S4-4 Damping Paper

7/25/2019 S4-4 Damping Paper

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EFFECT OF DAMPING MODELLING

ON RESULTS OF TIME-HISTORYANALYSIS OF R.C. BRIDGES

Nigel Priestley, Michele Calvi,Lorenza Petrini, Claudio Maggi

IUSS/University of Pavia


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3. INITIAL STIFFNESS DESIGN

DISPLACEMENT-EQUIVALENCE RULES

Basic Tenet: Displacement of designed inelastic structure can be

related to the elastic displacement of the initial-stiffness model bya displacement-equivalence rule

e.g. Newmark and Hall, 1982:

T=0: Equal accelerationT>0.5s: Equal displacement

T 0.2s: Equal energy

elastic

inelastic

Equal disp.

Equal energy

equal acc

Displacement

Strength

Design Codes from differentcountries use different rules


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19

INFLUENCE OF ELASTIC DAMPING ON

DISPLACEMENT-EQUIVALENCE RULEDisplacement-equivalence rules are based on Inelastic Time HistoryAnalysis (ITHA). The representation of the initial response has aconsiderable influence on the results:

Elastic damping is typically added to represent the initialstages,and expressed as a % of critical damping typically 5%.There are two main ways this could be modelled: as initial-stiffnessproportional damping, or tangent-stiffness proportional damping:

Initial-stiffness:damping alwaysproportional to thisslope

Tangent stiffness:damping force reduceswhen stiffness reduces;

increases whenstiffness increases


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F

gmakcm =++ &&&

mkmc 22 ==

SDOF MODEL

kinitial

Initial Stiffness damping: Constant Coefficient c, k=kinitial

Tangent Stiffness damping: Damping coefficient varies in

proportion to instantaneous stiffness: ck/kinitialNote: Peak displacement decreases as c increases

MOST ANALYSES USE CONSTANT COEFFICIENT DAMPING


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REASONS FOR ELASTIC DAMPING

LINEAR HYSTERESIS RULE UP TO FIRST YIELD: Norepresentation of hysteretic damping in the elastic range

FOUNDATION DAMPING: Soil flexibility, nonlinearity and

radiation damping generally not modelled in analysis

NON-STRUCTURAL DAMPING: Hysteretic response of non-structural elements; relative movement between structural andnon-structural elements may provide an effective damping force


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DISCUSSION OF REASONS FOR ELASTIC DAMPING

HYSTERESIS RULE: These are normally calibrated to model thefull hysteretic response in the non-linear range. Therefore the

elastic damping should stop after yieldFOUNDATION DAMPING: After structural yield, the foundationforces remain essentially constant as structure deformsinelastically. Therefore foundation damping should cease.

NON-STRUCTURAL ELEMENTS: Contribution is generally small(less than 1%) (particularly for bridges).

CONCLUSION: TANGENT-STIFFNESS DAMPINGIS MOST APPROPRIATE

(OTANI, 1981)


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-0.1 0 0.1Displacement (m)

-2000

-1000

0

1000

2000

StiffnessForce

(kN)


-2000

-1000

0

1000

2000

DampingForce(kN)


-2000

-1000

0

1000

2000

Stiffnes

sForce(kN)


-2000

-1000

0

1000

2000

DampingForce(kN)

(a) Analysis with Initial Stiffness Damping

(b) Analysis with Tangent Stiffness Damping

SDOF, T=0.5 sec

FORCED SINUSOIDAL

INPUT OF T=1.0sec

TAKEDA HYSTERESIS

Initial-StiffnessDamping Ad=0.83Ah

Tangent-StiffnessDamping Ad=0.15Ah


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0 4 8 12 16 20Time (seconds)

-0.1

-0.05

0

0.05

0.1

0.15

Displacement

(m)

Tangent Stiffness

Initial Stiffness

RESPONSE OF SDOF MODEL, T=0.5sec

TO 1.5*EL CENTRO 1940 NS

max,TS=1.43

max,IS


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EXAMPLES OF HYSTERETIC MODELS USED

TO TEST EQUAL-DISPLACEMENT

TAKEDA BILINEAR E-P FLAG-SHAPE

TAKEDA HYSTERESIS


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0 0.4 0.8 1.2 1.6 2Period (seconds)

0.8

1.2

1.6

2

2.4

D

isplacementRatio(/

elastic

)

0 0.4 0.8 1.2 1.6 2Period (seconds)

0.8

1.2

1.6

2

2.4

0 0.4 0.8 1.2 1.6 2Period (seconds)

0.8

1.2

1.6

2

2.4

0 0.4 0.8 1.2 1.6 2Period (seconds)

0.8

1.2

1.6

2

2.4

DisplacementR

atio(/

elastic

)

0 0.4 0.8 1.2 1.6 2Period (seconds)

0.8

1.2

1.6

2

2.4

0 0.4 0.8 1.2 1.6 2Period (seconds)

0.8

1.2

1.6

2

2.4

TS

IS

TS

IS

TS

IS

TS

IS

TS

IS

TS

IS

(a) r = 0.002

(b) r = 0.05

R=2R=4 R=6

R=2 R=4 R=6

TAKEDA HYSTERESIS

BILINEAR ELASTO PLASTIC


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0 0.4 0.8 1.2 1.6 2Period (seconds)

0.8

1.2

1.6

2

2.4

Dis

placementRatio(/

e

lastic

)

0 0.4 0.8 1.2 1.6 2Period (seconds)

0.8

1.2

1.6

2

2.4

0 0.4 0.8 1.2 1.6 2Period (seconds)

0.8

1.2

1.6

2

2.4

0 0.4 0.8 1.2 1.6 2Period (seconds)

0.8

1.2

1.6

2

2.4

DisplacementRatio(/

elastic

)

0 0.4 0.8 1.2 1.6 2Period (seconds)

0.8

1.2

1.6

2

2.4

0 0.4 0.8 1.2 1.6 2Period (seconds)

0.8

1.2

1.6

2

2.4

TS

IS

TS

IS

TS

IS

TS

IS

TS

IS

TS

IS

(a) r = 0.002

b r = 0.05

R=2 R=4 R=6

R=2 R=4 R=6

BILINEAR ELASTO-PLASTIC

FLAG HYSTERESIS


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0 0.4 0.8 1.2 1.6 2Period (seconds)

0.8

1.2

1.6

2

2.4

DisplacementRatio(/

elastic

)

0 0.4 0.8 1.2 1.6 2Period (seconds)

0.8

1.2

1.6

2

2.4

0 0.4 0.8 1.2 1.6 2Period (seconds)

0.8

1.2

1.6

2

2.4

0 0.4 0.8 1.2 1.6 2Period (seconds)

0.8

1.2

1.6

2

2.4

DisplacementRat

io(/

elastic

)

0 0.4 0.8 1.2 1.6 2Period (seconds)

0.8

1.2

1.6

2

2.4

0 0.4 0.8 1.2 1.6 2Period (seconds)

0.8

1.2

1.6

2

2.4

TS

IS

TS

IS

TS

IS

TS

IS

TS

IS

TS

IS

(a) = 0.35

R=2 R=4 R=6

R=2 R=4 R=6

FLAG HYSTERESIS


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0 0.4 0.8 1.2 1.6 2Period (seconds)

1

1.1

1.2

1.3

1.4

1.5

Displacement

RatioTS

/IS

0 0.4 0.8 1.2 1.6 2Period (seconds)

1

1.1

1.2

1.3

1.4

1.5

0 0.4 0.8 1.2 1.6 2Period (seconds)

1

1.2

1.4

1.6

1.8

2

(a) Takeda (dash:r=0.002, solid:r=0.05) (b) Bilinear (dash:r=0.002,solid:r=0.05) (c) Flag (dash:=0.35,solid:=0.70)

R=6

R=6R=4

R=4

R=2

R=6

R=4

R=2

R=4

R=6

R=6

R=4

R=2

TANGENT/INITIAL STIFFNESS DISPLACEMENTRATIOS

TAKEDA BILINEAR E-P FLAG


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SHAKE-TABLE TESTS ON BRIDGE PIERS

Static test to calibrate Takeda hysteresis rule

Chose accelerogram to emphasize differencebetween initial and tangent stiffness response

Dynamic test with high ductility demand

cm


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Pier cross-section

Column

50cm

200cm

288

cm

88cm

Spiral 6/3 cm

Spiral 6/6 cm

G

Footing

Deck mass

156cm

88cm

45cm

28cm

1810

Spiral 6

MODEL DIMENSIONS

fc = 39MPa

fy = 514MPa

Mass = 7.8tonnes


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-100

-80

-60

-40

-20

0

20

40

60

80

100

-0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

Displacement [m]

Force[kN]

SeismoStruct simulation

Experimental test

STATIC TEST + SEISMOSTRUCT SIMULATION


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STATIC TEST + TAKEDA SIMULATION

-100

-80

-60

-40

-20

0

20

40

60

80

100

-0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

Displacement [m]

Force

[kN]

Experimental test

Ruaumoko simulation

1 6


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-1.6

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

1.6

0 5 10 15 20 25 30 35 40

Time [s]

cceerato

n

g

MORGAN HILL RECORD SCALED TO 1.2g PGA


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VIDEO


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DYNAMIC TEST + TAKEDA SIMULATION

0 4 8 12 6 10Time (sec)

-0.1

0

0.1

-0.05

0.05

0.15

-0.15

Disp

lacement(m

)Experiment

Tangent Stiffness

Initial Stiffness

DYNAMIC TEST + SEISMOSTRUCT SIMULATION


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0 4 8 12 6 10

Time (sec)

-0.1

0

0.1

-0.05

0.05

0.15

-0.15

Displac

ement(m)

Experiment

Tangent-Stiffness

Initial Stiffness


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0 4 8 12 6 10Time (sec)

-0.1

0

0.1

-0.05

0.05

0.15

-0.15

Disp

lacement(m

)

Experiment

No damping

PEAK DISPLACEMENTS INITIAL 10 2sec


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PEAK DISPLACEMENTS, INITIAL 10.2sec

Positive Peak

(mm)

Negative Peak

(mm)

Experiment 110 109

SeismoStruct0% damping

99 108

Takeda

Tangent-Stiff.

114 112

TakedaInitial-Stiff.

60 74


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CONCLUSIONS

Common sense indicates that initial-stiffness is

inappropriate for elastic damping If tangent-stiffness is appropriate for elasticdamping, then the equal-displacement approximation

is non-conservative Shake-table testing supports tangent-stiffnessproportional damping for Takeda modelling

S4-4 Damping Paper

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