(b)
Rigid diaphragms.
When rigid
thus be expressed by the formula Ft = M T kd/Σ kd ,
2
diaphragms rotate, they develop shears in all of the
where k is the stiffness of a vertical-resisting element,
vertical-resisting elements. In the example (Figure 7-
d is the distance of the element from the center of
50) there is an eccentricity in both directions, and all
five walls develop resisting forces via the diaphragm.
inertia. For the wall forces, the direct components
due to Fpx at the cr are combined with the torsional
(c) Deformational compatibility. When a
components due to MΤ. In the example shown on
diaphragm rotates, whether it is rigid or flexible, it
Figure
7-50,
the
torsional
moment
is
causes displacements in all elements attached to it.
counterclockwise, and the diaphragm rotation will be
For example, the top of a column will be displaced
counterclockwise around the cr.
The direct
with respect to the bottom. Such displacements must
component of the load is shared by walls A and B,
be recognized and addressed.
while the torsional component of the load is resisted
by walls A, B, D, C, and E. Where the direct and
(d) Analysis for torsion. The method of
torsional components of wall force are the same
determining torsional forces is indicated in Figure 7-
direction, as in wall A, the torsional component adds
50. The diaphragm load, Fpx, which acts through the
to the direct component; where the torsional
cm, is replaced by an equivalent set of new forces.
component is opposite to the direct component, as in
By adding equal and opposite forces at the cr, the
wall B, the torsional component subtracts from the
diaphragm load can now be described as a
direct.
Walls C, D, and E carry only torsional
combination of a force component, Fpx (which acts
components; in fact, their design will most likely be
through the cr) and a moment component (which is
governed by direct forces in the east-west direction.
formed by the couple of the two remaining forces Fpx
separated by the eccentricity e). The moment, called
(e) Accidental torsion. Accidental torsion is
the torsional moment, MT, is equal to Fpx times e.
intended to account for uncertainties in the
The torsional moment is often called the "calculated"
calculation of the locations of the cm and the cr. The
torsion, because it is based on a calculated
accidental torsional moment, MA, is obtained using an
eccentricity; also this name distinguishes it from the
eccentricity, eacc, equal to 5 percent of the building
"accidental" torsion, which is described below. In
dimension perpendicular to the direction of the lateral
the modified loading, the force Fpx acts through the cr
forces; in other words, MA = Fpx x eacc.
For the
instead of cm; therefore, it causes no rotation and it is
example of Figure 7-50, the accidental torsion for
distributed to the walls, which are parallel to Fpx in
forces in the north-south direction is MT = Fpx x
proportion to their relative rigidities. The torsional
0.05L. In hand calculations, MA is treated like MT,
moment is resolved into a set of equivalent wall
except that absolute values of the resulting forces are
forces by a procedure similar to that used for finding
forces on bolts in an eccentrically loaded group of
bolts.
The formula is analogous to the torsion
formula τ = Tc/J. The torsional shear forces can
7 -118
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