Structural Concrete - Hassoun

422 Chapter 12 Slender Columns
is, at the two hinged ends. Therefore, the factor K = l u /l u is 1.0. If a column is fixed at both
ends and sidesway is prevented, the points of inflection (points of 0 moment) are at l u /4 from
each end. Therefore, K = 0.5l u /l u = 0.5 (Fig. 12.2). To evaluate the proper value of K, two
main cases are considered.
When structural frames are braced, the frame, which consists of beams and columns, is
braced against sidesway by shear walls, rigid bracing, or lateral support from an adjoining
structure. The ends of the columns will stay in position, and lateral translation of joints is
prevented. The range of K in braced frames is always equal to or less than 1.0. The ACI
Code, Section 6.2.5, recommends the use of K = 1.0 for braced frames.
When the structural frames are unbraced, the frame is not supported against sidesway,
and it depends on the stiffness of the beams and columns to prevent lateral deflection. Joint
translations are not prevented, and the frame sways in the direction of lateral loads. The range
of K for different columns and frames is given in Fig. 12.2, considering the two cases when
sidesway is prevented or not prevented.
12.3 EFFECTIVE LENGTH FACTOR (K)
The effective length of columns can be estimated by using the alignment chart shown in Fig. 12.3
[10]. To find the effective length factor K, it is necessary first to calculate the end restraint factors
ψ A and ψ B at the top and bottom of the column, respectively, where
∑ EI∕lc of columns
ψ = ∑ (12.1)
EI∕l of beams
Figure 12.2 (a) Effective lengths of columns and length factor K and (b) effective
lengths and K for portal columns.