Civil Engineering Reference
In-Depth Information
Figure 8.6
Variation of stirrup forces in compatibility truss model
top to bottom. Finally, the sequence of stirrup forces on the bottom faces of the five bottom
subelements from left to right are 0
.
1
w
s
,0
.
3
w
s
,0
.
5
w
s
,0
.
7
w
s
and 0
.
9
w
s
.
(Figure 8.4b and Figure 5) can now
be added to the uniform tensile stress field of stirrup forces due to
V
(Figure 8.4a), which is
a constant of
V
The nonuniform stress field of stirrup forces due to
w
5. The summation of the two stress fields is shown in Figure 8.6. This figure
clearly shows that the forces in the stirrups increase linearly downward and that the maximum
forces are located at the bottom of the bars. Stirrup forces at these lowest locations vary linearly
along the beam length according to the
conventional triangular shear diagram
. This pattern
of stress distribution in the stirrups has been verified by tests (Belarbi and Hsu, 1990).
In summary, the nonuniform stirrup forces of Figure 8.6 are based on the two assumptions
which satisfy not only the equilibrium condition, but also the compatibility conditions and the
linear stress-strain relationship of materials. This nonuniform continuous stress field derived
from compatibility truss model is quite different from the discontinuous banded stress field
assumed in the equilibrium (plasticity) truss model of Figure 8.2(b). The difference between
these two models has two significant consequences. First, while the equilibrium (plasticity)
truss model gives the upper bound solution as far as material is concerned, the compatibility
truss model provides the lower bound solution. Second, while the equilibrium (plasticity) truss
model suggests a staggered shear diagram for design of stirrups, the compatibility truss model
requires the conventional triangular shear diagram.
/
8.2.3 Longitudinal Web Steel Forces
Distribution of horizontal forces in the main body of the beam element can also be derived
from the two stress fields in Figure 8.4(a) and (b). For the stress field due to
V
(Figure 8.4a), the
shear force on the vertical face of each subelement is
V
5. This vertical force can be resolved
into a diagonal compressive force and a longitudinal tensile force
V
/
3. This longitudinal
tensile force should be taken by one longitudinal web bar at the center of each subelement.
/
