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CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -1-
EDCE: Civil and Environmental Engineering CIVIL 706 - Advanced Earthquake Engineering
Displacement-based methods
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -2-
Content
•! Link to force-based methods
•! Assumptions
•! Reinforced concrete: chord rotation
•! Capacity curve
•! Examples: RC shear walls and frames
•! Unreinforced masonry
•! Example: masonry shear walls
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -3-
Link to force-based methods •! Application of behavior factor q specified in
the codes SIA 262-266
•! For existing buildings, non ductile behavior (q = 1.5 or q = 2.0 for reinforced concrete)
•! Displacement-based method: more realistic approach of real seismic behavior
•! Allows to justify an higher value of q
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -4-
Displacement-based Method (DbM) •! Applicable to deformable structures
•! Brittle failure modes excluded:
- shear failure
- bending with concrete failure before steel yielding
- bending with steel failure without sufficient plastic deformation
- failure of overlapping zones or anchorage zones
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -5-
Displacement-based Method (DbM) •! More realistic approach of real behavior
Joe s
Beer! Food!
lateral displacement
lateral force
Joe s
Beer! Food!
small change in force level
large change in deformation level
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -6-
Method included in SIA 2018 •! Slides prepared by
Prof. Alessandro Dazio, IBK- ETH Zürich
i b k Erdbebeningenieurwesen und Baudynamik
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -7-
Reinforced concrete: chord rotation
V
V
M
!
shear wall
M
M V
V
column
!
Vl
Ml
Vr
Mr
beam
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -8-
Reinforced concrete: chord rotation •! Angle formed by the tg to element axis at
Mmax location and the chord joining this point to the zero moment point
•! Represents the element sollicitation (removal of rigid displacements)
•! Verification by the comparison (demand-capacity) of chord rotations at the element level
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -9-
Chord rotation !: definitions
rigid body rotation
V
V
M
!
shear wall
M
M V
V
column
!
Vl
Ml
Vr
Mr
beam
= chord = tangent to element axis = zero moment point
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -10-
Yielding chord rotation !y
Fy
Vy
My
!y
Lv
moment
My "y
curvature Fy
Vy
My
#y
vyvv
y
2vy
3vy
y LL3L
3L
EIM
EI3LF
!=""#="==$
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -11-
Ultimate chord rotation !u
Fu
Vu
Mu
!u
Lv
moment
Mu "y "p
"u
curvature
Lpl
Lpl = plastic hinge
( )blsvstpl dfLaL !!+!= 022.008.0
hsp
hsp = strain penetration
hpl
hpl = plastic zone
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -12-
Ultimate chord rotation !u
Fu
Vu Mu
#y "y "p
"u
Lpl Lpl
!y
Lv
( ) )L5.0L(LLL plvplyuvyvuu !"!!#"#+$=$=%
#p
#u
!p
!u
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -13-
#
F
Non-linear force-displacement relationship
F
V
M
!
Lv
#
reality
#y #u
Fy
Fu
approximation
#y = f("y), Fy = f(My), #u = f("y, "u), Fu = f(Mu),
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -14-
Bending moment-curvature relationship
"’y
M’y
"y
Mn
"y = "’y Mn / M’y
M
"
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -15-
Bending moment-curvature relationship •! shear wall
N = -1099 kN
200
4000
36 Ø82 Ø20 2 Ø20
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -16-
Bending moment-curvature relationship
First yield $s = $y $c = 0.002
"’y
M’y
Nominal strength $s = 0.015 $c = 0.004
"y
Mn Nominal yield
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 curvature [10 -3 m -1 ]
bend
ing
mom
ent
[MN
m] Ultimate
$s = $s,max, $c = $c,max
"u
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -17-
Bending moment-curvature relationship
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 curvature [10 -3 m -1 ]
bend
ing
mom
ent [
MN
m]
"’y "y "u
Mn
M’y
1st approximation
modified approximation
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -18-
Bending moment-curvature relationship •! nominal yield curvature "y
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -19-
Bending moment-curvature relationship •! Material mechanical properties
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -20-
Material mechanical properties
0
5
10
15
20
25
0 1 2 3 4 5 strain [‰]
stre
ss [M
Pa]
a) concrete BH300, SIA 168/68
1) f cd acc. to SIA 262 2) f ck acc. to SIA 2018 3) f ck (t) acc. to SIA 2018
1)
2)
$ cu
3)
f ck (t)
0
100
200
300
400
500
600
0 10 20 30 40 50 strain [‰]
stre
ss [M
Pa]
b) steel IIIa, SIA 168/68
1) f sd acc. to SIA 262 2) f sk approx. 3) f sk type
1)
2) 3)
$ uk
f tk
f sk
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -21-
Material mechanical properties •! Characteristic values
•! Partial factor for deformation capacity %D = 1.3
•! Ultimate concrete deformation $cmax = $cu = 0.004 (in general)
•! Peak steel strain $smax = &$su - in general: & = 1.0 - if Mu<2Mcr: & = 0.5 - special cases: & = ?
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -22-
Application of DbM •! Non-linear element behavior
•! Equivalent SDOF
•! Force-displacement relationship of structure
•! Capacity curve
•! Seismic displacement demand
•! Shear strength verification
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -23-
Example: reinforced concrete
•! 5 stories
•! Building end of 60
•! 1 x direction « wall »
•! 1 x direction « frame »
•! Zone 3b, CS C, CO I
Main properties
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -24-
Example RC: dimensions and structure m5=213t
m4=380t
m3=380t
m2=380t
380t
A
B
C
1 2 3 4 5
11 12 13 14 15
16 17 18 19 20
6
7 8 9
10
mtot = 1733t
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -25-
Example RC: element dimensions shear wall cross-section
beam at mid span beam at connection
column type A column type B
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -26-
Example RC: shear walls direction
Fd
wStructure réelle
m 1
m 2
m 3
m 4
m 5
F d
w (MDOF)
k MDOF = ' k walls
m*=1163t
F d
h*=1
1.95
m
k*
w/ (
(SDOF)
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -27-
Example RC: shear walls direction •! Force-displacement relationship
0 100 200 300 400 500 600 700 800 900
0.00 0.05 0.10 0.15 0.20 horizontal displacement, w/ ( [m]
glo
bal h
oriz
onta
l for
ce F
d [kN
m]
shear walls
building
F u =828kN F y =744kN
F u =414kN F y = 372kN
w y / ( =0.039m w u / ( =0.119m
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -28-
Example RC: shear walls direction •! ADRS spectrum with capacity curve
0
1
2
3
4
5
0.00 0.05 0.10 0.15 0.20 S ud , w / ( [m]
S ad ,
F d /m
* [m
/s 2 ]
elastic design spectrum (Z3b, CO I, CS C)
T=1.55s
w d / ( w u / ( w R,d / ( capacity curve
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -29-
Example RC: frame direction •! Force-displacement rel.: !frames A, B and C
•! Torsion neglected
A
B
C
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -30-
Example RC: frame direction •! Modelisation
Elément traverse
Lpl Lpl
L
!EIg (var.)
Elément pilier
Lpl
Lpl
L !EIg
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -31-
Example RC: frame direction •! Column element type A
M
M V
V L pl /2
= plastic hinge M/ ! (rigid-plastic with softening)
L pl /2
N
N
0 100 200 300 400 500
0 10 20 30 40 curvature [10 -3 m -1 ]
bend
ing
mom
. [K
Nm
]
N=-1500kN -1000kN -500kN
0kN
0 100 200 300 400 500
0 5 10 15 plastic rotation [10 -3 ]
bend
ing
mom
. [K
Nm
] -1000kN
N=0kN -500kN
-1500kN
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -32-
Example RC: frame direction •! Beam element
-400
-300
-200
-100
0
100
200
-100 -50 0 50 100 150 200
Courbure [10-3 m-1]
Mom
ent [
KN
m]
= Rotule plastique M/! (rigide-plastique avec adoucissement)
Type 2Type 8
Type 6
Type 4
V
M
V
M
Lpl/2 Lpl/2
-400-300-200-100
0100200
-20 0 20 40 60
Rotation plastique [10-3]
Mom
ent [
KN
m]
Type 2Type 8
Type 6
Type 4
2
Type 8
6
4
2
6
4
Type 8
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -33-
Example RC: frame direction •! Force-displacement relationship
0
500
1000
1500
2000
2500
0.00 0.05 0.10 0.15 0.20 0.25 0.30 horizontal displacement W d [m]
gl
obal
hor
izon
tal f
orce
F d [
kN]
frame A
frame B
building = 2 x frame A + 1 x frame B F y =2032kN
w y =0.098m w u =0.182m
onset of plastic hinge softening
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -34-
Example RC: frame direction •! Ultimate deformation of frame B
Fd
wd
= Rotule= Rotule (rupture atteinte)
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -35-
Example RC: frame direction •! ADRS spectrum with capacity curve
w u / ( w R,d / ( 0
1
2
3
4
5
0.00 0.05 0.10 0.15 0.20 S ud , w d / ( [m]
S ad ,
F d /m
* [m
/s 2 ]
elastic design spectrum (Z3b, CO I, CS C)
T=1.52s
w d / (
capacity curve
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -36-
Unreinforced masonry
Yverdon-Les-Bains
Vernier D
örfli
ngen
dou
ane
O
berriet
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -37-
DbM: unreinforced masonry •! Assumptions
(FEMA 356) plastic deformations concentrated in some elements (piers)
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -38-
DbM: unreinforced masonry •! Limitation: out-of-plane failure excluded
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -39-
Masonry: out-of-plane failure •! Control with h/t ratio (FEMA 356)
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -40-
Masonry: deformation capacity •! Compression depending (tests EPFZ)
V
)
0
100
200
300
400
500
600
0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008
Schiefstellung
Schu
bkra
ft [k
N]
W6
W4
W1
W7 W2
drift
late
ral f
orce
[kN
]
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -41-
Masonry: deformation capacity •! Influence of compression
0
100
200
300
400
500
600
0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008
Schiefstellung
Schu
bkra
ft [k
N]
W6
W4
W1
W7 W2
large compression
low compression
drift
late
ral f
orce
[kN
]
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -42-
Masonry: deformation capacity •! In function of normal force (K. Lang, 2002)
based on test results
nu !" #$= 25.08.0
maximum interstorey drift
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Normalspannung [MPa]
Sto
ckw
erks
schi
efst
ellu
ng [%
]
compression stress [MPa]
in
ters
tore
y dr
ift [%
]
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -43-
Strength of unreinforced masonry •! In function of the inclination of *2
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -44-
Lateral strength of URM elements •! Determination with stress fields
Principle: superposition of a
vertical field
and an
inclined field
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -45-
Lateral strength of URM elements •! Determination with stress fields
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -46-
Lateral strength of URM elements •! Determination with stress fields
– équilibre: ! !Nv + Nn = !! ! ! !Nv·e1v + Nn·e1n = M1
! ! ! !Nv·e2v + Nn·e2n = M2 = M1 + V·h
– limitation des sollicitations:
f! = ! fmy l2v·t·(cos!)Nv
2
fv = ! (fmx " fmy)l2n·tNn
tan! ! tan#
" "l2v = lw " 2·e2v
avec
" "l2n = lw " 2·e2n
– résistance: ! !V = Nv·tan!
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -47-
Failure modes •! Rocking
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -48-
Failure modes •! Shear
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -49-
Lateral strength of URM elements •! Simplified formulae (FEMA, EC8)
rocking (EC8): shear (EC8):
VRd,S = fvd · t · lc
with fvd = fvd0 + 0,4 · Nxd / (lc · t)
fvd ! 0,065 · fb
!
VRd ,R =lw " Nxd
2 " h0" 1#1,15 " Nxd
lw " tw " f xd
$
% &
'
( )
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -50-
Masonry: deformation capacity •! According to EC8
rocking: shear:
[%]8.0 0
wu l
h!="
[%]4.0=u!
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -51-
Unreinforced masonry buildings •! Swiss typical building (Yverdon)
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -52-
Example: unreinforced masonry •! Simplified typology
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -53-
Example: unreinforced masonry •! Lateral strength in transversal direction
Nd [kN] without coupling
effect VRd [kN]
with total coupling effect
VRd [kN]
lw = 2.0 m 180 22 96
lw = 5.0 m 360 117 216
lw = 6.0 m 1520 367 624
total 690 1440
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -54-
Example: unreinforced masonry •! Displacement determination (Lang 2002)
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -55-
Example: unreinforced masonry •! Capacity curve in transversal direction
0 100 200 300 400 500 600 700
0 5 10 15 20 25 top building displacement [mm]
horiz
onta
l str
engt
h [
kN]
wall 2m wall 5m wall 6m building y
capacity curve in y direction
wR
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -56-
Unreinforced masonry buildings •! Swiss typical building (“in line” building) RC rigid floors, diaphragm effect Very regular, no torsion 3 shear walls, long. direction Zone Z1, agd = 0.6 m/s2 Soil E, S = 1.40 Hstorey = 2.85 m Mstorey = 470 t VM,base = 880 kN Top, wy = 20 mm Top, wu = 35 mm
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -57-
Unreinforced masonry buildings •! Swiss typical building (“in line” building)
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -58-
Unreinforced masonry buildings •! Swiss typical building (“in line” building)
!
"!!
#!!
$!!
%!!
&!!
'!!
(!!
)!!
*!!
"!!!
! & "! "& #! #& $! $& %!
+,-./0121345/657822145922:
;,7<74/3015=5./5>/7159?@:
!"!
!"#
$"!
$"#
%"!
%"#
! $! %! &! '! #! (!
)*+,*-Γ .//0
)1+,2/-/3,./-4% 0
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -59-
Masonry: lot of uncertainties •! Real element deformation capacity?
•! Effective coupling effect?
•! Effective element stiffness (crack pattern)?
•! Research efforts needed
CIVIL 706 - displacement-based methods EDCE-EPFL-ENAC-SGC 2016 -60-
Real behavior ? •! Coupling effect (frame effect)
flexible floors (without coupling effect): very stiff lintels (total coupling effect):
Htot
hst
lo
lo
hst
h0
Htot
hst
lo
lo
hst
h0
F2
F1
F2
F1
!
h0 " 2 3 # htot
!
h0 " 12 # hst
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