Derivation of minimum steel ratio: based on service applied stresses

Reinforced concrete beams undergo several design stages starting with the immediate application of loading and till the ultimate failure of the beam. In many practical applications, the beam is designed to function in the service stage during which steel plays a major role in the determination of the location of the Neutral axis. The ACI code implements several provisions on the minimum steel ratio in order to encourage ductile behaviour of the beam and to prevent against the unfavourable scenario of a sudden failure. The ACI 318-08 provisions encompass a derived expression to calculate the minimum amount of flexural reinforcement that is independent of concrete strength. This paper suggests an expansion to the formula by deriving the minimum steel ratio based on the modulus of rupture and the applied service stresses. The special case for a cracking bending moment is extrapolated from the suggested formula and the result is compared to different minimum steel area formulas.


Introduction
Tensile steel reinforcement is indispensable to the concrete beam should the applied loads produce stresses that exceed the modulus of rupture. Steel becomes active then providing the necessary tensile strength and forming the cracked transformed section of the beam. The transformed section consists of uncracked concrete that is limited to the area above the neutral axis and an "equivalent" area of concrete to the steel located below the neutral axis. Upon the determination of the depth of the neutral axis, the cracked moment of inertia of the beam is calculated, and consequently the stresses in both concrete and steel are calculated and compared to the applied service stress limits. The process is usually conducted with the area of steel given from the start or based on the ACI code provisions [1].
However, the equation provided according to the ACI code is intended of the case of a prevailing concrete strength. Salmon [2] have updated and expanded the formula to account for the modulus of rupture and introduced his equation into ACI 318-95 [3]. In addition to the ACI several codes, specifications and researches have come up with their criteria for the minimum steel area. Some of these are listed in Table 1.
Freyermuth and Aalami [4] 1997 ‫ܥ‬ is a multiplier hat adjusts the section modulus for different beam shapes Note: ‫ܣ‬ ௦, = minimum area of nonprestressed flexural tension reinforcement; ݄ = height of the section; ܾ ௪ = web width; ݀ ௦ = distance from extreme compression fiber to centroid of nonprestressed flexural tension reinforcement; ݂ ሖ = specified compressive strength of concrete; ݂ ௧ = direct tensile strength of concrete; ݂ ௦௨ = specified tensile strength of nonprestressed flexural tension reinforcement; ݂ = modulus of rupture of concrete; ݂ ௬ = specified minimum yield strength of nonprestressed flexural tension reinforcement; ݊ = is the ratio of modulus of rupture to allowable compressive strength; ݊ ௦ = is the ratio of modulus of rupture to allowable tensile strength; ݊ = is the ratio of section height to depth of reinforcement; ‫ܥ‬ and ߙ are variables defined in the context of this paper.

Methodology of derivation of minimum steel ratio
The derivation of minimum steel ratio in this paper proceeds in the following order: a. Derivation of the Neutral axis depth ratio () and solving for the corresponding steel ratio b. Expressing the cracked moment of inertia in terms of c. Solving for in the flexural stress formula of concrete and steel and dismissing the unrealistic/unfeasible solutions d. Extrapolating the special cracking moment case and obtaining the minimum feasible limit of e. Substituting in the steel ratio formula and solving to obtain expression for minimum steel ratio The derived formula assumes a linear stress-strain relation and a perfect bond between the concrete and steel. However, the procedure is limited to the case of a rectangular beams or T beams with neutral axis in the flange.

Location of neutral axis
Once the beam has beam cracked, the neutral axis of the beam immediately deviates from the centroid of the beam, and the transformed section is formulated as shown in Figure 1.
The neutral axis location is defined as the point of which the first moment of the section about the neutral axis is zero. This leads to the formulation of the following Equation 1: Solving for ‫ݔ‬ leads to: - Despite the fact that ݇ is limited to a maximum value of 1, the steel ratio in unbounded, this is evident in Figure 2 as shown.

Evaluating cracked moment of inertia
The cracked moment of inertia is then evaluated according to Equation 5

Applying the concrete service stress limit
The stresses applied in the extreme compression fibers of concrete must be limited to the service stress limit as displayed according to Equation 6: - Substituting Equation 2 and Equation 5 into Equation 6 and rearranging the terms, a quadratic form in terms of ݇ results as Equation 6 dictates: - Solution of Equation 7 produces two solutions: The second solution is dropped because it yields ݇ that is always greater than one. Therefore, the feasible solutions of the inequality presented in Equation 7 lie within the range [݇ ଵ , 1]. It must be noticed that while the interval [݇ ଵ , 1] is a solution to the inequality presented in Equation 7, the interest of this paper is the value of ݇ ଵ itself because according to Figure 2 the minimum feasible value of ݇, results in the corresponding minimum feasible value of ߩ. Furthermore, by imposing the limitation that ݂ ݂ , it can be shown that ݇ ଵ is limited to not exceed an approximate value of 0.53.
This obtained feasible range however, is based solely on the compression limit of the concrete. Thus, the steel tensile limit must be treated similarly.

Applying the steel service stress limit
The stresses applied in the tensile reinforcement steel must be limited to the service stress limit as displayed according to Equation 10: -

Solution of Equation 11
produces three solutions: - Where The first solution is dropped because it yields ݇ that is always greater than one. The last solution is also dropped because it turns out to be negative for all practical cases. Therefore, the feasible solutions of the inequality presented in Equation 11 lie within the range [݇ ସ , 1].
This formulation presented in Equations 18-24 renders the definition of the minimum k dependent upon the three dimensionless ratios n ୡ , n ୱ , and n ୦ exclusively. k is then defined as: - Finally, the minimum steel ratio is computed as: - Accordingly, the minimum area of steel is determined by multiplying ߩ ୫୧୬ with the section width and effective depth.

Comparison of proposal with other formulas
In order to validate the derivation of the minimum steal ratio formula that has been provided in this paper, a comparison to other models is made. The comparison assumes fixed parameters as indicated in Table 2 and evaluates the minimum area of steal according to each formula that has been presented in Table 1. Accordingly, the results are tabulated as shown in Tables 3 and 4.

Discussion
The pattern of the results as displayed in Figure 3 indicates complete equivalence in the general trend of the minimum steal area resulting from all the formula, with the ACI 318-08 being the most conservative of all. Among the investigated formulas, only two of them are defined as a compound function of two criteria: steel and concrete stress requirement. Namely, the ACI-318-08 and the proposed formula. From the comparison, it can be noted that the first criteria of the ACI-318-08 formula namely ଶ ܾ ௪ ݀ ௦ is far more conservative than the second criteria.
Furthermore, the proposed formula seems to deviate from the general trend at values of concrete compressive strength roughly around 2000 ‫݅ݏ‬ , throughout which the concrete stress limit controls. It should be noted that the proposed formula is the only one that clearly accounts for both the stress limits of concrete and steel resulting in the possibility to define the best match of concrete and steel stress limits. Figures 3 and 4 indicate that f ୷ = 40000 psi is matching more harmonically with the results as the trend of the proposed formula exhibits less disparity when compared to f ୷ = 60000 psi.

Conclusion
The proposed formula is clearly in match with the other formulas but it results in a minimum area of steel that is less than the ones resulting from all the other formulas. The computations of the proposed formula is a little more complicated, however, this issue is of no concern to programs as the formula can be computed easily in a straight forward algorithm without any need for any sort of cyclic routines. The formula is however (like the rest), limited to the assumptions presented in the methodology section. Further work is expected to expand the range of applicability of the proposed formula.