Self-sensing concrete with nanomaterials - Nanotechnology in Eco-efficient Construction - page 67

˜ t in

the damage in the tension area is analysed. The effective tensile stress σ

the tension area is expressed by:

σ

t

( )

σ = = E εσ , ≥ 0 , D ≥ 0 [4.6]

t t t

1 − D

where D and E are respectively the degree of damage and the elastic
modulus;

σ t are respectively the strain and stress of the extreme
tension fi bre at the bottom of the concrete beam.
The effective compressive stress

ε t and

˜ c in the compressive area is given by:

σ

( ) σσ == E εσ , ≤ 0 D = 0 [4.7]

c c c c

where

c are respectively the strain and stress of the extreme com-
pression fi bre at the top of the concrete beam.

ε

c and

σ

σ , where k

˜

When the effective stress σ > 0 (tension stress), we have D =

k

˜

is the damage modulus. When σ ≤ 0, we have D = 0. Based on the strain

equivalence hypothesis, the following is obtained:

σ E

k

ε

⎛
⎝

⎞
⎠

⎛
⎝

⎞
⎠

( ) ( ) σ =− 1 D σ =− 1 D E ε =− 1 E ε =− 1 E ε [4.8]

k

where are respectively the strain and stress of the concrete matrix.

If the beam is in the elastic stage, the degree of damage in the concrete

ε and σ

D . After the elastic stage, the stress state of the tension

zone tends to be elasto-plastic due to the redistribution of stress before
cracking. The correlation between

= 0, then σ = E ε

ε and σ may be derived from Eq. [4.8]

and expressed by:

d
d

σ
ε

2 E

k

2

ε

=− E [4.9]

At the peak value of stress, the fi rst derivative of stress should be equal
to zero as described in Eq. [4.10]:

￿ ￿ ￿ ￿ ￿ ￿

d

d σ

= 0 [4.10]

ε

In this event, the critical cracking stress is

cr and the degree of damage
D is equal to D cr . From Eq. [4.9] and [4.10], we get E

σ

k
ε = 2 . Replacing E

ε

k
= 4 .

in Eq. [4.8] gives the critical cracking stress σ cr

σ ,

k

( )

When D = σ cr can be evaluated as σ cr ==− 1 DDk , from which

cr cr

k 4

˜ cr may

D cr = 0.5 is obtained. Therefore, the effective critical cracking stress σ

be expressed as: