Civil Engineering Reference
In-Depth Information
uniformly between points b
and c
, the required shear capacity should also be constant from
b
and c
. This constitutes the first step of the staggered shear diagram plotted in Figure 8.2(d).
In the second step, the uniform stirrup force at the bottom stringer from b
to c
is transmitted
vertically to the top stringer within b and c. This stirrup force at the top stringer level remains
at
w/
8. At this level between b and c, however, the beam will receive an additional load of
w/
4 between b and
c should then be resisted by the truss actions of the concrete struts and the top stringer. The
force in the concrete struts from b to c should be
8 from the externally applied load
w
. The combined vertical force of
w/
α
r
. This strut force is transmitted
diagonally to the bottom stringer within c
and d
, while crossing section c-c
en route. At the
bottom stringer level between c
and d
, the diagonal force in the concrete struts is resisted by
a stringer force as well as a stirrup force of
w/
4 cos
w/
4. This stirrup force is equal to the shear force
at section c-c
and is designated as
V
c
−
c
=
w/
4. Since the stirrup force is uniform between
c
and d
the shear diagram is a constant in this region. This constitutes the second step of the
staggered shear diagram in Figure 8.2(d).
Using the same logic, the stirrup force in the region from d
to e
should be 3
w/
8, which
is equal to the shear force at section d-d
,
V
d
−
d
=
8. This third step completes the entire
staggered shear diagram. It should be noted that the plasticity truss model predicts zero forces
in the stirrup near the midspan from a to b. In practice, however, a minimum amount of stirrup
should always be provided.
In Figure 8.2(d), it is interesting to compare the staggered shear diagram with the conven-
tional shear diagram, which has a triangular shape. Obviously, the staggered shear diagram
based on the plasticity truss model is much less conservative than the conventional shear
diagram. In order to elucidate this difference, the conventional shear diagram will be studied
in Section 8.2.2 using the compatibility truss model.
3
w/
8.1.2.2 Forces in Bottom and Top Stringers
The stress flow from midspan to support also induces forces in the bottom and top stringers.
These forces are plotted in Figure 8.2(e) and (f), respectively. Because the stirrup force is no
longer constant along the length of this beam, the stringer force at midspan,
F
s
, will be used
as a reference force in studying the variation of forces in the stringers. Noting that the bending
moment at midspan is
w/
8 and that
/
d
v
=
40/3, the forces in the bottom and top stringers
at midspan (section a-a
)is:
2
F
s
=
w
5
3
w
=
(8.3)
8
d
v
Similarly, the stringer forces calculated from the conventional moment diagram at sections
b-b
,c-c
,d-d
and e-e
16)
F
s
and 0, respectively. These stringer
forces, designated as M-conventional, are plotted in Figure 8.2(e) and (f).
Now, let's find the forces in the top stringers according to the plasticity truss model. Since
the vertical force from the external load between a and b is
are (15
/
16)
F
s
,(3
/
4)
F
s
,(7
/
w/
8, the
t
otal force increment in
the top stringer from a to b is (
w/
8) tan
α
r
=
(5
/
24)
w
=
(1
/
8)
F
s
. Therefore, the top stringer
force at point b is
F
s
-(1
/
8)
F
s
=
(7
/
8)
F
s
, as shown in Figure 8.2(f).
Further
down
the
route
of
stress
flow,
the
total
vertical
load
on
the
top
stringer
frombtocis
w/
4, and the total force increment in the top stringer from b to c is
