Air leakage characteristics in cracked concrete

In this paper a mathematical formulation for the air leakage rate through cracks in concrete is given. The formula works well as a good approximation for crack widths up to 1.30 mm and overpressures up to 0.80 MPa. The formula was found by means of systematic air leakage tests using unreinforced test specimens with one “defined single crack”. For thermodynamic formulation, isothermal changes in the gas state during the leakage through the crack was estimated and experimentally proved. By additional leakage tests using reinforced test specimens, practical usability of the leakage formula for reinforced panels with a “typical crack pattern” was checked.     

Research works on contamination transfers through cracked concrete walls

This study takes place within the framework of nuclear facilities containment assessment. The aims are to determine gaseous flow, two-phase flow and aerosol deposition models through a crack network by using the most realistic and representative crack network characteristics. For this, the crack network is considered as two infinite parallel plates. First of all, airflow experiments are performed on three concrete walls (128 cm in width, 75 cm in height and 10 cm in thickness), cracked by shear stresses. The results enable determining aeraulic crack network characteristics, thanks to the Poiseuille model in laminar compressible flow, and elaborating an experimental friction factor correlation for the transition flow, validated up to a Reynolds number of 250. Next, aerosol deposition experiments were performed with one of the previous concrete walls in order to determine a global aerosol deposition model in a crack network. The first experiments with an aerosol diameter of 60 nm showed that the aeraulic crack network characteristics are suitable for the aerosol physics. Therefore, geometrical crack network characteristics were determined by using several types of experiment and enable – thanks to aerosol deposition experiments – constituting and validating a global model of aerosol deposition in a crack network.     

Nonlinear dynamic response of cracked reinforced concrete nuclear containment vessels

In the design of reinforced concrete nuclear vessels horizontal cracks are assumed to exist as a result of pressurization. Seismic shear forces must be transmitted across these cracks. The nonlinear dynamic response of cracked vessels is studied. The force-displacement relationship across the cracks are taken from the experimental investigation that included the shear transferred by the concrete but not by dowel action of the vertical steel. The stiffness is highly nonlinear, hysteretic, and degrading. A modal analysis technique, based on an eigenvalue reanalysis procedure, is developed and it is compared with a direct numerical integration solution. Only typical response values are given for particular values of the variables and for one particular earthquake input.     

Air–steam leakage through cracks in concrete walls

In the context of a severe accident in a PWR nuclear plant, the evaluation of the leakage through the containment wall remains a key point of the safety analysis. Here we calculate the leakage of an air steam mixture through a traversing crack taking into account condensation. A 40 h test has been performed on a representative concrete slab with measurements of crack openings and flow rates. The CAST3M code enables us to simulate this test by making thermo-mechanical calculations and calculation of the leakage flow rate. Thermo-mechanical calculations provide data needed by the leakage calculations which are not measurable in the experiment. These are the internal crack profiles (variation of the opening with the curvilinear coordinate of the crack inside the concrete slab). Thermo-mechanical calculations are difficult to perform because boundary conditions of the test are complicated. Leakage calculations are performed with various hypotheses for the internal cracks profiles. A coefficient is applied on the friction factor to take into account additional complexity of the crack geometry.     

Gas leakage rate through reinforced concrete shear walls: Numerical study

Unlined reinforced concrete shear walls are often used as ‘tertiary boundaries’ in the United States Department of Energy (DOE) to house dangerous gases. An unanticipated event, such as an earthquake, may cause gases stored inside the walls to disperse into the environment resulting in excess pollution. To address this concern, in this paper, a methodology to numerically predict the gas leakage rate through these shear walls under lateral loading conditions is proposed. This methodology involves finite element and flow rate analysis. Strain distributions are obtained from the finite element analysis, and then used to simulate the crack characteristics on the concrete specimen. The flow rate through the damaged concrete specimen is then estimated using flow rate formulas available from the literature. Results from an experimental specimen are used to evaluate the methodology, and particularly its robustness in the flow rate estimation.     

Dynamic behaviour of anchors in cracked and uncracked concrete: a progress report

In early 1993, the US Nuclear Regulatory Commission began a research program at The University of Texas at Austin, dealing with the dynamic behavior of anchors in cracked and uncracked concrete. In this paper, the progress of that research program is reviewed. The test program is summarized, and work performed to date is reviewed, with emphasis on the dynamic and static behavior of single tensile anchors in uncracked concrete. General conclusions from that work are discussed, and future plans are presented.     

Analysis of nonlinear fluid-structure interaction in LMFBR containment

A generalized Eulerian method has been incorporated into ICECO for analyzing the nonlinear fluid-structure interaction in the primary containment of an LMFBR, consisting of complicated structural components such as the radial shield, core barrel, core-support structure, and the primary vessel. The method employs a Poisson equation to determine the hydrodynamic pressure in the fluid region, while using a relaxation equation to compute the pressure adjacent to the structure. A generalized coupling scheme is developed for treating the sliding condition at the fluid-structure interface, modeling the perforated structure, and analyzing the fluid motion at the geometrical discontinuities. Detailed formulations are given. Sample problems concerning wave propagation in a typical reactor containment are presented. It is shown from the results that this implicit, iterative method is unconditionally stable, and is especially suitable for excursions involving large material distortions.     

Refined cracked concrete analysis of concrete containment structures subject to operational and environmental loadings

Recent commercial nuclear power plant containment concepts involve the use of large reinforced concrete structures to form pressure boundaries. Where these structures are not provided with an integral steel liner, excessive cracking of the concrete under loads could result in the loss of the pressure boundary integrity with the risk of over-pressurization of other structures. Cracking of concrete is a local phenomenon and considerable detail must be included in any analytical model to obtain sufficiently refined results for the prediction of crack size and propagation. This imposes severe limitations on the overall size of structures or structural components for which detailed cracking analysis can be considered directly. To overcome this restriction, a two step procedure was developed in which linear analyses were performed to obtain the gross response, and nonlinear cracking analyses were performed for selected portions of the structure to evaluate local cracking in detail. Through iteration, compatibility of behavior between the linear and nonlinear analyses was achieved with the gross response being used to extrapolate the local cracking results to predict cracking over the entire structure. This paper discusses the analysis procedures for the detailed evaluation of cracking in large reinforced concrete structures and components. Analyses performed for an actual unlined reinforced concrete containment structure using these procedures are discussed and results are presented.     

Homogenized model for fluid-structure interaction of the pressurized water reactor core internals during blowdown

A method for the numerical simulation of the pressurized water reactor core internal's behaviour during a blowdown accident is described, by which the motion of the reactor core and the interaction of the fuel elements with the core barrel and the coolant medium is calculated. Furthermore, some simple models for the support columns, lower and upper core support and the grid plate are provided. In order to investigate the global core motion during the blowdown accident, the core model describes the coupled fluid-rod motion with Homogenization methods. The heterogeneous fluid-rod mixture thus is treated as a special continuum with anisotropic material properties. Furthermore, the core model considers elastical rod forces against bending and axial straining and the direct interaction of neighbouring fuel elements, which is a highly nonlinear process due to the finite gaps. Because this effect is very important, two simulation models have been developed and are compared. All these models have been implemented into the blowdown code FLUX-4. With the new code version FLUX-5 the PWR-blowdown is parametrically investigated.     

Significance of fluid-structure interaction phenomena for containment response to ex-vessel steam explosions

When studying the structural response of a containment building to ex-vessel steam explosion loads, a two-step procedure is often used. In the first step of this procedure the structures are treated as rigid and the pressure-time history generated by the explosion, at the rigid wall, is calculated. In the second step the calculated pressure is applied to the structures. The obvious weakness of the two-step procedure is that it does not correspond to the real dynamic behaviour of the fluid-structure system. The purpose of this paper is to identify and evaluate the relevant fluid-structure interaction phenomena. This is achieved through direct treatment of the explosion process and the structural response. The predictions of a direct and two-step treatment are compared for a BWR Mark II containment design, consisting of two concentric walls interacting with water masses in the central and annular pools. It is shown that the two-step approach leads to unrealistic energy transfer in the containment system studied and to significant overestimation of the deflection of the containment wall. As regards the pedestal wall, the direct method analysis shows that the flexibility of this wall affects the pressure-time history considerably. Three load types have been identified for this wall namely shock load, water blow as a result of water cavitation, and hydrodynamic load. Reloading impulse due to cavitation phenomena plays an important role as it amounts to ≈40% of the total impulse load. Investigation of the generality of the cavitation phenomena in the context of ex-vessel steam explosion loads was outside the scope of this work.     

The behaviour of concrete in prestressed concrete pressure vessels

In the design of prestressed concrete pressure vessels, long term concrete property data are required by the designer such that realistic estimates can be made of the vessels' 30-year stresses and deformations under the various operating conditions to which it will be subject. To achieve this aim, the shrinkage, short and long term deformation under load and thermal expansion behaviour of the vessel concrete has to be determined under conditions simulating those likely in the structure. In this paper, therefore, concrete properties are examined in relation to vessel design. Results obtained from the test programmes carried out for the Wylfa and Hartlepool nuclear power stations are presented in relation to our understanding of each property obtained from a detailed literature analysis.

The effect of temperature on three concrete properties of major importance in vessel design, e.g. compressive strength, thermal expansion and long term deformation under load (creep), is discussed at operational temperature up to 70°C. Consideration is also given, in the light of experimental data, on the effect of higher temperatures on these properties.     

Homogenisation method for the dynamic analysis of a complete nuclear steam generator with fluid-structure interaction

The present paper deals with the dynamic analysis of a steam generator tube bundle with fluid-structure interaction modelling. As the coupled fluid-structure problem involves a huge number of degrees of freedom to account for the tube displacements and the fluid pressure evolutions, classical coupled method cannot be applied for industrial studies. In the present case, the three-dimensional fluid-structure problem is solved with an homogenisation method, which has been previously exposed and successfully validated for FSI modelling in a nuclear reactor [Sigrist, J.F., Broc, D., 2007a. Homogenisation method for the modal analysis of a nuclear reactor with internal structures modelling and fluid-structure interaction coupling. Nuclear Engineering and Design 237, 431-440]. Formulation of the homogenisation method for general two- and three-dimensional cases is exposed in the paper. Application to a simplified, however representative, model of an actual industrial nuclear component (steam generator) is proposed. The problem modelling, which includes tube bundle, primary and secondary fluids and pressure vessel, is performed with an engineering finite element code in which the homogenisation technique has been implemented. From the practical point of view, the analysis highlights the major fluid-structure interaction effects on the dynamic behaviour of the steam generator; from the theoretical point of view, the study demonstrates the efficiency of the homogenisation method for periodic fluid-structure problems modelling in industrial configurations.     

Early-age behaviour of concrete nuclear containments

A numerical model has been developed to predict early-age cracking for massive concrete structures. Taking into account creep at early-age is essential if one wants to predict quantitatively the induced stresses if autogenous or thermal strains are restrained. Because creep strains may relax internal stresses, a creep model which includes the effects of hydration and temperature is used. For the prediction of cracking, a simple elastic damage model is used. Numerical simulations are performed in order to predict the behaviour of a massive wall and a concrete containment of a nuclear power plant. They show that significant relaxation of stresses (due to creep) occurs only after about 10 days, after cracking occurs. Moreover, since temperature in concrete may reach important values in massive concrete structures, it appears that effect of temperature on creep must be taken into account for an accurate prediction of cracking.     

Elastic-plastic analysis of local and integral straining behaviour in a cracked plate

For components of the primary coolant system of the German LMFBR prototype reactor SNR-300, integrity against anticipated accidents (Bethe-Tait) has to be shown for a cracked structure. Within this programme a number of tests with cracked wide plate specimens yielding overall limit strains of approximately 15% have been run; finite element calculations have been initiated for the wide plate geometry. The paper discusses the straining behaviour of a cracked plate by considering the numerical simulation of structures strained up to such high levels.The stress-strain diagram of the weldment of the austenitic stainless steel X6 CrNi 18 11 at 450°C has been used. Plane strain and stress conditions have been prescribed. The original plate dimensions (t=thickness=40 mm; h=height=400 mm) have been used as well as a similar, but smaller plate of t=8.8 mm width. The crack length is defined as 0.1t.The results show that for a cracked plate under high plastic strain the near-crack-tip-field values still govern the structural mechanical behaviour. Concerning the absolute dimensions the effects known for elasticity retain their influence in the plastic regime; however, the crack location becomes more unimportant with increasing strain, i.e. the appropriate pure geometry factor tends to unity in the plastic regime. The center-crack, defined as 2a=0.1t, corresponds to an equivalent edge crack of depth a=0.05t in the elastic case. It can be shown that for high plastic strains this correspondence remains fully valid.     

Comportement local des enceintes en beton sous l'impact d'un projectile rigide: Local behaviour of reinforced concrete walls under missile impact

Since 1974 CEA and EDF have developed in France a large programme with the aim to work out a means of computation reliable enough for the knowledge of the behaviour of reinforced concrete walls under missile impacts. The shots were performed on reinforced concrete slabs, the thickness of which were chosen to represent in a realistic way the thickness of the wall of a reactor containment. The scales used in this modeling were mainly one-half and one-third. The following parameters were kept constant: the properties of the concrete and the geometric shape of the missile (flat nose). Also studied were the effects of variation of parameters like missile velocity (25–450 m/s), its mass (20–300 kg), ratio of the missile diameter to the thickness of the slab (0.24–2.9) and characteristics of the steel reinforcement.The results of these tests may be summarized in a homogeneous perforation formula in the case of a velocity lower than 200 m/sec: where Vc is the minimum velocity for perforation, ø diameter of the missile, M its mass, σ the ultimate compressive strength of the concrete, its density and ρ its thickness.     

Prediction of air and steam leak rate through cracked reinforced concrete panels

The leak rate prediction of air and steam through a cracked concrete wall is an extremely important issue in assessing the safety of a nuclear reactor containment building. Such a problem requires a multidisciplinary approach involving both the non-linear analysis of the structure, and the thermodynamics aspects related to the flow of a gas through a conduit. In the present paper, some of the available leak rate evaluation formulae are reviewed, and an application to the prediction of the leak rate of either dry air or air+steam mixture through a cracked concrete panel is presented. Finally, in order to validate the numerical procedure herein adopted and to give some indication on the relative merit of the different leak rate formulae considered, the results of the numerical application are compared against leak rate values measured during an experimental test carried out at the ISMES laboratory.     

Parametric sensitivity of fracture behaviour of concrete

The fictitious crack method (FCM) is applied to determine the load-deflection diagrams of notched plain concrete beams under three-point bending using various forms of strain softening in the stress-deformation relationship. The results indicate that there is a need to determine a more realistic relationship.     

Numerical modeling of reinforced concrete containment walls under cyclic shear

This paper provides an overview of research in numerical modeling of reinforced concrete containment walls subjected to cyclic shear. Bases for the development of the model are discussed, and application of the model is shown. Further research needs and interests are suggested for improved analysis capabilities and design.     

Nonlinear dynamic analysis of reinforced concrete structures

Dynamic ultimate load calculations mainly for reinforced concrete beams and plates, are discussed. Starting from the corresponding differential equations, the calculations also include the rotational inertia of single beam or plate elements as well as the shear deformations. With actual structural dynamic problems in nuclear power plants, the shear behaviour of reinforced concrete beams and plates is more important than it is usually, as is shown by examples. The finite propagation velocity of bending and shear waves are taken into account. Solution of the equations of motion is obtained by numerical intergration using finite time and space intervals. The calculations are performed using time dependent bending and shear laws for reinforced concrete up to the point of failure with realistic deformations. These latest scientific developments are of great significance for dynamic ultimate load analysis in practice.Elastic-plastic examples of application are compared with corresponding linear-elastic solutions. It is shown that the design of construction members based on elastic-plastic dynamic stress calculations in general is economically advantageous. This important conclusion is proven by numerical results. Also the relation to the approximation of a one-degree-of-freedom dynamic system, including or excluding the plastic ductility of the structural member, is demonstrated.Finally, lumped-mass multi-degree systems calculated by integrating numerically the corresponding equations of motion, are dealt with briefly. A nonlinear dynamic calculation of a foundation of a recently built reactor building is presented as an example for blast resistant analysis.     

On the multiaxial behaviour of concrete exposed to high temperature

An attempt is made to formulate a multiaxial constitutive model for concrete in the temperature range up to 800°C. The proposed model can be characterized as isotropic, elastic-viscoplastic-plastic in the compression region. Brittle failure is assumed in the tensile region.The thermal strain increment is assumed to be a function of both temperature and the current stress tensor. This assumption implies that the thermal strain may have deviatoric components.The volumetric thermal strain is used as a scalar damage measure instead of temperature itself. The corresponding softening function is obtained from isothermal, uniaxial tests. Also the elastic properties are taken as functions of the volumetric thermal strain.The response of the model is illustrated and compared with experimental results.