A simplified model for concrete at low confining pressures

The paper describes a model for the response of concrete that is subjected to essentially monotonic straining at low confining pressures. We assume that, under these conditions, the response of the concrete is dominated by cracking when the stress state is predominantly tensile, and by gross inelastic deformation under compressive stress. The model uses a “crack detection surface” in stress space to determine when cracking takes place and the orientation of the cracking at a point, together with a damaged elasticity approach to describe the post-failure behavior of the concrete with open cracks. A yield/flow surface (associated flow) model is used to define the concrete's response in compressive states of stress. The model is simple enough that it can be implemented so as to operate effectively in an implicit finite element code: modeling accuracy is sacrificed for this purpose. Preliminary studies with the model indicate that it can give useful predictions in cases of interest.     

Homogenisation method for the modal analysis of a nuclear reactor with internal structures modelling and fluid–structure-interaction coupling

A homogenisation method is presented and validated in order to perform the dynamic analysis of a nuclear pressure vessel with a “reduced” numerical model accounting for inertial fluid–structure coupling and describing the geometrical details of the internal structures, periodically embedded within the nuclear reactor. Homogenisation techniques have been widely used in nuclear engineering to model confinement effects in reactor cores or tubes bundles. Application of such techniques to rector internals is investigated in the present paper. The theory bases of the method are first recalled. Adaptation of the homogenisation approach to the case of rector internals is then exposed: it is shown that in such case, confinement effects can de modelled by a suitable modification of classical fluid–structure symmetric formulation. The method is then validated by comparison of 3D and 2D calculations. In the latter, a “reduced” model with homogenised fluid is used, whereas in the former, a full finite element model of the nuclear pressure vessel with internal structures is elaborated. The homogenisation approach is proved to be efficient from the numerical point of view and accurate from the physical point of view. Confinement effects in the industrial case can then be highlighted.     

Chemo-mechanical coupling behaviour of leached concrete: Part II: Modelling

The paper is concerned with a coupled chemo-mechanical model describing the interaction between the calcium leaching and the mechanical damage in concrete materials. On the one hand, the phenomenological chemistry is described by the nowadays well-known simplified calcium leaching approach. It is based on the dissolution–diffusion process together with the chemical equilibrium relating the calcium concentration of the solid's skeleton and the calcium in the pore solution. For concrete, a homogenization approach using asymptotic expansions is used to take into account the influence of the presence of the aggregates leading to an equivalent homogeneous medium. On the other hand, the continuum damage mechanics is used to describe the mechanical degradation of concrete. The modelling accounts for the fact that concrete becomes more and more ductile as the leaching process grows. The model also predicts the inelastic irreversible deformation as damage evolves. The growth of inelastic strains observed during the mechanical tests is described by means of an elastoplastic-like model. The coupled nonlinear problem at hand is addressed within the context of the finite element method. And finally, numerical simulations are compared with the experimental results of first part of this work.     

Thermophysical properties of MgO, UO2, their eutectic solution and slurry of liquid-solid mixtures, concrete, sodium, stainless steel and debris beds for use in molten pool penetration of MgO substrate

We have prescribed various thermophysical and transport properties to describe various thermal states of the materials of interest such as MgO, UO₂, stainless steel, sodium, and concrete undergo during post accident heat removal (PAHR) in an ex-vessel cavity lined with MgO bricks. A number of properties, especially of molten MgO, had no experimental determination and therefore, by necessity, these were prescribed through available “best” estimates. We have also included the equivalent properties of various “composite” materials such as debris beds with a prescribed composition, solutions, and slurries to describe their participation in various thermophysical phenomena of interest in PAHR.     

Mechanics and mechanisms of creep deformation and damage

In recent years, several important advances in the understanding of the mechanics and mechanisms of creep deformation, damage and fracture in polycrystalline alloys have been achieved through the synergistic efforts of materials scientists and applied mechanicians. Current understanding, while far from complete, nonetheless provides a physically-based framework for developing computationally tractable continuum constitutive relations which capture major features of the intrinsic mechanisms of creep damage and deformation. Such models, when applied to the analysis of scientifically and technologically relevant boundary value problems, provide a basis for a local approach to high temperature fracture. The first section of the paper reviews certain aspects of the phenomenology, mechanisms, and mathematical models of creep fracture. It is concluded that an important internal damage parameter influencing macroscopic tertiary creep in conventional polycrystalline materials is the density of grain boundary facet cracks. The development of a “damage” constitutive model can conceptually be broken into two parts. One aspect is to quantify the effect of an (instantaneous) state of damage on the mechanical behavior. In the second part of the paper, this topic is explored for the case of a (generally non-dilute) distribution of aligned facet cracks in a power law creeping matrix. The remaining aspect of the model is to provide evolution equations for the damage variable(s). In the third section of the paper, we develop a simplified model for the evolution of facet crack density under conditions of creep-constrained cavitation. A central feature of the model is the variability in cavity nucleation potency over the grain boundary population.     

The use of energy absorbers to protect structures against impact loading

When a flying missible impacts a fixed structure, the interface loading is dependent on the deformation characteristics of both impacting and impacted bodies. If both are too rigid to accommodate the amount of gross deformation required to neutralize the incoming kinetic energy, or if such energy absorption has a chance to proceed in uncontrolled and unreliable ways, then there is a need to interpose a specifically designed “energy absorber” between missile and structure, from which a well-defined load time history can be derived during the course of impact.

The required characteristics of such an energy absorption material are:

• the capability to accommodate large permanent deformation without structural failure; and

• the reliable and controlled “load-deformation” (or “stress-strain”) behaviour under dynamic conditions, with an aim at an optimal square shape curve.

Consideration must also be given to environmental or other disturbing effects, like temperature, humidity, and “out of plane” loading. A short survey is presented of the wide range of energy absorbers already described in technical papers or used in a number of practical safety applications within varied engineering fields (from vehicle crash barriers to high energy pipe whipping restraints). However, with such open a literature, information is usually lacking in the specific data required for design analysis.

The following “energy absorption” materials and processes have thus been further experimentally investigated, with an a aim at pipe whipping restraint application for nuclear power plants:

1. (1) plastic extension of austenitic stainless steel rods;

2. (2) plastic compression of copper bumpers; and

3. (3) punching of lightweight concrete structures.

Dynamic “stress-strain” characteristics have been established for stainless steel bars at several temperatures under representative loading conditions. For this purpose, a test rig has been specifically designed to incorporate a number of adjustable parameters and to behave as a representative “slice” of an actual pipe whipping restraint; typical strain rates are in the 10 sec⁻¹ range. The behaviour of copper bumpers has been compared under static and dynamic conditions (using a conventional drop weight test (DWT) machine); as no significant strain rate effects were emphasized, only static tests have been further developed. The DWT rig was used again to investigate crushing or punching of cellular concrete under varying geometries and loading conditions. To remedy certain deficiencies of the regular commercial grades of cellular concrete, special lightweight mixtures have been studied to optimize material toughness and provide a wider range of specific resistance.Results of this experimental program are presented and discussed. The use of energy absorbers is then illustrated for a few typical pipe whipping restraints. The design of restraints is based on real dynamic characteristics of “energy absorption” material as produced by the test program. To derive design loads of restraints, a number of methods can be used ranging from a simplified “energy balance” graph to sophisticated plastodynamic computer analysis. Typical results are presented and discussed to compare the efficiency of these alternative methods.     

Failure and fracturing analysis of concrete structures

Some aspects of fracture analysis of concrete structures are discussed in this article. In particular it is shown that when localized failure occurs (by macrofracture propagation or localization of strain) structural size effects come into play. Mesh dependent finite element solutions are then observed unless size effects are correctly accounted for.Tensile fracture is examined first. The “classical” discrete and smeared crack approaches are reviewed and their extension to nonlinear fracture models like the fictitious crack model and the crack band model is illustrated. The smeared crack approach coupled first with a tensile strength criterion, second with a linear elastic fracture mechanics criterion is then applied to the failure mode analysis of a PCRV.Plastic fracturing with localization into shear bands, strain softening, mesh dependence and its correction are examined next. The use of plasticity for tensile fracture simulation is also discussed.Finally numerical difficulties inherent to the modeling of softening behavior are investigated.     

A dynamic model for element reliability

A dynamic model is developed for a system element reliability distribution over a generalized strength space. A differential equation is obtained describing the time-dependence of the reliability distribution function (RDF). The equation covers a wide class of power reactor system components which perform under intense stress conditions where a standard subdivision into a “burn-in” period, a “chance failures” range and a “wear-our” period is inapplicable.The hazard distribution function (HDF) over strength is introduced within the model and it is shown that a standard hazard rate is a strength-averaged failure intensity parameter with the RDF as a weighting function.It is shown that a well-known “bathtub” form of the hazard rate function corresponds to an analytical solution of the principal RDF transfer equation under some simplifying assumptions.     

Optimized aspect ratios of restrained thick-wall cylinders by virtue of Poisson's ratio selection. Part two: Temperature application

Analytical and numerical modelling have been employed to show that the choice of Poisson's ratio is one of the principal design criteria in order to reduce thermal stress build-up in isotropic materials. The modelling procedures are all twofold; consisting of a solution to a steady-state heat conduction problem followed by a linear static solution. The models developed take the form of simplistic thick-wall cylinders such model systems are applicable at macro-structural and micro-structural levels as the underlining formulations are based on the classical theory of elasticity. Generally, the results show that the Poisson's ratio of the material has a greater effect on the magnitude of the principal stresses than the aspect ratio of the cylinders investigated. Constraining the outside of these models significantly increases the thermal stresses induced. The most significant and original finding presented is that the for both freely expanding and constrained thick-wall cylinders the optimum Poisson's ratio is minus unity.     

A review of modeling concepts for sodium-concrete reactions and a model for liquid sodium transport to the unreacted concrete surface

A review of existing modeling concepts and studies of sodium-concrete reactions is presented. Consistent with experimental observations, the current modeling study being conducted at Hanford Engineering Development Laboratory assumes for hydrated concretes the presence of liquid layer of reaction products intervening between sodium pool and concrete surface. Primary liquid component in this layer is NaOH which has a low melting point. This liquid component dissolves the reaction products such as silicates, aluminates and forms a very viscous liquid more dense than sodium. As this layer assumes a significant thickness, the only mechanism available for transport of sodium to fresh concrete surface is the motion and agitation induced by gas bubbles consisting of hydrogen, water vapor, CO₂ and sodium vapor. However, to date there exists no satisfactory model that describes this transport mechanism. To rectify this shortcoming, we propose a mass “iffusion” model for sodium transport. The model reduces the sodium transport process by bubble motion to a single unknown parameter which has the appearance of a diffusion coefficient and consequently can be determined by solving an inverse problem in conjunction with measured “concentration” distributions in simulant material experiments.     

Artificial neural networks in prediction of mechanical behavior of concrete at high temperature

The behavior of concrete structures that are exposed to extreme thermo-mechanical loading is an issue of great importance in nuclear engineering. The mechanical behavior of concrete at high temperature is non-linear. The properties that regulate its response are highly temperature dependent and extremely complex. In addition, the constituent materials, e.g. aggregates, influence the response significantly. Attempts have been made to trace the stress–strain curve through mathematical models and rheological models. However, it has been difficult to include all the contributing factors in the mathematical model. This paper examines a new programming paradigm, artificial neural networks, for the problem. Implementing a feedforward network and backpropagation algorithm the stress–strain relationship of the material is captured. The neural networks for the prediction of uniaxial behavior of concrete at high temperature has been presented here. The results of the present investigation are very encouraging.     

A mixed implicit/explicit procedure for soil-structure interaction

This paper describes an efficient method for the solution of dynamic soil-structure interaction problems. The method which combines implicit and explicit time integration procedures is ideally suited to problems in which the structure is considered linear and the soil non-linear. The equations relating to the linear structures are integrated using an unconditionally stable implicit scheme while the non-linear soil is treated explicitly. The explicit method is ideally suited to non-linear calculations as there is no need for iterative techniques. The structural equations can also be integrated explicitly, but this generally requires a time step that is much smaller than that for the soil. By using an unconditionally stable implicit algorithm for the structure, the complete analysis can be performed using the time step for the soil. The proposed procedure leads to economical solutions with the soil non-linearities handled accurately and efficiently.     

Extreme value risk analysis in the structural design of reactor components

The tendency to confuse “uncertainties” associated with design assumptions and parameters and compensated by the safety factor with objective ‘risks of failure’ implicit in the design, has been characteristic of the approach to probability-based structural design on all levels. However, a clear differentiation between uncertainty and risk is required to remove the lack of correlation between design safety analysis and risk analysis implicit in the present approach to the design of major structural and mechanical components of nuclear reactors as well as other structures.In a recent paper [5] the author has used the definition of the safety factor as a random variable (distribution of a quotient) to construct a probability model that justifies the introduction of the asymptotic distributions of extreme values as the physically relevant distributions of the design parameters governing ultimate load failure on which a realistic risk assessment can be based. Realistic reliability and risk assessment of reactor components subject to fatigue and creep, i.e. design conditions that exceed in practical importance that of ultimate load failure, can be based on the use of the third asymptotic distribution of smallest values.In the case of structural components working under complicated conditions it becomes necessary to perform full-scale tests reproducing, as closely as possible, the anticipated operational and, whenever necessary, critical limiting conditions to be provided in the design as well as in the associated reliability and risk assessment. The economic necessity of keeping the number of such full-scale tests to a minimum which, in the case of larger components, is usually a single or very small number of tests, raises the problem of integration of the test results into the framework of a reliability and risk assessment.     

The frequency-dependent elements in the code SASSI: A bridge between civil engineers and the soil–structure interaction specialists

After four decades of the intensive studies of the soil-structure interaction (SSI) effects in the field of the NPP seismic analysis there is a certain gap between the SSI specialists and civil engineers. The results obtained using the advanced SSI codes like SASSI are often rather far from the results obtained using general codes (though match the experimental and field data). The reasons for the discrepancies are not clear because none of the parties can recall the results of the “other party” and investigate the influence of various factors causing the difference step by step. As a result, civil engineers neither feel the SSI effects, nor control them. The author believes that the SSI specialists should do the first step forward (a) recalling “viscous” damping in the structures versus the “material” one and (b) convoluting all the SSI wave effects into the format of “soil springs and dashpots”, more or less clear for civil engineers. The tool for both tasks could be a special finite element with frequency-dependent stiffness developed by the author for the code SASSI. This element can represent both soil and structure in the SSI model and help to split various factors influencing seismic response. In the paper the theory and some practical issues concerning the new element are presented.     

Heat transfer during molten corium-concrete interactions

Severe accidents in light water-cooled nuclear power plants involved heat transfer from molten reactor core materials or “corium” penetrating the reactor pressure vessel and coming to rest upon the containment building concrete floor covered by water. This paper discusses the difficulties of getting good information about the properties of the components and the flow structure during molten corium-concrete-water interactions. Also, potential heat transfer mechanisms are described and available prototypical tests are utilized to show that the enhancement in heat transfer by rising gas bubbles is the most likely mechanism, particularly if heat transfer by iradiation across the gas bubbles is included.     

Applicability of structural wall test results to seismic design of nuclear facilities

A review of tests on earthquake-resistant reinforced concrete structural walls is presented. Laboratory tests of isolated walls and construction joints are discussed. Where appropriate, design recommendations are given. The review indicates only few experimental data are available for short walls which are directly applicable to nuclear power plant design. In particular, tests of short rectangular walls subjected to load reversals are needed. Tests are also needed to determine the damping and frequency characteristics of cracked short walls. Analytical and experimental results should be correlated so that the hysteretic response observed in tests can be realistically related to the analytical response “demand” of nuclear power plant structures.     

Iodine removal by condensation from containment atmosphere in post external event conditions

An integral solution is derived for the problem of passive removal rate of a typical fission product (iodine), from a gas-vapor mixture, by condensate liquid film adjacent to a containment wall. The analytical model consists of a coupled set of five conservation equations: momentum, energy and three matter conservation equations for each individual component of the gas mixture: air, steam and elemental iodine. The set is solved in conjunction with two balance equations for the mass and energy transport at the interface with the condensate layer. The model accounts for free convection due to temperature and concentration gradients, for mass and thermal diffusion and for variable properties in both the liquid and the gas-vapor regions. An economic solution procedure of this model is presented and employed for a wide range of parameters. The computational results of this study are used to derive an efficient correlation which provides a quick and simplified means of the calculation of the iodine mass removal coefficient, as a function of the bulk conditions. Some results are compared to other theoretical and experimental works showing good agreement within about 10%. The significance of the removal process in the “external event” scenario is analyzed and found to be much higher than in scenarios that start with a mechanical failure in the primary system.     

Collection, processing and use of data

The proliferation in the past few years of data banks related to safety and reliability of nuclear reactors has somewhat alleviated the problem of lack of information when carrying out safety analysis; but it has succeeded meanwhile to identify several crucial aspects in the overall procedures of collection, processing and use of data. These aspects are reviewed in the paper with special emphasis on the interaction of the information and the various interpretative models of component and system behaviour. The need for the analyst of not only validating stated models but also of attempting new interpretations of the facts is particularly stressed. Eventually the role of new Artificial Intelligence techniques in supporting man in his search for model structures — that is the “history” underlying the facts, is mentioned.