SEISMIC RESPONSE OF BUILDINGS WITH NON-UNIFORM STIFFNESS MODELLED AS SHEAR BEAMS ANDRÉS ALONSO-RODRIGUEZ Universidad de Valparaíso, Chile EDUARDO MIRANDA Stanford University, CA, USA.

Closed form solutions are worth considering because: Motivation Closed form solutions are worth considering because: they allow to fully describe many aspects of the dynamic characteristics of complex multi-degree-of-freedom systems (e.g., buildings) with a very small number of parameters They are based on fundamental principles (e.g., equilibrium, mechanics) and therefore lead to a transparent approach to describing the model (i.e., they are not a black box model such as a complex FEM models) They provide valuable insight into the problem. For example they allow the identification of structural parameters that have significant effects on seismic response of buildings Ideal for performing parametric studies and assessing large building portfolios. July 22th 2014

Distribution of Story Shear Demands on Buildings Designed According to Response Spectrum Modal Analysis (RSMA) A Parabolic variation effectively represents how shear demands degrade along height. 𝑆 𝑥 =1− 1−𝛿 𝑥 2 d d d July 22th 2014

Shear Beam with Parabolic Stiffness Equation of Motion Legendre Differential Equation (Bielak Shear Beam) Frequency Definition in terms of Legendre parameter Mode Shape July 22 2014

Boundary Conditions Derivative of mode shape (property of Legendre functions) Boundary conditions in Matrix form Modal characteristic equation Mode Shape July 22th 2014

1000 finite element discretization Solution Validation Beam with δ=0.05 Mode shapes on top, Mode shapes derivatives at the bottom 1000 finite element discretization July 22th 2014

Modal Periods and Modal Participation Factors Period lengthening and MPF’s Change drastically for stiffness Reductions up to 80% of base Values… STIFFNESS REDUCTION DISPROPORTIONATELLY AFFECTS HIGHER MODES! Fundamental Period and period ratios Modal Participation Factors (MPF’s) July 22th 2014

Effects of stiffness Reduction in Mode Shapes July 22th 2014

Effects of Stiffness Reduction in Mode Shape Derivatives Drift for the first mode becomes evenly distributed, changes cluster at the last third of the beam July 22th 2014

Study Case SAC 20 Building First three modal periods, SAC 20 Building and its Parabolic Shear Beam Representation Mode Shapes, from the SAC 20 building and the non uniform shear beam July 22th 2014

Ground Motion Response Chi-Chi, 1999 earthquake, TCU 056 record NS Christchurch 2011 earthquake, Cathedral College record N64E July 22th 2014

Summary and Conclusions A closed-form solution has been developed to describe the dynamic characteristics of a non-uniform shear beam with a parabolic distribution of lateral stiffness. The dynamic characteristics are defined only by three parameters. The fundamental period of vibration T1, the damping ratio x, and d, the ratio of lateral stiffness at the top with respect to that at the base. The reduction in lateral stiffness has a larger effects on inter-story drift ratio demands than on peak floor acceleration demands. Validations using the SAC 20-story building suggest that the closed-form solution permits relatively good estimates of inter-story drift ratios and peak floor accelerations of moment-resisting frame buildings responding elastically. Acknowledgments July 22th 2014