1. 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.

2. 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

3. 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
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4. Shear Beam with Parabolic Stiffness
Equation of Motion
Legendre Differential Equation
(Bielak Shear Beam)
Frequency Definition in terms of
Legendre parameter
Mode Shape
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5. Boundary Conditions
Derivative of mode shape (property of Legendre functions)
Boundary conditions in Matrix form
Modal characteristic equation
Mode Shape
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6. 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
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7. 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)
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8. Effects of stiffness Reduction in Mode Shapes
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9. 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
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10. 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
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11. Ground Motion Response
Chi-Chi, 1999 earthquake, TCU 056 record NS
Christchurch 2011 earthquake, Cathedral College record N64E
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12. 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