Column Moment Formulas Under Lateral Story Drift (Δ): A Quick Reference Guide for Different End Conditions

In day-to-day structural design, we often focus on gravity loads — but it is the lateral loads (wind and earthquake) that truly test a column’s design. When a building sways under lateral forces, each story undergoes a horizontal displacement known as story drift (Δ). This drift forces columns to resist bending moments — and the magnitude of those moments depends entirely on how the column ends are restrained.

This post gives you a ready-to-use reference for the moment formulas in columns under lateral drift, broken down by end conditions. Whether you’re doing a quick hand check or verifying ETABS output, this guide will save you time.

Why Column End Conditions Matter

Imagine two identical columns — same cross-section, same material, same height — but one is fixed at both ends and the other is pinned at both ends. Under the same story drift, the fixed-fixed column will attract 100% of the moment, while the pinned-pinned column attracts zero moment. This is the fundamental reason why modeling boundary conditions correctly in software like ETABS or SAP2000 is critical — a wrong assumption here directly affects reinforcement design.

Moment Formulas for Columns Under Lateral Drift (Δ)

(All moments resist the drift direction; CW = Clockwise, CCW = Counter-Clockwise)

Variable Definitions

• E — Modulus of elasticity of the column material (e.g., 200 GPa for steel, ~25 GPa for M25 concrete)
• I — Moment of inertia of the column cross-section about the bending axis
• Δ — Story drift: the relative lateral displacement between the top and bottom of the column
• L — Clear height (length) of the column
• CW / CCW — Direction of the induced moment (clockwise or counter-clockwise), both acting to resist drift

Key Observations for Designers

• 🔒 Fixed-Fixed Columns attract the highest moments (6EIΔ/L²) — both ends resist rotation, making the column very stiff and highly stressed under drift. These are common in moment frame structures.
• 📌 Pinned-Fixed Columns carry half the moment (3EIΔ/L²), concentrated entirely at the fixed end. Common in braced frames where the top is connected to a pinned beam.
• 🔓 Pinned-Pinned Columns carry zero moment from drift — all lateral resistance must come from bracing or shear walls. These columns are designed for axial load only.
• ↕️ Guided-Fixed Columns (sway-prevented at top but rotation allowed) behave like cantilevers with equal moments at top and bottom — unusual but found in special frame configurations.

Worked Example

Consider a concrete column (M25 grade) in a multi-storey building with the following properties:

• E = 25 GPa = 25×10⁶ kN/m² (M25 concrete, E ≈ 5000√f_ck per BNBC/IS)
• I = 5,000 cm⁴ = 5,000×10⁻⁸ m⁴ = 5×10⁻⁵ m⁴
• Δ = 2 cm = 0.02 m (story drift)
• L = 3 m (story height) → L² = 9 m²
• EI = 25×10⁶ × 5×10⁻⁵ = 1,250 kN·m²

Substituting into the formulas (M = k × EI × Δ / L², where k = 6 for Fixed-Fixed, k = 3 for others):

Quick Calculation Check (Fixed-Fixed):
EI = 25×10⁶ kN/m² × 5×10⁻⁵ m⁴ = 1,250 kN·m²
M = (6 × 1,250 × 0.02) / (3)² = 150 / 9 = 16.67 kNm ✓

Conclusion

The moment induced in a column by story drift is not just a textbook concept — it directly affects how you design and detail column reinforcement, beam-column connections, and lateral load resisting systems. The stiffer the boundary conditions, the greater the moment demand. Use the formulas and tables above as a first-principles check alongside your structural analysis software to ensure your designs are both safe and efficient.