Civil Engineering Reference
In-Depth Information
The transverse shear steel calculated by Equation (2.109) is require
d in the v
ertical legs of
the cross-section.
The spa
cing
s
is limited to
d
33
f
c
(MPa)
b
w
d
, and
d
/
2 when
V
s
≤
0
.
/
4
33
f
c
(MPa)
b
w
d
. The angle
is taken as 45
◦
in the ACI code for shear. It would
when
V
s
>
0
.
θ
45
◦
be more logical to use the same
θ
=
in Equation (2.99) for torsion, especially in the case
of combined shear and torsion.
2.3.2.3 Longitudinal Shear Steel
According to the equilibrium (plasticity) truss model, shear stress also demands longitudinal
shear steel according to Equation (2.38). In the 2008 ACI Code, however, longitudinal shear
steel continues to be designed indirectly by the so-called shift rule in Section 8.1.1.3 (Chapter
8). In this indirect method, the bending moment diagram is shifted toward the support by a
distance
d
, the effective depth. As a result, the longitudinal bars required by bending are each
extended by a length
d
to take care of the longitudinal shear steel.
2.3.3 Maximum Shear and Torsional Strengths
2.3.3.1 Maximum Shear Strength
The maximum shear strength of a beam cross-section can be derived from Equation (2.40) by
taking
h
30
◦
. The compressive strength of concrete
σ
d
in Equation
(2.40) was found to be softened by the principal tensile strain in the perpendicular direction.
Quantifying the softening effect by a softening coefficient
=
b
w
,
d
v
=
0
.
9
d
, and
θ
=
f
c
ζ
, then
σ
d
=
ζ
and the maximum
shear strength becomes:
f
c
b
w
d
V
n
,
max
=
.
ζ
0
39
(2.110)
The softening coefficient
ζ
i
s s
hown in Section 6.1.7.2 and in Figure 6.9 (Chapter 6) to
be inversely proportional to
f
c
for concrete stre
ngth up
to
f
c
=
100 MPa (15 000 psi).
2.13/
f
c
(MPa), we arrive at the long-time ACI
provision for maximum shear stress of nonprestressed members:
Assuming a very conservative value of
ζ
=
83
f
c
(MPa)
10
f
c
(psi)
V
n
,
max
b
w
d
=
0
.
or
(2.111)
2.3.3.2 Maximum Torsional Strength
The maximum torsional strength of a cross-section can be derived from Equation (2.49) by
taking
h
30
◦
and
f
c
:
=
0
.
9
A
oh
/
p
h
,
A
o
=
0
.
85
A
oh
,
θ
=
σ
d
=
ζ
A
oh
p
h
f
c
T
n
,
max
=
0
.
66
ζ
(2.112)
2.13/
f
c
(MPa) gives the maximum
Again, assuming a very conservative value of
ζ
=
torsional strength:
41
f
c
(MPa)
17
f
c
(psi)
T
n
,
max
p
h
A
oh
=
1
.
or
(2.113)
