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5.2.3 Geometric Imperfection Methods

Geometric imperfection modeling combined with the Column Research Council (CRC) tangent modulus model is discussed in what follows. There are three methods: the explicit imperfection modeling method, the equivalent notional load method, and the further reduced tangent modulus method.

5.2.3.1 Explicit Imperfection Modeling Method

### 5.2.3.1.1 Braced Frames

The refined plastic-hinge analysis implicitly accounts for the effects of both residual stresses and spread of yielded zones. To this end, refined plastic-hinge analysis may be regarded as equivalent to the plastic-zone analysis. As a result, geometric imperfections are necessary only to consider fabrication error. For braced frames, member out-of-straightness, rather than frame out-of-plumbness, needs to be used for geometric imperfections. This is because the P-D effect due to the frame out-of-plumbness is diminished by braces. The ECCS [21,22], AS [23], and Canadian Standard Association (CSA) [24,25] specifications recommend an initial crookedness of column equal to 1/1000 times the column length. The AISC Code recommends the same maximum fabrication tolerance of Lc/1000 for member out-of-straightness. In this study, a geometric imperfection of Lc/1000 is adopted.

The ECCS [21,22], AS [23], and CSA [24,25] specifications recommend the out-of-straightness varying parabolically with a maximum in-plane deflection at the mid-height. They do not, however, describe how the parabolic imperfection should be modeled in analysis. Ideally, many elements are needed to model the parabolic out-of-straightness of a beam-column member, but it is not practical. In this study, two elements with a maximum initial deflection at the mid-height of a member are found adequate for capturing the imperfection. Figure 5.10 shows the out-of-straightness modeling for a braced beam-column member. It may be observed that the out-of-plumbness is equal to 1/500 when the half-segment of the member is considered. This value is identical to that of sway frames as discussed in recent papers by Kim and Chen [9,26,27]. Thus, it may be stated that the imperfection values are essentially identical for both sway and braced frames. It is noted that this explicit modeling method in braced frames requires the inconvenient imperfection modeling at the center of columns although the inconvenience is much lesser than that of the conventional LRFD method for frame design.

### 5.2.3.1.2 Unbraced Frames

The CSA [23,24] and the AISC Code of Standard Practice [1] set the limit of erection out-of-plumbness at Lc/500. The maximum erection tolerances in the AISC are limited to 1 in. toward the exterior

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