and load-bearing cross-sectional area. Since $\sigma =F/A$ and $\sigma =E\epsilon$,Thus $EA$ has units of force. It relates force to strain; axial stiffness $EA/L$ relates force to extension over a specified length $L$. The uniform-bar force-extension relation makes this distinction explicit.^[ref]

An original hollow-tube example keeps the area convention visible. Let the outer radius be $2\mathrm{mm}$, the inner radius $1\mathrm{mm}$ and the stipulated modulus $70\mathrm{GPa}$. The material area and rigidity areThe empty bore contributes no material area. An axial force $F=210\pi \mathrm{N}$ gives strain $\epsilon =0.001$. Tubes of lengths $50$ and $100\mathrm{mm}$ have this same rigidity, but extensions $0.050$ and $0.100\mathrm{mm}$ respectively. Their axial stiffnesses differ by a factor of two.

For a slowly varying area or modulus and constant axial force, extension instead accumulates locally:A single arithmetic average of $EA$ generally cannot replace that compliance integral. The stepped-bar calculation in axial stiffness is its piecewise-constant counterpart.

A useful experiment measures axial force and strain between defined specimen points, over the proportional regime. The slope $dF/d\epsilon$ estimates rigidity under that loading model; converting a crosshead displacement directly into strain can mix in machine compliance or grip motion. The model also assumes that lateral constraints and end effects do not replace the intended uniaxial material response.