## Abstract

This paper presents probabilistic analysis of structural capacity of pre-stressed concrete containments subjected to internal pressure. The conventional design methods for containments are based on allowable stress codes which ensure certain factor of safety between expected load and expected structural strength. Such an approach may give different values of structural reliability in different situations. In recent years, two international round robin exercises have been conducted aimed at predicting the capacity of lined and unlined pre-stressed concrete containments used in nuclear industry. These exercises involved experimental testing and numerical analysis of the models. The first exercise involved 1/4 scale steel-lined Pre-stressed Concrete Containment Vessel (PCCV) which was tested at Sandia National Laboratories (SNL) in USA. The second used an unlined containment being tested by the Bhabha Atomic Research Centre (BARC), Tarapur, India. These studies are essentially deterministic studies that have helped validate the analysis methodology and modelling techniques that can be used to predict pre-stressed concrete containment capacity and failure modes. The paper uses these two examples to apply structural reliability method to estimate the probability of failure of the containment.

The two international round robin exercises have already established the ultimate structural collapse mode of the containments under internal pressure loading which indicate that the failure takes place in the general field of the containment wall around mid-height and away from any major structural discontinuities like the penetrations. This is because robust design procedures have been used to avoid structural failure at discontinuities by providing adequate compensation. Based on these experimental studies and the attendant numerical analyses a failure function is presented that assumes first yielding in the hoop direction at mid-height of the cylinder wall. A failure function equating the free-field membrane hoop stress to the hoop strength as a function of cross-sectional area (per unit height) and yield stresses of concrete, rebar, liner plate and tendons is developed.

First Order Reliability Method (FORM) is applied to predict probability of failure of the containments. Probability of failure vs internal pressure is presented for both types of containments. The paper presents a simple method to establish structural reliability of a pre-stressed concrete containment which can be useful for probabilistic safety assessment when considering extreme events that lead to over-pressurisation of the containment. The FORM approach was validated by comparison to the results of analogous calculations using Subset Simulation and Importance Sampling techniques for Monte Carlo simulation. It was found that at high pressures the Advanced FORM approach yields a good approximation to the true probability of failure.

The sensitivity of the probability of failure to the assumed coefficients of variation of properties of the containment was studied using the Sobol and Total Sensitivity Indices. At design pressure it was found that the coefficients of variation of the tendon yield and tendon area are the most important parameters followed by the applied pressure and containment radius. At higher pressures it was found that the coefficients of variation of the applied pressure and containment radius are the most important parameters. The variability of the probability of failure is decreased at higher pressures, but the coefficients of variation still play an important role.

The two international round robin exercises have already established the ultimate structural collapse mode of the containments under internal pressure loading which indicate that the failure takes place in the general field of the containment wall around mid-height and away from any major structural discontinuities like the penetrations. This is because robust design procedures have been used to avoid structural failure at discontinuities by providing adequate compensation. Based on these experimental studies and the attendant numerical analyses a failure function is presented that assumes first yielding in the hoop direction at mid-height of the cylinder wall. A failure function equating the free-field membrane hoop stress to the hoop strength as a function of cross-sectional area (per unit height) and yield stresses of concrete, rebar, liner plate and tendons is developed.

First Order Reliability Method (FORM) is applied to predict probability of failure of the containments. Probability of failure vs internal pressure is presented for both types of containments. The paper presents a simple method to establish structural reliability of a pre-stressed concrete containment which can be useful for probabilistic safety assessment when considering extreme events that lead to over-pressurisation of the containment. The FORM approach was validated by comparison to the results of analogous calculations using Subset Simulation and Importance Sampling techniques for Monte Carlo simulation. It was found that at high pressures the Advanced FORM approach yields a good approximation to the true probability of failure.

The sensitivity of the probability of failure to the assumed coefficients of variation of properties of the containment was studied using the Sobol and Total Sensitivity Indices. At design pressure it was found that the coefficients of variation of the tendon yield and tendon area are the most important parameters followed by the applied pressure and containment radius. At higher pressures it was found that the coefficients of variation of the applied pressure and containment radius are the most important parameters. The variability of the probability of failure is decreased at higher pressures, but the coefficients of variation still play an important role.

Original language | English |
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Pages (from-to) | 235-244 |

Number of pages | 10 |

Journal | Nuclear Engineering and Design |

Volume | 323 |

Early online date | 20 Dec 2016 |

DOIs | |

Publication status | Published - 30 Nov 2017 |

## Keywords

- concrete containments
- structural reliability
- containment capacity
- probabilistic safety assessment
- fragility
- sensitivity analysis
- coefficient of variation