# Formulas for Minimum Reinforcement Based on Fracture Mechanics

In most cases, the minimum reinforcement must ensure that the ultimate (collapse) load (point T in Fig. 10.4.1 d) be equal to the first peak load (point M in Fig. 10.4.Id). Based on purely experimental grounds, Bosco and Carpinteri (1992) proposed a formula which correlates the brittleness number iVj, miil at which the minimum reinforcement condition is met, to the compressive strength of the concrete, fc:

NPmm 01 + °’23” > °4 = 100 MPa (10.4.22)

From this and the definition (10.4.6), we can solve for pmi„ and get

Note that this formula is purely empirical since the first peak of the load cannot be adequately predicted with the model of Bosco and Carpinteri (1992).

Baluch, Azad and Ashmawi (1992) proposed the following formula for minimum reinforcement based on the model described in Section 10.4.2:

1.9134ЛГ?,82

Pmin "

where Kjc and fy must be expressed, respectively, in MPa./m and MPa. Note that the lack of dimensional consistency indicates that there is a certain degree of empiricism in this equation. (In the original paper, there is a misprint in the formula using units of N and mm —the factor in the numerator should appear in the denominator; here the standard IS units arc used, and the formula has been checked against the tabulated values in that paper.)

Gcrstle et al. (1992) pushed the definition of /9m;„ further by requiring that the load increase mono – tonically all the time during the test (e. g., curve for pEs/Ec — 0.10 in Fig. 10.4.6f). The formula they proposed is

Note that this formula, due to the particular definition of the authors, does not depend on the strength of the reinforcement, but only on its elastic properties. We will see that this formula gives values far larger than the other models and larger than the currently accepted values in the codes.

Hawkins and Hjorsetet (1992) also proposed a formula for minimum reinforcement based on the cohesive crack model as well as some concepts derived from Carpinteri’s approach. Their final formula reads

LD

Pmn = 0.175 , /г – r (10.4.26)

JyU <-s)

where fr is the rupture modulus for an unreinforced beam of the same dimensions as the actual reinforced beam. According to these authors the modulus of rupture can be computed using a cohesive crack model with the appropriate softening curve. Therefore, it is possible to use the equation (9.3.12) to get a closed-form expression as

where £ — 1.046 for three-point bending and 1 for four-point bending.

None of the preceding formulas take into account the bond strength. Ruiz, Plunas and Elices (1996) and Ruiz (1996) have proposed a formula taking this effect into account. The formula is based on the cohesive crack model, more specifically on the expression (10.4.20) giving the first peak load. The final plastic collapse load is obtained from elementary considerations of equilibrium of moments as

°NP – pfy 6 (l – ~y) (10.4.28)

Then, setting ct/vc cr/vp, solving for p = pmin, and inserting (9.3.12), one gels

С 1 + (0.85 – I – 2.3Z9/^,)_1

Pmin ~ ~77 Tr\ ~ Г і /4 …….. V (10.4.29)

6(1 – cs/D) fjf, _ p [(ДА?,)17 _ 3.61cs/f|j

Comparing the aforementioned models is not straightforward because they tire based on different assumptions and depend on different parameters. This means that the predictions can be similar for certain conditions and differ for other conditions; no exhaustive comparison has been done to date. Fig. 10.4.16 shows the dependence on size of the minimum reinforcement for the various models in a particular case defined as follows:

1. The concrete is assumed to be characterized by f’t = 4 MPa, fc = 40 MPa, Ec — 30 GPa, Gy = 160 N/m (total fracture energy, as determined from the work-of-fracture test). For Carpinteri’s and Baluch’s model, it is assumed that Irwin’s relation holds; K{c — fEcGy — 2.19MPa/m. For Gerstle’s formula, linear softening is assumed with wc — uq = 2Gу/f’t — 80 gm. For Hawkins’ formula, Petersson’s bilinear softening is assumed with uq Gy/f[ ~ 48 pm. For Ruiz’s models, bilinear softening is assumed with uq = Gy/f[ = 40 pm.

2. The steel is assumed to have Es — 210 GPa, fy = 480 MPa.

3. The cover of concrete reinforcement is assumed to be of constant thickness c3 — 24 mm.

4. For Ruiz’s model, the bond parameter p is taken to be constant. This is achieved by using bars of the same diameter for all beam sizes. Two values are considered: p — 10 (weak bond,16 mm diameter bars with rc rs 0.4 f’t) and p = 40 (strong bond, 8 mm diameter bars with rc as 3 ft).

Fig. 10.4.16a shows the entire set of curves with a vertical logarithmic scale making it clear that the formulas of Baluch, Azad and Ashmawi (1992) and of Gerstle et al. (1992) give too large “Values. Fig. 10.4.16b shows an enlarged plot (with a linear scale) in which some code specifications have been included for comparison. From the plots it is evident that, for small beam depths, the models ofBosco and Carpinteri, Hawkins and Hjorsetet, and Ruiz, Planas and Elices give very similar results, slightly higher than those specified by ACI 318(92), and higher that the specifications of the Model Code and Eurocode 2. For medium and large sizes, the model of Bosco and Carpinteri gives values sharply below those of ACI and the other models, while the formula of Hawkins and Hjorsetet gives values very close to those given

by the formula of Ruiz et al. in the case of weak bond, both giving values between those in the Model Code and the ACI Code. For a strong bond, (lie model of Ruiz et al. predicts a minimum reinforcement that first decreases with the size and then increases, with values between those recommended by the ACI code and the Spanish Code.

Exercises

10.4 Show that in the plot of Fig. 10.4.5a the position of the asymptotes on the right is given by T — 6(1 — 7)Np, where 1 —7= 1 — c.,/D is the relative depth of the reinforcement.

10.5 Derive Eq. (10.4.28) in detail.