Z. P. Bažant

Justification and refinements of model B3 for concrete creep and shrinkage 1. statistics and sensitivity

Model B3 for creep and shrinkage prediction in the design of concrete structures, presented as a RILEM Recommendation inMater Struct.28 (1995) 357–365, is calibrated by a computerized data bank comprising practically all the relevant test data obtained in various laboratories throughout the world. The coefficients of variation of deviations of the model from the data are distinctly smaller than for the latest CEB model, and much smaller than for the previous ACI model (which was developed in the mid-1960s). The effect of concrete composition and design strength on the model parameters is identified as the main source of error. The model is simpler than the previous models (BP and BP-KX) developed at Northwestern University, yet it has comparable accuracy and is more rational.

NUMERICAL SMEARED FRACTURE ANALYSIS: NONLOCAL MICROCRACK INTERACTION APPROACH

SUMMARY A recently proposed new nonlocal concept based on microcrack interactions is discussed, its implementation in a smeared cracking finite element code for concrete is presented, numerical studies are reported, and comparisons with experimental results are made. The nonlocality is not merely a mathematical device to prevent excessive spurious localization into a zone of zero volume but is a necessary physical consequence of microcrack interactions. Since the constitutive law itself is strictly local, the new nonlocal concept can be combined with any type of constitutive law for strain-softening nonlocal damage, which is here chosen to be the micro plane model. A simple method is formulated to approximately identify the material parameters in the model from the basic characteristics of concrete such as the tensile strength, fracture energy and maximum aggregate size. The results of finite element analysis are shown to be mesh insensitive, and good convergence is obtained. Cracking damage is found to localize into a volume whose size and shape depend on the macroscopic concrete properties as well as the current stress-strain state. Although the damage is considered to be tensile on the microlevel, due solely to mode I microcracks, the new non local model can describe well not only mode I fracture tests but also complex shear-dominated and mixed-mode types of failure such a diagonal shear, and can do so for the same values of material parameters (which was not the case for previous nonlocal models). Most importantly, the new nonlocal model can correctly capture the size effect of quasibrittle fracture, in approximate agreement with Bazanfs size effect law.

STRAIN-SOFTENING MATERIALS AND FINITE ELEMENT SOLUTIONS.

Abstract Closed form and finite-element solutions are examined for several problems with strain-softening materials. In the closed form solutions, strain-softening causes localization of the strain which is accompanied by an instantaneous vanishing of the stress. The finite-element solutions agree closely with analytic solutions in many cases and exhibit a rate of convergence only slightly below that for linear problems. The main difficulty which has been identified in strain-softening constitutive models for damage is the absence of energy dissipation in the strain-softening domain, and this can be corrected by a nonlocal formulation. Finite-element solutions for the converging spherical wave problem exhibit multiple points of localization which change dramatically with mesh refinement. With a nonlocal material formulation, this pathology is eliminated.

VARIABLE-NOTCH ONE-SIZE TEST METHOD FOR FRACTURE ENERGY AND PROCESS ZONE LENGTH

Based on the generalized theory of the size effect law allowing dissimilar specimens, this paper proposes a new version of the size effect method for determining the fracture energy Gr and effective process zone length cr, which permits using specimens of only one shape and one size, but with different notch lengths. Cutting notches of different lengths on specimens of the same shape and size is an easy way to obtain specimens of different brittleness numbers, as required by the size effect method. Either linear or nonlinear regression of measured maximum loads of these specimens can give the material parameters G~ and cr. An experimental program is conducted to verify the proposed method. The notched holed split-tension cylinder is found to be a suitable specimen shape, while the notched eccentric compression specimen is found to provide a barely sufficient range of brittleness numbers. The analysis in the paper also indicates that there exists an upper limit of notch (or initial crack) length for some specimen geometries as the limit for validity of the definition of brittleness number. Copyright

CONCRETE MODEL WITH NORMALITY AND SEQUENTIAL IDENTIFICATION

Abstract To facilitate numerical finite element analysis, it is desirable to endow the constitutive model with normality, associatedness, continuity, convexity and absence of corners. Although these mathematical conditions represent only crude approximations of the actual behavior of concrete, it is of interest to find the best possible constitutive model which meets these conditons. This is one objective of the present paper. The second objective is to develop a model which permits a simple identification of material parameters from test data. The material parameters need not be obtained by simultaneous nonlinear optimization of the fits of all data. Rather, they are obtained in sequence through a precisely defined procedure which involves solving two systems of linear equations. The model describes not only hardening but also post-peak softening under various triaxial stress states. The model agrees well with the available basic test data from monotonic loading tests.

Analysis and application of prestressed concrete reactor vessels for LMFBR containment

Abstract An analytical model of a prestressed concrete reactor vessel (PCRV) for LMFBR and the associated finite element computer code, involving an explicit time integration procedure, is described. The model is axisymmetric and includes simulations of the tensile cracking of concrete, the reinforcement, and a prestressing capability. The tensile cracking of concrete and the steel reinforcement are both modeled as continuously distributed within the finite element. The stresses in the reinforcement and concrete are computed separately and combined to give an overall stress state of the composite material. The reinformcement is assumed to be elastic, perfectly-plastic; the concrete is taken to be elastic, with tensile and compressive stress limits. Cracking of concrete is based on the criterion of maximum principal stress; a crack is assumed to form normal to the direction of the maximum principal stress. Attention is also given to the fact that cracks do not form instantaneously, but develop gradually. Thus, after crack initiation the normal stress is reduced to zero gradually as a function of time. Residual shear resistance of cracks due to aggregate interlock is also taken into account. An existing crack is permitted to close. Prestressing of the PCRV is modeled by special structural members which represent an averaged prestressing layer equivalent to an axisymmetric shell. The internal prestressing members are superimposed over the reinforced concrete body of the PCRV; they are permitted to stretch and slide in a predetermined path, simulating the actual tendons. The validity of the code is examined by comparison with experimental data. Both static and dynamic data are compared with code predictions, and the agreement is satisfactory. A preliminary design has been developed for both pool and loop-type PCRVs. The code was applied to the analysis of these designs. This analysis reveals that the critical locations in such a design would be the head cover and the junction between the cover and the vessel wall and indicates the pattern of crack development. The results show that the development of a design adequate for current HCDA loads is quite feasible for pool-type or loop-type PCRVs.

Practical formulation of shrinkage and creep of concrete

A set of algebraic formulas is proposed to describe the shrinkage and creep of concrete over the entire range of time durations of interest. The formulas cover: the effects of drying at various environmental relatire humidities, the size and shape of cross section, aging (due to hydration), the effect of time lag of loading after the start of drying, creep of specimens predried to various humidities, the nonlinear dependence of stress, the increase of nonlinearity at simultaneous drying, and the decrease of strength for long-time loads. Simplification in the form of a linear dependence on stress is obtained as special case. The formulation is an extension of the double power creep law, which has been recently proposed for creep in absence of moisture exchange. The shape of the time curves of creep depends on the humidity. A rather close agreement with the extensive experimental data available in the literature is demonstrated.RésuméOn propose une série de formules algébriques qui traduisent le retrait et le fluage du béton considéré pour toutes les durées intéressantes. Ces formules prennent en compte: les effets du séchage à diverses humidités relatives ambiantes, dimensions et formes de la section, le vieillissement (dû à l’hydratation), l’effet retardé du chargement après le début du séchage, le fluage d’éprouvettes préséchées à diverses humidités, la fonction non linéaire de contrainte, l’accroissement de la non-linéarité durant le séchage, et la diminution de la résistance sous charge de longue durée. On a obtenu comme un cas particulier une simplification sous forme de fonction linéaire de la contrainte. La formulation est une extension de la loi de fluage dite «à double puissance» qui a été récemment proposée pour l’étude du fluage en l’absence d’échange d’humidité. La forme des courbes de fluage en fonction du temps dépend des conditions d’humidité. On montre qu’on obtient une assez bonne concordance avec les nombreuses données expérimentales disponibles.

Improved prediction model for time-dependent deformations of concrete: Part 3-Creep at drying

Part 3 of the present series gives the prediction formulae for the average compliance function in the cross-section of a specimen exposed to drying at constant temperature. The formulae describe the additional creep due to drying by means of the shrinkage function, which automatically introduces the consequences of diffusion theory, such as the dependence of creep on cross-section thickness and shape. The prediction formulae are compared with 19 different data sets from the literature, which reveal relatively good agreement, better than that with previous models. The main source of error is insufficient knowledge of the effect of mix composition and concrete strength. It is advisable to avoid this error by carrying out short-time measurements whenever possible.ResumeDans le troisième volet de cette série, on présente les formules de prédiction pour la fonction de compliance moyenne dans la section transversale d’une éprouvette soumise au séchage à température constante. Les formules décrivent le fluage additionnel causé par le séchage au moyen de la fonction de retrait qui introduit automatiquement les aboutissements de la théorie de diffusion, tels ceux qui font dépendre le fluage de l’épaisseur et de la forme de la section transversale. On compare les formules de prédiction à 19 séries de données prises dans la littérature, qui présentent une assez bonne concordance, supérieure à celle obtenue avec les modèles précédents. La source principale d’erreur réside dans une connaissance insuffisante du dosage et de la résistance du béton. On conseille d’éviter cette erreur en effectuant des mesures à court terme chaque fois que c’est possible.

Size Effect on Biaxial Flexural Strength of Concrete

The size effect on the tensile strength of concrete is investigated experimentally for the case of equi-biaxial tension. Tests of tensile strength under uniaxial tension were carried out for comparison using four-point bend beams. For measuring the biaxial tensile strength, the ASTM C1550 test and the biaxial flexure test were examined. To study the size effect, unreinforced circular plates of three different sizes are tested, with 13 specimens per size. The size effect on the equi-biaxial tensile strength is found to be stronger than it is on the uniaxial tensile strength, and to exhibit the characteristics of the deterministic Type I size effect. The detailed experimental procedure and the results are reported in this paper. Under the assumption that a distinct continuous crack develops only after the peak load, the approximate law of size effect is derived from the stress redistribution due to a boundary layer of cracking. The analysis leads to a deterministic Type I size effect.

Computation of Probability Distribution of Strength of Quasibrittle Structures Failing at Macrocrack Initiation

AbstractEngineering structures must be designed for an extremely low failure probability, Pf<10-6. To determine the corresponding structural strength, a mechanics-based probability distribution model is required. Recent studies have shown that quasibrittle structures that fail at the macrocrack initiation from a single representative volume element (RVE) can be statistically modeled as a finite chain of RVEs. It has further been demonstrated that, based on atomistic fracture mechanics and a statistical multiscale transition model, the strength distribution of each RVE can be approximately described by a Gaussian distribution, onto which a Weibull tail is grafted at a point of the probability about 10-4 to 10-3. The model implies that the strength distribution of quasibrittle structures depends on the structure size, varying gradually from the Gaussian distribution modified by a far-left Weibull tail applicable for small-size structures, to the Weibull distribution applicable for large-size structures. Com...