## 20

End condition code

y r~i ?

Rotation fixed and translation fixed Rotation free and translation fixed Rotation fixed and translation free Rotation free and translation free

FIGURE 31.2 Theoretical and recommended K-factors for individual columns with idealized end conditions (AISC 1999).

FIGURE 31.2 Theoretical and recommended K-factors for individual columns with idealized end conditions (AISC 1999).

analysis include the slope deflection method (Winter et al. 1948; Galambos 1968; Chen and Lui 1991), the three-moment equation method (Bleich 1952), and energy methods (Johnson 1960). In practice, however, such an analysis is not practical, and simple models are often used to determine the effective length factors for framed columns (Kavanagh 1962; Lu 1962; Gurfinkel and Robinson 1965; Wood 1974). One such practical procedure that provides an approximate value of the elastic K-factor is the alignment chart method (Julian and Lawrence 1959). This procedure has been adopted by the AISC (1989, 1999), ACI (2002), AASHTO (2004), and CSA (1994) Specifications, among others. At present, most engineers use the alignment chart method in lieu of an actual stability analysis.

### 31.4.1 Alignment Chart Method

The structural models employed for determination of K-factor for framed columns in the alignment chart method are shown in Figure 31.3. The assumptions used in these models are (Chen and Lui 1991; AISC 1999)

1. All members have constant cross-section and behave elastically.

2. Axial forces in the girders are negligible.

### 3. All joints are rigid.

4. For braced frames, the rotations at the near and far ends of the girders are equal in magnitude and opposite in direction (i.e., girders are bent in single curvature).

5. For unbraced frames, the rotations at the near and far ends of the girders are equal in magnitude and direction (i.e., girders are bent in double curvature).

6. The stiffness parameters, L^JP/EI, of all columns are equal.

7. All columns buckle simultaneously.

8. Joint restraint is distributed to the column above and below the joint in proportion to I/L of the two columns.

FIGURE 31.3 Subassemblage models for K-factors of framed columns: (a) braced frames and (b) unbraced frames.

Using the slope deflection equation method and stability functions, the effective length factor equations of framed columns are obtained as

For columns in braced frames:

gagB

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