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18 2.3.1 Flexural Resistance The nominal moment capacity (Mn) for non-prestressed members is commonly calculated by assuming a constant yield stress for the steel. For bars without a well-defined yield plateau, several approaches may be used to define the yield stress. In order to examine these methods, parametric stud- ies were performed to assess the flexural resistance of mem- bers reinforced with various grades of steel reinforcement that do not have a well-defined yield plateau. The moment capacity was calculated by a number of methods ranging from Figure 3. Ramberg-Osgood Curve simple design-oriented procedures to complex fiber analysis. and definition of parameters. In fiber analysis, a cross section is divided into layers (fibers). The cross sectional and material properties for each layer are defined, and strain compatibility between the layers is enforced. A critical objective of the present work is to identify an Realistic complete stress-strain relationships for concrete and appropriate steel strength and/or behavior model to ade- steel layers are employed as opposed to simplified relation- quately capture the behavior of high-strength reinforcing ships typically used in the strain compatibility method. There- steel while respecting the tenets of design and the needs of fore, complex analyses can be performed by fiber analysis the designer. As will be described throughout this report, a technique. Comparing the results from the range of models value of yield strength, fy, not exceeding 100 ksi was found to made it possible to evaluate whether approximate methods be permissible without requiring significant changes to the are appropriate for members reinforced with reinforcing bars LRFD specifications or, more critically, to the design philos- with no clear yield plateau and what material properties to ophy and methodology prescribed therein. Some limita- use in these cases. tions to this increase in permissible yield strength were identified and also are discussed. Based on the stress-strain diagrams obtained as part of the reported project and all pre- 2.3.1.1 Members and Parameters vious studies, A1035 reinforcing steel easily meets a yield Sections modeled were deck slabs, rectangular beams, and value of 100 ksi using the 0.2% offset method or for the `stress T-beams with varying steel types, amounts of steel, and con- at a strain' method for strains exceeding 0.004. All available crete compressive strengths. The variables considered are test data exhibit nonlinear behavior at stresses greater than summarized in Table 5. A total of 286 cases were examined. 70 ksi. Thus, it is felt that assumptions of a linear stress-strain Three different amounts of tensile reinforcement were incor- relationship made for calculating service load displacements porated in the rectangular beams. A maximum area of steel, and crack width are likely adequate since service load stresses As,max was determined based on the minimum steel strain of are traditionally taken as fs = 0.60fy. However, deflection or 0.004 imposed by ACI 318-08 (ACI 2008). A minimum area serviceability considerations at loads greater than this must of steel, As,min, was established to satisfy AASHTO 5.7.3.3.2 account for the nonlinear nature of the reinforcement at high (i.e., to ensure that the flexural resistance with As,min is at least stresses. 1.2Mcr, where Mcr is the cracking moment of the section). The Post-yield behavior, particularly when employing a plastic average of As,min and As,max also was considered. Rectangular design methodology, will also be affected by both the lack of beams with As,min are in the tension-controlled region. Rec- a well-defined yield plateau and the nonlinear post yield tangular beams reinforced with As,max have the lowest steel behavior. This behavior is most critical in seismic applica- strains allowed by ACI 318-08. The average of As,min and As,max tions, which are beyond the scope of the present work. results in cross sections with strains between these limits. Because of the additional compression strength provided by the flanges of the T-beams, the calculated amount of steel 2.3 Flexural Reinforcement required to provide As,max (i.e., to ensure a minimum strain of Flexural behavior and design of members reinforced 0.004) was found to be excessive and impractical. Therefore, with A1035 reinforcement and other grades of reinforcing the values of As,max determined for the rectangular beams were bars that do not exhibit well-defined yield plateaus were provided in the corresponding T-beams. Nonetheless, the examined analytically and experimentally. Different aspects selected values of As,max resulted in members that fell well into of this component of the research are presented in this the tension-controlled region. Providing more steel to obtain section. members in the transition region was impractical; hence, only

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19 Table 5. Variables for parametric studies. Parameter Deck Slab Rectangular Beam T-Beam Dimensions 7 in. and 10 in. thick 12 in.16 in., 12 in.28 in., 12 in.28 in., 12 in.36 in., 12in.36 in., 16 in.28 in., 16 in.36 in., and 16 in. 16 in.36 in., and 16 in. 40 in. with 96 in. effective 40 in. flange width and 7 in. flange thickness Concrete Strength, f' c 5 ksi 5, 10, and 15 ksi Reinforcement Grades A706, A496 & A82, A955 (3 grades), and A1035 Bar Sizes #4, #5, and #6 #6 for 12 in.-wide beams; and #8 for 16 in.-wide beams. All beams are assumed to have #4 stirrups with 2 in. of clear cover. Tension Reinforcement Based on AASHTO As,min, As,max, As,max from corresponding spacing limitations 0.5(As,min+As,max) rectangular beams one amount of reinforcement was used for the T-beams. 2.3.1.3 Results The amount of steel provided in the slabs was determined The moment capacity for each section computed based on based on spacing limitations prescribed in AASHTO LRFD the aforementioned methods was normalized with the corre- Bridge Design Specifications, 5.10.3.1, 5.10.3.2, and 5.10.8 sponding capacity calculated from the fiber analyses. Table 7 (AASHTO 2004). summarizes the results of the strain compatibility analyses conducted using the Ramberg-Osgood function for the rec- 2.3.1.2 Capacity Calculation Procedures tangular beams, T-beams, and slabs for all of the steel types and the selected concrete strengths considered. The com- The nominal moment capacity of each section was calcu- puted capacities are below or nearly equal to those calculated lated both by a strain compatibility procedure using different based on fiber analysis (i.e., the ratios are close to, or slightly methods for modeling the steel stress-strain relationships and less than, unity). The exceptional estimates of the expected a fiber analysis procedure. A commercial computer program capacity based on the Ramberg-Osgood function in conjunc- XTRACT (2007) was used to perform the fiber analyses. The tion with strain compatibility analysis should be expected concrete was modeled using the unconfined concrete model since this function closely replicates the measured stress-strain proposed by Razvi and Saatcioglu (1999). The measured curves that were used in the fiber analyses. Additionally, the stress-strain data (refer to Appendix A) for each type of rein- good correlation suggests that well-established procedures can forcing steel were input directly into the XTRACT program. be used to calculate the flexural capacity of members rein- By using the experimentally obtained data, a more accurate forced with bars that do not have a well-defined yield plateau capacity can be determined. Moment-curvature analyses were so long as the stress-strain relationship is modeled accurately. run in which the concrete strain was limited to 0.003, the level In spite of its success, the use of Ramberg-Osgood func- of strain used in the strain compatibility analyses. The results tions is not appropriate for routine design. Most designers are from fiber analyses are deemed to predict the most accurate familiar with using a single value of reinforcing bar yield, fy. flexural capacity. For this reason, further strain compatibility analyses were An Excel program (Shahrooz 2010) was used to compute carried out using the yield strength values given in Table 6. flexural capacities based on strain compatibility analysis. The The results are summarized in Table 8. For the beams having constitutive relationship of the reinforcing bars was modeled 5 ksi concrete, the ratios from any of the values of yield (1) as elastic-perfectly plastic with the yield point obtained by strength are less than unity (i.e., the flexural strength can be the 0.2% offset method and the stress at both strain = 0.0035 conservatively computed based on any of three methods used and strain = 0.005; and (2) by the Ramberg-Osgood (1943) to establish the yield strength). The same conclusion cannot function determined to best fit the experimentally obtained be drawn for the beams with 10 and 15 ksi concrete. For a data. The analyses utilized data from the measured stress- strain relationships of 102 samples of A706, A496 and A82, A955, and A1035 reinforcing bars. The measured relation- Table 6. Average and standard deviations of fy (ksi). ships are presented in Appendix A. Table 6 summarizes the Method for Establishing the Yield Strength yield strengths obtained from each method. 0.2% Offset Method Strain @ 0.005 Strain @ 0.0035 Bar An equivalent stress block for high-strength concrete, Avg. Std. Dev. Avg. Std. Dev. Avg. Std. Dev. developed as part of NCHRP 12-64 (Rizkalla et al. 2007), was (ksi) (ksi) (ksi) (ksi) (ksi) (ksi) A496 & A82 93 6.02 93 5.71 88 5.95 used to compute the concrete contribution to section behav- A706 68 3.30 68 3.83 67 3.05 ior. Additional information is provided in Appendix B and A995 78 5.21 78 5.21 72 3.53 Ward (2009). A1035 127 7.25 115 4.59 93 4.01

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20 Table 7. Ratios of flexural capacity Table 9. Cases where elastic- determined from Ramberg-Osgood strain plastic analysis overestimated compatibility analysis to that determined flexural capacity. from fiber model. Steel Type f'c (ksi) g Ratio 10 3.84% 1.014 Section Average Minimum Maximum Standard Deviation 15 3.67% 1.006 Rectangular 0.944 0.835 0.999 0.037 15 3.67% 1.005 T-beam 0.962 0.925 0.999 0.017 A706 15 4.06% 1.022 Slab 0.875 0.668 0.955 0.107 15 4.11% 1.023 Note: Ratio less than 1 is conservative. 15 2.88% 1.072 15 4.07% 1.023 A995 15 3.43% 1.020 A1035 15 2.65% 1.007 limited number of cases (given in Table 9) involving relatively large longitudinal reinforcement ratios (g), the strength ratio exceeds unity if the capacity is based on an idealized elastic- crete compressive stress of 0.003 is reached. Thus, the higher perfectly plastic stress-strain relationship with the yield yield strength from the elastic-perfectly plastic model over- strength taken as the stress at a strain of 0.005 or determined estimates the actual flexural capacity. based on the 0.2% offset method. That is, the yield strengths In the case of T-beams and slabs, any of the aforemen- based on these two methods may result in slightly unconser- tioned methods for establishing the yield strength result in vative estimates of the expected capacity in cases with large acceptable, conservative flexural capacities. As is evident from reinforcement ratios and high-strength concrete. Table 10, the ratios of the flexural capacity based on simple The aforementioned behavior can be understood with ref- elastic-perfectly plastic models to the corresponding values erence to Figure 4, which depicts a measured stress-strain curve from fiber analysis are less than one. The trend of data is for an A706 bar along with the idealized elastic-perfectly plastic expected, as the longitudinal strain in a T-beam will be higher model based on the yield strength taken as the value determined than that in an equivalent rectangular beam because of the addi- from the 0.2% offset method and the stress at strain equal to tional compressive force that can be developed in the flange. 0.005. Note that in this case, these two methods result in the The smaller depths of the slabs will also increase the strain in same values of yield strength. Between points "a" and "b" (see the longitudinal bars. In both these cases, the larger strains Figure 4) the elastic-perfectly plastic model deviates from the will correspond to cases beyond the strain at point "b" in Fig- measured stress-strain diagram. The stresses based on this ure 4, where the elastic-plastic assumption underestimates model exceed the actual values. For strains below point "a" the real stress developed in the steel. and strains above "b," the stresses from the idealized model are equal to, or less than, the measured values. As the rein- 2.3.1.4 Summary and Recommendations forcement ratio increases (i.e., as the amount of longitudinal steel becomes larger), the strain in the reinforcing bars at any Considering the presented results, the use of Ramberg- given applied moment will become less. For the cases involv- Osgood functions for defining the stress-strain characteristics ing the large reinforcement ratios shown in Table 9, the steel of reinforcing bars without a well-defined yield plateau will strains fall between points "a" and "b" when the extreme con- produce the most accurate estimate of the actual flexural Table 8. Ratios of rectangular beam flexural capacity calculated from elastic-plastic analyses to that from fiber model. Yield Point f'c (ksi) Average Minimum Maximum Standard Deviation 5 0.820 0.578 0.958 0.094 @ Strain = 0.0035 10 0.815 0.603 0.964 0.100 15 0.825 0.596 0.991 0.108 5 0.884 0.727 0.977 0.070 @ Strain = 0.005 10 0.880 0.652 1.014 0.084 15 0.891 0.688 1.072 0.092 5 0.909 0.789 0.966 0.057 0.2% offset 10 0.884 0.756 0.971 0.075 15 0.890 0.749 1.007 0.092 Note: Ratio less than 1 is conservative.