Civil Engineering Reference
In-Depth Information
Hognestad et al's tests
The shape of the concrete stress-strain curve was found by the 'dog bone' tests of Hognestad
et al.
(1955) to have two additional characteristics. First, the coefficients
k
1
and
k
2
de-
pend on the compressive strength of the concrete
f
c
. Apparently, the stress-strain curve
becomes more linear in the ascending portion and more steep in the descending portion
when the concrete strength is increased. In other words, the coefficients
k
1
and
k
2
should de-
crease with an increase of
f
c
. Second, the maximum stress of concrete in the stress block
of a beam is somewhat less than the maximum stress of a 152
12 in.)
standard concrete cylinder,
f
c
, because of size effect, shape effect, loading rate effect, etc. The
maximum stress of concrete in the stress block is defined as
k
3
f
c
, where
k
3
is also function
of
f
c
.
The three coefficients,
k
1
,
k
2
and
k
3
, are given for concrete with
f
c
<
×
305 mm (6
×
55 MPa (8000 psi)
as follows:
f
c
(MPa)
179
f
c
(psi)
26 000
k
1
=
0
.
94
−
or
k
1
=
0
.
94
−
(3.66)
f
c
(MPa)
552
f
c
(psi)
80 000
k
2
=
0
.
50
−
or
k
2
=
0
.
50
−
(3.67)
051
f
c
(MPa)
3
.
9
+
0
.
k
3
=
3
.
0
+
0
.
115
f
c
(MPa)
−
0
.
00081
f
c
(MPa)
35
f
c
(psi)
3
,
900
+
0
.
or
(3.68)
3 000
+
0
.
82
f
c
(psi)
−
0
.
000038
f
c
(psi)
Whitney's equivalent rectangular stress block
To simplify the analysis and design, ACI Code (ACI 318-08) allows the curved concrete stress
block to be replaced by an equivalent rectangular stress block as shown in Figure 3.9(d). The
replacement is such that the magnitude and the location of the resultant
C
remain unchanged.
The ACI rectangular stress block has a uniform stress of 0
85
f
c
.
and a depth,
a
=
β
1
c
, where
β
1
is determined to be:
f
c
β
1
≤
27.6 MPa (4000 psi)
0.85
34.5 MPa (5000 psi)
0.80
41.4 MPa (6000 psi)
0.75
48.3 MPa (7000 psi)
0.70
≥
55.2 MPa (8000 psi)
0.65
β
1
is 0.85 when
f
c
is less than
4000 psi, and reduces by 0.05 for every 1000 psi, up to 8000 psi.
Based on this rectangular stress block (a
In terms of the US conventional units, this table shows that
=
β
1
c
), the coefficients
k
1
=
0
.
85
β
1
and
k
2
=
β
1
/
2.
85
f
c
ba
85
f
c
b
The magnitude of the resultant
C
β
1
c
. The force equilibrium equation
(3.55) and the moment equilibrium equation (3.56) become
=
0
.
=
0
.
85
f
c
ba
A
s
f
s
=
.
0
(3.69)
A
s
f
s
d
a
2
M
u
=
ϕ
−
(3.70)
