Dissertation2002-Amaris

7/25/2019 Dissertation2002-Amaris

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Istituto Universitario

di Studi Superiori di Pavia

Universit degli Studi

di Pavia

EUROPEAN SCHOOL OF ADVANCED STUDIES IN

REDUCTION OF SEISMIC RISK

ROSE SCHOOL

DYNAMIC AMPLIFICATION OF SEISMIC MOMENTS AND

SHEAR FORCES IN CANTILEVER WALLS

A Dissertation Submitted in Partial

Fulfilment of the Requirements for the Master Degree in

EARTHQUAKE ENGINEERING

By

ALEJANDRO DARIO AMARIS MESA

Supervisor: Dr NIGEL PRIESTLEY

June, 2002


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The dissertation entitled Dynamic Amplifications of Seismic Moments and Shear Forces in cantileverwalls, by Alejandro Dario Amaris Mesa, has been approved in partial fulfilment of the requirements

for the Master Degree in Earthquake Engineering.

M. J. Nigel Priestley

M.J. Kowalsky


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Abstract

i

ABSTRACT

In recent years, Displacement based design procedure has been used to achieve a specified

acceptable level of damage under the design earthquake. The inappropriateness of the Forcebased design assumptions of initial stiffness and ductility capacity suggests that results of basemoments and shear reached in a structure when inelastic response had occurred are not valid. For that

reason, it is proposed in this analysis determine appropriate dynamic amplification factor for flexure

and shear for a wide range of cantilever wall buildings of 2, 4, 8, 12, 16 and 20 stories which were

design using the fundamental of displacement based design principles and compare the results withForce Based Design analysis and time history dynamic analyses.

In addition, the relationship between ductility demand and the dynamic amplification factor in each

wall system was investigated. This was carried out using time history analyses for five different

earthquakes intensities for each wall, and analysing the bending moment and shear force envelopes.

The effects of some of the issues discussed above were analysed through the use of the inelastic

dynamic analysis program, Ruaumoko and the results were compared with existing code requirements.

It was found that dynamic amplification of both shear and moment envelopes became more severe asthe initial elastic period of the structure increased, and also as the ductility increased (effected by

increasing the seismic intensity). Since all the walls were designed to the same drift limit of 0,02, the

level of ductility corresponding to the design seismic intensity decreased as the number of stories

increased.

It was further found that most of the dynamic amplification resulted from second mode response, andthat existing design equations for dynamic amplification for walls were grossly non-conservative.


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Acknowledgments

ii

ACKNOWLEDGEMENTS

A man is the promise of all the things he can be and the farthest aims depend on our interest to make

them true.

To my parents, because without your support and breadth in your advice I could not have reached this

dream that today I devote to you with all my heart. THANK YOU FOR ALWAYS BEING BY MYSIDE WHEN I NEEDED YOU.

To you Luisa, because day after day you always are the engine of my search, the aim of my thoughts,the motivation of my life. You always are my support and one reason of my life, all my

reasonsTHANKS FOREVER.

To Professor Nigel Priestley, because your lessons have always guided me to reach my goals, to

understand with clarity the concepts and I have benefited a great deal from your wealth of knowledge.THANKS FOR YOUR INVALUABLE HELP.

Today and always I will remember with affection all those people who taught me to cultivate theknowledge and to dream how to reach it. Each one of you has left me a piece of sky of yourknowledge in my heart and with it, I must follow my way to shine as a star guide and to discover that

knowledge is not that reality that we want to reach, knowledge is a dream which we can share witheach other to find the truth that ties us.


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Acknowledgments

iii

AGRADECIMIENTOS

Un hombre sera la promesa de lo que todos sus sueos puedan llegar a ser. Sin duda, las metas ms

lejanas dependen de nuestro interes para lograrlo.

A mis padres, porque sin su apoyo, sin su aliento en sus consejos no hubiera podido alcanzar estesueo que hoy les dedico con mi corazn. GRACIAS POR ESTAR SIEMRE A MI LADO CUANDO

LO HE NECESITADO.

A ti Luisa, porque da tras da eres el motor de mi bsqueda, el fin de mis pensamientos, la motivacin

de mi vida. Tu siempre eres mi apoyo y una de las razones de mi vida, todas mis razones....GRACIAS

ETERNAS.

Al Profesor Nigel Priestley, que con sus enseanzas siempre me orient a cmo lograr los objetivos, aentender con claridad los conceptos, a darme su opinin y criterio con su conocimiento. GRACIAS

POR SU INVALUABLE AYUDA.

Hoy y siempre recordar con cario a todas aquellas personas que me ensearon a cultivar elconocimiento, a soar cmo alcanzarlo cada una de ellas me ha dejado un trozo de firmamento de

su conocimiento en mi corazn y con l debo seguir mi camino para brillar como una estrella gua ydescubrir que conocimiento no es esa realidad que queremos alcanzar, conocimiento es un sueo que

queremos compartir con cada uno para encontrar esa verdad que nos ata.


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Index

iv

DYNAMIC AMPLIFICATION OF SEISMIC MOMENTS AND

SHEAR FORCES IN CANTILEVER WALLS

INDEX

ABSTRACT...

ACKNOWLEDGEMENTS ..

AGRADECIMIENTOS

INDEX ......

LIST OF TABLES

LIST OF FIGURES .......

1. INTRODUCTION ...

1.1 OBJECTIVES AND SCOPE .....

1.2 ORGANIZATION OF THE DISSERTATION

2. CURRENT DESIGN PROVISIONS ......

2.1 INTRODUCTION..

2.2 LATERAL FORCE DISTRIBUTION ......

2.3 BENDING MOMENT DISTRIBUTION AND DYNAMIC AMPLIFICATION

FACTOR

2.4 SHEAR FORCE DISTRIBUTION AND DYNAMIC AMPLIFICATION

page

i

ii

iii

iv

vii

viii

1

2

2

4

4

4

5


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Index

v

FACTOR.

3. DESIGN OF THE STRUCTURAL WALLS...

3.1 INTRODUCTION .

3.2 DIRECT DISPLACEMENT BASED DESIGN ....

3.2.1 Performance Limit State ...

3.2.2 Design Displacement Spectra ...3.2.3 Design displacement .

3.2.4 Effective Mass ..

3.2.5 Effective Damping 3.2.6 Effective Period and Effective Stiffness ...

3.2.7 Distribution Of Base Shear ...

3.3 DIRECT DISPLACEMENT BASED DESIGN..

4. PROCEDURE ANALYSIS .

4.1 INTRODUCTION .....

4.2 FORCE BASED DESIGN PROCEDURE.....

4.2.1 Equivalent Lateral Force Profile....

4.2.2 Multi Mode Analysis.

4.2.3 Paulay And Priestley Approach.....

4.3 NON LINEAR ANALYSIS PROCEDURE...

4.3.1 Dynamic Analysis Modelling Assuptions.

4.3.2 Moment - Curvature Properties.....4.3.3 Plastic Hinge length...

4.3.4 Hysteresis Rule..

4.3.5 Viscous Damping..

4.3.6 Shear Deformation

4.3.7 Input Ground Motion.

5. DISCUSSION...

5.1 COMPARISON TIME HISTORY DYNAMIC ANALYSISWITH

DISPLACEMENT BASED DESIGN ANALYSIS...

5.1.1 Shear distribution.......

5.1.2 Moment distribution......5.1.3 Displacement profile......5.1.4 Interstory Drift.......

5.2 FORCE BASED DESIGN PROCEDURE.....5.1.1 Shear distribution.......

5.1.2 Moment distribution......

6. CONCLUSIONS .

page7

10

10

10

11

1112

13

1314

15

18

23

23

23

23

28

40

40

41

4142

42

43

4444

58

58

58

626666

6769

70

78


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Index

vi

7. BIBLIOGRAPHY ..

page78


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Index

vii

LIST OF TABLES

3.1 Dimensions of the walls and shear force and bending moment at the base designed according to

displacement based design principles.3.2 a Distribution of forces, moments and displacement along the height of the walls A, B and Cdesigned according to displacement based design principles.

3.2 b Distribution of forces, moments and displacement along the height of the walls D, E and F

designed according to displacement based design principles.

3.3 Aspect ratio for each wall.3.4 Conditions used for determination of yield, nominal and ultimate moments and curvatures in

bilinear moment-curvature relations.

3.5 Bilinear moment-curvature parameters obtained from RECMAN2 analysis.3.6 Bilinear moment-curvature parameters obtained from RECMAN2 analysis just for the base of the

wall.

4.1 a Distribution of forces and moments along the height of the walls A, B and C designed

using Equivalent lateral force according to EUROCODE 8.

4.1 b Distribution of forces and moments along the height of the walls D, E and F designed usingEquivalent lateral force according to EUROCODE 8.

4.2 Shear force and bending moment at the base using Equivalent lateral force according to

EUROCODE 8.

4.3 Vibration modes for each wall using 0.5Ig.4.4 Periods and total mass participation for each wall.

4.5 a Distribution of forces and moments along the height of the walls A, B and C designed using

Modal Combination Analysis SRSS.

4.5 b Distribution of forces and moments along the height of the walls D, E and F designed usingModal Combination Analysis SRSS.

4.6 Shear force and bending moment at the base using modal Analysis SRSS according to

EUROCODE 8.

4.7 Plastic hinges lengths values for each structural wall from equation (3.2a) and (3.2b).

5.1 Maximum ductility and force reduction factor.


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Index

viii

LIST OF FIGURES

2.1 Modeling of the lateral forces and the structure for equivalent lateral force method.

2.2 Recommended design moment envelope for structural wall Paulay and Priestley, 1992.2.3 Recommended design moment envelope for structural wall EUROCODE8.2.4 Different distributions of shear from static lateral force analysis, under beam over strength, and

under dynamic amplification.

3.1 Idealization of different structural walls.

3.2 Elastic Displacement Response Spectrum (5% Damping).3.3 a Shear along the height for Displacement Based Design method.

3.3 b Moments along the height for Displacement Based Design method.

3.3 c Displacement profile for Displacement Based Design method.

3.4 d Interstory Drift for Displacement Based Design method.3.5 a Moment Curvature diagram for Walls A.

3.5 b Moment Curvature diagram for Walls B.3.5 c Moment Curvature diagram for Walls C.

3.5 d Moment Curvature diagram for Walls D.3.5 e Moment Curvature diagram for Walls E.3.5 f Moment Curvature diagram for Walls F.

4.1 Design Response Spectrum with PGA = 0.40g

4.2 a Shear along the height for Forced Based Design method.4.2 b Moments along the height for Forced Based Design method.

4.3 Elastic Response Spectrum with PGA=0.4g.

4.4 a Shear along the height for Modal Combination Analysis Wall A.4.4 b Moments along the height for Modal Combination Analysis Wall A.

4.4 c Shear along the height for Modal Combination Analysis Wall B.

4.4 d Moments along the height for Modal Combination Analysis Wall B.

4.4 e Shear along the height for Modal Combination Analysis Wall C.

4.4 f Moments along the height for Modal Combination Analysis Wall C.

4.4 g Shear along the height for Modal Combination Analysis Wall D.4.4 h Moments along the height for Modal Combination Analysis Wall D.4.4 i Shear along the height for Modal Combination Analysis Wall E.

4.4 j Moments along the height for Modal Combination Analysis Wall E.

4.4 k Shear along the height for Modal Combination Analysis Wall F.

4.4 l Moments along the height for Modal Combination Analysis Wall F.4.5 a Shear along the height for Modal Combination Analysis SRSS.

4.5 b Moments along the height for Modal Combination Analysis SRSS.

4.5 c Displacement profile for Modal Combination Analysis SRSS.4.5 d Interstory drift for Modal Combination Analysis SRSS.


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Index

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4.6 The Modified Takeda hysteretic rule.4.7 a Comparisons between EC8 and artificial accelerograms matching Elastic acceleration response

spectrum.

4.7 b Comparisons between EC8 and artificial accelerograms matching Elastic displacement response

spectrum.4.8 a Shear along the heights for time history dynamic analysis Wall A.

4.8 b Moments along the heights for time history dynamic analysis Wall A.

4.8 c Displacement profile for time history dynamic analysis Wall A.4.8 d Interstory drift for time history dynamic analysis Wall A.

4.9 a Shear along the heights for time history dynamic analysis Wall B.

4.9 b Moments along the heights for time history dynamic analysis Wall B.4.9 c Displacement profile for time history dynamic analysis Wall B.

4.9 d Interstory drift for time history dynamic analysis Wall B.

4.10 a Shear along the heights for time history dynamic analysis Wall C.4.10 b Moments along the heights for time history dynamic analysis Wall C.

4.10 c Displacement profile for time history dynamic analysis Wall C.

4.10 d Interstory drift for time history dynamic analysis Wall C.

4.11 a Shear along the heights for time history dynamic analysis Wall D.4.11 b Moments along the heights for time history dynamic analysis Wall D.

4.11 c Displacement profile for time history dynamic analysis Wall D.

4.11 d Interstory drift for time history dynamic analysis Wall D.

4.12 a Shear along the heights for time history dynamic analysis Wall E.4.12 b Moments along the heights for time history dynamic analysis Wall E.

4.12 c Displacement profile for time history dynamic analysis Wall E.4.12 d Interstory drift for time history dynamic analysis Wall E.

4.13 a Shear along the heights for time history dynamic analysis Wall F.

4.13 b Moments along the heights for time history dynamic analysis Wall F.

4.13 c Displacement profile for time history dynamic analysis Wall F.4.13 d Interstory drift for time history dynamic analysis Wall F.

5.1 Relation of shears at each floor for Time history dynamic analysis and Displacement base design

at intensity 1.05.2a Shear distribution considering higher mode effects from equation 5.2 at intensity 1.0 Wall A

5.2b Shear distribution considering higher mode effects from equation 5.2 at intensity 1.0 Wall B5.2c Shear distribution considering higher mode effects from equation 5.2 at intensity 1.0 Wall C5.2d Shear distribution considering higher mode effects from equation 5.2 at intensity 1.0 Wall D

5.2e Shear distribution considering higher mode effects from equation 5.2 at intensity 1.0 Wall E

5.2f Shear distribution considering higher mode effects from equation 5.2 at intensity 1.0 Wall F

5.3 Relation of moments at each floor for Time history dynamic analysis and displacement base

design at intensity 1.05.4a Moment distribution considering higher mode effects from equation 5.3 at intensity 1.0 - Wall A

5.4b Moment distribution considering higher mode effects from equation 5.3 at intensity 1.0 - Wall B5.4c Moment distribution considering higher mode effects from equation 5.3 at intensity 1.0 - Wall C

5.4d Moment distribution considering higher mode effects from equation 5.3 at intensity 1.0 - Wall D

5.4e Moment distribution considering higher mode effects from equation 5.3 at intensity 1.0 - Wall E

5.4f Moment distribution considering higher mode effects from equation 5.3 at intensity 1.0 - Wall F

5.5 Intensity of the level of earthquake versus interstory drift at the top of each wall.5.6 Equal Displacements and Energy approximation to estimate the displacement demand.5.7 Relation of shears at each floor for Time history dynamic analysis and Modal combination and

displacement ductility reduction factor applied only to 1stvibration mode at intensity 1.0

5.8 Relation of moments at each floor for Time history dynamic analysis and Modal combination anddisplacement ductility reduction factor applied only to 1stvibration mode at intensity 1.0

5.9a Shear Distribution for time history dynamic analysis compared with Modal combination

considering force reduction factor taken from maximum ductility of THA - Wall A.

5.9b. Moment Distribution for time history dynamic analysis compared with Modal combination

considering force reduction factor taken from maximum ductility of THA - Wall A


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Index

x

5.10a Shear Distribution for time history dynamic analysis compared with Modal combinationconsidering force reduction factor taken from maximum ductility of THA - Wall B.


considering force reduction factor taken from maximum ductility of THA - Wall B.

5.11a Shear Distribution for time history dynamic analysis compared with Modal combinationconsidering force reduction factor taken from maximum ductility of THA - Wall C.


considering force reduction factor taken from maximum ductility of THA - Wall C.5.12a Shear Distribution for time history dynamic analysis compared with Modal combination

considering force reduction factor taken from maximum ductility of THA - Wall D.

5.12b. Moment Distribution for time history dynamic analysis compared with Modal combinationconsidering force reduction factor taken from maximum ductility of THA - Wall D.


considering force reduction factor taken from maximum ductility of THA - Wall E.5.13b. Moment Distribution for time history dynamic analysis compared with Modal combination

considering force reduction factor taken from maximum ductility of THA - Wall E.


considering force reduction factor taken from maximum ductility of THA - Wall F.5.14b. Moment Distribution for time history dynamic analysis compared with Modal combination

considering force reduction factor taken from maximum ductility of THA - Wall F.


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Chapter 1. Introduction

1

1. INTRODUCTION

The behavior of structural wall systems can be relatively complex, particularly when walls of different

sizes and stiffness are coupled. Even when this is not the case, the behavior of a simple cantilever

wall is dependent on a number of properties of the wall. In particular, sectional shapes, the aspect

ratio of wall height to length, and base yield moments are important.

Modeling issues are also important in cantilever walls. Plastic hinge lengths, hysteresis rules,

effective stiffness, viscous damping, input motion and the consideration of shear deformation, can all

be important in a time history analysis of such a wall.

An assumed first mode response is the basis of most structural wall designs with modifications for

higher mode effects on moments and base shear. Current code provisions [NZS 4203, 1992] are based

on elastoplastic time history analyses, with a limited earthquake database [Blakely et al., 1975].

Others intensive studies [Portland Cement Association, 1980] were carried out to determine the

influence of structural wall response to seismic excitation. Parameters such as intensity, frequency

content, characteristic and duration of accelerograms, were studied. Also, base yield moment and

fundamental period of vibration of the structure were considered.

The basic criteria that the designer will aim to satisfy are the provision of adequate stiffness, strength,

and ductility. Also, the designer must take into account carefully detailed walls designed for flexural

ductility and protected against a shear failure by capacity design principles.

All these studies were analysed under principles of force based design procedures of the current

seismic design philosophy, which is based on a required minimum strength, related to initial stiffness,

seismic intensity, and a force reduction factor or ductility factor. It is imperative to do an examination


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of these proceedings based on the direct displacement based-design principle. The basis of the design

assumption is to achieve a specified acceptable level of damage under the design earthquake, starting

by using the displacement corresponding to a given limit, and the initial elastic properties including

stiffness, strength and period are the end product.

1.1 OBJECTIVES AND SCOPE

The main aim of this project is to determine appropriate dynamic amplification factor for flexure and

shear for a wide range of cantilever wall buildings of 2, 4, 8, 12, 16 and 20 stories which were design

using the fundamental of displacement based design principles.

In addition, the relationship between ductility demand and the dynamic amplification factor in each

wall system will be investigated. This will be carried out using time history analyses for five different

earthquakes intensities for each wall, and analysing the bending moment and shear force envelopes.

The effects of some of the issues discussed above are going to be analysed through the use of the

inelastic dynamic analysis program, Ruaumoko [Carr, 1996] and the results will be compared with

existing code requirements.

1.2 ORGANIZATION OF THE DISSERTATION

This dissertation is organized in six chapters covering some important issues of the topics related

above.

An introduction, objectives and the main purpose of this dissertation are presented in Chapter 1.

Chapter 2 refers to the current design provision for moments and shear forces and their dynamic

amplification factors.

The design of the structural walls according to the displacement-based design is shown in chapter 3.

Definition of the performance limit state, design displacement spectra, design displacement, effective

mass, damping, period and stiffness are shown. Finally, the distribution of moment and shear at the

based are shown.


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In chapter 4 different methods of analysis are carried out in order to compare moment amplification

factors for structural walls. Forced based design method, and modal combination analysis using the

EUROCODE 8 principles (equivalent lateral force profile) were carried out. Non linear analysis is

carried out in order to obtain a more precise response of the structure considering inelastic properties

of the section at each time history in a generated ground record which match the EUROCODE 8

acceleration and displacement response spectrum.

Chapter 5 presents the discussion of results from the different analysis methods and finally Chapter 6

presents the most relevant conclusions and discussions of the code implications resulting from this

dissertation and shows some important topics for further analysis.


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Chapter 2. Current design provisions

4

2. CURRENT DESIGN PROVISIONS

2.1 INTRODUCTION

Most code approaches to the design of buildings for earthquake resistance specify a distribution of

static lateral loads, which the structure must be able to resist safely. These static forces point out to

give a force distribution equivalent to that induced in the structure by seismic base excitation but of

reduced magnitude, if it is acceptable that inelastic deformations and associated hysteretic damping

will occur.

The form of such code lateral load distributions typically has load increasing linearly with height and

is an approximation of the loading pattern associated with the first mode of vibration of the structure

(inverted triangular distribution). Usually this first mode response dominates the deflection response

of a structure, but the participation of 2nd

and higher modes is often of significance in force patterns in

individual members.

The effect of higher mode participation (revealed in a quantitative manner by dynamic time history

analyses) is to vary the height of the centroid of action of the inertia forces acting on a wall.

2.2 LATERAL FORCE DISTRIBUTION

Within the context of force-based design, current literature [Paulay & Priestley, 1992] and design

codes [SANZ, 1992; UBC, 1997, EUROCODE8, 1992] suggest the use of an approximation of the

first mode response of the wall, with a correction for higher mode effects. This is the familiar

equivalent lateral force method of analysis (figure 2.1), with an inverted triangular distribution of

forces and a concentrated force at the top story. Although the use of this concentrated force could be


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questioned, it has been used here, as dynamic amplification of code distributions is being investigated.

The percentage of the total base shear applied as a concentrated force to the top story varies, with 10%

used in Paulay and Priestley, 8% in NZS 4203 [SANZ, 1992], can be 0 up to 25% for UBC, and the

Eurocode8 does not consider this force at the top of the height.

= +F

F

1

2

Specified forces Substitute forces

= +F

F

1

2

Specified forces Substitute forces

Figure 2.1 Modeling of the lateral forces and the structure for equivalent lateral force method.

2.3 BENDING MOMENT DISTRIBUTION AND DYNAMIC AMPLIFICATION FACTOR

An inverted triangular distribution of applied loads to the wall will result in a cubic distribution of

bending moment in the wall, with a maximum value, EM at the base. This distribution is shown in

Figure 2.2

Research into distribution of bending moments in walls from inelastic time history analyses has

suggested that a linear distribution of bending moments, with EM at the base, should be used for

design. This takes into account the contribution of higher modes to the bending moments up the wall.

This distribution is used if curtailment of longitudinal steel is desired at some location, which is of

particular interest for tall walls. In addition to this, the tension shift effect must be taken intoaccount, but this is independent of the time history analysis, and of no interest here. The design

distribution is compared with the moments consistent with the triangular load distribution in Figure

2.2.

When an appropriate amount of longitudinal steel has been provided such that the nominal moment,

NM , exceeds the design moment, EM (with any reduction factors required by the code taken into


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account), an overstrength moment, oM , can be calculated. This is the moment that will actually

develop at the base of the wall if material strengths are larger than specified values, and strain

hardening results in an increase in moment carried by the section. For design, oM can be taken as

NM , where 25.1= [Paulay & Priestley, 1992]. The actual bending moment distribution when

this over strength moment is developed could be expected to follow the linear distribution used for

design, increased uniformly by up the height of the wall. This is also shown in Figure 2.2.

Moment due to static lateralforces (1st Mode only)

Linear distribution (assumed

dynamic amplificacion)

Distribution of moment at over

strength

h

w

M = M (assumed)NE0M = M N

lw

Nominal minimum

Moment due to static lateralforces (1st Mode only)




strength

h

w

M = M (assumed)NE0M = M N

lw

Nominal minimum

Figure 2.2. Recommended design moment envelope for structural wall Paulay and Priestley, 1992.

EUROCODE 8 specifies thata distribution of moments along the height of the wall shall be given by

an envelope of the calculated bending moment diagram (obtain from the structural analysis), vertically

displaced (tension shift) by a distance equal to the height crh of the critical region of the wall. The

envelope curve may be assumed linear, if the structure does not exhibit important discontinuities of

mass, stiffness or resistance over its height. (Figure 2.3).

The height of the critical region crh above the base of the wall may be estimated as

= 6

,max nwcr hlh but greater than wl2 or the story height sh for structures lesser than 6 stories and

sh2 for structures with 7 or more stories.


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7

Moment due to static lateral

forces (1st Mode only)




strength

h w

M = M (assumed)NE

cr

h

Moment due to static lateral

forces (1st Mode only)




strength

h w

M = M (assumed)NE

cr

h

Figure 2.3. Recommended design moment envelope for structural wall EUROCODE8.

2.4 SHEAR FORCE DISTRIBUTION AND DYNAMIC AMPLIFICATION FACTOR

As with the bending moment, a shear force distribution consistent with the assumed linear distribution

of lateral forces can be derived. This will be parabolic in nature, with a non-zero value at the top due

to the 10% concentrated load, as shown in Figure 2.3. Also shown is the shear obtained by

multiplying the distribution obtained from the equivalent static analysis by the over strength factor,

EwoW MM /,,0 = . In this case, assuming no strength reduction factor is used, and longitudinal steel

is provided to ensure NE MM = exactly, then 25.1,0 == W .

As with the bending moments, shear forces obtained from analysis and over strength considerations

must be amplified to account for higher mode effects. In fact, the higher modes affect the shears a lot

more than they affect the moments, particularly for taller walls. A dynamic amplification factor, V ,

is applied to the over strength shear forces, as shown as the third curve in Figure 2.3. This factor has

been obtained from previous research, and is presented in Paulay and Priestley (1992) in the following

form:

15630/3.1

610/9.0


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8

1st Mode only

Distribution of shear for

beam overstrength

Shear with dynamic amplification

h w

V = VoV = V = V

VE

o,w E

EVU o V o,w

1st Mode only

Distribution of shear for

beam overstrength

Shear with dynamic amplification

h w

V = VoV = V = V

VE

o,w E

EVU o V o,w

Figure 2.4. Different distributions of shear from static lateral force analysis, under beam over strength,

and under dynamic amplification

EUROCODE 8 specifies a simplified procedure based on capacity design criterion to take into account

increase of shear forces after yielding at the base of the wall. A design envelope of the shear forces

UV along the height of the wall shall be

OU VV = (2.2)

where OV is the shear force along the height of the wall, obtained from the analysis and is the

magnification factor depending on the ductility class of the structure. For structures with low ductility

capacity the magnification factor may be taken equal to 1.3. For structures with high and medium

ductility capacity, can be obtained

qTSe

TSe

M

M

qq c

n

uo

+

=

2

1

2

)(

)(1.0

(2.3)

where q is the behavior factor which takes into account the energy dissipation capacity of the

structure. nM is the design bending moment at the base of the wall, uM is the design flexural

resistance at the base of the wall, o over strength ratio of steel and when precise data is not available

it can be taken as 1.25 for structures with high ductility capacity and 1.15 for medium ductility

capacity structures.


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9

1T is the fundamental period of vibration of the building along the direction of the wall and cT is the

upper limit period of the constant spectral acceleration branch. Se(T) is the ordinate of the elastic

response spectrum.

Finally, EUROCODE 8 classify squat walls are whose with a height to length ratio ww lh / not greater

than 2.0 and special provisions are given but not comments in this work. Further references see

EUROCODE 8 section 2.11.


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Chapter 3. Design Of Structural Walls

10

3. DESIGN OF THE STRUCTURAL WALLS

3.1 INTRODUCTION

In recent years, Displacement based design procedure has been used to achieve a specified acceptable

level of damage under the design earthquake. The starting point of the design is to obtain the

distribution of moment and base shear and therefore the dynamic amplification factors corresponding

to a given damage limit state.

The inappropriateness of the Force based design assumptions of initial stiffness and ductility capacity

suggests that results of base moments and shear reached in a structure when inelastic response had

occurred are not valid. For that reason, it is proposed in this analysis to design the structure according

to Displacement Based Design principles and compare the results with Force Based Design analysis

and time history dynamic analyses.

3.2 DIRECT DISPLACEMENT BASED DESIGN

Direct displacement-based design characterizes the structure by secant stiffness at a maximum

displacement and a level of equivalent viscous damping appropriate to the hysteretic energy absorbed

during the inelastic response. This reasonable assumption is closer to the real behavior of the structure

under an earthquake.

A series of walls of different height labeled A to F (Figure 3.1) were designed using Displacement

Based Design principles. They were assumed to be braced at the base.


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12

Using the displacement spectrum of 5% a linear distribution was considered until 4.0 sec instead of 3.0

sec, which is the recommendation of the EUROCODE8 because it seems more convenient for soil

class type B. This suggestion was very useful in the designing of Walls D to F because the structural

period of the substitute structures were longer than those for the elastic structure. [Priestley and

Kowalsky, 2000] and [Kowalsky, 2001].

3.2.3 Design displacement

In order to obtain the moment and shear forces for Displacement-Based design procedure, it is

required to initially determine the design displacement, and the effective mass and damping of the

equivalent single-degree of freedom substitute structure.

A maximum interstorey drift of 0.02 controlled the design at the top of the structure and, assuming a

linear distribution of curvature along the height, the design displacement profile is found thus

[Priestley and Kowalsky, 2000].

+

=

225.1

3

22

p

i

w

nsy

d

n

i

w

iyi

lh

l

h

h

h

l

h (3.1)

Where :

wl is the length of the wall, ih is the Story height at each floor, d the maximum interstorey drift, and

sy is the yield strain of the longitudinal reinforcement in the section and pl is the Plastic hinge length

which can be taken as

nwp hll 03.02.0 += (3.2a)

bynp dfhl 022.0054.0 += (3.2b)

Where nh is the total height of the wall, bd diameter and yf yield stress of the wall vertical

reinforcement.

The plastic hinges basically affect curvature ductility in a structural wall. Its magnitude depends on

the length of the wall wl , the moment gradient at the base, and axial load intensity.

Plastic hinge lengths for walls are expected to be much greater than those used for beams, due to the

large depth of section, and effective height. The plastic hinge length, pl , is directly related to the

extent of diagonal cracking, which is controlled by the section depth, wl .


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13

Equation (3.2a) takes the wall length into consideration, while Equation. (3.2b), originally obtained for

columns, does not. However, Eq. (3.2a) does not take into account strain penetration into the

foundation. It is thought that an equation taking all three parameters into account (wall length, height

and strain penetration) would be most appropriate, but the maximum of Eq. (3.2a) and (3.2b) should

be sufficient.

Having found the design displacement profile, the design displacement for the equivalent single-

degree of freedom system was obtained by the expression

( )

( )

=

=

=

n

i ii

n

i ii

d

m

m

1

1

2

(3.3)

Where im are the story masses.

3.2.4 Effective Mass

From consideration of the mass participating in the first inelastic mode, the effective system mass for

the equivalent single-degree of freedom system was calculated using the equation (3.4).

( )

d

n

i ii

e

mm

= =1 (3.4)

3.2.5 Effective Damping

Priestley & Kowalsky, 2000 proposed the use an appropriate level of viscous damping for inelastic

analysis in order to capture the assumptions made in the design as closely as possible. In the design

phase, damping was comprised of two components: elastic viscous damping of 5%, and hysteretic

damping, converted to equivalent viscous damping.

heff += 05.0 (3.5)

where

=

rr

h

11

(3.6)

ris a post-yield force-displacement stiffness ratio and is the design ductility which can be

determined from


26/87


14

y

d

= (3.7)

where y is the yield displacement and

3

)7.0( 2nyy

h

= (3.8)

where y is the yield curvature which can be calculated by

wsyy l/2 = (3.9)

The curvature ductility is determined by

( )( ))/(5.01)/(3

11

npnp hlhl

+=

(3.10)

3.2.6 Effective Period and Effective Stiffness

For a design displacement d and design damping the effective period at peak response is

2/1

)5,( 7

2

+

=

P

dpe TT (3.11)

Where )5,(P is the displacement at peak period PT for displacement corresponding at 5% damping.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Period (sec)

Displacement(m)

Figure 3.2 Elastic Displacement Response Spectrum (5% Damping)

The effective stiffness at peak response is


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15

+

=

2

7422

2

)5,(

dP

Pe

eT

mK (3.12)

3.2.7 Distribution Of Base Shear

The base shear was calculated as vertically distributed in proportion to the vertical mass and

displacement profiles and no additional force at the top is recommended

( )=

=

n

i

ii

iiBi

m

mVF

1

(3.13)

Where BV is the shear force from (3.13) and Moments are obtained multiplying the iihF for each

floor.

deB KV = (3.14)

The dimensions for each wall are shown in Table 3.1. Transverse reinforcing was assumed to be

2D12 legs at 75mm spacing where a minimum amount of transversal reinforcement is required to

confine the longitudinal reinforcement.

Table 3.1 Dimensions of the walls and shear force and bending moment at the base designed according to

displacement based design principles.

Wall b (m) Lw (m) l db (mm) Lp (m) Teff (sec) Vb (kN) Mb (kNm)

A 0.20 2.0 0.0046 14 0.58 6.4 20.6 1.2 242 1232

B 0.20 2.5 0.0080 14 0.86 3.4 12.6 1.8 312 2917

C 0.20 3.3 0.0162 20 1.49 1.9 6.0 2.6 446 8114D 0.25 4.0 0.0172 28 2.22 1.3 2.7 3.1 590 16222

E 0.25 5.0 0.0161 24 2.83 1.2 2.2 3.7 664 24372

F 0.30 5.6 0.0177 28 3.52 1.0 1.0 3.9 830 38739

Table 3.2a Distribution of forces, moments and displacement along the height of the walls A, B and C

designed according to displacement based design principles.

Story

Fi (kN) Vi (kN) Mi (kNm)

i (m) Fi (kN) Vi (kN) Mi (kNm)


i (m)

20

19

18

17

16

15

1413

12

11

10

9

8 120 120 0 0.346

7 99 218 359 0.287

6 79 297 1014 0.228

5 60 357 1906 0.173

4 138 138 0 0.193 42 399 2977 0.123

3 95 233 414 0.134 27 426 4175 0.078

2 169 169 0 0.103 56 289 1114 0.078 14 441 5454 0.042

1 73 242 507 0.044 22 312 1982 0.031 5 446 6777 0.014

0 0 242 1232 0.000 0 312 2917 0.000 0 446 8114 0.000

Wall A Wall B Wall C


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16

Table 3.2b Distribution of forces, moments and displacement along the height of the walls D, E and F

designed according to displacement based design principles

Story

Fi (kN) Vi (kN) Mi (kNm)



i (m)

20 114 114 0 0.725

19 105 219 342 0.665

18 95 314 998 0.606

17 86 400 1941 0.54716 106 106 0 0.617 77 477 3141 0.489

15 95 201 317 0.557 68 545 4572 0.433

14 85 286 919 0.497 59 605 6207 0.378

13 75 361 1777 0.439 51 656 8020 0.326

12 118 118 0 0.477 65 427 2861 0.382 43 699 9988 0.276

11 104 222 355 0.417 56 483 4140 0.327 36 735 12085 0.229

10 89 311 1021 0.358 47 530 5588 0.275 29 764 14291 0.185

9 75 386 1954 0.301 39 568 7176 0.226 23 787 16584 0.145

8 61 447 3110 0.246 31 599 8881 0.180 17 804 18946 0.109

7 48 495 4450 0.195 24 623 10678 0.138 12 817 21359 0.077

6 37 532 5935 0.148 17 640 12546 0.101 8 825 23809 0.050

5 26 558 7530 0.106 12 652 14466 0.069 4 829 26283 0.028

4 17 575 9204 0.069 7 659 16421 0.042 2 831 28770 0.012

3 10 585 10929 0.040 4 663 18398 0.022 0.2 831 31263 0.001

2 4 589 12683 0.018 1 664 20386 0.008 -1 831 33757 -0.003

1 1 590 14451 0.004 0.2 664 22379 0.001 -0.2 830 36248 -0.002

0 0 590 16222 0.000 0 664 24372 0.000 0 830 38739 0.000

Wall D Wall E Wall F

Distribution of shear

Displacement Based Design Analysis

0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850

Shear (kN)

Heigh

t(m)

Wall A

Wall B

Wall C

Wall D

Wall E

Wall F

Figure 3.3 a Shear along the height for Displacement Based Design method.


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17

Distribution of Moments


0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

0 2500 5000 7500 10000 12500 15000 17500 20000 22500 25000 27500 30000 32500 35000 37500 40000

Moment (kNm)

Height(m)

Wall A

Wall B

Wall C

Wall D

Wall EWall F

Figure 3.3 b Moments along the height for Displacement Based Design method.

Displacement profile


0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75

Displacement (m)

Height(m)

Wall A

Wall B

Wall CWall D

Wall E

Wall F

Figure 3.3c Displacement profile for Displacement Based Design method.


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18

Interstory Drift


0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Interstory Drift (%)

Height(m)

Wall A

Wall B

Wall C

Wall D

Wall EWall F

Figure 3.3 d Interstory Drift for Displacement Based Design method.

Aspect ratio WW lh / (wall height/length) for each wall is shown in Table 3.3. A factor

recommendation to increase the lateral design force specified for ordinary structural wall 1Z is

suggested when the aspect ratio is less than 3.0 [Paulay & Priestley, 1992]. In this project the lateral

design forcesfactor is not considered.

Table 3.3. Aspect ratio for each wall.

Wall hw/lw

A 3.0

B 4.8

C 7.3

D 9.0

E 9.6

F 10.7

3.3 ANALYSIS OF THE SECTION

A bilinear approximation to the actual curve was derived, using the yield, nominal and ultimate

conditions given [Paulay & Priestley, 1992].


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19

Using the GW-BASIC program, RECMAN2. These values are summarised in Table 3.4 below, where

the compression strength'

ccf of confined circular or rectangular section with equal confining stress

'

lf in the orthogonal x and y directions is related to the unconfined strength by the relationship

[Mander, Priestley & Park, 1988]

Table 3.4 Conditions used for determination of yield, nominal and ultimate moments and curvatures in

bilinear moment-curvature relations.

Concrete, c Steel, s

"First yield" (Myand y) 0.002 00225.0/ =sy Ef

Nominal (MN) 0.004 0.015

"Yield" of bilinear (y) Extrapolation from origin through first yield

Ultimate (Muand

u)

06.06.0 =su

Where s ,'

ccf , and cu are shown in Table 3.5

'

'

'

'

'' 294.71254.2254.1 c

c

l

c

lcc f

f

f

f

ff

++= (3.15)

where yhxelx fKf ='

, and''

lylx ff = .. Using 6.0=eK the confinement effectiveness coefficient

[Paulay & Priestley, 1992] and x is the ration of tension reinforcement.

Table 3.5. Bilinear moment-curvature parameters obtained from RECMAN2 analysis

Wall rx f'cc (kN) ecu

A 0.029 65.2 0.032

B 0.028 65.2 0.032

C 0.028 64.4 0.031

D 0.019 55.9 0.025

E 0.019 55.9 0.025

F 0.015 51.6 0.022

Use of the conditions from Table 3.4 and 3.5, along with the output from RECMAN2, values of initial

stiffness, k0, and the bilinear factor, r, were obtained for each level of axial load and for each wall (see

Table 3.6). The r-value, or ratio of the post-yield stiffness to initial stiffness, was only required for

sections expected to behave inelastically, therefore, it was assumed at the base of each wall. Table 3.6

presents a summary of the values obtained. Moment curvature diagram for each wall are shown in

figure 3.3.

cc

suyhs

f

f

'

4.1004.0

+


32/87


20

Table 3.6. Bilinear moment-curvature parameters obtained from RECMAN2 analysis just for the base of

the wall.

Wall N(kN) My(kNm) y' (/m) MN(kNm) y(/m) Mu(kNm) u(/m) k0(kNm2) r

A 400 813 0.00148 1158 0.00211 1246 0.03302 5.48E+05 0.00522

B 800 1883 0.00126 2749 0.00184 3014 0.03039 1.50E+06 0.00620

C 1600 5692 0.00104 8401 0.00154 9199 0.02625 5.47E+06 0.00591

D 2400 10896 0.00086 16139 0.00128 17853 0.01948 1.27E+07 0.00744

E 3200 16372 0.00069 24263 0.00102 27152 0.01678 2.39E+07 0.00767

F 4000 25456 0.00061 38407 0.00092 43762 0.01470 4.16E+07 0.00935

Moment Curvature Diagram Wall A

0

200

400

600

800

1000

1200

1400

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Curvature (1/m)

Mome

nt(kNm)

My= 813

y' = 0.00148

Mn = 1158

y = 0.00883

Mu = 1246

u = 0.033

M=20.6 = 1276

=20.6 = 0.044

Figure 3.5a. Moment Curvature diagram for Wall A.

Moment Curvature Diagram Wall B

0

500

1000

1500

2000

2500

3000

3500

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

curvature (1/m)

Moment(kNm)

Analysis of the section

Bilinear Aproximation

My = 1883

y' = 0.00126

Mn = 2749

y = 0.00184

Mu = 3014

u = 0.0304

M=12.6 = 2947

=12.6 = 0.0231

Figure 3.5b. Moment Curvature diagram for Wall B.


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21

Moment Curvature Diagram Wall C

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 0.005 0.01 0.015 0.02 0.025 0.03

Curvature (1/m)

Moment(kNm)


Bilinear aproximation

My = 5692

y' = 0.00104

Mn = 8401

y = 0.00154 Mu = 9199

u = 0.02625

M=6 = 8649

=6 = 0.00922

Figure 3.5c. Moment Curvature diagram for Wall C.

Moment Curvature Diagram Wall D

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

Curvature (1/m)

Moment(kNm)



My = 10896

' = 0.00086

Mn = 16139

y = 0.00128

Mu = 17853

u = 0.01948

M=2.7 = 16343

=2.7 = 0.00344

Figure 3.5d. Moment Curvature diagram for Wall D.


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22

Moment Curvature Diagram Wall E

0

5000

10000

15000

20000

25000

30000

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018

Curvature (1/m)

Moment(kNm)



My = 16372

' = 0.00069

Mn = 24263

y = 0.0010

Mu = 27152

u = 0.0167M=2.2 = 24486

=2.2 = 0.00223

Figure 3.5e. Moment Curvature diagram Wall E.

Moment Curvature Diagram Wall F

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

curvature (1/m)

Moment(kNm)



My = 25456

' = 0.00061

Mn = 38407

y = 0.00092

Mu = 43762

u = 0.0147

Figure 3.5f. Moment Curvature diagram Wall F.


35/87

Chapter 4. Procedure analysis

23

4. PROCEDURE ANALYSIS

4.1 INTRODUCTION

Analysis tools for the seismic response quantification of structural walls and the study of the moment

amplification factor for structural walls require the comparison of shear forces and bending moment

obtained from simple linear elastic analysis to dynamic nonlinear response history analysis.

Equivalent Lateral Force Profile is a simple and useful method, which replaces the seismic force for an

equivalent static lateral force but these lateral forces do not represent the real response of the structure,

under a seismic event. These forces represent the behavior of the first mode of vibration and when

higher mode effects are involved a coefficient is introduced which will increase the shear or bending

moment of the structure.

A very common procedure is Multi-mode analysis, which is an elastic dynamic analysis where the

maximum forces for each mode of response of the structure are from the design spectrum and later

combined with each other using CQC or SRSS procedure to obtain the maximum response of the total

structure.

An inelastic dynamic analysis can also be carried out in order to obtain a more precise response of the

structure considering inelastic properties of the section at each time history in the ground motion

record.

In the following, shear and bending moment distributions are found using these types of analyses.


36/87


24

4.2 FORCE BASED DESIGN PROCEDURE

For the last few decades, seismic design has been performed with what is often termed forced-based

design, which has worked well and generally met the objective of achieving a safe design. In order to

compare results, the distribution of base moments and shear are obtained using force-based design

procedure.

Three different procedures are using in the analysis, the Equivalent Lateral Force Profile, Multi-Mode

Analysis and Paulay and Priestley approach.

4.2.1 Equivalent Lateral Force Profile

This method replaces the seismic lateral force for an equivalent static lateral force at the base of the

structure. Using EUROCODE8, the force can be obtained

WTSV dB )( 1= (4.1)

where )( 1TSd is the ordinate of the design spectrum at period 1T which is the fundamental period of

vibration for translational motion in the direction considered. W is the total weight of the structure.

The design spectrum at period 1T can be calculated by

+= 11)(0 0

qT

TSTSTT

B

dB

(4.2a)

qSTSTTT dCB

0)( = (4.2b)

=

2.0)(

)(1

0

TS

T

T

qSTS

TTT

d

K

cd

DC

d

(4.2c)

=

2.0)(

)(21

TS

T

T

T

T

qSTS

TT

d

K

D

K

D

Cod

D

dd

(4.2d)


37/87


25

Where )(TSd is the ordinate of the design spectrum which is normalized by g, is the ratio of the

design ground acceleration ga to the acceleration of gravity g ( gag /= ) equal to 0.4, 0 is the

spectral acceleration amplification factor for 5% viscous damping determined by EURCODE 8 as 2.5,

CB TT , are limits of the constant spectral acceleration branch equal to 0,15 and 0.60 respectively, DT

is a value defining the beginning of the constant displacement range of the spectrum equal to 3.0,

3/5,3/2 21 == dd kk are exponents which influence the shape of the design spectrum for a vibration

period greater than DCTT , respectively, q is the behavior factor and Sis the soil parameter type B.

The behavior factor q takes into account the energy dissipation capacity of a structure. This can be

obtained by

5.1= WRDo kkkqq (4.3)

where oq is the basic value of the behavior factor, dependent on the structural type, assuming wall

system with uncoupled walls oq is taken as 4.0. Dk Is the factor reflecting the ductility class.

Assuming ductility class type H and equal to 1.0, which corresponds to structures whose design,

dimensioning and detailing provision ensure a stable mechanism associated with large dissipation of

hysteretic energy, Rk is the factor referring to the structural regularity in elevation and equal to 1.0 for

regular structures, and Wk is the factor referring to the prevailing failure mode in structural system

with walls and is equal to

oWk 5.05.2

1

= (4.4)

where o is the prevailing aspect ratio of the walls of the structural system wwo lh /= and can be

taken from the Table 3.3. In all cases 0.1=Wk , since 3o . For further references, see section 2

of the EUROCODE 8.

The intention of this analysis is to achieve a wall with identical moment capacity at the base to the

Displacement Based Design walls of the same height. See chapter 3 for more details.

For purposes of determining the fundamental vibration periods 1T of planar models of buildings,

approximate expressions based on methods of structural dynamics may be used. Then using an

approximation formula to obtain the fundamental period of the structural can be used according to

EUROCODE 8.

4/3

1 HCT t= (4.5)


38/87


26

where H is the height of the building and tC is a factor equal to 0.075 for moment resistant space

concrete frames.

The distribution of the horizontal seismic forces can be obtained assuming the entire mass of the

structure as a substitute mass of the fundamental mode of vibration and considering that thefundamental mode shape is approximated by horizontal displacement increasing linearly along the

height

0.0

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Period (sec)

SpectralAcceleration(g)

Figure 4.1 Design Response Spectrum with PGA = 0.40g

=

jj

iiBi

WZ

WZVF (4.6)

where iF is the horizontal force acting on story i, BV is the seismic base shear according to (4.1),

ji ZZ , are the heights of masses ji mm , above the level of application of the seismic action

(foundation).

The horizontal forces iF determined in (4.6) can be distributed to the lateral load resisting system

assuming a rigid floor. The results from this analysis are shown in Table 4.1 and Figure 4.2


39/87


27

Table 4.1a Distribution of forces and moments along the height of the walls A, B and C designed using

Equivalent lateral force according to EUROCODE 8.

story Fi (kN) Vi (kN) Mi (kNm) Fi (kN) Vi (kN) Mi (kNm) Fi (kN) Vi (kN) Mi (kNm)

20

19

18

17

16

15

1413

12

11

10

9

8 214 214 0

7 187 401 641

6 160 561 1843

5 134 694 3526

4 235 235 0 107 801 5609

3 177 412 7 06 80 881 8013

2 196 196 0 118 530 1942 53 935 10657

1 98 294 589 59 589 3532 27 962 13462

0 0 294 1472 0 589 5297 0 962 16346

Wall A Wall B Wall C

Table 4.1b Distribution of forces and moments along the height of the walls D, E and F designed using

Equivalent lateral force according to EUROCODE 8.

story Fi (kN) Vi (kN) Mi (kNm) Fi (kN) Vi (kN) Mi (kNm) Fi (kN) Vi (kN) Mi (kNm)

20 145 145 019 137 282 434

18 130 412 1281

17 123 536 2518

16 160 160 0 116 651 4125

15 150 310 480 109 760 6079

14 140 450 1409 101 861 8358

13 130 580 2759 94 955 10942

12 181 181 0 120 700 4498 87 1042 13807

11 166 347 543 110 810 6597 80 1122 16934

10 151 498 1585 100 910 9025 72 1194 20299

9 136 634 3079 90 999 11754 65 1259 23881

8 121 755 4981 80 1079 14753 58 1317 27658

7 106 860 7246 70 1149 17991 51 1368 31609

6 91 951 9827 60 1209 21439 43 1411 35712

5 75 1026 12680 50 1259 25067 36 1447 39946

4 60 1087 15760 40 1299 28845 29 1476 44288

3 45 1132 19020 30 1329 32744 22 1498 48716

2 30 1162 22417 20 1349 36732 14 1512 53210

1 15 1177 25904 10 1359 40779 7 1520 57748

0 0 1177 29436 0 1359 44857 0 1520 62307

Wall D Wall E Wall F

Distribution of Shear

Forced Based Design Analysis

0

3

6

9

12

15

1821

24

27

30

33

36

39

42

45

48

51

54

57

60

63

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600

shear (kN)

Height(m)

Wall A - FBD

Wall B - FBD

Wall C - FBD

Wall D - FBD

Wall E - FBD

Wall F - FBD

Figure 4.2a. Shear along the height for Forced Based Design method.


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28

Distribution of Moment

Forced Based Design Analysis

0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000 55000 60000 65000

Moment (kNm)

Height(m)

Wall A - FBD

Wall B - FBD

Wall C - FBD

Wall D - FBD

Wall E - FBD

Wall F - FBD

Figure 4.2 b. Moments versus normalised heights for Forced Based Design method.

Table 4.2 shows the shear force and bending moment at the base of each structural wall.

Table 4.2 Shear force and bending moment at the base using Equivalent lateral force according to

EUROCODE 8

Wall T (sec) Vb (kN) M (kNm) b (m) Lw (m) l

A 0.29 294 1472 0.20 2.0 0.0084

B 0.48 589 5297 0.20 2.9 0.0178

C 0.81 962 16346 0.20 5.1 0.0182

D 1.10 1177 29436 0.25 6.0 0.0194

E 1.37 1359 44857 0.25 7.5 0.0187

F 1.62 1520 62307 0.30 8.0 0.0193

4.2.2 Multi-Mode Analysis

The multi mode system analysis is used to determine the response of individual modes, which are then

combined in some way.

Periods of vibration were determined using Ruaumoko (see Table 4.3) with an effective inertia taken

as 0.5Ig

A portion of the total mass participates in each mode as follows:


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==

=

=N

i

i

N

mi

iim

N

mi

iim

m

mm

m

11,

2

2

1, 1

(4.9)

Table 4.3 Vibration modes for each wall using 0.5Ig

Story 1 2 1 2 3 1 2 3 1 2 3 1 2 3 4 1 2 3 4

20 1.000 1.000 1.000 1.000

19 0.932 0.785 0.678 0.590

18 0.864 0.569 0.352 0.166

17 0.796 0.356 0.040 -0.209

16 1.000 1.000 1.000 -0.906 0.729 0.149 -0.230 -0.461

15 0.913 0.668 0.412 -0.102 0.662 -0.046 -0.431 -0.538

14 0.827 0.342 -0.132 0.552 0.595 -0.222 -0.541 -0.428

13 0.741 0.032 -0.561 0.867 0.530 -0.373 -0.550 -0.175

12 1.000 1.000 0.953 0.656 -0.248 -0.811 0.758 0.466 -0.494 -0.460 0.139

11 0.884 0.541 0.159 0.572 -0.486 -0.847 0.296 0.404 -0.582 -0.289 0.414

10 0.769 0.101 -0.499 0.491 -0.669 -0.671 -0.307 0.344 -0.633 -0.065 0.563

9 0.655 -0.291 -0.860 0.412 -0.789 -0.331 -0.787 0.287 -0.647 0.174 0.540

81.000 1.000 0.800 0.544 -0.602 -0.832 0.338 -0.842 0.096 -0.936 0.234 -0.625 0.392 0.349

7 0.826 0.265 -0.301 0.437 -0.806 -0.451 0.268 -0.829 0.514 -0.685 0.184 -0.572 0.555 0.049

6 0.654 -0.381 -0.889 0.337 -0.888 0.127 0.204 -0.756 0.832 -0.138 0.139 -0.493 0.638 -0.272

5 0.488 -0.825 -0.651 0.245 -0.850 0.678 0.146 -0.634 0.985 0.479 0.100 -0.396 0.632 -0.518

4 1.000 -0.721 0.419 0.336 -0.991 0.166 0.164 -0.710 1.000 0.097 -0.481 0.949 0.913 0.066 -0.290 0.545 -0.620

3 0.651 0.516 -0.906 0.202 -0.872 0.901 0.097 -0.502 0.987 0.057 -0.317 0.747 1.000 0.038 -0.185 0.397 -0.558

2 1.000 -0.331 0.333 1.000 0.222 0.096 -0.553 1.000 0.045 -0.275 0.678 0.026 -0.165 0.447 0.737 0.018 -0.094 0.226 -0.368

1 0.331 1.000 0.096 0.548 1.000 0.026 -0.193 0.481 0.012 -0.086 0.256 0.007 -0.050 0.156 0.302 0.005 -0.028 0.076 -0.142

WALL D WALL E WALL FWALL B WALL CWALL A

Considering that there are the same vibration modes as number of masses (number of stories), im , is

the mode shape at floor i and mode m and im is the mass in each floor i to N number of stories of

the structure. EUROCODE 8 recommends that the response of all modes of vibration contributing

significantly to the global response shall be taken into account, and to satisfy this criteria it requires

that the sum of the effective modal masses for the modes considered amounts to at least 90% of the

total mass of the structure and/or by demonstrating that all modes with effective modal masses greater

than 5% of the total mass are considered.

The modal base shear for each mode is obtained then

=

=

N

i

immBm mgSaV1

(4.10)

where mSa is the pseudo acceleration corresponding at mode m , and g is the gravity acceleration (see

Figure 4.3).

Modal base shear is distributed up the structure as follows

=

=N

mi

iim

iimBmim

m

mVF

1,

(4.11)


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0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

0.0 0.3 0.5 0.8 1.0 1.3 1.5 1.8 2.0 2.3 2.5 2.8 3.0 3.3 3.5 3.8 4.0

Period (sec)

spectralAcceleration(g)

Figure 4.3 Elastic Response Spectrum with PGA=0.4g

Modal displacements may be found from the spectra modal displacements

4

2

mmm

TgSa= (4.12)

where mT is the vibration period at mode m and modal displacement for floor i and m is found.

Table 4.4 Periods and total mass participation for each wall.

Wall T1 (sec) T2 (sec) T3 (sec) T4 (sec) m (%)

A 0.340 0.059 - - 100.0

B 0.796 0.131 0.052 - 98.3

C 1.878 0.306 0.114 - 93.0

D 2.721 0.441 0.162 - 90.9

E 3.392 0.549 0.201 0.106 93.6

F 3.649 0.590 0.216 0.114 91.4

=

=

=N

mi iim

N

mi

miim

imim

m

m

1,

2

1,

(4.13)

Since the modal maximum does not occur simultaneously, it is excessively conservative to combine

the modal quantities by direct addition; generally SRSS or CQC is used. EUROCODE 8 express that

the response in two vibration modes i and j may be considered as independent of each other when their

periods iTand jT satisfy the condition that ij TT 9.0 . If the periods are independent then SRSS can

be used or else, a more accurate procedure for combination such as Complete Quadratic Combination


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can be used which involves cross modal coefficients when modes are close and the modal separation

implied by modal analysis is doubtful. SRSS was used in this project, where forces, shear and

displacement at each floor were obtained using the following equations and in Figure 4.4 the results

are shown and Table 4.5 shows the shear force and bending moment at the base of each structural

wall.

=

=N

mi

imi FF1,

2,

=

=N

mi

imi VV1,

2and

=

=N

mi

imi

1,

2 (4.14)

Table 4.5a Distribution of forces and moments along the height of the walls A, B and C designed using

Modal Combination Analysis SRSS

Story Fi (kN) Vi (kN) Mi (kNm) i (m) Fi (kN) Vi (kN) M i (kNm) i (m) Fi (kN) Vi (kN) Mi (kNm) i (m)

20

19

18

17

16

15

14

13

12

11

10

9

8 122 122 0 0.404

7 63 173 366 0.334

6 71 189 880 0.264

5 90 207 1405 0.198

4 163 163 0 0.160 96 244 1913 0.136

3 111 248 489 0.104 94 292 2446 0.083

2 177 177 0 0.034 101 305 1214 0.053 71 334 3075 0.0401 81 238 532 0.011 59 333 2079 0.015 29 351 3839 0.011

0 0 238 1236 0.000 0 333 3028 0.000 0 351 4715 0.000

WALL A WALL B WALL C

Table 4.5b Distribution of forces and moments along the height of the walls D, E and F designed usingModal Combination Analysis SRSS

Story Fi (kN) Vi (kN) Mi (kNm) i (m) Fi (kN) Vi (kN) Mi (kNm) i (m) Fi (kN) Vi (kN) Mi (kNm) i (m)

20 138 138 0 0.821

19 102 240 414 0.765

18 68 305 1132 0.709

17 42 338 2047 0.653

16 134 134 0 0.762 38 344 3057 0.597

15 85 217 401 0.695 51 331 4074 0.542

14 53 257 1050 0.629 62 311 5035 0.488

13 53 266 1812 0.563 67 292 5901 0.434

12 127 127 0 0.604 65 258 2584 0.499 65 280 6658 0.382

11 71 195 380 0.533 77 245 3295 0.436 59 275 7317 0.332

10 48 217 963 0.463 86 238 3906 0.375 56 275 7901 0.283

9 65 213 1599 0.395 94 248 4403 0.316 59 280 8438 0.237

8 82 210 2192 0.328 99 279 4795 0.261 67 294 8959 0.194

7 90 224 2700 0.265 99 327 5117 0.209 75 321 9496 0.154

6 94 256 3136 0.205 96 382 5435 0.161 77 360 10078 0.1175 97 303 3545 0.151 92 438 5831 0.117 73 404 10740 0.084

4 93 357 4000 0.102 82 489 6386 0.079 61 446 11507 0.056

3 76 406 4580 0.061 65 529 7153 0.047 44 478 12396 0.033

2 48 437 5339 0.029 40 553 8145 0.022 25 496 13406 0.015

1 17 449 6275 0.008 15 562 9330 0.006 8 503 14517 0.004

0 0 449 7343 0.000 0 562 10654 0.000 0 503 15702 0.000

WALL D WALL E WALL F


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Distribution of Shears - Wall A

Modal Combination Analysis

0

1

2

3

4

5

6

7

-50 -25 0 25 50 75 100 125 150 175 200 225 250 275

Shear (kN)

Height(m)

Vibration mode 1

vibration mode 2

Modal combination SRSS

Figure 4.4a. Shear along the height for Modal Combination Analysis Wall A.

Figure 4.4b. Moments along the height for Modal Combination Analysis Wall A.


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Distribution of Shear - Wall B


0

2

4

6

8

10

12

14

-100 -50 0 50 100 150 200 250 300 350 400

Shear (kN)

Height(m)

Vibration mode 1

Vibration mode 2

Vibration mode 3


Figure 4.4c. Shear along the height for Modal Combination Analysis Wall B.

Distribution of Moments - Wall B


0

3

6

9

12

15

-500 0 500 1000 1500 2000 2500 3000 3500

Moment (kNm)

Height(m)

Vibration mode 1

Vibration mode 2

vibration mode 3


Figure 4.4d. Moments along the height for Modal Combination Analysis Wall B.


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Distribution of shear - Wall C


0

3

6

9

12

15

18

21

24

27

-200 -100 0 100 200 300 400

Shear (kN)

Height(m)

Vibration mode 1

Vibration mode 2

Vibration mode 3


Figure 4.4e. Shear along the height for Modal Combination Analysis Wall C.

Distribution of moments - Wall C


0

3

6

9

12

15

18

21

24

27

-3000 -2000 -1000 0 1000 2000 3000 4000 5000 6000

Moment (kNm)

Height(m)

Vibration mode 1

Vibration mode 2

Vibration mode 3


Figure 4.4f. Moments along the height for Modal Combination Analysis Wall C.


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Distribution of shear - Wall D


0

3

6

9

12

15

18

21

24

27

30

33

36

39

-300 -200 -100 0 100 200 300 400 500

Shear (kN)

Height(m)

Vibration mode 1

Vibration mode 2

Vibration mode 3Modal combination SRSS

Figure 4.4g. Shear along the height for Modal Combination Analysis Wall D.

Distribution of moments - Wall D


0

3

6

9

12

15

18

21

24

27

30

33

36

39

-2000 -1000 0 1000 2000 3000 4000 5000 6000

Moment (kNm)

Height(m)

Vibration mode 1

Vibration mode 2

Vibration mode 3


Figure 4.4h. Moments along the height for Modal Combination Analysis Wall D.


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Distribution of shear - Wall E


0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

-200 -150 -100 -50 0 50 100 150 200 250 300 350 400 450 500 550 600

Shear (kN)

Height(m)

Vibration mode 1

Vibration mode 2

Vibration mode 3

Vibration mode 4


Figure 4.4i. Shear along the height for Modal Combination Analysis Wall E.

Figure 4.4j. Moments along the height for Modal Combination Analysis Wall E.


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Distribution of shear - Wall F


0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

-300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300 350 400 450 500 550

Shear (kN)

Height(m)

Vibration mode 1

Vibration mode 2

Vibration mode 3

Vibration mode 4

Modal combination

Figure 4.4k Shear along the height for Modal Combination Analysis Wall F.

Distribution of moments - Wall F


0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

-6000 -4500 -3000 -1500 0 1500 3000 4500 6000 7500 9000 10500 12000 13500 15000 16500 18000

Moment (kNm)

Height(m)

Vibration mode 1

Vibration mode 2

Vibration mode 3

Vibration mode 4


Figure 4.4l. Moments along the height for Modal Combination Analysis Wall F.


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Table 4.6 Shear force and bending moment at the base using using modal Analysis SRSS according to

EUROCODE 8

Wall Vb (kN) M(kNm)

A 237 1236

B 240 3028

C 309 4715

D 413 7343E 478 10654

F 405 15702

Results of the Shear and moment for modal combination SRSS are given in Figure 4.5

Distribution of shear

Modal Combination Analysis - SRSS

0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

0 50 100 150 200 250 300 350 400 450 500 550 600

Shear (kN)

Height(m)

Wall A - Modal combination SRSS

Wall B - Modal combination SRSS

Wall C - Modal combination SRSS

Wall D - Modal combination SRSS

Wall E - Modal combination SRSS

Wall F - Modal combination SRSS

Figure 4.5a. Shear along the height for Modal Combination Analysis SRSS.

Distribution of Moments


0

3

6

9

12

15

18

2124

27

30

33

36

39

42

45

48

51

54

57

60

63

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 13000 14000 15000 16000

Moment (kNm)

Height(m)







Figure 4.5b. Moments along the height for Modal Combination Analysis SRSS.


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Displacement Profile


0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

4548

51

54

57

60

63

0.0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

Displacement (m)

Height(m)







Figure 4.5c. Displacement profile for Modal Combination Analysis SRSS.

Interstory Drift


0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Interstory Drift (%)

Height(m)


Wall B - Modal combination SRSSWall C - Modal combination SRSS




Figure 4.5d. Interstory drift for Modal Combination Analysis SRSS.


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4.2.3 Paulay and Priestly approach

Paulay and Priestly suggest that in order to take into account high mode effects, a linear distribution

can be assumed for both the moment distribution and the shear distribution, and it is suggested that

shear magnification will increase with the number of stories (see section 2.3).

To isolate the contribution of higher mode effects to the amplification of base bending moment and

base shear, the value of wo, was not calculated from the usual design recommendations. The actual

overstrength moment obtained in the time history dynamic analysis, M o, (which only includes the

effects of strain hardening, and not material overstrength) was used and NM the moment obtained

from the a static analysis, giving

E

N

oo V

M

MV = (4.15)

This allowed the dynamic magnification factor to be applied directly to the appropriate overstrength

shear, to give the ultimate design shear

E

N

oovu V

M

MVV == (4.16)

where V is the dynamic shear magnification factor to be taken as suggested in section 2.4 such as

15630/3.1

610/9.0


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In this project a frame type element member, Tangent Stiffness Rayleigh Damping and Takeda

hysteretic rule were considered. Some artificial accelerograms were generated to match the elastic

response spectrum of Eurocode 8.

4.3.1 Dynamic Analysis Modeling Assumptions

Fundamental assumptions were required in order to analyze and obtain a good interpretation of the

results:

The structural properties of the reinforced concrete element (e.g. stiffness, strength and

ductility) are idealized under some well-established theories.

To avoid elastic and inelastic deformations in the foundation structure, all the structural walls

modeled were assumed to have a fixed base foundation.

It was assumed that the floor system provides an efficient diaphragm action (rigid diaphragm)

in order to introduce the inertia forces to the structural wall and an adequate connection to the

diaphragm.

Due to in-plane forces, the floor system remained elastic.

All the lateral forces were resisted by the structural wall.

It was assumed that the sectional properties were concentrated in the vertical centerline of the

each wall model.

4.3.2 Moment - Curvature Properties

Before the model could be input into Ruaumoko, moment-curvature relations were required for each

wall. In Ruaumoko, use of either beam-columns or simple, Giberson-type frame members were

considered. Use of the former would result in great inefficiency in the analysis. At each time step, an

interaction curve would be consulted to determine member properties for the current value of axial

load. Because the axial load is constant in each wall, member properties can be obtained directly for

the initial load, and input directly for each beam segment. For each wall moment curve analysis was

carried out for different level of axial load.


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Table 3.6 shows the ultimate moment uM and the ratios of the post-yield stiffness to initial stiffness r

required to Ruaumoko input file.

4.3.3 Plastic Hinge Lengths

As was shown in section 3.2.2 two recommendations were used, and the maximum value taken Paulay

& Priestley, 1992 and Priestley & Kowalsky, 2000

bynp

nwp

dfhl

hll

022.0054.0

03.02.0

+=

+= (3.2)

Where nh is the total height of the wall, bd diameter and yf yield stress of the wall vertical

reinforcement. Typical values of the plastic hinge length are such that 8.03.0


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F

Ko

Ku

Ku=Ko(dy/dm)

rKo

rKoFy -

Fy +

KL

KL

Ku =Ku(Ko,)

=0, Ku=ko

Increasing a softens unloading

KL = KL(Ko,)

=oKL points to B

1KL tend to point to A

Increasing stiffnes reloading

dp

dp

dy dm

F

Ko

Ku

Ku=Ko(dy/dm)

rKo

rKoFy -

Fy+

KL

KL

Ku =Ku(Ko,)

=0, Ku=ko

Increasing a softens unloading

KL = KL(Ko,)

=oKL points to B

1KL tend to point to A

Increasing stiffnes reloading

dp

dp

dy dm

Figure 4.6 The Modified Takeda hysteretic rule.

4.3.5 Viscous Damping

In Ruaumoko, a Rayleigh damping model can be used with either initial stiffness or tangent stiffness,

where the only difference is either the use of initial elastic stiffness or the current tangent stiffness.

Rayleigh viscous damping is determined from a sum of proportional mass and stiffness Kterms.

KMC += (4.17)

The coefficients and are computed to give the required level of viscous damping at two different

frequencies. This results in increased damping as frequency increases, and therefore higher damping

applied to higher modes. As highlighted in the Ruaumoko documentation (Carr, 1998), this can lead

to very unrealistic damping values being applied to higher modes. This is especially important for

structures subjected to strong-ground motion, such as the records used in this analysis, described in

Section 4.3.7.

A viscous damping value of 5% is typically assumed for the analysis. This may be appropriate for

elastic analysis, but when hysteresis models are incorporated into an inelastic analysis, this may

overestimate the actual damping. Hysteresis rules are based on laboratory testing, and should be

representative of the total structural damping, in both the elastic and inelastic ranges. Non-structural

damping will also contribute, and a value of 4% seems reasonable to take this into account.


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Carr (1998) recommends use of uniform damping, or Rayleigh damping based on the tangent stiffness

matrix. In the latter case, values of damping for the higher modes should be less than 100% of critical

damping. Use of the tangent stiffness matrix results in an unrealistic increase in viscous damping

when a member becomes inelastic. However, it is thought that this may help to offset the large

damping that may result in higher modes through inappropriate use of the Rayleigh model.

In this project, a Rayleigh damping model based on the tangent stiffness matrix has been used as a

recommendation (Carr, 1997). A value of 5% damping was applied to the first mode, and 4% to the

second. Ruaumoko generates the proportionality constants and from these values.

4.3.6 Shear Deform

Documents

Dissertation2002-Amaris