Civil Engineering Reference
In-Depth Information
Figure 2.9
Superposition of stringer forces due to torsion and bending
one-half of the total tensile force, (
Tp
o
/
α
r
. In the transverse direction, the torsional
moment will also produce a transverse force per unit length, (
T
4
A
o
)tan
/
2
A
o
) cot
α
r
, in the hoop steel.
2.2.2.1 First Failure Mode
Failure of a beam under torsion and bending also occurs in two modes. In the first failure
mode, failure occurs due to yielding of the
bottom stringer
and the
transverse steel
.Reviewing
Figure 2.9 reveals that the force in the bottom stringer
N
b
and the force per unit length in the
transverse steel
n
t
,are:
M
d
v
Tp
o
4
A
o
N
b
=
+
tan
α
r
(2.65)
T
2
A
o
n
t
=
cot
α
r
(2.66)
Substituting
α
r
from Equation (2.66) into Equation (2.65) to eliminate
α
r
results in
T
2
4
A
o
(2
N
b
/
M
N
b
d
v
+
p
o
)
n
t
=
1
(2.67)
Equation (2.67) expresses the interaction relationship between
M
and
T
. Assume that yielding
occurs in the bottom stringer and in the transverse steel, then
N
b
=
N
b
y
and
n
t
=
n
ty
.Also
define the pure torsional strength
T
o
, according to Table 2.1 as follows:
2
N
t
y
p
o
n
ty
T
o
=
2
A
o
(2.68)
Notice that the pure torsional strength
T
o
is defined based on the top stringer force at yield
N
t
y
, rather than the bottom stringer force at yield
N
b
y
. This definition gives the lowest
positive value for
T
o
, assuming the top stringer force at yield is less than the bottom stringer
force at yield. The total longitudinal force due to torsion
N
is then equal to 2
N
t
y
.
