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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.

With discussion by Theodore Zsutty, Jack R. Benjamin, C. Allen Cornell, and Milik Tichy and Milos Vorlicek. Because the ultimate strength and deformation ability of critical sections are random variables, the ultimate strength of a structure must likewise be a random variable. If the structure is subjected to load from one source and there is only one possible collapse mechanism, the determination of the ultimate strength ZU of the structure is simple. If the structure is subjected to load from one source but there are m possible collapse mechanisms, it becomes necessary to analyze the structure with the aid of equations of the type given herein. The ultimate strengthZUj, for j = 1, 2, . . . , m of the structure is determined by means of each of these equations assuming the occurrence of the j-th collapse mechanism. The probability pUj that the structure will change into the jth mechanism may be ascertained for a definite value of the load for each random variable ZUj But the actual probability of failure must be expressed with the aid of the so-called conditional probabilities since the individual mechanisms are not always statistically independent. If the structure is subjected to load from v sources and there are m possible collapse mechanisms an equation for the jth mechanism will graphically be represented by an interaction diagram. For a given population of structures, identical according to the design, there exists a number of possible combinations of load with a corresponding probability of failure pU. Geometrically speaking, they are points in the v - dimensional space. Their locus is the so called boundary of the safe domain IImin. When the deformation ability of a structure is considered, the system of equations forms the starting point. In this instance the random variable Zuj is a linear combination of ultimate moments MUi and the ultimate plastic rotation 0U of the section. The statistical solution is analogous with the previous one. It may be demonstrated that the variability in ultimate strength of a redundant structure is lower than that of a statically determinate one in all cases. Consequently, the application of the statistical method must result in savings of material in redundant structures.

With discussion by Milik Tichy and Milos Vorlicek; and Herbert A. Sawyer, Jr. Because structural failure generally occurs in successively more severe stages at successively less probable loads, design should ideally account for all stages and be based on comprehensive analysis utilizing a comprehensive, non-linear, force-strain relationship. The criterion for optimum design, using the failure-stage-versus-load profile, is derived. For frames, a method of comprehensive analysis based on a multilinear moment-curvature relationship, using critical moments and "plasticity factors," is presented. Procedures and the relative economics of comprehensive design and its special cases, elastic, plastic, and ultimate strength designs, are compared. A bilinear design procedure for concrete frames, based on two failure stages, is presented.

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 Milik Tichy. Evolution of moments distribution in reinforced concrete indeterminate structures is followed by means of real moment-rotation curves and imposition of compatibility conditions. Theory shows that such a redistribution begins at appearance of first crack and that its amount is already considerable at service load. Redistribution is present also if the structure is designed for bending moments by elastic theory; Therefore in this case its effect is unfavorable. Tests on 2 continuous beams (with measure of reactions) confirm the results of theory and show that the assumption of an elastic distribution of moments can lead to an overestimation of carrying capacity of structures. This danger is particularly important when a high percentage of reinforcement or the presence of axialload considerably reduce the rotation capacity of individual sections (brittle sections). The real behavior of such structures can be easily followed when they are not too complex. The method of "imposed rotations" applied to the tested continuous beams-involves considering inelastic rotations as rotations artificially imposed on critical sections of the structure, which is still considered to be acting elastically. The conclusion is that elastic distribution of moments is not a suitable basis for limit design of reinforced concrete structures and that inelastic calculations seem necessary in all cases. If certain conditions are fulfilled avoiding brittle sections, a great freedom in design seems possible, without any control of compatibility. In the other cases, the proposed method can be used if structures are not too complex; for complex frames simple rules can be found by further research.