Civil Engineering Reference
In-Depth Information
A beam under shear and bending may fail in two modes, because failure may be caused by
the yielding of bottom stringer or by the yielding of top stringer. The first and second modes
of failure are presented in Sections 2.2.1.1 and 2.2.1.2, respectively.
2.2.1.1 First Failure Mode
In the first failure mode, failure occurs due to yielding of the
bottom stringer
and the
transverse
steel
. Looking at Figure 2.7(a), the force in the bottom stringer,
N
b
, and in the force per unit
length in the transverse steel,
n
t
,are:
M
d
v
V
2
N
b
=
+
tan
α
r
(2.52)
V
d
v
n
t
=
cot
α
r
(2.53)
Substituting
α
r
from Equation (2.53) into Equation (2.52) to eliminate
α
r
,wehave
V
2
M
N
b
d
v
+
d
v
)
n
t
=
1
(2.54)
d
2
v
(2
N
b
/
Equation (2.54) expresses the interaction relationship between
M
and
V
. 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 bending strength
M
o
and the pure shear strength,
V
o
, according to
Table 2.1 as follows:
M
o
=
N
b
y
d
v
(2.55)
2
N
t
y
d
v
n
ty
V
o
=
d
v
(2.56)
where
N
t
y
=
force in the top stringer at yielding,
N
b
y
=
force in the bottom stringer at yielding,
Notice that the pure shear strength
V
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
V
o
, assuming the top stringer force at yield is less than the bottom stringer force at
yield. The total longitudinal force due to shear,
N
, is then equal to 2
N
t
y
. To replace
N
b
y
by
N
t
y
in Equation (2.54) we introduce the ratio
R
:
N
t
y
N
b
y
R
=
(2.57)
Substituting the definitions of
M
o
,
V
o
and
R
from Equations (2.55)-(2.57) into Equation
(2.54), the interaction equation for
M
and
V
is now expressed in a nondimensional form:
V
V
o
2
M
M
o
+
R
=
1
(2.58)
Equation (2.58) is plotted in Figure 2.8 for
R
=
0.25, 0.5 and 1.
