## R1 12r1 X [p21 r2 12r2

Equation 23.19 yields K factor values having a maximum error of less than 4% compared to corresponding theoretically correct values.

The effective-length factor concept is considered to be an essential part of many analysis procedures, and a corresponding alignment chart approach is recommended by almost all current design standards [1-4]. The idealizations and assumptions associated with the approach are as follows:

1. All beams and columns are purely elastic.

2. All members have constant cross-section.

### 3. All joints are rigid.

4. For braced (sidesway inhibited or sway prevented) frames, rotations at opposite ends of the restraining beams are equal in magnitude, producing single-curvature bending.

5. For unbraced (sidesway uninhibited or sway permitted) frames, rotations at opposite ends of the restraining beams are equal in magnitude, producing reverse-curvature bending.

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

7. Joint restraint is distributed to the column above and below the joint in proportion to the I/L values for the two columns.

8. All columns buckle simultaneously.

9. No significant axial compression force exists in the beams.

Based on the foregoing assumptions, the effective-length factor K of a column in a braced frame is determined from the following equation:

For an unbraced frame, the current Canadian standard CSA S16  takes sway effects into account directly by using K = 1.0 for columns and performing a second-order elastic analysis under actual and notional loads (i.e., in lieu of using the traditional effective-length factor to account for frame stability, as is permitted by the American design specifications [1,2]). Following the alignment chart approach, the K factor for a column in an unbraced frame is determined from the following equation:

In Equations 23.20 and 23.21, GU and GL are the stiffness factors for the upper and lower ends of the column, respectively. A stiffness factor G is defined as

where the summation is taken over all members connected to the joint, Ic is the moment of inertia of the column section corresponding to the plane of buckling, Lc is the unsupported length of the column, Ib is the moment of inertia of the beam/girder section corresponding to the plane of buckling, and Lb is the unsupported length of the beam/girder. Having the stiffness factors GU and GL evaluated in accordance with Equation 23.22, the effective-length factor K of a column in a braced or unbraced frame can be obtained from the applicable alignment charts developed from Equations 23.20 and 23.21.

The evaluation of the effective-length factor K for columns in semirigid frames can be conveniently achieved by pursuing the foregoing approach for rigid frames with a necessary modification to account for connection flexibility when calculating a stiffness factor as

where ab is a modification factor applied to the moment of inertia Ib of the restraining beams that can be expressed as a function of the end-fixity factors associated with each restraining beam depending on whether the frame is of braced or unbraced frames. For braced semirigid frames

while for unbraced semirigid frames r1(2 + r2)

where the end-fixity factors r1 and r2 in Equations 23.24 and 23.25 correspond to the so called ''near end'' and ''far end,'' respectively, of the beam. In the case where the ''far end'' of the restraining beam is connected to a rotation-restrained support instead of a column, the corresponding modification factors for braced and unbraced semirigid frames are ab = ——— (23.26)

As the modification factor ab is expressed by the end-fixity factors, beams with pinned, semirigid, and rigid connections can all be considered. For instance, Equation 23.24 will yield a value of 1.5 for the modification factor ab for a beam in a braced frame with a rigid connection at the ''near end'' (r1 = 1) and a pinned connection at the ''far-end'' (r2 = 0). If the ''near end'' connection is a pinned connection (r1 = 0), the corresponding modification factor ab becomes zero, which indicates the beam does not provide any rotational restraint to the connected column.

Having the modification factor value ab based on the end-fixity of the restraining beams, the stiffness factors for the upper and lower ends of the column can be calculated according to Equation 23.23. The corresponding column effective-length factor is then obtained from the applicable alignment chart. However, unlike that for the rigid frame case, in which the effective-length factor of a column is ab —

evaluated regardless of the applied loads, the rotational stiffnesses of beam-to-column connections in semirigid frames are interrelated to the loads and, hence, so are the end-fixity factors rj (j = 1, 2) and modification factors ab. As a frame is loaded, the connection stiffness and, thus, the restraint provided to the columns gradually decreases, causing the column effective-length factor to increase. Consequently, the effective-length factor of the column in a semirigid frame must be evaluated for each applied load case.

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