Report - PEER - University of California, Berkeley

The strength reduction factor due to ductility R µ , which will be denoted in this paperas R (R ≡ R µ ), can be determined as the ratio between the accelerations correspondingto the elastic and inelastic system. The ductility demand µ is then calculated frominelastic spectra, which are defined by the period dependent relation betweenreduction factor and ductility (R-µ-T relation), and the inelastic displacement demandS d is computed as S d = (µ/R)S de . The target displacement, which represents theseismic demand of the MDOF model, is obtained as D t = ΓS d , where Γ is thetransformation factor from the MDOF to the SDOF system.In principle any R-µ-T relation can be used. A very simple and fairly accurate R-µ-T relation is based on the equal displacement rule in the medium- and long-periodrange. This relation is used in the basic variant of the N2 method. It has beenimplemented in Eurocode 8 and is discussed below. The application of the N2 methodcan be extended also to complex structural systems, for example to infilled frames(Chapter 4), provided that an appropriate specific R-µ-T relation is known.For many years, the ductility factor method has been used in seismic codes. Thebasic assumption of this method is that the deformations of a structure produced by agiven ground motion are essentially the same, whether the structure respondselastically or yields significantly. This assumption represents the “equal displacementrule”. Using this rule, the ductility dependant reduction factor R is equal to ductilityfactor µ. The simple chart in Fig.1 is essential for understanding of the concept ofreduction factors and of the ductility factor method. The educational value of thefigure can be greatly increased by using the AD format, introduced by Freeman. InAD format, Fig.1 (force has to be divided by mass) can be combined with demandspectra (Fig.2). Fig.2, which enables a visualisation of the basic variant of the N2method, resembles to the basic chart in capacity spectrum method. The maindifference is in inelastic demand, which is defined by an inelastic spectrum ratherthan by an equivalent highly damped elastic spectrum. Inelastic spectrum in mediumandlong-period range in Fig.2 is based on the equal displacement rule.f maxForce, δDuctility factors:δmaxFmaxµ = =δyFyS aeT = T CT > T Cµ = 1 (elastic)f yS aS dd S dy S dS ayS adµδ yδ maxElastic or nonlinearDeformation, δS d=S deFigure 1. Basic diagram explainingthe ductility factor method (re-plottedfrom Clough and Penzien 1975, p.603).Figure 2. Elastic and inelastic demandspectra versus capacity diagram.359