L

2. Using the alignment chart without modification

A direct use of Equations 31.7 and 31.8 with an average section (0.75h) results in

From the alignment chart in Figure 31.4b, K = 1.49 or (1.49 — 1.59)/1.59 = — 6% in error on the less conservative side.

31.5.4 Unsymmetrical Frames

When the column sizes or column loads are not identical, adjustments to the alignment charts are necessary to obtain correct K-factors. Figure 31.9 presents a set of curves for a modification factor b developed by Chu and Chow (1969).

Kadjusted bKalignment chart (31. 79)

If the K-factor of the column under the load IP is desired, further modifications to K are necessary. Denoting K0 as the effective length factor of column with 10 = a1c subjected to the axial load P0 = IP as shown in Figure 31.9, we have

Equation 31.80 can be used to determine K-factors for columns in adjacent stories with different heights L0.

31.5.5 Effects of Axial Forces in Restraining Members

Compressive axial load in a restraining girder reduces its flexural stiffness and then affects adversely the K-factor of the column (AISC 1999). To account for any compression axial load in a girder, the girder stiffness parameter (£g1g/Lg) in Equations 31.7 and 31.8 should be modified by the factor [1 — (Q/Qcr)], where Q is the axial compression load in the girder, and Qcr is the in-plane buckling load of the girder based on K = 1.0. Tensile axial load in the girder can be ignored when determining the G-factor.

Bridge and Fraser (1987) observed that K-factors of a column in a braced frame may be greater than unity due to ''negative'' restraining effects. Figure 31.10 shows the solutions obtained by considering the both ''positive'' and ''negative'' values of G-factors. The shaded portion of the graph corresponds to the alignment chart shown in Figure 31.4a when both GA and GB are positive.

To account for the effect of axial forces in the restraining members, Bridge and Fraser (1987) proposed a more general expression for G-factor:

(1 /L) stiffness of member i under investigation

En(1 /L)ngnmn stiffness of all rigidly connected members where g is a function of the stability functions S and C (Equations 31.3 and 31.4), m is a factor to account for the end conditions of the restraining member (see Figure 31.11), and subscript n represents the other members rigidly connected to member i. The summation in the denominator is for all members meeting at the joint.

By using Figure 31.10, Figure 31.11, and Equation 31.81, the effective length factor Ki for the ith member can be determined by the following steps:

1. Sketch the buckled shape of the structure under consideration.

2. Assume a value of K; for the member being investigated.

FIGURE 31.9 Chart for the modification factor b in an unsymmetrical frame.

3. Calculate values of Kn for each of the other members that are rigidly connected to the ith member using the equation

4. Calculate g and obtain m from Figure 31.11 for each member.

5. Calculate G; for the ith member using Equation 31.81.

6. Obtain from Figure 31.10 and compare with the assumed at step 2.

7. Repeat the procedure by using the calculated as the assumed until calculated at the end of the cycle is approximately (say, 10%) equal to the at the beginning of the cycle.

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