Short Answer:

Elongation of a bar under an axial load is the increase in its length when a force acts along its axis. For a straight prismatic bar of original length , cross-sectional area , and Young’s modulus , carrying an axial tensile load  that is uniform along the bar, the elongation  is

This formula comes from Hooke’s law (stress ∝ strain) and simple geometry of strain = change in length / original length.

For non-uniform situations (varying area, varying load, or composite bars), we use the same idea on small elements and integrate: .

Detailed Explanation :

Elongation of a Bar under Axial Load

Goal and assumptions
We want the increase in length (elongation) of a straight bar when an axial force is applied. We assume: the bar is straight, the load is axial (along the centroidal axis), material is linear elastic and homogeneous (Hooke’s law applies), deformation is small, and temperature effects are ignored unless stated. Cross sections remain plane and perpendicular to the axis (no bending or buckling in the basic case).

1. Basic definitions

•  = axial force (N) — tensile positive (pulling), compressive negative (pushing).
•  = original length of the bar (m).
•  = cross-sectional area (m²).
•  = elongation (m).
•  = normal stress (N/m² or Pa).
•  = axial (longitudinal) strain (dimensionless).
•  = Young’s modulus or modulus of elasticity (Pa).

2. Relate stress and strain
   Normal stress is load divided by area,

For a linear elastic material Hooke’s law gives

Thus the axial strain is

3. Strain definition and elongation
   Axial (engineering) strain is change in length per original length,

Combining with the expression for  gives

This is the standard formula for elongation of a prismatic bar under a uniform axial load.

4. Units check (quick verification)
   in N, in m,  in m²,  in N/m² gives  in m because .
5. Small element derivation (useful for variable cases)
   Take a small slice of the bar of thickness at position . Let that slice undergo elongation . Its strain is . The local stress is . Using Hooke’s law,

Integrate from  to :

This general integral covers many practical cases:

• If , ,  are constant, the integral reduces to .
• If the area varies (tapered bar) or  varies (composite or graded material), plug the functions  and  and integrate.
• If the axial force varies along the length (e.g., due to distributed axial load or internal reactions), use .

6. Composite bar (multiple segments)
   If the bar is made of segments in series (different materials or areas), each segment of length , area , modulus  carries the same axial force  (series, same axial load). Total elongation is sum:

This is practical for stepped shafts and joined bars.

7. Special cases and notes

• Compression: formula applies with  negative;  is then negative (shortening). Magnitude is given by same expression.
• Large deformations or nonlinear material: Hooke’s law not valid; must use true stress–strain relations or more complex constitutive models.
• Thermal expansion: if temperature change  is present, add thermal elongation  where  is coefficient of thermal expansion. If axial restraint exists, thermal stress arises.
• Eccentric load: if load does not pass through centroid, bending couples with axial deformation and simple axial formula is insufficient.
• Bar with axial distributed load  (N/m): the internal axial force  (sign convention) and substitute into the general integral.

8. Example (simple numeric)
   A steel rod , (), GPa, under tensile  kN:

9. Physical meaning
   The formula shows elongation grows with applied load and length and decreases with stiffness and cross-section . Intuitively, a longer, thinner, or softer bar stretches more under the same load.

Conclusion:

The elongation of a bar under axial load is derived by combining the definitions of stress, Hooke’s law, and axial strain. For a uniform prismatic, linear-elastic bar carrying an axial load , the elongation is . For non-uniform bars, variable loads, or composite sections, divide the bar into small elements and integrate: . This result is fundamental in mechanics of materials and is widely used in design, analysis, and material selection.