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CIVIL-706 - Time history non-linear analysis

EPFL-ENAC-SGC 2009 -1-

Non-linear time history analysisHysteretic models - accelerograms -

linearization methods

Ecole doctorale StructuresCIVIL-706 Advanced Earthquake Engineering

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Content

Equation of motion

Hysteretic models - Experimental tests

Numerical Modelling

Recorded/synthetic accelerograms

Equal Displacement Rule - Linearization

methods

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Earthquake Engineering assessmentmethods

Non-linear time history computation is themost sophisticated method

non-linear

static dynamic

elastic

structure

action

EquivalentForce Method

ResponseSpectrum Meth.

Non-Linear

Dynamic

Pushover

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Non-linear time history analysis

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Content

Equation of motion


Numerical Modelling


Equal Displacement Rule - Linearization methods

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Equation of motion

Single-Degree-of-Freedom (SDOF) System

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Equation of motion

Linear case

Non-linear case

Non-linearity causes:- Coulomb damping force, R(x,xA) or other non-viscous

damping- Variable stiffness, FS(x)

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Equation of motion

Resolution

Numerical Method Linear Non-linearFourier (frequency domain resolution) X ( )

Step by step time domain integration:

- I n t e r polation of excitation X ( )-central difference-Newmark, Wilson

X

X

Models:

-hysteretic model- macro-model- fibres

Finite Elements X

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Content

Equation of motion


Numerical Modelling



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Observed non-linear behaviour

Experimental tests on a RC wall

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Experimental tests on a RC wall

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RC wall hysteresis loop

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Dynamic tests on URM wall (ElGawady, ETHZ-EPFL, 2004) rocking

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Dynamic tests on URM wall (ElGawady, ETHZ-EPFL, 2004) rockingHysteresis loops few energy dissipation

-20

-10

0

10

20

-10.0 -5.0 0.0 5.0 10.0

dplacement relatif [mm]

force[kN]

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Static-cyclic tests on URM wall (ElGawady,EPFL, 2004) shear and sliding

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Static-cyclic tests on URM wall (ElGawady,EPFL, 2004) shear and slidingHysteresis loops larger energy dissipation

-50

-40

-30

-20

-10

0

10

20

30

40

50

-10 -5 0 5 10

dplacement relatif [mm]

force[kN]

relative displacement [mm]

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Concept of Ductility

Definition

deformation

force strength

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Hysteretic models

Elastoplastic (bi-linear) modelF

x

K

K

1

1

K

1

rK1

M

x(t)

K

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Hysteretic models

Takeda model (behaviour for large loops)

M

x(t)

K

F

x

xeK0

K/K0 = f(xp/xe)

! (xp/xe)

xp

rK0

1

1

1

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Hysteretic models

Takeda model (behaviour for small loops)

F

x

AB

Rmin

RmaxR

X

petits cycles avec plastification

F

x

C

C

petites amplitudesSmall loops with plastic behaviour Small amplitudes

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Hysteretic models

Takeda model - Account for stiffness degradation

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Hysteretic models

Experimental Observations: crossing loops

relative displacement

force

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Hysteretic models

ModelF

x

K

K

K

1

1

1

! Fy

! FyFy

M

x(t)

K

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ETHZ dynamic tests

Comparison models with dynamic tests

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ETHZ dynamic tests

From the real building to the laboratory test setup

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ETHZ dynamic tests

Modelling of the test

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ETHZ dynamic tests

Tested RC walls

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ETHZ dynamic tests

EC 8-compatible synthetic ground motion

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ETHZ dynamic tests

EC 8-compatible synthetic ground (table) motion

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ETHZ dynamic tests

Recorded relative displacements WDH3 & WDH5

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ETHZ dynamic tests

Recorded Hysteresis loops

WDH3

-200

-150

-100

-50

0

50

100

150

200

-80 -60 -40 -20 0 20 40 60 80

drel 3rd floor [mm]

Mbase[kNm

]

! ,m= 3.4

WDH5

-200

-150

-100

-50

0

50

100

150

200

-80 -60 -40 -20 0 20 40 60 80

drel 3rd floor [mm]

! ,m= 3.2

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Experimental/Model comparison

Time histories WDH3 (, EP models)!-model: f0=1.25 Hz; r=10%; xe=22 mm; !=0.35

-100

-50

0

50

100

0 5 10 15 time [s]

drel3

rdfloor[mm]

measured

computed

EP-model: f0=1.25 Hz; r=10%; xe=22 mm

-100

-50

0

50

100

0 5 10 15 time [s]

dre

l3

rdfloor[mm]

measured

computed

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Time histories WDH3 (, Takeda models)

Takeda-model: f0=1.9 Hz; r=6%; xe=8.5 mm; !=0.35; "=0

-100

-50

0

50

100

0 5 10 15 time [s]

drel3

rdfloor[mm]

measured

computed

!-model: f0=1.25 Hz; r=10%; xe=22 mm; !=0.35

-100

-50

0

50

100

0 5 10 15 time [s]

drel3

rdfloor[mm]

measured

computed

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!-Model

-1

0

1

-80 -60 -40 -20 0 20 40 60 80

drel 3rd floor [mm]

M/Mmax;F/Fmax[-]

measured

computed

" ,p= 3.3

EP-Model

-1

0

1

-80 -60 -40 -20 0 20 40 60 80

drel 3rd

floor [mm]

M/Mmax;F/F

max[-]

measured

computed

! ,p= 3.5


Hysteresis loops WDH3, ,m = 3.4 (, EP models)

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!-Model

-1

0

1

-80 -60 -40 -20 0 20 40 60 80

drel 3rd floor [mm]

M/Mmax;F/Fm

ax[-]

measured

computed

",p

= 3.3

Takeda-Model

-1

0

1

-80 -60 -40 -20 0 20 40 60 80

drel 3rd floor [mm]

M/Mmax;F/Fm

ax[-]

measured

computed

!,p= 8.8


Hysteresis loops WDH3, ,m = 3.4 (, Takeda models)

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Time histories WDH5 (, Takeda models)!-model: f0=1.25 Hz; r=25%; xe=24 mm; !=0.45

-100

-50

0

50

100

0 5 10 15 time [s]

drel3

rdfloor[mm]

measured

computed

Takeda-model: f0=1.25 Hz; r=25%; xe=24 mm; !=0; "=0

-100

-50

0

50

100

0 5 10 15 time [s]

drel3

rdfloor[mm]

measured

computed

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!-Model

-1

0

1

-80 -60 -40 -20 0 20 40 60 80

drel 3rd floor [mm]

M/Mmax;F/Fma

x[-]

measured

computed

" ,p= 3.0

Takeda-Model

-1

0

1

-80 -60 -40 -20 0 20 40 60 80

drel 3rd

floor [mm]

M/Mmax;F/Fmax[-]

measured

computed

!,p= 2.9


Hysteresis loops WDH5, ,m

= 3.2 (, Takedamodels)

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Content

Equation of motion


Numerical Modelling



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Finite Element modelling

Macro-elements with hysteretic behaviour

(Reclosing of cracks)

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Fibres semi-local models

2D beam elements(Bernoulli, Timoshenko)

1D material behaviour

BtonF

rettNon

Frett

01!

1! !O 2!

Perte de lenrobage

u0!

BBton en compressionton en compression"

!

Dformation

sE

"

Contrainte

v

v

v

v

Discrtisation en couches accoles

vvvvvvvvvvvvvv

ArmaturesBton dmeBton frett de bordsBton denrobage

Side by side layers discretization

Concretecover

Fretted sideconcrete

Innerconcrete

Rebars

Strain

Stress

Concrete in compression

Frettedconcrete

Nonfretted

concre

te

Lost of cover

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Local models

3D elements

Smeared or discrete rebars

Too large computations for a whole structure fordynamic behaviour

Started to be usedfor construction elements

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Ex: Grenoble City Hall building (El Arem, Desprez,

Kotronis, Mazars - ARVISE project)LxlxH =

43x12x52 m

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Kotronis, Mazars - ARVISE project)Modelling:

- Slabs, prestressed beams and last floor walls linear

- Frames and walls non-linear

- Cast3M software- ~19000 elements

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La Borderie concrete modelMenegotto-Pinto steel model


Kotronis, Mazars - ARVISE project)Multi-fibre modelling with sections following the rebar plans

- Concrete following the La Borderie damage model(progressive stiffness restoration during reclosing)

- Steel following the Menegotto-Pinto model

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Kotronis, Mazars - ARVISE project)Linear validation using a weak earthquake recorded in the

structure

Correlation around 95%

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Application to a strong earthquakeML=5.5 at 15 km simulated byEmpirical Greens Functions

including site effectsPGA=3.5 m/s2 (~LAquila April2009)

Traction damage in concrete

Some plastic rebars at the first

floor


Kotronis, Mazars - ARVISE project)

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Drawback:

- 7250 command lines

- 60 hours of CPU time

- 120 Go data saved

Only for important structures


Kotronis, Mazars - ARVISE project)

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Content

Equation of motion Hysteretic models - Experimental tests

Numerical Modelling



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Seismic loading

Ductility demand varies a lot with seismic loading

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Seismic loading

Recorded accelerograms

- real events- number limited but exponentially increasing- limited variability due to the limited recorded EQs- not conservative as design spectra

Synthetic

- stationnary simulation- non-stationnary simulation- conservative following

the design codes

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Seismic loading - Recordings

Strong motion networks- for engineering purposes (attenuation relationships, soilamplifications, site effects, structure monitoring)- in urbanized areas- on various soils- accelerometers (more noise, no clipping, broadband)

Seismological networks- for seismological purposes (source location, internal earthstudies)- in quiet areas

- on hard rock- velocimeters (more sensitive, clipping, short period orbroadband)

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Source: S. Godeyhttp://www.neries-eu.org/

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Where to find accelerometric data ? Selected datasets (CD or website):

Earthquake Strong Motion Collection (US and more)http://www.ngdc.noaa.gov/hazard/fliers/se-0308.shtml

European Strong Motion database (Ambraseys et al., 2002)http://www.isesd.cv.ic.ac.uk/ESD/

Alpine Accelerometric Database (2006)

Complete databases including recent events (websites with searchengines)

Worldwide: Center for engineering strong motion data includingCOSMOS (worldwide) http://www.strongmotioncenter.org/

Switzerland: http://seispc2.ethz.ch/strong_motion/home_en.jsp

France: RAP http://www-rap.obs.ujf-grenoble.fr

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Ground motion parameters

To classify or select ground motions for experimental

tests or modelling. To quantify the seismic demand in the design codes

Should represent the potential danger for structures

Peak Ground Acceleration (PGA) most commonly usedbut high frequency parameter

Now standardization from the accelerometric networks

(NERIES project http://www.neries-eu.org/)

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Direct parameters based on acceleration:

Raw PGA (cm/s2) from unfiltered record

PGA (cm/s2) from filtered record at 0.1 Hz

Arias intensity AI (cm/s)

Trifunac duration TD (s)Cumulative Absolute Velocity CAV (cm/s)

Based on velocity or displacement:

PGV (cm/s) representative around 1 Hz

PGD (cm) above 1m/s2 low frequency parameter

AI="

2g[a(t)]2dt

0

#

$

TD = t95%

" t5%

t" #"

AI=

$

2g [a(t)]

2

dt0

t"

%

CAV= a(t) dt0

"

#

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Spectral parameters computed for SDOF systems:

PSV (5%) (from 0.1Hz-50Hz) (cm/s)

Housner intensity or Response Spectrum Intensity(cm) found to be well correlated with ductilitydemand

Computed from PSV

PSA (5%)

PSD (5%)

IH=

Sv (5%,T)dT0.1

2.5

"

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Macroseismic Intensity

Also estimating the ground motion amplitude

Based on people feelings, objects motion and damage

Only parameter available for strong historical earthquakes

Correlation with PGA exist (crude approximations)

Drawbacks:

Not fully objective

Circular definition (based on damage)

Not precise

Discrete values (fuzzy logic)

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Seismic loading

Recorded or synthetic accelerograms ?

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Seismic loading - Stationary simulation

SIMQKE softwareBased on random vibrations theory (Gasparini & Vanmarcke, 1976)

Time windows of an random stationary function

Poor quality simulation of real earthquakes

OK for linear behaviour, non-linear??

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Seismic loading - Stationary simulation

Poor quality simulation

of real EQ Non-varying frequency

content ( sinus)

Based on a statisticalrelationship betweenFourier and Response

spectra Random Phases

Time-envelope tosimulate non-stationarity

Iterations on Fourierspectrum to converge to

the target Responsespectrum

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Seismic loading - Non-stationary simulation

Sabetta & Pugliese method

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Influence of magnitude (M)

- amplitude increases with M

- period of maximal amplitude increases with M

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Influence of epicentral distance (R)

- amplitude decreases with increasing R

- shape unchanged

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Influence of soil conditions (S)

- frequency/period shift between stiff and deep

- amplification for shallow

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Sabetta & Pugliese method, compatible with

design spectra on average (SIA 261 Z3b soil A)

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Sabetta & Pugliese method, compatible with

design spectra on average (SIA 261 Z3b soil E)

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Seismic loading - Comparison

Systematic comparison recorded/synthetic

Schwab & Lestuzzi (2007)

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9 recordings of ESMD

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Average response spectrum (green) and

design spectrum

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5 simulation

techniquesSIMQKE

SIMQKE withoutiterations

SIMQKE recorded

SIMQKE recordedwithout iterations

Sabetta & Pugliese

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Average spectra for 100 generated

accelerograms for each simulationtechnique

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SDOF results (various hysteretic models)

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MDOF results

confirms SDOF results

does not depend on the selected accelerograms in thesimulations

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Conclusions:

non-stationnary simulation performs clearly better

- ductility demand- energy demand

do not use convergence procedures

- lost of demand and variability- not conservative results

properly define target spectrum

- realistic

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Content

Equation of motion


Numerical Modelling


Equal Displacement Rule - Linearization

methods

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Linearization of non-linear behaviour

Which linear SDOF represents the best the

hysteretic behaviour ?- Main parameter: Period T

Which value ? Elastic (initial), intermediate or secant ?

- Also important: Damping Constant value ? Increasing value ?

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Experimental record

Elastic period

Intermediate period

and increased damping

Secant periodand increased damping

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Equal Displacement Relationship

Empirical statement (Veletsos & Newmark 1960)

Base of current seismic design codes

Many numerical/experimental evidences

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Equal Displacement Relationship does not work

at low periodsR--T relationships (EC8 approach)

e.g. so called Equal Energy Rule (actuallywrong)

Source: Fajfar, 1999

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ATC 40 approach

T is the secant stiffness

is a function of and T (many availablerelationships)

e.g. Dwairi et al. (2007):

Te= T

"e = "+C#1

$

"Te>1 C= 0.5

"Te# 1 C= 0.5+ 0.4 $(1% T

e)

&'(

More knowledge in the model, should be better onaverage and decrease the uncertainty

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Intermediate approach

T is an intermediate period function of

e.g. Iwan (1980)

is a function of and T

More parameters fitted from non-linear time historyanalyses. Again, more knowledge in the model,should be better on average and decrease the

uncertainty

Te= T "(1+ 0.121"(#1)

0.939)

"e= "+ 0.0587#($1)0.371

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Comparison of linearization methods

164 ground motions extracted from the ESM

database

- M>5

- PGA>0.6 m/s2

- Ep. D. free

4.5

5

5.5

6

6.5

7

7.5

8

0 20 40 60 80 100 120 140 160 180 200

Epicentral Distance [Km]

Magnitude

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Comparison of linearization methods

Methodology

Non-linear SDOF considered as the true responseDistribution of the error with respect to the non-linear maximum

response

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Comparison on average (bias)

All methods diverge more or less at low periods

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Comparison of variability

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Comparison good estimates (less than 30%

error)

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CIVIL 706 Tim hi t li l i

Conclusion

Seismic loading implies taking into account many new

parameters due to the dynamic behaviour

Best available tool=experimental tests

Simplified numerical implementation validated using tests

Synthetic accelerograms do not always represent real

seismic loading - Variability has to be considered

Linearization of non-linear behaviour is a crude but

necessary approximation for design and assessment

Not possible at low periods

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