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 Moment-Resisting Frames Ductile behavior Free-Standing Rocking Structures One-hinge mechanism Four-hinge mechanism zero ductility

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

The Free-Standing Rocking Column

h b   tan

=size =slenderness g

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

6

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

standing

• verturning

g

u    g R Small blocks

• r long-duration

pulses

Large blocks or high-frequency pulses

p p

2 T    3 4 g p R  ,

Conclusion reached from studies motivated from the destruction

• bserved

after the May 1960 earthquake in Chile.

2

p

T

Time Scale and Length Scale of Pulse-Like Ground Motions

) ( )], ( cos[ ) ( )] ( sin[ ) ( ) ( )], ( cos[ ) ( )] ( sin[ ) (               t t a R t u m t a mgR t I t t a R t u m t a mgR t I

g g

               

R g p 4 3 

Review of the dynamics of the free-standing rocking block

Frequency Parameter

)] ( )) ( sgn( cos[ )] ( )) ( sgn( {sin[ ) (

2

t t a g u t t a p t

g

             

2 2 2 1 2 2 2 1 1

] sin 2 3 1 [ ) sin( 2 a r r I a bR m I               

Energy dissipation happens

• nly during impact, while the

ductility of the system is zero Coefficient of Restitution:

Rocking Structure

g a p

Earthquake Excitation

p p

T a ) , , , , (

p p T

a g a p f  

The six (6) variables appearing 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

2 1 1 2

] ][ [ , ] [ , ] [ , ] ][ [ [],

   

     T L g T p T T L a

p p

      

4 2 6  

) ), tan( , ( ) ( ) tan( g a a p t g a a p

g p g g a p

    

 

        

Dimensionless Products

Overturning spectra of a rigid block standing free on a monolithic base

• a

14 

A Notable Limitation of the Equivalent Static Lateral Force Analysis

The “equivalent static” Lateral Force analysis indicates that the stability

• f

a free-standing column depends solely

• n

the slenderness (gtanα) and is independent to the size



resistance demand mgb h u m

g

    



     resistance tan demand  g h b g ug  

• r

2 2

h b R  

Seismic Resistance of Free-Standing Columns subjected to Dynamic Loads

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.

                          resistance )] ( sin[ ) ( demand )] ( cos[ ) ( t a mgR t I t a R t u m

• g

                               resistance seismic )] ( sin[ ) ( 3 4 demand seismic )] ( cos[ ) (

2

t a gR t R t a R t ug             For rectangular blocks, Io=(4/3)mR2

TRADITIONAL EARTHQUAKE RESISTANCE DESIGN  Moment Resisting Frames  Braced Frames SEISMIC ISOLATION ROCKING ISOLATION

Strength

Moderate to Appreciable

0.10g-0.25g

Low

0.03g-0.09g

Low to Moderate

Stiffness

Positive and Variable due to Yielding Positive, Low and Constant Negative, Constant

Ductility

Appreciable μ=3-6 Very Large/Immaterial* LRB†: μ=10-30 CSB‡: μ=1000-3000 Zero

Damping

Moderate Moderate to High Low (only during impact)

Seismic Resistance Originates from:

Appreciable Strength and Ductility Low Strength and Low Stiffness in association with the capability to accommodate Large Displacements Low to Moderate Strength and Appreciable Rotational Inertia

Equivalent Static Lateral Force Analysis is Applicable?

YES YES NO

Design Philosophy

Equivalent Static Equivalent Static Dynamic   m Q u y

g

    m Q u y

g

  a g h b g uup

g

tan    

Basic design concepts and response-controlling quantities associated with: (a) the traditional earthquake resistant (capacity) design; (b) seismic isolation; and (c) rocking isolation.

*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.

   

2 sin sin u R       

 

2 cos u R       

    

 

2

2 sin cos u R              

   

2 cos cos v R      

 

2 sin v R        

    

 

2

2 cos sin v R              

Relations of the horizontal and vertical displacements with the angle of rotation

   d du u     d dv v 

d dT dT Q dt d d  

     

   dW Q d   

 

   

 

2 2 2

1 1 2 2

• b

T N I m u v       

 

2

b c g

N δW m m u δu gδv   

     



   

 

2 cos sin 2

b c g

dW N R m m u g d     

     

     

   

2 2 sin cos 1 2

• g

c

I R u m R a g g       

                 

       

θ(t)>0

Lagrange’s Equation: Generalized force acting on the system Kinetic Energy: Variation of the Work: Equation of Motion:

Equation of Motion: Variational Formulation

,    d dW W 

Equation of Motion of the Rocking Frame

Equation of Motion of the Rocking Column              )] ( )) ( sgn( cos[ )] ( )) ( sgn( sin[ ) (

2

t t a g u t t a p t

g

                        )] ( )) ( sgn( cos[ )] ( )) ( sgn( sin[ 3 1 2 1 ) (

2

t t a g u t t a p t

g

          

, 3 1 2 1 ˆ p p     

R R R                   2 1 1 2 1 3 1 ˆ

Important Finding:

The equation of motion of the rocking frame indicates that the heavier the cap beam is, the more stable is the free-standing rocking frame despite the rise of the center of gravity of the cap beam

c b

Nm m  

2 2 2 1 2

3 1 2 cos 3 sin 2 3 1                           

 

      r

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

φ

The Attraction of Variational Formulation Limit Equilibrium Analysis of Arches

Gravity loads and Earthquake loads Gravity Loads

Geometric formulation Variational formulation

) (     d dV

 V   V 

) , , , (

1 3 2 1

         V ) , , , (

2 3 2 1

         V ) , , , (

3 3 2 1

         V

φ1 φ2 φ3

t

Recent publications on Limit Equilibrium of Arches

• Makris N, Alexakis H (2013) The effect of stereotomy on the shape of the thrust-

line and the minimum thickness of semicircular masonry arches. Arch Appl Mech 83:1511-1533. DOI: 10.1007/s00419-013-0763-4

• Alexakis H, Makris N (2013) Minimum thickness of elliptical masonry arches.

Acta Mech 224:2977-2991. DOI: 10.1007/s00707-013-0906-2

• Alexakis H, Makris N (2014) Limit equilibrium analysis and the minimum

thickness of circular masonry arches to withstand lateral inertial loading. Arch Appl Mech , published online, DOI: 10.1007/s00419-014-0831-4

On-Going Research: The Rocking Frame for Bridges

tanα=1/6

Financial support from the ‘Aristeia’ grant, co-funded by the European Union and the Greek State.

Rotation, vertical and horizontal displacement histories of the free standing rocking frame

Experimental program on the shaking table of the University of Patras

Financial support from the ‘Aristeia’ grant, co-funded by the European Union and the Greek State.

One of the few applications of rocking isolation

Aim of this work: To develop the theoretical/technical background (from variational methods to shake table experiments) in an effort to accept and establish rocking isolation and the associated hinging mechanism not just as a limit-state mechanism; but, as an operational state (seismic protection mechanism for large, slender structures)

South Rangitīkei, New Zealand (78m tall piers)

The challenge remains: Most modem tall bridges with tall, slender piers are protected form the seismic action via base (shear) isolation after designing massive pile foundations to prevent uplift rather than adopting the most natural rocking isolation Views of the Polymylos Bridge in Greece

Vertically Restrained Rocking Bridges

Mander, J. B., & Cheng, C. T. (1997). Seismic resistance of bridge piers based

• n damage avoidance design. Technical Report NCEER, 97.

Vertically Restrained Rocking Columns

Vertically Restrained Rocking Columns

Mahin, S., Sakai, J., & Jeong, H. (2006, September). Use of partially prestressed reinforced concrete columns to reduce post-earthquake residual displacements of

• bridges. In Fifth National Seismic Conference on Bridges & Highways, San Francisco,

California.

The Dynamics of the Vertically Restrained Rocking Column

Lagrange’s Equation d dT dT dW dV dt d d d d                 

2

1 2

• T

I t       

cos sin ,

c g

dW m R u g d                

sin sin sin 2cos 2 2cos dV EA Po R d               

Kinetic Energy: Potential Energy from Field Forces: Potential Energy from Elastic Forces:

Governing Equation of Motion

     

2

sin sgn cos sgn 1 1 sin sin tan 2 2 2cos

g

• c

c

u g P EA t p m g m g elasticity prestressing                                                  

Dimensionless Products

 

,tan , , ,

p p

• c

c

a P EA t p g m g m g           

Time Scale and Length Scale of Pulse-Like Ground Motions

1992 Erzincan Turkey

ap=0.35g, Tp=1.44sec, ωp=4.36rad/sec

1971 San Ferdando Pacoima Dam/164

ap=0.38g, Tp=1.27sec, ωp=4.95rad/sec

1994 Northridge Rinaldi Receiving St.

ap=0.71g, Tp=0.8sec, ωp=7.85rad/sec

55 . 3  p

p

 . 4  p

p

 38 . 6  p

p

 60 . 5  p

p

 36 . 6  p

p

 10  p

p



Vertically Restrained Rocking Columns

The Dynamics of the Vertically Restrained Rocking Frame

Governing Equation of Motion Dimensionless Products      

2 2 sin

1 2 sin sgn cos sgn 1 3 2 1 sin sin tan 1 3 2 2cos

g

• c

c elasticity prestres g

u t p g P EA p m g m g                                                   

 

,tan , , , ,

p p

• c

c

a P EA t p g m g m g            

2 2 2 1 2

3 1 2 cos 3 sin 2 3 1                           

 

      r

Synthesis and Erection of Prefabricated Bridges

Traditional Prefabricated Concrete Pier System

Growing Accelerated Bridge Construction Technology

Ductile Connections

From: TRB Research Proposal Webinar, available on the internet

The aim remains the same: To create ductile connections

The aim remains the same: To create ductile connections

Hybrid Precast Concrete Pier System

Washington State Transportation Center University of Washington

Unbonded vertical tendons Only a portion of the mild steel reinforcement of the hybrid frame, extends into the footing and the crossbeam.

Expected Seismic Behavior of a Hybrid Frame

Wacker, JM, Hieber, D.G, Stanton JF and Eberhard, MO (2005) “Design of Precast Concrete piers for Rapid Bridge Construction in Seismic Regions”, Research Report, Federal Highway Administration

Two Column Bent Three Column Bent Challenges with current approach:

• Buckling of the longitudinal reinforcing bars
• Fracture of the longitudinal reinforcing bars
• Spalling of the concrete cover
• Ensure adequate anchorage of embedded

connections

Sizing the slenderness of freestanding columns

a=?

The rocking structure has to be slender enough to undergo rocking; while, it has to be wide enough to remain stable.

tan 1

p p p

a pT α g pT  

Makris, N. and M.F. Vassiliou, "Sizing the Slenderness of Free-Standing Rocking Columns to Withstand Earthquake Shaking", Archive of Applied Mechanics, published on line June 2012 DOI: 10.1007/s00419-012-0681-x.

• 1. Free-Standing Rocking Frame
• 2. Damage Avoidance Design
• 3. Pre-fabricated bridges with

ductile connections

• 4. The Hybrid Rocking Frame
• 5. Free-Standing Rocking Frame

• The larger of two geometrically similar columns can survive

the excitation that will topple the smaller column

• Out of two same acceleration-amplitude pulses the one with

the longest duration is more capable to reduce overturning

• A closed-form expression has been derived to size the

slenderness of a free-standing column with a given size

Conclusions (a) Freestanding Rocking Structures

• While the stiffness of a free standing rocking frame is negative, the

stiffness of a vertically restrained rocking frame can be anywhere from negative to positive depending on the stiffness of the tendons.

• Vertical tendons are effective in suppressing the response of smaller

column subjected to long-period excitations. As the size of the column

• r the frequency of the excitation increases, the effect of the

vertical tendons becomes increasingly immaterial given that most of the seismic resistance of tall rocking frames originates primarily from the mobilization of the rotational inertia of their columns.

• For medium size rocking frames where the concept of vertical

restrainers may be attractive, there is a merit for the vertical tendons to be flexible enough so that the overall lateral stiffness of the system remains negative.

Conclusions (b) Vertically Restrained Rocking Structures

Sustainable Engineering: The design and construction of structures that meet acceptable performance levels at present and in the years to come without compromising the ability of future generations to use them, maintain them and benefit from them.