Keywords: torsionally coupled structures, graphic dynamic modal analysis, graphic response history analysis, graphic response spectrum analysis, directional modal participation radii, directional spectral influence circles..

The dynamic behaviour of two-way plan-asymmetric linear systems cannot be characterized in terms of principal directions of deflection as it is normally done with static non-eccentric response. The two-way eccentricity results instead in a dynamic response decomposed in terms of modal rotational kinematics about three modal centers of rotation, referred to as modal torsional 'pivots'. In this paper we make a graphical derivation of the modal decomposition of the bi-directional coupled response. The ellipse of elasticity modal analysis is used to graphically identify the modal torsional pivots and the corresponding lines of action of the modal forces. A graphical response history analysis is performed to decompose the unidirectional seismic forces through the triangle of modal torsional pivots or forces. The concept of directional modal participation radii is introduced and a displacement-based unidirectional graphical response spectrum analysis is discussed, introducing graphical rational rules for directional modal response envelopes.

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