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

The portal method is used for buildings of intermediate or shorter height. In this method,Â 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 asÂ 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, whileÂ each exterior column is assigned half the shear for an interior column, since exterior columnsÂ do not share the loads of adjacent portals. If the bays are unequal, shear may be apportionedÂ to each column in proportion to the lengths of the girders it supports. When bays are equal,Â 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 andÂ midspan of beams. This approximates the deflected shapes and moment diagrams of thoseÂ members under lateral loads. The location of the inflection points may be adjusted for specialÂ cases, such as fixed or pinned base columns, or roof beams and top-story columns, or otherÂ special situations. On the basis of the preceding assumptions, member forces and bendingÂ moments can be determined entirely from the equations of equilibrium. As an example, Fig.Â 9.17 indicates the geometry and loading of an eight-story moment-resisting frame, and Fig.Â 9.18 illustrates the use of the portal method on the upper stories of the frame. The frameÂ has two interior columns. So one-third of the shear in each story is distributed to the interiorÂ columns and half of this, or one-sixth, is distributed to the exterior columns (Fig. 9.18). TheÂ other member forces are computed by equations of equilibrium on each subassemblage. ForÂ example, for the subassemblage at the top of the frame in Fig. 9.18, setting the sum of theÂ moments equal to zero yields

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

The cantilever method is used for tallÂ buildings. It is based on the recognition thatÂ axial shortening of the columns contributesÂ to much of the lateral deflections of suchÂ buildings (Fig. 9.19). In this method, theÂ floors are assumed to remain plane, and theÂ axial force in each column is assumed to beÂ proportional to the distance of the columnÂ from the centroid of the columns. InflectionÂ points are assumed to occur at midheight ofÂ the columns and at midspan of the beams.

The internal moments and forces are determinedÂ from equations of equilibrium, as withÂ the portal method. The determination of theÂ forces and moments in the members at theÂ top floors of the frame in Fig. 9.17 is illustratedÂ in Fig. 9.20. The lateral forces causeÂ overturning moments, which induce axialÂ tensile and compressive forces in the columns.

Therefore,

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

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

While many major, existing structures were designed without benefit of computer analysisÂ techniques, it is not advantageous to design modern buildings for wind and earthquakeÂ loading without this capability. It is needed to predict realistic structural response to windÂ loading and to evaluate occupant comfort, as well as for dynamic design for seismic loading,Â especially for buildings of unusual geometry. Both the seismic and wind load provisions inÂ the â€˜â€˜Uniform Building Codeâ€™â€™ result in local anomalies in the distribution of design forcesÂ due to the distribution of mass, stiffness, or local wind pressure, and many elements suchÂ as slabs and diaphragms may distribute large forces from one load element to another. TheÂ 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 forÂ seismic design for several reasons. First, while the seismic-design provisions of variousÂ building codes rely on linear-elastic concepts, they are based on inelastic response. SeismicÂ behavior of structures during major earthquakes depends on nonlinear material behaviorÂ caused by yielding of steel and cracking of concrete. The reduced stiffness due to yieldingÂ makes the stability of structures of great concern, and ensuring stability requires considerationÂ of geometric nonlinearities. Nonlinear analysis permits treatment of these stability effectsÂ with P moments (Fig. 9.21).

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

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