Civil Engineering Reference
In-Depth Information
Figure 2.12
Forces in box section subjected to shear, torsion, and bending
2.2.3.1 First Failure Mode
In the first failure mode, failure is caused by yielding in the
bottom stringer
and in the
transverse reinforcement
on the side where shear flows due to shear and torsion are additive
(i.e. left wall).
Equating the external moment to the internal moment about the top wall (Figure 2.12) gives:
d
2
−
d
2
−
M
=
N
b
d
v
−
q
w
d
v
tan
α
w
q
r
w
d
v
tan
α
r
w
q
b
w
b
v
tan
α
b
w
d
v
(2.81)
Substituting
q
w
,
q
r
w
,
q
b
w
from Equations (2.74)-(2.76) and tan
α
w
,tan
α
r
w
,tan
α
b
w
from
Equations (2.78)-(2.80) into Equation (2.81) gives
V
2
d
v
2
2
T
2
A
o
2
d
2
v
2
n
t
d
2
v
2
n
t
T
2
A
o
V
2
d
v
T
2
A
o
b
v
d
v
n
t
M
=
N
b
d
v
−
+
−
−
+
−
(2.82)
When the square terms are multiplied in Equation (2.82) we notice that the two mixed terms
of
VT
can be cancelled. The two
V
2
terms and the three
T
2
terms can be grouped. Making
these simplifications and dividing the whole equation by
N
b
d
v
results in:
V
2
d
v
2
T
2
A
o
2
M
N
b
d
v
d
v
N
b
1
n
t
+
(
b
v
+
d
v
)
N
b
1
n
t
=
+
1
(2.83)
