Civil Engineering Reference
In-Depth Information
be constant along its own length (transverse direction), as well as throughout the length of the
beam (longitudinal direction).
By definition, the force in one stirrup
F
t
should be
n
t
s
. Recalling
n
t
=
(
V
/
d
v
) cot
α
r
from
beam shear equation in Table 2.1 we have
V
s
d
v
cot
F
t
=
n
t
s
=
α
r
(8.1)
Adopting the assumed geometric relationship of cot
α
r
=
3/5 and
s
=
d
v
/
3, a very simple
equation for
F
t
results:
V
5
F
t
=
(8.2)
This simple value of
F
t
can also be obtained by taking vertical equilibrium of the free
body shown in Figure 8.1(b). Since
F
t
is a constant throughout the beam, we will use it as a
reference force to measure other forces in the beam.
8.1.1.2 Forces in Bottom and Top Stringers
Forces in the bottom stringers are contributed by the bending moment and the shear force
according to Equation (2.52):
N
b
=
α
r
. Since the bending moment varies
linearly along the length of a beam subjected to a midspan load, the force caused by bending
should have a triangular shape in one half-span as shown by the solid line AB in Figure 8.1(c).
Adopting the length of one half-span as (20
/
d
v
+
/
M
(
V
2) tan
/
3)
d
v
, the maximum stringer force at midspan is
V
(
3)
F
t
.
Since the shear force
V
is a constant along the beam, the stringer force due to shear should
also be a constant and equal to (
V
/
2)
/
d
v
=
5
F
t
(20
/
3)
d
v
/
d
v
=
(100
/
6)
F
t
. The sum of the two
stringer forces due to bending and shear is then represented by the dotted line CD, which is
displaced vertically from the solid line by a distance of (25/6)
F
t
. In actuality, of course, the
stirrups are not uniformly smeared, but are concentrated at discrete points with spacing
s
.
Therefore, the stringer force contributed by shear should change at each stirrup and should
have a stepped shape as indicated. Each step of change should introduce a stringer force of
F
t
tan
/
2) tan
α
r
=
(5
F
t
/
2)(5/3)
=
(25
/
α
r
. For the main region of the beam, tan
α
r
=
5/3 and each step is (5
/
3)
F
t
.
α
r
gradually decreases, and the last five
steps in the local region decrease in the following sequence: (9
When the midspan is approached, however, tan
/
6)
F
t
,(7
/
6)
F
t
,(5
/
6)
F
t
,(3
/
6)
F
t
and (1
6)
F
t
. This stepped curve near the midspan can be approximated conservatively by a
horizontal dotted line DB, which is commonly used in design.
Using the same logic, the forces in the top stringer are plotted in Figure 8.1(d). It can be
seen that the compressive force in the top stringer due to bending is reduced by the tensile
forces due to shear.
/
8.1.1.3 Shift Rule
As stated above, the dotted line CD in Figure 8.1(c) for bottom stringer force is displaced
downward by a distance (25
6)
F
t
from the solid line AB. It is also possible to view the
dotted line CD as having displaced horizontally toward the support by a distance (5
/
/
6)
d
v
from
