Civil Engineering Reference
In-Depth Information
Figure 8.7
Variation of longitudinal steel forces (compability truss model)
This longitudinal tensile force
V
/3 should be uniform throughout the main body of the beam
element.
For the stress field due to
s
is shown to act
on the right face of each of the five subelements in the far right-hand column. Each of these
five forces can be resolved into a diagonal tensile force and a longitudinal tensile force of
w
w
(Figures 8.4b and 8.5) an upward shear force
w
s
. This longitudinal tensile force should be taken by each of the longitudinal
web bar on the right face of the element, and should decrease linearly from (5
s
tan
α
r
=
(5
/
3)
w
/
3)
w
s
at the
right face to zero at the left face of the element.
The summation of the longitudinal steel forces in the two stress fields due to
V
and
w
is
given in Figure 8.7. It can be seen that the tensile force in each longitudinal web bar is
V
/
3
on the left face of the element. This tensile force increases linearly from left to right until it
becomes
V
/
+
/
w
s
on the right face of the element. The tensile forces at the centers of
the five subelements in each row increases linearly in the following sequence:
V
3
(5
3)
/
3
+
(1
/
6)
w
s
,
V
/
3
+
(1
/
2)
w
s
,
V
/
3
+
(5
/
6)
w
s
,
V
/
3
+
(7
/
6)
w
s
and
V
/
3
+
(3
/
2)
w
s
.
8.2.4 Steel Stresses along a Diagonal Crack
Now that the forces in the stirrups and in the longitudinal bars of a beam element are clarified,
we can observe the variation of these forces along a diagonal crack. A triangular free body of
the beam element is shown in Figure 8.8 and all the forces acting on the free body are indicated.
Along the diagonal crack, the stirrup forces increase from left to right in the following linear
sequence:
V
/
5
−
0
.
8
w
s
,
V
/
5
−
0
.
4
w
s
,
V
/
5,
V
/
5
+
0
.
4
w
s
,
V
/
5
+
0
.
8
w
s
. The longitudinal
web steel forces increase linearly from top to bottom in the sequence
V
/
3
+
(1
/
6)
w
s
,
V
/
3
+
(1
s
.
If the five longitudinal web rebars are not available, a redistribution of the forces will occur.
The longitudinal forces in the web rebars can be replaced by the longitudinal forces in the top
and bottom stringers to ensure the equilibrium of the total longitudinal forces. This replacement
can easily take care of the uniform longitudinal forces (
V
/
2)
w
s
,
V
/
3
+
(5
/
6)
w
s
,
V
/
3
+
(7
/
6)
w
s
and
V
/
3
+
(3
/
2)
w
3 in each web rebar) produced by
the shear force
V
. The equilibrium of the nonuniform longitudinal forces produced by the
/
