The design loads for wind and seismic effects are applied to structures in accordance with the guidelines in Arts. 9.2 to 9.5. Next, the structure must be analyzed to determine forces and moments for design of the members and connections. Member and connection design proceeds quite normally for wind load design after these internal forces are determined, but seismic design is also subject to the detailed ductility considerations described in Arts. 9.6 and 9.7. This is required for preliminary design and for interpretation and evaluation of computer results. Approximate methods are based on physical observations of the response of structures to applied loads. Two such methods are the portal and cantilever methods, often used for analyzing moment-resisting frames under lateral loads.

The portal method is used for buildings of intermediate or shorter height. In this method, a bent is treated as if it were composed of a series of two-column rigid frames, or portals.

Each portal shares one column with an adjoining portal. Thus, an interior column serves as both the windward column of one portal and the leeward column of the adjoining portal.

Horizontal shear in each story is distributed in equal amounts to interior columns, while each exterior column is assigned half the shear for an interior column, since exterior columns do not share the loads of adjacent portals. If the bays are unequal, shear may be apportioned to each column in proportion to the lengths of the girders it supports. When bays are equal, the axial load in interior columns due to lateral load is zero.

Inflection points (points of zero moment) are placed at midheight of the columns and midspan of beams. This approximates the deflected shapes and moment diagrams of those members under lateral loads. The location of the inflection points may be adjusted for special cases, such as fixed or pinned base columns, or roof beams and top-story columns, or other special situations. On the basis of the preceding assumptions, member forces and bending moments can be determined entirely from the equations of equilibrium. As an example, Fig. 9.17 indicates the geometry and loading of an eight-story moment-resisting frame, and Fig. 9.18 illustrates the use of the portal method on the upper stories of the frame. The frame has two interior columns. So one-third of the shear in each story is distributed to the interior columns and half of this, or one-sixth, is distributed to the exterior columns (Fig. 9.18). The other member forces are computed by equations of equilibrium on each subassemblage. For example, for the subassemblage at the top of the frame in Fig. 9.18, setting the sum of the moments equal to zero yields

The remaining axial and shear forces can be determined by this procedure, and bending moments can be determined directly from these forces from equilibrium equations.

The cantilever method is used for tall buildings. It is based on the recognition that axial shortening of the columns contributes to much of the lateral deflections of such buildings (Fig. 9.19). In this method, the floors are assumed to remain plane, and the axial force in each column is assumed to be proportional to the distance of the column from the centroid of the columns. Inflection points are assumed to occur at midheight of the columns and at midspan of the beams.

The internal moments and forces are determined from equations of equilibrium, as with the portal method. The determination of the forces and moments in the members at the top floors of the frame in Fig. 9.17 is illustrated in Fig. 9.20. The lateral forces cause overturning moments, which induce axial tensile and compressive forces in the columns.

Therefore,

Analysis of Dual Systems. Approximate analysis of braced frames can be performed as if the bracing were a truss. However, many braced structures are dual systems that combine moment-resisting-frame behavior with braced-frame behavior. Under these conditions, an approximate analysis can be performed by first distributing the lateral forces between the braced-frame and moment-resisting-frame portions of the structure in proportion to the relative stiffness of the components. Braced frames are commonly very stiff and normally would carry the largest portion of the lateral loads.

Once the initial distribution of member and connection forces and moments is completed, a preliminary design of the members can be performed. At this time, it is possible to reanalyze the structure by any of a number of linear-elastic, finite-element methods, for which computer programs are available.

While many major, existing structures were designed without benefit of computer analysis techniques, it is not advantageous to design modern buildings for wind and earthquake loading without this capability. It is needed to predict realistic structural response to wind loading and to evaluate occupant comfort, as well as for dynamic design for seismic loading, especially for buildings of unusual geometry. Both the seismic and wind load provisions in the ‘‘Uniform Building Code’’ result in local anomalies in the distribution of design forces due to the distribution of mass, stiffness, or local wind pressure, and many elements such as slabs and diaphragms may distribute large forces from one load element to another. The combination of these factors results in the requirement for finite-element analysis.

## Nonlinear Analysis of Structural Frames

7Although nonlinear analysis is not commonly used for structural design, it is important for seismic design for several reasons. First, while the seismic-design provisions of various building codes rely on linear-elastic concepts, they are based on inelastic response. Seismic behavior of structures during major earthquakes depends on nonlinear material behavior caused by yielding of steel and cracking of concrete. The reduced stiffness due to yielding makes the stability of structures of great concern, and ensuring stability requires consideration of geometric nonlinearities. Nonlinear analysis permits treatment of these stability effects with P moments (Fig. 9.21).

Second, design methods such as load-and-resistance-factor design encourage use of flexible, partly restrained (PR) connections. Such connections are inherently nonlinear in their response. Hence, it is necessary to analyze structures with attention to the contribution of connection flexibility. Further nonlinearity may occur due to the effects of connection flexibility on frame stability and P moments. These nonlinear effects are not commonly considered in design at present. However, computer programs are available to model nonlinear frame behavior and their use is growing.

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