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Rocking of Structures During Earthquakes: From Collapse of Masonry to Modern Design
Matt DeJong
Lecturer Department of Engineering
University of Cambridge
27 February, 2013
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L’Aquila, Italy (2009)
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L’Aquila, Italy (2009)
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Christchurch Spires
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Challenges
Collapse of Masonry – Can we explain collapse?
– Can we predict collapse?
– Can we improve seismic performance?
What benefits can rocking provide in modern design?
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Arch Structures: Static Analysis
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Typical Assumptions [1]
– Masonry has no tensile strength. – Masonry blocks are rigid. – Friction sufficient to prevent sliding.
Results of Assumptions:
– Geometry alone determines stability. – Results are scalable.
Graphical Methods [2]
Thrust Line Analysis
[1] Heyman, The Stone Skeleton, 1995.
[2] Huerta, Arcos, bóvedas y cúpulas, 2004.
Thrust Line Analysis
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Tilting Thrust Line Analysis
β
αcr
tan 0.37hcr
v
u gu
γ α= = =
β
β
α
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Buttressed Barrel Vault
0
15
30
45
0 5 10 15Vault height / Buttress width [-]
Tilt
angl
e [d
egre
es]
unstable
stable
Arch alone
Buttress alone
Fixed: Vault height Vault thickness Vault span Variable: Buttress width
( )tan 7.7 0.13gγ = =
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Church with Side Aisles
0
2
4
6
8
0 0.5 1 1.5Side aisle buttress width / Side aisle span [-]
Tilt
angl
e [d
egre
es]
Main aisle alone
Side aisle alone
Mechanism 1
Mechanism 2
Mechanism 3
Mechanism 1:
Mechanism 2:
Mechanism 3:
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Dynamics: The masonry arch
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The masonry arch
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Immediate questions result
When will it collapse?
Can we model this “rocking”
behavior?
What happens at impact?
How would the arch respond to an earthquake?
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The rigid rocking block
– Inverted pendulum G.W. Housner (1963) [1]
– Harmonic ground acceleration
Spanos (1984)
– Slide-rock response H.W. Shenton (1991) A. Sinopoli (1992)
– Bouncing response
Lipscombe (1990)
– Earthquake loading: Primary impulse response Makris (2001, 2004) [2]
[1] Housner. Bull of the Seism Soc of America, v.53(2), 1963.
[2] Makris & Black. J of Engineering Mechanics, v.130(9), 2004.
Previous work
H
x
z
R
α
O
B
θ
c.g.
gu
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Rocking Block: Impulse response
Pulse
acc
eler
atio
n, a
p[g
]
Pulse duration, Tp [s]
No impact collapse
Rocking and Recovery
No Rocking
One impact collapse
Governing failure curve
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The rocking arch [1]
Differential equation of motion:
Initial conditions:
After impact:
( ) ( ) ( ) ( ) gxPgFRLRM
θθθθθθ =++ 2
0)0( θθ =
0)0( =θ)(tθ
[1] De Lorenzis, DeJong & Ochsendorf, Earth Eng and Struct Dyn, v.36, 2007.
( ) ( ) ( ) ( ) gxPgFRLRM
θθθθθθ ′−=′+′′+′′ 2
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The impact problem
At impact, assume the impulsive force at the point of closing hinges:
5 Unknowns: – 2 components of FA – 2 components of FD – Rotational velocity after impact:
5 Equations: – Linear momentum along x and along y (2) – Angular momentum about O – Angular momentum about B of the left portion – Angular momentum about C of the right portion
Coefficient of restitution
– Referred to the angular velocities:
)()(
−
+′=
i
iv t
tcθθ
)( +′ itθ
)( +′ itθA
B
C
D
αA
αBαC
αD
y
xO
A
BC
D
FA FD
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Hinge reflection
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Failure domain plot – R = 10 m, t = 1.5 m, β = 157.5o
Analytical modeling results
β
αcr
0.37gγ =0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Impulse duration [s]
Impu
lse
acce
lera
tion
[*g]
No impact collapse
Rocking and Recovery
No Rocking
One impact collapse
Static Solution
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Analytical model predictions
Effect of scale Effect of thickness
Failure domain scales with the square root of R
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1Impulse duration [s]
Impu
lse
acce
lera
tion
[*g]
t/r = 0.11t/r = 0.15t/r = 0.17t/r = 0.19
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5Impulse duration [s]
Impu
lse
acce
lera
tion
[*g] l
R = 20 mR = 10 mR = 5 mR = 1 m
Larger Thicker
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Seismic loading: Hinge reflection
El Centro Earthquake
-3
0
3
0.0 0.4 0.8 1.2Time [s]
Acc
eler
atio
n [g
]
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0
0.2
0.4
0.6
0.8
1
0 0.05 0.1 0.15 0.2 0.25Impulse period [s]
Impu
lse
acce
lera
tion
[*g]
0
0.2
0.4
0.6
0.8
1
0 0.05 0.1 0.15 0.2 0.25Impulse period [s]
Impu
lse
acce
lera
tion
[*g]
Experimental program
Extract the “primary impulse” from the earthquake time history.[1] Increase acceleration magnitude and test repeatedly until failure occurs. Repeat for several earthquakes.
Time [s]
Acce
lera
tion
[*g]
Acceleration DataPrimary Impulse
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Testing results: Primary Impulse
-1.5
-1
-0.5
0
0.5
1
1.5
2.7 3.2 3.7
Time [s]
Acce
lera
tion
[g]
Acceleration dataPrimary impulse
-1.5
-1
-0.5
0
0.5
1
1.5
1.2 1.7 2.2
Time [s]Ac
cele
ratio
n [g
]
Acceleration dataPrimary impulse
-1.5
-1
-0.5
0
0.5
1
1.5
2.4 2.9 3.4
Time [s]
Acce
lera
tion
[g]
Acceleration dataPrimary impulse
-1.5
-1
-0.5
0
0.5
1
1.5
2.1 2.6 3.1
Time [s]
Acce
lera
tion
[g]
Acceleration dataPrimary impulse
-1.5
-1
-0.5
0
0.5
1
1.5
1.7 2.2 2.7
Time [s]
Acce
lera
tion
[g]
Acceleration dataPrimary impulse
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.05 0.1 0.15 0.2 0.25Pulse Period, t p [s]
Pul
se A
ccel
erat
ion,
ap /
g
ParkfieldEl CentroGolden gate (no failure)NorthridgeHelenaAnalytical Model
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.05 0.1 0.15 0.2 0.25Pulse Period, t p [s]
Pul
se A
ccel
erat
ion,
ap /
g
ParkfieldEl CentroGolden gate (no failure)NorthridgeHelenaAnalytical Model
[1] DeJong et al., Earthquake Spectra, v.24(4), 2008
Static solution
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Computational Modelling
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The Arch
El Centro earthquake
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The Arch
Failure domain plot – R = 10 m, t = 1.5 m, β = 157.5o
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4Impulse duration [s]
Impu
lse
acce
lera
tion
[*g]
analytical modelUDEC
No rocking
Rocking and
recovery
Mode 2 collapse
Mode 1 collapse
Static solution
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The Buttressed Barrel Vault
0
0.2
0.4
0.6
0.8
1
1.2
0 0.4 0.8 1.2 1.6Impulse duration [s]
Impu
lse
acce
lera
tion,
*g
Buttressed barrel vaultArch only (Mode I)Arch only (Mode II)
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Stone Masonry Spires
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Christchurch Spires
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Lincolnshire Earthquake, 2009
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Stone Masonry Spire, Waltham on the Wolds, UK
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[1] RMW Musson, Annals of GeoPhysics, v.47(2/3), 2004.
Essex Earthquake, 1884
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[1] P Hanning, The Great English Earthquake, 1976.
Essex Earthquake, 1884
“…It was clearly noticed that the fall of chimneys to the south-west proceeded the fall of the spire… Moreover the debris of the spire and of the chimneys nearly all over Colchester has tumbled on the north east sides of the buildings, pointing to the conclusion that something like a wave of upheaval was felt approaching from the south-west, and causing a fall in the opposite direction.”
-Dr Alexander Wallace, Colchester, 1884
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Analytical Rocking Model
x
y
H
rb
CM
H/3
O
λMcg
Mcg
CM
hc
β
φ
Geometry Tilt Test
( )
3 2 2 32 2
3 2 3
1 2 2 2 2332 2
c c cb
c c
h h H h H Hr
H h h H Hπ π
λ
− − − + =− +
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Analytical Rocking Model
x
y
H
rb
CM
H/3
O
R
α
θ
φ
Geometry Rocking
2 2
2 2
0
0
g
g
up p
g
up p
g
θ θ α θ
θ θ α θ
− = − + → >
− = − − → <
Op MgR I=where:
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Analytical Rocking Model
x
y
H
rb
CM
H/3
O
λMcg
Mcg
CM
O
hc R
α
θ β
φ
Geometry Rocking Tilt Test
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Stone Spire, Waltham on the Wolds
O 9.4 m
3.4 m
β
βmax H
O’
O’’
Analytical DEM Physical
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Tilt Test
DEM: ag = 0.17g
Analytical: ag = 0.19g (perfect hollow cone)
Physical: ag = 0.16g
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Impulse rocking response
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Seismic response
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5
Pulse
acc
eler
atio
n, a
p[g
]
Pulse duration, Tp [s]
Damage (DEM)
Collapse (DEM)
Collapse (DEM), Tilt Test
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Rocking Spire
Pulse
acc
eler
atio
n, a
p[g
]
Pulse duration, Tp [s]
No impact collapse
Rocking and Recovery
No Rocking
One impact collapse
Governing failure curve
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Impulse rocking response - Analytical
O
O
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5
Pulse
acc
eler
atio
n, a
p[g
]
Pulse duration, Tp [s]
Collapse (Analytical), entire spireCollapse (Analytical), Solid spire tipCollapse (DEM)
Damage (DEM)
Collapse (DEM), Tilt Test
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0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
λ
hc / ho
10 m
2 m
Puls
e Acc
eler
atio
n, A
p [g]
Pulse Duration, Tp [s]
λcs,min
Spire mechanisms
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0
1
2
3
4
3.5 5.5 7.5 9.5
p[1
/s]
H [m]
Impulse rocking response
O
β
H
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5
Pulse
acc
eler
atio
n, a
p[g
]
Pulse duration, Tp [s]
Collapse (DEM)
Collapse (Analytical), Cracked spire tip
2 2
2 2
0
0
g
g
up p
g
up p
g
θ θ α θ
θ θ α θ
− = − + → >
− = − − → <
Op MgR I=where:
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Impulse rocking response
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Seismic response
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5
Pulse
acc
eler
atio
n, a
p[g
]
Pulse duration, Tp [s]
Damage (DEM)
Collapse (DEM)
Collapse (DEM), Tilt Test
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Seismic Response
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Rocking Equivalence
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Essential rocking parameters
x
z
gx
R
α
+θ
c.g.
O’
O
, tang upliftxg
λ α= =
231 sin2
after
before
θη α
θ= = −
Uplift:
Frequency:
Damping:
( )3 4p g R=
( ) ( )2 sin cosgxp
gθ α θ α θ= − −
− −
4 Parameters
Equation of Motion (EOM):
crθ α=Critical rotation: (unstable equilibrium position)
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Essential rocking parameters: Linearized system
x
z
gx
R
α
+θ
c.g.
O’
O
tanλ α α= ≈
231 sin2
after
before
θη α
θ= = −
Uplift:
Frequency:
Damping:
( )3 4p g R=
4 Parameters
Linearized EOM:
crθ α=Critical rotation:
2 gxp
gθ θ α= − −
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-0.1
-0.05
0
0.05
0.1
0 2 4 6 8 10
Equivalent rocking response
H = 1.17 m B = 0.092 m θcr = 0.078 rad p = 2.50 rad/s
r = 5 m β = 160º t / r = 0.15 φcr = 0.078 rad p = 2.50 rad/s
C
D
B
A
t [s]
Geometry:
Response: blockarch
θ , φ [rad]
-0.5
0
0.5
0 0.4 0.8 1.2
blockarch
t [s]
gx
g
Loading:
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Four essential rocking parameters
0
0.05
0.1
0.15
0.2
0.25
150 160 170 180β [degrees]
φ cr
[rad
]
0
0.1
0.2
0.3
0.4
150 160 170 180β [degrees]
a scal
e
0
0.5
1
1.5
2
2.5
150 160 170 180
0.20.180.160.140.12
β [degrees]
p [r
ad/s
]
ta / r
** Plus coefficient of restitution (not shown)
C
D
B
A
Uplift: Frequency:
Damping:
Critical rotation:
β
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Can we improve seismic performance?
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Retrofit solutions
2b
α
C.M.
R
2h
O t
üg(t) O΄
üg(t)
x
y
z
T
θ > 0
T0
Tt N
tendon
Typical: Alternate:
θ
P
θ > 0
α mg
mag
üg
damper
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Experimental -vs- Analytical Results
0 0.05 0.1 0.15 0.20
0.2
0.4
0.6
0.8
1
γ
θ max
/ α
experimental
analytical
0 2 4 6 8 10-0.4
-0.2
0
0.2
0.4
t [s]
acc
[g]
dashpot
block
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Added Damping
θ
P
θ > 0
α mg
mag
üg
damper
Increased Damping
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Added Damping
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Retrofit solutions
cable
pulley
dashpot
A
B
C
D
dashpot cable guides
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
γ
θ max
/ α
5mm14mm23mm32mm
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Retrofit solutions
2b
α
C.M.
R
2h
O t
üg(t) O΄
üg(t)
x
y
z
T
θ > 0
T0
Tt N
tendon
Typical: Alternate:
Optimal Combination?
θ
P
θ > 0
α mg
mag
üg
damper
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Modern Design
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Large and Flexible Rocking Structures
[1] Elevated Water Tank, retrieved from http://www.flickr.com/photos/jmbower/2869069172.
[2] South Rangitikei Railway Bridge, New Zealand
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Equations of Motion
2DOF: Coupled motion
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Maximum Response During Earthquakes Non-Pulse-type earthquake: Pulse-type earthquake:
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Wavelet Pulse Fitting
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Overturning envelope – Acceleration Pulse
0 1 2 3 4 5 6 7 8 9 100
2
4
6
8
10
12
14
16
18
20
A/(g
tan α
)
ω/p
α = 0.2 ζ = 0.050
More flexible structures
Increasing size
Dim
ensio
nles
s Pu
lse A
mpl
itude
Dimensionless Frequency
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Rocking Demand Maps
Maximum Rocking Demand ( θmax / α )
ω/p
Ap/
(gta
n(α
)
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
ωn/p=50, α=0.15, ζ=0.05
γ=1, φ=0
Dim
ensio
nles
s Am
plitu
de
Dimensionless Frequency
Increasing size
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Acknowledgments
Collaborators: Dr. John Ochsendorf, MIT Dr. Laura De Lorenzis, Lecce, Italy Dr. Andrei Metrikine, TU Delft Dr. Elias Dimitrakopoulos, HKUST
Research Students: Mr. Sinan Acikgoz, Cambridge University Mr. James McInerney, Cambridge University Mr. Simon Cattell, Cambridge University Mr. Christopher Vibert, Cambridge University Mr. Stuart Adams
Financial Support: EPSRC (UK) Research Grant Cambridge University Trust
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Publications
http://www-civ.eng.cam.ac.uk/struct/mjd/publications.html