bar-extension-uses-e-a-over-l - Elasticity and Nonlinear Dynamics Step by Step

A bar of length L and area A stretches by delta-L under a force F, tied together by F = E * A * delta-L / L in the linear elastic regime. The arithmetic is exact only for the stated Young modulus, area, length, and extension in the ideal linear elastic model. Real materials add nonlinear response, geometry tolerance, temperature/loading history, fixture compliance, calibration, and measurement uncertainty.

A uniform bar derives its stiffness from Young modulus, area, and length. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A bar force starts from an accepted Young source

The bar does not choose stiffness directly. It cites Young modulus, then combines area, length, and extension.

E=16\ \text{Pa}\qquad \Delta L=1\ \text{m}

Stiffness and force respond to area and extension

The table separates stiffness from force. Area changes stiffness; extension changes force after stiffness is set.

\begin{array}{c|c|c|c|c|c}E&A&L&\Delta L&k&F\\16\ \text{Pa}&2\ \text{m}^{2}&8\ \text{m}&1\ \text{m}&4\ \text{N/m}&4\ \text{N}\\16\ \text{Pa}&2\ \text{m}^{2}&8\ \text{m}&2\ \text{m}&4\ \text{N/m}&8\ \text{N}\\16\ \text{Pa}&4\ \text{m}^{2}&8\ \text{m}&1\ \text{m}&8\ \text{N/m}&8\ \text{N}\\\end{array}

A uniform bar uses E A over L before force is trusted

The bar stiffness is derived from the cited Young modulus source, then multiplied by the checked extension.

k=\frac{EA}{L}=\frac{16\ \text{Pa}\cdot2\ \text{m}^{2}}{8\ \text{m}}=4\ \text{N/m};\quad F=k\Delta L=4\ \text{N}