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With discussion by Peter R. Barnard, George Pincus, Charles A. Rich, and Gerald Sturman, Surendra P. Shah, and George Winter. Inelastic behavior of concrete was studied by direct observations of internal microcracking. Thin slices were made from strained specimens and internal cracks were examined by X-ray and microscope techniques. Bond cracks at the interface between coarse aggregates and mortar, exist in concrete even before any load is applied. Analytical and experimental studies showed that tensile stresses are present at the mortar-aggregate interface because of volume changes of mortar and may be partly responsible for bond cracks in virgin concrete. These bond cracks begin to propagate noticeably at applied compression stresses of one-quarter to one-third of the ultimate strength. At this level the stress-strain curve begins to deviate from a straight line. At about 70% to 90% of ultimate strength cracks through mortar begin to increase noticeably and bridge between bond cracks to form a continuous crack pattern. Upon further load increase this condition eventually leads to a descending stress-strain curve and failure. Other investigators have noted that in that same load range, the volume of concrete begins to increase rather than decrease. An hypothesis explaining this volume expansion and propagation of bond cracks in terms of shear bond strength of the interface and microcracking has been presented. In order to investigate the influence of flexural strain gradients, microcracking and the stress-strain relation of eccentrically loaded specimens were compared with those of concentrically loaded specimens, It was found that a flexural strain gradient definitely retards microcracking, especially mortar cracking as compared to cracking at the same strain in axial compression. The stress-strain curve for eccentric compression, which was computed by an experimental-statistical approach was found to differ materially from that for concentric compression. The peak of the flexural curve was located at a strain about 50% larger and at a stress about 20% larger than the peak of the curve for concentric compression. Structural implications of these findings are briefly examined.

With discussion by Leonard G. Tulin and Kurt H. Gerstle, Ralph M. Richard and Stanley D. Hansen, and Peter R. Barnard. The purpose of this paper is to explain, in the light of recent research into the concrete stress-strain relationship in compression, the flexural behavior of statically indeterminate reinforced conrete beams when loaded to collapse. Based on the concept of concrete as a strain-softening material, it is shown that a length of a beam can continue to rotate when moment is falling off and that rupture will not occur unless the energy balance in the beam ceases to be satisfied. In a comparison between the inelastic behavior of structural steel and reinforced concrete beams, it is shownthat in the latter there is a distinct maximum load which such a beam can withstand; that hinging regions tend to contract rather than spread as in steel; that it is possible for some regions of a beam to be falling off in moment while the total load on the beam is increasing; and that moment redistribution occurs through falloff in moment at some sections as well as through inelastic action. Finally, the possible development of true collapse methods for the analysis or design of indeterminate reinforced concrete beams is discussed.

With discussion by M. Z. Cohn, Milik Tichy and Milos Vorlicek, and Herbert A. Sawyer, Jr. It is proposed that statically indeterminate beams and frames be designed for suitably low probabilities of failure for two failure stages. One stage would be wide cracking, using an elastic analyses for stresses at a section and for distribution of moments. The other stage would be crushing-spalling, for which the conventionalultimate strength analysis wouldbe used at sections, and an analysis based on a bilinear moment-curvature relationship and plasticity factors would be used for the distribution of moments. The required moment-curvature relationships and plasticity factors are derived and presented quantitatively. The design procedure based on these analyses is outlined, and revisions in present load factors, based on both a critical re-examination of simplebeam test results and the special characteristics of bilinear analysis, are recommended. Finally, the quantitative evidence available on the validity of the proposed method from the experimental investigations of continuous beams by Glanville and Thomas, Mattock, Ray and Nilsen, and Petcu and Cohn, is presented. Agreement is good within the limited range of these investigations.

With discussion by Peter R. Barnard, HerbertlA. Sawyer,,M.Z. Cohn, and Emilio Rosenblueth and Roger Diaz de Cossio. Failure due to crushing is a case of instability. The traditional nonlinear moment-curvature approach does not hold for systems exhibiting descending (unstable) portions in their force-deformation characteristics. A method based on a moment-rotation approach and an assumed contaminated zone is presented, which takes into account descending branches.

The aim of this work is to predict both the average value and the variance of the creep deflection of reinforced concrete beams under sustained loads. Two quite distinct problems emerge, the determination of a probabilistic model to predict the creep behavior of a concrete prism under axial compression, and the introduction of this description of material behavior into an analysis of the bending of a beam under an arbitrary vertical loading. The model of the creep mechanism of concrete is a simplified version of an earlier model suggested by one of the authors. Stochastic processes, namely varieties of the Markov birth process, are employed to represent both the viscous flow of the cement paste and the delayed-elastic effects caused by fluids -- water and viscous paste-initially trapped within the elastic skeleton of crystals and aggregate. In a manner similar to that developed by another of the authors for the bending of homogeneous beams of stochastically viscoelastic material, the bending of a reinforced concrete beam is formulated. The creep response of a unit length of concrete to a unit stress is assumedto be a stochastic process of the type presented in the first part of the paper. These arguments lead to the desired results, formulas which predict the mean and variance of the deflection of any point on the beam at any time. In addition, spatial and temporal covariance functions are obtained; the latter permits the engineer to take advantage of an early observation of the creep deflection to alter his prediction of later deflections and to reduce the variance of these predictions.