Background Knowledge - Strengthening Design of Reinforced Concrete with FRP - page 10

φ

φ = 0.75 + (ε t - 0.002) (50)

0.90

Spiral

0.75

φ = 0.65 + (ε t - 0.002) (250/3)

0.65

Other

Compression
controlled

Tension
controlled

Transition

ε t = 0.002 ε t = 0.005

c c = 0.375

= 0.600

d t d t

Interpolation on c / d t :Spiral φ = 0.75 + 0.15[(1/ c / d t ) - (5/3)]
Other φ = 0.65 + 0.25[(1/ c / d t ) - (5/3)]

FIGURE 2.1 Change of ϕ factor with net tensile strain in extreme steel bars (ε t ) or neutral-
axis depth ratio ( c / d t ) for Grade 60 reinforcement. (Courtesy of ACI 318-11.)

as seen in Figure 2.2b. The strain profile at steel yielding and ultimate capacity are
illustrated in Figure 2.2. At a level of tensile steel strain (ε t = 0.004) when compres-
sive extreme concrete fiber strain reaches concrete crushing (ε cu = 0.003), the value
of c / dt = 0.429 results from similar triangles (Figure 2.2b), which corresponds to a
factor ϕ = 0.817 for members other than those with spiral steel (Figure 2.1).

2.2.2 F orCe e quilibrium
To determine the location of the neutral axis in beams, force equilibrium needs to
be satisfied as follows:
Analysis problem:

C = T (2.1)

ε cf

0.003

c y c

α f ´ c b c y

α f ´ c b c

φ y

φ n

d t

d t

h A s f y

ε y

A s f y

h

ε s ≥0.004

(a) (b)

FIGURE 2.2 Strain and force profile at (a) first steel yielding and (b) ultimate capacity.