Axial load in bars

A bar loaded along its axis carries a uniform stress P / A over its cross-section and extends by P L / (A E). Real problems add three twists: the cross-section changes along the length, the bar is made of more than one material, or the bar is held at both ends so the forces cannot be found from equilibrium alone.

For a stepped bar, treat each segment separately and add the extensions: ΔL = Σ Pᵢ Lᵢ / (Aᵢ Eᵢ). Draw an axial force diagram first, because the internal force may change where loads are applied along the bar. For a tapered bar, integrate; a circular bar tapering from diameter D₁ to D₂ extends by 4 P L / (π E D₁ D₂).

Composite sections, such as a reinforced concrete column, share load in proportion to axial stiffness because both materials shorten by the same amount. With modular ratio m = Es/Ec, the steel stress is m times the concrete stress, and the load is P = σc (Ac + m As). Statically indeterminate bars fixed at both ends with a load in between are solved by compatibility: the total change in length must be zero.

Self-weight matters for long hanging members such as mine cables or deep piles: a uniform bar hanging under its own weight extends by γ L² / (2E), half what it would if its whole weight hung at the bottom. In design, axial members are checked for stress and, in compression, for buckling, which is covered under columns.