Civil Engineering Reference
In-Depth Information
Assume yielding in the bottom stringer and in the transverse steel, then
N
b
=
N
b
y
and
n
t
=
n
ty
. Also recall the definitions:
M
o
=
N
b
y
d
v
2
N
t
y
d
v
n
ty
V
o
=
2
d
v
for two webs of a box
2
N
t
y
p
o
n
ty
T
o
=
2
A
o
noting
d
v
+
b
v
=
p
o
/
2
N
t
y
N
b
y
R
=
Substituting these definitions of
M
o
,
V
o
,
T
o
and
R
into Equation (2.83) we obtain a nondi-
mensional interaction relationship for
M
,
V
and
T
in the first mode:
V
V
o
2
T
T
o
2
M
M
o
+
R
+
R
=
1
(2.84)
Equation (2.84) shows that the interaction of
V
and
T
is circular for a constant
M
. A typical
V
-
T
interaction curve for
M
1 is shown in Figure 2.10. When
M
is
varied, Equation (2.84) represents an interaction surface. This surface intersects the vertical
V
-
M
plane to form an interaction curve expressed by Equation (2.58) and intersects the
horizontal
T
-
M
plane to form an interaction curve expressed by Equation (2.69).
/
M
o
=
0.75 and
R
=
2.2.3.2 Second Failure Mode
In the second failure mode, failure is caused by yielding in the
top stringer
and in the
transverse
reinforcement
on the side where shear flows due to shear and torsion are additive (i.e. left-hand
side).
Equating the external moment to the internal moment about the bottom wall (Figure
2.12) gives:
d
2
−
d
2
−
−
M
=
N
t
d
v
−
q
w
d
v
tan
α
w
q
r
w
d
v
tan
α
r
w
q
t
w
b
v
tan
α
t
w
d
v
(2.85)
Substituting
q
w
,
q
r
w
,
q
t
w
from Equation (2.74)-(2.76) and tan
α
w
,tan
α
r
w
,tan
α
t
w
from
Equations (2.78)-(2.80) into Equation (2.85) and simplifying gives
V
2
d
v
2
T
2
A
o
2
M
N
t
d
v
d
N
t
1
n
t
+
(
b
v
+
d
v
)
1
n
t
=
−
+
1
(2.86)
N
t
Assume yielding 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
,
V
o
,
T
o
and
R
into Equation (2.86) we obtain
a nondimensional interaction relationship for
M
,
V
and
T
in the second mode:
M
M
o
V
V
o
2
T
T
o
2
1
R
+
−
+
=
1
(2.87)
