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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.

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.

With discussion by Peter R. Barnard, George Pincus, Charles A. Rich, and Gerald Sturman, Surendra P. Shah, and George Winter. Inelastic behavior of concrete was studied by direct observations of internal microcracking. Thin slices were made from strained specimens and internal cracks were examined by X-ray and microscope techniques. Bond cracks at the interface between coarse aggregates and mortar, exist in concrete even before any load is applied. Analytical and experimental studies showed that tensile stresses are present at the mortar-aggregate interface because of volume changes of mortar and may be partly responsible for bond cracks in virgin concrete. These bond cracks begin to propagate noticeably at applied compression stresses of one-quarter to one-third of the ultimate strength. At this level the stress-strain curve begins to deviate from a straight line. At about 70% to 90% of ultimate strength cracks through mortar begin to increase noticeably and bridge between bond cracks to form a continuous crack pattern. Upon further load increase this condition eventually leads to a descending stress-strain curve and failure. Other investigators have noted that in that same load range, the volume of concrete begins to increase rather than decrease. An hypothesis explaining this volume expansion and propagation of bond cracks in terms of shear bond strength of the interface and microcracking has been presented. In order to investigate the influence of flexural strain gradients, microcracking and the stress-strain relation of eccentrically loaded specimens were compared with those of concentrically loaded specimens, It was found that a flexural strain gradient definitely retards microcracking, especially mortar cracking as compared to cracking at the same strain in axial compression. The stress-strain curve for eccentric compression, which was computed by an experimental-statistical approach was found to differ materially from that for concentric compression. The peak of the flexural curve was located at a strain about 50% larger and at a stress about 20% larger than the peak of the curve for concentric compression. Structural implications of these findings are briefly examined.

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.