Civil Engineering Reference
In-Depth Information
k
= 0.70
u
Points
of
inflection
Point
of
inflection
u
k
=
k
= 0.50
u
u
u
u
u
u
k
= 1.0
(a)
k
= 0.50
k
= 0.70
(b)
(c)
Figure 11.1
Effective lengths for columns in braced frames (sidesway prevented).
Obviously, the smaller the effective length of a particular column, the smaller its dan-
ger of buckling and the greater its load-carrying capacity. Figure 11.1(c) shows a column
with one end fixed and one end pinned. The
k
factor for this column is theoretically 0.70.
The concept of effective lengths is simply a mathematical method of taking a col-
umn—whatever its end and bracing conditions—and replacing it with an equivalent
pinned end braced column. A complex buckling analysis could be made for a frame to de-
termine the critical stress in a particular column. The
k
factor is determined by finding the
pinned end column with an equivalent length that provides the same critical stress. The
k
factor procedure is a method of making simple solutions for complicated frame-buckling
problems.
Reinforced concrete columns serve as parts of frames, and these frames are some-
times
braced
and sometimes
unbraced
. A braced frame is one for which sidesway or joint
translation is prevented by means of bracing, shear walls, or lateral support from adjoin-
ing structures. An unbraced frame does not have any of these types of bracing supplied
and must depend on the stiffness of its own members to prevent lateral buckling. For
braced frames
k
values can never be greater than 1.0, but for unbraced frames the
k
values
will always be greater than 1.0 because of sidesway.
An example of an unbraced column is shown in Figure 11.2(a). The base of this par-
ticular column is assumed to be fixed, whereas its upper end is assumed to be completely
free to both rotate and translate. The elastic curve of such a column will take the shape of
the elastic curve of a pinned end column of twice its length. Its effective length will there-
fore equal 2
u
, as shown in the figure. In Figure 11.2(b) another unbraced column case is
illustrated.
The Code (10.12.1) states that the effective length factor is to be taken as 1.0 for
compression members in frames braced against sidesway unless a theoretical analysis
shows that a lesser value can be used. Should the member be in a frame not braced against
sidesway, the value of
k
will be larger than 1.0 and must be determined with proper con-
sideration given to the effects of cracking and reinforcing on the column stiffness. ACI-
ASCE Committee 441 suggests that it is not realistic to assume that
k
will be less than 1.2
