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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.

With discussion by E. Burnett, D. B. Beal, R. H. Wood, and A. L. L. Baker. The moment-rotation results are presented of tests on beams carried out by a number of laboratories working under the auspices of the European Concrete Committee. Idealized diagrams are plotted for comparison, and the basis of these diagrams is given as defining fundamental moment curvature relationships which may be used in ultimate load calculations of frameworks. A simple trial and adjustment method of design is explained in which compatibility of bending moment values and end-slopes can be established by joint by joint procedure. Simplification is effected by separating the sway angle from the total rotation at hinges. This simplification can either be made by joint trial and adjustment procedure or by using the Miiller-Breslau compatibility equations, separating the sway angle, which greatly simplifies these equations.

With discussion by Chan W. Yu and M. T. Soliman, and Alan H. Mattock. Limit design theories for reinforced concrete statically indeterminate structures require a knowledge of the rotational capacity of hinging regions in reinforced concrete members. An investigation is reported of this rotational capacity in reinforced concrete beams. Thirty-seven beams were tested involving the following variables: concrete strength, depth of beam, distance from point of maximum moment to point of zero moment, and amount and yield point of reinforcement. The data are analyzed and a method is proposed whereby the rotational capacity of a hinging region in a reinforced concrete beam may be calculated.