Short columns fail primarily by **crushing** rather than buckling. The allowable load is calculated using:

Here, σ_allow is the permissible compressive stress (derived from material strength with safety factors) and A is the column's cross-sectional area. For example, for a concrete column (f_ck=25 MPa), σ_allow is typically 0.4 × f_ck.

Eccentric loading induces **bending stress** in addition to axial stress. The combined stress is given by:

Where e is the eccentricity of load (distance from center), y is the fiber distance, and I is the moment of inertia. In practice, even minor eccentricities due to construction tolerances are considered for safety.

Base plates distribute the column load safely to the foundation concrete. The area is determined by:

For practical design, bearing stress (σ_bearing) is taken based on concrete strength (e.g., 0.45 × f_ck for M20 concrete). Plate thickness is then checked for bending.

Long, slender columns fail by **buckling** rather than crushing. Euler’s formula is used:

Where K is the effective length factor based on end conditions: - Fixed-Fixed: K = 0.5 - Fixed-Free: K = 2.0 - Pinned-Pinned: K = 1.0

The **Limit State Design** for concrete columns considers both concrete and steel reinforcement:

This accounts for 40% of concrete capacity and 67% of steel yield strength per IS 456:2000. This ensures ductile failure governed by reinforcement yield.

For steel columns designed using ASD (Allowable Stress Design), the load is:

Here, σ_allow is obtained from AISC tables or IS 800, factoring in slenderness effects (λ ≤ 180).