Civil Engineering Reference
In-Depth Information
In calculating the parameter
B
in Equation (6.88), the percentage of steel
ρ
should not
be less than 0.15%. This
0.15% should not impose any difficulty, because it is the
minimum specified by the ACI Code for deformed bars in walls.
Equations (6.85) and (6.86) are quite simple to use, because they are functions of only
one parameter
B
defined in Equation. (6.88). These two equations are applicable to both
longitudinal and transverse steel as follows:
ρ
min
=
f
s
=
f
or
f
t
when applied to longitudinal steel or transverse steel, respectively,
¯
ε
s
=
ε
¯
or ¯
ε
t
when applied to longitudinal steel or transverse steel, respectively,
The application of Equation (6.86) is illustrated by the following example:
Given:
ρ
=
0
.
75%
,
f
cr
=
2
.
07% MPa (300 psi);
f
y
=
413 MPa (60 ksi);
E
s
=
200 000 MPa (29 000 ksi)
Find: the smeared steel stress
f
s
at a smeared tensile strain ¯
ε
s
=
0.01
f
cr
f
y
1
.
5
2
1
.
5
1
ρ
1
07
413
.
10
3
)(0
10
−
3
)
B
=
=
=
(0
.
133
×
.
353
×
=
0
.
0469
0
.
0075
f
y
=
(0
.
93
−
2B)
f
y
=
(0
.
93
−
2
×
0
.
0469)
f
y
=
0
.
836 (413)
=
345 MPa
ε
y
=
f
y
/
E
s
=
/
,
=
.
345
200
000
0
00173
ε
y
), use Equation (6.86):
Since ¯
ε
s
=
0.01
>
0.00173 (
f
s
=
(0
.
91
−
2B)
f
y
+
(0
.
02
+
0
.
25B)
E
s
¯
ε
s
=
(0
.
91
−
2
×
0
.
0469)
f
y
+
(0
.
02
+
0
.
25
×
0
.
0469)
E
s
¯
ε
s
=
0
.
816(413)
+
(0
.
0317)(200 000)(0
.
01)
=
337
+
63
=
400 MPa
.
When the steel stress
f
s
reaches a peak stress
f
p
and starts to unload, the unloading
stress-strain curve is assumed to be a straight line with a slope of
E
s
. This unloading straight
line can be expressed as follows:
f
s
=
f
p
−
E
s
(¯
ε
p
−
ε
s
)
¯
ε
s
<
¯
ε
p
¯
(6.89)
Equation (6.89) is also shown in Figure 6.20 as a bold dotted straight line.
6.1.10 Smeared Stress-Strain Relationship of Concrete in Shear
A rational and simple shear modulus has been derived theoretically by Zhu, Hsu, and Lee
(2001) for a stress and strain analysis based on the smeared-crack concept. Referring to the
Mohr circles for stresses and strains in Figure 6.22(a) and (b), and assuming that the direction of
the principal stress of concrete coincides with the principal strain, the following two equations
can be obtained:
c
c
2
12
=
σ
1
−
σ
c
τ
tan 2
β
(6.90)
2
γ
12
=
(
ε
1
−
ε
2
) tan 2
β
(6.91)
