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8 modulus varies from steel grade to steel grade. High-strength deflections and minimal cracking while at higher loads the steel does, however, exhibit a "proportional limit" where the member should display large deflections and sufficient crack- modulus begins to decrease as is evident in Figure 1a. Although ing to provide warning before reaching its ultimate capacity. this limit is partially a function of the steel capacity, it has been Both deflection and cracking are primarily a function of steel observed that A1035 steel behaves in an essentially linear man- strain near the tension face of the member and, in general, ner to at least 70 ksi regardless of ultimate capacity (Mast et al. desirable behavior of a member is related to ductility, which 2008 and this study). It is noted that while some empirical relates to yielding or inelastic deformation of the steel rein- A1035 stress-strain relationships capture the behavior at large forcement. For lower strength reinforcing materials, the only strains reasonably well, they fail to capture the initial linear way to obtain high strains near the tension face at nominal behavior accurately and therefore may not be appropriate strength is to ensure yielding of the tension steel; however, for for design. An R-O formulation or a piece-wise formulation high-strength reinforcement, yielding is no longer necessary (Mast et al. 2008) overcomes this issue. (Mast et al. 2008). The objective of the work reported by Mast et al. (2008) was to assess the adequacy of a proposed 100 ksi reinforcement 1.3.2.2 Fatigue Performance of High-Strength stress-strain relationship for A1035-compliant steel in order Steel Reinforcement to establish acceptable strain limits for tension-controlled and DeJong and MacDougal (2006) and DeJong (2005) pres- compression-controlled sections reinforced with this high- ent a study of the fatigue behavior of high-strength reinforc- strength steel. Mast et al. studied the behavior of concrete ing steel. DeJong conducted fatigue tests of ASTM A1035 beams subject to flexural and axial loads at service level and steel having a reported (0.2% offset) yield value of 116 ksi and nominal strength and determined the section behavior using ultimate tensile strength of 176 ksi. Tests on #3, #4, and #5 a cracked section analysis that satisfied equilibrium and com- bars demonstrated a fatigue strength (at N = 1 million cycles) patibility. They assumed an elastic concrete stress distribu- of 45 ksi. Companion tests on Grade 60 reinforcing bars had tion under service load and used the ACI rectangular stress a fatigue life of 24 ksi. block to model concrete at nominal strength. Although they El-Hacha and Rizkalla (2002) report fatigue tests of proposed a more complex empirical relationship for A1035 #4 and #6 A1035 reinforcing bars having a nominal yield stress-strain behavior, Mast et al. adopted an elastic-perfectly strength, fy = 120 ksi. The endurance limit is not established plastic steel stress-strain relationship in their analysis. They in this study (no tests having N > 500,000 were conducted) used a steel modulus, Es = 29,000 ksi and defined a plastic although the behavior is reported to be generally superior yield plateau at fy = 100 ksi. This approach is equivalent to to that expected for A615 bars. The tests were run with simply increasing the current code-prescribed limits on steel fmin = 0.2fy and the lowest stress range was S = 0.45fy = 54 ksi reinforcement yield strength to 100 ksi. at which the observed fatigue life was approximately N = For the nominal strength, Mast et al. performed a numer- 500,000 and 360,000 for the #4 and #6 bars, respectively. ical analysis considering a rectangular, singly reinforced- Projecting these S-N results to greater values of N would concrete section having a number of different reinforcement lead to results similar to that reported by DeJong (2005) and ratios. They considered a concrete compressive strength fc = superior to the behavior predicted by the present AASHTO 6500 psi and an ultimate compression strain cu = 0.003. Mast requirements (Appendix E). et al. considered elastic-perfectly plastic steel behavior hav- No other known studies have examined the fatigue per- ing yield strengths of fy = 60, 80, and 100 ksi. They calculated formance of high-strength reinforcing steel, although a num- balanced reinforcement ratios, b = 3.95%, 2.60%, and ber of studies have reported fatigue properties of reinforcing 1.85% for the values of fy = 60, 80, and 100 ksi, respectively. steel having fy < 60 ksi. These investigations are summarized For b values greater than these limits, the section capacity in Appendix E. Based on the data presented in Appendix E, it was controlled by concrete compression and was therefore is seen that no studies report an endurance limit less than 24 ksi unaffected by the steel grade used. For sections with b < in tension-tension (i.e., fmin positive) tests. 1.75%, the use of the 100 ksi elastic-plastic model typically underestimated the nominal moment capacity of the section with respect to the actual behavior. On the other hand, for 1.3.3 Flexural Reinforcement 1.75% < b < 2.7%, the use of the 100 ksi limitation over- To apply the higher material resistance factor, = 0.9 allowed estimated the capacity of the section by only a marginal by AASHTO (and ACI 318) in the design of tension-controlled amount (about 2.5%), which was considered insignificant reinforced-concrete flexural members, a member should for design purposes. exhibit a desirable ductile behavior. A desirable behavior Through a series of moment-curvature and deflection implies that at service loads, the member should display small analyses, Mast el al. demonstrated that a simple beam designed