Civil Engineering Reference
In-Depth Information
the properties of the transverse steel in Equation (2.5):
A
t
f
ty
s
q
=
tan
α
r
(7.67)
Inserting
q
from Equation (7.67) into Equation (7.66) we have
A
t
f
ty
tan
α
r
T
n
=
(2
A
o
)
(7.68)
s
Equation (7.68) is the fundamental equation for torsion in the truss model. Since
A
o
is
defined by the centerline of shear flow, the crucial problem of finding
A
o
is to determine the
thickness of the shear flow zone,
t
d
.
The analysis of torsion shown above is analogous to the analysis of bending in a prismatic
member discussed in Section 3.2; in Figure 3.9, a rectangular cross-section is subjected to a
nominal bending moment
M
n
. This external moment
M
n
is resisted by an internal bending
moment which is the product of the resultant of the compressive stresses
C
in the compression
zone of depth
c
and the lever arm
jd
:
M
n
=
C
(
jd
)
(7.69)
The equilibrium of the forces requires that
C
T
, where
T
is the tensile force of longitudinal
steel
A
s
f
y
after the cracking of the flexural member and the yielding of steel. Inserting
C
=
=
A
s
f
y
into Equation (7.69) gives:
M
n
=
A
s
f
y
(
jd
)
(7.70)
Equation (7.70) is the fundamental equation for bending. Since
jd
is defined by the position
of the resultant
C
, the crucial problem of finding
jd
is to determine the depth of the compression
zone
c
.
Equation (7.70) shows that the bending moment capacity
M
n
is equal to the longitudinal
steel force
A
s
f
y
times the resultant lever arm
jd
. Similarly, in Equation (7.68) the torsional
moment capacity
T
n
is equal to a certain stirrup force per unit length (
A
t
f
ty
/
α
r
times
twice the lever arm area 2
A
o
. In other words, the term of twice the lever arm area 2
A
o
in
torsion is equivalent to the resultant lever arm
jd
in bending, and the shear flow
q
is similar
to the resultant of compressive stresses
C
.
In bending, an increase of the nominal bending strength
M
n
due to increasing reinforcement
results in an increase of the depth of the compression zone
c
, and a reduction of the resultant
lever arm
jd
. The relationships among
M
n
,
c
and
jd
can be derived from equilibrium,
compatibility and constitutive relationships of materials. Similarly in torsion, an increase of
the nominal torsional strength
T
n
due to increasing reinforcement results in an increase of the
thickness of shear flow zone
t
d
and a reduction of the lever arm area
A
o
. The relationships
among
T
n
,
t
d
and
A
o
can also be derived from the equilibrium, compatibility and constitutive
relationships of materials, as shown in Section 7.1. In this section we will derive
A
o
for torsion
design through the determination of
t
d
.
s
)tan
