Civil Engineering Reference
In-Depth Information
c
(
ρ
=
ρ
t
), both the angle
β
and the concrete shear stress
τ
12
are equal to zero. With increasing
12
are increased. An increase of
12
reduces the compressive capacity
ρ
/ρ
t
ratio, both
β
and
τ
τ
of the concrete struts through the reduction of
f
3
(
β
) and, in turn, the softening coefficient
ζ
.
Substituting
f
3
(
β
) into Equation (6.46), the softened coefficient
ζ
becomes:
5
9
1
8
√
f
c
≤
.
1
|
β
|
24
◦
ζ
=
0
.
√
1
−
(6.52)
+
400¯
ε
1
In the fixed angle theory,
β
can be calculated from the three strains
ε
1
,
ε
2
and
γ
21
using the
compatibility equation:
2
tan
−
1
1
γ
12
β
=
(6.53)
(
ε
1
−
ε
2
)
Equation (6.53) can also be derived from the Mohr strain circle, as will be shown later in
Figure 6.22(b).
6.1.8 Smeared Stress-Strain Relationship of Concrete in Tension
6.1.8.1 Smeared Stress
c
1
ε
1
of Concrete
In the measured Mohr circle for concrete stresses, as shown in Figure. 6.6, it can be seen that
the tensile stress of concrete
σ
and Smeared Strain ¯
c
1
c
σ
is small compared with the compressive stress
σ
2
, but not
c
zero. This stress
1
is an uniform tensile stress of concrete, representing the stiffening of the
steel bars by concrete in tension.
Figure 6.11 shows a typical tensile stress
-
strain curve of concrete. The curve consists of
two distinct branches. Before cracking the stress
-
strain relationship is essentially linear. After
cracking, however, a drastic drop of strength occurs and the descending branch of the curve
becomes concave. In the descending branch, the concrete is cracked and the concept of concrete
tensile stress
σ
1
and concrete tensile strain ¯
σ
ε
1
are quite different from those before cracking.
1
is defined as the
smeared
(or
average
) concrete tensile stress and ¯
σ
ε
1
is the
smeared
(or
average
) concrete tensile strain. These terms will be elaborated in Section 6.1.8.2.
Based on the tests of 35 full-size panels, Belarbi and Hsu (1994) and Pang and Hsu (1995)
proposed the following analytical expressions for the
1
-¯
σ
ε
1
curve:
Ascending branch (¯
ε
1
≤
ε
cr
)
1
σ
=
E
c
¯
ε
1
(6.54)
where:
where
E
c
=
modulus of elasticity of concrete, taken as 3875
f
c
(MPa), where
f
c
and
f
c
are in
MPa;
ε
cr
=
cracking strain of concrete, taken as 0.00008 mm/mm,
Descending branch (¯
ε
1
>ε
cr
)
f
cr
ε
cr
¯
0
.
4
c
1
σ
=
(6.55)
ε
1
31
f
c
(MPa).
where
f
cr
=
cracking stress of concrete, taken as 0
.
