Report - PEER - University of California, Berkeley

freedom (SDOF) system. In this regard pushover analysis serves for the approximateconstruction of the backbone curve of the SDOF hysteresis (Aydinoglu 2003), whichis called the capacity diagram (Chopra and Goel 1999) or capacity spectrum (ATC,1996). Thus seismic demand can be estimated in a simple manner using inelasticresponse spectrum concept (Fajfar 1999). Note that the capacity diagram is notexplicitly used in the so-called Displacement Coefficient Method of FEMA 356document (ASCE 2000), but its coordinates are implicitly considered in defining thecoefficients.It has to be admitted that the above-described intuition-driven approach has someserious problems and limitations. Firstly, the backbone curve of the SDOF hysteresis,i.e., the capacity diagram cannot be developed directly. Instead an auxiliary capacitycurve, i.e., the so-called pushover curve is needed, but its coordinates are definedsomewhat arbitrarily. The base shear and the roof displacement are traditionallyselected for buildings, but it is problematic as to which displacement component tochoose, for example, in bridges. On the other hand, it is not clear which mode shape isto be considered in the conversion process from the pushover curve to the capacitydiagram. In some applications invariant linear elastic mode shape is adopted while inthe others instantaneous deformed shapes due to invariant load patterns are used as ifthey were similar to instantaneous mode shapes.Selecting the pushover curve coordinates arbitrarily and assuming an artificialmode shape for capacity diagram conversion may lead to inconsistent, even erroneousresults. In this regard a typical but lesser known example is the misrepresentation ofP-delta effects in buildings through conventional pushover curve (Aydinoglu 2004).The problem deals with the contribution of equivalent P-delta forces to the baseshear. Note that generally linear shape functions are adopted for an approximatedevelopment of the geometric stiffness matrix, which represents P-delta effects(Clough and Penzien 1993). In a two-dimensional response of a building structurewith rigid floor diaphragms, for example, this approximation leads to a story P-deltamoment at each story (total story axial force times the story drift), which is thendivided to the story height and thus converted to an equivalent force couple.Resultants of those forces help define a tri-diagonal geometric stiffness matrix, whichis commonly used in most analysis software. It is clear that in calculating thecontribution of P-delta forces to the base shear, the sum of those equivalent forcecouples vanishes at every story except in the first story (since the force at the bottomis not counted). This leads to an awkward situation, meaning that the ordinate of theconventional pushover curve actually represents the P-delta effect of only the firststory! It may be argued that had the base overturning moment been selected as theordinate of the pushover curve instead of the base shear, P-delta effects could havebeen represented more correctly.It becomes evident that the main source of the above-mentioned problems is theinvariant seismic load patterns intuitively used in the conventional pushover analysis,which in turn requires the development of a conventional pushover curve and theselection of an artificial mode shape for conversion to the capacity diagram. Actually,346