Analyses are used in several aspects of seismic design, including the determination of required strengths, calculation of the fundamental period of vibration of the structure, and the evaluation of seismic drift limits. Any of the elastic methods in Specification Chapter C or Appendix 7 can be used to assess the stability of frames in high seismic regions; however, these methods have predominantly been developed and validated for nonseismic applications. As a result, some of the requirements of the elastic methods are not appropriate for all aspects of seismic design. Note that while underestimates of stiffness are generally deemed to be conservative in the determination of required strengths or evaluation of drift limits, an underestimate of stiffness that leads to an overestimate of the fundamental period of vibration can be unconservative in the determination of seismic loads.
The analysis to determine the required strengths should use stiffnesses in accordance with these Provisions and the Specification. For analyses to determine other aspects of behavior, appropriate stiffnesses should be used. For example, when using the equivalent lateral load procedure for seismic design and the direct analysis provisions in Specification Chapter C, the reduced stiffness and notional load provisions should be included for calculation of required strengths, but typically would not be included in the calculation of the fundamental period of vibration or the evaluation of seismic drift limits. Ongoing research may provide more definitive guidance.
For nonseismic applications, story drift limits, like deflection limits, are commonly used in design to ensure the serviceability of the structure. These limits vary because they depend upon the structural usage and building contents. As an example, for wind loads such serviceability limit states are regarded as a matter of engineering judgment rather than absolute design limits (West et al., 2003), and no specific design require ments are given in the Specification.
The situation is somewhat different when considering seismic effects. Research has shown that story drift limits improve frame stability ( effects) and seismic performance because of the resulting strength and stiffness. Although some building codes, load standards, and resource documents contain specific seismic drift limits, there are major differences among them as to how the limit is specified and applied. Nevertheless, drift control is important to both the serviceability and the stability of the structure. As a minimum, the designer should use the drift limits specified in the applicable building code.
The analytical model used to estimate building drift should accurately account for the stiffness of the frame elements and connections and other structural and nonstructural elements that materially affect the drift. Research on steel moment frame connections indicates that, in most cases, the effect of panel-zone deformations on elastic drift can be adequately accounted for by modeling beams to extend between column centerlines without rigid end offsets and that explicit panel-zone modeling is not required (FEMA, 2000d). In cases where nonlinear element deformation demands are of interest, panel-zone shear behavior should be represented in the analytical model whenever it significantly affects the state of deformation at a beam-to-column connection. Mathematical models for the behavior of the panel zone in terms of shear force-shear distortion relationships have been proposed by many researchers. FEMA 355C presents a good discussion of how to incorporate panel-zone deformations into the analytical model (FEMA, 2000e).
Adjustment of connection stiffness is usually not required for connections traditionally considered as fully restrained, although FEMA 350 (FEMA, 2000a) contains recommendations for adjusting calculated drift for frames with reduced beam sections. Nonlinear models should contain nonlinear elements where plastic hinging is expected to properly capture the inelastic deformation of the frame. Where partially restrained connections are used, analytical models must adequately reflect connection stiffness in both the elastic and inelastic range.
For composite systems that include composite members or steel members combined with reinforced concrete, the properties of the composite and concrete members should be modeled to represent the effects of concrete cracking. For design by elastic analysis, the composite and concrete member properties should reflect the effective stiffness of the members at the onset of significant yielding. The following guidance is provided for calculating effective stiffness values for design by elastic analysis:
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(a) In concrete beam and column members, stiffness properties for elastic analysis are typically specified as a fraction of the flexural stiffness, , where is the elastic modulus of concrete and is the gross moment of inertia. For concrete frames, ACI 318, Section 6.6.3.1.1 (ACI, 2019), recommends alternative moments of inertia in the range of 0.25 to for beams and 0.35 to for columns, or as justified by rigorous analysis. More detailed recommendations that account explicitly for axial load are given in Seismic Evaluation and Retrofit of Existing Buildings (ASCE/SEI 41) (ASCE, 2017), which recommends effective stiffness values of (a) for columns with unfactored gravity compressive loads that are greater than or equal to , where is the gross member area and is the specified compressive strength of the concrete, and (b) for columns and beams with axial gravity loads less than or equal to . Linear interpolation of stiffness is suggested for axial loads between 0.1 and .
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(b) For concrete walls, ACI 318, Section 6.6.3.1.1, recommends alternative moments of inertia, , between and , or as justified by rigorous analysis. The walls above the hinged region are typically expected to remain essentially elastic. For these regions and walls that are anticipated to remain in the elastic range, the effective stiffness based on cracked section properties for the walls may be taken as and . ASCE/SEI 41 also includes recommendations, which are deemed to be conservative for new composite ordinary shear walls.
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(c) For encased or filled beam-columns, the effective stiffness may be specified based on the use of a cracked transformed section (Ricles and Paboojian, 1994; Varma et al., 2002). Equations for the effective stiffness have also been proposed by Denavit et al. (2018). Attention should be paid to the relative values of the girder versus beam-column effective stiffnesses.
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(d) For steel beams with composite slabs in which the shear connection between the beam and slab is such that the contribution of the composite slab can be included in the stiffness and subject to reverse curvature due to earthquake loading, a reasonable assumption is to specify a flexural stiffness that is equal to the average of the composite beam stiffness in positive bending and bare steel beam stiffness in negative bending. Assuming that the beams are designed to have full composite action, it is suggested to take the effective stiffness as equal to , where is the modulus of elasticity of steel, is the moment of inertia of the bare steel beam, and is the transformed moment of inertia of the beam and slab. The effective width of the slab can be determined in accordance with Specification Chapter I.
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(e) For composite plate shear walls, the effective stiffness may be specified based on the use of a cracked transformed section. Equations for the effective stiffness have also been proposed (Varma et al., 2022).
Like most of the provisions in the Specification, the stability requirements are intended for cases where the strength limit state is based on the nominal elastic-plastic limit in the most critical members and connections, not to ensure stability under seismic loads where large inelastic deformations are expected. Thus, the provisions of Specification Chapter C do not alone ensure stability under seismic loads. Stability under seismic loads is synonymous with collapse prevention, which is provided for in the prescriptive design requirements given for each system, including such elements as the following:
- (1) The basic determination of the seismic design force using R factors, site effects, ρ factors, etc.
- (2) The drift limits under the seismic lateral load, a factor of both the limiting drift and the specified factor
- (3) The "theta" limits, which address sidesway stability collapse prevention
- (4) Other design requirements, such as strong-column/weak-beam requirements, limitations on bracing configurations, etc.