5. Probabilistic analysis of the concrete containment

The NPP buildings with the reactor VVER 440/213 consist of the turbine hall, middle building, reactor building and bubble condenser [3]. The building of the power block was idealized with a FEM model consisting of 28 068 elements with 104 287 DOF (Figures 12 and 13).

The roof plate of the bubbler tower was defined as the critical area of the containment failure. The probability of the containment failure was determined for various levels of the overpressures Δp = 250, 300, 350, 400, 450 kPa. The probability of the failure was determined by Eqs. (21) and (22) for 10<sup>6</sup> Monte Carlo simulations in program FReET [32].

## 5.1. Nonlinear deterministic analysis

Containment may fail at different locations under different failure modes (see Figure 11). Consider two failure modes A and B, each with n fragility curves and respective probabilities pi (i = 1, …, n) and qj (j = 1, …, n). Then, the union C = A∪B, the fragility FCij(x) is given by

Figure 11. Family of fragility curves showing modelling uncertainty.

84 Recent Improvements of Power Plants Management and Technology

where the subscripts i and j indicate one of the n fragility curves for the failure modes and x denotes a specific value of the pressure within the containment. The probability pij associated with fragility curve FCij(x) is given by pi. qj if the median capacities of the failure modes are independent. The result of the intersection term in Eq. (32) is FAj(x).FBj(x) when the randomness in the failure mode capacities is independent and min [FAi(x), FBj(x)] when the failure modes

The flow is the consequence of an accident that depends on the total leak area. Multiple leaks at different locations of the containment (e.g. bellows, hatch and airlock) may contribute to the total leak area. Using the methodology described earlier, we can obtain the fragility curves for

For a given accident sequence, the induced accident pressure probability distribution, h(x), is known. This is convolved with the fragility curve for each leak location to obtain the probability of leak from that location (PLi). It is understood that there is no break or containment rupture at

> pLi ¼ ð ∞

> > 0

are perfectly dependent.

leak at each location.

this pressure.

FCijðxÞ ¼ FAiðxÞ þ FBjðxÞ � FAiðxÞ ∩ FBjðxÞ ð33Þ

hðxÞ½1 � FbðxÞ�FlðxÞdx ð34Þ

On the basis of the nonlinear analysis due to the monotone increasing of overpressure inside the hermetic zone, the critical sections of the structure were determined [3]. The resistance of these critical sections was analysed taking into account the design values of the material characteristics and the load. The slab at the top of the bubbler tower building was defined as

Figure 12. Section plane of the condenser containment.

Figure 13. Calculation model of NPP VVER 440/213.

the critical area of hermetic zone failure. The cracking process started at the concrete slab along the middle wall due to concentration of the temperature and overpressure effects. There is the effective temperature gradient equal to 60–90�C in the middle plane of the wall and the plate. The mean value of the critical overpressure was equal to 352.5 kPa, and the max. strain is lower than 0.002 in the middle plane of the reinforced concrete panel.

The cracking process (ε<sup>1</sup> ≥ εt¼\_ 0:0001) at the bottom or top section of the reinforced concrete panels started when the overpressure was equal to 250 kPa.

The interior structures of the hermetic zone are loaded with the accident temperature equal to 150�C and the outside structures in the contact with the exterior are loaded by �28�C. The difference between the interior end of the exterior temperatures has significant influences to the peak strain in the structures.

The comparison of the strain shape from the linear and nonlinear solutions is presented in Figure 14 and the stress shape in Figure 15. The strain increases and the stress decreases in the nonlinear solution in comparison with the linear solution.

Figure 14. Strain intensity from the linear and nonlinear analysis.

Risk Assessment of NPP Safety in Case of Emergency Situations on Technology http://dx.doi.org/10.5772/intechopen.68772 87

Figure 15. Stress intensity from the linear and nonlinear analysis.

#### 5.2. Evaluation of the fragility curve

the critical area of hermetic zone failure. The cracking process started at the concrete slab along the middle wall due to concentration of the temperature and overpressure effects. There is the effective temperature gradient equal to 60–90�C in the middle plane of the wall and the plate. The mean value of the critical overpressure was equal to 352.5 kPa, and the max. strain is lower

The cracking process (ε<sup>1</sup> ≥ εt¼\_ 0:0001) at the bottom or top section of the reinforced concrete

The interior structures of the hermetic zone are loaded with the accident temperature equal to 150�C and the outside structures in the contact with the exterior are loaded by �28�C. The difference between the interior end of the exterior temperatures has significant influences to

The comparison of the strain shape from the linear and nonlinear solutions is presented in Figure 14 and the stress shape in Figure 15. The strain increases and the stress decreases in the

than 0.002 in the middle plane of the reinforced concrete panel.

panels started when the overpressure was equal to 250 kPa.

nonlinear solution in comparison with the linear solution.

Figure 14. Strain intensity from the linear and nonlinear analysis.

the peak strain in the structures.

Figure 13. Calculation model of NPP VVER 440/213.

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The probability of the containment failure was determined for this critical structure on the basis of the nonlinear deterministic analysis of the containment for various levels of the overpressure. The function of the failure was considered by Eqs. (28) and (29) for 106 Monte Carlo simulations in program FReET [32]. The probability of containment failure (Figure 16) is calculated from the probability of the reliability function RF in the form

$$P\_f = P(RF < 0) \text{ and } RF = 1 - F\_u(I\_{\varepsilon 1}; I\_{\varepsilon 2}; \varepsilon\_u) / \varepsilon\_u \tag{35}$$

where the reliability condition RF is defined depending on a concrete failure condition (30).

Figure 16. Fragility curve of failure pressure in the critical area of the hermetic zone under the accidental internal temperature T<sup>i</sup> = 150�C and external temperature T<sup>e</sup> = �28�C.
