## F2

FIGURE 31.33 Types of precast frames. Copyright 2005 by CRC Press strengths for members, connections, and other structural elements are obtained from the second-order elastic analysis under load and resistance factor design (LRFD) combinations. Its appendix provides the direct second-order analysis method for steel moment frames.

### 31.17.1 Second-Order Elastic Analysis

The second-order analysis required for using K = 1.0 in the interaction equations should capture both the P-d and the P-A effects. The analysis may be performed either using a direct second-order analysis or by modifying the results of a first-order analysis using the B1 and B2 amplification factors (AISC 1999), provided that the B1 and B2 factors are based on the reduced flexural stiffness. Approximate P-A methods should be permitted when the factored axial loads in all columns are less than 15% of their respective Euler buckling loads. When the second-order displacement based on 20% reduced elastic stiffness is greater than six times the first-order displacement, it is recommended to increase the frame stiffness to limit the amplification to 6. Otherwise, the high nonlinearity would be such that small changes in gravity load or member stiffness will result in large changes in the calculated second-order effects.

31.17.2 Direct Second-Order Analysis

The direct second-order analysis uses the notional load concept and the reduced flexural stiffness principle.

### 31.17.2.1 Notional Load Concept

Notional loads are lateral loads that are applied at each floor level and are specified in terms of the gravity loads applied at that floor level to account for the effects of geometric imperfections, inelasticity, or both. A notional load applied at floor i, N{ = 0.002 Y should be added to the factored lateral load in all LRFD load combinations. Y; is the gravity load from the load combination acting on floor i and should be equal to or greater than the gravity load associated with the load combination being evaluated. The notional load should be applied in the direction that adds to the destabilizing effects under the specified load combination.

31.17.2.2 Reduced Stiffness Principle

A reduced flexural stiffness, (EI)*, should be used for all members.

Alternatively, where Pu > 0.5Py for any column in the moment frame, an additive notional load of N = 0.001Y; should be added to the required notional load discussed in Section 31.17.2.1.

This chapter summarizes the state-of-the-art use of effective length factors for individual columns, framed columns, diagonal bracing systems, latticed and built-up members, tapered columns, crane columns, gable frames, columns in fire, space frames, truss-type highway sign support structures, precast concrete skeletal frames, and steel moment frames. Design implementation with formulas, charts, tables, various modification factors adopted in current codes and specifications, as well as those used in engineering practice are described. Several examples are given to illustrate the steps of practical applications of various methods.

0.8tEI for columns 0.8EI for other members

0.8tEI for columns 0.8EI for other members for Pu/Py < 0.5 for Pu/Py > 0.5

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