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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.

The chairman of this symposium has asked the writer to prepare a general and critical discussion of inelastic reinforced concrete design. This forced him to study more thanadozenof the preprinted papers, in considerable detail, in an attempt to assess the present state of knowledge and of the art.

The behavior of long restrained concrete columns as part of a building frame is much more complicated than that of long hinged concrete columns under eccentric load. A theoretical analysis for determining the critical column length for long hinged concrete columns has been derived previously by the writer. A method for determining the critical column length for long concrete column as part of a box frame is presented here. A long concrete column may buckle laterally as the critical section of the column reaches material failure; but the material failure of a column cannot be used as the criterion to determine the criticalcolumn length. Plastic hinges may be developed in a frame, but a long column may become unstable without developing plastic hinges. An analog computer was used as a tool to determine the critical column lengthfor the following reasons: (1) The problems involve differential equations which are particularly suitable for analog computer solutions (involving typically about 30 sec of computer time for a solution of adequate design accuracy); (2) the plotter, which is a standard unit of the computer, will plot the column or beam deflection curves on graph paper for visual reference; (3) the programmer can more readily make designdecisions by selection of proper constants for each preliminary trail of the problem. Concrete columns, subjected to eccentric loads at the ends will deflect laterally. As the columndeflects laterally the column moment along the column length will be increased by an amount equal to the product of column load and lateral displacement. This increment of moment becomes very important for the analysis of long columns. As the column deflects laterally, cracks will usually appear at the convex side of the column near the region of maximum moment. The error in using a constant EI (modulus of elasticity x moment of inertia) approximation to determine critical column length may be of substance. In considering variable E and I along the deflected column, moment versus edge-strain relationships was derived for a given column with a given column load. A nonlinear second order differential equation can then be obtained from each moment versus edge-strain curve. An analog computer was used to solve the differential equation and the column deflection curves and angle of rotation curves were plotted on graph paper by the computer plotter for a given column with given column load P. For any given values of end moment ME and the column load P, the critical column length for eccentrically loaded hinged column can be easily determined from the column deflection curves. The long column as part of a symmetrical box frame was further studied. It is assumed that all joints are rigid and that the joints do not move laterally. The end rotation 0E of the column must be equal to the end rotation of the beam, and the end moment ME of the column must equal to the end moment of the beam. For a given box frame with given column and beam loads, the critical column height can be determined. It is found that the co-tangency criterion for determining the critical column length for eccentrically loaded hinged column is not always applicable for determining the critical column length for restrained column.

SP12 Contains the proceedings of the 1964 International Symposium on Flexural Mechanics of Reinforced Concrete. In addition to providing a more basic understanding of the complex, non-ideal flexural behavior of reinforced concrete, this publication aims to further both immediate and long-range objectives in improving the analytical and statistical basis for the flexural design of reinforced concrete.

With discussion by Edward G. Nawy, H.A. Sawyer, M.Z. Cohn, and E.F.P. Burnett and C.W. Yu. An attempt is made to evaluate our present knowledge with regard to the analysis and design of reinforced concrete linear structural systems at ultimate load. The fundamental difference between the moment curvature concept and moment rotation concept is emphasized and discussed in detail. The authors have attempted to outline previous significant work, to underline a few basic principles, bearing in mind the difference between these two concepts, and to indicate the present extent of our knowledge of this subject with an appreciation of the assumptions and simplifications that are entailed. Readers are assumed to have some basic knowledge of some of the better known work on the subject, such as Sawyer’s or Baker’s work.