dissertação de mestrado de novelli

8/11/2019 Dissertao de mestrado de Novelli

1/87

THE UNLOADING STIFFNESS OF

REINFORCED CONCRETE MEMBERS

A Dissertation Submitted in Partial Fulfillment of the Requirements

for the Master Degree in

Earthquake Engineering & Engineering Seismology

By

Viviana Iris Novelli

Supervisors: Dr. T.J. Sullivan, Dr. R. Pinho, Prof. Calvi G. M.

December, 2008

Istituto Universitario di Studi Superiori di Pavia

Universit degli Studi di Pavia


2/87

i

The dissertation entitled The unloading stiffness of reinforced concrete members, by

Viviana Iris Novelli, has been approved in partial fulfillment of the requirements for the

Master Degree in Earthquake Engineering and Engineering Seismology.

Timothy Sullivan

Name of Reviewer 2

Name of Reviewer 3


3/87


4/87

Acknowledgements

iii

ACKNOWLEDGEMENTS

I am grateful to Professor T.J. Sullivan for his valuable guidance throughout the duration of this

dissertation. I want also to thank Professor R. Pinho and Professor G.M. Calvi for their important

reviews of the present research and suggestions. I am thankful to MEEES (Masters in Earthquake

Engineering and Engineering Seismology) consortium for providing the financial support to take part

and complete the MEEES program.

Moreover, I would like to express few words for all nice people that were with me in this short time of

my life .Everything started receiving the admission letter for this master, and Ro, you were with me,

how many jumped of happiness. September came very fast and I got on that ferry for Patrawith my

Vale.. I felt safe with you but I was alone too early and you came.. Rajesh. Your strength, enthusiasmand passion impressed me, although I felt very little in front of you. In March I was in Pavia, courses,

home-works but when I was with you, Myrto, my life smiled to me. The time to work for my

dissertation arrived too fast and I thought that the most complicated moment of my life was coming,

but Lena mou arrived making unforgettable each moment lived together. Today I am here. The first

acknowledgement is for my team. It would be impossible to arrive at the end of this travel without

you where I didnt learn to format a word document but I understood that nothing is without a

solution if I have people that love me.

And so thanks to:

Lenuccia.. for your single word.

Igor.. ino.. for your suggestions.. and if the dinners with us were good excuses to complain about you

it was a pleasure for me.

Juan.. become a perfect young house husband.

Myrto.. for your sweetness that made simpler my experience.

Rajesh.. for your energy, current in this my work.


5/87

Acknowledgements

iv

my friends of my life..

Vale.. always with me in each small step.

Andrea.. for your cute reproaches.

Daniele.. for the happiness that u feel for my success.

Manu.. for her way to look at the world

R: for the important position that you gave to me in your life.

Pat: for your passion.

Daniel.. because you are always on my side.

My parents.. for the patience to understand my choices.

Kika.. this success is dedicated to you, because without you my life doesnt make sense.

Sono grata al Professor T.J. Sullivan per il suo prezioso giudizio durante il periodo della mia tesi. Inoltre vorrei

ringraziare i Professori R. Pinho e Professor , G.M . Calvi per le loro importanti revisioni e suggerimenti

riguardanti il mio lavoro di ricerca. Sono riconoscente alla MEEES (Masters in Earthquake Engineering and

Engineering Seismology) per aver provveduto al supporto finanziario e per avermi dato lopportunit dicompletare il mio corso di studio.

Ora mi piacerebbe dedicare alcune parole a tutte le belle persone che mi hanno accompagnato in questo piccolo

pezzo di strada, per me tanto importante. Tutto incominci ricevendo la lettera di ammissione per questo

master.. momento indimenticabile.. Ro tu eri con me, quanti salti di gioia. Settembre arriv velocemente e su

quella nave per Patrasso io ci salii con la mia Vale.. con te mi sentivo al sicuro ma presto rimasi sola e le

difficolt non furono poche, ma arrivisti tu. Rajesh, la tua forza, il tuo entusiasmo e la tua passione mi colp

particolarmente, anche se a volte mi sentivo molto piccola davanti a te. In marzo ero in Pavia.. corsi,

homeworks, ma quando ero con te Myrto la mia vita mi sorrideva. Arriv troppo velocemente il tempo di

lavorare alla mia tesi e quando pensavo che avrei trascorso uno dei periodi pi complicati della mia vita arrivata Lena mou.. rendendo indimenticabili ogni singolo momento trascorso insieme. Ed oggi sono qui.. al

termine di questo percorso e il primo ringraziamento speciale per la mia squadra. Senza di voi non c lavrei

mai fatta di certo non ho imparato a formattare un documento word ma ho capito che niente impossibile se

intorno a me ci sono persone che mi vogliono bene

.E quindi grazie a...

Lenuccia.. per ogni tua singola parola

Igor.. ino.. per i tuoi consigli.. e anche se ogni cena da noi era un modo per lamentarmi sappi che ogni sera era

un piacere aspettarti


6/87


7/87

Table of contents

vi

TABLE OF CONTENTS

Page

ABSTRACT............................................................................................................................................. ii

LIST OF TABLES ................................................................................................................................ xiii

LIST OF SYMBOLS ............................................................................................................................ xiv

1 INTRODUCTION ........................................................................................................................... 1

1.1 General ..................................................................................................................................... 1

1.2 Literature review ...................................................................................................................... 2

1.3 Relationships between Force-Displacement (F-) and Moment-Curvature (M-)................. 32 EXPERIMENTAL OBSERVATIONS ........................................................................................... 6

2.1 Description ............................................................................................................................... 6

2.2 Description of test .................................................................................................................... 6

2.3 Analysis procedure ................................................................................................................... 7

2.4 Experimental results ............................................................................................................... 13

3 ANALYTICAL PREDICTIONS ................................................................................................... 16

3.1 Numerical model .................................................................................................................... 16

3.2 Numerical results ................................................................................................................... 18

3.3 Influence of the Aspect ratio, the Section ratio and the axial load ratio on the alpha-factor. 23

3.4 Computation of the yielding displacement ............................................................................ 30

3.5 Alpha- factor as a function of curvature ductility .................................................................. 35

4 SMALL DISPLACEMENT NONLINEAR TIME-HISTORY ANALYSES ............................... 40

4.1 Description ............................................................................................................................. 40

4.2 Accelerograms ....................................................................................................................... 40


8/87

Table of contents

vii

4.3 Ground- Motion Time Histories ............................................................................................ 40

4.4 Response Spectra ................................................................................................................... 42

4.5 Ground Motion Spectra Comparison and Comment .............................................................. 43

4.6 Modelling ............................................................................................................................... 44

4.7 Case-study: structural periods ................................................................................................ 45

5 CONCLUSIONS ........................................................................................................................... 64

6 APPENDIX .................................................................................................................................... 66

7 REFERENCES .............................................................................................................................. 69


9/87

List of figures

viii

LIST OF FIGURES

Page

Figure 1.1.Takeda hysteresis model Ref:Hysteresis Models of Reinforced Concrete for Earthquake

Response Analysis by Otani [May 1981]................................................................................................. 2

Figure 1.2. Effect of element slenderness on unloading displacements for =0.50 ................................ 5

Figure 2.1. The experimental specimen used for the present work is reported to show the method used

for the computation of the alpha-factor. .................................................................................................. 6

Figure 2.2. Geometrical characteristics and reinforcement detailing (in mm) of pier TP-01 Ref:

website of the Kawashima Laboratory (http://seismic.cv.titech.ac.jp) .................................................... 7

Figure 2.3. The time-history of lateral displacement and the cyclic loading correspondence of pier TP-

01. ............................................................................................................................................................ 7

Figure 2.4. Hysteresis loop of the TP-01st : the loading cycles in black are used for the computation

of alpha-factor. ......................................................................................................................................... 8

Figure 2.5. Correction of initial force of the loading test ........................................................................ 8

Figure 2.6. Estimation of the yielding force Fy and displacement Dy of the case study. ...................... 9

Figure 2.7. Definition of maximum and unloading force and displacement amplitude....................... 10

Figure 2.8. Alpha factor and displacement ductility plot consideringFunlequal to 0.50Fm................ 11

Figure 2.9. Alpha factor and displacement ductility plot consideringFunlequal to 0.25Fm................ 11

Figure 2.10. Alpha factor and displacement ductility plot considering Funl equal to 0.50 Fm ............. 12

Figure 2.11. Alpha factor and displacement ductility plot considering Funlequal to 0.25 Fm............... 12

Figure 2.12. Alpha factor and displacement ductility plot. .................................................................. 15


10/87

List of figures

ix

Figure 3.1. Idealization of curvature distribution [Ref: Priestley, M.J.N. Calvi G.M. Kowalsky M.J.

(2007)]. .................................................................................................................................................. 17

Figure 3.2. Effect of P-delta on lateral resistence [Ref: Priestley, M.J.N. Calvi G.M. Kowalsky M.J.

(2007)] ................................................................................................................................................... 18

Figure 3.3. Cyclic loading response: comparison between the experimental and numerical results (TP-

01). ......................................................................................................................................................... 19

Figure 3.5. Value of yielding displacement and Force in the numerical model (TP-01) ...................... 19

Figure 3.6. Alpha factor and displacement ductility comparison between the experimental and

numerical results (TP-01) ...................................................................................................................... 20

Figure 3.7. Geometrical characteristics and reinforcement detailing (in mm) for TP-31 and TP-32 -

Ref: website of the Kawashima Laboratory (http://seismic.cv.titech.ac.jp) .......................................... 20

Figure 3.8. The time-history of lateral displacement for TP-31 a) and TP-32 b) - Ref: website of the

Kawashima Laboratory (http://seismic.cv.titech.ac.jp) ......................................................................... 21

Figure 3.9. Cyclic loading response: comparison between the experimental and numerical results (TP-

31). ......................................................................................................................................................... 21

Figure 3.10. Yielding displacement and Force in the numerical model (TP-31). ................................. 21

Figure 3.11. Alpha factor and displacement ductility comparison between the experimental andnumerical results (TP-31) ...................................................................................................................... 22

Figure 3.12. Cyclic loading response: comparison between the experimental and numerical results ... 22

Figure 3.13. Yielding displacement and Force in the numerical model (TP-32). ................................. 22


numerical results (TP-32) ...................................................................................................................... 23

Figure 3.15. The aspect ratioH/dand the section ratio used in the numerical computation. ............... 24

Figure 3.16. Cyclic loading response used to obtain to numerical results for H/d=3; H/d=7and

H/d=10. .................................................................................................................................................. 24

Figure 3.17. Alpha factor as a function of the displacement ductility: comparison for H/d=3; H/d=7and

H/d=10. .................................................................................................................................................. 25

Figure 3.18. The time histories of the lateral displacement for the different section ratio ................... 26

Figure 3.18. Cyclic loading response for numerical analyses forL/d=1;L/d=3and;L/d=7 andL/d=10.

............................................................................................................................................................... 27


11/87

List of figures

x

Figure 3.19. Alpha factor as a function of the displacement ductility: comparison from numerical

analyses for l/d=1; l/d=5; l/d=7; L/d=10. ............................................................................................... 27

Figure 3.20. The axial load used in the numerical computation. .......................................................... 28

Figure 3.21. Cyclic loading response: comparison between the experimental results for different values

of the axial load ratio. ............................................................................................................................ 28

Figure 3.22. Alpha factor as a function of the displacement ductility: comparison obtained from

numerical analyses for =0; =0.05; =0.10; =0.15; =020 ............................................................... 29

Figure 3.23. Comparison a) Aspect ratio , b) Section ratio and c) Axial load ratio. ............................ 29

Figure 3.24. Yielding displacement for analytical models with H/d=3; H/d=7and H/d=10 computed

according to method 1 and method 2 ..................................................................................................... 31

Figure 3.25. Yielding displacement obtained for analytical models with a) l/d=1; b) l/d=5 according to

both methods, c1) l/d=7; d1) L/d=10 according to method 1 and c2) l/d=7; d2) L/d=10 according to the

method 2. ............................................................................................................................................... 32

Figure 3.26. Alpha factor as a function of the displacement ductility: comparison relative to analytical

models for l/d=1; l/d=5; l/d=7; L/d=10, where the yielding displacement is computed according to

method 2. ............................................................................................................................................... 33

Figure 3.27. Yielding displacement relative to a1) =0.0; according to method 1; a2) =0.0; according

to method 2; b) =0.05, c) =0.10, d) =0.15, e) =0.20 according to both methods. ........................ 34

Figure 3.28. Alpha factor as function of the displacement ductility: comparison relative to analytical

models for =0; =0.05; =0.10; =0.15; =020, where the yielding displacement is computed

according to method 2. ........................................................................................................................... 35

Figure 3.29. Alpha factor as a function of the curvature ductility: comparison for analytical results for

H/d=3; H/d=7and H/d=10. ..................................................................................................................... 37


l/d=1; l/d=5; l/d=7; L/d=10. ................................................................................................................... 38

Figure 3.31. Alpha factor as function of the curvature ductility: comparison relative to numerical

results for =0; =0.05; =0.10; =0.15; =020. ................................................................................. 38

Figure 4.1. Time histories of earthquake ground motions .................................................................... 41

Figure 4.2. Linear elastic response spectra for the LA09 record. ......................................................... 42

Figure 4.3. Linear elastic response spectra for the LA19 record. ......................................................... 42

Figure 4.4. Linear elastic response spectra for the EC record. ............................................................. 43


12/87

List of figures

xi

Figure 4.5.Comparison of the linear elastic response spectra for the 3 ground motions use. Damping is

5% of critical .......................................................................................................................................... 43

Figure 4.6. Modeling SDOF column and spring ................................................................................... 44

Figure 4.7. Modified Takeda Model Ref: Ruaumoko 2D, Appendix [Carr, 2004] .......................... 44

Figure 4.8. Effect of using different alpha ratios - SDOF: H/Lp=13 LA09....................................... 47

Figure 4.9. Effect of using different alpha ratios - SDOF: H/Lp=15 LA09....................................... 47

Figure 4.10. Effect of using different alpha ratios - SDOF: H/Lp=13 ELCE .................................... 49

Figure 4.11. Effect of using different alpha ratios - SDOF: H/Lp=15 ELCE ................................... 49

Figure 4.12. Effect of using different alpha ratios Ar=5 H1=15m; H2=30m; H3=45m ................. 51

Figure 4.13. Effect of using different alpha ratios Ar=10 H1=15m; H2=30m; H3=45m ............... 51






Figure 4.19. Effect of using different alpha ratios Ar=10 H1=15m; H2=30m; H3=45m .............. 54


Figure 4.21. a) Comparison of displacements of Column and SDOF spring models; b) Comparison of

hysteresis loop in terms of Force and displacement of Column and SDOF spring models - System 1

=0 ......................................................................................................................................................... 57


hysteresis loops in terms of Force and displacement of Column and SDOF spring models - System 1

=0.5. ..................................................................................................................................................... 57



=0 ......................................................................................................................................................... 58



=0.5. ..................................................................................................................................................... 58


13/87

List of figures

xii


hysteresis loop s in terms of Force and displacement of Column and SDOF spring models - System 3

a=0 ......................................................................................................................................................... 59


hysteresis loops in terms of Force and displacement of column and SDOF spring models - System 3

=0.5. ..................................................................................................................................................... 59

Figure 4.27. a) Time-history response in terms of a)Moment-Curvature and b) Force-Displacement

for ........................................................................................................................................................... 60

Figure 4.28. Hysteresis loop in terms of Moment and curvature relative to SDOF column ................. 62


14/87

List of tables

xiii

LIST OF TABLES

Page

Table 2.1. Identifications numbers of the cyclic test on the bridge piers Ref: website of the

Kawashima Laboratory (http://seismic.cv.titech.ac.jp) ......................................................................... 14

Table 4.1. Ground motion characteristic parameters. ........................................................................... 41

Table 4.2. Height and Plastic hinge used in SDOF column .................................................................. 45

Table 4.3. Moment of Inertia corresponding to different undamped natural periods of the SDOF

column. .................................................................................................................................................. 45

Table 4.4. Height and Plastic hinge used in SDOF column .................................................................. 50

Table 4.5. Moment of Inertia estimated fixing the fundamental period of the SDOF column. ............ 50

In the following section the SDOF systems having the properties indicated in the Table 4.6 with

ductility equal to 4 and damping equal to zero are analyzed. ................................................................ 55

Table 4.7. Properties of the SDOF column. .......................................................................................... 55


15/87

List of symbols

xiv

LIST OF SYMBOLS

Kun Unloading stiffness

ki Initial stiffness

ky Yielding stiffness

E Young's modulus of elasticity

Iunl(i) Unloading moment of inertia for each cycle

Iin Initial moment of inertia

f*M-(i) EIun(i)/ EIin

Lp Plastic hinge length

H Height of column

Dunl Unloading displacement

Dm Maximum displacement

Dy Yielding displacement

Dt Total displacement demand

De Elastic displacement component

Dp Plastic displacement component

kr(i) unloading stiffness of the loading cycle i

Lsp strain penetration length


16/87

List of symbols

xv

yielding strengthdbl diameter of the longitudinal reinforcement

fc0 unconfined concrete strength in compression

ft0 concrete strength in tension

kc confinement factor

fcc confined concrete strength in compression

f* the ratio between kr(i)the unloading stiffness of the loading cycle i

and the unloading stiffness of the system after reaching the first

maximum displacement amplitude when the ductility is greater than

one

Ec0 Concrete Youngs modulus of elasticity is estimated, according to

Priestley et al. [1996]

Feq,max Equivalent maximum force amplitude for each cycle without P-

delta effect

P Axial load

d section depth

L section length

Axial load ratio

Ag gross section

a*(i) logf*/log(1/(i)) Unloading stiffness degradation parameter in

terms of the displacement ductility

a*M-(i) Unloading stiffness degradation parameter in terms of the curvature

ductility

co unconfined concrete strain

cc confined concrete strain

Displacement ductility

Curvature ductility


17/87

List of symbols

xvi

(i) Displacement ductility for each cyclei

(i) Curvature ductility for each cycle i

Unloading stiffness degradation parameter

t Plastic curvature

e Elastic curvature

unl Unloading curvature

y Yielding curvature

m Total curvature

m(i) Maximum curvature for each cyclei


18/87

Chapter 1: Introduction

1

1 INTRODUCTION

1.1 General

Concrete structures experience a reduction in stiffness as function of the ductility. Recent

work by Tuan H. P., Sullivan T. J., Calvi G. M. et al (2008) highlighted a potentialinconsistency in the use of the unloading factor in lumped plasticity non-linear time history

analyses. For this reason the goal of the present research is to undertake a review of the

behaviour of RC structures, considering how the unloading stiffness varies as a function of the

ductility and other characteristics of the structure.

The scope of the work is:

To undertake a literature review on the hysteresis behaviour of concrete structure

(introduction)

To review experimental results available from the Kawashima Laboratory of the

Tokyo Institute of Technology (chapter 2)

To develop models of various single-degree-of-system (SDOF) oscillators in

SeismoStruct to simulate the hysteretic behaviour observed in experimental analyses

(chapter 3)

To investigate the behaviour of the unloading stiffness as a function of the

displacement ductility of the SDOF systems modeled in SeismoStruct with different

section ratio, aspect ratio and axial load ratio (chapter 3)

To create a lumped-plasticity model , that follows the unloading rules defined by

Takeda for hysteresis cycles, to determine the behaviour of the unloading stiffness as a

function of the curvature ductility of SDOF systems having different section ratio,

aspect ratio and axial load ratio (chapter 3)

To model an equivalent SDOF spring having a hysteretic force-displacement

behaviour that replicates the cantilever column defined by a hysteretic moment-


19/87


2

curvature relationship and compare the responses of the two models with different

unloading stiffnesses.

Make conclusions & recommendations.

1.2 Literature review

The Takeda hysteresis model was developed by Takeda, Sozen and Nielsen [1970], Otani

[1981] and Kabeyasawa, Shiohara, Otani, Aoyama [1983]to represent the force-displacement

hysteretic properties of RC structures.

The Takeda model according to Otani (1981) includes (a) stiffness changes at flexural

cracking and yielding, (b) rules for inner hysteresis loops inside the outer loop, and (c)

unloading stiffness degradation with deformation. The hysteresis rules are extensive and

comprehensive (Figure 1.1). In this work the modified Takeda Model [Ref: Kabeyasawa,

Shiohara, Otani, Aoyama; May 1983. Analysis of the full-scale Seven storey Reinforced

Concrete Test structure] is considered, in which the initial elastic branch up until cracking is

neglected. Instead the response is linear up until yield with the unloading stiffness defined as

(1.1)

in which (Dy, Fy): yielding point deformation and resistance, Dm: maximum deformation

amplitude greater than Dy, : unloading stiffness degradation parameter (normally between0.0 and 0.6).

Figure 1.1.Takeda hysteresis model Ref: Hysteresis Models of Reinforced Concrete for Earthquake

Response Analysis by Otani [May 1981]


20/87


3

According to the following literature reviews:

Reinforced concrete response to simulated earthquakes by Takeda, Sozen and Nielsen,

December [1970]:the unloading stiffness degradation parameter is equal to 0.4.

Hysteresis Models of Reinforced Concrete for Earthquake Response Analysis by Otani [May

1981]: the unloading stiffness degradation parameter, alpha, is normally between 0.0 and 0.5.

Analysis of the full-scale Seven storey Reinforced Concrete Test structure by Kabeyasawa,

Shiohara, Otani, Aoyama [May 1983]: the unloading stiffness degradation parameter is

normally between 0.0 and 0.6.

1.3 Relationships between Force-Displacement (F-) and Moment-Curvature (M-

)

In this section the relationship between Force-Displacement (F-) and Moment-Curvature

(M-) is explained.By specifying a plastic hinge length, Lp, increasing curvature demands on a SDOF cantilever

system with height H can be translated to an equivalent displacement response in accordance

with Equation (1.2).

D D D

H

3 LH

(1.2)

where De is the elastic displacement component, Dp is the plastic deformation component

associated with the inelastic rotation of a plastic hinge, t isthe total curvature at the plastic

hinge location and eis the elastic curvature. Note that the ratio of the total displacement to

the yield displacement (i.e. the displacement ductility demand) can be expressed for a

cantilever in terms of the curvature ductility demand by Equation (1.3).

1 3 1 (1.3)

After reaching a total displacement of t, the Takeda model instructs the structure to unload

with a reduced stiffness given by Equation (1.1).

If we assume, for simplicity, that there is no strain hardening and note that the Takeda model

is specified for NLTHAs in a Moment-Curvature environment, then the elastic curvature

recovered in unloading the structure from a total displacement demand of D t is given by

Equation (1.4).


21/87


4

(1.4)

The ratio of the elastic displacement recovered in unloading to the yield displacement of acantilever is therefore given by Equation (1.5).

3

3

(1.5)

Dividing Equation (1.2) by Equation (1.5), we obtain Equation (1.6) which expresses the ratio

of the total displacement demand to the unloading displacement as a function of the curvature

ductility demand, the ratio Lp/H, and the alpha factor.

1 3 1

(1.6)

This has interesting implications if we consider the displacement ratios predicted for

structures with different ratios of Lp/H, unloading from different levels of ductility demand

As shown in Figure 1.1., for an alpha of 0.5 and for low values of Lp/H (i.e. for tall slenderstructures), the ratio of unloading displacement to the peak displacement reduces below a

value of 1. This is equivalent to saying that a slender structure subject to a big push into the

inelastic range would be predicted to have residual displacements in the opposite direction.

This essentially highlights a potentially serious problem with application of the Takeda model

in a moment-curvature format and accounts for observations made by Sullivan et al (2008).


22/87


5

Figure 1.2. Effect of element slenderness on unloading displacements for =0.50

Before proposing solutions to this matter, the next chapters will consider how the unloading

stiffness varies , as observed in experimental testing.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 5 10 15 20 25

Curvature ductility demand

Ratioofpeakdisplace

mentto

unloadingdisplace

ment

Lp/H = 0.05

Lp/H = 0.10

Lp/H = 0.15

Lp/H = 0.20


23/87

Chapter 2: Experimental observations

6

2 EXPERIMENTAL OBSERVATIONS

2.1 Description

In order to obtain a better understanding of the unloading stiffness of concrete columns, first

of all we study the behavior the unloading coefficient as a function of the displacementductility by examining the experimental results available from the Kawashima Laboratory of

the Tokyo Institute of Technology as described in the following paragraphs.

2.2 Description of test

The validity of the value of the alpha-factor is carried out here by considering experimental

results available from the Kawashima Laboratory of the Tokyo Institute of Technology.

Several results of experimental tests for the study of the cyclic behavior of reinforced concrete

bridge piers are available at the website of the Kawashima Laboratory

(http://seismic.cv.titech.ac.jp). These experiments involved the simultaneous application ofvertical and horizontal loads to reinforced concrete specimens. Figure 2.2 depicts the

experimental set-up and the corresponding simplified structural model.

Figure 2.1. The experimental specimen used for the present work is reported to show the method used for

the computation of the alpha-factor.


24/87


7

The data taken is identified with the number TP-01; the general geometrical characteristics, as

well as the reinforcement detailing, are presented in Figure 2.2. The cylinder strength of

concrete is 35.9 MPa and the yield strength of the longitudinal reinforcement is 363 MPa. The

vertical load is constant and equal to 163 kN. Figure 2.3 includes the time-history of lateral

displacement and the cyclic loading and corresponding response in terms of Force-

Displacement.

Figure 2.2. Geometrical characteristics and reinforcement detailing (in mm) of pier TP-01 Ref: website

of the Kawashima Laboratory (http://seismic.cv.titech.ac.jp)

-30

-20

-10

0

10

20

30

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5

-200

-150

-100

-50

0

50

100

150

200

-80 -60 -40 -20 0 20 40 60 80Force(kN)

Dis lacement mm Figure 2.3. The time-history of lateral displacement and the cyclic loading correspondence of pier TP-01.

2.3 Analysis procedure

Using the hysteresis loop defined by the force-displacement relationship in Figure 2.3 the

alpha factor for each loading cycle ican be defined as in formula (2.1)

i log KiK

log 1i

(2.1)


25/87


8

where the displacement ductility is given by equation (2.2)

i Dmi

Dy

(2.2)

In order to avoid considering systems prone to shear failure the values of alpha-factors are

not consideared when the lateral resistence drops below 80% of the yileding force. For this

reason the unloading stiffness degradetion parameters are computed for the loading cycles of

Figure 2.6 in black and neglected for the ones in grey.

Figure 2.4. Hysteresis loop of the TP-01st : the loading cycles in black are used for the computation of

alpha-factor.

As shown in Figure 2.5 at the beginning of cyclic loading the initial force is not equal to zero

(grey line), so to have a null value of it, the hysteresis loop has been offsetted (black line).

Figure 2.5. Correction of initial force of the loading test

-160-140-120-100-80-60-40-20

020406080

100120140160

-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90Force(kN)

Displacement (mm)

-140

-110

-80

-50

-20

10

40

70

100

130

160

190

-6 -3 0 3 6 9 12F

orce(kN)

Displacement (mm)


26/87


9

Figure 2.6. Estimation of the yielding force Fy and displacement Dy of the case study.

The yielding stiffness

/for the case study presented in Figure 2.6, is computed

through a trend line used to approximate the first cycle until the yielding point of the system.

It is not always very simple to evaluate the yielding displacement, because the structure

doesnt yield from the first cycle.

In this specific case Fy=157 kN; Dy=8mm.

-140

-110

-80

-50

-20

10

4070

100

130

160

190

-6 -3 0 3 6 9 12Force(kN)

Displacement (mm)


27/87


10

The unloading stiffness for each cycle ican be obtained from equation (2.3) between themaximum (m) and unloading (unl) force and deformation amplitude.

Ki Fi FiDi Di (2.3)

Figure 2.7. Definition of maximum and unloading force and displacement amplitude.

Clearly there is some uncertainty in the exact unloading stiffness due to the very non-linear

response obtained with Fibre-Element, distributed plasticity modelling. For this reason, two

different unloading stiffness definitions were considered. As is shown in Figure 2.7, the

unloading force considered in the first case is taken equal to 0.50 and in the second case to

0.25 of the maximum force amplitude.


28/87


11

The following figures indicate the values of alpha factor as a function of different values of

displacement ductility.

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0,16

2 2,5 3 3,5 4 4,5 5 5,5

alpha-factora

ductility

Figure 2.8. Alpha factor and displacement ductility plot considering Funlequal to 0.50 Fm

0

0,05

0,1

0,15

0,2

0,25

0,3

2 2,5 3 3,5 4 4,5 5 5,5

alpha-factora

ductility


Analyzing the data it observed that after reaching a maximum displacement Dm in the first

cycle of loading, the structure unloads with stiffness greater than the initial one, therefore Kr

doesnt depend on the alpha-factor. Observing such independence between alpha-factor, Krfor ductility values less than 3.5 in Figure 2.8 and 2.0 in Figure 2.9, gives negative values of

the alpha-factor. The negative values are neglected in the plots, and are assumed equal to

zero.

The value of the alpha-factor starts to increase for higher values of ductility (=0.11. in Figure

2.8 and =0.28 in the Figure 2.9 form=5.1).


29/87


12

0

0,05

0,1

0,15

0,2

0,25

0,3

0,350,4

2 2,5 3 3,5 4 4,5 5 5,5

alpha-factora

*

ductility

Rather than consider the ratio of the unloading to initial stiffness, it appears more reasonable

to consider the ratio at the unloading stiffness (at cycle i) to the initial unloading stiffness (at

yield). For this reason, in the following figures the alpha-factor is computed by using equation

(2.4)

logflog 1i (2.4)

wheref*is defined equal to the ratio betweenKr(i)the unloading stiffness of the loading cycle

loading i and the unloading stiffness of the system after reaching the first maximum

displacement amplitude for which the ductility is greater than one.

(3,1)

Figure 2.10. Alpha factor and displacement ductility plot considering Funl equal to 0.50 Fm

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

2 2,5 3 3,5 4 4,5 5 5,5

alpha-factora

*

ductility


Although for Figure 2.10 and Figure 2.11 the previous considerations about alpha-factor are

still valid for low ductility values, it has to be noted that the values obtained from the new

definition of alpha-factor have a trend to remain constant as a function of ductility as


30/87


13

compared to the ones observed calculated from equation (2.1), which is more in accordance to

the Takeda model, that proposes the same unloading stiffness degradation parameter (between

0.0 and 0.6) to compute the unloading stiffness in consecutive loading cycles. Moreover with

this formulation of the alpha-factor, the initial stiffness of the reinforced concrete bridge piers

for m=1 under cyclic loading can be modelled successfully.

For these reasons, for the experimental results reported in the next section, the alpha-factor is

computed from equation (2.4) considering an unloading force equal to 0.25 of the maximum

force amplitude in each cycle.

2.4 Experimental results

Table 2.1 indicates the legend used in the plot (Figure 2.12) computed to evaluate the

dependence of the alpha-factor (calculated by formula (2.4)) on the ductility. The same legend

classifies the general geometrical characteristics of the sections analyzed and the several

identifications of the cyclic test on the bridge piers as they are referred to on the website of

the Kawashima Laboratory.

For each test examined, the axial load ratio is computed as (=P/fcc*Ag, wherePis the axial

load, fccthe confined compressive strength andAgthe gross section area) to be equal to 0.04.


31/87


14

Table 2.1. Identifications numbers of the cyclic test on the bridge piers Ref: website of the Kawashima Laboratory

(http://seismic.cv.titech.ac.jp)

Test # ID Number Section Section Size

(mm)TP-001 Square 400400

TP-002 Square 400400





TP-007 Oval 400900

TP-008 Oval 400900

TP-009 Oval 400901

TP-010 Square 400400TP-011 Square 400400





























Four reinforced concrete specimens were loaded to

evaluate the seismic performance of reinforced

concrete arch ribs with hollow section.


evaluate the effectiveness of densely arranged spiral

confinement zone in reinforced concrete section

columns with hollow section

Sixreinforcedconcrete specimens withsame sizeand

strength were loaded under different loading

hystereses to evaluate the effect of loading hystereses.

8

9

Six reinforced concrete specimens were loaded to

evaluate the seismic performance of reinforced

concrete bridge columns under bi-directional flexural

loading.

Six reinforced concrete specimens were loaded to

evaluate the seismic performance of C-bent columns

under bi-lateral seismic excitation based on a hybrid

loading test.

1

2

3

4

5

6

7

Threereinforcedconcrete specimens withovalsection

for bridge columns were loaded to evaluate the

confinement effect of interlocking hoops.


evaluate the effect of a longitudinal reinforcement

diameter on a plastic hinge length.


evaluate the effect of aspect ratio on a plastic hinge

length.


evaluate the seismic performance under varyingaxialforce.


32/87


15

Figure 2.12. Alpha factor and displacement ductility plot.

The results are very interesting and from the plot (Figure 2.12) the variation of alpha-factor

as a function of ductility can be understood.

It can be noted that for low ductility alpha-factor assumes the highest values (a=1.04 for

m=1.30) obtained from formula (2.4). The exception mentioned happens for m-range from 1

to 2.5, whenf* is very close to 1 (obviously lower than unity).

In addition it can be highlighted that the alpha-factor is always below 0.5 when the ductility is

from 2.5 to 10. Moreover it is observed for each cyclic test on the bridge piers that the alpha-

factor tends to remain constant which is in accordance with Takeda Hysteresis Model, that

proposes the same (normally between 0.0 and 0.6) to compute the unloading stiffness in

the consecutive loading cycles.

Finally, it is observed that the increasing values of the ductility yield a regular increase ofalpha-factor, expect for low mwhen aassumes very high values when constant a-notion is no

longer valid. At low values of ductility, the alpha value appears to be sensitive to the exact

value of ductility chosen.

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

1,1

1 2 3 4 5 6 7 8 9 10

alpha-factora

ductility

test 1

test 2

test 3

test 4

test 5

test 6

test 7

test 8

test 9


33/87

Chapter 3: Analytical predictions

16

3 ANALYTICAL PREDICTIONS

3.1 Numerical model

Numerical models were developed and analyzed in SeismoStruct [Seismo soft 2008] for the

same geometrical and loading characteristics of the experimental test TP-01 previouslypresented. The 1.25m high pier was modeled by four finite elements, the first two from the

bottom are 1/6 and the last two 1/3 of the column height. Two integration sections per element

were used (Gauss quadrature), each one containing around 250 integration points. The column

is also modeled with the length of the plastic hinge Lpover which strain and curvature are

considered to be equal to the maximum value at the base column. The plastic hinge length

incorporates the strain penetration length Lsp as shown in Figure 3.1. Further, the curvature

distribution higher up the column is assumed to be linear, in accordance with the SDOF

model being examined.

The strain penetration length, Lspmay be taken as:

0.022 (3.1)

Where fye and dbl are the expected yield strength and diameter of the longitudinal

reinforcement.


34/87


17

Figure 3.1. Idealization of curvature distribution [Ref: Priestley, M.J.N. Calvi G.M. Kowalsky M.J.

(2007)].

For the constitutive relation for concrete in compression, the well known model of Mander et

al. [1988] was adopted, with the improvements later introduced by Martnez-Rueda and

Elnashai [1997]. A linear behavior for the concrete in tension was assumed, followed by an

abrupt reduction after exceeding the tension resistance. This is achieved by setting ft0 = 0.34

fc01/2.where fc0 is the unconfined concrete strength in compression ([Vinagre, 1997], [Lin and

Scordelis, 1975]). Youngs modulus of elasticity for concrete is estimated according to

Priestley et al. [1996], asEc0 = 4700fc01/2. In order to account for the effect of confinement due

to the presence of stirrups, the compressive strength and the corresponding strain were

modified using thefollowing confinement factor (kc):

(3.2)

1 5 1 (3.3)

The unconfined concrete strain (co) corresponding to the maximum compression strength is

taken as 0.002. while the value for the confinement factor kc was 1.161 for the confined

concrete and 1.0 for the concrete cover.

The model of Giuffr, Menegotto and Pinto ([Giuffr and Pinto, 1970], [Menegotto and Pinto,

1973]) was applied for the longitudinal reinforcement, along with the subsequent

improvements introduced by Filippou et al. [1983]. In order to account for the cyclic

degradation of steel strength depicted by the experimental results without changing the steel

model, a negative value of the parameter a3 was considered [Ref: Seismostruct help,


35/87


18

Seismosoft (2008)]. Youngs modulus of elasticity for steel was taken equal to 205 GPa,

while the hardening and cyclic behavior parameters were calibrated in order to better

reproduce the experimental results: b = 0.015, R0 = 20, a1 = 19.3. a2 = 0.15, a3 = -0.025 and a4

=15.

3.2 Numerical results

Before presenting the comparison of the unloading stiffness during the cyclic loading between

numerical and experimental results, it is important to consider that the single-degree-of-

freedom systems experience gravity-load-induced overturning moments in addition to those

resulting from lateral inertia forces. Therefore according to the behavior of the structure with

reference to Figure 3.2 it can be seen that for a SDOF system with a given level of lateral

strength, P-delta effects effectively cause a reduction in the lateral resistance. The reduced

effective stiffness implies that the maximum and unloading force amplitude obtained for each

cyclic loading (Figure 2.7) estimated with a numerical model in SeismoStruct, are affected bythe P-delta effects. It becomes necessary to compute through equations (3.4) and (3.5) the

equivalent maximum and unloading lateral force amplitude for each cyclic loading.

, (3.4)

Feq,unliFunliPDunliH

(3.5)

Figure 3.2. Effect of P-delta on lateral resistence [Ref: Priestley, M.J.N. Calvi G.M. Kowalsky M.J.

(2007)]

In Figure 3.3, which compares the numerical and experimental results, it can be observed that

for each loading cycle and maximum resistance, the loading and unloading stiffnesses are

very similar.


36/87


19

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

20

40

6080

100

120

140

160

180

-40 -30 -20 -10 0 10 20 30 40Force(kN)

Displacement (mm)

Numerical results

Experimental results

Figure 3.3. Cyclic loading response: comparison between the experimental and numerical results (TP-01).

Moreover the values of alpha-factors computed from equation (2.4) for each cycle of the

hysteresis loop obtained from the numerical results are very close to the unloading stiffness

degradation *previously calculated with the experimental results (Figure 3.5) and as it was

noted previously from the data of the cyclic test on the bridge piers, they have a trend to

remain constant with increasing values of ductility, which is in accordance to the Takeda

Model.

In the following Figure 3.5 the value of the yielding displacement evaluated according to

method explained in the section 2.3 is reported.

Figure 3.4. Value of yielding displacement and Force in the numerical model (TP-01)

-140

-110

-80

-50

-20

10

40

70

100

130

160

190

-6 -3 0 3 6 9 12F

orce(kN)

Displacement (mm)

dy= 8mm; Fy=157 kN


37/87


38/87


21

a) -50-40

-30

-20

-10

0

10

20

30

40

50

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7

b) -40

-30

-20

-10

0

10

20

30

40

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7

Figure 3.7. The time-history of lateral displacement for TP-31 a) and TP-32 b) - Ref: website of the

Kawashima Laboratory (http://seismic.cv.titech.ac.jp)

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

20

40

6080

100

120

140

160

180

-50-45-40-35-30 -25 -20-15-10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60Force(kN)

Displacement (mm)

Numerical results


Figure 3.8. Cyclic loading response: comparison between the experimental and numerical results (TP-31).

Figure 3.9. Yielding displacement and Force in the numerical model (TP-31).

-200

-150-100

-50

0

50

100

150

200

-20 -15 -10 -5 0 5 10 15 20

Force(kN)

Displacement (mm)

dy=8 mm; Fy=163 kN


39/87


22


numerical results (TP-31)

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

20

40

60

80

100

120

140

160

180

-60-55-50-45-40-35-30 -25-20 -15-10 -5 0 5 10 15 20 25 30 35 40 45 50Force(kN)

Displacement (mm)

Numerical results


Figure 3.11. Cyclic loading response: comparison between the experimental and numerical results

(TP-32).

Figure 3.12. Yielding displacement and Force in the numerical model (TP-32).

0

0,1

0,2

0,3

0,4

0,5

0,6

1 1,5 2 2,5 3 3,5 4 4,5 5

alpha-factora

ductility

Experimental resultsNumerical results

-150

-100

-50

0

50

100

150

-25 -20 -15 -10 -5 0 5 10 15 20

Force(kN)

Displacement (mm)

dy=9.2 mm; Fy=104 kN


40/87


23

Figure 3.13. Alpha factor and displacement ductility comparison between the experimental and numerical

results (TP-32)

Figure 3.8 to Figure 3.13 show the cyclic response according to numerical and experimental

results and the alpha-factor trend as a function of the displacement ductility corresponding to

the tests identified with TP-31 and TP32 respectively.

The previous considerations about the alpha-factor explained for test TP-01 are still valid, and

therefore it can be concluded that the results obtained from the comparison between the

experimental and the numerical results are satisfactory.

3.3 Influence of the Aspect ratio, the Section ratio and the axial load ratio on the alpha-

factor.

After obtaining satisfactory results from the comparison between the experimental and the

numerical results, in this section the behavior of the alpha-factor is investigated through

numerical computation. The goal is to evaluate how the values of the alpha-factor change with

varyingH/d,aspect ratio,andL/d, section ratio of the pier identified with the number TP-31.

in Figure 3.14, whereHis the column height, dthe section depth andL is the section length

of the pier.

In each numerical model studied an axial load is applied on the pier so that the axial load ratio

(=P/fcc*Ag, where P is the axial load, fccconfined compressive strength and Ag the gross

section area) is equal to 0.05.

Moreover when L/d changes, the aspect ratio is fixed (H/L=3) and the longitudinal

reinforcement ratio is fixed to 1.5% of the gross section area Ag.

0

0,1

0,2

0,3

0,4

0,5

0,6

1 2 3 4 5

alpha-factor

a

ductility

Experimenta l results

Numerica l results


41/87


24

Figure 3.14. The aspect ratioH/dand the section ratio used in the numerical computation.

For the time-histories of cyclic loading in the numerical model when H/d=7 and H/d=10 the

displacement in each time step used for H/d=3. shown in Figure 3.7a is increased as the

square of H/d so that the structure can reach displacement ductility values around 6.

The following plots (Figure 3.15) show the cyclic loading response for the numerical models

with different values of the aspect ratioH/d.

H/d=3. =0.05 H/d=7; =0.05

-50-40

-30

-20

-10

0

10

20

30

40

50

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

Force(kN)

Displacement (mm)

H/d=10, =0.05

Figure 3.15. Cyclic loading response used to obtain to numerical results forH/d=3;H/d=7andH/d=10.

H

d

H/d=3

H/d=7

H/d=10

L

d

3L

5L

10L

d

d

d

- =0.05 =0.05-H/L=3-Aspect ratio Sectional ratio

load direction

-160

-140

-120

-100-80

-60

-40

-20

0

20

40

60

80

100

120

140

160

-50 -40 -30 -20 -10 0 10 20 30 40 50Force(kN)

Dis lacement (mm)-500

-400

-300

-200

-100

0

100

200

300

400

500

-140-120-100 -80 -60 -40 -20 0 20 40 60 80 100 120 140Force(kN)

Dis lacement mm


42/87


25

The alpha-factor is estimated in the three models according to the formula (2.4).

Figure 3.16. Alpha factor as a function of the displacement ductility: comparison for H/d=3; H/d=7andH/d=10.

In Figure 3.16 the plot shows that increasing the slenderness of the structure, or to be more

precise changing the aspect ratio from 3 to 10, the alpha-factor does not change a function of

the displacement ductility and instead assumes values of around 0.07. Therefore observing a

constant trend of the alpha-factor it can be concluded that the aspect ratio doesnt have a

significant influence on the cases studied.

0

0,05

0,1

0,15

0,2

0,25

0,3

0,350,4

0,45

0,5

2 2,5 3 3,5 4 4,5 5 5,5 6

alpha-factora

ductility

H/d=3 H/d=7 H/d=10


43/87


26

The time histories of the lateral displacement used to investigate the effect of the section ratio

on the alpha factor are shown in the Figure 3.17 for L/d=3.L/d=5,L/d=10 and in the Figure

3.7a forL/d=1.

L/d=3

L/d=5

L/d=10

Figure 3.17. The time histories of the lateral displacement for the different section ratio

-150

-50

50

150

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7

-1000

-500

0

500

1000

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 6,5 7

-1000

-500

0

500

1000

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5


44/87


27

The following plots (Figure 3.18) show the cyclic loading response of the numerical models

with different values of the section ratioL/d.

-160

-140

-120

-100

-80

-60

-40

-20

0

20

40

60

80

100120

140

160

-50 -40 -30 -20 -10 0 10 20 30 40 50Force(kN)

Displacement (mm) L/d=1;H/d=3; =0.05 L/d=5;H/d=3; =0.05

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

-700-600-500-400-300-200-100 0 100 200 300 400 500 600 700Force(kN)

Displacement (mm) L/d=7; H/d=3; =0.05 L/d=10; H/d=3; =0.05

Figure 3.18. Cyclic loading response for numerical analyses forL/d=1;L/d=3and;L/d=7 andL/d=10.

The alpha-factor is estimated in the three models according to the formula (2.4).

Figure 3.19. Alpha factor as a function of the displacement ductility: comparison from numerical analyses

for l/d=1; l/d=5; l/d=7; L/d=10.

From the plot in Figure 3.19 it is observed that the values of alpha-factor are very low (less

than 0.15) and demonstrate a slightly increasing slope.

-500

-400

-300

-200

-100

0

100

200

300

400

500

-140-120-100 -80 -60 -40 -20 0 20 40 60 80 100 120 140Force(kN)

Displacement (mm)

-1500

-1200

-900

-600

-300

0

300

600

900

1200

1500

-1000 -800 -600 -400 -200 0 200 400 600 800 1000Force(kN)

Dis lacement mm

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

2,5 3 3,5 4 4,5 5

alpha-factora

ductility

L/d=1 L/d=3 L/d=5 L/d=10


45/87


28

Figure 3.21 shows the plot of cyclic loading response obtained from the numerical

computations of the pier having values of the axial load ratio reported in Figure 3.20. For

each model the aspect ratio (H/d=3) and the section ratio (L/d=1) are fixed.

Figure 3.20. The axial load used in the numerical computation.

The load history of the lateral displacement used to investigate the effect of the axial load

ratio on the alpha factor is shown in the Figure 3.7a.

-200

-150

-100

-50

0

50

100

150

200

-50 -40 -30 -20 -10 0 10 20 30 40 50Force(kN)

Displacement (mm)

=0

=0.05

=0.10

=0.15

=0.20

Figure 3.21. Cyclic loading response: comparison between the experimental results for different values of

the axial load ratio.

-H/L=3-Axial load ratio L/d=1

=0.10=0.15

=0.20

=0.05=0.00


46/87


29

The alpha-factor is estimated in each of the three models according to equation (2.4).

Figure 3.22. Alpha factor as a function of the displacement ductility: comparison obtained from numericalanalyses for =0; =0.05; =0.10; =0.15; =020

From Figure 3.22 it is observed that increasing the axial load on the structure gives higher

values of the alpha-factor.

a) Aspect ratio

b) Section ratio c) Axial load ratio

Figure 3.23. Comparison a) Aspect ratio , b) Section ratio and c) Axial load ratio.

From Figure 3.23 the variation of the alpha factor as a function of aspect ratio and axial load

ratio can be observed.

0

0,1

0,2

0,3

0,4

0,5

0,6

2 2,5 3 3,5 4 4,5 5 5,5 6

alpha-factora

ductility

=0 =0.05 =0.10 =0.15 =0.20

0

0,05

0,10,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

a

lpha-factora

H/d=3 H/d=7 H/d=10

0

0,05

0,1

0,150,2

0,25

0,3

0,35

0,4

0,45

0,5

alpha-factora

L/d=1 L/d=3L/d=5 L/d=10

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

alpha

-factora

=0 =0.05 =0.10 =0.15 =0.20


47/87


30

First of all, Figure 3.20.a and Figure 3.20b record lower values of the alpha-factor compared

to that presented in the Figure 3.20.c.

In particular it is observed that the trend line related to the slenderness of the system could be

considered constant (Figure 3.20.a) while the second one (Figure 3.20.b) related to the sectionratio is slightly increasing with increased values of section ratio.

Figure 3.20.c illustrates that varying the axial load ratio from 0.00 to 0.20 the alpha-factor

tends to increase significantly.

Moreover, it is very interesting to note that the values of the alpha-factor as a function of the

displacement ductility for axial load ratio equal to 0.20 are very close to the unloading

stiffness degradation parameter provided in the Takeda model (usually equal to 0.5), in fact

it even reaches a value of 0.52.

From these considerations it can be said that alpha-factor as a function of displacement

ductility depends in particular on the axial load ratio.

3.4 Computation of the yielding displacement

The section presents the computation of the yielding displacement by two methods.

The first approach (method 1) is explained in paragraph 2.3. where the yielding displacement

is determined using a trend line set by eye, that approximates the necessary loading cycles

to yield the structure. This method has been used in the previous analyses preformed to study

the influence of the aspect ratio, section ratio and axial load ratio on the alpha-factor

The second method (method 2) is introduced by Priesltey, M.J.N. Calvi G.M. Kowalsky M.J.

[2007] (pag 76), through the following equations.

Equation 3.6 is approxiamted for design propose and defines the yielding displacement:

3 (3.6)

The yielding curvature is provided for different section shapes by the following equations for

rectangular concrete column:

2.10 (3.7)


48/87


31

and for concrete walls:

2.00

(3.8)

where y, hcand lware the yield strain of the flexural reinforcement (=fy/Es), the section depth

of the rectangular column and the length of the rectangular walls respectively.

In the numerical models where the goal is to evaluate the influence of the aspect ratio on the

alpha-factor, the yielding displacement obtained was the same for the two different methods.

H/d=3. =0.05 H/d=7; =0.05

H/d=10; =0.05

Figure 3.24. Yielding displacement for analytical models with H/d=3; H/d=7and H/d=10 computed

according to method 1 and method 2

-200

-150

-100

-50

0

50

100

150

200

-15 -10 -5 0 5 10 15

Force(kN)

Dis lacement mm

dy=6 mm; Fy=144 kN

-500

-400

-300

-200

-100

0

100

200

300

400

500

-25 -20 -15 -10 -5 0 5 10 15 20 25

Force(kN)

Dis lacement (mm)

dy=17 mm; Fy=380 kN

-50

-40

-30

-20

-10

010

20

30

40

50

-100 -75 -50 -25 0 25 50 75 100

Force(kN)

Displacement (mm)

dy=49 mm; Fy=39 kN


49/87


32

The following plots are related to the analytical models with different section ratio. It is noted

that by computing the yielding displacement through the two formulations, the obtained

values are the same for L/d=1 and L/d=5, while they are very similar for L/d=7 and L/d=10.

a) b)

L/d=1; H/d=3; =0.05 L/d=5; H/d=3; =0.05

c1) c2)

L/d=7; H/d=3; =0.05

d1) d2)

L/d=5; H/d=3; =0.05

Figure 3.25. Yielding displacement obtained for analytical models with a) l/d=1; b) l/d=5 according to both

methods, c1) l/d=7; d1) L/d=10 according to method 1 and c2) l/d=7; d2) L/d=10 according to

the method 2.

-200

-150

-100

-50

0

50

100

150

200

-15 -10 -5 0 5 10 15

Force(kN)

Dis lacement mm

dy=6 mm; Fy=144 kN

-500

-400

-300

-200

-100

0

100

200

300

400

500

-25 -20 -15 -10 -5 0 5 10 15 20 25

Force(kN)

Dis lacement (mm)

dy=17 mm; Fy=380 kN

-800

-600

-400

-200

0

200

400

600

800

-50 -40 -30 -20 -10 0 10 20 30 40 50

Force(kN)

Dis lacememt mm

dy=32mm; Fy=670 kN

-800

-600

-400

-200

0

200

400

600

800

-50 -40 -30 -20 -10 0 10 20 30 40 50

Force(kN)

Dis lacememt mm

dy=29mm; Fy=670 kN

-1500

-1000

-500

0

500

1000

1500

-100 -75 -50 -25 0 25 50 75 100

Force(kN)

Dis lacement mm

dy= 51 mm; Fy=1250 kN

-1500

-1000

-500

0

500

1000

1500

-100 -75 -50 -25 0 25 50 75 100

Force(kN)

Dis lacement mm

dy= 55 mm; Fy=1250 kN


50/87


33

Observing such differences regarding the yielding displacement, the values of alpha factor

are recomputed according the formula 2.4.

Figure 3.26. Alpha factor as a function of the displacement ductility: comparison relative to analytical

models for l/d=1; l/d=5; l/d=7; L/d=10, where the yielding displacement is computed

according to method 2.

Comparing the plot in the Figure 3.26 with one in the Figure 3.19 it can observed that the

results are very similar. For this reason the previous conclusions about the influence of the

aspect ratio on the alpha-factor are still valid, and therefore it is possible to neglect the small

difference and to take in account only of the value of alpha-factor computed according to the

method 1.

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

2 3 4 5

alpha-factora

ductility

L/d=1 L/d=3 L/d=5 L/d=10


51/87


52/87


35

Figure 3.28. Alpha factor as function of the displacement ductility: comparison relative to analytical

models for =0; =0.05; =0.10; =0.15; =020, where the yielding displacement is

computed according to method 2.

Comparing the plot in the Figure 3.28 with Figure 3.19 it can be observed that the results are

very similar. For that reason the previous conclusions about the influence on the axial load

ratio on the alpha-factor are still valid, and therefore it is possible to neglect the small

difference and to take into account only the value of the alpha-factor computed according to

method 1.

3.5 Alpha- factor as a function of curvature ductility

In order to obtain an equivalent force-displacement response for the intended Takeda model

using a plastic hinge model described in terms of moment and curvature, the alpha-factor is

defined as a function of curvature ductility.

Using lumped-plasticity modeling it could be interesting to define later the hinge

characteristics and initial stiffness in terms of moment-curvature by using a non linear time

history analysis program (e.g. Ruaumoko).

Recalling the alpha-factor definition, first of all it is necessary to define the curvature ductility

for each loading cycle, given by equation (3.9)

(3.9)

Where m(i) and y are the maximum curvature for each loading cycle and the yielding

curvature respectively. The curvatures are known because they depend on the maximum

deformation amplitudeDm(i)and the yielding displacementDydefined in the paragraph 2.3.

0

0,1

0,2

0,3

0,4

0,5

0,6

2 2,5 3 3,5 4 4,5 5 5,5 6

alpha-factora

ductility

=0 =0.05 =0.10 =0.15 =0.20


53/87


36

Dm(i)can also be expressed by equation (3.10)

(3.10)

where H andLp( Lsp, whereLspis defined in the formula (3.1)) are the height and the length

of the plastic hinge of the cantilever system defined with formula (3.11), respectively.

knowing that k=0.08 and Lc is the length of the critical section to the point of contraflexure in

the member,the length of the plastic hinge can be defined by equation 3.11.

2

whileDycan be defined through the following equation

(3.11)

(3.12)

In this section the alpha-factor is defined as a function of the curvature ductility, according tothe following equation

1

(3.13)

where is defined by equation (3.14) specifying that E is the elastic modulus of thesystem,

is the unloading moment of inertia for each loading cycle, formula (3.15), and

Iinis the initial moment of inertia of the single-degree-of-freedom system, formula (3.16).

(3.14)

(3.15)


54/87


37

(3.16)

In equation (3.15) (=Fm*H) and (=Funl*H) are the maximum and theunloading moment for each loading cycle (i) respectively.

The unloading curvature for each cyce of loading , is known because it can beobtained by the unloading deformation amplitude through equation (3.17), defined inFigure 2.7 and also given by the following equation

(3.17)

and the residual curvature is defined by

(3.18)

By defining the alpha-factor aM-as function of the curvature ductility

,the following plots

(Figure 3.29 to Figure 3.31) show how the values of the alpha-factor, formula (3.13), changewhen varying H/d, the aspect ratio and L/d, the section ratio and , the axial load ratio

according to the analyses done for the alpha-factor as a function of the displacement ductility.

0,0

0,20,4

0,6

0,8

1,0

1,2

1,4

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

alpha

-factor*M-

Ductility

H/L=3 H/L=7 H/L=10


H/d=3; H/d=7and H/d=10.


55/87


56/87


39

From this study it is observed that in the Moment-curvature Takeda model, in particular in the

plots related to the aspect ratio and section ratio, the unloading stiffness degradation

parameter could only be given using a non-linear expression, surely not simple to implement.

Therefore this study supports the definition of the alpha-factor introduced by the force-

displacement Takeda Model.


57/87

Chapter 4: Small displacement nonlinear time-history analyses

40

4 SMALL DISPLACEMENT NONLINEAR TIME-HISTORYANALYSES

4.1 Description

The effect of using different alpha ratios is investigated in this section by using non-lineartime-history analyses of single-degree-of-freedom (SDOF) systems. The analysis are carried

out using the computer program Ruaumoko [Carr, 2004]

4.2 Accelerograms

Non linear time-history analyses are performed for three ground motions; two of them are

selected from a suite of historical recordings from magnitude M=6 to M=7.3 earthquakes

which were scaled to match the uniform hazard spectrum for Los Angeles at an uniform

hazard level of 10% probability of exceedence in 50 years (SAC Joint Venture 1997), and the

third one is a historical record for California that have been used extensively in past.

The three ground motions are listed below:

LA09- Landers EQ, 28 Jun. 92. Yermo Fire Station, fault normal component.

LA19- North Palm Springs EQ, 8 Jul. 86, fault normal component.

El Centro- El Centro EQ, 1940, S00E component.

4.3 Ground- Motion Time Histories

The acceleration time-history of the three ground motions is presented indicating the peak

ground acceleration PGA. All the time histories are scaled to a similar intensity.

The peak ground acceleration points are highlighted in the plots and general properties are

also reported in Table 4.1. For every record the PGA value, the time at which it occurs and

other duration properties are reported. Regarding the duration, this considers the first and the

last time at which a threshold acceleration value is crossed in the record, which is taken as

0.05g and 0.10g respectively.


58/87


41

Table 4.1. Ground motion characteristic parameters.

PGA Interval ag>0.05g Interval ag>0.10g

t

[s]

acc.

[g]

ti

[s]

tf

[s]

ti - tf

[s]

ti

[s]

tf

[s]

ti - tf

[s]

LA09 16.30 0.5196 2.96 43.96 41.00 12.30 31.32 19.02

LA19 2.48 1.0190 0.32 22.68 22.36 0.32 14.44 14.12

El

Centro

2.12 0.3483 0.90 26.76 25.86 1.40 26.08 24.68

Figure 4.1. Time histories of earthquake ground motions

As a general comment in terms of PGA, it is clear that the LA19 record is significantly more

severe than the others, being almost twice in amplitude than LA09 and three times than El

Centro.

Regarding the duration parameters, if the threshold of 0.05g acceleration is considered the

LA09 record is two times longer than the others whilst for a higher acceleration threshold of

0.10g the differences in duration interval are small.


59/87


42

4.4 Response Spectra

For each earthquake ground motion, the relative displacement and acceleration response

spectrum for 5% of critical damping are presented below for a period range of 0 to 4 s with an

increment of 0.01s.

Fi

gure 4.2. Linear elastic response spectra for the LA09 record.

Figure 4.3. Linear elastic response spectra for the LA19 record.


60/87


43

Figure 4.4. Linear elastic response spectra for the EC record.

4.5 Ground Motion Spectra Comparison and Comment

To compare and comment on the differences between the different ground motions, different

spectra have been plotted in the same graph, as shown in Figure 4.5.

Regarding the frequency content and the spectral responses it is noted that:

The most severe condition in terms of acceleration response is given by the LA19

ground motion, for which the low period spectral acceleration from 0.0 to 0.5s is aboutthree times higher than that of other ground motions whilst for a period range of 0.5 to

2.0s LA09becomes critical in terms of spectral acceleration.

Figure 4.5.Comparison of the linear elastic response spectra for the 3 ground motions use. Damping is 5%

of critical


61/87


44

4.6 Modelling

To investigate the effect of using different alpha values for practical purposes, a cantilever

column exhibiting hysteretic moment-curvature relationship and its equivalent SDOF (spring)

with the same initial stiffness have been modeled and analyzed in Ruaumoko 2D [Carr, 2004],

as shown in Figure 4.6.

Figure 4.6. Modeling SDOF column and spring

The hysteretic behavior used is the modified Takeda hysteretic model [Otani 1974] reported

in Figure 4.7, where the unloading stiffness degradation parameter assumes values [0.0;

0.25; 0.50] and the reloading stiffness is set equal to 0.0.

Figure 4.7. Modified Takeda Model Ref: Ruaumoko 2D, Appendix [Carr, 2004]

The small displacement analysis regime and an integration time step of 0.005s is utilized in

the analyses.

The models analyzed are prepared with a series of natural periods of 0.5s, 1.0s, 2.0s and 3.0

respectively. The intensity of the accelerograms are adjusted to achieve displacement ductility

demands of 2, 4 and 6 respectively.

m

EIH

max

m

K=3EI

H3


62/87


45

4.7 Case-study: structural periods

Non-linear time history analysis of the modeled column requires the definition of a plastic

hinge length, which can be defined as the region of concentrated plasticity near the base of the

cantilever and can be calculated by using Equation 4.1 [Priestley et al 2007]

0.10 (4.1)where K and Lsp are respectively equal to 0.04 and 0.25 and the factor is related to theaspect ratio of the cantilever by the equation Ar(=H/LW). Assuming the aspect ratio of 5 and

10, the corresponding values of and the length of plastic hinge are presented in Table 4.2.Table 4.2. Height and Plastic hinge used in SDOF column

The moment of inertia of the sections are fixed so as to achieve undamped natural period of

0.5s, 1s, 2s and 3s respectively. The corresponding values are reported in Table 4.3

Table 4.3. Moment of Inertia corresponding to different undamped natural periods of the SDOF column.

The Young's elasticity modulus E is equal to 25740 MPa.

The lateral stiffness and yield strength of the equivalent spring can be calculated by using the

relation kx=3EI/H3and Fy=M/H respectively for a known moment of resistance, M and the

height of the cantilever, H.

In the following plots it is possible to observe the effect of using different alpha ratios on the

SDOF running the non-linear time history analyses with the accelerograms identified with

La09 and Elce.

The figures show the variation of the ratio of the maximum displacement demand of the

equivalent spring and the cantilever as a function of the undamped natural period of vibration.

The results obtained by non linear time history analyses with LA09 ground motion and are

reported in Figure 4.8 to 4.11, are calculated with a ratio between the column height,Hand

the length of the plastic hinge,Lpequal to 13 in the first group and 15 in the second group

respectively.

height Lw Lpm m m

dissertação de mestrado de novelli

Documents