Civil Engineering Reference
In-Depth Information
w
l
f
M
l
f
w
l
f
+
M
l
f
w
l
f
2
M
2
2
82
M
M
l
s
l
f
w
l
f
l
s
w
l
s
+
T =
+
M
4
M
2
l
f
2
8
4
(a) MOMENT DIAGRAM
(b) SHEAR DIAGRAM
(c) TORQUE DIAGRAM
Figure 7.16
Moment, shear and torque diagrams of test specimen
Notice that the joint moment is a function of the stiffness ratio
K
f
/
K
ts
.If
K
f
/
K
ts
=
0,
2
f
. This is the special case in which the spandrel beam is infinitely rigid in torsion,
and the joint moment becomes the fixed end moment. If
K
f
/
M
=
(1
/
8)
w
0. This gives the
other extreme case, in which the spandrel beam has no torsional stiffness. The floor beam is
then simply supported, and the joint moment must be zero.
The joint moment
M
, calculated by Equation (7.86) using the uncracked elastic stiffness
properties, should be applicable to the specimen before cracking. After cracking, however,
K
ts
drops drastically to a small fraction of the pre-cracking value (say 5%) while
K
f
only drops
to say, 50% of the pre-cracking value. Therefore, the
K
f
/
K
ts
=∞
,
M
=
K
ts
ratio increases by an order of
magnitude after cracking. According to Equation (7.86), this change will cause a large decrease
of the joint moment
M
compared with the uncracked elastic value. A decrease of
M
means
that the torsional moment in the spandrel beam is reduced, accompanied by a corresponding
increase of flexural moment in the floor beam near the midspan. This phenomenon, which can
be viewed as a redistribution of moments from the spandrel beam to the floor beam, is known
as 'moment redistribution after cracking of a spandrel beam.'
Moment redistribution after torsional cracking of a spandrel beam can be utilized to achieve
economy and is the basis of the ACI torsional limit design method. This phenomenon will be
illustrated by the tests of T-shaped test specimens, B1, B2 and B5, taken from series B in Hsu
and Burton (1974). Figure 7.17 shows a photograph of beam B1 subjected to four concentrated
loads evenly spaced to simulate a uniformly distributed load.
A typical T-shaped specimen in Figure 7.17 consists of a spandrel beam connected at
midspan to a floor beam. The spandrel beam has a cross-section of 152
×
305 mm (6
×
12 in.)
and a span of 2.74 m (9 ft). The floor beam has a cross section of 152
9in.)
and a span of 2.74 m (9 ft). A steel torsion arm with a load cell at its tip was attached to one
end of a spandrel beam in order to measure the torsional moment. Torsional moment is the
load cell force that maintains zero torsional rotation times the distance from the load cell to
the center line of the spandrel beam. The angle of twist is the rotation measured by a rotational
LVDT at the midspan of spandrel beam divided by one-half of the span.
The three specimens, B1, B2 and B3, were all subjected to a design load
×
229 mm (6
×
w
f
of 114 kN (25.6
kips), uniformly distributed along the floor beam. The design torques, however, are different
for the three specimens. Specimen B1 was designed by elastic analy
sis based
on u
ncracke
d
sections. This method requires a nominal torsional stress,
74
f
c
(MPa) (8
9
f
c
(psi)),
τ
n
=
0
.
.
