dynamic non-linear analysis hysteretic models€¦ · indicated that damping ratios were of a...
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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
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Content
• Equation of motion
• Resolution methods
• Non-linear behaviour
• Hysteretic models
• Comparison with experimental tests
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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
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Non-linear time-history analysis
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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)
CIVIL 706 - Hysteretic models EDCE-EPFL-ENAC-SGC 2016 -7-
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
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Resolution : Central difference method • Approximation by
finite differences :
• Equation of motion at time step ti :
• Next time step ti+1 :
CIVIL 706 - Hysteretic models EDCE-EPFL-ENAC-SGC 2016 -9-
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
CIVIL 706 - Hysteretic models EDCE-EPFL-ENAC-SGC 2016 -10-
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
CIVIL 706 - Hysteretic models EDCE-EPFL-ENAC-SGC 2016 -11-
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
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Observed non-linear behaviour Experimental tests on a RC wall
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Observed non-linear behaviour Experimental tests on a RC wall
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Observed non-linear behaviour RC wall hysteresis loop
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Observed non-linear behaviour Dynamic tests on URM wall (ElGawady, ETHZ-
EPFL, 2004) → rocking
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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]
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Observed non-linear behaviour Static-cyclic tests on URM wall (ElGawady, EPFL,
2004) → shear and sliding
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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]
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Observed non-linear behaviour Concept of Ductility Definition
deformation
forc
e strength
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Hysteretic models S-model
F
x
K 1
r K 1
M
x(t)
K
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Hysteretic models Elastoplastic (bi-linear) model
F
x
K
K1
1
K1
r·K1
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)
β ·(xp/xe)
xp
r·K0
1
1
1
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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
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Hysteretic models Takeda model - Account for stiffness degradation
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Hysteretic models Experimental Observations: crossing loops
relative displacement
forc
e
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Hysteretic models γ-Model
F
x
K
K
K
1
1
1
γ·Fy
γ·Fy
FyM
x(t)
K
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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
CIVIL 706 - Hysteretic models EDCE-EPFL-ENAC-SGC 2016 -28-
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
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ETHZ dynamic tests Comparison models with dynamic tests
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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
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
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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
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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
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
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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)
CIVIL 706 - Hysteretic models EDCE-EPFL-ENAC-SGC 2016 -39-
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
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)
CIVIL 706 - Hysteretic models EDCE-EPFL-ENAC-SGC 2016 -40-
Experimental/Model comparison Time histories WDH5 (γ, Takeda models)
γ-model: f0=1.25 Hz; r=25%; xe=24 mm; γ=0.45
-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
CIVIL 706 - Hysteretic models EDCE-EPFL-ENAC-SGC 2016 -41-
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= 3.0
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)
CIVIL 706 - Hysteretic models EDCE-EPFL-ENAC-SGC 2016 -42-
Experimental/Model comparison OOP, h= 1.5 m; t= 110 mm, Δmax = 44 mm (50%)
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Experimental/Model comparison OOP, h= 1.5 m; t= 110 mm, overturning for 66%
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Finite Element modelling Macro-elements with hysteretic behaviour
(Reclosing of cracks)
CIVIL 706 - Hysteretic models EDCE-EPFL-ENAC-SGC 2016 -45-
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
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Non-linear behaviour simulation Viscous damping increase (linear behaviour)
Takeda model with β=0 Elastoplastic model
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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 :
CIVIL 706 - Hysteretic models EDCE-EPFL-ENAC-SGC 2016 -48-
• Comparison with dynamic tests (WDH3)