ACI Structural Journal/July-August 2013 681 Title no. 110-S56 ACI STRUCTURAL JOURNAL TECHNICAL PAPER ACI Structural Journal, V. 110, No. 4, July-August 2013. MS No. S-2011-270.R1 received August 29, 2011, and reviewed under Institute publication policies. Copyright © 2013, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the May-June 2014 ACI Structural Journal if the discussion is received by January 1, 2014. On the Probable Moment Strength of Reinforced Concrete Columns by José I. Restrepo and Mario E. Rodriguez The probable moment strength (or flexural overstrength, as it is also known) is the theoretical maximum flexural strength that can be calculated for the critical section of a member, with or without axial load, subjected to bending in a given direction. In ACI 318, this strength is needed to capacity-design beams, columns of special-moment frames, and columns not designated as part of the seismic-resisting system. Supported on a column database, this paper provides evidence that the current method prescribed by ACI 318 to calculate this strength has a clear nonconservative bias and explains the reasons for this. To improve predictability, the authors propose a very simple, statistically calibrated mechanics model for determining the probable moment strength of rectan- gular and circular columns. An extension of the concept is made for computing the probable moment strength of rectangular columns subjected to bending along the two principal axes. Keywords: biaxial bending; capacity design; codes; confinement plastic hinges; long-term concrete strength; probable moment strength; reinforced concrete columns; seismic design. INTRODUCTION The probable moment strength (or flexural overstrength, as it is also termed by other codes 1-3 and textbooks 4 ) is the theoretical maximum flexural strength that can be calculated for the critical section of a member, with or without axial load, subjected to bending in a given direction. The prob- able moment strength is needed to calculate design forces to capacity protect any member where plastic hinges may develop, particularly if the kinematics of the mechanism of inelastic deformation indicates so. Examples of the former are the bases of first-level columns in buildings and building columns not designated as part of the seismic-resisting system framing into strong beams. For instance, in ACI 318-11, 5 the probable moment strength is needed to calculate the design shear forces of beams of special-moment frames. This is done to capacity protect these members by reducing the potential for shear failure during a rare but intense earth- quake. Moreover, ACI 318-11 5 specifies that all columns of special-moment frames in buildings and columns not desig- nated as part of the seismic-resisting system be capacity designed. Furthermore, this code specifies that when plastic hinges will likely develop in columns, the design shear force has to be determined using the column end probable moment strengths, regardless of the shear forces obtained from the structural analysis. Other codes 1-3 have similar requirements. In ACI 318, the probable moment strength is calculated using a simplified theory for flexure, where an elasto-plastic stress-strain relationship is assumed for the steel reinforce- ment, a rectangular stress block is assumed for concrete in compression, and strain compatibility is enforced, accepting the hypothesis that plain sections before bending remain plane after bending. In this analysis, the yield strength of the reinforcement is made equal to 1.25f y , where f y is the speci- fied yield strength of the reinforcement. The ACI 318 approach does not account for the likely increase in the concrete compressive strength over the speci- fied strength in the computation of the probable moment strength. The compressive strength of concrete batched, delivered to a construction site, and placed in a member following accepted quality control procedures should be similar to—if not greater than—the specified strength at the specified date, typically at 28 days. However, most concrete types continue to gain significant strength over time, 6-8 even in a dry environment 9 or in harsh environments subjected to freezing-and-thawing cycles. 10,11 The presence of passive confinement, by way of closely spaced hoops, also causes an additional strength increase. Moreover, the presence of an elastic member, such as a footing or beam-column joint, at the framing end of a member results in additional local concrete strength gain. 12-15 This is because this elastic element effec- tively confines the compressed concrete by preventing it from expanding transversely. The greatest manifestation of this local effect is the reduction in concrete cover spalling at the member end and a shift of the critical section away from the end. 16,17 In lightly axially loaded columns, a signifi- cant increase in the concrete compressive strength has only a minor influence on the probable moment strength. For this reason, the increase in the concrete compressive strength can be ignored in calculations. However, as the axial load increases, the probable moment strength becomes more sensitive to the compressive strength of the concrete. In the context of capacity design, an underestimation of the prob- able moment strength can result in a reduction of the defor- mation capacity of a hinging column, as the intended ductile mode of response may be hampered by the development of another behavioral mode associated with reduced ductility. RESEARCH SIGNIFICANCE ACI 318-11 5 specifies that columns in special-moment frames shall be capacity-designed. To achieve this objective when hinging is likely to occur in the columns, this code requires the computation of the probable moment strength at the column ends. This paper shows that the current approach in ACI 318 for computing the probable moment strength has a clear nonconservative bias. To improve predictability, the authors propose a very simple, statistically calibrated mechanics model for determining the probable moment