Background Knowledge - Strengthening Design of Reinforced Concrete with FRP - page 14

0.006

0.004

Design maximum, ε cu = 0.003

0.002

Columns
Beams

0 0

2000 4000 6000 8000

Compressive Strength, f c (psi)

FIGURE 2.6 Tests of reinforced concrete members showing selection of the design maxi-
mum compressive strain. (Courtesy of Portland Cement Association [2013].)

However, Young's modulus for concrete is taken as the secant value at 0.4 f c . This is

given by the following ACI 318-11 equation:

E = 57 f (US customary units, E c in ksi, f c in psi) (2.15a)

c c

E = 4700 f (SI units, E c and f c in MPa) (2.15b)

c c

For concrete between 3000-5000 psi, E ci ≅ 3000 - 5000 ksi, E c = 3122 - 4031 ksi,
so the two moduli, are comparable in this range of f ′ c values. In flexural calculations,
ACI allows the usage of Whitney's rectangular stress block to replace Hognestad's
parabola. At the concrete crushing failure point, ACI assigns the block a height fac-
tor of γ = 0.85 based on tests of columns (Hognestad 1951) and the effect of sustained
load on the strength of concrete (Rüsch 1960). Also, the effective depth factor is

β= 0.85 if f

f

≤
≤

4000 psi

1 c

(2.16)

or 30 MPa

c

f

c

β= − 1.05 0.05 1000 if 4000 psi < f < 8000 psi

1 c

(2.17)

β= − 1.09 0.008 f if 30 MPa < f < 55MPa

1 c c

β= 0.65 if f

f

≥
≥

8000 psi

1 c

(2.18)

or 55 MPa

c

This was selected by comparison with experimental data points (Kaar, Hanson, and
Capell 1978), as seen in Figure 2.7.