Minimum flexural reinforcement requirements have been a source of controversy for many years. The purpose of such provisions is to encourage ductile behavior in flexural members with sufficient cracking and deflection to warn of impending failure. Historically, these minimum reinforcement requirements have been intended to achieve one of the following two results:

• to avoid sudden failure of a flexural member at first cracking, and
• to permit such a failure only at a resistance sufficiently higher than the factored moments resulting from the specified strength load combinations.

These criteria are applicable to both nonprestressed and prestressed concrete flexural members. Our study focuses on reinforced concrete, which for purposes of our original paper on this topic (published in the Summer 2010 PCI Journal) includes only mild tensile reinforcement and no prestressing.

The first minimum reinforcement criterion is strictly a function of the member shape and material properties. The important parameters include the section modulus at the tension face, concrete strength, and stress-strain characteristics of the tensile steel. This criterion is not related to the actual loading on the beam. For our purposes, this type of criterion will be referred to as a sectional provision because it relates to behavior of the section rather than to actual loading.

In some cases, such as T-beams with the flange in tension, the section modulus at the tension face can become quite large, resulting in a substantial amount of sectional minimum reinforcement. Under these circumstances, the second criterion provides some relief in that the amount of minimum reinforcement can be derived directly from the applied factored load, which can be significantly smaller than the load that theoretically causes flexural cracking. This type of criterion will be referred to as an overstrength provision.

The primary objectives of our study were to summarize the apparent origin of current minimum reinforcement provisions, examine the margin of safety provided by existing provisions for reinforced concrete members of different sizes and shapes, and propose new requirements when they provide more consistent results than existing provisions. Parametric studies were performed to compare the proposed provisions with current American Association of State Highway and Transportation Officials (AASHTO) LRFD Bridge Design Specifications, American Concrete Institute (ACI) Building Code Requirements for Structural Concrete (ACI 318-08) and Commentary (ACI 318R-08), and requirements proposed by Freyermuth and Aalami and the American Segmental Bridge Institute (ASBI). Concrete strengths up to 15 kips per square inch (ksi, 103 MPa) and high-strength steels were included. The provisions in our paper apply to determinate members only, such as simple spans and cantilevers. These results are not included in this article, but can be found in the original article (see page 30 for link).

Flexural failure at minimum reinforcement levels can be initiated either by fracture of the tensile steel or crushing of the concrete at first cracking. There appears to be a misconception that minimum reinforcement is strictly intended to prevent fracture of the reinforcement at first cracking. In many cases, particularly in T-beams with a tension flange, sufficient reinforcement must be provided to engage enough compression area so the concrete will not crush at first cracking.

Summary of minimum reinforcement provisions
Tables 1 and Table 2 summarize the requirements for minimum flexural reinforcement from the AASHTO LRFD specifications, ACI 318-08, Freyermuth and Aalami, ASBI, and the new requirement proposal. The apparent origins of these provisions are described in the sections below.

AASHTO LRFD specifications provisions
Article 5.7.3.3.2 of the AASHTO LRFD specifications states the sectional requirement satisfy Eq. (1), where

In Eq. (1), M_n = nominal flexural resistance, φ = resistance factor = 0.9, and the cracking moment,

In Eq. (2), S_c = section modulus for the extreme fiber of the composite section where tensile stress is caused by externally applied loads; f_r = modulus of rupture of concrete; f_cpe = compressive stress in concrete due to effective prestress forces only (after allowance for all prestress loss) at extreme fiber of section where tensile stress is caused by externally applied loads; M_dnc = total unfactored dead load moment acting on the monolithic or noncomposite section; S_nc = section modulus for the extreme fiber of the monolithic or noncomposite section where tensile stress is caused by externally applied loads;

and f'_c = specified compressive strength of concrete.

Because AASHTO LRFD specifications are in a format that unifies the design of prestressed and nonprestressed concrete, this provision is applicable to both and in any combination. Eq. (1) is consistent with the ACI 318-08 provision for prestressed concrete except for the modulus of rupture used to calculate the cracking moment. In ACI 318-08, the coefficient used in Eq. (3) is 7.5 instead of 11.7. This difference has a significant impact on minimum reinforcement and is discussed in detail in the original paper.

AASHTO LRFD specifications allow the sectional requirement of Eq. (1) to be waived if

where M_u = factored moment. This overstrength criterion is consistent with the ACI 318-08 provision for reinforced concrete.

ACI 318-08 provisions
In 1963, Eq. (5) was introduced into ACI 318 to provide a minimum amount of flexural reinforcement.

where A_s,min = minimum area of nonprestressed flexural tension reinforcement; b_w = web width; d_s = distance from extreme compression fiber to centroid of nonprestressed flexural tension reinforcement; and f_y = specified minimum yield stress of nonprestressed flexural tension reinforcement.

Eq. (5) was said to be derived by equating the capacity of a reinforced section with a plain concrete section. However, because concrete strength is not a variable in this equation and the modulus of rupture depends on the concrete strength, this equation was apparently intended to provide minimum flexural reinforcement for the prevailing concrete strengths in use at the time. This equation was updated and expanded by C. G. Salmon, who derived an equation introduced into ACI 318-95 as shown below.

In Eq. (7) A_s = area of nonprestressed flexural tension reinforcement; and j = modifier for d_s to estimate the moment arm between the centroids of the compressive and tensile forces in a flexural member. The cracking moment,

where S_t = section modulus at the tension face of the member under consideration; C = multiplier that adjusts the section modulus for different beam shapes; and H = overall depth of member.

For rectangular members C is 1.0, and Salmon determined a range of 1.3 to 1.6 for T-beams with the flange in compression. Equating Eq. (7) and (8) and taking j equal to 0.95 and φ equal to 0.9 results in Eq. (9) or (10).

For rectangular members where H/d_s is assumed to vary from 1.05 to 1.2, K ranges from 1.6 to 2.1. For T-beams with the flange in compression, using a value for C of 1.5 and H/d_s between 1.05 and 1.2, K ranges from 2.4 to 3.2. Eq. (11) is the sectional expression for reinforced concrete that was adopted into ACI 318-95, and remains in ACI 318-08.

The adopted coefficient of 3 is at the upper end of Salmon's range. The lower limit is a hold-over from previous editions of ACI 318 and will govern only if the concrete strength in the compression zone is about 4,400 psi (30.3 MPa) or less.

For T-beams with the flange in tension, Salmon found C to be in the range of 3.0 to 4.0. Using a value for C of 3.5 and H/d_s between 1.05 and 1.2 leads to K values between 5.6 and 7.4. ACI 318-08 specifies the use of Eq. (11) for T-beams with the flange in tension, except that b_w is replaced by 2b_w or the width of the flange b, whichever is smaller. For most realistic T-beams with the flange in tension, Eq. (12) is the governing expression.

The coefficient of 6 is closer to the bottom of Salmon's range. C. P. Siess argued that a coefficient of 7 should have been chosen for a more conservative requirement.

The advantage of Eq. (11) and (12) is that the minimum quantity of tension reinforcement can be determined directly with a simple closed-form solution. However, in our opinion, choosing single coefficients to represent significant ranges of values will inevitably lead to variability in the margin of safety provided.

ACI 318-08 allows Eq. (11) or (12) to be waived if overstrength Eq. (4) is satisfied.

Freyermuth and Aalami provisions
The derivation of the requirements proposed by Freyermuth and Aalami begins with Eq. (13), which is an equation for minimum flexural reinforcement taken from the CEB-FIP Model Code for Concrete Structures.

where b_t = average width of the concrete zone in tension.

The logic behind this expression is not explained, but the derivation continues by increasing Eq. (13) by 1/3 to resolve some deficiencies in the CEB-FIP model and by substituting b_w for b_t to derive Eq. (14).

Freyermuth and Aalami do not discuss the deficiencies in the CEB-FIP model. To account for variations in concrete strength, the coefficient 0.002 is divided by the square root of 4,000 psi (28 MPa) to give Eq. (15).

This normalizes the equation for 4,000 psi (28 MPa) concrete, and requires more minimum reinforcement for higher strength levels. The strength of steel is addressed in Eq. (16) by multiplying Eq. (15) by 90,000 psi (620 MPa), the tensile strength of ASTM International's A615 Grade 60 (420 MPa) reinforcement, where f_su = specified tensile strength of nonprestressed flexural tension reinforcement.

This normalizes the equation for the grade of steel most commonly used in the United States. More or less reinforcement will be required for steels with lesser or greater tensile strengths, respectively. By rounding the coefficient up to 3, the proposed Freyermuth and Aalami sectional equation is Eq. (17).

This is similar to Eq. (11) from ACI 318-08 except that the tensile strength of the steel is used in the denominator rather than the yield strength and there is no lower limit.

The Freyermuth and Aalami proposal retains overstrength Eq. (4), which, if satisfied, allows Eq. (17) to be waived. The proposal also waives the sectional requirement of Eq. (17) for T-beams with the flange in tension and simply recommends that overstrength Eq. (4) be satisfied for those types of members.

ASBI provisions
The provisions in Eq. (18) through (20) were adapted from F. Leonhardt and proposed to AASHTO Subcommittee T-10 by the American Segmental Bridge Institute (ASBI). The concept is that the quantity of reinforcement must be sufficient to withstand the release of the tensile force resisted by the concrete prior to cracking. The direct tensile stress of concrete f_ct at cracking is estimated by Eq. (18).

This stress is assumed to cause cracking at the extreme concrete fiber in tension and vary linearly to zero at the center of gravity of the gross, uncracked concrete cross section. For a rectangular section, the total tension force F_ct can then be calculated by Eq. (19).

For nonrectangular members, the linearly varying stress must be integrated over the appropriate area bounded by the extreme tension fiber and the center of gravity of the gross, uncracked concrete cross section. The minimum sectional flexural reinforcement is then proposed to be Eq. (20).

It should be noted that the 1.2 coefficient in Eq. (20) is not part of Leonhardt's procedure and was added to the ASBI proposal only to make it more compatible with the existing AASHTO LRFD specifications. Minimum flexural reinforcement requirements calculated by the ASBI provisions can be reduced 20 percent to match Leonhardt's recommendations.

The overstrength provisions of Eq. (4) are retained as part of ASBI's proposal.

Proposed provisions
In his report to ACI Committee 318, Siess argued that the flexural capacity M_n of a reinforced concrete section should simply be the same or larger than a plain section of the same dimensions and concrete strength. Siess indicated that the margin of safety is provided by strain hardening of the mild reinforcement and the resistance factor. The primary disadvantage of this method is that the section modulus at the tension face of the member must be calculated, which is not necessary with simplified Eq. (11) and (12). However, the automated calculation methods employed today should mute such arguments.

We consider the argument by Siess to be persuasive with the following two modifications:

• Modern codes and design specifications are increasingly allowing more choices of reinforcing steel materials and strengths. Differences in the behavior of these materials should be reflected in minimum reinforcement provisions.
• The factor of safety represented by the yield-to-tensilestrength ratio should be kept constant for all grades of reinforcement.

The design strength of a flexural member is typically based on the nominal yield strength of the reinforcement, while the actual flexural strength includes strain hardening. For purposes of this study, strain hardening is the portion of the stress-strain curve where the steel stress increases beyond the yield stress with increasing strain. The peak stress is generally known as the tensile strength. Introducing a ratio of the yield to tensile-strength can increase the applicability of the equation to most grades of reinforcement including high-strength steels. Eq. (21) is the proposed sectional expression:

where M_cr is defined by Eq. (2) except that Eq. (22) determines the modulus of rupture f_r, such that:

The 1.5 coefficient in Eq. (21) normalizes the ratio of yield strength to tensile strength to 1.0 for ASTM A615 Grade 60 (420 MPa) reinforcement, as recommended by Seiss. Although Eq. (21) is designed to provide a consistent margin between cracking and collapse when the failure mode is fracture of the tensile reinforcement, the parametric study will show that the margins for failure by crushing of concrete are also reasonable, though full strain hardening of the steel is not achieved.

By substituting the applicable expressions into Eq. (21), a direct calculation of the quantity of minimum reinforcement can be derived. The derivation that produces Eq. (23) is provided in our original paper.

For the overstrength provision, we propose to waive Eq. (21) if Eq. (24) is satisfied.

This is a modified version of Eq. (4), which again includes the ratio of yield to tensile-strength of the reinforcing steel. The coefficient of 2 normalizes the modifier to the traditional 1.33 for ASTM A615 Grade 60 (420 MPa) reinforcement. Eq. (24) ensures a consistent margin between the design strength and the actual strength for all grades of reinforcement.

Conclusion
Ductility is an important aspect of structural design. The goal is to provide a reasonable margin of safety between first cracking and flexural failure or, alternatively, a reasonable amount of overstrength beyond the applied factored loads.

The method proposed in our paper is based on Seiss's recommendations and provides the most reasonable margins of safety among the methods examined. It is applicable to both normal- and high-strength concrete up to 15 ksi (103 MPa) and to the types and grades of reinforcement commonly allowed in the various codes and specifications. For beams used in structures, and for both normal- and high-strength concretes, the ACI 318-08 modulus of rupture of 7.5 √ f_c is recommended.

The following two important aspects of minimum flexural reinforcement should be emphasized:

• The provisions in our paper are intended to apply to determinate members only, such as simple spans and cantilevers. Indeterminate structures have redundancy and ductility inherent in their ability to redistribute moments. As such, we anticipate that less minimum reinforcement will be necessary for indeterminate structures; this goal requires a different approach than presented in this paper.
• Flexural failure at minimum reinforcement levels can be initiated either by fracture of the tension steel or crushing of the concrete at first cracking. There appears to be a misconception that minimum reinforcement is strictly intended to prevent fracture of the reinforcement at first cracking. This paper presents many cases where, using the proposed method, SM_cr is less than 1.66, indicating that the primary mode of failure is concrete crushing. With respect to minimum flexural reinforcement, this consideration applies to all concrete members, determinate or indeterminate, nonprestressed or prestressed, bonded or unbonded.

This article is a portion of the paper, "Making sense of minimum flexural reinforcement requirements for reinforced concrete members," by Stephen J. Seguirant, Richard Brice, and Bijan Khaleghi, which was printed in the Summer 2010 PCI Journal, reprinted with permission. Read the complete paper online at http://www.pci.org/publications/journal/back_view.cfm?year=2010&season=Summer.

Stephen J. Seguirant, P.E., FPCI, is the vice president and director of engineering for Concrete Technology Corporation in Tacoma, Wash., and he can be reached at 253-383-3545. Richard Brice, P.E., is a bridge software engineer for the Bridge and Structures Office at the Washington State Department of Transportation in Olympia, Wash. Bijan Khaleghi, Ph.D., P.E., S.E., is the bridge design engineer for the Bridge and Structures Office at the Washington State Department of Transportation.