Cracked Section Analysis

Cracked-section analysis (also called cracked-section moment of inertia, Ieff) accounts for the reality that concrete cracks under flexural tension at service load — and the cracked region contributes minimal stiffness to the section. Per IS 456:2000 Annex C, cracked section is used to compute deflection and crack width at serviceability limit state (SLS). The cracked Ieff is significantly less than the gross section moment of inertia Ig (typically 30-60% of Ig for normally-reinforced beams), causing larger deflections than uncracked analysis predicts.

The Bischoff equation (used in IS 456 Annex C and ACI 318) computes Ieff from a transition between Ig (gross uncracked) and Ie (cracked transformed): Ieff = Ig × (Mcr/Ma)³ + Icr × (1 − (Mcr/Ma)³), where Mcr is the cracking moment and Ma is the actual applied moment. For a typical beam at service load with Ma ≈ 1.5 × Mcr: Ieff ≈ 0.3 × Ig + 0.7 × Icr ≈ 50-65% of Ig depending on reinforcement percentage.

Applications: (1) Beam and slab deflection check per IS 456 Cl. 23.2 — using Ieff in standard deflection formulas (e.g., for simply-supported beam with UDL: δ = 5wL⁴/(384EIeff)). (2) Long-term deflection multiplier — short-term elastic × creep multiplier × shrinkage curvature factor. (3) Crack width estimation per IS 456 Cl. 35.3.2. (4) Pre-stressed concrete losses estimation per IS 1343. (5) Building drift check under service loads. The most-overlooked aspect: many designers use gross section moment of inertia (Ig) in deflection calculations, which underestimates deflection by 30-50%. Software (ETABS, SAFE, STAAD) defaults to cracked-section analysis for deflection; users sometimes switch to gross section for 'simpler' analysis without realising the consequence. Always use cracked Ieff for deflection and crack width SLS checks.