50.2: Longitudinal (Normal) Stress

In longitudinal (or normal) stress, the applied force is normal (perpendicular) to the surface.

Imagine a metal rod, for example: pulling on both ends of the rod (so as to stretch it to a longer length) is called tensile stress. If instead we push the ends of the rod together (so as to compress the rod to a shorter length), it is called compressional stress. In either case, the area $A$ in Eq. 50.1.2 is the cross-sectional area of the rod; the longitudinal stress is then the force applied to either end of the rod divided by the rod's cross-sectional area.

Strain

When applying a longitudinal stress to the rod, it changes from its original length $L_{0}$ to a new deformed length $L$. Then the longitudinal strain $\varepsilon$ is defined by

$$\varepsilon=\frac{\Delta L}{L_{0}}$$

where $\Delta L=L-L_{0}$ is the change in the length of the rod from its original length, and will be positive for tensile stress and negative for compressional stress.

Young's Modulus

In the case of a longitudinal stress, the appropriate elastic modulus is the Young's modulus $Y$ :

$$Y=\frac{F_{n} L_{0}}{A \Delta L}$$

Here $F_{n}$ is the force applied normal to the area $A, L_{0}$ is the original (unstressed) length of the rod, $L$ is the stressed length of the rod, and $\Delta L=L-L_{0}$.