Background Knowledge - Strengthening Design of Reinforced Concrete with FRP - page 13

steel in tension to bridge cracks and carry tension. Concrete and steel are thermally
compatible, making them ideal for a composite material. Reinforcing steel has a
coefficient of thermal expansion of 6.5 × 10 -6 /°F (11.7 × 10 −6 /°C), while concrete
has a coefficient of thermal expansion of 5.5 × 10 -6 /°F (9.9 × 10 -6 /°C) (Beer and
Johnston 1992). The constitutive behavior of reinforced concrete constituent materi-
als is described in the following sections.

2.2.4.1 Behavior of Concrete in Compression
The stress-strain response of concrete in compression is nearly linear at the begin-
ning of loading up to approximately

0.7 c . Beyond that stress, the response becomes
highly nonlinear up to failure. One of the simplest models to effectively capture the
stress-strain response of concrete in compression is that of Hognestad's parabola
(Hognestad 1951), shown in Figure 2.5.

f

2

ε
ε

ε
ε

c c

σ= f 2 − 0 <ε <ε = 0.003 (2.13)

c c c cu

c c

where ε c is the strain corresponding to f c , typically equal to 0.002 for normal-
strength concrete or, more accurately, ε=
c

f
E

1.71 (MacGregor 1992). The variable
f c is the 28-day compressive strength of standard 6 × 12 in. (150 × 300 mm) cylin-
ders, and ε cu is the limit of useful compression strain of concrete before crushing,
identified as 0.003 by ACI 318-11, as demonstrated in Figure 2.6.
The initial tangent modulus from Hognestad's parabola is

c
c

d
d

σ
ε

22 ε

c

(2.14)

== ε Ef − ≅ 1000 f

ci c c

2

ε

c c c

ε= 0

c

6

5

4

3000

3

2000

2

1000

1

0

0 .001 0 .002

Strain, ε c (in/in)

0.003

0 .004

FIGURE 2.5 Typical stress-strain curves for concrete of various strengths. (Courtesy of
Portland Cement Association [2013].)