This current research goal is to develop a framework for the performance-based analysis and design of the nuclear containment structures subject to missile impact. Proposed performance levels are linked to impact levels to estimate the reliability of the containment structure for the desired performance objectives.
Introduction
- General
- Motivation
- Existing Studies
- Objectives
- Organisation of the Thesis
First of all, three different levels of structure performance will be identified and calculated based on the damages that occur in the RC and PC panels subjected to missile impact. Chapter eleven details the development of probabilistic capacitance models for three performance levels of PC panels subjected to missile impact.
State of Practice For Missile Impact
Impact Loading
Configuration of the Missile
Types of Missile Impact on Structures
The effect of aircraft impacts on prestressed nuclear containment and LNG storage tanks is presented in (Frano and Forasassi 2011; Lee et al., 2013; Zhang et al., 2017). In order to understand the effect of projectile impact on structures, experimental and empirical research is carried out on concrete slabs (Orbovic et al., 2015;.
Empirical Formulation of Effect of Missile Impact on Structures
Velocity of the projectile, hper – the penetration limit of the concrete target, H – the thickness of the target. Vr - Projectile residual velocity, V0 - Projectile velocity, d - Projectile diameter, hper.
Nuclear Power Plant
Nuclear Energy
Calamities of Nuclear Power Plants
Nuclear Containment Structures
The occurrence of a missile impact scenario on the containment structure becomes a horrifying possibility in the coming days due to the very large increase in the number of nuclear power plants worldwide.
Performance Levels Of Nuclear Containment Structures
- Performance-Based Design (PBD)
- Bayesian Approach
- Fragility Analysis
- PBD of Nuclear Containment Structures
- Behaviour of Target During Missile Impact
- Performance Levels for Missile Impact
- Methodology for Estimating Probabilistic Models
Performance levels of the target structure where the panel is subject to missile impacts have been identified to define the damage states. The damage levels increase in intensity from minimal damage to total failure of the target structure. Performance level three is considered to be the prevention of collapse of the missile penetration of the target object to the full depth of the target panel, as shown in Figure 4.3.
A performance-based analysis and design can help mitigate the damage of the structure due to missile impact.
Finite Element Model for RC Members Subject to Missile Impact
- Numerical Approach for RC Members
- Material Models
- Finite Element Configuration
- Finite Element Validation
Many of the numerical results made a desirable match with results obtained by experimentation with the same configuration. To model the impact of the missile with the RC member, a three-dimensional solid model is used. It can be seen that the formulation is able to capture the deflection of the structure with allowable accuracy.
In Figure 5.9 it can be seen that the FE approach is able to capture the deflection profile of the structure subject to missile impact with great accuracy.
Finite Element Model For PC Members Subject to Missile Impact
Numerical Approach for PC Members
This is one of the most widely used techniques in the literature to date (Jiang and Chorzepa, 2015) (Figure 6.2).
Finite Element Validation
Initial prestress of 10% is applied with unconfined compressive strength of 48 MPa, as can also be deduced from Figure 6.7, i.e. parameters considered from experiments are displacement time history at the center of a beam and the impact force on concrete of the impactor is compared with FE numerical simulation in Figure 6.16 and Figure 6.17. This validation considers displacement obtained from analytical equations (Euro code), FE simulation results obtained from (Johansson and Fredberg, 2015) and FE simulation results obtained from this research as shown in Figure 6.19.
The analytically used axial force is 164 kN and is validated in the same way as in numerical analysis, as shown in Figure 6.20.
Finite Element Design
Introduction
Selection of Variables and Range
D-Optimal Point Selection Scheme
Based on the range considered, the best set of cases are selected using the D-optimal point selection scheme (Myers et al., 1995). FE models of 50 combinations of RC panels and 100 combinations of concrete core PC bearing with plane loading scenarios of strong missile impacts are simulated to obtain the database for the development of probabilistic models. The thickness range of concrete panels is based on actual data available in the literature.
The yield strength of reinforcing bars and cladding is determined based on the availability of a commercially available grade of steel.
Illustration of Curved PC Panel
In this chapter, probabilistic models are developed for parameters such as plate penetration depth, target structure perforation limit, and projectile residual velocity for PC and RC plates subjected to projectile impact. The construction of the current formulation of probabilistic models is by adding the required correction terms to the mechanistic models. The process for developing probabilistic models for RC and PC panels is given below in the thesis.
Validation with existing formulas as shown in Figure 5.6 and Figure 5.7 (Due to the significant error in the results, a new probabilistic formulation has been developed).
Probabilistic Models of RC Panels
The residual velocity of the missile for all FE simulations is recorded as shown in Figure 8.1. A plot of the residual velocity of a missile is shown in Figure 8.3, and rest of the FE simulations are recorded the same by selecting a node of the missile. The newly proposed probability equation in this thesis by the author to estimate the residual velocity (Vr) of missile after impacting an RC.
Cov is 9.5%, which describes the accuracy of the developed probabilistic model to estimate the residual velocity of missile to RC target.
Probabilistic Models for PC Panels
The cov is 4% which describes the accuracy of the developed probability model for estimating the penetration depth of missile in the PC target. The same procedure is followed to capture the residual velocity of the missile as shown in Figure 8.1. For PC panels, the residual velocity variation compared to the initial velocity of the missile is large due to massiveness in the PC structure.
The cov is 0.2%, which describes the accuracy of the developed probabilistic model for estimating the residual velocity of a missile hitting the PC panel.
Inference
Probabilistic model predictions show that parity data lie both above and below the 1:1 line.
Performance-Based Capacity Models For RC Panels
- Mechanical Model for Performance Level P1
- Mechanical Model for Performance Level P2
- Mechanical Model for Performance Level P3
- Model Correction
- Model Assessment
- Parameter Estimation for Performance Level P1
- Parameter Estimation for Performance Level P2
- Parameter Estimation for Performance Level P3
- Inference
The estimate of the dynamic capacity for performance level 1 (P1) is based on the resistive energy of the target structure. The estimate of the dynamic capacity for performance level 2 (P2) is based on the resistive energy of the target structure. The estimate of the dynamic capacity for performance level 3 (P3) is based on the resistive energy of the target structure.
Cov is 6.3%, which describes the accuracy of the developed probabilistic model to estimate the capacity of RC panels subjected to missile impact at performance level P1.
Performance-Based Demand Models for RC Panels
- Mechanical Model
- Model Correction
- Model Assessment
- Parameter Estimation
- Fragility Estimates
- Inference
The dynamic demand estimate 𝑃̂𝐷is based on the kinetic energy of the projectile loaded on the RC plate. The fragility of an RC nuclear containment plate subjected to a missile impact is formulated as. The exceedance contours for Performance Level – 2 (P2) show a good spread, as shown by the distribution of fragility lines from 0.1 to 0.9 (Figure 10.5).
Exceedance contours for performance level – 3 (P3) show the well-distributed nature, as evidenced by the distribution of the fragility lines from 0.1 to 0.9 (Figure 10.7).
Performance-Based Capacity Models for PC Panels
- Mechanical Model for Performance Level P1
- Mechanical Model for Performance Level P2
- Mechanical Model for Performance Level P3
- Model Correction
- Model Assessment
- Parameter Estimation for Performance Level P1
- Parameter Estimation for Performance Level P2
- Parameter Estimation for Performance Level P3
- Inference
The dynamic performance rating for performance level 1 (P1) is based on the resistive energy of the target structure. In this method, diagnostic plots are generated between the explanatory function and the dynamic performance residual of the FE model and the mechanical model. The Cov is 10%, which details the accuracy of the developed probabilistic model for estimating the performance of a PC board subjected to projectile impact at the P1 performance level.
The Cov is 1.88%, which details the accuracy of the developed probabilistic model for estimating the performance of a PC board subjected to projectile impact at the P3 performance level.
Performance-Based Demand Models of PC Panels
Mechanical Model
The estimate of the dynamic demand 𝑃̂𝐷is based on the kinetic energy of the missile placed on the computer panel. Considered terms for this mechanical model are mass of the missile and velocity of the missile. 𝑃̂𝐷 = dynamic demand of missile placed on computer sheet, M = Mass of the missile, V0 = Velocity of the missile.
Model Correction
Where Ls = Length of panel, tc = thickness of concrete target, ∆ = Hump of target from center, Lm = Length of missile, d= diameter of missile, M = mass of missile, V0 = Velocity of missile, fc = concrete compressive strength, 1/T = frequency, Mc = Moment Capacity of target structure, Xpen = penetration at different levels, n = 1,2,3. The first explanatory function h1(x) is considered to capture a potential constant bias present in the mechanical model. The effect of the natural frequency of the system is explained by the sixth explanatory function h6(x).
Model Assessment
Parameter Estimation
𝑃𝐷 = Probabilistic dynamic demand imposed on the PC panel by missile impact, 𝑃̂𝐷 = dynamic demand imposed on the RC panel by missile impact from the mechanical model. The mechanical and probabilistic models are both plotted against the demand in terms of the kinetic energy of the rocket impacting the nuclear containment RC panel in terms of values estimated from the FE simulations. The cov is 0.0001%, which describes the accuracy of the developed probabilistic model for estimating demand on a PC panel subject to missile impacts.
Fragility Estimates
The contour plot shows the fragility variation in the mass and velocity domain. Exceedance contours for performance level – 1 (P1) show the well-distributed nature, as evidenced by the distribution of the fragility lines from 0.1 to 0.9 (Figure 10.3). Exceedance contours for performance level – 2 (P2) show the well-distributed nature, as evidenced by the distribution of the fragility lines from 0.1 to 0.9 (Figure 12.5).
Exceedance contours for performance level – 3 (P3) show the well-distributed nature, as evidenced by the distribution of the fragility lines from 0.1 to 0.9 (Figure 12.7).
Inference
Most of the structures are designed for static loads, at maximum quasi-static loads, but the proposed work moves the available methodology to the effect of local damage based on dynamic loads, which is a realistic assessment of the phenomena in the case of a projectile impact. The research develops a framework for performance-based analysis and design of nuclear containment structures subject to missile impact that considers inherent uncertainty in system properties, material and geometric configurations, and structure-missile interaction. Three different levels of structural performance are quantified based on the damage produced in the RC and PC panels subjected to projectile impact.
The developed probabilistic capacity models accurately estimate the capacity of the RC and PC panel at the given performance level.
