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TitleEarthquake Engineering
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Table of Contents
                            Preface Earthquake Engineering
Section 1 Seismic Risk, Hazard, Wave Simulation and Geotechnical Aspects
Chapter 1 Seismic Risk of Structures and the Economic Issues of Earthquakes
Chapter 2 Assessment of Seismic Hazard of Territory
Chapter 3 A Cognitive Look at Geotechnical Earthquake Engineering: Understanding the Multidimensionality of the Phenomena
Chapter 4 Three-Dimensional Wavefield Simulation in Heterogeneous Transversely Isotropic Medium with Irregular Free Surface
Chapter 5 Full-Wave Ground Motion Forecast for Southern California
Chapter 6 Soil-Structure Interaction
Section 2 Seismic Performance and Simulation of Behavior of Structures
Chapter 7 Seismic Performance of Historical and Monumental Structures
Chapter 8 Bridge Embankments - Seismic Risk Assessment and Ranking
Chapter 9 Finite Element Analysis of Cable-Stayed Bridges with Appropriate Initial Shapes Under Seismic Excitations Focusing on Deck-Stay Interaction
Chapter 10 Dynamic Behaviour of the Confederation Bridge Under Seismic Loads
Chapter 11 Seismic Performance Evaluation of Corroded Reinforced Concrete Structures by Using Default and User-Defined Plastic Hinge Properties
Mechanical Characterization of Laminated Rubber Bearings and Their Modeling Approach
Document Text Contents
Page 1


Edited by Halil Sezen

Page 2

Earthquake Engineering
Edited by Halil Sezen

Afshin Kalantari, V. B. Zaalishvili, Silvia Garcia, Haiqiang Lan, Zhongjie Zhang, En-Jui Lee,
Po Chen, Alexander Tyapin, Halil Sezen, Adem Dogangun, Wael A. Zatar, Issam E. Harik,
Ming-Yi Liu, Pao-Hsii Wang, Lan Lin, Nove Naumoski, Murat Saatcioglu, Hakan Yalçiner,
Khaled Marar, A. R. Bhuiyan, Y.Okui

Published by InTech
Janeza Trdine 9, 51000 Rijeka, Croatia

Copyright © 2012 InTech
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any damage or injury to persons or property arising out of the use of any materials,
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Publishing Process Manager Marijan Polic
Typesetting InTech Prepress, Novi Sad
Cover InTech Design Team

First published August, 2012
Printed in Croatia

A free online edition of this book is available at
Additional hard copies can be obtained from [email protected]

Earthquake Engineering, Edited by Halil Sezen
p. cm.
ISBN 978-953-51-0694-4

Page 174

Earthquake Engineering 162

structure will be solved exactly, providing U0(x) in the whole upper part. So, this check is
OK. However, in “problem A2” exact displacement UA(-H) given by (27) is different from
U0(-H) given by (22). As a result, both upcoming wave and the wave coming down in Vint
will be different from the exact solution. If this new solution is marked with upper index
”3”, then the displacement of the mass is given by

3 3 3 01 2 / [cos( / ) sin( / )]

U U U U H H 

    (29)

Note that this solution depends on H, while the exact solution does not (see (21)). As our
boundary is artificial, such dependence cannot be physical.

We can calculate the “error coefficient” relating approximate solution (29) to the exact
solution (21). With dimensionless frequency

m m

 
  (30)

and dimensionless depth of Q

h H

 (31)

this “error coefficient” makes

3 1

cos( ) sin( )A
U i

h hU

  

 


Curves for different µ(ω) for different h are shown in Fig.9.

We see that the solution in not satisfactory. The increase of the boundary depth h does not
improve the situation. General conclusion is that one must be very careful in placing lower
boundary in the direct approach when there is no rock seen in the depth.

6.3. Impedance approach

Now let us apply the impedance approach to the same system. As our structure rests on the
surface, we can use Fig.6. If the displacement of the mass is Ub, then the equation of motion
(10) turns to

2 0[ ( ) ] ( ) ( ) ( )bC m U C U      (33)

Impedance is the same as in Fig.4: C(ω)=iωρc. So, from (33) we at once get the ultimate result
which turns to be similar to the exact one (21):

0 0/ [1 ] / [1 ] Ab
m m

U U U i U
i c

 

     (34)

Page 175

Soil-Structure Interaction 163

Figure 9. “Error coefficient” for direct approach after changing the lower boundary condition

General conclusion is that the correct application of the impedance method provides the
exact solution.

The decisive factor in getting the exact solution both in direct and in impedance
approaches for our 1D model was the ability to get exact model for substituting Vext
(dashpot in our case).

Unfortunately, this example is of methodology value only. While “problem B“ (without
structure) is often treated in practice with 1D models (though not homogeneous), “problem
A” (with structure) is principally different. Great part of energy is taken from the structure
by surface waves, which are not represented in 1D model.

7. Impedances in the frequency domain

Historically, the first problems for impedances considering soil inertia were solved semi-
analytically for homogeneous half-space without the internal damping and for circular
surface stamps. It turned out that horizontal impedances behaved more or less like pairs of
springs and viscous dashpots (as mentioned above, in reality the dissipation of energy had
completely different nature; mechanical energy was not converted in heat, as in dashpot, but
taken away by elastic waves). One more “good news” was that the impedance matrix
appeared to be almost diagonal. Though non-diagonal terms coupling horizontal translation
with rocking in the same vertical plane were non-zero ones, their squares were considerably
less in module than products of the corresponding diagonal terms of the impedance matrix.










0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

Безразмерная частота




Page 347

Mechanical Characterization of Laminated Rubber Bearings and Their Modeling Approach 335

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[28] Amin A F M S, Lion A, Sekita S and Okui Y. Nonlinear Dependence of Viscosity in
Modeling the Rate-Dependent Response of Natural and High Damping Rubbers in

Compression and Shear: Experimental Identification and Numerical Verification.

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Page 348

Earthquake Engineering 336

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