Civil Engineering Reference
In-Depth Information
SOLUTION
Section Properties
1
12
I
(12)(24)
3
13,824 in.
4
A
(12)(24)
288 in.
2
M
(3)(20)
2
8
150 ft- k
Stresses at Beam Centerline
288
(250)(9)(12)
(12)(150)(12)
13,824
P
A
Pec
Mc
I
250
f
top
I
13,824
0.868
1.953
1.562
0.477 ksi
P
A
Pec
Mc
I
f
bottom
0.868
1.953
1.562
1.259 ksi
I
Stresses at Beam Ends
P
A
Pec
f
top
0.868
1.953
1.085 ksi
I
P
A
Pec
f
bottom
0.868
1.953
2.821 ksi
I
In Example 19.1 it was shown that when the prestress tendons are straight, the tensile
stress at the top of the beam at the ends will be quite high. If, however, the tendons are
draped, as shown in Figure 19.7, it is possible to reduce or even eliminate the tensile
stresses. Out in the span, the centroid of the strands may be below the lower kern point, but
if at the ends of the beam, where there is no stress due to dead-load moment, it is below the
kern point, tensile stresses in the top will be the result. If the tendons are draped so that at
the ends they are located at or above this point, tension will not occur in the top of the beam.
In posttensioning, the sleeve or conduit is placed in the forms in the curved position
desired. The tendons in pretensioned members can be placed at or above the lower kern
points and then can be pushed down to the desired depth at the centerline or at other points.
In Figure 19.7 the tendons are shown held down at the one-third points. Two alternatives to
draped tendons that have been used are to use straight tendons, located below the lower
kern point but which are encased in tubes at their ends, or have their ends greased. Both
methods are used to prevent the development of negative moments at the beam ends.
Example 19.2 shows the calculations necessary to locate the kern point for the beam
of Example 19.1. In addition, the stresses at the top and bottom of the beam ends are com-
puted. It will be noted that, according to these calculations, the kern point is 4 in. below
the mid-depth of the beam, and it would thus appear that the prestress tendons should be
Figure 19.7
Draped tendons.
