rocking of structures during earthquakes: from collapse of

Rocking of Structures During Earthquakes: From Collapse of Masonry to Modern Design

Matt DeJong

Lecturer Department of Engineering

University of Cambridge

27 February, 2013

L’Aquila, Italy (2009)

Christchurch Spires

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?

Arch Structures: Static Analysis

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

Tilting Thrust Line Analysis

β

αcr

tan 0.37hcr

v

u gu

γ α= = =

β

β

α

http://web.mit.edu/masonry/mdejong/tilt/TiltingArch.html

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 = =

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:

Dynamics: The masonry arch

The masonry arch

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?

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

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

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

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

Hinge reflection

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


No Rocking

One impact collapse

Static Solution

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

Seismic loading: Hinge reflection

El Centro Earthquake

-3

0

3

0.0 0.4 0.8 1.2Time [s]

Acc

eler

atio

n [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]

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

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

]


-1.5

-1

-0.5

0

0.5

1

1.5

2.4 2.9 3.4

Time [s]

Acce

lera

tion

[g]


-1.5

-1

-0.5

0

0.5

1

1.5

2.1 2.6 3.1

Time [s]

Acce

lera

tion

[g]


-1.5

-1

-0.5

0

0.5

1

1.5

1.7 2.2 2.7

Time [s]

Acce

lera

tion

[g]


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

Computational Modelling

The Arch

El Centro earthquake

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

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)

Stone Masonry Spires

Christchurch Spires

Lincolnshire Earthquake, 2009

Stone Masonry Spire, Waltham on the Wolds, UK

[1] RMW Musson, Annals of GeoPhysics, v.47(2/3), 2004.

Essex Earthquake, 1884

[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

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

− − − + =− +


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:


x

y

H

rb

CM

H/3

O

λMcg

Mcg

CM

O

hc R

α

θ β

φ

Geometry Rocking Tilt Test

Stone Spire, Waltham on the Wolds

O 9.4 m

3.4 m

β

βmax H

O’

O’’

Analytical DEM Physical

Tilt Test

DEM: ag = 0.17g

Analytical: ag = 0.19g (perfect hollow cone)

Physical: ag = 0.16g

Impulse rocking response

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

]


Damage (DEM)

Collapse (DEM)

Collapse (DEM), Tilt Test

Rocking Spire

Pulse

acc

eler

atio

n, a

p[g

]


No impact collapse


No Rocking

One impact collapse

Governing failure curve

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

]


Collapse (Analytical), entire spireCollapse (Analytical), Solid spire tipCollapse (DEM)

Damage (DEM)


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

0

1

2

3

4

3.5 5.5 7.5 9.5

p[1

/s]

H [m]


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

]


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:

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

]


Damage (DEM)

Collapse (DEM)


Seismic Response

Rocking Equivalence

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)

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= − −

-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:

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

Can we improve seismic performance?

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

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

Added Damping

θ

P

θ > 0

α mg

mag

üg

damper

Increased Damping

Added Damping

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

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

Modern Design

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

http://www.flickr.com/photos/jmbower/2869069172

Equations of Motion

2DOF: Coupled motion

Maximum Response During Earthquakes Non-Pulse-type earthquake: Pulse-type earthquake:

Wavelet Pulse Fitting

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

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

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

Publications

http://www-civ.eng.cam.ac.uk/struct/mjd/publications.html

http://www-civ.eng.cam.ac.uk/struct/mjd/publications.html

rocking of structures during earthquakes: from collapse of

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