The Dynamics of Rocking Isolation Nicos Makris Professor of - PowerPoint PPT Presentation

The Dynamics of Rocking Isolation Nicos Makris Professor of Structures and Applied Mechanics University of Central Florida email: nicos.makris@ucf.edu London 18-3-2015

Fundamental Differences between Articulated “Ancient” and Modern Structural Systems Statically Intermediate Free-Standing Rocking Structures Moment-Resisting Frames Ductile behavior One-hinge mechanism zero ductility Four-hinge mechanism

Free-standing rocking structures have survived the most severe earthquakes level (2% probability in 50 years)

The Free-Standing Rocking Column =size b =slenderness   tan h g

Parameters of the linear oscillator and the free-standing rocking block.

Fundamental size-frequency effect 1963 George W. Housner (a) The larger of two geometrically similar blocks can survive excitation that will topple the smaller block (b) Out of two same acceleration amplitude pulses the one with longer duration is more capable to induce overturning Conclusion reached from studies motivated from the destruction observed after the May 1960 earthquake in Chile. T p g 2 standing overturning  R  u  Small blocks Large blocks or g high-frequency or long-duration  3 g 2 pulses    pulses , p p T 4 R p 6

Time Scale and Length Scale of Pulse-Like Ground Motions

Review of the dynamics of the free-standing rocking block Frequency Parameter 3 g  p 4 R               I ( t ) mgR sin[ a ( t )] m u   ( t ) R cos[ a ( t )], ( t ) 0 0 g             I ( t ) mgR sin[ a ( t )] m u ( t ) R cos[ a ( t )], ( t ) 0   0 g u     g    2        ( t ) p {sin[ a sgn( ( t )) ( t )] cos[ a sgn( ( t )) ( t )] g

        I m 2 bR sin( a ) I 0 1 1 0 2 Coefficient of Restitution:  2   2 r Energy dissipation happens  2  only during impact, while the 1 ductility of the system is zero 3 2 2   r [ 1 sin a ] 2

Rocking Structure p Earthquake Excitation a a p g T p   f ( p , a , g , a p T , ) p The six (6) variables appearing            2 1 1 2 [], a [ L ][ T ] , [ T ] , p [ T ] , g [ L ][ T ]      p p involve only 2 reference dimensions; that of length [L] and time [T]. According to Buckingham’s Π -theorem the number of dimensionless Π - products are   6 2 4

Dimensionless Products      p    p   tan( a ) a a g   g g  a p g    ( t ) ( , tan( a ), ) p g

Overturning spectra of a rigid block standing free on a monolithic base  o a 14

A Notable Limitation of the Equivalent Static Lateral Force Analysis  m u h mgb    g   resistance demand or b    u g g g tan       h resistance demand The “equivalent static” Lateral Force analysis indicates that the stability of a free-standing column depends solely on the slenderness ( g tan α ) and is independent to the size   2 2 R b h

Seismic Resistance of Free-Standing Columns subjected to Dynamic Loads           m u ( t ) R cos[ a ( t )] I ( t ) mgR sin[ a ( t )]    g  o                     resistance demand   0 For rectangular blocks, I o =(4/3) mR 2 4       2      u g  ( t ) R cos[ a ( t )] R ( t ) gR sin[ a ( t )]         3            seismic demand seismic resistance   0 Simply stated, Housner’s size effect uncovered in 1963 is merely a reminder that a quadratic term eventually dominates over a linear term regardless the values of their individual coefficients.

Basic design concepts and response-controlling quantities associated with: (a) the traditional earthquake resistant (capacity) design ; (b) seismic isolation ; and (c) rocking isolation . TRADITIONAL EARTHQUAKE RESISTANCE DESIGN SEISMIC ISOLATION ROCKING ISOLATION  Moment Resisting Frames  Braced Frames Moderate to Appreciable Low Low to Moderate Q b Q Strength u y  m  u y u up    m    0.10g-0.25g 0.03g-0.09g   g g tan a   g g g h Positive, Low Stiffness Positive and Variable due to Yielding Negative, Constant and Constant Very Large/Immaterial * Appreciable Ductility LRB † : μ =10-30 Zero μ =3-6 CSB ‡ : μ =1000-3000 Damping Moderate Moderate to High Low (only during impact) Low Strength and Low Stiffness in Seismic Resistance association with the capability to Low to Moderate Strength and Appreciable Strength and Ductility accommodate Large Appreciable Rotational Inertia Originates from: Displacements Equivalent Static Lateral YES YES NO Force Analysis is Applicable? Design Philosophy Equivalent Static Equivalent Static Dynamic * Makris and Vassiliou (2011) † LRB=Lead Rubber Bearings ‡ CSB=Concave Sliding Bearings

The Dynamics of the Rocking Frame

The Rocking Frame A one-degree-of-freedom structure g

Direct vs Variational Formulation Direct Approach: Derivation of the equations of motion by employing Newton’s law of dynamic equilibrium. There is a need to calculate the internal forces. Indirect Approach: The average kinetic energy less the average potential energy is a minimum along the true path from one position to another: Variational formulation – No need to calculate internal forces.

Relations of the horizontal and vertical displacements with the angle of rotation du    u           u 2 R sin sin   d         u  2 R cos          2            u  2 R sin cos  dv       v        v 2 R cos cos  d         v  2 R sin           2              v 2 R cos sin

Equation of Motion: Variational Formulation   d dT dT Lagrange’s Equation:     Q      dt d d dW Generalized force acting on the system   Q  d   1 1       Kinetic Energy:  2 2 2     T N I m u  v  o b 2 2   N   Variation of the Work:      δ W m m u δ u  g δ v b c g   2 dW   dW N        W   ,            2 R m m u  cos g sin    b c g d  d 2   I   o 2 R   u  2 m R              g    c sin a cos Equation of Motion:   1   g     g       2 θ (t) >0

Equation of Motion of the Rocking Frame     u   1 2           g     2 ( t ) p sin[ a sgn( ( t )) ( t )] cos[ a sgn( ( t )) ( t )]     1 3 g   Equation of Motion of the Rocking Column   u             g     2 ( t ) p sin[ a sgn( ( t )) ( t )] cos[ a sgn( ( t )) ( t )]   g   Important Finding:   1 2 m    The equation of motion of the rocking ˆ b p p ,   Nm 1 3 frame indicates that the heavier the c cap beam is, the more stable is the      free-standing rocking frame despite 1 3 ˆ      R R 1 R   the rise of the center of gravity of the     1 2 1 2   cap beam

2   3 2         2 1 sin 3 cos 2      2     r 2        1 3        1  

Makris, N. and Vassiliou, M. (2013). Planar rocking response and stability analysis of an array of freestanding columns capped with a freely supported rigid beam, EESD, 42(3): 431-449.

Church of St. Marko in Gaio, Italy (from S. Lagomarsino, July 2008) Formation of rocking frame offered dynamic stability which led to collapse prevention