Torsion - Unified Theory of Concrete Structures - page 304

σ σ

Eq. (5-100)

Eq. (5-100)

Eq. (5-101)

σ p

σ p

k 1 σ p

k 1 σ p

ε ds ε p ε ε p ε ds 2 ε 0 ε

0 0

(a) ASCENDING REGION (b) DESCENDING REGION

ε ds / ε p ≤ 1 ε ds / ε p > 1

Figure 7.5 Integration of softened stress-strain curve of concrete to determine k 1

in Figure 7.5(a) and (b), respectively. The two expressions for coefficient k 1 , one for the
ascending branch and another for the descending branch, will be derived separately below:

Ascending branch 1 (Figure 7.5a)

Inserting Equation (5.100) into (7.32) gives

ε ds /ε p ≤

2 ε
ε p

2 d ε ds ε

ε p

1
σ p ε ds

k 1 = σ p − ε (7.36)

o

σ p and peak strain ε p are used

For convenience of integration, the symbols of peak stress

, rather than
ε d , to avoid confusing with the symbol for the average strain. Integrating Equation (7.36)
results in

ζ

f c

and ζε o . The dummy variable is written as ε

in Equation (7.36), rather than

1

k 1 = ε ds
ε p

1

3 ε ds

− (7.37)

ε p

Letting ε p = ζε o and expressing k 1 in terms of ζ , Equation (7.37) becomes, when

ε ds /ζ ε o ≤ 1

1

ε ds
ζε o

1

3 ε ds

k 1 = − (7.38) or [13 a ]

ζε o

Descending branch 1 (Figure 7.5b)

Inserting Equations (5.100) and (5.101) into (7.32) gives

ε ds /ε p >

2 ε
ε p

2 d

1

2 d

ε p ε

ε p

ε ds

1
σ p ε ds

1
σ p ε ds

ε − ε p

k 1 = σ p − ε + σ p − ε

2 ε o − ε p

o ε p

(7.39)