Optimization of CFRP and Steel Reinforced Concrete Nuclear Shell Roofs using Excel Solver

Abstract

Nuclear containment shell roofs must remain leak tight over service lives approaching 100 years, yet they are difficult to inspect and impossible to replace, so the corrosion of steel reinforcement is a recognised durability concern. Carbon fibre reinforced polymer (CFRP) reinforcement is immune to electrochemical corrosion but has a lower modulus and no ductility, so its design must be reformulated rather than substituted. This study developed an optimization framework that designed the reinforcement of a nuclear shell roof with steel and with CFRP on a strictly common basis; corrosion resistance motivated the comparison, but its long-term benefit was not quantified here. The dome was idealised as a spherical cap, the membrane and edge bending forces were derived analytically, and the weighted material consumption was minimised over the shell thickness and reinforcement ratio subject to strength, crack width, deflection, buckling, durability and severe accident constraints drawn from the governing Eurocode 2 provisions. The optimisation model was solved in Microsoft Excel Solver with the generalised reduced gradient algorithm, the shell solution was verified against a finite element analysis in Abaqus, and the reliability of each optimum was computed by the first order reliability method (FORM) and checked by Monte Carlo simulation (MCS) in Python. The steel optimum was a 0.31 m shell with a 1.18% reinforcement ratio governed jointly by strength and crack control; the CFRP optimum was a 0.32 m shell with a 0.52% ratio governed by crack control alone, a 56% reduction in reinforcement ratio at essentially the same thickness. The analytical forces agreed with the finite element results to within 3%. FORM returned reliability indices of 4.44 for the steel strength state and 3.76 for the CFRP crack width state, and MCS confirmed them with failure probabilities of 4.4 × 10-6 from 50 million samples and 1.0 × 10-4 from 10 million samples, equivalent to indices of 4.45 and 3.72. Both designs satisfied their respective ultimate and serviceability targets for nuclear structures, and the optimisation model itself was run in a standard spreadsheet in which every cell and relationship remained open to audit.