Flexural Reliability of an IS 456 Reinforced-Concrete Beam: FORM by the Hasofer–Lind–Rackwitz–Fiessler Algorithm, Importance-Sampling Verification, and Reliability-Based Sizing

I compute the flexural reliability index of a singly reinforced rectangular beam (230 × 450 mm effective depth, M25 concrete, Fe500 steel) designed to IS 456:2000 limit-state rules at 90% of the limiting moment. The limit state is written in physical (unfactored) strengths, derived from the IS 456 rectangular-parabolic stress block, with lognormal strengths, a lognormal model-uncertainty factor, normal dead load and Gumbel live load. All statistical parameters are stated assumptions. FORM via the HL–RF iteration converges in 5 steps to β = 3.32 (pf = 4.5 × 10⁻⁴). Importance sampling centred on the design point gives β = 3.30, and crude Monte Carlo with 4 × 10⁶ samples gives pf = 4.8 × 10⁻⁴. Live-load uncertainty accounts for 79% of the squared sensitivity vector and model uncertainty for 16%. β ranges from 2.7 to 5.3 across plausible live-load models. Reaching β = 3.8 needs 15% more steel than the IS 456 design and brings the neutral axis to the code's ductility limit. Reaching β = 4.3 (the EN 1990 50-year target for class RC3) is not possible in this section without becoming over-reinforced. The computed β depends on the load model, and that model is the input most in need of Indian data.