Report - PEER - University of California, Berkeley

(i) (i)in which M denotes mass matrix, K and KGrepresent instantaneous (tangent) firstorderstiffness and geometric stiffness matrices, respectively, the combination ofwhich defines the instantaneous second-order stiffness matrix. Geometric stiffnessmatrix (Clough and Penzien 1993) accounts for P-delta effects with compressive axial(i)forces taken positive. ωnis the instantaneous natural frequency.On the other hand, static-equivalent seismic load vector increment correspondingto the displacement vector increment given by Eq.1 can be written aswhere ∆∆f = ( K − K ) ∆u = MΦ Γ ∆a(3)(i)a n(i) (i) (i) (i) (i) (i) (i)Sn G n n xn nrefers to the modal pseudo-acceleration increment:∆a= (ω ) ∆ d(4)(i) (i) 2 (i)n n nModal displacement and modal pseudo-acceleration developed at the end of the (i)’thpushover step are calculated by adding their increments to those obtained at the end ofthe previous pushover step:d = d + ∆d ; a = a + ∆ a(5)(i) (i 1) (i) (i) (i−1) (i)n n n n n n2.1 Modal Capacity DiagramsA hypothetical nonlinear time-history analysis based on a piecewise linear modesuperpositionmethod has led to a conclusion that modal pseudo-acceleration versusmodal displacement diagrams, i.e., a n – d n diagrams can be defined for each mode,which may be interpreted as modal hysteresis loops (Aydinoglu, 2003). The backbonecurves of those loops, i.e. the envelopes of peak response points in the first quadrant,as shown in Fig. 1, are called modal capacity diagrams. According to Eq.4, theinstantaneous slope of a given diagram is equal to the eigenvalue of the correspondingmode at the piecewise linear step concerned. By definition, first-mode capacitydiagram is essentially identical to the capacity spectrum defined in the CapacitySpectrum Method (ATC 1996) of the conventional single-mode pushover analysis.Note that instantaneous slope of the first-mode capacity diagram or those of the fewlower-mode diagrams could turn out to be negative due to P-delta effects whenaccumulated plastic deformations result in a negative-definite second-order stiffnessmatrix. A negative slope means a negative eigenvalue and thus an imaginary naturalfrequency, which leads to a modal response that resembles the non-vibratory responseof an over-damped system (Aydinoglu and Fahjan 2003). The corresponding modeshape has a remarkable physical significance, representing the post-bucklingdeformation state of the structure under gravity loads and instantaneous staticequivalentseismic loads. Although structural engineers are not familiar with thenegative (or zero) eigenvalues due to negative-definite (or singular) stiffness matrices,those quantities are routinely calculated by matrix transformation methods ofeigenvalue analysis, such as the well-known Jacobi method (Bathe 1996).348