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 object of the paper is to provide a simple, rational technique to check the rotation compatibility of plastic hinges in limit designed reinforced concrete continuous beams proportioned basing on optimum considerations. The relationship between the plastic adaptability and the rotation compatibility is outlined, expressing conveniently both the inelastic rotations and the rotation capacities of critical sections. It is concluded that the compatibility requirement implies only limited adaptability tobe used in the design of concrete structures. Since a similar conclusion can be derived with regard to the serviceability conditions of limit designed structures, adoption of convenient upper bounds for the redistribution factors (or lower bounds for the yield safety parameters) of critical sections will implicitly provide adequate solutions for ultimate safety, compatibility, and serviceability as well. From the practical viewpoint, the significant result follows that for given ( 1) properties of materials, (2) loading conditions, and (3) amount of accepted redistribution, the rotationcompatibility condition to an upper limitation of the steel percentages at critical sections.

Plastic analysis is applied to evaluation of the membrane action in transversally loaded reinforced concrete slabs with edges restrained against lateral movement. Relations of the large deflection theory of flexure together with the yield condition, appropriate for reinforced concrete slabs, are used in order to obtain the load-deflection curves both in the compressive and tensile membrane action. The membrane action is found to influence considerably the actual carrying capacities of slabs. The developed method yields a continuous transition from the compressive membrane response to the tensile one.

With Discussion by D. H. Clyde, M. P. Nielsen, and R. H. Wood. Yield-line theory for slab design as pioneered by Johansen, has always presented the designer with two alternative methods. The first method is to evaluate the dissipation of energy belonging to any chosen mode of collapse, from which the corresponding collapse load is obtained, the layout of yield lines for the worst mode being found by trial and error. This is known as the "work method" and is on a firm mathematical foundation, even if sometimes slow in application. The second method is the "equilibrium" method using "nodal" forces where yield lines meet, or where they meet edges. This quick method has been popular with designers, but the foundations of the theory are in dispute, and on occasions it gives false results or else provides no results at all. The reasons for breakdown are discussed herein and new techniques are evolved for overcoming the difficulties. In this new outlook there are not, in fact, two separate methods, but merely two mathematical rearrangements of the same approach. The argument brings out the observation that there is a disturbing lack of information on the yield criterion for bending of slabs.

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.