Deformation of Bars in Tension and Compression

Under centric tension or compression, a bar changes its length. This change is described by the absolute axial deformation $\Delta l$ and the axial strain $\varepsilon$.

Absolute and relative deformation

The absolute deformation is the difference between final and initial length: $\Delta l=l-l_0$. It is positive for elongation and negative for shortening. Axial strain is the change in length divided by the initial length: $\varepsilon=\Delta l/l_0$.

Deformation of a prismatic bar

The product $EA$ is the axial rigidity. For the same $N$ and $L$, increasing $E$ or $A$ reduces deformation, while increasing $L$ increases it.

Stepped bar

If axial force, cross-sectional area, or Young's modulus changes along the bar, divide it into segments. Determine $N_i$, $L_i$, $E_i$, and $A_i$ for each segment and algebraically sum the changes in length. Tensile segments contribute positive elongation and compressed segments negative shortening under the adopted convention.

Example

A steel bar has $N=20\ \text{kN}$, $L=1000\ \text{mm}$, $A=200\ \text{mm}^2$, and $E=200000\ \text{MPa}$. Using N and mm consistently:

$$\Delta l=\frac{20000\cdot1000}{200000\cdot200}=0.5\ \text{mm}.$$

The bar therefore elongates by $0.5\ \text{mm}$ in tension.

Limits of applicability

The formula $\Delta l=NL/(EA)$ in this form assumes centric axial loading, linearly elastic behavior, and constant $N$, $E$, and $A$ within the considered segment. Variable quantities require segmentation or the corresponding integral form.