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85 4.1 Introduction On the basis of the experimental and analytical inves- tigation of the research described herein, modifications to the AASHTO LRFD Bridge Design Specifications, 8th edition (AASHTO 2017), minimum flexural reinforcement provisions are proposed. This section describes the methodology used to examine the current provisions and to propose changes or updates. A wide range of updates was evaluated, from complete replacement of the specifications with a new methodology to a recalibration of the existing specification on the basis of the analytical and experimental investigations. The consideration of alternative specifications was limited to methodologies that were not considered in NCHRP Web-Only Document 149: Recommended LRFD Minimum Flexural Reinforcement Requirements (Holombo and Tadros 2010). 4.2 Criteria for Evaluation The criteria for proposing modifications and updates to the AASHTO LRFD minimum reinforcement requirements are based on safety, economy, reliability, ease of use, and consistency with other provisions within the AASHTO LRFD specifications. This evaluation includes consistency in order to ensure uniform safety for all concrete elements covered in the AASHTO LRFD specifications. The criteria are as follows: â¢ Safety. All updates to specifications will be required to meet the minimum safety requirements for all concrete elements covered within the AASHTO LRFD specifica- tions. The proposed provisions are to continue to provide ultimate flexural strength (Mo) greater than the flexural cracking strength (mcr). Sections that do not meet this requirement would need additional flexural strength, which is similar to the resistance factor for non-tensioned controlled sections. â¢ Economy. A comparison of the amount of minimum flexural reinforcement is performed for each proposed specification update. Of particular concern are nega- tive bending regions of heavily prestressed structures, where a wide tension flange could result in a compression- controlled section that does not meet minimum reinforce- ment requirements. â¢ Consistency. The provisions should be consistent with the overall reliability methodology woven throughout the AASHTO LRFD. As emphasized, the provisions should provide a uniform level of safety for all concrete elements. â¢ Ease of use. This is a qualitative factor to compare the analytical and computational effort to determine whether the code requirements have been met. More importantly, if the requirement is easily derivable, it can easily be checked by using hand calculations. â¢ Reliability. The provisions should provide a uniform reliability for all concrete members covered with AASHTO LRFD specifications. In particular, the probability that Mo will be greater than mcr should be the same for all members. 4.3 Specification Update Evaluation Updates to the specifications evaluated herein include the development of a variable-depth factor, recalibration of the flexural strength factor, and replacement of the minimum reinforcement requirements with a maximum net tensile strain requirement. Most of the beams tested (except for one) as part of this investigation were designed with less than the AASHTO LRFD minimum flexural reinforcement require- ment. Since all of these beams demonstrated ductile behavior with Mo/mcr > 1, relaxation of the requirement is justified. For UNB1, it was noted that it did not have noticeable reserve capacities. However, this beam failed prematurely at the anchorage, and, therefore, Girder UNB1 is assumed to meet the expected ductility condition provided that premature anchorage failure is avoided. C H A P T E R 4 Specifications Update
86 4.3.1 Moment Resistance Greater than 1.33 Factored Moment Minimum reinforcement provisions are intended to reduce the probability of brittle failure by providing flexural capac- ity greater than the cracking moment. Article 188.8.131.52 of the AASHTO LRFD allows the designer to forgo this require- ment if resistance (Mr) is greater than 1.33 times the factored moment. Therefore, 33% additional resistance is necessary if sufficient ductility is not provided and the section is tension controlled. This amount is consistent with the resistance factor Ï = 0.75 specified for compression-controlled sections in Article 184.108.40.206 of the AASHTO LRFD. It is noted that the inverse of 0.75 is equal to 1.33. Both 1.33 and Ï address ductility or the lack thereof. The current AASHTO LRFD requires that compression- controlled sections meet the minimum reinforcement requirements of Article 220.127.116.11. Therefore 33% more strength is required if Mr is less than Mcr, even though additional strength is required for compression-controlled sections with reduced Ï. Negative bending regions of segmental box girder bridges can be subjected to this doubling effect because they are typically heavily prestressed and top flanges are relatively wide as compared with the bottom flanges. Reducing and/or eliminating the 1.33 additional strength requirement in compression-controlled and transition sec- tions could remedy the doubling effect. However, the require- ment also applies to transition sections. For consistency, the following underlined update is proposed: Mr â¥ 1.33Î±Mu or Mr â¥ Mcr where Mr = factored moment resistance, Mcr = factored cracking moment (AASHTO LRFD 18.104.22.168), Î± = strength factor for minimum reinforcement, 1.0 1.0 0.33 1.33 cl tl cl t( ) ( ) â¤ Î± = + e â e e â e â¤ et = net tensile strain in the extreme tension steel at nomi- nal resistance, per AASHTO LRFD Article 22.214.171.124, ecl = compression-controlled strain limit in the extreme tension steel, and etl = tension-controlled strain limit in the extreme tension steel. Using Î± instead of 1.33 is consistent with variable resis- tance factor (Ï) of AASHTO LRFD Article 126.96.36.199. The factor requires additional strength if the resistance is less than the factored cracking moment and the section is not com- pression controlled. For compression-controlled sections, additional strength is required with a lower resistance factor. For transition sections, Î± increases with increasing net tensile strain in proportion to the resistance factor, as illustrated in Figure 4-1. 4.3.2 Member Depth and Flexural Cracking Strength The recognition that depth influences flexural cracking strength was one of the main reasons for reducing the modulus of rupture ( fr) in AASHTO LRFD Article 188.8.131.52 from 0.37 f câ² to 0.24 f câ² . According to the commentary in AASHTO LRFD Article C184.108.40.206: Figure 4-1. Strength factor (`) for minimum reinforcement (Grade 60 rebar and prestress shown).
87 The flexural cracking stress of concrete members has been shown to significantly reduce with increasing member depth.â Shioya et al. (1989) observed that the flexural cracking strength is proportional to h-0.25, where h is the overall depth of the flex- ural member in inches. Based on this observation, a 36.0 in. deep girder should achieve a flexural cracking stress that is 36% lower than that of a 6.0 in. deep modulus of rupture test. The effects of member depth are included in the fib Model Code 2010 for Concrete Structures (International Federation for Structural Concrete 2010) for calculation of flexural cracking strength. The ratio of direct tension strength and flexural cracking strength (Afl) is calculated using the fol- lowing equation: 1 (4-1)fl f l 0.7 f l 0.7 A h h = Î± + Î± h = height of the beam [mm (ft)] Î±fl = 0.06 mm (3.27 ft) Development of this relationship is also based on fracture mechanics. However, no sources are referenced. A combination of the theoretical recommendation of Carpinteri and Corrado (2011), which suggested that flex- ural cracking strength is proportional to h-0.15, with the flexural cracking strength of full-sized members, including that obtained in the current project, is presented in Figure 4-2 as a function of the square root of the compressive strength. Members included in this plot have depths ranging from 8 in. to 10 ft. Members with depths of less than 8 in. were excluded from the set to intentionally eliminate modulus of rupture tests, which may not represent field- or plant-constructed concrete systems because of curing conditions and the effects of member size, as discussed in the commentary of AASHTO LRFD Article 220.127.116.11 and in NCHRP Web-Only Document 149 (Holombo and Tadros 2010). The data in Figure 4-2 were used to examine flexural strength/depth relationships. The procedure employed was as follows: 1. Normalize the large-scale specimen data sets by using the strength/depth formulas discussed previously. 2. Compute the mean, standard deviation, and coefficient of variation for the normalized data sets. 3. Compare coefficient of variation values for the five flexural strength/depth relationships and the constant depth data and make a recommendation as to whether a depth vari- able should be introduced, and which depth-dependent expression represents the data sets most closely. A summary of the statistical parameter for full-size test data based on an assumed normal distribution is shown in Table 4-1, where depth models proposed by various authors are examined. The first column of this table includes all data assuming no depth influence. The second column includes hâ0.15, as suggested by Carpinteri and Corrado (2011), the third includes hâ0.25, as suggested by Shioya et al. (1989), the fourth column includes hâ0.33, as suggested by Wright and Garwood (1952). The fifth includes the depth relationship specified in the fib Model Code 2010 for Concrete Structures (International Federation for Structural Concrete 2010). f r /â (f c ) (p si ) Figure 4-2. Observed fr / fcâ² test data for full-depth monolithic concrete members versus depth.
88 As shown, the average flexural cracking stress is below 0.24 f câ² (ksi) for all depth-specified models, except for one case. All methods of including depth improve the coeffi- cient of variation significantly. It appears that hâ0.15 and hâ0.25 equally reduce the coefficient of variation by the greatest amount. Use of hâ0.15 as a depth factor results in an average flexural cracking stress that is approximately 0.24 f câ² (ksi) [7.5 f câ² (psi)], corresponding to a depth of 1.0 ft, for normal weight concrete. Hence, using hâ0.15 instead of hâ0.25 as a depth factor to define the variation in flexural cracking stress has merit because it is consistent with the historical code-specified value for the modulus of rupture and would require minimal changes to the AASHTO specifications. As a result, the following relation is proposed for the purposes of evaluating minimum reinforcement: 12 (4-2)cr 0.15f f h r ( ) = where fcr = anticipated flexural cracking strength of full-depth member, fr = modulus of rupture as defined in Article 18.104.22.168, and h = member depth (in.) 4.3.3 Recalibration of Flexural Cracking Factors The factors influencing flexural cracking strength have been recalibrated based on the data developed in NCHRP 12-94 and other sources. The flexural cracking factors include: Î³1, the flexural cracking variability factor; Î³2, the prestress vari- ability factor; and Î³3, the ratio of specified minimum yield strength to ultimate tensile strength of the reinforcement. Since the Î³3 is not based on variability, and is simply a ratio, no calibration is necessary. 22.214.171.124 Flexural Cracking Variability Factor The modulus of rupture calculated in Article 126.96.36.199 is a mean or average value. This is not consistent with the specified f câ², which is a lower-bound value. Therefore, the flexural cracking variability factor, Î³1, accounts for the vari- ability of both the specified compressive strength and the flexural cracking strength. The difference between the specified f câ² and an average or mean compressive strength ( fcm) is not explicitly addressed in the AASHTO LRFD code or commentary. Chapter 5 of the ACI Building Code Requirements for Reinforced Concrete (ACI Committee 318 2014) suggests that the difference between the two variables varies from 1,000 to 1,400 psi, depending on the compressive strength, as shown in Box 4-1. Further, this chapter indicates that the nominal (specified) compressive strength is 1.34 standard deviations from the mean, which implies that 10% of all possible strength mea- surements may be expected to fail. Conversely, the fib Model Code 2010 for Concrete Structures (International Federation for Structural Concrete 2010) defines the specified (charac- teristic) strength as the strength at which 5% of all possible strength measurements may be expected to fail. Statistic fib Model Code 2010 Average 6.29 7.23 7.97 5.72 5.35 Standard deviation 1.44 1.17 1.25 1.01 0.96 Coefficient of variation 0.23 0.16 0.16 0.18 0.18 Note: and in pounds per square inch; h in feet. Table 4-1. Statistical parameters of full-size concrete member flexural cracking stress when normal distribution is assumed. Box 4-1. Code requirements for mean compressive strength. Building Code Requirements for Reinforced Concrete fcm = fcâ² + 1,000 (fcâ² < 3,000 psi) = fcâ² + 1,200 (3,000 â¤ fcâ² â¤ 5,000 psi) = fcâ² + 1,400 (fcâ² > 5,000 psi) = fcâ² + 1.34s Source: ACI Committee 318 2014. fib Model Code 2010 for Concrete Structures fcm = fcâ² + 8 MPa (1,160 psi) = fcâ² + 1.65s Source: International Federation for Structural Concrete 2010.
89 The full-scale test data, discussed previously, is based on compressive strength measured on the day of testing, and not based on the specified strength. Therefore, the statistical analysis is not indicative of the actual relationship between specified compressive strength and flexural cracking strength. For the ACI 318 (2014), the ratio of factored-to-mean flex- ural cracking strength is 2.0 for reinforced concrete members, which is implied by the direct minimum steel area formula. For prestressed concrete, the corresponding ratio is 1.2, in recognition of the large contribution of the prestressing on the flexural cracking moment and the corresponding reduced variability. It should be noted that in both cases, the specified flexural cracking strength is used and not the corresponding mean value. The fib Model Code 2010 for Concrete Structures (Inter- national Federation for Structural Concrete 2010) provides calculations for the mean flexural cracking stress and the maxi- mum, or upper bound, value with a factor of 1.3. This upper bound corresponds to a 95% confidence interval. This maxi- mum flexural cracking stress compares well with the factored cracking strength calculated using Equation 4.2 and a flexural cracking variability factor Î³1 of 1.6, as shown in Figure 4-3. For precast segmental structures connected with epoxy joints, flexural cracking may develop at critical section, which may be at the joints or away from the joints where the maximum moment occurs. Due to the laitance layer effect described by Megally et al. (2002) and observed in the current project, cracks developing between two adjacent segments, will not form at the joint interface. Instead, they will form at the ends of a segment adjacent to the joint where neither longitudinal reinforcement nor large aggregates are typically present, as shown in Figure 4-4. At these sections, members may possess a somewhat lower modulus of elasticity. In addition to the observations made during testing, the pres- ence of a weak layer adjacent to epoxy joints was repeatedly observed during demolition of the test units. Following the formation of flexural cracks, a segmental beam with unbonded prestressing will concentrate damage at one or more locations adjacent to epoxy joints because of the absence of mild steel reinforcement. As noted above, the modulus of rupture value at these locations may be smaller than what may be expected away from the ends of segments. Although a critical section at a nonâjoint location could experience flexural cracking at that location, as reported in Chapter 3, the eventual failure will not occur at that location. Additionally, segmental construction tends to involve longer spans where dead load typically dominates the demands. As a result, the positive bending regions typically have a low moment gradient, and cracking is unlikely to develop away from the joint. At negative bending locations, joints are typically located at critical locations. For these reasons, the flexural cracking factor (Î³1) of 1.2 should be used for precast segmental structures to be consistent with the AASHTO LRFD prior to the 2005 Interim Revisions (AASHTO 2005) and until significantly more data become available. On the basis of these observations, the following update for the flexural cracking variability factor (Î³1) of Article 188.8.131.52 is shown in the underlined text of the following: Î³1 = flexural cracking variability factor, = 1.2 (h/12)â0.15 for precast segmental structures, and = 1.6 (h/12)â0.15 for all other concrete structures, where h is the member depth (in.). In summary, the AASHTO LRFD flexural cracking strength variability factor Î³1 of 1.6 for monolithic members does not Depth (ft) fr /s qr t( f'c ) ( ps i) Figure 4-3. Comparison of the proposed method and the fib Model Code (International Federation for Structural Concrete 2010).
90 represent four standard deviations above the mean flexural cracking strength because the requirement is based on the specified compressive strength rather than the mean compres- sive strength. The additional strength gain with age beyond the 28-day strength could also affect and lower this apparent large factor of safety. Finally, considering that the factored AASHTO LRFD flexural cracking stress compares well with the upper-bound fib Model Code, no modification to this factor is recommended. 184.108.40.206 Prestress Variability Factor The level of prestress (Î³2) has a significant impact on the flexural cracking strength of concrete members, as discussed previously. Methods and research on anticipated prestress and the amount of prestress loss that is anticipated to occur over the life of the bridge are covered in detail in the PCI Bridge Design Manual, 3rd edition (Precast/Prestressed Concrete Institute 2011), for pre-tensioned members. The variability of prestress losses in pretensioned mem- bers has been evaluated by Steinberg (1995), Gilbertson and Ahlborn (2004), Tadros et al. (2003), and Al-Omaishi et al. (2009). Results of these studies are based on the variability of parameters, including jacking force, initial and final con- crete strengths, relative humidity, dimensional tolerances, age at jacking, and others. In all of these studies, Monte Carlo simulations were used to evaluate the overall variability of prestress losses. Using the AASHTO LRFD method for calculating prestress losses, Gilbertson and Ahlborn (2004) demonstrated that, for a 70-in. I-girder, prestress loss deviated from nominal levels by less than 4% within a 95% confi- dence interval. (a) Crack formation adjacent to an epoxy joint (b) Crack concentration adjacent to a joint (c) Separation of a segment during demolition (d) Concrete surface adjacent to a joint Epoxy Joint Epoxy Joint Figure 4-4. Formation of cracks adjacent to epoxy joints.
91 Tadros et al. demonstrated that long-term prestress loss due to creep, shrinkage, and relaxation can vary by as much as 30% from the mean value. Considering that prestress loss is about 17%, the variation in the prestress force can be as much as 0.3 î° (0.17) = 0.05. Consequently, a reduction in the prestress factor for pretensioned members could be justified. Variability of frictional losses in post-tensioned structures can be significant in structures with long tendons. This is especially true for cast-in-place, post-tensioned box girder structures. These structures can have frames that exceed 500 ft in length with continuous tendons. In summary, although a reduction in the prestress factor could be justified for pretensioned members, the current value of 1.1 is retained because of the variability in frictional losses in post-tensioned structures with internal (bonded) tendons. The experimental study conducted in the current NCHRP project also did not provide sufficient data to alter this value. 4.3.4 Maximum Net Tensile Strain The current study also evaluated the maximum net tensile strain (etm) as a replacement for minimum flexural reinforcement. In this method, etm is the net tensile strain corresponding to a factored nominal resistance (Mr) that equals the factored flexural cracking strength (Mcr), accord- ing to AASHTO LRFD Article 220.127.116.11.2. It should be noted that etm is unique to each cross section. Net tensile strains beyond etm require a resistance factor (Ï) of 1/1.33 Ã 0.9 = 0.7 for reinforced concrete members. This method treats the minimum flexural reinforcement in a manner similar to compression-controlled sections. Although this method is attractive because it eliminates the calculation of Mcr, the method does not provide a consis- tent level of safety because it does not consider pre-cracking behavior. For example, a tee girder would have the same minimum reinforcement as a rectangular girder that had the same width and depth but would have substantially different cracking moments. Therefore, adoption of the maximum net tensile strain method is not recommended. 4.4 Parametric Study of Proposed Updates to Specifications To evaluate effects of updating the minimum reinforce- ment provisions, a parametric study was performed on the current AASHTO LRFD provisions and the proposed updated provisions, which are referred to as the âproposed method.â The criteria for this evaluation included reliability (defined as providing a consistent level of safety for all con- crete bridge members covered in the LRFD specifications), economy, and ease of use, as described previously. On the basis of this evaluation, recommended changes to the LRFD specifications are provided. The parametric study included computation of the required minimum reinforcement and/or prestress with both the AASHTO LRFD and the proposed method for the members listed in the concrete member database. The data- base included both the reinforced concrete and prestressed concrete members. The procedure used in the parametric study included the following steps: Step 1. Compute the required minimum reinforcement and/or prestress by using both methods. For each con- crete member in the database, the minimum reinforce- ment methods were applied to find either the minimum area of reinforcement (As,min.) or the minimum area of prestressed steel (Aps,min.). Step 2. Compute the theoretical cracking moment (mcr). On the basis of either the computed As,min. or the computed Aps,min., mcr was calculated by using a theoretical cracking stress that included h-0.15 as part of the calculation of Î³1. Although mcr is the same for reinforced concrete members, prestressed concrete members differ because mcr is depen- dent on the amount of prestress. Step 3. Compute the flexural strength at the ultimate moment capacity (Mo). With the use of either As,min. or Aps,min. for each of the minimum flexural reinforcement provisions, Mo was calculated by using strain compatibility. Step 4. Compare As,min. with Aps,min. for each of the methods. The consistency between the two methods was also com- pared for the full range of concrete members. 4.4.1 Concrete Structures Database The concrete structures database was taken from the report for NCHRP Project 12-80 (Holombo and Tadros 2010) and was intended to represent the range of structures com- monly used for construction covered by the AASHTO LRFD Bridge Design Specifications (Table 4-2). Parameters that have a significant effect on the minimum flexural reinforcement provisions were of particular interest. This study used typical design assumptions, whereas the parametric study in Chapter 3 used more realistic material behavior. In developing this database of concrete structures, the range of spans, girder types, spacing, and concrete strengths was based on recommended practices found in the PCI Bridge Design Manual, 3rd edition (Precast/Prestressed Concrete Institute 2001), and the American Segmental Bridge Institute (ASBI) AASHTO Segmental Box Girder Standards (https:// www.asbi-assoc.org/index.cfm/resources/aashto). Depart- ment of Transportation (DOT) guidelines, including those from the Florida DOT (2018), the California DOT (2018), the
92 Nebraska Department of Roads (2016), and the Washington State DOT (2018) were also evaluated. On the basis of this review, Table 4-2 was developed to capture the range of practical applicability regarding struc- ture dimensions. By using these guidelines, the structures database captures the upper and lower bounds of each structure classification. 4.4.2 Assumptions 18.104.22.168 Reinforced Concrete Members For reinforced concrete members, the minimum flexural reinforcement requirement was calculated directly without iteration. The following assumptions were made in preparing this study, as routinely done in design: 1. The reinforcement is A615 Grade 60. 2. The moment resistance is calculated as Mr = ÏAsfy(d â a/2). 3. The strength reduction factor Ï = 0.9. These assumptions differ from those used in the parametric study in Chapter 3, in that those analyses relied on moment- curvature responses of the sections, and thereby captured the section responses more accurately. 22.214.171.124 Prestressed Concrete Members The difficulty in calculating minimum reinforcement provisions for prestressed sections is that the cracking moment and the subsequent post-cracking resistance are dependent on the amount of prestress. Therefore, iteration was required. For evaluation of prestressed concrete sections, the follow- ing assumptions were made: 1. The cracking moment included the use of composite, transformed section properties and a theoretical flexural cracking stress of ( )â² â0.24 12 (ksi, in.)0.15f hc . 2. The cracking moment and minimum prestress were deter- mined on the basis of iteration with an assumed prestress loss of 30 ksi to account for anchor set, friction, and long- term prestress losses. 3. For composite sections, an assumed noncomposite moment of zero was used. Since the parametric study was intended to represent a wide range of prestressed sections, the zero-moment assumption was considered conservative. 4. The nominal moment at overstrength (Mo) included strain hardening of the reinforcement corresponding to Span Length (ft) Depth/Span (L) Girder Spacing (ft) No. Bridge Type Min. Max. Simple Continuous Min. Max. Cast-in-Place Bridges 2 Slab 20 45 0.07 0.06 2 Reinforced concrete box 60 120 0.06 0.055 6.5 14 2 Post-tensioned slab 40 70 0.03 0.027 2 Post-tensioned concrete boxa 80 250 0.045 0.040 6 20 Precast Concrete Bridges 2 Slab 20 50 0.03 0.03 2 Double-tee 30 60 0.05 0.05 4 4 2 Box beam 50 120 0.033 0.030 3 4 2 I-girder 70 200 0.045 0.040 6 12 2 U-beam 80 200 0.045 0.04 12 26 Segmental Bridges (precast) 2 Span Ã spanb 100 150 0.045 0.040 28b 45b 2 Balanced cantileverc 100 200 N/A 0.025 28b 45b Concrete Substructure Elements 2 Footingsd 12d 35d 2 Cap beamse 20e 60e 0.045L 0.04 26 Total aPractical upper limit for prismatic members. bSegment width. cUpper limit noted in the ASBI AASHTO Segmental Box Girder Standards (https://www.asbi-assoc.org/index.cfm/resources/aashto). dFooting width. eCap beam span measured from centerline column to centerline column. Table 4-2. Concrete member database structural dimension limits.
93 either rupture of the prestressed strand (esu = 0.04) or a peak compressive strain of 0.003. 5. Sections were analyzed under positive bending only, since negative bending regions will crack prior to positive bend- ing regions in most continuous spans, which allows for redistribution of load. With the exception of spans with hinges and cantilevered bridges during construction, negative bending regions are not critical for minimum reinforcement. 6. Compression-controlled or transition sections were not considered as part of this study. Examples of sections that are compression controlled and do not meet the mini- mum reinforcement requirements are negative bending regions of continually prestressed bridges with relatively wide top flanges. Sections in this category require greater strength with the variable resistance factor (Ï) to account for reduced ductility. Therefore, an additional factor of safety is not required. 4.4.3 Parametric Study Results The data and calculations developed as part of the para- metric study are shown in Table 4-3. This includes a ratio of Mo/Mcr, which is an indicator of the level of safety provided by AASHTO LRFD Proposed Method Type Section h (ft) (ksi) As,min. (in.2) Mo (kip-ft) mcr (kip-ft) Mo/mcr As,min. (in.2) Mo (kip-ft) mcr (kip-ft) Mo/mcr As,prop/ As,LRFD Reinforced Concrete Slab CRS1 0.88 3.6 0.28 14.9 8.5 1.74 0.28 15.2 8.5 1.78 1.02 CRS2 1.79 3.6 0.46 62.4 32.1 1.94 0.42 57.1 32.1 1.78 0.91 Box BRC1 3.20 3.6 3.53 906 428 2.12 2.96 761 428 1.78 0.84 BRC2 6.60 3.6 11.31 6,327 2,681 2.36 8.50 4,798 2,681 1.79 0.75 Cap CAP1 4.00 4 9.51 3,065 1,400 2.19 7.70 2,489 1,400 1.78 0.81 CAP2 10.00 4 32.98 28,190 11,226 2.51 23.30 19,985 11,226 1.78 0.71 Footing F1 5.00 4 17.73 7,167 3,167 2.26 13.87 5,625 3,167 1.78 0.78 F2 10.00 4 74.70 61,468 24,467 2.51 52.60 43,495 24,467 1.78 0.70 Prestressed Concrete Cast-in place slab CPS1 1.25 4 0.29 62 43 1.46 0.28 60 41 1.44 0.96 CPS2 2.00 4 0.43 171 116 1.48 0.38 152 106 1.43 0.87 CIP box BPT1 3.20 4 4.02 2,807 1,951 1.44 3.16 2,257 1,632 1.38 0.79 BPT2 10.00 4 27.43 67,019 46,517 1.44 17.80 44,521 32,783 1.36 0.65 I-girder PCI1 3.63 5 0.95 825 554 1.49 0.78 678 493 1.38 0.82 PCI2 8.71 10 4.79 10,555 7,029 1.50 3.40 7,549 5,597 1.35 0.71 U-beam PUB1 3.58 7 3.62 2,962 2,057 1.44 2.89 2,404 1,780 1.35 0.80 PUB2 8.58 10 6.71 14,608 9,665 1.51 4.80 10,508 7,760 1.35 0.71 Box beam PBB1 2.25 5 2.16 962 709 1.36 1.77 812 615 1.32 0.82 PBB2 3.92 5 1.8 1,595 1,129 1.41 1.39 1,257 944 1.33 0.77 Voided slab PPS1 1.00 4 0.79 124 91 1.35 0.79 124 91 1.35 1.00 PPS2 1.79 4 1.33 452 315 1.43 1.16 402 287 1.40 0.87 Spliced girder PSP1 6.63 7 2.95 4,605 3,032 1.52 2.19 3,438 2,517 1.37 0.74 PSP2 15.71 10 5.8 22,608 14,436 1.57 3.79 14,858 10,935 1.36 0.65 Segmental (Bonded) SBC1 6.00 4 7.71 11,536 8,949 1.29 5.82 8,756 7,647 1.14 0.75 SBC2 10.00 7 14.7 36,044 26,786 1.35 10.35 25,449 22,127 1.15 0.70 Segmental (Unbonded) SBS1 6.00 7 12.45 13,802 11,515 1.20 10.73 11,929 10,479 1.14 0.86 SBS2 8.00 7 16.16 24,669 20,227 1.22 13.30 20,345 17,877 1.14 0.82 Note: section = concrete structure callout from database listed in Appendix B: Design Examples [see NCHRP Project 12-94 on the TRB website (trb.org)]; h = height of section; As,min. = area of flexural reinforcement or prestressed steel required to meet respective method; Mo = nominal moment at overstrength, including effects of strain hardening; mcr = unfactored theoretical cracking moment based on assumed cracking stress of ; and Mo/Mcr = ratio of the nominal moment at overstrength to the cracking moment. This term is the effective factor of safety, or brittleness ratio. This ratio allowed for evaluation of the minimum reinforcement methods regarding safety. However, for prestressed concrete members, the comparison becomes meaningless because of the separately factored prestressed and concrete cracking stress components. c Table 4-3. Parametric study results.
94 Figure 4-5. Ratio of required minimum reinforcement versus depth. each of the methods for each concrete member. The required minimum reinforcement for both methods is shown along with the proposed method (As,prop) and the AASHTO LRFD (As,LRFD) for direct comparison. The parametric study results indicated that adding the variable depth factor can significantly reduce the amount of minimum reinforcement for deeper members without sacri- ficing safety. For reinforced concrete members, the proposed method gave a uniform Mo/mcr ratio, as one would expect, whereas the AASHTO LRFD method indicated an increase in the factor of safety with depth. For prestressed concrete members, the prestress influence on the theoretical cracking moment was variable and dependent on the section geometry. Therefore, the Mo/mcr ratio was meaningless. As shown in Figure 4-5, the savings were approximately proportional to h-0.15 for both reinforced and prestressed con- crete. The prestress savings were greater because the flexural cracking moment was dependent on the amount of prestress and, therefore, plots were below the hâ0.15 line. The two points above the hâ0.15 line are the two precast segmental units with unbonded tendons. In summary, the parametric study demonstrated that significant savings can be realized with the inclusion of a variable depth component to the flexural cracking stress. 4.5 Recommendations On the basis of the results of the parametric study and related documentation, it is recommended that the mini- mum reinforcement provisions in the AASHTO LRFD Bridge Design Specifications be changed to those of the proposed method, as discussed in the previous sections. This change is recommended because (a) the proposed method provides a consistent level of safety for all components in the database of concrete structures and (b) a significant savings can be real- ized. This savings is the result of the incorporation of widely accepted observations that the flexural cracking strength is depth dependent. 4.6 Proposed Revisions to the AASHTO LRFD Specifications The changes the project team recommends regarding the minimum flexural reinforcement provisions in the LRFD specifications are shown below. The recommended code changes are presented first, and then changes to the com- mentary are provided. Note that changes are recommended to Article 126.96.36.199 only. Deletions are shown as a single strike- through. Additions are shown as underlined.
95 LRFD. 188.8.131.52 Modulus of Rupture Unless determined by physical tests, the modulus of rupture, , for lightweight concrete with specified compressive strengths of up to 10.0 ksi and normal weight concrete with specified strengths up to 15.0 ksi may be taken as 0.24 where is the concrete density modification factor as specified in Article 184.108.40.206. Where physical tests are used to determine modulus of rupture, the tests shall be performed in accordance with AASHTO T 97 and shall be performed on concrete using the same proportions and materials as specified for the structure. The test units shall be cured in the same manner as the structure. LRFD. C220.127.116.11 Most modulus of rupture test data on normal weight concrete are between and (ksi) (Walker and Bloem 1960; Khan, Cook and Mitchell 1996). A value of has been recommended for the prediction of the tensile strength of high-strength concrete (ACI 1992). However, the modulus of rupture is sensitive to curing methods, and nearly all of the test units in the data set mentioned previously were moist cured until testing. Carrasquillo et al. (1981) noted a 26-percent reduction in the 28-day modulus of rupture if high-strength units were allowed to dry after 7 days of moist curing over units that were moist cured until testing. The flexural cracking stress of concrete members has been shown to significantly reduce with increasing member depth. Shioya et al. (1989) observed Past research has suggested that the flexural cracking strength is proportional to may be considered to be proportional to to (Shioya et al. 1989; Carpinteri and Corrado 2011), where is the overall depth of the flexural member. Based on this observation, a 36.0-in. deep girder should achieve a flexural cracking stress that is 36 31 to 57 percent lower than a 6.0-in. deep modulus of rupture test specimen. Since modulus of rupture units are either 4.0 or 6.0 in. deep and typically moist cured up to the time of testing, the modulus of rupture should be significantly greater than the flexural cracking strength of an average size bridge member composed of the same concrete. Therefore, 0.24 is appropriate for checking minimum reinforcement in Article 18.104.22.168. The properties of higher-strength concretes are particularly sensitive to the constitutive materials. If test results are to be used in design, it is imperative that tests be made using concrete with not only the same mix proportions, but also the same materials as the concrete used in the structure and curing procedures. The given values may be unconservative for tensile cracking caused by restrained shrinkage, anchor zone splitting and other tensile forces caused by effects other than flexure. The direct tensile strength stress should be used for these cases.
96 LRFD. 22.214.171.124 Minimum Reinforcement Unless otherwise specified, at any section of a non-compression-controlled flexural component, the amount of prestressed and non-prestressed tensile reinforcement shall be adequate to develop a factored flexural resistance, Mr, greater than or equal to the lesser of the following at least equal to the lesser of: 1.33 times the factored moment required by the applicable strength load combination specified in Table 3-4.1-1; )1-126.96.36.199( where = strength factor for minimum reinforcement 1.0 = 1.0 + 1.33 = net tensile strain in the extreme tension steel at nominal resistance, per AASHTO LRFD fr = modulus of rupture of concrete specified in Article 188.8.131.52 fcpe = compressive stress in concrete due to effective prestress forces only (after allowance for all prestress losses) at extreme fiber of section where tensile stress is caused by externally applied loads (ksi) Mdnc = total unfactored dead load moment acting on the monolithic or noncomposite section (k-in.) Sc = section modulus for the extreme fiber of the composite section where tensile stress is caused by externally applied loads (in.3) LRFD. C184.108.40.206 Minimum reinforcement provisions are intended to reduce the probability of brittle failure by providing flexural capacity greater than the cracking moment. If this condition is not met, additional flexural strength is required by multiplying the required factored moment by . For tension-controlled sections, is 1.33, which is equivalent to the inverse of the resistance factor ( ) for compression-controlled sections. For compression-controlled and transition sections, is reduced to avoid doubling the additional strength requirement for reduced ductility that is already accounted for in . Based on the completed tests, a member having the minimum reinforcement is expected to possess a displacement capacity of 1.0% the span length at the minimum. Further, the minimum curvature ductility for tension-controlled sections of flexural members can be approximated as the minimum net tensile strain ( ) divided by the yield strain ( ), . Sources of variability in computing the cracking moment and resistance are appropriately factored (Holombo and Tadros 2010). The factor applied to the modulus of rupture ( 1) is greater than the factor applied to the amount of prestress ( 2) to account for greater variability. For precast segmental construction, cracking generally starts at the segment joints. Research at the University of California, San Diego, has shown that flexure cracks occur adjacent to the epoxy bonded match cast face, where the accumulation of fines reduces the tensile strength (Megally et al. 2003). Based on this observation, a reduced 1 factor of 1.2 is justified. Snc = section modulus for the extreme fiber of the monolithic or non-composite section where tensile stress is caused by externally applied loads (in.3) Appropriate values for Mdnc and Snc shall be used for any intermediate composite sections. Where the beams are designated for the monolithic or noncomposite section to resist all loads, substitute Snc for Sc in the above equation for the calculation of Mcr. The flexural cracking stress of concrete members has been shown to decrease with increasing member depth. Sritharan et al. (2019)* observed that the flexural cracking strength is proportional to . A similar equation for estimating the flexural cracking strength on the basis of the depth is found in fib Model Code 2010 for Concrete Structures. *Note: This reference is to the present report, NCHRP Research Report 906: LRFD Minimum Flexural Reinforcement Requirements.
97 For prestressing steel, 3 shall be taken as 1.0. The provisions of Article 5.10.6 shall apply. The following factors account for variability in the flexural cracking strength of concrete, variability of prestress and the ratio of nominal yield stress of reinforcement to ultimate. 1 flexural cracking variability factor 0.67 for ASTM M 334 (ASTM A1035), Grade 100 reinforcement 1.2 (h/12)-0.15 for precast segmental structures 1.6 (h/12)-0.15 for all other concrete structures, where h is the member depth (in.) 2 prestress variability factor 1.1 for bonded tendons 1.0 for unbonded tendons 3 ratio of specified minimum yield strength to ultimate tensile strength of the nonprestressed reinforcement 0.67 for AASHTO M 31 (ASTM A615), Grade 60 reinforcement 0.75 for ASTM M 31 (ASTM A615), Grade 75 reinforcement 0.76 for ASTM M 31 (ASTM A615), Grade 80 reinforcement 0.75 for A706, Grade 60 reinforcement 0.80 for A706, Grade 80 reinforcement = = = = = = = = = = = = =