Seismic fragility of RC frame and wall-frame dual ... · PDF fileSeismic fragility of RC frame and wall-frame dual buildings designed to EN ... frame dual buildings designed to EN-Eurocodes

Seismic fragility of RC frame and wall-frame

dual buildings designed to EN- Eurocodes

A Dissertation Submitted in Partial Fulfilment of the Requirements

for the Master Degree in

Earthquake Engineering &/or Engineering Seismology

By

Kyriakos Antoniou

Supervisor(s): Prof. Michael N. Fardis

February, 2013

University of Patras

The dissertation entitled “Seismic fragility of RC frame and wall-frame dual buildings

designed to EN-Eurocodes” by Kyriakos Antoniou, has been approved in partial fulfilment of

the requirements for the Master Degree in Earthquake Engineering

Professor M. N. Fardis ________________

Abstract

1

ABSTRACT

Fragility curves are constructed for structural members of regular reinforced concrete frame

and wall-frame buildings designed according to Eurocode 2 and Eurocode 8. Prototype plan-

and height- wise very regular buildings are studied with parameters including the height of the

building, the level of Eurocode 8 design (in terms of design peak ground acceleration and

ductility class) and for dual systems the percentage of seismic base shear taken by the walls.

Member fragility curves are constructed based on the results of nonlinear static (pushover)

analysis (SPO) and incremental dynamic analysis (IDA) using 14 spectrum-compatible semi-

artificial accelerograms. Analysis is performed using three-dimensional structural models of

the full buildings. These results are compared to fragility curves obtained from previous

studies for a simplified analysis method using the lateral force method (LFM).

The fragility curves are addressed on two member limit states; yielding and the ultimate

deformation in bending or shear. The peak chord rotation and peak shear force demands at

member ends are taken as damage measures; the peak ground acceleration (PGA) is used as

seismic intensity measure. The probability of exceedance of each limit state is computed from

the probability distributions of the damage measures (conditional on intensity measure) and of

the corresponding capacities.

The alternative methods yield results that are in good agreement for beams and columns in

both frame and dual buildings and for the flexural behavior of walls. Results from the

simplified procedure using the LFM shows that Medium Ductility Class walls are likely to

fail in shear even before their design PGA. The dynamic analysis confirms to a certain extend

the inelastic amplification of shear forces due to higher mode effects and shows that the

relevant rules of Eurocode 8 are on the conservative side.

Keywords: Concrete buildings; Concrete walls; Eurocode 8; Fragility curves; Seismic Design

Acknowledgements

2

ACKNOWLEDGEMENTS

I would like to sincerely thank my supervisor Professor M. N. Fardis for his guidance and the

time dedicated and G. Tsionis for his continuous support for the project.

Index

3

TABLE OF CONTENTS

ABSTRACT ............................................................................................................................................ 1

ACKNOWLEDGEMENTS ..................................................................................................................... 2

TABLE OF CONTENTS......................................................................................................................... 3

LIST OF FIGURES ................................................................................................................................. 6

LIST OF TABLES ................................................................................................................................. 12

LIST OF SYMBOLS ............................................................................................................................. 14

1. INTRODUCTION ........................................................................................................................ 20

2. DEFINITIONS AND BACKGROUND ....................................................................................... 22

2.1. Building codes ..........................................................................................................22

2.2. Performance-based requirements ..............................................................................22

2.3. Intensity Measure ......................................................................................................23

2.4. Damage measures .....................................................................................................25

2.5. Seismic Vulnerability Assessment Methodologies ...................................................26

2.5.1. Empirical Fragility Curves ................................................................................26

2.5.2. Expert Opinion method .....................................................................................27

2.5.3. Analytical Fragility Curves ...............................................................................28

2.5.4. Hybrid methods .................................................................................................30

2.6. Seismic safety assessment of RC buildings designed to EC8...................................30

3. DESCRIPTION OF BUILDINGS ................................................................................................ 32

3.1. Typology of buildings ...............................................................................................32

Index

4

3.2. Geometry of buildings ..............................................................................................32

3.3. Materials ...................................................................................................................34

4. DESIGN OF BUILDINGS ........................................................................................................... 35

4.1. Actions on structure and assumptions.......................................................................35

4.2. Behaviour factors and local ductility ........................................................................36

4.3. Design procedure ......................................................................................................37

4.3.1. Sizing of beams and columns in frame systems ...............................................37

4.3.2. Sizing of beams, columns and walls in wall-frame (dual) systems ..................38

4.4. Dimensioning of Beams ............................................................................................39

4.5. Dimensioning of Columns ........................................................................................40

4.6. Dimensioning of Walls .............................................................................................42

5. ANALYSIS METHODS AND MODELLING ASSUMPTIONS ................................................ 46

5.1. Nonlinear Static “Pushover” Analysis ......................................................................46

5.2. Incremental Dynamic Analysis .................................................................................47

5.3. Structural modelling for IDA and SPO .....................................................................51

5.4. Linear Static Analysis - “Lateral Force Method” .....................................................53

6. ASSESMENT OF BUILDINGS ................................................................................................... 57

6.1. Limit State of Damage Limitation (DL) ...................................................................57

6.2. Limit State of Near Collapse (NC) ...........................................................................60

6.3. Estimation of damage measure demands ..................................................................63

7. METHODOLOGY OF FRAGILITY ANALYSIS ....................................................................... 64

7.1. Damage Measures .....................................................................................................64

7.2. Exclusion of unrealistic results for IDA ...................................................................65

7.3. Determination of variability ......................................................................................65

7.4. Construction of fragility curves ................................................................................69

8. RESULTS AND DISCUSSION ................................................................................................... 71

8.1. Modal analysis results ...............................................................................................72

8.2. Median PGAs at attainment of the damage state for the three methods ...................74

8.3. Fragility curve results for wall-frame dual systems ..................................................76

8.4. Fragility curve results for frame systems ..................................................................91

8.5. Comparison between analysis methods ....................................................................96

8.6. Fragility results of walls in the ultimate state .........................................................111

Index

5

9. SUMMARY AND CONCLUSIONS ......................................................................................... 116

REFERENCES .................................................................................................................................... 119

APPENDIX A ....................................................................................................................................... A1

APPENDIX B ....................................................................................................................................... B1

APPENDIX C ....................................................................................................................................... C1

Index

6

LIST OF FIGURES

FIGURE 2.1 DEFINITION OF CHORD ROTATION [ADAPTED FROM FARDIS, 2009] ................................... 26

FIGURE 2.2 FLOWCHART TO DESCRIBE THE COMPONENTS OF THE CALCULATION OF ANALYTICAL

VULNERABILITY CURVE [ADAPTED FROM DUMOVA-JOVANOSKA (2004)] ................................... 29

FIGURE 3.1 PLAN OF WALL-FRAME (DUAL) BUILDINGS [PAPAILIA, 2011] ............................................ 33

FIGURE 3.2 GEOMETRY OF FRAME BUILDINGS [PAPAILIA, 2011] .......................................................... 33

FIGURE 3.3 STRUCTURAL 3D MODEL TAKEN FROM ANSRUOP FOR FIVE – STOREY DUAL SYSTEM ...... 34

FIGURE 4.1CAPACITY DESIGN VALUES OF SHEAR FORCES ON BEAMS [CEN, 2004] .............................. 40

FIGURE 4.2 CAPACITY DESIGN SHEAR FORCE IN COLUMNS [CEN 2004] ............................................... 42

FIGURE 4.3: DESIGN ENVELOPE FOR BENDING MOMENTS IN THE SLENDER WALLS (LEFT: WALL

SYSTEMS ; RIGHT: DUAL SYSTEMS ) [CEN 2004] .......................................................................... 43

FIGURE 4.4 DESIGN ENVELOPE OF THE SHEAR FORCES IN THE WALLS OF A DUAL SYSTEM [CEN 2004]

....................................................................................................................................................... 44

FIGURE 5.1 PSEUDO-ACCELERATION SPECTRA FOR THE SEMI-ARTIFICIAL INPUT MOTIONS COMPARED

TO THE SMOOTH TARGET SPECTRUM (SHOWN WITH THICK BLACK LINE) ..................................... 49

FIGURE 5.2 TIME-HISTORIES OF ACCELEROGRAMS USED IN THE ANALYSIS .......................................... 50

FIGURE 5.3 TAKEDA MODEL MODIFIED BY LITTON AND OTANI ............................................................ 51

FIGURE 5.4 STRUCTURAL MODEL FOR A FIVE – STOREY DUAL BUILDING TAKEN FROM ANSRUOP ..... 53

FIGURE 5.5 STRUCTURAL MODEL FOR AN EIGHT – STOREY DUAL BUILDING TAKEN FROM ANSRUOP 53

FIGURE 7.1 EXCLUSION OF UNREALISTIC RESULTS IN IDA (DAMAGE INDICES ABOVE CONTINUOUS

LINE ARE NEGLECTED) .................................................................................................................. 65

FIGURE 7.2 COEFFICIENT OF VARIATION (COV) OF DM-DEMANDS FOR FIVE-STOREY FRAME BUILDING

DESIGNED TO DC M AND PGA=0.20G .......................................................................................... 67

Index

7

FIGURE 7.3 COEFFICIENT OF VARIATION (COV) OF DM-DEMANDS FOR FIVE-STOREY FRAME-

EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.20G .................................................... 68

FIGURE 8.1 FRAGILITY CURVES FOR FIVE-STOREY WALL-EQUIVALENT BUILDING DESIGNED TO

PGA=0.20G AND DC M ANALYZED USING IDA METHOD ............................................................ 77

FIGURE 8.2 FRAGILITY CURVES OF WALLS FOR EIGHT-STOREY FRAME-EQUIVALENT (LEFT) AND WALL-

EQUIVALENT BUILDING (RIGHT) DESIGNED TO PGA=0.20G AND DC M ANALYZED USING IDA

METHOD ......................................................................................................................................... 78

FIGURE 8.3 FRAGILITY CURVES OF WALLS FOR EIGHT-STOREY FRAME-EQUIVALENT (LEFT) AND WALL-

EQUIVALENT BUILDING (RIGHT) DESIGNED TO PGA=0.25G AND DC M ANALYZED USING IDA

METHOD ......................................................................................................................................... 78

FIGURE 8.4 FRAGILITY CURVES OF WALLS FOR FIVE-STOREY FRAME-EQUIVALENT BUILDING DESIGNED

TO PGA=0.20G AND DC M (LEFT) AND WALL BUILDING DESIGNED TO DC H AND PGA=0.25G

(RIGHT) ANALYZED USING IDA METHOD ...................................................................................... 78

FIGURE 8.5 FRAGILITY CURVES OF WALLS FOR FIVE-STOREY FRAME-EQUIVALENT (LEFT) AND WALL-

EQUIVALENT (RIGHT) BUILDINGS DESIGNED TO DC H AND PGA=0.25G ANALYZED USING IDA

METHOD ......................................................................................................................................... 79

FIGURE 8.6 BEAM FRAGILITY CURVES FOR A) YIELDING AND B) ULTIMATE STATE OF A FIVE-STOREY

FRAME-EQUIVALENT (LEFT), WALL-EQUIVALENT (MIDDLE) AND WALL SYSTEM (RIGHT)

BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITH IDA ...................................... 80

FIGURE 8.7 COLUMN FRAGILITY CURVES FOR C) YIELDING AND D) ULTIMATE STATE OF A FIVE-STOREY



FIGURE 8.8 BEAM FRAGILITY CURVES FOR A) YIELDING AND B) ULTIMATE STATE OF A EIGHT-STOREY



FIGURE 8.9 COLUMN FRAGILITY CURVES FOR C) YIELDING AND D) ULTIMATE STATE OF A EIGHT-

STOREY FRAME-EQUIVALENT (LEFT), WALL-EQUIVALENT (MIDDLE) AND WALL SYSTEM (RIGHT)


FIGURE 8.10 FRAGILITY CURVES FOR MOST CRITICAL MEMBERS OF FIVE–STOREY FRAME-EQUIVALENT

BUILDING DESIGNED TO PGA=0.25G AND DC M ANALYZED USING IDA METHOD...................... 83

FIGURE 8.11 FRAGILITY CURVES FOR MOST CRITICAL MEMBERS OF FIVE–STOREY WALL-EQUIVALENT

BUILDING DESIGNED TO PGA=0.25G AND DC M ANALYZED USING IDA METHOD...................... 84

FIGURE 8.12 FRAGILITY CURVES FOR MOST CRITICAL MEMBERS OF FIVE–STOREY WALL BUILDING

DESIGNED TO PGA=0.25G AND DC M ANALYZED USING IDA METHOD ...................................... 85

Index

8

FIGURE 8.13 MEMBER FRAGILITY CURVES OF FRAME-EQUIVALENT DUAL SYSTEMS DESIGNED TO

PGA=0.25G AND DC M FOR: (TOP) FIVE – STOREY; (BOTTOM) EIGHT-STOREY USING IDA

METHOD ......................................................................................................................................... 86

FIGURE 8.14 MEMBER FRAGILITY CURVES FOR WALL SYSTEMS DESIGNED TO PGA=0.25G AND DC M

CURVES OF: (TOP) FIVE – STOREY; (BOTTOM) EIGHT-STOREY USING IDA METHOD ...................... 87

FIGURE 8.15 MEMBER FRAGILITY CURVES FOR A FIVE-STOREY FRAME-EQUIVALENT (FE), WALL-

EQUIVALENT (WE), WALL DUAL (WS) SYSTEM DESIGNED TO PGA=0.20G AND DC M USING SPO

METHOD FOR MOST CRITICAL STOREY MEMBERS. ........................................................................ 88

FIGURE 8.16 FRAGILITY CURVES OF EIGHT–STOREY FRAME-EQUIVALENT BUILDING DESIGNED TO DC

M AND: (TOP) PGA=0.20G; (BOTTOM) PGA=0.25G ANALYZED USING IDA METHOD ................. 89

FIGURE 8.17 MEMBER FRAGILITY CURVES FOR A EIGHT-STOREY WALL-EQUIVALENT SYSTEM

DESIGNED TO DC M AND FOR PGA=0.20G AND PGA=0.25G USING IDA METHOD FOR MOST

CRITICAL STOREY MEMBERS. ........................................................................................................ 90

FIGURE 8.18 MEMBER FRAGILITY CURVES FOR A FIVE-STOREY FRAME SYSTEM DESIGNED DC M AND

TO PGA=0.20G AND PGA=0.25G USING IDA METHOD FOR MOST CRITICAL STOREY MEMBERS. 91

FIGURE 8.19 MEMBER FRAGILITY CURVES FOR A FIVE-STOREY FRAME SYSTEM DESIGNED PGA=0.25G

AND TO DC M AND DC H USING IDA METHOD FOR MOST CRITICAL STOREY MEMBERS. ............ 92

FIGURE 8.20 FRAGILITY CURVES OF FIVE-STOREY BUILDINGS DESIGNED TO PGA=0.25G AND DC M

ANALYZED USING IDA METHOD: (TOP) FRAME BUILDINGS; (BOTTOM) FRAME-EQUIVALENT

BUILDINGS ..................................................................................................................................... 93


ANALYZED USING IDA METHOD: (TOP) FRAME BUILDINGS; (BOTTOM) WALL-EQUIVALENT

BUILDINGS ..................................................................................................................................... 94


ANALYZED USING IDA METHOD: (TOP) FRAME BUILDINGS; (BOTTOM) WALL BUILDINGS ............ 95

FIGURE 8.23 BEAM FRAGILITY CURVES IN YIELDING STATE FOR FIVE-STOREY FRAME-EQUIVALENT

BUILDING DESIGNED TO DC M AND PGA=0.20G (LEFT) AND WALL-EQUIVALENT BUILDING

DESIGNED TO DC H AND PGA=0.25G (RIGHT). ............................................................................. 96

FIGURE 8.24 BEAM FRAGILITY CURVES IN YIELDING STATE FOR EIGHT-STOREY FRAME-EQUIVALENT


DESIGNED TO DC M AND PGA=0.25G (RIGHT). ............................................................................ 97

FIGURE 8.25 BEAM FRAGILITY CURVES IN YIELDING STATE FOR FIVE-STOREY FRAME BUILDING

DESIGNED TO PGA=0.25G AND DC M (LEFT) AND DC H (RIGHT). ............................................... 97

Index

9

FIGURE 8.26 BEAM FRAGILITY CURVES IN ULTIMATE STATE FOR FIVE-STOREY FRAME-EQUIVALENT



FIGURE 8.27 BEAM FRAGILITY CURVES IN ULTIMATE STATE FOR EIGHT-STOREY FRAME-EQUIVALENT



FIGURE 8.28 BEAM FRAGILITY CURVES IN ULTIMATE STATE FOR FIVE-STOREY FRAME BUILDING

DESIGNED TO PGA=0.25G AND DC M (LEFT) AND DC H (RIGHT). ............................................... 98

FIGURE 8.29 COLUMN FRAGILITY CURVES IN YIELDING STATE FOR FIVE-STOREY FRAME-EQUIVALENT



FIGURE 8.30 COLUMN FRAGILITY CURVES IN YIELDING STATE FOR EIGHT-STOREY FRAME-

EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.20G (LEFT) AND WALL BUILDING


FIGURE 8.31 COLUMN FRAGILITY CURVES IN YIELDING STATE FOR FIVE-STOREY FRAME BUILDING

DESIGNED TO DC M AND PGA=0.20G AND (LEFT) PGA=0.25G (RIGHT). ................................... 100

FIGURE 8.32 COLUMN FRAGILITY CURVES IN ULTIMATE STATE FOR FIVE-STOREY WALL -EQUIVALENT

BUILDING DESIGNED TO DC M AND PGA=0.25G (LEFT) AND WALL BUILDING DESIGNED TO DC

M AND PGA=0.25G (RIGHT). ....................................................................................................... 100

FIGURE 8.33 COLUMN FRAGILITY CURVES IN ULTIMATE STATE FOR FIVE-STOREY FRAME-EQUIVALENT

BUILDING DESIGNED TO DC H AND PGA=0.25G (LEFT) AND WALL-EQUIVALENT BUILDING

DESIGNED TO DC H AND PGA=0.25G (RIGHT). ........................................................................... 101

FIGURE 8.34 COLUMN FRAGILITY CURVES IN ULTIMATE STATE FOR EIGHT-STOREY FRAME-

EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.25G (LEFT) AND WALL-EQUIVALENT

BUILDING DESIGNED TO DC M AND PGA=0.25G (RIGHT). ......................................................... 101

FIGURE 8.35 COLUMN FRAGILITY CURVES IN ULTIMATE STATE FOR FIVE-STOREY FRAME BUILDING

DESIGNED TO DC M AND PGA=0.20G AND (LEFT) DC H AND PGA=0.25G (RIGHT). ................. 101

FIGURE 8.36 WALL FRAGILITY CURVES IN YIELDING STATE FOR FIVE-STOREY FRAME-EQUIVALENT


M AND PGA=0.20G (RIGHT). ....................................................................................................... 102

FIGURE 8.37 WALL FRAGILITY CURVES IN YIELDING STATE FOR FIVE-STOREY FRAME-EQUIVALENT


M AND PGA=0.20G (RIGHT). ....................................................................................................... 102

Index

10

FIGURE 8.38 WALL FRAGILITY CURVES IN ULTIMATE STATE IN FLEXURE FOR FIVE-STOREY FRAME-

EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.25G (LEFT) AND WALL-EQUIVALENT

BUILDING DESIGNED TO DC H AND PGA=0.25G (RIGHT). .......................................................... 103

FIGURE 8.39 WALL FRAGILITY CURVES IN ULTIMATE STATE IN FLEXURE FOR FIVE-STOREY WALL

BUILDING DESIGNED TO DC M AND PGA=0.25G (LEFT) AND EIGHT-STOREY WALL BUILDING

DESIGNED TO DC M AND PGA=0.20G (RIGHT). .......................................................................... 103

FIGURE 8.40 BEAM FRAGILITY CURVES FOR A) YIELDING AND B) ULTIMATE STATE FOR FIVE-STOREY

FRAME-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.25G ANALYZED WITH IDA

(LEFT), SPO (MIDDLE) AND LFM (RIGHT). .................................................................................. 104


WALL-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.25G ANALYZED WITH IDA



WALL BUILDING DESIGNED TO DC M AND PGA=0.25G ANALYZED WITH IDA (LEFT), SPO

(MIDDLE) AND LFM (RIGHT). ....................................................................................................... 105

FIGURE 8.43 BEAM FRAGILITY CURVES FOR A) YIELDING AND B) ULTIMATE STATE FOR EIGHT-STOREY

FRAME-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITH IDA


FIGURE 8.44 BEAM FRAGILITY CURVES FOR A) YIELDING AND B) ULTIMATE STATE FOR EIGHT -STOREY

WALL-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITH IDA


FIGURE 8.45 BEAM FRAGILITY CURVES FOR A) YIELDING AND B) ULTIMATE STATE FOR EIGHT -STOREY

WALL BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITH IDA (LEFT), SPO


FIGURE 8.46 BEAM FRAGILITY CURVES FOR A) YIELDING AND B) ULTIMATE STATE FOR FIVE -STOREY

FRAME BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITH IDA (LEFT), SPO


FIGURE 8.47 COLUMN FRAGILITY CURVES FOR C) YIELDING AND D) ULTIMATE STATE FOR FIVE-

STOREY FRAME-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITH

IDA (LEFT), SPO (MIDDLE) AND LFM (RIGHT). .......................................................................... 108


STOREY WALL-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.25G ANALYZED WITH


Index

11


STOREY WALL BUILDING DESIGNED TO DC M AND PGA=0.25G ANALYZED WITH IDA (LEFT),

SPO (MIDDLE) AND LFM (RIGHT). .............................................................................................. 109

FIGURE 8.50 COLUMN FRAGILITY CURVES FOR C) YIELDING AND D) ULTIMATE STATE FOR EIGHT-

STOREY FRAME-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITH


FIGURE 8.51 COLUMN FRAGILITY CURVES FOR C) YIELDING AND D) ULTIMATE STATE FOR EIGHT -

STOREY WALL-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITH


FIGURE 8.52 COLUMN FRAGILITY CURVES FOR C) YIELDING AND D) ULTIMATE STATE FOR EIGHT -

STOREY WALL BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITH IDA (LEFT),


FIGURE 8.53 COLUMN FRAGILITY CURVES FOR C) YIELDING AND D) ULTIMATE STATE FOR FIVE -

STOREY FRAME BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITH IDA (LEFT),


FIGURE 8.54 FRAGILITY CURVES OF WALLS FOR THE ULTIMATE DAMAGE STATE IN SHEAR OF A FIVE-

STOREY WALL-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.20G. ......................... 114

FIGURE 8.55 FRAGILITY CURVES OF WALLS FOR THE ULTIMATE DAMAGE STATE IN SHEAR OF A FIVE-

STOREY WALL BUILDING DESIGNED TO DC M AND PGA=0.20G. ............................................... 114

FIGURE 8.56 FRAGILITY CURVES OF WALLS FOR THE ULTIMATE DAMAGE STATE IN SHEAR OF A EIGHT-

STOREY WALL BUILDING DESIGNED TO DC M AND PGA=0.20G. ............................................... 115

Index

12

LIST OF TABLES

TABLE 3.1: MATERIAL FACTORS AND VALUES ...................................................................................... 34

TABLE 4.1 BASIC VALUES OF THE BEHAVIOUR FACTOR, QO................................................................... 36

TABLE 4.2 BASIC FACTORED VALUES OF THE BEHAVIOR FACTOR, QO ................................................... 37

TABLE 4.3 DEPTHS OF BEAMS (HB) AND COLUMNS (HC) FOR FIVE-STOREY FRAME BUILDINGS [ADAPTED

FROM PAPAILIA, 2011] .................................................................................................................. 38

TABLE 4.4 DEPTHS OF BEAMS (HB) AND COLUMNS (HC) AND WALL LENGTHS (LW) FOR WALL-FRAME

DUAL BUILDINGS [ADAPTED FROM PAPAILIA, 2011] .................................................................... 39

TABLE 5.1: ACCELEROGRAM RECORDS USED IN THE ANALYSIS............................................................ 48

TABLE 7.1 VALUES OF COEFFICIENT OF VARIATION FOR DM-CAPACITY VALUES ................................ 70

TABLE 7.2 VALUES OF COEFFICIENT OF VARIATION FOR DM-DEMAND VALUES .................................. 70

TABLE 8.1 MODAL PERIODS AND PARTICIPATING MASSES FOR FRAME SYSTEMS ................................. 72

TABLE 8.2 MODAL PERIODS AND PARTICIPATING MASSES FOR FRAME-EQUIVALENT DUAL SYSTEMS . 72

TABLE 8.3 MODAL PERIODS AND PARTICIPATING MASSES FOR WALL-EQUIVALENT DUAL SYSTEMS ... 73

TABLE 8.4 MODAL PERIODS AND PARTICIPATING MASSES FOR WALL DUAL SYSTEMS ......................... 73

TABLE 8.5 MEDIAN PGA (G) AT ATTAINMENT OF THE DAMAGE STATE IN 5-STOREY FRAME SYSTEMS 74

TABLE 8.6 MEDIAN PGA (G) AT ATTAINMENT OF THE DAMAGE STATE IN 5-STOREY FRAME-

EQUIVALENT SYSTEMS .................................................................................................................. 74

TABLE 8.7 MEDIAN PGA (G) AT ATTAINMENT OF THE DAMAGE STATE IN 5-STOREY WALL-

EQUIVALENT DUAL SYSTEMS ........................................................................................................ 75

TABLE 8.8 MEDIAN PGA (G) AT ATTAINMENT OF THE DAMAGE STATE IN 5-STOREY WALL SYSTEMS . 75

TABLE 8.9 MEDIAN PGA (G) AT ATTAINMENT OF THE DAMAGE STATE IN 8-STOREY FRAME-


Index

13

TABLE 8.10 MEDIAN PGA (G) AT ATTAINMENT OF THE DAMAGE STATE IN 8-STOREY WALL-


TABLE 8.11 MEDIAN PGA (G) AT ATTAINMENT OF THE DAMAGE STATE IN 8-STOREY WALL SYSTEMS

....................................................................................................................................................... 76

TABLE 8.12 MEDIAN PGA (G) AT ATTAINMENT OF THE ULTIMATE DAMAGE STATE FOR WALLS IN 5-

STOREY BUILDINGS ..................................................................................................................... 112

TABLE 8.13 MEDIAN PGA (G) AT ATTAINMENT OF THE ULTIMATE DAMAGE STATE FOR WALLS IN 8-

STOREY BUILDINGS ..................................................................................................................... 112

TABLE 8.14 MEDIAN PGA (G) AT ATTAINMENT OF THE ULTIMATE DAMAGE STATE IN SHEAR FOR

WALLS IN 5-STOREY BUILDINGS .................................................................................................. 113

TABLE 8.15 MEDIAN PGA (G) AT ATTAINMENT OF THE ULTIMATE DAMAGE STATE IN SHEAR FOR

WALLS IN 8-STOREY BUILDINGS .................................................................................................. 113

Index

14

LIST OF SYMBOLS

Ac cross section area

Ecd design value of the modulus of elasticity of concrete

Ecm secant modulus of elasticity of concrete

(EI)b,i effective rigidity of the beams in storey i

(EI)c,i effective rigidity of the columns in storey i

Fb total lateral seismic shear (“base shear”)

Fi seismic horizontal force in storey i

FV,Ed total vertical load

G permanent (dead) load

Hcl clear height of a column

Hi transverse storey forces which represent the effect of the inclination φi

Hst storey height

Ic the moment of inertia of concrete cross section

Is the second moment of area of reinforcement, about the centre of area of the

concrete

Iw second moment of area (uncracked concrete section) of shear wall

Kc factor for effects of cracking, creep etc.

Ks factor for contribution of reinforcement

Lb bay length

Lcl,i beam clear span in storey i

Ls shear span of a member

MEb seismic bending moment at beam ends

MEc seismic bending moment at column ends

MEdo bending moment at the base of a wall, as obtained from the elastic analysis for

the design seismic action

Index

15

Mel elastic seismic moment at the end of the element

MRd,b,i- design value of negative beam moment resistance at end

MRd,b,j+ design value of positive beam moment resistance at end

MRdo flexural capacity at the base section of a wall

My yield moment

N axial force

NEd design value of the applied axial force

PGA Peak ground acceleration

PGV Peak ground velocity

Q imposed (live) load

Qd load for the persistent and transient design situation

QEq Combination of actions for seismic design situations

S soil factor according to EC8

Sa Spectral acceleration

Sa,ds spectral acceleration necessary to cause the certain damage state to occur

SD Spectral displacement

Sd(T) Design spectrum

Se(T) elastic response spectrum

T vibration period of a single-degree-of-freedom system

T1 fundamental period of vibration of a building

Tc corner period at the upper limit of the constant acceleration region of the

elastic spectrum

Teff effective period of vibration

VCD,c capacity-design shear of the columns

VEc seismic shear force at column ends

Vg+ψq,o shear force at end regions of interior beams due to quasi-permanent gravity

loads

VN contribution of the element axial load to its shear resistance

Vo shear force due to gravity loads

VR,c shear force at diagonal cracking of a member

VR,cycl shear resistance under cyclic loading

VR0 shear capacity before plastic hinging

VRs the contribution of transverse reinforcement to shear resistance

VS shear demand before plastic hinging

Vtot,base total base shear of the building

Vwall,base the fraction of the building total base shear taken by the walls

X random variable

Index

16

al tension shift

a effectiveness factor for confinement by transverse reinforcement

α1 is the value by which the horizontal seismic design action is multiplied in

order to first reach the flexural resistance in any member in the structure,

while all other design actions remain constant

acy zero-one variable for the type of loading

aem ratio of elastic moduli (steel-to-concrete)

αg design ground acceleration on type A ground according to EC8

ah reduction factor for height

am reduction factor for number of members

asl zero-one variable accounting for the slippage of longitudinal bars from the

anchorage zone beyond the end section

av zero-one variable

b width of compression zone

bi the centreline spacing of longitudinal bars (indexed by i) laterally restrained

by a stirrup corner or a cross-tie along the perimeter of the cross-section

bo width of confined core of a column or in the boundary element of a wall

bwo wall web thickness

cv coefficient of variation

d effective depth of a section

d1 distance of the center of the compression reinforcement from the extreme

compression fibres

dbL mean tension bar diameter

fbc normalised compressive strength of the masonry units

fcd design value of concrete compressive strength

fck characteristic value of concrete compressive strength

fcm mean value of concrete compressive strength

fmc specified compressive strength of the mortar

fyd design value of steel yield strength

fyk characteristic value of steel yield strength

fyL yield stress of the longitudinal bars

fym mean value of steel yield strength

fyw yield stress of transverse steel

h depth of a cross section

hb beam depth

hc column depth

ho depth of confined core of a column or in the boundary element of a wall

Index

17

hw wall height

ig radius of gyration of the uncracked concrete section

k1; k2 relative flexibilities or rotational restrains at member ends 1 and 2

l clear height of compression member between end restrains

l0 effective length of a member

lw wall length

m mean of the non-logarithmized variables of a lognormal distribution

maxVi,d,b capacity design shear at the end regions of interior beams

meff effective mass of a building

mi mass of floor i

n relative normal force for the design value of the applied axial force

nst number of storeys

nflx number of flexible frames per one stiff

q behaviour factor

qo basic values of the behaviour factor

s standard deviation of the non-logarithmized variables of a lognormal

distribution

xy neutral axis depth at flexural yielding

z length of the internal lever arm of a member

zi the height of the mass, , above the level of application of the seismic action

(foundation or top of a rigid basement)

Δδi interstorey drift from mid-height of the storey i to the mid-height i+1 of the

frame

Θ rotation of restraining member for bending moment M

ΣMRd,b sum of beam design flexural capacities

ΣMRd,c sum of column design flexural capacities

αu the value by which the horizontal seismic design action is multiplied in order

to form plastic hinges in a number of sections sufficient for the development

of overall structural instability, while all other design actions remain constant

β the normalised composite log-normal standard deviation

βD lower bound factor for the horizontal design spectrum

βR dispersion of the capacity (in terms of standard deviation)

βS dispersion of the demand (in terms of standard deviation)

βSp dispersion of the spectral value (in terms of standard deviation)

γc partial factor for concrete

γg partial factor for permanent action

γq partial factor for variable action

Index

18

γRd factor accounting for steel strain hardening

γs partial factor for steel

δi the displacement of floor from an elastic analysis of the structure for the set of

lateral forces

ε capacity design magnification factor

εRV uncertainty factor for shear capacity

εsV,el demand uncertainty factor for shear failure (prior to the formation of a plastic

hinge)

εsζ uncertainty factor of the chord rotation demand

εu capacity uncertainty factor

εy uncertainty factor for the yielding chord rotation

ε damping correction factor with a reference of for 5% viscous damping

ζ member chord rotation

ζs mean chord rotation demand

ζu ultimate chord rotation

ζum the expected chord rotation capacity

ζy chord rotation at yielding

ζym the expected chord rotation value at yielding

κ1 factor which depends on concrete strength class

κ2 factor which depends on axial force and slenderness

λ slenderness ratio

μ normal distribution mean

μζpl ratio of the plastic part of the rotation demand at the end of the member to the

value at yielding

μφ curvature ductility factor

ν axial load ratio, positive for compression

ξ reduction factor for unfavourable permanent actions

ξy neutral axis depth at yielding

π geometric reinforcement ratio

π1 ratio of the tension reinforcement

π2 ratio of the compression reinforcement

πd steel ratio of diagonal reinforcement in each diagonal direction

πs ratio of transverse steel parallel to the loading direction

πw the transverse reinforcement ratio

πν ratio of “web” reinforcement

δ normal distribution standard deviation

φ0 basic value of the inclination taking account for the geometric imperfections

Index

19

φeff effective creep ratio of concrete

φi inclination taking account for the geometric imperfections

φy yield curvature

ψ2 factor for quasi-permanent value of a variable action

ψο factor for combination value of a variable action

ω1 mechanical reinforcement ratio of tension and “web” longitudinal

reinforcement

ω2 mechanical reinforcement ratio of compression longitudinal reinforcement

Introduction

20

1. INTRODUCTION

This study deals with the seismic fragility of members for frame and wall-frame buildings

designed in accordance to EN-Eurocode 2 and 8. Prototype plan- and height-wise very regular

buildings are studied. Parameters include the number of storeys, the level of Eurocode 8

design (in terms of design peak ground acceleration and ductility class) and for wall-frame

dual systems the percentage of seismic base shear taken by the walls.

The fragility curves relate seismic ground motion to structural damage which is important in

order to denote the damage probability of the members in a structure. Fragility curves are

important for estimating the risk from potential earthquakes and for predicting the economical

impact for future earthquakes. They can be used for emergency response and disaster

planning by national agencies and by insurance companies for estimating the overall loss after

an earthquake event. Fragility curves can be used to mitigate risk by improving the seismic

codes.

Fragility curves are constructed for generic members for each building assuming a lognormal

distribution. The probability of exceedance of each limit state is computed from the

probability distributions of the damage measures (conditional on intensity measure) and of the

corresponding capacities. The intensity measure (IM) used for the construction of the fragility

curves is the peak ground acceleration (PGA) and the damage measures are the peak chord

rotation and the peak shear force demands at member ends. Seismic performance is addressed

on two damage states; the yielding and the ultimate deformation in bending or shear. The

estimations for the peak response quantities and capacities for each member are according to

Eurocode 8 – Part 3 [CEN, 2005].

The fragility curves are developed by using the analysis results obtained from three-

dimensional structural models of the full buildings using nonlinear incremental dynamic

analysis (IDA) and nonlinear static (pushover) analysis (SPO). IDA is carried out using

fourteen semi-artificial spectrum-compatible ground motion records scaled in order to cover a

range of ground motion intensities. SPO is carried out using the inverted triangular

distribution pattern and the N2 method [Fajfar et. al., 2000] is being employed to combine the

results of the static pushover analysis with the response spectrum analysis of an equivalent

single degree-of-freedom system to compute the IM for each step of the analysis.

Introduction

21

Dispersions used for the construction of fragility curves from IDA take into account explicitly

model uncertainties for the estimation of the damage measure demands taken from the

analysis. Estimates for the dispersions of the damage measure demands for the SPO method

are taken from previous studies. Both methods use estimates for the damage measure

capacities based on previous studies.

The results of a simplified method using the lateral force method (LFM) taken from Papailia

[2011] is compared against the results from SPO and IDA. The LFM is performed by using

simplified models under the assumption that all beam ends in a storey have the same elastic

seismic moments and inelastic chord rotation demands. Vertical elements are considered to

have negligible bending moments due to gravity loads and the axial force variation due to

seismic action is neglected in interior columns. The shear force demands taken from the LFM

are amplified to take into account higher mode effects.

Discussion will focus on the differences between geometric and design parameters of the

buildings and the differences between the alternative analysis methods. The walls of buildings

designed according to Eurocode 8 for Medium Ductility Class is an important point of the

discussion since according to the results using the lateral force method they fail in shear

before their design PGA.

Chapter 2: Definitions and Background

22

2. DEFINITIONS AND BACKGROUND

A brief introduction for various definitions and a review of previous studies is found in this

Chapter.

2.1. Building codes

The analysis, design and assessment of the buildings were performed in accordance to the

European Standards; Eurocode 2 [CEN, 2004a], Eurocode 8 - Part 1 [CEN, 2004b] and Part 3

[CEN, 2005]. Eurocode 2 and Eurocode 8 – Part 1 were published by the European

Committee for Standardization (CEN) in December of 2004. Eurocode 2 is for the design of

concrete structures and Eurocode 8 – Part 1 is for the seismic design of new buildings.

Eurocode 8 – Part 3 was published by CEN in June 2005 for the seismic retrofit and

assessment of structures. Since March 2010 all CEN member countries use the EN-

Eurocodes.

2.2. Performance-based requirements

Performance-based earthquake engineering allows for design to meet more than one

performance level thus replacing the traditional design against collapse. The performance

level is the condition of the facility or structure after a seismic event. The seismic event is

identified by the annual probability of exceedence known as the “seismic hazard level”.

In EN-Eurocodes the performance levels are associated to the Limit States of the structure.

The Ultimate Limit State concerns the safety of people and the Serviceability Limit State

concerns the comfort of its occupants and the function and use of the structure. According to

Eurocode 8 – Part 1 [CEN, 2004] the following two Limit States (or performance levels) are

considered:

1. “No-(local)- collapse”: It is considered as the Ultimate Limit State. This limit state

protects life against rare seismic events by preventing the collapse of structural

members. The seismic action associated with this limit state is the “design seismic

action” having 10% probability of being exceeded in 50 years (mean return period of

475 years).

2. “Damage Limitation”. It is considered as the Serviceability Limit State, where the

structural or non structural damage is limited under frequent seismic events. The

structure is expected not to have any permanent deformations and should retain its


23

strength and stiffness. The seismic action associated with this limit state is the

“damage limitation seismic action” with 10% probability of being exceeded in 10

years (mean return period of 95 years).

Eurocode 8 – Part 3 [CEN, 2005] for the assessment and retrofitting of structures has fully

adopted the performance-based approach for three performance levels:

1. “Damage Limitation” (DL), structural elements are not significantly yielded and retain

their strength and stiffness and the structure has negligible permanent drifts and no

repairs are required. It is recommended that the performance objective should be

reached for a 20% probability of exceedence in 50 years (return period of 225 years).

2. “Significant Damage” (SD), which corresponds to the “no-(local)-collapse” according

to EC8-Part 1, where the structure is significantly damaged but retains some residual

lateral strength and stiffness and its vertical load bearing capacity. Non-structural

components are damaged and moderate drifts are present. The structure will be able to

survive aftershocks of moderate intensity. It is recommended that the performance

objective should be reached for a 10% probability of exceedence in 50 years (return

period of 475 years).

3. “Near Collapse” (NC), the structure is heavily damaged with large permanent drifts

and little residual lateral strength or stiffness is retained although the vertical elements

are still able to retain vertical loads. The structure would most probably not be able to

survive another earthquake. It is recommended that the performance objective should

be reached for a 2% probability of exceedence in 50 years (return period of 2475

years).

This study is addressed on two limit states; the yielding and the ultimate. The yielding

corresponds to the “Damage Limitation” limit state and the ultimate corresponds to the “Near

Collapse” limit state as defined by Eurocode 8 – Part 3 [CEN, 2005].

2.3. Intensity Measure

An Intensity Measure (IM) is the ground motion parameter that is being used in order to relate

the ground motion to the damage of the building. The selected parameter should be able to

correlate the ground motion to the damage of the buildings. Intensity measures can be divided

into instrumental IM and non-instrumental IM.

For non-instrumental IM, macroseismic data are used in computing the empirical

vulnerability of structures. Macroseismic data is expressed in different macroseismic intensity

scales, which identify the effects of ground motion, and is taken from observation of damage

due to earthquake ground motion and its effects on the earth’s surface, people and structures.

Macroseismic intensity scale is a qualitative scale expressed in terms of Roman numerals

representing different intensity levels. An advantage of this type of intensity measure is that it

is directly related to the vulnerability of the buildings and there is no requirement to take

instrumental measurements. The gathered data depends on the area where it is collected and

how far away this area is from the epicenter.


24

The most important IMs for non-instrumental seismicity are the MSK: Medvedev-Sponheur-

Karnik Intensity scale [Medvedev and Sponheuer, 1969], the MMI: Modified Mercalli

Intensity Scale [Wood and Neumann, 1931], the European Macroseismic Sclae (EMS98)

[Grünthal, 1998] and the MCS: Mercalli – Cancani – Sieberg [Sieberg, 1923]. The MCS was

proposed as the development of the Mercalli scale and includes twelve degrees from I

“Instrumental” to XII “Cataclysmic”. MMI scale is composed of twelve degrees. MSK goes

from I “No perceptible” to XII “Very catastrophic”.

Previous studies made use of the non-instrumental intensity measures using the empirical

vulnerability procedures to produce post-earthquake damage statistics [Calvi et al., 2006].

Such studies include Braga et al. [1982] where the damage probability matrices have been

developed based on damage data obtained from the Irpinia 1980 earthquake. The buildings

were separated in three classes and the matrices were based on the MSK scale for each class.

Di Pasquale et al. [2005] updated Braga’s study and changed the MSK scale to the MCS scale

because the Italian seismic catalogue is based on this intensity measure. Dolce et. al. [2003]

have adapted the damage probability matrices with an additional vulnerability class using the

EMS98 scale, which takes into account the buildings constructed after 1980. Singhal and

Kiremidjian [1996] developed fragility curves and damage probability matrices using the

Modified Mercalli Intensity.

In instrumental intensity measures, instruments are used in order to record the ground motion

and then recorded accelerograms are processed to get the appropriate measurement. The

instrumental intensity measures include the Peak ground Velocity (PGV), the Peak Ground

Acceleration (PGA), the Peak Ground Displacement (PGD), the Spectral Acceleration at the

first mode of vibration Sa(T1,5%) and the spectral displacement Sd. PGV correlates well with

the earthquake magnitude and gives useful information on the ground-motion frequency

content and strong-motion duration which influence the seismic demands of the structure

[Akkar and Őzen, 2006]. The Spectral Acceleration at the first mode of vibration Sa(T1) is

often used since it is well suited for structures that are sensitive to the strength of the

frequency content near its first mode frequency [Vamvatsikos and Cornell, 2002].

These instrumental intensity measures were used in reference studies such as Kircil and Polat

[2006] where elastic pseudo-spectral acceleration was considered as an intensity measure in

developing fragility curves for RC frame buildings. Akkar et al. [2005] constructed fragility

functions for RC buildings using PGV as the IM since maximum inelastic displacements are

better correlated with PGV than with PGA and PGV has a good correlation with MMI for

large amplitude earthquakes. Borzi et al. [2006] used PGA as the intensity measure for the

vulnerability analysis of RC buildings. PGA was used since it is consistent with the parameter

used in seismic hazard maps in the current codes.

More complicated IMs have been introduced such as the vector-valued IMs by Baker [2005]

which consists of two parameters; the spectral acceleration and epsilon. Epsilon is found to be

able to predict the structural response. It is defined as the difference between spectral

acceleration of a record and the mean of the ground motion prediction equation at a given


25

period. Neglecting the effect of epsilon gives conservative estimates on the response of the

structure.

The ground motion IM that is being used in this study is the Peak Ground Acceleration

(PGA). The reason for this choice is due to the simplicity of its use and due to the fact that the

results can be easily compared against the design acceleration of the structures.

2.4. Damage measures

Damage measure (DM) is a scalar quantity that can be deducted from the analysis and

characterizes the response of the structural model due to seismic loading. Selecting a suitable

DM depends on the application and the structure.

The damage measures for members that are used in reference studies include:

The peak chord rotation demand at member end

The peak shear force demand

The local Park and Ang Damage Index [1985].

The node rotations

Displacement ductility, μ

The Park and Ang Damage index takes into account the damage due to maximum

deformation and the damage due to repeated cycles of inelastic deformation. The

displacement ductility is associated with the inelastic response and is defined as the ratio of

the maximum displacement to the yield displacement.

Common damage measures selected for the assessment of buildings as a whole include:

The residual deformation

The global Park and Ang Damage Index [1985]

Maximum base shear

The peak roof drift

Interstorey drift ratio

The peak interstorey drift angle , 𝜃𝑚𝑎𝑥 = 𝑚𝑎𝑥(𝜃1 ……𝜃𝑛)

Peak floor accelerations

The peak interstorey drift angle is used for structural damage of buildings and relates well to

joint rotations. The peak floor accelerations are used for damage to non-structural components

in multi-storey buildings. [Vamvatsikos and Cornell, 2002]. The Interstorey drift ratio is the

ratio of the maximum storey displacement over the storey height. It gives significant

information on the structural and non-structural damage.

Examples of reference studies that used the DMs above include Singhal and Kiremidjian

[1996], where the global damage index based on Park and Ang [1985], in order to develop

fragility curves and damage probability matrices for RC frame structures. Őzer and Erberik

[2008] developed fragility curves for the damage measure of the maximum interstorey drift


26

ratio and a softening index (SI) which was originally proposed by DiPasquale and Cakmak

[1987]. SI takes a value according to the stiffness change due to inelastic action. In another

reference study, Borzi et al. [2006] based the building limit conditions on displacements

which are well correlated with building damage.

For the purposes of this study the damage measures used are the peak chord rotations at a

member end and the peak shear force demands. The chord rotation at a member end is defined

as the angle between the tangent to the member section there and the chord connecting the

two members ends as shown in Figure 2.1. When plastic hinge forms in the member end, the

chord rotation is equal to the plastic hinge rotation.

Figure 2.1 Definition of chord rotation [adapted from Fardis, 2009]

2.5. Seismic Vulnerability Assessment Methodologies

Different methodologies for the seismic vulnerability assessment of buildings are used

according to the data available and the uncertainties considered. These methods include the

empirical, expert opinion, analytical and hybrid methods.

2.5.1. Empirical Fragility Curves

Empirical methods for the vulnerability assessment of buildings are based on the damage

observed after a seismic event. The two main types of empirical methods are the damage

probability matrices (DPM) and the continuous vulnerability functions. DPM is a form of

conditional probability of obtaining a damage level due to the IM. The continuous

vulnerability functions illustrate the probability of exceeding a given damage state as a

function of the seismic IM. The advantages of using empirical fragilities are that the observed

damage from the earthquakes is the most realistic way to model fragility and takes into

account many uncertainties such as soil-structure-interaction and variability of the structural


27

capacity. The disadvantages are that the empirical vulnerability functions require that the

survey forms are not incomplete and the way post-processing is done with the data should not

be deficient. These curves need to be derived for buildings in the same region and should

account for damage subjected after a specific earthquake event. Often undamaged buildings

are not recorded so when deriving the vulnerability analysis it is difficult to assess the total

number of buildings in the analysis [SYNER-G, 2012]. Empirical vulnerability cannot model

the evaluation of retrofit options and do not cover all building types and values of IM. [Calvi

et al. 2006].

Sabetta et al. [1998] developed vulnerability curves from post earthquake damage surveys and

estimated ground motion. The damage surveys of nearly 50000 buildings after earthquake

events in Italy together with estimates of strong ground motion parameters from attenuation

relationships was used for the development of fragility curves. The binomial distribution of

the damage was plotted as a function of PGA, Arias Intensity and Effective Peak Acceleration

for three structural classes and six damage levels according to the MSK macroseismic scale.

Effective Peak Acceleration is defined as the mean response spectral acceleration divided by a

factor of 2.5.

Sarabandi et. al. [2004] developed empirical fragility functions from recent earthquakes with

data taken from the Northridge, California earthquake in 1994 and the Chi-Chi earthquake in

1999 in Taiwan. Buildings situated near the strong motion recording stations were used in the

assessment and were divided into two groups according to their distance from the recording

station. Empirical fragility curves are produced for steel moment frames, concrete frames,

concrete shear walls, wood frame and unreinforced masonry buildings.

Rota et al. [2006] developed typological fragility curves from post-earthquake survey data on

the damage observed on the buildings after Italian earthquakes from the past three decades.

150,000 survey building records have been post processed to define the empirical damage

probability matrices for different building typologies. Typological fragility curves have been

obtained using advanced nonlinear regression methods. Typological risk maps were then

developed for both single damage state and for average loss parameters after combining the

hazard definitions, fragility curves and inventory data.

2.5.2. Expert Opinion method

Exert opinion method is a method to construct fragility curves based on the judgment and

information taken by experts. The probability of damage for different building typologies

covering a range of ground motion intensities are taken from the opinion of experts. The

advantage of the method is that it is not affected by the quantity and quality of the structural

damage data and statistics. The main disadvantage is that the method is restricted on the

knowledge and experience of the experts consulted. The study of Kostov et a. [2007]

produced damage probability matrices for buildings in Sofia according to the EMS-98. The

damage probability matrices were then converted in vulnerability curves.


28

2.5.3. Analytical Fragility Curves

This method features a more detailed vulnerability assessment with direct physical meaning.

The analytical fragility curves are computed by constructing appropriate structural models

which express the probability of damage computed under increasing seismic intensity. Figure

2.2 summarizes the basic procedures that are being followed in order to calculate the

analytical vulnerability curves or damage probability matrices. The advantage of this method

is that it provides results that are very close to reality. One of the main disadvantages of

analytical vulnerability curves is that they are computationally demanding and time

consuming. Also the capability of modelling the structure significantly affects the reliability

of the results.

Eurocode 8 - Part 3 [CEN, 2005] provides guidelines for the assessment of existing buildings

which may be used to develop analytical fragility curves. The methods of analysis include the

lateral force analysis, the modal response spectrum analysis, the nonlinear static pushover

analysis, the nonlinear time-history dynamic analysis. The nonlinear static method applies

forces to the model which includes the nonlinear properties of the elements. The nonlinear

dynamic analysis although time consuming gives results that are closer to reality. Also it

allows the influence of the variability of the accelerogram to be taken into account. These

methods are performed in order to compute the seismic action effects.

In order to choose the type of analysis to be performed and the appropriate confidence factor

values EC8 - Part 3 defines three knowledge levels:

KL1: Limited Knowledge

KL2: Normal Knowledge

KL3: Full Knowledge

The factors that determine the knowledge levels are the geometrical properties of the

structural system and non structural elements, the details (regarding the reinforcement in

reinforced concrete members, the connections between steel members, the floor diaphragm

connection to lateral resisting structure etc.) and the mechanical properties of the constituent

materials used.

For the purpose of this study analytical fragility curves have been developed using nonlinear

time-history dynamic analysis and nonlinear static (pushover) analysis. The buildings

assessed belong to the Full knowledge level (KL3) of Eurocode 8 – Part3 since all

geometrical properties, details and mechanical properties of the materials are known.


29

Figure 2.2 Flowchart to describe the components of the calculation of analytical vulnerability curve

[adapted from Dumova-Jovanoska (2004)]

Existing studies for the computation of seismic fragility curves for RC buildings that are

based on the analytical method include the following.

Singhal and Kiremidjian [1996] developed fragility curves and damage probability matrices

using Monte Carlo simulation for low-rise, mid-rise and high-rise RC frames using Park and

Ang (1985) damage index to identify different degrees of damage. The analysis was based on

nonlinear dynamic analysis where the ground motion is characterized by spectral acceleration.

For the computation of damage probability matrices the modified Mercalli intensity was used

as the ground motion parameter.

B. Borzi et. al. [2006] use analytical methods where the nonlinear behavior of a random

population of RC buildings was defined with simplified pushover and displacement based

procedures. The vulnerability curves were generated by comparing the displacement

capacities by the pushover analysis with the displacement demands obtained from response

spectrum of each building in the random population. The vulnerability curves were

formulated using the conditional probability of exceeding a certain damage limit state in terms

of the IM.

Dumova et.al [2000] evaluated the vulnerability curves/ damage probability matrices using

analytical methods for frame-wall RC buildings designed according to the Macedonian design


30

code. Two sets of buildings were analyzed; six storey frame buildings and sixteen storey

frame-wall buildings. Nonlinear time-history analysis was performed for a set of synthetic

time histories and the response of the structure to the earthquake excitation was defined

according to modified Park and Ang (1985) damage model using five damage states to

express the condition of damage. The probability of occurrence of damage was assumed to be

normal probabilistic distribution.

Masi [2003] employed analytical methods for the seismic vulnerability assessment of existing

RC frame buildings (bare, regularly infilled and pilotis) designed only to gravity loads for

buildings representative of the Italian building block of the past 30 years designed according

to the building codes at the period of their construction. The analysis was performed using

nonlinear time-history analysis using artificial and natural accelerograms. The vulnerability

was characterized through the use of European Macroseismic Scale.

Kirçil and Polat [2006] evaluated the behavior of mid-rise RC frame buildings using

analytical methods. The building stock represented buildings of 3, 5 and 7 storeys that were

designed according to the (1975) Turkish seismic code. In this study only yielding and

collapse damage levels are considered and they were determined analytically under the effect

of twelve artificial accelerograms using incremental dynamic analysis. The yielding and

collapse capacities are evaluated by statistical methods to develop fragility curves in terms of

elastic pseudo-spectral acceleration. Lognormal distribution is assumed for the construction of

the fragility curves.

2.5.4. Hybrid methods

Hybrid damage probability matrices and vulnerability functions combine damage observed

after earthquakes with damage obtained from analytical methods. This method is

advantageous when there is lack of observational data. Also post-earthquake damage data can

be used to calibrate the analytical model. Observational data can reduce the computational

effort that would normally be required to perform complete analytical analysis.

Kappos et. al. [1998] developed the damage probability matrices using a hybrid procedure

where data from past earthquakes was combined with results of nonlinear dynamic analysis

for typical Greek buildings designed for the 1959 codes. The results of the dynamic analysis

were used in order to obtain a global damage index and correlated with loss in terms of cost of

repair. Observational damage from the 1978 Thessaloniki earthquake was combined with the

analytical damage results.

2.6. Seismic safety assessment of RC buildings designed to EC8

The efficacy of Eurocode 8 and design provisions and the expected performance has been

evaluated in the past. The following studies were performed for the seismic safety assessment

of RC buildings.

Panagiotakos and Fardis [2004] evaluated the performance of RC buildings designed

according to Eurocode 8 using nonlinear analysis. RC frames of 4, 8 and 12 storeys were


31

designed for a PGA of 0.2g or 0.4g and to the three ductility classes. The limit states are

considered as in EC8 for the life-safety (475 years) and the damage limitation (95 years) and

are evaluated through nonlinear seismic response analysis. It was found that the design to

Ductility Class High (DC H) or Medium (DC M) is more cost effective than DC Low even in

moderate seismicity and more cost effective than the 2000 Greek national codes. It was also

found that the large differences in material quantities and detailing of the alternative designs

do not translate into large differences in performance.

Rivera and Petrini [2011] investigate the efficacy of the Eurocode 8 force-based design

provisions for RC frames. This study evaluates whether the RC buildings that are designed

according to the EC8 provisions have the expected performance. Four, eight and sixteen

storey RC frame buildings were designed and analyzed using the EC8 response spectrum

analysis. Nonlinear time-history analysis was performed to determine the seismic response of

the structures and validate the EC8 forced base designs. The results indicate that the design of

flexural members in medium-to-long period structures is not significantly influenced by the

choice of effective member stiffness. However the interstorey drift demands calculated are

significantly affected. Design storey forces and interstorey drift demands found using the

code’s force base procedure varied substantially from the results of the nonlinear time-history

analysis. From the results it was concluded that EC8 may yield life-safe designs. Also the

seismic performance of RC frame buildings of the same type and ductility class can be highly

non-uniform.

Rutenberg and Nsieri [2005] evaluated the seismic shear demand in ductile cantilever wall

systems. Two aspects were considered; (1) Single walls or a system of equal-length walls and

(2) resisting system consisting of walls of different length. The results of the parametric

studies showed that DC M and DC H walls designed to EC8 provisions are in need of revision

since for DC M walls the inelastic amplification which takes into account the higher mode

effects as required in EC8 is under-conservative whereas the amplification used for DC H

walls according to the detailed procedure per Keintzel [1990] overestimates the shear demand

in walls for most cases..

Chapter 3: Description of Buildings

32

3. DESCRIPTION OF BUILDINGS

For the scope of this study pure frame and wall-frame (dual) reinforced concrete buildings

were analyzed and assessed. Two analysis methods were performed: nonlinear static and

nonlinear dynamic analysis of the structures comprising different design and geometric

parameters. The parameters, methods and assumptions made when modelling the structures

are explained and discussed in this section.

3.1. Typology of buildings

The design and detailing of the frame and the wall-frame (dual) buildings correspond to

certain design parameters including:

Number of storeys: 5 and 8 storeys

Seismic Design level per EC8 for Ductility class

o Medium Ductility Class (DC M)

o High Ductility Class (DC H)

Seismic Design level per EC8 for design PGA

o 0.20g

o 0.25g

For wall-frame dual buildings, the fraction of the seismic base shears taken by the walls:

Frame-equivalent dual system 0.35Vtot,base≤ Vwall,base≤ 0.50Vtot,base

Wall-equivalent dual system 0.50Vtot,base ≤Vwall,base≤ 0.65Vtot,base

Wall system Vwall,base≥ 0.65Vtot,base

3.2. Geometry of buildings

The buildings are regular in plan and in elevation having storey height of Hst=3.0m, where all

storeys are of the same height. The buildings consist of five bays along the two horizontal

directions of bay length Lb=5.0m with the same bay length throughout the plan.

The buildings consist of square columns, beams of width 0.3m and slab thickness of 150mm.

The size of columns is constant throughout all storeys and the size of beams is constant

throughout each storey. The perimeter beams and exterior columns have half the elastic

rigidity of interior ones and corner columns have one quarter of elastic rigidity of interior

ones.


33

In wall-framed dual systems two walls on each direction are placed as shown in Figure 3.1

and Figure 3.3 sharing the same displacements with the frame. The geometry of the frame

building is illustrated in Figure 3.2 for a five- and eight-storey building. The beam and

column depths and wall lengths for wall-frame buildings are shown in Table 4.4 and the beam

and column depths for frame buildings are shown in Table 4.3.

Figure 3.1 Plan of wall-frame (dual) buildings [Papailia, 2011]

Figure 3.2 Geometry of frame buildings [Papailia, 2011]


34

Figure 3.3 Structural 3D model taken from ANSRuop for five – storey dual system

3.3. Materials

The material strengths and partial factors are taken according to Annex C of Eurocode 2

[CEN,2004a]. The structural materials consist of concrete of class C25/30, having a nominal

strength of 25MPa and Tempocore steel of grade S500 (Class C). The following table

provides the material properties for steel and concrete and their partial factors.

Table 3.1: Material factors and values

Partial factors

Partial factor for Concrete c 1.5

Partial factor for Steel s 1.15

ConcreteC25/30

Concrete compressive strength fck 25 MPa

Design compressive strength fcd=ccfck/ c 16.67MPa

Mean concrete compressive strength fcm=fck+8MPa 33MPa

Mean axial tensile concrete strength fctm 2.56 MPa

Secant modulus of elastic of concrete Ecm 30470 MPa

Design value of modulus of elasticity Ecd=Ecm/ cE 25392 MPa

Concrete Cover cnom 30mm

SteelS500

Characteristic yield strength of reinforcement fyk 500MPa

Design yield strength of reinforcement fyd= fyk/s 434.78MPa

Mean yield strength of reinforcement fym=1.15 fyk 575 MPa

Design value of modulus of elasticity of steel Es 200000 MPa

For the seismic vulnerability assessment the mean values for material strengths are being used

(fym=575MPa for reinforcing steel and fcm=33MPa for concrete).

Chapter 4: Design of Buildings

35

4. DESIGN OF BUILDINGS

4.1. Actions on structure and assumptions

The actions considered in the analysis correspond to the seismic design situation and the

persistent and transient design situation according to EN1990.

The combination of vertical actions for the seismic design situation is:

QEQ = G + ψ2 Q ( 4.1)

Where,

ψ2 quasi-permanent value of a variable action factor (=0.3)

G permanent load (=7 kN/m2)

Q imposed load (=2 kN/m2)

The combination for the persistent and transient design situation according to EN1990 is

given by :

Qd=max(ξγg G+ γg Q ; γg G+ ψo γg Q) ( 4.2)

where:

ξ is the reduction factor for unfavourable permanent actions (=0.85)

ψ0 is the factor for combination value of a variable action (=0.7)

γg is the partial factor for permanent action (=1.35)

γq is the partial factor for variable action (=1.5)

The permanent load acting on the structure is 7kN/m2, which includes the weight of the slab,

finishing, partitions and facades and the weight of the beams, columns and walls. The

occupancy loads (live loads) amount to 2kN/m2.


36

The design of the building was taken from Papailia [2011] where the “lateral force method” is

used to proceed with the design according to EC8 [CEN,2004b]. In order to compute the base

shear force, as required by the lateral force method, the design spectrum and the fundamental

period is used. The design spectrum is computed by the use of the behaviour factor q obtained

as explained in the section below and the fundamental period of the structure is obtained by

the Rayleigh quotient.

In concrete buildings the stiffness of the load bearing elements are evaluated by taking into

account the effects of cracking. The cracking effect corresponds to the yielding initiation of

the reinforcement. In Eurocode 8 [CEN, 2004], this simplification can be taken into account

by assuming that the flexural and shear stiffness properties are one half of the initial

uncracked stiffness of the element.

4.2. Behaviour factors and local ductility

In force-based design according to EC8 [CEN,2004b], the use of the behaviour factor

accounts for a simplification in design where the forces found by elastic analysis are reduced.

The values of the basic behaviour factor for buildings designed to DC M and DC H are given

in Table 4.1 for frame systems, wall-frame systems and uncoupled wall systems. Uncoupled

wall systems are defined as wall systems which are linked by a connecting medium which is

not effective in flexure.

Table 4.1 Basic values of the behaviour factor, qo

DC M DC H

Frame system, wall-frame system 3.0 αu/α1 4.5 αu/α1

Uncoupled Wall system 3.0 4.0 αu/α1

Where,

α1 the value by which the horizontal seismic design action is multiplied to reach the

flexural resistance in any member in the structure while other design actions remain constant.

αu the value by which the horizontal seismic design action is multiplied to form plastic

hinges in a number of sections sufficient for the development of structural instability, while

all other design actions remain constant.

The ratio of αu/α1 for frame or frame-equivalent dual system may be taken equal to 1.3, for

wall-equivalent systems equal to 1.2 and for wall system with two uncoupled walls per

horizontal direction equal to 1.0. Thus the basic values of the behaviour factor, qo, are:


37

Table 4.2 Basic factored values of the behavior factor, qo

Frame–equivalent / Frame systems Wall-equivalent Wall systems

DC M 3.9 3.6 3.0

DC H 5.85 5.4 4.0

4.3. Design procedure

This section describes the procedure that was followed for the sizing of beams, columns and

walls.

4.3.1. Sizing of beams and columns in frame systems

The sizing of beams and columns in frame systems was performed according to Eurocode 8

[CEN,2004b] and Eurocode 2 [CEN,2004a]. The sizing of the beams and the columns was

taken from Papailia [2011]. The procedure to size the member is described in this section.

Eurocode 2 [CEN,2004a] gives a simplified criterion for the slenderness ratio of isolated

columns:

λ =lo

ig ≤ λlim = 20

A B C

n ( 4.3)

Where,

ig is the radius of gyration of the uncracked concrete section

l0 is the effective length

n Is the normalised axial force taken as n=Ned/ Ac fcd and Ned is the design value of the

applied axial force.

The default values for A, B and C are A=0.7, B=1.1 and C=0.7.

The effective length is given by:

𝑙𝑜 = 𝐻𝑐𝑙 .𝑚𝑎𝑥 1 + 10𝑘1𝑘2

𝑘1+𝑘2; 1 +

𝑘1

1+𝑘1 1 +

𝑘2

1+𝑘2 ( 4.4)

Where,

ki is the column rotational stiffness at the end node i relative to the total restraining

stiffness of the members framing in the plane of bending.

𝑘𝑖 =𝜃𝑖

𝑀𝑖

𝐸𝐼𝑐 ,𝑒𝑓𝑓

𝐻𝑐𝑙=

𝐸𝐼𝑐 ,𝑒𝑓𝑓

𝐻𝑐𝑙

4 𝐸𝐼𝑐 ,𝑒𝑓𝑓

𝐻𝑐𝑙+4

𝐸𝐼𝑏 ,𝑒𝑓𝑓

𝐿𝑐𝑙

( 4.5)


38

Where,

Lcl is the clear length of a beam framing into node i

𝐸𝐼𝑏 ,𝑒𝑓𝑓 is the cracked flexural rigidity, taking into account creep

𝐸𝐼𝑐 ,𝑒𝑓𝑓 = 𝐸𝑠𝐼𝑠 + 𝐸𝑐𝑑 𝑓𝑐𝑘 (𝑀𝑃𝑎 )

20

𝐾2𝐼𝑐

1+𝜑𝑒𝑓𝑓 ( 4.6)

Es and Is are the elastic modulus and the moment of inertia of the sections reinforcement with

respect to the centroid of the section. Ic is the moment of inertia of the uncracked gross

concrete section and K2 is :

𝐾2 =𝑛𝜆

170=

1

170

𝑁𝐸𝑑

𝐴𝑐𝑓𝑐𝑑

𝑙𝑜

𝑖𝑐≤ 0.20 ( 4.7)

The effective length of the column and the size of the section are both unknown at the

beginning, thus iterations are performed after dimensioning of the top beam reinforcement at

the supports.

In pure frame systems the depths of the columns and beams are chosen iteratively as the

minimum values meeting the requirements of Eurocode 2 [CEN,2004a] and Eurocode 8

[CEN,2004b]. This takes into account the above implementation for the slenderness limit to

meet the negligible second order effects and the 0.5% storey drift limit per EC8 under the

damage limitation seismic action, where the 50% of the design seismic action is taken.

In the following table the sizes of the beams and columns are presented for different design

parameters (ductility class and design PGA)

Table 4.3 Depths of beams (hb) and columns (hc) for five-storey frame buildings [adapted from Papailia,

2011]

Design

PGA

DC

hb (m) hc (m)

0.20g M/H 0.40 0.55

0.25g M/H 0.45 0.55

4.3.2. Sizing of beams, columns and walls in wall-frame (dual) systems

In dual (wall-frame) buildings the lateral force procedure according to EC8 [CEN,2004b] was

performed and iterated until certain criteria were met. The sizing of the members is taken

from Papailia [2011]. The depths of columns (hc) and beams (hb) and the length of the walls

(lw) were chosen iteratively to meet the following requirements according to EC8

[CEN,2004b]:


39

Meet the storey drift ratio of 0.5% according to Eurocode 8 [CEN, 2004b].

To cover the three cases for the requirements of the wall to total base shear fraction

following the different behavior factors and design rules per EC8:

o Frame-equivalent dual system 0.35Vtot,base≤ Vwall,base≤ 0.50Vtot,base

o Wall-equivalent dual system 0.50Vtot,base ≤Vwall,base≤ 0.65Vtot,base

o Wall system Vwall,base≥ 0.65Vtot,base

In the following table the sizes of the beams and columns and the length of the walls are

presented for different design parameters (wall base shear fraction, ductility class and design

PGA)

Table 4.4 Depths of beams (hb) and columns (hc) and wall lengths (lw) for wall-frame dual buildings

[adapted from Papailia, 2011]

Design DC 5 storeys 8 storeys

PGA hb (m) hc (m) lw (m) Vwall,b (%) hb (m) hc (m) lw (m) Vwall,b (%)

0.20g M/H a 0.40 0.40 1.5 37 0.45 0.45 2.0/- 42/-

2.0 53 3.0/3.0 b 63/73

2.5 65 4.0/- 76/-

0.25g M/H a 0.45 0.45 2.0 44 0.50 0.45 2.0/- 40/-

2.5 57 3.0/- 61/-

3.5/3.5 b 73/81 4.0/5.5

b 74/90

a When DC M and DC H have different fraction of base shear and wall length, this is distinguished

with a slash, where the left hand side is the DC M and the right hand side the DC H.

b Wall width is 0.5m. In all other cases wall width is 0.25m.

4.4. Dimensioning of Beams

The longitudinal reinforcement for ULS in bending in beams is designed for the persistent-

and-transient and the seismic design situations using the lateral force method. The

reinforcement in the effective beam flange was taken to be 500mm2.

For the seismic design situation, the dimensioning of the end regions of the beams is done in

accordance to the capacity design rules computed using the design base shears at the member

ends, according to EC8 [CEN,2004b]. The beam design shear forces were determined under

the transverse load through the seismic design situation and the end moments, Mi,d, which

correspond to the formation of plastic hinges.

The end moments Mi,d depend on the moment resistances of the columns it is connected to

and the moment resistance of the beams itself. It can be found using:

𝑀𝑖 ,𝑑 = 𝛾𝑅𝑑𝑀𝑅𝑏 ,𝑖 min(1, 𝑀𝑅𝑐

𝑀𝑅𝑏) ( 4.8)


40

Where,

γRd factor accounting for steel strain hardening, equal to 1.0 of DC M and 1.2 for DC H.

MRb,i design value of the beam moment resistance at end i

ΣMRc sum of the column design moment of resistance.

ΣMRb sum of the beam design moment of resistance, framing to the point.

Thus the capacity design shear at the member ends corresponds to:

𝑉𝐸𝑑 ,𝑖 =𝑀1,𝑑+𝑀2,𝑑

𝑙𝑐𝑙+ 𝑉𝑔+𝜓𝑞 ,0 ( 4.9)

Where,

VEd,i capacity design shear at the member ends.

𝑉𝑔+𝜓𝑞 ,0 Shear force at the end regions due to the transverse quasi-permanent loads

under the design seismic situation.

Figure 4.1Capacity design values of shear forces on beams [CEN, 2004]

4.5. Dimensioning of Columns

The vertical reinforcement of the columns for the ULS in bending was designed for the axial

load taken from the actions of the seismic design situation. The detailing rules according to

Eurocode 8 [CEN, 2004] are taken into account for each seismic design level.

The dimensioning for the end regions of the columns is computed in accordance to the

capacity design rule through the design shear forces. The design shear forces are based on the

element equilibrium under the end moments Mi,d which correspond to the formation of plastic

hinges as shown in Figure 4.2. The end moments are computed by taking into account the

moment resistances of the beams to which it is connected and the moment resistances of the

column itself.


41

The end moments Mi,d are determined through:

𝑀𝑖 ,𝑑 = 𝛾𝑅𝑑𝑀𝑅𝑐 ,𝑖 min(1, 𝑀𝑅𝑏

𝑀𝑅𝑐) ( 4.10)

Where,

γRd factor accounting for steel strain hardening and the confinement of the concrete of the

compression zone of the section, equal to 1.1.

MRc,i design value of the column moment resistance at end i

ΣMRc sum of the column design moment of resistance.

ΣMRb sum of the beam design moment of resistance, framing to the point.

Thus the capacity design shear at the member ends corresponds to:

𝑉𝐸𝑑 ,𝑖 =𝑀1,𝑑+𝑀2,𝑑

𝐻𝑐𝑙 ( 4.11)

Where,

VEd,i capacity design shear at the end regions.

𝐻𝑐𝑙 clear height of column.


42

Figure 4.2 Capacity design shear force in columns [CEN 2004]

4.6. Dimensioning of Walls

The design shear force and moments for the walls are according to the capacity design

principles and their calculation is explained below according to EC8 [CEN,2004b]. The

values for the axial force are computed from the analysis of the structure in the seismic design

situation using the lateral force method.

The design bending moment diagram along the height of slender walls should be given by an

envelope of the bending moment diagram from analysis, with a tension drift, as shown in

Figure 4.3. Slender walls are defined as walls having a height to length ratio greater than 2.0.

The envelope is assumed to be linear since there are no discontinuities over the height of the

building. It takes into account potential development of moments due to higher mode inelastic

response after the formation of plastic hinge at the bottom of the wall, thus the region above

this critical height is designed to remain elastic.


43

KEY:

a moment diagram from analysis

b design envelope

a1 tension drift

Figure 4.3: Design envelope for bending moments in the slender walls (left: wall systems ; right: dual

systems ) [CEN 2004]

The design envelope of shear forces, as shown in Figure 4.4, takes into account the

uncertainties of higher modes. The flexural capacity at the base of the wall MRd exceeds the

seismic design bending moment derived from the analysis, MEd. Thus the design shear found

for the analysis, 𝑉𝐸𝑑′ , is magnified by the magnification factor i.e. the ratio of MRd/MEd. The

magnification factor depends on the ductility class of the structure. The design base shear is

thus computed by:

VEd = 𝑉𝐸𝑑′ ( 4.12)

Where,

For walls in DC M buildings the magnification factor, is taken as 1.5

For walls in DC H buildings the magnification factor, is taken as:

휀 = 𝑞 . 𝛾𝑅𝑑

𝑞.𝑀𝑅𝑑

𝑀𝐸𝑑

2

+ 0.1 𝑆𝑒(𝑇𝑐)

𝑆𝑒(𝑇1)

2

≤ 𝑞 ( 4.13 )

Where,

γRd overstrength factor taken as 1.2

Se(T1) ordinate of the elastic response spectrum at fundamental period

Se(TC) ordinate of the elastic response spectrum at corner period


44

KEY:

a shear diagram from analysis

b magnified shear diagram

c design envelope

A Vwall,base

B Vwall,top≥Vwall,base/2

Figure 4.4 Design envelope of the shear forces in the walls of a dual system [CEN 2004]

At the critical regions of the wall the curvature ductility factor μφ is required in order to

calculate the confining reinforcement within boundary elements. The curvature ductility

factor is now the product of the basic behaviour factor qo found in Section 4.2 and the ratio of

the design bending moment from the analysis MEd, to the design flexural resistance MRd. This

confining reinforcement should extend vertically up to a height hcr of the critical region and

horizontally along the length lc of the boundary element.

The length of this boundary element is the measure from extreme compression fibre to the

point where spalling occurs in concrete due to large compressive strains. As a minimum the

boundary region should be taken as being larger than 0.15.lw or 1.5.bw. The wall critical

region height, hcr, is estimated using the following relationship:

hcr=max lw,hw

6 ≤

2lwHcl for : nst≤6

2 Hcl for : nst

≤7

( 4.14)


45

Where,

nst the number of storeys

hw the wall height

Hcl is the clear storey height. The base is defined as the level of the foundation or the top

of the basement storey.

lw is the length of the cross section of the wall

Above the height of the critical region, hcr, the rules of EN1992 apply for the dimensioning of

vertical and horizontal reinforcement.

Chapter 5: Structural modelling and analysis methods

46

5. ANALYSIS METHODS AND MODELLING ASSUMPTIONS

For the construction of the fragility curves different analysis methods were performed each

following different modelling assumptions. For the purpose of this study two methods were

performed; the nonlinear static pushover analysis and the nonlinear dynamic analysis. The

results from these methods were then compared against a simplified method following the

lateral force analysis method by Papailia [2011]. The following section explains the procedure

and assumptions for the analysis methods and structural models.

5.1. Nonlinear Static “Pushover” Analysis

“Static pushover” (SPO) analysis is performed for the evaluation of the buildings according to

Eurocode 8 – Part 1 [CEN,2004b]. SPO is performed using the structural model assumptions

determined in Chapter 5.3 and using the computational software of ANSRuop.

SPO is essentially an extension of the “lateral force method” of static analysis, but in the

nonlinear regime. This method simulates the inertial forces due to a horizontal component of

the seismic action. These lateral forces Fi increase throughout the analysis and are applied in

small steps on the mass mi in proportion to the pattern of horizontal displacements, Φi. The

magnitude of the lateral loads is controlled by 𝑎 and magnified in each step.

𝐹𝑖 = 𝑎 𝑚𝑖Φi (5.1)

According to EC8 [CEN,2004b], pushover analysis can be performed using the “modal

pattern” which simulates the inertial forces of the first mode shape in the elastic regime. Since

the buildings in the current study meet the conditions of the linear static analysis an “inverted

triangular” lateral load pattern is applied. In this method the horizontal displacements Φi are

such that Φi = zi, where zi is the height of the mass mi above the level of the application of

the seismic action.

The N2 method is employed according Fajfar et. al. [2000] as adopted in EC8 [CEN,2004b].

This method combines the pushover analysis of the multi-degree-of-freedom (MDOF) model

with the response spectrum analysis of an equivalent single-degree-of-freedom (SDOF)

system. This method is formulated in the acceleration – displacement format thus it enables

the visualization of the relations between various quantities controlling the seismic response.

Thus using this method the ground accelerations at the top of the soil are related to seismic


47

demands for every step of the analysis. The demands are then compared against the limit

states according to Eurocode 8 – Part 3 [CEN, 2005], therefore the PGA value that causes

yielding and ultimate chord rotations and the ultimate shear force for each member on the

structure is computed. Also the damage indices (ratio of the damage measure demand to the

damage measure capacity for a member) can be easily obtained for every step of the analysis

and used to construct fragility curves.

5.2. Incremental Dynamic Analysis

Incremental dynamic analysis (IDA) is a method by Vamvatsikos and Cornell [2002] where

seismic demands are estimated accurately through a series of nonlinear time-history analyses

using several ground motion records scaled to multiple levels of intensity. IDA is used in

order to uncover the structural model’s behavior in the elastic phase, the yielding and the

nonlinear inelastic phase. The damage measures that are of interest are the peak chord rotation

demands and the shear force demands at member ends. IDA is performed using the structural

model assumptions determined in Chapter 5.3 and using the computational software of

ANSRuop.

As defined by Vamvatsikos and Cornell [2002], the scale factor (SF) is the scalar λ used in

order to uniformly scale up or down the amplitude of the accelerogram. The accelerograms

are scaled by a scalable Intensity Measure (IM) (i.e. excitation PGA).

𝜶𝝀 = 𝝀 . 𝜶𝟏 (5.2)

Where,

𝛼𝜆 is the scaled accelerogram time-history record

𝛼1 is the unscaled accelerogram time-history record

λ is the scale factor

The records were scaled so that they cover a range of PGA values which range from 0.05g to

0.95g with a step of 0.05g. The total number of analyses performed for each building sums up

to 266 having 14 analyses for each of the 19 selected IM points.

Eurocode 8-Part 3 [CEN, 2005] requires at least seven nonlinear dynamic analyses and then

the average response quantities from these analyses are used as the damage measure damands.

For this study 14 records have been selected as shown in Table 5.1 and Figure 5.2 in order to

take into account the differences in the characteristics of the ground motion. Seven historic

earthquakes were used to get semi-artificial bidirectional ground motion records for two

horizontal directions X and Y. Each accelerogram is modified to be compatible with a smooth

5%- damped elastic response spectrum. The spectrum consists of an acceleration sensitive

part for the periods of 0.2 to 0.6 sec, a velocity controlled part from 0.6 to 2 sec and a

displacement control part from 2 and beyond. The pseudo-acceleration spectra for the 14


48

accelerogram records are compared to the smooth 5%-damped elastic spectrum for a PGA of

1g as shown in Figure 5.1.

The damping matrix C is taken to be of Rayleigh type where C=aoM+a1K. ao and a1 are the

mass and stiffness proportional damping coefficients respectively. These are obtained using

the modal periods of the first and the second periods of the structure with the highest

participating mass in the horizontal direction. A damping ratio of 5% is used and thus with the

use of Rayleigh damping the viscous damping ratio is lower than 5% between the range of ω1

and ω2 and higher outside this range.

The numerical integration of the equation of motion was performed using the Newmark

method and the Newton-Rapson algorithm for the solution algorithm for the nonlinear

analysis problem.

Table 5.1: Accelerogram records used in the analysis

No Event Station Component

1 Imperial Valley, 1979 BondsCorner 140

2 Imperial Valley, 1979 BondsCorner 230

3 Loma Prieta, 1989 Capitola 000

4 Loma Prieta, 1989 Capitola 090

5 Kalamata, 1986 Kalamata X

6 Kalamata, 1986 Kalamata Y

7 Montenegro, 1979 Herceg Novi X

8 Montenegro, 1979 Herceg Novi Y

9 Friuli, 1976 Tolmezzo X

10 Friuli, 1976 Tolmezzo Y

11 Montenegro, 1979 Ulcinj (2) X

12 Montenegro, 1979 Ulcinj (2) Y

13 Imperial Valley, 1940 Elcentro Array #9 180

14 Imperial Valley, 1940 Elcentro Array #9 270


49

.

Figure 5.1 Pseudo-acceleration spectra for the semi-artificial input motions compared to the smooth target

spectrum (shown with thick black line)


50

Figure 5.2 Time-histories of accelerograms used in the analysis


51

5.3. Structural modelling for IDA and SPO

ANSRuop is the computational tool that is used in order to perform the modelling, seismic

response analysis and evaluation of the structures [Kosmopoulos et al., 2005]. It is an

improved and expanded version of ANSR-I which was developed at UC Berkeley [Mondkaret

al., 1975]. The software is used for the analysis of reinforced concrete structures and consists

of a user interface where the user can perform the various tasks. ANSRuop was used to

perform nonlinear time-history analysis and nonlinear static pushover analysis. This section

will explain the modelling assumptions taken for the members and the structure.

Key points of the modelling of the reinforced concrete members are:

For the modelling of all the reinforced concrete members inelasticity is lumped at the

ends. For monotonic loading the reinforced concrete members follow a bilinear Moment –

curvature envelope and for the cyclic loading the members follow the Takeda hysteretic

rules [Takeda et. al., 1970], modified to Litton [1975] and Otani [1974]. The chord

rotations and moments are calculated in accordance to the EC8 [CEN,2004b], taking into

account the confinement of the members.

Figure 5.3 Takeda model modified by Litton and Otani

Element elastic stiffness is taken as equal to the secant stiffness at yielding (EIeff). In order

to find this value the shear span at the yielding end of the element is required. The shear

span of the columns and the beams is taken as half the clear length between the beam-to-

column joints within the plane of bending. In positive or negative bending it is the average

secant-to-yielding stiffness at the two end sections. For walls the secant-to-yielding

stiffness of the bottom section is used with a shear span ratio of one-half the height from

the bottom of the section to the top of the wall in the building.

The walls are modelled as cantilever walls. Axial load acts on the walls due to its self-

weight and the floor loads. No mass is assigned due to its self-weight since it is taken into

account by the mass taken from the floor loads.

Masses for beams and columns are lumped at the nearest node of the element and are

taken from the action of the permanent and imposed loads acting uniformly on the floors.

No self-weight is assigned to the frame since it is taken into account in the floor loads.


52

Key points of the modelling of the structure are:

The perimeter beam and exterior columns are modelled such that they have half the elastic

rigidity of interior ones. Thus both interior and exterior beam and columns have the same

seismic chord rotations demands whereas perimeter beam and exterior columns have half

the elastic seismic moments of interior ones. Corner columns have one-quarter of elastic

rigidity of interior ones thus the corner columns have one quarter of the elastic seismic

moments of interior ones. This was modelled by applying an elastic seismic moment

modification factor equal to 0.5 or 0.25 accordingly.

One component of seismic action is considered along the X-axis direction.

The translational degree of freedom (DOF) parallel to the direction of the seismic action

(UX) is constrained for all nodes on each floor such that walls and frame share the same

displacements. Since the building is symmetric with no torsional effects, the translational

horizontal DOF perpendicular to the direction of the seismic action (UZ) and the

rotational DOF in the vertical axis (RY) and the horizontal axis parallel to the direction of

the seismic action (RX) are restrained. The translational DOF in the vertical axis (UY) and

the rotational DOF in the horizontal axis perpendicular to the direction of the seismic

action (RZ) are free.

Prismatic beams are used where effective beam width is used for the contribution of the

stiffness of the slab. The effective flange width of the T- beams on either side of the beam

is taken to be 0.6m having a constant width over the whole span of the beam. The flange

width is determined according to Eurocode 2 [CEN 2004a].

The strength and stiffness of the columns or walls are modelled independently in the two

bending planes. The axial load variation is taken into account for the variation of the

flexural properties.

Columns support the gravity loads within a tributary area extending up to beam mid-span.

All permanent and imposed loads per unit floor produce triangular distribution of loads on

beams.

P-δ effects are considered in the analysis through the linearized geometric stiffness matrix

of columns.

Due to the building’s symmetry only half of the building was used in the analysis to

reduce computational demands having a building plan of 25m x 12.5m. The beams

perpendicularly connected to the line of symmetry have half their length (2.5m) and no

columns are located on the line of symmetry. (see Figure 5.4 and Figure 5.5)

Columns and walls are assumed fixed at ground level.

Joints are considered rigid.


53

Figure 5.4 Structural model for a five – storey dual building taken from ANSRuop

Figure 5.5 Structural model for an eight – storey dual building taken from ANSRuop

5.4. Linear Static Analysis - “Lateral Force Method”

The linear elastic (equivalent) static analysis “lateral force method” was performed by

Papailia [2011] in order to carry out the design and the evaluation of the buildings for the

construction of the fragility curves. The method was performed according to Eurocode 8 –

Part 1 [CEN,2004b], where the horizontal component of the seismic action is distributed with

an assumed linear mode shape along the height of the building. This method is applied to

buildings which are both regular in plan and in elevation, if the building response is not

affected by higher modes. The base shear of the structure is determined according to the mass

of the building and the design or elastic spectrum at the 1st translational mode of the structure.

The design spectrum is used for the design of the buildings and the elastic spectrum for the

assessment.

𝑽𝒃 = 𝒎𝒆𝒇𝒇 𝑺𝒆,𝒅 𝑻𝟏 (5.3)

Where

meff is the effective mass of the building associated with the gravity loads


54

Se(T1) the elastic horizontal ground acceleration response spectrum at the fundamental period

Sd(T1) the design spectrum at the fundamental period

According to EC8, the elastic response spectrum Se(T) is defined by:

𝟎 ≤ 𝑻 ≤ 𝑻𝑪 ∶ 𝐒𝐞 𝐓 = 𝒂𝒈𝑺 𝟏 +𝑻

𝑻𝑩. (𝜼 𝟐. 𝟓 − 𝟏) (

5.4)

𝑻𝑩 ≤ 𝑻 ≤ 𝑻𝑪 ∶ 𝐒𝐞 𝐓 = 𝟐. 𝟓 𝑺𝒂𝒈𝜼 (

5.5)

𝑻𝑪 ≤ 𝑻 ≤ 𝑻𝑫: 𝐒𝐞 𝐓 = 𝟐. 𝟓 𝑺𝒂𝒈𝜼𝑻𝒄

𝑻 ( 5.6)

𝑻𝑫 ≤ 𝑻 ≤ 𝟒𝒔: 𝐒𝐞 𝐓 = 𝟐. 𝟓 𝑺𝒂𝒈𝜼 𝑻𝑪𝑻𝑫

𝑻𝟐 ( 5.7)

Where,

S is the soil factor

𝜂 is the damping correction factor

T the period of vibration of linear SDOF system

Tc the corner period of the constant spectral acceleration branch

𝑎𝑔 the design ground acceleration on type A ground

The design response spectrum Sd(T1) is defined by:

𝟎 ≤ 𝑻 ≤ 𝑻𝑪 ∶ 𝐒𝐝 𝐓 = 𝑺 𝒂𝒈 𝟐

𝟑+

𝑻

𝑻𝑩. (𝟐.𝟓

𝒒−

𝟐

𝟑) ( 5.8)

𝑻𝑩 ≤ 𝑻 ≤ 𝑻𝑪 ∶ 𝐒𝐝 𝐓 = 𝟐. 𝟓 𝑺𝒂𝒈

𝒒 ( 5.9)

𝑻𝑪 ≤ 𝑻 ≤ 𝑻𝑫: 𝐒𝐝 𝐓 = 𝟐. 𝟓 𝑺

𝒂𝒈

𝒒 𝑻𝒄

𝑻

≥ 𝛃 . 𝒂𝒈

( 5.10)

𝑻𝑫 ≤ 𝑻 ≤ 𝟒𝒔: 𝐒𝐝 𝐓 = 𝟐. 𝟓 𝑺

𝒂𝒈

𝒒 𝑻𝑪𝑻𝑫

𝑻𝟐

≥ 𝛃 . 𝒂𝒈

(

5.11)

𝑞 is the behaviour factor

β is the lower bound factor for the horizontal design spectrum


55

All spectrums are computed by taking spectrum as Type 1 of soil class C, thus TC=0.6sec and

the soil factor is 1.15.

For the computation of the fundamental period of the structure the Rayleigh quotient is being

used:

𝑻𝟏 = 𝟐𝝅 𝒎𝒊𝜹𝒊

𝟐

𝑭𝒊𝜹𝒊 (5.12)

Where,

i is the index of the degree of freedom,

mi is the mass of the floors

Fi is the lateral force applied to the corresponding degree of freedom

δi is the displacement obtained from the elastic analysis.

The base shear calculated in (5.3) is distributed along the height of the building. The

distribution of the lateral forces is given by:

𝑭𝒊 = 𝑽𝒃𝐳𝐢𝐦𝐢

𝐳𝐣𝐦𝐣

(5.13)

Where zi , zj is the height of the masses mi, mj above the level of application of the seismic

action. According to EC8, if T1<2Tc and the building has more than two storeys then Fi is

multiplied by a factor of 0.85.

The simplifying assumptions for the lateral force analysis method are as follows:

Members considered in wall-frame or frame buildings are the interior columns and

beams and walls in wall-frame buildings.

All beam ends in a storey of a frame have the same elastic seismic moments and

inelastic chord rotation demands.

Perimeter beams have one-half the rigidity of interior ones for the same storey but the

same inelastic seismic chord rotation demands.

Interior columns have the same elastic seismic moments and inelastic chord rotation

demands.

Exterior columns have one-half the elastic seismic moments of the interior columns

for the same storey but the same inelastic seismic chord rotation demands.

The axial force variation due to seismic action is neglected in interior columns.

The assessment of the seismic response is performed using the secant-to-yield-point

rigidity according to CEN [2005].

Vertical elements are considered fixed at ground level and as having negligible

bending moments due to gravity loads.

Beam column joints and floor diaphragms are taken as rigid.


56

P-δ effects are considered in the analysis.

Chapter 6: Assessment of Buildings

57

6. ASSESMENT OF BUILDINGS

The assessment of the buildings is done by the procedure according to Eurocode 8 - Part 3

[CEN, 2005]. The estimation of the damage measure capacities for each limit state and

computation of the damage measure demands for each analysis method are described in this

chapter. Two limit states are considered as specified by EC8 Part3 [CEN,2005] for “Damage

Limitation” which accounts for the yielding of the elements and the “No collapse” state which

accounts for the ultimate or collapse limit of the elements. The equations given in this chapter

are used for the assessment of the buildings and are adopted in the computational software

ANSRuop and by Papailia [2011] for the simplified analysis using the lateral force method.

6.1. Limit State of Damage Limitation (DL)

According to Eurocode 8 Part 3 [CEN, 2005], the capacity used for this limit state is the

yielding bending moment under the design value of the axial force. In order to compute the

yielding moment of the members, first the yield curvature should be calculated, which is

identified with the yielding of the tension reinforcement.

The yield curvature, 𝜑𝑦 , is given by:

𝜑𝑦 =𝑓𝑦𝐿

𝐸𝑠 1−𝜉𝑦 𝑑 ( 6.1)

Where,

fyL is the yield stress of the longitudinal bars

𝜉𝑦 is the neutral axis depth at yielding (normalized to the section effective depth, d),

given by:

𝜉𝑦 = (𝑎2𝐴2 + 2𝑎𝐵)1/2 − 𝑎𝐴 ( 6.2)

Where,

𝑎 is the ratio of the elastic moduli (steel to concrete) , Es

Ec

A and B are given by:


58

𝛢 = 𝜌1 + 𝜌2 + 𝜌𝑣 +𝑁

𝑏𝑑𝑓𝑦 𝑎𝑛𝑑 𝛣 = 𝜌1 + 𝜌2𝛿1 +

𝜌𝑣(1+𝛿1)

2+

𝑁

𝑏𝑑𝑓𝑦 ( 6.3)

Where,

𝜌1 and 𝜌2 the ratios of the tension and compression reinforcement respectively. The area

of any diagonal steel reinforcement is added multiplied by the cosine of their

angle.

𝜌𝑣 the ratio of the web reinforcement

𝛿1 the ratio of the distance of the centre of compression reinforcement from the

extreme compression fibre to the width of the compression zone, 𝑑1

𝑏.

N is the axial load

For members of high axial load ratio, ν=Ν/Αcfc, the curvature is:

𝜑𝑦 =1.8𝑓𝑐

𝐸𝑐 𝜉𝑦 𝑑 ( 6.4)

where the neutral axis depth at yielding, 𝜉𝑦 , is the same as before, but A and B becomes

𝛢 = 𝜌1 + 𝜌2 + 𝜌𝑣 −𝑁

1.8𝛼𝑏𝑑 𝑓𝑐 𝑎𝑛𝑑 𝛣 = 𝜌1 + 𝜌2𝛿1 +

𝜌𝑣(1+𝛿1)

2 ( 6.5)

The lower of the two 𝜑𝑦 values becomes the yield curvature. Thus the yield moment can be

computed as:

𝛭𝑦

𝑏𝑑3 = 𝜑𝑦{𝐸𝑐𝜉𝑦

2

2

1+𝛿1

2−

𝜉𝑦

3 +

𝐸𝑠 1−𝛿1

2 1 − 𝜉𝑦 𝜌1 + 𝜉𝑦 − 𝛿1 𝜌2 +

𝜌𝑣

6 1 − 𝛿1 ( 6.6)

The chord rotation at yielding according to Biskinis and Fardis [2010], adopted in Eurocode 8

– Part 3 is evaluated by:

For beams and columns with rectangular sections,

𝜃𝑦 = 𝜑𝑦𝐿𝑉+𝑎𝑣𝑧

3+ 0.0014 1 + 1.5

𝑕

𝐿𝑉 + asl

𝜑𝑦𝑑𝑏𝑓𝑦

8 𝑓𝑐 ( 6.7)

For walls,

𝜃𝑦 = 𝜑𝑦𝐿𝑣+𝑎𝑣𝑧

3+ 0.0013 + asl

𝜑𝑦𝑑𝑏𝑓𝑦

8 𝑓𝑐 ( 6.8)

Where,


59

φy is the yield curvature of the end section

avz is the tension drift of the bending moment diagram where:

o av = 1, if yield moment at the section exceeds the product of LV and the shear

resistance of the member considered without shear reinforcement according to

Eurocode 2 (CEN 2004). My > VR,c Lv. av = 0 if otherwise.

o z is the length of the internal lever arm taken equal to z = d-d1 in beams and

columns, z = 0.8h in walls with rectangular section.

asl asl=1 if slippage of longitudinal bars from anchorage zone beyond the

end section is possible. The contribution of bar pull-out from joints to the fixed

end rotation at member ends is considered when asl=1.

asl=0 if slippage is not possible

fy and fc steel yield stress and concrete strength respectively

d the effective depth of the full section.

Ls/h shear span ratio

dbL the mean diameter of the tension reinforcement.

The first term of the above equations relate to the theoretical yield curvature. It takes into

account the shift rule where the yielding of the tension reinforcement shifts up to the point of

the first diagonal crack leading to an increase in yield chord rotation. The second term of the

above expression relates to the experimental chord rotation at flexural yielding and the third

term of the expression accounts for the slippage of the longitudinal bars from the anchorage

zone to the end of the section.

For verifications carried out in terms of deformations, deformation demands obtained from

the analysis of the structural model require the use of the estimation of the effective cracked

stiffness of concrete at yielding. Thus according to EC8 [CEN,2005] the secant stiffness to the

member yield-point is used:

𝐸𝐼𝑒𝑓𝑓 =𝑀𝑦 𝐿𝑉

3𝜃𝑦 ( 6.9)

Where,

My is the yield moment using the mean material strengths.

LV is the member shear span which is the ratio of M/V at the member ends, thus it is the

distance of the member end to the point of zero moments.

θy is the yield chord rotation


60

6.2. Limit State of Near Collapse (NC)

The value of the total chord rotation capacity at ultimate of concrete members under cyclic

loading is taken from Biskinis and Fardis [2010] which is also adopted in Eurocode 8- Part 3.

The flexure-controlled ultimate chord rotation is equal to:

𝜃𝑢 = 𝜃y + 𝜃𝑢𝑚𝑝𝑙

( 6.10)

Where

𝜃𝑢𝑚𝑝𝑙

the plastic part of the chord rotation capacity of concrete members under cyclic

loading

𝜃𝑢𝑚𝑝𝑙 =

𝑎𝑠𝑡𝑕𝑏𝑤 1 − 0.525acy 1 + 0.6asl 1 −

0.052max 1.5; min 10;h

bw 0.2v

max 0.01;𝜔2

max 0.01;𝜔1 min(9;

Lv

h)

1/3

𝑓𝑐0.225

𝛼𝜌𝑠𝑥𝑓𝑦𝑤

𝑓𝑐 1.225100𝜌𝑑

( 6.11)

Where,

𝑎𝑠𝑡𝑕𝑏𝑤 is equal to 0.022 for heat-treated (Tempcore) steel

acy is equal to zero for monotonic loading and one for cyclic loading.

asl is equal to one if there is slip in the longitudinal reinforcement bars from their

anchorage beyond the section of maximum moment or zero if there is not.

h is the depth of the member

LV=M/V is the shear span ratio at the end of the section

ν =N/bhfc where b is the width of compression zone and N is the axial force

𝜔1, 𝜔2 is the mechanical reinforcement ratio of the tension and compression

longitudinal reinforcement respectively, including web reinforcement

Where, 𝜔1 = 𝜌1 + 𝜌𝑣 𝑓𝑦𝐿/𝑓𝑐 and 𝜔2 = 𝜌2𝑓𝑦𝐿/𝑓𝑐

𝑓𝑐 and 𝑓𝑦𝑤 the concrete compressive strength and the stirrup yield strength (MPa)

respectively obtained as mean values.

𝜌𝑠𝑥=Asx/bwsh is the ratio of transverse steel parallel to the direction x of the loading,


61

sh is the stirrup spacing.

𝜌𝑑 the steel ratio of the diagonal reinforcement in each diagonal direction

𝛼 the confinement effectiveness factor which is equal to:

𝛼 = 1 −𝑠𝑕

2𝑏𝑜 1 −

𝑠𝑕

2𝑕𝑜 1 −

𝑏𝑖2

6𝑕𝑜𝑏𝑜 ( 6.12)

Where,

𝑕𝑜 𝑎𝑛𝑑 𝑏𝑜 the dimension of confined core to the centreline of the hoop

bi the centreline spacing of longitudinal bars laterally restrained by a stirrup

corner or a cross tie along the perimeter of the cross section.

According to Eurocode 8 - Part 3 [CEN, 2005], the cyclic shear strength, VR as controlled by

the stirrups, for beams, columns and walls is according to the following expression. (units are

MN and meters).

𝑉𝑅 = 𝑕−𝑥

2𝐿𝑉min 𝑁; 0.55𝐴𝑐𝑓𝑐 + 1 − 0.05𝑚𝑖𝑛 5; 𝜇∆

𝑝𝑙 . 0.15𝑚𝑎𝑥 0.5; 100𝜌𝑡𝑜𝑡 1 −

0.16𝑚𝑖𝑛5;𝐿𝑉𝑕𝑓𝑐𝐴𝑐+𝑉𝑊 ( 6.13)

Where,

h is the depth of the cross section

x is the compression zone depth

LV=M/V is the ratio of moment/shear at the end of the section

N is the compression axial force

Ac is the cross sectional area taken as bwd for a rectangular web of width bw and

structural depth of d.

𝑓𝑐 the concrete compressive strength (MPa) obtained as mean values. For primary

seismic elements it is divided by a partial factor for concrete.

𝜌𝑡𝑜𝑡 the longitudinal reinforcement ratio

𝜇∆𝑝𝑙

the plastic demand of ductility demand, which is the ratio of the plastic part of

the chord rotation, ζ, normalized to the chord rotation at yielding, ζy.

Vw is the contribution of the transverse reinforcement to shear resistance taken

equal to


62

𝑉𝑊 = 𝜌𝑤𝑏𝑤𝑧𝑓𝑦𝑤 ( 6.14)

Where,

𝜌𝑤 the transverse reinforcement ratio

z length of the internal lever arm

𝑓𝑦𝑤 yield stress of the transverse reinforcement. For primary seismic elements it is divided

by the partial factor for steel

The shear strength of a concrete wall, 𝑉𝑅, should not exceed the value which corresponds to

the failure due to web crushing, 𝑉𝑅,𝑚𝑎𝑥 . This limit under cyclic loading is given by the

following expression (units are in MN and meters)

𝑉𝑅,𝑚𝑎𝑥 =

0.85 1 − 0.06𝑚𝑖𝑛 5; 𝜇∆𝑝𝑙 1 + 1.8𝑚𝑖𝑛 0.15;

𝑁

𝐴𝑐𝑓𝑐 1 +

0.25max 1.75;100𝜌𝑡𝑜𝑡)1−0.2𝑚𝑖𝑛2;𝐿𝑉𝑕𝑓𝑐𝑏𝑤𝑧

( 6.15)

fc is in MPa, bw and z are in meters and VR,max is in MN.

If web crashing occurs prior to flexural yielding then the shear strength under cyclic loading is

obtained when 𝜇∆𝑝𝑙 = 0.

If the shear span ratio at the end section in a concrete column is less than or equal to 2 (Ls/h ≤

2.0) then its shear strength, 𝑉𝑅, should not exceed the value which corresponds to failure by

the crushing of the web along the diagonal of the column after flexural yielding, 𝑉𝑅,𝑚𝑎𝑥 ,

which under cyclic loading may be calculated as:

𝑉𝑅,𝑚𝑎𝑥 =

4

7 1 − 0.02𝑚𝑖𝑛 5; 𝜇∆

𝑝𝑙 1 + 1.35𝑁

𝐴𝑐𝑓𝑐 1 + 0.45(100𝜌𝑡𝑜𝑡 ) 𝑚𝑖𝑛 40; 𝑓𝑐 𝑏𝑤 𝑧 𝑠𝑖𝑛2𝛿

( 6.16)

Where 𝛿 is the angle between the diagonal and the axis of the column: tan 𝛿 =𝑕

2𝐿𝑉.


63

6.3. Estimation of damage measure demands

Demands are obtained from the analysis of the structural model for the seismic action

depending on the analysis method. It is reminded that the damage measure demands in this

study are the peak chord rotation and shear force demands. The peak chord rotation is defined

as the member drift ratio; the deflection at the end of the shear span divided by the shear span.

In the nonlinear time-history analysis the wall shear force demands in wall-frame buildings

are not amplified to capture the effects of higher modes since they are taken into account in

the analysis. In the lateral force method by Papailia [2011] once plastic hinge starts forming in

the base of the wall the shear force demands are amplified to take into account higher mode

effects according to the proposal in Keintzel [1990] adopted also by CEN [2004a] for DC H

walls. Once plastic hinges starts forming in the structure the shear forces in beams and

columns are calculated from the plastic mechanism and the yield moments of the sections.

Chapter 7: Methodology of Fragility Analysis

64

7. METHODOLOGY OF FRAGILITY ANALYSIS

The seismic fragility curves of regular reinforced concrete frame and wall-frame buildings are

studied. Three-dimensional models of the full buildings are used in order to construct the

fragility curves using the nonlinear static “pushover” analysis (SPO) and the dynamic analysis

(IDA). These results are then compared against the fragility curves obtained using the “lateral

force method” by Papailia [2011]. The results are presented in terms of fragility curves for

two member limit states of yielding and ultimate deformation in bending or shear.

7.1. Damage Measures

The damage measures (DM) used in order to obtain the fragility curves in this study are the

chord rotations and the shear force demands. The chord rotations are found for the two

damage states of yielding and ultimate conditions and the shear forces are found for the

ultimate condition due to shear failure. The shear forces are taken from outside or inside the

plastic hinge. The mean values for the capacities of the two damage states are obtained using

Eurocode 8 Part 3 [CEN,2005] and are consistent with the capacities for flexure of Biskinis

and Fardis [2010a,b] and for shear of Biskinis et al. [2004] as presented in Chapter 6.

The values for DM-demand for each member are obtained through the deterministic seismic

analysis (for the three seismic analysis methods).

The damage measure demands obtained from the LFM are taken for each IM through

a deterministic static analysis using an inverted triangular pattern as presented in

Papailia [2011].

In the dynamic analysis (IDA) the mean damage measure demands from the 14 semi-

artificial accelerogram dynamic analyses are obtained for each IM (i.e. the excitation

PGA).

The damage measure demands obtained for the SPO analysis are obtained from

deterministic nonlinear static analysis using the inverted triangular distribution pattern.

All analysis procedures follow the methods and approaches provided by CEN (2005) and

the mean material properties were used (fcm=fck+8MPa and fym=1.15fyk, see Section 3.3).


65

7.2. Exclusion of unrealistic results for IDA

Certain damage measure demands obtained from IDA are much higher than the capacities of

the members. This may lead to erroneous response estimates. This error comes from

numerical instability thus this may lead to unrealistic response values. These values need to be

neglected when calculating the mean and variance values of these damage indices which are

required to construct the fragility curves. Therefore, damage indices (ratio of the DM-

demands to DM-capacities) larger than a threshold of 200% of the mean damage indices per

IM (i.e. excitation PGA) are neglected when calculating the statistical parameters. Figure 7.1

shows an example where the damage indices above the continuous line on the plot (i.e. the

threshold) are neglected. The zero-value damage indices are due to incomplete analyses and

are also neglected.

Figure 7.1 Exclusion of unrealistic results in IDA (damage indices above continuous line are

neglected)

7.3. Determination of variability

The coefficient of variation (CoV) reflects all the variability and uncertainty regarding the

used models, materials and geometries and the characteristics of seismic input.

The variation of the DM-capacities reflects the uncertainty in the models that are used to

estimate the mean capacity values and the scatter of the material and the geometric properties.

These CoV values are taken from Biskinis et al. [2004] and Biskinis and Fardis [2010a,b], and

are presented in Table 7.1.

The CoV values for the DM-demands used for the SPO and LFM are different than the ones

taken for IDA:


66

The CoV values for the damage measure demands used for IDA are found explicitly

from the analysis. In the dynamic analysis the variability of the DM-demand of the 14

semi-artificial accelerograms cover the variability of the ground motion and of damage

measure demand.

For the computation of the fragility curves using the LFM and SPO the CoV values

for DM-demands cannot be found explicitly from the analysis. Thus the CoV values

for the chord rotation demands are based on extensive comparisons of inelastic to their

elastic estimates of chord rotation demands in height wise regular multi-storey

buildings by Panayiotakos and Fardis [1999], Kosmopoulos and Fardis [2007]. The

coefficient of variation values for the shear force demands listed are based on

parametric studies. These CoV values are presented in Table 7.2.

The CoV values per storey in terms of intensity measure (i.e. PGA) obtained from IDA are

presented in Appendix C1. On the same plots the straight line represents the CoV values for

the damage measure demands and the CoV values of the spectral value taken from Table 7.2.

Figure 7.2 and Figure 7.3 illustrate examples of the dispersion values per IM (i.e. PGA) for a

frame and a wall-frame dual building. It can be observed that the CoV-values determined

through IDA are lower than the ones determined from previous studies (shown in a straight

line on the plot representing CoV-values of DM-demand and spectral value). Also the mean

dispersions of DM-demands for beams and columns are slightly higher in wall-frame

buildings than in frames. There is a larger scatter of CoV-values in the storeys of dual

buildings compared to pure frame.


67

Figure 7.2 Coefficient of variation (CoV) of DM-demands for five-storey frame building designed to DC M

and PGA=0.20g


68

Figure 7.3 Coefficient of variation (CoV) of DM-demands for five-storey frame-equivalent building

designed to DC M and PGA=0.20g


69

7.4. Construction of fragility curves

For the construction of fragility curves the probability of a damage measure (DM) demand to

exceed a certain DM-capacity is expressed in terms of Peak Ground Acceleration (PGA).

PGA was used instead of other intensity measures in order to be consistent with the use of the

design acceleration as a design parameter.

As mentioned previously the members’ fragilities are expressed for the damage states of

yielding and ultimate. The member yielding or ultimate damage state in flexure is reached

when the chord rotation at the member end exceeds the yielding or ultimate flexural capacity.

The shear failure is when the shear force exceeds the shear capacity of the member, where the

shear capacity is a function of the rotation ductility demand at the member end.

The fragility of the member is obtained for each IM (i.e. PGA) from deterministic analysis

and is the conditional-on-IM probability that the demand of the given damage measure will

exceed its capacity. It is assumed that the fragility curves are expressed in log-normal

distribution. Based on this assumption the cumulative probability of occurrence can be

expressed as:

𝑃 𝐷′ ≥ 𝐶 = 1 −Φ 𝑙𝑛 𝜆

𝛽𝐷2 +𝛽𝑃𝐺𝐴

2 +𝛽𝐶2 ( 7.1)

Where,

D’ is the damage measure demand (DM- demand)

C is the threshold damage measure capacity for a limit state (DM- capacity)

λ is the mean damage index for each IM. The damage indices obtained using

IDA are the mean of the 14 damage indices per IM (i.e. PGA). The damage

indices for the SPO and LFM are found from the analysis.

Φ is the standard normal distribution

𝛽𝐶 𝑎𝑛𝑑 𝛽𝐷 are the standard deviation for the capacity and the demand, such that 𝛽𝐶 =

ln(1 + 𝛿𝐶2 ) and 𝛽𝐷 = ln(1 + 𝛿𝐷

2 ).

𝛽𝑃𝐺𝐴 it is the standard deviation for the spectral value (𝛿𝑃𝐺𝐴 ) given in Table 7.2.

𝛽𝑃𝐺𝐴 = ln(1 + 𝛿𝑃𝐺𝐴2 ). It is not used for IDA, since dispersion is taken

explicitly from the analysis; i.e. 𝛽𝑃𝐺𝐴 = 0 for IDA.

𝛿𝐶 is the coefficient of variation for the DM-capacity values found in Table 7.1.

𝛿𝐷 is the coefficient of variation for the DM-demand values and values from

Table 7.2. For the IDA method they are found explicitly from the analysis.


70

𝛿𝐷 is the coefficient of variation for the spectral value.

Table 7.1 Values of coefficient of variation for DM-capacity values

Capacity CoV

𝛿𝐶1 Beam or column chord rotation at yielding 0.33

𝛿𝐶2 Beam or column chord rotation at ultimate 0.38

𝛿𝐶3 Shear resistance in diagonal tension (inside or outside

plastic hinge)

0.15

𝛿𝐶4 Wall chord rotation at yielding of the base 0.40

𝛿𝐶5 Wall chord rotation at ultimate of the base 0.32

𝛿𝐶6 Wall shear resistance in diagonal compression 0.175

Table 7.2 Values of coefficient of variation for DM-demand values

Demand CoV

𝛿𝐷1 Beam chord rotation demand, for given spectral

value at the fundamental period

0.25

𝛿𝐷2 Column chord rotation demand, for given spectral


0.20

𝛿𝐷3 Wall chord rotation demand, for given spectral


0.25

𝛿𝐷4 Beam shear force demand, for given spectral value

at the fundamental period

0.10

𝛿𝐷5 Column shear force demand, for given spectral


0.15

𝛿𝐷6 Wall shear force demand, for given spectral value

at the fundamental period

0.20

𝛿𝑃𝐺𝐴 Spectral value, for given PGA and fundamental

period

0.25

Chapter 8: Results and Discussion

71

8. RESULTS AND DISCUSSION

The results from the analysis of the structural models and the member fragility curves for the

types of buildings examined are discussed in this chapter. Section 8.1 presents the modal

analysis results from the three-dimensional structural models. Section 8.2 indicates the

median PGAs (g) at attainment of the damage states of each member for the three analysis

methods. Section 8.3 discusses the fragility results for wall-frame dual systems and Section

8.4 for frame systems. The differences in fragility curves according to the different design

parameters (see Section 3.1) are further discussed. Section 8.5 presents the comparison of the

member fragility curves for the three different analysis methods. It is reminded that the

methods of analysis include the Incremental Dynamic Analysis (IDA) and the Static Pushover

Analysis (SPO) and these were compared against a simplified method using the Lateral force

method (LFM) by Papailia [2011].

In the current chapter only indicative results will be shown in order to draw conclusions on

the results of the analysis. Appendix A1 presents the member fragility curves of all the

examined buildings analysed using IDA and in Appendix A2 using the SPO analysis.

Appendix A3 presents the wall member fragility curves for shear ultimate state for the

different methods. In LFM analysis wall fragility curves for shear failure include results with

and without inelastic amplifications to take into account higher mode effects.

Appendix B1 presents the member fragility curves for the three different methods and

Appendix B2 presents the comparison of the member fragility curves for the most critical

member for the three methods. Appendix C1 presents the coefficient of variation values per

IM (i.e. excitation PGA) used in the construction of the fragility curves and Appendix C2

presents the damage indices (ratio of DM-demands to DM-capacities) per IM for each

member.


72

8.1. Modal analysis results

The modal periods and participating masses of the structural model used for the three modes

with the largest modal participation mass percentage is given in the following tables. The

modal periods of the structure are obtained using the effective stiffness of the members using

the structural software of ANSRuop.

Table 8.1 Modal periods and participating masses for frame systems

Storeys Design

PGA

DC Mode T (sec) Effective modal

mass (%)

5 0.20g M 1

2

3

1.91

0.56

0.27

78.17

12.72

5.31

5 0.25g M 1

2

3

1.72

0.52

0.26

79.21

11.88

5.35

5 0.25g H 1

2

3

1.69

0.51

0.26

80.08

11.46

4.98

Table 8.2 Modal periods and participating masses for frame-equivalent dual systems

Storeys Design

PGA


mass (%)

5 0.20g M 1

2

3

1.99

0.56

0.26

75.42

13.22

6.03

5 0.25g M 1

2

3

1.66

0.45

0.20

73.60

14.37

5.62

5 0.25g H 1

2

3

1.63

0.46

0.22

74.4

13.44

6.32

8 0.20g M 1

2

3

2.61

0.72

0.33

70.87

13.92

6.22

8 0.25g M 1

2

3

2.50

0.70

0.34

71.72

13.03

5.98


73

Table 8.3 Modal periods and participating masses for wall-equivalent dual systems

Storeys Design

PGA


mass (%)

5 0.20g M 1

2

3

1.83

0.48

0.23

73.12

14.59

6.63

5 0.25g M 1

2

3

1.46

0.37

0.15

71.52

15.84

6.64

5 0.25g H 1

2

3

1.50

0.40

0.17

72.39

15.19

6.64

8 0.20g M 1

2

3

2.49

0.67

0.29

69.62

14.58

6.50

8 0.25g M 1

2

3

2.32

0.67

0.33

69.13

13.68

6.45

Table 8.4 Modal periods and participating masses for wall dual systems

Storeys Design

PGA


mass (%)

5 0.20g M 1

2

3

1.62

0.39

0.16

70.66

16.46

6.88

5 0.25g M 1

2

3

1.24

0.28

0.11

69.22

17.65

7.19

5 0.25g H 1

2

3

1.15

0.25

0.10

68.56

18.14

7.30

8 0.20g M 1

2

3

2.11

0.51

0.21

67.57

16.12

6.77

8 0.25g M 1

2

3

1.92

0.46

0.19

67.34

16.45

6.81

8 0.25g H 1

2

3

1.57

0.33

0.13

65.22

17.77

7.17


74

The modal analysis results illustrate that the buildings designed for higher design peak ground

acceleration or ductility class generally slightly reduces the fundamental period of the

structure is; i.e. making the structure stiffer. As the proportion of total base shear taken by the

walls increases, the effective modal mass percentage decreases at the first mode and increases

for higher modes. Design for a higher PGA reduces effective modal mass percentage at the

fundamental period and increases at the higher modes.

8.2. Median PGAs at attainment of the damage state for the three methods

The median PGAs at attainment of the damage states for the members indicate clearly the

differences between the three analysis methods and the differences when designing to

different design parameters (Table 8.5 to Table 8.11). The median PGAs indicate the PGA

values for 50% probability of exceeding a certain damage state in each member. A dash (-)

indicates that the median PGA is larger than 1g. Member median PGA at attainment of the

damage state is presented for members in flexure and in shear. Discussion on the results

shown in these tables is made in Section 8.3, Section 8.4 and Section 8.5.

Table 8.5 Median PGA (g) at attainment of the damage state in 5-storey frame systems

design

PGA

DC

Analysis

method

Beam

Yielding

Beam

Ultimate

(flex)

Beam

Ultimate

(shear)

Column

Yielding

Column

Ultimate

(flex)

Column

Ultimate

(shear)

0.20g M LFM 0.14g 0.65g - 0.84g - -

SPO 0.12g 0.70g - 0.69g - -

IDA 0.14g 0.74g - 0.85g - -

0.25g M LFM 0.16g 0.79g - 0.74g - 0.95g

SPO 0.16g 0.78g - 0.70g - -

IDA 0.19g 0.74g - 0.81g - -

0.25g H LFM 0.13g 0.70g - 0.68g - -

SPO 0.17g 0.80g - 0.60g - -

IDA 0.19g 0.83g - 0.64g - -

Table 8.6 Median PGA (g) at attainment of the damage state in 5-storey frame-equivalent systems

design

PGA

DC

Analysis

method

Beam

Yielding

Beam

Ultimate

(flex)

Column

Yielding

Column

Ultimate

(flex)

Wall

Yielding

Wall

Ultimate

(flex)

Wall

Ultimate

(shear)

0.20g M LFM 0.14g 0.62g 0.35g - 0.09g 0.35g 0.25g

SPO 0.13g 0.75g 0.42g 0.82g 0.06g 0.29g -

IDA 0.18g 0.82g 0.52g - 0.08g 0.38g 0.94g

0.25g M LFM 0.18g 0.83g 0.39g 0.94g 0.11g 0.43g 0.19g

SPO 0.19g 0.68g 0.52g - 0.09g 0.39g -

IDA 0.22g 0.75g 0.46g - 0.10g 0.43g 0.39g

0.25g H LFM 0.14g 0.83g 0.38g - 0.10g 0.44g 0.38g

SPO 0.16g 0.77g 0.44g - 0.09g 0.41g -

IDA 0.19g 0.94g 0.44g - 0.10g 0.47g 0.90g


75

Table 8.7 Median PGA (g) at attainment of the damage state in 5-storey wall-equivalent dual systems

design

PGA

DC

Analysis

method

Beam

Yielding

Beam

Ultimate

(flex)

Column

Yielding

Column

Ultimate

(flex)

Wall

Yielding

Wall

Ultimate

(flex)

Wall

Ultimate

(shear)

0.20g M LFM 0.14g 0.73g 0.38g 0.98g 0.10g 0.38g 0.19g

SPO 0.18g 0.87g 0.46g 0.95g 0.09g 0.37g -

IDA 0.19g 0.71g 0.45g - 0.11g 0.45g 0.42g

0.25g M LFM 0.19g 0.95 0.43g - 0.11g 0.44g 0.18g

SPO 0.25g - 0.53g - 0.11g 0.43g -

IDA 0.25g 0.84g 0.46g - 0.13g 0.46g 0.37g

0.25g H LFM 0.14g 0.83g 0.38g - 0.10g 0.44g 0.38g

SPO 0.20g 0.81g 0.48g - 0.09g 0.45g -

IDA 0.22g 0.76g 0.40g - 0.11g 0.45g 0.53g

Table 8.8 Median PGA (g) at attainment of the damage state in 5-storey wall systems

design

PGA

DC

Analysis

method

Beam

Yielding

Beam

Ultimate

(flex)

Column

Yielding

Column

Ultimate

(flex)

Wall

Yielding

Wall

Ultimate

(flex)

Wall

Ultimate

(shear)

0.20g M LFM 0.17g 0.84g 0.42g - 0.11g 0.41g 0.17g

SPO 0.19g 0.78g 0.47g - 0.11g 0.39g -

IDA 0.20g 0.65g 0.38g - 0.13g 0.39g 0.33g

0.25g M LFM 0.22g - 0.47g - 0.13g 0.53g 0.19g

SPO 0.29g - 0.61g - 0.16g 0.54g -

IDA 0.24g - 0.51g - 0.18g 0.39g 0.29g

0.25g H LFM 0.25g - 0.57g - 0.17g 0.71g 0.51g

SPO 0.24g 0.95g 0.72g - 0.18g 0.62g -

IDA 0.28g - 0.54g - 0.20g 0.57g 0.72g

Table 8.9 Median PGA (g) at attainment of the damage state in 8-storey frame-equivalent dual systems

design

PGA

DC

Analysis

method

Beam

Yielding

Beam

Ultimate

(flex)

Column

Yielding

Column

Ultimate

(flex)

Wall

Yielding

Wall

Ultimate

(flex)

Wall

Ultimate

(shear)

0.20g M LFM 0.14g 0.64g 0.35g - 0.09g 0. 31g 0.58g

SPO 0.25g 0.86g 0.76g - 0.07g 0.29g -

IDA 0.27g 0.84g 0.77g - 0.09g 0.39g 0.88g

0.25g M LFM 0.14g 0.78g 0.33g - 0.09g 0.33g 0.51g

SPO 0.28g 0.77g 0.71g - 0.06g 0.30g -

IDA 0.31g 0.94g 0.77g - 0.09g 0.40g 0.77g


76

Table 8.10 Median PGA (g) at attainment of the damage state in 8-storey wall-equivalent dual systems

design

PGA

DC

Analysis

method

Beam

Yielding

Beam

Ultimate

(flex)

Column

Yielding

Column

Ultimate

(flex)

Wall

Yielding

Wall

Ultimate

(flex)

Wall

Ultimate

(shear)

0.20g M LFM 0.13g 0.70g 0.44g - 0.11g 0.43g 0.18g

SPO 0.24g 0.88g 0.83g - 0.10g 0.47g -

IDA 0.29g 0.70g 0.67g - 0.12g 0.53g 0.61g

0.25g M LFM 0.20g 0.92g 0.38g - 0.12g 0.48g 0.18g

SPO 0.27g 0.82g 0.83g - 0.12g 0.50g -

IDA 0.30g 0.81g 0.57g - 0.14g 0.54g 0.67g

Table 8.11 Median PGA (g) at attainment of the damage state in 8-storey wall systems

design

PGA

DC

Analysis

method

Beam

Yielding

Beam

Ultimate

(flex)

Column

Yielding

Column

Ultimate

(flex)

Wall

Yielding

Wall

Ultimate

(flex)

Wall

Ultimate

(shear)

0.20g M LFM 0.17g 0.89g 0.49g - 0.12g 0.47g 0.18g

SPO 0.22g 0.84g 0.84g - 0.13g 0.47g -

IDA 0.24g 0.74g 0.64g - 0.14g 0.39g 0.29g

0.25g M LFM 0.20g - 0.44g - 0.14g 0.53g 0.19g

SPO 0.37g 0.94g 0.56g - 0.13g 0.50g -

IDA 0.35g 0.84g 0.45g - 0.14g 0.40g 0.35g

8.3. Fragility curve results for wall-frame dual systems

The fragility curves of members for prototype plan- and height-wise very regular reinforced

concrete wall-frame buildings are discussed in this section for the results obtained from IDA

and SPO. Parameters that were studied include the number of storeys, the level of Eurocode 8

design (in terms of design peak ground acceleration and ductility class) and the percentage of

seismic base shear taken by the walls. The member fragility curves of all the buildings

examined for the analysis performed using nonlinear dynamic analysis are presented in

Appendix A1 and for nonlinear static pushover analysis in Appendix A2.

Figure 8.1, Figure 8.13, Figure 8.14, Figure 8.16, Figure 8.20, Figure 8.21 and Figure 8.22

refer to examples of wall-frame buildings. The first column in each set concerns the beams,

the second the column and the third the walls. The first row in each set is for yielding and the

second for ultimate state. The fragility curves of beams and columns for the wall-frame

systems are presented for the ultimate state in flexure since it is more critical than shear

failure. In frame systems the envelope of flexural and shear ultimate damage state for beams

and columns is presented. Fragility curves of walls in the ultimate state are the envelope of the

ultimate damage state in flexure and shear.


77

The conclusions for the wall-frame (dual) systems are:

Beams are much more likely to reach the ultimate damage state than columns. (see

Figure 8.1).

Figure 8.1 Fragility curves for five-storey wall-equivalent building designed to PGA=0.20g and DC M

analyzed using IDA method

Walls are the most critical members in every design scenario for both yielding and

ultimate damage state.

For the analysis performed using IDA, wall failure is usually more critical in shear

than in flexure, except in the following cases:

o Eight-storey frame-equivalent and wall-equivalent dual systems (see Figure

8.2 and Figure 8.3).

o Five-storey frame-equivalent system designed to 0.20g and DC M (see Figure

8.4 (left)).

o Five-storey buildings designed to 0.25g and DC H. (see Figure 8.4 (right) and

Figure 8.5).

The wall fragility curves of wall-equivalent dual buildings in the ultimate state are

similar for shear and flexure failure. (see Figure 8.2 to Figure 8.4)

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th


78

Figure 8.2 Fragility curves of walls for eight-storey frame-equivalent (left) and wall-

equivalent building (right) designed to PGA=0.20g and DC M analyzed using IDA method

Figure 8.3 Fragility curves of walls for eight-storey frame-equivalent (left) and wall-

equivalent building (right) designed to PGA=0.25g and DC M analyzed using IDA method

Figure 8.4 Fragility curves of walls for five-storey frame-equivalent building designed to PGA=0.20g and

DC M (left) and wall building designed to DC H and PGA=0.25g (right) analyzed using IDA method


79

Figure 8.5 Fragility curves of walls for five-storey frame-equivalent (left) and wall-equivalent

(right) buildings designed to DC H and PGA=0.25g analyzed using IDA method

The fragility curves of beams and columns, for the results taken from IDA method, are

presented for frame-equivalent, wall-equivalent and wall systems in Figure 8.6, Figure

8.7, Figure 8.8 and Figure 8.9. As it can be observed, the fragility results of beams for

both damage states show that the middle-storey beams have the highest fragility. The

top-storey beams have the lowest in the yielding state and the first-storey beams the

lowest in the ultimate state (see Figure 8.6 and Figure 8.8). The first-storey columns

are the most critical in five-storey buildings (see Figure 8.7) and the first- and top-

storey columns in eight-storey buildings (see Figure 8.9). The middle-storey columns

are the least fragile in all buildings for both damage states (see Figure 8.7 and Figure

8.9).

As the proportion of the total base shear taken by the wall increases, the fragility of the

middle-storey columns in both damage states increases. This observation holds for

five- and eight-storey wall-frame buildings (see Figure 8.7 and Figure 8.9).

As the proportion of the total base shear taken by the wall increases, the fragility of the

lower- and top-storey beams does not significantly change in the yielding state and

increases in the ultimate state. This observation holds for five- and eight-storey wall-

frame buildings (see Figure 8.6 and Figure 8.8).


80

Figure 8.6 Beam fragility curves for a) yielding and b) ultimate state of a five-storey frame-equivalent

(left), wall-equivalent (middle) and wall system (right) building designed to DC M and PGA=0.20g

analyzed with IDA

Figure 8.7 Column fragility curves for c) yielding and d) ultimate state of a five-storey frame-

equivalent (left), wall-equivalent (middle) and wall system (right) building designed to DC M and

PGA=0.20g analyzed with IDA

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th


81

Figure 8.8 Beam fragility curves for a) yielding and b) ultimate state of a eight-storey frame-equivalent


analyzed with IDA

Figure 8.9 Column fragility curves for c) yielding and d) ultimate state of a eight-storey frame-equivalent


analyzed with IDA

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

1st 2nd 3rd 4th5th 6th 7th 8th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4



82

The differences between fragility curves for different design parameters are:

1) Design ductility class

Design to DC M in lieu of DC H the fragility of beams may reduce against

yielding and increase against ultimate state. However, such effects are neither

systematic nor marked (see Figure 8.10, Figure 8.11 and Figure 8.12).

Design to DC M in lieu of DC H column fragility is reduced in frame-equivalent

and wall-equivalent systems (see Figure 8.10 and Figure 8.11) but increased in

wall systems (see Figure 8.10, Figure 8.11 and Figure 8.12).

Wall fragility in the yielding damage state does not significantly change and in the

ultimate damage state (the envelope of flexure and shear collapse) is higher for DC

M walls. (see Figure 8.10, Figure 8.11 and Figure 8.12)


83

Figure 8.10 Fragility curves for most critical members of five–storey frame-equivalent

building designed to PGA=0.25g and DC M analyzed using IDA method

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

DC M DC H


84

Figure 8.11 Fragility curves for most critical members of five–storey wall-equivalent building

designed to PGA=0.25g and DC M analyzed using IDA method

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

DC M DC H


85

Figure 8.12 Fragility curves for most critical members of five–storey wall building designed

to PGA=0.25g and DC M analyzed using IDA method

2) Height of the building

Taller buildings exhibit lower fragilities for beams and columns and similar

fragilities for walls in both damage states (see Figure 8.13). The latter is observed

except in the case of the eight-storey wall building designed to PGA=0.25g and

DC M which has lower fragility for beams but higher fragility for columns and

walls. (see Figure 8.14)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

DC M DC H


86

Figure 8.13 Member fragility curves of frame-equivalent dual systems designed to PGA=0.25g and DC M

for: (top) five – storey; (bottom) eight-storey using IDA method

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

0 0.2 0.4 0.6 0.8 10

0.51

1st 2nd 3rd 4th

5th 6th 7th 8th


87

Figure 8.14 Member fragility curves for wall systems designed to PGA=0.25g and DC M curves of: (top)

five – storey; (bottom) eight-storey using IDA method

3) Proportion of total base shear taken by the wall

Wall-equivalent dual and wall buildings have similar fragilities for beams and

columns but lower than frame-equivalent dual systems since the deformation

demand of the frame is higher. Wall-equivalent dual systems are in-between but

closer to wall dual systems. (see Figure 8.15 - Wall ultimate damage state is

presented for both (f) flexure and (g) shear collapse ).

As the proportion of total base shear taken by the wall increases, the walls in

yielding and ultimate damage state in flexure have lower fragilities but higher for

the ultimate state in shear. (see Figure 8.15).

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

0 0.2 0.4 0.6 0.8 10

0.51

1st 2nd 3rd 4th

5th 6th 7th 8th


88

Figure 8.15 Member fragility curves for a five-storey frame-equivalent (FE), wall-equivalent (WE), wall

dual (WS) system designed to PGA=0.20g and DC M using SPO method for most critical storey members.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

FE WE WS


89

4) Design peak ground acceleration (PGA)

Design for a higher PGA reduces fragility of beams in both damage states and may

slightly increase fragility of columns; however this effect is neither systematic nor

marked. Wall fragility is not significantly changed when designing for higher

PGA. (see Figure 8.16 and Figure 8.17)

Figure 8.16 Fragility curves of eight–storey frame-equivalent building designed to DC M and: (top)

PGA=0.20g; (bottom) PGA=0.25g analyzed using IDA method

0 0.2 0.4 0.6 0.8 10

0.51

1st 2nd 3rd 4th

5th 6th 7th 8th


90

Figure 8.17 Member fragility curves for a eight-storey wall-equivalent system designed to DC M and for

PGA=0.20g and PGA=0.25g using IDA method for most critical storey members.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

PGA=0.20g PGA=0.25g


91

8.4. Fragility curve results for frame systems

The construction of fragility curves of plan- and height-wise very regular reinforced concrete

frame buildings were also examined. Parameters that were studied include the level of

Eurocode 8 design (in terms of ductility class and design PGA). The following conclusions

can be drawn based on the results of the frame systems:

Frames give satisfactory fragility results even beyond their design PGAs. (see Table

8.5)

Beams yield before their design PGA whereas the columns remain elastic well beyond

the design PGA. Also beams are much more likely to reach the ultimate damage state

than columns. (see Figure 8.18 to Figure 8.22 and Table 8.5).

Design for higher PGA reduces only slightly the fragilities of beams and columns in

yielding damage state and may increase fragility in the ultimate damage state (see

Figure 8.18).

Design to DC M instead of DC H may reduce slightly the fragility of beams and

columns against yielding, but may increase that of beams against ultimate. (see Figure

8.19)

Wall-frame (dual) systems have, in general, higher fragility than frame systems for

columns and lower for beams (see Figure 8.20, Figure 8.21 and Figure 8.22).

Figure 8.18 Member fragility curves for a five-storey frame system designed DC M and to PGA=0.20g and

PGA=0.25g using IDA method for most critical storey members.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

PGA=0.20g PGA=0.25g


92

Figure 8.19 Member fragility curves for a five-storey frame system designed PGA=0.25g and to DC M and

DC H using IDA method for most critical storey members.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

DC M DC H


93

Figure 8.20 Fragility curves of five-storey buildings designed to PGA=0.25g and DC M analyzed using

IDA method: (top) frame buildings; (bottom) frame-equivalent buildings

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th


94


IDA method: (top) frame buildings; (bottom) wall-equivalent buildings

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th


95


IDA method: (top) frame buildings; (bottom) wall buildings

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th


96

8.5. Comparison between analysis methods

As described in previous chapters the methods of analysis performed for the construction of

member fragility curves are the Incremental Dynamic Analysis (IDA) and the Static Pushover

Analysis (SPO). These fragility curves were then compared against results taken from a

simplified analysis using the lateral force method (LFM) by Papailia [2011].

Conclusions and observations about the comparison between each analysis method can be

made from the median PGA at the attainment of each damage state (see Table 8.5 to Table

8.11) and the fragility curves as illustrated in Appendix B1 for all the examined buildings.

Also Appendix B2 presents the fragility comparisons of the three methods for the most

critical members.

Comparing the three methods the following observations can be made:

Examples of beam fragility curves in the yielding damage state for the three methods

(LFM, SPO and IDA) for the most critical members are presented in Figure 8.23 and

Figure 8.24 for wall-frame buildings and Figure 8.25 for frame buildings. It can be

observed that the three methods yield similar results.

In five-storey buildings the fragility results of beams in the yielding damage state

taken from LFM are slightly higher than SPO and IDA (see Figure 8.23 and Figure

8.25). In eight-storey buildings there is a larger difference between fragility results

taken from LFM and the other two methods. (see Figure 8.24)

Figure 8.23 Beam fragility curves in yielding state for five-storey frame-equivalent building designed to

DC M and PGA=0.20g (left) and wall-equivalent building designed to DC H and PGA=0.25g (right).

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

LFM (ε>1) LFM (ε=1) SPO IDA

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1



97

Figure 8.24 Beam fragility curves in yielding state for eight-storey frame-equivalent building designed to

DC M and PGA=0.20g (left) and wall-equivalent building designed to DC M and PGA=0.25g (right).

Figure 8.25 Beam fragility curves in yielding state for five-storey frame building designed to PGA=0.25g

and DC M (left) and DC H (right).

The fragility curves of beams in the ultimate damage state, for the most critical

members, illustrate that the three methods have similar results (Figure 8.26 and Figure

8.27 for wall-frame buildings and Figure 8.28 for frame buildings). The method with

the highest or lowest fragility is neither marked nor systematic.

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1



98

Figure 8.26 Beam fragility curves in ultimate state for five-storey frame-equivalent building designed to


Figure 8.27 Beam fragility curves in ultimate state for eight-storey frame-equivalent building designed to


Figure 8.28 Beam fragility curves in ultimate state for five-storey frame building designed to PGA=0.25g

and DC M (left) and DC H (right).

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1



99

Examples of the fragility curves for the columns in yielding damage state for the most

critical members taken from results of the three methods are presented in Figure 8.29

and Figure 8.30 for wall-frame buildings and in Figure 8.31 for frame buildings. The

results obtained for the columns in yielding damage state using IDA and SPO in five-

storey buildings match well for both frame and wall-frame buildings (see Figure 8.29

and Figure 8.31). In eight-storey wall-frame buildings the fragilities obtained through

LFM have higher fragilities than the other two methods (see Figure 8.30).

Figure 8.29 Column fragility curves in yielding state for five-storey frame-equivalent building designed to


Figure 8.30 Column fragility curves in yielding state for eight-storey frame-equivalent building designed

to DC M and PGA=0.20g (left) and wall building designed to DC M and PGA=0.20g (right).

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1



100

Figure 8.31 Column fragility curves in yielding state for five-storey frame building designed to DC M and

PGA=0.20g and (left) PGA=0.25g (right).

Examples of column fragility curves in ultimate damage state for the most critical

members are shown for the three analysis methods in Figure 8.32, Figure 8.33 and

Figure 8.34 for wall-frame buildings and in Figure 8.35 in frame buildings. In five-

storey wall-frame buildings the fragility curves for the columns in their ultimate

damage state obtained using IDA and SPO match well, whereas the ones taken from

the LFM are slightly lower (see Figure 8.32, Figure 8.33). In eight-storey wall-frame

buildings the three methods yield similar fragility results (see Figure 8.34). In five-

storey frame buildings the fragility curves for the columns in their ultimate damage

state obtained using IDA and LFM match well, whereas the ones taken from the SPO

are slightly higher (see Figure 8.35).

Figure 8.32 Column fragility curves in ultimate state for five-storey wall -equivalent building designed to

DC M and PGA=0.25g (left) and wall building designed to DC M and PGA=0.25g (right).

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1



101

Figure 8.33 Column fragility curves in ultimate state for five-storey frame-equivalent building designed to

DC H and PGA=0.25g (left) and wall-equivalent building designed to DC H and PGA=0.25g (right).

Figure 8.34 Column fragility curves in ultimate state for eight-storey frame-equivalent building designed

to DC M and PGA=0.25g (left) and wall-equivalent building designed to DC M and PGA=0.25g (right).

Figure 8.35 Column fragility curves in ultimate state for five-storey frame building designed to DC M and

PGA=0.20g and (left) DC H and PGA=0.25g (right).

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1



102

Fragility curves in wall-frame buildings for walls in the yielding damage state taken

for the three methods are shown in Figure 8.36 and Figure 8.37. The three methods

yield similar results; the results taken from IDA have the lowest fragilities and the

ones taken from SPO and LFM match well. (see Figure 8.36 and Figure 8.37).

Figure 8.36 Wall fragility curves in yielding state for five-storey frame-equivalent building designed to


Figure 8.37 Wall fragility curves in yielding state for five-storey frame-equivalent building designed to


Examples of fragility curves for the walls in the ultimate damage state in flexure are

illustrated in Figure 8.38 and Figure 8.39. The three methods yield similar results (see

Figure 8.38) except in five- and eight-storey wall dual systems where fragilities

obtained from IDA are slightly higher. (see Figure 8.39).

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1



103

Figure 8.38 Wall fragility curves in ultimate state in flexure for five-storey frame-equivalent building

designed to DC M and PGA=0.25g (left) and wall-equivalent building designed to DC H and PGA=0.25g

(right).

Figure 8.39 Wall fragility curves in ultimate state in flexure for five-storey wall building designed to DC

M and PGA=0.25g (left) and eight-storey wall building designed to DC M and PGA=0.20g (right).

Figure 8.40 to Figure 8.53 present the beam and column fragility curves for the three methods

in the yielding and ultimate damage state for both wall-frame and frame buildings. The first

column in each set concerns the fragility curves obtained from IDA the second from SPO and

the third are obtained from LFM. The first row in each set is for yielding and the second is for

ultimate damage state.

The fragility results of beams in the yielding damage state show that the middle-storey

beams have the highest fragility whereas the top-storey beams have the lowest

fragility for the three analysis methods in all the buildings examined (see Figure 8.40

to Figure 8.45).

The fragility results for the beams in the ultimate damage state show that middle-

storey beams have highest fragility for all three methods. The first-storey beams in the

ultimate damage states for IDA results and top-storey beams for SPO and LFM results

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1



104

have the lowest fragilities. These observations hold for five- and eight-storey buildings

and for the three ranges of the ratio of total base shear taken by the walls in wall-frame

buildings (see Figure 8.40 to Figure 8.45) and for all frame buildings (see Figure

8.46).

Figure 8.40 Beam fragility curves for a) yielding and b) ultimate state for five-storey frame-equivalent

building designed to DC M and PGA=0.25g analyzed with IDA (left), SPO (middle) and LFM (right).

Figure 8.41 Beam fragility curves for a) yielding and b) ultimate state for five-storey wall-equivalent


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th


105

Figure 8.42 Beam fragility curves for a) yielding and b) ultimate state for five-storey wall building

designed to DC M and PGA=0.25g analyzed with IDA (left), SPO (middle) and LFM (right).

Figure 8.43 Beam fragility curves for a) yielding and b) ultimate state for eight-storey frame-equivalent


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4



106

Figure 8.44 Beam fragility curves for a) yielding and b) ultimate state for eight -storey wall-equivalent


Figure 8.45 Beam fragility curves for a) yielding and b) ultimate state for eight -storey wall building


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4



107

Figure 8.46 Beam fragility curves for a) yielding and b) ultimate state for five -storey frame building


The fragility curves of columns in five-storey wall-frame and frame buildings, for the

results taken from IDA and SPO, show that the first-storey columns have the highest

fragility and the top-storey columns have the lowest. The fragility curves for the

results taken from the LFM shows that the middle-storey columns have the highest

fragilities and the top-storey columns the lowest. These observations hold for the three

ranges of the ratio of total base shear taken by the wall (see Figure 8.47, Figure 8.48,

Figure 8.49 for wall-frame building and Figure 8.53 for frame buildings).

The fragility results of columns in eight-storey wall-frame buildings for the results

taken from IDA and SPO show that the first- and top-storey columns have the highest

fragility and the middle-storey columns have the lowest. The fragility curves of eight-

storey buildings for columns in yielding damage state show that the results taken from

LFM are higher than IDA and SPO. These observations hold for all three ranges of the

ratio of total base shear taken by the walls (see Figure 8.50, Figure 8.51 and Figure

8.52).

Fragility curves of columns for the non-critical members in wall-frame buildings for

results taken from LFM have higher fragilities than for the results taken from IDA and

those taken from SPO are lower than the ones obtained from IDA. These observations

hold for five- and eight- storey wall-frame buildings for all three ranges of the ratio of

total base shear taken by the walls (see Figure 8.47, Figure 8.48 and Figure 8.49)

Fragilities of columns for the non-critical members in frame buildings for results taken

from LFM have higher fragilities than for the results taken from SPO and those taken

from SPO are higher than the ones obtained from IDA (see Figure 8.53).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th


108

Figure 8.47 Column fragility curves for c) yielding and d) ultimate state for five-storey frame-equivalent


Figure 8.48 Column fragility curves for c) yielding and d) ultimate state for five-storey wall-equivalent


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th


109

Figure 8.49 Column fragility curves for c) yielding and d) ultimate state for five-storey wall building


Figure 8.50 Column fragility curves for c) yielding and d) ultimate state for eight-storey frame-equivalent


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4



110

Figure 8.51 Column fragility curves for c) yielding and d) ultimate state for eight -storey wall-equivalent


Figure 8.52 Column fragility curves for c) yielding and d) ultimate state for eight -storey wall building


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4



111

Figure 8.53 Column fragility curves for c) yielding and d) ultimate state for five -storey frame building


8.6. Fragility results of walls in the ultimate state

As previously explained the fragility curves are obtained from nonlinear dynamic analysis

(IDA), nonlinear static analysis (SPO) and from a simplified analysis using the lateral force

method (LFM) by Papailia [2011]. The fragility curves of walls for the ultimate damage state

in shear obtained using the SPO is the envelope of the fragility curves for each storey. Higher

mode effects on wall shear demands are already taken into account in IDA.

The shear demands obtained from the LFM are amplified by a factor ε (eq. 4.13) which takes

into account higher mode effects. This amplification is used for both DC M and DC H walls

following a detailed procedure according to Keintzel [1990] also adopted in Eurocode 8

[CEN, 2004b].

The median PGAs at attainment of the ultimate damage state of walls are presented in Table

8.12 and Table 8.13. The ultimate damage state indicates the envelope of the shear and

flexural failure for IDA and LFM; i.e. the lowest median PGA at attainment of the ultimate

damage state between flexure and shear failure. LFM results presented use inelastic

amplification due to higher mode effects.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th


112

Table 8.12 Median PGA (g) at attainment of the ultimate damage state for walls in 5-storey buildings

design PGA DC

Analysis

method

Frame-

equivalent

Wall-

equivalent

Wall

systems

0.20g M LFM 0.25g 0.19g 0.17g

IDA 0.38g 0.42g 0.33g

0.25g M LFM 0.19g 0.18g 0.19g

IDA 0.39g 0.37g 0.29g

0.25g H LFM 0.38g 0.38g 0.51g

IDA 0.47g 0.45g 0.57g

Table 8.13 Median PGA (g) at attainment of the ultimate damage state for walls in 8-storey buildings

design PGA DC Analysis

method

Frame-

equivalent

Wall-

equivalent

Wall

systems

0.20g M LFM 0.31g 0.18g 0.18g

IDA 0.39g 0.53g 0.29g

0.25g M LFM 0.33g 0.18g 0.19g

IDA 0.40g 0.54g 0.35g

The conclusions on the walls for the ultimate damage state (maximum of flexure and shear

failure) in wall-frame buildings are:

Wall fragilities for the results obtained from LFM (where inelastic amplification for the

higher mode effects is used) show that the DC M walls in wall-equivalent and wall dual

systems may fail in shear before their design PGA. DC M walls in frame-equivalent

systems fail beyond their design PGA in most cases. (see Table 8.12 and Table 8.13)

Results taken from dynamic analysis (IDA) show that DC M walls fail at PGA values 1.6

to 1.9 times their design PGA in frame-equivalent systems, 1.5 to 2.5 times their design

PGA in wall-equivalent systems and 1.2 to 1.4 times their design PGA in wall systems

(see Table 8.12 and Table 8.13)

The fragility results obtained for the DC H walls show that for the results obtained from

the LFM they fail at PGA values 1.5 to 2 times higher than their design PGA and 1.8 to

2.2 times their design PGA for results obtained from IDA. (see Table 8.12 and Table

8.13).

The median PGAs at attainment of the shear failure of walls are presented in Table 8.14 and

Table 8.15. The values shown include the results obtained from IDA and the LFM. The

results taken from the LFM are shown for both amplified LFM(ε>1) and non-amplified

LFM(ε=1) shear demands.

Examples of wall fragilities in the ultimate damage state in shear are presented in Figure 8.54,

Figure 8.55 and Figure 8.56 for the results taken from (a) LFM with inelastic amplification of

shear demands due to higher modes, b) LFM without the inelastic amplification of shear


113

demands, c) incremental dynamic analysis (IDA) and d) the static pushover analysis (SPO).

These figures are fully presented in Appendix A3 for all the examined buildings.

The conclusions on walls for the ultimate damage state in shear for wall-frame buildings are:

The results taken from the dynamic analysis are in-between the LFM results with and

without inelastic amplification of shear demands due to higher modes. (see Figure 8.54,

Figure 8.55, Figure 8.56, Table 8.14 and Table 8.15). The latter applies for both DC M

and DC H walls, except in five- and eight-storey frame-equivalent buildings and eight-

storey wall-equivalent systems (see Table 8.14 and Table 8.15)

The dynamic analysis (IDA) confirms to a certain extent the inelastic amplification of

shear forces due to higher modes in both DC M and DC H walls and show that the

relevant rules of Eurocode 8 are on the conservative side. The latter was also observed for

DC H walls in Ruttenberg and Nsieri [2006] and Kappos and Antoniadis [2007].

The fragility curve results taken from SPO match well with the other two methods up to

yielding. Beyond yielding there is no significant increase in shear force demands. (see

Figure 8.54, Figure 8.55 and Figure 8.56 ).

Table 8.14 Median PGA (g) at attainment of the ultimate damage state in shear for walls in 5-storey

buildings

design

PGA

DC Analysis

method

Frame-

equivalent

Wall-

equivalent

Wall

system

0.20g M LFM (ε>1) 0.25g 0.19g 0.17g

IDA 0.94g 0.42g 0.33g

LFM (ε=1) 0.41g 0.52g 0.54g

0.25g M LFM (ε>1) 0.19g 0.18g 0.19g

IDA 0.39g 0.37g 0.29g

LFM (ε=1) 0.39g 0.52g 0.74g

0.25g H LFM (ε>1) 0.38g 0.38g 0.51g

IDA 0.90g 0.53g 0.72g

LFM (ε=1) 0.61g 0.61g 0.98g

Table 8.15 Median PGA (g) at attainment of the ultimate damage state in shear for walls in 8-storey

buildings

design

PGA

DC Analysis

method

Frame-

equivalent

Wall-

equivalent

Wall

system

0.20g M LFM (ε>1) 0.58g 0.18g 0.18g

IDA 0.88g 0.61g 0.29g

LFM (ε=1) 0.58g 0.20g 0.44g

0.25g M LFM (ε>1) 0.51g 0.18g 0.19g

IDA 0.77g 0.67g 0.35g

LFM (ε=1) 0.52g 0.20g 0.60g


114

Figure 8.54 Fragility curves of walls for the ultimate damage state in shear of a five-storey wall-equivalent

building designed to DC M and PGA=0.20g.

Figure 8.55 Fragility curves of walls for the ultimate damage state in shear of a five-storey wall building

designed to DC M and PGA=0.20g.


115

Figure 8.56 Fragility curves of walls for the ultimate damage state in shear of a eight-storey wall building


Chapter 8: Conclusions

116

9. SUMMARY AND CONCLUSIONS

This study deals with the seismic fragility curves of reinforced concrete frame and wall-frame

(dual) buildings designed according to Eurocode 2 and Eurocode 8 [CEN, 2004a,b]. Prototype

plan- and height- wise very regular buildings are studied with parameters including the height

of the building, the level of Eurocode 8 design (in terms of design peak ground acceleration

and ductility class) and for dual systems the percentage of seismic base shear taken by the

walls.

The member fragilities were constructed using two different methods; incremental nonlinear

dynamic analysis (IDA) and nonlinear static (pushover) analysis (SPO). These methods were

performed using a three-dimensional structural model of the full buildings. IDA is performed

using 14 spectrum-compatible semi-artificial accelerograms and SPO is performed using

inverted triangular load pattern. The N2 method is employed to combine the results of the

SPO with the response spectrum analysis of an equivalent single degree-of-freedom system to

relate the damage measure demands for each analysis step to the intensity measure (i.e. peak

ground acceleration). A simplified analysis using the lateral force method (LFM) by Papailia

[2011] was compared against SPO and IDA.

The results of the three analysis methods are presented in the form of fragility curves for two

member limit states; yielding and ultimate deformation in bending or shear. The structural

damage of members is expressed in terms of peak ground acceleration (PGA) as the intensity

measure (IM) since it is easier to compare it against the design PGA of the buildings. The

damage measures (DM) taken are the peak chord rotation and the shear force demands at

member ends. The probability of exceedance of each limit state is computed from the

probability distributions of the damage measures (conditional on intensity measure) and of the

corresponding capacities.

Dispersions for DM-demands are taken explicitly from the analysis for IDA method and

estimates of dispersions of DM-demands are taken from previous studies for the SPO and the

LFM. All three methods use estimates for the damage measure capacities based on previous

studies. It can be observed that the CoV-values determined through IDA are slightly lower

than the ones determined from previous studies. The dispersions of DM-demands taken from

IDA for beams and columns have a larger scatter in the storeys in dual buildings than in frame

buildings and their mean is slightly higher.


117

The results taken from Papailia [2011] using the LFM indicate that the walls in buildings

designed according to Eurocode 8 and Medium Ductility Class are likely to fail in shear even

before the design PGA. The shear force demands taken from the LFM in concrete walls are

amplified to consider higher modes effects. Results from the nonlinear static (pushover) and

dynamic analysis were used to better understand the seismic behavior of regular dual or frame

buildings and the inelastic amplification of shear force demands in concrete walls due to

higher modes.

In wall-frame dual buildings the following conclusions and observations are made for the

results obtained from IDA and SPO:

Walls are the most critical members for both yielding and ultimate damage states.

Beams are much more likely to reach ultimate damage state than columns.

Design to DC M in lieu of DC H may reduce the fragility of beams against yielding

and increase fragility of columns. However, these effects are neither systematic not

marked. Wall fragility against yielding does not significantly change and is higher in

DC M walls in the ultimate damage state.

Design to a higher PGA reduces the fragilities of beams and slightly increases those of

columns. Wall fragilities do not significantly change.

As the proportion of the total base shear taken by the wall increases the beams,

columns and walls in flexure have lower fragilities but walls in shear ultimate state

have higher fragilities.

Taller buildings generally exhibit lower fragilities for beams and columns and similar

fragilities for walls.

The conclusions made on frame buildings for the results obtained from IDA and SPO are:

Frame buildings give satisfactory fragility results even beyond their design PGA.

Beams yield before their design PGA whereas the columns remain elastic well beyond

the design PGA.

Beams are much more likely to reach yielding and collapse than in columns.

Design to DC M instead of DC H may reduce slightly the fragility of beams and

columns against yielding, but may increase that of beams against ultimate

Design for higher PGA reduces only slightly the fragilities of beams and columns in

yielding state and may increase fragility in the ultimate state.

Wall-frame (dual) systems have, in general, higher fragility than frame systems for

columns and lower for beams.

The conclusions when comparing the alternative analysis methods are:

The alternative methods yield results that are in good agreement with either damage

state of columns and beams in both frame and dual buildings and to the flexural

behavior of walls. Larger differences are observed in eight-storey wall-frame dual

buildings for columns in yielding state where the fragility results taken from LFM are

higher than IDA and SPO.


118

Wall fragilities for the results obtained from LFM (where inelastic amplification for

the higher mode effects is used) show that the DC M walls in wall-equivalent and wall

dual systems may fail in shear before their design PGA. Walls in frame-equivalent

systems are likely to fail at PGA values beyond the design PGA.

Results taken from dynamic analysis (IDA) show that DC M walls fail at PGA values

1.6 to 1.9 times their design PGA in frame-equivalent systems, 1.5 to 2.5 times their

design PGA in wall-equivalent systems and 1.2 to 1.4 times their design PGA in wall

systems.

The fragility results obtained for the DC H walls show that they fail at PGA values 1.5

to 2 times higher than the design PGA for the results obtained from the LFM and 1.8

to 2.2 times the design PGA for results obtained from dynamic analysis (IDA).

The fragility curve results for wall shear failure taken from SPO match well with the

other two methods up to yielding. Beyond yielding there is no significant increase in

shear force demands.

The results taken from the dynamic analysis (IDA) are in-between the LFM results

with and without inelastic amplification of shear demands due to higher modes. The

latter applies in both DC M and DC H walls, except in five- and eight-storey frame-

equivalent buildings and eight-storey wall-equivalent systems.

The dynamic analysis confirm to a certain extent the inelastic amplification of shear

forces due to higher modes in both DC M and DC H walls and show that the relevant

rules of Eurocode 8 are on the conservative side. The latter was also observed for DC

H walls in Ruttenberg and Nsieri [2006] and Kappos and Antoniadis [2007].

References

119

REFERENCES

Akkar S., Sucuaglu H., M.EERI and Yakut. A. [2005] “Displacement based fragility

functions for Low- and Mid- rise ordinary concrete buildings,” Earthquake Spectra Vol. 21,

No. 4.

Akkar S. and Őzen Ő., [2006] “Effect of peak ground velocity on deformation demands for

SDOF systems,” Earthquake Engineering and Engineering Dynamics, 34:1551-1571.

Baker J. W. and Cornell C.A. [2005] “A vector valued ground motion intensity measure

consisting of spectral acceleration and epsilon,” Earthquake Engineering and Structural

Dynamics, Vol. 34, 1193-1217.

Baker J. W. [2007] “Probabilistic Structural Response Assessment using vector-valued

Intensity Measures,” Earthquake Engineering and Structural Dynamics, Vol. 36, 1861-1883.

Biskinis D. and Fardis M. N. [2010] “Flexure-controlled ultimate deformations of members

with continuous or lap-spliced bars,” Structural Concrete Journal of the fib, Vol. 11, No. 2,

93-108

Biskinis D. and Fardis M. N. [2010] “Deformations at flexural yielding of members with

continuous or lap-spliced bars,” Structural Concrete Journal of the fib, Vol. 11, No. 3,

September 2010, 127-138

Biskinis D.E., Roupakias G.K. and Fardis M.N. [2004], “Degradation of Shear Strength of

RC Members with Inelastic Cyclic Displacements,” ACI Structural Journal, Vol. 101, No. 6,

pp.773-783

Borzi B., Pinho R., and Crowley H. [2006] “Simplified pushover based vulnerability analysis

for large scale assessment of RC buildings,” Engineering Structures, Vol.30, 804-820

Braga, F., Dolce M., and Liberatore D. [1982] “A statistical study on damaged buildings and

an ensuing review of the MSK-76 scale,” Proceedings of the 7th European Conference on

Earthquake Engineering, Athens, Greece, Vol. 7.

CEN [2002] EN1990 Eurocode – Basis of structural design, European Committee for

Standardization, April 2002, Brussels, Belgium

References

120

CEN [2004] EN1992-1-1 Eurocode 2: Design of concrete structures – Part 1-1: General rules

and rules for buildings, European Committee for Standardization, December 2004, Brussels,

Belgium

CEN [2004] EN1998-1 Eurocode 8: Design of structures for earthquake resistance – Part1:

General rules, seismic actions and rules for buildings, European Committee for

Standardization, December 2004, Brussels, Belgium

CEN [2005] EN1998-3 Eurocode 8: Design of structures for earthquake resistance – Part3:

Assessment and retrofitting of buildings. European Committee for Standardization, June

2005, Brussels, Belgium

Calvi G. M., R.Pinho, Magenes G., Bommer J. J., Restrepo-Velez L. F. and Crowley H.

[2006] “Development of seismic vulnerability assessment methodologies over the past 30

years,” Journal of Earthquake Technology, Paper No. 472, Vol. 43, No. 3, September 2006,

pp 75-104

DiPasquale E., Cakmak A.S. [1987] “Detection and Assessment of seismic structural

damage” Technical Report,NCEER-87-0015, State University of New York at Buffalo

Dumova-Jovanoska E. [2000] “Fragility curves for reinforced concrete structures in Scopje

region,” Soil Dynamics and Earthquake Engineering, 19(6), 455-466, 2000

Dolce M., Masi A., Marino M. and Vona M., [2003] “Earthquake damage scenarios of the

building stock of Potenza (Southern Italy) including site effects,” Bulletin of Earthquake

Engineering, 1: 115 – 140

Fardis M. N. [2009] “Seismic Design, Assessment and Retrofitting of concrete buildings,”

Spinger, London New York

Fardis M.N. [1994] “Eurocode 8: reinforced concrete, special session on EC8” Proceedings of

10th European conference on earthquake engineering, Vienna, pp 2945–2950

Grünthal G. [1998] “Cahiers du Centre Europėen de Gėodynamique et de Sėismologie:

Volume 15 – European Macroseismic Scale 1998”, European Centre for Geomechanics and

Seismology, Luxemburg

Kappos A.J., Antoniadis P. [2007] “A contribution to seismic shear design of RC walls in

dual structures”. Bull Earthq Eng, 5(3):443–466

Kappos AJ, Stylianidis KC, Pitilakis K. [1998] “Development of seismic risk scenarios based

on a hybrid method of vulnerability assessment,” Nat Hazards, 17(2):177–192

Keintzel E. [1990] “Seismic design shear forces in RC cantilever shear wall structures,”

European Journal of Earthquake Engineering, Vol. 3, 7-16.

References

121

Kirçil M. S. and Polat Z. [2006] "Fragility Analysis of Mid-Rise R/C Frame Buildings",

Engineering Structures, 28(9):1335-1345

Kosmopoulos A. and Fardis M.N. [2007], “Estimation of Inelastic Seismic Deformations in

Asymmetric Multistory RC Buildings”, Journal of Earthquake Engineering and Structural

Dynamics, Vol. 36, No. 9, 1209-1234

Kostov M. Vaseva E., Kaneva A. Koleva N, Varbanov G. Stefanov D., Darvarova E. [2007]

“An advanced approach to earthquake risk scenarios with application to different European

towns” RISK-UE WP13

Litton R.W. [1975] “A contribution to the analysis of concrete structures under cyclic

loading,” PhD Thesis, Dept Civil Engineering, Univ. of California, Berkeley, CA

Masi A. [2003] “Seismic vulnerability assessment of gravity load designed RC frames”,

Bulletin of Earthquake Engineering, 1: 371-395, 2003

Mondkar D.P., Powell G.H. [1975] “ANSR-1, general purpose program for nonlinear

structural response” EERC 75-37 (2nd ed.) Univ. California, Earthquake Eng. Res. Center,

Berkeley, CA

Özer B. AY and Erberik M. A. [2008] “Vulnerability of Turkish low-rise and mid-rise reinforced

concrete frame structures” Journal of Earthquake Engineering, 12(S2):2-11,2008.

Otani S. [1974] “Inelastic analysis of R/C frame structures”, ASCE J. Structural Div, 100

pp.1433-1449.

Papailia A. [2011] “Seismic fragility curves of RC buildings”, MSc Thesis, University of

Patras

Park Y.J. and Ang Ah-S [1985] Mechanic seismic damage model for reinforced concrete,

Journal of Structural Engineerring, 722-739.

Panagiotakos T. B. and Fardis M. N. [1999] “Estimation of Inelastic Deformation Demands in

Multistory RC Buildings”, Journal of Earthquake Engineering and Structural Dynamics, Vol.

28, Feb. 1999, 501-528

Panagiotakos T. B. and Fardis M. N. [2004] “Seismic performance of RC frames designed to

Eurocode 8 or to the Greek Codes 2000,” Bulletin of Earthquake Engineering, 2: 221-259

Rota M., Penna A., Strobbia C. L. [2007] “Processing Italian damage data to derive

typological fragility curves”, Soil dynamics and Earthquake Engineering, 28 933-947

Rutenberg A, Nsieri E [2006] “The seismic shear demand in ductile cantilever wall systems

and the EC8 provisions,” Bull Earthquake Engineering, 4(1):1–21

References

122

Rivera J. and Petrini L. [2011], “On the design and seismic response of RC frame buildings

desgined to Eurtocode 8”, Bull Earthquake Engineering, 9:1593-1616 2011

Sabetta F., Goretti A. and Lucantoni A. [1998]. “Empirical Fragility Curves from Damage

Surveys and Estimated Strong Ground Motion”, 11th European Conference on Earthquake

Engineering, Balkema, Rotterdam.

Sarabandi P., Pachakis D., King S. and Kirmidjian A. [2004], “Empirical fragility functions

from recent earthquakes”, 13th

World Conference on Earthquake Engineering, Vancouver

Canada.

Singhal and Kiremidjian [1996], “Method of Probabilistic Evaluation of Seismic Structural

Damage”, ASCE Journal of Structural Engineering. Vol.122. No. 12.

Sieberg [1923] “Erdbebenkunde”, Fisher, Jena 102-104

SYNER-G [2011] “Systematic seismic vulnerability and risk analysis for buildings, lifeline

networks and infrastructures safety gain” Project No. 244061

Vamvatsikos, D. and Cornell, C.A. [2002]. “Incremental dynamic analysis,” Journal. of

Earthquake Eng and Structural Dynamics, 1-23.

Vamvatsikos, D. and Cornell, C.A. [2002] “The incremental dynamic analysis and its

application to performance-based earthquake engineering,” 12th

European Conference on

Earthquake Engineering,London

Appendices

A1

APPENDIX A

A1. Fragility curves using Incremental Dynamic analysis

Appendix A1 presents the member fragility curves of each examined buildings for the two

damage states of yielding and ultimate for the analysis performed using Incremental dynamic

analysis (IDA). The sub-figures (a) and (b) refer to the beam yielding and ultimate damage

state in flexure, (c) and (d) refer to column yielding and ultimate damage state in flexure. In

wall-frame dual buildings (e) and (f) refer to wall yielding and ultimate damage state in

flexure and (g) refers to wall ultimate damage state in shear. In frame buildings (e) and (f)

refer to beam and column ultimate damage state in shear respectively.

Appendices

A2

Figure A. 1 Fragility curves of five-storey frame building designed to DC M and PGA=0.20g using IDA

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A3

Figure A. 2 Fragility curves of five-storey frame building designed to DC M and PGA=0.25g using IDA

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A4

Figure A. 3 Fragility curves of five-storey frame building designed to DC H and PGA=0.25g using IDA

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A5

Figure A. 4 Fragility curves of five-storey frame-equivalent building designed to DC M and PGA=0.20g

using IDA

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A6


using IDA

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A7

Figure A. 6 Fragility curves of five-storey frame-equivalent building designed to DC H and PGA=0.25g

using IDA

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A8

Figure A. 7 Fragility curves of five-storey wall-equivalent building designed to DC M and PGA=0.20g

using IDA

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A9


using IDA

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A10

Figure A. 9 Fragility curves of five-storey wall-equivalent building designed to DC H and PGA=0.25g

using IDA

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A11

Figure A. 10 Fragility curves of five-storey wall building designed to DC M and PGA=0.20g using IDA

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A12

Figure A. 11 Fragility curves of five-storey wall building designed to DC M and PGA=0.25g using IDA

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A13

Figure A. 12 Fragility curves of five-storey wall building designed to DC H and PGA=0.25g using IDA

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A14

Figure A. 13 Fragility curves of eight-storey frame-equivalent dual system designed to DC M and

PGA=0.20g using IDA

0 0.2 0.4 0.6 0.8 10

0.51

1st 2nd 3rd 4th

5th 6th 7th 8th

Appendices

A15


PGA=0.25g using IDA

0 0.2 0.4 0.6 0.8 10

0.51

1st 2nd 3rd 4th

5th 6th 7th 8th

Appendices

A16

Figure A. 15 Fragility curves of eight-storey wall-equivalent dual system designed to DC M and

PGA=0.20g using IDA

0 0.2 0.4 0.6 0.8 10

0.51

1st 2nd 3rd 4th

5th 6th 7th 8th

Appendices

A17


PGA=0.25g using IDA

0 0.2 0.4 0.6 0.8 10

0.51

1st 2nd 3rd 4th

5th 6th 7th 8th

Appendices

A18

Figure A. 17 Fragility curves of eight-storey wall building designed to DC M and PGA=0.20g using IDA

0 0.2 0.4 0.6 0.8 10

0.51

1st 2nd 3rd 4th

5th 6th 7th 8th

Appendices

A19

Figure A. 18 Fragility curves of eight-storey wall building designed to DC M and PGA=0.25g using IDA

0 0.2 0.4 0.6 0.8 10

0.51

1st 2nd 3rd 4th

5th 6th 7th 8th

Appendices

A20

A2. Fragility curves using Static Pushover Analysis

Appendix A2 presents the member fragility curves of each examined building for the two

damage states of yielding and ultimate for the analysis performed using nonlinear static

(pushover) analysis (SPO). The sub-figures (a) and (b) refer to the beam yielding and ultimate

damage state in flexure, (c) and (d) refer to column yielding and ultimate damage state in

flexure. In wall-frame dual buildings (e) and (f) refer to wall yielding and ultimate damage

state in flexure and (g) refers to wall ultimate damage state in shear. In frame buildings (e)

and (f) refer to beam and column ultimate damage state in shear respectively.

Appendices

A21

Figure A. 19 Fragility curves of five-storey frame building designed to DC M and PGA=0.20g using SPO

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A22

Figure A. 20 Fragility curves of five-storey frame building designed to DC M and PGA=0.25g using SPO

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A23

Figure A. 21 Fragility curves of five-storey frame building designed to DC H and PGA=0.25g using SPO

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A24


using SPO

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A25


using SPO

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A26

Figure A. 24 Fragility curves of five-storey frame-equivalent building designed to DC H and PGA=0.25g

using SPO

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A27


using SPO

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A28


using SPO

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A29

Figure A. 27 Fragility curves of five-storey wall-equivalent building designed to DC H and PGA=0.25g

using SPO

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A30

Figure A. 28 Fragility curves of five-storey wall building designed to DC M and PGA=0.20g using SPO

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A31

Figure A. 29 Fragility curves of five-storey wall building designed to DC M and PGA=0.25g using SPO

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A32

Figure A. 30 Fragility curves of five-storey wall building designed to DC H and PGA=0.25g using SPO

0 0.2 0.4 0.6 0.8 10

0.5

1

1st 2nd 3rd 4th 5th

Appendices

A33


PGA=0.20g using SPO

0 0.2 0.4 0.6 0.8 10

0.51

1st 2nd 3rd 4th

5th 6th 7th 8th

Appendices

A34


PGA=0.25g using SPO

0 0.2 0.4 0.6 0.8 10

0.51

1st 2nd 3rd 4th

5th 6th 7th 8th

Appendices

A35


PGA=0.20g using SPO

0 0.2 0.4 0.6 0.8 10

0.51

1st 2nd 3rd 4th

5th 6th 7th 8th

Appendices

A36


PGA=0.25g using SPO

0 0.2 0.4 0.6 0.8 10

0.51

1st 2nd 3rd 4th

5th 6th 7th 8th

Appendices

A37

Figure A. 35 Fragility curves of eight-storey wall building designed to DC M and PGA=0.20g using SPO

0 0.2 0.4 0.6 0.8 10

0.51

1st 2nd 3rd 4th

5th 6th 7th 8th

Appendices

A38

Figure A. 36 Fragility curves of eight-storey wall building designed to DC M and PGA=0.25g using SPO

0 0.2 0.4 0.6 0.8 10

0.51

1st 2nd 3rd 4th

5th 6th 7th 8th

Appendices

A39

A3. Fragility curves of walls in shear

Appendix A3 presents the fragility curves of walls for the ultimate damage state in shear

using the alternative methods of analysis. The sub-figures (a) and (b) refer to the lateral force

method with and without inelastic amplification respectively which takes into account the

higher mode effects. (ε=1) indicates that the shear demand taken from LFM is not amplified

for higher mode effects and (ε>1) indicates that the shear demands taken from LFM are

amplified for higher mode effects. (c) and (d) refer to the incremental dynamic analysis (IDA)

and the static pushover analysis (SPO) respectively.

Figure A. 37 Fragility curves of walls for the ultimate damage state in shear of a five-storey frame-

equivalent building designed to DC M and PGA=0.20g.

Appendices

A40




equivalent building designed to DC H and PGA=0.25g.

Appendices

A41

Figure A. 40 Fragility curves of walls for the ultimate damage state in shear of a five-storey wall-




Appendices

A42



Figure A. 43 Fragility curves of walls for the ultimate damage state in shear of a five-storey wall building


tsionis

Cross-Out

tsionis

Inserted Text

H

Appendices

A43




designed to DC H and PGA=0.25g.

Appendices

A44

Figure A. 46 Fragility curves of walls for the ultimate damage state in shear of a eight-storey frame-


Figure A. 47 Fragility curves of walls for the ultimate damage state in shear of a eight-storey frame-


Appendices

A45

Figure A. 48 Fragility curves of walls for the ultimate damage state in shear of a eight-storey wall-


Figure A. 49 Fragility curves of walls for the ultimate damage state in shear of a eight-storey wall-


Appendices

A46

Figure A. 50 Fragility curves of walls for the ultimate damage state in shear of a eight-storey wall


Figure A. 51 Fragility curves of walls for the ultimate damage state in shear of a eight-storey wall


Appendices

B1

APPENDIX B

B1. Fragility curves - comparison of three methods

Appendix B1 presents the fragility curves of members for the three methods of analysis. The

methods of analysis include 1) the Incremental Dynamic Analysis, 2) Static Pushover

Analysis and 3) the lateral force method. The plots illustrate the two damage states of yielding

and ultimate. The sub-figures (a) and (b) refer to the beam yielding and ultimate state (c) and

(d) refer to column yielding and ultimate state. In wall-frame systems (e) and (f) refer to wall

yielding and ultimate state respectively. The ultimate state for all members is the envelope of

the ultimate damage state in shear and in flexure.

Appendices

B2

1) Incremental Dynamic Analysis

2) Static Pushover Analysis

3) Lateral Force Method

Figure B. 1 Coefficient Fragility curves of five-storey frame systems designed to DC M and PGA=0.20g

using IDA, SPO and LFM analysis

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

Appendices

B3




Figure B. 2 Fragility curves of five-storey frame systems designed to DC M and PGA=0.25g using IDA,

SPO and LFM analysis

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

Appendices

B4




Figure B. 3 Fragility curves of five-storey frame systems designed to DC H and PGA=0.25g using IDA,


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

Appendices

B5




Figure B. 4 Fragility curves of five-storey frame-equivalent dual systems designed to DC M and

PGA=0.20g using IDA, SPO and LFM analysis

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

Appendices

B6




Figure B. 5 Fragility curves of five-storey frame-equivalent dual systems designed to DC M and


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

Appendices

B7




Figure B. 6 Fragility curves of five-storey frame-equivalent dual systems designed to DC H and


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

Appendices

B8




Figure B. 7 Fragility curves of five-storey wall-equivalent dual systems designed to DC M and PGA=0.20g


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

Appendices

B9




Figure B. 8 Fragility curves of five-storey wall-equivalent dual systems designed to DC M and PGA=0.25g


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

Appendices

B10




Figure B. 9 Fragility curves of five-storey wall-equivalent dual systems designed to DC H and PGA=0.25g


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

Appendices

B11




Figure B. 10 Fragility curves of five – storey wall dual systems designed to DC M and PGA=0.20g using

IDA, SPO and LFM analysis

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

Appendices

B12




Figure B. 11 Fragility curves of five-storey wall systems designed to DC M and PGA=0.25g using IDA,


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

Appendices

B13




Figure B. 12 Fragility curves of five-storey wall systems designed to DC H and PGA=0.25g for IDA, SPO

and LFM analysis

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8x 10

-3

1st 2nd 3rd 4th 5th

Appendices

B14




Figure B. 13 Fragility curves of eight-storey frame-equivalent dual system designed to DC M and

PGA=0.20g for IDA, SPO and LFM analysis

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


Appendices

B15




Figure B. 14 Fragility curves of eight-storey frame-equivalent dual system designed to DC M and


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


Appendices

B16




Figure B. 15 Fragility curves of eight-storey wall-equivalent dual systems designed to DC M and


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


Appendices

B17




Figure B. 16 Fragility curves of eight-storey wall-equivalent dual systems designed to DC M and


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


Appendices

B18




Figure B. 17 Fragility curves of eight-storey wall systems designed to DC M and PGA=0.20g for IDA,


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


Appendices

B19




Figure B. 18 Fragility curves of eight-storey wall systems designed to DC M and PGA=0.25g for IDA, SPO

and LFM analysis

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


Appendices

B20

B2. Fragility curves - comparison of three methods for most critical members

Appendix B2 presents the comparison of fragility curves of the most fragile members for the

three methods of analysis. The methods of analysis include the lateral force method (LFM),

nonlinear static Pushover Analysis (SPO) and the Incremental Dynamic Analysis (IDA). The

two damage states of yielding and ultimate are presented in each sub-figures where (a) and (b)

refer to the beam yielding and ultimate state in flexure (c) and (d) refer to column yielding

and ultimate state in flexure (e) and (f) refer to wall yielding and ultimate state in flexure and

(g) refers to the wall ultimate state in shear.

Appendices

B21

Figure B. 19 Fragility curves for most critical members for results taken from LFM, SPO and IDA for a

five-storey frame system designed to DC M and PGA=0.20g.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

LFM SPO IDA

Appendices

B22


five-storey frame system designed to DC M and PGA=0.25g.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

LFM SPO IDA

Appendices

B23


five-storey system designed to DC H and PGA=0.25g.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

LFM SPO IDA

Appendices

B24

Figure B. 22 Fragility curves for most critical members for results taken from LFM with (ε>1) and

without (ε=1) inelastic amplification to higher modes, SPO and IDA for a five-storey frame-equivalent

dual system designed to DC M and PGA=0.20g.

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1


Appendices

B25



dual system designed to DC M and PGA=0.25g.

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1


Appendices

B26



dual system designed to DC H and PGA=0.25g.

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1


Appendices

B27


without (ε=1) inelastic amplification to higher modes, SPO and IDA for a five-storey wall-equivalent dual

system designed to DC M and PGA=0.20g

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1


Appendices

B28




0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1


Appendices

B29



system designed to DC H and PGA=0.25g

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1


Appendices

B30


without (ε=1) inelastic amplification to higher modes, SPO and IDA for a five-storey wall building


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1


Appendices

B31




0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1


Appendices

B32



designed to DC H and PGA=0.25g

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1


Appendices

B33


without (ε=1) inelastic amplification to higher modes, SPO and IDA for a eight-storey frame-equivalent

dual system designed to DC M and PGA=0.20g

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1


Appendices

B34


without (ε=1) inelastic amplification to higher modes, SPO and IDA for a eight-storey frame-equivalent


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1


Appendices

B35


without (ε=1) inelastic amplification to higher modes, SPO and IDA for a eight-storey wall-equivalent


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1


Appendices

B36


without (ε=1) inelastic amplification to higher modes, SPO and IDA for a eight-storey wall-equivalent


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1


Appendices

B37


without (ε=1) inelastic amplification to higher modes, SPO and IDA for a eight-storey wall building


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1


Appendices

B38


without (ε=1) inelastic amplification to higher modes, SPO and IDA for a eight-storey wall building

designed to DC M and PGA=0.25

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1


Appendices

C1

APPENDIX C

C1. Coefficient of variation per Intensity measure

Appendix C1 presents the Coefficient of variation (CoV) of DM-demands as a function of

intensity measure (i.e. PGA). The CoV values illustrated are taken from the nonlinear time-

history analysis (IDA) and are shown for each floor. On the same plot the dispersion values

used in LFM and the SPO methods are shown in a straight line. These are the member

dispersions due to damage measure demands and the dispersion of the spectral value. (see

Table 7.2).

In frame buildings the sub-figures (a) and (b) refer to the CoV values for beam yielding and

ultimate state in flexure (c) for beam ultimate state in shear. (e) and (f) refer to the CoV values

for column yielding and ultimate state in flexure and (g) for column ultimate state in shear.

In wall-frame buildings the sub-figures (a) and (b) refer to the CoV values for beam yielding

and ultimate state in flexure and (c) and (d) for column yielding and ultimate state in flexure.

(e) and (f) refer to the CoV values for wall yielding and ultimate state in flexure and (g) for

wall ultimate state in shear.

Appendices

C2

Figure C. 1 Coefficient of variation (CoV) of DM-demands determined through IDA and CoV used in

SPO and LFM (straight line) for a five-storey frame system designed to DC M and PGA=0.20g

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

1st 2nd 3rd 4th 5th

Appendices

C3


SPO and LFM (straight line) for a five-storey frame system designed to DC M and PGA=0.25g

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

1st 2nd 3rd 4th 5th

Appendices

C4


SPO and LFM (straight line) for a five-storey frame system designed to DC H and PGA=0.25g

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

1st 2nd 3rd 4th 5th

Appendices

C5


SPO and LFM (straight line) for a five-storey frame-equivalent dual system designed to DC M and

PGA=0.20g

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

1st 2nd 3rd 4th 5th

Appendices

C6


SPO and LFM (straight line) for a five-storey frame-equivalent dual system designed to DC M and

PGA=0.25g

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

1st 2nd 3rd 4th 5th

Appendices

C7


SPO and LFM (straight line) for a five-storey frame-equivalent dual system designed to DC H and

PGA=0.25g

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

1st 2nd 3rd 4th 5th

Appendices

C8

Figure C. 7 Coefficient of variation (CoV) of DM-demands determined through in IDA and CoV values

used for SPO and LFM (straight line) for a five-storey wall-equivalent dual system designed to DC M and

PGA=0.20g

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

1st 2nd 3rd 4th 5th

Appendices

C9


used for SPO and LFM (straight line) for a five-storey wall-equivalent dual system designed to DC M and

PGA=0.25g

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

1st 2nd 3rd 4th 5th

Appendices

C10


used for SPO and LFM (straight line) for a five-storey wall-equivalent dual system designed to DC H and

PGA=0.25g

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

1st 2nd 3rd 4th 5th

Appendices

C11


used for SPO and LFM (straight line) for a five-storey wall dual buildings designed to DC M and

PGA=0.20g

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

1st 2nd 3rd 4th 5th

Appendices

C12


SPO and LFM (straight line) for five-storey wall building designed to DC M and PGA=0.25g

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

1st 2nd 3rd 4th 5th

Appendices

C13


SPO and LFM (straight line) for a five-storey wall dual system designed to DC H and PGA=0.25g

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

1st 2nd 3rd 4th 5th

Appendices

C14


SPO and LFM (straight line) for eight-storey frame-equivalent dual system designed to DC M and

PGA=0.20g

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


Appendices

C15


SPO and LFM (straight line) for eight-storey frame-equivalent dual system designed to DC M and

PGA=0.25g

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


Appendices

C16


SPO and LFM (straight line) for an eight-storey wall-equivalent dual system designed to DC M and

PGA=0.20g

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


Appendices

C17


SPO and LFM (straight line) for an eight-storey wall-equivalent dual system designed to DC M and

PGA=0.25g

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


Appendices

C18


SPO and LFM (straight line) for a eight-storey wall building designed to DC M and PGA=0.20g

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


Appendices

C19


SPO and LFM (straight line) for a eight-storey wall building designed to DC M and PGA=0.25g

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4


Appendices

C20

C2. Damage indices per Intensity measure

Appendix C2 presents the damage indices (ratio of the damage measure demands to the

damage measure capacities) taken explicitly from IDA as a function of the intensity measure

(i.e. PGA). Each point on the plots represents the damage index for one record accelerogram

for either the yielding or ultimate damage state of the member. Each record is presented in

different markers as shown in Figure C. 19. The sub-figures (a) to (e) refer to the storey of the

member from the first- to fifth-storey in a five-storey building and (a) to (h) refer to the storey

of the member from the first- to the eighth-storey in an eight-storey building.

Figure C. 19 Legend for damage index plots where each point represents a single earthquake record

Appendices

C21

Figure C. 20 Damage indices for each floor for a beam member at yielding damage state of five-storey

frame system designed to DC M and PGA=0.20g

Appendices

C22

Figure C. 21 Damage indices for each floor for a beam member at ultimate damage state of five-storey


Appendices

C23

Figure C. 22 Damage indices for each floor for a column member at yielding damage state of five-storey


Appendices

C24

Figure C. 23 Damage indices for each floor for a column member at ultimate damage state of five-storey


Appendices

C25

Figure C. 24 Damage indices for each floor for a beam member at shear damage state of five-storey frame


Appendices

C26

Figure C. 25 Damage indices for each floor for a column member at shear damage state of five-storey


Appendices

C27



Appendices

C28



Appendices

C29



Appendices

C30



Appendices

C31



Appendices

C32



Appendices

C33


frame system designed to DC H and PGA=0.25g

Appendices

C34



Appendices

C35



Appendices

C36



Appendices

C37



Appendices

C38



Appendices

C39


frame-equivalent dual system designed to DC M and PGA=0.20g

Appendices

C40



Appendices

C41



Appendices

C42



Appendices

C43

Figure C. 42 Damage indices for each floor for a wall member at a) yielding b) ultimate c) shear ultimate

damage state of five-storey frame-equivalent system designed to DC M and PGA=0.20g

Appendices

C44



Appendices

C45



Appendices

C46



Appendices

C47



Appendices

C48



Appendices

C49


frame-equivalent dual system designed to DC H and PGA=0.25g

Appendices

C50



Appendices

C51



Appendices

C52



Appendices

C53


damage state of five-storey frame-equivalent system designed to DC H and PGA=0.25g

Appendices

C54


wall-equivalent dual system designed to DC M and PGA=0.20g

Appendices

C55



Appendices

C56



Appendices

C57



Appendices

C58


damage state of five-storey wall-equivalent system designed to DC M and PGA=0.20g

Appendices

C59



Appendices

C60



Appendices

C61



Appendices

C62



Appendices

C63


damage state of five-storey wall-equivalent system designed to DC M and PGA=0.25g

Appendices

C64


wall-equivalent dual system designed to DC H and PGA=0.25g

Appendices

C65


wall -equivalent dual system designed to DC H and PGA=0.25g

Appendices

C66



Appendices

C67



Appendices

C68


damage state of five-storey wall -equivalent system designed to DC H and PGA=0.25g

Appendices

C69

Figure C. 68 Damage indices for each floor for a beam member at yielding damage state of five-storey wall


Appendices

C70


wall system designed to DC M and PGA=0.20g

Appendices

C71



Appendices

C72



Appendices

C73


damage state of five-storey wall system designed to DC M and PGA=0.20g

Appendices

C74



Appendices

C75



Appendices

C76



Appendices

C77



Appendices

C78


damage state of five-storey wall system designed to DC M and PGA=0.25g

Appendices

C79



Appendices

C80


wall system designed to DC H and PGA=0.25g

Appendices

C81



Appendices

C82



Appendices

C83


damage state of five-storey wall system designed to DC H and PGA=0.25g

Appendices

C84

Figure C. 83 Damage indices for each floor for a beam member at yielding damage state of eight-storey

frame-equivalent system designed to DC M and PGA=0.20g

Appendices

C85

Figure C. 84 Damage indices for each floor for a beam member at ultimate damage state of eight-storey


Appendices

C86

Figure C. 85 Damage indices for each floor for a column member at yielding damage state of eight-storey


Appendices

C87

Figure C. 86 Damage indices for each floor for a column member at ultimate damage state of eight-storey


Appendices

C88



Appendices

C89



Appendices

C90



Appendices

C91



Appendices

C92



Appendices

C93


damage state of eight-storey frame-equivalent system designed to DC M and PGA=0.25g

Appendices

C94


wall-equivalent system designed to DC M and PGA=0.20g

Appendices

C95



Appendices

C96



Appendices

C97



Appendices

C98


damage state of eight-storey wall-equivalent system designed to DC M and PGA=0.20g

Appendices

C99



Appendices

C100



Appendices

C101



Appendices

C102

Figure C. 101 Damage indices for each floor for a column member at ultimate damage state of eight-

storey wall-equivalent system designed to DC M and PGA=0.25g

Appendices

C103



Appendices

C104



Appendices

C105



Appendices

C106



Appendices

C107


storey wall system designed to DC M and PGA=0.20g

Appendices

C108



Appendices

C109



Appendices

C110


wall – system designed to DC M and PGA=0.25g

Appendices

C111



Appendices

C112


storey wall system designed to DC M and PGA=0.25g

Appendices

C113


damage state of eight-storey wall system designed to DC M and PGA=0.25g

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