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6 2.1 Introduction This section presents the current state of knowledge on minimum reinforcement requirements, including philosoph- ical and historical background, factors affecting minimum reinforcement requirements, and the behavior of structural members designed with minimum reinforcement. 2.2 Philosophical Background As previously noted, the main purpose of ensuring mini- mum longitudinal reinforcement in a reinforced or pre- stressed concrete member is to provide a sufficient level of protection for the member from failing suddenly in a brittle manner immediately following the formation of first flexural cracks (Figure 2-1). In this case, the member experiences a limited number of cracks and concentrated damage in the critical region, as shown in Figure 2-2. These flexural cracks develop when the flexural tensile stress in the extreme con- crete tension fiber exceeds the modulus of rupture of con- crete. With a sufficient amount of minimum reinforcement, additional flexural cracks will develop along the member length, increasing its moment resistance and deflection. If the amount of minimum reinforcement is below a threshold value, it will not have the capacity to resist the tension carried by the concrete, causing the reinforcement to rupture follow- ing formation of the crack and the member to experience a wide localized crack and sudden failure. To ensure satisfactory behavior of flexural members with minimum reinforcement, different measures have been sug- gested. These include (1) ductility, (2) brittleness number, (3) deflection, and (4) Mo/Mcr ratio, where Mo is the flexural strength at the ultimate limit state and Mcr is the moment that produces the first flexural cracks. However, there is no consensus on what the best measure is for ensuring adequate safety of flexural members with minimum reinforcement. In general, ductility has been more frequently used to define member displacement capacity, while the Mo/Mcr ratio has been commonly used for ensuring the safety of flexural members with minimum reinforcement in design codes. The use of Mo/Mcr ensures that failure will not commence upon the first flexural cracks but does not guarantee a particular amount of ductility. The AASHTO LRFD Bridge Design Specifications, 8th edition (2017) prescribes the amount of minimum flexural reinforce- ment for both reinforced and prestressed concrete members, with the intention of minimizing the probability of brittle failure of both member types. This is ensured by providing an adequate amount of minimum reinforcement, so that the flexural capacity of the member will be sufficiently greater than the cracking moment, or 33% greater than that required by the applied factored loading of the member, whichever is smaller. Consequently, the members are protected from experiencing sudden failure without warning or redistribu- tion of loads. In addition, for tension-controlled members, other provisions in the specifications ensure that these mem- bers will possess adequate ductility, so that the members will provide visual warning before experiencing a complete fail- ure. The code infers that if a section is tension controlled, the member will experience observable warnings of failure before collapse. However, similar behavior is not possible for compression-controlled members, and, therefore, such members are expected to be designed with a lower resistance factor, which increases the ratio between the flexural capacity and cracking moment. The minimum flexural reinforcement requirement typi- cally consists of an explicit consideration of the cracking moment in the equation, which is a function of section geometry and modulus of rupture for reinforced concrete members. For prestressed concrete members, the cracking moment also depends on the amount of effective prestress and its location at the critical section. Among these vari- ables, the modulus of rupture has large variations, and a typical value of 0.24 â²fc (ksi) is used (Holombo and Tadros C H A P T E R 2 State of Knowledge on Minimum Reinforcement Requirements
7 2010). However, large-scale flexural tests have shown that the amount of minimum reinforcement required in a flex- ural beam depends on the height of the member [e.g., Shioya et al. (1989)]. This finding has also been supported analyti- cally using the fracture mechanics theory (Hillerborg et al. 1976; Bosco and Carpinteri 1992; Hawkins and Hjorteset 1992; BaÅ¾ant 1999; Ince et al. 2003). Accordingly, the modulus of rupture is not considered a material property; rather, the characteristic length, which is defined by the modulus of elas- ticity, modulus of rupture and fracture energy per unit area, is suggested to be the material property. This implies that the minimum amount of reinforcement could be decreased as the member depth increases. 2.3 Flexural Member Behavior with Minimum Reinforcement One of the earliest tests on lightly reinforced concrete beams was carried out by Lash (1953) by limiting the longi- tudinal reinforcement ratio to 1% or smaller. In this study, it was shown that the cracking moment depends on the concrete strength and the amount of reinforcement. The results also suggested that deflection of a beam when the tension steel yields is independent of the amount of steel reinforcement, as long as the provided reinforcement ratio is more than 0.7%. However, in this test the accuracy of capturing the first crack- ing event in the beam was low because of the capabilities of the equipment available at that time. The first cracking was assumed to have occurred when it was visible, rather than at the initiation point, if it happened subtly. Therefore, the determination of the modulus of rupture was suspected to have been affected by this inaccuracy. The size effect in reinforced concrete beams was observed in a series of tests completed at the University of Toronto (Collins and Kuchma 1999; Angelakos et al. 2001; Collins et al. 2015). The beams tested ranged in depth from 5 in. up to 13 ft 1 in. Although these tests were not specifically intended to determine the flexural capacity of the beams nor the impact of minimum reinforcement, the flexural reinforce- ment ratio in the beams varied from 0.5% up to 2.09%. The beam depth was found to affect the cracking moment of the beams, a result that confirmed the expected outcome from the fracture mechanics theory. Research performed at the University of Illinois by Warwaruk et al. (1962) was noted as the experimental foun- dation for the AASHTO LRFD Bridge Design Specifications, 8th edition (2017). A review of this study is presented in order to understand its philosophical and experimental conclusions. Warwaruk et al. (1962) tested 82 simply supported prestressed concrete beams. Of these beams, 74 were post-tensioned and eight were pre-tensioned. Segmental girders were excluded from the experimental program as they were not prevalent at the time of testing. Of the 74 post-tensioned beams, 26 had unbonded strands, 33 had only bonded strands through grouting of the strands, and 15 had supplementary bonded mild steel reinforcement. The simply supported span length remained constant at 9 ft. Two point loads varying from Â¼ to 5â12 of the span or one point load at midspan was used for testing. The dimensions of the nominal cross section were 12 in. deep and 6 in. wide. The 28-day compressive concrete strength varied from 1,060 psi to 8,320 psi. The typical test set up implemented in the research is shown in Figure 2-3. Only straight reinforcement was used for both the strands and the reinforcing bars. The diameter of the single-wire strand ranged from 0.191 to 0.199 in., the effective prestress of the strands varied from 19 to 151.3 ksi, and the stress at failure varied from 186 to 267 ksi. The mild steel reinforce- ment used was #3 bars with a yield strength and ultimate strength of 48 ksi and 75 ksi, respectively. The depth of the steel used for testing as a percentage of the member height varied and, for some cases, was less than that anticipated for use in full-scale design. Table 2-1 gives an overview of the material properties of the specimens. Figure 2-2. Brittle failure of a beam with insufficient minimum reinforcement under four-point bending (Murray et al. 2007). Figure 2-1. Illustrative example of ductile and brittle responses. Mcr1 My Mo1 Mcr2 Mo2 0 20 40 60 80 100 120 140 0 1 2 3 4 5 Ap pl ie d M om en t Displacement Ductile Response Brittle Response
8 Figure 2-3. Typical details for post-tensioned test specimen (after Warwaruk et al. 1962). Post-tensioned Bonded Beam (n = 33) Max. 8.32 0.35 151.3 257.0 245.0 80 Avg. 4.75 0.21 101.8 241.8 209.2 72 Min. 1.27 0.07 19.0 186.0 148.0 66 Post-tensioned Unbonded Beam (n= 26) Max. 7.60 0.35 127.5 255.0 214.0 70 Avg. 3.95 0.23 120.1 251.5 210.5 63 Min. 1.53 0.11 111.0 250.0 199.0 58 Post-tensioned Unbonded with Supplementary Bonded Mild Reinforcement (n = 15) Max. 5.43 0.42 124.4 255.0 214.0 65 Avg. 4.03 0.22 120.3 252.9 206.0 62 Min. 1.06 0.10 117.0 251.0 199.0 59 Pre-tensioned (n = 8) Max. 5.28 0.27 118.2 267.0 220.0 76 Avg. 4.83 0.18 114.4 267.0 220.0 75 Min. 3.97 0.12 112.1 267.0 220.0 74 Value ( ksi) (ksi) fse (ksi) fpu (ksi) fpy (ksi) ds/h (%) Table 2-1. Beam properties tested by Warwaruk et al. (1962).
9 Warwaruk et al. (1962) noted that prestressed beams can experience three phases in their load-deflection behavior. The first is the linear elastic phase without flexural cracking. Once flexural cracking occurs, the second phase of a constantly changing slope is observed while the steel reinforcement grad- ually increases its stress. The third phase is categorized by the very slow, almost linear, increase in load as deflection grows. In this final phase, the inelastic steel strain dominates the response. The effect of the r/f câ² ratio was observed, where r is the reinforcement ratio (As/bd), where As is the total reinforce- ment area, b is the beam width, d is the beam effective depth, and f câ² is the 28-day concrete compressive strength. It was noted that for beams with low values of r/fcâ², an increase in effective prestressing decreased the second-phase behavior because the steel reached its inelastic range more quickly and the prestress variability was found to have a negligible effect on the moment capacity for the practical ranges of prestressing appli- cation. For high r/fcâ² values, the increase in prestressing led to a quicker progression from the second phase to the third phase and a slightly higher moment capacity. Bonded beams with high values of r/fcâ² and all unbonded beams did not undergo the third phase of the load-deflection behavior and therefore pro- duced a more brittle response. Thus, to maintain a certain level of ductility Warwaruk et al. (1962) suggested an upper limit of 0.25 sur â² =f fc where fsu is the stress in the prestressed steel at failure. This was defined as the limit for the degree to which a section was com- pression controlled. Second, to avoid collapse upon initiation of flexural cracking, the lower limit of Mo â¥ Mcr was suggested, where Mo is the moment capacity of the beam and Mcr is the moment at which the section first develops flexural cracks. Freyermuth and Aalami (1997) noted that the abrupt fail- ure of beams after flexural cracks were developedâwhich minimum flexural reinforcement attempts to preventâhas not been observed in real-world conditions. As these authors unified the minimum reinforcement requirements for pre- stressed and reinforced concrete beams in the American Concrete Institute (ACI) 318 specifications, they relied on data from the testing completed by Warwaruk et al. (1962) in establishing their conclusions. They observed from Warwaruk et al.âs test specimens that for all beams with bonded reinforce- ment, there was no decrease in load-carrying capacity beyond the flexural cracking limit state of the member and that the tests did not suggest any concern for beams with reinforce- ment ratios as small as 0.101%. Freyermuth and Aalami further noted that the greatest ductility was found with the most lightly reinforced member; however, it was acknowl- edged that under a load-controlled testing scheme, these members could have failed abruptly. Of the unbonded post- tensioned beams with the lowest Mo/Mcr, the actual applied moment during the test was substantially greater than the design moment of the beam from the code. In addition, the steel depth of the unbonded beams was less effective than that of their bonded counterparts by a factor of about 1.2. This effect contributed to the less-favorable results for the unbonded beams. Ozcebe et al. (1999) evaluated the minimum reinforcement by testing six T-beams while varying the concrete strength and reinforcement ratio. The simply supported beams were tested under four-point loading, and once the beams reached a Do/L ratio of around 1â40 to 1â30, the tests were terminated. This precaution was taken because the stability of the test setup was endangered by the possible collapse of the beam. Figure 2-4 shows a linear relationship of the overstrength 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.000 0.200 0.400 0.600 0.800 1.000 O ve rs tr en gt h M om en t R ati o Ïl % (As/bwd) Ozcebe (1999) Test Specimens 1.2Mcr Figure 2-4. Test specimen overstrength moment ratios observed by Ozcebe et al. (1999) and a linear trend line.
10 moment ratio and the percentage of minimum flexural rein- forcement. The overstrength moment ratio remains above the conventionally accepted ratio of 1.2Mcr, despite having less than the minimum allowable flexural reinforcement in two of the specimens. The linear relationship indicates that as the percentage of the minimum flexural reinforcement increased, the overstrength moment ratio increased propor- tionally. Table 2-2 summarizes the response of the beams, which shows that the deflection ductility and the Do/L ratios were quite high for all beams, despite somewhat premature termination of the tests. Brenkus and Hamilton (2014) proposed a minimum rein- forcement requirement for ACI based on a modified Leonhardt approach, which is presented below in Section 2.4.10. The modifications of the Leonhardt approach came from the assumptions used in the original formulation, which are the location of the steel depth at the resultant of the tensile force of the beam, the geometry of the beam under tension having a rectangular section, and the ignoring of the possible composite nature of a beam and deck section. It was noted that these assumptions are conservative for most cases, but that greater accuracy could be achieved by using a refined set of assumptions. Rather than keeping the requirement based on the forces in the steel, Brenkus and Hamilton proposed a moment-based approach. The net cracking moment was defined as the moment beyond the decompression condition required to cause flexural crack- ing in the member. Therefore, the design moment must be greater than the sum of the decompression and net cracking moments. The factors of Ï and 1.2 were used for the design moment and the net cracking moment, respectively, where Ï is the resistance factor. The design moment was con- structed from an internal moment arm philosophy with an assumed j and Ï of 0.9 each, where j is the internal moment arm ratio. The equation proposed avoids an iterative pro- cess and consistently provides the overstrength moment ratio at a minimum of 1.17 times the cracking moment for the parametric study completed by the authors. From a fracture mechanics perspective, several experi- ments have shown that depth has an influence on the frac- ture energy. One such experiment was carried out by Bosco et al. (1990). Thirty reinforced concrete beams were tested by varying the longitudinal reinforcement ratio, rl, and beam depth. The span/depth ratio was kept constant at 6, and the dimensions shown in Figure 2-5 were used for the beams. A brittleness number, Np, was designed to evaluate the mini- mum amount of flexural reinforcement for high-strength T601 8.83 0.323 1.42 2.52 155.3 T602 9.33 0.486 1.78 3.33 138 T603 8.92 0.646 2.22 2.68 143.2 T901 12.55 0.395 1.49 2.81 136.5 T902 11.52 0.465 1.8 3.34 195.6 T903 12.3 0.698 2.5 3.25 130.3 Beam ID (ksi) l (%) M L o o /Mcre (%) = cro Table 2-2. Response summary of T-beams tested by Ozcebe et al. (1999). Figure 2-5. Dimensions of specimens tested by Bosco et al. (1990).
11 concrete (HSC) beams. The equation for the brittleness number is N (2-1) 1 2 IC = f h K A A p y s where A = gross area of section, As = area of steel reinforcement, h = beam height, fy = yield stress of reinforcement, and KIC = fracture toughness of the concrete. The equation defining the concrete fracture toughness in terms of the fracture energy, GF, and the modulus of elasticity of the concrete, Ec, is (2-2)IC =K G EF c The experiments showed that the brittleness number, Np, increased when the amount of reinforcement decreased or the beam depth increased. Additionally, the study concluded that if Np is held constant, the reinforcement ratio is inversely proportional to the beam depth. This suggested that lower minimum reinforcement ratios are needed in deeper beams. Figure 2-6 shows graphs of the midspan load deflection for the beams and highlights the detrimental behavior associated with not having sufficient reinforcement in concrete beams. The consistent negative post-cracking stiffness leads to brittle failure. It was shown that even beams with seemingly ade- quate reinforcement could experience a slight negative stiff- ness before recovering to a positive stiffness. Ferro et al. (2007) used a numerical model with linear elastic fracture mechanics (LEFM) concepts to assess the minimum reinforcement in concrete members by the bridged crack model. They defined maximum and minimum brittle- ness numbers that represented two types of collapse failure expected in a beam under flexure: concrete crushing in the compressive zone and steel rupturing upon development of flexural cracks. With LEFM, it was concluded that the mini- mum reinforcement ratio requirement would decrease with increase in the beam depth. Shin (1989) tested 12 reinforced concrete beams to evalu- ate the performance of HSC. It was shown that the maximum deflection for HSC members increased marginally and that the initial flexural stiffness for the members with HSC was 0 1 2 3 4 5 6 0 0.01 0.02 0.03 0.04 Lo ad (k ip s) Midspan Deflection for Beam Size A (in.) 0 1 2 3 4 0 2 4 6 8 10 0 0.01 0.02 0.03 0.04 Lo ad (k ip s) Midspan Deflection for Beam Size B (in.) 0 1 2 3 4 0 5 10 15 0 0.02 0.04 0.06 0.08 Lo ad (k ip s) Midspan Deflection for Beam Size C (in.) 0 1 2 3 4 0 = brittleness number Np = 0.00 (no reinforcement) 1 = brittleness number Np = 0.10 (average) 2 = brittleness number Np = 0.26 (average) 3 = brittleness number Np = 0.53 (average) 4 = brittleness number Np = 0.87 (average) Figure 2-6. Midspan load-deflection graphs for beams tested by Bosco et al. 1990.
12 higher. It was noted that the members with low reinforce- ment ratios underwent large deformation without a sudden decrease in the load-carrying capacity. There were sharp decreases in the load-carrying capacities of the beams with larger reinforcement ratios; however, the load rebounded after experiencing crushing of concrete that transferred the compression into compression reinforcement. Therefore, for the reinforcement ratios tested, it was concluded that as the reinforcement ratio decreased, the ductility increased. Shin (1989) observed the ductility of the higher-strength concrete specimens being generally greater than the specimens with moderate concrete strength. Rashid and Mansur (2005) evaluated the effect of HSC on 16 reinforced concrete beams. On the basis of the tests and analytical evaluation, it was noted that as the concrete compressive strength was increased up to 15.2 ksi, the ductil- ity of the beams increased. Increasing the concrete compres- sive strength beyond 15.2 ksi decreased the ductility in the member. Rashid and Mansur concurred with the modulus of rupture of 0.24 ksi assigned by ACI to concrete. Lambotte and Taerwe (1990) tested reinforced concrete beams and slabs, with concrete strengths and the reinforce- ment ratio as the main test variables. It was shown within the stabilized cracking region that crack widths and the crack spacing are independent of concrete strength. It was also observed that the flexural stiffness before cracking was higher for higher-strength concrete and that the post-crack stiffness was not significantly influenced by the concrete strength. The lowest reinforcement ratio that was tested was 0.48%, and those beams behaved in a ductile manner. Fayyad and Lees (2015) used the fracture mechanicsâbased model to explore the minimum reinforcement ratio as it related to concrete strength and beam depth. The authors noted that there was much disagreement about the correlation between the minimum reinforcement ratio and beam depth. The minimum reinforcement ratio, rmin., was defined as the minimum amount of steel required to avoid unstable crack growth divided by the concreteâs compressive width, b, and the depth of steel, d. The model was used to conclude again that as beam depth increased, the required rmin. decreased. It was also concluded that an increase in the concrete strength increased rmin.. Figure 2-7 shows the proposed values for min- imum reinforcement in relation to an increase in beam depth compared with ACI 318-11 (ACI Committee 318 2011) and Eurocode 2 (European Committee for Standardization 2004). Mattock et al. (1971) tested 10 specimens to observe the effects of bonded versus unbonded tendons and the use of unprestressed reinforcement. On the basis of the tests, the use of seven-wire strand as an effective bonded unprestressed reinforcement was verified. It was found that the unbonded beams with additional unprestressed bonded reinforcement had ductility and strength qualities equal to or better than those of comparable bonded post-tensioned beams. Mattock and colleagues recommended providing bonded steel of 0.4% of the area between the flexural tension face to the neutral axis of the gross section. This requirement for bonded rein- forcement is to ensure serviceability requirements and suf- ficient post-cracking strength. Rabbat and Sowlat (1987) tested three segmental girders to understand the differences in the behavior of beams with external versus internal tendons. The three test units included one with wholly bonded internal tendons, one with external unbonded tendons, and one that was a modified combina- tion. The modified specimen had external tendons covered with concrete in a second stage cast to produce a bonded-like condition. Figure 2-8 illustrates the different bonding condi- tions by showing the cross sections and their reinforcement. There was no mild reinforcement crossing the joints in the test units. The simply supported spans were loaded in four-point bending. In the first cycle, the beams were loaded to a dis- placement of around 3 in., where significant nonlinear deformation was observed. In the second cycle, the beams were forced to fail after the end anchorages of one strand on each side of the web were torched. The anchors were torched to represent a seismic-induced failure. Figure 2-9 shows the applied moment and deflection at the midspan of the beams. For a given moment, the external tendons had a high deflection, but the internal tendons reached a larger displacement. The torching of the tendons did not affect the bonded tendon in the second cycle, whereas the behavior of the other two specimens was clearly influenced. The bonded test specimen had an initiation of failure with concrete crushing, and then the strands ruptured, whereas the unbonded and modified unbonded specimens failed in 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 10 20 30 40 50 M in im um R ei nf or ce m en t R ati o (% ) Beam Depth (in.) Proposed Values ACI318(2011) EC2 = 4.35 ksi Figure 2-7. Minimum reinforcement requirements in design standards compared with values proposed by Fayyad and Lees (2015).
13 shear with the shear keys breaking off. Table 2-3 summa- rizes the results of these tests. Hindi et al. (1995) studied the performance of a three- span segmental structure. An objective of this study was to observe the influence of the number of deviators to which the external tendons were bonded on the strength and ductil- ity of the member. The authors found that as the number of bonded locations increased, the ductility and strength capac- ity of the member increased. The increase in bonded loca- tions increased the number of joints that opened under the loading. The joint openings allowed the length in the tendon to change over a shorter distance, which resulted in a higher stress change in the strands. Figure 2-10 shows the load- deflection plot that exemplifies this trend. Figure 2-8. Cross section of specimens tested by Rabbat and Sowlat (1987). Figure 2-9. Midspan load-deflection graphs from Rabbat and Sowlat (1987). 0 1 2 3 4 5 6 7 8 0 2 4 6 8 10 12 Ap pl ie d M om en t ( 1, 00 0 ki ps .in .) Deflection (in.) Bonded Tendon Modified Unbonded Tendon Unbonded Tendon Cycle 1 Cycle 2
14 Aparicio et al. (2001) flexurally tested unbonded segmen- tal and monolithic girders to failure. For the monolithic beams, failure was defined on the basis of when mild steel reinforcement yielded; for the segmental beam, it was taken when physical failure occurred. The physical failure of the segmental girder occurred as a result of concrete crushing in the top flange. Although low reinforcement ratios were used, adequate ductility was shown. The ductility is shown as Do/L in Table 2-4. The authorsâ determined that as the external tendon length was decreased, the change in stress, from initial conditions to final, increased. Intuitively, the higher stress increases both the moment capacity and ductility because of the elongation in the strands. While studying the beam depth effect on the minimum reinforcement requirement, Bruckner and Eligehausen (1998) concluded that an increase in beam depth resulted in a decrease in ductility. The three sets of beams were reinforced with rl = 0.15%. The beam depths were 4.92 in., 9.84 in., and 19.7 in., while the span/depth ratio remained constant at 6. Each set behaved in a ductile manner, but as the beam depth was increased, the ductility decreased, which is exemplified in the Do/L ratio in Table 2-5. Despite the low reinforcement ratio, the beamsâ maximum moment capacity surpassed the 1.2Mcr threshold. Wafa and Ashour (1997) studied the effect that HSC has on the minimum flexural reinforcement requirement. They tested 20 beams by varying the amount of steel and the strength of the concrete. As shown in Table 2-6 and Figure 2-11, despite the varying of the amount of reinforcement, the overstrength moment ratio always remained above 1.4. This finding suggests that low reinforcement ratios can still produce adequate strength capacity beyond the flexural cracking state. The experiments showed that after flexural cracking, the stiffness of the beam decreased, and this decrease was affected by the amount of reinforcement provided. Wafa and Ashour observed experimentally that as concrete strength was increased, a higher minimum reinforcement ratio was required to achieve a specific moment reserve capacity. The moment reserve capacity was defined as the ratio between (a) the moment where the reinforcement first yielded and (b) the moment at which the section cracked. In general, as found in various codes, standards, and speci- fications, the minimum reinforcement requirements are directly related to the modulus of rupture of concrete, fr, since this determines the cracking moment needed for the calcula- tion of the Mn/Mcr ratio. In the current AASHTO LRFD Bridge Design Specifications and ACI Code, fr is defined as 0.24 â²fc (AASHTO 2017; ACI Committee 318 2014). The modulus of rupture of concrete is directly related to the tensile strength of concrete, which ranges from approximately 0.13 â²fc if obtained from direct tension tests to approximately 0.38 â²fc if obtained from modulus of rupture tests on beams curing under specific conditions. It is also important to note that for a brief period of time, fr was defined as 0.37 â²fc , which was deemed overly conservative and considered to not represent real-world scenarios (Holombo and Tadros 2010; Seguirant et al. 2010). A more recent study by Gamble (2017) suggests that the minimum reinforcement requirements should explic- itly include an additional d/h term, where d is the depth to the steel reinforcement and h is the total depth of the beam, to obtain a more consistent Mn/Mcr ratio for all cases, as shown in Figure 2-12. A number of studies have suggested that the influence of member depth on cracking of flexural members, which is not recognized in the current AASHTO LRFD Bridge Design Specifications (2017), should be expected. Table 2-3. Summary of tests by Rabbat and Sowlat (1987). Bonded 0.103 2.13 28.4 2.08 Unbonded 0.103 1.20 18.8 1.90 Modified unbonded 0.103 1.21 16.5 2.18 a Beam Type (%) (%) Mo L /Mcra 1 o Mo is from first cycle loading and Mcr is first joint opening. Figure 2-10. Midspan load-deflection response established by Hindi et al. (1995). 0 2 4 6 8 10 12 0 1 2 3 4 Lo ad (L L+ I) Displacement for Span with Epoxy Joints (in.) Bonded at 4 Deviators Bonded at 10 Deviators Table 2-4. Test results and characteristics from Aparicio et al. (2001). Beam Beam Type Longitudinal Reinforcement D2 Segmental 0.078 1.36 4 (0.6 in. diameter) M2 Monolithic 0.078 0.74 4 (0.6 in. diameter) M3 Monolithic 0.117 1.01 6 (0.6 in. diameter) M4 Monolithic 0.156 0.78 8 (0.6 in. diameter) l (%) o/L (%)
15 2.4 Methodologies and Code Approaches In this section, a comparison of current international standards for minimum flexural reinforcement is presented to highlight the philosophical differences in the methodolo- gies and different purposes for their use. This review will aid in the illustration of the varying approaches taken to specify minimum flexural reinforcement. 2.4.1 AASHTO LRFD Bridge Design Specifications The current AASHTO LRFD methodology for calculating minimum reinforcement is a reliability-based approach in which the factored resistance (Mr) is required to be greater than a factored cracking moment (Mcr) (AASHTO 2017). In this method, components of the factored cracking moment account for more realistic variability of key parameters. The true benefit of this method is that sources of variability in computing the factored cracking moment and the resis- tance are appropriately factored. The cracking stress factor is applied to the modulus of rupture, which has far greater vari- ability than the amount of prestress ( fcpe) at the extreme fiber. Set (%) o/L (%) Mo/Mcr h (in.) b (in.) Span (in.) Longitudinal Reinforcement A 0.15 4.0 1.93 4.921 11.811 29.528 2 (1/4 in. diameter) B 0.15 2.3 2.53 9.843 11.811 59.055 4 (1/4 in. diameter) C 0.15 1.5 1.62 19.685 11.811 118.110 2 (1/2 in. diameter) l Table 2-5. Summary of test results reported by Bruckner and Eligehausen (1998). Beam (%) Mo/Mcr (ksi) F1 0.21 2.14 6.30 F2 0.32 2.33 6.45 F3 0.37 3.08 6.71 F4 0.40 2.97 6.69 F5 0.32 1.61 8.55 F6 0.37 2.59 8.56 F7 0.48 3.48 8.81 F8 0.59 3.63 8.78 F9 0.37 1.70 11.09 F10 0.48 2.10 11.10 F11 0.59 2.38 11.12 F12 0.77 3.14 11.12 F13 0.32 1.42 11.20 F14 0.48 1.96 11.19 F15 0.67 3.26 11.16 F16 0.88 3.85 11.16 F17 0.32 1.85 13.09 F18 0.48 1.70 13.04 F19 0.67 2.24 12.98 F20 0.88 3.09 13.04 l Table 2-6. Summary of test results reported by Wafa and Ashour (1997). Figure 2-11. Overstrength moment ratio of experimental beams tested by Wafa and Ashour (1997). 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.00 0.20 0.40 0.60 0.80 1.00 O ve rs tr en gt h M om en t R ati o Ïl % (As/bd) Wafa and Ashour (1997) Test Specimens 1.2Mcr
16 For prestressed beams with unbonded tendons, the mini- mum area of bonded reinforcement shall be computed by As,min. = 0.004Act, where Act is the area of the cross section between the flexural tension face and center of gravity of the gross section. This requirement for prestressed members with unbonded tendons is implemented so the provided bonded reinforcement ensures a capacity of at least 1.2 times the cracking moment calculated by using the modulus of rup- ture. This requirement for the unbonded case is to facilitate acceptable flexural performance at the ultimate limit state while limiting the width and spacing of flexural cracks at service load levels. This is similar to the regular reinforced concrete case when tensile stresses in the concrete exceed the modulus of rupture of the concrete. 2.4.3 New Zealand Standard The New Zealand standard (NZS) states that for all beam sections (in SI units), the minimum reinforcement provided shall be greater than that given by 4 (2-4)= â²A f f b ds c y w and shall be equal to or greater than 1.4bwd/fy (Concrete Design Committee 2006). This requirement is somewhat comparable to that required by ACI 318. Both ACI 318 and NZS requirements are expressed as a reinforcement ratio defined as r = AS/(bwd), treating r as independent of the beam size. This requirement also aims to prevent sudden flexural failure. For prestressed members, NZS does not allow the use of unbonded tendons without requiring bonded rein- forcement in the amount of As = 0.004A, where A is the area of concrete between the extreme flexural tension face of the member and the centroid of the uncracked section. 2.4.4 British Standards Table 2-7 presents the minimum percentage of steel required under different circumstances as set forth in BS 8110 (British Standards Institution 2007). These percentages are based on the yield strength of the reinforcement, section shape, and different loading conditions. There is also a requirement for prestressing tendons that the cracking of concrete must occur before the beam fails. 2.4.5 Japan Society of Civil Engineers According to Article 13.4.1-ii of the Standard Specifications for Concrete Structures of the Japan Society of Civil Engineers (2010), at least 0.2% of the concrete area must be provided as Figure 2-12. Effect of proposed minimum reinforcement equation for slabs (Gamble 2017). For convenience, Article 18.104.22.168 of the AASHTO LRFD is reproduced in Box 2-1 and AASHTO LRFD Article 22.214.171.124 is shown in Box 2-2. 2.4.2 ACI 318 For reinforced concrete members, ACI 318-14 states (in U.S. customary units) that the reinforcement provided shall not be less than 3 (2-3),min. = â² A f f b ds c y w and not less than 200bwd/fy (ACI Committee 318 2014). These requirements may be waived if the provided area of steel is one-third greater than the area of steel required by analysis, primarily to avoid large members having excessive reinforce- ment. It is noted that these requirements are for preventing sudden failure of the members and do not account for any effects from the size of the member. For prestressed members with bonded reinforcement, the total amount of As and Aps is required to produce a fac- tored load greater than 1.2 times the cracking moment, which is dependent on the modulus of rupture. The crack- ing moment equation is not provided, but rather computed from the modulus of rupture, which is 7.5Î» â²fc (psi) or 0.24Î» â²fc (ksi), where Î» is taken as 1 for normal weight concrete. Similar to the case of reinforced concrete mem- bers, this provision is required to prevent sudden failure upon flexural cracking. Interestingly, it is included in the commentary that this provision does not apply to beams with unbonded tendons because abrupt flexural failure upon flexural cracking is not expected to occur.
17 Box 2-1. AASHTO LRFD Article 126.96.36.199 Unless otherwise specified, at any section of a nonâcompression-controlled flexural component, the amount of prestressed and non-prestressed tensile reinforcement shall be adequate to develop a factored flexural resistance, Mr, 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; and ( )= Î³ Î³ + Î³ â âï£«ï£ ï£¶ ï£¸ ï£® ï£°ï£¯ ï£¹ ï£»ï£º M f f S M s s r c c AASHTOLRFD 188.8.131.52-1cr cpe dnc nc 1 ( )3 1 2 where Mcr = cracking moment (kip-in.), fr = modulus of rupture of concrete specified in Article 184.108.40.206., 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 (kip-in.), Sc = section modulus for the extreme fiber of the composite section where tensile stress is caused by externally applied loads (in.3), and Snc = section modulus for the extreme fiber of the monolithic or noncomposite 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 calcu- lation of Mcr. The following factors account for variability in the flexural cracking strength of concrete, variability of prestress, and ratio of nominal yield stress of reinforcement to ultimate. Î³1 = flexural cracking variability factor = 1.2 for precast segmental structures = 1.6 for all other concrete structures Î³2 = prestress variability factor = 1.1 for bonded tendons = 1.0 for unbonded tendons Î³3 = ratio of specified minimum yield strength to ultimate tensile strength of the reinforcement = 0.67 for AASHTO M 31 (ASTM A615), Grade 60 reinforcement = 0.75 for AASHTO M 31 (ASTM A615), Grade 75 reinforcement = 0.76 for AASHTO M 31 (ASTM A615), Grade 80 reinforcement = 0.75 for A706, Grade 60 reinforcement = 0.80 for A706, Grade 80 reinforcement = 0.67 for AASHTO M 334 (ASTM A1035), Grade 100 reinforcement For prestressing steel, Î³3 shall be taken as 1.0. The provisions for Article 5.10.8 shall apply.
18 Box 2-2. AASHTO LRFD Article 220.127.116.11 Unless determined by physical tests, the modulus of rupture, fr, 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 0.24Î»âfcâ², where Î» is the concrete density modification factor as specified in Article 18.104.22.168. When 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. Situation Definition of Percentage Minimum Percentage fy = 250 N/mm2 fy = 500 N/mm2 Tension Reinforcement Sections subjected mainly to pure tension 100As/Ac 0.80 0.45 Sections subjected to flexure: (a) Flanged beams, web in tension: (1) bw/b < 0.4 100 As/bwh 0.32 (2) bw/b 4.0 100 As/bwh 0.24 0.13 (b) Flanged beams, flange in tension: (1) T-beam 100 As/bwh 0.48 0.26 (2) L-beam 100 As/bwh 0.36 0.20 (c) Rectangular sectiona 100 As/Ac 0.24 0.13 Compression Reinforcementb General rule 100Asc/Acc 0.40 0.40 Simplified rules for particular cases: (a) rectangular column or wall 100Asc/Ac 0.40 0.40 (b) flanged beam: (1) flange in compression 100Asc/bhf 0.40 0.40 (2) web in compression 100Asc/bwh 0.20 0.20 (c) rectangular beam 100Asc/Ac 0.20 0.20 Transverse reinforcement in flanges or flanged beamsc 100Ast/hf L 0.15 0.15 Note: aIn solid slabs, this minimum should be provided in both directions. bWhere such reinforcement is required for the ultimate limit state. cProvided over full effective flange width near top surface to resist horizontal shear. Ac = total area of concrete, Asc = area of steel in compression, Acc = area of concrete in compression, Ast = area of transverse steel in a flange, hf = flange depth, and L = span length. As = minimum recommended area of reinforcement, Table 2-7. Minimum reinforcement requirements in British standards.
19 longitudinal reinforcement to prevent brittle failure. When HSC is used, the minimum reinforcement must be provided to satisfy 0.058 (2-5)min. 2 2 3( )r = ï£« ï£ï£¬ ï£¶ ï£¸ï£· â²h d f f c y The code stipulates that a specific percentage of the gross area must be longitudinal reinforcement for normal strength concrete. The ratio for HSC is dependent on the ratio of beam height to the depth of reinforcement. The reinforcement requirement may be lowered if high-strength reinforcement is used. For prestressed members, 0.1% of the gross concrete area must be provided as reinforcing steel, which can combine both the deformed reinforcing steel and bonded prestressing strands. This minimum flexural rein- forcement steel provision is not applied to precast segmental structures because the minimum amount of reinforcement for prestressed concrete members in this code is primarily to control harmful cracks resulting from shrinkage or tem- perature gradient. 2.4.6 Eurocode 2 Eurocode 2 states that if crack control is required, the mini- mum amount of bonded reinforcement may be estimated by equilibrating the tensile force in concrete just before crack- ing and the tensile force in reinforcement at yielding or at a lower stress value if necessary to limit the crack width (Euro- pean Committee for Standardization 2004). The equation provided in Eurocode 2 (in SI units) is (2-6),min. ct,eff cts =A k kf As s c where As,min. = minimum area of reinforcing steel within the tensile zone. Act = area of concrete within tensile zone (tensile zone = that part of the section calculated to be in tension just before formation of the first crack). ss = absolute value of maximum stress permitted in reinforcement immediately after formation of the crack. (This may be taken as the yield strength of the reinforcement, fyk. A lower value may, how- ever, be needed to satisfy the crack width limits according to the maximum bar size or spacing.) fct,eff = mean value of tensile strength of concrete effective at time when cracks may first be expected to occur: = fctm (mean value of axial tensile strength of con- crete) or lower [fctm (t), mean value of axial tensile strength of concrete at time being considered], if cracking is expected earlier than 28 days. k = coefficient that allows for effect of nonuniform self-equilibrating stresses, which lead to a reduc- tion of restraint forces; = 1.0 for webs with h â¤ 300 mm [11.81 in.] or flanges with widths < 300 mm; = 0.65 for webs with h â¥ 800 mm [31.50 in.] or flanges with widths > 800 mm (intermediate values may be interpolated). kc = coefficient that takes account of stress distribu- tion within section immediately prior to crack- ing and of change of lever arm. For pure tension, kc = 1.0. For bending or bending combined with axial forces, the following values are suggested: â¢ For rectangular sections and webs of box sec- tions and T-sections, 0.4 1 1 (2-7) 1 ct,eff k k h h fc c î°( )= â sï£® ï£° ï£¯ ï£¯ ï£¹ ï£» ï£º ï£º â¤ â¢ For flanges of box sections and T-sections, 0.9 0.5 (2-8) cr ct ct,eff k F A f c = â¥ where sc = mean stress of concrete acting on part of section under consideration, that is, (2-9) EdN bh cs = where NEd = axial force at serviceability limit state acting on part of cross section under consideration (compressive force posi- tive); NEd should be determined con- sidering the characteristic values of prestress and axial forces under the relevant combination of actions. hî° = h for h < 1.0 m [39.37 in.] = 1.0 m for h â¥ 1.0 m [39.37 in.]. k1 = coefficient considering effects of axial forces on stress distribution = 1.5 if NEd is a compressive force = 2 3 h h î° if NEd is a tensile force. Fcr = absolute value of tensile force within flange immediately prior to cracking due to cracking moment calculated with fct,eff. The Eurocode provides another requirement for minimum reinforcement that states that the crack control requirement
20 discussed above shall also be checked. This second require- ment is given by 0.26 ; but not less than 0.0013 (2-10),min. ctm yk =A f f b d b ds t t Eurocode 2 requires the minimum reinforcement to control cracking. This requirement is based on many factors but does not directly include any size effects. The only inclusion related to size is the h/h* factor in the calculation for the stress dis- tribution factor. Bonded tendons may also be used to control cracking according to the code. On the basis of the Eurocode recommendation, in certain prestressed members, no minimum reinforcement is required if the tensile stress in the tendon is less than the mean value of the tensile strength of the concrete. 2.4.7 fib Model Code 2010 The minimum reinforcement requirement for beams in the fib Model Code 2010 for Concrete Structures (International Federation for Structural Concrete 2010) is 0.20 0.001 (2-11)min ctm ykr = â¥f f where fctm = average value of the tensile strength of the concrete matrix = 0.3(fck)2/3 for concrete grades â¤ C50 [7.25 ksi] = 2.12 * ln(1 + 0.1(fck + Df)) for concrete grades > C50 [7.25 ksi], Df = 8 MPa [1.16 ksi], fck = specified characteristic cylinder compressive strength, and fyk = characteristic value of tension yield stress of non- prestressing reinforcement. This requirement is simply to prevent brittle failure and is based only on material properties. 2.4.8 Norwegian Standard The Norwegian standard (Norwegian Standards Associa- tion 2003) requires a minimum amount of reinforcement (in SI units) such that 0.25 (2-12)tk skâ¥A k A f fs w c where 1.5 1.0 (2-13) 1 k h h w = â â¥ h1 = 1.0 m [39.37 in], ftk = tensile strength of concrete, and fsk = steel yield strength. This standard is no longer utilized, as Eurocode 2 is currently used as a standard in Norway. However, this requirement is included in this report because it considers the beam depth in the computation. 2.4.9 Japanese Specifications for Highway Bridges The required minimum reinforcement (in SI units) set in the Japanese Specifications for Highway Bridges (Japan Road Association 2012) is 0.005 . (2-14)st â¥A b dw where Ast is the area of steel in tension. This specification appears to request the least amount of minimum reinforce- ment as compared with other codes and standards. 2.4.10 Leonhardtâs Method Leonhardt (1964) proposed a method for solving for the minimum reinforcement by equating the tensile forces in the concrete beam to the change in the steel stresses, as shown in the equations below. The stresses are assumed to vary linearly across the section, and, thus, the tensile force, Tcr, is due to the concreteâs tensile strength. The tensile force is modified by a shape factor, k, to represent the area under tension in the section. The change in the steel stresses arises from its elongation after the beam has cracked. Figure 2-13 shows the uncracked state stress distribution of the beam and the beam at its flexural capacity. 1 2 2 (2-15)cr = kï£«ï£ï£¬ ï£¶ ï£¸ï£·T b h fw r (2-16)ps ps pe cr( )â + â¥A f f A f Ts y This method is advantageous due to its simplistic nature. The procedure for calculating the minimum required steel is noniterative and it doesnât necessitate the calculation of a cracking moment. However, if a nonrectangular section is uti- lized, as in most situations, the computing of the tensile force can become complex. It should also be noted that Leonhardtâs proposed method does not consider the depth of the steel and the concreteâs compression capacity is neglected. This assumes the concrete will not crush as the steel increases its stress and that the steel lies at a depth at which it will yield before the concrete crushes.
21 2.4.11 Comparison of a Reinforced Concrete Beam Design with Various Codes A comparative study of different codes was carried out by using an inverted-tee reinforced concrete beam cross section, Girder RC1, shown in Figure 2-14. For this study, Girder RC1 reinforcement was designed on the basis of 11 minimum reinforcement equations previously presented in this section. The required minimum reinforcements from different codes or methodologies are presented in Figure 2-15. It is shown that the variability of the required minimum reinforcement is quite large, ranging from 0.63 in.2 to 2.91 in.2 with a mean of 1.89 in.2 and standard deviation of 0.69. Two cases repre- senting British standards are denoted in the graph as ââ1â and ââ2,â representing the reinforcement yield strength of 250 MPa (36.3 ksi) and 500 MPa (72.5 ksi), respectively. The AASHTO requirement is above the average of all codes. The provided steel reinforcement for this girder test was chosen to be 1.77 in.2, below the average value of 1.89 in.2. More details about Girder RC1, including the test results are discussed in Chapter 3. 2.5 Past Research on AASHTO Minimum Reinforcement Requirements The report for NCHRP Project 12â80, NCHRP Web-Only Document 149: Recommended LRFD Minimum Flexural Rein- forcement Requirements (Holombo and Tadros 2010), refined the AASHTO requirement and established the equation included in the current version of AASHTO specification (2017). Its objective was to properly assign levels of uncer- tainty on the basis of the different equation variables. This was achieved in creating three factors to account for the variabil- ity in the modulus of rupture, the prestress, and the overstress steel ratio of fu/fy. A statistical analysis on the modulus of rup- ture was performed. The authors drew from data on full-size members to make their conclusions on the variability and rec- ommended the value for the modulus of rupture of 0.24 â²fc . The prestress variability was determined from the unknown nature of prestress losses. The authors acknowledged that for pre-tensioned members, the prestress variability factor could be 1.05, on the basis of the literature review on this topic. How- ever, for uniformity with post-tensioned members, the factor was determined to be 1.1 on the basis of the unknown nature of friction losses, which can vary from 15% to 25%. The NCHRP 12â80 report notes that for compression- controlled and transition sections, the increased margin of safety is provided through the resistance factor, Ï. However, it is observed that there is a lack of consistency in the applica- bility of the margin of safety, specifically, in the region where the net tensile strain shifts from tension-controlled to the transition. In this region, Ï is such that the nominal moment capacity needs to increase incrementally, while perhaps a larger factor of safety would have been provided following the minimum reinforcement provision. To address this inconsis- tency, the report recommends that Ï not be reduced for the Figure 2-13. Concrete and reinforcement forces (Brenkus and Hamilton 2014). Figure 2-14. Girder RC1 cross section.
22 minimum reinforcement. It is observed that lightly reinforced concrete members can reach their full nominal moment in their post-cracked state and behave in a ductile manner. 2.6 Factors Affecting Minimum Reinforcement Requirements A discussion on the key factors affecting minimum rein- forcement requirements based on the literature is presented herein. 2.6.1 Material Strengths It is widely accepted that minimum reinforcement require- ments depend on both concrete and steel strengths. The higher the strength, the more brittle the member behavior becomes. Most of the currently used codes were developed and calibrated with lower concrete and steel strengths that were commonly used at the time (Gamble 2017). The mini- mum reinforcement requirements may therefore need to be recalibrated with the more common concrete and steel strengths currently used as the baseline. 2.6.2 Modulus of Rupture Cracking in concrete is controlled by its tensile strength. However, quantifying concrete tensile strength may not be straightforward. There are three tests that can be done to quantify this parameter: direct tension, split cylinder, and small-scale modulus of rupture (four-point bending) beam tests. The concrete stress at failure obtained from these tests varies from approximately 0.13 â²fc ksi for direct tension tests to approximately 0.38 â²fc ksi for modulus of rup- ture tests. This range is caused mostly by the differences in the stress gradients in the cross section. The modulus of rupture test results are also affected by the curing method (Seguirant et al. 2010). Some studies proposed higher values of the modulus of rupture, depending on whether the specimens were moist cured or heat cured (Carrasquillo et al. 1981; Mokhtarzadeh and French 2000), and a study by Tuchscherer et al. (2007) proposed a coefficient based on a more rational approach. The current modulus of rup- ture adopted in AASHTO is 0.24 â²fc (ksi) or 7.5 â²fc (psi), which is independent of the beam depth and is the same as the one adopted by ACI. 2.6.3 Depth Influence Experimental research on minimum flexural reinforce- ment mostly utilizes scaled beams for reasons of economy. Yet, as previously noted, the depth of the member influences the modulus of rupture, as has been supported by frac- ture mechanics theory and some experimental work (Bosco et al. 1990; Bruckner and Eligehausen 1998; Ferro et al. AA SH TO (2 01 7) A CI (2 01 4) N ZS (2 00 6) BS I ( 20 07 ) - 1 BS I ( 20 07 ) - 2 JC (2 00 7) E C2 (2 00 4) FI B (2 01 0) N S (2 00 3) JR A (2 00 2) Le on ha rd t 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 Different Methodologies Avg of Steel Required = 1.89 Area of Steel Provided = 1.77 A s ,re q' d (in 2 ) Figure 2-15. Comparison of minimum reinforcement requirements for Girder RC1.
23 2007; Rao et al. 2008; Carpinteri and Corrado 2011). How- ever, owing to lack of sufficient experimental data, the code requirements generally ignore the influence of the member depth, which can lead to an unnecessarily high amount of minimum flexural reinforcement in real-world examples. Including this effect of depth may significantly influence the minimum flexural reinforcement because of the basis in the cracking moment, which is dependent on the modulus of rupture. Figure 2-16 shows Carpinteri and Corradoâs (2011) suggested correlation with member depth and the modulus of rupture along with backcalculated fr/ â²fc ratios from dif- ferent tests. Additionally, segmentally constructed girders appear to have a lower modulus of rupture (Megally et al. 2002). This research concluded that the lower modulus of rupture is due to the soft layer of concrete at the ends of the segments adjacent to the epoxied joints, where large aggregates are hardly present. Thus, if a primary goal in prescribing mini- mum flexural reinforcement is to ensure the stability of the structure after flexural cracking, it is imperative to consider the influence of the depth and type of construction of the member for accuracy. Segmental girders tested for minimum flexural reinforcement are absent from the experimental literature. Pre-tensioned girders are more present, though these tests are not always conducted on full-scale specimens. There has been much research on mildly reinforced concrete girders; however, these tests were often performed on scaled specimens for increased economy, and thereby introduced scale effects. NCHRP 12â80 addressed the variability of the flexural cracking strength owing to the member height but con- cluded that, because of the limited data, the accuracy gained by including the effect was not worth the complexity added to the formulation. 2.6.4 Strength Ratio The historical strength requirements in the AASHTO LRFD Bridge Design Specifications were based on two equations: 1.2 (2-17)crM M n â¥ Ï 1.33 (2-18)M M n uâ¥ Ï However, the adequacy of these coefficients has not been thoroughly studied. The purpose of these strength require- ments is to ensure that the beam possesses positive stiffness after cracking and thus can still resist load beyond experienc- ing flexural cracking. 2.6.5 Ductility In the AASHTO LRFD Bridge Design Specifications, duc- tility requirements are not addressed as part of the mini- mum flexural reinforcement. However, as previously noted, Figure 2-16. fr / â²fc ratios from full-depth concrete members by Holombo and Tadros (2010) together with a recommendation from Carpinteri and Corrado (2011). (Carpinteri and Corrado 2011) f r/ (f c ) (p si )
24 ductility is implicitly addressed in other parts of specifica- tions for tension-controlled members. A structural member can behave in several ways after it reaches its âyieldâ point, as illustrated in Figure 2-17. The post-âyieldâ performance determines the ductility, which can be loosely defined as the ratio between the displacement at failure and the displace- ment at âyield.â A specific ductility quantity in terms of how much ductility is sufficient for flexural members, however, is not provided in the AASHTO LRFD Bridge Design Specifica- tions. While a significant negative stiffness and a significant positive stiffness beyond yielding can be characterized as unacceptable and acceptable performances, respectively, it is unclear whether elasticâperfectly plastic performance is acceptable, although it can be argued as unacceptable per- formance from a theoretical viewpoint. This leads to a sug- gestion that the system must have a positive stiffness beyond yielding and that the required ductility should be function of a ratio of moments such as Mn/Mcr. Requiring the mini- mum reinforcement to be based on the moment strengths and ductility would make the design process complex. Therefore, to keep the design process simple and ensure a sufficient safety margin against brittle failure, it is envi- sioned that drastic changes to code specifications should be avoided. 2.7 Summary of Findings Important findings from the state of knowledge presented in this chapter that are relevant to this project are summa- rized below. â¢ There has been no extensive experimental research to study the minimum flexural reinforcement requirements on bridge girders, in particular, research involving prestressed concrete and segmentally constructed girders subjected to large-scale testing. â¢ Past research on minimally reinforced structures has used different measures to quantify satisfactory performance, such as moment ratios, ductility, brittleness number, and deflection; however, there has been no consensus on what the best measure is to ensure the adequate safety of flexural members designed with minimum reinforcement after they experience flexural cracking. The most commonly used parameters are strength and ductility. â¢ Analysis on beams with minimum flexural reinforcement has shown inconclusive results with regard to whether beam depth has considerable influence on member ductility. â¢ Use of the fracture mechanics concept has led to better understanding of the influence of member depth on flexural cracking. However, it has not been widely used in routine design practice. â¢ Three parameters have been shown to limit minimum flexural reinforcement in flexural members: ductility and strength ratio (mainly for safety) and crack width (mainly for serviceability). Because of the emphasis on safety, crack width is not considered a main controlling parameter in the current project, since different provisions are used in the AASHTO LRFD Design Specifications to evaluate this. â¢ Minimum flexural reinforcement requirements in the cur- rent codes, standards, and specifications vary significantly and are based on different assumptions. The variability of the material properties from specified to actual values is not addressed in the codes and standards. This variation should be given consideration in establishing the minimum reinforcement; however, it is outside of the scope of this project. Figure 2-17. Illustration of various force versus displacement curves.