CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -1- EDCE: Civil and Environmental Engineering CIVIL 706 - Advanced Earthquake Engineering N2 method and Rocking behaviour NL Time History Analysis check
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -1-
EDCE: Civil and Environmental Engineering CIVIL 706 - Advanced Earthquake Engineering
N2 method and Rocking behaviour NL Time History Analysis check
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -2-
Content
• Equal displacement rule
• Target displacement, EC 8 part 1, Annex B
• Approximate graphical assessment
• N2 displacement demand prediction accuracy
• Rocking (S-model and Griffith model)
• Fast assessment for cantilever out-of-plane
• Amplification with floor response spectrum
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -3-
Equal displacement rule
• Favorable effect, behaviour factor q • Displacement ductility = behaviour factor value • Avoid brittle failures (shear failures)
displacement
elastic
xe = xel / q xp = xel
Fy = Fel / q
Fel
plastic
force
Empirical rule: equal displacements
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -4-
Equal displacement rule, check NLTHA • 164 recorded earthquakes, methodology
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -5-
Equal displacement rule, check NLTHA • 164 recorded earthquakes, results
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -6-
EC 8 P1, Annex B : Target displacement • Long period, equal displacement rule
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -7-
EC 8 P1, Annex B : Target displacement • Short period, relative larger displacement
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -8-
Determination of target displacement • Alternative procedure used in USA, NZ, etc.
0.0
1.0
2.0
3.0
4.0
5.0
0.00 0.02 0.04 0.06 0.08 0.10
Sd [m]
S a [
m/s
2 ]
Performance point Sd = 35 mm
0.0
1.0
2.0
3.0
4.0
5.0
0.00 0.05 0.10 0.15 0.20 Sd
Sa
Damping according to « Substitute structure approach » ( βéquivalent , ksecant )
F
x
β=5%
β=10%
β=20%
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -9-
EC 8 P1, Annex B : Target displacement • Short period, used assumption (Tlim=TC)
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -10-
EC 8 P1, Annex B : Target displacement • Short period, relative larger displacement
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -11-
EC 8 P1, Annex B : Target displacement • Short period, graphical approximation
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -12-
EC 8 P1, Annex B : Target displacement • Short period, graphical approximation
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -13-
Parameters for EC8 (SIA 261) soil classes
classe description S TB[s] TC[s] TD[s]
A massifs rocheux 1,00 0,15 0,4 2,0
B gravier ou sable très compact 1,20 0,15 0,5 2,0
C compact ou semi-compact 1,15 0,20 0,6 2,0
D lâche à semi-compact 1,35 0,20 0,8 2,0
E couche C ou D sur A ou B 1,40 0,15 0,5 2,0
F sensible, organique - - - -
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -14-
Elastic response spectra in EC 8 (SIA 261)
0
1
2
3
4
0.01 0.1 1 10période T[s]
Se/
a gd
classe de sol A
classe de sol B
classe de sol C
classe de sol D
classe de sol E
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -15-
Displacement demand prediction accuracy • Sets of 12 recorded TH, Abrahamson matching
10-2 10-1 100 1010
1
2
3
4
5Sa
[m/s
2]SIA 261 soil class A, before modification
10-2 10-1 100 1010
1
2
3
4
5
Sa [m
/s2]
SIA 261 soil class A, after modification
10-2 10-1 100 1010
1
2
3
4
5
Sa [m
/s2]
SIA 261 soil class B, before modification
10-2 10-1 100 1010
1
2
3
4
5
Sa [m
/s2]
SIA 261 soil class B, after modification
10-2 10-1 100 101
period [s]
0
1
2
3
4
5
Sa [m
/s2]
SIA 261 soil class C, before modification
10-2 10-1 100 101
period [s]
0
1
2
3
4
5
Sa [m
/s2]
SIA 261 soil class C, after modification
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -16-
0 0.1 0.2 0.3 0.4 0.5 0.60
0.005
0.01
0.015di
spl.
dem
and
[m]
12 modified ESMD-earthquakes for soil class A, R=2
Prop1
Prop2
EC8
0 0.1 0.2 0.3 0.4 0.5 0.60
0.005
0.01
0.015
disp
l. de
man
d [m
]
Takeda model (alpha=0.4, beta=0.0), R=3
Prop1
Prop2
EC8
0 0.1 0.2 0.3 0.4 0.5 0.6period [s]
0
0.005
0.01
0.015
disp
l. de
man
d [m
]
strength reduction factor R=4
Prop1
Prop2
EC8
0 0.1 0.2 0.3 0.4 0.5 0.6period [s]
0
0.005
0.01
0.015
disp
l. de
man
d [m
]
strength reduction factor R=5
Prop1
Prop2
EC8
Displacement demand prediction accuracy
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -17-
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.005
0.01
0.015
0.02
0.025
0.03di
spl.
dem
and
[m]
12 modified ESMD-earthquakes for soil class B, R=2
Prop1
Prop2
EC8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
0.005
0.01
0.015
0.02
0.025
0.03
disp
l. de
man
d [m
]
Takeda model (alpha=0.4, beta=0.0), R=3
Prop1
Prop2
EC8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7period [s]
0
0.005
0.01
0.015
0.02
0.025
0.03
disp
l. de
man
d [m
]
strength reduction factor R=4
Prop1
Prop2
EC8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7period [s]
0
0.005
0.01
0.015
0.02
0.025
0.03
disp
l. de
man
d [m
]
strength reduction factor R=5
Prop1
Prop2
EC8
Displacement demand prediction accuracy
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -18-
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04di
spl.
dem
and
[m]
12 modified ESMD-earthquakes for soil class C, R=2
Prop1
Prop2
EC8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
disp
l. de
man
d [m
]
Takeda model (alpha=0.4, beta=0.0), R=3
Prop1
Prop2
EC8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8period [s]
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
disp
l. de
man
d [m
]
strength reduction factor R=4
Prop1
Prop2
EC8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8period [s]
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
disp
l. de
man
d [m
]
strength reduction factor R=5
Prop1
Prop2
EC8
Displacement demand prediction accuracy
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -19-
0 0.2 0.4 0.6 0.8 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08di
spl.
dem
and
[m]
12 modified ESMD-earthquakes for soil class D, R=2
Prop1
Prop2
EC8
0 0.2 0.4 0.6 0.8 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
disp
l. de
man
d [m
]
Takeda model (alpha=0.4, beta=0.0), R=3
Prop1
Prop2EC8
0 0.2 0.4 0.6 0.8 1period [s]
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
disp
l. de
man
d [m
]
strength reduction factor R=4
Prop1
Prop2
EC8
0 0.2 0.4 0.6 0.8 1period [s]
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
disp
l. de
man
d [m
]
strength reduction factor R=5
Prop1
Prop2
EC8
Displacement demand prediction accuracy
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -20-
Site effect
Earthquake =
source * propagation * site
Site effect due to the reflexions of seismic waves in the "soft » soil deposits
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -21-
Microzonation : H/V measurements
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -22-
Microzonation : SASW measurements
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -23-
Microzonation : Yverdon
0
1
2
3
4
0.01 0.1 1 10période T[s]
Se [
m/s
2 ]
Yverdon, S1
Yverdon, S2
Yverdon, S3
Yverdon, S4
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -24-
Microzonation : Yverdon, response spectra
0
1
2
3
4
0.01 0.1 1 10période T[s]
Se [
m/s
2 ]Yverdon, S1
Yverdon, S2
Yverdon, S3
Yverdon, S4
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -25-
• ADRS format
0
1
2
3
4
0 0.02 0.04 0.06 0.08 0.1
Sd [m]
Se [
m/s
2 ]
soil class A, zone 2
soil class E, zone 2
Yverdon, Zone S1
Yverdon, Zone S2
Yverdon, Zone S3
Yverdon, Zone S4
Microzonation : Yverdon, response spectra
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -26-
• Zone S1 : are usual procedures valid ?
0
1
2
3
4
0 0.01 0.02 0.03 0.04 0.05 0.06
Sd [m]
Se [
m/s
2 ]
soil class A, zone 2
Yverdon, Zone S1
Microzonation : Yverdon, response spectra
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -27-
• Zone S1 : are usual procedures valid ?
Microzonation : Yverdon, response spectra
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -28-
• Zone S1 : are usual procedures valid ?
Microzonation : Yverdon, response spectra
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -29-
• Zone S1 : are usual procedures valid ?
Microzonation : Yverdon, response spectra
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -30-
• Zone S1 : are usual procedures valid ?
Microzonation : Yverdon, response spectra
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -31-
Conclusions – N2 method
• Equal displacement rule from Tlim=TC
• Target displacement in Part 1, Annex B
• Approximate graphical assessment
• Accuracy (R=3 ✔, R<3 !, R>3 ")
• Procedure also valid for special response spectra from microzonation studies
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -32-
Rocking behaviour, check NLTHA • 164 recorded earthquakes, methodology
duct
ility
dem
and
frequency
non-linear behavior
164 recorded earthquakes
acce
lera
tion
time
disp
lace
men
t
time
non-linear dynamic response
12 MDOF
MDOF
13 f09 Rfo
rce
displacement
4 hysteretic models
SDOF
disp
lace
men
t
time
non-linear dynamic response
statistical
analysis
statistical
analysis
4 R
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -33-
Rocking behaviour, check NLTHA • Hysteretic models
force
S-modelforce
displacement displacement
modified Takeda-modelelastoplastic (EP-model)
Flag-model
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -34-
Rocking behaviour, check NLTHA • 164 recorded earthquakes, results
initial frequency [Hz]
disp
lace
men
t duc
tility
dem
and
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
123456789
10S-model
initial frequency [Hz]di
spla
cem
ent d
uctil
ity d
eman
d
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
123456789
10modified Takeda-model
R = 4
R = 3.5
R = 3
R = 2.5
R = 2
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -35-
Rocking behaviour, check NLTHA • 164 recorded earthquakes, results
frequency [Hz]
disp
lace
men
t duc
tility
dem
and
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
123456789
10S-model
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00123456789
10Flag-model
frequency [Hz]di
spla
cem
ent d
uctil
ity d
eman
d R = 4
R = 3.5
R = 3
R = 2.5
R = 2
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -36-
Rocking behaviour, check NLTHA • Slightly modified R – µΔ relationships
frequency [Hz]
disp
lace
men
t duc
tility
dem
and
0.0 0.5 1.0 1.5 2.0 2.5 3.0
1
2
3
4
5
6
7S-model
0.0 0.5 1.0 1.5 2.0 2.5 3.00
1
2
3
4
5
6
7Flag-model
frequency [Hz]di
spla
cem
ent d
uctil
ity d
eman
d R = 4
R = 3.5
R = 3
R = 2.5
R = 2
= 4 /3 ·R - 1 /3 = 3 /2 ·R - 1 /2
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -37-
Masonry out-of-plane behaviour • Griffith : rigid body 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 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -38-
Griffith model : rigid body behaviour • Force – displacement relationships 838 K. DOHERTY ET AL.
Forc
e
∆ f=2/3 t
F0=M egt/h F0= 4M egt/hKo
KoF0= 4(1+ψ)M egt/h
loadbearing
Non-loadbearing
Ψ=overburden weight/(Mg/2)
Displacement
Forc
e
Displacement
−∆ f −∆ f∆ f=2/3 t
(a) Parapet Wall (b) Simply Supported Wall
Figure 4. Force–displacement relationships of rigid URM walls.
3. MODELLING OF CRACKED UNREINFORCED MASONRY WALLSAS RIGID BLOCKS
The spring force function F(!e) can be obtained by determining the total horizontal reaction(or base shear) at di"erent displacements using basic principles of static equilibrium. Forexample, the overturning equilibrium of a parapet wall about the pivot point at the base ofthe wall can be used to determine F(!e).For a parapet wall at the point of incipient rocking (i.e. !e = 0+ or alternatively !t = 0+),
moment equilibrium leads to (refer Figure 3(a)) the expression:
Mgt=2=F0(2=3)h (6a)
Solving for F0 (F at !e = 0+) and substituting Equation (3) into Equation (6a) gives
F0 =Me(gt=h) (6b)
For a parapet wall at the point of instability (!e = 2=3t or alternatively !t = t), the force Frequired for static equilibrium of the wall is given by
F =0 (6c)
Therefore, the F(!e) function for a parapet wall can be constructed in accordance withEquations (6b) and (6c) as shown in Figure 4(a).Similarly, moment equilibrium can also be used to determine F(!e) at the point of incipient
rocking (!e = 0+) for a wall simply supported at the top and bottom. By considering momentequilibrium of the upper half of a simply supported wall (of height = h=2 and mass=M=2)about the pivot point in the cracked cross-section at the mid-height of the wall leads to
(Mg=2)t=2=R(h=2)− (F0=2)(h=6) (7a)
Copyright ? 2002 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2002; 31:833–850
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -39-
Griffith model : rigid body behaviour • Tri-linear hysteretic model 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
UNREINFORCED MASONRY WALLS 843
Table II. Empirically derived trilinear F–! de"ning displacements.
State of degradation at cracked joint !1=!f !2=!f
New 6% 28%Moderate 13% 40%Severe 20% 50%
deformations that occur in the mortar joint. However, there is relatively little deviation fromthe original bilinear model at large displacements.This curvilinear pro"le can be idealized by a trilinear model that is de"ned by three dis-
placement parameters: !e;1, !e;2 and !e;max and the force parameter F0 (refer Figure 6). Toconstruct the trilinear model, the bilinear model is "rst constructed in accordance with F0and K0. The amplitude of the force plateau is, therefore, controlled by the ratio !2=!f . Fordisplacements in the range exceeding !2, the trilinear and the bilinear models coincide. Fordisplacements between !1 and !2, the force is constant. The initial slope of the trilinearmodel is governed by the force amplitude of the plateau and the value of !1.The ratios !1=!f and !2=!f are related to the material properties and the state of degra-
dation of the mortar joints at the pivot points. Data recorded during many quasi-static anddynamic tests of 14 simply supported walls suggests nominal values for the ratios of !1=!fand !2=!f for walls in ‘new’, ‘moderately degraded’ and ‘severely degraded’ condition asshown in Table II. The interpretation of the ‘moderately degraded’ and ‘severely degraded’conditions are highly subjective. From the experimental tests, the e#ective width of the mortarin the cracked bedjoint for walls classi"ed as severely degraded was approximately 90 percent of the original width. Moderately degraded walls had e#ective bedjoint widths that wereessentially equal to their original widths. However, the exposed vertical faces of the mortarjoints had rounded due to some rocking having taken place. Full details of these tests aregiven in Reference [13]. This trilinear F–! relationship proved to be e#ective for the wallstested in this study over the full range of degradation.The traditional method of selecting a secant sti#ness for use with a substitute structure rep-
resentation of a multi-degree-of-freedom system is not straightforward for non-ductile systemssuch as URM. One method commonly used is to adopt the secant sti#ness from the system’snon-linear force–displacement curve corresponding to the point of maximum (permissible)displacement. For ductile systems, this is often associated with a point on the post-peak soft-ening section of the non-linear force–displacement curve where the force has reduced to somefraction (75–80 per cent is common) of the peak force value. In this study, and for masonryin general, it was not simple to de"ne this point due to material strength variability and a lackof de"nitive yield and=or softening points. However, it was observed that the sti#ness corre-sponding to a line going through the point on the trilinear force–displacement curve where!=!2 as shown in Figure 7 was reasonably consistent with this notion.The e#ective secant sti#ness, Ks-e# , for the semi-rigid wall obtained in this manner can be
expressed mathematically in generic terms as
Ks-e# = K0!
1− 1!2=!1
"
(11)
Copyright ? 2002 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2002; 31:833–850
CIVIL 706 – N2, Rocking, check NLTHA EDCE-EPFL-ENAC-SGC 2016 -40-
Out-of-Plane : rigid body behaviourSeveral stability conditions Static conditions: rocking onset
italian code Overturning : Betbeder-Matibet
agd ≤α ⋅ g
a0* = PGA 2 ≤α ⋅ g
V ≈agd10
≤12⋅α ⋅ g ⋅ r
⇒ agd ≤ 5 ⋅twhw⋅ g ⋅ r
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Casestudy,defini.onsCantilever non bearing wall Dimensions: hw = 3 m (3rd storey)
tw = 0.1 m - 0.4 m Seismic: Z2
CO I soil CSC
Building: T1 = 0.3 s
T1 = 1.2 s T1 = 1.0 s T1 = 1.67 s
844 K. DOHERTY ET AL.
∆f
F0
∆1
(∆2-∆f)Ko
Ks-eff= Ko (∆2-∆f) / ∆2
∆2
Note : Only the positivedisplacement range isshown
Figure 7. E!ective secant sti!ness (Ks-e! ) of semi-rigid walls.
where K0 is de"ned as shown in Figures 5 and 6 and values for #2=#1 are given byTable II. The e!ective undamped natural frequency, fs-e! , for the equivalent SDOF system isaccordingly given by the following equation:
fs-e! =!
Ks-avg=Me
2!(12)
The experimentally observed ‘resonance’ frequency for each of the test walls was found toagree well with estimates given by Equation (12) using e!ective secant sti!ness values asde"ned above. (Note: using the approach of Section 3 where the e!ective sti!ness was takenas the slope of the line going through the centroid of the area under the force–displacementcurve gives similar results.)
5. DISPLACEMENT DEMAND PREDICTION BY SUBSTITUTE STRUCTUREIDEALIZATION
The DB analysis methodology provides a rational means for determining seismic design ac-tions as an alternative to the more traditional ‘quasi-static’ force-based approach. In the DBmethod, the dynamic lateral displacement capacity of a structure, subjected to an excitationis determined based on a comparison of the displacement demand imposed on the structureduring a seismic event with a pre-determined critical displacement capacity. The ‘substitutestructure’ methodology proposed by Shibata and Sozen [14] was adopted to simplify highlynon-linear systems into a linearized DB procedure. An elastic SDOF oscillator is selected withlinear properties that characterize those of the real non-linear structure. The e!ectiveness ofthe linearized DB procedure is reliant on the assumption that both the ‘substitute structure’and real system will reach the same critical displacement under the same excitation.It was observed from the parametric studies using Guassian Pulses (described in Section 3)
that incipient instability most likely occurs as a consequence of the large displacement am-pli"cations associated with resonance of the wall. The e!ective resonant frequency, fe! , isrelated to a particular e!ective secant sti!ness. It appears that the displacement demand ofURM walls arising from rocking can be predicted using the linearized DB analysis procedure
Copyright ? 2002 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2002; 31:833–850
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Griffith model : rigid body behaviour • Sets of 12 recorded TH, methodology
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Abrahamson matching for 12 TH – EC8/SIA
10-2 10-1 100 1010
1
2
3
4
5Sa
[m/s
2]SIA 261 soil class A, before modification
10-2 10-1 100 1010
1
2
3
4
5
Sa [m
/s2]
SIA 261 soil class A, after modification
10-2 10-1 100 1010
1
2
3
4
5
Sa [m
/s2]
SIA 261 soil class B, before modification
10-2 10-1 100 1010
1
2
3
4
5
Sa [m
/s2]
SIA 261 soil class B, after modification
10-2 10-1 100 101
period [s]
0
1
2
3
4
5
Sa [m
/s2]
SIA 261 soil class C, before modification
10-2 10-1 100 101
period [s]
0
1
2
3
4
5
Sa [m
/s2]
SIA 261 soil class C, after modification
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Non linear time history analysis • Examples
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Non linear time history analysis • Determination using incremental analysis
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Non linear time history analysis • Ultimate increasing value : results assessment
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Non linear time history analysis
tw[m]
tw / Hw
[-]α g
[m/s2]2 α g[m/s2]
NL (mean 12 TH > tw/2)ζ = 1% - 3% - 5% [m/s2]
NL (mean 12 TH > tw)ζ = 1% - 3% - 5% [m/s2]
0.10 0.033 0.3 0.7 0.29 – 0.38 – 0.43 0.35 – 0.51 – 0.610.12 0.040 0.4 0.8 0.35 – 0.46 – 0.52 0.40 – 0.61 – 0.730.15 0.050 0.5 1.0 0.44 – 0.57 – 0.65 0.50 – 0.73 – 0.910.18 0.060 0.6 1.2 0.53 – 0.68 – 0.78 0.60 – 0.93 – 1.090.20 0.067 0.7 1.3 0.58 – 0.76 – 0.86 0.67 – 1.02 – 1.210.25 0.083 0.8 1.6 0.73 – 0.94 – 1.08 0.81 – 1.27 – 1.510.30 0.100 1.0 2.0 0.87 – 1.13 – 1.29 1.00 – 1.46 – 1.810.40 0.133 1.3 2.6 1.16 – 1.51 – 1.72 1.33 – 1.95 – 2.42
• Aggregate results for different wall thicknesses
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Amplification with floor response spectra • Methodology
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• Limit case : stiff building
Amplification with floor response spectra
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• Limit case : flexible building
Amplification with floor response spectra
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T1 = 1.2 s T1 = 1.67 s
Amplification with floor response spectra
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Amplification with floor response spectra
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Conclusions – Rocking (S and Griffith)
• Slight modifications of R – µΔ relationships for S-model and flag-model
• Onset rocking condition on the safe side for cantilever walls with rigid body behaviour
• Approach also valid for walls in upper stories by taking into account adequate floor response spectra