Showing papers by "Zdenek P. Bazant published in 1984"

Bayesian Statistical Prediction of Concrete Creep and Shrinkage

Abstract: In present design practice, the statistical approach is used for strength but not for deformations, including creep and shrinkage. However, predicting concrete creep properties from design strength and concrete composition involves a large uncertainty, much larger than that of strength. It is shown that by carrying out some short-time creep measurements, even rather limited ones, the uncertainty can be drastically reduced, and extrapolation of short-time measurements can be made much more reliable. This is accomplished by developing a Bayesian approach to creep prediction. Prior information consists of the coefficient of variation of deviations from the creep law for concrete in general, as determined in a recent statistical analysis of the numerous creep data that exist in literature. This information is combined, according to Bayes' theorem, with the probability of a given concrete's creep values to yield the posterior probability distribution of the creep values for any load duration and age at loading. Only a linear creep case is considered, and a normal distribution of errors is assumed for the given concrete as well as for the prior information. To demonstrate and verify the method developed, various creep data reported in literature are considered. Predictions made on the basis of only a part of the test data are compared with the rest of the data, and very good agreement is found. The effects of various amounts of measured data, and of various degrees of uncertainty in the prior information, are also illustrated. The present approach is recommended for concrete structures for which the creep deflections, creep-induced cracking, or creep buckling are of special concern, e.g., nuclear reactor vessels and containments, certain very large bridges, shells, or building frames.

Rock fracture via strain-softening finite elements

Abstract: The fracture of rock is assumed to arise from propagation of a blunt crack band with continuously distributed (smeared) microcracks or continuous cracks. This approach, justified by material heterogeneity, is convenient for finite element analysis, and allows analyzing fracture on the basis of triaxial stress‐strain relations which cover the strain‐softening behavior. A simple compliance formulation is derived for this purpose. The practical form of the theory involves two independent material parameters, the fracture energy and the tensile strength. The width of the crack band front is considered as a fixed material property and can be taken as roughly five‐times the grain size of rock. The theory is shown to be capable of satisfactorily representing the test data available in the literature. In particular, good fits are demonstrated for the measured maximum loads, as well as for the measured resistance curves (R‐curves). Statistical analysis of the deviations from the test data is also presented.

Deformation of progressively cracking reinforced concrete beams

Abstract: A consistent theory for the analysis of curvature and deflections of reinforced concrete beams in the cracking stage is presented. The theory assumes concrete to have a nonzero tensile carrying capacity, characterized by a uniaxial stress-strain diagram which characterizes progressive microcracking due to strain softening. The tensile stressstrain properties are the same as those which are obtained in direct tensile tests and those which have recently been used with success in modeling fracture test results for concrete. The theory agrees well with the simpler formula of Branson within the range for which his formula is intended. The value of the proposed theory is its much broader applicability. Aside from demonstrating a good agreement with available test data for short-time deformations up to the ultimate load, it is shown that the theory also correctly predicts the longtime creep deformations of cracked beams. To this end, the average creep coefficient for tensile response including peak stress and strain softening needs to be taken about three times larger than that for compression states. The theory also predicts the reduction of creep deflections achieved by the use of compression reinforcement, and a comparison of modeling this effect is made with an ACI formula. As a simplified version of the model, it is proposed to replace the tensile strain-softening behavior by the use of an equivalent tensile area of concrete at the level of tensile steel, behaving linearly. Assuming this area to be a constant, realistic predictions for shorttime as well as longtime deformations in the service stress range can still be obtained.

Imbricate continuum and progressive fracturing of concrete and geomaterials

Abstract: After giving an overview of the recent results at Northwestern University on mathematical models for fracturing heterogeneous materials, the lecture addresses the problem of a continuum model for strain-softening. In a classical, local continuum, strain-softening leads to unrealistic unstable response, such that failure localizes into a layer of vanishing thickness and occurs at vanishing energy dissipation. While the classical nonlocal continuum does not resolve the problem, solution is found in the form of a new type of nonlocal continuum, called the imbricate continuum, which represents the limiting case of a system of overlapping (imbricated) finite elements of a certain fixed characteristic size that is a material property.