Seismic damage mitigation in existing structures by ... · PDF fileSeismic damage mitigation in existing structures by superelastic dissipators Alessandro Baratta1 and Ottavia Corbi1

Seismic damage mitigation in existing structures bysuperelastic dissipators

Alessandro Baratta1 and Ottavia Corbi1

1University of Naples “Federico II”, Dept. of “Structural Engineering”via Claudio 21, 80125, Naples, Italy

[email protected]

In the paper the dynamics of frame structures subject to some vertical and horizontal loads are analysed, in the hypothesis that some superelastic behaviour is included or they consist of an elastic-plastic structure coupled with pseudo-elastic SMA members.

In both cases, the results are compared with those relevant to the corresponding elastic-plastic oscillator. The adoption of SMAs is shown to always improve the performance of the structural system, in terms of either dissipative skill or re-centering capacity or stabilizing capacity for P-Δ effects.

Shape Memory Alloys (SMAs) (Graessel and Cozzarelli, 1991, 1994) may undergo reversible phase transformations. Two main features induced by their martensitic phase: Theshape memory effect provides the SMAs with a high dissipative capacity with comparison to ordinary metals, achieving large hysteretic loops without incurring plastic deformation, while thesuper-elastic behaviour produces an energy-absorbing effect combined with a theoretically zero residual strain upon unloading. An application of SMAs in seismic engineering is the realisation of energy dissipating members, based on their super-elastic behaviour and able to mitigate the effects induced by eventually incoming dynamic excitations.

Furukawa's Shape Memory and Super-elastic alloys (Furukawa NT Alloys)

SIMPLE OSCILLATOR

The equation ruling the motion of the model is

where is the restoring force. For a simple elastic-plastic oscillator

(2)

( )Hm

w ;auu,ugu 22

⋅=θ=θ−+ &&&

( )u,ug &

( ) ( )u,uUu2u,ug 2oo &&& ω+ζω=

( )⎪⎩

⎪⎨

⎧

′′≤≤′′′≥≥′

′≤≤′′−

=uu 0,u if uuu 0,u if u

uuu if uu

u,uU

o

o

p

&

(1)

oo u,u ′′′ u,u ′′′

pu

with and the initial and instantaneous yield displacements, and

the cumulated plastic displacement.

where

The yield limits of the shear stress standardised variable T=k·u and the P-Δ failure displacements are

.0m

Tu ,0m

Tu 2o

c2o

c <θ⋅′′

=′′>θ⋅′

=′

0ukT ,0ukT oooo <′′=′′>′=′(3)

T

u

T Io

TIIo

uIIo

uIo

tan-1(k)

T

u

TIo

TIIo

uIIo

uIo uI

c

tan -1(mθ2)

tan-1(k)uIIc

(a) (b)

Elastic perfectly plastic model behaviour (a) without and (b) with vertical load.

SIMPLE OSCILLATOR WITH SMA PSEUDO-ELASTIC PILESIn the model with SMA members (the piles could be covered with SMAs or

include SMA reinforcements or SMA connections) the restoring force (1) is modified, while (3) holds with ks of the SMA piles instead of k and T = ks·u..

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡ Θ−Θ−−=

−

c

1n

cs T

TT

TuukT &&&

( )[ ] ,uu ,uutanu2kTuk q31p

ss =

⎭⎬⎫

⎩⎨⎧

⋅ξ⋅⋅ξπψ

+−γ=Θ −

s

c

os

co

kTc ,

TkTT

ψ=ψ−ψ

−=γ

By adopting a one-dimensional SMA constitutive relation based on modified plasticity models able to develop inner hysteretic cycles (Graessel and Cozzarelli, 1991, Baratta and Corbi, 2000) and applying the isothermal stress-strain rate relation to the frame, one can write

where Θ is the evolutionary back-shear, q<1, all other symbols represents material constants and To and Tc are the limit and critical shear stress values

(4)

where ν is a prefixed percentage. By assuming the SMA piles with the same viscous damping capacity of the elastic-plastic ones, the restoring force is

(6) with T given by simultaneous integration of (4) and (1).

.0T if T0T if T

T ;0T if T0T if T

To

oc

o

oo

⎩⎨⎧

<′′ν>′ν

=⎩⎨⎧

<′′>′

=

( )mTu

mk2u,ug s +⋅ζ= &&

(5)

MN

T

T

N

b

N1=M/b N2 = − M/b

central wedge

SMA bars SMA bars

Fig. Model with SMA hinges: the central massive wedge equilibrates the normal and shear forces, while the peripheral SMA plates accomplish the task to react to the bending moment.

The frame subject to a(t)= ao⋅sin(ωt), with pulsation ωf=1 rad⋅sec-1 and amplitude ao is considered for numerical investigation.It is designed in order to overcome a ground acceleration up to 0.2g; with:

m=100 kg⋅sec2 cm-1, H=300 cm, ζ=0.05, ωo= 37.42 rad⋅sec-1

θ =1.81 sec-1, k=ks=140000 kg⋅m-1, .

In case of symmetric resistance, one assumes for both the elastic-plastic and pseudo-elastic model

while, for the no-hardening SMA members, the following parameters are assumed:

cm 1420kg, 20000 .uu TT oooo =′′=′=′′=′

NUMERICAL RESULTS:SIMPLE OSCILLATOR WITH SMA PSEUDO-ELASTIC PILES

(7)

(8)

(9)3/4q 25.9,= 2000 200c 0=p ,120n cm 0u =ξ=ν=== ψ ,.,,,ˆ

NUMERICAL RESULTS: SMA PSEUDO-ELASTIC PILES

u-0.6 -0.4 -0.2 0.0 0.2 0.4

-2E+4

-1E+4

0E+0

1E+4

2E+4T T

,o

T ,,o

-0.4

0.0

0.4

-4E+2

-2E+2

0E+0

2E+2

4E+2a(t)u(t)

0.0 2.0 4.0 6.0 8.0

-0.4 0.0 0.4u

T

-2E+4

-1E+4

0E+0

1E+4

2E+4

backshear

T ,o

T ,,o

u(t)

-0.4

0.0

0.4

t0.0 2.0 4.0 6.0 8.0

El-Pl. Oscillator.

Oscillator withSMA inclusions.

SYMMETRY: ao=0.2g


El-Pl. Oscillator.


SYMMETRY: ao=0.8g

-2.0 -1.0 0.0 1.0 2.0u

backshear

T ,,o

-2E+4

-1E+4

0E+0

1E+4

2E+4T T ,

o u(t)

-2.0

-1.0

0.0

1.0

2.0

t0.0 4.0 8.0 12.0

u(t)

-20.0

0.0

20.0

40.0

60.0

-1E+3

0E+0

1E+3

2E+3

3E+3a(t)

t =11.98, u =61.16FAILURE POINT:

0.0 4.0 8.0 12.0

u

T

-4E+4

-2E+4

0E+0

2E+4

4E+4

-20.0 0.0 20.0 40.0 60.0

T ,o

T ,,o

FAILURE POINT:T=0, u =61.16


El-Pl. Oscillator.


ASYMMETRY: ao=0.2g

u-4.0 0.0 4.0 8.0 12.0

-2E+4

0E+0

2E+4

T

T ,o

T ,,o -10.0

-5.0

0.0

5.0

10.0

-4E+2

-2E+2

0E+0

2E+2

4E+2a(t)u(t)

0.0 2.0 4.0 6.0 8.0

-0.4 0.0 0.4u

T

-2E+4

0E+0

2E+4

backshear

T ,o

T ,,o

u(t)

-0.4

0.0

0.4

t0.0 2.0 4.0 6.0 8.0


El-Pl. Oscillator.


ASYMMETRY: ao=0.8gT

4E+4

-2E+4

0E+0

2E+4

4E+4

-20.0 0.0 20.0 40.0 60.0FAILURE POINT:

T=0, u =61.16

u

T ,o

T ,,o

u(t)

-20.0

0.0

20.0

40.0

60.0

-1E+3

0E+0

1E+3

2E+3

3E+3a(t)

t =7.78, u =61.16FAILURE POINT:

0.0 2.0 4.0 6.0 8.0

u

T

backshear

-2.0 -1.0 0.0 1.0 2.0

-2E+4

0E+0

2E+4T

,o

T ,,o

u(t)

-2.0

-1.0

0.0

1.0

2.0

t0.0 2.0 4.0 6.0 8.0

u

w/2w/2L

f

φ

αo α

l ol

Δl

H

N s

N s

β

N p’’

N p’’

N p’

N p’

T ’p

mu..mu..mu..

T ’’p

A

B C

D

SIMPLE OSCILLATOR WITH ELASTIC-PLASTIC TENDONSFor positive counter-clockwise rotations, one can write the geometric relations instantaneously expressing the angle α of inclination of the stretched stay-rod and its length

By the relations

one gets the dynamic equation of motion

with the SMA tendon elongation

( ) ( )αφ

=φΔ+=φsincosH o lll( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛φ+

φ=φα= − ,

sinHLcostan 1

( ) ( ) ( ) ( )[ ]mH mHsin c N sin N cH T f ws pcos & & cos & cos cos

cos& tanφ φ φ φ α φ α φ φ φ

φφ

φ φ φ⋅ − ⋅ + ⋅ + ⋅ − ⋅ ′′ ++⎛

⎝⎜

⎞⎠⎟ ⋅ ⋅ + = +2

212

Δl

( ) [ ]φφ−φφ=φ sincosHu 2&&&&&( ) φφ=φ ,cosHu &&( ) φ=φ ,sinHu

( ) ( ) ( ) ( ) φ⋅αφ

⋅

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

φ⋅α⋅+

⋅

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

φ+

φ+

⎟⎟⎠

⎞⎜⎜⎝

⎛φ+

−α⋅−=⎥⎦⎤

⎢⎣⎡

αφ

φ=φΔ−φ=φΔ &l&lll

22

2

1

1

sinsincoscos

HL

sinHL

cos

sinHLH

sinHsincos

ddH ,o

(12)

(10)

(11)

(13)

The restoring force portion T(φ) of the simple elastic perfectly plastic oscillator in the motion equation can be expressed as a function of the piles’ rotation φ

( ) ( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

φ ′′≤φ≤φφ ′′

φ′≥φ≥φφ′

φ′≤φ≤φ ′′φ−φ

=φφφ=φ

, if

, if

if

R ,sinkHRT

o

o

p

0

0&

&

with k the piles stiffness, and respectively the initial and instantaneous positive and negative yielding values of the Lagrangian parameter, and the plastic rotation cumulated throughout the loading history.

The shear stress yield limits in the piles are then

oo ,φ ′′φ′ φ ′′φ′,pφ

oo T ,T ′′′

0 T 0 <φ ′′φ ′′−=′′φ′φ′=′ oooooo sinkH,>sinkHT (15)

(14)

u

w/2w/2L

f

φ

αo α

l ol

Δl

H

N s

N s

β

N p’’

N p’’

N p’

N p’

T ’p

mu..mu..mu..

T ’’p

A

B C

D

SIMPLE OSCILLATOR WITH SMA PSEUDO-ELASTIC TENDONSWhen applying the isothermal material stress-strain rate relation to the stretched SMA tie-rod inserted in the considered model, one can get the rate relation between the N(φ) portion of the SMA rod restoring force and its elongation

with

( ) ( ) ( ) ( ) ( ) ( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ φΘ−φφΘ−φφΔ−φΔ=φ

−

c

n

cs N

NN

NkN1

l&l&&

( ) ( ) ( ) ( ) ( )[ ]Θ Δ Δφ γ φφ ψ

πφ= − +

⎧⎨⎩

⎫⎬⎭

−kN

kxs

s

p$ tan ,l l2 1

( ) ( ) ( )Δ Δ Δφ φ ξ φ= = ⋅ = ⋅ <x x x q$ , , ,l l l 13 x q

(16)

Where Θ(φ) is the evolutionary one-dimensional backstress that accounts for the metastable forms of SMA behaviour, Nc is the critical normal force and all other symbols represent material constants.

(12)

(17)

One can notice that, for specified rotations (and thereafter tie length variations), the force-elogantion relationship for the stretched SMA tie-rod can be obtained by numerical integration of (16) while N(φ)can be deduced by simultaneous integration of the motion equation (12) and (16). Furthermore, by assuming that the mechanical model shows an isoresistant behaviour, the following relations hold

with No, Nc the limit and critical normal forces related to the relevant yield values , ν a given prefixed percentage and cψ a given constant.

( )( )

( )( )⎩

⎨⎧

<φ′ν>φ′ν

=⎩⎨⎧

<φ′′>φ′

=00

00

N if NN if N

N ,N if NN if N

No

oc

o

oo

s

o

os

co

kN

c ,NkNN

ψ=ψ−ψ−

=γ

′ ′′N No o,

(18)

The frame subject to a(t)=ao⋅sin(ωft), with pulsation ωf and amplitude ao is considered for numerical investigation. Its characteristics are:

m=100 kg⋅sec2 cm-1, H=300 cm, L=500 cm, k=140000 kg⋅cm-1

ωo=37.42 sec-1, c=187 kg⋅sec⋅cm-1 , Rtot =To+No⋅cosαo=40000⋅cosαo kg

The SMA tendons parameters are:

ks=140000 kg⋅cm-1, cs= 69 kg⋅sec⋅cm-1

The equivalent shear stress in the piles =T(φ)⋅cosφ and normal stress in the stretched SMA tendon =N(φ)⋅cosαo versus the frame response in terms of piles’ rotation, and the phase diagrams for the two forcing amplitude cases are depicted in the following.

NUMERICAL RESULTS:SIMPLE OSCILLATOR WITH SMA PSEUDO-ELASTIC TENDONS

3/4q 20.5,= ,200.0 ,200c ,0=p ,120n ,cm 0ˆ =ξ=ν===Δ χl

TN

(19)

(20)

NUMERICAL RESULTS: ELASTIC-PLASTIC TENDONS

ao= 0.8 g

ωf=ωo

ao= 1.6 g

φ

T ,,o

T ,o

-1.5E+4

-5.0E+3

5.0E+3

1.5E+4

-1.0E+4

0.0E+0

1.0E+4

-0.015 -0.005 0.005 0.015-0.010 0.000 0.010

N,o

N,,o

N = 564 = -9.62 E-3φ

T, N

T = -755 = -9.62 E-3φ

φ

N,o

T, N

T ,o

T ,,o

,,oN

N = 1638 = 1.11 E-2φ

T = -4611 = 1.11 E-2φ

-1.5E+4

-5.0E+3

5.0E+3

1.5E+4

-1.0E+4

0.0E+0

1.0E+4

-0.150 -0.050 0.050 0.150-0.100 0.000 0.100

NUMERICAL RESULTS: SMA PSEUDO-ELASTIC TENDONS

ao= 0.8 g

ωf=ωo

φ

T ,,o

T ,o

-1.5E+4

-5.0E+3

5.0E+3

1.5E+4

-1.0E+4

0.0E+0

1.0E+4

N,o

T, N

T = -1115 = 3.42 E-4φ

N = 2086 = 3.42 E-4φ

N,,o

-0.006 -0.002 0.002 0.006-0.004 0.000 0.004φ

N = 2733 = 1.25 E-3φ

-0.020 -0.010 0.000 0.010 0.020

N,o

T, N

T ,o

T ,,o

-1.5E+4

-5.0E+3

5.0E+3

1.5E+4

-1.0E+4

0.0E+0

1.0E+4

,,oN

T = -1304 = 1.25 E-3φ

ao= 1.6 g


0 20 40 60 80 1001.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

1.0E+0

SMA (%)

a = 1200

a = 1600

a = 2000

a = 800

ω =ωf o|φ( )|Tf

0 20 40 60 80 100

1.0E-3

1.0E-2

1.0E-1

1.0E+0

SMA (%)

a = 1200

a = 1600

a = 2000

a = 800

ω =ωf o|φ|max

0 20 40 60 80 1001.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

SMA (%)

|φ( )|Tf

ω =ωf c

a = 1200a = 1600a = 2000

a = 800

0 20 40 60 80 1001.0E-3

1.0E-2

1.0E-1

SMA (%)

ω =ωf c

a = 1200

a = 1600

a = 2000

a = 800

|φ|max

ωc = overall pulsation of the structure endowed with SMA tendons ωo = pulsation of the elastic-plastic frame


0 20 40 60 80 100

1.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

SMA (%)

|φ( )||φ|

fTmax

a = 800, ω =ωf o

SMA tendonsUnilateral elastic-plastic tendonsElastic-plastic tendons

0 20 40 60 80 1001.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

1.0E+0

SMA (%)

|φ( )||φ|

fTmax

a = 1600 ω =ωf o,


0 20 40 60 80 1001.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

SMA (%)

|φ( )||φ|

fTmax

ω =ωf ca = 800,


0 20 40 60 80 1001.0E-6

1.0E-5

1.0E-4

1.0E-3

1.0E-2

1.0E-1

1.0E+0

SMA (%)

|φ( )||φ|

fTmax

a = 1600 ω =ωf c,


ao= 0.8 g

ao= 0.8 g

ao= 1.6 g

ao=1.6 g

THE MDOF SHEAR FRAME EQUIPPED WITH SMA PROVISIONSSimple shear frame

Sketches of the shear frame during the motion

a) without P-Δ effect, b) with P-Δ effect.

One considers cases where SMA tendons contribute to the total storey resistance by 25%, 50% and 75%. Under simply horizontal loads, SMAs prevent the structure from entering the plastic phase. In the presence of P-Δ effects,SMAs prevent the frame from reaching the failure condition.

THE MDOF SHEAR FRAME EQUIPPED WITH SMA PROVISIONS

SMA tendons contribute to the total storey resistance by absorbing a 25%, 50% and 75%.

floor mi ci ki u′oi , u″oi T′oi , T″oi

1 20 200.000 200000 ± 0.056898 ± 11379.62 16 185.714 185714 ± 0.058357 ± 10837.713 16 171.439 171439 ± 0.060059 ± 10295.834 16 157.142 157142 ± 0.062070 ± 9753.9435 16 142.857 142857 ± 0.064484 ± 9212.0586 16 128.571 128571 ± 0.067435 ± 8670.1727 16 114.286 114286 ± 0.071122 ± 8128.286

PILES TENDONS

ith u′oi , u″oi T′oi , T″oi Δl′oi⋅cosα N′oi⋅cosα

1 ± 0.028449 ± 5689.800 ± 0.028449 ± 5689.800

2 ± 0.029178 ± 5418.857 ± 0.029178 ± 5418.857

3 ± 0.060059 ± 10295.83

4 ± 0.062070 ± 9753.943

5 ± 0.064484 ± 9212.058

6 ± 0.067435 ± 8670.172

7 ± 0.071122 ± 8128.286

Simple shear frame

Shear frame with SMA tendons absorbing: 50% SMA percentage

THE MDOF SHEAR FRAME EQUIPPED WITH SMA PROVISIONSP-

Δef

fect

No

P-Δ

effe

ctshear frame without and with 50% SMA provision

Max plastic drift

Max plastic drift

Hysteresis loops at the 1st floor


Hysteresis loops at the 7th floor


0.00 2.00 4.00 6.000.00

0.40

0.80

1.20

1.60

2.00 Δ[ u (t)]p max

shear framewithout SMA provisions

t

50% SMA provisions

1T (t)

u (t)1


-20000

-10000

0

10000

20000

-2 -1 0 1

50% SMA provisions

7T (t)

-0.08 -0.04 0.00 0.04 0.08

-10000

-5000

0

5000

10000


u (t)7Δ

50% SMA provisions

0.00 2.00 4.00 6.000.00

10.00

20.00

30.00

40.00

50.00 Δ[ u (t)]p max


t

50% SMA provisions

1T (t)

50% SMA provisions


Δu (t)1-6.00 -4.00 -2.00 0.00 2.00

-20000

-10000

0

10000

20000 7T (t)

-10000

-5000

0

5000

10000

-0.12 -0.08 -0.04 0.00 0.04 0.08

50% SMA provisions


Δu (t)7

THE MDOF SHEAR FRAME EQUIPPED WITH SMA PROVISIONSP-

Δef

fect

No

P-Δ

effe

ctshear frame without and with 50% SMA provision

Max plastic drift

Max plastic drift





0.00 2.00 4.00 6.000.00

0.40

0.80

1.20

1.60 Δ[ u (t)]p max

t

75% SMA

50% SMA

25% SMA

1T (t)

u (t)1


25% SMA

50% SMA

75% SMA

-20000

-10000

0

10000

20000

-2 -1 0 1Δ

7T (t)

u (t)7Δ

25% SMA shear framewithout SMA provisions

75% SMA

50% SMA

-0.08 -0.04 0.00 0.04 0.08

-10000

-5000

0

5000

10000

0.00 2.00 4.00 6.000.00

0.40

0.80

1.20

1.60 Δ[ u (t)]p max

t

75% SMA

50% SMA

25% SMA

1T (t)

u (t)1


-20000

-10000

0

10000

20000

-6 -4 -2 0 2

25% SMA

50% SMA75% SMA

Δ

7T (t)

u (t)7


50% SMA

75% SMA

Δ

25% SMA

-10000

-5000

0

5000

10000

-0.10 -0.05 0.00 0.05 0.10

A SMA ISOLATION DEVICE FOR A MDOF SHEAR FRAME

Model characteristics

1 20 200.000 200000 22759.20 -22759.202 16 185.714 185714 21675.43 -21675.433 16 171.439 171439 20591.66 -20591.664 16 157.142 157142 19507.89 -19507.895 16 142.857 142857 18424.12 -18424.126 16 128.571 128571 17340.34 -17340.347 16 114.286 114286 16256.57 -16256.57

ith

levelmi

(kg·sec2/cm)ci

(kg·sec/cm)ki

(kg/cm)T′oi(kg)

T″oi(kg)

Shear frame without SMA isolation device Shear frame with SMA isolation device

The objective of the approach is to provide the main structure with a dissipation device able to attenuate the effects induced by the incoming dynamic excitation.Obviously, the exploitation of the pseudo-elastic character of the SMA members requires a suitable tuning of the alloy parameters on the basis of the structure mechanical and geometrical characteristics. Therefore, the analysis of the dynamics of the main structure is necessary in order to set up the isolation device in such a manner to make the SMA elements exhibit hysteresis loops, i.e. the classical super-elastic behaviour.

Red arrows indicate directions of plastic excursions when they occur. Most of the energy of dynamic excitation is dissipated by the isolator which prevents the superstructure from entering the plastic phase.


By comparing the maximum absolute inter-storey plastic drift in the un-isolated shear frame with the value attained in case of SMA isolation, one can notice that no plastic excursion occurs in the latter case.

By looking at the inter-storey drifts at the 1st

and 7th levels for structure equipped or not with SMA isolation device, one can notice a wide response attenuation in case of adoption of SMA provision; moreover this can be achieved with a zero final residual deformation.

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

shea

r fra

me

t

shear framewithout isolation

0.0

1.0

2.0

3.0

4.0

isol

ator

[ u (t)]Δ p max

SMA isolator

Δ| u (t)|p isol

shear framewith SMA isolation

Max

imum

pla

stic

drif

t

0 1 2 3 4 5-1.0

-0.5

0.0

0.5

1.0

shea

r fra

me

t

shear framewithout SMA isolation

-4.0

-2.0

0.0

2.0

4.0

isol

ator

u (t)isol

SMA isolator

u (t)Δ 1 Δ


Drif

t at

the

1stflo

or

0 1 2 3 4 5-0.2

-0.1

0.0

0.1

0.2

shea

r fra

me

t


-4.0

-2.0

0.0

2.0

4.0

isol

ator

u (t)isol

SMA isolator


u (t)Δ 7 Δ

Drif

t at

the

7thflo

or


N (t)pos

-4 -2 0 2 4u (t)pos

-4E+3

-2E+3

0E+0

2E+3

4E+3

u (t)neg-4 -2 0 2 4

-4E+3

-2E+3

0E+0

2E+3

4E+3 N (t)neg

-1.0 -0.5 0.0 0.5 1.0

T(t)1

-4E+4

-2E+4

0E+0

2E+4

4E+4

u (t)1Δ



-0.2 -0.1 0.0 0.1 0.2

-2E+4

-1E+4

0E+0

1E+4

2E+4


u (t)7Δ

T(t)7


Hys

tere

sisl

oops

in th

e sh

ear f

ram

eH

yste

resi

sloo

ps in

the

isol

ator

BIDIRECTIONAL SHAKING vs. THREE-DIMENSIONAL STRUCTURE

( )tug&&

( )tvg&&

BIDIRECTIONAL SHAKING vs. THREE-DIMENSIONAL STRUCTURE

a),b) 3-d.o.f. frame; c) base acceleration components, d) deformation pattern.

x

yz

x

y

G

O

1 23

54

fx(t)

fy(t)

t t

t

c)

a) b)

d)

In the considered model, the Lagrangian coordinates are the two components of floor rigid translation, u(t) and v(t), that can be identified in the displacement components of the reference frame origin point G, and the xy-axes rotation φ(t). The piles are assumed to show an elastic-perfectly-plastic behaviour with indefinite ductility, with Txi, Tyi the components of the shear absorbed by the i-th one, Toxi, Toyi its limit shearsand kxi, kyi its stiffnesses.

For simplicity, the plasticity domain of each pile is assumed to have an elliptical shape, with the principal axes parallel to the edges of the rectangular cross-sections. Tx

Ty

T

u.

G

y

x

1 23

4

5

Tx

Ty

T

u.

Tx

Ty

T u.

Tx

Ty

T u.

Tx

Ty

Tu.

3

BIDIRECTIONAL SHAKING vs. THREE-DIMENSIONAL STRUCTUREElastic-Plastic


Floor centroid trajectory


Piles’ response. Three-dimensional; Plane model

BIDIRECTIONAL SHAKING vs. THREE-DIMENSIONAL STRUCTURESuper elastic

σ1, p1

σ2, p2

ϕ = ϕoe

ϕ = ϕoi

O

stress path

p.

p.p.

p.p.p.

Γi

Γe

Assuming that the two limit contours Γi and Γe act also as anelastic strain potentials, and that they are expressed by the functions

( )( )

( ) 0oee

oii

≥ϕϕ=ϕΓϕ=ϕΓ

σσσ

::

The inequalities ϕ(σ) < ϕoi and ϕ(σ) < ϕoe denote the interiors of Di and De. The total strain ε is set, as usual, in the form of the sum of the linearly elastic strain e = Cσplus the anelastic component p; the anelastic strain rate is assumed to be ruled by the following equations, that are somewhat derived from Plasticity

ϕ∇λ= &&p

with

i

e

on0

on0

Γ≤λ

Γ≥λ&

&

λ > 0, and ( )

( )[ ] ( )[ ] 0

0

oeoi =λϕ−ϕϕ−ϕ

=λϕ&

&&

σσ

σ

In the analysis of the response of a structural system it is fundamental to state the relationships between ϕ(σ) and λ, in way that the above equations automatically hold by following the evolution of stress, strain and displacements.


σ 1, p 1

σ 2, p 2

ϕ = ϕ o e

ϕ = ϕ o i

O

stress p a th

p.

p.p.

p.p.p.

Γi

Γe

In the analysis of the response of a structural system it is fundamental to state the relationships between ϕ(σ) and λ, in way that the above equations automatically hold by following the evolution of stress, strain and displacements.


σ1, p1

σ2, p2

ϕ = ϕoe

ϕ = ϕoi

O

stress path

p.

p.p.

p.p.p.

Γi

Γe

To this aim, a evolutionary pattern can be set by somewhat manipulating the basic Bingham model, which is based on the introduction of a internal evolutionary variable β, driving the proceeding of ϕ with respect to λ. Put

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

ϕβ−ϕ

ϕβ−ϕ

λ−λ=ϕ−

βoc

1n

ocE &&&

( )[ ]3oc atnq2

αλϕπ

=β ** After calibration of the parameters

n = 2 x 102 , α = 8 x 104 ; q = 4 ; ϕoc = q + 4 x 103

the result in Figure is obtained, showing that all conditions set for the development of anelastic strain are checked

CONCLUSIONS

In the paper, the influence of SMA elements contributing to the overall strength of a sdof and mdof structural model undergoing horizontal

shaking and subject to vertical loads has been investigated. Pseudo-elastic SMA members are shown to decisively improve the

dynamic response capacity of the structure with respect to the analogous simply elastic-plastic cases, either in terms of response reduction, or of re-

centering capacity, or of P-Δ effects attenuation. The effectiveness of SMA inclusions in improving the structure

response also in the presence of geometrical non-linearities and in annealing its sensitivity to possible strength (or load) asymmetry, further factor of early collapse in ordinary elastic-plastic structures, is proved.

The chance for bidimensional analysis is outlined and basic ideas are set forth

REFERENCES1. Armstrong, W.D., 1996. “A One-Dimensional Model of a Shape Memory Alloy Fiber Reinforced

Aluminium Metal Matrix Composite”, Journal of Intelligent Material Systems and Structures, 7: 448-454.

2. Baratta, A., 1974. “Plastic Frames Under Strong Earthquakes: A Simplified Treatment”, Journal of Structural Mechanics, 3(2): 197-220.

3. Baz, A., Iman, K., McCoy, J., 1990. “Active Vibration Control of Flexible Beams sing Shape Memory Actuators”, Journal of Sound and Vibration, 140: 437-456.

4. Duerig, T.W., Melton, K.N., Stockel, D., Wayman, C.M., 1990. “Engineering Aspects of Shape Memory Alloys”, Boston: Butteworth-Heinmann.

5. Feng, Z.C., Li, D-Z., 1996. “Dynamics of a Mechanical System with a Shape Memory Alloy Bar”, Journal of Intelligent Material Systems and Structures, 7: 399-410.

6. Graesser, E.J., Cozzarelli, F.A., 1991. “Shape memory Alloys as New Materials for AseismicIsolation”, Journal of Engineering Mechanics, ASCE, 117: 2590-2608.

7. Graesser, E.J., Cozzarelli, F.A., 1994. “A Proposed Three-Dimensional Constitutive Model for Shape Memory Alloys”, Journal of Intelligent Material Systems and Structures, 5: 78-89.

8. Liang, C., Rogers, C.A., 1990. “One-Dimensional Thermomechanical Constitutive Relations for Shape Memory Materials”, Journal of Intelligent Materials Systems and Structures, 1: 207-234.

9. Liang, C., Rogers, C.A., 1992. “Design of Shape Memory Alloy Actuators”, Journal of Mechanical Design, 114: 223-230.

10.Maclean, B.J., Patterson, G.J., Misra, M.S., 1991. “Modeling of a Shape Memory Integrated Actuator for Vibration Control of Large Space Structures”, Journal of Intelligent Material Systems and Structures, 2: 71-94.

11.Rogers, C.A., Liang, C., 1990. “Design of Shape Memory Alloy Springs with Applications in Vibration Control”, Journal of Vibration and Acoustics, 115: 129-135.

12.Tanaka, K., Hayashi, T., Itoh, Y., Tobushi, H., 1992. “Analysis of Thermomechanical Behaviour of Shape Memory Alloys”, Journal of Mechanics Materials, 13: 207-215.

Documents

Seismic damage mitigation in existing structures by ... · PDF fileSeismic damage mitigation in existing structures by superelastic dissipators Alessandro Baratta1 and Ottavia Corbi1