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CIVIL 706 - Hysteretic models EDCE-EPFL-ENAC-SGC 2016 -1-

EDCE: Civil and Environmental Engineering CIVIL 706 - Advanced Earthquake Engineering

Dynamic non-linear analysis Hysteretic models


Content

•  Equation of motion

•  Resolution methods

•  Non-linear behaviour

•  Hysteretic models

•  Comparison with experimental tests


Assessment methods Non-linear time-history computation is the

most sophisticated method

non-linear

static dynamic

elastic

structureaction

Equivalent Force Method

Response Spectrum Meth.

Non-Linear Dynamic

Pushover


Non-linear time-history analysis


Equation of motion Single-Degree-of-Freedom (SDOF) System


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)


Equation of motion Resolution

Numerical Method Linear Non-linear Fourier (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 - m a c r o-model - fibres…

Finite Elements X


Resolution : Central difference method •  Approximation by

finite differences :

•  Equation of motion at time step ti :

•  Next time step ti+1 :


Central difference method : MatLab

I=1; ppsurM=Erdbeben(I)-asurM*uzero-bsurM*u(I); u(I+1)=ppsurM/kksurM; v(I)=u(I+1)-uzero/2/dt; a(I)=(u(I+1)-2*u(I)+uzero)/dt^2; for I=2:nb-1,

ppsurM=Erdbeben(I)-asurM*u(I-1)-bsurM*u(I); u(I+1)=ppsurM/kksurM; v(I)=(u(I+1)-u(I-1))/2/dt; a(I)=(u(I+1)-2*u(I)+u(I-1))/dt^2;

end


Resolution : Newmark’s method •  Approximation of relative displacement and

velocity at time step ti+1 :

•  Main distinction for non-linear analysis Central difference method : explicit Newmark’s method : implicit


Newmark’s method (linear) : MatLab

for I=1:nb-1, dpsurM=Erdbeben(I+1)-Erdbeben(I)+asurM*v(I)+bsurM*a(I); duI=dpsurM/kksurM; dvI=gama/beta/dt*duI-gama/beta*v(I)+dt*(1-gama/2/beta)*a(I); daI=1/beta/dt^2*duI-1/beta/dt*v(I)-1/2/beta*a(I); u(I+1)=u(I)+duI; v(I+1)=v(I)+dvI; a(I+1)=a(I)+daI;

end


Observed non-linear behaviour Experimental tests on a RC wall


Observed non-linear behaviour RC wall hysteresis loop


Observed non-linear behaviour Dynamic tests on URM wall (ElGawady, ETHZ-

EPFL, 2004) → rocking


Observed non-linear behaviour Dynamic tests on URM wall (ElGawady, ETHZ-

EPFL, 2004) → rocking Hysteresis loops → few energy dissipation

-20

-10

0

10

20

-10.0 -5.0 0.0 5.0 10.0

dépl acement relatif [mm]

forc

e [k

N]

relative displacement [mm]


Observed non-linear behaviour Static-cyclic tests on URM wall (ElGawady, EPFL,

2004) → shear and sliding


Observed non-linear behaviour Static-cyclic tests on URM wall (ElGawady,

EPFL, 2004) → shear and sliding Hysteresis loops → larger energy dissipation

-50

-40

-30

-20

-10

0

10

20

30

40

50

-10 -5 0 5 10

déplacement relatif [mm]

forc

e [k

N]

relative displacement [mm]


Observed non-linear behaviour Concept of Ductility Definition

deformation

forc

e strength


Hysteretic models S-model

F

x

K 1

r K 1

M

x(t)

K


Hysteretic models Elastoplastic (bi-linear) model

F

x

K

K1

1

K1

r·K1

M

x(t)

K


Hysteretic models Takeda model (behaviour for large loops)

M

x(t)

K

F

x

xeK0

K/K0 = f(xp/xe)

β ·(xp/xe)

xp

r·K0

1

1

1


Hysteretic models Takeda model (behaviour for small loops)

F

x

AB

Rmin

RmaxR

X

petits cyc les avec pl astifi cation

F

x

C

C

petites ampl itudesSmall loops with plastic behaviour Small amplitudes


Hysteretic models Takeda model - Account for stiffness degradation


Hysteretic models Experimental Observations: crossing loops

relative displacement

forc

e


Hysteretic models γ-Model

F

x

K

K

K

1

1

1

γ·Fy

γ·Fy

FyM

x(t)

K


Hysteretic models Griffith-Model: masonry out-of-plane behaviour UNREINFORCED MASONRY WALLS 837

pivot

2/3 h

∆e = 2/3 t

F0

Mgt/2 pivots

pivotInertia force distribution

F0/2Mg/2

R=F0/2-Mgt/2h

h/6

∆e = 2/3 t

F=0

∆e = 0 ∆e = 0

F=0

Inertia forcedistribution

R’=F0/2+Mgt/2h

R’=F0

F0/2

(a) Parapet Wall at incipient Rockingand Point of Instability

(b) Simply-Supported Wall at Incipient Rockingand Point of Instability

Figure 3. Inertia forces and reactions on rigid URM walls.

A similar expression, Equation (4), also derived using standard modal analysis procedures,is used to de!ne the e"ective displacement (#e).

#e =!n

i=1 mi!2i

!ni=1 mi!i

(4)

It can be shown from Equation (4) that

#e = 2=3#t (for a parapet wall) and (5a)

#e = 2=3#m (for simply-supported wall) (5b)

where #t and #m are the top of wall and mid-height wall displacements, respectively.Note that both Equations (3) and (5) are based on the assumption of a triangular-shaped

relative displacement pro!le. This can be justi!ed for a rocking wall where the displacementsdue to rocking far exceed the imposed support displacements. The accuracy of this assumptionhas been veri!ed with shaking table tests and THA as described in Reference [12]. Thus, theresultant inertia force is applied at two-thirds of the height of a parapet wall, and one-third ofthe upper half of the simply supported wall measured from its mid-point (Figures 3(a) and3(b)).

Copyright ? 2002 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2002; 31:833–850


Hysteretic models Griffith-Model: masonry out-of-plane behaviour 842 K. DOHERTY ET AL.

Forc

e

Rigid body(bi-linear model)

Experimentalnon-linear

Tri-linear model

F0

∆1 ∆2 ∆fDisplacement

Note : Only the positivedisplacement range isshown

Figure 6. Force–displacement relationship of deformable URM walls.

value of ! for parapet walls to be in the order of 3 per cent using this technique. The viscousdamping factor can also be calculated from dynamic equilibrium as the net di!erence betweenthe experimentally determined inertia force and the restoring force (according to the recordedacceleration and displacement, respectively) at any instant of time during the rocking response.Subsequent free-vibration experiments carried out on a range of simply supported walls [12]indicated that damping ratios were of a similar order. This critical damping ratio can betranslated into a viscous damping factor using the following equation to carry out non-linearTHA:

C = 2!!Me = 4"f!Me (10)

where ! is the angular velocity of the linearized system. Further details considering thefrequency dependence (and hence amplitude dependence) is provided in Reference [12].

4. MODELLING OF CRACKED UNREINFORCED MASONRY WALLSAS DEFORMABLE (SEMI-RIGID) BLOCKS

The bilinear force–displacement relationship described in the previous section is based onthe assumption that URM walls behave essentially as rigid bodies which rock about pivotpoints positioned at cracks. It has been con"rmed by experimental static push-over tests thatthe individual blocks of the URM wall can deform signi"cantly when subjected to highpre-compression. This results in: (i) pivot points possessing "nite dimensions (rather thanbeing in"nitesimally small) so that the resistance to rocking is associated with a lever armsigni"cantly less than half the wall thickness (as for a rigid wall) and (ii) the wall possessing"nite lateral sti!ness (rather than being rigid) prior to incipient rocking. Importantly, thethreshold resistance to rocking is reduced signi"cantly from the original level associated witha rigid wall, to a ‘force plateau’ as shown in Figure 6. It can be further seen from Figure 6that the F–# relationship observed during the experiment deviates signi"cantly from thisbilinear relationship and assumes a curvilinear pro"le. This is largely due to the non-linear

Copyright ? 2002 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2002; 31:833–850


ETHZ dynamic tests Comparison models with dynamic tests


ETHZ dynamic tests Modelling of the test


ETHZ dynamic tests Tested RC walls


ETHZ dynamic tests EC 8-compatible synthetic ground motion


ETHZ dynamic tests EC 8-compatible synthetic ground (table) motion


ETHZ dynamic tests Recorded relative displacements WDH3 & WDH5


ETHZ dynamic tests

WDH3

-200

-150

-100

-50

0

50

100

150

200

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

drel 3rd floor [mm]

Mba

se [k

Nm

]

µΔ,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]

µΔ,m= 3.2

Recorded Hysteresis loops


Experimental/Model comparison Time histories WDH3 (γ, EP models)

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

-100

-50

0

50

100

0 5 10 15 time [s]

drel 3

rd fl

oor [

mm

]

measuredcomputed

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

-100

-50

0

50

100

0 5 10 15 time [s]

drel 3

rd fl

oor [

mm

] measuredcomputed


Experimental/Model comparison Time histories WDH3 (γ, Takeda models)

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

-100

-50

0

50

100

0 5 10 15 time [s]

drel 3

rd fl

oor [

mm

]

measuredcomputed


-100

-50

0

50

100

0 5 10 15 time [s]

drel 3

rd fl

oor [

mm

]

measuredcomputed


Experimental/Model comparison

γ-Model

-1

0

1

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

drel 3rd floor [mm]

M/M

max

; F

/Fm

ax [

-]

measuredcomputed

µΔ,p= 3.3

EP-Model

-1

0

1

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

drel 3rd floor [mm]

M/M

max

; F

/Fm

ax [-

]

measuredcomputed

µΔ,p= 3.5

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



γ-Model

-1

0

1

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

drel 3rd floor [mm]

M/M

max

; F

/Fm

ax [

-]

measuredcomputed

µΔ,p= 3.3

Takeda-Model

-1

0

1

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

drel 3rd floor [mm]

M/M

max

; F

/Fm

ax [

-]

measuredcomputed

µΔ,p= 8.8

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


Experimental/Model comparison Time histories WDH5 (γ, Takeda models)


-100

-50

0

50

100

0 5 10 15 time [s]

drel 3

rd fl

oor [

mm

]

measuredcomputed

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

-100

-50

0

50

100

0 5 10 15 time [s]

drel 3

rd fl

oor [

mm

]

measuredcomputed



γ-Model

-1

0

1

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

drel 3rd floor [mm]

M/M

max

; F

/Fm

ax [

-]

measuredcomputed

µΔ,p= 3.0

Takeda-Model

-1

0

1

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

drel 3rd floor [mm]

M/M

max

; F

/Fm

ax [-

]

measuredcomputed

µΔ,p= 2.9

Hysteresis loops WDH5, µΔ,m = 3.2 (γ, Takeda models)


Experimental/Model comparison OOP, h= 1.5 m; t= 110 mm, Δmax = 44 mm (50%)


Experimental/Model comparison OOP, h= 1.5 m; t= 110 mm, overturning for 66%


Finite Element modelling Macro-elements with hysteretic behaviour

(Reclosing of cracks)


Finite Element modelling Fibres semi-local models 2D beam elements

(Bernoulli, Timoshenko) 1D material behaviour

v

v

v

v

D is crétis ation en couches accolées

vvvvvvvvvvvvvv

ArmaturesBéton d’âmeBéton fretté de bordsBéton d’en robage

Side by side layers discretization

Concrete cover

Fretted side concrete

Inner concrete

Rebars

Strain

Stre

ss

Concrete in compression

Lost of cover


Non-linear behaviour simulation Viscous damping increase (linear behaviour)

Takeda model with β=0 Elastoplastic model


Non-linear behaviour simulation •  Linear elastic substitute-SDOF with equivalent

stiffness and equivalent damping coefficient

displacement ductility : equ

ival

ent d

ampi

ng c

oeff.

: EP-model :


•  Comparison with dynamic tests (WDH3)

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