Dottorato di Ricerca in Ingegneria delle Strutture e del recupero edilizio e urbano - IX ciclo N. S....

Dottorato di Ricerca in Ingegneria delle Strutturee del recupero edilizio e urbano - IX ciclo N. S.

Presentazione del lavoro di tesi

Analisi non lineare di pareti murarie sotto azioni orizzontali: modellazione a

telaio equivalente

Fisciano, 6 Maggio 2011

Dipartimento di Ingegneria Civile – Università degli Studi di Salerno

Dottorando: Ing. Riccardo Sabatino Tutor: Prof. Vincenzo PilusoCo-Tutor: Prof. Gianvittorio Rizzano

All exact science is dominated by the idea of All exact science is dominated by the idea of

approximationapproximation

Bertrand RussellBertrand Russell

PhD Dissertation Talk – Fisciano, 6th May 2011

R. Sabatino – Non-linear analysis of masonry walls under horizontal loads: equivalent frame modelling

Science is organized knowledgeScience is organized knowledge

Herbert SpencerHerbert Spencer

Chapter 1: IntroductionChapter 1: Introduction

Chapter 2: FEM modellingChapter 2: FEM modelling

Chapter 3: Masonry Buildings Modelling StrategiesChapter 3: Masonry Buildings Modelling Strategies

Chapter 4: Mechanical Behaviour of masonry panelsChapter 4: Mechanical Behaviour of masonry panels

Chapter 5: Matrix Analysis of structuresChapter 5: Matrix Analysis of structures

Chapter 6: The FREMA codeChapter 6: The FREMA code

OutlineOutline









OutlineOutline



First Stone Dwellings (8350 b.C. – Jericho, First Stone Dwellings (8350 b.C. – Jericho,

Tell-es-Sultan)Tell-es-Sultan)

IntroductionIntroductionFresco found in Rekhamara’s Tomb (1500 b.C. – Egypt)Fresco found in Rekhamara’s Tomb (1500 b.C. – Egypt)

Djoser Pyramid (2600 b.C. – Egypt)Djoser Pyramid (2600 b.C. – Egypt)



Performance-based Earthquake Engineering Performance-based Earthquake Engineering Non-linear static procedures (NLP) Non-linear static procedures (NLP)

IntroductionIntroduction

Non-linear Static AnalysisNon-linear Static Analysis



Strategies for Modelling Masonry BuildingsStrategies for Modelling Masonry Buildings

IntroductionIntroduction

FEM modelsFEM models Simplified modelsSimplified modelsspandrel

pierrigid offest

√√ Very accurate predictionVery accurate prediction

√√ Any kind of structure may be analysedAny kind of structure may be analysed

XX Time-consuming Time-consuming

XX Amount of input data Amount of input data

XX High Analytical Skills requiredHigh Analytical Skills required

√√ Suitable for professional purposesSuitable for professional purposes

√√ Quick analysesQuick analyses

XX Regular geometry needed Regular geometry needed

XX Simplifications need to evaluated Simplifications need to evaluated

VSVS









OutlineOutline



Accurate Modelling: mesoscale modelAccurate Modelling: mesoscale modelBlocks are modelled using continuum elements, while mortar and brick-mortar Blocks are modelled using continuum elements, while mortar and brick-mortar interfaces are modelled by means of nonlinear interface elements (Lourenço & interfaces are modelled by means of nonlinear interface elements (Lourenço & Rots, 1996).Rots, 1996).

Solid and interface elements account for large displacements, while only interface elements represent cracks in mortar and bricks.



Material modelMaterial modelMulti-surfaceMulti-surface

nonassociated plasticity nonassociated plasticity

Elastic responseElastic response

0σ = k u

0

0

0

0 0

0 0

0 0

t

t

n

k

k

k

0kElastic stiffness

Mortar jointsMortar joints

0m

tj

Gk

h 0

mn

j

Ek

h

2 22 21 tan tan 0x y tF C C

2 22 22 tan tan 0x y cF D D

2 22 21 tan tan 0x y Q Q Q t QQ C C

Yield functions F1 - F2

Plastic potentials Q1 - Q2

A novel 2D nonlinear interface elementA novel 2D nonlinear interface elementPhD Dissertation Talk – Fisciano, 6th May 2011


A novel 2D nonlinear interface elementA novel 2D nonlinear interface elementMaterial modelMaterial model

Nonassociated plasticity Nonassociated plasticity Elastic responseElastic response

0σ = k u

0

0

0

0 0

0 0

0 0

t

t

n

k

k

k

0kElastic stiffness

0 0t nk k penalty factor

2 22 21 tan tan 0x y tF C C

2 22 21 tan tan 0x y Q Q Q t QQ C C

Brick interfaceBrick interface

Yield function F1

Plastic potential Q1



A novel 2D nonlinear interface elementA novel 2D nonlinear interface element

Material propertiesMaterial properties

t - tensile strength

C - cohesion

Gf,II - mode II fracture energy

C - compressive strength

Gf,I - mode I fracture energy

Gf,I

uz

t

- friction angle

<0

Gf,I

I ux(y)

c

tan

GC - crushing energy

c

uz

Gc



A novel 2D nonlinear interface elementA novel 2D nonlinear interface elementWork-softening plasticity Work-softening plasticity

0 0 rA A A A

** *

*

* *

11 cos 0

2

1

plpl f

f

pl f

WW G

G

W G

t cA C, ,tan ,D, ,tan

Evolution of the material parametersEvolution of the material parameters

with

Wpl1 - plastic work related to F1

Wpl2 - plastic work related to F2

Evolution of the surfacesEvolution of the surfaces



A novel 2D nonlinear interface elementA novel 2D nonlinear interface elementTraction deformation responseTraction deformation response

tension-compression cyclic behaviour

shear cyclic behaviour

tensionshear



In-plane behaviourIn-plane behaviourVermeltfoort AT, Raijmakers TMJ (1993)Vermeltfoort AT, Raijmakers TMJ (1993)

J4DJ4D J5DJ5D

pv=0.3 MPamortar mortar

interfaceinterface

mortar mortar interfaceinterface

brick brick interfaceinterface




J4DJ4D J5DJ5D

Wpl1Wpl1Wpl1Wpl1

Wpl2

pv=0.3 MPa

pv=0.3 MPa

pv=2.12 MPa

dynamic analysisdynamic analysis



In-plane behaviourIn-plane behaviour

Mesh assessmentMesh assessment

• Mesh refinement

• Number of integration points over the interface




Wpl1Wpl1

Wpl2Wpl2



Retrofitting: Influence of spandrelsRetrofitting: Influence of spandrels

Numerical Modelling (STRAUS7): 144 panels

Panels have been obtained by assembling a

reference module whose geometry (piers,

spandrels) has been properly varied.

The parameter , ratio between the shear

stiffness of piers and spandrels, has been used to

take into account the panels geometry.

500

500

H=

400

B=500

30

30 bm

bf

hm

bm

hf

1-1 1-2 1-4

2-22-1 2-3 2-4

3-3 4-4

3

,3

3 12

12

f

em

f

f

m

m

f

m

EI

b

h

EI

k

k

f

ff h

b

m

mem b

h,



Panels have been modelled by assuming two limit schemes:

infinite stiffness spandrels and “unreinforced” spandrels.

The parameter represents the expected

improvement of relative shear strength achievable by means of

spandrels retrofitting.

nc

nc

T

TT




The comparison shows a

significant improvement in

the field <1.5, i.e. for weak

spandrels.

In such field the average

strength improvement can

be estimated as:

0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5 6,0

1-1 1-2 1-4

2-1 2-2 2-3

2-4 3-3 4-4

59.015.0 :5.1

0,20

0,30

0,40

0,50

0,60

0,70

0,50 0,75 1,00 1,25 1,50

1-1 1-2 1-4

2-1 2-2 2-3

2-4 3-3 4-4




RetrofittingRetrofitting

Then the panels 1-1 and 4-4 have been extensively investigated by considering

the retrofitting approach suggested in Italian Building Code.

For each panel, the weak spandrel (=0.70) and the strong spandrel (=5.35)

schemes have been analysed.

Three different kinds of reinforcements have been taken into account:

Injection Grouts;

Reinforced plasters;

Ring beams / Jack Arch.




The shear resistance of the panels has been evaluated by means of a

non-linear static analysis. Both the unreinforced and the reinforced

wall have been analyzed.

The improvement deriving from the reinforcement has been

summarized into the parameter

where T is the reinforced wall resistance, Tnc the unreinforced wall

resistance.

nc

nc

T

TT






Retrofitting: Wall 1-1Retrofitting: Wall 1-1

0,024 0,031 0,042

0,718

1,015

0

0,2

0,4

0,6

0,8

1

1,2

Efficacia 0,024 0,031 0,042 0,718 1,015

CordoloCordolo +

piattabandaFascia rigida Iniezioni

Intonaco armato

Parete 1-1, =5,35

0,328

0,5260,582

0,749

1,052

0

0,2

0,4

0,6

0,8

1

1,2

Efficacia 0,328 0,526 0,582 0,749 1,052

CordoloCordolo +

piattabandaFascia rigida Iniezioni

Intonaco armato

Parete 1-1, =0,70

For weak spandrels walls, the spandrel improvement gives the same results of injection

grouts/reinforced plasters.

For strong spandrels walls, best improvements have been achieved with injection

grouts/reinforced plasters.



Retrofitting: Wall 4-4Retrofitting: Wall 4-4

0,056

0,140 0,153

0,430 0,421

0

0,2

0,4

0,6

0,8

1

Efficacia 0,056 0,140 0,153 0,430 0,421

CordoloCordolo +

piattabandaFascia rigida

Iniezioni (1° piano)

Intonaco armato (1° piano)

Parete 4-4, =5,35

0,286

0,414

0,502

0,742 0,771

0

0,2

0,4

0,6

0,8

1

Efficacia 0,286 0,414 0,502 0,742 0,771

CordoloCordolo +

piattabandaFascia rigida

Iniezioni (1° piano)

Intonaco armato (1°

piano)

Parete 4-4, =0,70




[Modena et al.][Modena et al.]

The expected improvement gets the same order of magnitude of data available in literatureThe expected improvement gets the same order of magnitude of data available in literature



Retrofitting: Parete 4-4Retrofitting: Parete 4-4 For the wall 4-4, further numerical simulations have been performed, by assuming the

reinforcement (reinforced plaster/injection ) applied to 1 to 4 storeys.

Consolidamento con intonaco armato - Parete 4-4

0,00

1,00

2,00

3,00

4,00

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5 6,0

1 piano 2 piani

3 piani 4 piani

Consolidamento con iniezioni - Parete 4-4

0,00

1,00

2,00

3,00

4,00

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5 6,0

1 piano 2 piani

3 piani 4 piani









OutlineOutline



Simplified ModelsSimplified Models









OutlineOutline



Description of the modelDescription of the model

F2

F1

Main featuresMain features

1. Displacement Control approach 1. Displacement Control approach NLP NLP

2. Global and local equilibrium2. Global and local equilibrium

3. Spread plasticity approach3. Spread plasticity approach

4. Quick Analysis and Easy Post-processing4. Quick Analysis and Easy Post-processing

Piers (HPiers (Heffeff after Dolce, 1991) after Dolce, 1991)SpandrelsSpandrels

Rigid OffsetsRigid Offsets

Equivalent Frame ModelEquivalent Frame Model



Piers - Constitutive Laws Piers - Constitutive Laws Generalized Uniaxial Compressive Stress-Strain RelationshipGeneralized Uniaxial Compressive Stress-Strain Relationship

C

u u u

A B

A=2, B=-1, C=2 A=2, B=-1, C=2 [Hendry, 1998][Hendry, 1998]

A=6.4, B=-5.4, C=1.17 A=6.4, B=-5.4, C=1.17 [Turnšek-Čačovič, 1980][Turnšek-Čačovič, 1980]

Accurate Moment-CurvatureAccurate Moment-Curvature

d

ud

[After Tomaževič, 1999]



Piers - Flexural BehaviourPiers - Flexural Behaviour

N

M

D

t

M

D

yc

N

GG

yc

t

N

M

D

t

M

D

yc

N

GG

yc

t

normalised neutral axis

normalised axial force

normalised bending moment

u

M

Cross-section Equilibrium EquationsCross-section Equilibrium Equations




Moment-Curvature relationship workflowMoment-Curvature relationship workflow

0

5

10

15

20

25

30

35

40

0,0 5,0 10,0 15,0 20,0 25,0 30,0 35,0

M [k

Nm

]

[mm-1 x 106]

D, t, ,

u

u

cr

M

YES

NOD [mm] 500t [mm] 250N [kN] 200

A 2B -1C 2

u 0,003

r 0,0045

u [MPa] 6,2

END


Piers - Shear BehaviourPiers - Shear Behaviour

[After Anthoine, Magenes, Magonette, 1994]

Experimental BehaviourExperimental Behaviour ModelModel

Ultimate drift Ultimate drift uu = 0.4% H = 0.4% Heffeff

[Italian Building Code][Italian Building Code]

u

V

Vu



Piers - Shear BehaviourPiers - Shear Behaviour




i=Vi/Ki

min Vu

i

el

Ki+1=Ki *Vu/Vi

NO

YES

el=Vu/Kel

V

Ksec,i

Ksec,i+1

Vi

Vu

el

Ki+1=Kel

Shear-strain relationship workflowShear-strain relationship workflow


Collapse condition when the desired value of Collapse condition when the desired value of drift (set by the user) is attained drift (set by the user) is attained

(Italian Building Code suggets (Italian Building Code suggets = 0.004 for = 0.004 for shear collapse)shear collapse)

Shear-strain relationship workflowShear-strain relationship workflow

u

V

Vu



Spandrels - Shear BehaviourSpandrels - Shear Behaviour


Residual Strength Residual Strength = 0.25= 0.25

[Magenes and Della Fontana, 1998][Magenes and Della Fontana, 1998]

u

V

Vu

Vu

0u vdV ht f



Spandrels - Flexural BehaviourSpandrels - Flexural Behaviour


[After Calderoni et al., 2008]

u

M

Mu



Spandrels - Flexural BehaviourSpandrels - Flexural Behaviour

Proposed formulations for MProposed formulations for Muu – – [Italian Building Code, 2008][Italian Building Code, 2008]

1.1. Stress-block approach (same equation of piers)Stress-block approach (same equation of piers)

2. If no tensile-resistant element is present 2. If no tensile-resistant element is present M Muu=0=0



Spandrels - Flexural BehaviourSpandrels - Flexural BehaviourProposed formulations for MProposed formulations for Muu [Schubert & Weschke, 1986][Schubert & Weschke, 1986]

Take into account an “equivalent Take into account an “equivalent strut” provided with a tensile strut” provided with a tensile strength fstrength ftutu

fftutu is the minimum between two is the minimum between two collapse mechanisms:collapse mechanisms:

a) bricks failurea) bricks failure

b) bed joints failureb) bed joints failure

,

joint 22y bt

tu a bt

y

ff f

t

,

joint22

x xtu b

yy

f c p c pt

a)a)

b)b)



Spandrels - Flexural BehaviourSpandrels - Flexural BehaviourSpandrels M-N Limit Domain Spandrels M-N Limit Domain [Cattari and Lagomarsino, 2008][Cattari and Lagomarsino, 2008]

= ratio between tensile strength f= ratio between tensile strength ftutu and compressive strength and compressive strength

fwc

/y-

c1

-

10.85

Constitutive LawConstitutive Law

Improvement of rocking Improvement of rocking resistance, also with low resistance, also with low

(or zero) values of N.(or zero) values of N.









OutlineOutline



The FREMA CodeThe FREMA Code






Masonry Panels – Anthoine, Magonette and Magenes (1998)Masonry Panels – Anthoine, Magonette and Magenes (1998)

Low PanelLow Panel High PanelHigh Panel

Cross-Section: 100 x 25 cmCross-Section: 100 x 25 cm22

Low panel high: 135 cmLow panel high: 135 cm

High panel high: 200 cmHigh panel high: 200 cm

Normal Load: 150 kNNormal Load: 150 kN




Pavia Door Wall – Calvi and Magenes (1994)Pavia Door Wall – Calvi and Magenes (1994)












Catania Project - Investigation on the seismic response of two masonry buildings (2000)Catania Project - Investigation on the seismic response of two masonry buildings (2000)

““Via Martoglio” 2D WallVia Martoglio” 2D Wall

Equivalent Frame model: 128 elements, 81 nodes, 219 DOF




Model 1: Masonry, NO R.C. Ring BeamsModel 1: Masonry, NO R.C. Ring Beams





Model 2: Masonry, Elastic R.C. Ring Beams (E=20,000 MPa)Model 2: Masonry, Elastic R.C. Ring Beams (E=20,000 MPa)





Model 3: Masonry, Elastic R.C. Ring Beams (E=4,000 MPa)Model 3: Masonry, Elastic R.C. Ring Beams (E=4,000 MPa)





Catania Project - Investigation on the seismic response of two masonry buildings (2000)Catania Project - Investigation on the seismic response of two masonry buildings (2000)

Preliminary validation of the modelPreliminary validation of the model

““Via Verdi” BuildingVia Verdi” Building

Wall 1Wall 1Wall 2Wall 2

Wall 3Wall 3



““Via Verdi” – Wall 1Via Verdi” – Wall 1












Mallardo et al. (2008) – Palazzo Renata di FranciaMallardo et al. (2008) – Palazzo Renata di Francia




Mallardo et al. (2008)Mallardo et al. (2008)




Salonikios et al. (2003)Salonikios et al. (2003)

Two-storey, 7-bay masonry wallTwo-storey, 7-bay masonry wall

Two lateral load distributions Two lateral load distributions considered:considered:

1.1. Uniform (ACC)Uniform (ACC)

F= {1.00; 0.59}F= {1.00; 0.59}

2.2. Inverse Triangular (LOAD)Inverse Triangular (LOAD)

F= {1.00; 1.19}F= {1.00; 1.19}




Salonikios et al. (2003) – 7B_UniformSalonikios et al. (2003) – 7B_Uniform

0

200

400

600

800

1000

0 2 4 6 8 10 12 14 16

top displacement [mm]

Tota

l Bas

e Sh

ear [

kN]

Proposed Model

SAP 2000

Discrete FEM model




Salonikios et al. (2003) – 7B_Inverse TriangularSalonikios et al. (2003) – 7B_Inverse Triangular

0

200

400

600

800

1000

0 2 4 6 8 10 12 14 16

top displacement [mm]

Tota

l Bas

e Sh

ear [

kN]

Proposed Model

SAP 2000

Discrete FEM model





Mesh RefinementMesh Refinement

=Log(Nc/x)



Salonikios et al. (2003) – 7B_Inverse TriangularSalonikios et al. (2003) – 7B_Inverse Triangular




Salonikios et al. (2003) – 7B_UniformSalonikios et al. (2003) – 7B_Uniform





Time-cost AnalysisTime-cost Analysis

=Log(Nc/x)



Time-cost AnalysisTime-cost Analysis



This dissertation deals with the seismic behaviour of masonry structures;This dissertation deals with the seismic behaviour of masonry structures;

The first part of the work is aimed at understanding the potentialities of The first part of the work is aimed at understanding the potentialities of very accurate FEM model very accurate FEM model in predicting masonry panels seismic response; the in predicting masonry panels seismic response; the panels simulated by means of ADAPTIC showed a very good prediction of the panels simulated by means of ADAPTIC showed a very good prediction of the experimental results, both in terms of force-displacement curve and in terms experimental results, both in terms of force-displacement curve and in terms of cracks path.of cracks path.

A further application of simplified (homogeneous) FEM models has been A further application of simplified (homogeneous) FEM models has been performed on masonry panels, aiming at evaluating the performed on masonry panels, aiming at evaluating the influence of influence of spandrels reinforcementspandrels reinforcement on the overall resistance; in the same application on the overall resistance; in the same application some some reinforcement techniques reinforcement techniques have been applied considering the Italian have been applied considering the Italian Building Code approach;Building Code approach;

ConclusionsConclusions



ConclusionsConclusions In the second part of the dissertation, a novel equivalent frame model has In the second part of the dissertation, a novel equivalent frame model has been developed. The main features of the model have been discussed, by been developed. The main features of the model have been discussed, by highlighting the main features of the proposed model (displacement control highlighting the main features of the proposed model (displacement control approach, accurate moment-curvature for piers behaviour, spandrels approach, accurate moment-curvature for piers behaviour, spandrels behaviour);behaviour);

A A validation and applicationvalidation and application of the model has been carried out of the model has been carried out comparison with experimental tests and accurate numerical simulationscomparison with experimental tests and accurate numerical simulations

The comparison showed a The comparison showed a good agreementgood agreement between the proposed model between the proposed model and both experimental and numerical results, showing that FREMA code is a and both experimental and numerical results, showing that FREMA code is a reliable tool for performing the non-linear static analysis of masonry panels.reliable tool for performing the non-linear static analysis of masonry panels.



Thank you very much!Thank you very much!



Dottorato di Ricerca in Ingegneria delle Strutturee del recupero edilizio e urbano - IX ciclo N. S.

Presentazione del lavoro di tesi

Analisi non lineare di pareti murarie sotto azioni orizzontali: modellazione a

telaio equivalente

Fisciano, 6 Maggio 2011

Dipartimento di Ingegneria Civile – Università degli Studi di Salerno

Dottorando: Ing. Riccardo Sabatino Tutor: Prof. Vincenzo PilusoCo-Tutor: Prof. Gianvittorio Rizzano

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Dottorato di Ricerca in Ingegneria delle Strutture e del recupero edilizio e urbano - IX ciclo N. S....

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