##### Document Text Contents

Page 1

EARTHQUAKE

ENGINEERING

Edited by Halil Sezen

Page 2

Earthquake Engineering

http://dx.doi.org/10.5772/1608

Edited by Halil Sezen

Contributors

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

All chapters are Open Access distributed under the Creative Commons Attribution 3.0 license,

which allows users to download, copy and build upon published articles even for commercial

purposes, as long as the author and publisher are properly credited, which ensures maximum

dissemination and a wider impact of our publications. After this work has been published by

InTech, authors have the right to republish it, in whole or part, in any publication of which they

are the author, and to make other personal use of the work. Any republication, referencing or

personal use of the work must explicitly identify the original source.

Notice

Statements and opinions expressed in the chapters are these of the individual contributors and

not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy

of information contained in the published chapters. The publisher assumes no responsibility for

any damage or injury to persons or property arising out of the use of any materials,

instructions, methods or ideas contained in the book.

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 www.intechopen.com

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( / )]

m

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

c

(30)

and dimensionless depth of Q

h H

m

(31)

this “error coefficient” makes

3 1

cos( ) sin( )A

U i

h hU

(32)

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

1

2

3

4

5

6

7

8

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

Безразмерная частота

h=1

h=2

h=3

Page 347

Mechanical Characterization of Laminated Rubber Bearings and Their Modeling Approach 335

[22] Amin A F M S, Wiraguna S I, Bhuiyan, A R and Okui Y. Hyperelasticity Model for

Finite Element Analysis of Natural and High Damping Rubbers in Compression and

Shear. Journal of Engineering Mechanics 2006; 132 54-64.

[23] Bergstrom J S and Boyce M C. Constitutive Modeling of the Large Strain Time-

Dependent Behavior of Elastomers. Journal of the Mechanics and Physics of Solids 1998;

46 931-954.

[24] Haupt P and Sedlan K. Viscoplasticity of Elastomeric Materials: Experimental Facts and

Constitutive Modeling, Archive of Applied Mechanics 2001; 71 89-109.

[25] Lion A. A Constitutive Model for Carbon Black Filled Rubber: Experimental

Investigations and Mathematical Representation. Continuum Mechanics and

Thermodynamics 1996; 8 153-169.

[26] Miehe C and Keck J. Superimposed Finite Elastic-Viscoplastic-Plasto-Elastic Stress

Response with Damage in Filled Rubbery Polymers. Experiments, Modeling and

Algorithmic Implementation. Journal of Mechanics and Physics of Solids 2000; 48 323-

365.

[27] Spathis G. and Kontou E. Modeling of Nonlinear Viscoelasticity at Large Deformations.

Journal of Material Science 2008; 43 2046-2052.

[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.

International Journal of Plasticity 2006; 22 1610-1657.

[29] Bergstrom J S and Boyce M C. Large Strain Time-Dependent Behavior of Filled

Elastomers. Mechanics of Materials 2000; 32 627-644.

[30] Mullins L. Softening of Rubber by Deformation. Rubber Chemistry and Technology

1969; 42 339-362.

[31] Gent A N. Relaxation Processes in Vulcanized Rubber I: Relation among Stress

Relaxation, Creep, Recovery and Hysteresis. Journal of Applied Polymer Science 1962; 6

433-441.

[32] International Organization of Standardization (ISO). Elastomeric Seismic-Protection

Isolators, Part 1: Test methods, Geneva, Switzerland; 2005.

[33] Bhuiyan A R. Rheology Modeling of Laminated Rubber Bearings for Seismic Analysis.

PhD thesis. Saitama University, Saitama, Japan; 2009

[34] Bhuiyan A R, Okui Y, Mitamura H and Imai T. A Rheology Model of High Damping

Rubber Bearings for Seismic Analysis: Identification of Nonlinear Viscosity.

International Journal of Solids and Structures 2009; 46 1778-1792.

[35] Imai T , Bhuiyan A R, Razzaq M K, Okui Y and Mitamura H. Experimental Studies of

Rate-Dependent Mechanical Behavior of Laminated Rubber Bearings. Joint Conference

Proceedings of 7th International Conference on Urban Earthquake Engineering

(7CUEE) & 5th International Conference on Earthquake Engineering (5ICEE), March 3-

5, 2010, Tokyo Institute of Technology, Tokyo, Japan; 2010.

[36] Bueche F. Moelcular Basis for the Mullins Effect. Journal of Applied Polymer Science

1960; 4 107-114.

Page 348

Earthquake Engineering 336

[37] Bueche F. Mullins Effect and Rubber-Filler Interaction. Journal of Applied Polymer

Science 1961; 5 271-281.

[38] Burtscher S and Dorfmann A. Compression and Shear Tests of Anisotropic High

Damping Rubber Bearings. Engineering Structures 2004; 26 1979-1991.

[39] Treloar L R G. The Physics of Rubber Elasticity. 3rd edition, Oxford University Press;

1975.

[40] Besdo D and Ihlemann J. Properties of Rubber Like Materials under Large

Deformations Explained By Self-Organizing Linkage Patterns. International Journal of

Plasticity 2003; 19 1001-1018.

[41] Ihlemann J. Modeling of Inelastic Rubber Behavior under Large Deformations Based on

Self-Organizing Linkage Patterns. Constitutive Models for Rubber, Balkema,

Rotterdam; 1999.

[42] Lion A. A Physically Based Method to Represent the Thermo-Mechanical Behavior of

Elastomers. Acta Mechanica 1997; 123 1-25.

[43] Kilian H G, Strauss M and Hamm W. Universal Properties in Filler-Loaded Rubbers.

Rubber Chemistry and Technology 1994; 67 1-16.

[44] Wolfram Research Inc. Mathematica Version 5.2, USA; 2005.

[45] Venkataraman P. Applied Optimization with Matlab Programming. John Wiley and

Sons, New York; 2002.

[46] Truesdell C. The Mechanical Foundations of Elasticity and Fluid Dynamics. Journal of

Rational Mechanics and Analysis 1952; 1 125–300.

EARTHQUAKE

ENGINEERING

Edited by Halil Sezen

Page 2

Earthquake Engineering

http://dx.doi.org/10.5772/1608

Edited by Halil Sezen

Contributors

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

All chapters are Open Access distributed under the Creative Commons Attribution 3.0 license,

which allows users to download, copy and build upon published articles even for commercial

purposes, as long as the author and publisher are properly credited, which ensures maximum

dissemination and a wider impact of our publications. After this work has been published by

InTech, authors have the right to republish it, in whole or part, in any publication of which they

are the author, and to make other personal use of the work. Any republication, referencing or

personal use of the work must explicitly identify the original source.

Notice

Statements and opinions expressed in the chapters are these of the individual contributors and

not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy

of information contained in the published chapters. The publisher assumes no responsibility for

any damage or injury to persons or property arising out of the use of any materials,

instructions, methods or ideas contained in the book.

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 www.intechopen.com

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( / )]

m

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

c

(30)

and dimensionless depth of Q

h H

m

(31)

this “error coefficient” makes

3 1

cos( ) sin( )A

U i

h hU

(32)

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

1

2

3

4

5

6

7

8

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

Безразмерная частота

h=1

h=2

h=3

Page 347

Mechanical Characterization of Laminated Rubber Bearings and Their Modeling Approach 335

[22] Amin A F M S, Wiraguna S I, Bhuiyan, A R and Okui Y. Hyperelasticity Model for

Finite Element Analysis of Natural and High Damping Rubbers in Compression and

Shear. Journal of Engineering Mechanics 2006; 132 54-64.

[23] Bergstrom J S and Boyce M C. Constitutive Modeling of the Large Strain Time-

Dependent Behavior of Elastomers. Journal of the Mechanics and Physics of Solids 1998;

46 931-954.

[24] Haupt P and Sedlan K. Viscoplasticity of Elastomeric Materials: Experimental Facts and

Constitutive Modeling, Archive of Applied Mechanics 2001; 71 89-109.

[25] Lion A. A Constitutive Model for Carbon Black Filled Rubber: Experimental

Investigations and Mathematical Representation. Continuum Mechanics and

Thermodynamics 1996; 8 153-169.

[26] Miehe C and Keck J. Superimposed Finite Elastic-Viscoplastic-Plasto-Elastic Stress

Response with Damage in Filled Rubbery Polymers. Experiments, Modeling and

Algorithmic Implementation. Journal of Mechanics and Physics of Solids 2000; 48 323-

365.

[27] Spathis G. and Kontou E. Modeling of Nonlinear Viscoelasticity at Large Deformations.

Journal of Material Science 2008; 43 2046-2052.

[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.

International Journal of Plasticity 2006; 22 1610-1657.

[29] Bergstrom J S and Boyce M C. Large Strain Time-Dependent Behavior of Filled

Elastomers. Mechanics of Materials 2000; 32 627-644.

[30] Mullins L. Softening of Rubber by Deformation. Rubber Chemistry and Technology

1969; 42 339-362.

[31] Gent A N. Relaxation Processes in Vulcanized Rubber I: Relation among Stress

Relaxation, Creep, Recovery and Hysteresis. Journal of Applied Polymer Science 1962; 6

433-441.

[32] International Organization of Standardization (ISO). Elastomeric Seismic-Protection

Isolators, Part 1: Test methods, Geneva, Switzerland; 2005.

[33] Bhuiyan A R. Rheology Modeling of Laminated Rubber Bearings for Seismic Analysis.

PhD thesis. Saitama University, Saitama, Japan; 2009

[34] Bhuiyan A R, Okui Y, Mitamura H and Imai T. A Rheology Model of High Damping

Rubber Bearings for Seismic Analysis: Identification of Nonlinear Viscosity.

International Journal of Solids and Structures 2009; 46 1778-1792.

[35] Imai T , Bhuiyan A R, Razzaq M K, Okui Y and Mitamura H. Experimental Studies of

Rate-Dependent Mechanical Behavior of Laminated Rubber Bearings. Joint Conference

Proceedings of 7th International Conference on Urban Earthquake Engineering

(7CUEE) & 5th International Conference on Earthquake Engineering (5ICEE), March 3-

5, 2010, Tokyo Institute of Technology, Tokyo, Japan; 2010.

[36] Bueche F. Moelcular Basis for the Mullins Effect. Journal of Applied Polymer Science

1960; 4 107-114.

Page 348

Earthquake Engineering 336

[37] Bueche F. Mullins Effect and Rubber-Filler Interaction. Journal of Applied Polymer

Science 1961; 5 271-281.

[38] Burtscher S and Dorfmann A. Compression and Shear Tests of Anisotropic High

Damping Rubber Bearings. Engineering Structures 2004; 26 1979-1991.

[39] Treloar L R G. The Physics of Rubber Elasticity. 3rd edition, Oxford University Press;

1975.

[40] Besdo D and Ihlemann J. Properties of Rubber Like Materials under Large

Deformations Explained By Self-Organizing Linkage Patterns. International Journal of

Plasticity 2003; 19 1001-1018.

[41] Ihlemann J. Modeling of Inelastic Rubber Behavior under Large Deformations Based on

Self-Organizing Linkage Patterns. Constitutive Models for Rubber, Balkema,

Rotterdam; 1999.

[42] Lion A. A Physically Based Method to Represent the Thermo-Mechanical Behavior of

Elastomers. Acta Mechanica 1997; 123 1-25.

[43] Kilian H G, Strauss M and Hamm W. Universal Properties in Filler-Loaded Rubbers.

Rubber Chemistry and Technology 1994; 67 1-16.

[44] Wolfram Research Inc. Mathematica Version 5.2, USA; 2005.

[45] Venkataraman P. Applied Optimization with Matlab Programming. John Wiley and

Sons, New York; 2002.

[46] Truesdell C. The Mechanical Foundations of Elasticity and Fluid Dynamics. Journal of

Rational Mechanics and Analysis 1952; 1 125–300.