S. Lagomarsino, A. Penna, A. Galasco e S. Cattari

Dipartimento di Ingegneria Strutturale e Geotecnica

Università degli Studi di Genova

Analisi sismica non lineare di edifici in muratura con il

programma TREMURI

Strategie di modellazioneAnalisi limite Metodo POR Elementi finiti

Macroelementispandrel beam

pier

joint

λ F1

λ F2

Como & Grimaldi Tomaževic, Braga & Dolce Gambarotta & Lagomarsino, Anthoine, Maier et al., Lourenço

Pagano et al. Magenes & Della FontanaD’Asdia & ViskovicBraga & Liberatore

(Podestà, 2001)

3D model of a masonry building

Hypotheses:Hypotheses: • Earthquake Resistant Structure: walls + floors

• Walls are bearing elements

• Floors share vertical loads to the walls and are planar stiffening elements (orthotropic membrane)

• Walls out-of-plane behaviour and flexural floors response negligible with respect to global one

Wall inWall in--plane model:plane model: • Frame-type model

• 2 nodes macro-elements: piers and lintels

• Joints: rigid bodies

• Tie-rods (no compression spar) and stringcourses (beam) elements included

Macro-element wall modelsNodoRigido

Fascia

Maschio

Earthquake Damage Observation

FEM Non-linear Continuum Model

The non-linear macro-element

h

∆

b

sui

ϕ1

wi

i

j uj

u1

u2

wj

w1

w2ϕ2

ϕi

ϕj

1

2

1

2

3∆

(a)

φδ

M1

T1

N1

1

2

j Tj

T2

Nj

N2

M2

Mj

23

2

T2

N2

M2

Ti

Ni

i

Mi

1

M1 T1

N1

1

(b)

nm

(Gambarotta & Lagomarsino, 1996)

The non-linear macro-element(Gambarotta & Lagomarsino, 1996)

Bending-rocking

Shear-sliding (friction)

b

x

z

?b

w f wo

wR

wmax = µ wR

s s R

*max max

1( , , )N w k bsw

µµ ζ ζ

µ−

=

* *max

1( , , )

3 2M w bN

ζµ ζ = −

*elN N N= −*elM M M= −

w

s

wR

s R

Crushing with degrade

ϕ

w

Vertical displacement – rotation interaction: with and w/o crushing

3D Building ModelPlane structuresPlane structures

Orthotropic floorsOrthotropic floors

5 dof 3D nodes5 dof 3D nodes

Mass sharingMass sharing

θ, E1, E2, G

3D Building ModelPlane structuresPlane structures

Orthotropic floorsOrthotropic floors

5 dof 3D nodes5 dof 3D nodes

Mass sharingMass sharing

3 dof nodes (2D)

5 dof nodes (3D)

3 dof nodes (2D)

5 dof nodes (3D)?

X Y

Z

ux uy

uz = w f x

f y f

u

3D Building ModelPlane structuresPlane structures

Orthotropic floorsOrthotropic floors

5 dof 3D nodes5 dof 3D nodes

Mass sharingMass sharing

X

ZY

My

My

MxJ

Mx

I

m

α

x

l

(1 cos )

(1 )

I Ix x

I Iy y

l xM M m

ll x

M M m sinl

α

α

−= + −

−= + −

TREMURI ProgramImplemented non-linear analysis procedures

STATIC INCREMENTAL (FORCE / DISPLACEMENT)DYNAMIC (Newmark integration, Rayleigh viscous damping)

-100000

-80000

-60000

-40000

-20000

0

20000

40000

60000

80000

100000

-80000

-60000

-40000

-20000

0

20000

40000

60000

80000

100000

-0.003 -0.0025 -0.002 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015 0.002 0.0025 0.003

Influence of the

vertical vibrations

TREMURI ProgramImplemented non-linear analysis procedures

STATIC INCREMENTAL (FORCE / DISPLACEMENT)DYNAMIC (Newmark integration, Rayleigh viscous damping)

PUSHOVER (analisi statica non lineare p.to 4.5.4)

mm m mk x fλλ

=

FF Fm FC F FT TFm Cm

CF Cm CC C C

K k K x fk kK k K x r

1 1 1 ... ... 0i i ii m im mm m in mn n

m m m

f f fk k x k k x k k x

f f f

− + + − + + − =

mm m mk x fλ

=

FF Fm FC FT TFm Cm

CF Cm CC C C

K k K x 0k kK k K x r

% % %

3D Pushover Analysis

Parete 1

Paret

e 4

Parete 2

Paret

e 3

3D Pushover Analysis

P1

P2

P3 P4

0

50000

100000

150000

200000

250000

300000

350000

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

Parete 1Parete 2Globale

Rigid floors

P1

P2

P3 P4

0

50000

100000

150000

200000

250000

300000

350000

0 0.002 0.004 0.006 0.008 0.01 0.012Spostamento 2° piano [m]

Tagl

io a

lla b

ase

[N]

Parete 1Parete 2Globale

Flexible floors

3D Dynamic Analysis

-3

-2

-1

0

1

2

3

0 2 4 6 8 10 12 14 16 18 20

T [s]

a [m/s 2]

-0,009

-0,006

-0,003

0

0,003

0,006

0,009

-300000

-200000

-100000

0

100000

200000

300000

-0,009 -0,006 -0,003 0 0,003 0,006 0,009 -0,009 -0,006 -0,003 0 0,003 0,006 0,009 -0,009 -0,006 -0,003 0 0,003 0,006 0,009

Wall 1 Wall 2 Global

Simulazione numerica delle prove sperimentali su un prototipo in scala reale di edificio in muratura

(Università di Pavia – Magenes & Calvi, 1997)

-25 -20 -15 -10 -5 0 5 10 15 20 25

Numerical results

Damage

n1 n2 n3

n4

n5

n6 n7

n8 n9

1 2 3

4 5 6

7 8

9 10

1 2

3 4

Earthquake response prediction

3D building model

Modal Analysis:T1 = 0.16 s

(= experimental value)

1p=P M?Load pattern:

Pushover Analysis:

Capacity Curve

Cyclic Pushover Analysis:

πξξ

2, += veffc

Equivalent hysteretic damping

Curva di Capacità ed energia dissipata

-300

-200

-100

0

100

200

300

-30 -20 -10 0 10 20

Second floor displacement [mm]

Bas

e sh

ear [

kN]

Window wall

Door wall

Global

Pushover

Analisi Dinamica

0.2 g

0.6 g

0.4 g

PGA

-300000

-200000

-100000

0

100000

200000

300000

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5Second floor displacement [cm]

Bas

e sh

ear [

N]

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10 12 14 16 18 20

Time [s]

Dis

plac

emen

t [cm

]

Risultati delle analisi dinamiche

0.2 g 0.6 g0.4 g

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.5 1 1.5 2 2.5 3 3.5

Sd [cm]

Sa [m

/s2]

Analisi sismica di edifici reali

Osservatorio Sismico delle StruttureServizio Sismico Nazionale – SSN

Dynamic Identification by Modal Testing

Modal analysis

P1

P2

P3P4P5P6

P7

P1

P2

P3P4P5P6

P7

P1

P2

P3P4P5P6

P7

P1

P2

P3P4P5P6

P7

P1

P2

P3P4P5P6

P7

P1

P2

P3P4P5P6

P7

0.139

T1

0.105

T3

0.119

T2

Modal testing

0.139

T1

0.095

T3

0.120

T2

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

0 0,005 0,01 0,015 0,02 0,025 0,03 0,035 0,04 0,045 0,05

Displacement [m]

Acc

eler

atio

n [m

s-2]

SDOF Structure

Seismic Analysis

Analisi sismica di edifici reali

Eixample district, Barcelona(Bonet et al., 2002) - RISK-UE Project

Analisi di edifici realiCastelnuovo Belbo Hall, Piedmont

Monferrato Earthquake 2000

Analisi di edifici realiMunicipio Castelnuovo BelboMonferrato Earthquake 2000

3 1

3 2

33

34

35

36 37

38

39

4 0

4 1

4 2

4 3

4 4

4 5

4 6

4 7

4 8

49

50

51

52

53

54

55

56

57

58

59

n25

n26

n27

n28

n36

n37

n32

n33

N7

N8

N9

N20 N29

N31

N34

N35

N38

N39

N40

N41

Seismic analysis of real buildings

Pushover Analysis

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50

Umedio sommità (mm)

Tbas

e (k

N)

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50

Umedio sommità (mm)

Tbas

e (k

N)

Dynamic Analysis Direzione X

-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

-30 -20 -10 0 10 20 30

s (mm)

T (k

N)

Direzione Y

-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

-30 -20 -10 0 10 20 30

s (mm)

T (

kN)

Municipio Castelnuovo BelboMonferrato Earthquake 2000

Analisi di edifici realiDamage simulation

n1210

n1211

n7760

n7761

n7762

n7759 N59

N60

N61

N62

N759

N760

N761

N762

N1000

N1006

N1007 N1010

N1011

154

155

156

157

158

159

160

161

162

1154

1155

1156

317 1317 318342

343

344

13

14

15

42

43

49

50

51

n1205n1206

n1207n1208

n1209

N13

N14

N15

N23 N71

N72

N73

N74

N91

N92

N93

N94

N95

N96

N97

N98

N1003

N1074N1097

207

208

209

210

211

212

213

214

215

216

217

218

219

220 221

222

301

322330 336 337

338339

340

5

6 7

20

3738

39 40

41

Municipio Castelnuovo BelboMonferrato Earthquake 2000

Simulazione del danno osservatoDamage simulation

n1210

n1211

n7760

n7761

n7762

n7759 N59

N60

N61

N62

N759

N760

N761

N762

N1000

N1006

N1007 N1010

N1011

154

155

156

157

158

159

160

161

162

1154

1155

1156

317 1317 318342

343

344

13

14

15

42

43

49

50

51

n1205n1206

n1207n1208

n1209

N13

N14

N15

N23 N71

N72

N73

N74

N91

N92

N93

N94

N95

N96

N97

N98

N1003

N1074N1097

207

208

209

210

211

212

213

214

215

216

217

218

219

220 221

222

301

322330 336 337

338339

340

5

6 7

20

3738

39 40

41

Numerical simulationObserved damage