Civil Engineering Reference
In-Depth Information
Substituting the definitions of
M
o
,
R
, and
T
o
from Equations (2.55), (2.57) and (2.68) into
Equation (2.67), the interaction equation for
M
and
T
is expressed in a nondimensional form:
T
T
o
2
M
M
o
+
R
=
1
(2.69)
2.2.2.2 Second Failure Mode
In the second failure mode, failure occurs due to yielding of the
top stringer
and the
transverse
steel
.
Looking at Figure 2.9, the force in the top stringer
N
t
and the force per unit length in
transverse steel
n
t
,are:
M
d
v
Tp
o
4
A
o
N
t
=−
+
tan
α
r
(2.70)
T
2
A
o
n
t
=
cot
α
r
(2.71)
Substituting
α
r
from Equation (2.71) into Equation (2.70) to eliminate
α
r
,wehave
T
2
4
A
o
(2
N
t
/
M
N
t
d
v
−
+
p
o
)
n
t
=
1
(2.72)
Assuming that yielding occurs in the top stringer and in the transverse steel, then
N
t
=
N
t
y
and
n
t
=
n
ty
. Substituting the definitions of
M
o
,
R
, and
T
o
from Equations (2.55), (2.57) and
(2.68) into Equation (2.72), the nondimensional interaction equation for
M
and
T
becomes
T
T
o
2
M
M
o
1
R
+
−
=
1
(2.73)
When the nondimensional interaction equation for the first mode of failure in torsion and
moment (Equation 2.69), is compared with that in shear and moment (Equation (2.58), it
can be seen that they are identical, if the nondimensional ratio
T
V
o
.
Similar observation can also be seen when the equations for the second mode of failure,
Equations (2.73) and (2.62), are compared. Therefore, the interaction curves for torsion and
moment can be illustrated by Figure 2.8, if the axis
V
/
T
o
is replaced by
V
/
T
o
.Both
the torsion-bending interaction curves and the shear-bending interaction curves are plotted
in Figure 2.10 using the three axes,
V
/
V
o
is replaced by an axis
T
/
M
o
. The shear-bending curves are
shown in the vertical plane and the torsion-bending curves in the horizontal plane.
In Section 2.2.1 we have discussed the peak point of maximum shear. Similar logic can
be used to find the peak point of maximum torsional moment, which corresponds also to the
simultaneous yielding of top and bottom stringers. This i
ntersection p
oint of two curves for the
first and second modes of failure is located at
T
/
V
o
,
T
/
T
o
and
M
/
T
o
=
√
(1
/
+
R
)
/
2
R
and
M
/
M
o
=
(1
−
R
)
/
2.
T
o
=
√
1
For the case of
R
=
0.5, this peak point is located at
T
/
.
5 and
M
/
M
o
=
0.25, which
are indicated in Figure 2.10.
