Seismic retrofitting of buildings using isolation and/or ... Ferraro.pdf · Seismic retrofitting of buildings using ... Timoshenko beam theory. •The stability of elastomeric bearings

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PRIN 2007 WORKSHOPPRIN 2007 WORKSHOP

26 26 JuneJune 20092009

Ferraro Gabriela – Oliveto Giuseppe

Department of Civil and Environmental Engineering - University of Catania

Seismic retrofitting of buildings using isolationand/or energy dissipation techniques :

Design, Modelling, Identification

On the stability of elastomeric bearings and isolation systems

On the stability of elastomeric bearings and isolation systems

2

IntroductionIntroduction

•Objective of seismic isolation is to limit damage of buildingsduring seismic events.•Objective of seismic isolation is to limit damage of buildingsduring seismic events.

•Today,different types of isolation systems exist: here onlyisolation systems that use elastomeric bearings will beconsidered.

•Today,different types of isolation systems exist: here onlyisolation systems that use elastomeric bearings will beconsidered.

•The assessment of the stability of elastomeric bearings is of great interest for the scientific and professional comunity.•The assessment of the stability of elastomeric bearings is of great interest for the scientific and professional comunity.

•The stability of elastomeric bearings and related base isolationsystems in the undeformed configuration is studied using the Timoshenko beam theory.

•The stability of elastomeric bearings and related base isolationsystems in the undeformed configuration is studied using the Timoshenko beam theory.

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•The two classical formulations are based on the followingassumption:the top surface is free to traslate but prevented torotate.

•The two classical formulations are based on the followingassumption:the top surface is free to traslate but prevented torotate.

•The critical load of elastomeric bearings with the top surfacefree to rotate as well as to traslate shall also be considered.•The critical load of elastomeric bearings with the top surfacefree to rotate as well as to traslate shall also be considered.

IntroductionIntroduction

•Two theories, derived from theTimoshenko beam theory, estimate the critical load of beams undergoing strong sheardeformation: The one due to Haringx provides reliable enoughresults while the other due to Engesser provides decidedlyunacceptable ones.

•Two theories, derived from theTimoshenko beam theory, estimate the critical load of beams undergoing strong sheardeformation: The one due to Haringx provides reliable enoughresults while the other due to Engesser provides decidedlyunacceptable ones.

4Timoshenko, S. P., Gere, J. M., Theory of elastic stability

( ) ( ) ( )zzdzdvz γϑϕ +==

Kinematic relations

Generalized constitutive equations

( )dzdEIzM ϑ

−=

( ) ( ) ( )⎟⎠⎞

⎜⎝⎛ −== z

dzdvkGAzkGAzT ϑγ

Timoshenko Timoshenko theorytheory

5

The shear force is perpendicularto the centreline of the beam in

deformed configuration

The shear force is perpendicularto the centreline of the beam in


The shear force is tangent to the cross-section of the beam in the


The shear force is tangent to the cross-section of the beam in the


Engesser Haringx

Timoshenko Timoshenko beambeam theoriestheories

T

N

v(z) v(z)

N

T

6

01 2

2

4

4

=+⎟⎟⎠

⎞⎜⎜⎝

⎛−

dzvdP

dzvd

PPEI

S

3

3

1dz

vdPPEIH

S⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

( ) 3

3

1dz

vdkGAEI

PP

dzdvz

S⎟⎟⎠

⎞⎜⎜⎝

⎛−+=ϑ

Equilibrium equation

End conditions

Engesser Engesser theorytheory

γϑ −=dzdv

H

P

7

013

3

=⎟⎟⎠

⎞⎜⎜⎝

⎛++

dzd

PPP

dzdEI

S

ϑϑ

( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+⎟⎟⎠

⎞⎜⎜⎝

⎛+−==

−

zPdzdEI

PPzHH

S

ϑϑ2

21

1

( ) ( ) ( ) zPHd

PPvzv

S

z

S

+⎟⎟⎠

⎞⎜⎜⎝

⎛++= ∫ ζζϑ

010

Equilibrium equation

Haringx Haringx theorytheory

End conditions

H

P

8

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

3

2

1

3

2

1

ϑϑϑ

δϑϑϑαβϑβα

hEI

MMM

Stiffness Stiffness MatrixMatrix

hHM =3 hvv 12

3−

=ϑEIhP 2

2 =σ 2kGAhEI

=ρ

P

P

H

M2

HM1

zh

v2

v1

2

1

1

2

SPPp =

9

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

3

2

1

3

2

1

ϑϑϑ

δϑϑϑαβϑβα

hEI

MMM

( ) Dcps /1 τσα −+=

( ) Dps /1 τσβ +−=

( ) Dc /1−=τσϑ

Ds /3σδ =

Stiffness Stiffness FunctionFunction

( ) σscpD −−+= 112

( ) Dcs /2στα −=

( ) Ds /2 τσβ −=

( ) Dc /12 −= σϑ

Ds //4 τσδ =

( ) τσ scD 212 −−=

HA

RIN

GX

ENGE

SSER

p+= 1στ21 ρσ

στ−

=

10

CP⇒= 0δ

PC = Critical load of Elastomeric bearing with the top surfacefree to traslate but prevented to rotate

CriticalCritical LoadLoadH

M2

H

M1

zh

1

2P

P

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

3

2

1

3

2

1

ϑϑϑ

δϑϑϑαβϑβα

hEI

MMM

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−⋅⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

hvv

hEIM

M

12

2

1

00

0 δϑϑϑαβϑβα

v2

v1

Pc

Pc

M2

M1

11

ESC PPP111

+=

ES

CC P

PP

P =⎟⎟⎠

⎞⎜⎜⎝

⎛+1

1

1−

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

S

E

E

C

PP

PP

1

1−

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

S

E

E

C

E

C

PP

PP

PP

Engesser

Haringx

22

hEIPE π=kGAPS =

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-5

-4

-3

-2

-1

0

1

PE/PS

P C/P

E

EngesserHaringx Pc>0

Haringx Pc<0

•Timoshenko, S.P., Gere, J.M., 1961. Theory of Elastic Stability, McGraw Hill International Book Company, New York.

1/0 ≤≤ SE PP

Haringx Haringx vsvs EngesserEngesser

12

1/ >SE PP

Haringx Haringx vsvs EngesserEngesser

100 101 102 103 104 10510-1

100

101

102

103

PE/PS

P C/P

S

EngesserHaringx

H

S

E

SolarinoElastomericbearing

H – Haringx critical load

S – Service load

E – Engesser critical load

13

In absence of shear deformation 02 == ρσp

( )EH D

cs ασσα =−

= ( )EH D

s βσσβ =−

=

( )EH D

c ϑσϑ =−

=12

EH Ds δσδ ==

3

The stiffness functions defined by Haringx and Engessercoincide.

The difference between the two theories is related exclusivelyto shear deformation.

Haringx Haringx theorytheoryLimit cases

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No Axial Force

ρρα

121314

++

=

ρρβ

121612

+−

=

ρϑ

1216

+=

ρδ

12112+

=

0=ρ

4=α

2=β

6=ϑ12=δ

∞→ρ1=α1−=β

0=ϑ0=δ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

ρ

α, β

, θ, δ

β

δ

ϑ

α

100

101

102

103

-1

-0.5

0

0.5

1

1.5

ρ

α, β

, θ, δ

α

β

δ

ϑ

0=σ

Haringx Haringx theorytheoryLimit cases

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Haringx Haringx theorytheoryStiffness Stiffness FunctionFunction

-100 -80 -60 -40 -20 0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

p

α

300

600

900

1200

1500

-100 -80 -60 -40 -20 0 20 40 60 80 100-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

p

β

300

600

900

1200

1500

-100 -80 -60 -40 -20 0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5x 10

-3

p

δ

300

900

12001500

600

-100 -80 -60 -40 -20 0 20 40 60 80 100-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

p

ϑ

600

300

900

1200

1500

a b

d q

16

02 =−ϑαδ

411 =⎟⎟

⎠

⎞⎜⎜⎝

⎛+

S

E

E

C

E

C

PP

PP

PP

ESC PPP21

±≅ESC PPP ±≅

11 =⎟⎟⎠

⎞⎜⎜⎝

⎛+

S

E

E

C

E

C

PP

PP

PP


0=δ

17

λδσϑαδ 22

21

=−

0 20 40 60 80 100 120 140 160 180 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

λ=H/hP C

/PC

0

300

1500

ρ = 300:300:1500

Two elastomeric bearings with an interposed rigid element

h

H

h

hH /=λ

If 0/ == hHλ 02 =−ϑαδ


18

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

λ=H/h

P c/Pc0

b/D =0

b/D =1

b/D =5

ρ = 300:300:1500

Isolated building on two rows of elastomeric bearings

2 b

2P 2P

2 H

The building is a rigid body!

λδσϑαδ 22

2 48 =⎟⎠⎞

⎜⎝⎛+−

Db

0/ =DbIf0/ == hHλ 02 =−ϑαδ


190 10 20 30 40 50 60 70 80 90 100

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

λ=H/h

P c/Pc0

b/D = 5

b/D = 1

ρ =300:300:1500

Isolated building on two rows of elastomeric bearings


20

h

H⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−++−

−+−+

III

IIPIP

IIPIP

kkk

kkk

kkk

λδδλλϑ

δλδλδϑλϑ

λϑϑλϑλαα

2

232

2

P

I

EIEI

k =hH

=λ

500 1000 1500 2000 2500 3000 3500 4000 4500 50000.85

0.875

0.9

0.925

0.95

0.975

1

EIP/EII

P/P C

0

ρ = 300

λ =20:10:60

500 1000 1500 2000 2500 3000 3500 4000 4500 50000.85

0.875

0.9

0.925

0.95

0.975

1

EIP/EII

P/P C

0

λ=20:10:60

ρ = 1500

Elastomeric bearing on the top of column


21

Stability of deformed configuration – literature

( ) vKvhEIH h== ρσδ ,3

h

P

P

H

H

D-vc

vc

( )vDPHh −=

Constantinou, M., C., et al., 2007. Performance of seismic isolation hardware under service and seismic loading, Technical Report MCEER – 07 – 0012.

0 50 100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300

350

400

450

500

v (mm)

H (k

N)

0 50 100 150 200 250 300 350 400 450 5000

50

100

150

200

250

300

350

400

450

500

v (mm)

H (k

N)

Stable Path Unstable Path

Haringx Haringx theorytheory –– deformeddeformed configurationconfiguration

22

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P/Pc

v c/D

ρ = 300:300:1500

Constantinou, M., C., et al., 2007. Performance of seismic isolation hardware under service and seismic loading, Technical Report MCEER – 07 – 0012.

( )ρσδσσ

,2

2

+=

DvC

Stability of deformed configuration – leterature


23

Stability of deformed configuration – literatureCo

nsta

ntin

ou, M

., C.

, et

al.,

2007

. Per

form

ance

of

seis

mic

isol

atio

n ha

rdwa

re u

nder

se

rvic

e an

d se

ism

ic lo

adin

g, T

echn

ical

Rep

ort

MCE

ER –

07 –

0012

.


⎟⎠⎞

⎜⎝⎛= −

Dv1cos2η

( )ηη sin4

2

−=DAr

AAPP r

crcr =′

v

D

Ar

v

D

Ar

v=D

D

Ar=0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Critical load - deformed configuration

v/D

P c'/P

24

LiteratureLiterature•Buckle, I., Nagarajaiah, S., Ferrell, K., 2002. Stability of Elastomeric Isolation Bearings: Experimetal Study. Journal of Structural Engineeering,128 (1), 3-11. •Constantinou, M., C., Whittaker, A., S., Kalpakidis, Y., Fenz, d., M., Warn, G., P., 2007. Performence of seismic isolation hardware under service and seismic loading, Technical Report MCEER – 07 – 0012.•Engesser, F., 1889. Die Knickfestigkeit Gerard Stäbe. Zentralblatt des Bauverwaltung.11, 483-486.•FIP Industriale, 2003. Verbale di prova relative agli isolatori sismici in elastomero armato N 24 SI 80/96 dis. A22247 per adeguamento sismico edifici contrada Tigna – Solarino (SR).•Haringx, J. A., 1948a. On highly compressible helical springs and rubber rods, and their application for vibration-free mountings, I. Philips Research Report, 3(6), 401-449.•Haringx, J. A., 1948b. On highly compressible helical springs and rubber rods, and their application for vibration-free mountings, II. Philips Research Report, 4, 49-80.•Haringx, J. A., 1948c. On highly compressible helical springs and rubber rods, and their application for vibration-free mountings, III. Philips Research Report, 4, 206-220.•Kelly, J. M., 2003. Tension Buckling in Multilayer Elastomeric Bearings. Journal of Engineering Mechanics, 129 (12), 1363-1368.•Nagarajaiah, S., Ferrell, K., 1999. Stability of Elastomeric Seismic Isolation Bearings, Journal of Structural Engineering, 125(9), 946 – 954.•Naim, F., Kelly, J., M., 1999. Design of Seismic Isolated Structures: From Theory to Practice, John Wiley & Sons, Inc.Oliveto, G., 1992. Dynamic Stiffness and Flexibility Functions for Axially Strained Timoshenko Beams. Journal of Sound and Vibration, 154(1), 1-23.•Oliveto, G., Granata, M., Buda. G., Sciacca, P., 2004. Preliminary result from full – scale free vibration tests on a four story reinforced concrete building after seismic rehabilitation by base isolation, JSSI 10th Anniversary Symposium on Performance of Response Controlled Buildings, November 17 – 19, Yokohama Japan.•Timoshenko, S.P., Gere, J.M., 1961. Theory of Elastic Stability, McGraw Hill International Book Company, New York.

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Documents

Seismic retrofitting of buildings using isolation and/or ... Ferraro.pdf · Seismic retrofitting of buildings using ... Timoshenko beam theory. •The stability of elastomeric bearings