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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 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.

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.