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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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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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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.